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[ [ "RNNoise-Ex: Hybrid Speech Enhancement System based on RNN and Spectral\n Features" ], [ "Abstract Recent interest in exploiting Deep Learning techniques for Noise Suppression, has led to the creation of Hybrid Denoising Systems that combine classic Signal Processing with Deep Learning.", "In this paper, we concentrated our efforts on extending the RNNoise denoising system (arXiv:1709.08243) with the inclusion of complementary features during the training phase.", "We present a comprehensive explanation of the set-up process of a modified system and present the comparative results derived from a performance evaluation analysis, using a reference version of RNNoise as control." ], [ "Introduction", "Signal Processing has undoubtedly a wide range of useful applications in the modern world.", "Narrowing our focus on the domain of Audio Signal Processing, Speech Enhancement is an especially interesting subfield, due to the number of its applications, such as telecommunication networks, online video conferencing [1], cochlear implants [2], speech-to-text systems [3], etc.", "Speech Enhancement is heavily dependent on the concept of denoising; that is the removal of undesired audio signals that degrade the speech signal which may result in reduction of quality and intelligibility.", "Noise Suppression is by no means a new field of study among scientists and engineers.", "The application, however, of modern techniques, ideas and innovations has enabled the field to grow and include some very promising denoising algorithms and systems.", "Such approaches can be divided to causal (e.g: [1]) and non-causal [7], depending on whether they exploit information in future signal frames to process the current.", "They can also be categorized in real time or non real time systems depending on their ability to process signal frames within a predefined time constraint.", "In the past, the focus of Noise Suppression was on the utilization of conventional signal processing techniques (filtering), which operate by estimating the statistical characteristics of the noise signal to be removed.", "Some commonly used such methods include Wiener [4] and Kalman [5] filters.", "Following the increase of interest for machine learning shown in recent years by the scientific community, a new realm of possibility was now available to researchers in the vein of Noise Suppression.", "In the last decade especially, no small number of works have been published that approach the denoising problem by employing neural network architectures and innovative deep learning (DL) techniques to counteract non-stationary noise signals [6].", "A yet more recent trend among researchers is the development of hybrid systems that combine both conventional and ML techniques.", "The motivations and advantages of such an approach appear to be: the exploitation of existing knowledge on the problem nature, leading to the design of concept-aware systems engagement of data-driven approaches with large models that give the flexibility to better model the complex acoustic patterns of speech an increase of system performance by balancing/counteracting each method's weaknesses with the strengths of the other a decrease in unnecessary complexity as compared to purely ML techniques the better handling of auditory artifacts, which constitute one of the greatest hindrances in Speech Enhancement to date.", "Elaborating on hybrid systems, in [8] the noisy audio signal of the current time frame is first processed using a suppression rule computed as a geometric mean of the clean speech estimation of the current frame using a conventional denoising technique and the result of the suppression rule of the previous frame which was determined by an LSTM deep-learning technique.", "This first step is used to remove quasi-stationary noise components.", "The intermediate enhanced signal that results from the previously described process is then used to estimate the clean speech signal and the current frame suppression rule, using an LSTM-based approach.", "The aim of the second step is to efficiently remove non-stationary noise signals.", "The approach taken in [9] follows a similar structure to that of [8].", "Namely, first the noisy signal is enhanced using the well-known Wiener filter.", "Afterwards, the resulting signal is further processed by a multi stream approach, which includes a number of denoising autoencoders and auto-associative memories, based on LSTM networks.", "Another example of a system that combines both conventional and deep learning techniques, and the base of our work, is RNNoise [1], implemented by Jean Marc Vallin with the support of Mozilla.", "RNNoise is a real-time system designed to run on simple hardware (e.g.", "Raspberry Pi).", "To achieve lower complexity, a Recurrent Neural Network (RNN) was employed for the portion of the spectral mask estimation process that was hard to tune and a conventional signal processing technique for the rest of it.", "In the following subsection we review some details of the RNNoise implementation that will better help the understanding of the work presented afterwards.", "In RNNoise, the denoising process is applied to 20 ms windows, overlapped by 50% and windowed by a Vorbis window.", "For each window, follows the extraction of certain features that will be analyzed in Section II that are afterwards used as an input for the RNN.", "RNNoise operates on 48kHz full-band audio input.", "The network computes an ideal ratio mask (IRM) $m = [m_1, m_2, ..., m_{22}]^T \\in \\mathbb {R}^{22} : m_i \\in [0, 1]$ , for 22 triangular bands derived from a modified version of the Bark scale that is similar to the Opus scale [10].", "The 22 gains in $[\\sqrt{m_1}, \\sqrt{m_2}, ..., \\sqrt{m_{22}}]^T$ , after an interpolation, can be applied to the Discrete Fourier Transform (DFT) magnitudes of each window.", "Before that, a pitch filter, namely a comb filter defined at the pitch period, is applied to each window.", "What this filter essentially implements is the addition of the original signal to its scaled and delayed by the pitch period version.", "The role of the pitch filter is to suppress noise between pitch harmonics of voiced speech, which is not feasible by the coarse 22-band gains mask produced by the neural network.", "After the application of both the gains and the pitch filter, the waveform of the processed DFT is calculated and the overlap-add method is used to produce the final denoised signal.", "Practically, the overlap-add method is applied gradually, after each new 10 ms samples arrive, to achieve better response time.", "For a more detailed analysis of the RNNoise system see [1].", "every picture/.style=line width=0.75pt Figure: RNNoise system architecture overview .Now that the basics of RNNoise and similar systems have been covered, what ensues is the presentation of our modifications to the system.", "In Section II, the input features –new and old- are described, as well as the training and evaluation datasets and toolchains.", "The assessment of the results of the new system, as well as a comparison with a retrained, reference version of the original RNNoise system, along with some comments, comprise Section III of the paper.", "Finally, in Section IV we conclude and summarize everything discussed in the previous sections." ], [ "Methodology", "The main objective in this study is to explore possible performance gains over the original RNNoise system [1] by modifying it so that it utilizes extended information regarding its input.", "We first review the input features, then train a reference RNNoise system and our extended system using our selected datasets, so that we can later evaluate them and make a fair comparison between the two, and finally present the toolchain we developed to aid us in this process.", "The original system by Valin (2018) uses 42 input features to perform speech enhancement [1].", "The first 22 are Bark Frequency Cepstral Coefficients (BFCCs) as derived from applying the Discreet Cosine Transformation (DCT) on the log spectrum of the previously mentioned modified Bark scale.", "The next 12 features are the first and second order temporal derivatives of the first 6 BFCCs.", "The following 6 features are calculated by applying the DCT on the pitch correlation across frequency bands and selecting the first 6 coefficients.", "The final two features are the pitch period and a spectral non-stationary metric that assists in speech detection.", "Given that the original system generally relies upon features related to pitch and BFCCs, we decided to explore the potential of combining them with characteristics of a different nature.", "Reviewing commonly used features in the literature [11] [12] [13] [14] [15], we chose to use the following, standardized to zero mean and unit variance, for the full spectral and temporal range of each 20 ms frame processed by the extended system: Spectral Centroid: Signal's spectral “center of mass” Spectral Bandwidth: Signal's highest minus lowest frequency Spectral Roll-Off: Threshold frequency over which 90% of the signal's energy is situated To calculate Spectral Centroid, first the Discrete Fourier Transform (DFT) for each frame is calculated using (REF ), where $k$ is the $k$ -th frequency for the $n$ -th frame, $x(m)$ is the input signal, $w(m)$ is a window function and $L$ is the window's length.", "Spectral Centroid is then calculated using (REF ) with $K$ being the DFT's order.", "$A(n,k) = |\\sum _{m=-\\infty }^{\\infty } x(m)w(nL-M)e^{-j(\\frac{2\\pi }{L}km)}|$ $SC(n) = \\frac{\\sum _{k=0}^{K-1}k\\cdot |A(n,k)|^2}{\\sum _{k=0}^{K-1}|A(n,k)|^2}$ Spectral Roll-Off is calculated using (REF ) where $N$ is the total number of frames, $K$ is the order of the DFT, $TH$ is a threshold (usually $\\approx 0.9$ ) and $A(n,k)$ is calculated using (REF ).", "$SRF(n) = max(h|\\sum _{k=0}^{h}A(n,k) < TH\\cdot \\sum _{k=0}^{K-1}|A(n,k)|^2)$ To train and evaluate our extended system we implemented a modified toolchain which reuses modified parts of [1].", "In the following paragraphs we present the components of our toolchain for feature extraction, training and evaluation.", "The full training and evaluation toolchain is visualized in Fig.", "REF .", "Our source code is publicly available Source Code: https://github.com/CedArctic/rnnoise-ex.", "every picture/.style=line width=0.75pt Figure: Training and Evaluation toolchain overview.For feature extraction we first use Sound eXchange (SoX) [16] to concatenate and convert the input clean speech and noise to RAW format files which we then process using the appropriate tool from [1] to generate the training samples, by mixing clean speech and noise tracks as shown in [1], and extract the original 42 features as well as additional features used for training.", "After, we process the training samples using a feature extraction tool to extract the additional features.", "We train the extended system using Keras with Tensorflow [17] through the training tool which we modified according to the extended system's parameters.", "Both reference and extended system are trained through the course of 120 epochs with 8 steps each using the Adam optimizer with the learning rate set to 0.001.", "We use the loss function (REF ) (as proposed in [1]), where $m$ is the ground truth IRM mask, $\\hat{m}$ is the mask calculated by the RNN, $\\gamma = \\frac{1}{2}$ is a parameter that tunes the suppression's aggressiveness and $N$ is the number of bands, which in our case is l to 22.", "During training, both systems process 3 600 000 audio frames, each with a non-overlapping 10 ms duration.", "$L(m, \\hat{m}) =\\\\\\multicolumn{1}{r}{\\frac{1}{N} \\cdot \\bigg (10 \\cdot \\sum _{i=1}^{N}\\Big (min(m_i + 1, 1) \\cdot (10\\cdot (m_i-\\hat{m_i})^4 \\\\+ (\\sqrt{\\hat{m_i}} - \\sqrt{m_i} )^\\gamma - 0.01 \\cdot m_i \\cdot log(\\hat{m_i}))\\Big )}\\\\- \\frac{1}{2} \\cdot \\sum _{i=1}^{N}\\Big (2 \\cdot |m_i-0.5| \\cdot m_i \\cdot log(\\hat{m_i})\\Big )\\bigg )$ To evaluate inputs to our trained extended system, we first extract the 42 features presented in [1] using the original feature extraction tool and then merge them with the additional features extracted using the tools previously described.", "We pass this data along with the input audio file to the evaluation tool which calculates, interpolates and applies the modified Bark scale gains along with a pitch filter to the audio file as described in [1].", "The architecture of the neural network used follows that of the original RNNoise system with the difference that the input layer creates a tensor whose size is modified to fit that of the increased number of features.", "The topology is presented in detail in Fig.", "REF and the system contains 215 units and 4 hidden layers.", "Figure: Deep Recurrent Neural Network Topology.As described in the original paper [1], the network is so designed that it follows the usual structure of many conventional noise suppression algorithms.", "The basic idea behind the design of the system is that it can be divided into three subsystems: a Voice Activity Detector (VAD), a noise spectral estimation and a spectral subtraction block.", "Each subsystem includes a recurrent layer and specifically a gated recurrent unit (GRU).", "Concerning VAD, it contributes significantly to the training process by helping the system differentiate noise from speech.", "It also outputs a voice activity probability even though it is not actively used in the inference process.", "To train our RNNoise and feature-extended systems we utilize the clean speech dataset included in the Edinburgh Dataset [20] which is comprised of audio recordings, sampled at 48 kHz, of 28 English speakers (14 men and 14 women) with similar pronunciation.", "For noise recordings, we used a subset of the acoustical environments available in the DEMAND dataset [21].", "These environments were then excluded from those used as the test set.", "The DEMAND dataset includes noise recordings corresponding to six distinct acoustic scenes (Domestic, Nature, Office, Public, Street and Transportation), which are further subdivided in multiple more specific noise sources [21].", "Note that while we used clean speech and noise included in the Edinburgh Dataset, the samples used for training the systems are not the noisy samples found in the noisy speech subset of the Edinburgh Dataset, but rather samples mixed using the method described in [1].", "The test set used is the one provided in the Edinburgh Dataset [20], which has been specifically created for speech enhancement applications and consists of wide-band (48kHz) clean and noisy speech audio tracks.", "The noisy speech in the set included four different SNR levels (2.5dB, 7.5dB, 12.5dB, 17.5dB).", "The clean speech tracks included in the set are recordings of two english language speakers, a male and a female.", "As for the noise recordings that were used in the mixing of the noisy speech tracks, those were derived from the DEMAND database [21] [22].", "More specifically, the noise profiles found in the testing set are: Living: noise inside a living room (Domestic) Office: noise from a small office with three people using computers (Office) Psquare: noise from a public town square with many tourists (Street) Cafe: noise from the terrace of a cafe at a public square (Street) Bus: noise a public transit bus (Transportation) The selection of the appropriate evaluation metrics is of great importance in the effort of regular evaluation of any system.", "In order to evaluate our system we used a metric that focuses on the sound quality (PESQ) and a metric that focuses on the intelligibility of the voice signal (STOI).", "The wide-band Perceptual Evaluation of Speech Quality (PESQ) [18] is an objective and generally used standard for measuring sound quality.", "It takes account of features such as sound sharpness, speech volume, ambient noise, interruptions and interferences [35].", "The PESQ scale calibration ranges from -0.5 to 4.5, with higher values corresponding to better quality.", "The Short-Time Objective Intelligibility (STOI) [19] is a metric that increases according to the average intelligibility of the processed signal, given the original signal.", "Average intelligibility (or comprehensibility) is the percentage of words that are properly understood by a group of users.", "This metric ranges from 0 to 1." ], [ "Results and Discussion", "It was deemed appropriate to present our results in a comparison between the reference RNNoise system and the modified version that makes use of the additional features.", "By comparing the two systems with regards to the PESQ quality metric, as seen in Fig.", "REF , it becomes apparent that the modified version falls short by significant margin in all acoustic environment settings and in all SNR levels, but especially in higher SNRs.", "Similarly, examining the STOI intelligibility measure, as depicted in Fig.", "REF , it is deduced that the modified version again falls severely short, but this time it is especially so for higher values of SNR.", "An exception to this appears to be the case of the \"Living\" audio scene, where, especially for low SNRs the performance of the two systems seems to be similar.", "Figure: Extended and Reference System PESQ performance in different acoustical environments under various SNR levels:a.", "Reference system and b.", "Extended systemFigure: Extended and Reference System STOI performance in different acoustical environments under various SNR levels:a.", "Reference system and b.", "Extended systemOverall, the modified system appears to have a generally worse performance than the reference version of RNNoise.", "Having also compared several pairs of spectrograms of both denoising system cases, it was observed that in general the modified version does indeed subtract less noise components.", "Having taken these results into consideration, we now discuss some avenues for further future development that will hopefully yield better performance results.", "Firstly, while for the base system [1] Valin notes that adding more hidden layers does not improve performance significantly, we believe that it might indeed be beneficial for our extended system.", "Given that we provide the system with more and diverse input information, the RNN might be able to better exploit the proposed features with additional hidden layers.", "Currently our extended system utilizes the additional features as calculated on the full spectrum of each window processed by the RNN.", "We believe that the system's performance can potentially be improved by calculating these features for each individual subband of the modified Bark scale or for a small selection of them.", "This will subsequently lead to an increase of the RNN's input features and as such the network's hidden layers will have to be adapted to properly accommodate this change.", "Studying samples processed by our extended system, we speculate that the system could benefit from changing how aggressively the noise suppression occurs.", "This can be achieved by fine-tuning the value of the $\\gamma $ parameter in the loss function (REF ), keeping in mind that smaller $\\gamma $ values lead to more aggressive suppression.", "According to [1], setting $\\gamma = \\frac{1}{2}$ is an optimal balance.", "We initially considered also using Root Mean Square (RMS), which is related to the signal's energy and its change over time, and Spectral Flatness which is used to discern tone-like from noise-like signals.", "However, when we calculated and visualized these features for our dataset, we discovered that they offered limited variance and had many outliers.", "This led us to omit them from our feature set as we believed that they would increase input dimensionality more than would benefit performance.", "Revisiting these features under the subbanding context described above might prove to improve the system.", "Finally, we believe that further research can be done regarding the performance of the base and extended systems as the training dataset increases in size and diversity." ], [ "Conclusion", "In this paper we have presented our efforts to extend and improve a hybrid speech enhancement system.", "We proposed features which we believed would further assist the denoising process and assessed them as inputs to the recurrent neural network.", "We illustrated our toolchain for training the system with extended input features and compared the system against a reference RNNoise instance trained using the same training parameters.", "We discussed our findings from this process, concluding that the extra features have no obvious positive effect on the system's performance for the training test size used.", "Finally, we laid out our thoughts on future avenues to be explored for further improvement of the base system using spectral features." ], [ "Acknowledgment", "We would like to thank our teachers, Dr. Charalampos A. Dimoulas (Associate Professor) and Dipl.", "Iordanis Thoidis (PhD candidate) (Laboratory of Electroacoustics and TV Systems, School of Electrical and Computer Engineering, Aristotle Univeristy of Thessaloniki) for their enthusiastic guidance and support throughout the research process and writing of this paper." ] ]
2105.11813
[ [ "Continuity of critical exponent of quasiconvex-cocompact groups under\n Gromov-Hausdorff convergence" ], [ "Abstract We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces." ], [ "Introduction", "In this paper we begin the study of convergence of actions by isometries on packed, Gromov-hyperbolic GCB-spaces, starting from the easier setting of quasiconvex-cocompact actions.", "A GCB-space is a couple $(X,\\sigma )$ , where $X$ is a complete metric space and $\\sigma \\colon X\\times X \\times [0,1] \\rightarrow X$ is a geodesically complete, convex geodesic bicombing on $X$ .", "This means that for every $x,y\\in X$ the map $\\sigma _{xy}(t) = \\sigma (x,y,t)$ is a geodesic (parametrized by arc-length) between $x$ and $y$ , that for every $x,y,x^{\\prime },y^{\\prime } \\in X$ the map $t\\mapsto d(\\sigma _{xy}(t), \\sigma _{x^{\\prime }y^{\\prime }}(t))$ is convex and that every $\\sigma $ -geodesic segment can be extended to a bigger $\\sigma $ -geodesic, where a $\\sigma $ -geodesic is a segment of the for $\\sigma _{xy}$ with $x,y\\in X$ .", "Classical examples of GCB-spaces are geodesically complete CAT$(0)$ -spaces, geodesically complete Busemann convex spaces and Banach spaces.", "The interest on this notion of non-positive curvature is due to its stability under limits, while for instance limit of Busemann convex spaces may be not uniquely geodesic and so not Busemann convex.", "The existence of a convex geodesic bicombing plays the role of a weak non-positive upper bound on the curvature.", "The uniform packing condition replaces in our context a lower bound on the curvature: we say that a metric space $X$ is $P_0$ -packed at scale $r_0$ if the maximal cardinality of a $2r_0$ -separated subset of any ball of radius $3r_0$ is at most $P_0$ .", "It gives a sort of homogeneity of the spaces under consideration at macroscopical scales.", "For more details about this notion see [7] and [6].", "It turns out that a $P_0$ -packing condition at scale $r_0$ implies a well-controlled packing condition at every scale, if $(X,\\sigma )$ is a GCB-space.", "This fact was used intensively in the previous works [7], [6] and it will be important also in this paper.", "An isometry $g$ of a GCB-space $(X,\\sigma )$ is called a $\\sigma $ -isometry if it sends $\\sigma $ -geodesics to $\\sigma $ -geodesics: that is if $\\sigma _{g(x)g(y)} = g\\sigma _{xy}$ for every $x,y\\in X$ .", "Many properties of groups of $\\sigma $ -isometries acting discretely on packed, Gromov-hyperbolic, GCB-spaces were studied by the author and A.Sambusetti in [6].", "A discrete group $\\Gamma $ of $\\sigma $ -isometries of a $\\delta $ -hyperbolic, GCB-space $(X,\\sigma )$ is quasiconvex-cocompact if it acts cocompactly on the quasiconvex-hull of its limit set $\\Lambda (\\Gamma )$ , namely QC-Hull$(\\Lambda (\\Gamma ))$ .", "In this case the codiameter is the diameter of the quotient metric space $\\Gamma \\backslash $ QC-Hull$(\\Lambda (\\Gamma ))$ .", "We denote by GCB$^\\textup {qc}(P_0,r_0,\\delta ; D)$ the class of 4-uples $(X,x,\\sigma ,\\Gamma )$ where $(X,\\sigma )$ is a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ , $\\Gamma $ is a discrete, non-elementary, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ and $x$ is a point of QC-Hull$(\\Lambda (\\Gamma ))$ .", "The critical exponent of a discrete group of isometries $\\Gamma $ of a proper metric space $X$ is defined by $h_\\Gamma = \\limsup _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\Gamma x \\cap \\overline{B}(x,T),$ where $x$ is any point of $X$ .", "Our main result is: Theorem A The class GCB$^\\textup {qc}(P_0,r_0,\\delta ; D)$ is compact with respect to the pointed equivariant Gromov-Hausdorff convergence and with respect to this convergence the critical exponent is continuous, i.e.", "if $(X_n,x_n,\\sigma _n,\\Gamma _n)$ is a sequence of spaces of $\\textup {GCB}^\\textup {qc}(P_0,r_0,\\delta ; D)$ converging to $(X_\\infty , x_\\infty , \\sigma _\\infty , \\Gamma _\\infty )$ then $h_{\\Gamma _\\infty } = \\lim _{n\\rightarrow +\\infty }h_{\\Gamma _n}.$ The precompactness of the class GCB$^\\textup {qc}(P_0,r_0,\\delta ; D)$ is an almost direct application of [6] because the upper nilradius is uniformly bounded above.", "The compactness then follows by the non-trivial fact that the limit group is again quasiconvex-cocompact.", "In order to do that we will show that in the situation above the Gromov boundary of the space $X_\\infty $ can be seen as the limit (in the Gromov-Hausdorff sense) of the Gromov boundaries of the spaces $X_n$ (Proposition REF ) and moreover that the limit sets of the groups $\\Gamma _n$ converge to the limit set of the group $\\Gamma _\\infty $ (Theorem REF ).", "The lower semicontinuity of the critical exponent is known in some cases (see [2] and [12]), but in those works several restrictions and assumptions on the class of groups are made.", "Instead the proof of the continuity part of Theorem REF is based on the following quantified version of a result of M.Coornaert ([5]): Theorem B Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ and let $\\Gamma $ be a discrete, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ .", "Then the Patterson-Sullivan measure $\\mu _{\\textup {PS}}$ on $\\Lambda (\\Gamma )$ is $(A,h_\\Gamma )$ -Ahlfors regular, i.e.", "for every $z\\in \\Lambda (\\Gamma )$ and every $0<\\rho \\le 1$ it holds $\\frac{1}{A}\\rho ^{h_\\Gamma } \\le \\mu _{\\textup {PS}}(B(z,\\rho )) \\le A\\rho ^{h_\\Gamma },$ where $A$ is a constant depending only on $P_0,r_0,\\delta $ and $D$ .", "The set $B(z,\\rho )$ is the so-called generalized visual ball of center $z$ and radius $\\rho $ , see Section REF .", "If $\\Gamma $ is elementary the proof is trivial, while in the non-elementary case it depends heavily on uniform estimates of the critical exponent.", "Indeed by Corollary 1.3 of [6] and Proposition REF we have $0<h^-\\le h_\\Gamma \\le h^+$ , where $h^-$ and $h^+$ depend only on $P_0,r_0,\\delta $ and $D$ .", "Let us outline how Theorem REF implies the continuity part of Theorem REF : the uniform estimates on the Ahlfors-regularity constants have two consequences: (i) first of all the critical exponent of $\\Gamma $ equals the visual Minkowski dimension of the limit set $\\Lambda (\\Gamma )$ , defined as $\\text{MD}(\\Lambda (\\Gamma )) = \\lim _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\text{Cov}(\\Lambda (\\Gamma ), e^{-T}),$ where $\\text{Cov}(\\Lambda (\\Gamma ), e^{-T})$ denotes the minimal number of generalized visual balls of radius $e^{-T}$ needed to cover $\\Lambda (\\Gamma )$ ; (ii) moreover Theorem REF yields a uniform rate of convergence to the limit in the definition (REF ) of the visual Minkowski dimension.", "This uniform convergence to the critical exponent of the expression (REF ) for every space in the class GCB$^\\text{qc}(P_0,r_0,\\delta ;D)$ will give the continuity part of Theorem REF .", "A consequence of Theorem REF , together with its proof, is a quantified equidistribution of the orbit (see again [5] for a non-quantified version).", "Theorem C Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ , let $\\Gamma $ be a discrete, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ and let $x\\in \\textup {QC-Hull}(\\Lambda (\\Gamma ))$ .", "Then there exists $K>0$ depending only on $P_0,r_0,\\delta $ and $D$ such that for all $T\\ge 0$ it holds $\\frac{1}{K}\\cdot e^{T\\cdot h_\\Gamma } \\le \\Gamma x\\cap \\overline{B}(x,T) \\le K\\cdot e^{T\\cdot h_\\Gamma }.$" ], [ "Gromov hyperbolic metric spaces", "Throughout the paper $X$ will denote a metric space and $d$ will denote the metric on $X$ .", "The open (resp.closed) ball of radius $r$ and center $x$ is denoted by $B(x,r)$ (resp.", "$\\overline{B}(x,r)$ ).", "A geodesic segment is an isometry $\\gamma \\colon I \\rightarrow X$ where $I=[a,b]$ is a a bounded interval of $\\mathbb {R}$ .", "The points $\\gamma (a), \\gamma (b)$ are called the endpoints of $\\gamma $ .", "A metric space $X$ is said geodesic if for all couple of points $x,y\\in X$ there exists a geodesic segment whose endpoints are $x$ and $y$ .", "We will denote any geodesic segment between two points $x$ and $y$ , with an abuse of notation, as $[x,y]$ .", "A geodesic ray is an isometry $\\gamma \\colon [0,+\\infty )\\rightarrow X$ while a geodesic line is an isometry $\\gamma \\colon \\mathbb {R}\\rightarrow X$ .", "Let $Y$ be any subset of a metric space $X$ : – a subset $S$ of $Y$ is called $r$ -dense if $\\forall y \\in Y$ $\\exists z\\in S$ such that $d(y,z)\\le r$ ; – a subset $S$ of $Y$ is called $r$ -separated if $\\forall y,z \\in S$ it holds $d(y,z)> r$ .", "The packing number of $Y$ at scale $r$ is the maximal cardinality of a $2r$ -separated subset of $Y$ and it is denoted by $\\text{Pack}(Y,r)$ .", "The covering number of $Y$ is the minimal cardinality of a $r$ -dense subset of $Y$ and it is denoted by $\\text{Cov}(Y,r)$ .", "Let $X$ be a geodesic metric space.", "Given three points $x,y,z \\in X$ , the Gromov product of $y$ and $z$ with respect to $x$ is defined as $(y,z)_x = \\frac{1}{2}\\big ( d(x,y) + d(x,z) - d(y,z) \\big ).$ The space $X$ is said $\\delta $ -hyperbolic if for every four points $x,y,z,w \\in X$ the following 4-points condition hold: $(x,z)_w \\ge \\min \\lbrace (x,y)_w, (y,z)_w \\rbrace - \\delta $ or, equivalently, $d(x,y) + d(z,w) \\le \\max \\lbrace d(x,z) + d(y,w), d(x,w) + d(y,z) \\rbrace + 2\\delta .$ The space $X$ is Gromov hyperbolic if it is $\\delta $ -hyperbolic for some $\\delta \\ge 0$ .", "This formulation of $\\delta $ -hyperbolicity is convenient when interested in taking limits (since they are preserved under ultralimits).", "We will also make use of another classical characterization of $\\delta $ -hyperbolicity.", "A geodesic triangle in $X$ is the union of three geodesic segments $[x,y], [y,z],$ $[z,x]$ and is denoted by $\\Delta (x,y,z)$ .", "For every geodesic triangle there exists a unique tripod $\\overline{\\Delta }$ with vertices $\\bar{x},\\bar{y},\\bar{z}$ such that the lengths of $[\\bar{x}, \\bar{y}], [\\bar{y}, \\bar{z}], [\\bar{z}, \\bar{x}]$ equal the lengths of $[x,y], [y,z], [z,x]$ respectively.", "There exists a unique map $f_\\Delta $ from $\\Delta (x,y,z)$ to the tripod $\\overline{\\Delta }$ that identifies isometrically the corresponding edges, and there are exactly three points $c_x \\in [y,z], c_y \\in [x,z], c_z\\in [x,y]$ such that $f_\\Delta (c_x) = f_\\Delta (c_y) = f_\\Delta (c_z) = c$ , where $c$ is the center of the tripod $\\overline{\\Delta }$ .", "By definition of $f_\\Delta $ it holds: $d(x,c_z) = d(x,c_y), \\qquad d(y,c_x) = d(y,c_z), \\qquad d(z,c_x)=d(z,c_y).$ The triangle $\\Delta (x,y,z)$ is called $\\delta $ -thin if for every $u,v \\in \\Delta (x,y,z)$ such that $f_\\Delta (u)=f_\\Delta (v)$ it holds $d(u,v)\\le \\delta $ ; in particular the mutual distances between $c_x,c_y$ and $c_z$ are at most $\\delta $ .", "It is well-known that every geodesic triangle in a geodesic $\\delta $ -hyperbolic metric space (as defined above) is $4\\delta $ -thin.", "Moreover, the last condition is equivalent to the above definition of hyperbolicity, up to slightly increasing the hyperbolicity constant $\\delta $ in (REF ).", "The following is a basic property of Gromov-hyperbolic metric spaces.", "Lemma 2.1 (Projection Lemma, cp.", "Lemma 3.2.7 of [4]) Let $X$ be a $\\delta $ -hyperbolic metric space and let $x,y,z \\in X$ .", "For every geodesic segment $[y,z]$ we have $(y,z)_x \\ge d(x, [y,z]) - 4\\delta .$ Let $X$ be a proper, $\\delta $ -hyperbolic metric space and $x$ be a point of $X$ .", "The Gromov boundary of $X$ is defined as the quotient $\\partial X = \\lbrace (z_n)_{n \\in \\mathbb {N}} \\subseteq X \\hspace{2.84526pt} | \\hspace{2.84526pt} \\lim _{n,m \\rightarrow +\\infty } (z_n,z_m)_{x} = + \\infty \\rbrace \\hspace{2.84526pt} /_\\approx ,$ where $(z_n)_{n \\in \\mathbb {N}}$ is a sequence of points in $X$ and $\\approx $ is the equivalence relation defined by $(z_n)_{n \\in \\mathbb {N}} \\approx (z_n^{\\prime })_{n \\in \\mathbb {N}}$ if and only if $\\lim _{n,m \\rightarrow +\\infty } (z_n,z_m^{\\prime })_{x} = + \\infty $ .", "We will write $ z = [(z_n)] \\in \\partial X$ for short, and we say that $(z_n)$ converges to $z$ .", "This definition does not depend on the basepoint $x$ .", "There is a natural topology on $X\\cup \\partial X$ that extends the metric topology of $X$ .", "The Gromov product can be extended to points $z,z^{\\prime } \\in \\partial X$ by $(z,z^{\\prime })_{x} = \\sup _{(z_n) , (z_n^{\\prime }) } \\liminf _{n,m \\rightarrow + \\infty } (z_n, z_m^{\\prime })_{x}$ where the supremum is taken among all sequences such that $(z_n) \\approx z$ and $(z_n^{\\prime })\\approx z^{\\prime }$ .", "For every $z,z^{\\prime },z^{\\prime \\prime } \\in \\partial X$ it continues to hold $(z,z^{\\prime })_{x} \\ge \\min \\lbrace (z,z^{\\prime \\prime })_{x}, (z^{\\prime },z^{\\prime \\prime })_{x} \\rbrace - \\delta .$ Moreover for all sequences $(z_n),(z_n^{\\prime })$ converging to $z,z^{\\prime }$ respectively it holds $(z,z^{\\prime })_{x} -\\delta \\le \\liminf _{n,m \\rightarrow + \\infty } (z_n,z_m^{\\prime })_{x} \\le (z,z^{\\prime })_{x}.$ The Gromov product between a point $y\\in X$ and a point $z\\in \\partial X$ is defined in a similar way and it satisfies a condition analogue of (REF ).", "Every geodesic ray $\\xi $ defines a point $\\xi ^+=[(\\xi (n))_{n \\in \\mathbb {N}}]$ of the Gromov boundary $ \\partial X$ : we say that $\\xi $ joins $\\xi (0) = y$ to $\\xi ^+ = z$ , and we denote it by $[y, z]$ .", "Moreover for every $z\\in \\partial X$ and every $x\\in X$ it is possible to find a geodesic ray $\\xi $ such that $\\xi (0)=x$ and $\\xi ^+ = z$ .", "Any such geodesic ray is denoted as $\\xi _{xz} = [x,z]$ even if it is possibly not unique.", "Analogously, given different points $z = [(z_n)], z^{\\prime } = [(z^{\\prime }_n)] \\in \\partial X$ there always exists a geodesic line $\\gamma $ joining $z$ to $z^{\\prime }$ , i.e.", "such that $\\gamma |_{[0, +\\infty )}$ and $\\gamma |_{(-\\infty ,0]}$ join $\\gamma (0)$ to $z,z^{\\prime }$ respectively (just consider the limit $\\gamma $ of the segments $[z_n,z^{\\prime }_n]$ ; notice that all these segments intersect a ball of fixed radius centered at $x_0$ , since $(z_n,z^{\\prime }_m)_{x_0}$ is uniformly bounded above).", "We call $z$ and $z^{\\prime }$ the positive and negative endpoints of $\\gamma $ , respectively, denoted $\\gamma ^\\pm $ .", "The relation between Gromov product and geodesic ray is highlighted in the following well known lemma.", "Lemma 2.2 Let $X$ be a proper, $\\delta $ -hyperbolic metric space, $z,z^{\\prime }\\in \\partial X$ and $x\\in X$ .", "Then (i) if $(z,z^{\\prime })_{x} \\ge T$ then $d(\\xi _{xz}(T - \\delta ),\\xi _{xz^{\\prime }}(T - \\delta )) \\le 4\\delta $ ; (ii) for all $b> 0$ , if $d(\\xi _{xz}(T),\\xi _{xz^{\\prime }}(T)) < 2b$ then $(z,z^{\\prime })_{x} > T - b$ .", "Assume $(z,z^{\\prime })_{x} \\ge T$ and suppose $d(\\xi _z(T - \\delta ),\\xi _{z^{\\prime }}(T - \\delta )) > 4\\delta $ .", "We fix $S\\ge T - \\delta $ and we consider the triangle $\\Delta (x, \\xi _z(S), \\xi _{z^{\\prime }}(S))$ .", "We know there exist $a\\in [x,\\xi _z(S)], b\\in [x,\\xi _{z^{\\prime }}(S)], c\\in [\\xi _z(S), \\xi _{z^{\\prime }}(S)]$ such that $d(a,b)<\\delta ,\\,\\, d(b,c)<\\delta ,\\,\\, d(a,c)<\\delta $ and $T_\\delta := d(x,a)=d(x,b),$$d(\\xi _{xz}(S),a) = d(\\xi _{xz}(S),c),\\,\\, d(\\xi _{xz^{\\prime }}(S),b) = d(\\xi _{xz^{\\prime }}(S),c).$ Since this triangle is $4\\delta $ -thin we conclude that $T - \\delta > T_\\delta $ .", "Moreover $d(\\xi _{xz}(S),\\xi _{xz^{\\prime }}(S)) = d(\\xi _{xz}(S),c) + d(c,\\xi _{xz^{\\prime }}(S)) = 2(S - T_\\delta ).$ Hence $\\begin{aligned}(z,z^{\\prime })_{x} \\le \\liminf _{S\\rightarrow + \\infty }\\frac{1}{2}\\big ( 2S - d(\\xi _{xz}(S),\\xi _{xz^{\\prime }}(S)) \\big ) + \\delta = T_\\delta + \\delta < T\\end{aligned}$ where we have used (REF ).", "This contradiction concludes the first part.", "Now we assume $d(\\xi _{xz}(T),\\xi _{xz^{\\prime }}(T)) < 2b$ .", "Using $d(\\xi _{xz}(S),\\xi _{xz^{\\prime }}(S)) < 2(S-T) + 2b$ for all $S\\ge T$ we obtain, again by (REF ), $(z,z^{\\prime })_{x} \\ge \\liminf _{S\\rightarrow + \\infty }\\frac{1}{2}\\big ( 2S - d(\\xi _{xz}(S),\\xi _{xz^{\\prime }}(S)) \\big ) > T + b.$ Remark 2.3 We remark that the computation above shows also that if $z\\in \\partial X$ , $y\\in X$ and $(y,z)_x \\ge T$ then $d(x,y)> T-\\delta $ and for every geodesic segment $\\gamma = [x,y]$ it holds $d(\\gamma (T-\\delta ), \\xi _{xz}(T-\\delta )) \\le 4\\delta $ .", "The following is a standard computation, see for instance [1].", "Lemma 2.4 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Then every two geodesic rays $\\xi , \\xi ^{\\prime }$ with same endpoints at infinity are at distance at most $8\\delta $ , i.e.", "there exist $t_1,t_2\\ge 0$ such that $t_1+t_2=d(\\xi (0),\\xi ^{\\prime }(0))$ and $d(\\xi (t + t_1),\\xi ^{\\prime }(t+t_2)) \\le 8\\delta $ for all $t\\in \\mathbb {R}$ .", "The quasiconvex hull of a subset $C$ of $\\partial X$ is the union of all the geodesic lines joining two points of $C$ and it is denoted by QC-Hull$(C)$ .", "We need the following approximation result.", "Lemma 2.5 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Let $C\\subseteq \\partial X$ be a subset with at least two points and $x\\in \\textup {QC-Hull}(C)$ .", "Then for every $z\\in C$ it exists a geodesic line $\\gamma $ with endpoints in $C$ such that $d(\\xi _{xz}(t), \\gamma (t)) \\le 14\\delta $ for every $t\\ge 0$ .", "In particular $d(\\xi _{xz}(t), \\textup {QC-Hull}(C)) \\le 14\\delta $ .", "Since $x\\in \\textup {QC-Hull}(C))$ it exists a geodesic line $\\eta $ joining two points $\\eta ^-, \\eta ^+$ of $C$ such that $x\\in \\eta $ .", "Of course we have $(\\eta ^+,\\eta ^-)_x \\le \\delta $ , so by (REF ) we get $\\delta \\ge (\\eta ^+,\\eta ^-)_x \\ge \\min \\lbrace (\\eta ^+,z)_x, (\\eta ^-,z)_x\\rbrace - \\delta .$ Therefore one of the two values $(\\eta ^+,z)_x$ , $(\\eta ^-,z)_x$ is $\\le 2\\delta $ .", "Let us suppose it is the first one.", "We consider a geodesic line $\\gamma $ joining $\\eta ^+$ and $z$ .", "By Lemma REF we get, for every $S\\ge 0$ , $d(x,\\gamma ([-S,S])) \\le (\\gamma (-S), \\gamma (S))_x + 4\\delta .$ Taking $S\\rightarrow +\\infty $ the points $\\gamma (-S)$ and $\\gamma (S)$ converge respectively to $\\eta ^+$ and $z$ .", "Therefore by (REF ) we get $d(x,\\gamma ) \\le 6\\delta $ .", "Now we parametrize $\\gamma $ so that $d(x,\\gamma (0)) \\le 6\\delta $ , then by Lemma REF we have $d(\\xi _{xz}(t), \\gamma (t)) \\le 14\\delta $ for every $t\\ge 0$ .", "The Busemann function associated to $z\\in \\partial X$ with basepoint $x$ is the map $B_z(x,\\cdot )\\colon X \\rightarrow \\mathbb {R},\\qquad y \\mapsto \\lim _{T\\rightarrow +\\infty } (d(\\xi _{xz}(T), y) - T).$ It depends on the choice of the geodesic ray $[x,z]$ but two maps obtained taking two different geodesic rays are at bounded distance and the bound depends only on $\\delta $ .", "Every Busemann function is 1-Lipschitz." ], [ "Visual metrics and Minkowski dimension", "When $X$ is a proper, $\\delta $ -hyperbolic metric space it is known that the boundary $\\partial X$ is metrizable.", "A metric $D_{x,a}$ on $\\partial X$ is called a visual metric of parameter $a\\in \\left(0,\\frac{1}{2\\delta \\cdot \\log _2e}\\right)$ and center $x \\in X$ if there exists $V> 0$ such that for all $z,z^{\\prime } \\in \\partial X$ it holds $\\frac{1}{V}e^{-a(z,z^{\\prime })_{x}}\\le D_{x,a}(z,z^{\\prime })\\le V e^{-a(z,z^{\\prime })_{x}}.$ A visual metric is said standard if for all $z,z^{\\prime }\\in \\partial X$ it holds $(3-2e^{a\\delta })e^{-a(z,z^{\\prime })_{x}}\\le D_{x,a}(z,z^{\\prime })\\le e^{-a(z,z^{\\prime })_{x}}.$ For all $a$ as before and $x\\in X$ there exists always a standard visual metric of parameter $a$ and center $x$ , see [11].", "The generalized visual ball of center $z \\in \\partial X$ and radius $\\rho \\ge 0$ is $B(z,\\rho ) = \\bigg \\lbrace z^{\\prime } \\in \\partial X \\text{ s.t. }", "(z,z^{\\prime })_{x} > \\log \\frac{1}{\\rho } \\bigg \\rbrace .$ It is comparable to the metric balls of the visual metrics on $\\partial X$ .", "Lemma 2.6 Let $D_{x,a}$ be a visual distance of center $x$ and parameter $a$ on $\\partial X$ .", "Then for all $z\\in \\partial X$ and for all $\\rho >0$ it holds $B_{D_{x,a}}\\left(z, \\frac{1}{V}\\rho ^a\\right) \\subseteq B(z,\\rho )\\subseteq B_{D_{x,a}}(z, V\\rho ^a ).$ For all $z^{\\prime }\\in B(z,\\rho )$ by definition it holds $(z,z^{\\prime })_{x} > \\log \\frac{1}{\\rho }$ and therefore $D_{x,a}(z,z^{\\prime })\\le Ve^{-a(z,z^{\\prime })_{x}} < V\\rho ^a.$ If $z^{\\prime }\\in B_{D_{x,a}}(z,\\frac{1}{V}\\rho ^a)$ then $\\frac{1}{V}e^{-a(z,z^{\\prime })_{x}} \\le D_{x_0,a}(z,z^{\\prime }) < \\frac{1}{V}\\rho ^a.$ This easily implies $z^{\\prime }\\in B(z,\\rho )$ .", "It is classical that generalized visual balls are related to shadows, whose definition is the following.", "Let $x\\in X$ be a basepoint.", "The shadow of radius $r>0$ casted by a point $y\\in X$ with center $x$ is the set: $\\text{Shad}_x(y,r) = \\lbrace z\\in \\partial X \\text{ s.t. }", "[x,z]\\cap B(y,r) \\ne \\emptyset \\text{ for all rays } [x,z]\\rbrace .$ Lemma 2.7 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Let $z\\in \\partial X$ , $x\\in X$ and $T\\ge 0$ .", "Then (i) $B(z,e^{-T}) \\subseteq \\textup {Shad}_{x}\\left(\\xi _{xz}\\left(T\\right), 7\\delta \\right)$ ; (ii) $\\textup {Shad}_{x}\\left(\\xi _{xz}\\left(T\\right), r\\right) \\subseteq B(z, e^{-T + r})$ for all $r> 0$ .", "Let $z^{\\prime }\\in B(z,e^{-T})$ , i.e.", "$(z,z^{\\prime })_{x}> T$ .", "By Lemma REF we know that $d(\\xi _{xz}(T - \\delta ), \\xi _{xz^{\\prime }}(T - \\delta )) \\le 4\\delta .$ So $d(\\xi _{xz^{\\prime }}(T), \\xi _{xz}(T)) \\le 6\\delta < 7\\delta $ .", "This implies $z^{\\prime }\\in \\text{Shad}_{x}(\\xi _{xz}(T),7\\delta )$ , showing (i).", "Now we fix $z^{\\prime }\\in \\text{Shad}_{x}(\\xi _{xz}(T),r)$ , which means that every geodesic ray $\\xi _{xz^{\\prime }}$ passes through $B(\\xi _{xz}(T), r)$ , so $d(\\xi _{xz^{\\prime }}(T),\\xi _{xz}(T)) < 2r$ .", "By Lemma REF we conclude $(z,z^{\\prime })_{x} > T - r$ implying the second containment.", "The upper and lower Minkowski dimension of a subset $C$ of $\\partial X$ with respect to a visual metric $D_{x,a}$ are respectively classically defined as $\\overline{\\text{MD}}_{D_{x,a}}(C) = \\limsup _{\\rho \\rightarrow 0} \\frac{\\log \\text{Cov}_{D_{x,a}}(C, \\rho )}{\\log \\frac{1}{\\rho }},$ $\\underline{\\text{MD}}_{D_{x,a}}(C) = \\liminf _{\\rho \\rightarrow 0} \\frac{\\log \\text{Cov}_{D_{x,a}}(C, \\rho )}{\\log \\frac{1}{\\rho }},$ where the covering is considered with respect to the metric $D_{x,a}$ .", "Covering $C$ with generalized visual balls we define the upper and lower visual Minkowski dimension as $\\overline{\\text{MD}}(C) = \\limsup _{\\rho \\rightarrow 0} \\frac{\\log \\text{Cov}(C, \\rho )}{\\log \\frac{1}{\\rho }}, \\qquad \\underline{\\text{MD}}(C) = \\liminf _{\\rho \\rightarrow 0} \\frac{\\log \\text{Cov}(C, \\rho )}{\\log \\frac{1}{\\rho }}$ respectively, where $\\text{Cov}(C, \\rho )$ denotes the minimal number of generalized visual balls of radius $\\rho $ needed to cover $C$ .", "Lemma REF implies that the visual Minkowski dimensions do not depend on $x$ , indeed: Lemma 2.8 Let $D_{x,a}$ be a visual metric of center $x$ and parameter $a$ .", "Then for every $C\\subseteq \\partial X$ : $\\overline{\\textup {MD}}(C) = a\\cdot \\overline{\\textup {MD}}_{D_{x,a}}(C), \\qquad \\underline{\\textup {MD}}(C) = a\\cdot \\underline{\\textup {MD}}_{D_{x,a}}(C).$ A compact metric space $Z$ is $(A,s)$ -Ahlfors regular if there exists a probability measure $\\mu $ on $Z$ such that $\\frac{1}{A}\\rho ^s \\le \\mu (B(z,\\rho )) \\le A\\rho ^s$ for all $z\\in Z$ and all $0\\le \\rho \\le \\text{Diam}(Z)$ , where Diam$(Z)$ is the diameter of $Z$ .", "In case $Z=\\partial X$ we say that $Z$ is visual $(A,s)$ -Ahlfors regular if there exists a probability measure $\\mu $ on $\\partial X$ such that $\\frac{1}{A}\\rho ^s \\le \\mu (B(z,\\rho )) \\le A\\rho ^s$ for all $z\\in Z$ and all $0\\le \\rho \\le 1$ , where $B(z,\\rho )$ is the generalized visual ball of center $z$ and radius $\\rho $ .", "From Lemma REF it follows immediately: Lemma 2.9 If $\\partial X$ is $(A,s)$ -Ahlfors regular with respect to a visual metric of center $x$ and parameter $a$ , then it is visual $(AV^s,as)$ -Ahlfors regular, where $V$ is the constant of (REF )." ], [ "Limit set and critical exponent", "Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "A group of isometries $\\Gamma $ of $X$ is discrete if for all $x\\in X$ and $R\\ge 0$ the set $\\Sigma _R(x) = \\lbrace g \\in \\Gamma \\text{ s.t. }", "g x\\in \\overline{B}(x,R)\\rbrace $ is finite.", "Every isometry of $X$ acts naturally on $\\partial X$ and the resulting map on $X\\cup \\partial X$ is a homeomorphism.", "The limit set $\\Lambda (\\Gamma )$ of a discrete group of isometries $\\Gamma $ is the set of accumulation points of the orbit $\\Gamma x$ on $\\partial X$ , where $x$ is any point of $X$ ; it is the smallest $\\Gamma $ -invariant closed set of the Gromov boundary (cp.", "[5], Theorem 5.1).", "The group $\\Gamma $ is called elementary if $\\# \\Lambda (\\Gamma ) \\le 2$ .", "The set $\\Lambda (\\Gamma )$ is $\\Gamma $ -invariant so it is its quasiconvex hull.", "A discrete group of isometries $\\Gamma $ is called quasiconvex-cocompact if its action on QC-Hull$(\\Lambda (\\Gamma ))$ is cocompact, i.e.", "if there exists $D\\ge 0$ such that for all $x,y\\in \\text{QC-Hull}(\\Lambda (\\Gamma ))$ it holds $d(gx,y)\\le D$ for some $g\\in \\Gamma $ .", "The smallest $D$ satisfying this property is called the codiameter of $\\Gamma $ .", "The critical exponent of $\\Gamma $ is $h_\\Gamma := \\inf \\bigg \\lbrace s \\ge 0 \\text{ s.t. }", "\\sum _{g \\in \\Gamma }e^{-sd(x,g x)} < +\\infty \\bigg \\rbrace .$ It does not depend on $x\\in X$ .", "We remark that for every $s \\ge 0$ the series $\\sum _{g \\in \\Gamma }e^{-sd(x,g x)}$ , which is called the Poincaré series of $\\Gamma $ , is $\\Gamma $ -invariant.", "In other words $\\sum _{g \\in \\Gamma }e^{-sd(x,g x)} = \\sum _{g \\in \\Gamma }e^{-sd(x^{\\prime },g x^{\\prime })}$ for all $x^{\\prime } \\in \\Gamma x$ .", "The critical exponent coincides with the exponential growth rate of the cardinality of the orbits of the group.", "Lemma 2.10 (Proposition 5.3 of [5]) Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries of $X$ .", "Then $h_\\Gamma = \\limsup _{T\\rightarrow + \\infty } \\frac{1}{T} \\log \\# \\Gamma x \\cap \\overline{B}(x,T).$ There is a canonical way to construct a measure on $\\partial X$ starting from the Poincaré series.", "For every $s>h_\\Gamma $ the measure $\\mu _s = \\frac{1}{\\sum _{g \\in \\Gamma }e^{-sd(x,g x)}}\\sum _{g \\in \\Gamma }e^{-sd(x,g x)}\\Delta _{g x},$ where $\\Delta _{g x}$ is the Dirac measure at $g x$ , is a probability measure on the compact space $X\\cup \\partial X$ .", "Then there exists a sequence $s_i$ converging to $h_\\Gamma $ such that $\\mu _{s_i}$ converges $*$ -weakly to a probability measure on $X\\cup \\partial X$ .", "Any of these limits is called a Patterson-Sullivan measure and it is denoted by $\\mu _{\\text{PS}}$ .", "Proposition 2.11 (Theorem 5.4 of [5].)", "Let $X$ be a proper, $\\delta $ -hyperbolic metric space and let $\\Gamma $ be a discrete group of isometries of $X$ with $h_\\Gamma <+\\infty $ .", "Then every Patterson-Sullivan measure is supported on $\\Lambda (\\Gamma )$ .", "Moreover it is a $\\Gamma $ -quasiconformal density of dimension $h_\\Gamma $ , i.e.", "it satisfies $\\frac{1}{Q}\\cdot e^{h_\\Gamma (B_z(x,x) - B_z(x,g^{-1}x))} \\le \\frac{d(g_* \\mu _{\\textup {PS}})}{d\\mu _{\\textup {PS}}}(z) \\le Q\\cdot e^{h_\\Gamma (B_z(x,x) - B_z(x,g^{-1}x))}$ for every $g \\in \\Gamma $ and every $z\\in \\Lambda (\\Gamma )$ , where $Q$ is a constant depending only on $\\delta $ and an upper bound on $h_\\Gamma $ .", "The quantification of $Q$ is not explicitated in the original paper, but it follows from the proof therein." ], [ "Thresholds and Ahlfors regularity of the limit set", "In the first part we will recall the properties of GCB spaces, later we will proceed to the proof of the quantified Ahlfors regularity Theorem." ], [ "GCB-spaces", "Let $X$ be a metric space.", "A geodesic bicombing is a map $\\sigma \\colon X\\times X\\times [0,1] \\rightarrow X$ with the property that for all $(x,y) \\in X\\times X$ the function $\\sigma _{xy}\\colon t\\mapsto \\sigma (x,y,t)$ is a (parametrized proportionally to arc-length) geodesic from $x$ to $y$ , i.e.", "$d(\\sigma _{xy}(t), \\sigma _{xy}(t^{\\prime })) = \\vert t - t^{\\prime } \\vert d(x,y)$ for all $t,t^{\\prime }\\in [0,1]$ and $\\sigma _{xy}(0)=x, \\sigma _{xy}(1)=y$ .", "When $X$ is equipped with a geodesic bicombing then for all $x,y\\in X$ we will denote by $[x,y]$ the geodesic $\\sigma _{xy}$ parametrized by arc-length.", "A geodesic bicombing is: convex if the map $t\\mapsto d(\\sigma _{xy}(t), \\sigma _{x^{\\prime }y^{\\prime }}(t))$ is convex on $[0,1]$ for all $x,y,x^{\\prime },y^{\\prime } \\in X$ ; consistent if for all $x,y \\in X$ , for all $0\\le s\\le t \\le 1$ and for all $\\lambda \\in [0,1]$ it holds $\\sigma _{pq}(\\lambda ) = \\sigma _{xy}((1-\\lambda )s + \\lambda t)$ , where $p:= \\sigma _{xy}(s)$ and $q:=\\sigma _{xy}(t)$ ; reversible if $\\sigma _{xy}(t) = \\sigma _{yx}(1-t)$ for all $t\\in [0,1]$ .", "Every Busemann convex metric space (so also any CAT$(0)$ metric space) admits a unique convex, consistent, reversible geodesic bicombing.", "Given a geodesic bicombing $\\sigma $ we say that a geodesic (segment, ray, line) $\\gamma $ is a $\\sigma $ -geodesic (segment, ray, line) if for all $x,y\\in \\gamma $ we have that $[x,y]$ coincides with the subsegment of $\\gamma $ between $x$ and $y$ .", "A geodesic bicombing is geodesically complete if every $\\sigma $ -geodesic segment is contained in a $\\sigma $ -geodesic line.", "A couple $(X,\\sigma )$ is said a GCB-space if $\\sigma $ is a convex, consistent, reversible, geodesically complete geodesic bicombing on the complete metric space $X$ .", "This notion is relevant when combined with a packing (or covering) condition.", "We recall that the packing and the covering functions of $X$ are respectively $\\text{Pack}(R,r)=\\sup _{x\\in X}\\text{Pack}(\\overline{B}(x,R),r), \\qquad \\text{Cov}(R,r)=\\sup _{x\\in X}\\text{Cov}(\\overline{B}(x,R),r).$ It is classical that $\\text{Pack}(R,2r)\\le \\text{Cov}(R,2r)\\le \\text{Pack}(R,r).$ Let $P_0,r_0>0$ .", "We say that a metric space $X$ is $P_0$ -packed at scale $r_0$ if Pack$(3r_0,r_0)\\le P_0$ , that is every ball of radius $3r_0$ contains no more than $P_0$ points that are $2r_0$ -separated.", "The packing condition has a controlled behaviour in GCB-spaces.", "Proposition 3.1 (Proposition 3.2 of [6]) Let $(X,\\sigma )$ be a GCB-space that is $P_0$ -packed at scale $r_0$ .", "Then: (i) for all $r\\le r_0$ the space $X$ is $P_0$ -packed at scale $r$ and is proper; (ii) for every $0<r\\le R$ and every $x\\in X$ it holds: $\\begin{aligned}\\textup {Pack}(R,r)&\\le P_0(1+P_0)^{\\frac{R}{r} - 1} \\text{, if } r\\le r_0;\\\\\\textup {Pack}(R,r)&\\le P_0(1+P_0)^{\\frac{R}{r_0} - 1} \\text{, if } r > r_0;\\\\\\textup {Cov}(R,r)&\\le P_0(1+P_0)^{\\frac{2R}{r} - 1} \\text{, if } r\\le 2r_0;\\\\\\textup {Cov}(R,r)&\\le P_0(1+P_0)^{\\frac{R}{r_0} - 1} \\text{, if } r > 2r_0.\\end{aligned}$ When $(X,\\sigma )$ is a GCB-space we will always consider $\\sigma $ -isometries, that is isometries $g$ of $X$ such that $g\\sigma _{xy} = \\sigma _{g(x)g(y)}$ for every $x,y \\in X$ .", "The supremum of such $\\varepsilon _x$ 's is denoted by sys$^\\diamond (\\Gamma ,x)$ and it is called the free-systole of $\\Gamma $ at $x$ .", "Proposition 3.2 Let $(X,\\sigma )$ be a GCB-space that is $P_0$ -packed at scale $r_0$ and let $\\Gamma $ be a discrete group of $\\sigma $ -isometries.", "Then (i) $h_\\Gamma \\le \\frac{\\log (1+P_0)}{r_0} =: h^+$ ; (ii) if $X$ is also $\\delta $ -hyperbolic and $\\Gamma $ is non-elementary and quasiconvex cocompact with codiameter $\\le D$ then $h_\\Gamma \\ge h^- > 0$ , where $h^-$ depends only on $P_0,r_0,\\delta $ and $D$ .", "The second statement is exactly Example 5.8 of [6], so we focus on the first one.", "Since $\\Gamma $ is discrete for every $x\\in X$ there exists $\\varepsilon _x >0$ such that $d(x,gx)\\le \\varepsilon _x$ implies $gx = x$ .", "Therefore two points of the orbit $\\Gamma x$ are $\\varepsilon _x$ -separated, implying $\\#\\Gamma x \\cap \\overline{B}(x,T) \\le \\text{Pack}\\left(\\overline{B}(x,T), \\frac{\\varepsilon _x}{2}\\right) \\le \\text{Cov}\\left(\\overline{B}(x,T), \\frac{\\varepsilon _x}{2}\\right).$ Moreover clearly $\\begin{aligned}\\text{Cov}\\left(\\overline{B}(x,T), \\frac{\\varepsilon _x}{2}\\right) &\\le \\text{Cov}\\left(\\overline{B}(x,T), 2r_0\\right) \\cdot \\sup _{y\\in X}\\text{Cov}\\left(\\overline{B}(y,2r_0), \\frac{\\varepsilon _x}{2}\\right)\\\\&= \\text{Cov}\\left(\\overline{B}(x,T), 2r_0\\right) \\cdot \\text{Cov}\\left(2r_0, \\frac{\\varepsilon _x}{2}\\right).\\end{aligned}$ Therefore we get: $\\begin{aligned}\\limsup _{T\\rightarrow +\\infty }\\frac{1}{T}\\log \\#\\Gamma x \\cap \\overline{B}(x,T) &\\le \\limsup _{T\\rightarrow +\\infty }\\frac{1}{T} \\log \\text{Cov}\\left(\\overline{B}(x,T), 2r_0\\right) \\cdot \\text{Cov}\\left(2r_0, \\frac{\\varepsilon _x}{2}\\right)\\\\&= \\limsup _{T\\rightarrow +\\infty }\\frac{1}{T} \\log \\text{Cov}\\left(\\overline{B}(x,T), 2r_0\\right)\\\\&\\le \\limsup _{T\\rightarrow +\\infty }\\frac{1}{T} \\log P_0(1+P_0)^{\\frac{T}{r_0} - 1} = \\frac{\\log (1+P_0)}{r_0},\\end{aligned}$ where in every step we used Proposition REF ." ], [ "Ahlfors-regularity", "Let $\\Gamma $ be a discrete, quasiconvex-cocompact group of isometries of a proper, $\\delta $ -hyperbolic metric space $X$ .", "Then it is proved in [5] that the Patterson-Sullivan measure on $\\Lambda (\\Gamma )$ is $(A,h_\\Gamma )$ -Ahlfors regular for some $A> 0$ .", "We want to precise this result quantifying the constant $A$ .", "Before we introduce the following notation: two functions $f,g\\colon [0,+\\infty ) \\rightarrow \\mathbb {R}$ have the same asymptotic behaviour if for all $\\varepsilon > 0$ there exists $T_\\varepsilon \\ge 0$ such that if $T\\ge T_\\varepsilon $ then $\\vert f(T) - g(T) \\vert \\le \\varepsilon $ .", "In this case we write $f\\asymp g$ .", "The function $T_\\varepsilon $ is called the threshold function.", "Moreover we write $f\\underset{P_0,r_0,\\delta ,\\ldots }{\\asymp } g$ meaning that the threshold function can be expressed only in terms of $\\varepsilon $ and $P_0,r_0,\\delta ,\\ldots $ For instance if $g$ is constantly equal to $g_0 \\in \\mathbb {R}$ and $f \\underset{P_0,r_0,\\delta ,\\ldots }{\\asymp } g_0$ then the function $f$ tends to $g_0$ when $T$ goes to $+\\infty $ and moreover the rate of convergence of the limit can be expressed only in terms of $P_0,r_0,\\delta ,\\ldots $ Theorem 3.3 Let $X$ be a proper, $\\delta $ -hyperbolic metric space.", "Let $\\Gamma $ be a discrete, quasiconvex-cocompact group of isometries of $X$ with codiameter $\\le D$ and $x$ be a point of $\\textup {QC-Hull}(\\Lambda (\\Gamma ))$ .", "Suppose $0 < h^- \\le h_\\Gamma \\le h^+ < +\\infty $ .", "Then: (i) $\\Lambda (\\Gamma )$ is visually $(A,h_\\Gamma )$ -Ahlfors regular with respect to any Patterson-Sullivan measure, where $A$ depends only on $h^-,h^+,\\delta $ and $D$ .", "(ii) it holds $\\frac{1}{T}\\log \\textup {Cov}(\\Lambda (\\Gamma ), e^{-T}) \\underset{h^-,h^+,\\delta ,D}{\\asymp } h_\\Gamma ;$ (iii) $\\overline{\\textup {MD}}(\\Lambda (\\Gamma )) = \\underline{\\textup {MD}}(\\Lambda (\\Gamma )) = h_\\Gamma $ .", "We observe that (iii) follows immediately from (ii), while (ii) is essentially straightforward once proved (i).", "Indeed we have Lemma 3.4 Suppose $C\\subseteq \\partial X$ is visually $(A,s)$ -Ahlfors regular.", "Then $\\frac{1}{T}\\log \\textup {Cov}(C, e^{-T}) \\underset{\\delta ,A,s}{\\asymp } s.$ We define the packing$^*$ number at scale $\\rho $ of a subset $C$ of $\\partial X$ as the maximal number of disjoint generalized visual balls of radius $\\rho $ with center in $C$ and we denote it by $\\text{Pack}^*(C, \\rho )$ .", "Lemma 3.5 For all $T\\ge 0$ it holds $\\textup {Pack}^*(C, e^{-T +\\delta }) \\le \\textup {Cov}(C, e^{-T})$ and $\\textup {Cov}(C, e^{-T +\\delta }) \\le \\textup {Pack}^*(C, e^{-T}).$ Let $z_1, \\ldots , z_N$ be points of $C$ realizing $\\textup {Cov}(C, e^{-T})$ .", "Suppose there exist points $w_1,\\ldots ,w_M$ of $C$ such that $B(w_i, e^{-T +\\delta })$ are disjoint, in particular $(w_i,w_j)_{x}\\le T - \\delta $ for every $i\\ne j$ .", "If $M> N$ then two different points $w_i,w_j$ belong to the same ball $B(z_k, \\rho )$ , i.e.", "$(z_k,w_i)_{x}> T$ and $(z_k,w_j)_{x}> T.$ By (REF ) we have $(w_i,w_j)_{x}> T - \\delta $ which is a contradiction.", "This shows the first inequality.", "Now let $z_1,\\ldots ,z_N$ be a maximal collection of points of $C$ such that $B(z_i, \\rho )$ are disjoint.", "Then for every $z\\in C$ there exists $i$ such that $B(z,\\rho ) \\cap B(z_i,\\rho ) \\ne \\emptyset $ .", "Therefore there exists $w\\in \\partial X$ such that $(z_i,w)_{x}> T$ and $(z,w)_{x}> T$ .", "As before we get $(z_i,z)_{x}> T- \\delta ,$ proving the second inequality.", "[Proof of Lemma REF .]", "Since the measure $\\mu $ in the definition of Ahlfors regularity is assumed to be of total measure one, we have $1 = \\mu (C) \\le Ae^{-sT}\\cdot \\text{Cov}(C, e^{-T}) \\text{ and } 1 = \\mu (C) \\ge \\frac{1}{A}e^{-sT} \\cdot \\text{Pack}^*(C, e^{-T})$ implying $\\text{Cov}(C, e^{-T}) \\ge \\frac{1}{A}e^{sT}$ and $\\text{Pack}^*(C, e^{-T}) \\le Ae^{sT}.$ Therefore $\\frac{1}{T}\\log \\text{Cov}(C, e^{-T}) \\ge s + \\frac{1}{T}\\log \\frac{1}{A}$ and $\\frac{1}{T}\\log \\text{Cov}(C, e^{-T}) \\le \\frac{1}{T}\\log \\text{Pack}^*(C, e^{-T - \\delta }) \\le s + \\frac{1}{T}\\log A + \\frac{s\\delta }{T}.$ [Proof of Theorem REF ] As observed (iii) follows from (ii) and (ii) follows from (i) applying Lemma REF and the assumption $h_\\Gamma \\le h^+$ .", "In order to prove (i) we consider two cases: if $\\Gamma $ is elementary then $\\#\\Lambda (\\Gamma )\\in \\lbrace 0,2\\rbrace $ and $h_\\Gamma =0$ .", "If this cardinality is 0 there is nothing to prove.", "If $\\Lambda (\\Gamma )=\\lbrace z^-,z^+\\rbrace $ then it is straightfoward to see that $\\mu _{\\text{PS}}(z^-) = \\mu _{\\text{PS}}(z^+) =\\frac{1}{2}$ .", "Step 1: $\\forall z\\in \\partial X$ and $\\forall \\rho >0$ it holds $\\mu _{\\textup {PS}}(B(z,\\rho ))\\le e^{h_\\Gamma (55\\delta + 3D)}\\rho ^{h_\\Gamma }$ .", "We suppose first $z\\in \\Lambda (\\Gamma )$ and we take the set $\\tilde{B}(z,\\rho )= \\left\\lbrace y \\in X \\cup \\partial X \\text{ s.t. }", "(y,z)_{x} > \\log \\frac{1}{\\rho } \\right\\rbrace .\\vspace{-8.53581pt}$ It is open (cp.", "Observation 4.5.2 of [8]) and $\\tilde{B}(z,\\rho )\\cap \\partial X = B(z,\\rho )$ , so $\\mu _{\\text{PS}}(\\tilde{B}(z,\\rho )) = \\mu _{\\text{PS}}(B(z,\\rho ))$ since $\\mu _{\\text{PS}}$ is supported on $\\Lambda (\\Gamma )\\subseteq \\partial X$ .", "Let $T=\\log \\frac{1}{\\rho }$ and $\\xi _{xz}$ be a geodesic ray $[x,z]$ .", "For every $y\\in \\Gamma x \\cap \\tilde{B}(z,\\rho )$ we have $d(x,y)\\ge T - \\delta \\,\\,\\,\\text{ and } \\,\\,\\, d(x,y) \\ge d(x,\\xi _{xz}(T)) + d(\\xi _{xz}(T), y) - 12\\delta .$ The first inequality is given by Remark REF .", "Let $\\gamma $ be any geodesic segment $[x,y]$ .", "Again by Remark REF we have $d(\\xi _{xz}(T), \\gamma (T)) \\le 6\\delta $ , therefore $d(x,y) = d(x,\\gamma (T)) + d(\\gamma (T), y) \\ge d(x,\\xi _{xz}(T)) + d(\\xi _{xz}(T), y) -12\\delta .$ Moreover we have $d(\\xi _{xz}(T), \\text{QC-Hull}(\\Lambda (\\Gamma ))) \\le 14\\delta $ by Lemma REF , since $x\\in \\text{QC-Hull}(\\Lambda (\\Gamma ))$ .", "By the cocompactness of the action on $\\text{QC-Hull}(\\Lambda (\\Gamma ))$ we can find a point $x_1 \\in \\Gamma x$ such that $d(\\xi _{xz}(T), x_1)\\le 14\\delta +D$ .", "This implies $d(x,y) \\ge d(x,x_1) + d(x_1, y) - 40\\delta - 2D$ for every $y\\in \\Gamma x \\cap \\tilde{B}(z,\\rho )$ .", "Therefore $\\begin{aligned}\\sum _{y \\in \\Gamma x \\cap \\tilde{B}(z,\\rho )} e^{-s d(x,y)} &\\le \\sum _{y \\in \\Gamma x \\cap \\tilde{B}(z,\\rho )} e^{-s (d(x, x_1) + d(x_1,y) - 40 \\delta - 2D)} \\\\&= e^{s(40\\delta + 2D)}e^{-sd(x,x_1)}\\cdot \\sum _{y \\in \\Gamma x \\cap \\tilde{B}(z,\\rho )}e^{-sd(x_1,y)} \\\\&\\le e^{s(54\\delta + 3D)}e^{-sd(x,\\xi _{xz}(T))}\\cdot \\sum _{g \\in \\Gamma }e^{-sd(x_1,g x_1)} \\\\&= e^{s(54\\delta + 3D)}\\cdot \\rho ^s\\cdot \\sum _{g \\in \\Gamma }e^{-sd(x,g x)}.\\end{aligned}$ In other words we have $\\mu _s(\\tilde{B}(z,\\rho )) \\le e^{s(54\\delta + 3D)}\\rho ^s$ , and by $\\ast $ -weak convergence we conclude that $\\begin{aligned}\\mu _{\\text{PS}}(B(z,\\rho )) = \\mu _{\\text{PS}}(\\tilde{B}(z,\\rho )) \\le \\liminf _{i\\rightarrow +\\infty } \\mu _{s_i}(\\tilde{B}(z,\\rho )) \\le e^{h_\\Gamma (54\\delta + 3D)}\\rho ^{h_\\Gamma }.\\end{aligned}\\vspace{-5.69054pt}$ In the general case of $z\\in \\partial X$ we observe that if $B(z,\\rho )\\cap \\Lambda (\\Gamma )=\\emptyset $ then the thesis is obviously true since $\\mu _{\\text{PS}}$ is supported on $\\Lambda (\\Gamma )$ .", "Otherwise there exists $w\\in \\Lambda (\\Gamma )$ such that $(z,w)_{x} > \\log \\frac{1}{\\rho }$ .", "It is straightforward to check that $B(w,\\rho ) \\subseteq B(z,\\rho e^\\delta )$ by (REF ).", "Then $\\mu _{\\text{PS}}(B(z,\\rho ))\\le e^{h_\\Gamma (55\\delta + 3D)}\\rho ^{h_\\Gamma }$ .", "Step 2: for every $R\\ge R_0 :=\\frac{\\log 2}{h_\\Gamma } + 55\\delta + 3D + 5\\delta $ and for every $g \\in \\Gamma $ it holds $\\mu _{\\textup {PS}}(\\textup {Shad}_{x}(g x, R)) \\ge \\frac{1}{2Q}e^{-h_\\Gamma d(x,g x)},$ where $Q$ is the constant of Proposition REF that depends only on $\\delta $ and $h^+$ .", "From the first step we know that for every $\\rho \\le \\rho _0 := 2^{-\\frac{1}{h_\\Gamma }}e^{-(55\\delta + 3D)}$ and for every $z\\in \\partial X$ it holds $\\mu _{\\text{PS}}(B(z,\\rho ))\\le \\frac{1}{2}.$ A direct computation shows that $R_0 = \\log \\frac{1}{\\rho _0} + 5\\delta $ .", "We claim that for every $R\\ge R_0$ and every $g \\in \\Gamma $ the set $\\partial X \\setminus g(\\text{Shad}_{x}(g^{-1}x,R))$ is contained in a generalized visual ball of radius at most $\\rho _0$ .", "Indeed if $z,w \\in \\partial X \\setminus g(\\text{Shad}_{x}(g^{-1}x,R))$ then there are geodesic rays $\\xi =[g x, z], \\xi ^{\\prime }=[g x, w]$ that do not intersect the ball $B(x,R)$ .", "Therefore we get $(\\xi (T),g x)_{x} \\ge d(x, [g x, \\xi (T)]) - 4\\delta \\ge R - 4\\delta $ by Lemma REF .", "This implies $(z,g x)_{x} \\ge \\liminf _{T\\rightarrow +\\infty } (\\xi (T),g x)_{x} \\ge R - 4\\delta $ and the same holds for $w$ .", "Thus by (REF ) we get $(z,w)_{x} \\ge R - 5\\delta ,$ proving the claim.", "By Proposition REF we get $\\frac{\\mu _{\\text{PS}}(\\text{Shad}_{x}(g x, R))}{\\mu _{\\text{PS}}(g^{-1}(\\text{Shad}_{x}(g x, R)))} \\ge \\frac{1}{Q}e^{-h_\\Gamma (B_z(x,x) - B_z(x,g x))}.$ Since $R\\ge R_0$ the measure of $g^{-1}(\\text{Shad}_{x}(g x, R))$ is at least $\\frac{1}{2}$ and the Busemann function is 1-Lipschitz, so $\\mu _{\\text{PS}}(\\text{Shad}_{x}(g x, R)) \\ge \\frac{1}{2Q}e^{-h_\\Gamma d(x,gx)}.$ Step 3.", "$\\mu _{\\textup {PS}}(B(z,\\rho ))\\ge \\frac{1}{2Q}e^{-h_\\Gamma (R_0+28\\delta +2D)}\\rho ^{h_\\Gamma }$ for every $z\\in \\Lambda (\\Gamma )$ and every $\\rho >0$ .", "For every $\\rho > 0$ we set $T=\\log \\frac{1}{\\rho }$ .", "If $z\\in \\partial X$ and $R\\ge 0$ then by Lemma REF we get Shad$_{x}(\\xi _{xz}(T+R), R) \\subseteq B(z,e^{-T})$ .", "We take $R=R_0 + 14\\delta + D$ , where $R_0$ is the constant of the second step and we conclude that Shad$_{x}(\\xi _{xz}(T+R), R)$ is contained in $B(z,\\rho )$ .", "Applying again Lemma REF and the cocompactness of the action we can find $g \\in \\Gamma $ such that $d(gx, \\xi _{xz}(T+R))\\le 14\\delta + D$ , implying Shad$_{x}(gx,R_0)\\subseteq \\text{Shad}_{x}(\\xi _{xz}(T+R), R) \\subseteq B(z,\\rho )$ .", "From the second step we obtain $\\mu _{\\text{PS}}(B(z,\\rho )) \\ge \\frac{1}{2Q}e^{-h_\\Gamma d(x,g x)}.$ Furthermore $d(x, g x) \\le T + R_0 + 28\\delta + 2D,$ so finally $\\mu _{\\text{PS}}(B(z,\\rho ))\\ge \\frac{1}{2Q}e^{-h_\\Gamma (R_0+28\\delta +2D)}\\rho ^{h_\\Gamma }.$ The explicit description of the constants shows as they depend only on $\\delta , h^-,h^+$ and $D$ , proving the theorem.", "A direct application of Proposition REF gives Theorem REF .", "Corollary 3.6 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic, GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a discrete, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ and $x$ be a point of $\\textup {QC-Hull}(\\Lambda (\\Gamma ))$ .", "Then: (i) $\\Lambda (\\Gamma )$ is visually $(A,h_\\Gamma )$ -Ahlfors regular with respect to any Patterson-Sullivan measure, where $A$ depends only on $P_0,r_0,\\delta $ and $D$ .", "(ii) it holds $\\frac{1}{T}\\log \\textup {Cov}(\\Lambda (\\Gamma ), e^{-T}) \\underset{P_0,r_0,\\delta ,D}{\\asymp } h_\\Gamma ;$ (iii) $\\overline{\\textup {MD}}(\\Lambda (\\Gamma )) = \\underline{\\textup {MD}}(\\Lambda (\\Gamma )) = h_\\Gamma $ .", "Following again the ideas of [5] we can prove a uniform distribution theorem for the orbit of a quasiconvex-cocompact group of isometries.", "Theorem 3.7 Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a discrete, non-elementary, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ and $x$ be a point of $\\textup {QC-Hull}(\\Lambda (\\Gamma ))$ .", "Then there exists $K > 0$ depending only on $P_0,r_0,\\delta $ and $D$ such that for all $T\\ge 0$ it holds $\\frac{1}{K}\\cdot e^{T\\cdot h_\\Gamma } \\le \\Gamma x \\cap \\overline{B}(x,T) \\le K\\cdot e^{T\\cdot h_\\Gamma }.$ We remark that for this theorem we need a control of the packing function, so the GCB-assumption seems necessary.", "We also need: Proposition 3.8 (Example 5.8 of [6]) Let $(X,\\sigma )$ be a $\\delta $ -hyperbolic GCB-space that is $P_0$ -packed at scale $r_0$ .", "Let $\\Gamma $ be a discrete, non-elementary, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ .", "Then there exists $s_0 = s_0(P_0,r_0,\\delta ,D) > 0$ depending only on $P_0,r_0,\\delta $ and $D$ such that for all $x\\in X$ and all $g\\in \\Gamma $ , if $d(x,gx)\\le s_0$ then $x=gx$ .", "Let $s_0 = s_0(P_0,r_0,\\delta , D)$ be the number of Proposition REF , $R_0 = R_0(P_0,r_0,\\delta , D)$ be the number of Step 2 of Theorem REF and $Q$ be the constant of Proposition REF .", "We recall that $R_0$ can be expressed in terms of $P_0,r_0,\\delta $ and $D$ by Proposition REF .", "Moreover we set $N_0 = \\text{Pack}\\left(4R_0+1,\\frac{s_0}{2}\\right)$ , which depends only on $P_0,r_0,\\delta ,D$ by Proposition REF .", "It is easy to check that if $[x,z]\\cap B(y,R_0) \\ne \\emptyset $ and $[x,z]\\cap B(y^{\\prime },R_0) \\ne \\emptyset $ , where $z\\in \\partial X$ and $y,y^{\\prime }$ are points of $X$ with $\\vert d(x,y) - d(x,y^{\\prime })\\vert \\le 1$ , then $d(y,y^{\\prime })\\le 4R_0 + 1$ .", "Therefore for every $j\\in \\mathbb {N}$ we have $\\#\\lbrace y\\in \\Gamma x \\text{ s.t. }", "y\\in A(x,j,j+1) \\text{ and } z\\in \\textup {Shad}_x(y,R_0) \\rbrace \\le N_0.$ Step 1.", "For all $k\\in \\mathbb {N}$ it holds $\\#\\Gamma x \\cap \\overline{B}(x,k) \\le 4QN_0e^{h_\\Gamma k}.$ Let $A_j = \\Gamma x \\cap A(x,j,j+1)$ .", "By the observation made before we conclude that among the set of shadows $\\lbrace \\text{Shad}_x(y,R_0) \\rbrace _{y\\in A_j}$ there are at least $\\frac{\\# A_j}{N_0}$ disjoint sets.", "Thus $1 \\ge \\mu _{\\text{PS}}\\left( \\bigcup _{y\\in A_j}\\text{Shad}_x(y,R_0) \\right) \\ge \\frac{\\# A_j}{N_0}\\cdot \\frac{1}{2Q}e^{-h_\\Gamma (j+1)},$ where we used Step 2 of Theorem REF .", "This implies $\\#A_j \\le 2QN_0e^{h_\\Gamma (j+1)}$ for every $j\\in \\mathbb {N}$ .", "Finally we have $\\#\\Gamma x \\cap \\overline{B}(x,k) \\le \\sum _{j=0}^{k-1} \\#A_j \\le 4QN_0e^{h_\\Gamma k}.$ Step 2.", "For all $T\\ge 0$ it holds $\\#\\Gamma x \\cap \\overline{B}(x,T) \\ge e^{-h_\\Gamma (84\\delta + 5D + 1)}e^{h_\\Gamma T}.$ We fix $z_1,\\ldots ,z_N \\in \\Lambda (\\Gamma )$ realizing Pack$^*(\\Lambda (\\Gamma ), e^{-T + 28\\delta + 2D + 1})$ : in particular $(z_i,z_j)_x \\le T - 28\\delta - 2D - 1$ for all $1\\le i \\ne j \\le N$ .", "By Lemma REF we deduce that $d(\\xi _{xz_i}(T-14\\delta -D), \\xi _{xz_j}(T-14\\delta -D)) \\ge 28\\delta + 2D + 1 > 28\\delta + 2D$ .", "Moreover for every $1\\le i \\le N$ we can find a point $y_i\\in \\Gamma x$ such that $d(\\xi _{xz_i}(T-14\\delta -D), y_i) \\le 14\\delta + D$ by Lemma REF .", "Therefore we have $d(x,y_i)\\le T$ and $d(y_i,y_j) > 0$ for every $1\\le i \\ne j \\le N$ .", "So $\\begin{aligned}\\#\\Gamma x \\cap \\overline{B}(x,T) &\\ge \\text{Pack}^*(\\Lambda (\\Gamma ), e^{-T+28\\delta + 2D + 1}) \\\\&\\ge \\text{Cov}(\\Lambda (\\Gamma ), e^{-T+29\\delta + 2D + 1}) \\\\&\\ge e^{-h_\\Gamma (84\\delta + 5D + 1)}e^{h_\\Gamma T}.\\end{aligned}$ The first inequality follows from the discussione above, while the second one is Lemma REF .", "The last inequality can be prove as in Lemma REF .", "Indeed, using Step 1 of Theorem REF , we get $\\begin{aligned}\\text{Cov}(\\Lambda (\\Gamma ), e^{-T + 29\\delta + 2D + 1}) &\\ge e^{-h_\\Gamma (55\\delta + 3D)}e^{-h_\\Gamma (-T + 29\\delta + 2D + 1)} \\\\ &=e^{-h_\\Gamma (84\\delta + 5D + 1)}e^{h_\\Gamma T}.\\end{aligned}$ The thesis follows by the bounded quantification of all the constants involved in terms of $P_0,r_0,\\delta $ and $D$ .", "We remark that the non-elementarity of $\\Gamma $ is essential to this result.", "Indeed if one consider $X=\\mathbb {R}$ and $\\Gamma _\\lambda = \\lambda \\mathbb {Z}$ , where $\\lambda $ is any positive real number, then $\\#\\lambda \\mathbb {Z} \\cap [-n,n] \\ge \\frac{2n}{\\lambda }$ showing that there is no a uniform upper bound when $\\lambda $ tends to 0." ], [ "Compactness and continuity", "We denote by GCB$(P_0,r_0)$ the class of pointed GCB-spaces $(X,x,\\sigma )$ that are $P_0$ -packed at scale $r_0$ and by GCB$(P_0,r_0,\\delta )$ its subclass made of $\\delta $ -hyperbolic metric spaces.", "We recall that both GCB$(P_0,r_0)$ and GCB$(P_0,r_0,\\delta )$ are closed under ultralimits, see [6].", "In [6] it is introduced the notion of ultralimit of groups: if $(X_n,x_n)$ is a sequence of pointed metric spaces and if $\\Gamma _n$ is a group of isometries of $X_n$ for every $n$ , then a sequence $\\lbrace g_n \\rbrace $ , where $g_n\\in \\Gamma _n$ , is said admissible if $\\sup _n d(x_n,g_n x_n) < +\\infty $ .", "For every non-principal ultrafilter $\\omega $ we have that an admissible sequence defines an isometry $g_\\omega = \\omega $ -$\\lim g_n$ of the ultralimit space $X_\\omega $ that acts as $g_\\omega y_\\omega = \\omega $ -$\\lim g_n y_n$ , where $(y_n)$ is any sequence such that $\\omega $ -$\\lim y_n = y_\\omega $ .", "The set of isometries of $\\Gamma _\\omega $ defined by admissible sequences is called the ultralimit group and it is denoted by $\\Gamma _\\omega $ .", "For further properties of the ultralimit group we refer to [6].", "We just recall that if $(X_n,x_n,\\sigma _n) \\subseteq \\text{GCB}(P_0,r_0)$ and if $\\Gamma _n$ is $\\sigma _n$ -invariant for $\\omega $ -a.e.$(n)$ then $\\Gamma _\\omega $ is $\\sigma _\\omega $ -invariant.", "Moreover in [3] the author proved that in the situation above there exists a subsequence $(X_{n_k}, x_{n_k}, \\sigma _{n_k}, \\Gamma _{n_k})$ that converges in the pointed equivariant Gromov-Hausdorff sense (as introduced in [9]) to $(X_\\omega , x_\\omega , \\sigma _\\omega , \\Gamma _\\omega )$ .", "Therefore in order to prove compactness of a class of 4-uples as above under equivariant Gromov-Hausdorff convergence it is enough to show closure under ultralimits (and it is actually an equivalent condition, see [3])." ], [ "Convergence of the boundaries and compactness", "The boundary is stable under ultralimits.", "Proposition 4.1 Let $(X_n,x_n,\\sigma _n)\\subseteq \\textup {GCB}(P_0,r_0,\\delta )$ and let $D_{x_n,a}$ be a standard visual metric of parameter $a$ and center $x_n$ on $\\partial X_n$ .", "Let $\\omega $ be a non-principal ultrafilter and let $(X_\\omega , x_\\omega , \\sigma _\\omega )$ be the ultralimit of the sequence $(X_n,x_n, \\sigma _n)$ .", "Then there exists a visual metric $D_{x_\\omega , a}$ of parameter $a$ and center $x_\\omega $ on $\\partial X_\\omega $ such that $\\omega $ -$\\lim (\\partial X_n, D_{x_n,a})$ is isometric to $(\\partial X_\\omega , D_{x_\\omega , a})$ .", "We observe that since the spaces $\\partial X_n$ are compact with diameter at most 1 then the ultralimit $\\omega $ -$\\lim \\partial X_n$ does not depend on the basepoints.", "A point of $\\omega $ -$\\lim \\partial X_n$ is a class of a sequence of points $(z_n) \\in \\partial X_n$ and each point $z_n$ is identified to the unique $\\sigma _n$ -geodesic ray $\\xi _n =[x_n,z_n]$ , see [6].", "The sequence of $\\sigma _n$ -geodesic rays $(\\xi _n)$ defines a $\\sigma _\\omega $ -geodesic ray $\\xi _\\omega $ of $X_\\omega $ with $\\xi _\\omega (0)=x_\\omega $ which provides a point of $\\partial X_\\omega $ .", "It is then defined the map $\\Psi \\colon \\omega $ -$\\lim \\partial X_n \\rightarrow \\partial X_\\omega $ that sends the sequence $(z_n)$ to the boundary point identified by the $\\sigma _\\omega $ -geodesic ray $\\xi _\\omega $ .", "Good definition.", "We need to show that $\\Psi $ is well defined.", "Let $(z_n^{\\prime })$ be another sequence of points equivalent to $(z_n)$ , i.e.", "$\\omega \\text{-}\\lim D_{x_n,a}(z_n,z_n^{\\prime }) = 0$ .", "Since $D_{x_n,a}$ is a standard visual metric for every $n$ this implies that for all $\\varepsilon > 0$ and for $\\omega $ -a.e.$(n)$ it holds $(z_n,z_n^{\\prime })_{x_n} > \\log \\frac{1}{\\varepsilon } =: T_\\varepsilon $ .", "By Lemma REF we have $d(\\xi _n(T_\\varepsilon - \\delta ), \\xi _n^{\\prime }(T_\\varepsilon - \\delta )) \\le 4\\delta $ and, by convexity of $\\sigma _n$ , we have that $d(\\xi _n(S_\\eta ), \\xi _n^{\\prime }(S_\\eta )) < \\eta $ , where $S_\\eta = \\eta \\cdot \\frac{T_\\varepsilon }{4\\delta }$ for all $\\eta >0$ .", "This means that for every $T\\ge 0$ and every $\\eta > 0$ we have $d(\\xi _n(T), \\xi _n^{\\prime }(T)) < \\eta $ for $\\omega $ -a.e.$(n)$ .", "Since $\\eta $ is arbitrary we obtain that $\\xi _\\omega $ and $\\xi _\\omega ^{\\prime }$ coincide up to time $T$ for every $T\\ge 0$ and therefore $\\xi _\\omega = \\xi _\\omega ^{\\prime }$ .", "Bijectivity.", "The next step is to show that $\\Psi $ is bijective.", "It is clearly surjective since the Gromov boundary of $X_\\omega $ coincides with the $\\sigma _\\omega $ -boundary of $X_\\omega $ and every $\\sigma _\\omega $ -geodesic ray of $X_\\omega $ is ultralimit of $\\sigma _n$ -geodesic rays of $X_n$ by definition, see again [6].", "Let us show it is injective: if two sequence of points $(z_n), (z_n^{\\prime })$ have the same image under $\\Psi $ then for all $T\\ge 0$ and for every $\\eta >0$ we have that for $\\omega $ -a.e.$(n)$ the $\\sigma _n$ -geodesic rays $\\xi _{x_nz_n}$ and $\\xi _{x_nz_n^{\\prime }}$ stay at distance less than $2\\eta $ up to time $T$ .", "By Lemma REF we conclude that $(z_n,z_n^{\\prime })_{x_n} > T - \\eta $ and therefore $D_{x_n,a}(z_n,z_n^{\\prime })\\le e^{-a(T - \\eta )}$ .", "Since this is true for $\\omega $ -a.e.$(n)$ we get $\\omega $ -$\\lim D_{x_n,a}(z_n,z_n^{\\prime }) \\le e^{-a(T - \\eta )}$ implying $\\omega $ -$\\lim D_{x_n,a}(z_n,z_n^{\\prime }) = 0$ , i.e.", "$(z_n) = (z_n^{\\prime })$ as elements of $\\omega $ -$\\lim \\partial X_n$ , by the arbitrariness of $T$ and $\\eta $ .", "Homeomorphism.", "Let us show $\\Psi $ is continuous.", "Both $\\omega $ -$\\lim \\partial X_n$ and $\\partial X_\\omega $ are metrizable, then it is enough to check the continuity on sequences of points.", "We take a sequence $(z_n^k)_{k\\in \\mathbb {N}}$ converging to $(z_n^\\infty )$ in $\\omega $ -$\\lim \\partial X_n$ .", "This means that for every $\\varepsilon > 0$ there exists $k_\\varepsilon \\ge 0$ such that if $k\\ge k_\\varepsilon $ then $\\omega $ -$\\lim D_{x_n,a}(z_n^k, z_n^\\infty ) <\\varepsilon $ .", "Arguing as before we obtain that for every $\\varepsilon > 0$ there exists $k_\\varepsilon \\ge 0$ such that for every fixed $k\\ge k_\\varepsilon $ it holds $(z_n^k,z_n^\\infty )_{x_n} \\ge \\log \\frac{1}{\\varepsilon } =: T_\\varepsilon $ for $\\omega $ -a.e.$(n)$ .", "Therefore by the same argument used before we conclude that for every $T\\ge 0$ there exists $k_T\\ge 0$ such that for every fixed $k\\ge k_T$ then $\\xi _{x_nz_n^k}$ and $\\xi _{x_nz_n^\\infty }$ stay at distance at most 2 up to time $T$ for $\\omega $ -a.e.$(n)$ .", "So the same conclusion holds for $\\xi _{x_\\omega z_\\omega ^k}$ and $\\xi _{x_\\omega z_\\omega ^\\infty }$ and by Lemma REF we have $(\\Psi (z_n^k), \\Psi (z_n^\\infty ))_{x_\\omega } \\ge T-1$ .", "This implies exactly that the sequence $\\Psi (z_n^k)$ converges to $\\Psi (z_n^\\infty )$ .", "To prove the continuity of the inverse map we suppose $\\Psi (z_n^k)$ converges to $\\Psi (z_n^\\infty )$ .", "By similar arguments used before we get that the $\\sigma _\\omega $ -geodesic rays $\\xi _{x_\\omega z_\\omega ^k}$ and $\\xi _{x_\\omega z_\\omega ^\\infty }$ stay at bounded distance up to time $T$ , provided $k\\ge k_T$ .", "So the same happens for $\\xi _{x_nz_n^k}$ and $\\xi _{x_nz_n^\\infty }$ for $\\omega $ -a.e.$(n)$ implying once again the convergence of $(z_n^k)$ to $(z_n^\\infty )$ .", "The metric on $\\partial X_\\omega $ .", "Since $\\Psi $ is an homeomorphism we can endow $\\partial X_\\omega $ with the metric induced by $\\Psi $ , i.e.", "$D(z_\\omega , z_\\omega ^{\\prime }) = \\omega $ -$\\lim D_{x_n,a}(z_n,z_n^{\\prime })$ , where $z_n$ and $z_n^{\\prime }$ are sequences such that $\\Psi (z_n)=z_\\omega $ and $\\Psi (z_n^{\\prime })=z_\\omega ^{\\prime }$ .", "It remains to show it is a visual metric.", "We show one of the two conditions since the other is similar.", "We take $z_\\omega = \\Psi (z_n), z_\\omega ^{\\prime }=\\Psi (z_n^{\\prime })$ and we set $D_n:= D_{x_n,a}(z_n,z_n^{\\prime })$ .", "By definition $D_\\omega = \\omega $ -$\\lim D_n = D(z_\\omega , z_\\omega ^{\\prime })$ .", "Since each $D_{x_n,a}$ is a standard visual metric we get $(z_n,z_n^{\\prime })_{x_n}\\le \\frac{1}{a}\\log \\frac{1}{D_n}=:T_n$ for every $n$ and by Lemma REF we conclude that $d(\\xi _{x_nz_n}(T_n + 3\\delta ), \\xi _{x_nz_n^{\\prime }}(T_n + 3\\delta )) \\ge 6\\delta $ for every $n$ .", "There are two possibilities: $T_\\omega :=\\omega $ -$\\lim T_n$ is either $+\\infty $ or a positive real number.", "In the first case we have $D_\\omega = 0$ and so there is nothing to prove.", "In the second case we know that $d(\\xi _{z_\\omega }(T_\\omega + 3\\delta ), \\xi _{z_\\omega ^{\\prime }}(T_\\omega + 3\\delta )) \\ge 6\\delta $ and so by Lemma REF we conclude that $(z_\\omega ,z_\\omega ^{\\prime })_{x_\\omega } < T_\\omega +\\delta = \\frac{1}{a}\\log \\frac{1}{D_\\omega } + \\delta $ , implying $D_\\omega < e^{\\delta }e^{-a(z_\\omega ,z_\\omega ^{\\prime })_{x_\\omega }}$ .", "We recall that GCB$^\\text{qc}(P_0,r_0,\\delta ;D)$ denotes the class of 4-uples $(X,x,\\sigma ,\\Gamma )$ such that $(X,x,\\sigma )\\in \\text{GCB}(P_0,r_0,\\delta )$ , $\\Gamma $ is a discrete, non-elementary, quasiconvex-cocompact group of $\\sigma $ -isometries of $X$ with codiameter $\\le D$ and $x\\in \\text{QC-Hull}(\\Lambda (\\Gamma ))$ .", "This class is closed under ultralimits.", "Theorem 4.2 Let $(X_n,x_n, \\sigma _n, \\Gamma _n) \\subseteq \\textup {GCB}^\\textup {qc}(P_0,r_0,\\delta ;D)$ , $\\omega $ be a non-principal ultrafilter and let $(X_\\omega , x_\\omega , \\sigma _\\omega , \\Gamma _\\omega )$ be the ultralimit 4-uple.", "Then $\\Psi (\\omega $ -$\\lim \\Lambda (\\Gamma _n)) = \\Lambda (\\Gamma _\\omega )$ , where $\\Psi $ is the isometry of Proposition REF .", "Moreover $\\Gamma _\\omega $ is a discrete, non-elementary, quasiconvex-cocompact group of $\\sigma _\\omega $ -isometries of $X_\\omega $ with codiameter $\\le D$ and $x_\\omega \\in \\textup {QC-Hull}(\\Lambda (\\Gamma _\\omega ))$ .", "We fix a sequence $z_n \\in \\Lambda (\\Gamma _n)$ and we observe that by Lemma REF and the cocompactness of the action of $\\Gamma _n$ on $\\text{QC-Hull}(\\Lambda (\\Gamma _n))$ we can find a sequence $(g_n^k)_{k\\in \\mathbb {N}}$ such that, denoted by $\\xi _{x_nz_n}$ the $\\sigma _n$ -geodesic ray $[x_n,z_n]$ , it holds: (a) $g_n^k x_n$ converges to $z_n$ when $k$ tends to $+\\infty $ ; (b) $g_n^0 = \\text{id}$ ; (c) $d(g_n^k x_n, g_n^{k+1}x_n)\\le 28\\delta + 2D$ ; (d) $d(g_n^k x_n, \\xi _{x_nz_n}(k)) \\le 14\\delta + D$ .", "For every $k \\in \\mathbb {N}$ the sequence $g_n^k$ is admissible by (b) and (c), so it defines a limit isometry $g_\\omega ^k \\in \\Gamma _\\omega $ .", "Moreover, if $\\xi _{x_\\omega z_\\omega }$ is the ultralimit of the sequence of $\\sigma _n$ -geodesic rays $\\xi _{x_nz_n}$ , we have $d(g_\\omega ^k x_\\omega , \\xi _{x_\\omega z_\\omega }(k)) \\le 14\\delta + D$ for every $k\\in \\mathbb {N}$ .", "We remark that $\\xi _{x_\\omega z_\\omega }^+ = \\Psi (z_n)$ by definition.", "As a consequence the sequence $g_\\omega ^k x_\\omega $ converges to $\\Psi (z_n)$ and so $\\Psi (z_n)\\in \\Lambda (\\Gamma _\\omega )$ .", "In other words $\\Psi (\\omega \\text{-}\\lim \\Lambda (\\Gamma _n)) \\subseteq \\Lambda (\\Gamma _\\omega )$ .", "It is easy to show that $\\Gamma _\\omega $ acts on $\\omega \\text{-}\\lim \\Lambda (\\Gamma _n)$ by $(g_n)(z_n) = (g_n z_n)$ and that the action commutes with $\\Psi $ .", "Moreover the set $\\omega \\text{-}\\lim \\Lambda (\\Gamma _n)$ is $\\Gamma _\\omega $ -invariant and closed, so it is $\\Psi (\\omega \\text{-}\\lim \\Lambda (\\Gamma _n))$ .", "The $\\Gamma _\\omega $ -invariance is trivial, while if $(z_n^k)_{k\\in \\mathbb {N}} \\in \\omega \\text{-}\\lim \\Lambda (\\Gamma _n)$ is a sequence converging to $(z_n^\\infty )$ and $z_n^\\infty \\notin \\omega \\text{-}\\lim \\Lambda (\\Gamma _n)$ then there exists $\\varepsilon _0 > 0$ such that for $\\omega $ -a.e.$(n)$ we have $D_{x_n,a}(z_n^\\infty , \\Lambda (\\Gamma _n)) \\ge \\varepsilon _0$ and this is a contradiction.", "Therefore the set $\\Psi (\\omega \\text{-}\\lim \\Lambda (\\Gamma _n))$ is a closed $\\Gamma _\\omega $ -invariant subset of $\\partial X_\\omega $ , that implies it contains $\\Lambda (\\Gamma _\\omega )$ and so the equality between these two sets.", "This also implies that $\\omega $ -$\\lim \\text{QC-Hull}(\\Lambda (\\Gamma _n)) = \\text{QC-Hull}(\\Lambda (\\Gamma _\\omega ))$ and so $x_\\omega \\in \\text{QC-Hull}(\\Lambda (\\Gamma _\\omega ))$ .", "By Theorem 6.13, Proposition 6.15 and Example 5.8 of [6] we know that $\\Gamma _\\omega $ is a non-elementary and discrete group.", "Moreover for every two points $y_\\omega , y_\\omega ^{\\prime } \\in \\text{QC-Hull}(\\Lambda (\\Gamma _\\omega ))$ there exist sequences of points $y_n, y_n^{\\prime } \\in \\text{QC-Hull}(\\Lambda (\\Gamma _n))$ such that $y_\\omega = \\omega $ -$\\lim y_n$ and $y_\\omega ^{\\prime } = \\omega $ -$\\lim y_n^{\\prime }$ and so there are $g_n \\in \\Gamma _n$ such that $d(g_ny_n,y_n^{\\prime })\\le D$ .", "The sequence $g_n$ is clearly admissible so it defines an element $g_\\omega = \\omega $ -$\\lim g_n$ of $\\Gamma _\\omega $ and $d(g_\\omega y_\\omega , y_\\omega ^{\\prime })\\le D$ , implying that the action of $\\Gamma _\\omega $ on $\\text{QC-Hull}(\\Lambda (\\Gamma _\\omega ))$ is cocompact with codiameter $\\le D$ ." ], [ "Continuity", "Finally we can show the continuity of the critical exponent of quasiconvex-cocompact groups.", "Theorem 4.3 Let $(X_n,x_n,\\sigma _n,\\Gamma _n) \\subseteq \\textup {GCB}^{\\textup {qc}}(P_0,r_0,\\delta ; D)$ and let $\\omega $ be a non-principal ultrafilter.", "Then $h_{\\Gamma _\\omega } = \\omega $ -$\\lim h_{\\Gamma _n}$ .", "We want to show $\\begin{aligned}&\\text{Pack}^*(\\Lambda (\\Gamma _\\omega ), e^{-T}) \\le \\omega \\text{-}\\lim \\text{Pack}^*(\\Lambda (\\Gamma _n), e^{-T-5\\delta })\\\\&\\text{Pack}^*(\\Lambda (\\Gamma _\\omega ), e^{-T}) \\ge \\omega \\text{-}\\lim \\text{Pack}^*(\\Lambda (\\Gamma _n), e^{-T+5\\delta })\\end{aligned}$ for every $T\\ge 0$ .", "Let $z_\\omega ^1,\\ldots , z_\\omega ^N$ be points realizing $\\text{Pack}^*(\\Lambda (\\Gamma _\\omega ), e^{-T})$ .", "In particular $(z_\\omega ^i,z_\\omega ^j)\\le T$ for every $i\\ne j$ .", "So by Lemma REF we have $d(\\xi _{x_\\omega z_\\omega ^i}(T+3\\delta ), \\xi _{x_\\omega z_\\omega ^j}(T+3\\delta ))\\ge 6\\delta $ for every $i\\ne j$ , where $\\xi _{x_\\omega z_\\omega ^i}$ is the unique $\\sigma _\\omega $ -geodesic ray joining $x_\\omega $ to $z_\\omega ^i$ .", "We know that $\\xi _{x_\\omega z_\\omega ^i}$ is the ultralimit of the $\\sigma _n$ -geodesic rays $\\xi _{x_n z_n^i}$ , where we can suppose $z_n^i \\in \\Lambda (\\Gamma _n)$ by Theorem REF .", "So for $\\omega $ -a.e.$(n)$ we get $d(\\xi _{x_n z_n^i}(T+3\\delta ), \\xi _{x_n z_n^j}(T+3\\delta ))> 4\\delta $ for every $i\\ne j$ : this is true since we have a finite set of limit conditions.", "Again by Lemma REF we get $(z_n^i,z_n^j)< T + 4\\delta $ for every $i\\ne j$ .", "Therefore, by (REF ), we know that the balls $B(z_n^i,e^{-T - 5\\delta })$ are all disjoint.", "Since this is true for $\\omega $ -a.e.$(n)$ we conclude $\\text{Pack}^*(\\Lambda (\\Gamma _\\omega ), e^{-T}) \\le \\omega \\text{-}\\lim \\text{Pack}^*(\\Lambda (\\Gamma _n), e^{-T-5\\delta }).$ Now let us show the other inequality.", "For every $n$ let $\\lbrace z_n^1,\\ldots , z_n^{N_n}\\rbrace $ be a set realizing $\\text{Pack}^*(\\Lambda (\\Gamma _n), e^{-T+5\\delta })$ .", "We consider the set $A_\\omega = \\lbrace \\Psi (z_n^{i_n}) \\text{ s.t. }", "1\\le i_n \\le N_n\\rbrace ,$ where $\\Psi $ is the isometry given by Proposition REF .", "By Theorem REF we know that $A_\\omega \\subseteq \\Lambda (\\Gamma _\\omega )$ .", "Let us take two points $z_\\omega = \\Psi (z_n^{i_n}), w_\\omega = \\Psi (z_n^{j_n})$ of $A_\\omega $ .", "If $i_n = j_n$ for $\\omega $ -a.e.$(n)$ then clearly $z_\\omega = w_\\omega $ .", "Otherwise we know that for $\\omega $ -a.e.$(n)$ it holds $(z_n^{i_n}, z_n^{j_n})_{x_n} \\le T - 5\\delta $ , so $d(\\xi _{x_nz_n^{i_n}}(T), \\xi _{x_nz_n^{j_n}}(T)) \\ge 10\\delta $ by Lemma REF .", "Here again $\\xi _{x_nz_n^{i_n}}$ is the unique $\\sigma _n$ -geodesic ray joining $x_n$ and $z_n^{i_n}$ .", "Passing this condition to the ultralimit we get $d(\\xi _{x_\\omega z_\\omega }(T), \\xi _{x_\\omega w_\\omega }(T)) \\ge 10\\delta $ , so $d(\\xi _{x_\\omega z_\\omega }(T -2\\delta ), \\xi _{x_\\omega w_\\omega }(T-2\\delta )) \\ge 6\\delta > 4\\delta $ .", "Applying again Lemma REF we have $(z_\\omega , w_\\omega )_{x_\\omega } < T-\\delta $ .", "Hence if $z_\\omega , w_\\omega \\in A_\\omega $ either $z_\\omega = w_\\omega $ or the balls $B(w_\\omega , e^{-T}), B(z_\\omega , e^{-T})$ are disjoint, by (REF ).", "We deduce that the cardinality of $A_\\omega $ , say $N_\\omega $ , is finite since $\\Lambda (\\Gamma _\\omega )$ is compact.", "We claim that the set $I=\\lbrace n\\in \\mathbb {N}\\text{ s.t. }", "N_n = N_\\omega \\rbrace $ satisfies $\\omega (I)=1$ .", "In order to prove it we rename the elements of $A_\\omega $ as $z^1_\\omega ,\\ldots ,z^{N_\\omega }_\\omega $ , where $z^k_\\omega = \\omega \\text{-}\\lim z_n^{i^k_n}$ for some $1\\le i^k_n \\le N_n$ .", "From what said before we know that for $k\\ne l$ we have $\\omega (\\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "i_n^k \\ne i_n^l \\rbrace ) = 1.$ So $\\begin{aligned}1 &= \\omega \\bigg (\\bigcap _{1\\le k<l\\le N_\\omega }\\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "i_n^k \\ne i_n^l \\rbrace \\bigg ) \\\\&=\\omega ( \\lbrace n \\in \\mathbb {N} \\text{ s.t. }", "i_n^k\\ne i_n^l \\text{ for all } 1\\le k < l \\le N_\\omega \\rbrace ) \\\\&\\le \\omega (\\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "N_n\\ge N_\\omega \\rbrace ) = \\omega (I\\cup J)\\end{aligned}$ where $J=\\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "N_n > N_\\omega \\rbrace $ .", "Then the claim is true if $\\omega (J)=0$ .", "If $\\omega (J)=1$ then for all $1\\le j \\le N_\\omega + 1$ we can define $z^j_\\omega = \\omega $ -$\\lim z_n^j$ if $n\\in J$ .", "From what said before they are $N_\\omega + 1$ distinct points of $A_\\omega $ , which is impossible.", "In this way we conclude that for $\\omega $ -a.e.$(n)$ it holds $\\text{Pack}^*(\\Lambda (\\Gamma _\\omega ), e^{-T})\\ge N_\\omega = N_n = \\text{Pack}^*(\\Lambda (\\Gamma _n), e^{-T + 5\\delta })$ implying the second inequality in (REF ).", "Clearly from Lemma REF we get $\\begin{aligned}&\\text{Cov}(\\Lambda (\\Gamma _\\omega ), e^{-T}) \\le \\omega \\text{-}\\lim \\text{Cov}(\\Lambda (\\Gamma _n), e^{-T-6\\delta })\\\\&\\text{Cov}(\\Lambda (\\Gamma _\\omega ), e^{-T}) \\ge \\omega \\text{-}\\lim \\text{Cov}(\\Lambda (\\Gamma _n), e^{-T+6\\delta }).\\end{aligned}$ Now let $\\tilde{h} = \\omega \\text{-}\\lim h_{\\Gamma _n} \\in \\left[0,\\frac{\\log (1+P_0)}{r_0}\\right]$ by Proposition REF .", "By Corollary REF .", "(ii) for every $\\varepsilon > 0$ there exists $T_\\varepsilon $ depending on $\\varepsilon , P_0,r_0,\\delta ,D$ but not on $n$ such that $\\bigg \\vert \\frac{1}{T}\\log \\textup {Cov}(\\Lambda (\\Gamma _n)) - h_{\\Gamma _n} \\bigg \\vert < \\varepsilon \\text{ for all } T\\ge T_\\varepsilon .$ We define the sets $B_1=\\bigg \\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "\\bigg \\vert \\frac{1}{T}\\log \\textup {Cov}(\\Lambda (\\Gamma _n), e^{-T}) - h_{\\Gamma _n} \\bigg \\vert < \\varepsilon \\text{ for all } T\\ge T_\\varepsilon \\bigg \\rbrace $ and $B_2=\\lbrace n\\in \\mathbb {N} \\text{ s.t. }", "\\vert h_{\\Gamma _n} - \\tilde{h}\\vert < \\varepsilon \\rbrace .$ Clearly they belong to $\\omega $ .", "Observe also that for all $T \\ge T_\\varepsilon + 6\\delta $ and every $n\\in B_1$ it holds $\\bigg \\vert \\frac{1}{T}\\log \\text{Cov}(\\Lambda (\\Gamma _n), e^{-T-6\\delta }) - \\frac{1}{T}\\log \\text{Cov}(\\Lambda (\\Gamma _n), e^{-T + 6\\delta })\\bigg \\vert < 2\\varepsilon .$ In conclusion for all $T\\ge T_\\varepsilon + 6\\delta $ and for all $n\\in B_1\\cap B_2$ , i.e.", "for $\\omega $ -a.e.$(n)$ , we have $\\bigg \\vert \\frac{1}{T}\\log \\text{Cov}(\\Lambda (\\Gamma _\\omega ), e^{-T}) - \\tilde{h} \\bigg \\vert < 3\\varepsilon .$ Finally, by Corollary REF and Theorem REF , we know that $\\frac{1}{T}\\log \\text{Cov}(\\Lambda (\\Gamma _\\omega ), e^{-T})$ tends to $h_{\\Gamma _\\omega }$ when $T$ tends to $+\\infty $ .", "We conclude that $\\tilde{h}=h_{\\Gamma _\\omega }$ .", "We conclude explicitating the fact that continuity under ultralimits implies continuity under pointed Gromov-Hausdorff convergence.", "Proposition 4.4 Let $\\mathcal {C}$ be a class of pointed, proper metric spaces and $h\\colon \\mathcal {C}\\rightarrow \\mathbb {R}$ be a function.", "Suppose that $\\mathcal {C}$ is closed under ultralimits and $h$ is continuous under ultralimits, i.e.", "for every non-principal ultrafilter $\\omega $ and every sequence $(X_n,x_n)\\in \\mathcal {C}$ it holds $h(X_\\omega ) = \\omega $ -$\\lim h(X_n)$ .", "Suppose that $(X_n,x_n) \\subseteq \\mathcal {C}$ converges in the pointed Gromov-Hausdorff sense to $(X_\\infty , x_\\infty )$ .", "Then $X_\\infty \\in \\mathcal {C}$ and $h(X_\\infty ) = \\lim _{n\\rightarrow +\\infty } h(X_n).$ We need the following lemma.", "Lemma 4.5 Let $a_n$ be a bounded sequence of real numbers.", "Let $a_{n_j}$ be a subsequence converging to $\\tilde{a}$ .", "Then there exists a non-principal ultrafilter $\\omega $ such that $\\omega $ -$\\lim a_n = \\tilde{a}$ .", "The set $\\lbrace n_j\\rbrace _j$ is infinite, then there exists a non-principal ultrafilter $\\omega $ containing $\\lbrace n_j\\rbrace _j$ (cp.", "[10], Lemma 3.2).", "Moreover for every $\\varepsilon > 0$ there exists $j_\\varepsilon $ such that for all $j\\ge j_\\varepsilon $ it holds $\\vert a_{n_j} - \\tilde{a} \\vert < \\varepsilon $ .", "The set of indices where the inequality is true belongs to $\\omega $ since the complementary is finite.", "This implies exactly that $\\tilde{a} = \\omega $ -$\\lim a_n$ .", "[Proof of Proposition REF .]", "We fix a non-principal ultrafilter $\\omega $ .", "Since the class $\\mathcal {C}$ is made of proper metric spaces then $X_\\omega $ is isometric to $X_\\infty $ (cp.", "[7], Proposition A.11).", "Therefore $h(X_\\infty ) = h(X_\\omega ) = \\omega \\text{-}\\lim h(X_n).$ This implies that $\\omega \\text{-}\\lim h(X_n)$ does not depend on the ultrafilter $\\omega $ .", "By the previous lemma we conclude that every converging subsequence of $h(X_n)$ has $h(X_\\infty )$ as a limit, i.e.", "$\\liminf _{n\\rightarrow +\\infty }h(X_n) = \\limsup _{n\\rightarrow +\\infty }h(X_n)=h(X_\\infty )$ ." ] ]
2105.11764
[ [ "A Survey on Complex Knowledge Base Question Answering: Methods,\n Challenges and Solutions" ], [ "Abstract Knowledge base question answering (KBQA) aims to answer a question over a knowledge base (KB).", "Recently, a large number of studies focus on semantically or syntactically complicated questions.", "In this paper, we elaborately summarize the typical challenges and solutions for complex KBQA.", "We begin with introducing the background about the KBQA task.", "Next, we present the two mainstream categories of methods for complex KBQA, namely semantic parsing-based (SP-based) methods and information retrieval-based (IR-based) methods.", "We then review the advanced methods comprehensively from the perspective of the two categories.", "Specifically, we explicate their solutions to the typical challenges.", "Finally, we conclude and discuss some promising directions for future research." ], [ "Introduction", "A knowledge base (KB) is a structured database that contains a collection of facts in the form (subject, relation, object).", "Large-scale KBs, such as Freebase [8], DBPedia [38] and Wikidata [56], have been constructed to serve many downstream tasks.", "Based on available KBs, knowledge base question answering (KBQA) is a task that aims to answer natural language questions with KBs as its knowledge source.", "Early work on KBQA [9], [16], [24], [36], [35] focuses on answering a simple question, where only a single fact is involved.", "For example, “Where was JK Rowling born?” is a simple question which can be answered using just the fact “(J.K. Rowling, birthplace, United Kingdom)”.", "Figure: An example of complex KBQA for the question “Who is the first wife of TV producer that was nominated for The Jeff Probst Show?”.We present the related KB subgraph for this question.The ground truth path to answer this question is annotated with colored borders.The topic entity and the answer entity are shown in the bold font and shaded box respectively.“multi-hop” reasoning, “constrained” relations and “numerical” operation are highlighted in black dotted box.", "We use different colors to indicate different reasoning hops to reach each entity from the topic entity.Recently, researchers start paying more attention to answering complex questions over KBs, i.e., the complex KBQA task [25], [41].", "Complex questions usually contain multiple subjects, express compound relations and include numerical operations.", "Take the question in Figure REF as an example.", "This example question starts with the subject “The Jeff Probst Show”.", "Instead of querying a single fact, the question requires the composition of two relations, namely, “nominee” and “spouse”.", "This query is also associated with an entity type constraint “(Jeff Probst, is a, TV producer)”.", "The final answer should be further aggregated by selecting the possible candidates with the earliest marriage date.", "Generally, complex questions are questions involving multi-hop reasoning, constrained relations, numerical operations, or some combination of the above.", "Tracing back to the solutions for simple KBQA, a number of studies from two mainstream approaches have been proposed.", "These two approaches first recognize the subject in a question and link it to an entity in the KB (referred to as the topic entity).", "Then they derive the answers within the neighborhood of the topic entity by either executing a parsed logic form or reasoning in a question-specific graph extracted from the KB.", "The two categories of methods are commonly known as semantic parsing-based methods (SP-based methods) and information retrieval-based methods (IR-based methods) in prior work [9], [16], [24], [19].", "They include different working mechanisms to solve the KBQA task.", "The former approach represents a question by a symbolic logic form and then executes it against the KB and obtains the final answers.", "The latter approach constructs a question-specific graph delivering the comprehensive information related to the question and ranks all the entities in the extracted graph based on their relevance to the question.", "However, when applying the two mainstream approaches to the complex KBQA task, complex questions bring in challenges on different parts of the approaches.", "We identify the main challenges as follows: Parsers used in existing SP-based methods are difficult to cover diverse complex queries (e.g., multi-hop reasoning, constrained relations and numerical operations).", "Similarly, previous IR-based methods may fail to answer a complex query, as their ranking is performed over small-scope entities without traceable reasoning.", "More relations and subjects in complex questions indicate a larger search space of potential logic forms for parsing, which will dramatically increase the computational cost.", "Meanwhile, more relations and subjects could prevent IR-based methods from retrieving all relevant entities for ranking.", "Both approaches treat question understanding as a primary step.", "When questions become complicated in both semantic and syntactic aspects, models are required to have strong capabilities of natural language understanding and generalization.", "It is expensive to label the ground truth paths to the answers (see the example in Figure REF ) for complex questions.", "Generally, only question-answer pairs are provided.", "This indicates SP-based methods and IR-based methods have to be trained without the annotation of correct logic forms and reasoning paths, respectively.", "Such weak supervision signals bring difficulties to both approaches.", "Regarding the related surveys, we observe [58] [[58]] and [12] [[12]] reviewed the existing work on simple KBQA.", "Furthermore, [18] [[18]] investigated the current advances on complex KBQA.", "They provided a general view of advanced methods only from the perspective of techniques and focused more on application scenarios in the e-commerce domain.", "Different from these surveys, our work tries to identify the challenges encountered in previous studies and extensively discusses existing solutions in a comprehensive and well-organized manner.", "Specifically, we categorize the methods for complex KBQA into two mainstream approaches based on their working mechanisms.", "We decompose the overall procedure of the two approaches into a series of modules and analyze the challenges in each module.", "We believe that this way is particularly helpful for readers to understand the challenges and how they are addressed in existing solutions to complex KBQA.", "Furthermore, we provide a thorough outlook on several promising research directions on complex KBQA.", "The remainder of this survey is organized as follows.", "We will first introduce the preliminary knowledge about the task formulation, multiple available datasets and evaluation protocol in Section .", "Next, we introduce the two mainstream categories of methods for complex KBQA in Section .", "Then following the categorization, we figure out typical challenges and solutions to these challenges in Section .", "Finally, we conclude and discuss some future research directions in Section .", "Table: Several complex KBQA benchmark datasets.", "“LF\" denotes whether the dataset provides Logic Forms, and “NL\" denotes whether the dataset incorporates crowd workers to rewrite questions in Natural Language." ], [ "Background", "In this section, we first give a task definition about complex KBQA, and then introduce available datasets and evaluation protocol for this task.", "[leftmargin=0cm, label=] Task.", "For the task of complex KBQA, a KB consisting of a set of facts is given as input, where the subject and object are connected by their relation.", "All the subjects and objects in the facts form the entity set of a KB.", "Given the available KB, this task aims to answer complex natural language questions in the format of a sequence of tokens.", "Specially, we assume the correct answers come from the entity set of the KB.", "Unlike answers of simple KBQA, which are entities directly connected to the topic entity, the answers of the complex KBQA task are entities multiple hops away from the topic entities or even some aggregation of them.", "Datasets.", "Generally, the answers of the questions should be provided to train a complex KBQA system.", "For this purpose, many efforts have been devoted to constructing datasets for complex KBQA.", "We list the available complex KBQA datasets in Table REF .", "Overall, these datasets are constructed with the following steps.", "Given a topic entity in a KB as question subject, simple questions are first created with diverse templates.", "Based on simple questions and the neighborhood of a topic entity in a KB, complex questions are further generated with predefined templates, and some work [51] also generates executable logic forms with templates.", "Meanwhile, answers are extracted with corresponding rules.", "In some cases, crowd workers are hired to paraphrase the template queries into natural language questions and refine the generated logic forms, making the question expressions more diverse and fluent.", "In order to serve realistic applications, these datasets typically create questions which require multiple KB facts to reason.", "Moreover, they might include numerical operations (e.g., counting, ranking operations for comparative or superlative questions) and constraints (e.g., entity, temporal keywords), which further increase the difficulty in reasoning the answers from KBs.", "Evaluation Protocol.", "The KBQA system usually predicts entities with the top confidence score to form the answer set.", "Note that there can be more than one answer to a question.", "In previous studies, there are some classical evaluation metrics such as precision, recall, $F_1$ and Hits@1.", "Some studies [62], [39], [1] use the precision, recall, $F_1$ score to evaluate the prediction.", "Precision indicates the ratio of the correct answers over all the predicted answers.", "Recall is the ratio of the correct predicted answers over all the ground truth.", "And $F_1$ score considers precision and recall simultaneously.", "Some other methods [43], [52], [60], [23] use Hits@1 to assess the fraction that the correct answers rank higher than other entities.", "Figure: Illustration of two mainstream approaches for complex KBQA." ], [ "Two Mainstream Approaches", "As introduced in Section , SP-based and IR-based methods are two mainstream approaches to solving complex KBQA task.", "SP-based methods parse a question into a logic form and execute it against KBs for finding the answers.", "IR-based methods retrieve a question-specific graph and apply some ranking algorithms to select entities from top positions.", "To summarize, the two approaches follow either a parse-then-execute paradigm or a retrieval-and-rank paradigm, which are illustrated in Figure REF .", "[leftmargin=0cm, label=] Semantic Parsing-based Methods.", "This category of methods aims at parsing a natural language utterance into a logic form [5], [48].", "They predict answers via the following steps: (1) They fully understand a question via a question understanding module, which is to conduct the semantic and syntactic analysis and obtain an encoded question for the subsequent parsing step.", "(2) A logical parsing module is utilized to transfer the encoded question into an uninstantiated logic form.", "The uninstantiated logic form is a syntactic representation of the question without the grounding of entities and relations.", "The grammar and constituents of logic forms could be different according to specific designs of a system.", "(3) To execute against KBs, the logic form is further instantiated and validated by conducting some semantic alignments to structured KBs via KB grounding.", "Note that, in some work [62], [39], the logical parsing and KB grounding are simultaneously performed, where logic forms are validated in KBs while partially parsed.", "(4) Eventually, the parsed logic form is executed against KBs to generate predicted answers via a KB execution module.", "Information Retrieval-based Methods.", "As another mainstream approach, IR-based methods directly retrieve and rank answers from the KBs considering the information conveyed in the questions [9], [16].", "They consist of the following steps: (1) Starting from the topic entity, the system first extracts a question-specific graph from KBs.", "Ideally, this graph includes all question related entities and relations as nodes and edges, respectively.", "Without explicitly generating an executable logic form, IR-based methods perform reasoning over the graph and then rank entities in the graph.", "(2) Next, the system encodes input questions via a question representation module.", "This module analyzes the semantics of the question and outputs reasoning instructions, which are usually represented as vectors.", "(3) A graph-based reasoning module conducts semantic matching via vector-based computation to propagate and then aggregate the information along the neighboring entities within the graph.", "The reasoning status, which has diverse definitions in different methods (e.g., distributions of predicted entities, representations of relations), is updated based on the reasoning instruction.", "Recently, several studies [29], [13] repeat Step (2) and (3) for multiple times to perform the reasoning.", "(4) An answer ranking module is utilized to rank the entities in the graph according to the reasoning status at the end of reasoning.", "The top-ranked entities are predicted as the answers to the question.", "Pros and Cons.", "Overall, SP-based methods can produce a more interpretable reasoning process by generating expressive logic forms.", "However, they heavily rely on the design of the logic form and parsing algorithm, which turns out to be the bottleneck of performance improvement.", "As a comparison, IR-based methods conduct complex reasoning on graph structure and perform semantic matching.", "Such a paradigm naturally fits into popular end-to-end training and makes the IR-based methods easier to train.", "However, the blackbox style of the reasoning model makes the intermediate reasoning less interpretable." ], [ "Challenges and Solutions", "Since the aforementioned approaches are developed based on different paradigms, we describe the challenges and corresponding solutions for complex KBQA with respect to the two mainstream approaches.", "A summary of these challenges and solutions is presented in Table REF ." ], [ "Semantic Parsing-based Methods", "In this part, we discuss the challenges and solutions for semantic parsing-based methods.", "[leftmargin=0cm, label=] Overview.", "As introduced in Section , SP-based methods follow a parse-then-execute procedure via a series of modules, namely question understanding, logical parsing, KB grounding and KB execution.", "These modules will encounter different challenges for complex KBQA.", "Firstly, question understanding becomes more difficult when the questions are complicated in both semantic and syntactic aspects.", "Secondly, logical parsing has to cover diverse query types of complex questions.", "Moreover, a complex question involving more relations and subjects will dramatically increase the possible search space for parsing, which makes the parsing less effective.", "Thirdly, the manual annotation of logic forms are both expensive and labor-intensive, and it is challenging to train a SP-based method with weak supervision signals (i.e., question-answer pairs).", "Next, we will introduce how prior studies deal with these challenges.", "Understanding Complex Semantics and Syntax.", "As the first step of SP-based methods, question understanding module converts unstructured text into encoded question (i.e., structural representation), which benefits the downstream parsing.", "Compared with simple questions, complex questions are featured with more complex query types and compositional semantics, which increases the difficulty in linguistic analysis.", "To better understand complex natural language questions, many existing methods rely on syntactic parsing, such as dependencies [1], [2], [41] and Abstract Meaning Representation (AMR) [31], to provide better alignment between question constituents and logic form elements (e.g., entity, relation, entity types and attributes).", "However, the accuracy of producing syntactic parsing is still not satisfying on complex questions, especially for those with long-distance dependency.", "To alleviate error propagation from syntactic parsing to downstream semantic parsing, [54] [[54]] leveraged the skeleton-based parsing to obtain the trunk of a complex question, which is a simple question with several branches (i.e., pivot word of original text-spans) to be expanded.", "Another line of work focuses on leveraging structural properties (such as tree structure or graph structure) of logic forms for ranking candidate parsing.", "They try to improve the matching between logic forms and questions by incorporating structure-aware feature encoder [68], applying fine-grained slot matching [42], and adding constraints about query structure to filter noisy queries out [14].", "Parsing Complex Queries.", "During parsing, traditional semantic parses (e.g., CCG [11], [33], [48]), which are developed without considering KB schemas, have shown their potential in parsing simple questions.", "However, these methods could be sub-optimal for complex questions due to the ontology mismatching problem [62].", "Thus, it is necessary to leverage the structure of KBs for more accurate parsing.", "To satisfy the compositionality of the complex questions, researchers have developed diverse expressive logic forms as parsing targets.", "[4] [[4]] designed three query templates as the parsing targets, which could cover questions querying 1-hop, 2-hop relations and single constraint involved relations.", "Although this piece of work can successfully parse several types of complex questions, it suffers from the limited coverage issue.", "[62] [[62]] proposed query graph as the expressive parsing target.", "A query graph is a logic form in graph structure which closely matches with the KB schemas.", "Such query graphs have shown strong expression capability in complex KBQA task.", "However, they are restrictedly generated with predefined manual rules, which is inapplicable to large-scale datasets and long-tail complex question types.", "The follow-up work tried to improve the formulation of query graphs.", "To generalize to unseen and long-tail question types, [15] [[15]] proposed to leverage frequent query substructure for formal query generation.", "[1] [[1]] utilized syntactic annotation to enhance the structural complexity of the query graph.", "[25] [[25]] applied more aggregation operators (e.g., “merging”) to fit complex questions, and conducted coreference resolution.", "Grounding with Large Search Space.", "To obtain executable logic forms, KB grounding module instantiates possible logic forms with a KB.", "As one entity in the KB could be linked to hundreds or even thousands of relations, it would be unaffordable to explore and ground all the possible logic forms for a complex question considering both computational resource and time complexity.", "Recently, researchers proposed multiple approaches to solving the problem.", "[66] [[66]] proposed to decompose a complex question into multiple simple questions, where each question was parsed into a simple logic form.", "Next, intermediate answers are generated via these simple logic forms and final answers are jointly obtained.", "This decompose-execute-join strategy could effectively narrow down the search space.", "A similar approach was studied by [7] [[7]] and they reduced human annotations by leveraging dependency structure.", "Meanwhile, a number of studies adopted the expand-and-rank strategies to reduce the search space by searching the logic forms with beams.", "[13] [[13]] first adopted the hopwise greedy search strategy to expand the most likely query graphs and stop until the best query graph was obtained.", "[37] [[37]] proposed an iterative matching module to parse the questions without revisiting the generated query graphs at each searching step.", "Such a sequential expansion process is only effective in answering multi-hop questions, while helpless for questions with constraints or numerical operations.", "[34] [[34]] defined more operations to support three typical complex queries, which can largely reduce the search space.", "Training under Weak Supervision Signals.", "To deal with the issue of limited or insufficient training data, Reinforcement Learning (RL) based optimization has been adopted to maximize the expected reward [39], [47].", "In such a way, SP-based methods can only receive the feedback after the execution of the complete parsed logical form, which leads to severe sparse positive rewards and data inefficiency issues.", "To tackle these issues, some research work adopted reward shaping strategies for parsing evaluation.", "[49] [[49]] rewarded the model by the additional feedback when the predicted answers are the same type as the ground truth.", "[27] [[27]] adopted a similar idea to evaluate the generated logic form by comparing it with the high-reward logic forms stored in the memory buffer.", "Besides rewards for the whole procedure, intermediate rewards during the semantic parsing process may also help address this challenge.", "Recently, [47] [[47]] formulated query graph generation as a hierarchical decision problem, and proposed a framework based on hierarchical RL with intrinsic motivations to provide intermediate rewards.", "To accelerate and stablize the training process, Qiu et al.", "[47] pre-trained model with pseudo-gold programs (i.e., high-reward logic forms generated by hand-crafted rules).", "As pseudo-gold programs can be also generated from the model, [39] [[39]] proposed to maintain pseudo-gold programs found by an iterative maximum-likelihood training process to bootstrap training." ], [ "Information Retrieval-based Methods", "Here, we summarize the main challenges brought by complex questions for different modules of IR-based methods.", "[leftmargin=0cm, label=] Overview.", "The overall procedure typically consists of the modules of retrieval source construction, question representation, graph based reasoning and answer ranking.", "These modules will encounter different challenges for complex KBQA.", "Firstly, the retrieval source construction module extracts a question-specific graph from KBs, which covers a wide range of relevant facts for each question.", "Due to unneglectable incompleteness of source KBs [44], the correct reasoning paths may be absent from the extracted graph.", "This issue is more likely to occur in the case of complex questions.", "Secondly, question representation module understands the question and generates instructions to guide the reasoning process.", "This step becomes challenging when the question is complicated.", "After that, reasoning on graph is conducted through semantic matching.", "When dealing with complex questions, such methods rank answers through semantic similarity without traceable reasoning in the graph, which hinders reasoning analysis and failure diagnosis.", "Eventually, this system encounters the same training challenge under weak supervision signals (i.e., question-answer pairs).", "The following parts illustrate how prior work deal with these challenges.", "Reasoning under Incomplete KB.", "IR-based methods first extract a question-specific graph from KBs, and conduct subsequent reasoning on it.", "Since simple questions only require 1-hop reasoning on the neighborhood of topic entity in KBs, IR-based methods are less likely to suffer from the inherent incompleteness of KBs [44].", "In comparison, it may be a severe problem for complex questions, where the correct reasoning path may be absent from the question-specific graph.", "Furthermore, this incompleteness reduces the neighborhood information used for encoding entities, which poses additional challenges for effective reasoning.", "To tackle this challenge, researchers utilize auxiliary information to enrich the knowledge source.", "Intuitively, large question-related text corpus retrieved from Wikipedia can provide a wide range of unstructured knowledge as supplementary evidence.", "[52] [[52]] and [53] [[53]] proposed to complement the subgraph extracted from incomplete KBs with extra question-related text sentences to form a heterogeneous graph and conduct reasoning on it.", "Instead of directly complementing sentences to question-specific graph as nodes, [60] [[60]] and [20] [[20]] proposed to fuse extra textual information into the entity representation to supplement knowledge.", "They first encoded sentences and entities conditioned on questions, and then supplemented the incomplete KB by aggregating representations of sentences to enhance corresponding entity representations.", "Besides extra text corpus, knowledge base embeddings have been adopted to alleviate the sparsity of KB by performing missing link prediction.", "Inspired by KB completion task, [50] [[50]] utilized pre-trained knowledge base embeddings to enrich the learned entity representations and address incomplete KB issue.", "Understanding Complex Semantics.", "In general, IR-based methods generate reasoning instructions by directly encoding questions as low-dimensional vectors through neural network (e.g., LSTM).", "Static reasoning instructions obtained through above approaches cannot effectively represent the compositional semantics of complex questions.", "In order to comprehensively understand questions, recent work dynamically updated the reasoning instructions during the reasoning process.", "To focus on the currently unanalyzed part of question, [43] [[43]], [67] [[67]] and  [61] [[61]] proposed to update the reasoning instruction with information retrieved along the reasoning process.", "Besides updating the instruction representation with the reasoned information, [23] [[23]] proposed to focus on different parts of the question with dynamic attention mechanism.", "This dynamic attention mechanism can promote the model to attend to other information conveyed by the question and provide proper guidance for subsequent reasoning steps.", "Instead of decomposing the semantics of questions, [52] [[52]] proposed to augment the representation of the question with contextual information from graph.", "They updated the reasoning instruction through aggregating information from the topic entity after every reasoning step.", "Uninterpretable Reasoning.", "Traditional IR-based methods rank answers by calculating a single semantic similarity between questions and entities in the graph, which is less interpretable at the intermediate steps.", "As the complex questions usually query multiple facts, the system is supposed to accurately predict answers over the graph based on a traceable and observable reasoning process.", "Even though some work repeated reasoning steps for multiple times, they cannot reason along a traceable path in the graph.", "To derive a more interpretable reasoning process, multi-hop reasoning is introduced.", "Specifically, [67] [[67]] and  [61] [[61]] proposed to make the relation or entity predicted at each hop traceable and observable.", "They output intermediate predictions (i.e., matched relations or entities) from predefined memory as the interpretable reasoning path.", "Nevertheless, it can not fully utilize the semantic relation information to reason edge by edge.", "Thus, [21] [[21]] constructed a denser hypergraph by pinpointing a group of entities connected via same relation, which simulated human’s hopwise relational reasoning and output a sequential relation path to make the reasoning interpretable.", "Training under Weak Supervision Signals.", "Similar to the SP-based methods, it is difficult for IR-based methods to reason the correct answers without any annotations at intermediate steps, since the model cannot receive any feedback until the end of reasoning.", "It is found that this case may lead to spurious reasoning [23].", "To mitigate such issues, [46] [[46]] formulated the reasoning process over KBs as expanding the reasoning path on KB and adopted reward shaping strategy to provide intermediate rewards.", "To evaluate reasoning paths at intermediate steps, they utilized semantic similarity between the question and the reasoning path to provide feedback.", "Besides evaluating the reasoning path at intermediate steps, a more intuitive idea is to infer pseudo intermediate status and augment model training with such inferred signals.", "Inspired by bidirectional search algorithm on graph, [23] [[23]] proposed to learn the intermediate reasoning entity distributions by synchronizing bidirectional reasoning process.", "While most of existing work focused on enhancing the supervision signals at intermediate steps, few work paid attention to the entity linking step.", "Researchers utilized off-the-shelf tools to locate the topic entity in question, which may cause error propagation to subsequent reasoning.", "In order to accurately locate the topic entity without annotations, [64] [[64]] proposed to train entity linking module through a variational learning algorithm which jointly modeled topic entity recognition and subsequent reasoning over KBs." ], [ "Conclusion and Future Directions", "This paper attempted to provide an overview of typical challenges and corresponding solutions on complex KBQA.", "We introduced commonly used datasets and summarized the widely employed SP-based methods as well as IR-based methods.", "Existing complex KBQA methods are generally summarized into these two categories.", "Besides them, some other methods [55] may not fall into these two categories.", "For example,  [55] [[55]] proposed to transform a complex question to a composition of simple questions through rule-based decomposition, which focused on question decomposition instead of KB based reasoning or logic form generation.", "We believe that complex KBQA will continue to be an active and promising research area with wide applications, such as natural language understanding, compositional generalization, multi-hop reasoning.", "Many challenges presented in this survey are still open and under-explored.", "Considering the challenges summarized in this paper, we point out several promising future directions for complex KBQA task: [leftmargin=0cm, label=] Evolutionary KBQA.", "As we can see, existing methods for complex KBQA task are usually learned on offline training datasets and then deployed online to answer user questions.", "Due to such clear separation, most of existing KBQA systems fail to catch up with the rapid growth of world knowledge and answer new questions.", "However, user feedback may provide deployed KBQA systems an opportunity to improve themselves.", "Based on this observation, [2] [[2]] leveraged the user feedback to rectify answers generated by the KBQA system and made further improvement.", "Despite verifying the correctness of system prediction, users may also play a more active role in the question answering process.", "[65] [[65]] designed an interactive method to engage users in the question parsing process of the KBQA system directly.", "In the future, an evolutionary KBQA system is imperative to get continuous improvement after online deployment.", "Robust and Interpretable Models.", "While existing methods have achieved promising results on benchmark datasets where i.i.d assumption is held, they may easily fail to deal with an out-of-distribution case.", "Few-shot setting is a scenario where the training data is limited.", "A few previous studies [26], [23] discussed related topics, but they are still far from comprehensive in terms of challenge anslysis and problem solving.", "Compositional generalization is another scenario where the novel combinations of component items seen in training should be inferred during testing.", "To support more research on such issue, [19] [[19]] and [32] [[32]] have introduced related datasets, namely GrailQA and CFQ.", "The models are supposed to handle out-of-distribution questions and obtain explainable reasoning process.", "Designing methods for KBQA with good interpretability and robustness may be a challenging but promising topic for future research.", "More General Knowledge Base.", "Due to KB incompleteness, researchers incorporated extra information (such as text [52], images [59] and human interactions [22]) to complement the knowledge base, which would further improve the complex KBQA performance.", "There are also some tasks (e.g., visual question answering and commonsense knowledge reasoning), which can be formulated as question answering based on specific KBs.", "For example, in visual question answering, the scene graph extracted from an image can be regarded as a special KB [28].", "Despite explicitly representing relational knowledge as the structural KB, some researchers proposed to reason on implicit “KB”.", "Petroni et al.", "[[45]] analyzed the relational knowledge in a wide range of pretrained language models, and some follow-up work [10], [30] further demonstrated its effectiveness to answer “fill-in-the-blank” cloze statements.", "While most of existing work focused on traditional structured KBs, a more general definition about KBs and flexible usage of KBs may help KBQA research topic show greater impact." ], [ "Acknowledgements", "This work is partially supported by the National Research Foundation, Singapore under its International Research Centres in Singapore Funding Initiative, the National Natural Science Foundation of China under Grant No.", "61872369 and 61832017, Beijing Academy of Artificial Intelligence (BAAI) under Grant No.", "BAAI2020ZJ0301 and Beijing Outstanding Young Scientist Program under Grant No.", "BJJWZYJH012019100020098.", "Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore.", "Wayne Xin Zhao is the corresponding author." ] ]
2105.11644
[ [ "Using Game Theory to maximize the chance of victory in two-player sports" ], [ "Abstract Game Theory concepts have been successfully applied in a wide variety of domains over the past decade.", "Sports and games are one of the popular areas of game theory application owing to its merits and benefits in solving complex scenarios.", "With recent advancements in technology, the technical and analytical assistance available to players before the match, during game-play and after the match in the form of post-match analysis for any kind of sport has improved to a great extent.", "In this paper, we propose three novel approaches towards the development of a tool that can assist the players by providing detailed analysis of optimal decisions so that the player is well prepared with the most appropriate strategy which would produce a favourable result for a given opponent's strategy.", "We also describe how the system changes when we consider real-time game-play wherein the history of the opponent's strategies in the current rally is also taken into consideration while suggesting." ], [ "Introduction", "There have been various advancements in the way in which technology is utilized in sports and games.", "The aid provided to players through various technological means has improved at a rapid pace over the past few years.", "Match analysis results in a high volume of statistical data which in turn is used by players and coaching team for match preparation.", "In this paper, we propose to use a game-theoretic approach to develop a tool that can potentially help players in any two-player sports to thoroughly investigate the opponent and prepare a strategic plan to maximize the chance of winning.", "For the purpose of proposing a viable and feasible solution that can be directly employed in the real-world, we have chosen badminton as the primary sport for experiments, analysis and discussion owing to our deep understanding, the common passion for the sport and since it is easy to understand.", "Badminton is a racket sport in which two players alternately hit a shuttlecock until a point is scored by one of them.", "In this work, we propose two models and both the models use the history of match data of the opponent and the player under consideration as the input to provide suggestions and recommendations for a player.", "We also discuss the necessary steps to develop a complete, end-to-end solution integrating different types of data and technology in order to create a dedicated software application/ program that can be customized for each sport.", "Our first model called the recommendation system takes into consideration the various shots that the player and the opponent have played throughout their career.", "The main purpose of this system is to help the players gain an understanding of the different shots that the opponent plays and to gain knowledge of the best possible shots that he/she can play such that the chances of gaining a point are maximum.", "This model could be used by the players and coaches before going into a match and is not intended to be used during match play.", "Our second model which we like to call the Simulation model is an extension of the recommendation system but it takes into consideration the history of the match when it is in use.", "We use a reward system to determine the best shots for the players.", "This model is intended to provide match practice to the players against opponents so that they can simulate match situation and gain experience before heading into the actual match.", "Also, we intend to utilize some of the recent revisions in Badminton laws, one of which allows for coach intervention during the match.", "The proposed tool can prove to be very handy here as the coach can influence the player and guide the player in the middle of the match through the use of this tool by quickly analysing the match up till that particular point in the game-play.", "Moreover, our method could prove valuable in saving players and coaching staff the huge amount time expended in going through hours of match videos of the player and opponents and can critical help them in quantitatively analyse the performance and recommend the necessary preparation from players historical data." ], [ "Relevant works", "Game theory has been used to study various strategic sports in the past and most of the prevalent work done so far has been towards studying specific parts of a sport and not the entire match or game.", "In soccer, the penalty kick has been modelled as a strategic game with imperfect information because of uncertainty about the kicker's type [1].", "Bayesian equilibrium concept was used and it was found out that the kickers adopt a mixed strategy equilibrium depending based on their strong foot.", "In cricket, a normal form game was modelled between the batsman and the bowler [2].", "The strategies of the batsmen depended upon the type of shot played and the bowler's strategies were the different types of deliveries that he can bowl.", "The utility values were derived based on the probability of the player to take a particular strategy.", "The study revealed that the probability distribution followed by the players in adopting different strategies in real-world cricket is very close to the Nash equilibrium conditions.", "Alpha Go, an artificial intelligence entity that can play the game of Go which was developed by Deep-Mind was able to defeat the best player in the world by 5 games to 0.", "This intelligent program uses a combination of Monte Carlo Simulation with value and policy networks [3] with the concept of Markov Decision Processes (MDP) [4][3][5] as its base.", "The best of the players in any sports around the world learn at a rapid pace as they progress in their professional careers.", "However, there are few areas where this natural learning process doesn't prove to be very effective.", "In such cases, advanced mathematical and computer modelling can come to aid and convert this slow time-consuming process into a rapid and results-oriented one.", "In our literature survey, we came across many instances where game theory was used to solve problems pertaining to sports.", "One such case traces back to 2012 in the Olympics encounter between Yu Yang and Wang Xiaoli of China and South Korean pairs Jung Kyun-eun/Kim Ha-na and Ha Jung-eun/Kim Min-jung (doubles).", "The Chinese team tried to lose on purpose in this group stage encounter to avoid playing against their teammates Tian Qing and Zhao Yunle so that China is assured both the gold as well as the silver medals.", "[6] presents a detailed analysis on badminton match throwing using this example through game theory.", "The study reveals that the reason for this kind of match throwing lies in the loopholes of the format that the competition adopts and any rational player would adopt this strategy in the interest of the team.", "Besides this, there have been various other similar cases that drove us towards using game theory for our problem.", "In [7], game theory is used to determine the optimal time during the match to play a risky serve and how the surprise factor plays a part, also studying how it affects the outcome for the player under consideration.", "Apart from that, it is found that it is beneficial for the player to play a risky serve during the critical points of the match rather than the less important ones.", "Highly motivated with the past work along these lines, through this project, we intend to contribute towards the game of badminton and develop a highly effective tool for player assistance with the aid of game theory concepts.", "In summary our contributions are: A recommendation tool to suggest the best shots for each of the possible shots of the opponent using the concept of best strategy from game theory.", "A simulation model that considers the history up to two shots while determining the favorable shot to be played at any particular stage of the match and the approach is modelled as an approximated finite non-zero sum extensive from game" ], [ "Data Collection", "Comprehensive data is essential to effectively model the capability and choice of players which is vital in sports.", "Badminton is not like board games where predefined moves or strategies can consistently help you win.", "Humans tend to think differently when it comes to physical capabilities especially in sports.", "We can’t expect a player to play the same shot with the same accuracy every time; rather, we understand that a player’s capability, stability and mentality changes during the course of a game.", "But to best model these factors, data plays a key role.", "Our data was manually collected by going through several full match videos of the players.", "We considered matches between two of the best and most consistently performing badminton players in the world – Lin Dan from China and Lee Chong Wei from Malaysia so that the inefficiency of the players won't tamper with the final results and also incorporate the variation of left-handed and right-handed player in our data.", "The matches we recorded are spanned over a period of 8 years (2011 - 2019) so that we cover the changing game plan and shot selection over a considerable period.", "The data was manually collected and annotated on a shot-by-shot basis for a comprehensive modelling, with their outcomes in terms of points and sets won.", "This format is essential to calculate the necessary parameters for the proposed models.", "The types of shots which we have considered while collecting our data contribute to efficient results.", "Figure REF presents the scope of the shots for our paper.", "We have tried to incorporate the position of the player on the court in terms of the type of shots.", "Also, there are certain exceptions to the types of shots that can be played against a particular shot of the opponent.", "None of the shots can be responded with a service which is quite obvious.", "Also, a smash and a block cannot be returned by a smash and a block respectively.", "Drops are usually quite difficult to smash.", "It is important to note that important factors like agility, fatigue and mental state of the player during the course of the game are not taken into account while modelling due to the complexity involved.", "Figure: Various badminton Shots under consideration" ], [ "Methodology", "The detailed workings of each of the proposed methods are discussed in this section along with the mathematical modelling.", "For the purpose of modelling and easy representation, the player under consideration who uses our proposed approaches is referred to as $X_p$ and the player's opponent as $X_o$" ], [ "Recommendation Tool", "The recommendation tool considers $X_p$ ’s choice and capabilities based on $X_p$ 's history with a particular opponent $X_o$ to offer the best suggestions to each shot of $X_o$ .", "The concept of best response from game theory is adopted which will help $X_p$ to be match ready with the best and safe shots to play given any shot from $X_o$ to maximize $X_p$ 's chances of winning each point and in turn winning the match.", "We model the recommendation tool as a normal form game.", "The reason being that we are only worried about $X_p$ ; meaning, we only care about $X_p$ 's strategy and how to maximize $X_p$ 's chances of winning the match and not $X_o$ 's.", "So it is enough to consider the game on a shot-by-shot basis rather than a sequential game.", "At any stage of the game, for a given shot $s_{-i}$ of $X_o$ , $X_p$ will have set of probable shots (strategies) $S$ to play and the recommendation tool outputs those best shot $s_{i}*$ from the available shots $S$ which is the best response to $s_{-i}$ .", "The game can be modelled as a tuple representing a normal form game as represented in REF where $I = {X_p, X_o}$ and $S$ is a set of all badminton shots $G = <I, (s_i)_{i \\epsilon I}, (u_i)_{i \\epsilon I}> \\forall s_i, s_{-i} \\epsilon S$ $u_{i}\\left(s_{i} ,s_{-i}\\right) = P\\left( s_{i}|s_{-i} \\right)* P_{success}\\left( s_{i}|s_{-i} \\right)$ where $P( s_{i}|s_{-i} )$ is the probability of playing a shot $s_{i}$ for a given shot $s_{-i}$ of the opponent $X_o$ , $P_{success}( s_{i}|s_{-i} )$ is the success rate of a shot $s_{i}$ for $s_{-i}$ .", "These probabilities are calculated taking into account the data of the previous matches between the same two players.", "We consider the number of times a particular shot has been played by $X_p$ to calculate the probability of playing that shot including the instances where the shot yielded a point, resulted in the loss of a point or continuation of the rally.", "Within these instances, we consider the number of times that particular shot has yielded a point for $X_p$ while calculating the probability of success for that shot.", "This is done specifically for every shot played by $X_o$ .", "For calculating the best response for a particular shot of $X_o$ , we find the shot $s_{i}$ for $X_p$ which yields the maximum value of the utility according to the equation REF .", "Shot recommendation $s_i$ Point not gained 1.", "Select $s_i$ for $X_p$ for given $s_{-i}$ from $X_o$ 2.", "Calculate utility $u_i (s_i | s_{-i}), \\forall s_i \\epsilon S$ according to equation REF 3.", "Pick $s_i$ with maximum utility $u_{max}$ 4.", "Play $s_{i_{u_{max}}}$ usepackage Algorithm for recommendation system This recommendation can be a return to a type of service or any other shot during the rally.", "It will help $X_p$ be prepared to face $X_o$ with confidence and certainly rule out few shots which have resulted in an immediate point loss in $X_p$ 's history.", "Now, we can extend this tool further where we consider $X_o$ as our primary player and make all the computations to find his best possible shots for all the shots of $X_p$ based on the same data set.", "This will return a set of most probable shots of a particular for each shot of $X_p$ .", "Now, this will help predict the return shot of $X_o$ for a particular shot $s_{i}$ of $X_p$ .", "Accurate prediction of the type of serve or the type of return of $X_o$ for a particular shot $s_{i}$ of $X_p$ can prove to be very important in winning a point in crucial situations like dues or the first point of the set.", "This approach is the base for our other two methods and proves to be a vital one in real-world scenarios.", "We can observe many cases both from recommendation and the datasets, where a player rightly predicts a return of $X_o$ and surprises with a trick shot and wins a point." ], [ "Simulator", "The simulator is an extension to the recommendation tool but modelled as an extensive form game instead of a normal form game.", "The most important value addition to this method is that the history of shots between the players $X_p, X_o$ is taken into consideration.", "In badminton, it may not be always the case that the last shot results in winning or losing a point; the earlier shots played during the rally can also be responsible for a particular outcome.", "We observed from the collected data that this dependency on the history needs to be considered a maximum for two earlier shots for best results.", "We have introduced a reward system for incorporating history.", "We consider four types of rewards as follows: A high positive reward $R_{hp}$ when a shot of $X_p$ results in a direct point.", "For instance, a smash resulting in a direct point; $R_{hp}$ = +5 A medium positive reward $R_{mp}$ when a shot of $X_p$ induces a poor return from $X_o$ and thereby yielding a point.", "For instance, a good lift to the back making $X_o$ make a poor clearance helping $X_p$ kill immediately and gain a point; $R_{mp}$ = +2 A medium negative reward $R_{mn}$ for a poor shot of $X_p$ which $X_o$ takes advantage of making $X_p$ lose a point; $R_{mn}$ = -2 A low negative reward $R_{ln}$ when a shot results in a direct point loss; $R_{ln}$ = -5 The rewards are hence considered with history up to two shots which will help the algorithm suggest the best possible outcome for $X_p$ .", "The total reward for a shot of player $X_p$ given an opponent's shot is stated as follows: $\\begin{split}R_{T} (s_{i}, s_{-i}) = (R_{hp} (s_{i}, s_{-i})) + (R_{mp}(s_{i}, s_{-i}, s_{i}:t-1)) \\\\+ (R_{mn}(s_{i}, s_{-i}, s_{i}:t-1)) + (R_{ln}(s_{i}, s_{-i}))\\end{split}$ The purpose of the simulator is to help $X_p$ by predicting the result of a rally up to a predefined number of steps through the match.", "It will help $X_p$ emulate the sequence of rallies in different ways to practice with another person before the match.", "Though the game of badminton is a perfect information zero sum game, the simulator is modelled as a a perfect-information infinite non-zero sum extensive form game as the end utilities according to equation REF won't be the same for both the players.", "It is not possible to solve an infinite game.", "Hence we make a few modifications to the above-defined game into a finite extensive form game to be able to solve it up to a predefined number of steps.", "We call it the approximated sequential representation of the original game.", "Here, we restrict the number of sequences to a predefined number depending on the type of sport we are applying to.", "In badminton, it is enough for a player to think about his next two moves with respect to one move of the opponent in between as he could rectify his mistakes within that else it would result in a point gain or loss within that.", "The scenario is illustrated in the figure REF Figure: Extensive form representation of the gameWe introduce a reward system which is inspired from reinforcement learning [8] [9] that helps us calculate the favorable outcome for the player while taking into consideration the opponent’s moves.", "The model of the game can be represented as in equation REF where $I = {X_p, X_o}$ is the set of agents (players), $S$ is the set of possible badminton shots, $H$ is the set of choice nodes, $Z$ is the set of terminal nodes, $\\alpha $ agent function, $\\beta $ action function and $\\rho $ successor function respectively.", "This model is treated as a tree $T(n)$ where n is the number of nodes and $T$ can be expanded to $n$ levels denoting history of past events (3 in our case) for simulation.", "The game for $n$ levels is solved using backward induction with data containing the updated reward values of shots depending on $(\\alpha , \\beta , \\rho )$ .", "$G = <I, S, H, Z, \\alpha , \\beta _{i}, \\rho , u_i>$ $u_{i}(s_{i} ,s_{-i} ) = P( s_{i}|s_{-i} )* P_{success}( s_{i}|s_{-i} )* R_{T} ( s_{i},s_{-i} )$ where $u_{i}$ is the utility of agent $i$ , $P( s_{i}|s_{-i} )$ is the probability of $X_p$ playing a shot $s_{i}$ for a given shot $s_{-i}$ of $X_o$ , $P_{success}( s_{i}|s_{-i} )$ is the success rate of a shot $s_{i}$ of $X_p$ for a given shot $s_{-i}$ of $X_o$ , $R_{T}( s_{i},s_{-i})$ is the reward for playing $s_{i}$ for $s_{-i}$ .", "By approximating the infinite extensive game into a finite extensive form game, based on the rewards and utilities of players, we can predict the progress in the game after each shot.", "The simulator then recommends the best favourable way the game could progress with the shots $X_p$ has to play along with the ones $X_o$ is expected to play.", "Though this model has the capability of producing very good results, it is often very hard to model the cognitive process of humans.", "In sports, humans tend to think differently, a move by a badminton player will involve a lot of factors like fatigue, ability, confidence, ability, condition in game, precision of opponents shot and even gut feeling.", "The algorithm for the simulation system is presented in algorithm REF Shot recommendation $s_i$ Maximum number of predictions not reached 1.", "Select $s_i$ for $X_p$ for given $s_{-i}$ from $X_o$ 2.", "Expand simulation tree for next $n$ moves incorporating all possible shot combinations 3.", "Calculate accumulated utilities for $X_p , X_o$ using equation REF 4.", "Use backward induction to determine favourable shots and discard rest of the nodes 5.", "Solve and reduce traversal path towards one optimal shot with maximum utility $s_{i_{u_{max}}}$ 6.", "Update values for all the shot types based on real time data 7.", "Get $S_{-i}$ of $X_o$ for $s_{i_{u_{max}}}$ of $X_p$ Algorithm for simulator system" ], [ "Results", "In this section, we discuss the results of our models.", "As there is no established metric to verify the results of our proposed models, an expert opinion or domain knowledge is the only way to check the correctness and accuracy of results.", "As the model operates on the history of two players from the data, the results are only relevant to those players and will be different for others.", "Also, it is possible to create a generic model to focus on one player completely, given the data of the player with different opponents.", "The accuracy of the model depends on the volume and consistency of the available data." ], [ "Results of the Recommendation System", "The recommendation system suggests the best response for an $X_o$ ’s shot without considering the history of shots in the rally and the condition of the player in the game.", "The results i.e.", "the recommendations of the model for the player $X_p$ are shown in Table 1 and that for the opponent $X_o$ are shown in Table 2.", "It can be seen from the data that the model has successfully identified the best shots for given shots and that it has discarded the shots that are not playable for the opponent’s shot successfully.", "The number of suggestions can be increased and we have fixed it to 2 so that $X_p$ always has an alternative.", "Also, from the suggestions above, it can be seen that the shot $forehand drop$ has been repeated the most.", "This coincides with the fact that both the players are successful and capable in playing forehand drop shot without any error which was evident from the matches.", "Using this model, a player can be mentally prepared on what shot to play to a given shot of the opponent that can either lead to a point or keep $X_p$ alive in the rally to avoid losing a point.", "As this is modelled directly from the capability and behavior of the players, this information will be of value to the players before the match.", "Table: Recommendations for X p X_pTable: Recommendations for X o X_oFigure: Frequency of shots for Player P 1 P_{1} and player P 2 P_{2}" ], [ "Results of the simulator", "The purpose of the simulator is to take the history of the ongoing match into consideration and to model the game strategy for $X_p$ based on the position in the match, to identify the feasible sequence of shots that has lead to point gains and to avoid the poor shots.", "The simulator can be fed with seed shots i.e.", "few inputs in the start on how the game should proceed and it predicts the next few shots.", "Since there is no information about a point gain or loss to the simulator, it will infinitely predict the sequence of the shots in the game.", "It is accurate when compared to following only the best strategy since in reality, the player should think 2-3 steps ahead in badminton to realize the after-effect of playing a shot as there is always the possibility of a poor shot leading to a point loss during the consecutive shots played.", "Hence, at any point of time, the simulator builds a tree for the next three shots (2 for the player and 1 for the opponent), does backward induction on the utility values according to equation REF , arrives at the most favorable shot for $X_p$ , discards the rest of the tree and the process is repeated.", "To check the working of the simulator, we seeded it with a few shots from a match's data which was not used for modelling.", "The actual sequence of shots from the match figure REF and the sequence of predicted shots from the simulator is given in Table REF .", "Figure: Data from an actual match between Lin Dan and Lee Chong WeiTable: Output from the SimulatorIt can be seen from the tables that the results of simulator are closer to the actual match.", "The results mostly differ only in the sub-category of shots and the actual type of shots are the same.", "This tool can help the players understand how the match will proceed after a shot is played which is crucial to analyze the repercussions of playing a shot.", "The simulator model is likely to work better when the data used for modelling is larger.", "Fig.", "REF shows the distribution of shots for both the players in the data collected.", "There is an unequal distribution of shots which affects the accuracy of predictions of the models.", "We assume that data from at least 20 full matches is required to precisely model the behavior as in our case with 3 matches, the frequency of most of the shots is very low.", "Almost equal distribution of shots is required to ensure the reliability of the results from the model." ], [ "Conclusion", "In this paper, we have successfully developed two novel approaches for the development of an assistance tool for the game of badminton based on the concepts of Game Theory.", "Our recommendation tool takes in match data for the player under consideration against a particular opponent and gives out the best possible set of strategies (shots) which the player can use.", "The simulator model is a generalized and robust extension of this recommendation tool which considers the history of shots played in the ongoing match along with match history to suggest the favorable strategy for the players.", "The results, analysis and comparison with the actual match data shows the effectiveness of the system and that it is well-rounded.", "Our current work is restricted by the availability of data and the using the manually annotated data from 3 matches for our experiments was to show the feasibility of our approaches.", "In the presence of a considerable amount of annotated data, our future works are to test our approaches for other two player sports, remodelling the approaches for team sports, building a complete pipeline of dedicated software application or program that can dynamically function by adapting to the real-time change during a course of a tournament or a match using computer vision to capture visual information and reinforcement learning approaches like Markov Decision Process (MDP) to mathematically model decisions for a system of advanced assistance." ] ]
2105.11650
[ [ "An experimental review of open heavy flavor and quarkonium production at\n RHIC" ], [ "Abstract Open heavy flavor and quarkonium are unique probes of the hot-dense medium produced in heavy-ion collisions.", "Their production in p+p collisions also provide important test of QCD.", "In this paper, we review the selected results on open heavy flavor and quarkonium in p+p and heavy-ion collisions achieved at RHIC.", "Physics implications are also discussed." ], [ "Introduction", "In the ultra-relativistic heavy-ion collisions, the smash of two colliding nucleus creates an extremely hot and dense medium, in which the quarks and gluons are liberated from confinement inside hadrons and form a new state of matter called Quark-Gluon Plasma (QGP) [1], [2].", "Since past twenty years, many experiment evidences from Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC), such as the jet quenching and strong particle flow of light flavor hadrons (consists of light quarks $u$ , $d$ , $s$ ), show that the QGP matter is strongly coupled and behaves like a liquid with small viscosity over entropy density [3], [4], [5], [6].", "However, most of the light flavor hadrons are produced late in the collisions with final state effects and the information of the QGP created in early stage of the collisions may be smeared.", "Heavy quark masses ($m_{c} \\sim $ 1.3 GeV/$c^{2}$ , $m_{b} \\sim $ 4.8 GeV/$c^{2}$ ) are much larger than light quarks and QCD energy scale ($\\Lambda _{\\rm QCD}$ ).", "They are predominately produced via initial hard processes at early stage in the ultra-relativistic heavy-ion collisions and the probability of thermal production is negligible, especially at RHIC energies.", "Thus heavy quarks experience the whole evolution of the QCD matter created in heavy-ion collisions and are ideal probes to study the QGP matter properties.", "Most of the heavy quarks hadronize into open heavy flavor mesons (e.g.", "$D^0$ , $D^{\\pm }$ , $D_{s}^{\\pm }$ , $B^0$ , $B^{\\pm }$ , $B_{s}^{\\pm }$ ) and baryons (e.g.", "$\\Lambda _{c}$ ), while a small fraction $<$ 1% of the total heavy quarks forms hidden heavy flavor, quarkonium states (e.g.", "J/$\\psi $ , $\\Upsilon $ ) and their families.", "Theoretic model predicts that heavy quarks loose less energy compared with light quarks due to the suppression of gluon radiation angle by their large masses [7].", "The measurement of open charm/bottom nuclear modification factor ($R_{AA}$ ), which is defined as the yield measured in Au+Au collisions divided by that in $p$ +$p$ collisions and scaled by the average number of binary collisions ($N_{\\rm coll}$ ), is commonly used to measure the medium effect characterizing as any deviation from unity.", "Strong suppression of open charm hadron $R_{AA}$ in central heavy-ion collisions at high transverse momentum ($p_T$ ) from recent measurements of STAR and ALICE experiments [8], [9] indicates strong interactions between charm and hot-dense medium.", "The similar suppression level of charmed and light hadrons can be explained by model calculations including both elastic and inelastic energy loss [10], [11].", "In the mean time, the open bottom hadron measurements are very challenging due to poor production rate and small hadronic decay branching ratios.", "The effective way to measure bottom is via its decay products.", "On the other hand, the second order coefficient of Fourier expansion of particle azimuth distribution in the momentum space, $v_2$ , is commonly used to measure medium bulk properties and how the medium transports the partons [12].", "Heavy quarks are more difficult to participate in the partonic collectivity due to their large masses.", "Recent measurements of large elliptic flow of $D$ mesons indicate that charm quark has similar flow as light quarks and may reach thermalization [13], [14], [15].", "By comparison of experimental data and theoretical model calculations, the transport diffusion coefficient of charm quark traversing in the medium can be obtained with large uncertainties [13].", "To well understand the interactions between heavy quarks and medium, experiments keep devoting efforts in upgrading detectors to pursue high spatial resolution and fast response for precise measurement of heavy flavor hadron production in new era.", "Quarkonium is a tightly bound state of a heavy quark and its anti-quark.", "Charmonium (bottomonium) refers the bound state of charm (bottom) quark and its anti-quark.", "Table REF shows the mass, binding energy and radius of various quarkonium states [16].", "If QGP is formed, the potential of heavy quark and its anti-quark is expected to be modified by the deconfined medium.", "The real part of the potential can get color-screened statically in the medium, resulting in a broadening of the wave function of the pair of heavy quark and its anti-quark.", "While the imaginary part of the potential is related to the dissociation of quarkonium arising from the scattering of quarkonium with the medium constituents such as gluons.", "The suppression of J/$\\psi $ due to color screening was proposed as a signature of the QGP formation [17] and was considered as a strong experimental evidence of deconfinement in the medium produced in Pb+Pb collisions at SPS [18].", "The temperature required to dissociate a quarkonium state (dissociation temperature, $T_d$ ) depends on the binding energy of the quarkonium state.", "More loosely bounded state has lower $T_d$ .", "In both charmonium and bottomonium sectors, $T_d$ decreases with increasing quarkonium mass and the excited states have lower $T_d$ than the 1S state.", "Based on the radius of the quarkonium as shown in Tab.", "REF , it is expected that $T_d^{\\Upsilon (1S)} > T_d^{\\chi _b} \\sim T_d^{J/\\psi } \\sim T_d^{\\Upsilon (2S)} > T_d^{\\chi _b^{\\prime }} \\sim T_d^{\\chi _c} \\sim T_d^{\\Upsilon (3S)} > T_d^{\\psi (2S)}$ .", "The systematical measurements of quarkonium suppression can also help to constrain the temperature profile and the dynamic evolution of the fireball produced in ultra-relativistic heavy-ion collisions.", "Table: The mass, binding energy and radius of charmonium and bottomonium states .In contrast to the color-screening, quarknoium production yield could be enhanced due to (re)combination of (un)associated heavy quark and its anti-quark during QGP evolution and/or hadronization.", "The dissociation rate and/or the recombination probability depend on the properties of QGP, such as the temperature profile and the evolution of the fireball etc, as well as the size of the quarknium.", "Although the (re)combination effect is competing with the QGP melting effect, both of them require deconfinement and can be used to search for QGP and study its properties.", "In additional to these two hot nuclear matter effects, quarkonium production in heavy-ion collisions is also affected by cold nuclear matter (CNM) effects, including modification of parton distribution function in nucleus (nPDF), breakup by hadrons, the scattering and/or energy loss of the parton evolved in quarkonium production etc.", "The CNM effects can be experimentally studied via the collisions of $p$ or light nucleus and heavy nucleus.", "There are other effects need to be taken into account when interpret the experimental results.", "One important effect is the feeddown contribution of the quarkonium production.", "Since the relative contributions of these effects have different dependences on various variables, such as initial energy density, system size, total heavy-quark cross-section, size and transverse momentum ($p_T$ ) of the quarkonium state etc, a comprehensive study of the quarkonium yield as a function of the collision energy, collision system, quarkonium $p_T$ and rapidity of different quarkonium states, as well as collectivity of heavy flavor hadron and quarkonium, is essential for a complete understanding of quarkonium production in heavy-ion collisions.", "In the following sections, we imply the average of particle and anti-particle when using the term of particle unless otherwise specified.", "The charm production cross sections in high energy $p$ +$p$ collisions can be evaluated by perturbative Quantum Chromodynamics (QCD) [19], [20].", "The differential transverse momentum ($p_{T}$ ) spectra of $D^0$ mesons in a wide energy range from $\\sqrt{s} = $ 200 GeV upto 7 TeV in $p$ +$p$ collisions measured by STAR [21], [22], CDF [23] and ALICE [24], [25], [26] experiments respectively are in good agreement with the upper limit of Fixed-Order-Next-to-Leading-Logarithm (FONLL) calculations [27], [28], [19], [29].", "In heavy-ion collisions, charm quarks interact with the hot-dense medium and their transverse momenta are modified via energy loss, collective flow or cold nuclear matter (CNM) effects.", "The charmed hadrons are formed via charm hadronization from fragmentation, coalescence or recombination until chemical freeze-out.", "After kinetic freeze-out, the final state interactions stop, the charmed hadrons spectra are fixed as what we measure.", "Figure REF shows the centrality dependence of charmed hadron $p_{T}$ spectra measured by STAR with the help of identifying secondary decay vertices of charmed hadrons utilizing recent developed state-of-the-art silicon pixel detector, Heavy Flavor Tracker (HFT) [30], [31].", "The $D^0$ $p_{T}$ spectra at mid-rapidity ($|y|<1$ ) in 0–10%, 10–20%, 20–40%, 40–60% and 60–80% Au+Au collisions [32] are shown in the left panel.", "The $D_s$ $p_{T}$ spectra (triangles) in 0–10%, 10–40%, 40–80% and $\\Lambda _c$ spectrum (stars) in 60–80% Au+Au collisions at $|y|<1$  [33], [34] are shown in the right panel.", "The spectra in some centrality bins are scaled with arbitrary factors indicated on the figure for clarity.", "Figure: Left Panel: The D 0 D^0 p T p_{T} spectra at |y|<1|y|<1 in 0–10%, 10–20%, 20–40%, 40–60% and 60–80% Au+Au collisions .", "Right Panel: The D s D_s p T p_{T} spectra (triangles) in 0–10%, 10–40%, 40–80% and Λ c \\Lambda _c spectrum (stars) in 60–80% Au+Au collisions at |y|<1|y|<1 , .", "The statistic and systematic uncertainties are represented by vertical error bars and brackets, respectively.The nuclear modification factor $R_{\\rm AA}$ is calculated as the ratio of $N_{\\rm bin}$ –normalized yields between Au+Au and $p$ +$p$ collisions.", "The $R_{\\rm AA}$ of $D^0$ mesons in 0-10% central Au+Au collisions at ${\\sqrt{s_{\\rm NN}} = \\rm {200\\,GeV}}$  [32] is compared to that of (a) average $D$ -meson from ALICE [35] and (b) charged hadrons from ALICE and $\\pi ^{\\pm }$ from STAR [36], [37], shown in Fig.", "REF .", "The $D^0$ $R_{\\rm AA}$ from this measurement is comparable to that from the LHC measurements in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV despite the large energy difference between these measurements.", "A significant suppression is seen at $p_{T}>$ 5 GeV/$c$ .", "The suppression level is similar to that of light flavor mesons, indicating strong interactions of charm with medium and energy loss.", "At $p_{T}$  $<$  5 GeV/$c$ , the $D^0$ $R_{\\rm AA}$ shows a characteristic bump structure.", "The Duke model and the Linearized Boltzmann Transport (LBT) model [38], [39] calculations that predict sizable collective motion for charm quarks during the medium evolution can qualitatively describe the STAR data.", "The uncertainties from the $p$ +$p$ reference [21] dominates the systematic uncertainty for STAR $R_{\\rm AA}$ .", "Figure: (color online) D 0 D^{0} R AA R_{\\rm AA} in 0–10% Au+Au collisions at s NN =200 GeV {\\sqrt{s_{\\rm NN}} = \\rm {200\\,GeV}} compared to the ALICE DD-meson result in 0–10% Pb + Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}} = 2.76 TeV (a) and charged hadrons from ALICE and π ± \\pi ^{\\pm } from STAR (b).", "Also shown in panel (a) are the model calculations from the LBT and Duke groups , .", "Notations for statistical and systematic uncertainties are the same as in previous figures.Several functions, such as Levy, power-law, $m_T$ exponential, Blast-wave [40], Tsallis Blast-wave [41], are used to fit the $D^0$ data in above centrality bins to extract the collectivity and thermal properties.", "In this paper, only the physics results and conclusions are discussed.", "The analysis details can be found in Ref. [32].", "The obtained slope parameter $T_{\\rm eff}$ for $D^0$ mesons is compared to other light and strange hadrons measured at RHIC.", "Left panel of Fig.", "REF summarizes the slope parameter $T_{\\rm eff}$ for various identified hadrons ($\\pi ^{\\pm }$ , $K^{\\pm }$ , $p$ /$\\bar{p}$ , $\\phi $ , $\\Lambda $ , $\\Xi ^-$ , $\\Omega $ , $D^0$ and J/$\\psi $ ) in central Au+Au collisions at $\\sqrt{s_{_{\\rm NN}}} {\\rm = 200\\,GeV}$  [42], [43], [44], [45].", "Point-by-point statistical and systematic uncertainties are added as a quadratic sum when performing these fits.", "All fits are performed up to $m_{T} - m_{0} <1\\,\\rm {GeV}/c^2$ for $\\pi ,\\ K,\\ p$ , $<2$  GeV/$c^2$ for $\\phi ,\\ \\Lambda ,\\ \\Xi $ , and $<3$  GeV/$c^2$ for $\\Omega ,\\ D^{0},\\ J/\\psi $ , respectively.", "The slope parameter $T_{\\rm eff}$ in a thermalized medium can be characterized by the random (generally interpreted as a kinetic freeze-out temperature $T_{\\rm fo}$ ) and collective (radial flow velocity $\\langle \\beta _{T}\\rangle $ ) components with a simple relation [3], [46], [47]: $\\begin{aligned}T_{\\rm eff} = T_{\\rm fo} + m_0 \\langle \\beta _{T}\\rangle ^2.\\end{aligned}$ Therefore, $T_{\\rm eff}$ will show a linear dependence as a function of particle mass $m_0$ with a slope that can be used to characterize the radial flow collective velocity.", "The data points of $\\phi ,\\ \\Lambda ,\\ \\Xi ^{-},\\ \\Omega ^{-},\\ D^0$ follow a linear dependence with different slope compared to that of $\\pi ,\\ K,\\ p$ , as represented by the dashed lines shown in left panel of Fig.", "REF .", "Light flavor hadrons, such as, $\\pi ,\\ K,\\ p$ gain radial collectivity through the whole system evolution, therefore the linear dependence exhibits a larger slope.", "On the other hand hadrons contains strangeness or heavy quarks, such as, $\\phi ,\\ \\Lambda ,\\ \\Xi ^{-},\\ \\Omega ^{-},\\ D^0$ may freeze out from the system earlier, and therefore receive less radial collectivity, resulting in a smaller slope of the linear dependence of $T_{\\rm eff}$ versus mass.", "Figure: (color online) Left Panel: Slope parameter T eff T_{\\rm eff} for different particles in 0-10% central Au+Au collisions , , , .", "The dashed lines depict linear function fits to π,K,p\\pi ,K,p and φ,Λ,Ξ - ,Ω - ,D 0 \\phi ,\\Lambda ,\\Xi ^{-},\\Omega ^{-},D^0 respectively.", "Right Panel: Results of T kin T_{\\rm kin} vs. 〈β〉\\langle \\beta \\rangle from the Blast-Wave model fits to different groups of particles.", "The data points for each group of particles present the results from different centrality bins.", "The one in most central collisions is at the largest 〈β〉\\langle \\beta \\rangle .Figure REF right panel summarizes the fit parameters $T_{\\rm kin}$ vs. $\\langle \\beta \\rangle $ from the Blast-Wave model fits to different groups of particles: black markers for the simultaneous fit to $\\pi ,\\ K,\\ p$ ; red markers for the simultaneous fit to $\\phi ,\\ \\Xi ^-$ and blue markers for the fit to $D^0$ .", "The data points for each group of particles represent the fit results from different centrality bins with the most central data point at the largest $\\langle \\beta \\rangle $ value.", "Similar as in the fit to the $m_{T}$ spectra, point-by-point statistical and systematic uncertainties are added in quadrature when performing the fit.", "The fit results for $\\pi ,\\ K,\\ p$ are consistent with previously published results [41].", "The fit results for multi-strangeness particles $\\phi ,\\ \\Xi ^{-}$ , and for $D^0$ show much smaller mean transverse velocity $\\langle \\beta \\rangle $ and larger kinetic freeze-out temperature, suggesting these particles decouple from the system earlier and gain less radial collectivity compared to light hadrons.", "The resulting $T_{\\rm kin}$ parameters for $\\phi ,\\ \\Xi ^-$ and for $D^0$ are close to the pseudocritical temperature $T_{c}$ calculated from a lattice QCD calculation at zero baryon chemical potential [48], indicating negligible contribution from the hadronic stage to the observed radial flow of these particles.", "Therefore, the collectivity that $D^0$ mesons obtain is mostly through the partonic stage re-scatterings in the QGP phase.", "Figure: (color online) (a) v 2 v_2 as a function of p T p_{T} and (b) v 2 /n q v_2/n_q as a function of (m T -m 0 )/n q (m_{\\rm T}-m_0)/n_q for D 0 D^0 in 10–40%\\% centrality Au+Au collisions compared with K S 0 K^0_S, Λ\\Lambda , and Ξ - \\Xi ^- .", "The vertical bars and brackets represent statistical and systematic uncertainties, and the grey bands represent the estimated non-flow contribution.Other observable to measure bulk collectivity is the elliptic flow characterized by the second order coefficient of particle azimuth distribution in the momentum space, $v_2$ .", "The elliptic flow measurement of multi-strange hadrons and $\\phi $ mesons indicates the partonic collectivity has been built up at the top energy heavy ion collisions of RHIC [50].", "Recently, with the help of the silicon vertex detector HFT, STAR experiment measured the $D^0$ $v_2$  [13] in Au+Au collisions at $\\sqrt{s_{_{\\rm NN}}}$ 200 GeV.", "Figure REF compares the measured $D^0$ $v_2$ from the event plane method in 10–40$\\%$ centrality bin with $v_2$ of $K_{s}^0$ , $\\Lambda $ , and $\\Xi ^-$  [49].", "The comparison between $D^0$ and light hadrons needs to be done in a narrow centrality bin to avoid the bias caused by the fact that the $D^0$ yield scales with number of binary collisions while the yield of light hadrons scales approximately with number of the participants [51].", "Panel (a) shows $v_2$ as a function of $p_{T}$ where a clear mass ordering for $p_{T}$  $<$  2 GeV/$c$ including $D^0$ mesons is observed.", "For $p_{T}$  $>$  2 GeV/$c$ , the $D^0$ meson $v_2$ follows that of other light mesons indicating significant charm quark flow at RHIC [52], [49], [50].", "Recent ALICE measurements show that the $D^0$ $v_2$ is comparable to that of charged hadrons in 0-50% Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV [14], [15] suggesting sizable charm flow at the LHC.", "Panel (b) shows $v_2/n_q$ as a function of scaled transverse kinetic energy, $(m_{\\rm T}-m_0)/n_q$ , where $n_q$ is the number of constituent quarks in the hadron, $m_0$ is the rest mass, and $m_{\\rm T}=\\sqrt{p_{\\rm T}^2+m_0^2}$ .", "We find that the $D^0$ $v_2$ falls into the same universal trend as all other light hadrons [52], [49], in particular for $(m_{\\rm T}-m_0)/n_q$  $<$  1 GeV/$c^2$ .", "This suggests that charm quarks have gained significant flow through interactions with the sQGP medium in 10–40% Au+Au collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 200 GeV.", "Figure: (color online) Integrated D 0 D^0 cross section per nucleon-nucleon collision at mid-rapidity in s NN =\\sqrt{s_{\\rm NN}} = 200 GeV Au+Au collisions for p T p_{T} > 0 (a) and p T p_{T} > 4 GeV/c (b) as a function of centrality Npart.", "The statistical and systematic uncertainties are shown as error bars and brackets on the data points.", "The green boxes on the data points depict the overall normalization uncertainties in pp+pp and Au+Au data respectively.In heavy-ion collisions, charm quark interacts with the QGP matter when traversing in the medium.", "The transverse momentum of charm quark is modified by the medium via energy loss or collective flow.", "However, the total number of charm quarks may keep conserved since they are produced in initial hard processes before the QGP formation and there is no more charm quark created later via thermal production at RHIC energies.", "Figure REF (a) and (b) shows the $p_{T}$ –integrated cross section for $D^0$ production per nucleon-nucleon collision $d\\sigma ^{\\rm NN}/dy|_{y=0}$ from different centrality bins in $\\sqrt{s_{_{\\rm NN}}} {\\rm = 200\\,GeV}$ 200 GeV Au+Au collisions for the full $p_{T}$ range and for $p_{T}$  $>$  4 GeV/$c$ , respectively [32].", "The result from the $p$ +$p$ measurement at the same collision energy is also shown in both panels [21].", "The high $p_{T}$ ($>$  4 GeV/$c$ ) $d\\sigma ^{\\rm NN}/dy|_{y=0}$ shows a clear decreasing trend from peripheral to mid-central and central collisions and the result in peripheral collisions is consistent with $p$ +$p$ collisions within uncertainties.", "This is consistent with charm loses more energy in more central collisions at high $p_{T}$ .", "However, the $d\\sigma ^{\\rm NN}/dy|_{y=0}$ integrated over full $p_{T}$ range shows approximately a flat distribution as a function of $N_{\\rm part}$ .", "The values for the full $p_{T}$ range in mid-central to central Au+Au collisions are smaller than that in $p$ +$p$ collisions with $\\sim 1.5\\sigma $ effect considering the large uncertainties from the $p$ +$p$ measurements.", "The total charm quark yield in heavy-ion collisions is expected to follow the number-of-binary-collision scaling since charm quarks are conserved at RHIC energies.", "However, the cold nuclear matter (CNM) effect including shadowing could also play an important role.", "In addition, hadronization through coalescence could alter the hadrochemistry distributions of charm quark in various charm hadron states which may lead to the reduction in the observed $D^0$ yields in Au+Au collisions [53].", "For instance, hadronization through coalescence can lead to an enhancement of the charmed baryon $\\Lambda _{c}^+$ yield over $D^0$ yield [54], [55], [56], and together with the strangeness enhancement in the hot QCD medium and sequential hadronization, can also lead to an enhancement in the charmed strange meson $D_{s}^+$ yield relative to $D^0$  [57], [58], [55], [56].", "Figure: (color online) Left Panels: The measured Λ c /D 0 \\Lambda _{c}/D^0 ratio at mid-rapidity (|y|<|y|< 1) as a function of p T p_{T} for Au+Au collisions at (s NN )=\\sqrt{(}s_{\\rm NN}) = 200 GeV in 10-80% centrality, compared to the baryon-to-meson ratios for light and strange hadrons (a) and various model calculations (b).", "The p T p_{T} integrated Λ c /D 0 \\Lambda _{c}/D^0 ratio from the THERMUS  model calculation with a freeze-out temperature of T ch =160T_{\\rm ch}=160 MeV is shown as a horizontal bar on the left axis of the plot.", "Right panels: (c) The integrated D s /D 0 D_{s}/D^{0} ratio (black solid circles) of 1.5 << p T p_{T} << 8 GeV/cc as a function of p T p_{T} compared to model calculation (curves) in 0-10% Au+Au collisions at s NN \\sqrt{s_{_{\\rm NN}}} = 200 GeV.", "(d) Same D s /D 0 D_{s}/D^{0} ratio as (c) but with 10-40% centrality.", "The vertical lines and brackets on the data points indicate statistical and systematic uncertainties respectively.The STAR Heavy Flavor Tracker (HFT) with a silicon pixel detector achieved $\\sim $ 30 $\\mu m$ spacial resolution of the track impact parameter to the primary vertex allows a topological reconstruction of the decay vertices of open charm hadrons.", "Figure REF left panels show the charmed baryon over meson ratio compared with light and strange baryon over meson ratios [60], [61] (a) and various models (b).", "The $\\Lambda _c/D^0$ ratio is comparable in magnitude to the $\\Lambda /K^0_s$ and $p$ /$\\pi $ ratios and shows a similar $p_{T}$ dependence in the measured region.", "A significant enhancement is seen compared to the calculations from the latest PYTHIA 8.24 release (Monash tune [62]) without (green solid curve) and without (magenta dot-dashed curve) color reconnections (CR) [63].", "The implementation with CR is found to enhance the baryon production with respect to mesons.", "However, both calculations fail to fully describe the data and its $p_{T}$ dependence.", "Figure REF (b) also shows the comparison to various models with coalescence hadronization of charm quarks [54], [56], [55], [57], [58].", "The comparisons suggest coalescence hadronization plays an important role in charm-quark hadronization in the presence of QGP.", "Also, the data can be used to constrain the coalescence model calculations and their model parameters.", "Figure REF right panel shows the $D_{s}/D^{0}$ ratio as a function of $p_{T}$ compared to coalescence model calculations for 0-10% (c) and 10-40% (d) collision centralities.", "Several models incorporating coalescence hadronization of charm quarks and strangeness enhancement are used to describe the $p_{T}$ dependence of $D_{s}/D^{0}$ ratio.", "Those models assume that $D_s^{\\pm }$ mesons are formed by recombination of charm quarks with equilibrated strange quarks in the QGP [54], [56], [55], [57], [58].", "In particular, the sequential coalescence model together with charm quark conservation [55] considers that more charm quarks are hadronized to $D_s^{\\pm }$ mesons than $D^{0}$ since the former is created earlier in the QGP, which results in further enhancement of $D_{s}/D^{0}$ ratio in Au+Au collisions relative to $p$ +$p$ collisions.", "STAR experiment extracted the total charm production cross section per binary nucleon collision at midrapidity in 200 GeV Au+Au collisions by summing all yields of the open charm hadron states [64], which is consistent with that in $p$ +$p$ collisions [21] within uncertainties.", "The numbers are reported as, $ AuAu: d\\sigma ^{\\rm NN}/dy|_{y=0} = 152 \\pm 13 (stat) \\pm 29 (sys) \\mu b, \\\\pp: d\\sigma /dy|_{y=0} = 130 \\pm 30 (stat) \\pm 26 (sys) \\mu b$ This result is consistent with charm quark conservation in heavy-ion collisions at RHIC top energy." ], [ "Open bottom production", "Theoretical calculations predict that the heavy quark energy loss is less than that of light quarks due to suppression of the gluon radiation angle by the quark mass.", "Bottom quark mass is a factor of three larger than charm quark mass, thus the less bottom quark energy loss is expected compared to charm quark when they traverse the hot-dense medium created in the heavy-ion collisions [7], [10], [11].", "However, the low production cross section of bottom quark in RHIC energy and very small hadronic decay branching ratio prevent direct measurement of open bottom hadrons in experiments at RHIC.", "Fortunately, different life time of open charm hadrons and open bottom hadrons allow us to separate their decay products utilizing STAR HFT to distinguish their decay vertices and provide the impact parameter (or the distance of closest approach to primary collision vertex, DCA) distributions.", "Figure: (color online) R AA R_{\\rm AA} of B→J/ψB{\\rightarrow }J/\\psi (blue solid squares), B→D 0 B{\\rightarrow }D^0 (red solid circles), b→eb{\\rightarrow }e (blue open circles) and c→ec{\\rightarrow }e (red open squares) at mid-rapidity in s NN =\\sqrt{s_{\\rm NN}} = 200 GeV Au+Au collisions from STAR experiment , , .", "Vertical bars and bands represent statistic and systematic uncertainties, respectively.", "Dashed and dot-dashed curves are DUKE model calculations  for b→eb{\\rightarrow }e and c→ec{\\rightarrow }e, respectively.Recently the non-prompt products from open bottom decays, $B{\\rightarrow }J/\\psi $ , $B{\\rightarrow }D^0$ and $b{\\rightarrow }e$ , were measured by the STAR experiment at mid-rapidity in $\\sqrt{s_{\\rm NN}} = $ 200 GeV Au+Au collisions via a template fit method using the different shape of impact parameters between signal and background [65], [66], [67].", "The results of $R_{AA}$ as a function of $p_{T}$ are shown in Fig.", "REF .", "The data of $B{\\rightarrow }J/\\psi $ , represented by blue solid squares, are observed suppressed in the whole $p_{T}$ region from 2 to 8 GeV/c.", "The similar suppression is also observed for $B{\\rightarrow }D^0$ (red solid circles) and $b{\\rightarrow }e$ (blue open circles) at high $p_{T}$ .", "These results indicate that interactions between bottom quark and the hot-dense nuclear matter lead to bottom quark energy loss in the medium.", "It should be noted that the non-prompt $J/\\psi $ , $D^0$ and electrons shown here are in 0-80%, 0-10% and 0-80% Au+Au collisions, respectively.", "Comparing with $c{\\rightarrow }e$ , shown as red open squares, $b{\\rightarrow }e$ is systematically less suppressed, which indicates bottom looses less energy than charm.", "The calculations of a transport model from Duke group [68] reproduce the data within uncertainties.", "The non-prompt $D^0$ at 4 GeV/c shows no suppression as well, again consistent with less energy loss of bottom quark due to its heavier mass comparing with charm and light quarks.", "These two independent measurements provide important evidence of mass-dependent parton energy loss in the hot QCD medium in high energy heavy-ion collisions.", "Recently an experimental data driven approach was applied to extract bottom elliptic flow from heavy flavor semileptonic decay channels [69].", "Taking the advantage of silicon vertex detectors, high precision open charm hadrons were measured by STAR experiment, which allows bottom contribution can be extracted by subtracting the contributions of open charm decays from the inclusive heavy flavor electron spectrum.", "Figure REF shows the $v_2$ results of electrons from open charm ($v_2^{\\rm c\\rightarrow e}$ ) and open bottom ($v_2^{\\rm b\\rightarrow e}$ ) decays as the blue solid curve with an uncertainty band and red circles, respectively.", "The $v_2$ of $\\rm \\phi \\rightarrow e$ ($v_2^{\\rm \\phi \\rightarrow e}$ is shown as red long-dashed curve with band.", "DUKE model predictions [68] are also shown as dot-dashed curves for comparison.", "The electron $v_2$ from beauty hadron decays at $p_T$ $>$ 3.0 GeV/$c$ is observed with an average of 4-sigma significance ($\\chi ^2/ndf$ = 29.7/6) deviating from zero.", "And it is consistent with electrons from charmed or strange hadron decays within uncertainties at $p_T$ $>$ 4.5 GeV/$c$ .", "This flavor independent $v_2$ at high $p_T$ could be attributed to the initial geometry anisotropy or the path length dependence of the energy loss in the medium.", "A smaller $v_2^{\\rm b\\rightarrow e}$ compared with $v_2^{\\rm c\\rightarrow e}$ is observed at $p_T$ $<$ 4.0 GeV/$c$ , which may be driven by the larger mass of beauty quark than that of charm quark.", "The $v_2^{\\rm b\\rightarrow e}$ deviates from the hypothesis of that B-meson $v_2$ follows the NCQ scaling (black curve) at 2.5 GeV/$c$ $<$ $p_T$ $<$ 4.5 GeV/$c$ with a confidence level of 99% ($\\chi ^2/ndf$ = 14.3/4), which favors that the beauty quark elliptic flow is smaller than that of light quarks, unlike the $D^0$ $v_2$ scaled with that of light flavor hadrons by dividing number of constituent quarks in both $v_2$ and $\\left(m_{\\rm T}-m_0\\right)$ as presented in Fig.", "REF previously.", "This suggests that beauty quark is unlikely thermalized and too heavy to be moved following the collective flow of lighter partons in heavy-ion collisions at RHIC energy.", "Figure: (color online) The elliptic flows (v 2 v_2) of electrons from open charm (blue band) and open bottom (red circles) decays at mid-rapidity (η\\left|\\eta \\right| << 0.7) in minimum bias Au+Au collisions at s NN \\sqrt{s_{_\\mathrm {NN}}} = 200 GeV.", "The v 2 b→e v_2^{\\rm b\\rightarrow e} with B-meson v 2 v_2 NCQ scaling assumption and the v 2 φ→e v_2^{\\rm \\phi \\rightarrow e} are shown as the dashed curve and open squares, respectively.", "Results from DUKE  model predictions are shown for comparison.", "Figure is taken from Ref.", "." ], [ "Quarkonium production in $p$ +{{formula:e5ea8805-c4f2-4924-8de0-4c02bf3c361f}} collisions", "Quarkonium production in $p$ +$p$ collisions is a crucial baseline of the study of quarkonium production in medium for various reasons.", "First of all, to quantify the modification of quarkonium production in heavy-ion collisions, we usually compare the production yield of quarkonium in heavy-ion collisions to that in $p$ +$p$ collisions by calculating the nuclear modification factor $ R_{\\textrm {AA}}=\\frac{y_{\\textrm {AA}}}{y_{pp}~N_{\\textrm {coll}}},$ where $y_\\textrm {AA}$ and $y_{pp}$ is the yield of quarkonium in heavy-ion collisions and $p$ +$p$ collisions, respectively, and $N_\\textrm {coll}$ is number of binary nucleon-nucleon collisions in heavy-ion collisions.", "If the heavy-ion collision is only superposition of nucleon-nucleon collisions, the yield of quarkonium should follow the $N_\\textrm {coll}$ -scaling and $R_\\textrm {AA}=1$ .", "Deviation of $R_{\\mathrm {AA}}$ from unity indicates modification of quarkonium production in heavy-ion collisions.", "Secondly, understanding the production mechanism of quarkonium in $p$ +$p$ collisions is essential to interpret the $R_{\\mathrm {AA}}$ measurements in heavy-ion collisions.", "For example, quarkonium has relative large formation time, whether the $q\\bar{q}$ interstate (before the quarkonium is fully formed) is color-singlet or color-octet may result in different modification of quarkonium yield in heavy-ion collisions when it traverses the medium.", "In fact, quarkonium production in hadron collisions also provides important test of QCD.", "The production of the pair of heavy quark and its anti-quark are dominantly from the initial hard scattering and can be calculated in the framework of perturbative QCD down to low $p_T$ .", "However, when the heavy-quark pair forms a physical quarkonium bound state, the process involves long distances and soft momentum scales and thus is a non-perturbative.", "The latter relies on modeling.", "The detailed study of quarkonium production with hadron collider and the comparison to theoretical calculations provides an important test ground of both perturbative and non-perturbative aspects of QCD calculations.", "Experimental study of the quarkonium production mechanism is usually conduced by measuring the production yield and polarization against different kinematic variables such as rapidity, $p_{_T}$ and compare to different theoretical calculations.", "It is also important to note that not all of the quarkonium are produced directly, but a large fraction of quarkonium are produced via the decay of other hadrons such as higher quarkonium states and $B$ -hadrons for charmonium.", "The contribution from the decay of other hadrons is called feeddown contribution.", "Because of different properties such as binding energy, the directly produced quarkonium and those from the decay of different hadrons should have different modification in heavy-ion collisions, the feeddown contribution has to be taken into account when extracting physics information from the measurements of quarkonium suppression in heavy-ion collisions.", "The feeddown contribution is usually studied in $p$ +$p$ collisions and/or small systems such as $p$ +A collisions." ], [ "J/$\\psi $ production cross-section", "Figure REF shows the inclusive J/$\\psi $ production cross-section at mid-rapidity ($|y|<1$ ) as a function of $p_{_T}$ in non-single-diffractive $p$ +$p$ collisions at $\\sqrt{s}$ = 200 GeV measured by the STAR Collaboration [70] via the di-electron decay channel by combining the events from minimum-bias trigger and triggered by the electromagnetic calorimeter with various thresholds, taken in 2012.", "The results are found to be consistent with STAR previously published results [72], [71] and PHENIX published results ($|y|<0.35$ ) [73] but with improved precision at $p_T>2 ~\\textrm {GeV}/c$ .", "The decay branching ratio is not corrected for.", "The total production cross-section per unit rapidity for inclusive J/$\\psi $ is extracted to be $B_{ee} \\frac{d\\sigma }{dy}|_{y=0} = 43.2 \\pm 3.0(\\textrm {stat.})", "\\pm 7.5(\\textrm {syst.", "})~\\textrm {nb}.$ The precise $p_{_T}$ spectrum is compared to theoretical calculations.", "The theory-to-data ratios are shown in the lower panels in Fig.", "REF .", "The green band represents the calculation from Color Evaporation Model (CEM) for prompt J/$\\psi $ at $|y|<0.35$  [74].", "The orange and magenta bands represent the calculation in the framework of Next-to-Leading Order (NLO) Non-Relativistic QCD (NRQCD) with different treatments, labeled as NRQCD A  [75] and NRQCD B [76].", "NRQCD A is for prompt J/$\\psi $ and NRQCD B is for direct J/$\\psi $ .", "The blue band shows the calculation from NRQCD A incorporating a Color-Glass Condensate (CGC) effective theory for small-$x$ resummation for prompt J/$\\psi $  [77].", "The CEM and NRQCD calculations describe the data in the applicable $p_{_T}$ ranges within uncertainties.", "The CGC+NRQCD calculations at low $p_{_T}$ are systematically higher than the data but the lower boundary touches the data.", "It is noted that the contribution from $B$ -hadron decay is not included in the theoretical calculations.", "As will be discussed in the next subsection, its contribution increases with increasing $p_T$ , but less than 20% at $p_{_T}$ below 5 GeV/$c$ .", "Figure: (a) Inclusive J/ψ\\psi production cross section times decay branching ratio as a function of p T p_{_T} at mid-rapidity in non-single diffractive pp+pp collisions at s\\sqrt{s} = 500 and 510 GeV measured by the STAR Collaboration .", "(b-d), Ratios of data and different model calculations to a fit to the data.", "See text for details.", "The figure is taken from .Figure REF shows inclusive J/$\\psi $ production cross-section in mid-rapidity as a function of $p_{_T}$ in non-single-diffractive $p$ +$p$ collisions at $\\sqrt{s}$ = 500 and 510 GeV measured by the STAR Collaboration [78] via both dimuon and di-electron decay channels.", "It covers a broad $p_{_T}$ range ($0<p_T<20~\\textrm {GeV}/c$ ).", "The data points depict the results from unpolarized-assumption and the gray band denote the polarization envelope.", "The total production cross-section per unit rapidity for inclusive J/$\\psi $ within $0<p_T<9~\\textrm {GeV}/c$ is $B_{\\mu \\mu } \\frac{d\\sigma }{dy}|_{y=0} = 67 \\pm 6(\\textrm {stat.})", "\\pm 10(\\textrm {syst.})", "^{+100}_{-18}(\\textrm {pol.})", "\\pm 7(\\textrm {lumi.})", "~\\textrm {nb}.$ The data is compared to NRQCD [75], CGC+NRQCD [77] and Improved CEM (ICEM) [79] calculations.", "All the model calculations are for prompt J/$\\psi $ , the feeddown from $B$ -hadrons is not included.", "In order to do a fair comparison between data and theoretical calculations, the feeddown contribution from $B$ -hadrons is estimated using FONLL calculations [19], [27], [28] and added to the theoretical calculations.", "Figure REF (b-d) shows the ratio of the theoretical calculations to a fit to the data using an empirical function.", "At high-$p_T$ , the NRQCD and ICEM describe the data reasonably good.", "At low-$p_T$ , both CGC+NRQCD and ICEM calculations are above the data, but within the huge uncertainty from the polarization envelope." ], [ "Feeddown contribution of J/$\\psi $", "The inclusive J/$\\psi $ production includes prompt J/$\\psi $ and non-prompt J/$\\psi $ .", "The former includes the directly produced J/$\\psi $ and the contribution from the decay of excited charmonium states such as $\\psi (2S)$ , $\\chi _{c0,1,2}$ etc.", "While non-prompt J/$\\psi $ refers to the contribution from the decay of $B$ -hadrons.", "It is important to understand the fraction of different components of inclusive J/$\\psi $ to interpret the measurements of inclusive J/$\\psi $ production mechanism in both $p$ +$p$ and A+A collisions.", "Figure: The ratio of the production cross section times branching ratio of ψ(2S)\\psi (2S) and J/ψ\\psi as a function of p T p_{_T} in p+p(p ¯p+p(\\bar{p}) or pp + A collisions from various experiments.", "The two panels of this figure are respectively taken from  and .$\\psi (2S)$ and J/$\\psi $ is usually reconstructed in the same di-lepton decay channel.", "The systematic uncertainties can be largely cancelled out when calculating the ratio of their yields.", "The measurement is actually very challenging because the study of J/$\\psi $ at RHIC is already statistics limited, the yield of $\\psi (2S)$ is even much lower than J/$\\psi $ and the di-lepton decay branching ratio is also much lower.", "This results in about a factor of 50 reduction of reconstructed $\\psi (2S)$ signal than the J/$\\psi $ , while the combinatorial and correlated background is similar.", "The yield ratios of inclusive $\\psi (2S)$ and inclusive J/$\\psi $ , after correcting for the difference of acceptance and efficiency, measured at RHIC by the STAR and the PHENIX Collaborations are shown in Fig.", "REF and compared to world data.", "The uncertainties are dominated by statistical uncertainties.", "Note that the decay branching ratio is not corrected for, which is about $\\frac{\\textrm {BR}_{\\psi (2S)}}{\\textrm {BR}_{J/\\psi }} \\sim 7.53 \\pm 0.16~(e^+e^-) \\textrm { or } 7.5 \\pm 0.6~(\\mu ^+\\mu ^-) ,$  [81].", "The ratio increases with $p_{_T}$ but shows no much energy dependence from center-of-mass energy of 40 GeV to 7 TeV.", "The ICEM calculations [79] at RHIC can describe the data.", "The branching ratio of $\\psi (2S) \\rightarrow J/\\psi + X$ is $(61.4 \\pm 0.6)\\%$  [81].", "The feeddown contribution of $\\psi (2S)$ to inclusive J/$\\psi $ can be roughly estimated by multiplying the ratio shown in Fig.", "REF by a factor of $\\sim 4.6 \\pm 0.5$ .", "The fraction is $<\\sim 10\\%$ at low $p_{_T}$ and increases to about 15% at $p_{_T}$ up to 10 GeV/c.", "Figure: The feeddown fraction of BB-hadron decays to inclusive J/ψ\\psi at a function of energy and p T p_{_T} in p+p(p ¯)p+p(\\bar{p}) collisions.", "The figure is taken from .The feeddown contribution of $\\chi _c$ to J/$\\psi $ is usually studied via the radiative decay of $\\chi _c$ ($\\chi _c \\rightarrow J/\\psi + \\gamma $ ).", "This measurement is also very challenging since the photon from the decay typically has very low energy, which requires the electromagnetic calorimeter having very good energy resolution and low threshold.", "At RHIC, so far only the PHENIX Collaboration successfully performed this measurement [83].", "The feeddown fraction of $\\chi _c$ decays in the inclusive J/$\\psi $ is measured to be $(32 \\pm 9)\\%$ .", "There is no enough statistics to study $p_{_T}$ dependence of the fraction.", "The LHCb Collaboration measured the feeddown fraction of $\\chi _c$ decays in prompt J/$\\psi $ as a function of $p_{_T}$ with good precision and found that the fraction is only 14% at $p_T=2$ GeV/$c$ and monotonically increases to about 25% at $p_T=10$ GeV/$c$ in $p$ +$p$ collisions at $\\sqrt{s}$ = 7 TeV at forward rapidity [84].", "The feeddown contribution of non-prompt J/$\\psi $ ($J/\\psi \\leftarrow B$ ) can be measured via two methods.", "The STAR Collaboration measured the feeddown fraction of $B$ -hadron decays at mid-rapidity ($|y|<1$ ) at $p_T > 5$ GeV/$c$ in $p$ +$p$ collisions at $\\sqrt{s}$ = 200 GeV via the correlation function of J/$\\psi $ and charged hadrons since the non-prompt J/$\\psi $ is usually associated with more charged hadrons than prompt J/$\\psi $  [71].", "The PHENIX Collaboration measured the fraction at forward rapidity (1.2<|y|<2.2) in $p$ +$p$ collisions at $\\sqrt{s}$ = 510 GeV using the silicon vertex detector to statistically distinguish J/$\\psi $ coming from primary vertex and secondary vertex [82].", "Figure REF shows the fraction as a function of energy and $p_T$ .", "The data from Tevatron and LHC experiments are also shown.", "The fraction shows no much center-of-mass energy dependence but significant $p_{_T}$ dependence.", "The fraction is $8.1\\% \\pm 2.3\\%$ (stat.)", "$\\pm ~1.9\\%$ (syst.)", "at low $p_{_T}$ ($0<p_T<5$ GeV/$c$ ) and $1.2<|y|<2.2$ .", "It increases to about 15% at $p_T=5$ GeV/$c$ and about 25% at $p_T=10$ GeV/$c$ .", "For more details of the feeddown contribution to quarkonium production, please refer to a recent review [85]." ], [ "J/$\\psi $ polarization", "The measurements of J/$\\psi $ polarization is important not only because the acceptance and efficiency used in the production cross section measurements depend on the polarization, but more important is that it helps to distinguish or constraint different theoretical models.", "Currently there is no model in the market can simultaneously describe J/$\\psi $ production cross section and polarization.", "J/$\\psi $ polarization (spin alignment) can be measured through the angular distribution of the di-lepton decay from J/$\\psi $ .", "It can be parameterized as $\\begin{split}W(\\cos \\theta ,\\varphi ) \\propto \\frac{1}{3+\\lambda _{\\theta }}(1+\\lambda _{\\theta }\\cos ^{2}\\theta \\\\+\\lambda _{\\varphi }\\sin ^{2}\\theta \\cos 2\\varphi +\\lambda _{\\theta \\varphi }\\sin 2\\theta \\cos \\varphi ).\\end{split}$ Where $\\lambda _{\\theta }$ , $\\lambda _{\\varphi }$ and $\\lambda _{\\theta \\varphi }$ are the polarization parameters.", "$\\theta $ and $\\varphi $ are the polar and azimuthal angle of a lepton in the J/$\\psi $ rest frame with respect to the chosen quantization axis.", "The coefficients depend on the chosen quantization axis.", "The name of the reference frames and the corresponding quantization axises are: Helicity (HX) frame: The direction along the J/$\\psi $ momentum in the center-of-mass system of the colliding beams; Collins-Soper (CS) frame: The bisector of the angle formed by one beam direction and the opposite direction of the other beam in J/$\\psi $ rest frame; Gottfried-Jackson (GJ) frame: The direction of the beam momentum boosted into J/$\\psi $ rest frame.", "Figure REF shows the measurements of polarization parameters for inclusive J/$\\psi $ in various reference frames at forward rapidity in $p$ +$p$ collisions at $\\sqrt{s}$ = 510 GeV [86].", "In all frames, the polarization parameter $\\lambda _\\theta $ is significantly negative at low $p_{_T}$ and consistent with no polarization at high $p_T$ .", "In contrast, the polarization parameter $\\lambda _\\phi $ is close to zero and becomes slightly negative at high $p_T$ .", "The theoretical calculation on $\\lambda _\\theta $ for prompt J/$\\psi $ in HX frame in the NRQCD factorization approach by H. S. Chung et al.", "[88] at $2<p_T<5$ GeV/$c$ and by H. Shao [89] at $p_{_T}$ above 5 GeV/$c$ are also shown for comparison.", "Both calculations are consistent with the data at high $p_T$ .", "However, there is discrepancy between the data and the theoretical calculation at low $p_T$ .", "The theory expects small but positive $\\lambda _\\theta $ , but the data is significantly negative.", "STAR recently measured J/$\\psi $ polarization parameters $\\lambda _\\theta $ and $\\lambda _\\phi $ at mid-rapidity in $p$ +$p$ collisions at $\\sqrt{s}$ = 200 GeV, as shown in Fig.", "REF  [87].", "In the HX frame, the measured $\\lambda _\\theta $ is slightly negative, but consistent with zero at $p_T<6$ GeV/$c$ .", "While $\\lambda _\\phi $ is slightly positive.", "In CS frame, $\\lambda _\\theta $ is slightly positive, but consistent with zero within uncertainties.", "The bands in Fig.", "REF depict NRQCD calculations using two sets of Long Distance Matrix Elements (LDME).", "The two calculations are labeled as NRQCD1 [90] and NRQCD2 [91].", "They show significant difference especially at low $p_T$ .", "Although with the current precision the data is consistent with both calculations, the measurements with improved precision in future should be able to constraint the LDMEs." ], [ "$\\Upsilon $ production cross section", "The production cross section of $\\Upsilon $ times the di-lepton decay branching ratio is about 3 orders of magnitude lower than that of J/$\\psi $ and is 9 orders of magnitude lower than that of inelastic $p$ +$p$ collision at RHIC.", "About half billion minimum bias $p$ +$p$ collision events is needed to produce one $\\Upsilon \\rightarrow l^+l^-$ decay at mid-rapidity.", "In order to enhance the recorded integral luminosity for $\\Upsilon $ study, a special trigger based on the barrel electromagnetic calorimeter is designed and make the measurement of $\\Upsilon $ at STAR possible.", "However, the statistics and momentum resolution are not good enough to separate the 1S, 2S and 3S states in $p$ +$p$ collisions.", "The 1S, 2S and 3S states are measured together.", "Figure REF shows the measured $\\Upsilon (1S+2S+3S)$ cross section per unit rapidity at mid-rapidity times the $\\Upsilon \\rightarrow e^+e^-$ branching ratio as a function of center-of-mass energy [87].", "Theoretical calculation from the Next-to-Leading Order (NLO) Color Evaporation Model (CEM) [74] describes the cross section from the center-of-mass energy of 20 GeV to 7 TeV and the RHIC data follow the world wide trend.", "PHENIX also measured $\\Upsilon (1S+2S+3S)$ at forward rapidity (1.2<|y|<2.2), thanks to the di-muon trigger [92].", "The rapidity distribution is obtained by combining the measurements from STAR (|y|<1) and PHENIX (1.2<|y|<2.2).", "It is narrower than the NLO CEM prediction [74].", "Figure: Υ(1S+2S+3S)\\Upsilon (1S+2S+3S) production cross section per unity rapidity per binary nucleon-nucleon collision at mid-rapidity times Υ→l + l - \\Upsilon \\rightarrow l^+l^- branching ratio in p+p(p ¯,A)p + p(\\bar{p}, \\textrm {A}) collisions.", "The figure is taken from .The binding energy of $\\Upsilon (1S)$ , $\\Upsilon (2S)$ and $\\Upsilon (3S)$ are very different.", "The modification of the production in nuclear medium is expect to be different for different $\\Upsilon $ states.", "It is of particular interest to separate the 1S, 2S and 3S states.", "Although it is not achievable at current stage in $p$ +$p$ collisions at 200 GeV, but it is possible to separate 1S from 2S+3S or even 1S, 2S and 3S in $p(d)$ + A and A + A collisions due to higher statistics and better momentum resolution thanks to better primary vertex resolution.", "However, the production cross section of 1S and 2S+3S (or 2S and 3S) state in $p$ +$p$ collisions is needed as a reference to study the different nuclear matter effects of different $\\Upsilon $ states.", "The authors of Ref.", "[93] performed a systematic study of $\\Upsilon $ production in $p + p(\\bar{p}, \\textrm {A})$ collisions from world wide experiments and developed a way to predict $\\Upsilon (2S)/(1S)$ and $\\Upsilon (3S)/(1S)$ in $p$ +$p$ collisions at given center-of-mass energy.", "The production cross section of 1S, 2S and 3S in $p$ +$p$ collisions can be obtained by combining the derived ratios and the measurement of $\\Upsilon (1S+2S+3S)$ production cross section.", "The STAR Collaboration also established a method to derive the $p_{_T}$ spectrum of $\\Upsilon (1S)$ , $\\Upsilon (2S)$ and $\\Upsilon (3S)$ in $p$ +$p$ at 200 GeV in order to study the $p_{_T}$ dependence of the nuclear modification factor of $\\Upsilon $ states.", "Recently, the STAR managed to measure the $p_{_T}$ spectrum of $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ in $p$ +$p$ collisions at 500 GeV [94].", "The newly installed iTPC in STAR will improve the mass resolution of $\\Upsilon $  [95].", "The possibility of further improving the mass resolution by changing the working gas of TPC to suppress transverse diffusion is under investigation.", "Although the production mechanism in $p$ +$p$ collisions is not fully understood, quarkonium in heavy-ion collisions is one of a few most important probes of QGP.", "The suppression of J/$\\psi $ production yield in relativistic heavy-ion collisions with respect to the yield in $p$ +$p$ collisions scaled by the number of binary nucleon-nucleon collisions has been proposed as a “smoking gun” signature of QGP formation by T. Matsui and H. Satz in 1986 [17].", "The suppression is the result of J/$\\psi $ dissociation due to the screening of the potential between charm quark and anti-charm quark in the deconfined hot, dense medium.", "J/$\\psi $ production in nucleus-nucleus collisions is extensively studied at CERN SPS since 1986.", "The pioneer experiment NA38 found that J/$\\psi $ production in S+U collisions is suppressed relative to $p$ +U collisions and as a function of the transverse energy $E_T$ , which is related to the collision centrality.", "However, it is found later that the suppression pattern was compatible with the extrapolation of the trend observed in $p$ +A collisions and can be accounted for normal nuclear absorption.", "The NA50 experiment collected high statistics data with $p$ beam with energy of 450 or 400 GeV on target ranging from Be, Al, Cu, Ag, W and Pb.", "The normal nuclear absorption of J/$\\psi $ production is obtained by systematically studying the $p$ +A data [96].", "The NA50 experiment also collected data with Pb beam with energy per nucleon of 158 GeV on Pb-target ($\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 GeV) in 1995, 1996, 1998 and 2000.", "The analysis of these data showed that J/$\\psi $ production, relative to Drell-Yan, is anomalously suppressed with respect to the normal nuclear absorption pattern, which is extracted by extrapolating the J/$\\psi $ suppression in $p$ +A collisions as aforementioned [18].", "In order to take $p$ +A data under the same condition as the A+A data, the NA60 experiment run with proton beam on nuclear targets (Be, Al, Cu, In, W, Pb and U) and In beam on In target with energy per nucleon of 158 GeV [97].", "The nuclear absorption is found to have beam energy dependence.", "The anomalous J/$\\psi $ suppression is calculated with the new nuclear absorption cross section and also take the difference of the parton distribution function in nucleus (nPDF) and nucleon (shadowing effect) into account.", "The observed suppression of J/$\\psi $ is compatible with the extrapolation of cold-nuclear-matter (CNM) effects upto $N_{\\textrm {part}} \\sim 200$ .", "When $N_{\\textrm {part}} > 200$ , there is an anomalous suppression of upto $\\sim 20\\%-30\\%$ in the most central Pb+Pb collisions.", "Figure: R AA R_{\\mathrm {AA}} as a function of N part N_{\\textrm {part}} for inclusive J/ψ\\psi at SPS and RHIC.", "The panels are taken from , .The PHENIX Collaboration at RHIC measures J/$\\psi $ production in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV, more than a factor of 10 higher than that of SPS, for rapidities |y|<0.35 and 1.2<|y|<2.2, through the di-electron and di-muon decay channel, respectively.", "The left panel of Fig.", "REF shows the nuclear modification factor $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ at mid-rapidity in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV measured by the PHENIX Collaboration [99] and in Pb+Pb, In+In and S+U collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 GeV measured by the NA50, NA60 and NA38 Collaborations [100], [18], [97], respectively.", "Even the center-of-mass energies differ by one order of magnitude, the amount of suppression is very similar at RHIC and SPS energies.", "This was puzzling as stronger suppression iss expected at RHIC due to high energy density and/or initial temperature.", "Another puzzle was that the suppression is observed to be much stronger at forward rapidity than at mid-rapidity as shown in the right panel of Fig.", "REF .", "It was expected that the suppression at mid-rapidity where the energy density is higher to be stronger.", "An additional J/$\\psi $ production mechanism, (re)combination of charm quark and anti-charm quark, is introduced [101], [102], [103], [104], [105].", "The idea of the (re)combination mechanism is that when the charm and anti-charm quark not initial produced as a bound state get close enough in space and momentum after transportation in QGP, they may form a bound state such as J/$\\psi $ .", "The production yield of J/$\\psi $ is only about 1% of the total number of charm and anti-charm quark pair in $p$ +$p$ collisions, the (re)combination mechanism may have sizable effect even though the (re)combination probability is small.", "The probability is proportional to the square of number of charm and anti-charm quark pair produced in one event.", "The probability is negligible in $p$ +$p$ collisions at RHIC energy.", "But since the number of charm and anti-charm quark pair is roughly scaled by the number of binary nucleon-nucleon collisions which can reach as large as 1000 in central Au+Au collisions, the probability is much larger in heavy-ion collisions than in $p$ +$p$ collisions.", "Unlike the color-screening (or QGP melting) mechanism, the (re)combination mechanism could result in enhancement of $R_{\\mathrm {AA}}$ in heavy-ion collisions.", "Since the charm and anti-quark charm pair production cross section increases dramatically with center-of-mass energy, the (re)combination mechanism play more important role in heavy-ion collisions at higher center-of-mass energy.", "Based on a theoretical calculation [106] on charm and anti-charm quark pair production cross section in $p$ +$p$ collisions, we estimated that the number of charm and anti-charm quark pair in a 0-10% Pb+Pb collisions at SPS energy ($\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 GeV) is $0.13 \\pm 0.03$ .", "The number increases to $18 \\pm 4$ in 0-10% Au+Au collisions at RHIC energy ($\\sqrt{s_{_\\mathrm {NN}}}$ = 200) and over 100 in 0-10% Pb+Pb collisions at LHC energies.", "The contribution of (re)combination is negligible at SPS but could have sizable effect at RHIC and LHC.", "The theoretical models (such as transport models) including both QGP melting and (re)combination can explain the $R_{\\mathrm {AA}}$ observed at RHIC at both mid-rapidity and forward rapidity, as well as at SPS.", "The J/$\\psi $ production in heavy-ion collisions at RHIC and LHC is the interplay of QGP melting, CNM effects as well as (re)combination effects.", "To study the properties of the QGP via J/$\\psi $ , we need good understanding of each of the three effects.", "The CNM effects are usually be studied experimentally in $p$ +A collisions or the collisions of light ions where the QGP melting and/or (re)combination effect is unlikely existed at least at RHIC energy.", "But the separation of QGP melting and (re)combination effects is very difficult.", "Fortunately these two have very different collision energy, collision system and $p_{_T}$ dependence.", "A systematically study of J/$\\psi $ production in heavy-ion collisions is helpful to understand the J/$\\psi $ production mechanism in heavy-ion collisions and study the properties of QGP using J/$\\psi $ ." ], [ "Collision energy dependence", "RHIC launched the Beam Energy Scan (BES) program in 2010 to explore the QCD phase diagram.", "Both STAR and PHENIX collected data in Au+Au collisions at 62.4 and 39 GeV in 2010 and at 27 and 19 GeV in 2011 in the phase-I of the BES program (BES-I).", "These center-of-mass beam energies fill the large gap between SPS energy and RHIC top energy.", "These BES data can be used to study the evolution of the CNM effects, QGP melting and (re)combination from SPS to RHIC.", "The production cross section of J/$\\psi $ decreases dramatically with decreasing center-of-mass energy and the luminosity of RHIC also decreases quickly with decreasing beam energy.", "We were only able to measure J/$\\psi $ production in the collisions at 39 and 62.4 GeV.", "To obtain $R_{\\mathrm {AA}}$ at these two energies, J/$\\psi $ cross section in $p$ +$p$ collisions is needed.", "There are several measurements from $p$ + A fixed-target experiments and $p$ +$p$ collider experiment at the Intersection Storage Ring (ISR) near these two energies performed in last century.", "But unfortunately some data at mid-rapidity are found to be not consistent with each other.", "At mid-rapidity, STAR uses J/$\\psi $ production cross section derived from world wide experimental data [108] to calculation J/$\\psi $ $R_{\\mathrm {AA}}$ in Au+Au collisions at 39 and 62.4 GeV [107].", "At forward-rapidity, PHENIX used the reference data derived based on the data at Fermilab fixed-target experiment, ISR collider experiment and CEM model calculations [109].", "The left panel of Fig.", "REF shows the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ as a function on $N_{\\textrm {part}}$ in Au+Au collisions at 39 and 62.4 GeV at both mid- and forward rapidity and compared to that at 200 GeV.", "It is consistent with no suppression in peripheral collisions but exhibit strong suppression towards central collisions.", "At mid-rapidity, no significant energy dependence is observed within uncertainties from SPS ($\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 GeV) to RHIC top energy ($\\sqrt{s_{_\\mathrm {NN}}}$ =200 GeV).", "However, at forward rapidity, the suppression seems less in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 39 and 62.4 GeV than at 200 GeV.", "The rapidity dependence of the suppression can be obtained by comparing the STAR data at mid-rapidity (|y|<1) and the PHENIX data at forward rapidity (1.2<|y|<2.2).", "The suppression is much stronger at forward rapidity than at mid-rapidity in Au+Au collisions at 200 GeV.", "However, at 39 and 62.4 GeV, the suppression of J/$\\psi $ shows no significant rapidity dependence within uncertainties.", "This could be due to the energy and rapidity dependence of the (re)combination contribution, which is larger at higher collision energy and mid-rapidity.", "In order to understand the collision energy dependence, the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ in central heavy-ion collisions is plotted as a function of center-of-mass energy.", "The $R_{\\mathrm {AA}}$ is flat at $\\sim 0.4$ from $\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 to 200 GeV then dramatically increase to $>0.6$ at LHC.", "The curves shown in Fig.", "REF are the theoretical calculations from a transport model [110] which implemented CNM effects, QGP melting and (re)combination.", "The dashed curve represent the $R_{\\mathrm {AA}}$ of primordially produced J/$\\psi $ , whose production yield suffers from the QGP melting and CNM effects.", "The $R_{\\mathrm {AA}}$ is fairly flat from $\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 to 62 GeV then decreases with center-of-mass energy.", "The trend is the result of counter-balance of the CNM effects and QGP melting.", "The nuclear absorption cross section decreases with increasing center-of-mass energy, resulting in increasing $R_{\\mathrm {AA}}$ with increasing center-of-mass energy.", "While the QGP melting results in decreasing trend due to the increasing energy density.", "The QGP melting plays significant role at RHIC top energy.", "The $R_{\\mathrm {AA}}$ with only CNM effects is estimated to about 0.6 in central Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV [110].", "The QGP melting brings down the $R_{\\mathrm {AA}}$ from 0.6 to 0.2.", "The dotted curve in Fig.", "REF shows the $R_{\\mathrm {AA}}$ for the J/$\\psi $ produced from (re)combination.", "It is negligible from SPS energy to RHIC BES at $\\sqrt{s_{_\\mathrm {NN}}}$ $<\\sim 50$ GeV and starts to play a role at higher center-of-mass energy.", "At RHIC top energy, the contribution of J/$\\psi $ from (re)combination is comparable to the survived primordial J/$\\psi $ and becomes dominant at LHC.", "The solid curve in Fig REF shows the sum of the two components.", "It can describe the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ in central heavy-ion collisions from SPS (CNM effects domain) to LHC ((re)combination domain) within uncertainties.", "It is particularly interesting to study the J/$\\psi $ suppression in heavy-ion collisions at center-of-mass energy around 50 GeV where the (re)combination contribution remains negligible as SPS, but the energy density is higher and the expected CNM effects such as nuclear absorption is smaller than at SPS.", "The STAR experiment has collected a large data sample of Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 54 GeV.", "The statistics is one order of magnitude higher than that of the 62 GeV data sample.", "This will allow more precise and differential measurement of the J/$\\psi $ suppression at $\\sqrt{s_{_\\mathrm {NN}}}$ around 50 GeV.", "A fixed-target experiment using the LHC beams (AFTER@LHC [111] or current LHC experiments in fixed-target mode) will be able to collect high statistics heavy-ion collision data at $\\sqrt{s_{_\\mathrm {NN}}}$ = 72 GeV with Pb beam of 2.76 TeV per nucleon.", "The $p$ +A data can also be taken at the same energy with Pb beam on proton target to study the CNM effects." ], [ "Collision system dependence", "The measurements in Pb+Pb collisions at SPS shows anomalous J/$\\psi $ suppression from semi-peripheral to central Pb+Pb collisions ($N_{\\textrm {part}}>\\sim 100$ ) at $\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 GeV.", "At RHIC energy, the energy density is expected to reach the energy density required for QGP formation based on Lattice QCD calculations at $N_{\\textrm {part}}$ below 100.", "However, the Au+Au data in this QGP transition threshold region is limited.", "In order to provide crucial information in such an important region, PHENIX has measured J/$\\psi $ suppression in Cu+Cu collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV.", "Figure REF .a and REF .b shows the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ in Cu+Cu collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV for mid-rapidity (|y|<0.35) and forward/backward rapidity (1.2<|y|<2.2), respectively.", "The results in Au+Au collisions at the same energy is also shown for comparison.", "The $R_{\\mathrm {AA}}$ in Cu+Cu and Au+Au are consistent with other within uncertainties at comparable $N_{\\textrm {part}}$ .", "The Cu+Cu data covers the $N_{\\textrm {part}}$ range upto 100 with much finer bins than in Au+Au collisions.", "The observed suppression has large contribution from CNM effects.", "In order to extract the possible QGP melting in Cu+Cu collisions, the CNM effects are estimated by projecting the suppression measurement in $d$ +Au collision at the same energy with nPDF and nuclear absorption.", "The nPDF is taken from the shadowing model EKS98 [114] or nDSg [115].", "The nuclear absorption cross section are optimized separately for mid-rapidity and forward rapidity from the fit to $d$ +Au data.", "The estimated CNM effects with EKS98 (method 1) is shown in Fig.", "REF .a and REF .b as solid lines.", "The dashed lines depict the calculation by varying the nuclear absorption cross section by $1\\sigma $ .", "The predicted CNM $R_{\\mathrm {AA}}$ show almost no difference in Cu+Cu and Au+Au collisions at the same $N_{\\textrm {part}}$ .", "Figure REF .c shows the measured $R_{\\mathrm {AA}}$ in Cu+Cu collisions divided by the predicted CNM $R_{\\mathrm {AA}}$ with EKS98 parameterization at both mid- and forward/backward rapidity.", "At $N_{\\textrm {part}}$ $<\\sim 50$ , the measured $R_{\\mathrm {AA}}$ in Cu+Cu collisions is seen to be consistent with the CNM projection within about 15% uncertainties.", "At $N_{\\textrm {part}}$ above 50, the centroid of the measured ratios at both mid-rapdity and forward rapidity are smaller than unity.", "But no strong conclusion can be drawn with the larger uncertainties.", "More precise measurement in $p(d)$ +A collisions and better understanding of how to project the CNM effects in $p(d)$ +A collisions to A+A collisions is needed.", "Nevertheless, the CNM effects dominate J/$\\psi $ production in Cu+Cu collisions and peripheral and semi-peripheral Au+Au collisions.", "RHIC collided Cu+Au collisions at 200 GeV in 2012.", "The rapidity dependence of J/$\\psi $ suppression in the asymmetric collision system may provide key insights on the balance of CNM effects and hot nuclear matter effects.", "The parton distribution functions are more strongly modified in heavier Au nucleus than in light nucleus.", "At forward rapidity (Cu-going direction), J/$\\psi $ probes gluons at lower Bjorken $x$ in the Au nucleus while higher $x$ in the Cu nucleus.", "This is reversed at backward rapidity.", "The shadowing effects are expected to be stronger at forward rapidity than at backward rapidity.", "On the other hand, the J/$\\psi $ produced at forward rapidity has a large rapidity relative to the Au nucleus thus have shorter proper time.", "This will result in less nuclear absorption of J/$\\psi $ or energy loss at forward rapidity.", "Furthermore, the energy density and hadron multiplicity are also asymmetric in Cu+Au collisions.", "It is higher at backward rapidity (Au-going direction).", "The asymmetric energy density and hadron multiplicity may result in different CNM and hot matter effects.", "The breakup of J/$\\psi $ by comovers depends on the density of comovers and is expected to be stronger at backward rapidity than at forward rapidity.", "The asymmetric hot nuclear matter effects are not so straightforward.", "The QGP melting effect is stronger at the backward rapidity thus results in smaller $R_{\\mathrm {AA}}$ at backward rapidity.", "However, the (re)combination effect is also stronger at the backward rapidity thus results in larger $R_{\\mathrm {AA}}$ at backward rapidity.", "The upper panel of Fig.", "REF shows the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ at forward (1.2<y<2.2) and backward (-2.2<y<-1.2) rapidity in Cu+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV.", "The forward and backward rapidity combined Au+Au results at the same center-of-mass energy is also shown for comparison.", "The $R_{\\mathrm {AA}}$ in Cu+Au collisions at backward rapidity (Au-going direction) is similar to that for Au+Au collisions at similar $N_{\\textrm {part}}$ .", "While $R_{\\mathrm {AA}}$ in Cu+Au collisions at forward rapidity (Cu-going direction) is systematically lower.", "The difference between forward rapidity and backward rapidity is more clear in the bottom panel of Fig.", "REF in which the ratio of $R_{\\mathrm {AA}}$ at forward rapidity to that at backward rapidity is shown.", "The ratios are 20%-30% lower than unity and no significant $N_{\\textrm {part}}$ dependence.", "The grey band depicts the prediction from a simple Glauber model incorporating gluon distribution function modification from the EPS09 [116] parameterization, and a rapidity-independent effective $c\\bar{c}$ breakup cross section of 4 mb to account for the nuclear absorption effect.", "The expected shadowing difference has the right sign as the data and the level of the difference is comparable with the data within uncertainties but systematically smaller than the data especially in peripheral collisions.", "The QGP melting are expected to result in the forward-to-backward ratio above unity.", "However, the expectation from (re)combination effect has the same sign as the data and decrease from peripheral collisions.", "It seems that both QGP melting and (re)combination can not explain the possible difference between the data and the estimated $R_{\\mathrm {AA}}$ from gluon shadowing effect.", "Figure: Inclusive J/ψ\\psi R AA R_{\\mathrm {AA}} as a function of N part N_{\\textrm {part}} at forward rapidity (1.2<|y|<2.2) in U+U collisions at s NN \\sqrt{s_{_\\mathrm {NN}}} = 193 GeV and compared to Au+Au collisions at s NN \\sqrt{s_{_\\mathrm {NN}}} = 200 GeV.", "The figure is taken from .The energy density or particle multiplicity dependence of J/$\\psi $ suppression can also be studied in U+U collisions.", "The energy density in U+U collisions is about 20% higher than that in Au+Au collisions with similar number of participants.", "Figure REF shows the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ in U+U collisions at 193 GeV at forward rapidity and compared to that in Au+Au collisions [117].", "The U+U data were taken in 2012.", "Unlike Au nucleus, U nucleus is deformed and the shape is not well understood.", "The number of participants and number of binary nucleon-nucleon collisions in U+U collisions depends on the shape of the U nucleus.", "The U+U results shown in the upper and lower panel of Fig.", "REF are from two parameterization of deformed Woods-Saxon distribution for U (set 1 [118] and set 2 [119]).", "The parameterization of set 2 has smaller surface diffuseness, resulting in a notably more compact nucleus (and larger number of binary nucleon-nucleon collisions).", "From both parameterizations, the observed $R_{\\mathrm {AA}}$ in U+U collisions is similar as in Au+Au collisions with the same number of participants in peripheral and semi-peripheral collisions, but exhibit less suppression than in Au+Au collisions in central collisions.", "The CNM effects due to shadowing are expected to be similar in Au+Au and U+U collisions.", "The difference in Au+Au and U+U are likely due to hot nuclear matter effects.", "The increase of $R_{\\mathrm {AA}}$ from Au+Au collisions to U+U collisions is consistent with a picture in which the enhancement due to (re)combination becomes more important than the suppression due to QGP melting." ], [ "Transverse momentum dependence", "The QGP melting, (re)combination and CNM effects not only have collision energy and collision system dependence, but also depend on the transverse momentum of J/$\\psi $ .", "The $R_{\\mathrm {AA}}$ from CNM effects usually exhibit an increasing trend as a function of J/$\\psi $ $p_T$ .", "The E866 [121] and HERA-B [122] experiments found J/$\\psi $ suppression factor $\\alpha $ , which is obtained by assuming the cross section dependence on nuclear mass, A, to be of the form $\\sigma _A = \\sigma _N \\times A^\\alpha $ , in fixed-target $p$ +A collisions has a clear increasing trend as a function of $p_T$ and cross unity at $p_{_T}$ around $2-3$ GeV/$c$ .", "The increasing trend is usually attributed to multiple scattering of the incident parton before the hard scattering and of the nascent $c\\bar{c}$ in the final state.", "This effect is sometimes also referred to the Cronin effect.", "At RHIC, the PHENIX Collaboration published J/$\\psi $ suppression as a function of $p_{_T}$ in $d$ +Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV at both mid- and forward/backward rapidity [123] and recently submitted the results in $p$ +Al, $p$ +Au and $^3$ He+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV for publication [124].", "The STAR Collaboration also measured the $p_{_T}$ dependence of J/$\\psi $ suppression in $p$ +Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV [125].", "In all the collision systems with Au, the suppression of inclusive J/$\\psi $ also shows an increasing trend.", "The suppression is on the level of about 30% at low-$p_{_T}$ but consistent with no suppression at $p_{_T}$ above $3-4$ GeV/$c$ .", "A transport model [110] predicts $R_{\\mathrm {AA}}$ of about 0.4 at $p_{_T}$ around zero and lies on unity at $p_{_T}$ from 4.5 to 10 GeV/$c$ at mid-rapidity in 0-20% central Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV with CNM effects including nuclear absorption and feeddown contribution from $B$ .", "The (re)combination effect is expected to decrease with $p_T$ .", "This is mainly because the yield of J/$\\psi $ from (re)combination is approximately proportional to the square of the number of charm quarks, which falls fast with $p_T$ .", "The transport models  [110], [126] show that the contribution of (re)combination is comparable with the primordial J/$\\psi $ at $p_{_T}$ below 1 GeV/$c$ and is negligible at $p_{_T}$ above 5 GeV/$c$ .", "The $p_{_T}$ dependence of QGP melting is not well understand.", "The formation time effect predicts increasing trend because J/$\\psi $ with higher $p_{_T}$ is more likely to form outside of the medium and less affected by the hot, dense medium.", "However, the dissociation temperature of J/$\\psi $ may depend on the relative velocity of J/$\\psi $ and the medium and its $p_{_T}$ dependence is model dependent.", "J/$\\psi $ with higher $p_{_T}$ may have higher or lower dissociation temperature in different models [127], [128].", "A detailed differential measurement of J/$\\psi $ suppression over a broad kinematic range can shed new lights on J/$\\psi $ production mechanism in heavy-ion collisions and the properties of QGP.", "The STAR Collaboration attempted to extend the J/$\\psi $ measurement in heavy-ion collisions to $p_{_T}$ beyond 5 GeV/$c$ since 2006 [129], [71].", "The J/$\\psi $ production at high $p_T$ , where the CNM effects and (re)combination is negligible, is found to be consistent with no suppression in Cu+Cu and peripheral Au+Au collisions but significantly suppressed in (semi-)central Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV.", "But limited by statistics, no firm conclusion was drawn.", "In 2014 and 2016, the STAR Collaboration collected large samples of Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV utilizing the Muon Telescope Detector (MTD).", "The MTD detector is designed to trigger on and identify muons and is completed in early 2014.", "Compare to the previous measurement at mid-rapidity through the di-electron channel [107], [71], [130], the new data allows to extend the kinematic reach towards high $p_{_T}$ with better precision.", "Figure REF shows the inclusive J/$\\psi $ $R_{\\mathrm {AA}}$ as a function of $p_{_T}$ in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV from the data taken in 2014.", "The filled stars represent the new results from STAR through di-muon channel and the open and filled circles depict the previous results through di-electron channel.", "The results at low $p_{_T}$ from PHENIX are also shown as hollow crosses [99].", "The new results are consistent with previous results in the overlapping kinematic range, but have better precision and covers a wider kinematic range.", "Within the uncertainties, J/$\\psi $ suppression shows little $p_{_T}$ dependence from $p_T \\sim 0$ upto 14 GeV/$c$ .", "For comparison, J/$\\psi $ suppression at mid-rapidity in Pb+Pb collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 2.76 TeV measured by the ALICE [131] and CMS [132] Collaborations are shown in panel (a).", "The $p_{_T}$ dependence at LHC and RHIC are significantly different.", "J/$\\psi $ $R_{\\mathrm {AA}}$ at RHIC is lower than at LHC at low $p_{_T}$ , but systematical higher than at LHC at high $p_T$ .", "The difference of the $R_{\\mathrm {AA}}$ is due to different (re)combination contribution in J/$\\psi $ production and initial temperature and lifetime of the QGP in heavy-ion collisions at different collision energy.", "The shaded bands and dashed lines represent two transport model calculations for Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV from Tsinghua (TM1) [126] and TAMU (TM2) [110] groups.", "The CNM effects, QGP meting and (re)combination are taken into account in both models, although the detailed treatments are different.", "The TM1 model describes the data reasonable well at low $p_T$ , but show a stepper increasing trend towards high $p_{_T}$ than data.", "The TM2 model has better $p_{_T}$ dependence as the data, but the absolute values are systematically lower than the data from intermediate to high $p_T$ .", "The two solid bands covering $3.5<p_T<15$ GeV/$c$ in panel (b) show theoretical calculations using vacuum J/$\\psi $ wave function without any color-screening, and including both radiative energy loss of color-octet $c\\bar{c}$ pairs and collisional dissociation of J/$\\psi $  [133].", "The two bands are corresponding to two different values of the J/$\\psi $ formation time.", "Both of them are consistent with data.", "All these calculations include feeddown contribution constrained by the measurements in $p$ +$p$ collisions as well as CNM effects constrained by the measurements in $p(d)$ +A collisions.", "Figure REF shows the centrality dependence of J/$\\psi $ suppression in heavy-ion collisions at both RHIC and LHC energies for low-$p_{_T}$ (upper) and high-$p_{_T}$ (lower) J/$\\psi $ .", "For low-$p_{_T}$ J/$\\psi $ , the suppression decreases towards central collisions at RHIC but more flat at LHC.", "The low-$p_{_T}$ J/$\\psi $ suppression is the interplay of all CNM effects, QGP melting and (re)combination.", "The much less suppression in central heavy-ion collisions at LHC than at RHIC is likely due to the different fraction of (re)combination at RHIC and LHC, expected from the different charm quark production cross section at these energies.", "The high-$p_{_T}$ J/$\\psi $ is suppressed by a factor of 3.1 with a significance of 8.1$\\sigma $ in 0-10% Au+Au collisions.", "The CNM effects and (re)combination contribution are expected to be minimal at this $p_{_T}$ range ($p_T>5$ GeV/$c$ ).", "The significant suppression of high-$p_{_T}$ J/$\\psi $ in central Au+Au collisions provides strong evidence for the color-screening effect in QGP.", "Unlike the low-$p_{_T}$ J/$\\psi $ , the high-$p_{_T}$ J/$\\psi $ is more suppressed at LHC than at RHIC.", "This could be because the temperature of the medium created at the LHC is higher than that at RHIC." ], [ "Collective flow", "The measurements on collective flow of J/$\\psi $ may also shed light on the relative contribution of primordial J/$\\psi $ and J/$\\psi $ from (re)combination.", "The primordial J/$\\psi $ is dominantly produced before the QGP formation thus do not have initial collective flow.", "In non-central collisions, the primordial J/$\\psi $ may have different suppression along different azimuthal angle with respect to the reaction plane due to the different path lengths in azimuthal.", "But the azimuthal anisotropy should be limited.", "On the other hand, the J/$\\psi $ produced from the (re)combination of charm quark and its antiquark should inherit the flow of charm quarks and may have considerable flow.", "As discussed in Sec.", "REF , the $D^0$ $v_2$ shown in Fig.", "REF is found to follow the mass ordering at low $p_T$ , expected from hydrodynamics, and NCQ-scaling as the light and strange hadrons in the intermediated $p_T$ , expected from quark coalescence.", "It is concluded that the charm quarks have gained significant flow in QGP.", "The radial flow of $D^0$ meson in heavy-ion collisions is also studed with the precise measurements of $p_{_T}$ or $m_T$ spectra in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV [32], as discussed in Sec.", "REF .", "The $T_{\\text{eff}}$ , slope parameter of an exponential fit to the $m_T$ spectra, as a function of particle mass for light flavor hadrons, strange hadrons and $D^0$ mesons clearly show two different systematic trends.", "The light flavor hadrons $\\pi $ , $K$ , and $p$ data follow one linear dependence while strange and charm hadrons $\\phi $ , $\\Lambda $ , $\\Omega $ and $D^0$ data follow another linear dependence.", "The $T_{\\textrm {fo}}$ and $\\langle \\beta _t \\rangle $ of $D^0$ , extracted by fitting the $p_{_T}$ spectra with blast-wave (BW) [134] or Tsallis blast-wave (TBW) [41] model, is found to group with the multi-strangeness particles $\\phi $ , $\\Xi $ and $\\Omega $ , showing much smaller $\\langle \\beta _t \\rangle $ and larger $T_{\\textrm {fo}}$ compared to light hadrons $\\pi $ , $K$ and $p$ .", "This suggests that the $D^0$ flows with the medium and the collectivity obtained is mostly through the partonic rescattering in the QGP phase.", "If J/$\\psi $ produced from the (re)combination of charm quark and and its antiquark is the dominant process, it should have significant $v_2$ and radial flow.", "Figure: J/ψ\\psi v 2 v_2 as a function of p T p_{_T} in 0-80% Au+Au collisions at s NN \\sqrt{s_{_\\mathrm {NN}}} = 200 GeV, compared to the results for φ\\phi and charged hadrons and theoretical calculations.", "The figure is taken from .The STAR Collaboration measured J/$\\psi $ $v_2$ in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV from the combination of various triggers operating in 2010.", "The inclusive J/$\\psi $ s are reconstructed through di-electron channel.", "The upper panel of Fig.", "REF shows $v_2$ as a function of $p_{_T}$ for inclusive J/$\\psi $ , $\\phi $ and charged hadrons which are dominated by $\\pi $ s. The grey box on the J/$\\psi $ data shows the estimated maximum possible range of $v_2$ if the influence of non-flow is corrected.", "It is estimated using the measurement of J/$\\psi $ -hadron correlation in $p$ +$p$ collisions at the same energy.", "Unlike the $D^0$ , the J/$\\psi $ $v_2$ is significantly lower than that for $\\phi $ and charged hadrons at $p_{_T}$ above 2 GeV/$c$ .", "The lower panel of Fig.", "REF shows the comparison of the J/$\\psi $ $v_2$ data and theoretical calculations.", "The solid line shows the calculation for J/$\\psi $ produced from the initial hard scattering.", "It is non-zero but limited in the $p_{_T}$ range of 0-5 GeV/$c$ .", "Although significant suppression of J/$\\psi $ is observed in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV, the azimuthally different suppression along the different path lengths in azimuth is limited, beyond the sensitivity of the current measurement.", "The dotted line shows the prediction for J/$\\psi $ produced by coalescence of fully thermalized charm quarks at the freeze-out ((re)combination).", "The maximum is similar as that of light and strange hadrons, but shifted to higher $p_{_T}$ due to significant larger mass of charm quark and J/$\\psi $ particle.", "The prediction for J/$\\psi $ from coalescence is systematic higher than the data at $p_{_T}$ > 2 GeV/$c$ .", "The $\\chi ^2$ /ndf is as large as 16.2/3, corresponding to a small $p$ -value of $1.0\\times 10^{-3}$ .", "The transport models including contribution from both primordial production and (re)combination predict a much smaller $v_2$ and are consistent with the data.", "The $p$ -values is $0.58$ and $0.38$ for the TAMU [136] and Tsinghua [137] transport model, respectively.", "The small $v_2$ in the transport model is due to the fact the $v_2$ of charm quark is small at low $p_T$ and the (re)combination contribution is small at high $p_T$ .", "Although both transport models describe the data reasonably well, there is sizable difference among the models.", "The TAMU model is more close to the $v_2$ for initially produced J/$\\psi $ .", "The measurement with improved precision will be helpful to distinguish or constrain the two transport models.", "The hydrodynamics model tuned to describe $v_2$ of light hadrons predicts a J/$\\psi $ $v_2$ that strongly increase with $p_{_T}$ below 4 GeV/$c$ and fails to describe the data.", "The $\\chi ^2$ /ndf is 7.0/3, corresponding to $p$ -value of 0.072 [138].", "Based on the data and model comparisons, it is concluded that the J/$\\psi $ $v_2$ data disfavor the scenario that J/$\\psi $ with $p_{_T}$ $>2$ GeV/$c$ are produced dominantly by coalescence from charm quarks and anti-charm quarks which are thermalized and flow with QGP.", "The J/$\\psi $ radial flow at SPS and RHIC is systematically study in [139] with the Tsallis blast-wave (TBW) model.", "The $p_{_T}$ spectra of light hadrons and strange hadrons in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV at RHIC as well as in Pb+Pb collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 17.3 at SPS are fit with the TBW model to extract the radial flow, kinetic freeze-out temperature and the non-extensive parameter.", "The $p_{_T}$ spectrum for J/$\\psi $ predicted from the TBW with the same set of parameters as light and strange hardons is much softer than the measurement.", "It overestimates the yield at low $p_{_T}$ and underestimates the yield at high $p_T$ .", "A fit to the J/$\\psi $ $p_{_T}$ spectrum alone shows that the radial flow of J/$\\psi $ at both RHIC and SPS is consistent with zero.", "This provides another evidence that J/$\\psi $ production at RHIC and SPS is not dominantly from (re)combination of thermalized charm quarks." ], [ "J/$\\psi $ photoproduction with nuclear overlap", "J/$\\psi $ can also be generated by the intense electromagnetic fields accompanied with the relativistic heavy ions [140].", "The intense electromagnetic field can be viewed as a spectrum of equivalent photons by the equivalent photon approximation [141].", "The quasi-real photon emitted by one nucleus could fluctuate into $c\\bar{c}$ pair, scatters off the other nucleus, and emerge as a real J/$\\psi $ .", "The coherent nature of these interactions gives the processes distinctive characteristics: the final products consist of a J/$\\psi $ with very low transverse momentum, two intact nuclei, and nothing else.", "Conventionally, these reactions are only visible and studied in Ultra-Peripheral Collisions (UPC), in which the impact parameter ($b$ ) is larger than twice the nuclear radius ($R_{A}$ ) to avoid any hadronic interactions.", "Several results of J/$\\psi $ production in UPC are already available at RHIC [142] and LHC [143], [144], [145], which provide valuable insights into the gluon distribution in the colliding nuclei [146].", "Can the coherent photon products also exist in Hadronic Heavy-Ion Collisions (HHIC, $b < 2R_{A}$ ), where the violent strong interactions occur in the overlap region?", "The story starts with the measurements from ALICE: significant excesses of J/$\\psi $ yield at very low $p_{T} (< 0.3$ GeV/c) have been observed in peripheral Pb+Pb collisions at $\\sqrt{s_{\\rm {NN}}} =$ 2.76 TeV [147], which can not be explained by the hadronic J$/\\psi $ production with the known cold and hot medium effects.", "STAR made the measurements of di-electron [148] in Au+Au collisions at $\\sqrt{s_{\\rm {NN}}} =$ 200 GeV, and also observed significant enhancement at very low $p_{T}$ in peripheral collisions.", "The anomaly excesses observed possess characteristics of coherent photon interaction and can be quantitatively described by the theoretical calculations with coherent photon-nucleus [149], [150], [151], [152] and photon-photon [153], [154], [155] production mechanisms, which points to evidence of coherent photon reactions in HHIC.", "The observed excesses may originate from coherent photon induced interaction, which impose great challenges for the existing models, e.g., how the broken nuclei satisfy the requirement of coherence.", "Measurements of J/$\\psi $ production at very low $p_{T}$ at different collision energies, collision systems, and centralities can shed new light on the origin of the excess.", "The STAR Collaboration measured J/$\\psi $ production yields at very low $p_{T}$ in Au+Au collisions at $\\sqrt{s_{\\rm {NN}}} =$ 200 GeV and U+U collisions at $\\sqrt{s_{\\rm {NN}}} =$ 193 GeV at mid-rapidity via the di-electron decay channel.", "Figure REF shows the J/$\\psi $ $R_{\\mathrm {AA}}$ as a function of $p_{T}$ in Au+Au collisions and U+U collisions for different centrality classes.", "Suppression of J/$\\psi $ production is observed for $p_{T} >$ 0.2 GeV/c in all collision centrality classes, which is consistent with the previous measurements [130], [71], [107], [99] and can be well described by the transport models [110], [126] incorporating cold and hot medium effects.", "However, in the extremely low $p_{T}$ range, i.e., $p_T < 0.2$ GeV/c, a large enhancement of $R_{\\rm {AA}}$ above unity is observed in peripheral collisions (40-80$\\%$ ) both for Au+Au and U+U collisions.", "In this $p_{T}$ range, the color screening and CNM effects would suppress J/$\\psi $ production, and the only gain effect, which is regeneration, is negligible in peripheral collisions [126].", "The overall effect would lead to $R_{\\rm {AA}} <$ 1 for hadronic production, which is far below the current measurement.", "For $p_T < 0.05$ GeV/c in the 60-80$\\%$ centrality class, the $R_{\\rm {AA}}$ is $24\\pm 5 (\\rm {stat.})", "\\pm 9 (\\rm {syst.", "})$ for Au+Au collisions and $52\\pm 18 (\\rm {stat.})", "\\pm 16 (\\rm {syst.", "})$ for U+U collisions, significantly deviating from the hadronic $p$ +$p$ reference with $N_{\\rm {coll}}$ scaling, which strongly suggests an additional production mechanism.", "Figure: The J/ψ\\psi R AA R_{\\rm {AA}} as a function p T p_{T} in Au+Au collisions at s NN =\\sqrt{s_{\\rm {NN}}} = 200 GeV and U+U collisions at s NN =\\sqrt{s_{\\rm {NN}}} = 193 GeV.", "The figure is taken from .Assuming that the excess observed originates from coherent photoproduction, STAR also reported the differential cross section $d\\sigma /dt$ , where $t$ is the negative momentum transfer squared $-t \\sim p_T^2$ , which reveals the distribution of interaction sites and is closely related to the parton distribution in the nucleus.", "Figure REF shows the J/$\\psi $ yield with the expected hadronic contribution subtracted as a function of $-t$ for the 40-80$\\%$ centrality class in Au+Au and U+U collisions in the low $p_{T}$ range.", "The shape of the $dN/dt$ distribution is very similar to that observed in UPC for $\\rho ^{0}$ meson [157].", "An exponential fit has been applied to the distribution in the $-t$ range of $0.001-0.015~(\\text{GeV}/c)^{-2}$ for Au+Au collisions.", "The slope parameter of this fit can be related to the position of the interaction sites within the target.", "The extracted slope parameter is $177 \\pm 23~(\\text{GeV}/c)^{-2}$ , which is consistent with that expected for an Au nucleus (199 $(\\text{GeV/c})^{-2}$ ) [158] within uncertainties.", "As shown in the figure the data point at $-t < 0.001 ~(\\text{GeV}/c)^{-2}$ is significantly lower ($3.0 \\sigma $ ) than the extrapolation of the exponential fit.", "This suppression may be a hint of interference, which has been confirmed by STAR [159] in the UPC case for $\\rho ^{0}$ meson.", "The theoretical calculation with interference from [150], shown as the blue curve in the plot, can describe the Au+Au data reasonably well ($\\chi ^{2}$ /ndf = 4.8/4) for $-t < 0.015~(\\text{GeV}/c)^{-2}$ .", "It should be aware that there also exists possible contribution from incoherent J/$\\psi $ photoproduction.", "The fitting $-t$ range is chosen to ensure that the coherent production ($\\langle -t \\rangle \\sim 0.005~(\\text{GeV}/c)^{-2}$ ) is dominant over the incoherent production ($\\langle -t \\rangle \\sim 0.250~(\\text{GeV}/c)^{-2}$ ).", "Due to the different nuclear profile, the $-t$ distribution in U+U collisions is expected to be different from that in Au+Au collisions, however, as shown in the figure, the difference is not observed due to the large uncertainties.", "Figure: The J/ψ\\psi yield as a function of the negative momentum transfer squared -t-t ( -t∼p T 2 -t \\sim p_{T}^{2} ) for the 40-80%\\% collision centrality class in Au+Au and U+U collisions.", "The figure is taken from .Figure REF shows $p_{T}$ -integrated J/$\\psi $ yields for $p_{T} <$ 0.1 GeV/$c$ with the expected hadronic contribution subtracted as a function of $N_{\\rm {part}}$ for 30-80$\\%$ Au+Au and 40-80$\\%$ U+U collisions.", "The expected hadronic contributions in Au+Au collisions are also plotted for comparison.", "As depicted in the figure, the contribution from hadronic production is not dominant for the low-$p_{T}$ range in the measured centrality classes.", "Furthermore, the hadronic contribution increases dramatically towards central collisions, while the measured excess shows no sign of significant centrality dependence within uncertainties.", "With the assumption of coherent photoproduction, the excess in U+U collisions should be larger than that in Au+Au collisions.", "Indeed the central value of measurements in U+U collisions is larger than that in Au+Au collisions.", "However, limited by the current experimental precision, the observed difference (2.0$\\sigma $ ) is not significant.", "The model calculations for Au+Au collisions with the coherent photoproduction assumption [150] are also plotted for comparison.", "In the model calculations, the authors consider either the whole nucleus or only the spectator nucleons as photon and Pomeron emitters, resulting in four configurations for photon emitter + Pomeron emitter: (1) Nucleus + Nucleus; (2) Nucleus + Spectator; (3) Spectator + Nucleus; (4) Spectator + Spectator.", "All four scenarios can describe the data points in the most peripheral centrality bins (60-80$\\%$ ).", "However, in more central collisions, the Nucleus + Nucleus scenario significantly overestimates the data, which suggests that there may exist a partial disruption of the coherent production by the violent hadronic interactions in the overlapping region.", "The measurements in semi-central collisions seem to favor the Nucleus + Spectator or Spectator + Nucleus scenarios.", "The approach used in the model effectively incorporates the shadowing effect, which can describe the UPC results in the $x$ -range probed by the RHIC measurement.", "However, the coherently produced J$/\\psi $ could be modified by hot medium effects, e.g.", "QGP meltingg, which is not included in the model.", "More precise measurements toward central collisions and advanced modeling with hot medium effects included are essential to distinguish the different scenarios.", "Figure: The p T p_{T}-integrated J/ψ\\psi yields (p T <p_{T} < 0.1 GeV/c) as a function of N part N_{\\rm {part}} for 30-80%\\% Au+Au collisions and 40-80%\\% U+U collisions.", "The figure is taken from .The $b\\bar{b}$ cross section is much smaller than that of $c\\bar{c}$ at both RHIC and LHC.", "Based on an FONLL calculation [27], [28], [19], the number of $b\\bar{b}$ pairs per event is estimated to be less than 0.1 in 0-10% Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV and $2.3 \\pm 0.4$ ($4.9 \\pm 0.9$ ) in 0-10% Pb+Pb collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 2.76 (5.5) TeV.", "And due to the large masses of $b$ quarks, it is difficult to reach the thermalization.", "Thus the contribution of (re)combination is negligible at RHIC for $\\Upsilon $ .", "In terms of CNM effects, Ref.", "[160] points out that the dissociation of $\\Upsilon (1S)$ by comovers is much smaller than that of J/$\\psi $ and can be neglected at RHIC.", "Thus, compared to J/$\\psi $ , $\\Upsilon $ provides a cleaner probe of QGP melting at least at RHIC.", "The binding energy of radius of $\\Upsilon (1S)$ , $\\Upsilon (2S)$ and $\\Upsilon (3S)$ are quite different.", "Since the dissociation temperature depends on the radius of the quarkonium states.", "The measurements on the suppression of various $\\Upsilon $ states can be used to study the properties of the color screening and the QGP." ], [ "$\\Upsilon $ production in {{formula:a6337e5b-9862-4a58-a2fb-641d3ce12004}} +Au collisions", "The CNM effects on $\\Upsilon $ production can be studied in $p$ +A or $d$ +A collisions.", "Figure REF shows the suppression of $\\Upsilon (1S+2S+3S)$ as a function of rapidity in $p$ +Au and $d$ +Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV.", "The STAR results are measured at mid-rapidity through di-electron channel [162], [161] and the PHENIX results are measured at the forward/backward rapidity through di-muon channel [92].", "The data point at $|y|<0.5$ in $p$ +Au collisions has the highest precision.", "It is $R_{pAu} = 0.82 \\pm 0.10~\\textrm {(stat.)}", "^{+0.08}_{-0.07}~\\textrm {(syst.)}", "\\pm 0.10~\\textrm {(global)}$ , indicating suppression of $\\Upsilon $ due to CNM effects.", "The shaded area shown in the figure represents the calculation from CEM with the EPS09 nuclear parton distribution function [74].", "It predicts enhancement at $y \\sim 0$ , mainly due to the anti-shadowing effect.", "The long dashed line depicts the calculation including parton energy loss only [163].", "It is more closer to the data.", "The dashed line shows the calculation including both EPS09 nPDF and parton energy loss [163].", "It is closer to the calculation with EPS09 alone at mid-rapidity, while is closer to the calculation with parton energy loss only at forward rapidity.", "The data is systematically lower than the calculations, particularly at mid-rapidity.", "This suggests that other CNM effects than the nPDF effect alone are needed to describe the data." ], [ "$\\Upsilon $ production in Au+Au collisions", "The $\\Upsilon $ measurements at RHIC is very challenging due to the small production cross section.", "The STAR Collaboration collected large data samples in 2014 and 2016 for $\\Upsilon $ study in Au+Au collisions using di-muon trigger utilizing the MTD detector completely installed in early 2014.", "Thanks to the large statistics and good momentum resolution in Au+Au collisions, the separation of $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ from the invariant mass spectrum of di-muon is possible.", "Figure REF shows $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ for $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV.", "The suppression of $\\Upsilon (1S+2S+3S)$ in $p$ +Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV is also shown for comparison.", "The suppression for both $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ increases towards central collisions.", "In central collisions, significant suppression for both $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ is observed.", "The observed suppression for inclusive $\\Upsilon (1S)$ could be due to the suppression of the feeddown contribution of higher bottomonium states.", "The fraction of direct $\\Upsilon (1S)$ in the inclusive $\\Upsilon (1S)$ is estimated to be $(71 \\pm 5)\\%$ at low-$p_{_T}$ and $(45.5 \\pm 8.5)\\%$ at high $p_T$ in $p$ +$p$ collisions [85].", "It is not clear whether the direct $\\Upsilon (1S)$ is suppressed with current precision.", "The suppression for $\\Upsilon (2S+3S)$ is found to be larger than $\\Upsilon (1S)$ in (semi-)central collisions, consisting with the “sequential” suppression picture." ], [ "Comparison between RHIC and LHC", "The results at RHIC are compared to the results at LHC in Pb+Pb collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 2.76 TeV measured by the CMS Collaboration [165].", "Both measurements are done at mid-rapidity through the di-muon channel.", "The upper panel of Fig.", "REF is for $\\Upsilon (1S)$ .", "The suppression for $\\Upsilon (1S)$ at RHIC and CMS is similar from peripheral to central heavy-ion collisions although the center-of-mass energies differ by one order of magnitude.", "It is plausible that the inclusive $\\Upsilon (1S)$ suppression arises mainly from the CNM effects and the suppression of the feeddown from excited bottomonium states while the direct $\\Upsilon (1S)$ remains unaffected by the deconfined medium at both RHIC and LHC.", "The lower panel of Fig.", "REF shows the comparison of $R_{\\mathrm {AA}}$ as a function of $N_{\\textrm {part}}$ for $\\Upsilon (2S+3S)$ in Au+Au collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV and for $\\Upsilon (2S)$ and $\\Upsilon (3S)$ in Pb+Pb collisions at $\\sqrt{s_{_\\mathrm {NN}}}$ = 2.76 TeV.", "The $\\Upsilon (2S+3S)$ seems less suppressed at RHIC than at LHC, especially in the peripheral collisions.", "This could be due to the different temperature profile of the medium produced in the heavy-ion collisions at RHIC and LHC.", "It is expected that the initial temperature is higher in central collisions and at higher center-of-mass energy.", "If the initial temperature reached in the heavy-ion collisions at LHC and RHIC are both well above the dissociation temperature of $\\Upsilon (2S)$ and $\\Upsilon (3S)$ , no significant difference is expected at RHIC and LHC.", "However, if the initial temperature is close to the dissociation temperature of $\\Upsilon (2S)$ , the suppression of inclusive $\\Upsilon (2S)$ will be sensitive to the temperature profile of the medium and could result in the different behavior at RHIC and LHC." ], [ "Comparison between experiment and theory", "For better understanding of the $\\Upsilon $ production and constraining the temperature of the medium produced in heavy-ion collision at RHIC, the $\\Upsilon $ suppression data is compared to two theoretical calculations.", "In the TAMU transport model (Rapp) [160], the QGP melting and (re)combination of the $\\Upsilon $ mesons are controlled by a kinetic-rate equation.", "The binding energies in the medium are predicted by thermodynamic microscopic T-matrix calculations using the internal energy from lattice QCD as potential.", "The space-time evolution of the fireball is dictated by a lattice-QCD-based equation of state.", "The initial temperature of the fireball is about 310 MeV in the most central Au+Au collisions.", "The CNM effects are also considered in this calculation.", "The model by Rothkopf and his collaborators [166] uses a lattice QCD vetted complex value heavy-quark potential coupled with a QGP background following anisotropic hydrodynamic evolution.", "The initial temperature used is about 440 MeV in the most central collisions.", "No (re)combination and CNM effects are included in Rothkopf calculation.", "Figure REF shows the comparison of the STAR measurements on $\\Upsilon (1S)$ and $\\Upsilon (2S+3S)$ and the corresponding theoretical calculations from the two models mentioned above.", "Both model calculations are consistent with the data for the ground and excited $\\Upsilon $ states within experimental and theoretical uncertainties.", "The precision of the data and the theoretical calculations are to be improved in order to extract the temperature reached in the heavy-ion collisions via the systematic study of quarkonium suppression." ], [ "Summary", "This paper presents a review of recent experiment measurements of open heavy flavor and quarkonium production at RHIC.", "Heavy quarks, due to their large masses, are expected to behave different from light flavors when interacting with the nuclear matter created in high energy heavy-ion collisions, such as the production mechanisms, hadronization, thermalization, interactions with medium and so on.", "Taking advantage of the silicon vertex detector technology development, precise measurement is achieved and will provide better constraints on theoretical calculations.", "In the following, we summarize few key points for open heavy flavor and quarkonium production.", "Open heavy flavor production: Open charm hadrons $p_{T}$ spectra, including $D^0$ , $D_s$ and $\\Lambda _c$ in various centrality bins at midrapidity $|y|<1$ in $\\sqrt{s_{_\\mathrm {NN}}}$ = 200 GeV Au+Au collisions are presented.", "Thermal parameters, $T_{eff}$ , $T_{kin}$ and radial flow velocity extracted from $D^0$ $p_{T}$ spectra show that $D^0$ freezes out early than light hadrons.", "The strong suppression of $D^0$ nuclear modification factor $R_{AA}$ at high $p_{T}$ and large elliptic flow $v_2$ follow the similar level of light flavor hadrons, indicating strong interactions between charm and medium and charm quark may be thermalized as light quarks.", "The enhancement of $\\Lambda _c/D^0$ and $D_{s}/D^{0}$ ratios in Au+Au collisions compared to that in $p$ +$p$ collisions provide important data for understanding the charm quark hadronization mechanisms.", "The comparison to various models with charm-quark coalescence suggest that coalescence mechanism plays an important role in charm-quark hadronization in the presence of QGP.", "The open bottom hadrons are indirectly measured via their decay products, $B{\\rightarrow }J/\\psi $ , $B{\\rightarrow }D^0$ and $b{\\rightarrow }e$ , by the STAR experiment at mid-rapidity in $\\sqrt{s_{\\rm NN}} = $ 200 GeV Au+Au collisions.", "The suppression observed for non-prompt $J/\\psi $ and non-prompt $D^0$ /e at high $p_{T}$ indicates bottom-medium interactions and bottom energy loss.", "The less suppression of $b{\\rightarrow }e$ compared with $c{\\rightarrow }e$ and no suppression of $B{\\rightarrow }D$ at 4 GeV/c indicating less energy loss of bottom quark due to its extremely large mass and consistent with flavor dependent parton energy loss mechanisms.", "Quarkonium production: The production of J/$\\psi $ in $p$ +$p$ and heavy-ion collisions are intensively studied at RHIC.", "In $p$ +$p$ collisions, the measured $p_{_T}$ spectra and polarization (spin alignment) is found to be consistent with the theoretical calculations from (I)CEM and NRQCD.", "The CEM calcualtion and NRQCD calculations with different Long Distance Matrix Elements predict different polarization for J/$\\psi $ , but the current data can not tell the difference.", "The measurement of J/$\\psi $ polarization with high statistics will be helpful to constraint models or LDMEs in NRQCD.", "The collision energy, collision size and $p_{_T}$ dependence of J/$\\psi $ production in heavy-ion collisions are measured at RHIC and compared to SPS and LHC.", "At low $p_T$ , it is consistent with the picture that QGP melting, (re)combination and CNM effects play important roles.", "Their relative contribution varies with the collision energy, collision centrality, system size and the kinematic variables of J/$\\psi $ .", "Based on a transport model calculation, low-$p_{_T}$ J/$\\psi $ suppression in central Au+Au collisions is about 0.6, QGP melting further suppresses it to 0.2, but (re)combination enhances it back to about 0.4.", "At high $p_T$ , where the contribution from CNM effects and (re)combination is negligible, significant suppression is observed, providing strong evidence of QGP melting at RHIC.", "The collectivity of J/$\\psi $ is studied in Au+Au collisions via radial and elliptic flow.", "The results disfavor that J/$\\psi $ at RHIC is dominantly produced via (re)combination of thermalized charm quarks.", "$\\Upsilon $ , a cleaner probe of QGP meting, is found to be suppressed significantly in central Au+Au collisions.", "Sequential suppression (stronger suppression of $\\Upsilon (3S)$ and $\\Upsilon (2S)$ than $\\Upsilon (1S)$ ) observed at LHC is confirmed at RHIC, which provided another strong evidence of QGP melting.", "The comparisons of the RHIC data to LHC data and theoretical calculations are useful to extract the properties of QGP.", "In around 2023, sPHENIX will start high luminosity runs with high-speed silicon vertex detector (MVTX), which is based on a state-of-art monolithic active pixel sensors (MAPS) technology.", "At least 100 times more statistics will be collected aiming for dedicated bottom measurement via hadronic decay channels.", "Such as precision measurement of nuclear modification factors and flows for B-mesons and b-tagged jets [167].", "With the high statistics Zr+Zr and Ru+Ru (3B events for each collision system) data taken in 2018, the J/$\\psi $ $R_{\\mathrm {AA}}$ and elliptic flow will be measured with good precision to deepen our understanding of the interplay of QGP melting, (re)combination and CNM effects on J/$\\psi $ production.", "The $Z$ dependence of J/$\\psi $ photoproduction can be also studied with the isobaric collision data and the comparison to Au+Au collisions.", "The Cu+Au collision data taken by STAR in 2012 is recently been fully produced, J/$\\psi $ production via di-electron decay channel will be measured.", "The STAR forward upgrade program, including finished inner Time Projection Chamber, Endcap Time-of-Flight upgrades and ongoing Forward Tracking System and Forward Calorimeter System upgrades will extend the rapidity coverage of quarkonium measurement at STAR upto $y=4$ .", "Many unique physics opportunities with quarkonium in $p$ +$p$ , $p(d)$ +A and A+A collisions at very forward rapidity will be enabled.", "With sPHENIX in high luminosity A+A runs, the precision of $\\Upsilon $ measurement are expected to be significantly improved.", "These facility upgrades will deepen our understanding of the interaction of heavy flavor and quarkonium with the hot dense medium created in heavy-ion collisions at RHIC." ] ]
2105.11656
[ [ "Minimal unique palindromic substrings after single-character\n substitution" ], [ "Abstract A palindrome is a string that reads the same forward and backward.", "A palindromic substring $w$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $w$ occurs only once in $T$ and any proper palindromic substring of $w$ occurs at least twice in $T$.", "MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018].", "Given a string $T$ of length $n$, all MUPSs of $T$ can be computed in $O(n)$ time.", "In this paper, we study the problem of updating the set of MUPSs when a character in the input string $T$ is substituted by another character.", "We first analyze the number $d$ of changes of MUPSs when a character is substituted, and show that $d$ is in $O(\\log n)$.", "Further, we present an algorithm that uses $O(n)$ time and space for preprocessing, and updates the set of MUPSs in $O(\\log\\sigma + (\\log\\log n)^2 + d)$ time where $\\sigma$ is the alphabet size.", "We also propose a variant of the algorithm, which runs in optimal $O(1+d)$ time when the alphabet size is constant." ], [ "Introduction", "Palindromes are strings that read the same forward and backward.", "Finding palindromic structures has important applications to analyze biological data such as DNA, RNA, and proteins, and thus algorithms and combinatorial properties on palindromic structures have been heavily studied (e.g., see [16], [3], [7], [17], [19], [12], [20] and references therein).", "In this paper, we treat a notion of palindromic structures called minimal unique palindromic substring (MUPS) that is introduced in [14].", "A palindromic substring $T[i.. j]$ of a string $T$ is called a MUPS of $T$ if $T[i..j]$ occurs exactly once in $T$ and $T[i+1..j-1]$ occurs at least twice in $T$ .", "MUPSs are utilized for solving the shortest unique palindromic substring (SUPS) problem proposed by Inoue et al.", "[14], which is motivated by an application in molecular biology.", "They showed that there are no more than $n$ MUPSs in any length-$n$ string, and proposed an $O(n)$ -time algorithm to compute all MUPSs of a given string of length $n$ over an integer alphabet of size $n^{O(1)}$ .", "After that, Watanabe et al.", "[23] considered the problem of computing MUPSs in an run-length encoded (RLE) string.", "They showed that there are no more than $m$ MUPSs in a string whose RLE size is $m$ .", "Also, they proposed an $O(m\\log \\sigma _R)$ -time and $O(m)$ -space algorithm to compute all MUPSs of a string given in RLE, where $m$ is the RLE size of the string, and $\\sigma _R$ is the number of distinct single-character runs in the RLE string.", "Recently, Mieno et al.", "[18] considered the problems of computing palindromic structures in the sliding window model.", "They showed that the set of MUPSs in a sliding window can be maintained in a total of $O(n\\log \\sigma _W)$ time and $O(D)$ space while a window of size $D$ shifts over a string of length $n$ from the left-end to the right-end, where $\\sigma _W$ is the maximum number of distinct characters in the windows.", "This result can be rephrased as follows: The set of MUPSs in a string of length $D$ can be updated in amortized $O(\\log \\sigma _W)$ time using $O(D)$ space after deleting the first character or inserting a character to the right-end.", "To the best of our knowledge, there is no efficient algorithm for updating the set of MUPSs after editing a character at any position so far.", "Now, we consider the problem of updating the set of MUPSs in a string after substituting a character at any position.", "Formally, we tackle the following problem: Given a string $T$ of length $n$ over an integer alphabet of size $n^{O(1)}$ to preprocess, and then given a query of single-character substitution.", "Afterwards, we return the set of MUPSs of the edited string.", "In this paper, we first show that the number $d$ of changes of MUPSs after a single-character substitution is $O(\\log n)$ .", "In addition, we present an algorithm that uses $O(n)$ time and space for preprocessing, and updates the set of MUPSs in $O(\\log \\sigma + (\\log \\log n)^2 + d) \\subset O(\\log n)$ time.", "We also propose a variant of the algorithm, which runs in optimal $O(d)$ time when the alphabet size is constant." ], [ "Related Work.", "This line of research was initiated by Amir et al. [2].", "They proposed an algorithm that computes the longest common factor of two strings after a single-character edit operation, i.e., insertion, deletion, and substitution.", "Also, there are several studies of the same settings for some notions of regularities of strings [22], [1], [10].", "In particular, regarding palindromic structures, Funakoshi et al.", "[10] proposed algorithms for computing the longest palindromic substring after single-character or block-wise edit operations.", "Let $\\Sigma $ be an alphabet of size $\\sigma $ .", "An element of $\\Sigma $ is called a character.", "An element of $\\Sigma ^\\ast $ is called a string.", "The length of a string $T$ is denoted by $|T|$ .", "The empty string $\\varepsilon $ is the string of length 0.", "For a string $T = xyz$ , then $x, y$ , and $z$ are called a prefix, substring, and suffix of $T$ , respectively.", "They are called a proper prefix, proper substring, proper suffix of $T$ if $x \\ne T$ , $y \\ne T$ , and $z \\ne T$ , respectively.", "For each $1 \\le i \\le |T|$ , $T[i]$ denotes the $i$ -th character of $T$ .", "For each $1 \\le i \\le j \\le |T|$ , $T[i.. j]$ denotes the substring of $T$ starting at position $i$ and ending at position $j$ .", "For convenience, let $T[i^{\\prime }.. j^{\\prime }] = \\varepsilon $ for any $i^{\\prime } > j^{\\prime }$ .", "A positive integer $p$ is said to be a period of a string $T$ if $T[i] = T[i+p]$ for all $1 \\le i \\le |T|-p$ .", "For strings $X$ and $Y$ , let $\\mathit {lcp}(X, Y)$ denotes the length of the longest common prefix (in short, lcp) of $X$ and $Y$ , i.e., $\\mathit {lcp}(X, Y) = \\max \\lbrace \\ell \\mid X[1.. \\ell ] = Y[1.. \\ell ]\\rbrace $ .", "For a string $T$ and two integers $1 \\le i \\le j \\le |T|$ , let $\\mathit {lce}_T(i, j)$ denotes the length of the longest common extension (in short, lce) of $i$ and $j$ in $T$ , i.e., $\\mathit {lce}_T(i, j) = \\mathit {lcp}(T[i..|T|], T[j..|T|])$ .", "For non-empty strings $T$ and $w$ , $\\mathit {beg}_T(w)$ denotes the set of beginning positions of occurrences of $w$ in $T$ .", "Also, for a text position $i$ in $T$ , $\\mathit {inbeg}_{T, i}(w)$ denotes the set of beginning positions of occurrences of $w$ in $T$ where each occurrence covers position $i$ .", "Namely, $\\mathit {beg}_T(w) = \\lbrace b \\mid T[b.. e] = w\\rbrace $ and $\\mathit {inbeg}_{T, i}(w) = \\lbrace b \\mid T[b.. e] = w \\textrm { and } i \\in [b, e]\\rbrace $ .", "Further, let $\\mathit {xbeg}_{T, i}(w) = \\mathit {beg}_T(w) \\setminus \\mathit {inbeg}_{T, i}(w)$ .", "For convenience, $|\\mathit {beg}_T(\\varepsilon )| = |\\mathit {inbeg}_{T, i}(\\varepsilon )| = |\\mathit {xbeg}_{T, i}(\\varepsilon )| = |T| + 1$ for any $T$ and $i$ .", "We say that $w$ is unique in $T$ if $|\\mathit {beg}_T(w)| = 1$ , and that $w$ is repeating in $T$ if $|\\mathit {beg}_T(w)| \\ge 2$ .", "Note that the empty string is repeating in any other string.", "Since every unique substring $u = T[i..j]$ of $T$ occurs exactly once in $T$ , we will sometimes identify $u$ with its corresponding interval $[i, j]$ .", "In what follows, we consider an arbitrarily fixed string $T$ of length $n \\ge 1$ over an alphabet $\\Sigma $ of size $\\sigma = n^{O(1)}$ .", "For a string $w$ , $w^R$ denotes the reversed string of $w$ .", "A string $w$ is called a palindrome if $w = w^R$ .", "A palindrome $w$ is called an even-palindrome (reps. odd-palindrome) if its length is even (resp.", "odd).", "For a palindrome $w$ , its length-$\\lfloor |w|/2 \\rfloor $ prefix (resp.", "length-$\\lfloor |w|/2 \\rfloor $ suffix) is called the left arm (resp.", "right arm) of $w$ , and is denoted by $\\mathsf {larm}_w$  (resp.", "$\\mathsf {rarm}_w$ ).", "Also, we call $\\mathsf {Larm}_w = \\mathsf {larm}_w\\cdot s_w$  (resp.", "$\\mathsf {Rarm}_w = s_w\\cdot \\mathsf {rarm}_w$ ) the extended left arm (resp.", "extended right arm) of $w$ where $s_w$ is the character at the center of $w$ if $w$ is an odd-palindrome, and $s_w$ is empty otherwise.", "Note that when $w$ is an even-palindrome, $\\mathsf {Rarm}_w = \\mathsf {rarm}_w$ and $\\mathsf {Larm}_w = \\mathsf {larm}_w$ .", "For a non-empty palindromic substring $w = T[i.. j]$ of a string $T$ , the center of $w$ is $\\frac{i+j}{2}$ and is denoted by $\\mathit {center}(w)$ .", "A non-empty palindromic substring $T[i.. j]$ of a string $T$ is said to be maximal if $i = 1$ , $j = n$ , or $T[i-1] \\ne T[j+1]$ .", "For a non-empty palindromic substring $w = T[i.. j]$ of a string $T$ and a non-negative integer $\\ell $ , $v = T[i-\\ell .. j+\\ell ]$ is said to be an extension of $w$ if $1 \\le i-\\ell \\le j+\\ell \\le n$ and $v$ is a palindrome.", "Also, $T[i+\\ell .. j-\\ell ]$ is said to be a shrink of $w$ .", "A non-empty string $w$ is called a 1-mismatch palindrome if there is exactly one mismatched position between $w[1..\\lfloor |w|/2 \\rfloor ]$ and $w[\\lceil |w|/2 \\rceil +1..|w|]^R$ .", "Informally, a 1-mismatch palindrome is a pseudo palindrome with a mismatch position between their arms.", "As in the case of palindromes, a 1-mismatch palindromic substring $T[i.. j]$ of a string $T$ is said to be maximal if $i = 1$ , $j = n$ or $T[i-1] \\ne T[j+1]$ .", "A palindromic substring $T[i.. j]$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $T[i..j]$ is unique in $T$ and $T[i+1..j-1]$ is repeating in $T$ .", "We denote by $\\mathsf {MUPS}(T)$ the set of intervals corresponding to MUPSs of a string $T$ .", "A MUPS cannot be a substring of another palindrome with a different center.", "Also, it is known that the number of MUPSs of $T$ is at most $n$ , and set $\\mathsf {MUPS}(T)$ can be computed in $O(n)$ time for a given string $T$ over an integer alphabet [14].", "The following lemma states that the total sum of occurrences of strings which are extended arms of MUPSs is $O(n)$ : Lemma 1 The total sum of occurrences of the extended right arms of all MUPSs in a string $T$ is at most $2n$ .", "Similarly, the total sum of occurrences of the extended left arms of all MUPSs in $T$ is at most $2n$ .", "It suffices to prove the former statement for the extended right arms since the latter can be proved symmetrically.", "Let $w_1$ and $w_2$ be distinct odd-length MUPSs of $T$ with $|w_1| \\le |w_2|$ .", "For the sake of contradiction, we assume that $T[j.. j+|\\mathsf {Rarm}_{w_1}|-1] = \\mathsf {Rarm}_{w_1}$ and $T[j.. j+|\\mathsf {Rarm}_{w_2}|-1] = \\mathsf {Rarm}_{w_2}$ for some position $j$ in $T$ .", "Namely, $\\mathsf {Rarm}_{w_1}$ is a prefix of $\\mathsf {Rarm}_{w_2}$ .", "Then, $\\mathsf {larm}_{w_1}$ is a suffix of $\\mathsf {larm}_{w_2}$ by palindromic symmetry.", "This means that $w_1$ is a substring of $w_2$ .", "This contradicts that $w_2$ is a MUPS of $T$ .", "Thus, all occurrences of the extended right arms of all odd-length MUPSs are different, i.e., the total number of the occurrences is at most $n$ .", "Similarly, the total number of all occurrences of the right arms of all even-length MUPSs is also at most $n$ ." ], [ "Tools", "This subsection lists some data structures used in our algorithm.", "Our model of computation is a standard word RAM model with machine word size $\\Omega (\\log n)$ ." ], [ "Suffix Trees.", "The suffix tree of $T$ is the compacted trie for all suffixes of $T$  [24].", "We denote by $\\mathsf {STree}(T)$ the suffix tree of $T$ .", "If a given string $T$ is over an integer alphabet of size $n^{O(1)}$ , $\\mathsf {STree}(T)$ can be constructed in $O(n)$ time [8].", "Not all substrings of $T$ correspond to nodes in $\\mathsf {STree}(T)$ .", "However, the loci of such substrings can be made explicit in linear time: Lemma 2 (Corollary 8.1 in [15]) Given $m$ substrings of $T$ , represented by intervals in $T$ , we can compute the locus of each substring in $\\mathsf {STree}(T)$ in $O(n + m)$ total time.", "Moreover, the loci of all the substrings in $\\mathsf {STree}(T)$ can be made explicit in $O(n + m)$ extra time.", "Also, this lemma implies the following corollary: Corollary 1 Given $m$ substrings of $T$ , represented by intervals in $T$ , we can sort them in $O(n + m)$ time.", "An LCE query on a string $T$ is, given two indices $i, j$ of $T$ , to compute $\\mathit {lce}_T(i, j)$ .", "Using $\\mathsf {STree}(T\\$)$ enhanced with a lowest common ancestor data structure, we can answer any LCE query on $T$ in constant time where $\\$$ is a special character with $\\$ \\notin \\Sigma $ .", "In the same way, we can compute the lcp value between any two suffixes of $T$ or $T^R$ in constant time by using $\\mathsf {STree}(T\\$T^R\\#)$ where $\\#$ is another special character with $\\# \\notin \\Sigma $ .", "The eertree (a.k.a.", "palindromic tree) of $T$ is a pair of rooted edge-labeled trees $\\mathcal {T}_{\\mathsf {odd}}$ and $\\mathcal {T}_{\\mathsf {even}}$ representing all distinct palindromes in $T$  [19].", "The roots of $\\mathcal {T}_{\\mathsf {odd}}$ and $\\mathcal {T}_{\\mathsf {even}}$ represent $\\varepsilon $ .", "Each non-root node of $\\mathcal {T}_{\\mathsf {odd}}$  (resp.", "$\\mathcal {T}_{\\mathsf {even}}$ ) represents an odd-palindrome (resp.", "even-palindrome) which occurs in $T$ .", "Let $\\mathit {pal}(v)$ be the palindrome represented by a node $v$ .", "For the root $r_{\\mathsf {odd}}$ of $\\mathcal {T}_{\\mathsf {odd}}$ , there is an edge $(r_{\\mathsf {odd}}, u)$ labeled by $a \\in \\Sigma $ if there is a node $u$ with $\\mathit {pal}(u) = a$ .", "For any node $v$ in the eertree except for $r_{\\mathsf {odd}}$ , there is an edge $(v, w)$ labeled by $a \\in \\Sigma $ if there is a node $w$ with $\\mathit {pal}(w) = a\\cdot \\mathit {pal}(v)\\cdot a$ .", "We denote by $\\mathsf {EERTREE}(T)$ the eertree of $T$ .", "We will sometimes identify a node $u$ in $\\mathsf {EERTREE}(T)$ with its corresponding palindrome $\\mathit {pal}(u)$ .", "Also, the path from a node $u$ to a node $v$ in $\\mathsf {EERTREE}(T)$ is denoted by $\\mathit {pal}(u) \\rightsquigarrow \\mathit {pal}(v)$ .", "If a given string $T$ is over an integer alphabet of size $n^{O(1)}$ , $\\mathsf {EERTREE}(T)$ can be constructed in $O(n)$ time [19].", "A path-tree LCE query is a generalized LCE query on a rooted edge-labeled tree $\\mathcal {T}$  [4]: Given three nodes $u$ , $v$ , and $w$ in $\\mathcal {T}$ where $u$ is an ancestor of $v$ , to compute the lcp between the path-string from $u$ to $v$ and any path-string from $w$ to a descendant leaf.", "The following result is known: Theorem 1 (Theorem 2 of [4]) For a tree $\\mathcal {T}$ with $N$ nodes, a data structure of size $O(N)$ can be constructed in $O(N)$ time to answer any path–tree LCE query in $O((\\log \\log N)^2)$ time.", "We will use later path-tree LCE queries on the eertree of the input string.", "Let $\\mathcal {I}$ be a set of $n$ intervals, each of which is a subinterval of the universe $U = [1, O(n)]$ .", "An interval stabbing query on $\\mathcal {I}$ is, given a query point $q \\in U$ , to report all intervals $I \\in \\mathcal {I}$ such that $I$ is stabbed by $q$ , i.e., $q \\in I$ .", "We can answer such a query in $O(1+k)$ time after $O(n)$ -time preprocessing, where $k$ is the number of intervals to report [21]." ], [ "Changes of MUPSs After Single Character Substitution", "In the following, we fix the original string $T$ of length $n$ , the text position $i$ in $T$ to be substituted, and the string $T^{\\prime }$ after the substitution.", "Namely, $T[i] \\ne T^{\\prime }[i]$ and $T[j] = T^{\\prime }[j]$ for each $j$ with $1 \\le j \\le n$ and $j \\ne i$ .", "This section analyzes the changes of the set of MUPSs when $T[i]$ is substituted by $T^{\\prime }[i]$ .", "For palindromes covering editing position $i$ , Lemma REF holds.", "All the proofs omitted due to lack of space can be found in Appendix .", "Lemma 3 For a palindrome $w$ , if $\\mathit {inbeg}_{T, i}(w) \\ne \\emptyset $ , then $\\mathit {inbeg}_{T^{\\prime }, i}(w) = \\emptyset $ .", "For a position $i$ , let $\\mathcal {W}_i$ be the set of palindromes $w$ such that $|\\mathit {inbeg}_{T, i}(w)| \\ge 1$ , $|\\mathit {xbeg}_{T, i}(w)|= 1$ , and $w$ is minimal, i.e., $|\\mathit {inbeg}_{T, i}(v)| = 0$ or $|\\mathit {xbeg}_{T, i}(v)| \\ge 2$ where $v = w[2..|w|-1]$ .", "This set $\\mathcal {W}_i$ is useful for analyzing the number of changes of MUPSs in the proof of Theorem REF .", "Lemma 4 For any position $i$ in $T$ , $|\\mathcal {W}_i| \\in O(\\log n)$ .", "Lemma 5 For each position $i$ in $T$ , the number of MUPSs covering $i$ is $O(\\log n)$ .", "By using Lemmas REF , REF , and REF , we show the following theorem: Theorem 2 $|\\mathsf {MUPS}(T) \\bigtriangleup \\mathsf {MUPS}(T^{\\prime })| \\in O(\\log n)$ .", "In the following, we consider the number of MUPSs to be removed.", "First, at most one interval can be a MUPS of $T$ centered at $i$ .", "Also, any other interval in $\\mathsf {MUPS}(T)$ covering position $i$ cannot be an element of $\\mathsf {MUPS}(T^{\\prime })$ since its corresponding string in $T^{\\prime }$ is no longer a palindrome.", "By Lemma REF , the number of such MUPSs is $O(\\log n)$ .", "Next, let us consider MUPSs not covering position $i$ .", "When a MUPS $w$ of $T$ not covering $i$ is no longer a MUPS of $T^{\\prime }$ , then either (A) $w$ is repeating in $T^{\\prime }$ or (B) $w$ is unique in $T^{\\prime }$ but is not minimal.", "Let $w_1$ be a MUPS of the case (A).", "Since $w_1$ does not cover $i$ , is unique in $T$ , and is repeating in $T^{\\prime }$ , $|\\mathit {inbeg}_{T^{\\prime },i}(w_1)| \\ge 1$ and $|\\mathit {xbeg}_{T^{\\prime },i}(w_1)| = 1$ .", "Let $v_1$ be the minimal shrink of $w_1$ such that $|\\mathit {inbeg}_{T^{\\prime },i}(v_1)| \\ge 1$ and $|\\mathit {xbeg}_{T^{\\prime },i}(v_1)| = 1$ .", "Contrary, $w_1$ is the only MUPS of the case (A) which is an extension of $v_1$ since $|\\mathit {xbeg}_{T^{\\prime },i}(v_1)| = 1$ .", "Namely, there is a one-to-one relation between $w_1$ and $v_1$ .", "By Lemma REF , the number of palindromes that satisfy the above conditions of $v_1$ is $O(\\log n)$ .", "Thus, the number of MUPSs of the case (A) is also $O(\\log n)$ .", "Let $w_2$ be a MUPS of the case (B).", "In $T^{\\prime }$ , $w_2$ covers some MUPS as a proper substring since it is not a MUPS and is unique in $T^{\\prime }$ .", "Let $v_2$ be the MUPS of $T^{\\prime }$ , which is a proper substring of $w_2$ .", "While $v_2$ is unique in $T^{\\prime }$ , it is repeating in $T$ since $w_2$ is a MUPS of $T$ .", "Namely, $|\\mathit {inbeg}_{T,i}(v_2)| \\ge 1$ and $|\\mathit {xbeg}_{T,i}(v_2)| = 1$ hold.", "Also, $v_2$ is actually minimal: Let $u_2 = v_2[2.. |v_2|-1]$ .", "If we assume that $|\\mathit {inbeg}_{T,i}(u_2)| \\ge 1$ and $|\\mathit {xbeg}_{T,i}(u_2)| = 1$ , then $u_2$ becomes unique in $T^{\\prime }$ , and this contradicts that $v_2$ is a MUPS of $T^{\\prime }$ .", "Furthermore, similar to the above discussions, there is a one-to-one relation between $w_2$ and $v_2$ .", "Again by Lemma REF , the number of palindromes that satisfy the above conditions of $v_2$ is $O(\\log n)$ .", "Thus, the number of MUPSs of the case (B) is also $O(\\log n)$ .", "Therefore, $|\\mathsf {MUPS}(T)\\setminus \\mathsf {MUPS}(T^{\\prime })| \\in O(\\log n)$ holds.", "Also, $|\\mathsf {MUPS}(T^{\\prime })\\setminus \\mathsf {MUPS}(T)| \\in O(\\log n)$ holds by symmetry.", "To summarize, $|\\mathsf {MUPS}(T) \\bigtriangleup \\mathsf {MUPS}(T^{\\prime })| = |\\mathsf {MUPS}(T)\\setminus \\mathsf {MUPS}(T^{\\prime }) \\cup \\mathsf {MUPS}(T^{\\prime })\\setminus \\mathsf {MUPS}(T)| = |\\mathsf {MUPS}(T)\\setminus \\mathsf {MUPS}(T^{\\prime })| + |\\mathsf {MUPS}(T^{\\prime })\\setminus \\mathsf {MUPS}(T)| \\in O(\\log n)$ ." ], [ "Algorithms for Updating Set of MUPSs", "In this section, we propose an algorithm for updating the set of MUPSs when a single-character in the original string is substituted by another character.", "We denote by $\\mathit {sub}(i, s)$ the substitution query, that is, to substitute $T[i]$ by another character $s$ .", "First, we define a sub-problem that will be used in our algorithm: Problem 1 Given a substitution query $\\mathit {sub}(i, s)$ on $T$ , compute the longest odd-palindromic substring $v$ of $T^{\\prime }$ such that $\\mathit {center}(v) = i$ and $v$ occurs in $T$ if it exists.", "Also, if such $v$ exists, determine whether $v$ is unique in $T$ or not.", "Furthermore, if $v$ is unique in $T$ , compute the shrink of $v$ that is a MUPS of $T$ .", "We show the following lemma: Lemma 6 After $O(n)$ -time preprocessing, we can answer Problem REF in $O(\\delta (n,\\sigma ) + (\\log \\log n)^2)$ time where $\\delta (n,\\sigma )$ denote the time to retrieve any child of the root of the odd-tree of $\\mathsf {EERTREE}(T)$ .", "When $\\sigma \\in O(n)$ , we can easily achieve $\\delta (n, \\sigma ) \\in O(1)$ with linear space, by using an array of size $\\sigma $ .", "Otherwise, we achieve $\\delta (n, \\sigma ) \\in O(\\log \\sigma )$ for a general ordered alphabet by using a binary search tree.", "In the rest of this paper, we propose an algorithm to compute the changes in MUPSs after a single-character substitution.", "Our strategy is basically to pre-compute changes in MUPSs for some queries as much as possible within linear time.", "The other changes will be detected on the fly by using some data structures." ], [ "Computing MUPSs to be Removed", "We categorize MUPSs to be removed into three types as follows: R1) A MUPS of $T$ that covers $i$ .", "R2) A MUPS of $T$ that does not cover $i$ and is repeating in $T^{\\prime }$ .", "R3) A MUPS of $T$ that does not cover $i$ and is unique but not minimal in $T^{\\prime }$ .", "See Fig.", "REF in Appendix  for illustration." ], [ "Type R1.", "All MUPSs covering editing position $i$ are always removed.", "Thus, we can detect them in $O(1+\\alpha _{\\mathit {rem}})$ time after a simple linear time preprocessing (e.g., using stabbing queries), where $\\alpha _{\\mathit {rem}}$ is the number of MUPSs of Type R1.", "Before describing our algorithm, we give a few observations about MUPSs of Type R2.", "Let $w$ be a MUPS of Type R2.", "Since $w$ is unique in $T$ and is repeating in $T^{\\prime }$ , $|\\mathit {inbeg}_{T^{\\prime }, i}(w)| \\ge 1$ .", "When $w$ occurs in $T^{\\prime }$ centered at editing position $i$ , we retrieve such $w$ by applying Problem REF .", "If it is not the case, we can utilize the following observations: Consider the starting position $j$ of an occurrence of $w$ in $T^{\\prime }$ such that $T^{\\prime }[j.. j+|w|-1] = w$ and $i \\in [j, j+|w|-1]$ .", "If position $i$ is covered in the right arm of $T^{\\prime }[j.. j+|w|-1]$ , then $\\mathsf {Larm}_w$ occurs at position $j$ in both $T$ and $T^{\\prime }$ .", "Further, the Hamming distance between $T[j+|\\mathsf {Larm}_w|.. j+|w|-1]$ and $w[|\\mathsf {Larm}_w|+1.. |w|] = \\mathsf {rarm}_w$ equals 1.", "Namely, for each occurrence at position $k$ of string $\\mathsf {Larm}_w$ in $T$ , $w$ can occur at $k$ in $T^{\\prime }$ only if the Hamming distance between $T[k+|\\mathsf {Larm}_w|.. k+|w|-1]$ and $w[|\\mathsf {Larm}_w|+1.. |w|]$ equals 1.", "In other words, if the Hamming distance is greater than 1, $w$ cannot occur at $k$ in $T^{\\prime }$ .", "See also Fig.", "REF in Appendix .", "In the preprocessing phase, we first apply the $O(n)$ -time preprocessing of Lemma REF for Problem REF .", "Next, we initialize the set $\\mathcal {A}_{R2} = \\emptyset $ .", "The set $\\mathcal {A}_{R2}$ will become an index of MUPSs of Type R2 when the preprocessing is finished.", "For each MUPS $w = T[b.. e]$ of $T$ , we process the followings: For the beginning position $j \\ne b$ of each occurrence of $\\mathsf {Larm}_w$ in $T$ , we first compute the lcp value between $T[j+|\\mathsf {Larm}_w|..|T|]\\$$ and $\\mathsf {rarm}_w$ with allowing one mismatch.", "Note that $T[j+|\\mathsf {Larm}_w|..|T|]\\$$ must have at least one mismatch with $\\mathsf {rarm}_w$ , since $T[j.. j+|\\mathsf {Larm}_w|-1] = \\mathsf {Larm}_w$ , $j \\ne b$ , and $T[b.. e]$ is unique in $T$ .", "If there are two mismatch positions between them, do nothing for this occurrence since $w$ cannot occur at $j$ after any substitution.", "We can check this by querying LCE at most twice.", "Otherwise, let $q = j+|\\mathsf {Larm}_w|-1+d$ be the mismatched position in $T$ .", "When the $q$ -th character of $T$ is substituted by the character $\\mathsf {rarm}_w[d]$ , $w = T[b.. e]$ occurs at $j \\ne b$ , i.e., it is a MUPS of Type R2.", "So we add MUPS $w = T[b.. e]$ into $\\mathcal {A}_{R2}$ with the pair of index and character $(q, \\mathsf {rarm}_w[d])$ as the key.", "In addition, symmetrically, we update $\\mathcal {A}_{R2}$ for each occurrence of $\\mathsf {Rarm}_w$ in $T$ .", "After finishing the above processes for every MUPS of $T$ , we sort the elements of $\\mathcal {A}_{R2}$ by radix sort on the keys.", "If there are multiple identical elements with the same key, we unify them into a single element.", "Also, if there are multiple elements with the same key, we store them in a linear list.", "By Lemma REF , the total number of occurrences of arms of MUPSs is $O(n)$ , and hence, the total preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we query Problem REF with the same pair $(i, s)$ as the input.", "Then, we complete checking whether there exists a MUPS of Type R2 centered at $i$ .", "Next, consider the existence of the remaining MUPSs of Type R2.", "First, an element in $\\mathcal {A}_{R2}$ corresponding to the key $(i, s)$ can be detected in $O(\\log \\sigma _i)$ time by using random access on indices and binary search on characters, where $\\sigma _i$ is the number of characters $s_i$ such that the key $(i, s_i)$ exists in $\\mathcal {A}_{R2}$ .", "After that, we can enumerate all the other elements with the key by scanning the corresponding linear list.", "Thus, the total query time is $O(\\delta (n,\\sigma ) + (\\log \\log n)^2 +\\log \\sigma _i + \\beta _{\\mathit {rem}})$ where $\\beta _{\\mathit {rem}}$ is the number of MUPSs of Type R2.", "Finally, we show $\\sigma _i \\in O(\\min \\lbrace \\sigma , \\log n\\rbrace )$ .", "Let us consider palindromes in $T^{\\prime }$ whose right arm covers position $i$ .", "Those whose left arms cover $i$ can be treated similarly.", "Any palindrome in $T^{\\prime }$ whose right arm covers $i$ is an extension of some maximal palindrome in $T$ ending at $i-1$ .", "It is known that the number of possible characters immediately preceding such maximal palindromes is $O(\\log n)$  [10].", "Therefore, $\\sigma _i \\in O(\\log n)$ holds, and thus, the query time is $O(\\delta (n,\\sigma ) + (\\log \\log n)^2 + \\beta _{\\mathit {rem}})$ .", "Let $w = T[b.. e]$ be a MUPS of $T$ and let $v = T[b+1..e-1]$ .", "Further let $T[b_{l1}.. e_{l1}]$ and $T[b_{r1}.. e_{r1}]$ be the leftmost and the rightmost occurrence of $v$ in $T$ except for $T[b+1.. e-1]$ .", "We define interval $\\rho _w = \\lbrace k \\mid k \\notin [b+1, e-1] \\text{ and } k \\in [b_{r1}, e_{l1}]\\rbrace $ .", "Note that $\\rho _w$ can be empty.", "See also Fig.", "REF in Appendix  for illustration.", "If the editing position $i$ is in $\\rho _w$ , then the only occurrence of $v$ in $T^{\\prime }$ is $T^{\\prime }[b+1.. e-1]$ , i.e., $v$ is unique in $T^{\\prime }$ .", "Thus, $w$ is a removed MUPS of Type R3.", "Contrary, if $i \\notin [b, e]$ and $i \\notin \\rho _w$ , there are at least two occurrences of $v$ in $T^{\\prime }$ , i.e., $w$ cannot be a MUPS of Type R3.", "In the preprocessing phase, we first compute the set of intervals $\\mathcal {R} = \\lbrace \\rho _w \\mid w \\text{ is a MUPS of }T \\rbrace $ .", "$\\mathcal {R}$ can be computed by traversing over the suffix tree of $T$ enhanced with additional explicit nodes, each of which represents a substring $T[b+1..e-1]$ for each MUPS $T[b..e]$ of $T$ .", "Also, we apply the preprocessing for stabbing queries to $\\mathcal {R}$ .", "The total time for preprocessing is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , compute all intervals in $\\mathcal {R}$ stabbed by position $i$ by answering a stabbing query.", "They correspond to MUPSs of Type R3.", "The query time is $O(1 + \\gamma _{\\mathit {rem}})$ , where $\\gamma _{\\mathit {rem}}$ is the number of MUPSs of Type R3.", "To summarize, we can compute all MUPSs to be removed after a single-character substitution in $O(\\delta (n,\\sigma ) + (\\log \\log n)^2 + \\alpha _{\\mathit {rem}} + \\beta _{\\mathit {rem}} + \\gamma _{\\mathit {rem}})$ time." ], [ "Computing MUPSs to be Added", "Next, we propose an algorithm to detect MUPSs to be added after a substitution.", "As in Section REF , we categorize MUPSs to be added into three types: A1) A MUPS of $T^{\\prime }$ that covers $i$ .", "A2) A MUPS of $T^{\\prime }$ that does not cover $i$ and is repeating in $T$ .", "A3) A MUPS of $T^{\\prime }$ that does not cover $i$ and is unique but not minimal in $T$ .", "Furthermore, we categorize MUPSs of Type A1 into two sub-types: A1-1) A MUPS of $T^{\\prime }$ that covers position $i$ in its arm.", "A1-2) A MUPS of $T^{\\prime }$ centered at editing position $i$ .", "See Fig.", "REF in Appendix  for illustration." ], [ "Type A1-1.", "A MUPS of Type A1-1 is a shrink of some maximal palindrome in $T^{\\prime }$ covering editing position $i$ in its arm.", "Further, such a maximal palindrome in $T^{\\prime }$ corresponds to some 1-mismatch maximal palindrome in $T$ , which covers $i$ as a mismatch position.", "Thus, we preprocess for arms of each 1-mismatch maximal palindrome in $T$ .", "For MUPSs of Type A1, we utilize the following observation: Observation 1 For any palindrome $v$ covering position $i$ in $T^{\\prime }$ , $v$ is unique in $T^{\\prime }$ if and only if $|\\mathit {inbeg}_{T^{\\prime }, i}(v)| = 1$ and $|\\mathit {xbeg}_{T^{\\prime }, i}(v)| = 0$ .", "In the preprocessing phase, we first consider sorting extended arms of 1-mismatch maximal palindromes in $T$ .", "Let $\\mathit {EA}$ be the multiset of strings consists of the extended right arms and the reverse of the extended left arms of all 1-mismatch maximal palindromes in $T$ .", "Note that each string in $\\mathit {EA}$ can be represented in constant space since it is a substring of $T$ or $T^R$ .", "Let $\\mathsf {MA}^{\\prime }$ be a lexicographically sorted array of all elements in $\\mathit {EA}$ .", "Here, the order between the same strings can be arbitrary.", "Also, for each string in $\\mathit {EA}$ , we consider a quadruple of the form $(\\mathit {par}, \\mathit {pos}, \\mathit {chr}, \\mathit {rnk})$ where $\\mathit {par} \\in \\lbrace \\mathsf {odd}, \\mathsf {even}\\rbrace $ represents the parity of the length of the corresponded 1-mismatch maximal palindrome, $\\mathit {pos}$ is the mismatched position on the opposite arm, $\\mathit {chr}$ is the mismatched character on the extended arm, and $\\mathit {rnk}$ is the rank of the extended arm in $\\mathsf {MA}^{\\prime }$ .", "Let $\\mathsf {MA}$ be a radix sorted array of these quadruples.", "It can be seen that for each triple $(p, i, s)$ of parity $p$ , mismatched position $i$ , and mismatched character $s$ , all elements corresponding to the triple are stored continuously in $\\mathsf {MA}$ .", "We denote by $\\mathsf {MA}_{p,i,s}$ the subarray of $\\mathsf {MA}$ consists of such elements.", "In other words, $\\mathsf {MA}_{p,i,s}$ is a sorted array of extended arms of maximal palindromes of parity $p$ covering position $i$ in $T^{\\prime }$ when the $i$ -th character of $T$ is substituted by $s$ .", "Now let us focus on odd-palindromes.", "Even-palindromes can be treated similarly.", "We construct the suffix tree of $T$ and make the loci of strings in $\\mathit {EA}$ explicit.", "We also make the loci of the extended right arm of every odd-palindrome in $T$ explicit.", "Simultaneously, we mark the nodes corresponding to the extended right arms and apply the preprocessing for the nearest marked ancestor (NMA) queries to the marked tree.", "We denote the tree by $\\mathcal {ST}_{\\mathit {odd}}$ .", "Next, we initialize the set $\\mathcal {A}_{A1,1} = \\emptyset $ .", "The set $\\mathcal {A}_{A1,1}$ will become an index of MUPSs of Type A1-1 when the preprocessing is finished.", "For each non-empty $\\mathsf {MA}_{\\mathsf {odd},i,s}$ and for each string $w$ in $\\mathsf {MA}_{\\mathsf {odd},i,s}$ , we do the followings: Let $x_w$ be the odd-palindrome whose extended right arm is $w$ when $T[i]$ is substituted by $s$ .", "Let $u$ and $v$ are the preceding and the succeeding string of $w$ in $\\mathsf {MA}_{\\mathsf {odd}, i, s}$ (if such palindromes do not exist, they are empty).", "Further let $\\ell _w = \\max \\lbrace \\mathit {lcp}(u, w), \\mathit {lcp}(w, v)\\rbrace $ .", "When $T[i]$ is substituted by $s$ , any shrink $y$ of $x_w$ such that $y$ covers position $i$ and the arm-length of $y$ is at least $\\ell _w$ has only one occurrence which covers position $i$ in $T^{\\prime }$ , i.e.,$|\\mathit {inbeg}_{T^{\\prime }, i}(y)| = 1$ .", "Next, we query the NMA for the node corresponding to $w$ on $\\mathcal {ST}_\\mathit {odd}$ .", "Let $\\ell ^{\\prime }_w$ be the length of the extended right arm obtained by the NMA query.", "When $T[i]$ is substituted by $s$ , any shrink $y^{\\prime }$ of $x_w$ such that the arm-length of $y^{\\prime }$ is at least $\\ell ^{\\prime }_w$ has no occurrences which do not cover position $i$ in $T^{\\prime }$ , i.e., $|\\mathit {xbeg}_{T^{\\prime }, i}(y^{\\prime })| = 0$ .", "Thus, by Observation REF , the shrink $y^\\star $ of $x_w$ of arm-length $\\max \\lbrace \\ell _w, \\ell ^{\\prime }_w\\rbrace $ is a MUPS of Type A1-1 for the query $\\mathit {sub}(i,s)$ , if such $y^\\star $ exists.", "In such a case, we store the information about $y^\\star $  (i.e., its center and radius) into $\\mathcal {A}_{A1,1}$ using $(\\mathsf {odd}, i, s)$ as the key.", "After finishing the above preprocessing for all strings in $\\mathsf {MA}$ , we sort all elements in $\\mathcal {A}_{A1,1}$ by their keys.", "Since each element in $\\mathit {EA}$ is a substring of $T\\$T^R\\#$ , they can be sorted in $O(n+|\\mathit {EA}|) = O(n)$ time by Corollary REF .", "Namely, $\\mathsf {MA}^{\\prime }$ can be computed in linear time, and thus $\\mathsf {MA}$ too.", "By Lemma REF , tree $\\mathcal {ST}_\\mathit {odd}$ can be constructed in $O(n)$ time.", "Also, we can answer each NMA query and LCP query in constant time after $O(n)$ time preprocessing.", "Hence, the total preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we compute all MUPSs of Type A1-1 by searching for elements in $\\mathcal {A}_{A1,1}$ with keys $(\\mathsf {odd}, i, s)$ and $(\\mathsf {even}, i, s)$ .", "An element with each of the keys can be found in $O(\\log \\min \\lbrace \\sigma ,\\log n\\rbrace )$ time.", "Thus, all MUPSs of Type A1-1 can be computed in $O(\\log \\min \\lbrace \\sigma ,\\log n\\rbrace + \\alpha ^{\\prime }_{\\mathit {add}})$ time where $\\alpha ^{\\prime }_{\\mathit {add}}$ is the number of MUPSs of Type A1-1.", "The MUPS of Type A1-2 is a shrink of the maximal palindrome in $T^{\\prime }$ centered at $i$ .", "By definition, there is at most one MUPS of Type A1-2.", "In the preprocessing phase, we again construct $\\mathsf {MA}$ and related data structures as in Type A1-1.", "Further, we apply the $O(n)$ -time preprocessing of Lemma REF for Problem REF .", "The total preprocessing time is $O(n)$ .", "Given substitution query $\\mathit {sub}(i,s)$ , we compute the MUPS centered at $i$ in $T^{\\prime }$ as follows (if it exists): It is clear that $T^{\\prime }[i..i] = s$ is the MUPS of Type A1-2 if $s$ is a unique character in $T^{\\prime }$ .", "In what follows, we consider the other case.", "Let $w$ be the maximal palindrome centered at $i$ in $T^{\\prime }$ .", "First, we compute the maximum lcp value $\\ell _w$ between $\\mathsf {Rarm}_w$ and extended arms in $\\mathsf {MA}_{\\mathsf {odd}, i, s}$ .", "Then, any shrink $y$ of $w$ such that the arm-length of $y$ is at least $\\ell _w$ has no occurrences which cover position $i$ in $T^{\\prime }$ , i.e., $|\\mathit {inbeg}_{T^{\\prime }, i}(y)| = 1$ .", "We can compute $\\ell _w$ in $O(\\log \\min \\lbrace \\sigma ,\\log n\\rbrace )$ time, combining LCE queries and binary search.", "Note that $\\mathsf {rarm}_w$ occurs at $i+1$ in both $T$ and $T^{\\prime }$ while $\\mathsf {Rarm}_w$ might be absent from $T$ .", "Next we compute the arm-length $\\ell ^{\\prime }_w$ of the shortest palindrome $v$ such that $\\mathit {center}(v) = i$ and $|\\mathit {xbeg}_{T^{\\prime }, i}(v)| = 0$ , i.e., $v$ is absent from $T$ .", "Since the shrink $\\tilde{v}$ of $w$ of arm-length $\\ell ^{\\prime }_w-1$ is the longest palindrome such that $\\mathit {center}(\\tilde{v}) = i$ and $\\tilde{v}$ occurs in $T$ , we can reduce the problem of computing $\\ell ^{\\prime }_w$ to Problem REF .", "Thus, we can compute $\\ell ^{\\prime }_w$ in $O(\\delta (n,\\sigma ) + (\\log \\log n)^2)$ time by Lemma REF .", "Similar to the case of Type A1-1, by Observation REF , the shrink $y^\\star $ of $x_w$ of arm-length $\\max \\lbrace \\ell _w, \\ell ^{\\prime }_w\\rbrace $ is a MUPS of Type A1-2, if such $y^\\star $ exists.", "Therefore, the MUPS of Type A1-2 can be computed in $O(\\delta (n,\\sigma ) + (\\log \\log n)^2)$ time." ], [ "Type A2.", "A MUPS of Type A2 occurs at least twice in $T$ , and there is only one occurrence not covering editing position $i$ .", "For a palindrome $w$ repeating in $T$ , let $T[b_{l1}.. e_{l1}]$ and $T[b_{l2}.. e_{l2}]$ be the leftmost and the second leftmost occurrence of $w$ in $T$ .", "Further let $T[b_{r1}.. e_{r1}]$ and $T[b_{r2}.. e_{r2}]$ be the rightmost and the second rightmost occurrence of $w$ in $T$ .", "We define interval $\\rho _w$ as the intersection of all occurrences of $w$ except for the leftmost one, i.e., $\\rho _w = \\lbrace k \\mid k \\notin [b_{l1}, e_{l1}] \\text{ and } k \\in [b_{r1}, e_{l2}]\\rbrace $  (see also Fig.", "REF in Appendix ).", "Similarly, we define interval $\\tilde{\\rho }_w$ as the intersection of all occurrences of $w$ except for the rightmost one.", "Note that $\\rho _w$ and $\\tilde{\\rho }_w$ can be empty.", "Then, $w$ is unique after the $i$ -th character is edited if and only if $i \\in \\rho _w \\cup \\tilde{\\rho }_w$ .", "Thus, any MUPS of Type A2 is a palindrome corresponding to some interval in $\\rho _w \\cup \\tilde{\\rho }_w$ stabbed by $i$ .", "To avoid accessing intervals that do not correspond to the MUPSs to be added, we decompose each $\\rho _w$ .", "It is easy to see that for any shrink $v$ of $w$ , $\\rho _v \\subset \\rho _w$ holds.", "Also, if $T[i]$ with $i \\in \\rho _{v}$ is edited, then both $w$ and $v$ become unique in $T^{\\prime }$ , i.e., $w$ cannot be a MUPS of $T^{\\prime }$ .", "For each unique palindrome $w$ in $T$ , we decompose $\\rho _w$ into at most three intervals $\\rho _w = \\rho ^1_w\\rho _{w^{\\prime }}\\rho ^2_w$ where $w^{\\prime } = w[2.. |w|-1]$ .", "Similarly, we decompose $\\tilde{\\rho }_w$ into $\\tilde{\\rho }_w = \\tilde{\\rho }^1_w\\tilde{\\rho }_{w^{\\prime }}\\tilde{\\rho }^2_w$ .", "Then, $w$ is a MUPS of Type A2 if and only if $i \\in \\rho ^1_w \\cup \\rho ^2_w \\cup \\tilde{\\rho }^1_w \\cup \\tilde{\\rho }^2_w$ .", "In the preprocessing phase, we first construct the eertree of $T$ and the suffix tree of $T$ enhanced with additional explicit nodes for all distinct palindromes in $T$ .", "Next, we compute at most four (leftmost, second leftmost, rightmost, second rightmost) occurrences of each palindrome in $T$ by traversing the enhanced suffix tree.", "At the same time, we compute $\\rho _w$ and $\\tilde{\\rho }_w$ for each palindrome in $w$ .", "Next, we sequentially access distinct palindromes by traversing $\\mathsf {EERTREE}(T)$ in a pre-order manner.", "Then, for each palindrome $w$ , we decompose $\\rho _w$ and $\\tilde{\\rho _w}$ based on the rules as mentioned above.", "Finally, we apply the preprocessing for stabbing queries to the $O(n)$ intervals obtained.", "The total preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we compute all intervals stabbed by position $i$ .", "The palindromes corresponding to the intervals are MUPSs of Type A2.", "Hence, the query time is $O(1 + \\beta _{\\mathit {add}})$ , where $\\beta _{\\mathit {add}}$ is the number of MUPSs of Type A2." ], [ "Type A3.", "A MUPS of Type A3 is unique but not minimal in $T$ .", "Such a unique palindrome $u$ in $T$ contains a MUPS $w \\ne u$ of $T$ as a shrink.", "Since $u$ is a MUPS of $T^{\\prime }$ , $w$ is repeating in $T^{\\prime }$ , i.e., $w$ is a removed MUPS of Type R2.", "Contrary, consider a MUPS $w$ of Type R2, which is repeating in $T^{\\prime }$ .", "Then, the shortest unique extension of $w$ in $T^{\\prime }$ is an added MUPS of Type A3, if it exists.", "The preprocessing for Type A3 is almost the same as for Type R2.", "We store a bit more information for Type A3 in addition to the information in $\\mathcal {A}_{R2}$ .", "In the preprocessing phase, we first apply the $O(n)$ -time preprocessing of Lemma REF for Problem REF .", "Next, we initialize the set $\\mathcal {A}_{A3} = \\emptyset $ .", "This set $\\mathcal {A}_{A3}$ will become an index of MUPSs of Type A3 when the preprocessing is finished.", "For each MUPS $w = T[b.. e]$ of $T$ , we process the followings: For the beginning position $j \\ne b$ of each occurrence of $\\mathsf {Larm}_w$ in $T$ , we compute the lcp value $\\ell _j$ between $T[j+|\\mathsf {Larm}_w|..|T|]\\$$ and $T[\\lceil c \\rceil ..|T|]\\$$ with allowing one mismatch where $c$ is the center of $w$ in $T$ .", "If $\\ell _j$ is smaller than $\\mathsf {rarm}_w$ , then we do nothing for this occurrence since $w$ cannot occur at $j$ after any single-character substitution.", "Otherwise, let $q = j+|\\mathsf {Larm}_w|-1+d$ be the first mismatched position in $T$ .", "When the $q$ -th character of $T$ is substituted by the character $\\mathsf {rarm}_w[d]$ , $w = T[b.. e]$ occurs at $j \\ne b$ , i.e., it is a MUPS of Type R2.", "Unlike for Type R2, we add the pair of MUPS and (1-mismatched) lcp value $(T[b.. e], \\ell _j)$ into $\\mathcal {A}_{A3}$ with the pair of index and character $(q, \\mathsf {rarm}_w[d])$ as the key.", "In addition, symmetrically, we update $\\mathcal {A}_{A3}$ for each occurrence of $\\mathsf {Rarm}_w$ in $T$ .", "After finishing the above processes for every MUPS of $T$ , we then sort the elements of $\\mathcal {A}_{A3}$ by radix sort on the keys.", "If there are multiple identical elements with the same key, we unify them into a single element.", "Also, if there are multiple elements with the same key, we store them in a linear list.", "By Lemma REF , the total number of occurrences of arms of MUPSs is $O(n)$ , and hence, the total preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we query Problem REF with the same pair $(i, s)$ as the input.", "Then, we complete checking whether there exists a MUPS of Type A3 centered at $i$ .", "For the remaining MUPSs of Type 3, we retrieve the MUPSs of Type A3 using the index $\\mathcal {A}_{A3}$ as in the query algorithm for Type R2.", "This can be done in $O(\\log \\min \\lbrace \\sigma , \\log n\\rbrace + \\gamma _{\\mathit {add}})$ time where $\\gamma _{\\mathit {add}}$ is the number of MUPSs of Type A3.", "Therefore, the total query time of Type A3 is $O(\\delta (n,\\sigma ) + (\\log \\log n)^2 + \\gamma _{\\mathit {add}})$ .", "To summarize, we obtain our main theorem: Theorem 3 After $O(n)$ -time preprocessing, we can compute the set of MUPSs after a single-character substitution in $O(\\delta (n, \\sigma ) + (\\log \\log n)^2 + d) \\subset O(\\log n)$ time where $d$ is the number of changes of MUPSs.", "With a little effort, we obtain the following (see Appendix  for the proof): Corollary 2 If $\\sigma \\in O(1)$ , after $O(n)$ -time preprocessing, we can compute the set of MUPSs after a single-character substitution in $O(d)$ time.", "We would like to thank Associate Professor Shunsuke Inenaga (Kyushu University) for the valuable discussions on simplifying our algorithms.", "Omitted Proofs In this appendix, we provide omitted proofs in Sections  and  that are omitted due to lack of space.", "First, we give the following corollary, which is immediately yielded by combining known results in [3], [11], [17]: Corollary 3 For a position $i$ , divide the set of palindromic suffixes of $T[1..i]$ into groups $G_1, G_2, \\ldots , G_{m_i}$ w.r.t their smallest periods.", "Then, $m_i \\in O(\\log i)$ holds.", "Also, for each group $G_k$ , the following properties hold: The difference between centers of any two palindromes in $G_k$ is an integer power of $0.5p_k$ , where $p_k$ is their smallest period.", "For all maximal palindromes $s_1, \\ldots , s_t$ in $T$ that are extensions of palindromes in $G_k$ , excluding at most one, their smallest period is also $p_k$ .", "Among $s_1, \\ldots , s_t$ , at most two palindromes can be longest, and the others are contained by one of them.", "We remark that symmetric arguments hold for palindromic prefixes as well.", "[Proof of Lemma REF ] For the sake of contradiction, we assume that there is a palindrome $w$ with $\\mathit {inbeg}_{T, i}(w) \\ne \\emptyset $ and $\\mathit {inbeg}_{T^{\\prime }, i}(w) \\ne \\emptyset $ .", "Let $c$  (resp.", "$c^{\\prime }$ ) be the center of an occurrence of $w$ in $T$  (resp.", "in $T^{\\prime }$ ) covering position $i$ .", "It is clear that $c \\ne c^{\\prime }$ since $T[i]$ is substituted by another character $T^{\\prime }[i]$ .", "Also, it suffices to consider when $c < c^{\\prime }$ from the symmetry of $T$ and $T^{\\prime }$ .", "Let $d$  (resp.", "$d^{\\prime }$ ) be the distance between $c$ and $i$  (resp.", "$c^{\\prime }$ and $i$ ), i.e., $d = |i-c|$ and $d^{\\prime } = |i-c^{\\prime }|$ .", "There are the following two cases: either (1) $i \\notin [c,c^{\\prime }]$ or (2) $i \\in [c,c^{\\prime }]$ .", "See also Fig.", "REF for illustration.", "(1) Now we consider the case when $c^{\\prime } < i$ .", "Another case ($i < c$ ) can be treated similarly.", "On the one hand, since $c$ and $c^{\\prime }$ are the centers of $w$ , $T^{\\prime }[c^{\\prime }+d] = T[c+d] = T[i]$ .", "Further, $T^{\\prime }[c^{\\prime }-d] = T[c^{\\prime }+d]$ by palindromic symmetry, and hence, $T^{\\prime }[c^{\\prime }-d] = T[i]$ .", "On the other hand, again, since $c$ and $c^{\\prime }$ are the centers of $w$ , $T[c-d^{\\prime }] = T^{\\prime }[c^{\\prime }-d^{\\prime }]$ .", "Further, $T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[c^{\\prime }+d^{\\prime }] = T^{\\prime }[i]$ by palindromic symmetry, and hence, $T[c-d^{\\prime }] = T^{\\prime }[i]$ .", "Also, $T^{\\prime }[c-d^{\\prime }] = T[c-d^{\\prime }] = T^{\\prime }[i]$ since $c-d^{\\prime } \\ne i$ .", "Since $c^{\\prime }-d = c-d^{\\prime }$ holds in this case, $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c-d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "(2) Similar to the first case, it can be seen that $T^{\\prime }[c^{\\prime }-d] = T[i]$ .", "If $d = d^{\\prime }$ , then $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "Hence $d \\ne d^{\\prime }$ holds, and thus, $T^{\\prime }[c+d^{\\prime }] = T[c+d^{\\prime }]$ .", "Also, $T[c+d^{\\prime }] = T^{\\prime }[c^{\\prime }+d^{\\prime }] = T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[i]$ holds.", "Finally, since $c^{\\prime }-d = c+d^{\\prime }$ holds in this case, $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c+d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "Figure: Illustration for the two cases of Lemma .Note that this illustration is for the sake of contradiction.", "[Proof of Lemma REF ] First, by minimality of palindromes in $\\mathcal {W}_i$ , centers of palindromes in $\\mathcal {W}_i$ are different from each other.", "Let $\\mathcal {W}_i^L \\subset \\mathcal {W}_i$  (resp.", "$\\mathcal {W}_i^R\\subset \\mathcal {W}_i$ ) be the set of palindromes in $\\mathcal {W}_i$ whose center is at most $i$  (resp.", "at least $i$ ).", "Let us consider the size of $\\mathcal {W}_i^L$ .", "Every palindrome in $\\mathcal {W}_i^L$ is an extension of some palindromic suffix of $T[1.. i]$ .", "From Corollary REF , the set of palindromic suffixes of $T[1.. i]$ can be divided into $m_i \\in O(\\log i)$ groups w.r.t.", "their smallest period.", "Let $G_1, G_2, \\ldots , G_{m_i}$ be such groups for palindromic suffixes of $T[1.. i]$ , and let $p_k$ be the smallest period corresponding to $G_k$ .", "Also, for each $k$ with $1 \\le k \\le m_i$ , let $E_k$ be the union set of all extensions of every palindrome in $G_k$ .", "Let $H_k = \\mathcal {W}_i^L \\cap E_k$ .", "Since $|\\mathcal {W}_i^L| = |\\bigcup _{k=1}^{m_i} H_k| = \\sum _{k=1}^{m_i}|H_k|$ and $m_i \\in O(\\log n)$ , it suffices to show that $|H_k|$ is at most a constant.", "For the sake of contradiction, we assume $|H_k|\\ge 4$ .", "From (REF ) of Corollary REF , at least three palindromes in $H_k$ has the same smallest period $p_k$ .", "Also, by (REF ) of Corollary REF , the difference between centers of any two palindromes in $H_k$ is a power of $0.5p_k$ .", "Therefore, at least two distinct palindromes in $H_k$ have a difference of power of $p_k$ in their center positions.", "Let $w_1$ , $w_2$ be such palindromes, and assume that $|w_1| \\le |w_2|$ w.l.o.g.. Then, the string between the centers of $w_1$ and $w_2$ can be represented by $x^j$ for positive integer $j$ and string $x$ of length $p_k$ .", "Since the smallest period of $w_1$ is $p_k$ , its extended right arm $\\mathsf {Rarm}_{w_1}$ can be written by $\\mathsf {Rarm}_{w_1} = x^{j_1}x^{\\prime }_1$ where $j_1$ is a non-negative integer and $x^{\\prime }_1$ is a proper prefix of $x$ .", "Similarly, the extended right arm $\\mathsf {Rarm}_{w_2}$ of $w_2$ can be written by $\\mathsf {Rarm}_{w_2} = x^{j_2}x^{\\prime }_2$ where $j_2$ is a non-negative integer and $x^{\\prime }_2$ is a proper prefix of $x$ .", "See also Fig.", "REF for illustration.", "If $|w_1| = |w_2|$ , then this leads $j_1 = j_2$ and $x^{\\prime }_1 = x^{\\prime }_2$ , i.e., $w_1 = w_2$ , a contradiction.", "If $|w_1| < |w_2|$ , then $j_1 < j_2$ or $j_1 = j_2$ and $|x^{\\prime }_1| < |x^{\\prime }_2|$ .", "In both cases, $\\mathsf {Rarm}_{w_1}$ is a proper prefix of $\\mathsf {Rarm}_{w_2}$ , i.e, $w_1$ is a shrink of $w_2$ .", "This contradicts the minimality of $w_2$ .", "Thus $|H_k| \\le 3$ holds, and hence, we obtain $|\\mathcal {W}_i^L| = \\sum _{k=1}^{m_i}|H_k| \\le 3m_i \\in O(\\log n)$ .", "Similarly, the size of $\\mathcal {W}_i^R$ is also $O(\\log n)$ .", "Therefore, $|\\mathcal {W}_i| \\in O(\\log n)$ .", "Figure: Example for Lemma , wherew 1 w_1 and w 2 w_2 have the same smallest period 5 and the difference between centers of them is a power of 5.Here x=𝚊𝚊𝚋𝚊𝚊x=\\mathtt {aabaa}, x 1 ' =𝚊𝚊𝚋𝚊x^{\\prime }_1=\\mathtt {aaba}, and x 2 ' =𝚊𝚊𝚋x^{\\prime }_2=\\mathtt {aab}.", "[Proof of Lemma REF ] By symmetry, it suffices to show that the number of MUPSs covering $i$ and centered before $i$ is $O(\\log n)$ .", "Each of such MUPSs is an extension of some palindromic suffix of $T[1..i]$ .", "Thus, similar to the proof of Lemma REF , we consider dividing the set of palindromic suffixes of $T[1..i]$ into $m_i \\in O(\\log i)$ groups, $G_1, G_2, \\ldots , G_{m_i}$ w.r.t.", "their smallest periods.", "In the following, we consider MUPSs that are extensions of palindromes in an arbitrary group $G_k$ .", "We show that the number of such MUPSs is at most two by contradiction.", "We assume the contrary, i.e., there are three MUPSs that are extensions of palindromes in $G_k$ .", "By (REF ) of Corollary REF , at least one of the three MUPSs is a substring of an extension of a palindrome in $G_k$ with a different center.", "This contradicts that a MUPS cannot be a substring of another palindrome with a different center.", "Thus, the number of MUPSs that are extensions of palindromes in $G_k$ is at most two, and we finish the proof.", "[Proof of Lemma REF ] In the preprocessing, we construct $\\mathsf {EERTREE}(T)$ and apply the preprocessing for the path-tree LCE queries to the odd-tree $\\mathcal {T}_{\\mathsf {odd}}$ of $\\mathsf {EERTREE}(T)$ .", "Also, we mark the nodes in $\\mathsf {EERTREE}(T)$ that corresponds to MUPSs of $T$ and apply the preprocessing for the nearest marked ancestor (NMA) queries.", "The preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we query the path-tree LCE between path $T[i] \\rightsquigarrow \\mathsf {larm}_wT[i]\\mathsf {rarm}_w$ and tree rooted at $s$ on $\\mathcal {T}_{\\mathsf {odd}}$ where $w$ is the maximal palindrome in $T^{\\prime }$ centered at $i$ .", "Let $\\ell _w$ be the depth of the LCE nodes.", "Then the shrink $v$ of $w$ with $|\\mathsf {Rarm}_v| = \\ell _w$ occurs in $T$ .", "Also, $v$ is the longest since any other shrink $u$ of $w$ longer than $v$ does not occur in $T$ .", "Further, we can determine whether $v$ is unique in $T$ or not by checking the existence of a mark on path $s \\rightsquigarrow v$ .", "It can be done by querying NMA, and the MUPS of $T$ contained in $v$ can be computed simultaneously, if $v$ is unique in $T$ .", "We can compute the value $\\ell _w$ in $\\delta (n,\\sigma )$ time for searching for the node $s$ in $\\mathcal {T}_{\\mathsf {odd}}$ , plus $O((\\log \\log n)^2)$ time for the path-tree LCE query.", "Alternative Algorithm for Problem  REF In this appendix, we present an alternative algorithm for solving Problem REF .", "The query time of Theorem REF is dominated by the time to answer Problem REF .", "Here, we introduce another solution for Problem REF utilizing nearest colored ancestor queries instead of path-tree LCE queries.", "NCA Queries.", "A nearest colored ancestor query (NCA query) on a tree $\\mathcal {T}$ with colored nodes is, given a query node $v$ and a color $C$ , to compute the nearest ancestor $u$ of $v$ such that the color of $u$ is $C$ .", "Noticing that the notion of NCA is a generalization of well-known nearest marked ancestor.", "For NCA queries, we will use the following known results: Lemma 7 ([13]) Given a tree $\\mathcal {T}$ with colored nodes, a data structure of size $O(N)$ can be constructed in deterministic $O(N\\log \\log N)$ time or expected $O(N)$ time to answer any NCA query in $O(\\log \\log N)$ time, where $N$ is the number of nodes of $\\mathcal {T}$ .", "Lemma 8 ([5], [6]) If the number of colors is $O(\\log N)$ , a data structure of size $O(N)$ can be constructed in $O(N)$ time to answer any NCA query in $O(1)$ time.", "Preprocessing for Problem REF .", "We first construct the suffix tree of $T\\$$ .", "Also, for each odd-palindrome in $T$ , we make the locus of the right arm explicit and label the node with the pair of the center character and the binary flag that indicates if the palindrome is a MUPS.", "We regard the pair as the color of the node.", "Furthermore, we apply a preprocessing for NCA queries to the colored treeThere can be a node with multiple colors in the tree.", "However, we can easily avoid such a situation by copying a node with $k$ colors to $k$ nodes.", "Also, in the case of Problem REF , the cumulative total number of colored nodes is $O(n)$ ..", "The preprocessing time is $O(n+c_{\\mathsf {nca}}(n,\\sigma ))$ , where $c_{\\mathsf {nca}}(n,\\sigma )$ is the preprocessing time for NCA queries.", "Query for Problem REF .", "Given a substitution query $\\mathit {sub}(i,s)$ , we start at the node corresponding to $\\mathsf {rarm}_w$ where $w$ is the maximal palindrome in $T$ centered at $i$ .", "We then compute the nearest ancestor $V$ colored with $(s, \\mathsf {0})$ by using NCA query.", "If such node $V$ exists, palindrome $P = \\mathit {str}(V)^R\\cdot s\\cdot \\mathit {str}(V)$ is the answer of the former part of Problem REF where $\\mathit {str}(V)$ denotes the string corresponding to $V$ in the enhanced suffix tree of $T\\$$ .", "Also, we query NCA $(s, \\mathsf {1})$ from $V$ .", "We can determine if $P$ is unique, and if it is unique, we can find the MUPS contained in $P$ .", "The query time is $O(q_{\\mathsf {nca}}(n,\\sigma ))$ where $q_{\\mathsf {nca}}(n,\\sigma )$ is the query time for NCA.", "Let $s_{\\mathsf {nca}}(n,\\sigma )$ denote the space for the NCA data structure.", "We obtain the following theorem: Theorem 4 After $O(n + c_{\\mathsf {nca}}(n,\\sigma ))$ -time and $O(n + s_{\\mathsf {nca}}(n,\\sigma ))$ -space preprocessing, we can compute the set of MUPSs after a single-character substitution in $O(q_{\\mathsf {nca}}(n,\\sigma ) + \\log \\min \\lbrace \\sigma ,\\log n\\rbrace + d)$ time.", "The results for NCA queries in Lemmas REF and REF can be plugged into the functions $c_\\mathsf {nca}$ , $q_\\mathsf {nca}$ , and $s_\\mathsf {nca}$ .", "In addition, even when a general case, we can handle $\\delta (n,\\sigma )$ as a constant by utilizing a perfect hashing [9] after $O(n\\log \\log n)$ -time or $O(n)$ -expected time preprocessing.", "Table REF lists different representations of the time/space complexities of Theorems REF and REF .", "We emphasize that our algorithm runs in optimal $O(d)$ time when $\\sigma $ is constant (i.e., Corollary REF holds).", "Table: Concrete complexities of our algorithms for the problem of computing MUPSs after a single-character substitution.All the above results require only linear space.Each query time is O(logn)O(\\log n) since logσ∈O(logn)\\log \\sigma \\in O(\\log n) and d∈O(logn)d \\in O(\\log n).", "Figures In this appendix, we present supplementary figures.", "Figure: Illustration for three types of MUPSs to be removed.The red arrows represent MUPSs.w 1 w_1, w 2 w_2, and w 3 w_3 are MUPSs of Type R1, Type R2, and Type R3 in TT, respectively.Also, vv is the MUPS of T ' T^{\\prime } that is a shrink of w 3 w_3.It is not unique in TT, but is unique in T ' T^{\\prime }.Figure: Example for observations about Type R2,where T=𝚊𝚊𝚋𝚊𝚊𝚌𝚊𝚊𝚋𝚊𝚌𝚊𝚊𝚋𝚋𝚊𝚊𝚊𝚋𝚌𝚋𝚌T = \\mathtt {aabaacaabacaabbaaabcbc} and w=𝚊𝚊𝚋𝚊𝚊w = \\mathtt {aabaa}.𝖫𝖺𝗋𝗆 w =𝚊𝚊𝚋\\mathsf {Larm}_w = \\mathtt {aab} occurs at position 7, 12, and 17 excluding the occurrence of ww.Since the Hamming distance between T[10..11]T[10..11] and w[4..5]w[4..5] equals 1, ww occurs at position 7 when T[11]T[11] is substituted by 𝚊\\mathtt {a}.Also, ww occurs at position 12 when T[15]T[15] is substituted by 𝚊\\mathtt {a}.Conversely, ww cannot occur at 17 after any single-character substitutionsince the Hamming distance between T[20..21]T[20..21] and w[4..5]w[4..5] equals 2.Figure: Illustration for ρ w \\rho _w of Type R3.The top arrow (resp.", "the middle arrow) represents the leftmost (resp.", "rightmost) occurrence of vv except for T[b+1..e-1]T[b+1..e-1].Also, the bottom arrow represents T[b+1..e-1]T[b+1..e-1].In this case, ρ w =[b r1 ,b]\\rho _w = [b_{r1},b].Figure: Illustration for four types of MUPSs to be added.The red arrows represent MUPSs.w 11 w_{1 1}, w 12 w_{1 2}, w 2 w_2, and w 3 w_3 are MUPSs of Type A1-1, Type A1-2, Type A2, and Type A3 in T ' T^{\\prime }, respectively.Also, vv is the MUPS of TT that is a shrink of w 3 w_3.It is unique in TT, but is not unique in T ' T^{\\prime }.Figure: Illustration for ρ w \\rho _w of Type A2.The top two arrows represent the leftmost and the second leftmost occurrence of ww.Also, the bottom two arrows represent the second rightmost and the rightmost occurrence of ww.In this case, ρ w =[b r1 ,e l2 ]\\rho _w = [b_{r1},e_{l2}]." ], [ "Omitted Proofs", "In this appendix, we provide omitted proofs in Sections  and  that are omitted due to lack of space.", "First, we give the following corollary, which is immediately yielded by combining known results in [3], [11], [17]: Corollary 3 For a position $i$ , divide the set of palindromic suffixes of $T[1..i]$ into groups $G_1, G_2, \\ldots , G_{m_i}$ w.r.t their smallest periods.", "Then, $m_i \\in O(\\log i)$ holds.", "Also, for each group $G_k$ , the following properties hold: The difference between centers of any two palindromes in $G_k$ is an integer power of $0.5p_k$ , where $p_k$ is their smallest period.", "For all maximal palindromes $s_1, \\ldots , s_t$ in $T$ that are extensions of palindromes in $G_k$ , excluding at most one, their smallest period is also $p_k$ .", "Among $s_1, \\ldots , s_t$ , at most two palindromes can be longest, and the others are contained by one of them.", "We remark that symmetric arguments hold for palindromic prefixes as well.", "[Proof of Lemma REF ] For the sake of contradiction, we assume that there is a palindrome $w$ with $\\mathit {inbeg}_{T, i}(w) \\ne \\emptyset $ and $\\mathit {inbeg}_{T^{\\prime }, i}(w) \\ne \\emptyset $ .", "Let $c$  (resp.", "$c^{\\prime }$ ) be the center of an occurrence of $w$ in $T$  (resp.", "in $T^{\\prime }$ ) covering position $i$ .", "It is clear that $c \\ne c^{\\prime }$ since $T[i]$ is substituted by another character $T^{\\prime }[i]$ .", "Also, it suffices to consider when $c < c^{\\prime }$ from the symmetry of $T$ and $T^{\\prime }$ .", "Let $d$  (resp.", "$d^{\\prime }$ ) be the distance between $c$ and $i$  (resp.", "$c^{\\prime }$ and $i$ ), i.e., $d = |i-c|$ and $d^{\\prime } = |i-c^{\\prime }|$ .", "There are the following two cases: either (1) $i \\notin [c,c^{\\prime }]$ or (2) $i \\in [c,c^{\\prime }]$ .", "See also Fig.", "REF for illustration.", "(1) Now we consider the case when $c^{\\prime } < i$ .", "Another case ($i < c$ ) can be treated similarly.", "On the one hand, since $c$ and $c^{\\prime }$ are the centers of $w$ , $T^{\\prime }[c^{\\prime }+d] = T[c+d] = T[i]$ .", "Further, $T^{\\prime }[c^{\\prime }-d] = T[c^{\\prime }+d]$ by palindromic symmetry, and hence, $T^{\\prime }[c^{\\prime }-d] = T[i]$ .", "On the other hand, again, since $c$ and $c^{\\prime }$ are the centers of $w$ , $T[c-d^{\\prime }] = T^{\\prime }[c^{\\prime }-d^{\\prime }]$ .", "Further, $T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[c^{\\prime }+d^{\\prime }] = T^{\\prime }[i]$ by palindromic symmetry, and hence, $T[c-d^{\\prime }] = T^{\\prime }[i]$ .", "Also, $T^{\\prime }[c-d^{\\prime }] = T[c-d^{\\prime }] = T^{\\prime }[i]$ since $c-d^{\\prime } \\ne i$ .", "Since $c^{\\prime }-d = c-d^{\\prime }$ holds in this case, $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c-d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "(2) Similar to the first case, it can be seen that $T^{\\prime }[c^{\\prime }-d] = T[i]$ .", "If $d = d^{\\prime }$ , then $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "Hence $d \\ne d^{\\prime }$ holds, and thus, $T^{\\prime }[c+d^{\\prime }] = T[c+d^{\\prime }]$ .", "Also, $T[c+d^{\\prime }] = T^{\\prime }[c^{\\prime }+d^{\\prime }] = T^{\\prime }[c^{\\prime }-d^{\\prime }] = T^{\\prime }[i]$ holds.", "Finally, since $c^{\\prime }-d = c+d^{\\prime }$ holds in this case, $T[i] = T^{\\prime }[c^{\\prime }-d] = T^{\\prime }[c+d^{\\prime }] = T^{\\prime }[i]$ , a contradiction.", "Figure: Illustration for the two cases of Lemma .Note that this illustration is for the sake of contradiction.", "[Proof of Lemma REF ] First, by minimality of palindromes in $\\mathcal {W}_i$ , centers of palindromes in $\\mathcal {W}_i$ are different from each other.", "Let $\\mathcal {W}_i^L \\subset \\mathcal {W}_i$  (resp.", "$\\mathcal {W}_i^R\\subset \\mathcal {W}_i$ ) be the set of palindromes in $\\mathcal {W}_i$ whose center is at most $i$  (resp.", "at least $i$ ).", "Let us consider the size of $\\mathcal {W}_i^L$ .", "Every palindrome in $\\mathcal {W}_i^L$ is an extension of some palindromic suffix of $T[1.. i]$ .", "From Corollary REF , the set of palindromic suffixes of $T[1.. i]$ can be divided into $m_i \\in O(\\log i)$ groups w.r.t.", "their smallest period.", "Let $G_1, G_2, \\ldots , G_{m_i}$ be such groups for palindromic suffixes of $T[1.. i]$ , and let $p_k$ be the smallest period corresponding to $G_k$ .", "Also, for each $k$ with $1 \\le k \\le m_i$ , let $E_k$ be the union set of all extensions of every palindrome in $G_k$ .", "Let $H_k = \\mathcal {W}_i^L \\cap E_k$ .", "Since $|\\mathcal {W}_i^L| = |\\bigcup _{k=1}^{m_i} H_k| = \\sum _{k=1}^{m_i}|H_k|$ and $m_i \\in O(\\log n)$ , it suffices to show that $|H_k|$ is at most a constant.", "For the sake of contradiction, we assume $|H_k|\\ge 4$ .", "From (REF ) of Corollary REF , at least three palindromes in $H_k$ has the same smallest period $p_k$ .", "Also, by (REF ) of Corollary REF , the difference between centers of any two palindromes in $H_k$ is a power of $0.5p_k$ .", "Therefore, at least two distinct palindromes in $H_k$ have a difference of power of $p_k$ in their center positions.", "Let $w_1$ , $w_2$ be such palindromes, and assume that $|w_1| \\le |w_2|$ w.l.o.g.. Then, the string between the centers of $w_1$ and $w_2$ can be represented by $x^j$ for positive integer $j$ and string $x$ of length $p_k$ .", "Since the smallest period of $w_1$ is $p_k$ , its extended right arm $\\mathsf {Rarm}_{w_1}$ can be written by $\\mathsf {Rarm}_{w_1} = x^{j_1}x^{\\prime }_1$ where $j_1$ is a non-negative integer and $x^{\\prime }_1$ is a proper prefix of $x$ .", "Similarly, the extended right arm $\\mathsf {Rarm}_{w_2}$ of $w_2$ can be written by $\\mathsf {Rarm}_{w_2} = x^{j_2}x^{\\prime }_2$ where $j_2$ is a non-negative integer and $x^{\\prime }_2$ is a proper prefix of $x$ .", "See also Fig.", "REF for illustration.", "If $|w_1| = |w_2|$ , then this leads $j_1 = j_2$ and $x^{\\prime }_1 = x^{\\prime }_2$ , i.e., $w_1 = w_2$ , a contradiction.", "If $|w_1| < |w_2|$ , then $j_1 < j_2$ or $j_1 = j_2$ and $|x^{\\prime }_1| < |x^{\\prime }_2|$ .", "In both cases, $\\mathsf {Rarm}_{w_1}$ is a proper prefix of $\\mathsf {Rarm}_{w_2}$ , i.e, $w_1$ is a shrink of $w_2$ .", "This contradicts the minimality of $w_2$ .", "Thus $|H_k| \\le 3$ holds, and hence, we obtain $|\\mathcal {W}_i^L| = \\sum _{k=1}^{m_i}|H_k| \\le 3m_i \\in O(\\log n)$ .", "Similarly, the size of $\\mathcal {W}_i^R$ is also $O(\\log n)$ .", "Therefore, $|\\mathcal {W}_i| \\in O(\\log n)$ .", "Figure: Example for Lemma , wherew 1 w_1 and w 2 w_2 have the same smallest period 5 and the difference between centers of them is a power of 5.Here x=𝚊𝚊𝚋𝚊𝚊x=\\mathtt {aabaa}, x 1 ' =𝚊𝚊𝚋𝚊x^{\\prime }_1=\\mathtt {aaba}, and x 2 ' =𝚊𝚊𝚋x^{\\prime }_2=\\mathtt {aab}.", "[Proof of Lemma REF ] By symmetry, it suffices to show that the number of MUPSs covering $i$ and centered before $i$ is $O(\\log n)$ .", "Each of such MUPSs is an extension of some palindromic suffix of $T[1..i]$ .", "Thus, similar to the proof of Lemma REF , we consider dividing the set of palindromic suffixes of $T[1..i]$ into $m_i \\in O(\\log i)$ groups, $G_1, G_2, \\ldots , G_{m_i}$ w.r.t.", "their smallest periods.", "In the following, we consider MUPSs that are extensions of palindromes in an arbitrary group $G_k$ .", "We show that the number of such MUPSs is at most two by contradiction.", "We assume the contrary, i.e., there are three MUPSs that are extensions of palindromes in $G_k$ .", "By (REF ) of Corollary REF , at least one of the three MUPSs is a substring of an extension of a palindrome in $G_k$ with a different center.", "This contradicts that a MUPS cannot be a substring of another palindrome with a different center.", "Thus, the number of MUPSs that are extensions of palindromes in $G_k$ is at most two, and we finish the proof.", "[Proof of Lemma REF ] In the preprocessing, we construct $\\mathsf {EERTREE}(T)$ and apply the preprocessing for the path-tree LCE queries to the odd-tree $\\mathcal {T}_{\\mathsf {odd}}$ of $\\mathsf {EERTREE}(T)$ .", "Also, we mark the nodes in $\\mathsf {EERTREE}(T)$ that corresponds to MUPSs of $T$ and apply the preprocessing for the nearest marked ancestor (NMA) queries.", "The preprocessing time is $O(n)$ .", "Given a query $\\mathit {sub}(i, s)$ , we query the path-tree LCE between path $T[i] \\rightsquigarrow \\mathsf {larm}_wT[i]\\mathsf {rarm}_w$ and tree rooted at $s$ on $\\mathcal {T}_{\\mathsf {odd}}$ where $w$ is the maximal palindrome in $T^{\\prime }$ centered at $i$ .", "Let $\\ell _w$ be the depth of the LCE nodes.", "Then the shrink $v$ of $w$ with $|\\mathsf {Rarm}_v| = \\ell _w$ occurs in $T$ .", "Also, $v$ is the longest since any other shrink $u$ of $w$ longer than $v$ does not occur in $T$ .", "Further, we can determine whether $v$ is unique in $T$ or not by checking the existence of a mark on path $s \\rightsquigarrow v$ .", "It can be done by querying NMA, and the MUPS of $T$ contained in $v$ can be computed simultaneously, if $v$ is unique in $T$ .", "We can compute the value $\\ell _w$ in $\\delta (n,\\sigma )$ time for searching for the node $s$ in $\\mathcal {T}_{\\mathsf {odd}}$ , plus $O((\\log \\log n)^2)$ time for the path-tree LCE query." ], [ "Alternative Algorithm for Problem ", "In this appendix, we present an alternative algorithm for solving Problem REF .", "The query time of Theorem REF is dominated by the time to answer Problem REF .", "Here, we introduce another solution for Problem REF utilizing nearest colored ancestor queries instead of path-tree LCE queries." ], [ "NCA Queries.", "A nearest colored ancestor query (NCA query) on a tree $\\mathcal {T}$ with colored nodes is, given a query node $v$ and a color $C$ , to compute the nearest ancestor $u$ of $v$ such that the color of $u$ is $C$ .", "Noticing that the notion of NCA is a generalization of well-known nearest marked ancestor.", "For NCA queries, we will use the following known results: Lemma 7 ([13]) Given a tree $\\mathcal {T}$ with colored nodes, a data structure of size $O(N)$ can be constructed in deterministic $O(N\\log \\log N)$ time or expected $O(N)$ time to answer any NCA query in $O(\\log \\log N)$ time, where $N$ is the number of nodes of $\\mathcal {T}$ .", "Lemma 8 ([5], [6]) If the number of colors is $O(\\log N)$ , a data structure of size $O(N)$ can be constructed in $O(N)$ time to answer any NCA query in $O(1)$ time.", "We first construct the suffix tree of $T\\$$ .", "Also, for each odd-palindrome in $T$ , we make the locus of the right arm explicit and label the node with the pair of the center character and the binary flag that indicates if the palindrome is a MUPS.", "We regard the pair as the color of the node.", "Furthermore, we apply a preprocessing for NCA queries to the colored treeThere can be a node with multiple colors in the tree.", "However, we can easily avoid such a situation by copying a node with $k$ colors to $k$ nodes.", "Also, in the case of Problem REF , the cumulative total number of colored nodes is $O(n)$ ..", "The preprocessing time is $O(n+c_{\\mathsf {nca}}(n,\\sigma ))$ , where $c_{\\mathsf {nca}}(n,\\sigma )$ is the preprocessing time for NCA queries.", "Given a substitution query $\\mathit {sub}(i,s)$ , we start at the node corresponding to $\\mathsf {rarm}_w$ where $w$ is the maximal palindrome in $T$ centered at $i$ .", "We then compute the nearest ancestor $V$ colored with $(s, \\mathsf {0})$ by using NCA query.", "If such node $V$ exists, palindrome $P = \\mathit {str}(V)^R\\cdot s\\cdot \\mathit {str}(V)$ is the answer of the former part of Problem REF where $\\mathit {str}(V)$ denotes the string corresponding to $V$ in the enhanced suffix tree of $T\\$$ .", "Also, we query NCA $(s, \\mathsf {1})$ from $V$ .", "We can determine if $P$ is unique, and if it is unique, we can find the MUPS contained in $P$ .", "The query time is $O(q_{\\mathsf {nca}}(n,\\sigma ))$ where $q_{\\mathsf {nca}}(n,\\sigma )$ is the query time for NCA.", "Let $s_{\\mathsf {nca}}(n,\\sigma )$ denote the space for the NCA data structure.", "We obtain the following theorem: Theorem 4 After $O(n + c_{\\mathsf {nca}}(n,\\sigma ))$ -time and $O(n + s_{\\mathsf {nca}}(n,\\sigma ))$ -space preprocessing, we can compute the set of MUPSs after a single-character substitution in $O(q_{\\mathsf {nca}}(n,\\sigma ) + \\log \\min \\lbrace \\sigma ,\\log n\\rbrace + d)$ time.", "The results for NCA queries in Lemmas REF and REF can be plugged into the functions $c_\\mathsf {nca}$ , $q_\\mathsf {nca}$ , and $s_\\mathsf {nca}$ .", "In addition, even when a general case, we can handle $\\delta (n,\\sigma )$ as a constant by utilizing a perfect hashing [9] after $O(n\\log \\log n)$ -time or $O(n)$ -expected time preprocessing.", "Table REF lists different representations of the time/space complexities of Theorems REF and REF .", "We emphasize that our algorithm runs in optimal $O(d)$ time when $\\sigma $ is constant (i.e., Corollary REF holds).", "Table: Concrete complexities of our algorithms for the problem of computing MUPSs after a single-character substitution.All the above results require only linear space.Each query time is O(logn)O(\\log n) since logσ∈O(logn)\\log \\sigma \\in O(\\log n) and d∈O(logn)d \\in O(\\log n)." ], [ "Figures", "In this appendix, we present supplementary figures.", "Figure: Illustration for three types of MUPSs to be removed.The red arrows represent MUPSs.w 1 w_1, w 2 w_2, and w 3 w_3 are MUPSs of Type R1, Type R2, and Type R3 in TT, respectively.Also, vv is the MUPS of T ' T^{\\prime } that is a shrink of w 3 w_3.It is not unique in TT, but is unique in T ' T^{\\prime }.Figure: Example for observations about Type R2,where T=𝚊𝚊𝚋𝚊𝚊𝚌𝚊𝚊𝚋𝚊𝚌𝚊𝚊𝚋𝚋𝚊𝚊𝚊𝚋𝚌𝚋𝚌T = \\mathtt {aabaacaabacaabbaaabcbc} and w=𝚊𝚊𝚋𝚊𝚊w = \\mathtt {aabaa}.𝖫𝖺𝗋𝗆 w =𝚊𝚊𝚋\\mathsf {Larm}_w = \\mathtt {aab} occurs at position 7, 12, and 17 excluding the occurrence of ww.Since the Hamming distance between T[10..11]T[10..11] and w[4..5]w[4..5] equals 1, ww occurs at position 7 when T[11]T[11] is substituted by 𝚊\\mathtt {a}.Also, ww occurs at position 12 when T[15]T[15] is substituted by 𝚊\\mathtt {a}.Conversely, ww cannot occur at 17 after any single-character substitutionsince the Hamming distance between T[20..21]T[20..21] and w[4..5]w[4..5] equals 2.Figure: Illustration for ρ w \\rho _w of Type R3.The top arrow (resp.", "the middle arrow) represents the leftmost (resp.", "rightmost) occurrence of vv except for T[b+1..e-1]T[b+1..e-1].Also, the bottom arrow represents T[b+1..e-1]T[b+1..e-1].In this case, ρ w =[b r1 ,b]\\rho _w = [b_{r1},b].Figure: Illustration for four types of MUPSs to be added.The red arrows represent MUPSs.w 11 w_{1 1}, w 12 w_{1 2}, w 2 w_2, and w 3 w_3 are MUPSs of Type A1-1, Type A1-2, Type A2, and Type A3 in T ' T^{\\prime }, respectively.Also, vv is the MUPS of TT that is a shrink of w 3 w_3.It is unique in TT, but is not unique in T ' T^{\\prime }.Figure: Illustration for ρ w \\rho _w of Type A2.The top two arrows represent the leftmost and the second leftmost occurrence of ww.Also, the bottom two arrows represent the second rightmost and the rightmost occurrence of ww.In this case, ρ w =[b r1 ,e l2 ]\\rho _w = [b_{r1},e_{l2}]." ] ]
2105.11693
[ [ "Guaranteed a posteriori local error estimation for finite element\n solutions of boundary value problems" ], [ "Abstract This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method.", "Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh.", "Such a result is applicable to problems without $H^2$-regularity, since it only utilizes the first order derivative of the solution.", "The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes." ], [ "Introduction", "This paper studies the finite element solution of the boundary value problem (BVP) of Poisson's equation, and proposes an a posteriori local error estimation method for finite element solution, based on the Hypercircle method, i.e., the Prager-Synge theorem.", "The motivation of this research originates from the error analysis of the four-probe method, which has been used for the resistivity measurement of semiconductors over the past century [7].", "The image of the four-probe method is illustrated in Figure REF : four probes $A, B, C$ , and $D$ are aligned on the surface of the sample; a constant current $I_{AD}$ is applied between $A$ and $D$ and the potential difference $V_{BC}$ between $B$ and $C$ is measured.", "The resistivity$\\rho $ is then calculated by $\\rho =F_c V_{BC}/I_{AD}$ , where $F_c$ is the correction factor.", "As an important quantity for high-precision measurement, $F_c$ is evaluated theoretically by considering the governing equation of the distribution of the potential.", "A well-used model for the potential distribution $u$ is described by the following boundary value problem of Poisson's equation (see, e.g., [7], [21]): $-\\Delta u = 2 \\rho \\: I_{AD} \\: (\\delta (A;x) - \\delta (D;x)) \\mbox{ in } \\Omega ;\\quad \\frac{\\partial u}{\\partial \\mathbf {n}}=0 \\mbox{ on }\\partial \\Omega \\:,$ where $\\delta (A;x)$ and $\\delta (D;x)$ are Dirac's delta functions located at $A$ and $D$ , respectively.", "Note that this model regards the current $I_{AD}$ as a point charge on the surface of the sample.", "By setting $\\rho \\: I_{AD}=1$ , the value of $F_c$ can be evaluated by $F_c = \\frac{1}{u(B) - u(C)}\\:.$ The calculation of $F_c$ only utilizes the potential $u$ at the probes $B$ and $C$ , i.e., the local information of the solution around the probes.", "To have a sharp estimation of the correction factor $F_c$ , we focus on the local error around the probes and develop the local error estimation for the FEM approximation to $u$ .", "In this study, a model problem $-\\Delta u=f$ with $f\\in L^2(\\Omega )$ is considered instead of the equation (REF ), the right-hand side of which does not belong to the $L^2$ space.", "The technique to solve equation (REF ) directly will be discussed in our subsequent papers.", "Figure: The four-probe methodThere is limited literature available on local error estimation for finite element solutions (see, e.g.,[5], [3], [4], [17], [1]).", "The local error estimation was first studied by Nitche and Schatz in [17].", "In the studies of [17], [20], for the subdomain $\\Omega _0$ of interest, an intermediate subdomain $\\Omega _1$ such that $\\Omega _0 \\subset \\subset \\Omega _1 \\subset \\subset \\Omega $ is utilized to deduce the following error estimation: for $u \\in H^l(\\Omega )$ , $\\Vert u-u_h \\Vert _{s,\\Omega _0} \\le C(h^{l-s} \\Vert u\\Vert _{l,\\Omega _1} + \\Vert u-u_h\\Vert _{-p,\\Omega _1})\\:,$ for $s = 0$ or 1 and $p$ as a fixed integer.", "In [5], [3], [4], Demlow relaxes the condition of [17] on the mesh and subdomain, and considers the a priori and a posteriori local error estimations.", "The existing results on local error estimation mainly focus on the qualitative analysis (convergence rate etc.)", "of local error terms, while the explicit bound for the local error estimation is not available.", "In this paper, we propose a quantitative error estimation method for the local error of the finite element solutions.", "Such a method is regarded as an extension of the explicit error estimation theorem developed by Liu [13], which inherits the idea of Kikuchi [11] to utilize the hypercircle method.", "The idea of local error estimation was also introduced in a concise manner in our previous work [15] published in Japanese.", "Here, thorough discussions along with detailed numerical examples are provided to describe this newly developed local error estimation method.", "The application of the hypercircle method to the a posteriori error estimation can also be found in [2], [16].", "Instead of developing $\\mathbf {p}_h\\in H(\\mbox{div\\:};\\Omega )$ by processing the discontinuity of $\\nabla u_h$ across the edges of elements [2], [16], the feature of Kikuchi's approach and our method is to construct the hypercircle by utilizing $\\mathbf {p}_h \\in H(\\mbox{div\\:};\\Omega )$ such that $\\mbox{div\\:} {\\mathbf {p}}_h +f_h=0$ holds exactly, where $f_h$ is the projection of $f$ to totally discontinuous piecewise polynomials.", "The hypercircle method, namely the Prager-Synge theorem, was developed more than fifty years ago by [19] for elastic analysis.", "Around the same time [19], based on the $T^*T$ theory of Kato [8], Fujita developed a method similar to the hypercircle method [6], which has been applied to develop the point-wise estimation method for boundary value problems with specially constructed base functions.", "The extension of Kato-Fujita's approach to finite element method for local error estimation will be considered in our succeeding work.", "The local error estimation proposed in this paper has the following features.", "(1) As a quantitative result, it provides explicit estimation for the energy error in the subdomain of interest.", "(2) The method deals with domains of general shapes in the natural way and is applicable to non-convex domains where a singularity may appear around the re-entry corner of the boundary.", "(3) There are no constraints on the mesh generation of the domain, compared to the stringent requirement of a uniform mesh in past studies on local error estimation.", "It even works for the case in which the subdomain of interest is connected to the boundary of the whole domain.", "The rest of the paper is organized as follows: In section 2, we provide preliminary information on the problem setting and basic knowledge about the finite element method spaces.", "In section 3, the global error estimation based on the hypercircle method developed in [13] is introduced.", "In section 4, the details of the local error estimation are described.", "In section 5, the numerical examples for Poisson's equation over the square domain and the L-shaped domain are presented.", "Finally, in section 6, we summarize the conclusions and discuss future studies." ], [ "Problem settings", "Throughout this study, the domain $\\Omega $ is assumed to be a bounded polygonal domain of $\\mathbb {R}^2$ .", "Thus, $\\Omega $ can be completely triangulated without any gap near the boundary.", "Standard symbols are used for the Sobolev spaces $H^m(\\Omega )(m>0)$ .", "The norm of $L^2(\\Omega )$ is written as $\\Vert \\cdot \\Vert _{L^2(\\Omega )}$ or $\\Vert \\cdot \\Vert _{\\Omega }$ .", "Symbols $|\\cdot |_{H^m(\\Omega )},\\Vert \\cdot \\Vert _{H^m(\\Omega )}$ denote semi-norm and norm of $H^m(\\Omega )$ , respectively.", "Let $(\\cdot ,\\cdot )$ be the inner product of $L^2(\\Omega )$ or $(L^2(\\Omega ))^2$ .", "Sobolev space $W^{1,\\infty }(\\Omega )$ is a function space where weak derivatives up to the first order are essentially bounded on $\\Omega $ .", "The standard vector valued function space $H(\\mbox{div};\\Omega )$ is defined as follows: $H(\\mbox{div};\\Omega ):= \\left\\lbrace \\mathbf {q} \\in (L^2(\\Omega ))^2 ;~ \\mbox{div } \\mathbf {q} \\in L^2(\\Omega ) \\right\\rbrace .$ In this paper, the finite element solution for the following model boundary value problem will be discussed: $-\\Delta u = f \\mbox{ in } \\Omega ,\\quad \\displaystyle \\frac{\\partial u }{\\partial \\mathbf {n}} = g_N\\mbox{ on } \\Gamma _N,\\quad u= g_D\\mbox{ on } \\Gamma _D.$ Here, $\\Gamma _N,\\Gamma _D$ are disjoint subsets of $\\partial \\Omega $ satisfying $\\Gamma _N \\cup \\Gamma _D = \\partial \\Omega $ ; $\\mathbf {n}$ is the unit outer normal direction on the boundary and $\\frac{\\partial }{\\partial \\mathbf {n}}$ is the directional derivative along $\\mathbf {n}$ on $\\partial \\Omega $ .", "Let $S$ be a subdomain of $\\Omega $ of interest.", "Suppose that $u_h$ is an approximate solution to the problem (REF ).", "The error of $(\\nabla u - \\nabla u_h)$ in the subdomain $S$ will be evaluated in this study.", "The weak form for the aforementioned problem is given by: $\\mbox{Find } u \\in V \\mbox{ s.t. }", "(\\nabla u,\\nabla v) = (f,v) + \\int _{\\Gamma _N} g_N v ~ds, \\quad \\forall v \\in V_0 .$ where $(g_N,v)_{\\Gamma _N} := \\int _{\\Gamma _N} g_N v ~ds .$ In case $\\Gamma _D$ is not an empty set, the function space $V$ of the trial function and the function space $V_0$ of the test function are defined by $V := \\left\\lbrace v \\in H^1(\\Omega ); v =g_D \\mbox{ on } \\Gamma _D \\right\\rbrace ,~V_0 := \\left\\lbrace v \\in H^1(\\Omega ); v =0 \\mbox{ on } \\Gamma _D \\right\\rbrace .$ For an empty $\\Gamma _D$ , the definition of $V$ and $V_0$ are modified as follows: $V = V_0 = \\left\\lbrace v \\in H^1(\\Omega ); \\int _\\Omega v ~dx =0 \\right\\rbrace .$" ], [ "Finite element space setting", "To prepare for the discussion on the newly developed local error estimation in §4, we review the standard FEM approaches to (REF ).", "To simplify the discussion, assume $g_D,g_N$ in the boundary conditions of the model problem (REF ) to be piecewise linear and piecewise constant at the boundary edges of $\\mathcal {T}^h$ , respectively.", "Let $\\mathcal {T}_h$ be a proper triangulation of the domain $\\Omega $ .", "Given an element $K \\in \\mathcal {T}_h$ , let $h_K$ denote the length of longest edge of $K$ .", "The mesh size $h$ of $\\mathcal {T}_h$ is defined as follows: $h := \\max _{K \\in \\mathcal {T}_h} h_K .$ On each element $K \\in \\mathcal {T}_h$ , the set of polynomials with degree up to $d$ is denoted by $P_d(K)$ .", "Let $V_h,V_{h,0}(\\subset H^1(\\Omega ))$ denote the finite element spaces consisting of piecewise linear and continuous functions, the boundary conditions of which follow the settings of $V$ and $V_0$ .", "The conforming finite element formulation of (REF ) is given by $\\mbox{Find } u_h \\in V_h \\mbox{ s.t. }", "(\\nabla u_h,\\nabla v_h) = (f,v_h) + (g_N,v_h)_{\\Gamma _N}, \\quad \\forall v_h \\in V_{h,0}.", "$ To provide the local error estimation for $(\\nabla u - \\nabla u_h)$ , let us introduce the following finite element spaces.", "(a) Piecewise constant function space: $X_h := \\left\\lbrace v_h \\in L^2(\\Omega ) ~:~ v_h|_K \\in P_0(K),~ \\forall K \\in \\mathcal {T}_h \\right\\rbrace .$ In case $\\Gamma _D$ is empty, it is further required that $\\int _\\Omega v dx =0$ for $v_h \\in X_h$ .", "(b) The Raviart-Thomas finite element space: $RT_h := \\left\\lbrace \\mathbf {p}_h \\in H(\\mbox{div};\\Omega ) ~:~ \\mathbf {p}_h|_K =(a_K+c_Kx,b_K+c_Ky) \\mbox{ for } K \\in \\mathcal {T}_h \\right\\rbrace .$ $RT_{h,0} =\\lbrace \\mathbf {p}_h \\in RT_h ~:~ \\mathbf {p}_h \\cdot \\mathbf {n} =0 \\mbox{ on } \\Gamma _N \\rbrace \\;.$ Here, $a_K,b_K,c_K \\in P_0(K)$ for $K \\in \\mathcal {T}_h$ .", "The standard mixed finite element formulation of (REF ) reads: Find $ (\\mathbf {p}_h,\\mu _h) \\in RT_h \\times X_h,~ \\mathbf {p}_h \\cdot \\mathbf {n} = g_N \\mbox{ on } \\Gamma _N$ , s.t.", "$(\\mathbf {p}_h,\\mathbf {q}_h) + (\\mbox{div } \\mathbf {q}_h,\\mu _h) +(\\mbox{div } \\mathbf {p}_h,\\eta _h) = (g_D , (\\mathbf {q}_h \\cdot \\mathbf {n}) )_{\\Gamma _D} -(f,\\eta _h)\\;,$ for $(\\mathbf {q}_h ,\\eta _h) \\in RT_{h,0} \\times X_h$ .", "Define the projection $\\pi _{h}:L^2(\\Omega ) \\rightarrow X_h $ such that for $f \\in L^2(\\Omega )$ , $(f-\\pi _{h} f,\\eta _h) =0, \\quad \\forall \\eta _h \\in X_h .$ The following error estimation holds for $\\pi _h$ , $\\Vert f- \\pi _h f\\Vert _{\\Omega } \\le C_0 h |f|_{H^1(\\Omega )}, \\quad \\forall f \\in H^1(\\Omega ).", "$ To give a concrete value of $C_0$ in (REF ), let us define $C_0(K)$ as a constant that depends on the shape of the triangle $K \\in \\mathcal {T}_h$ and satisfies $ \\Vert f- \\pi _h f\\Vert _{K} \\le C_0(K) |f|_{H^1(K)}, \\quad \\forall f \\in H^1(K).", "$ By using $C_0(K)$ , the constant $C_0$ that depends on triangulation can be defined by $C_0:=\\max _{K \\in \\mathcal {T}_h} \\frac{C_0(K)}{h}.", "$ The previous studies [9], [10], [14] reported that the optimal value of $C_0(K)$ is given by $C_0(K):=h_K/j_{1,1} (\\le 0.261 h_K) $ using positive minimum root $j_{1,1}\\approx 3.83171$ of the first kind Bessel's function $J_1$ ." ], [ "Global ", "In this section, we introduce the global error estimation developed in [13], [18], [12], which will be used in Theorem REF for local error estimation.", "We focus on the global a priori error estimation for problems with homogeneous boundary value conditions, which fits the needs in the proof for Theorem REF .", "For global a priori error estimation of non-homogeneous boundary value problems, refer to [12].", "As a preparation for Theorem REF , let us consider the following boundary value problem.", "$-\\Delta \\phi = f \\mbox{ in } \\Omega ,\\quad \\displaystyle \\frac{\\partial \\phi }{\\partial \\mathbf {n}} = 0 \\mbox{ on } \\Gamma _N,\\quad \\phi = 0 \\mbox{ on } \\Gamma _D.$ The weak formulation of (REF ) seeks $\\phi \\in V_0$ , s.t., $(\\nabla \\phi ,\\nabla v) = (f,v), \\quad \\forall v \\in V_0 .$ The Galerkin projection operator $P_h:V_{0} \\rightarrow V_{h,0}$ satisfies, for $ v \\in V_{0}$ $(\\nabla (v-P_h v),\\nabla v_h) =0 , \\quad \\forall v_h \\in V_{h,0}\\:.$ In [13], the following quantity $\\kappa _h$ is introduced for the purpose of a priori error estimation to the Galerkin projection $P_h \\phi $ : $\\kappa _h :=\\max _{f_h \\in X_h} ~~\\min _{ \\begin{array}{c}v_h \\in V_{h,0} ,~ {\\mathbf {q}}_h \\in RT_{h,0}, \\\\ \\text{div } {\\mathbf {q}}_h + f_h =0\\end{array} } \\frac{\\Vert \\nabla v_h -{\\mathbf {q}}_h\\Vert }{\\Vert f_h\\Vert } .$ The theorem below provides an a priori error estimation using $\\kappa _h$ and the Prager-Synge theorem.", "Theorem 3.1 (Global a priori error estimation [13]) Given $f \\in L^2(\\Omega )$ , let $\\phi $ be the solution to (REF ).", "Then, the following error estimation holds.", "$&& |\\phi - P_h \\phi |_{H^1(\\Omega )} \\le C(h) \\Vert f\\Vert _{\\Omega }, \\\\&&\\Vert \\phi - P_h \\phi \\Vert _{\\Omega } \\le C(h) |\\phi - P_h \\phi |_{H^1(\\Omega )} \\le C(h)^2 \\Vert f\\Vert _{\\Omega }\\:,$ where $C(h):=\\sqrt{\\kappa _h^2+(C_0h)^2}$ ; $C_0$ is the quantity defined in (REF ).", "Remark 3.2 When the exact solution $\\phi $ belongs $H^2(\\Omega )$ , the constant $C(h)$ appearing (REF ), () can be replaced for the error constant of the Lagrange interpolation.", "For example, in §5.2, it is possible to use $C_h=0.493h$ as $C(h)$ for the right-angled triangle mesh.", "Remark 3.3 Calculation of $\\kappa _h$ : For given $f_h \\in X_h$ , let $R_h:X_h \\rightarrow V_h$ , $T_h:X_h \\rightarrow RT_h$ be the linear operators, which map $f_h$ to the Lagrange FEM approximation to $\\nabla \\phi $ and the Raviart-Thomas FEM approximation to $\\nabla \\phi $ , respectively.", "Then, $\\kappa _h$ is characterized by the following maximum formulation $\\kappa _h = \\max _{f_h \\in X_h} \\frac{\\Vert (R_h - T_h) f_h\\Vert _{\\Omega } }{\\Vert f_h\\Vert _{\\Omega }}\\:,$ which reduces to a matrix eigenvalue problem.", "See [13], [12] for a more detailed discussion on the calculation of $\\kappa _h$ ." ], [ "Weighted hypercircle formula and main result", "In this section, we propose a posteriori local error estimation for the finite element solutions.", "Let $S(\\subset \\Omega )$ be the subdomain of interest.", "In §4.1, the weighted inner product and weighted norm corresponding to $S$ will be introduced through a cutoff function $\\alpha $ .", "In §4.2, we show the weighted Hypercircle formula as an extension of Theorem REF .", "The result of the local error estimation will be provided in §4.3." ], [ "The weight function", "Let $\\Omega ^{\\prime }$ be the extended domain of $S$ with a band of width $\\varepsilon $ , that is, $\\Omega ^{\\prime }:= \\left\\lbrace x \\in \\Omega ~;~ \\mbox{dist}(x, S) < \\varepsilon \\right\\rbrace $ .", "Denote the band surrounding $S$ by $B_S$ .", "Refer to Figure REF -(a),(b) for two examples of $S$ and $B_S$ .", "The weight function $\\alpha \\in W^{1,\\infty }(\\Omega )$ is defined as a piecewise polynomial with the following property.", "$\\alpha (x,y) =\\left\\lbrace \\begin{array}{l} 1 \\quad (x,y) \\in S \\\\ 0 \\quad (x,y) \\in (\\Omega ^{\\prime })^c \\end{array}\\right.", ", \\quad 0 \\le \\alpha (x,y) \\le 1, \\mbox{ For } (x,y) \\in B_S\\:.$ Figure: Definition of the weight function α\\alpha .To construct a concrete weight function $\\alpha (x,y)$ , let us define $\\alpha _{(a,b)}$ over interval $(a,b)$ as follows.", "$\\alpha _{(a,b)}(x) =\\left\\lbrace \\begin{array}{ll}1+ (x-a)/\\varepsilon \\quad & x \\in (a-\\varepsilon ,a] \\\\1 \\quad & x \\in (a,b) \\\\1- (x-b)/\\varepsilon \\quad & x \\in [b,b+\\varepsilon ) \\\\0 \\quad &\\mbox{otherwise}\\end{array}\\right.\\:.$ Refer to Figure REF -(c) for the graph of $\\alpha _{(a,b)}$ .", "For $S$ being a rectangular subdomain constructed by the Cartesian product of two open intervals $(x_a, x_b)$ , $(y_a, y_b)$ , the weight function $\\alpha $ can be defined by $\\alpha (x,y) =\\min {\\lbrace \\alpha _{(x_a, x_b)}(x), \\alpha _{(y_a,y_b)}(y)\\rbrace }$ .", "The weighted inner product and norm are defined by using $\\alpha $ as follows.", "(a) Weighted inner product $(\\cdot ,\\cdot )_{\\alpha }$ :   For $f,g \\in L^2(\\Omega )$ or $f,g \\in (L^2(\\Omega ))^2$ , $(f,g)_{\\alpha } := \\int _{\\Omega ^{\\prime }} \\alpha f \\cdot g ~dx.$ (b) Weighted norm $\\Vert \\cdot \\Vert _{\\alpha }:~$ For $f \\in L^2(\\Omega )$ , $\\Vert f\\Vert _{\\alpha } : = \\sqrt{(f,f)_{\\alpha }}= \\sqrt{\\int _{\\Omega ^{\\prime }} f^2 \\alpha ~dx} \\quad (=\\left\\Vert f\\sqrt{\\alpha } \\right\\Vert _{\\Omega }).$ The following inequalities hold.", "$ \\Vert f \\Vert _S \\le \\Vert f \\Vert _{\\alpha } \\le \\Vert f \\Vert _{\\Omega ^{\\prime }} \\le \\Vert f \\Vert _{\\Omega }.", "$" ], [ "Weighted hypercircle formula", "In this sub-section, a weighted hypercircle formula is proposed, which can be regarded as an extention to the classical Prager-Synge theorem below.", "Theorem 4.1 (Prager-Synge's theorem[19]) Let $\\phi $ be the solution of (REF ).", "For any $v \\in V_0$ and $\\widetilde{\\mathbf {p}} \\in H(\\mbox{\\em div};\\Omega )$ satisfying $\\mbox{\\em {div} }\\widetilde{\\mathbf {p}}+f=0 ,~ \\widetilde{\\mathbf {p}} \\cdot \\mathbf {n} = 0 \\mbox{ on } \\Gamma _N \\:,$ we have, $\\Vert \\nabla \\phi - \\nabla v \\Vert _{\\Omega }^2 + \\Vert \\nabla \\phi - \\widetilde{\\mathbf {p}}\\Vert _{\\Omega }^2 = \\Vert \\nabla v - \\widetilde{\\mathbf {p}} \\Vert _{\\Omega }^2.", "$ For weighted norms introduced in the previous section, we have the following extended formulation of the hypercircle (REF ).", "Theorem 4.2 Let $u$ be the solution of (REF ).", "For any $v \\in V_0$ and $\\mathbf {p} \\in H(\\mbox{\\em div};\\Omega )$ satisfying $ \\mbox{\\em div }\\mathbf {p}+f=0 \\mbox{ in } \\Omega , \\quad \\mathbf {p} \\cdot \\mathbf {n} = g_N \\mbox{ on } \\Gamma _N .$ Then, $\\Vert \\nabla u - \\nabla v \\Vert _{\\alpha }^2 + \\Vert \\nabla u- \\mathbf {p}\\Vert _{\\alpha }^2 \\le \\Vert \\nabla v - \\mathbf {p} \\Vert _{\\alpha }^2 + 2 \\sqrt{2} \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\Vert u-v \\Vert _{\\Omega ^{\\prime }} \\Vert \\nabla u - \\mathbf {p} \\Vert _{\\Omega ^{\\prime }} .$ The expansion of $\\Vert \\nabla v - \\mathbf {p} \\Vert _{\\alpha }^2=\\Vert (\\nabla v - \\nabla u )+ (\\nabla u- \\mathbf {p}) \\Vert _{\\alpha }^2$ tells that $\\Vert \\nabla v - \\mathbf {p} \\Vert _{\\alpha }^2 = \\Vert \\nabla v - \\nabla u \\Vert _{\\alpha }^2 + \\Vert \\nabla u- \\mathbf {p}\\Vert _{\\alpha }^2 + 2 (\\nabla v - \\nabla u ,\\nabla u- \\mathbf {p})_{\\alpha }.$ Let $w: = v- u$ .", "Below, we show the estimation for the cross-term of (REF ), i.e., $(\\nabla w, \\nabla u)_{\\alpha } -(\\nabla w, \\mathbf {p})_{\\alpha } $ .", "To deal with $(\\nabla w ,\\nabla u)_{\\alpha } $ , let us take the test function as $\\alpha w$ in (REF ) and apply the chain rule to $\\alpha w$ , i.e., $\\nabla (\\alpha w) = w \\nabla \\alpha +\\alpha \\nabla w$ .", "Then we have $(\\nabla w, \\nabla u)_{\\alpha } = -\\int _{\\Omega } w \\nabla \\alpha \\cdot \\nabla u ~dx + \\int _{\\Omega } f \\cdot (\\alpha w) ~dx +\\int _{\\Gamma _N} g_N \\cdot (\\alpha w) ds.", "$ For $(\\nabla w ,\\mathbf {p})_{\\alpha }$ , Green's formula tells that $(\\nabla w,\\mathbf {p})_{\\alpha } & = &\\int _{\\Omega } \\nabla w \\cdot (\\alpha \\mathbf {p}) ~dx = \\int _{\\Gamma _N} g_N\\cdot (\\alpha w) ds - \\int _{\\Omega } w \\mbox{ div} (\\alpha \\mathbf {p}) ~dx\\nonumber \\\\&=&\\int _{\\Gamma _N} g_N\\cdot (\\alpha w) ds- \\int _{\\Omega } (\\alpha w) \\mbox{ div } \\mathbf {p} ~dx - \\int _{\\Omega } w \\nabla \\alpha \\cdot \\mathbf {p} ~dx.", "\\nonumber \\\\&=& - \\int _{\\Omega } w \\nabla \\alpha \\cdot \\mathbf {p} ~dx + \\int _{\\Omega } (\\alpha w) f ~dx + \\int _{\\Gamma _N} g_N\\cdot (\\alpha w) ds.$ By taking (REF )-(REF ) and noticing that $\\alpha = 0$ on $\\Omega \\setminus \\Omega ^{\\prime }$ , we have $(\\nabla v - \\nabla u ,\\nabla u- \\mathbf {p})_{\\alpha } = - \\int _{\\Omega } (v-u) \\nabla \\alpha \\cdot ( \\nabla u- \\mathbf {p} )~dx = - \\int _{\\Omega ^{\\prime }} (v-u) \\nabla \\alpha \\cdot ( \\nabla u- \\mathbf {p} )~dx.", "$ By applying Hölder's inequality, we have $|(\\nabla v - \\nabla u ,\\nabla u- \\mathbf {p})_{\\alpha }| \\le \\sqrt{2} \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\Vert u-v\\Vert _{\\Omega ^{\\prime }} \\Vert \\nabla u -\\mathbf {p} \\Vert _{\\Omega ^{\\prime }}.$ For (REF ) and (REF ), we draw the conclusion.", "Remark 4.3 Theorem REF holds no matter $\\partial S \\cap \\partial \\Omega = \\emptyset $ or not, as can be confirmed in the proof." ], [ "As a preparation to the argument of the main result, let us follow the idea of Kikuchi [11] to introduce auxiliary functions $\\overline{u} \\in V$ and $ \\overline{u}_h \\in V_h$ as the solutions to the following equations.", "$(\\nabla \\overline{u},\\nabla v ) = (\\pi _h f,v) + (g_N,v)_{\\Gamma _N}, ~\\forall v \\in V_0\\:; $ $(\\nabla \\overline{u}_h,\\nabla v_h ) = (\\pi _h f,v_h) + (g_N,v_h)_{\\Gamma _N}, ~\\forall v_h \\in V_{h,0}\\:.", "$ Both the functions are introduced only for error analysis in a theoretical way, and the above equations do not need to be solved explicitly.", "Lemma 4.4 For functions $\\overline{u}$ and $\\overline{u}_h$ , the following estimations hold.", "$|u-\\overline{u}|_{H^1(\\Omega )} \\le C_0h \\Vert f-\\pi _hf\\Vert _{\\Omega } ,\\\\|u_h - \\overline{u}_h |_{H^1(\\Omega )} \\le C_0 h \\Vert f-\\pi _h f\\Vert _{\\Omega }.", "$ Here, $C_0$ is the projection error constant defined in (REF ).", "According to the definitions of $u$ and $\\overline{u}$ , $(\\nabla (u-\\overline{u}),\\nabla v) = (f-\\pi _h f,v) = (f-\\pi _h f,v-\\pi _h v), \\quad v \\in V_0.$ By taking $v = u - \\overline{u}$ in the above equation and using the error estimation of projection $\\pi _h$ in (REF ), we have $|u-\\overline{u}|_{H^1(\\Omega )} ^2 = |(\\nabla (u-\\overline{u}),\\nabla (u-\\overline{u}))| \\le \\Vert f-\\pi _hf \\Vert _{\\Omega } \\cdot C_0h |u-\\overline{u}|_{H^1(\\Omega )}.$ Hence, $ |u-\\overline{u}|_{H^1(\\Omega )} \\le C_0h \\Vert f-\\pi _hf \\Vert _{\\Omega } .$ Estimation () is obtained in the same way.", "With the selected $\\pi _h f \\in X_h$ , the property $\\mbox{div}(RT_h) = X_h$ makes it possible to find $\\mathbf {p}_h \\in RT_h$ such that $\\mbox{div }\\mathbf {p}_h+\\pi _h f=0,~ \\mathbf {p}_h \\cdot \\mathbf {n} = g_N \\mbox{ on } \\Gamma _N\\:.$ In this case, the following hypercircle equation holds: $\\Vert \\nabla \\overline{u} - \\nabla v_h \\Vert ^2_{\\Omega } +\\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert ^2_{\\Omega } =\\Vert \\nabla v_h - \\mathbf {p}_h \\Vert ^2_{\\Omega }\\:.$ Remark 4.5 The estimation (REF ) along with the hypercircle (REF ) leads to an a posteriori estimation of the global error.", "$\\Vert \\nabla (u-u_h)\\Vert _\\Omega \\le \\Vert \\nabla u_h - \\mathbf {p}_h \\Vert _\\Omega + C_0h\\Vert f-\\pi _h f\\Vert _\\Omega \\:.$ Here, $\\mathbf {p}_h$ can be chosen freely to approximate $\\nabla u$ under the condition (REF ).", "Below, we apply Theorem REF to the current function settings.", "Lemma 4.6 Let $\\overline{u}$ and $\\overline{u}_h$ be the solutions of (REF ) and (REF ), respectively.", "For $\\mathbf {p}_h \\in RT_h$ satisfying (REF ), the following estimation holds, $\\Vert \\nabla (\\overline{u} - \\overline{u}_h)\\Vert _{\\alpha }^2 \\le \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _\\alpha ^2+2\\sqrt{2} \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\Vert \\overline{u}- \\overline{u}_h\\Vert _{\\Omega } ~ \\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega } .$ Take $f := \\pi _h f ,~\\mathbf {p} := \\mathbf {p}_h,~ u :=\\overline{u} $ and $ v := \\overline{u}_h$ in Theorem REF , then the following inequality holds.", "$\\Vert \\nabla (\\overline{u} - \\overline{u}_h)\\Vert _{\\alpha }^2 + \\Vert \\nabla \\overline{u} - \\mathbf {p}_h \\Vert _\\alpha ^2 \\le \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _\\alpha ^2+2\\sqrt{2} \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\Vert \\overline{u}- \\overline{u}_h\\Vert _{\\Omega ^{\\prime }} ~ \\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega ^{\\prime } } .$ Because $\\Omega ^{\\prime }\\subset \\Omega $ , we conclude by replacing $\\Vert \\cdot \\Vert _{\\Omega ^{\\prime }}$ with $\\Vert \\cdot \\Vert _{\\Omega }$ .", "To state the results in Theorem REF and REF , let us define the following four computable quantities $\\overline{E}_1, ~\\overline{E}_2, ~E_1$ and $E_2$ .", "$&&\\overline{E}_1 := \\Vert \\nabla \\overline{u}_h -\\mathbf {p}_h\\Vert _{\\alpha }, \\quad E_1 := \\Vert \\nabla u_h - \\mathbf {p}_h\\Vert _{\\alpha } + C_0h\\Vert f-\\pi _h f\\Vert _\\Omega ,\\\\&&\\overline{E}_2 := \\left\\lbrace 2\\sqrt{2} C(h)\\cdot \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\right\\rbrace ^{1/2}\\cdot \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _{\\Omega } , \\\\&&E_2 := \\left\\lbrace 2\\sqrt{2} C(h)\\cdot \\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )} \\right\\rbrace ^{1/2} \\cdot \\Vert \\nabla u_h - \\mathbf {p}_h\\Vert _{\\Omega }\\:.", "~ \\\\$ Theorem 4.7 (a posteriori local error estimation for $\\overline{u}$ ) Let $u$ and $\\overline{u}_h$ be the solutions of (REF ), (REF ), respectively.", "For $\\mathbf {p}_h \\in RT_h$ satisfying $ \\mbox{\\em {div} }\\mathbf {p}_h+\\pi _h f=0,~ \\mathbf {p}_h \\cdot \\mathbf {n} = g_N \\mbox{ on } \\Gamma _N .$ Then, the following local error estimation holds.", "$\\Vert \\nabla u - \\nabla \\overline{u}_h \\Vert _{S} \\le \\sqrt{\\overline{E}_1^2+\\overline{E}_2^2} + C_0h \\Vert f - \\pi _h f\\Vert _{\\Omega }.$ With $\\overline{u}$ defined in (REF ) and the triangle inequality, we obtain: $\\Vert \\nabla (u-\\overline{u}_h) \\Vert _S \\le \\Vert \\nabla (u-\\overline{u}) \\Vert _S + \\Vert \\nabla (\\overline{u} - \\overline{u}_h) \\Vert _S \\le \\Vert \\nabla (u-\\overline{u}) \\Vert _{\\Omega }+ \\Vert \\nabla (\\overline{u} - \\overline{u}_h) \\Vert _{\\alpha }.$ By applying the estimation of $\\Vert \\nabla (u-\\overline{u})\\Vert _\\Omega $ in Lemma REF and the estimation of $\\Vert \\nabla (\\overline{u}-\\overline{u}_h)\\Vert _\\alpha $ in Lemma REF , the following estimation holds.", "$\\Vert \\nabla (u-\\overline{u}_h) \\Vert _S&\\le &\\left\\lbrace \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _\\alpha ^2 +2 \\sqrt{2} \\Vert \\nabla \\alpha \\Vert _{\\infty } \\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega } ~ \\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega } \\right\\rbrace ^{\\frac{1}{2}} \\\\&& + C_0 h \\Vert f - \\pi _h f \\Vert _{\\Omega } \\: .$ Next, we give the estimation for $\\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega },~\\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega } $ in (REF ).", "(a) From Theorem REF , the hypercircle below is available for $\\overline{u}$ defined in (REF ), $\\Vert \\nabla \\overline{u} - \\nabla v_h \\Vert _{\\Omega }^2 + \\Vert \\nabla \\overline{u}- \\mathbf {p}_h\\Vert _{\\Omega }^2 = \\Vert \\nabla v_h - \\mathbf {p}_h \\Vert _{\\Omega }^2, \\quad \\forall v_h \\in V_h \\:.$ By taking $v_h:=\\overline{u}_h$ , we obtain the estimation of $\\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega } $ : $\\Vert \\nabla \\overline{u}-\\mathbf {p}_h \\Vert _{\\Omega } \\le \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h \\Vert _{\\Omega }\\:.", "$ (b) To give the estimation of $\\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega }$ , let us define the dual problem.", "$\\mbox{Find } \\phi \\in V_0 \\mbox{ s.t. }", "(\\nabla \\phi ,\\nabla v) = (\\overline{u}-\\overline{u}_h,v), \\quad \\forall v \\in V_0.", "$ By applying $P_h$ defined in (REF ) along with the a priori estimation (REF ) in Theorem REF , we have, $\\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega }^2&\\le &\\Vert \\nabla (\\phi -P_h\\phi ) \\Vert _{\\Omega } \\cdot \\Vert \\nabla (\\overline{u}-\\overline{u}_h) \\Vert _{\\Omega } .\\\\&\\le & C(h) \\: \\Vert \\nabla (\\overline{u}-\\overline{u}_h) \\Vert _{\\Omega } \\:\\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega }.$ Notice that (REF ) with $v_h:=\\overline{u}_h$ implies $\\Vert \\nabla \\overline{u} - \\nabla \\overline{u}_h \\Vert _{\\Omega } \\le \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h \\Vert _{\\Omega }\\:.", "$ Thus, we have the estimation of $ \\Vert \\overline{u}-\\overline{u}_h \\Vert _{\\Omega } $ : $\\Vert \\overline{u}-\\overline{u}_h \\Vert _{\\Omega } \\le C(h) \\Vert \\nabla (\\overline{u} - \\overline{u}_h) \\Vert _{\\Omega }\\le C(h) \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h \\Vert _{\\Omega },.$ Apply (REF ) and (REF ) to the first term of the right-hand side of (REF ), $&\\quad &\\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _\\alpha ^2 +2\\sqrt{2} \\:\\Vert \\nabla \\alpha \\Vert _{\\infty } \\Vert \\overline{u}-\\overline{u}_h\\Vert _{\\Omega } ~ \\Vert \\nabla \\overline{u} - \\mathbf {p}_h\\Vert _{\\Omega } \\\\&&\\le \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _\\alpha ^2 + 2\\sqrt{2} \\:\\Vert \\nabla \\alpha \\Vert _{\\infty } \\:C(h)\\: \\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h\\Vert _{\\Omega } ^2\\\\&& = {\\overline{E}_1^2 + \\overline{E}_2^2}.$ Now, we draw the conclusion by sorting the estimation of (REF ).", "Theorem 4.8 Under the assumptions of Theorem REF , the following estimation holds.", "$ \\Vert \\nabla u - \\nabla u_h \\Vert _{S} \\le \\sqrt{E_1^2 + E_2^2} + 2 C_0h \\Vert f - \\pi _h f\\Vert _{\\Omega } \\:.$ First, we apply the triangle inequality to $(u-\\overline{u}_h) + (\\overline{u}_h-{u}_h )$ and the estimation () in Lemma REF to have, $\\Vert \\nabla (u - {u}_h) \\Vert _S\\le \\Vert \\nabla (u_h - \\overline{u}_h ) \\Vert _S+\\Vert \\nabla ( u - \\overline{u}_h ) \\Vert _S\\le C_0h \\Vert f-\\pi _h f\\Vert _{\\Omega } +\\Vert \\nabla ({u} - \\overline{u}_h) \\Vert _{S}\\:.$ Next, we apply the result in Theorem REF to $\\Vert \\nabla ({u} - \\overline{u}_h) \\Vert _{S}$ and process the term $\\overline{u}_h$ in $\\overline{E}_1$ and $\\overline{E}_2$ .", "Because $\\overline{u}_h$ is the best approximation to $\\overline{u}$ in $V_h$ , the hypercircle (REF ) with respect to $\\pi _h f$ leads to $\\Vert \\nabla \\overline{u}_h - \\mathbf {p}_h \\Vert _\\Omega \\le \\Vert \\nabla u_h - \\mathbf {p}_h \\Vert _\\Omega \\:.$ For the term $\\Vert \\nabla \\overline{u}_h-\\mathbf {p}_h \\Vert _{\\alpha }$ , apply the triangle inequality and () to obtain $\\Vert \\nabla \\overline{u}_h-\\mathbf {p}_h \\Vert _{\\alpha } \\le \\Vert \\nabla (\\overline{u}_h-u_h) \\Vert _{\\alpha } + \\Vert \\nabla u_h - \\mathbf {p}_h \\Vert _{\\alpha }\\le C_0h \\Vert f-\\pi _h f\\Vert _{\\Omega } +\\Vert \\nabla u_h - \\mathbf {p}_h \\Vert _{\\alpha }\\: .$ Now, we can conclude as in (REF ).", "Remark 4.9 In Theorem REF , REF , for a solution with $H^2$ -regularity, the convergence rates of the involved terms are expected to be $\\overline{E}_1, E_1 =O(h^1),\\quad \\overline{E}_2, E_2=O(h^{1.5}), \\quad C_0h\\Vert f-\\pi _h f\\Vert _\\Omega =O(h^2)\\:.$ Because the global terms have higher convergence rates, the terms $\\overline{E}_1, E_1$ involved with the local error become dominant when the mesh is refined.", "Thus, the proposed local error estimation will provide a sharper bound to the local error itself, compared with the global error estimation, for example, (REF )." ], [ "Preparation", " The selection of bandwidth of the $B_{S}$ is important in the local error estimation.", "A large bandwidth of $B_{S}$ leads to a large value of $E_1$ , while a small bandwidth of $B_{S}$ results in a large value of $\\Vert \\nabla \\alpha \\Vert _{L^{\\infty }(\\Omega )}$ in $E_2$ .", "Therefore, in each example, we first investigate the impact of the bandwidth of $B_{S}$ , and then take an appropriate width of $B_{S}$ for subsequent computation.", "Besides the symbols $E_1, E_2$ in (REF ), we introduce new symbols as follows: The local error and its estimation in (REF ) are denoted by $E_{L} := \\Vert \\nabla u - \\nabla u_h \\Vert _{S},\\quad \\overline{E}_{L} := \\sqrt{E_1^2+E_2^2 } + 2C_0h \\Vert f - \\pi _h f\\Vert _{\\Omega }\\: .$ The global error and its estimation in (REF ) are denoted by $E_{G} := \\Vert \\nabla u - \\nabla u_h \\Vert _{\\Omega },\\quad \\overline{E}_{G} := \\Vert \\nabla u_h - \\mathbf {p}_h \\Vert _{\\Omega } + C_0h \\Vert f -\\pi _h f \\Vert _{\\Omega }.$ Here, $u_h \\in V_h$ and $\\mathbf {p}_h \\in RT_h$ are finite element solutions of the objective problems; $\\mathbf {p}_h$ also satisfies the condition (REF )." ], [ "Square domain", "The error estimation proposed in this paper is applicable to problems with different boundary conditions.", "To illustrate this feature, let us consider the following Poisson equations over the unit square domain $\\Omega =(0,1)^2$ , where the subdomain is selected as $S=(0.375,0.625)^2$ .", "Dirichlet boundary condition (exact solution $u = \\sin (\\pi x)\\sin (\\pi y)$ ).", "$- \\Delta u = 2 \\pi ^2 \\sin (\\pi x) \\sin (\\pi y) \\mbox{ in } \\Omega , \\quad u = 0 \\mbox{ on } \\partial \\Omega .$ Neumann boundary condition (exact solution $u = \\cos (\\pi x)\\cos (\\pi y)$ ).", "$- \\Delta u = 2 \\pi ^2 \\cos (\\pi x) \\cos (\\pi y) \\mbox{ in } \\Omega ,\\quad \\frac{\\partial u}{\\partial \\mathbf {n}}= 0 \\mbox{ on } \\partial \\Omega ,\\quad \\int _\\Omega u d x =0 \\:.$ Figure: Dependency of local error estimation on the bandwidth of B S B_{S} (Neumann BVP and square domain).For Dirichlet and Neumann boundary conditions, the dependencies of the local error estimator $\\overline{E}_L$ on the bandwidth of $B_S$ are shown in Figure REF and Figure REF , respectively.", "The relative variation of local error estimator with respect to bandwidth selection is displayed for two problems.", "It is noteworthy that the local error estimation is not significantly sensitive to variations in bandwidth.", "For example, in Figure REF , for $h=1/64$ , the relative variation in error estimation with respect to a bandwidth in the range $[0.125,0.275]$ is less than 5%.", "In the following discussion, the bandwidth of $B_{S}$ is selected as $0.15$ for the Dirichlet boundary condition and $0.10$ for the Neumann boundary condition.", "Figure: Error estimators for Neumann BVP (square domain).Table: Error estimate for Dirichlet BVP (square domain)Table: Error estimate for Neumann BVP (square domain).A detailed discussion on each component of the error estimators is also presented; see Table REF and Figure REF for Dirichlet boundary condition, Table REF and Figure REF for Neumann boundary condition.", "From the numerical results, we confirm that for both the problems the main term $E_1$ of the error estimation (REF ) becomes dominant when $h \\le 1/128$ , which agrees with the analysis in Remark REF ." ], [ "L-shaped domain", "The proposed error estimation (REF ) is applicable to problems with a singular solution and the even case in which the subdomain $S$ and $\\Omega $ share a common part of the boundary.", "In this sub-section, we consider the boundary value problem over an L-shaped domain $\\Omega := (-0.5,0.5)^2 \\setminus [-0.5,0]^2$ ; see Figure REF .", "The error estimation on two subdomains $S=\\Omega \\cap (-0.125,0.125)^2$ and $S^{\\prime }=(0.25,~0.5)^2$ will be considered.", "Let $u=r^{\\frac{2}{3}} \\sin {\\left( \\frac{2}{3} (\\theta + \\frac{\\pi }{2})\\right)} \\cos {(\\pi x)}\\cos {(\\pi y)}$ , where $r$ and $\\theta $ are the variables under the polar coordinates.", "Define $f=-\\Delta u$ .", "Then $u$ is the solution of the following equation.", "$- \\Delta u = f \\mbox{ in } \\Omega ,\\quad u = 0 \\mbox{ on } \\partial \\Omega .$ It is easy to confirm that $u \\notin H^2(\\Omega )$ due to the singularity around the re-entry corner point of the domain.", "Selection of the bandwidth of $B_{S}$ the dependency of $\\overline{E}_{L}$ on the bandwidth of the $B_S$ is shown in Figure REF .", "In the following computation, the bandwidth of $B_S, B_{S^{\\prime }}$ is selected as $0.375$ , $0.25$ , respectively.", "Figure: Dependency of local error estimation on the bandwidth of B S B_{S} (L-shaped domain).For subdomain $S$ , the asymptotic behavior of $\\overline{E}_{L}$ with respect to mesh size $h $ is shown in the Table REF and Figure REF .", "The numerical results tell that the local error component $E_1$ in $\\overline{E}_L$ gradually becomes dominant as the mesh is refined.", "Table: Error estimators for subdomain SS (uniform mesh of L-shaped domain).Figure: Error estimators for subdomain SS (uniform mesh of L-shaped domain).We also compare $E_L$ , $\\overline{E}_L$ with $E_G$ , $\\overline{E}_G$ in Table REF .", "Denote $\\beta :=E_L/E_G$ and $\\overline{\\beta }:=\\overline{E}_L$ /$\\overline{E}_G$ .", "It is observed that the approximation error concentrates in the subdomain $S$ around the re-entry corner as the mesh is refined.", "For $h = 1/256$ , the local error in $S$ is about $91\\%$ of the global error in the whole domain.", "Table: Comparison of the estimated local error on SS and S ' S^{\\prime }.Finally, we consider the local error estimation for a non-uniform mesh; see computation results Table REF and Figure REF .", "It is observed that for the subdomain $S$ , both $\\beta $ and $\\overline{\\beta }$ become smaller compared to the results in the case of uniform meshes, which implies that a denser mesh around the re-entry corner improves the quality of local approximation.", "Table: Error estimators for subdomain SS (non-uniform mesh of L-shaped domain)Figure: Error estimators for subdomain SS (non-uniform mesh of L-shaped domain)." ], [ "Conclusion and future work", "In this study, we proposed an explicit local a posterior error estimation for the finite element solutions and performed numerical experiments on the boundary value problem of the Poisson equation, defined on the square and L-shaped domains.", "The numerical results show that the proposed method provides an efficient estimation of the local error, especially compared to the general overestimated global error estimation.", "In future, we will further apply the local error estimation to the four-probe method used in resistivity measurement.", "Another promising approach for the explicit point-wise error estimation needed by the four-probe method is to apply the idea of [6] and the hypercircle method to the finite element method." ] ]
2105.11777
[ [ "Dense Regression Activation Maps For Lesion Segmentation in CT scans of\n COVID-19 patients" ], [ "Abstract Automatic lesion segmentation on thoracic CT enables rapid quantitative analysis of lung involvement in COVID-19 infections.", "However, obtaining a large amount of voxel-level annotations for training segmentation networks is prohibitively expensive.", "Therefore, we propose a weakly-supervised segmentation method based on dense regression activation maps (dRAMs).", "Most weakly-supervised segmentation approaches exploit class activation maps (CAMs) to localize objects.", "However, because CAMs were trained for classification, they do not align precisely with the object segmentations.", "Instead, we produce high-resolution activation maps using dense features from a segmentation network that was trained to estimate a per-lobe lesion percentage.", "In this way, the network can exploit knowledge regarding the required lesion volume.", "In addition, we propose an attention neural network module to refine dRAMs, optimized together with the main regression task.", "We evaluated our algorithm on 90 subjects.", "Results show our method achieved 70.2% Dice coefficient, substantially outperforming the CAM-based baseline at 48.6%." ], [ "Introduction", "THE coronavirus disease 2019 (COVID-19) has been declared a global pandemic since March of 2020.", "The total number of infected cases has reached over 83 million worldwide, with 1.8 million deaths by 2020.", "Unfortunately, both numbers are still increasing.", "To reduce the fatality rate, effective diagnosis and treatment planning are essential.", "As COVID-19 mainly damages the lungs of infected subjects, chest Computed Tomography (CT) plays a critical role in rapid diagnosis and progression monitoring of the COVID-19 infections.", "Based on chest CT analysis, standardized CT scoring systems, such as the COVID-19 Reporting and Data System (CO-RADS) [1], were defined to quantify the degree of suspicion of COVID-19 according to CT findings into 1-5 scores with an increasing level of suspicion.", "Similarly, a CT severity scoring system [2] was designed to assess the extent of parenchymal involvement of the disease.", "These scoring systems may be applied more accurately and rapidly when the automatic segmentation of infected areas (lesions) would be available.", "Therefore, this work aims at developing an algorithm that can automatically segment lesions related to COVID-19 on chest CT scans.", "One of the major obstacles of semantic segmentation is the need to acquire a large amount of voxel-wise annotations for training the networks, which is particularly challenging when facing a new problem such as COVID-19.", "The lack of training data makes state-of-the-art supervised methods impractical.", "Therefore, in this work, we present a novel weakly-supervised segmentation method that only requires lobe-wise severity scores as the input reference for training and can produce dense and precise localized lesion maps that can be used as lesion segmentations.", "Weakly-supervised semantic segmentation has been extensively studied in recent years.", "In the weakly-supervised setting, reference annotations can be provided using scribbles [3], or surface points [4].", "Both these approaches seek a trade-off between annotation efforts and the amount of training information provided to the network regarding shapes and locations of target objects.", "However, because typical COVID-19 CT abnormalities often have bilateral lung involvement with a peripheral and diffuse distribution [5], manually annotating scribbles or extreme points could still be very demanding.", "To reduce the annotation cost further, it is favorable to use only image or region-level labels.", "Early weakly-supervised segmentation methods using image-level labels were based on multi-instance learning frameworks [6] and expectation-maximization algorithm [7].", "The current state-of-the-art weakly supervised segmentation methods using image-level labels were based on class activation maps [8], [9], [10], [11] (see recent results on PASCAL VOC2012 benchmark).", "CAMs correspond to the regions responsible for distinguishing image categories in a classification task.", "Because CAMs naturally only represent discriminative regions and may not fully cover or detect all objects, iterative approaches [11], [12] were proposed to erase already-found object maps in the previous iteration and force the network to discover new and complement regions at later iterations.", "One major drawback of CAMs is that they are generated by taking high-level convolution features (at the bottom of the convolution neural network, usually before global pooling and linear layers) and multiplying them with class-specific weights in the linear layer.", "These high-level features contain rich semantic information but are generally at a low resolution compared with the input.", "The use of low-resolution features causes CAMs to lose local details, which is problematic since segmentation requires dense voxel-wise predictions.", "In addition, CAMs intrinsically reflect classification decisions, which are not necessarily aligned with the object segmentation task.", "Instead of using low-resolution features, BagNet [13] resorted to features in the earlier layers of convolution neural networks for extracting CAMs.", "Their method may indeed produce fine-resolution CAMs.", "However, low-level features do not suffice to represent complex objects without high-level semantics, leading to possibly very noisy CAMs.", "Another research direction is to use CAMs generated at a low resolution only as the initial seed regions.", "Extra steps were needed to refine CAMs for generating object segmentations.", "A seeded region growing module was proposed in [8] to expand CAMs towards the complete object boundaries in an iterative manner.", "AffinityNet [9] exploited local inter-pixel affinities as the transition probability matrix and applied random walks to revise CAMs.", "Many of these CAM refinement methods were implemented as post-processing steps.", "Therefore their hyperparameters were tuned separately from the neural network training.", "For instance, random walks based on trained voxel-wise affinities were executed in separate post-processing steps to refine CAMs in AffinityNet [9].", "Instead of relying on early layer features or refine CAMs in post-processing steps, we propose to train a segmentation network directly to generate high-resolution dense regression activation maps (dRAMs).", "We present a network trained for regressing the per-lobe lesion percentage.", "We used implied lesion percentage information from the lobe-wise severity scores, as typically provided by radiologists.", "When annotating lobe-wise severity scores, radiologists measure the lesion percentage per lobe and assign a corresponding score if the ratio falls in a specified range (Table REF (b)).", "This lobe-level supervision limits lesion searching to lobes, which is considerably easier than that using scan-level labels in [14].", "Meanwhile, the per-lobe lesion percentage is a richer type of information regarding the lesion volume, and such an approach was not used in previous CAMs-based classification approaches based on categorical labels.", "Because the per-lobe lesion percentage was defined as an interval given a lobe-wise severity score, we propose an interval regression loss to enforce the predicted percentage to fall in a particular range.", "Furthermore, we introduce an attention module for revising dRAMs, trained together with the regression task.", "The refinement of dRAMs is necessary because the regression target does not provide information regarding the object boundary.", "Inspired by AffinityNet, we intended to capture local voxel-wise affinities in the attention module, which enriches object semantics using neighboring information in revising dRAMs.", "Our key contributions are as follows: 1) we propose a lesion segmentation framework that produces fine-resolution segmentation maps using only lobe-wise labels in training; 2) we convert the lesion segmentation problem to regression of the per-lobe lesion percentage defined by the lobe-wise severity score.", "The regression problem is solved using a proposed interval regression loss.", "These ideas are generic and can be extended to other weakly-supervised segmentation problems if specific statistics of the object segmented are available as the regression target; 3) we refine the dense regression activation maps using an attention neural network module and dense conditional random field trained together with the main regression target.", "There are recent works on COVID-19 lesion segmentation that attempted to reduce the demand for voxel-level supervision in training.", "Fan et al.", "[15] proposed a semi-supervised training strategy that requires a few labeled images to train the initial segmentation model and leverages primarily unlabeled data to fine-tune the model progressively.", "Laradji et al.", "[16] proposed to use point-level labels in an active learning schema to generate lesion segmentation maps.", "Yao et al.", "[17] superimposed synthesized lesions on healthy CT scans for their network to learn to separate high-intensity structures such as vessels from possible COVID-19 lesions.", "Xu et al.", "[18] proposed a generative adversarial learning framework to segment COVID-19 lesions, which primarily relied on scan-level labels and used a small amount of voxel-level labeled data to initialize training.", "Wang et al.", "[14] proposed to train a binary CNN classifier based on the presence of COVID-19 on CT scans and used the classifier to generate CAMs for lesion localization.", "However, without the refinement of CAMs, their approach was limited to lesion localization rather than segmentation.", "Our approach is closely related to CAMs-based weakly supervised segmentation approach.", "We differ in three general building blocks of these methods: 1) the generation of CAMs by training a convolution neural network, often in a classification task.", "2) regularization, as an ill-defined problem, weakly-supervised semantic segmentation based on image labels requires networks to localize objects, whereas such information is not given in the supervision.", "The loss of location information may cause training to converge to a trivial solution.", "Therefore, regularization is necessary.", "3) CAM refinement is needed because CAMs do not necessarily reflect object shapes and align with the object boundaries.", "In these three aspects, our approach generates dense class regression maps (dRAMs) by training a segmentation network towards a regression target, which is the major novelty of this paper.", "In terms of regularization, we use entropy loss to ensure the dRAMs differentiate foreground and background as confident as possible.", "Meanwhile, we use equivariant regularization proposed in [10] to improve training consistency in a self-supervised learning schema.", "The idea is to introduce an inherent constrain that dRAM produced by an affine-transformed input should be similar to affine-transformed dRAM produced by the original input.", "In terms of the refinement, our method is motivated by the use of local voxel affinities in [9], where authors proposed a network to predict semantic affinities among local pixel pairs.", "Then CAMs were refined by iteratively running random-walks in which the probability transitional matrix was derived from semantic affinities.", "A similar but end-to-end solution can be found in [10] where an attention module was designed for capturing global pixel affinities, which can be trained together with the main classification task.", "The dRAMs refinement process in our method is closely related to [10] and [9], while the major difference is that we capture voxel affinities via attention maps inspired by [10], but computed within a local neighborhood similar to that in [9].", "The use of local affinities is because we intended to rely on local details in revising dRAMs." ], [ "Data", "In this study, we used CT scans from patients who presented at the emergency wards of the Radboud University Medical Center, the Netherlands, from March to September 2020.", "Patients were referred for CT imaging because of suspicion of moderate to severe COVID-19 pneumonia.", "The ethical review board approved the retrospective and anonymous collection of this data (Radboudumc CMO2016-3045, Project 20027).", "All CT scans were obtained with a low-dose thin slice protocol without administration of contrast.", "Further details can be found elsewhere [19].", "Following the guidelines of the Dutch Radiollgical Society [1], the radiology report for each scan contained CO-RADS and lobe-wise severity scores.", "CO-RADS 1 is defined as a scan that is normal or has non-infectious etiologies, and thus a very low level of suspicion for COVID-19.", "CO-RADS 2 indicates the CT-scan has features typical for infections other than COVID-19.", "CO-RADS 3 indicates equivocal findings: features compatible with COVID-19 but also with other diseases.", "CO-RADS 4 and 5 indicate a high and very high level of COVID-19 suspicion, respectively.", "CO-RADS 6 was given to scans from patients that were already known to be positive for COVID-19 with reverse transcription-polymerase chain reaction (RT-PCR) tests at the time of reporting.", "Lobe-wise severity scores indicate the extent of lobar involvement of the COVID-19 infection.", "A score from 0 to 5 is assigned to each lobe according to the visually assessed lesion percentage of that lobe.", "The total CT severity score is the summation of lobe-wise severity scores.", "The mapping between lobar severity score and lesion percentage per lobe can be found in Table REF (b).", "We used lobe-wise severity scores as the weak labels in training our models." ], [ "Data Selection and Partitioning", "For this study, we selected 391 subjects (randomly split into 322 for training and 69 for testing).", "This selection included all subjects that were available when this project started.", "A single scan was used for each subject.", "Thirty subjects in the training set were used as the validation set during model development to prevent over-fitting.", "The distribution of CO-RADS and lobe-wise severity scores is provided in Table REF (a and b).", "In addition to this primary data set, we later randomly selected another 435 subjects not included in the primary data collection.", "Their baseline CT scans were all reported with a total severity score of 0 and CO-RADS 1.", "These 435 CT scans were used as an auxiliary data collection for training our vessel segmentation network (see Sect.", "REF )." ], [ "Reference Standard", "For evaluating our method, lesion segmentation references on 69 test scans in the primary collection were obtained from Thirona (Nijmegen, the Netherlands), a medical image analysis service company specializing in chest CT analysis.", "First, lung parenchyma regions with a higher attenuation were identified by thresholding and morphologic operations.", "Automatic methods were used to suppress vessels and airways.", "Following the radiology report, lesion candidates in lobes not affected by COVID-19 were then removed.", "A certified image analyst with at least one year of experience reviewed the remaining lesion candidates and corrected segmentations where necessary.", "The analysts also labeled segmented lesions into ground-glass, consolidation, and mixed to evaluate segmentation performance for different lesion subtypes.", "During the annotation process, the analyst could consult a radiologist in cases of doubt." ], [ "Weakly-supervised segmentation framework", "The overview of the proposed weakly-supervised lesion segmentation framework is shown in Fig.", "REF .", "We first trained a regression network to predict the lesion percentage per lobe and in the process we generate the dense regression activation maps (dRAMs).", "Since the lobe-wise severity scores (Table REF (b)) represent a lesion percentage per lobe in a range (a severity score 1 indicates the percentage of lesion involvement in the lobe is within the range of 1%-5%, e.g.", "), we propose an interval regression loss for training the regression network.", "In addition to the regression, dRAMs are refined in an auxiliary training task that employs a dense conditional random field and an attention mechanism.", "Also, an independently trained vessel segmentation model was used in the refinement process to suppress false detected vessels.", "Moreover, regularization techniques were used to stabilize the regression training.", "Finally, we used the refined dRAMs as pseudo segmentation references for training a segmentation network from scratch.", "The following subsections elaborate on each of these steps.", "The dense regression activation map is generated by training a regression network for predicting the lesion percentage per lobe.", "We used standard 3D U-Net [20] as the regression network (detailed parameters in Fig.", "REF (a)) because of its simplicity and robustness in various medical segmentation tasks.", "The 3D U-Net has three down-sampling layers in the encoding path, and each layer consists of two convolutions and a max-pooling operation.", "Following the down-sampling path, two more convolutions are used to double the number of convolution filters.", "In the up-sampling path, three layers are used to reconstruct the resolution, and each contains one tri-linear interpolation, followed by two convolutions to reduce the interpolation artifacts.", "Before the final one, convolution kernels have $3\\times 3 \\times 3$ kernel size, a stride of 1 voxel, and zero-padding.", "The last convolution is a $1\\times 1 \\times 1$ convolution to squeeze features to have a single channel before applying sigmoid activation.", "The network takes an $80\\times 80\\times 80$ chunked image as the input, which is cropped around each segmented lobe and resized.", "The segmentation of pulmonary lobes was done using a publicly-available algorithm [21].", "The input and the output size of the 3D U-Net are identical as we used zero-padding.", "For each lobe chunk input, the region outside the lobe of interest was set to zero.", "The output of this network is referred to as the dense regression activation map (dRAM).", "The lobe chunk image as the input allowed us to compute the lesion percentage for the given lobe by simply averaging the dRAM over all voxels within the lobe (lobe-wise mean pooling).", "The reference for regression training was the lobe-wise severity score reflecting only a particular interval of the per-lobe lesion percentage (see Table REF (b) for the mapping between lobe-wise severity scores and lesion percentage per lobe).", "Therefore, we propose an interval regression loss that enforces the predicted percentage fall into a corresponding predefined range.", "Denoting the predicted lesion percentage as $\\hat{y}$ , the lower bound and the upper bound of the percentage range defined by the severity score as $r_{l}$ and $r_{u}$ , we defined the interval regression loss function to minimize as $\\begin{array}{l}max(0, (\\hat{y}–0.5*(r_{l} + r_{u}))^2-K ), \\\\\\mbox{where~} K = (0.5 * (r_{l} – r_{u}))^2.\\end{array}$ This can also be interpreted as the quadratic version of the piecewise linear loss function that minimizes $|\\hat{y}–r_{l}| + |\\hat{y}-r_{u}|-|r_{u}-r_{l}|$ .", "The interval regression loss is weighted on each instance (a lobe image chunk) by the reverse frequency of the corresponding severity score in the training set." ], [ "Regularization techniques for regression training", "Training on weak labels may converge to a trivial solution because information on the location of objects or, in our case, abnormal regions, is not available.", "Therefore, regularization techniques are commonly used to stabilize training.", "Wang et al.", "[10] introduced an implicit equivariant constraint for training their weakly supervised segmentation networks based on class activation maps (CAMs).", "Their basic idea was to enforce CAMs produced by an affine-transformed input be similar to the affine-transformed CAMs produced by the original input.", "Denote the 3D U-Net for regression training as $F(\\cdot )$ , a predefined spatial affine transformation as $A(\\cdot )$ , and an input image to the network as $I$ .", "Then $F(I)$ represents the dRAM.", "Equivariant regularization can be formulated as $R_{ER}=||F(A(I)) - A(F(I)) ||_{1}.$ Equivariant regularization can also be interpreted as a way to introduce the self-supervising correspondences between affine-transformed objects.", "The affine transformations used in this paper are resizing and rotation at a random scale or angle.", "Additionally, we introduce entropy regularization to reduce uncertainty in the generated dRAM.", "Given dRAM as $F(I)$ , which is already rescaled into a probability distribution by a sigmoid activation before lobe-wise average pooling, we introduce an entropy regularization term that minimizes $R_{E}=-F(I) * \\log (F(I)) - (1.0 - F(I)) * \\log (1.0 - F(I)).$" ], [ "Vessel segmentation", "During our initial experiments, we observed that raw dRAMs may erroneously include vessels, possibly caused by the interval regression target since this only defines a range of acceptable per-lobe lesion percentages.", "To suppress vessels in our framework, we trained a separate 3D U-Net segmentation network (using the same architecture as our regression network) on the auxiliary data collection of 435 CT scans without COVID-19 CT signs (details in REF ) for segmenting vessels.", "Here the training references were generated by applying Otsu's threshold [22] on vessel maps generated by a Frangi filter [23].", "Using this model, we generated vessel segmentations for all 322 training scans in the primary data collection, taking voxels with confidences above 0.7 as vessels from the model prediction.", "The predicted vessels were used as object cues in our framework to suppress false lesions.", "Such low-level features were commonly exploited in weakly-supervised semantic segmentation.", "As an example, Wei et al.", "[24] used saliency maps based on low-level image features to train convolutional segmentation networks in the weakly supervised settings in a progressive manner." ], [ "Refinement of dense regression activation maps", "As dRAMs were obtained by training on a regression target and in this process, no voxel-wise supervision was provided, raw dRAMs may not suffice to delineate lesions accurately.", "To alleviate this issue, we proposed to refine dRAMs using an auxiliary training objective.", "We post-processed dRAMs by suppressing vessels using vessel segmentations (REF ) and applying dense conditional random field (denseCRF) on dRAMs.", "The post-processed dRAMs were used as the training target to provide voxel-wise supervision in the refinement step.", "As shown in Fig.", "REF , vessels were suppressed in the dRAM, and dense conditional random field helped to refine the lesion borders in the refinement target (the 4th row).", "We used a bootstrapping loss [25] for the refinement training because both dRAMs and vessel segmentations were generated by automatic methods and may contain noise.", "The bootstrapping loss function minimizes $\\begin{array}{l}\\sum _{k=0}^{L}[\\beta t_{k} + (1-\\beta )z_{k}]log(q_{k}) \\\\z_{k} = 1[k=argmax(q_{i})],i=0,1,..,L\\end{array}$ where $L$ is the number of classes (3 in our case, including background, vessel, and lesion), for the class label $k$ , $t_{k}$ is one-hot encoding pseudo reference and $z_{k}$ is the bootstrapping reference.", "$q_{k}$ is the softmax probability of assigning a voxel into the class $k$ .", "$\\beta $ is set to 0.8.", "Note that we detached the computation of the bootstrapping reference $z_{k}$ in the gradient back-propagation.", "The idea of this loss is to leverage the knowledge learned by the network during training to provide hints for the true labels.", "To further improve the dRAM refinement, we added an attention module on top of the generated dRAMs from the regression training.", "This module calculated local affinities using low-level convolution features from the regression network and image intensities.", "First, the input image was concatenated with convolution features before pooling at the first and second layers of the regression network (up-scaled to the same size as the input image).", "This concatenation is for capturing the low-level information, including raw voxel intensities.", "We detached these concatenated features out of the back-propagation computation.", "We then reduced the concatenated features to have eight channels via a $1\\times 1\\times 1$ convolution filter to save computational memory.", "Denote this reshaped feature map as $x$ , $x \\in R^{D\\times W\\times H\\times 8}$ .", "The affinity $a(x_{i}, x_{j})$ between two locations $i$ and $j$ in $x$ can be computed in an embedded Gaussian function $a(x_{i}, x_{j}) = e^{(W_{\\theta }x_{i})^T(W_{\\phi }x_{j})}$ , where $W_{\\theta }$ and $W_{\\phi }$ are linear transformations without changing dimensions of the input.", "Local affinities for a location $i$ were measured by computing pairwise $a(x_{i}, x_{j})$ between $i$ and surrounding locations $j$ in a $3\\times 3\\times 3$ spatial window taking $i$ as the center with a connectivity of 2, resulting in total 19 neighboring pairs including the self-connection.", "Computing local affinities for all locations in $x$ resulted in a attention map $A$ , $A \\in R^{D\\times W\\times H\\times 19}$ .", "Local affinities for location $i$ were normalized over neighboring locations $\\Omega {j}$ of $i$ .", "The normalizing factor $\\zeta (x)$ was simply a summation $\\zeta (x)=\\sum _{j\\in \\Omega {j}}a(x_{i}, x_{j})$ .", "In matrix form, this normalization is equivalent as applying softmax function over the last dimension on $A$ .", "To revise dRAM $y$ , we first project dRAM into a subspace by $g(y)$ implemented as the linear projection $W_{g}^{T}y$ , and apply matrix multiplication between the projected dRAM and the attention map $A$ .", "This matrix multiplication can be seen as each location in dRAM selectively collecting information from its local neighbors.", "The impact from local neighbors in updating a dRAM location was determined by their pairwise affinities.", "This is very similar to propagate messages from local neighbors using random-walks for refining CAMS in [9].", "Since the updated dRAM is in a subspace, we projected it back using a linear transformation $r(\\cdot )$ or $W_{r}$ in the form of matrix multiplication.", "The whole process of using local affinities to refine dRAM $y$ in the location $i$ can be formulated as follows $\\hat{y_{i}} = r(\\frac{1}{\\zeta (x)}\\sum _{\\Omega {j}}a(x_{i}, x_{j})g(y_{j})) + y_{i}.$ The use of residual connections allows gradients to flow through a network directly if the dRAM already provided a good segmentation.", "Note that dRAM refinement branch in Fig.", "REF (b) and the segmentation branch in Fig.", "REF (a) were trained simultaneously along with the regularization loss terms.", "The total loss is the weighted average of the regression term, regularization terms, and the refinement term.", "The weight for the main regression target was set to 2.0, and the rest of the learning objectives were weighted by 1.0." ], [ "Context aggregation", "In this step, a lesion segmentation network was trained from scratch using pseudo lesion and vessel labels.", "This segmentation network is a standard 3D U-Net, the same as the regression network.", "This step is necessary because the regression network may overlook features across lobes due to the use of lobe chunk images as the network input.", "Therefore, this step constructs a scan-level dRAM by filling lobe-wise refined dRAMs back to the scan from which the lobe chunks were cropped.", "The scan-level dRAM was used as the pseudo lesion label.", "We re-sampled scans and pseudo labels into an isotropic spacing of 1.5 millimeters to set the receptive field of the network to 132 millimeters (88 voxels in the 3D U-Net).", "The input to the network is a mini-batch of two $132\\times 132 \\times 132$ 3D image chunks, randomly cropped from the scan during training, and the corresponding output chunks in size of $44\\times 44 \\times 44$ due to valid-padding in convolutions (this is the only difference compared to what we use in the regression network).", "The pseudo vessel and lesion labels were stacked together channel-wise, where 0 indicates background, 1 denotes the vessels, and 2 indicates lesions.", "Due to the possible noise in the pseudo labels, the final segmentation training adopted the bootstrapping loss function (Eq.", "REF ).", "Because the final segmentation network was trained in a patch-based fashion, the softmax probabilities of all 3D output chunks are tiled together by sliding over the entire scan without overlap to build up a scan-level probability map.", "The lesion prediction was assigned if a maximum probability was found on the 2nd channel in the softmax probability map." ], [ "Lesion segmentation in different subcategories", "Given the lesion segmentation results, we further labeled each connected component in the segmentation into one of three subcategories: ground-glass, consolidation, and mixed.", "We adopted a non-parametric approach based on Kullback–Leibler divergence (KLD) for the similarity measurements.", "We had analysts manually segment and label COVID-19 lesions in six selected CT scans from the validation set in our main data collection, including lesions with all three subcategories.", "We performed connected component analysis for these reference scans using the segmentation references and computed the mean and standard deviation for all the components.", "For each test scan, the same connected component analysis was applied to the segmentation map of our method.", "Moreover, we looped through all test components to compute the mean and standard deviation.", "Assuming that intensity values were Gaussian distributed for each component, we can compute pairwise KLDs between a test component and all components in the labeled six segmentation references to measure the similarity between components.", "Due to the impact of the component size in computing the statistics, we first shortlisted $K$ components in the references with the smallest differences in lesion volumes ($K$ =10) to a test component.", "Among shortlisted components in the references, components with the smallest $N$ KLD were selected for weighted label voting ($N$ =5).", "Weights were determined by their rankings in their KLD similarities to the test component.", "KLD for two Gaussians is defined as: $KLD(p,q) = log(\\frac{\\sigma _{2}}{\\sigma _{1}}) + \\frac{\\sigma _{1}^{2}+(\\mu _{1}-\\mu _{2})^{2}}{2\\sigma ^{2}}-\\frac{1}{2},$ where $q$ and $p$ are distributed under $N(\\mu _{1},\\sigma _{1})$ and $N(\\mu _{2},\\sigma _{2})$ , respectively." ], [ "Training and testing details", "Training, validation, and testing of each experiment were carried out on a machine with an NVidia TitanX GPU with 12 GB memory.", "The methods were implemented using Python 3.6 and the Pytorch 1.1.0 library [26].", "The trainable parameters of each method were initialized using Kaiming He initialization [27] and were optimized using stochastic gradient descent with a momentum of 0.9, and the initial learning rate is set to $10^{-5}$ .", "Dense conditional random field was implemented using the Pydensecrf python library [28]https://github.com/lucasb-eyer/pydensecrf.", "During training and testing, CT scans were standardized by clamping intensity values to the $\\left[ -1200\\sim 300\\right]$ range before re-scaling into $\\left[ 0\\sim 1\\right]$ .", "We segmented lobes using an automated algorithm proposed in [21] on all the CT scans, used for masking out regions outside the lobes during the training and the testing.", "We applied random flip, resampling, and contrast stretching as data augmentation methods during all methods training.", "We resampled input scans into an isotropic spacing of 1.2 millimeters (with a small random jittering) by tri-linear interpolation for training regression networks.", "Input chunk images were rescaled using tri-linear interpolation by a factor of 0.8, 1.0, 1.2, and 1.5 when running the regression network at a test time to generate multi-scale dRAMs.", "These dRAMs were merged by averaging.", "For training the final segmentation network, we resampled the scans and pseudo labels into a fixed isotropic spacing of 1.5 millimeters by tri-linear interpolation for both training and testing.", "Not using multi-scale input images and test ensembles guarantied the runtime efficiency of our final model." ], [ "Evaluation Metrics", "The Intersection over Union (IoU), also known as the Jaccard index, between predictions and segmentation references, was used to evaluate segmentation performance.", "The IOU between two binary masks $X, Y$ is defined as: $IoU(X, Y) = \\frac{|X\\cap Y|}{|X\\cup Y|}.$" ], [ "The fully-supervised method", "To evaluate the segmentation performance of a model trained on voxel-wise labels, analysts annotated lesions on 108 scans from our training set in the main data collection using the same protocol as defined in REF .", "We trained a 3D U-Net based on the nn-UNet [29] framework, which has shown superior performance in many medical image segmentation challenges.", "The framework itself is an implementation of U-Net but took advantage of model ensembles (2D, 3D, and Cascading U-Nets), rich data augmentation techniques, and the combination of state-of-the-art segmentation loss functions.", "In their framework, all of these configurations are automatically adapted during training to data at hand.", "We resampled scans into 1.5 millimeters isotropic spacing by tri-linear interpolation for training and testing the nn-UNet." ], [ "Weakly supervised methods", "We used the standard CAM-based weakly-supervised segmentation as the baseline approach, where the network is only the down-sampling path of the 3D U-Net.", "During training, CAMs were generated by applying the fully-connected layer on the lowest resolution features before the global average pooling.", "The network was trained to classify each lobe chunk image into two classes: positive if lesions are present and negative if not.", "Training and testing hyperparameters (data augmentation, resampling of input images, test ensembles and model initialization, e.g.)", "were the same as those used in the proposed method.", "During testing, lobe-wise CAMs were tiled back to become a scan-level CAM, which was rescaled into $\\left[ 0\\sim 1\\right]$ by subtracting the minimum and dividing the maximum.", "The lesion segmentation was obtained by binarizing the rescaled scan-level CAM using Otsu's threshold.", "Because of using features at a low resolution, results from raw CAMs were far from the segmentation expectation.", "To obtain a reasonable performance, we applied Otsu's segmentation method within the lung area to obtain high-attenuation in the lung as the lesion proposal after excluding vessels predicted by the trained vessel segmentation network (see Sect.", "REF ).", "Then CAMs were refined by excluding regions outside the lesion proposal.", "The result from the refined CAMs was denoted as CAM in Table (a).", "Results from both raw and refined CAMs can be found in the ablation study in Table (b).", "The baseline result refers to the result using refined CAMs.", "To evaluate the effectiveness of each component in the proposed weakly supervised segmentation framework based on dRAMs, we conducted the following ablation study.", "Raw dRAMs (denoted as raw in Table (b)) were trained using only the regression loss.", "This demonstrates the benefits of using fine resolution features from a segmentation network and training towards the regression target rather than the classification training using CAMs.", "The importance of proper regularization was measured by adding entropy and equivariant losses (denoted as regularizer) to raw dRAMs training.", "On top of using regularizers, refinement steps with or without attention module were both evaluated.", "Additional advantages of using vessel suppression in the refinement step were also assessed on top of the contribution of the attention module.", "Finally, we reported the performance of the proposed method, which was the segmentation model in context aggregation step trained using pseudo lesion and vessel labels (denoted as context in Table (b) and dRAM in Table (a)).", "A Wilcoxon signed-rank test was employed to assess whether the performance difference was statistically significant (p <0.01 with Bonferroni correction).", "The best weakly-supervised approach was significantly better than all ablated alternatives and is shown in bold in Table (b)." ], [ "Quantitative Results", "From the results in Table , the proposed method reaches 0.495 overall IoU compared with 0.336 IoU achieved by the baseline method in weakly-supervised settings trained using only lobe-wise severity scores.", "This performance improvement can also be seen in the segmentation of all lesion sub-categories.", "Meanwhile, both weakly-supervised segmentation methods are outperformed by the nnU-Net method trained on voxel-wise labels because of the rich semantic information embedded in voxel labels.", "The ablation study demonstrates that adding regularizers improves the performance dramatically from 0.395 to 0.435, benefiting from self-supervised training and entropy minimization.", "The refinement auxiliary training task further improves the performance to 0.452, which is caused by the refinement of lesion borders under the guidance of dense conditional random field.", "The attention module in the refinement process further boosts the performance to 0.469 by learning voxel-wise affinities.", "Vessel suppression provided additional improvement by reducing false lesions, forcing the regression training to discover other regions associated with the per-lobe lesion percentage.", "Finally, the context aggregation step recollects contextual features across lobes in a patch-based training, resulting in an IoU of 0.495." ], [ "Prediction of the CT Severity Score", "Based on the segmentation prediction of abnormal regions in each lobe, the algorithm can output the lesion percentage per lobe, which can be translated into lobe-wise severity scores via the mapping in Table REF (b).", "We computed linear weighted kappa scores for the baseline method, the proposed method, and the nnU-Net method trained with voxel-wise labels against the lobe-wise severity scores assigned by the radiologist.", "The kappa scores were categorized as slight, fair, moderate, good, or excellent based on $k$ values of 0.20 or less, 0.21–0.40, 0.41–0.60, 0.61–0.80, and 0.81 or higher, respectively, following [30].", "The $k$ value was 0.392 (95% CI: 0.334, 0.449) for the baseline method, 0.461 (95% CI: 0.399, 0.524) for the proposed, and 0.514 (95% CI: 0.459, 0.570) for the nnU-Net method.", "There was a moderate agreement between the predicted scores by the proposed method and manual scores, but a fair agreement between the scores predicted by the baseline method and manual scores." ], [ "Qualitative Results", "As shown in Fig.REF , the result from raw CAMs often exhibited from substantial over-segmentation (1st row).", "This is because CAMs were computed from the low-resolution features and resampled to the original resolution by interpolation.", "The baseline method (2nd row) using refined CAMs often missed lesions (4th and fifth columns) due to the difficulties of finding optimal thresholds in post-processing CAM refinement steps.", "The proposed method (4th row) generally performed well on lesion segmentation.", "Compared with the nn-UNet results, the proposed method produced more false positives in regions near vessels with low attenuation.", "One reason is that the lobe-wise severity scores only represent an interval of the per-lobe lesion percentage, and not the precise percentage, which potentially allows the network to tolerate certain mistakes.", "This can also cause the method to be less precise for small ground glass lesions (see the subpleural region of the left upper lung in the 5th column).", "On the other hand, the regions near vessels appeared in a similar intensity range as the ground-glass opacities and were possibly related to inflammation caused by COVID-19.", "In general, ground-glass opacities may create challenges in visual recognition.", "This challenge may also cause measurement errors for radiologists in labeling severity scores, further contributing to confusion regarding ground-glass opacities in our method.", "Figure: Segmentation results for six representative test cases represented in columns.", "The 1st row shows six scans represented in a coronal slice.", "At the same slice, the 2nd, 3rd, 4th, and 5th rows show the segmentation results of the raw CAM, the baseline (refined CAM), the proposed and nnU-Net method, respectively.", "The last row shows the segmentation reference." ], [ "Discussion and Conclusion", "We proposed a novel weakly-supervised segmentation method.", "This method is able to train a segmentation network using only severity scores provided for individual lobes, where these scores correspond to a range of percentages of affected regions in these lobes.", "The use of such visually assessed percentages of affected regions is common in radiological scoring systems.", "From only these lobe scores, the network is able to generate dense regression activation maps (dRAMs).", "These dRAMs were refined by aligning with the outputs from dense conditional random fields.", "We also proposed an attention module that enriches the semantic representation at each voxel based on its local neighbors (affinities).", "Pseudo labels were generated based on refined dRAMs and an additional step to remove false responses on vessels.", "The final segmentation was trained from scratch based on the pseudo labels using a bootstrapping loss to handle possible noise in the pseudo labels.", "The proposed method achieved significant improvements in segmentation performance compared with the baseline approach.", "In terms of the prediction of lobe-wise severity scores, the proposed method reached a moderate agreement with the scores assigned by the radiologist, while the baseline method only reached a fair agreement.", "As we showed in our results, the proposed model sometimes produced false positives in the regions near vessels and may miss small ground glass lesions.", "However, as weak labels are cheap to collect, more advanced approaches can be built upon our model using our methods as the initial seed for interactive (e.g., adaptive learning scenarios) or iterative refinement (e.g., knowledge distillation).", "The proposed method is generic and can be easily adapted to other weakly-supervised segmentation problems if specific object statistics are given and can be used as the regression target.", "We believe that the proposed weakly-supervised segmentation framework can be used for many segmentation problems in medical imaging, where automatic segmentation is often used for quantification analysis.", "In these scenarios, visually assessed quantification results from radiological scoring systems can be directly used in our framework as the regression target.", "[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Since 2012, he is full professor of Medical Image Analysis at Radboud University Medical Center and chairs the Diagnostic Image Analysis Group.", "He also works for Fraunhofer MEVIS in Bremen, Germany, and is a founder of Thirona, a company that develops software and provides services for medical image analysis.", "He is member of the Editorial Board of Medical Image Analysis.", "He pioneered the concept of challenges in medical image analysis." ] ]
2105.11748
[ [ "The simple emergence of complex molecular function" ], [ "Abstract At odds with a traditional view of molecular evolution that seeks a descent-with-modification relationship between functional sequences, new functions can emerge {\\it de novo} with relative ease.", "At early times of molecular evolution, random polymers could have sufficed for the appearance of incipient chemical activity, while the cellular environment harbors a myriad of proto-functional molecules.", "The emergence of function is facilitated by several mechanisms intrinsic to molecular organization, such as redundant mapping of sequences into structures, phenotypic plasticity, modularity, or cooperative associations between genomic sequences.", "It is the availability of niches in the molecular ecology that filters new potentially functional proposals.", "New phenotypes and subsequent levels of molecular complexity could be attained through combinatorial explorations of currently available molecular variants.", "Natural selection does the rest." ], [ "Introduction", "Half a century ago, the idea that gene specificity could rely on a unique protein sequence raised concerns regarding the come into being of functional genes.", "Natural selection would be ineffective if the raw material on which it had to act were random sequences, given that a myriad of universes as old and as large as ours, where random sequences would be systematically generated, appeared absolutely insufficient to produce the tiniest functional molecule [1].", "The apparent paradox, whose assumptions echoed intelligent design arguments [2], was obviating the overwhelming number of neutral and quasi-neutral mutations [3], [4], [5] and the fact that partial functionality is better than no functionality at all [6].", "Actually, molecular complexity might be relatively straight to achieve.", "Molecular function is highly redundant.", "The same molecular phenotype can be obtained from an astronomically large number of different genotypes, as revealed by computational and empirical studies of complete genome spaces for short sequences [7], [8], [9].", "Such genotypes are organized as networks-of-networks [10], [11], a non-trivial topology with important dynamic consequences [11] that also points to the need of an updated metaphor to represent adaptive landscapes [12].", "The abundance of phenotypes is not homogeneous in sequence space [13], [14], [8], [15], since most sequences are mapped into a small fraction of very large phenotypes.", "This fraction, however, suffices to guarantee a spectrum of different and efficient enough functions so as to sustain life as we know it.", "The high dimensionality of genotype spaces and the fact that the networks of abundant phenotypes percolate the space of sequences under very general conditions [16] further ensures that different functions may be awaiting just a few mutations apart [17], [18].", "The genotype-to-function map is redundant in several different ways beyond neutral and quasi-neutral mutations.", "The many layers of expression from genotype to function [19] act towards increasing the ensemble of possible genotypes coding for comparable phenotypes [20], [21].", "Further, function is flexible, so phenotypes admit a range of variation, and phenotypes are plastic, so their expression adapts to different environments [22], [23].", "Beyond multiple inconsequential variations in genotypes, also changes in molecular structure or composition might be irrelevant for the functionality of a phenotype.", "Molecular mimicry [24], protein moonlighting [25], [26] or enzyme promiscuity [27] are three widespread expressions of phenotypic redundancy and functional flexibility.", "Molecular mimicry was originally defined in the context of immunology: occasionally, self-peptides are sufficiently similar in sequence so as to be mistaken by pathogen-derived peptides and, consequently, trigger an autoimmune response [24].", "The concept, however, can be extended to embrace the many instances where molecules of dissimilar composition disguise to resemble others' structure [28].", "In protein moonlighting, the same protein may have multiple, context-dependent functions; this property has been also reported for RNA molecules [29].", "Enzyme promiscuity is a conceptually related notion that refers to the ability of an enzyme to catalyze reactions other than those for which it was in principle selected.", "RNA promiscuity naturally appears when structures compatible with a given sequence, but different from the minimum-free-energy secondary structure, are considered [30].", "The former mechanisms illustrate complementary properties of relevance in evolution: on the one hand, molecules of different origins might perform similar functions; on the other hand, the same molecule can be recruited to perform a different function.", "Phenotype tolerance to endogenous and exogenous variation implies that genes and, in general, any sort of functional molecule may not need initial adjustments to engage into a secondary function [31].", "But, once in place, natural selection can act towards optimization, if needed.", "The skewed distribution of phenotype sizes conditions what is visible to evolution [32] but may, under certain circumstances, also facilitate the appearance of simple molecular functions de novo.", "The emergence of function has in all likelihood been relevant in the origin of an RNA world [33], [34] and in the genesis of simple replicators [35].", "Still, single gene or single molecule redundancy, and the selective improvement of their function through point mutations, only represents the fine-tuning of molecular evolution.", "Once originated and optimized, small functional sequences might act as the basic bricks of multi-purpose molecules [36] through a modular constructive principle that applies from proteins [37] to organisms [38], [39].", "Ensembles of agents that replicate independently —which, in a certain sense, may act as competitors— can integrate to form a new, more complex entity [40] through what is known as a major evolutionary transition [41].", "Many of these transitions are cooperative [42], a paradigmatic example at the molecular level being the emergence of chromosomes.", "However, there are multiple examples, especially in the viral world, where flexible cooperation, that is, without irreversible fusion of the parts, appears as a successful adaptive strategy.", "In viral quasispecies, ecological roles can be allocated in different mutant classes [43], and collective cooperation may be needed to maintain pathogenesis [44].", "In viruses, horizontal gene transfer (HGT) is not only a common mechanism for adaptation, but probably also a way to generate new viral species [45].", "A remarkable form of distributed cooperation is that of multipartite viruses, whose genes are propagated in independent capsids [46], [47].", "Viruses are a powerful system of generation of new molecular function.", "Together with other mobile elements, they function as transporters of genomic sequences and may assist their integration in higher organisms.", "In the coevolution of viruses with hosts, the former may even promote increases in host complexity [48] and spur major evolutionary transitions [49].", "In the forthcoming sections, we will focus on specific examples that illustrate how some of the mechanistic principles outlined in this introduction can be used to devise scenarios where complex molecular function could parsimoniously emerge.", "The conceptual framework integrates low-cost pieces with incipient functionality, different cooperative schemes and an ecological context (or an organized molecular environment) responsible for the selective pressures that act on such incipiently functional systems.", "Our aim is to link robust features of the molecular genotype-to-structure map, contingency, and selective evolution, and to discuss how, at the next level, innovation can emerge through distributed cooperation." ], [ "Phenotypic bias: the free lunch of molecular function", "RNA has been studied in depth as a paradigmatic example of the sequence-to-structure map [50].", "It has been shown that the frequency distribution of secondary structures follows a log-normal distribution [14], [15], thus quantifying the bias in phenotype (structure) abundance.", "Accordingly, common phenotypes are orders of magnitude more frequent than average-sized or small phenotypes (which are, by any reasonable measure, invisible to evolution).", "RNA structure is tightly related to molecular function and, as such, it may have played a main role at the early stages of chemical evolution and, especially, in an RNA world predating modern cells [51].", "The repertoire of secondary structures in large populations of short RNA polymers is limited [52]: topologically simple RNA modules are abundant.", "In open RNA chains, there is a predominance of stem–loops (composed of a stem and a hairpin loop) and hairpin structures [53], while closed RNA chains preferentially fold into rod-like structures (a stem closed by two hairpin loops) [54].", "This fact is solely based on thermodynamic principles [55] and implies that, by default, short RNA sequences will predominantly yield a handful of structures.", "The relevant implication of this high redundancy is that some of those abundant structures might be the end result of random RNA polymerization (as it could have been the case in prebiotic environments [56]).", "With no previous selection, random polymers might have thus covered an array of incipient functions.", "Such is the case of a variety of hairpin-like structures able to promote ligation reactions [57], [58], or of hammerhead structures involved in cleavage [59].", "RNA self-ligation might indeed be instrumental in the modular construction of more complex ribozymes, as theoretically proposed [33] and empirically shown [60], [61].", "There are reasons to believe that the severe phenotypic bias of the sequence-to-structure RNA map, as quantified through the log-normal distribution of phenotype sizes, is a general property of genotype-to-phenotype maps [15], [21], [9], [19].", "If this is the case, the scenario described for RNA should hold broadly, and apply to other polynucleotides, to peptides, and to polymers at large.", "The next step in the construction of chemical complexity regards the emergence of cooperative behaviour of some kind [62], such that the proto-functionalities provided by such molecules do not get lost.", "Eigen's hypercycles [63] were an early proposal in this respect that has been deeply explored [64].", "Indeed, some experiments indicate that networks of interacting molecules may have formed early in chemical evolution, pointing also at an intrinsic ability of RNA to evolve complexity through cooperation [65]." ], [ "Emergence of viroid-like replicators", "Genotype-to-phenotype redundancy is an intrinsic property of natural systems [66], [67].", "This redundancy should have been important in the prebiotic origin of chemical function, but also all along evolution: as first enunciated in geology [68], chemical processes should have acted in the same manner and with essentially the same intensity in the past as they do in the present.", "We must add that the effect of such processes is different because it is the molecular ecosystem that has changed.", "For instance, it seems reasonable to assume that hairpin-like RNAs with potential ligase activity are continuously produced in the RNA-rich cellular environment [69], where multiple RNA sequences of various origins might be present: fragments of RNA transcription [70], transient products of RNA degradation [71], pieces of viral genomes [72] or rod-shaped microRNAs [73], [74].", "Natural processes should be inadvertently and steadily generating raw proto-functional sequences.", "However, such proposals will be filtered by purifying selection, by competition with (probably fitter) existing functions or simply by degradation so, at odds with what happens in geological systems, they mostly disappear without a trace.", "With an exception: when an ecological niche is available.", "Viroids are small, non-coding, circular RNA molecules [75].", "Despite having a small genome of a few hundred nucleotides, they behave as competent and persistent replicators in higher plants, apparently their only natural hosts.", "The evolutionary origin of viroids has been a matter of discussion, and various hypotheses have been put forward [76], [77].", "It has been suggested that viroids may be related to other extant cellular RNAs [78], could have originated from retroelements or retroviruses [79], or be ancient relics of a precellular RNA World [80], [81].", "All these possibilities seek the origin of viroids in a previously functional system, assuming shared ancestry and a broad phylogeny linking viroids to other extant functional RNAs [82], [83].", "This assumption has been criticized on the basis that the observed sequence similarity might be spurious [84] (see also referee reports in [85]), attending as well to the high mutation rates of viroids [86], the diversity of their populations [87] and their low sequence conservation [88].", "Actually, it cannot be discarded that modern viroids emerged once eukaryotic cells had evolved [89], [35].", "Inspection of the two families of viroids reveals important differences in their structure and replication cycle, as well as in the interactions with host proteins [90], [91].", "The two families are so different that a polyphyletic origin cannot be ruled out [84].", "In a context of a de novo emergence of rudimentary replicons, phenotypic bias may have played a role: for example, all circular RNAs of length 20 and lower fold into rod-like structures [54], [35].", "Remarkably, the rod-like structure of viroids could be a case of molecular mimicry, since that structure resembles dsDNA and facilitates the recognition by RNA polymerases [81].", "Hence, a small rod could be \"recognized\" by the replicating machinery of the cell, triggering its replication and therefore its differential selection.", "Viroids exhibit a modular structure that has prompted their description as a “collection of structural motifs which play specific functional roles in viroid replication, processing, transport, and pathogenesis” [92].", "The first step in their emergence, as described in the previous paragraph, could have been relatively straightforward, as that of other potentially useful functions in the same or nearby environments.", "Once kick-started, their modularity strongly supports the possibility that different functional features of viroids could have been acquired through recombination with RNAs of different origins transiently present in the molecular ecology where they evolved.", "This is more than a conjecture, since some viroids are known to arise from other viroids through recombination [93], [94].", "A highly plausible case of modular recombination is provided by hepatitis $\\delta $ virus [95], which has a viroid-like non-coding domain [83] linked to a coding domain of independent evolutionary origin [96].", "Despite justified criticism of proposals that assume an unbroken phylogenetic connection between extant viroids and their potential ancestors in an RNA world, there is reason to believe that the molecular niche occupied by viroid-like molecules may have been continuously available (and likely occupied) given a minimal chemical complexity.", "It will be extremely difficult, if not impossible, to solve this question empirically.", "However, it comes to reason that any self-replicating system, however rudimentary, is subject to the emergence of parasites [97], [98], of non-cooperative defectors that use system's resources for their sole benefit.", "The cellular environment is an extremely rich and varied ecology [99] where multiple control mechanisms, up to cell death [100] are acting and, indirectly, thus limiting the selfish escape of functional molecules.", "Another mechanism contributing to the preservation of system's integrity might be the difficulty of newcomers to invade a functional ecology [101].", "Molecules that may potentially occupy a given niche may find it difficult to succeed if the system already has an optimized solution for that function.", "This sort of non-invasibility principle implies a degree of phylogenetic continuity, and endows first-comers with an implicit advantage.", "Extant viroids, as a possible example of such process at a molecular level, might be a combination of contingency (a frozen accident) and continuity as a side-effect of non-invasibility.", "In a viroid-free situation, it is highly likely that similar viroid-like replicators would emerge in short evolutionary time, occupying the vacant niche in the same way that radiating species do [102].", "Macroecological niches enjoy long periods of functional stasis, even in the face of taxonomic variability [103].", "At odds with macroevolution and macroecology [104], an integration of molecular ecology and evolution (and phylogeny) is yet to be worked out." ], [ "Viral gene sharing", "Viruses are extremely abundant, diverse, strict molecular parasites that, perhaps with rare exceptions, infect all cellular organisms on Earth.", "It is difficult to overstate the role that viruses may have played in the construction of our complex biosphere [105], [106].", "They are motors of biodiversity [107], regulate ecosystems and global biogeochemical cycles [108], and constitute a huge reservoir of genetic diversity [109].", "Metagenomic techniques are enormously enlarging the quantity and quality of previously described viral species, and strongly suggest that we have only grasped the surface of viral gene diversity [110], [111].", "Viruses mutate much faster than cellular genes [112]: a viral gene explores in a few thousand years a sequence space comparable to that explored by a typical nuclear gene since the Cambrian explosion [113].", "Single viral populations are organized in viral quasispecies, swarms of mutants where each sequence may differ from each other in at least one mutation [114], allowing a much more efficient exploration of sequence spaces [115].", "Not surprisingly, viral sequences found in metagenomic studies are dominated by rare genes, with up to 90% of DNA reads encoding proteins not found in other cellular organisms, or in other viruses [116].", "The evolutionary freedom enjoyed by viruses may turn them into cradles of functional diversity.", "A substantial part of organismal evolution could be virus driven, since viruses contribute essential (functional) pieces that promote complexity increases in organisms [62].", "Viruses harbor protein domains with folds unknown in cellular organisms, and some of these domains have been transferred to the host [113].", "Massive transfers of genes from virus to host are not uncommon [117], [118].", "Viruses are extreme examples of mosaicism.", "They are truly chimeric in their composition, most often the emergent result of broad and wide viral gene sharing [45] and occasionally puzzles of pieces assembled from the most distant origins [119].", "The identification of common elements with a shared phylogenetic history in large viral groups is relatively limited, with exceptions [120].", "Recent studies suggest that a hierarchical taxonomy of large viral groups is possible [121] though, more often, viral phylogenies are limited to viral cohorts [122].", "The recognition of a phylogenetic signal speaks for the evolutionary continuity of the shared element, but does not inform on the actual composition of viral genomes or on the viral phenotype: the former are dominated by extensive HGT and divergent evolution, the latter by the viral ecological niche, both being interdependent.", "Actually, it is important to distinguish between bona fide phylogenetic elements and genes that have been recently acquired by HGT from contemporary bacteria or from the viral host [119] which, instead of revealing common origin, might represent examples of recent, likely fast, viral adaptation to new niches.", "Modularity is a driving force of viral genome evolution [123] and, probably, also a mechanism for the generation of new viral species.", "Actually, the proneness of viruses to loss, gain, and exchange genes has prompted the representation of large viral groups as bipartite networks with viral genomes and viral genes as the two classes of nodes [45], [124].", "Each genome has as many links as genes it contains, while genes are linked to all of the genomes where they are found.", "From an evolutionary perspective, genes can be in principle classified into different groups, as signature genes (characteristic of one particular group of viruses), hallmark genes (encoding key proteins and shared by overlapping sets of diverse viruses), or orphan genes (found in a single genome) [45].", "However, the bipartite network representation or viral genes and genomes allows the use of multiple quantitative measures that, in the framework of complex networks [125], might both reveal the intimate architecture of gene sharing and point at dominant evolutionary mechanisms.", "For example, the observed scale-free distribution of the number of genomes that contain a given gene suggests a unique underlying generating mechanism.", "Community analyses [126] of the bipartite network corresponding to the double-stranded DNA viruses reveal, however, the existence of non-trivial correlations among genomes that translate into the consistent identification of major viral groups [45].", "Our current knowledge of the virosphere is, as of yet, poor and biased [127], a fact that certainly limits our understanding of its role as source of functional diversity.", "The apparent phylogenetic discontinuity between viral groups that we observe may also result from that poor sampling: a more comprehensive knowledge of viral diversity could bring about a more parsimonious understanding of how viral evolution has unfolded [128].", "With obvious differences, this suggestion recalls the interpretative difficulties (and even the disdain) that surrounded paleontology, due to the incompleteness of the fossil record, until well into the 20th century.", "A history of the world imperfectly kept, in Darwin's words [129], severely delayed the incorporation of paleontology as a discipline proper of evolutionary biology [130].", "This is certainly not the case of virology, but suggests that evolutionary principles inferred from too sparse data might require substantial revision when data becomes more abundant and complete.", "Evolutionary theory is on the move." ], [ "Flexible cooperation and multipartite virus", "The viral world harbors amazing examples of competition and cooperation, both among kin and non-kin.", "Quasispecies are ensembles of genetically related genomes where multiple associations and interactions are possible, and whose composition depends on the joint action of endogenous antagonistic interactions [131], [132].", "Replication at high mutation rates favors the generation of diversity, thus facilitating, in principle, adaptation [133].", "However, too high a mutation rate might hinder the fixation of beneficial mutations [134] and produce an excess of defective genomes.", "Defective interfering particles were originally considered as artifacts of in vitro evolution with a detrimental effect on viral fitness [135].", "But these particles are produced in vivo and play a role, among others, in viral adaptation and in disease progression [136].", "High mutation rates also permit the coincident appearance of mutations with similar beneficial effect in independent genomes, causing clonal interference and potentially delaying adaptation [137].", "The previous effects notwithstanding, the mutation rate is itself subject to selection along evolutionary time, so it is sensible to assume that it has been tuned to favor viral survivability [138].", "Quasispecies diversity is actually needed to maintain specific viral phenotypes which, in agreement with conceptual hypotheses [139], are a collective property of the ensemble.", "A decrease in quasispecies diversity limits its adaptive ability and attenuates its pathogenic potential [44], and hinders the production of new phenotypes through cooperative interactions [140].", "Even within a population, therefore, different genotypes interact non-linearly: the whole is more than the sum of its parts.", "A quasispecies bears an enormous innovative potential that can be explored through genotype combinatorics.", "Population bottlenecks, which are common in viral propagation and facilitate the fixation of mutants [141], could act as filters to explore many random combinations of few genotypes simultaneously, thus benefiting viral phenotypic innovation and, eventually, viral persistence.", "Multipartite viruses take flexible cooperation to the extreme.", "These viruses have their genomes fragmented in a variable number of pieces, from two to eleven, that are encapsidated and propagated in independent particles [46], [47].", "This lifestyle faces the risk of loosing genomic information due to the seemingly small number of viral particles that are transmitted from host to host.", "Until now, the advantages of such genomic organization remain unclear [142], [143], though there are two important factors that may have contributed to the repeated emergence of multipartite viruses in evolution: the possibility to adapt to new hosts through gene copy number variation [144] and the advantage conferred by fast adaptation through rapid associations with non-kin when new niches become available [47].", "The scenario where multipartitism emerges as a successful adaptive strategy of the type first-come first-served is supported by a number of empirical observations.", "First, there was a fast radiation of multipartite viruses when agriculture became common practice [145]; second, the genome of some multipartite viruses has genes originated in different viral families [146]; third, many viruses undergo transient associations with subviral particles, such as virus satellites, that change the viral phenotype [147].", "This flexible cooperation might be a first step to permanent associations in the form of a bipartite virus: there are examples of viral families with virus-satellite associations and bipartite species, as Geminiviruses [148].", "Fourth, multipartite viruses rapidly modify the copy number of each genomic fragment from one host species to another [149].", "Multipartite viruses infect mostly plants, which are often simultaneously infected by viruses of different families [150].", "This permissiveness may underlie the exploration of new associations among viral genes.", "The route to multipartitism, however, should not be unique as, beyond de novo associations, gene duplication or genome fragmentation may also promote the emergence of multipartite viral species [47].", "Multipartitism has appeared multiple times in evolution and, as such, it has to be understood as a successful evolutionary strategy [151]." ], [ "Conclusion", "The genotype-to-function map is many-to-many.", "Many genotypes can code for similar phenotypes and each genotype has the potential to express a variety of phenotypes.", "As a consequence, the map is proto-functional and highly adaptable.", "The exploration of evolutionary innovations, further, is a process that runs in parallel under many different selective conditions.", "Also, what is not useful in a certain context may provide an advantage and thrive in another.", "HGT in its many expressions promotes this distributed assay-select-share-combine process.", "Once a variety of pieces is in place, spontaneous cooperative associations can give rise to new levels of complexity.", "Evolution operates at such long time scales that even detailed observations of the here and now turn out to be insufficient to educate our intuition on the diversity of complex molecular organizations possible, and on the underlying mechanisms.", "It happens often that certain evolutionary pathways are disregarded only because we never considered them as a possibility or never looked for their products.", "An example is the idea of a de novo generation of function: only in the last decade have we uncovered how genes can be generated from non-genic sequences [152], [153], how transposable elements can be \"domesticated\" to perform specific functions in their host [154], how promoters emerge from random sequences [155] and how this can happen even in the absence of sequence diversity, simply through successive cycles of mutation, enrichment and selection [34].", "Evolution is a powerful tinkerer.", "It will use any mechanism that is available, low-cost, and constructive in a very generic way.", "It uses from phenotypic redundancy to gene combinatorics.", "When all these elements are taken into consideration, what comes as a difficulty is to imagine a world devoid of molecular complexity." ], [ "Acknowledgments", "The author is grateful to José A. Cuesta, Ester Lázaro, and Luis F. Seoane for their insightful comments.", "This work has been funded by the Spanish Ministerio de Ciencia, Innovación y Universidades-FEDER funds of the European Union support, under project MiMevo (FIS2017-89773-P).", "The Spanish MICINN has also funded the “Severo Ochoa” Centers of Excellence to CNB, SEV 2017-0712." ] ]
2105.11784
[ [ "On the Rank, Kernel, and Core of Sparse Random Graphs" ], [ "Abstract We study the rank of the adjacency matrix $A$ of a random Erdos Renyi graph $G\\sim \\mathbb{G}(n,p)$.", "It is well known that when $p = (\\log(n) - \\omega(1))/n$, with high probability, $A$ is singular.", "We prove that when $p = \\omega(1/n)$, with high probability, the corank of $A$ is equal to the number of isolated vertices remaining in $G$ after the Karp-Sipser leaf-removal process, which removes vertices of degree one and their unique neighbor.", "We prove a similar result for the random matrix $B$, where all entries are independent Bernoulli random variables with parameter $p$.", "Namely, we show that if $H$ is the bipartite graph with bi-adjacency matrix $B$, then the corank of $B$ is with high probability equal to the max of the number of left isolated vertices and the number of right isolated vertices remaining after the Karp-Sipser leaf-removal process on $H$.", "Additionally, we show that with high probability, the $k$-core of $\\mathbb{G}(n, p)$ is full rank for any $k \\geq 3$ and $p = \\omega(1/n)$.", "This partially resolves a conjecture of Van Vu for $p = \\omega(1/n)$.", "Finally, we give an application of the techniques in this paper to gradient coding, a problem in distributed computing." ], [ "Abstract", "We study a natural question about sparse random matrices which arises from an application in distributed computing: what is the distance from a fixed vector to the column span of a sparse random matrix $A \\in \\mathbb {R}^{n \\times m}$ ?", "We answer this question for several ensembles of sparse random matrices in which the average number of non-zero entries per column, $d$ , is smaller than $\\log (n)$ .", "Key to our analysis is a new characterization of linear dependencies in sparse random matrices.", "We show that with high probability, in certain random matrices, including rectangular matrices with i.i.d.", "Bernoulli entries and $m \\ge (1 + \\epsilon )n$ , and symmetric random matrices with Bernoulli entries, any linear dependency must be caused by one of three specific combinatorial structures.", "We show applications of our result to analyzing and designing gradient codes, replication schemes used in distributed machine learning to mitigate the effect of slow machines, called stragglers.", "We give the first known construction for a gradient code that achieves near-optimal error for both random and adversarial choices of stragglers." ], [ "Introduction", "We study a natural question about sparse random matrices which arises from an application in distributed computing: Question 1 What is the distance from a fixed vector to the column span of a sparse random matrix $A \\in \\mathbb {R}^{n \\times m}$ ?", "That is, for $v \\in \\mathbb {R}^n$ , what is $\\min _{w \\in \\mathbb {R}^m}{|Aw - v|_2}$ ?", "While related questions such as the rank of a sparse random matrix [14], [7] or the distance from a random vector to the span of a random matrix [20] have received considerable attention in the discrete random matrix literature, surprisingly little is understood about this question.", "Our motivation comes from gradient coding in distributed computing, in which the fixed vector is the all-ones vector.", "In this application, the matrix $A$ defines a redundant distribution of $n$ tasks to $m$ machines, such that machine $j$ completes task $i$ iff $A_{ij} \\ne 0$ .", "We are interested in matrices where on average each machine completes $d$ tasks, where $d$ is small.", "This translates to matrices $A$ with $d$ non-zero entries per column.", "We elaborate more on this application and the connection to Question REF shortly.", "Our work addresses Question REF for several ensembles of random matrices where $m \\ge n$ and where $d$ , the average number of non-zero entries per column, is smaller than $\\log (n)$ .", "We emphasise that due to the sparsity of the matrix, with high probability, the matrix does not have full row-rank, and hence Question REF is nontrivial.", "Our main tool in analyzing this distance is a new characterization of all linear dependencies that occur with constant probability in sparse random matrices.", "This characterization builds upon a phenomenon that has been explored in prior work on sparse random matrices: linear dependencies come from small combinatorial structures [10], [9], [23].", "Connection to Gradient Coding Gradient codes are data replication schemes that can be used to provide robustness against stragglers, machines that are slow or unresponsive [21], [19].", "Specifically, such replication schemes can be used to approximately compute the sum $\\sum _i f(X_i)$ of the evaluations of a function $f: \\mathcal {X} \\rightarrow \\mathbb {R}$ over a set of data points $X_1, \\ldots , X_n \\in \\mathcal {X}$ .", "Typically in machine learning, the function $f$ represents the gradient of a loss function.", "To distribute the computation of $\\sum _i f(X_i)$ while maintaining robustness against stragglers, the data points $\\lbrace X_i\\rbrace $ are distributed redundantly among $m$ machines according to an assignment matrix $A_0 \\in \\mathbb {R}^{n \\times m}$ .", "Each machine $j$ is tasked with computing $g_j := \\sum _i (A_0)_{ij}f(X_i)$ , a weighted sum of $f$ over the data is stores.", "A central server then collects these values $g_j$ for all $j$ which are not stragglers, and linearly combines the $g_j$ using weights $w_j$ to best approximate $\\sum _i f_i(X_i)$ .", "Formally, we define the decoding error of an assignment $A_0$ along with a set of stragglers $S \\subset [m]$ to be the squared distance between the all-ones vector, $\\mathbb {1}$ , and the span of the columns $(A_0)_j$ for $j \\notin S$ : $\\textnormal {err}(A_0, S) := \\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|A_0w - \\mathbb {1}|_2^2.$ The decoding error immediately gives us a bound on the approximation error of $\\sum _i f(X_i)$ .", "Let $f^*$ be the vector in $\\mathbb {R}^n$ , where $f^*_i := f(X_i)$ .", "Then for any $w \\in \\mathbb {R}^m$ , $\\sum _j w_j g_j - \\sum _i f(X_i) = f^{*T}A_0w - f^{*T}\\mathbb {1} = f^{*T}(A_0w - \\mathbb {1}).$ Hence for any $S$ , there exists a vector $w \\in \\mathbb {R}^{[m] \\setminus S}$ such that $\\left(\\sum _{j \\in [m] \\setminus S} w_j g_j - \\sum _i f(X_i)\\right)^2 \\le \\textnormal {err}(A_0, S)|f^*|_2^2.$ In this work, we will design assignment matrices which have small error when the set of stragglers $S$ is chosen either adversarially or randomly.", "In the case where $S$ is an adversarially chosen $p$ fraction of the machines, we consider the adversarial decoding error of $A_0$ : $\\max _{S \\in \\binom{[m]}{pm}}\\left( \\textnormal {err}(A_0, S)\\right).$ If $S$ is a random $p$ fraction of the machines, we consider the expected decoding error of $A_0$ : $\\mathbb {E}_{S \\sim \\binom{[m]}{pm}}{ \\textnormal {err}(A_0, S)}.$ Analyzing the decoding error under a random set of stragglers amounts to understanding Question REF when the matrix $A$ is the random matrix achieved by deleting $pm$ random columns from $A_0$ .", "Contributions Our work considers three ensembles of sparse random matrices in the regime $d \\le \\log (n)$ .", "The first is an $n \\times \\gamma n$ rectangular matrix with i.i.d.", "Bernoulli entries with parameter $d/n$ for $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ .", "The second is a symmetric matrix with i.i.d.", "Bernoulli entries with parameter $d/n$ in the upper diagonal portion, for $d = \\omega (1)$ .", "The third ensemble arises from deleting random columns from the bi-adjacency matrix of a random $(d\\gamma , d)$ -biregular bipartite graph for $d \\ge d_0(\\gamma )$ .", "The analysis of the third ensemble is useful for constructing improved gradient codes.", "Our contributions are as follows.", "A tight combinatorial characterization of linear dependencies in the three ensembles of random matrices listed above.", "Our results consider the structure of all minimal linear row dependencies in a matrix, sets of rows which are linearly dependent, but for which any strict subset of the rows is linearly independent (see Definition REF ).", "We show in Propositions REF , REF and REF that with high probability, for the three ensembles of matrices we consider, all minimal linear dependencies among $k$ rows have exactly $2k - 2, 2k - 1,$ or $2k$ total non-zero entries among them.", "We further characterize the structure of these dependencies by viewing the linearly dependent rows as the incidence matrix of a hypergraph (see Figure REF ).", "Progress on Question REF for random matrices with $m \\ge n$ .", "We show how to bound with high probability the distance from the all-ones vector, or more generally, from any incompressible vector, to the column span of any random matrix where our characterization of row dependencies applies.", "A novel gradient code construction with improved decoding error guarantees.", "We construct a gradient code called the Augmented Biregular Code (ABC), and prove that it achieves an expected decoding error of $np^{d - o(d)}$ and adversarial decoding error of $\\Theta (np/d)$ .", "To our knowledge, the ABC is the first gradient code proved to simultaneously have $O(np/d)$ adversarial decoding error and $np^{d - o(d)}$ expected decoding error.", "Existing lower bounds show that the adversarial error of a code must be $\\Omega (np/d)$ , while the expected decoding error under random stragglers is at least $np^{d}$ .", "We describe this construction and result in more detail in Section .", "Related Work In the discrete random matrix theory literature, understanding the probability of invertibility and the rank of discrete random matrices is an active area of research [23], [9], [10], [7], [14], [22], [16], [15].", "A phenomenon core to many of these works is that small structures are the primary reason for linear dependencies in discrete random matrices.", "For instance, in matrices with Bernoulli entries with parameter $p$ , for $\\log (n)/n \\le p \\le 1/2$ , [22] and [16] show that the probability of invertibility is nearly exactly the probability that there is an all-zero row or column (or two equivalent rows and columns when $p = 1/2$ ).", "Further work [7] shows that the probability of having corank $k$ is nearly the probability that $k$ rows are all zero.", "Most similar to our work are two papers by Costello and Vu [10], [9], which consider the adjacency matrices of sparse graphs where $p = \\frac{c\\log (n)}{n}$ for a constant $c \\le 1$ .", "In this line of work, they show that with high probability, all minimal dependencies are due to non-expanding sets, small groups $k < 1/c$ vertices which are incident to at most $k - 1$ distinct vertices.", "Our result for symmetric random matrices, Proposition REF , refines this result.", "In particular, our result pertains to more broadly to any $p = \\omega \\left(\\frac{1}{n}\\right)$ , and we refine the type of minimal dependencies that exist to “tree-dependencies” (Figure REF (a)), which are a subclass of non-expanding sets.", "The question of the distance between a fixed unit vector and the span of a random matrix has received little attention in the literature.", "It is well understood that for rotationally invarant matrices — such as matrices with Gaussian entries — the distance squared is close to the the corank of the matrix with high probability.", "Some work on universality for deformed Wigner matrices [18] suggests that this should extend more generally to mean-zero random matrices with subgaussian entries.", "However, we are unaware of any progress on this question for sparse random matrices.", "Gradient coding, introduced in [21], originally considered regimes where $\\sum _i f(X_i)$ could be recovered exactly under adversarial stragglers.", "For assignment matrices $A \\in \\mathbb {R}^{n \\times n}$ that can recover $\\sum _i f(X_i)$ exactly under any choice of $pn$ stragglers, the number of data points at each machine, $d$ , must exceed $pn$ .", "Closer to our work is a line of work on approximate gradient codes, introduced in [19], which considers the adversarial or expected decoding error in regimes where $A$ is too sparse to recover $\\sum _i f(X_i)$ exactly [5], [24], [25], [17], [1], [4].", "One important assignment matrix used here is a block matrix based on a Fractional Repetition Code [21], which achieves the optimal expected decoding error of $np^{d}$ .", "Despite its optimality under random stragglers, the FRC of [21] achieves an adversarial decoding error of $np$ .", "This is a significantly larger than the adversarial error of $\\Theta (np/d)$ achieved by the expander-graph based assignment matrix of [19].", "Several works [13], [4] have aimed to design assignment matrices that have small adversarial and expected decoding error, but to our knowledge, no existing assignments have achieved adversarial error that decays in $d$ (at any rate) while simultaneously achieving random error that decays faster than inversely linearly in $d$ .", "In Section , we include a table (Table REF ) comparing the adversarial and expected decoding errors of our work to other work on approximate gradient coding.", "Organization In Section , we formally define the three ensembles of random matrices we consider in this work.", "In Section , we formally state our results.", "In Section , we give a high level overview of the proof.", "We first show how to reduce our theorems on the distance from $v$ to the span of $A$ to our results on characterizing linear dependencies of $A$ .", "We give an overview of the proof of our characterization results, which breaks down into two main cases: a “small” case which concerns kernel vector with small support, and a “large\" case which concerns kernel vectors with large support.", "In Section , we state some lemmas that will be used in the small and large cases for more than one matrix ensemble.", "In Sections , , and , we prove our main result characterizing dependencies for rectangular Bernoulli matrices, symmetric Bernoulli matrices, and the ABC ensemble.", "In Section , we interpret our result for gradient codes, and show how the ABC ensemble achieves near-optimal decoding error for both random and adversarial stragglers.", "Notation and Formal Set-up In this work, we consider the following three ensembles of random matrices.", "We refer to two of these ensembles as “codes\" to follow the gradient coding literature.", "Rectangular Bernoulli Matrix (Bernoulli Gradient Code) Let $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ denote a random matrix in $\\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ where each entry of $A$ is Bernoulli with parameter $d/n$ .", "We will consider this ensemble for any $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ .", "Symmetric Bernoulli Matrix Let $A \\sim \\textnormal {SB}(n, d)$ denote a random symmetrix matrix in $\\lbrace 0, 1\\rbrace ^{n \\times n}$ whose upper diagonal entries are i.i.d.", "Bernoulli random variables with parameter $d/n$ , and whose diagonal is 0.", "We will consider this ensemble of matrices for $d = \\omega (1)$ .", "Augmented Biregular Code The Augmented Biregular Code (ABC) is based on the adjacency matrix of a random biregular graph generated from the configuration model.", "Formally, consider the following process to generate a random matrix $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ from the distribution $\\textnormal {ABC}(n, \\gamma , d)$ , for $\\gamma > 1$ .", "Create $n$ row-nodes and $\\gamma n$ column-nodes and associate to each row-node $\\gamma d$ half-edges and to each column node $d$ half-edges.", "Create a multigraph $G$ by choosing a uniformly random pairing of the $\\gamma dn$ half-edges from the row-nodes to the $\\gamma dn$ half-edges from the column-nodes.", "Given this bipartite graph, we will take $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ to be the matrix where $(A_0)_{ij} = 1$ iff there is at least one edge from node $i$ to $j$ .", "We will study the ensemble of random matrices obtained by removing a random $p$ fraction of columns from this ABC matrix $A_0$ .", "Formally, we call this ensemble $\\textnormal {ABC}_p(n, \\gamma , d)$ such that $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ is an $n \\times \\gamma n(1-p)$ matrix formed by deleting $\\gamma n p$ random columns from an ABC matrix $A_0 \\sim \\textnormal {ABC}(n, \\gamma , d)$ , which has dimensions $n \\times \\gamma n$ .", "For a positive integer $n$ , we use $[n]$ to denote the set $\\lbrace 1, \\ldots , n\\rbrace $ .", "All big-O notation denotes limiting behaviour as $n \\rightarrow \\infty $ .", "We use $\\mathrm {Span}(A)$ to mean the span of the columns of a matrix $A$ .", "For a vector $v \\in \\mathbb {R}^n$ and a set $S \\subset [n]$ , let $v_S$ denote the vector $v$ restricted to entries indexed by elements of $S$ .", "Similarly, for a matrix $A \\in \\mathbb {R}^{n \\times m}$ and a set $S \\subset [n]$ , let $A_S$ denote the matrix $A$ restricted to the rows in the set $S$ .", "For $j \\le m$ , let $A^{:j}$ denote the matrix restricted to the first $j$ columns of $A$ .", "Let $A_j$ denote the $j$ th column of $j$ .", "For a symmetric matrix $A \\in \\mathbb {R}^{n \\times n}$ , for $j \\in [n]$ , let $A^{(j)}$ denote the $n - 1 \\times n - 1$ matrix equal to $A$ with its $j$ -th row and column removed.", "We refer to a matrix $A$ as the vertex-edge incidence matrix of a graph $G$ if each there is a bijection between edges of $G$ and columns of $A$ that maps an edge $(i, j)$ to a column with ones in locations $i$ and $j$ and zeros elsewhere.", "Let $e_i$ denote the $i$ th canonical basis vector with a 1 in position $i$ .", "We use the notation $D_{KL}(p||q)$ to refer to the KL divergence of two Bernoulli random variables with parameters $p$ and $q$ respectively.", "Statement of Results In this section, we state our main results highlighted in the contributions.", "We begin with a few definitions, which we summarize in Table REF .", "Definition 1 A matrix in $B \\in \\mathbb {R}^{k \\times m}$ is a minimal linear dependency if it satisfies the following two properties: There exists $x \\in \\mathbb {R}^k$ such that $x^TB = 0$ and $\\mathrm {supp}(x) = [k]$ .", "$B$ has rank $k-1$ .", "Let $\\mathcal {M}_k \\subset \\bigcup _{m \\ge 1} \\mathbb {R}^{k \\times m}$ denote matrices of height $k$ which are minimal dependencies.", "We will distinguish among these matrices three particular types of linear dependencies, which are illustrated in Figure REF .", "Figure: The three structures of minimal dependencies that occur with constant probability in the random matrices in Propositions , and for k=5k = 5.", "(a) An element of 𝒯 k \\mathcal {T}_k.", "(b) An element of 𝒯 k + \\mathcal {T}_k^+.", "(c) An element of 𝒯 k C \\mathcal {T}_k^C.Definition 2 (Tree dependency) Define $\\mathcal {T}_k$ as the set of matrices $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with exactly $2k - 2$ non-zero entries such that the non-zero columns of $B$ form the vertex-edge incidence matrix of a tree on $k$ vertices.", "Definition 3 (Two-forest dependency) Let $\\mathcal {T}_k^+$ be the set of matrices $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with exactly $2k - 1$ non-zero entries satisfying the following: $B$ has $k-1$ non-zero columns: $k-2$ columns supported on 2 entries and one column supported on 3 entries.", "The submatrix of $B$ restricted to the columns of support 2 is the vertex-edge incidence matrix of a forest $F$ with two connected components $F_1,F_2$ .", "The column of support 3 contains 1's at rows $a,b,c$ where $a,b \\in F_i$ are connected by an even-length path, and $c \\in F_j$ for $\\lbrace i, j\\rbrace = \\lbrace 1, 2\\rbrace $ .", "Definition 4 (Tree-with-added-edge dependency) Define $\\mathcal {T}_k^C$ as the set of $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with $2k$ non-zero entries such that the non-zero columns of $B$ form the vertex-edge incidence matrix of a tree on $k$ vertices, with an added edge between two vertices in the tree of odd distance from each other.", "The additional edge may create a multi-edge in the this graph.", "Table: Notation in this workCharacterization of Linear Dependencies Our main technique in addressing Question REF is a new characterization of the linear dependencies that occur among the rows of a random matrix $A$ .", "The following three theorems describe our main results: [Characterization BGC]theoremcharbgc There exists a universal constant $c$ such that for any $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ , for $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ , with probability $1 - o(1)$ : All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^{+} \\cup \\mathcal {T}_k^C$ .", "The number of rows involved in a linear dependency of $A$ is at most $ne^{-\\gamma d + c\\log (\\gamma d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le ne^{-\\gamma d + c\\log (\\gamma d)}.$ [Characterization Symmetric Bernoulli]theoremcharsquare Let $A \\sim \\textnormal {SB}(n, d)$ , where $d = \\omega (1)$ .", "Then with probability $1 - o(1)$ , All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k$ .", "The number of rows involved in a linear dependency of $A$ is at most $ne^{-d + o(d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le ne^{- d + o(d)}.$ [Characterization ABC]theoremcharabc There exist universal constants $c$ and $\\gamma _0$ such that for any constant $p < 1/2$ , $\\gamma > \\gamma _0$ , and $d \\ge d_0(\\gamma )$ , for $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ , with probability $1 - o(1)$ : All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^{+} \\cup \\mathcal {T}_k^C$ .", "The number of rows involved in a linear dependency of $A$ is at most $np^{\\gamma d - c\\log (\\gamma d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le np^{\\gamma d - c\\log (\\gamma d)}.$ Remark 1 With high probability, one can show that under the conditions of Theorem REF , $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ does have minimal dependencies in $\\mathcal {T}_k$ for $k \\lesssim \\frac{\\log (n)}{\\gamma d}$ .", "We expect the same to hold for the symmetric matrix and for the ABC.", "For constant $d$ , with constant probability, $A \\sim \\text{BGC}(n, \\gamma , d)$ has minimal dependencies in $\\mathcal {T}_k^+$ and $\\mathcal {T}_k^C$ for $k$ sufficiently small, including $k = 5$ .", "We expect the same to hold for the ABC.", "Remark 2 While our initial motivation was Question REF , we believe these characterization theorems may have implications for understanding the exact rank of a sparse random matrices.", "For instance, in Corollary REF in the appendix, we show for $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ , with probability $1 - o(1)$ , the rank of $A$ is exactly equal to the graph-theoretic 2-core rank bound.", "Progress on Question  REF The following three bound the distance from $\\mathbb {1}$ to the column span of the random matrices we study.", "Theorem 1 (BGC Distance) Let $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "For $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ , there exists a constant $c$ such that with probability $1 - o(1)$ , $(1 - o(1))e^{-\\gamma d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le e^{-\\gamma d + c\\log (\\gamma d)}.$ Theorem 2 (Square Bernoulli Distance) Let $A \\sim \\textnormal {SB}(n, d)$ .", "For $d = \\omega (1)$ , with probability $1 - o(1)$ , $(1 - o(1))e^{-d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le e^{-d + o(d)}.$ Theorem 3 (ABC Distance) Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ .", "For any $p < 1/2$ , there exists constants $c$ , $\\gamma _0$ and $d_0$ such that for $\\gamma \\ge \\gamma _0$ and $d \\ge d_0$ , with probability $1 - o(1)$ , $(1 - o(1))p^{\\gamma d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le p^{\\gamma d + c\\log (\\gamma d)}.$ Remark 3 (Distances to arbitrary vectors: good news) These results can be extended beyond $\\mathbb {1}$ to unit vectors $v \\in \\mathbb {R}^n$ whose mass is well-distributed among the coordinates of $v$ .", "Formally, suppose $v$ is a distance of at least $\\rho $ from any $\\delta n$ -sparse unit vector, that is, $v$ is $(\\delta , \\rho )$ -incompressible.", "Then under the conditions of Theorem REF , with probability $1 - o(1)$ , $\\frac{\\rho ^4\\delta }{8}e^{-d} \\le \\min _{w}|A w - v|_2^2 \\le \\max (n|v|_{\\infty }^2e^{-d + o(d)}, 1)$ Similar results hold for $A \\sim \\textnormal {BGC}$ or $A \\sim \\textnormal {ABC}_p$ , with the exponent of $e$ begin the same exponent as in the respective theorems.", "Remark 4 (Distances to arbitrary vectors: bad news) For arbitrary vectors $v$ , we cannot hope to prove similar high probability bounds as for $\\mathbb {1}$ .", "For instance, if $v$ is a $O(1)$ -sparse unit vector, then under the conditions of Theorems REF , REF , or REF , with constant probability, the distance from $v$ to the span of $A$ is 0.", "Similarly, under the conditions of Theorems REF or REF , with constant probability (that may depend on $d$ ), the distance from $v$ to the span of $A$ is 1.", "The statements in these remarks are evident from the proofs of the theorems above, described in Section REF .", "Gradient Coding Results We are able to use Theorem REF on the distance between $\\mathbb {1}$ and $A \\sim \\textnormal {ABC}_p$ to analyze the expected decoding error of an assignment matrix based on the ABC ensemble.", "Since most results in the gradient coding literature concern a square assignment matrices where $m = n$ , we design an $n \\times n$ assignment matrix $B \\sim \\textnormal {ABC}_{\\text{stacked}}(n, \\gamma , d)$ by stacking together $\\gamma $ copies of a rectangular matrices $A_0 \\sim \\textnormal {ABC}(n/\\gamma , \\gamma , d/\\gamma )$ for an appropriate choice of $\\gamma $ .", "Full details of the construction are given in Section .", "We prove the following theorem about the assignment matrix $B$ : []theoremabcstacked Let $c,\\gamma _0,d_0$ be the universal constants from Theorem $\\ref {abc_random}$ .", "Choose any $\\gamma ,d\\in \\mathbb {Z}^+$ such that $\\gamma \\ge \\gamma _0$ , $\\gamma \\mid d$ and $\\frac{d}{\\gamma }\\ge d_0$ .", "For any sufficiently large $n$ divisible by $\\gamma $ , let $B \\sim \\textnormal {ABC}_{\\text{stacked}}(n, \\gamma , d)$ .", "Then with constant probability over the choice of $B$ : $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)} \\le p^{d - c\\log (d)}+o(1),$ and $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\left( \\textnormal {err}(B, S)\\right) \\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1).$ Overview of Proofs In the next subsection, we give an overview of how we bound the distance from $\\mathbb {1}$ to the span of $A$ to yield Theorems REF , REF , and REF .", "In the following subsection, we give an overview of our proof of the characterization results.", "Bounding the Distance from $\\mathbb {1}$ to the Span of $A$ Given a matrix $A$ , we will partition its rows into two sets: $D$ , the set of all rows involved in a linear dependency, and $[n] \\setminus D$ .", "Formally, $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x).$ We will use the following lemma to bound the distance from a vector $v$ to the column span of $A$ .", "Lemma 1 Let $A \\in \\mathbb {R}^{n \\times m}$ , and let $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x)$ be the set of rows which are involved in a linear dependency.", "Then for any $v \\in \\mathbb {R}^n$ , we have $\\min _{w \\in \\mathbb {R}^m}|Aw - v|_2^2 \\le |v_D|_2^2.$ This lemma follows from applying the following lemma, which we prove in Appendix .", "lemmabasisone Let $A \\in \\mathbb {R}^{n \\times m}$ , and let $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x)$ .", "Then for any $i \\notin D$ we have $e_i \\in \\mathrm {Span}(A)$ .", "To see how Lemma REF follows from Lemma REF , for any $v$ , by Lemma REF , the vector $v^{\\prime } := \\sum _{i \\in [n] \\setminus D}v_ie_i \\in \\mathrm {Span}(A)$ .", "Hence $\\min _w |Aw - v|_2^2 \\le |v - v^{\\prime }|_2^2 = |v_D|_2^2,$ establishing Lemma REF .", "For $v = \\mathbb {1}$ , we have $\\min _w |Aw - \\mathbb {1}|_2^2 \\le |D|.$ Plugging in the bound on $|D|$ given in the characterization theorems from Section REF yields the upper bounds in the distance theorems in Section REF .", "The lower bounds on the distance are given by counting the number of of all-zero rows in $A$ .", "Notice that the squared distance between $\\mathbb {1}$ and the span of $A$ is at least the number of all-zero rows in $A$ .", "We formally prove these lower bounds using standard concentration tools in Section .", "Overview of Proof of Characterization Our proof of each characterization result is divided into either two or three main cases: a “small\" case, a “large\" case, and sometimes a “medium\" case.", "The small case proves that for some constant $c_{\\text{s}}$ , for $k \\lesssim \\frac{n}{d^{c_{\\text{s}}}}$ , with high probability, all minimal dependencies of $k$ rows in $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "The large case proves that for a second constant $c_{\\ell }$ , for $k \\gtrsim \\frac{n}{d^{c_{\\ell }}}$ , with high probability there are no minimal dependencies of $k$ rows in $A$ .", "The exact constants $c_{\\text{s}}$ and $c_{\\ell }$ depend on the particular ensemble of random matrices.", "We require a medium case when $c_{\\text{s}} > c_{\\ell }$ , in order to account for all $k$ .", "While our proofs are organized by the specific ensemble of random matrices, we give here a short overview of the techniques in the small and large cases, as they are similar among all three ensembles we study.", "The main idea in proving this characterization of dependencies in the small and medium cases comes from the following two trivial observations.", "Observation 1 Let $B \\in \\lbrace 0, 1\\rbrace ^{k \\times m}$ be a matrix.", "Then $B$ cannot be a minimal dependency if some column of $B$ contains exactly one 1.", "Recall that our matrices are $0/1$ valued, and thus, if there is only one row with value 1 at index $j$ , that row is linearly independent to all other rows in our subset.", "Hence, no such dependency exists.", "Observation 2 Let $B \\in \\lbrace 0, 1\\rbrace ^{k \\times m}$ be a matrix.", "If $B$ has fewer than $2k-2$ entries that are 1, then $B$ is not a minimal dependency.", "The requirement that $B$ is rank $k-1$ means that at least $k-1$ columns must have at least one 1 in them and it follows from the previous observation that for a minimal dependency to exist, each of these columns must have at least two ones in them.", "Thus, we need at least $2k-2$ entries which are 1 in $B$ before a minimal dependency can exist.", "Small Case The goal of the small case is to prove that with high probability all small minimal row dependencies ($k \\lesssim n/d^{c_s}$ ) are contained in $\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "We begin by selecting an arbitrary set $S$ of $k$ rows, which induces a submatrix $A_S$ .", "Then, by conditioning on $L$ , which is the number of 1s which appear in $A_S$ , we consider a random process derived from the distribution of $A$ which places these $L$ 1s in the submatrix one by one.", "Due to Observation REF , we only must consider the case when $L \\ge 2k - 2$ .", "By Observation 1, $A_S$ does not have a minimal dependency if there exists a column with exactly one 1 in $A_S$ .", "To lower bound the probability of this event, we consider a random walk which increases by 1 every time our random process places a 1 in an already occupied column and stays constant otherwise.", "As long as the value of this random walk is less than $L/2$ at the time we have placed of the $L$ 1s in the submatrix, we know that $A_S$ does not have a minimal dependency.", "For the case of symmetric matrices, we modify this argument slightly by coupling with a random walk which also increases every time a 1 is placed in the “symmetric\" portion of $A_S$ , that is, a column of $A_S$ indexed by an element of $S$ .", "Large Case The goal of the large case is to show that with high probability, for $k \\gtrsim n/d^{c_\\ell }$ , there are no minimal dependencies of $k$ rows.", "Our main tool in ruling out large dependencies is the following set of anti-concentration results, most of which are standard in the literature.", "Roughly speaking, these results state that the dot product of a random vector and a deterministic vector will not concentrate on any one value with too large probability.", "Each ensemble of random matrices we study requires an anti-concentation lemma tailored to the distribution of a column vector from that matrix.", "In the BGC matrix, we use the following version of the Littlewood-Offord theorem for sparse random vectors.", "[c.f.", "[10] Lemma 8.2]lemmalosparse Let $v \\in \\mathbb {R}^n$ be a deterministic vector with support at least $m$ .", "Let $z \\in \\mathbb {R}^n$ be the random vector with i.i.d.", "Bernoulli entries with parameter $p \\le 1/2$ .", "Then for any fixed $c$ , $\\Pr \\left[v^Tz=c\\right] \\le \\left(\\frac{1}{\\sqrt{\\pi /2}}\\right)\\frac{1}{\\sqrt{mp}}+\\left(e^{(\\ln (2)-1)mp}\\right).$ In particular, for $mp\\ge 9$ , we have: $\\Pr \\left[ v^Tz=c\\right] \\le \\frac{1}{\\sqrt{mp}}.$ In the symmetric matrix case, we will additionally use a quadratic Littlewood-Offord result, originally due to [8].", "We state a version for sparse random vectors from [10].", "Lemma 2 (c.f.", "[10] Lemma 8.4) Let $M \\in \\mathbb {R}^{n \\times n}$ be a deterministic matrix with a least $m$ non-zero entries in each of $m$ distinct columns of $M$ .", "Let $z \\in \\mathbb {R}^n$ be the random vector with i.i.d.", "Bernoulli entries with parameter $p \\le 1/2$ .", "Then for any fixed $c$ , $\\Pr \\left(z^TMz = c\\right) = O\\left(\\frac{1}{(mp)^{1/4}}\\right).$ For the ABC matrix, in which each column of $A$ is close to $d$ -regular, we will use the following weaker anti-concentration lemma, which we prove in Section .", "Lemma 3 (Anti-concentration for Sparse Regular Vectors) Let $v \\in \\mathbb {R}^n$ be an arbitrary vector whose most common entry is $a$ .", "Then for any $d \\le \\sqrt{\\frac{n}{2}}$ , if $z \\in \\lbrace 0, 1\\rbrace ^n$ is sampled uniformly from the set of vectors with exactly $d$ 1s, we have: $\\Pr \\left[v^Tz = c\\right]\\le 1/2+\\frac{d^2}{n}.$ for all $c \\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "Our general strategy for the rectangular BGC and ABC matrices is as follows.", "Fix a set $S \\subset [n]$ of size $k$ .", "We consider the random process $A^{:1}_S, A^{:2}_S, \\cdots A^{:m}_S = A_S$ , where we add columns of $A_S$ one at a time.", "(Recall that $A^{:1}_S$ is the submatrix of $A$ given by restricting to the rows and $S$ and the first $i$ columns.)", "At each step $i$ , we keep track of the left kernel of $A^{:i}$ .", "Since our goal is to show that the left kernel of $A_S$ contains no vectors with support size $k$ , we leverage the anti-concentration results above as follows: if the kernel of $A^{:i}_S$ contains a vector $v$ with support $k$ , then it is likely that the next column added, $(A_S)_{i + 1}$ , will not be orthogonal to $v$ .", "After adding enough columns, we show that with high probability, we “knock out” all candidate kernel vectors with large support.", "While this approach is relatively straightforward for the BGC matrix where the columns are independent, we must be more careful for the ABC since the columns are not independent.", "In this case, for each column added, we consider the pairing of the $d$ half-edges of the corresponding column-node, and we show that for at least half of the columns, these pairings are “sufficiently random\".", "We use a similar approach to rule out some large dependencies ($n/d^{c_\\ell } < k < \\Theta (n)$ ) for the symmetric Bernoulli matrices.", "However, this approach breaks down for when $k$ becomes close to $n$ since the matrix is square.", "For instance, in the extreme case when $k = n$ , we would need to “knock-out\" a kernel vector at every single step of adding columns to $A$ .", "This certainly won't occur with high probability.", "Our strategy for ruling out dependencies on the order of $\\Theta (n)$ rows is inspired by a combination of the approaches in [12] and [8].", "Using Markov's law, we show that if there exists a kernel vector of $A$ with large support, then $A$ must contain many columns $A_i$ which are in the span of the remaining columns $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "We bound this probability that $A_i \\in \\mathrm {Span}(\\lbrace A_j\\rbrace _{j \\ne i})$ by conditioning on $A^{(i)}$ , the matrix formed by removing the $i$ th row and column of $A$ , and leveraging the randomness of $A_i$ .", "We consider two main cases: one in which $A^{(i)}$ has a kernel vector with large support, and one in which it doesn't.", "If $A^{(i)}$ has a kernel vector $v$ with large support (on the order of $\\Theta (n)$ ), we use Lemma REF to show that with probability $1 - O(1/\\sqrt{d})$ , $A_i$ is orthogonal to $v$ , and hence $A_i$ is not in the span of the remaining columns $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "If $A^{(i)}$ has no kernel vectors with large support, we are able to construct a dense “pseudoinverse” $B$ for $A^{(i)}$ , for which $A_i^TBA_i \\ne 0$ implies that $A_i$ is not in the span of $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "Lemma REF guarantees this occurs with probability $1 - O(1/\\@root 4 \\of {d})$ .", "Notice that unlike the results for the BGC and ABC which hold with probability that decays at least polynomially in $n$ , our results for square matrices hold with probability that decays polynomially in $d$ .", "For this reason, our results only hold with high probability if $d$ goes to infinity with $n$ .", "Medium Case We sometimes need an extra medium case to rule out dependencies of $k$ rows where $n/d^{c_s} < k < n/d^{c_\\ell }$ .", "This is easily accomplished using Observation REF by showing the existence of a column with a single 1.", "Useful Lemmas We gather here a few definitions and lemmas that we use in our proofs for multiple of the random matrix ensembles we study.", "Their proofs are technical, so we defer them to the appendix.", "The first lemma allow us to show that the linear dependencies we encounter with constant probability have the graph-structures depicted in Figure REF .", "Definition 5 Define $\\mathcal {S}_{L, k}$ to be union over all integers $m$ of the set of matrices $B\\in \\lbrace 0,1\\rbrace ^{k\\times m}$ such that $B$ forms a minimal dependency and $B$ has exactly $L$ entries which are 1.", "Definition 6 Define $\\mathcal {S}_{L, k}^{\\prime } \\subset \\mathcal {S}_{L, k}$ to be the subset of matrices in $\\mathcal {S}_{L, k}$ which have exactly $k$ non-zero columns.", "The following lemma relates these sets to the sets $\\mathcal {T}_k, \\mathcal {T}_k^+$ and $\\mathcal {T}_k^C$ introduced earlier in Definitions REF , REF , and REF .", "[Classification of Dependencies]lemmalemmaclassification We have the following three equivalences: $\\mathcal {S}_{2k-2, k} = \\mathcal {T}_k$ .", "$\\mathcal {S}_{2k-1, k} = \\mathcal {T}_k^+$ .", "$\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ .", "We will use the next lemma to bound the correlation between the existence of linear dependencies in intersecting submatrices $A_S$ and $A_T$ .", "Lemma 4 Suppose we have two sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ where $A_S \\in \\mathcal {M}_{|S|}$ and $A_T \\in \\mathcal {M}_{|T|}$ .", "Let $\\ell $ be the number of non-zero entries in $A_{S \\cup T}$ .", "Then there are at least $\\max \\left(|S \\cup T| - 1, \\frac{\\ell }{2}\\right)$ non-zero entries in $A_{S \\cup T}$ that are not the first (top) non-zero entry in their column.", "Lemmas  and REF are proved in Appendix .", "Several of our proofs use the following tail bound on Binomial distributions with a small parameter.", "[Tail Bound on Binomial]lemmataillemma If $t \\ge 2np$ , then $\\Pr \\left[\\mathrm {Bin}(n, p) \\ge t\\right] \\le 2\\left(\\frac{enp}{t}\\right)^t.$ Several of our proofs in the small case will use the following black-box calculation to bound the probability of encountering minimal dependencies.", "[Small Case Binomial Calculation]lemmamastersmall For constants $\\gamma , d > 0$ , for $k \\le \\frac{n}{8e^4\\gamma d^2}$ , there exists a constant $c_{\\ref {masterlemma:small}}$ such that for any $j \\in \\lbrace k - 1, k, k + 1\\rbrace $ and $\\gamma \\ge 1/2$ , we have $\\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^j.$ Similarly, several of our large cases will use the following black-box calculation.", "[Large Case Binomial Calculation]lemmalargegeneral There exists constants $C$ , $d_0$ , and $c_{\\ref {large_general}}$ , such that for all $d \\ge d_0$ , for any positive integer $\\frac{4n}{d} \\le k \\le n/C$ , we have $\\sum _{k = \\frac{2n}{d}}^{n/C}\\binom{n}{k}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\le e^{-c_{\\ref {large_general}}n}$ Lemmas , and are proved in Appendix .", "Bernoulli Gradient Codes In this section, we prove Theorem REF on the characterization of linear dependencies in matrices $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "As discussed in the overview, we will divide the proof of this result into small, medium, and large cases.", "Small Case The goal of this section will be to prove the following lemma.", "Recall that $\\mathcal {M}_k$ is the set of matrices with $k$ rows that are minimal dependencies.", "Lemma 5 (BGC Small Case) Let $A\\sim \\textnormal {BGC}(n,\\gamma ,d)$ .", "Let $S \\subset [n]$ be a set of size $k$ for $k \\in [1,\\frac{n}{8e^4\\gamma d^2}]$ .", "There exist universal constants $c_{\\ref {masterlemma:small}}, d_0$ such that if $d > d_0$ , then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_{k}^+ \\cup \\mathcal {T}_{k}^C \\right)] =\\left(e^{-\\gamma d + c_{\\ref {masterlemma:small}}\\log (d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k+1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k]\\le \\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k-1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^+]\\le \\left(\\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{k}{n}\\right)\\right)^k$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^C]\\le \\left(\\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{k}{n}\\right)\\right)^k$ .", "Further if $S, T \\subset [n]$ with $S \\cap T \\ne \\emptyset $ and $j := |S \\cup T| \\le \\frac{n}{8e^4\\gamma d^2}$ , then $\\Pr [A_S \\in \\mathcal {M}_{|S|} \\wedge A_T \\in \\mathcal {M}_{|T|}] \\le \\left( \\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{j}{n}\\right)\\right)^{j-1}.$ [Proof of Lemma REF ] Let $A\\sim BGC(n,\\gamma ,d)$ .", "For a row subset $S$ where $|S|=k$ , let $E_S$ denote the event that the submatrix $A_S$ induced by the row subset $S$ does not contain a column with exactly one 1.", "Let $L_S\\sim \\mathrm {Bin}(\\gamma k n,d/n)$ denote the number of 1s in $A_S$ .", "By Observation $\\ref {observation1}$ , we know the following: $\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]\\le \\Pr \\left[E_S|L_S=\\ell \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]$ By Observation $\\ref {observation2}$ , we already know the expression on the left hand-side is 0 when $L_S<2k-2$ .", "Thus, we only need to address the case where $L_S\\ge 2k-2$ .", "Recall that each entry in our matrix is an independent Bernoulli random variable.", "Thus, after conditioning on the event that $L_S=\\ell $ for some $\\ell \\in [2k-2, \\gamma k n]$ , the $\\ell $ ones in $A_S$ are distributed uniformly at random throughout the matrix $A_S$ .", "We can arbitrarily enumerate the 1s from 1 to $\\ell $ and consider the random process that places each 1 into $A_S$ sequentially starting from an $k \\times \\gamma n$ all zeros matrix.", "On each step of this process, we can query whether a column with exactly one 1 has been created.", "If such a column is created, we will call this step good.", "All other steps are bad.", "As we add exactly $\\ell $ ones into the matrix, and there are $\\gamma n$ columns, we conclude that the probability of each step being a good event is at least $1-\\frac{\\ell }{\\gamma n}$ .", "As each bad step can remove at most one column with exactly one 1 in it, it is clear that if more than half of the steps are good events, then our resulting submatrix $A_S$ must contain a column with exactly one 1.", "Let $\\lbrace X_i\\rbrace _{i = 1}^{\\ell }$ be the random process which counts the number of bad events that have occurred after $i$ ones have been added to the matrix.", "Thus we have: $\\Pr \\left[E_S|L_S=\\ell \\right]\\le \\Pr \\left[X_\\ell \\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right].$ Since the probability of $X_i$ increases at each step is at most $\\frac{\\ell }{\\gamma n}<\\frac{\\ell +k}{\\gamma n}$ , we can define $Y^{(\\ell )}\\sim \\mathrm {Bin}\\left(\\ell , \\frac{\\ell +k}{\\gamma n}\\right)$ , such that $\\Pr \\left[E_S|L_S=\\ell \\right]\\le \\Pr \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right].$ This implies that $\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]\\le \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]$ We now employ Lemma , which we restate here for convenience.", "* Using this Lemma , we conclude the following results.", "For the second statement in the lemma, we have $\\Pr \\left[A_S\\in \\mathcal {S}_{2k-2,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k-1\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k-1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma k n, d/n) = \\ell ]\\\\&\\le e^{-\\gamma d k + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k-1}.$ Similarly for the third and fourth statements, we have $\\Pr \\left[A_S\\in \\mathcal {S}_{2k-1,k}\\cup \\mathcal {S}_{2k,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k}.$ For the first statement in the lemma, observe that by Lemma , we have $\\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_{k}^+ \\cup \\mathcal {T}_{k}^C\\right) = \\left(\\bigcup _{\\ell \\ge 2k + 1} {\\mathcal {S}_{\\ell , k}}\\right) \\cup \\left(\\mathcal {S}_{2k, k} \\setminus \\mathcal {S}_{2k, k}^{\\prime }\\right).$ Now $\\sum _{\\ell =2k+1}^{\\gamma nk}\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k+1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k+1}.$ Finally, we note that if $A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }$ , then there must be at least one column with three ones.", "Hence, the number of bad events cannot be exactly $k$ as an equal number of good and bad events means that each column has of $A_S$ has exactly 2 ones in it.", "Thus, we find: $\\Pr \\left[A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k+1\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k+1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k+1}.$ Combining these two equations yields the first statement in the lemma.", "For the final setting, where we have two sets $S$ with $S \\cap T \\ne \\emptyset $ , by Lemma REF , it must be the case that if $A_S$ and $A_T$ are minimal dependencies, then $A_{S \\cup T}$ contains at least $\\max \\left(\\frac{\\ell }{2}, |S \\cup T| - 1\\right)$ ones that are not the first in their column, where $\\ell $ is the total number of ones in $A_{S \\cup T}$ .", "Plugging in Lemma  again proves that $\\Pr [A_S \\in \\mathcal {M}_{|S|}, A_T \\in \\mathcal {M}_{|T|}] \\le \\left(\\frac{|S \\cup T|}{n}\\right)^{|S \\cup T| - 1}e^{-\\gamma d|S \\cup T| + c_{\\ref {masterlemma:small}}|S \\cup T|\\log (\\gamma d)}.$ Finally, the lemma follows by using the classification lemma, Lemma , to replace $\\mathcal {S}_{2k - 2, k}$ with $\\mathcal {T}_k$ , $\\mathcal {S}_{2k - 1, k}$ with $\\mathcal {T}_k^+$ , and $\\mathcal {S}_{2k, k}^{\\prime }$ with $\\mathcal {T}_k^C$ .", "Medium Case To fill the small gap between where our small case ends and our large case begins, our proof requires the following lemma which shows that, with high probability, medium-sized minimal dependencies do not occur in our matrix $A$ .", "Unlike in the small case, tree-like dependency structures no longer occur with significant probability.", "Thus, it is sufficient to explicitly calculate the probability that $A_S$ has no columns with exactly 1 one without conditioning on the number of ones in $A_S$ .", "However, one should observe that this strategy will begin to fail as the expected number of 1s in each column begins to exceed 1.", "Thus, a more careful bound on the existence of a minimal dependency will be needed when $k$ is greater than approximately $n/d$ .", "This case is covered in the next subsection.", "Lemma 6 (BGC Medium Case) Let $A\\sim \\textnormal {BGC}(n,\\gamma ,d)$ .", "For any constants $\\alpha ,\\beta \\in \\mathbb {R}^+$ , there exists $d(\\alpha ,\\beta )$ , such that for any $d > d(\\alpha ,\\beta )$ , $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d]\\right] = o(1).$ We will take a union bound over all possible sets $S$ of size $k \\in [\\alpha n/d^2,\\beta n/d]$ of the probability that $A_S$ is a minimal dependency.", "For a row subset $S$ , let $E_S$ be the event that the matrix $A_S$ has a no columns with exactly one 1.", "Then by Observation REF , $\\Pr \\left[\\exists x: x^TA = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d] \\right]\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d} \\binom{n}{k}\\Pr \\left[E_{[k]}\\right].$ For sufficiently large $n$ , it follows: $\\Pr [E_{[k]}]&= \\left(1-k\\left(\\frac{d}{n}\\right)\\left(1-\\frac{d}{n}\\right)^{k-1}\\right)^{\\gamma n}\\\\&\\le \\left(1-k\\left(\\frac{d}{n}\\right)e^{-2dk/n}\\right)^{\\gamma n}\\\\&\\le e^{-\\gamma dk\\left(e^{-2dk/n}\\right)}.$ So we find: $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d]\\right] &\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d} \\binom{n}{k}\\Pr \\left[E_{[k]}\\right]\\\\&\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d}\\left(\\frac{en}{k}\\right)^ke^{-\\gamma dk\\left(e^{-2dk/n}\\right)}\\\\&\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d}\\left(e^{-\\gamma d\\left(e^{-2dk/n}\\right)+\\ln (en/k)}\\right)^k\\\\&\\le n\\left(e^{-\\gamma d\\left(e^{-2\\beta }\\right)+\\ln (ed^2/\\alpha )}\\right)^k\\\\&=o(1).$ for sufficiently large $d$ depending only on $\\alpha $ and $\\beta $ .", "Large Case The goal of the this section is to prove the following lemma: Lemma 7 (BGC Large Case) Let $A\\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "For any $\\gamma > 1$ , there exists $d_0$ such that for all $d > d_0$ we have, $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [9n/d, n]\\right] = o(1).$ Our main tool in proving this Lemma is the following following Littlewood-Offord Theorem due to Costello and Vu [10].", "However, their paper only states the result up to an implied constant.", "As our theorems rely on this implied constant, we reprove this result in Appendix using the same strategy as Costello and Vu, but with an explicit constant and sharper lower order terms.", "* We now prove Lemma REF .", "[Proof of Lemma REF ] We union bound over all $k \\in [9n/d, n]$ and all sets $S$ of size $k$ of the probability that $A_S$ is a minimal dependency.", "Fix a set $S$ of size $k$ .", "We will consider the random process where we generate the $\\gamma n$ independent columns $(A_S)_i$ for $i = 1\\ldots \\gamma n$ one at a time.", "For $i \\le \\gamma n$ , let $\\mathcal {N}_i \\in \\mathbb {R}^k$ be the nullspace of the first $i$ columns drawn, and let $\\mathcal {D}_i \\subset \\mathcal {N}_i$ be the span of the set of vectors in $\\mathcal {N}_i$ which have no zeros.", "Let $R_i$ be the dimension of $\\mathcal {D}_i$ .", "If $R_i > 0$ , then we can choose an arbitrary vector $v$ in $\\mathcal {D}_i$ with support $k$ , and by Lemma REF , with probability at least $1 - \\frac{1}{\\sqrt{kd/n}}$ , the $(i + 1)$ th column drawn is not orthogonal to $v$ .", "In this case $R_{i + 1} = R_i - 1$ .", "If at any point $R_i$ becomes 0, then this means there can be no dependency involving all the rows.", "It follows that since $R_0 = k$ , we have $\\Pr [A_S \\in \\mathcal {M}_k] \\le \\Pr \\left[R_{\\gamma n} \\ne 0\\right] \\le \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].$ Let $C$ be the universal constant in Lemma , which we restate for the reader's convenience: * We first handle when $k \\le n/C$ .", "Assuming $d$ is larger than the universal constant in Lemma , we can apply Lemma to compute the following union bound: $\\begin{split}\\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [9n/d, n/C]\\right] &\\le \\sum _{k = 9 n/d}^{n/C} \\Pr \\left[A_{[k]} \\in \\mathcal {M}_{|S|}\\right] \\\\&\\le \\sum _{k=9 n/d}^{n/C} \\binom{n}{k} \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\\\& \\le \\sum _{k=9 n/d}^{n/C} \\binom{n}{k} \\Pr \\left[\\mathrm {Bin}\\left(n-k-1, 1-\\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\\\&\\le e^{-c_{\\ref {large_general}} n} = o(1).\\end{split}$ Now we handle the case where $k \\ge n/C$ .", "Choose $d$ large enough, dependent on $\\gamma $ and $C$ to satisfy $\\left(1 - \\frac{2}{\\sqrt{d/C}}\\right)\\gamma > 1.$ This implies that for $\\frac{n}{C} \\le k \\le n$ , we have $\\gamma n - k \\ge \\frac{2\\gamma n }{\\sqrt{(d/n)k}}.$ .", "The inequality in this form allows us to use the the Binomial Tail bound in Lemma , which establishes: $\\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{(d/n)k}}\\right) \\le k\\right] &= \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, \\frac{1}{\\sqrt{(d/n)k}}\\right) \\ge \\gamma n - k\\right] \\\\&\\le 2 \\left(\\frac{e \\gamma n}{\\sqrt{(d/n) k} (\\gamma n - k)}\\right)^{\\gamma n- k} \\\\&\\le 2 \\left(\\frac{e \\gamma n}{\\sqrt{d/C} (\\gamma n - k)}\\right)^{\\gamma n-k} \\\\&\\le 2 \\left(\\frac{e\\gamma }{\\sqrt{d/C} (\\gamma - 1)}\\right)^{(\\gamma -1)n}.", "\\\\$ By choosing $d$ large enough depending on $\\gamma $ and the universal constant $C$ , we may get $(\\frac{e\\gamma }{\\sqrt{d/C} (\\gamma -1)})^{\\gamma -1} < 1/3$ , so $\\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{(d/n) k}}\\right) \\le k\\right] \\le (1/3)^n.$ This overcomes the union bound over at most $2^n$ sets: $ \\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [n/C, n]\\right] \\le 2^n (1/3)^n = o(1).", "$ Thus, we may combine Equations REF and REF to obtain our result $\\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [9n/d, n]\\right] \\le o(1).$ Proof of Theorem  REF We are now ready to prove Theorem REF , which we restate here for the reader's convenience.", "* [Proof of Theorem REF ] It follows immediately by combining Lemmas REF , REF (with $\\alpha = \\frac{1}{8e^4\\gamma }, \\beta = 9$ ), and Lemma REF that with probability $1 - o(1)$ , there are no sets $S$ where $A_S$ is a minimal dependencies $A_S$ , but $A_S$ is not in $\\cup _{k \\le \\log (n)}\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "We bound the number of rows involved in minimal linear dependencies using the second moment method.", "For $S \\subset [n]$ , let $X_S$ be the indicator random variable of the event that $A_S$ is a minimal dependency.", "Then with probability $1 - o(1)$ , the number of rows involved in minimal dependencies is at most $X := \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|X_S.$ First we consider the expectation of $X$ .", "Combining the probability that $A_S$ is in $\\mathcal {T}_k$ , $\\mathcal {T}_{k}^+$ , or $\\mathcal {T}_k^C$ , we have from Lemma REF that for any $S$ with $|S| = k$ , for the constant $c_{\\ref {masterlemma:small}}$ from that lemma, $\\mathbb {E}[X_S] \\le \\left(\\frac{k}{n}\\right)^{k - 1}e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}$ Hence $\\begin{split}\\mathbb {E}[X] &= \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|\\mathbb {E}[X_S] \\\\&\\le \\sum _{k \\le n}\\binom{n}{k}k\\left(\\frac{k}{n}\\right)^{k - 1}e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)} \\\\&\\le \\sum _{k \\le n}ne^ke^{-\\gamma d k + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)} \\\\& \\le \\sum _{k \\le n}ne^{-\\gamma dk + (c_{\\ref {masterlemma:small}} + 1)k\\log (\\gamma d)}\\\\& \\le ne^{-\\gamma d + c_1\\log (\\gamma d)}\\end{split}$ for some constant $c_1$ .", "Here the final inequality is given by the sum of geometric series.", "Next we bound the variance of $X$ .", "Let $S, T \\subset [n]$ with $|S|, |T| \\le \\log (n)$ .", "If $S \\cap T = \\emptyset $ , by the independence of $A_S$ and $A_T$ , we have $\\mathbb {E}[X_SX_T] = \\mathbb {E}[X_S]\\mathbb {E}[X_T].$ Alternatively, if $S \\cap T \\ne \\emptyset $ , then with $R := S \\cup T$ , by Lemma REF , we have $\\mathbb {E}[X_SX_T] \\le \\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}.$ Hence we can compute $\\begin{split}\\text{Var}(X) &= \\mathbb {E}[X^2] - \\mathbb {E}[X]^2 \\\\&= \\sum _{S: |S| \\le \\log (n)}\\sum _{T: |T| \\le \\log (n)} |S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&= \\sum _{S}\\sum _{T : T \\cap S = \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) + \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\mathbb {E}[X_SX_T] \\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}\\sum _{S, T \\subset R}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}\\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}2^{2|R|}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}\\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + (c_{\\ref {masterlemma:small}} + 1)|R|\\log (\\gamma d)}\\\\&\\le \\sum _{k \\ge 1}\\binom{n}{k}\\left(\\frac{k}{n}\\right)^{k- 1}e^{-\\gamma dk + (c_{\\ref {masterlemma:small}} + 1)k\\log (\\gamma d)} \\\\&\\le ne^{-\\gamma d + c_2\\log (\\gamma d)}\\end{split}$ for some constant $c_2$ .", "Using Markov's law, it follows that $\\begin{split}\\Pr [X \\ge \\mathbb {E}[X] + t] &\\le \\Pr [(X - \\mathbb {E}[X] )^2 \\ge t^2] \\le \\frac{\\text{Var}(X)}{t^2}.\\end{split}$ Plugging in $t = ne^{-\\gamma d}$ , we obtain that $\\begin{split}\\Pr [X \\ge ne^{-\\gamma d + c_1\\log (\\gamma d)} + ne^{-\\gamma d}] &\\le \\frac{ne^{-\\gamma d + c_2\\log (\\gamma d)}}{\\left(ne^{-\\gamma d}\\right)^2} \\le \\frac{1}{ne^{-\\gamma d - c_2\\log (\\gamma d)}}.\\end{split}$ The theorem follows by choosing $c \\ge c_1 + \\log (2)$ .", "Symmetric Bernoulli Matrices In this section, we prove Theorem REF on the characterization of minimal linear dependencies in symmetric Bernoulli matrices.", "As before, we break down our proof into small, medium, and large cases.", "The small and medium cases are similar to the last section on the BGC.", "However, Unlike for the BGC, however, the large case is not self contained: showing that there are no kernel vectors with large support relies on understanding the image of $Ax$ over vectors $x$ of small support (see Lemma REF below).", "This will require understanding small and medium dependencies in $A$ and in the matrix $A$ with one column removed, which is why our small case and medium case lemmas contain additional statements to this affect.", "Small Case Our main goal in the small case is to prove the following lemma.", "Lemma 8 (Symmetric Small Case) There exists a universal constant $c_{\\ref {lemma:small}}$ such that the following holds.", "Let $A\\sim \\textnormal {SB}(n, d)$ for any $d \\ge 1$ .", "Let $S \\subset [n]$ be any set of size $k\\in [1,\\frac{n}{8e^4d^2}]$ .", "Then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] = e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^k.$ $\\Pr [A_S \\in \\mathcal {T}_{k}]\\le e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1}.$ The same result applies if $A$ has one column removed, i.e.", "$A = B^{:n - 1}$ , where $B \\sim \\textnormal {SB}(n, d)$ .", "We introduce some notation to prove this, pictured in Figure REF .", "Fix a set $S$ of $k$ rows and consider the submatrix $A_S$ induced by these rows.", "Let $E_{Sym}$ be the set of entries of $A_S$ whose columns are indexed by values in $S$ .", "Let $E_{SymAD}$ be subset of entries in $E_{Sym}$ that are above the diagonal of $A$ , and hence mutually independent.", "Let $E_{Asym}$ be the set of entries who columns are not in $S$ , and finally, let $E = E_{Asym} \\cup E_{SymAD}$ be the full set of mutually independent entries that determine $A_S$ .", "Formally: $\\begin{split}&E_{Sym} := \\lbrace (i, j): i, j \\in S \\rbrace \\\\&E_{SymAD} := \\lbrace (i, j): i, j \\in S, i < j \\rbrace \\\\&E_{Asym} := \\lbrace (i, j): i \\in S, j \\notin S \\rbrace \\\\&E := E_{SymAD} \\cup E_{Asym}\\end{split}$ Figure: Regions of A S A_S in a symmetric matrix.We will couple the process of putting non-zero entries in these rows with a random walk that counts the number of times a non-zero entry is inserted in $E_{Asym}$ in a column that already contains a non-zero entry or into $E_{SymAD}$ .", "We condition on $L_S$ , the number of non-zero entries in $E$ .", "Note that $L_S \\sim \\mathrm {Bin}(|E|, d/n)$ .", "Conditioned on $L_S = \\ell $ , the process of choosing random entries in $A_S$ is equivalent to randomly choosing $\\ell $ locations in $E$ for these non-zero entries without replacement.", "(Note that this is true even if we are considering a matrix with the last column removed, even though $E_{SymAD}$ may include some entries which are not repeated below the diagonal if the index of the last column is in $S$ ).", "Let $(X_{i})_{i \\in \\ell }$ be the random walk that increases by 1 if the $i$ th random location chosen is in $E_{SymAD}$ or if the $i$ th random location is in $E_{Asym}$ and is not the first non-zero entry placed in its column.", "Otherwise, let $X_i = X_{i - 1}$ .", "The following claims say that $X_{\\ell }$ must be large for $A_S$ to be a minimal dependency not in $\\mathcal {T}_k$ .", "Claim 1 If $X_{L_S} < \\max \\left(k, \\frac{L_S}{2}\\right)$ and there are at least $2k - 1$ non-zero entries total in the rows, then there is a column (whose index is not in $S$ ) with a single non-zero entry.", "Let $P$ be the number of non-zeros entries in $E_{Asym}$ , and let $M$ be the number of non-zero entries in $E_{SymAD}$ such that $M + P = L_S$ and $2M + P \\ge 2k - 1$ .", "Let $Y:= X_{L_S} - M$ be the number of non-zero entries in $E_{Asym}$ which are not the first in their column.", "Now the number of columns not in $S$ which have exactly one non-zero entry is at least $ (P - Y) - Y = P - 2X_{L_S} + 2M \\ge \\max (2k - 1, L_S) - 2X_{L_S} \\ge 1.$ The proof of the following claim is nearly identical.", "Claim 2 If $X_{L_S} < \\max \\left(k - 1, \\frac{L_S}{2}\\right)$ and there are at least $2k - 2$ non-zero entries total in the rows, then there is a column (whose index is not in $S$ ) with a single non-zero entry.", "Recall from Observations REF that any minimal dependency $A_S$ must have at least $2k - 2$ non-zero entries total.", "Further, any minimal dependency not in $\\mathcal {T}_k$ must have at least $2k - 1$ non-zero entries.", "Hence by Claim REF , $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] \\le \\Pr [X_{L_S} < \\max \\left(k, L_S/2\\right)]$ Further by Claim REF , $\\Pr [A_S \\in \\mathcal {T}_k] \\le \\Pr [X_{L_S} < \\max \\left(k - 1, L_S/2\\right)].$ To bound the probability that $X_{L_S}$ is large, we couple $X_i$ with a random walk $(Y_i)_{i \\in L_S}$ which increases by 1 with probability $\\frac{k(k + L_S)}{|E|}$ and otherwise stays constant.", "Observe that $Y_i$ stochastically dominates $X_i$ , because there are at most $k + L_S$ columns — and hence $k(k + L_S)$ locations in $E$ — in which placing a non-zero entry will increase $X_i$ .", "Then conditioned on $L_S = \\ell $ , for any $j$ , we have $\\begin{split}\\Pr \\left[X_{\\ell } \\ge j\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{k + \\ell }{n - k}\\right) \\ge j\\right].\\end{split}$ We now use Lemma , which we restate here, to sum this probability over all values of $L_S$ .", "* We employ this lemma with $\\gamma = \\frac{|E|}{nk} \\ge 1 - k/n$ and $j = \\max \\left(k, \\ell /2\\right)$ , to achieve $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] \\le e^{-dk + (c_{\\ref {masterlemma:small}}+1)klog(d)}\\left(\\frac{k}{n}\\right)^k.$ Further, setting $j = \\max \\left(k - 1, L_S/2\\right)$ , we achieve $\\Pr [A_S \\in \\mathcal {T}_k] \\le e^{-dk + (c_{\\ref {masterlemma:small}}+1)k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1}.$ Putting $c_{\\ref {lemma:small}} = c_{\\ref {masterlemma:small}} + 1$ proves the lemma.", "Medium Case Our main goal of this section is to prove the following lemma.", "Lemma 9 (Symmetric Medium Case) Let $A \\sim \\textnormal {SB}(n, d)$ with $d$ at least some universal constant $d_0$ .", "Then $\\Pr \\left[\\exists x: A^Tx = 0, \\frac{n}{8e^4d^2} \\le \\mathrm {supp}(x) < \\frac{9n}{d}\\right] = o(1).$ The same result applies if $A$ has one column removed, ie.", "$A = B^{:n - 1}$ where $B \\sim \\text{SB}(n, d)$ .", "We will take a union bound over all possible sets $S$ of size $k \\in \\left[\\frac{n}{8e^4d^2}, \\frac{9n}{d}\\right]$ of the probability that $A_S$ is a minimal dependency.", "By Observation REF , it suffices to show that with probability $1 - o(1)$ , for all such sets $S$ , there is a column in $A_S$ with exactly one 1.", "Since the columns $(A_S)_i$ for $ i \\in [n] \\setminus S \\setminus \\lbrace n\\rbrace $ are mutually independent, we have $\\Pr [A_S \\in \\mathcal {M}_{|S|}] \\le \\left(1 - k\\frac{d}{n}\\left(1 - \\frac{d}{n}\\right)^{k - 1}\\right)^{n - k - 1}$ The calculation in Lemma REF gives the result that $\\sum _{k = \\frac{n}{8e^4d^2}}^{\\frac{9n}{d}}\\binom{n}{k} \\Pr [A_S \\in \\mathcal {M}_{k} \\text{ where $|S| = k$}] = o(1).$ Large Case The first lemma in our large case rules out with high probability minimal dependencies of $k$ rows for $\\frac{9n}{d} \\le k < \\frac{n}{C}$ for some constant $C$ .", "The proof is similar to the large case for the BGC matrix.", "Lemma 10 (Symmetric Large Case 1) Let $A \\sim \\textnormal {SB}(n, d)$ .", "There exist constants $d_0$ and $C$ such that for all $d > d_0$ , $\\Pr \\left[\\exists x: A^Tx = 0, \\frac{9n}{d} \\le \\mathrm {supp}(x) < \\frac{n}{C}\\right] = o(1).$ The same result applies if $A$ has one column removed, ie.", "$A = B^{:n - 1}$ where $B \\sim \\text{SB}(n, d)$ .", "We union bound over all $k \\in [9n/d, n]$ and all sets $S$ of size $k$ of the probability that $A_S$ is a minimal dependency.", "Fix a set $S$ of size $k$ .", "We will consider the random process where we generate the $n - k - 1$ columns $(A_S)_i$ for $i \\in [n - 1] \\setminus S$ one at a time.", "Note that these columns are all mutually independent since they do not include columns indexed by $S$ .", "Further, they do not include the last column of $A$ .", "Consider the following process, where we draw these independent columns one at a time.", "For $i \\le n - k - 1$ , let $\\mathcal {N}_i \\in \\mathbb {R}^k$ be the nullspace of the first $i$ columns drawn, and let $\\mathcal {D}_i \\subset \\mathcal {N}_i$ be the span of the set of vectors in $\\mathcal {N}_i$ which have no zeros.", "Let $R_i$ be the dimension of $\\mathcal {D}_i$ .", "If $R_i > 0$ , then we can choose an arbitrary vector $v$ in $\\mathcal {D}_i$ with support $k$ , and by Lemma REF , with probability at least $1 - \\frac{1}{\\sqrt{kd/n}}$ , the $(i + 1)$ th column drawn is not orthogonal to $v$ .", "In this case $R_{i + 1} = R_i - 1$ .", "If at any point $R_i$ becomes 0, then this means there can be no dependency involving all the rows.", "It follows that since $R_0 = k$ , we have $\\Pr [R_{n - k - 1} \\ne 0] \\le \\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].$ Next we take a union bound over all $k$ in the desired range and all $\\binom{n}{k}$ sets $S$ of size $k$ .", "Employing the calculation from Lemma , for universal constants $c_{\\ref {large_general}}$ , $C$ , and $d_0$ , for $d \\ge d_0$ , we have $\\Pr [\\exists x: Ax = 0, 9n/d \\le \\mathrm {supp}(x) \\le n/C] \\le e^{-c_{\\ref {large_general}}n}.$ This concludes the proof of the lemma.", "The remainder of the large case is based off of ideas from the works of Ferber, Kwan, and Sauermann [12](see Lemma 2.1), and Costello, Tau, and Vu [8](see proof of Lemma 2.8).", "We will use the following two linear-algebraic lemmas, proved in Appendix .", "lemmanullspace Let $A$ be a matrix with columns $A_i$ for $i \\in [n]$ .", "Let $H_i$ be the space spanned by the column vectors $A_1, A_2, \\cdots A_{i - 1}, A_{i + 1}, \\cdots A_n$ .", "Let $S$ be the set of all $i$ such that $A_i \\in H_i$ .", "Then there exists some $y$ with $\\text{supp}(y) = S$ such that $Ay = 0$ .", "lemmabasis With the terminology of the previous lemma, $e_i \\in \\mathrm {Span}(A^T)$ , if and only if $A_i \\notin H_i$ .", "The following lemma is our main tool in ruling out large dependencies and proving Theorem REF .", "It breaks down the probability that there is a large linear dependency consisting of more than $t$ rows into the sum of the probabilities of several other events.", "Two of these probabilities (lines 1 and 5 of the right hand side of Equation REF in the lemma) we can show to be small via anti-concentration lemmas.", "Two of the probabilities (lines 2 and 4 of the right hand side of Equation REF in the lemma), we can show to be small using lemmas we proved in the small case.", "This is unlike the previous large case lemma, where weren't concerned with any small structures that might exist in $A$ .", "Lemma 11 (Symmetric Large Case Main) Let $A \\sim \\text{SB}(n, d)$ .", "Let $A^{(i)}$ denote the submatrix of $A$ given by the $i$ th row and column removed.", "Let $A_i$ denote the $i$ th column of $A$ , and let $A_i^\\prime $ be this vector with the $i$ th entry removed.", "For any $u , r, s, t \\in [n]$ with $r < s$ , $\\begin{split}\\Pr \\left[\\exists x: Ax = 0, \\mathrm {supp}(x) > t\\right] &\\le \\frac{n}{t}\\max _{x \\in \\mathbb {R}^{n - 1}: \\mathrm {supp}(x) \\ge s}\\Pr \\left[x^TA^{\\prime }_n = 0\\right]\\\\&\\quad +\\frac{n}{t}\\Pr \\left[\\exists x: A^{(n)}x = 0, r < |\\mathrm {supp}(x)| < s \\right]\\\\&\\quad + \\frac{n}{t} \\frac{dr}{n}\\\\&\\quad + \\frac{n}{t}\\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right]\\\\&\\quad + \\frac{n}{t}\\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[A^{\\prime T}_nXA^{\\prime }_n = 0\\right],\\end{split}$ where $\\mathcal {T}^{m}_{p, q}$ denotes the set of matrices in $\\mathbb {R}^{m \\times m}$ with some set of $p$ columns that each have at least $q$ non-zero entries.", "For $i \\in [n]$ , let $H_i$ be the space spanned by the column vectors $A_1, A_2, \\cdots A_{i - 1}, A_{i + 1}, \\cdots A_n$ .", "Then $Ax = 0$ for some $x$ implies that for all $i \\in \\text{supp}(x)$ , $A_i \\in H_i$ .", "Let $\\mathcal {E}_i$ denote the event that $A_i \\in H_i$ .", "Let $X_i$ be the indicator of this event, and let $X = \\sum _i X_i$ .", "Then by Markov's inequality and the exchangeability of the columns, $\\Pr \\left[\\exists x : Ax = 0, |\\mathrm {supp}(x)| \\ge t \\right] = \\Pr \\left[X \\ge t\\right] \\le \\frac{\\mathbb {E}[X]}{t} = \\frac{n\\Pr [\\mathcal {E}_n]}{t}.$ We will break down the probability $\\Pr [\\mathcal {E}_n]$ into several cases, depending on the size of the support of vectors in the kernel of $A^{(n)}$ .", "Let $S \\subset [n - 1]$ be the set of all $i$ such that $e_i \\in \\mathrm {Span}(A^{(n)})$ , such that by Lemma REF and Lemma REF , $k:= \\max (\\mathrm {supp}(x): A^{(n)}x = 0) = n - 1 - |S|.$ .", "Case 1: $A^{(n)}$ has a kernel vector $x$ with large support, that is, $k \\ge s$ .", "Case 2: $A^{(n)}$ has a kernel vector $x$ with medium support, that is $r < k < s$ .", "Case 3: $A^{(n)}$ does not have any kernel vectors with large or medium support vectors in its kernel, that is, $k \\le r$ .", "We can expand $\\begin{split}\\Pr [\\mathcal {E}_n] &= \\Pr [\\mathcal {E}_n| k \\ge s]\\Pr [k \\ge s]\\\\&\\qquad + \\Pr [\\mathcal {E}_n|r < k < s]\\Pr [r < k < s]\\\\&\\qquad + \\Pr [\\mathcal {E}_n| k \\le r]\\Pr [k \\le r].\\end{split}$ For simplicity, define $\\textbf {a} := A_n^{\\prime }$ to be the first $n - 1$ entries of the column $A_n$ .", "To evaluate the probability of the first case, we condition on $A^{(n)}$ and let $x$ be any vector of support at least $s$ in the kernel of $A^{(n)}$ .", "Observe that $\\mathcal {E}_n$ cannot hold if $x^T\\textbf {a}$ is non-zero.", "Indeed, if $x^T\\textbf {a} \\ne 0$ , then let $x^{\\prime } = (x_1, x_2,\\ldots ,x_{n-1},0)/(x^T\\textbf {a})$ such that $Ax^{\\prime } = e_n$ .", "Then by Lemma REF , $A_n \\notin H_n$ and hence $\\mathcal {E}_n$ does not occur.", "Since $\\textbf {a}$ is independent from $x$ , we have $\\Pr [\\mathcal {E}_n| k \\ge s]\\Pr [k \\ge s] \\le \\max _{x: \\mathrm {supp}(x) \\ge s}\\Pr [x^T\\textbf {a} = 0].$ Combined with Equation REF , the contribution from this case yields the first term in the right hand side of Equation REF .", "For the second case, we bound $\\Pr [\\mathcal {E}_n|r < k < s]\\Pr [r < k < s] \\le \\Pr [r < k < s] \\le \\Pr \\left[\\exists x: A^{(n)}x = 0, r < |\\mathrm {supp}(x)| < s \\right].$ Combined with Equation REF , the contribution from this case yields the second term in the right hand side of Equation REF .", "The third case will lead to the final three terms in the right hand side of Equation REF .", "In this case, we will show conditions under which we can algebraically construct a vector $v$ such that $Av = e_n$ .", "This will imply by Lemma REF that $A_n \\notin H_n$ .", "Recall that $S \\subset [n - 1]$ is the set of all $i$ such that $e_i \\in \\mathrm {Span}(A^{(n)})$ .", "For $i \\in S$ , let $w_i $ be any vector such that $A^{(n)}{w_i } = e_i$ .", "We next construct a sort of “pseudoinverse\" matrix $B \\in \\mathbb {R}^{n - 1 \\times n - 1}$ as follows: For $i \\in S$ , define $B_{ij}$ to be the $i$ th entry of $w_i $ .", "That is, for $i \\in S$ , the $i$ th column of $B$ is $w_i $ .", "Define all other entries of $B$ to be zero.", "The following claim shows a condition for $\\mathcal {E}_n$ not holding.", "Claim 3 If $\\mathrm {supp}(\\textbf {a}) \\subset S$ and $\\textbf {a}^TB\\textbf {a} \\ne 0$ , then $e_n \\in \\mathrm {Span}(A)$ .", "Let ${w^{\\prime }} := B\\textbf {a} = \\sum _{i \\in S}{\\textbf {a}_iw_i }$ such that $A^{(n)}{w^{\\prime }} = \\sum _{i \\in S}{\\textbf {a}_ie_i}.$ Hence if $\\mathrm {supp}(\\textbf {a}) \\subset S$ , $A^{(n)}{w^{\\prime }} = \\textbf {a}.$ In this case, define $w \\in \\mathbb {R}^n$ to be the vector with $w^{\\prime }$ in the first $n - 1$ entries and $-1$ in the final entry.", "Then the first $n - 1$ entries of $Aw$ are 0, and the last entry is ${\\bf {a}}^T w^{\\prime } = {\\bf {a}}^T B \\bf {a}$ .", "Evidently, if ${\\bf {a}}^T B {\\bf {a}} \\ne 0$ , then $\\frac{Aw}{{\\bf {a}}^T B {\\bf {a}}} = e_n,$ so $e_n \\in \\mathrm {Span}(A)$ .", "By definition, in the third case, we have $|S| \\ge n - 1 - r$ .", "Hence by Claim REF , $\\begin{split}\\Pr [\\mathcal {E}_n \\wedge k \\le r]&\\le \\Pr \\left[\\mathrm {supp}(\\textbf {a}) \\lnot \\subset S \\wedge |S| \\ge n - 1 - r\\right]\\\\&\\quad + \\Pr [\\textbf {a}^TB\\textbf {a} = 0 \\wedge |S| \\ge n - 1 - r].\\end{split}$ Notice that $S$ is a function of $A^{(n)}$ and so $\\textbf {a}$ is independent from $S$ .", "It is easy to check that for any set $S$ of size at least $n - 1 - r$ , $\\Pr \\left[\\mathrm {supp}(\\textbf {a}) \\lnot \\subset S\\right] \\le 1 - \\left(1 - \\frac{d}{n}\\right)^r \\le \\frac{dr}{n}.$ We will break up the second term in Equation REF by conditioning on whether the support of $B$ has many entries or not, and using the independence of $\\textbf {a}$ from $B$ : $\\Pr [\\textbf {a}^TB\\textbf {a} = 0] \\le \\Pr \\left[B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s} \\wedge |S| \\ge n - 1 - r\\right] + \\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[\\textbf {a}^TX\\textbf {a} = 0\\right]$ To further bound the first probability on the right hand side, observe that if $|S| \\ge n - 1 - r$ and $B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}$ , there must exist at least $u$ different $i \\in S$ such that $\\mathrm {supp}(w_i ) \\le s$ .", "So $\\begin{split}\\Pr \\left[B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s} \\wedge |S| \\ge n - 1 - r\\right] &\\le \\Pr \\left[|\\lbrace i: \\exists x \\ne 0: A^{(n)}x = e_i, |\\mathrm {supp}(x)| < s\\rbrace | \\ge u \\right] \\\\ &\\le \\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right],\\end{split}$ where the last inequality follows by Markov's inequality.", "Combining this with Equations REF , REF , and REF yields $\\Pr [\\mathcal {E}_n \\wedge k \\le r] \\le \\frac{dr}{n} + \\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[\\textbf {a}^TX\\textbf {a} = 0\\right] + \\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right].$ Plugging this and Equations REF and REF into Equation REF and finally Equation REF yields the lemma.", "We instantiate Lemma REF with $t = \\frac{n}{C}$ , $s = \\frac{n}{C}$ , $r = \\frac{n}{d\\log (d)}$ , $u = \\frac{n}{2}$ , where $C$ is the constant from Lemma REF , to obtain the following lemma.", "Lemma 12 Let $A \\sim \\textnormal {SB}(n, d)$ for $d = \\omega (1)$ .", "With $C$ equal to the constant from Lemma REF , $\\begin{split}\\Pr [\\exists x: Ax = 0, \\mathrm {supp}(x) > n/C] &\\le C\\Pr [\\exists x: A^{(n)}x = 0, \\frac{n}{d\\log (d)} \\le |\\mathrm {supp}(x)| \\le n/C] \\\\&\\qquad + 2C\\Pr [\\exists x: A^{(n)}x = e_1, |\\mathrm {supp}(x)| \\le n/C] \\\\&\\qquad + o(1).\\end{split}$ This follows immediately from plugging in these values of $t, s, r$ and $u$ into Lemma REF and applying the anti-concentration results in Lemmas REF and REF to the first and last terms.", "Indeed, Lemmas REF shows that $\\frac{n}{t}\\max _{x \\in \\mathbb {R}^{n - 1}: \\mathrm {supp}(x) \\ge s}\\Pr \\left[x^TA^{\\prime }_n = 0\\right] \\le \\frac{n}{t}\\frac{1}{\\sqrt{sd/n}} = o(1).$ Lemma REF shows that $\\frac{n}{t}\\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[A^{\\prime T}_nXA^{\\prime }_n = 0\\right] \\le O\\left(\\frac{1}{\\@root 4 \\of {\\min (s, n - 1 - r - u)d/n}}\\right) = o(1).$ .", "The third term is at most $O(1/\\log (d))$ which is also $o(1)$ .", "Proof of Theorem  REF We are now ready to put the results of the small, medium, and large cases together to prove Theorem REF .", "For the convenience of the reader, we restate this theorem: * We will need the following two lemmas to show that the first two terms in the right hand size of Lemma REF are $o(1)$ .", "Lemma 13 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Then $\\Pr \\left[\\exists x: Ax = 0, \\frac{n}{d\\log (d)} < |\\mathrm {supp}(x)| < \\frac{n}{C} \\right] = o(1).$ This is immediate from the medium and first large case Lemmas REF and REF and the fact that $d = \\omega (1)$ , which implies that $\\frac{n}{d\\log (d)} \\ge \\frac{n}{8e^4d^2}$ .", "Lemma 14 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Then $\\Pr \\left[\\exists x: Ax = e_1, |\\mathrm {supp}(x)| < \\frac{n}{C} \\right] = o(1)$ We reduce Lemma REF to Lemma REF , which are better suited to prove with our medium and small case lemmas.", "Lemma 15 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Let $K = \\lbrace x: Ax = 0, |\\mathrm {supp}(x)| \\le \\frac{n}{C}\\rbrace $ .", "Then $\\Pr \\left[\\left|\\bigcup _{x \\in K}{\\mathrm {supp}(x)}\\right| \\ge \\frac{n}{d\\log (d)} \\right] = o(1).$ Similarly, suppose $A^{\\prime }$ is $A$ with the first row removed.", "Let $K^{\\prime } = \\lbrace x: A^{\\prime }x = 0, |\\mathrm {supp}(x)| \\le \\frac{n}{C}\\rbrace $ .", "Then $\\Pr \\left[1 \\in \\bigcup _{x \\in K^{\\prime }}{\\mathrm {supp}(x)} \\right] = o(1).$ We first prove Lemma REF from Lemma REF : [Proof of Lemma REF ] First we consider vectors $x$ with $1 \\notin \\mathrm {supp}(x)$ .", "Applying the first part of Lemma REF to $A^{(1)}$ , with probability $1 - o(1)$ , there exists some set $T \\subset [n] \\setminus \\lbrace 1\\rbrace $ with $|T| \\le \\frac{n}{d\\log (d)}$ such that $\\mathrm {supp}(x) \\subset T$ for all $x$ satisfying that $A^{(1)}x = 0$ and $\\mathrm {supp}(x) \\le \\frac{n}{C}$ .", "With probability $1 - o(1)$ , $\\mathrm {supp}(A_1) \\cap T = \\emptyset $ , so for all vectors $x$ with support in $[n] \\setminus \\lbrace 1\\rbrace $ and of size less than $\\frac{n}{C}$ , we do not have $Ax = e_1$ .", "Next we consider vectors $x$ with $1 \\in \\mathrm {supp}(x)$ .", "If such an $x$ exists, that is, $Ax = e_1$ and $1 \\in \\mathrm {supp}(x)$ , then it must be the case that $A^{\\prime }x = 0$ , where $A^{\\prime }$ is $A$ with the first row removed.", "By the second part of Lemma REF , the probability that such an $x$ exists is $o(1)$ .", "[Proof of Lemma REF ] Let $\\mathcal {L}_2$ be the event that $\\exists x: Ax = 0, \\frac{n}{d\\log (d)} < |\\mathrm {supp}(x)| < \\frac{n}{C}.$ Recall that by Lemma REF , this event occurs with probability $o(1)$ .", "To prove the first part, we will show that conditioned on $\\mathcal {L}_2$ not occurring, we have $\\left|\\bigcup _{x \\in K}{\\mathrm {supp}(x)}\\right| < \\frac{n}{d\\log (d)}$ .", "Index the set $K$ as follows: $K = \\lbrace x^{(1)}, \\cdots , x^{(|K|)}\\rbrace $ .", "For $i = 1$ to $|K|$ , let $y^{(i)} = \\sum _{j = 1}^i{c_i x^{(i)}}$ , where $c_i$ is chosen uniformly from the interval $[0, 1]$ .", "It follows that with probability 1, for all $i \\le |K|$ , $\\mathrm {supp}(y^{(i)}) = \\bigcup _{j \\le i}{\\mathrm {supp}(x^{(i)})}.$ If $\\mathcal {L}_2$ does not occur, then $|\\mathrm {supp}(x^{(i)})| \\le \\frac{n}{d\\log (d)}$ for all $i$ , so we have that $|\\mathrm {supp}(y^{(i + 1)})| \\le |\\mathrm {supp}(y^{(i)})| + \\frac{n}{d\\log (d)}$ .", "It follows that if $|\\mathrm {supp}(y^{(|K|)})| \\ge \\frac{n}{d\\log (d)}$ , then there exists some $i \\le |K|$ such that $|\\mathrm {supp}(y^{(i)})| \\in [\\frac{n}{d\\log (d)}, \\frac{2n}{d\\log (d)}]$ .", "However, this would imply that $\\mathcal {L}_2$ holds, which is a contradiction.", "For the second part, by Lemmas REF , REF , and REF , with probability $1 - o(1)$ , the only possible minimal dependencies in the columns of $A$ must have a total of between $2(k - 1)$ and $2k$ non-zero entries.", "(Note that we are applying the results of these lemmas to columns instead of rows).", "There are $\\binom{n}{k-1}$ possible sets of $k$ columns which include the first column.", "By Lemma REF , the probability of a dependency occurring in one of those sets of columns is at most $ \\left(\\frac{k}{n}\\right)^{k-1}e^{-kd + c_{\\ref {lemma:small}}k\\log (d)}.$ Hence by a union bound, with probability at most $ \\binom{n}{k - 1}\\left(\\frac{k}{n}\\right)^{k-1}e^{-kd + c_{\\ref {lemma:small}}k\\log (d)} = o(1),$ there are no small dependencies involving the first column.", "We restate Theorem REF for the reader's convenience before proving it.", "* [Proof of Theorem REF ] Combining Lemma REF with the Lemmas REF and REF presented above, with probability $1 - o(1)$ , for any $k \\ge \\frac{n}{d\\log (d)}$ , there are no minimal dependencies in $A$ of $k$ rows.", "Further applying Lemma REF , we see that with probability $1 - o(1)$ , all minimal dependencies of of size $k$ must be in $\\mathcal {T}_k$ .", "This proves the first part of the theorem.", "It remains to bound with high probability the number of rows involved in minimal dependencies in $\\mathcal {T}_k$ .", "For $S \\subset [n]$ , let $X_S$ be the indicator variable that $A_S \\in \\mathcal {T}_{|S|}$ .", "Then the total number of rows in minimal dependencies is at most $X = \\sum _{S \\subset [n]}|S|X_S.$ By Lemma REF , for the universal constant $c_{\\ref {lemma:small}}$ , we have $\\begin{split}\\mathbb {E}[X] &= \\sum _{k \\le \\frac{n}{8e^4d^2}}\\binom{n}{k}{k}e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1} \\\\&\\le \\sum _{k \\le \\frac{n}{8e^4d^2}}\\left(\\frac{en}{k}\\right)^ke^{-dk + c_{\\ref {lemma:small}}\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1} \\\\&= \\sum _{k \\le \\frac{n}{8e^4d^2}}ne^ke^{-dk + c_{\\ref {lemma:small}}k\\log (d)} \\\\&= \\sum _{k \\le \\frac{n}{8e^4d^2}}ne^{-dk + c_{\\ref {lemma:small}}k\\log (d) + k} \\\\&\\le n\\frac{e^{-d + 1 + c_{\\ref {lemma:small}}\\log (d)}}{1 - e^{-d + 1 + c_{\\ref {lemma:small}}\\log (d)}}\\\\&\\le ne^{-d + c\\log (d)}\\end{split}$ for some constant $c$ .", "By Markov's law, we have $\\Pr \\left[X \\ge ne^{-d + (c + 1)\\log (d)} \\le \\frac{\\mathbb {E}[X]}{ne^{-d + (c + 1)\\log (d)}}\\right] \\le \\frac{1}{d}.$ Since $d = \\omega (1)$ , this proves the theorem.", "Augmented Biregular Codes In this section, we prove Theorem REF on the characterization of minimal linear dependencies in matrices from $\\textnormal {ABC}_p$ .", "Our proof is broken down into a small and large case.", "The ABC distribution via the Configuration Model Recall that we use the following process to generate a random matrix $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ from the distribution $\\textnormal {ABC}(n, \\gamma , d)$ : Create $n$ row-nodes and $\\gamma n$ column-nodes and associate to each row-node $\\gamma d$ half-edges and to each column node $d$ half-edges.", "Create a multi-graph $G$ by choosing a uniformly random pairing of the $\\gamma d n$ half-edges from the-row nodes to the $\\gamma d n$ half-edges from the columns-nodes.", "Given this bipartite graph, we will take $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ to be the matrix where $(A_0)_{ij} = 1$ iff there is at least one edge from node $i$ to $j$ .", "Because we will study the resulting matrix $A_0$ via the half-edge pairing process, in our proofs we will sometimes consider the permutation $\\rho \\in \\mathcal {S}_{\\gamma d n}$ which defines the random pairing of half-edges.", "From $\\rho $ , we can can construct an additional matrix $H_0 = H_0(\\rho )$ , which we call the row half-edge occupancy matrix .", "Let $P \\in \\lbrace 0, 1\\rbrace ^{\\gamma d n \\times \\gamma d n}$ be the permutation matrix of $\\rho $ .", "From $P$ , construct $H_0 \\in \\lbrace 0, 1\\rbrace ^{\\gamma d n \\times \\gamma n}$ by summing adjacent columns of $P$ corresponding to half-edges of the same column nodes.", "Symbolically, $(H_0)_{ij} = \\sum _{k=(j-1)d + 1}^{jd} P_{ik}.$ Note that $H_0$ has $\\gamma n$ columns; column $i$ represents column-node $i$ in the configuration model.", "$H_0$ has $\\gamma d n$ rows: rows $(j-1)\\gamma d + 1$ through $j\\gamma d$ represent the $\\gamma d$ half edges of row-node $j$ , for $1 \\le j \\le n$ .", "Each column contains exactly $d$ 1's, corresponding to that column-node's half edges.", "We may write $A_0 = A_0(\\rho )$ in terms of the random matrix $H_0$ .", "Because $(A_0)_{ij} = 1$ iff there is at least one edge from vertex $i$ to $j$ , we have $(A_0)_{ij} = \\mathbb {1}(\\exists k, d(i-1) < k \\le di, H_{kj} = 1 ).$ When generating a matrix $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ , we can without loss of generality generate $A_0 \\sim \\textnormal {ABC}(n, \\gamma , d)$ and take $A := A_0^{:\\gamma n(1 - p)}$ to be the matrix containing the first $\\gamma n(1 - p)$ columns of $A_0$ .", "We define $H := H_0^{:\\gamma n(1 - p)}$ to be the first $\\gamma n(1-p)$ columns of the row half-edge occupancy matrix associated with $A_0$ .", "For a set $S \\subset [n]$ with $|S| = k$ , let $H(S) \\in \\mathbb {R}^{\\gamma d k \\times \\gamma n(1-p)}$ be the matrix $H$ restricted to the $\\gamma d k$ rows corresponding to half-edges of the row-nodes in $S$ .", "Small Case The goal of this section will be to prove the following lemma: Lemma 16 Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ for $\\gamma \\ge 1$ .", "Let $S\\subset [n]$ be any set of size $k\\in [1,\\frac{n}{18e\\gamma d^2}]$ .", "There exists universal constants $c_{\\ref {lemma:abc_small}},d_0$ such that if $d>d_0$ , then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C \\right)] = O\\left(e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\right)$ .", "$\\Pr [A_S \\in \\mathcal {T}_k]\\le \\left(p^{\\gamma d+ c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k-1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^+]\\le \\left(p^{\\gamma d+c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^C]\\le \\left(p^{\\gamma d+a_0\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k}$ .", "Further if $S, T \\subset [n]$ with $S \\cap T \\ne \\emptyset $ .", "Let $|S| = k$ , $|T| = j$ , and $R := S \\cup T$ , then $\\Pr [A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C \\wedge A_T \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C] \\le \\left( \\left(p^{\\gamma d+c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)\\left(\\frac{|R|}{n}\\right)\\right)^{|R|-1}.$ To prove Lemma REF , we shall condition on $L_S$ , the number of 1s in $H(S)$ .", "We observe that $L_S\\sim \\text{HyperGeom}(\\gamma d n, \\gamma d(1-p)n, \\gamma dk)$ .", "Note that $L_S$ is greater than or equal to the number of entries which are 1 in the submatrix $A_S$ , with equality if and only if there does not exist a row node in $S$ which pairs multiple half-edges to the same column-node in $[\\gamma n (1 - p)]$ .", "We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Let $E_S$ denote the event that there are no columns in $H(S)$ with exactly 1 one.", "It follows from Observation REF that $\\Pr [A_S\\in \\mathcal {S}_{\\ell ,k}]\\le \\Pr [E_S |L_S=\\ell ]\\cdot \\Pr [L_S=\\ell ]$ Our general strategy to bound $\\Pr [E_S|L_S = \\ell ]$ relies on the following claim.", "Claim 4 Let $X$ be the number of non-zero columns in $H(S)$ , or equivalently, the number of 1s in $H(S)$ which are the first (top) 1 in their column.", "If $E_S$ occurs, then $X \\le \\lfloor {L_S/2}\\rfloor $ .", "If $X > \\lfloor {L_S/2}\\rfloor > L_S/2$ , then there are strictly less than $L_S/2$ ones in $H(S)$ which are not the first 1 in their column.", "By the pigeonhole principle, this means there must be at least one column with a “top\" 1 but no 1s below it, ie.", "this column has exactly one 1.", "We will bound the probability that $X$ is small by considering a random walk which counts the number of non-zero columns as half-edges are paired one at time.", "We formalize this random walk as follows.", "Conditioned on $L_S$ , the matrix $H(S) \\in \\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ is distributed like a uniformly random matrix on the set of all matrices in $\\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ with exactly $L_S$ ones.", "We construct $H(S)$ via the following random process $M^0, M^1, \\ldots , M^{L_S} = H(S)$ on matrices in $\\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ , in which the $L_S$ half-edges represented in $H(S)$ are paired one at a time.", "Formally, at each step $i$ , we construct $M^i$ from $M^{i - 1}$ by placing a 1 in a uniformly random location in $M^{i - 1}$ without a 1.", "For $i \\in 0, 1, \\ldots L_S$ , let $X_i$ equal the number of non-zero columns in $M^i$ , such that $X_{L_S} = X$ .", "When placing the $i$ -th 1, there are at most $i-1$ non-zero columns so far.", "In particular, since $i \\le L_S$ , there are at most $L_S$ non-zero columns throughout this process.", "Because $H(S)$ has $\\gamma (1-p)n$ columns, we thus have the bound $\\Pr [X_i - X_{i - 1} = 1] \\ge 1 - \\frac{L_S}{\\gamma (1-p)n}$ .", "It follows that the random variable $X$ first-order stochastically dominates the random variable $\\sum _{i=1}^{L_S} \\textnormal {Bernoulli}(1 - \\frac{L_S}{\\gamma (1-p)n})$ .", "So in particular, we have: $\\Pr \\left[X\\le \\lfloor L_S/2\\rfloor \\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(L_S, 1 - \\frac{L_S}{\\gamma (1-p)n}\\right) \\le \\lfloor L_S/2\\rfloor \\right],$ which implies $\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}]&\\le \\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1 - \\frac{\\ell }{\\gamma (1-p)n}\\right) \\le \\lfloor \\ell /2\\rfloor \\right]\\cdot \\Pr [ L_S=\\ell ].$ We will use the following claim proved in Appendix : claimclaimabc Let $p<1/2$ .", "Let $K\\le \\frac{3}{2}pN$ .", "There exists constants $c_{\\ref {claim:abc1}}$ , and $d_0$ such that for all $\\gamma >1$ and $d>d_0$ , the following two bounds hold.", "For $\\ell \\in \\lbrace 2K-2,2K-1,2K\\rbrace $ , and $j \\le \\lfloor \\ell /2\\rfloor $ , we have: $\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1-p)N}\\right)\\le j\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma d N,\\gamma d(1-p)N,\\gamma d K)=\\ell ]\\le \\left(p^{\\gamma d - c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{\\ell -j}.$ Further, $\\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{4K}{\\gamma (1 - p)N}\\right)\\ge K-1\\right]\\cdot \\Pr \\left[\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell \\right]\\le \\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{K-1}.$ Since $L_S \\sim \\textnormal {HyperGeom}(\\gamma d n, \\gamma d(1-p)n, \\gamma d k)$ , it must be the case that $L_S \\ge \\gamma d k - p\\gamma dn.$ Hence if $L_S \\in \\lbrace 2k-2,2k-1,2k\\rbrace $ , then necessarily $k \\le \\frac{p\\gamma d n}{\\gamma d - 2} \\le \\frac{3}{2}pn,$ for $\\gamma d \\ge 6$ , which is guaranteed if $d \\ge 6$ .", "Thus we can apply the first bound in Claim REF with $K = k$ and $N = n$ to study $\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}] \\le \\Pr \\left[Y\\le \\lfloor \\ell /2\\rfloor ]\\cdot \\Pr [ L_S=\\ell \\right]$ , yielding: $\\Pr [A_S\\in \\mathcal {S}_{2k-2,k} ]&=\\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k-1}.\\\\\\Pr [A_S\\in \\mathcal {S}_{2k-1,k} ]&=\\left(p^{\\gamma d- c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k}.\\\\\\Pr [A_S\\in \\mathcal {S}_{2k,k}^{\\prime }]\\le \\Pr [A_S\\in \\mathcal {S}_{2k,k} ]&=\\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k}.$ We use the equivalences of $\\mathcal {S}_{2k - 2, k} = \\mathcal {T}_k$ , $\\mathcal {S}_{2k - 1, k} = \\mathcal {T}_k^+$ , and $\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ from Lemma  to yield statements 2, 3, and 4 in this lemma.", "Next we obtain a bound on the event that $A_S \\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)$ .", "Observe that $A_S \\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C) \\Rightarrow A_S \\in \\left(\\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }\\right) \\cup \\bigcup _{\\ell \\ge 2k + 1}\\mathcal {S}_{\\ell , k}.$ If $A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }$ , there must be a column with at least three 1s.", "So by a similar argument to Claim REF , it must be the case that $X \\le k - 1$ .", "Hence $\\Pr [A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }]&\\le \\Pr \\left[\\mathrm {Bin}\\left(2k, 1 - \\frac{2k}{\\gamma (1-p)n}\\right) \\le k-1\\right]\\cdot \\Pr [L_S=2k]$ Another application of the Claim REF gives us: $\\Pr [A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k, k}^{\\prime }]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k+1}.$ It follows that $\\Pr [A_S\\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right) \\left(\\frac{k}{n}\\right)^{k+1}+\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}]\\\\&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right) \\left(\\frac{k}{n}\\right)^{k+1}+\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1 - \\frac{\\ell }{\\gamma (1-p)n}\\right) \\le \\left\\lfloor {\\frac{\\ell }{2}}\\right\\rfloor \\right]\\cdot \\Pr [L_S = \\ell ]$ To bound the final term, we use the following claim, which we prove in Appendix .", "claimClaimabc Let $p<1/2$ .", "There exists $\\gamma _0$ and $d_0$ such that for all $\\gamma >\\gamma _0$ and $d>d_0$ , if $k\\le \\frac{n}{18e\\gamma d^2}$ , we have: $\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr \\left[\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor \\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma d n,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\le e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}$ This gives us the desired result for sufficiently large $\\gamma d$ : $\\Pr [A_S\\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k+1}+ e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\\\&\\le O\\left(e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\right).$ This yields the first statement in the lemma.", "Finally we prove the final statement about pairs of sets $S$ and $T$ of size $k$ and $j$ respectively.", "Recall that $R = S \\cup T$ .", "Since $A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ , and $A_S \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C$ , $A_S$ contains at most $2k$ non-zero entries and $A_T$ contains at most $2j$ non-zero entries.", "Hence $A_R$ contains at most $2(k + j) \\le 4|R|$ non-zero entries.", "By Lemma REF , $A_{R}$ must contain at least $|R| - 1$ non-zero entries that are not the first in their column.", "By the same argument as before, conditioned on $L_R = \\ell $ , the probability that $A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ , and $A_S \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C$ is at most $\\Pr [\\mathrm {Bin}(\\ell , \\frac{4|R|}{\\gamma (1 - p)n}) \\ge |R| - 1]$ .", "We check that the conditions of Claim REF hold with $K =|R|$ and $N = n$ .", "Since we only consider the case when $L_R \\le 4|R|$ , we must have $K = |R| \\le \\frac{p\\gamma d n}{\\gamma d - 4} = \\frac{p\\gamma d N}{\\gamma d - 4} \\le \\frac{3}{2}pN$ for $d \\ge 12$ .", "Hence employing Claim REF yields the lemma for $d_0 \\ge 12$ and $c_{\\ref {lemma:abc_small}} = c_{\\ref {claim:abc1}}$ .", "Large Case The main goal of the large case is to prove the following lemma.", "Lemma 17 (ABC Large Case) Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ for some constant $\\gamma \\ge 16$ .", "Then there exists a constant $d_0(\\gamma )$ such that for $d \\ge d_0$ , $\\Pr \\left[\\exists x: x^TA = 0, \\mathrm {supp}(x) \\ge \\frac{n}{18e\\gamma d^2}\\right] \\le o(1).$ Our main tool in proving this Lemma is an anti-concentration Lemma that is based on the sparse Littlewood-Offord Theorem from [9].", "While typically such anti-concentration lemmas concern the dot product of a deterministic vector and a random vector with independent entries, we derive a weaker result which concerns the dot product of a deterministic vector with a vector in which there are a fixed number of non-zero entries whose positions are random.", "Lemma 18 (Anti-concentration for Sparse Regular Vectors (Same as Lemma REF )) Let $v\\in \\mathbb {R}^N$ be an arbitrary vector whose most common entry is $a$ .", "Then for any $d \\le \\sqrt{\\frac{N}{2}}$ , if $x \\in \\lbrace 0, 1\\rbrace ^N$ is sampled uniformly from the set of vectors with exactly $d$ 1s, we have: $\\Pr \\left[x\\cdot v=w\\right]\\le 1/2+\\frac{d^2}{N}.$ for all $w\\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "Remark 5 As $d \\rightarrow \\infty $ , the anti-concentration probability approaches the smaller value of $1/e$ , though $1/2$ is tight for $d = 2$ .", "For $i \\in \\lbrace 1,...,d\\rbrace $ , let $j_i \\sim \\text{Uniform}([N])$ , and let $X_i := v_{j_i}$ .", "Then $\\Pr [x\\cdot v=w] &= \\Pr \\left[\\sum _{i=1}^d X_i =w\\ | \\text{All $j_i$ are unique}\\right] \\\\&\\le \\frac{\\Pr \\left[\\sum _{i=1}^d X_i =w\\right]}{\\Pr \\left[\\text{All $j_i$ are unique}\\right]}.$ Claim 5 $\\Pr \\left[\\text{All $j_i$ are unique}\\right] \\ge 1 - \\frac{d^2}{N}.$ By a union bound, $\\Pr \\left[\\exists i \\ne \\ell : j_i = j_{\\ell } \\right] &\\le \\binom{d}{2}\\cdot \\Pr \\left[j_1 = j_2\\right]\\\\&=\\binom{d}{2}\\left(\\frac{1}{N}\\right)\\\\&\\le \\frac{d^2}{N}.$ Next we show by induction on $d$ that $\\Pr \\left[\\sum _{i=1}^d X_i = w\\right] \\le 1/2$ for all $w\\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "For $d=1$ , we note that the chosen element $w$ cannot be the most common element $a$ .", "Thus, $w$ is at worst equally as common as the most common element $a$ .", "This implies that the number of times $w$ appears in $v$ is at most $\\frac{N}{2}$ , hence $\\Pr [X_1 = w]\\le \\frac{1}{2}$ .", "Now assume that $\\Pr \\left[\\sum _{i=1}^{d-1} X_i = u\\right]\\le 1/2$ holds for all $u\\ne (d-1)a$ .", "For $w\\ne da$ , we write: $\\Pr \\left[\\sum _{i=1}^{d} X_i = w\\right] &=\\sum _{x\\in \\text{Supp($X_d$)}} \\Pr \\left[X_d = x\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1} X_i = w-x\\right]\\\\&=\\Pr \\left[X_d=a\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1}X_i=w-a\\right] +\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1} X_i = w-x\\right]$ Let $p_x := \\Pr \\left[\\sum _{i=1}^{d-1}X_i=w-x\\right]$ such that by the induction hypothesis $p_a \\le 1/2$ as $w-a=(d-1)a$ if and only if $w=da$ .", "Thus, we conclude: $\\Pr \\left[\\sum _{i=1}^{d} X_i = w\\right] &\\le \\max _{p_a \\le 1/2, \\sum _{x}p_x \\le 1}\\left(\\Pr \\left[X_d=a\\right]\\cdot p_a + \\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot p_x\\right) \\\\&\\le \\max _{\\sum _{x \\ne a}p_x \\le 1/2}\\left(\\Pr \\left[X_d=a\\right]\\cdot \\left(1/2\\right) + \\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot p_x\\right) \\\\&\\le \\max _{\\sum _{x \\ne a}p_x \\le 1/2}\\left(\\Pr \\left[X_d=a\\right]\\cdot (1/2) + \\left(\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} P(X_d = x)\\right)\\cdot \\left(\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}}p_x\\right)\\right)\\\\&= \\Pr \\left[X_d=a\\right]\\cdot (1/2)+(1-\\Pr [X_d=a])\\cdot (1/2)\\\\&= 1/2.$ Here the second inequality follows from the fact that the optimum over the $p_x$ is achieved by putting the maximum possible mass on $p_a$ , that is, $p_a = 1/2$ .", "Returning to Equation REF , we have for all $w \\ne ad$ and $d \\le \\sqrt{\\frac{N}{2}}$ , $\\Pr [x\\cdot v=w] &\\le \\frac{\\Pr \\left[\\sum _{i=1}^d X_i =w\\right]}{\\Pr \\left[\\text{All $j_i$ are unique}\\right]} \\le \\frac{1/2}{1 - \\frac{d^2}{N}} \\le 1/2 + \\frac{d^2}{N},$ where the final inequality follows from the fact that $\\frac{1}{1 - x} \\le 1 + 2x$ for $0 \\le x \\le 1/2$ .", "To prove Lemma REF , we take a similar approach to the large case for the BGC.", "For a fixed set $S$ of size $k$ , we consider the stochastic process $A_S^{:1}, A_S^{:2}, \\ldots $ in which we add the columns of $A_S$ one by one.", "We define the space: $D(A_S^{:j})=\\mathrm {Span}\\left(\\lbrace v\\in (\\mathbb {R}\\backslash \\lbrace 0\\rbrace )^k: v A_S^{:j} =0 \\rbrace \\right),$ and let $R_j := \\text{Rank}(D(A_S^{:j})).$ If $R_{\\gamma n(1-p)} =0$ , then $A_S$ is not a minimal dependency.", "Each time we add a new column, $R_j$ either stays constant or decreases by at least 1 (note that $R_j$ can decrease by more than 1: if a new column $j$ has exactly one non-zero entry, then $R_j = R_{j+1} = \\ldots = 0$ no matter what $R_{j-1}$ was).", "We will use the following lemma to show for $j \\le \\gamma n/2$ , each column we add is close to random.", "Then, using Lemma REF , we show that with decent probability, $R_j$ decreases.", "We can then apply a Chernoff Bound to show that $R_{\\gamma n/2} = 0$ with high probability.", "More formally, we consider the process of constructing $H \\in \\lbrace 0,1\\rbrace ^{\\gamma d n\\times \\gamma n(1-p)}$ (the row half-edge occupancy matrix) one column at a time by pairing the $d$ half-edges from each column-node at each step to a random set of $d$ unpaired row-half-edges.", "Define $e(S, j) := \\gamma d k - |H^{:j}(S)|_1$ , that is the number of unpaired half-edges among the $k$ row-nodes in $S$ after the first $j$ column-nodes have randomly paired their half edges.", "The following lemma uses standard concentration bounds to show that for the first $\\gamma n/2$ columns, there are many unpaired half-edges out of row-nodes in $S$ .", "Lemma 19 Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ .", "Let $\\Omega $ be the event in which for any set $S$ with $|S| \\ge \\frac{n}{18e\\gamma d^2}$ and for any $j\\in [1,\\frac{\\gamma }{2}n]$ , we have $\\frac{e(S,j)}{\\gamma dn-dj}\\ge \\frac{k}{2n}$ .", "For $\\gamma \\ge 2$ , there exists a constant $d_0(\\gamma )$ such that for $d \\ge d_0$ , we have $\\Pr [\\Omega ] \\ge 1 - \\gamma ne^{-n/d^3}$ .", "For $d$ larger than some constant $d_0(\\gamma )$ , we have $\\frac{n}{d^3} \\le \\frac{n}{18e\\gamma d^2}$ , hence it suffices to prove the result for all $k\\in [n/d^3,n)$ .", "Fix $k$ and let us define $\\eta _k$ to be the ratio $k/n$ .", "Choose a subset $S$ of size $k$ .", "Furthermore, fix $j\\le \\frac{\\gamma }{2}n$ .", "We proceed by applying the following tail bound on Hypergeometric distributions to $e(S,j)$ which is clear from [6].", "Lemma 20 (Hypergeometric Tail Bound from [6]) Let $X$ be given by a hypergeometric distribution with parameters $A$ , $B$ , $c$ .", "Then for all $t < \\frac{B}{A}$ , $\\Pr \\left[X \\le tc \\right] \\le e^{-cD_{KL}(t||B/A)}.$ Similarly, letting $Y$ be given by a hypergeometric distribution with parameters $A$ , $A - B$ , and $c$ , for $t > \\frac{B}{A}$ , we have $\\Pr \\left[X \\ge tc \\right] = \\Pr [Y \\le (1 - t)c] \\le e^{-cD_{KL}(1 - t||1 - B/A)} = e^{-cD_{KL}(t||B/A)}.$ Using this bound, we have $P[e(S,j)\\le ({\\eta _k-t})(\\gamma d n-dj)]&\\le e^{-D_{KL}\\left(\\eta _k-t||\\eta _k\\right)(\\gamma d n-dj)}\\\\&\\le e^{-D_{KL}\\left(\\eta _k-t||\\eta _k\\right)\\frac{\\gamma dn}{2}}$ for all $t\\in (0,\\eta _k)$ .", "The following claims expands this KL-divergences for $t = \\eta _k/2$ .", "Claim 6 $\\begin{split}D_{KL}\\left(\\eta _k-\\eta _k/2||\\eta _k\\right) \\ge \\frac{\\eta _k}{12}.\\end{split}$ $\\begin{split}D_{KL}\\left(\\eta _k/2||\\eta _k\\right) &= \\frac{\\eta _k}{2}\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\ln \\left(\\frac{1 - \\eta _k/2}{1 - \\eta _k}\\right) \\\\&\\ge \\frac{\\eta _k}{2}\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(\\frac{\\eta _k}{2} + \\frac{3\\eta _k^2}{8}\\right) \\\\&= \\frac{\\eta _k}{2}\\left(\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(1 + \\frac{3\\eta _k}{4}\\right)\\right) \\\\& \\ge \\frac{\\eta _k}{12}.\\end{split}$ Here the first inequality follows by using the Taylor expansion for $\\ln (1 - x)$ and the final inequality follows by noting that the quadratic $\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(1 + \\frac{3\\eta _k}{4}\\right)$ achieves its minimum over $\\eta _k \\in [0, 1]$ at $\\eta _k \\in \\lbrace 0, 1\\rbrace $ .", "Union bounding over all $\\binom{n}{k}$ sets $S$ of size $k$ , we have $\\Pr \\left[\\exists S, k := |S| \\in [n/d^3, n] : e(S,j)\\le \\frac{\\eta _k}{2}(\\gamma dn-dj)\\right]&\\le \\sum _{k=n/d^3}^n \\binom{n}{k}e^{-\\frac{\\eta _k\\gamma dn}{24}}\\\\&\\le \\sum _{k=n/d^3}^n \\left(\\frac{en}{k}\\right)^ke^{-\\frac{k\\gamma d}{24}} \\\\&= \\sum _{k=n/d^3}^n e^{-k\\left(\\frac{\\gamma d}{24}- \\ln (1/\\eta _k) - 1\\right)} \\\\&\\le \\sum _{k=n/d^3}^n e^{-k\\left(\\frac{\\gamma d}{24}- \\ln (d^3) - 1\\right)} \\\\&\\le \\sum _{k=n/d^3}^n e^{-k} \\\\& \\le 2e^{-n/d^3}.$ We take a union bound over $j \\in [1, \\gamma n/2]$ to achieve $\\Pr \\left[\\exists S, k := |S| \\in [n/d^3,n] : e(S,j)\\le \\frac{\\eta _k}{2}(\\gamma dn-dj)\\right] \\le \\gamma ne^{-n/d^3}.$ Conditioned on $\\Omega $ , we can use our anti-concentration bound, Lemma REF , to show that for $j \\le \\gamma n/2$ , $R_j$ often decreases.", "Lemma 21 For $j \\le \\gamma n/2$ , conditioned on $\\Omega $ , if $R_j \\ge 1$ , then with probability at least $\\mu _k := \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k},$ we have $R_{j} \\le R_{j-1} - 1.$ Suppose $R_{j - 1} \\ge 1$ , and let $v$ be any vector in $(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )^k$ such that $A_S^{:j - 1}v=0$ .", "Let $v^{\\prime } \\in \\mathbb {R}^{\\gamma dk}$ be the vector which repeats each coordinate of $v$ $\\gamma d$ times: that is, $v^{\\prime }_i = v_{\\lceil \\frac{i}{\\gamma d}\\rceil }$ .", "Let $h_j \\in \\lbrace 0, 1\\rbrace ^{\\gamma dn}$ be $j$ th column of $H$ , which has exactly $d$ ones indicating the half-edges matched to the $d$ half-edges from the column-node $j$ .", "Let $h_j(S) \\in \\lbrace 0, 1\\rbrace ^{\\gamma d k}$ be the restriction of $h_j$ to the entries corresponding to half-edges from row-nodes in $S$ .", "Claim 7 If at most one half-edge is matched from column-node $j$ to a single row-node in $S$ and $h_j(S) \\cdot v^{\\prime } \\ne 0$ , then $R_{j} \\le R_{j-1} - 1$ .", "If at most one half-edge is matched from column-node $j$ to a single row-node in $S$ , then $v \\cdot (A_S)_j = \\sum _{i \\in S}v_iA_{ij} = \\sum _{i \\in S}v_i\\mathbb {1}(\\exists \\ell , \\gamma d(i-1) < \\ell \\le \\gamma di, H_{\\ell j} = 1 ) = \\sum _{i \\in S}v_i\\sum _{\\ell = \\gamma d(i - 1) + 1}^{\\gamma di} H_{\\ell j} = v^{\\prime } \\cdot h_j(S) \\ne 0.$ Hence $v \\notin D(A_S^{:j})$ , so the rank of $D(A_S^{:j})$ is strictly smaller than that of $D(A_S^{:j-1})$ .", "Claim 8 $\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | \\Omega ] \\le \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) + \\frac{d}{\\gamma k}.$ We condition on the number of non-zero entries in $h_j(S)$ , which we denote $s$ .", "Observe that conditioned on $s$ , the vector $h_j(S)$ is a uniformly random vector from the set of all vectors in $\\lbrace 0, 1\\rbrace ^{\\gamma d k}$ with $s$ 1s.", "This holds even when we conditioned on $\\Omega $ , because this event says nothing about which half-edges among the nodes in $S$ have been paired.", "By Lemma REF , since $v^{\\prime }$ contains no zeros (and hence its most common element is not zero), we have $\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | s \\ge 1, \\Omega ] \\le 1/2 + \\frac{d}{\\gamma k}.$ It remains to consider the probability that $s = 0$ , since in this case, we always have $h_j(S) \\cdot v^{\\prime } = 0$ .", "We know that $s$ is distributed like a hypergeometric random variable $\\text{HyperGeom}(\\gamma d n - dj, d, e(S, j))$ .", "Indeed, there are $d$ half-edges that are paired with the addition of the $(j + 1)$ -th column, there are $e(S, j)$ unpaired half-edges among the nodes in $S$ , and there are $\\gamma d n - dj$ total unpaired half-edges among the row nodes.", "Since we have conditioned on $\\Omega $ , we know that $\\frac{e(S,j)}{\\gamma d n-dj}\\ge \\frac{k}{2n}$ .", "Hence we can compute $\\Pr [s = 0 | \\Omega ] \\le \\left(1 - \\frac{k}{2n}\\right)^{d} \\le e^{-\\frac{dk}{2n}}.$ Combining Equations REF and REF , it follows that $\\begin{split}\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | \\Omega ] &= \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | s \\ge 1, \\Omega ]\\\\&\\le \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\left(1/2 + \\frac{d}{\\gamma k}\\right)\\\\&\\le \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\left(1/2\\right) + \\frac{d}{\\gamma k} \\\\&= \\frac{1}{2}\\left(1 + \\Pr [s = 0| \\Omega ]\\right) + \\frac{d}{\\gamma k}\\\\&\\le \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) + \\frac{d}{\\gamma k}.\\end{split}$ Claim 9 The probability that more than one half-edge is matched from a column-node $j$ to a single row-node in $S$ is at most $\\frac{2\\gamma d^3}{k}$ .", "Conditioned on $\\Omega $ , we have at each step $j$ , there are at least $\\frac{k(\\gamma d n - dj)}{2n}$ open half-edges out of $S$ , and hence at least $\\frac{k(\\gamma d n - dj)}{2n\\gamma d} \\ge \\frac{k}{4}$ nodes in $S$ with at least one open half-edge.", "Since each node in $S$ has at most $\\gamma d$ open half-edges, each pair of half-edges from the $j$ th column collide with a row-node with probability at most $\\frac{\\gamma d}{k/4}$ .", "Hence by a union bound, the probability of collision is at most $\\binom{d}{2}\\left(\\frac{\\gamma d}{k/4}\\right) \\le \\frac{2\\gamma d^3}{k}.$ It follows from the previous two claims that the probability that $h_j(S) \\cdot v^{\\prime } \\ne 0$ and at most one half-edge is match from a column-node $j$ to a single row-node in $S$ is at least $\\begin{split}1 - \\Pr [\\textnormal {Event in Claim~\\ref {claim:double_match} occurs}] - \\Pr [h_j(S)\\cdot v^{\\prime } = 0 | \\Omega ]&\\ge 1 - \\frac{2\\gamma d^3}{k} - \\frac{d}{\\gamma k} - \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) \\\\&= \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{2\\gamma d^3}{k} - \\frac{d}{\\gamma k} \\\\&\\ge \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k}.\\end{split}$ Using Claim REF , this proves the lemma.", "We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Recall that our goal is to show that with high probability, for all $k \\ge \\frac{n}{18e\\gamma d^2}$ , for all sets $S$ of size $k$ , we have $R_{\\gamma n/2} = 0$ .", "Throughout the rest of the proof, we assume that we have conditioned on $\\Omega $ , since $\\Pr [\\Omega ] = 1 - o(1)$ .", "For a fixed set $S$ of size $k$ , by Lemma REF , conditioned on each term being positive, the random process $R_1, R_2, \\cdots , R_{\\gamma n/2}$ is stochastically dominated by the random process $Y_1, Y_2, \\ldots ,Y_{\\gamma n/2}$ , where $Y_{i + 1} = Y_i - \\mathrm {Ber}(\\mu _k)$ and $\\mu _k = \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k}.$ Hence $\\Pr [R_{\\gamma n/2} > 0] \\le \\Pr [Y_{\\gamma n/2} > 0].$ By a Chernoff Bound, since $Y_0 = k$ , for any $\\mu \\le \\mu _k$ , we have $\\begin{split}\\Pr [Y_{\\gamma n/2} > 0] &= \\Pr [Y_{0} - Y_{\\gamma n/2} < k]\\\\&\\le e^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k.\\end{split}$ Define $\\eta _k := k/n$ .", "Taking a union bound over all sets $S$ of size $k$ , conditioned on $\\Omega $ , the probability that at least one set $S$ has $R{\\gamma n/2} > 0$ is at most $\\begin{split}\\binom{n}{k}e^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k &\\le \\left(\\frac{en}{k}\\right)^ke^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k \\\\&= e^{-\\gamma n\\mu /2}\\left(\\frac{e^2\\gamma n^2\\mu }{2k^2}\\right)^k \\\\&= e^{-\\gamma n\\mu /2+ k\\log \\left(\\frac{e^2\\gamma n^2\\mu }{k^2}\\right)} \\\\&= e^{k\\left(-\\frac{\\gamma \\mu }{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu }{2\\eta _k^2}\\right)\\right)}\\end{split}$ We consider two cases: Case 1: $\\frac{n}{18e\\gamma d^2} \\le k \\le \\frac{5n}{d}$ .", "Case 2: $k \\ge \\frac{5n}{d}$ .", "In the first case, since $1 - e^{-x} \\ge x/2$ for $0 < x < 1$ , for $n$ large enough, we have $\\mu _k \\ge \\frac{1}{2}\\left(\\frac{d\\eta _k}{4}\\right) - \\frac{3\\gamma d^3}{k} \\ge \\frac{d\\eta _k}{10}.$ Hence for $\\mu = \\frac{d\\eta _k}{10}$ , we have $-\\frac{\\gamma \\mu }{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu }{2\\eta _k^2}\\right) \\le -\\frac{\\gamma d}{20} + \\log \\left(\\frac{e^2\\gamma d}{20\\eta _k^2}\\right) \\le -\\frac{\\gamma d}{20} + \\log \\left(\\frac{18^2e^3\\gamma ^3d^5}{20}\\right) \\le -0.25$ for $d$ larger than some constant $d_0$ .", "In the second case, we have $\\mu _k \\ge 0.45$ , and hence for $\\mu = 0.45$ and $\\gamma \\ge 16$ , we have $-\\frac{\\gamma \\mu _k}{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu _k}{2\\eta _k^2}\\right) \\le \\max _{\\eta \\le 1} -\\frac{3.6}{\\eta } + \\log \\left(\\frac{3.6e^2}{\\eta ^2}\\right) \\le -0.25.$ Returning to Equation REF , conditioned on $\\Omega $ , the probability that there is a set $S$ of size $k$ for which $R_{\\gamma n/2} > 0$ is at most $e^{-k/4}$ .", "Summing over all $k \\ge \\frac{n}{168\\gamma d^2}$ yield a probability of failure among any $k$ of at most $\\frac{e^{-\\frac{n}{64e\\gamma d^2}}}{1 - e^{-1/4}} \\le 5e^{-\\frac{n}{64e\\gamma d^2}}.$ Unioning with the probability that $\\Omega $ doesn't occur from Lemma REF yields Lemma REF .", "Proof of Theorem  REF We are now ready to prove Theorem REF , which we restate here for the reader's convenience.", "* [Proof of Theorem REF ] By combining the result of the small case, Lemma REF , with the result of the large case, Lemma REF , we observe that $1 - o(1)$ , there are no minimal dependencies in $A$ that are not in $\\bigcup _{k \\le \\log (n)}\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C .$ This proves the first statement in the theorem.", "Next we bound the size of $D$ , the set of rows involved in linear dependencies: $D := \\bigcup _{x: x^TA = 0}{\\mathrm {supp}(x)}.$ Define $\\tilde{\\mathcal {T}_k} := \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ For a set $S \\subset [n]$ , let $X_S$ be the indicator of the event that $A_S \\in \\tilde{\\mathcal {T}_k}$ .", "Notice that with probability $1 - o(1)$ , $|D|$ is at most $X := \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|X_S.$ We will bound $X$ with high probability via the second moment method with Lemma REF as our main tool.", "First we compute the expectation of $X$ .", "By Lemma REF , for some constant $c_{\\ref {lemma:abc_small}}$ , we have $\\begin{split}\\mathbb {E}[X] &= \\sum _{S: |S| \\le \\log (n)}{|S|\\Pr [A_S \\in \\tilde{\\mathcal {T}_k}]} \\\\&\\le \\sum _{k \\ge 1}k\\binom{n}{k}\\left(\\frac{k}{n}\\right)^{k - 1}p^{\\gamma dk + c_{\\ref {lemma:abc_small}}k\\log (\\gamma d)} \\\\&\\le n \\sum _{k \\ge 1}p^{\\gamma d k + (c_{\\ref {lemma:abc_small}} + 1)k\\log (\\gamma d)} \\\\&\\le n p^{\\gamma d - c_1\\log (\\gamma d)}\\end{split}$ for some constant $c_1$ .", "If $T, S \\subset [n]$ are such that $|S| \\le |T| \\le \\log (n)$ and $S \\cap T \\ne \\emptyset $ , then by Lemma REF , with $R := S \\cup T$ $\\mathbb {E}[X_SX_T] \\le \\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)}.$ In the next claim, we show that if $T \\cap S = \\emptyset $ , then the events $X_S$ and $X_T$ are almost uncorrelated.", "Claim 10 Let $T, S \\subset [n]$ and $|T|, |S| \\le \\log (n)$ .", "If $T \\cap S = \\emptyset $ , then $\\mathbb {E}[X_SX_T] \\le \\left(1 + \\frac{6\\gamma d\\min (|S|, |T|)}{n}\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T].$ [Proof of Claim REF ] Let $|S| = k$ and $|T| = j$ and without loss of generality assume $k \\le j$ .", "Let $\\rho $ be the permutation defining the mapping of half-edges used to generate $A$ .", "Let $\\mathcal {P}_{\\gamma dk, \\gamma dn}$ be the set of injections from $[\\gamma dk] $ to $[\\gamma dn]$ .", "For $\\pi \\in \\mathcal {P}_{\\gamma dk, \\gamma dn}$ , recall that $A(\\pi ) \\in \\lbrace 0, 1\\rbrace ^{k \\times \\gamma n(1-p)}$ is the first $k$ rows of the $\\textnormal {ABC}_p$ matrix which is generated from mapping half-edges according to $\\pi $ and then dropping the last $\\gamma pn$ columns.", "Let $\\mathcal {R}_k = \\lbrace \\pi \\in \\mathcal {P}_{\\gamma dk, \\gamma dn}: A(\\pi ) \\in \\tilde{\\mathcal {T}_k}\\rbrace .$ Since each element of $\\mathcal {P}_{\\gamma dk, \\gamma dn}$ is equally likely to be the restriction of $\\rho $ to $[\\gamma dk]$ , we have $\\mathbb {E}[X_S] = \\frac{|\\mathcal {R}_k|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}.$ Our goal is to compute $\\begin{split}\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T] &= \\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_S \\in \\mathcal {R}_k \\wedge \\rho |_T \\in \\mathcal {R}_j \\right] - \\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_S \\in \\mathcal {R}_k\\right]\\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_T \\in \\mathcal {R}_j \\right] \\\\&= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\Pr \\left[\\rho |_S = \\pi _1 | \\rho |_T = \\pi _2\\right] - \\Pr \\left[\\rho |_S = \\pi _1\\right]\\right)\\left(\\Pr \\left[\\rho |_T = \\pi _2\\right]\\right).\\end{split}$ Here we have abused notation to interpret the restriction of $\\rho $ to a set $U$ as being an element of $\\mathcal {P}_{\\gamma d|U|, \\gamma dn}$ .", "If the images of $\\pi _1$ and $\\pi _2$ intersect, then $\\Pr _{\\rho }[\\rho |_S = \\pi _1 | \\rho |_S = \\pi _2] = 0.$ Otherwise, if the images do not intersect, we have $\\Pr _{\\rho }[\\rho |_S = \\pi _1 | \\rho |_S = \\pi _2] = \\frac{1}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|} = \\Pr [\\rho |_S = \\pi _1]\\frac{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|}$ It follows that $\\begin{split}\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T] &= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\Pr \\left[\\rho |_S = \\pi _1 | \\rho |_T = \\pi _2\\right] - \\Pr \\left[\\rho |_S = \\pi _1\\right]\\right)\\left(\\Pr \\left[\\rho |_T = \\pi _2\\right]\\right)\\\\&= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\frac{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|} - 1\\right)\\Pr \\left[\\rho |_T = \\pi _2\\right]\\left(\\Pr \\left[\\rho |_S = \\pi _1\\right]\\right) \\\\&= \\left(\\frac{(\\gamma dn)!", "(\\gamma dn - 2\\gamma dk)!", "}{(\\gamma dn - \\gamma dk)!", "(\\gamma dn - \\gamma dk)!", "}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&= \\left(\\left(\\prod _{i = 1}^{\\gamma dk}\\frac{\\gamma dn - i + 1}{\\gamma dn - \\gamma dk - i + 1}\\right)-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\left(1 + \\frac{2\\gamma dk}{\\gamma dn - \\gamma dk}\\right)^{\\gamma dk}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\left(1 + \\frac{3k}{n}\\right)^{\\gamma dk}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\frac{6\\gamma dk^2}{n}\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T].\\end{split}$ where the first inequality follows from the fact that $\\frac{1}{1 - x} \\le 1 + 2x$ for $x \\le \\frac{1}{4}$ .", "We can now bound the variance of $X$ : $\\begin{split}\\text{Var}(X) &= \\mathbb {E}[X^2] - \\mathbb {E}[X]^2 \\\\&= \\sum _{S: |S| \\le \\log (n)}\\sum _{T: |T| \\le \\log (n)}|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\sum _{S}\\sum _{T : T \\cap S = \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) + \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _S \\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\mathbb {E}[X_SX_T] \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _S \\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\frac{|S \\cup T|}{n}\\right)^{|S \\cup T| - 1}p^{\\gamma d|S \\cup T| - c_{\\ref {lemma:abc_small}}|S \\cup T|\\log (\\gamma d)}\\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _{R: 1 \\le |R| \\le 2\\log (n)} \\sum _{S, T \\subset R}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)}\\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _{R: 1 \\le |R| \\le 2\\log (n)} 2^{2|R|}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)} \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + np^{\\gamma d - c_2\\log (\\gamma d)} \\\\&\\le 6\\gamma d\\log (n)^2np^{2\\gamma d - 2c_1\\log (\\gamma d)} + np^{\\gamma d - c_2\\log (\\gamma d)}.\\end{split}$ for some constant $c_2$ (the second to last line follows from a similar calculation as used when computing $\\mathbb {E}[X]$ ).", "It follows by Markov's law that for any $t$ , $\\Pr [X \\ge \\mathbb {E}[X] + t] \\le \\frac{\\text{Var}(X)}{t^2}$ Plugging in $t = \\left(np^{\\gamma d}\\right)^{3/4}$ , we have $\\begin{split}\\Pr [X \\ge \\mathbb {E}[X] + n^{3/4}p^{-\\gamma d}] &\\le \\frac{\\text{Var}(X)}{\\left(np^{\\gamma d}\\right)^{3/2}} \\\\&\\le \\frac{6\\gamma d\\log (n)^2np^{2\\gamma d - 2c_1\\log (\\gamma d)} + np^{\\gamma d - c_2\\log (\\gamma d)}}{\\left(np^{\\gamma d}\\right)^{3/2}} \\\\&\\le b^{-1/2}\\left(6\\gamma d\\log (n)^2p^{\\gamma d/2-2c_1\\log (\\gamma d)} + p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\right)\\\\&\\le n^{-1/2}p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\log (n)^2\\end{split}$ It follows that with probability at most $n^{-1/2}p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\log (n)^2 = o(1)$ , we have $X \\le np^{\\gamma d - c_1\\log (\\gamma d)} + \\sqrt{np^{\\gamma d}}\\log (n)^2p^{-c_2\\log (\\gamma d)} \\le np^{\\gamma d - c\\log (\\gamma d)},$ for some constant $c$ .", "This proves the theorem.", "Applications to Gradient Coding In the following section we will address the motivating application of our work: the design of gradient codes with small decoding error.", "The literature on gradient codes largely focuses on the special case where the assignment matrix is a square.", "Hence, to align our results with this standard, we introduce the following stacked $ABC$ construction, which will us to apply our results on wide matrices to square $n\\times n$ assignment matrices.", "Definition 7 For $\\gamma ,d,n\\in \\mathbb {Z}^+$ such that $\\gamma \\mid d$ and $\\gamma \\mid n$ , we define the $\\gamma $ -stacked Augmented Biregular Code $B$ to be an $n\\times n$ matrix formed by sampling $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ and stacking $\\gamma $ identical copies of $A_0$ .", "We will denote the distribution of such matrices as $ABC_{\\gamma -stacked}(n,d)$ (see Figure REF ).", "Figure: The stacked ABCRemark 6 We note that these stacked designs can be viewed as a natural generalization of the Fractional Repetition Code (FRC) of [21].", "In particular, any $B\\sim \\textnormal {ABC}_{stacked}(N,d,d)$ is the $N\\times N$ FRC matrix with $d$ ones in each column, up to a permutation of the rows and columns.", "The key improvement is that our generalization allows the stacking variable $\\gamma $ — which increases the adversarial decoding error — to stay constant for arbitrarily large $d$ .", "Using these stacked ABC matrices, we will prove the following theorem: * While the first bound will be a direct corollary of Theorem $\\ref {abc_random}$ , the second bound relies on some external lemmas that will be introduced below.", "Thus, for the sake of clarity, we will split the proof of Theorem $\\ref {abc_stacked}$ into the following two lemmas.", "Lemma 22 Let $c,\\gamma _0,d_0$ be the universal constants from Theorem $\\ref {abc_random}$ .", "Choose any $\\gamma , d\\in \\mathbb {Z}^+$ such that $\\gamma geq \\gamma _0$ , $\\gamma \\mid d$ , and $\\frac{d}{\\gamma }\\ge d_0$ .", "For sufficiently large $N$ such that $\\gamma \\mid n$ , let $B\\sim \\textnormal {ABC}_{stacked}(n, \\gamma ,d)$ .", "Then with probability $1-o(1)$ over the choice of $B$ , we have that for any $p<\\frac{1}{2}$ : $\\frac{1}{n}\\mathbb {E}_{S\\sim \\binom{[n]}{pn}} \\textnormal {err}(B, S) \\le p^{d + c\\log (d)}+o(1)$ Lemma 23 Choose any $\\gamma ,d,N\\in \\mathbb {Z}^+$ such that $\\gamma \\mid d$ and $\\gamma \\mid n$ .", "Let $B\\sim \\textnormal {ABC}_{stacked}(n,\\gamma , d)$ .", "With constant probability, we have: $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\left( \\textnormal {err}(B, S)\\right) \\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1).$ Temporarily assuming these lemmas, we note that Theorem REF follows immediately.", "We will now prove Lemma $\\ref {stacked_prop1}$ .", "[Proof of Lemma REF ] Let $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ denote the ABC matrix which is stacked $\\gamma $ times to generate $B$ , then it follows that: $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}&= \\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|Bw - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|A_0w - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\mathbb {E}\\left[\\min _{w}|Aw - \\mathbb {1}|_2^2\\right]$ where $A\\sim \\textnormal {ABC}_p(n/\\gamma ,\\gamma ,d/\\gamma )$ .", "First observe trivially that $ \\frac{\\gamma }{n}\\min _{w}|Aw - \\mathbb {1}|_2^2$ is always at most 1.", "This can be seen by taking $w$ to be the vector of all zeros.", "By Theorem REF , there is a $1-o(1)$ chance that $A_0$ has the property that the following holds with probability $1-o(1)$ over the choice of $S$ : $\\frac{\\gamma }{n}\\min _{w}|Aw - \\mathbb {1}|_2^2\\le p^{d-c\\log (d)}+o(1).$ For $A_0$ where this holds, we can calculate the expectation as: $\\gamma \\mathbb {E}\\left[\\min _{w}|Aw - \\mathbb {1}|_2^2\\right]\\le (1-o(1))\\left(p^{\\gamma d-c\\log (\\gamma d)}+o(1)\\right)n+o(n).$ This gives us the desired result: $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}\\le p^{d-c\\log (d)}+o(1)$ for a matrix $B\\sim \\textnormal {ABC}_{stacked}(n,\\gamma , d)$ with probability $1-o(1)$ .", "Lemma REF requires a bit more machinery.", "We are going employ the following lemma from [13] which allows us to bound the adversarial error of a gradient code as a function of the second largest singular value.", "Formally, we have the following lemma.", "Lemma 24 (Proposition 4.1 of [13]) Let $A\\in \\lbrace 0,1\\rbrace ^{N\\times M}$ be an assignment matrix such that each row has exactly $D$ ones.", "Let $\\sigma _2$ be the largest singular value of $A$ .", "Then for any set of stragglers $S$ such that $|S|=s$ , we have: $\\frac{1}{N}\\textnormal {err}(A,S)\\le \\frac{1}{N}\\left(\\frac{\\sigma _2}{D}\\right)^2\\frac{sM}{M-s}$ To calculate an upper bound on the second largest singular value of an $ABC$ matrix, we first need a result from [2] which states that with constant probability, the configuration model we use to generate our $ABC$ matrix has no rows which map to the same column node more than once ie., the bipartite graph produced is simple.", "Let $\\mathcal {G}(n, \\gamma n, d, \\gamma d)$ denote the uniform distribution on simple $(d, \\gamma d)$ -biregular bipartite graphs with $n$ left nodes and $\\gamma n$ right nodes.", "Lemma 25 ([2]) Let $A_0\\sim ABC(n,\\gamma ,d)$ , then the probability that $\\begin{pmatrix}0 & A_0\\\\A_0^T & 0\\end{pmatrix}$ is the adjacency matrix of a bipartite, biregular random graph $G$ is at least $\\varepsilon (d)>0$ .", "Furthermore, if we condition on this event occurring, then $G\\sim \\mathcal {G}(n,\\gamma n,d,\\gamma d)$ .", "The previous lemma allows us to apply the following result of [3] to bound the second largest singular value of these well behaved ABC matrices.", "Lemma 26 (Theorem 4 of [3]) Let $A=\\begin{pmatrix}0 & X\\\\X^T & 0\\end{pmatrix}$ be the adjacency matrix of a bipartite, biregular random graph $G\\sim \\mathcal {G}(n,\\gamma n,d,\\gamma d)$ .", "Then, with probability $1-o(1)$ , $A$ 's second largest eigenvalue $\\lambda _2$ satisfies $\\lambda _2 \\le \\sqrt{d_1-1}+\\sqrt{d_2-1}+o(1).$ We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Let $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ denote the $ABC$ matrix which is stacked $\\gamma $ times to generate $B$ .", "It follows that: $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}&= \\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|Bw - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|A_0w - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}} err(A_0,S)$ By Lemma $\\ref {stacked_lemma2}$ , with constant probability, $A^{\\prime }=\\begin{pmatrix}0 & A_0\\\\A_0^T & 0\\end{pmatrix}$ is the adjacency matrix of a bipartite, biregular random graph $G$ .", "Condition on this event occurring.", "Then $G$ is uniformly sampled from $\\mathcal {G}(n/\\gamma ,N,d/\\gamma ,d)$ .", "By Lemma $\\ref {stacked_lemma3}$ , with probability $1-o(1)$ , the second largest eigenvalue of $A^{\\prime }$ is $\\lambda _2\\le \\sqrt{d/\\gamma -1}+\\sqrt{d-1}+o(1)$ .", "Thus, the second largest singular value of $A_0$ is $\\sigma _2 \\le \\sqrt{d/\\gamma -1}+\\sqrt{d-1}+o(1)$ .", "Hence, by Lemma $\\ref {stacked_lemma1}$ , we have: $\\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}} err(A_0,S)&\\le \\frac{\\gamma }{n}\\left(\\frac{\\gamma \\sigma _2}{d}\\right)^2\\frac{pn^2}{(1-p)n}\\\\&\\le \\gamma ^3\\left(\\frac{2\\sqrt{d}+o(1)}{d}\\right)^2\\frac{p}{(1-p)}\\\\&\\le \\left(\\frac{4\\gamma ^3p}{d(1-p)}\\right)+o(1)\\\\&\\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1)$ as desired.", "Table: Comparison of Related Work.", "We have normalized the decoding error by 1/N1/N.", "Acknowledgements The authors thank Mary Wootters for helpful comments on this manuscript.", "Proof of Lemmas  and REF for classification of minimal dependencies Recall Lemma : * We break the proof of Lemma  into three lemmas, which we prove independently.", "Lemma 27 $\\mathcal {S}_{2k-2, k} = \\mathcal {T}_k$ .", "Further, for $B \\in \\mathcal {T}_k = \\mathcal {S}_{2k - 2, k}$ , there is a unique (up to constant multiple) non-zero vector $v$ satisfying $B^T v = 0$ , where $v_i = (-1)^{d_G(i,j)} v_j$ , and $d_G(i,j)$ is the path length from vertex $i$ to $j$ in the graph $G$ encoded by $B$ .", "We prove this by showing both inclusions in the following two claims.", "Claim 11 $\\mathcal {T}_k \\subset \\mathcal {S}_{2k-2, k}$ .", "Let $B \\in \\mathcal {T}_k$ .", "Since $B$ has $k-1$ non-zero rows, it has rank at most $k-1$ , and so the nullspace of $B^T$ has dimension at least 1.", "Let $v$ be any vector such that $B^Tv = 0$ .", "Suppose $(i,j)$ is an edge in the tree $G$ encoded by $\\mathcal {T}_k$ , then there is a column $C$ of $B$ such that $B$ has 1's exactly at coordinates $i$ and $j$ .", "We have $C \\cdot v = 0$ thus $v_j = -v_i$ .", "Applying this to paths of multiple edges gives that $v_i = (-1)^{d_G(i,j)} v_j$ where $d_G$ is the path length from $i$ to $j$ in $G$ (which is unique and well-defined because $G$ is a tree).", "By this formula, if any $v_i$ is non-zero, then all other entries of $v$ are non-zero, and are uniquely determined by $v_i$ .", "Thus $B^T$ (and $B$ ) have rank $k-1$ .", "We also know that $B$ has $k-1$ non-zero columns, each with 2 1's, so it has $2k-2$ non-zero entries.", "Thus $B \\in \\mathcal {S}_{2k - 2, k}$ .", "In addition, by the description of $v$ , this establishes the second statement of the lemma.", "Claim 12 $\\mathcal {S}_{2k - 2, k} \\subset \\mathcal {T}_k$ .", "Suppose $B \\in \\mathcal {S}_{2k - 2, k}$ .", "Because $B$ has rank $k-1$ it must have $\\ge k-1$ non-zero columns.", "Recall that by Observation REF , no column of $B$ can have exactly one 1.", "Thus, $B$ must have exactly $k-1$ non-zero columns, each with exactly 2 1's.", "Let $G$ be be the graph on vertices $[1,\\ldots ,k]$ encoded by incidence matrix given by the non-zero columns of $B$ .", "We claim $G$ is connected.", "Suppose otherwise, and without loss of generality let $[1,\\ldots ,L], [L+1,\\ldots ,k]$ be two disconnected components of $G$ for $1 \\le L \\le k-1$ .", "By definition of $\\mathcal {S}_{2k-2,k}$ there is non-zero $v \\in \\mathbb {R}^k$ with $B^T v = 0$ , and $v$ has all non-zero entries.", "Let $v^{(L)}, v^{(R)} \\in \\mathbb {R}^k$ with $v^{(L)} = (v_1,v_2,\\ldots ,v_L,0,0,\\ldots ,0)$ and $v^{(R)} = (0,0,\\ldots ,0,v_{L+1},v_{L+2},\\ldots ,v_k)$ .", "$B^T(v^{(L)} + v^{(R)}) = B^T v = 0$ .", "By the disconnectedness of $[1,\\ldots ,L]$ and $[L+1,\\ldots ,k]$ , the set of columns of $B$ having non-zero entries in rows $[1,\\ldots ,L]$ is disjoint from the set of columns of $B$ having non-zero entries in rows $[L+1,\\ldots , k]$ .", "Thus $B^T v^{(L)}$ and $B^T v^{(R)}$ are non-zero in disjoint coordinates.", "Thus $B^T(v^{(L)} + v^{(R)}) = 0 \\rightarrow B^T v^{(L)} = B^T v^{(R)} = 0$ .", "This implies the null space of $B^T$ has rank at least 2, which contradicts that $B$ has rank $k-1$ .", "Thus $G$ is a connected graph.", "Because $G$ is a connected graph on $k$ vertices with at most $k-1$ edges, $B$ must encode a tree, and all its $k-1$ non-zero columns encode distinct edges.", "Thus $B \\in \\mathcal {T}_k$ .", "Call a column of a $\\lbrace 0,1\\rbrace $ -matrix an $n$ -column if it has exactly $n$ non-zero entries.", "Lemma 28 $\\mathcal {T}_k^+ = \\mathcal {S}_{2k - 1, k}$ We prove this by showing both inclusions in the following two claims.", "Claim 13 $\\mathcal {T}_k^+ \\subset \\mathcal {S}_{2k - 1, k}$ .", "Suppose $B \\in \\mathcal {T}_k^+$ .", "$B$ is the incidence matrix of a forest with two trees connected by a 3-hyperedge; without loss of generality we may assume the first column of $B$ encodes this hyper-edge.", "Further, by relabeling rows appropriately, we may assume $B_{11} = B_{21} = B_{k1} = 1$ , with vertices 1 and 2 in the same tree and connected by an even path, and vertex $k$ in the other tree.", "Then we may further relabel rows such that rows $1,2,\\ldots ,L$ correspond to one tree, and $L+1,\\ldots ,k$ correspond to the other, for some $L$ with $2 \\le L \\le k-1$ .", "Let $B^{^{\\prime \\prime }}$ be $B$ without its first column.", "The entries of $B^{^{\\prime \\prime }}$ , restricted to rows $1,\\ldots ,L$ , define a tree in $T_L$ .", "So by the second statement of Lemma REF , there is a unique (up to constant multiple) non-zero vector $v_\\ell \\in \\mathbb {R}^L$ satisfying $(B^{^{\\prime \\prime }}_{[L]})^T v_\\ell = 0$ .", "Similarly, there is a unique (up to constant multiple) non-zero vector $v_r \\in \\mathbb {R}^{k-L}$ satisfying $(B^{^{\\prime \\prime }}_{[k] \\setminus [L]})^T v_r = 0$ .", "Because the non-zero columns of rows $1,\\ldots ,L$ are disjoint from the non-zero columns of rows $L+1,\\ldots ,k$ , the null space of $(B^{^{\\prime \\prime }})^T$ is the direct sum of the null spaces of $(B^{^{\\prime \\prime }}_{[L]})^T$ and $(B^{^{\\prime \\prime }}_{[k] \\setminus [L]})^T$ .", "Concretely, let $v^{(\\ell )}, v^{(r)} \\in \\mathbb {R}^k$ be defined $v^{(\\ell )} = v_\\ell || 0^{k-L}$ and $v^{(r)} = 0^L || v_r$ , where $||$ means concatenation.", "Then the null space of $B^{^{\\prime \\prime }}$ are precisely the vectors $\\alpha v^{(\\ell )} + \\beta v^{(r)}$ for $\\alpha ,\\beta \\in \\mathbb {R}$ .", "Recall $B$ is $B^{^{\\prime \\prime }}$ , with the additional first column with three 1's in $B_{11}, B_{21}$ , and $B_{k1}$ .", "Thus the null space of $B$ is: $\\alpha v^{(\\ell )} + \\beta v^{(r)}: \\alpha , \\beta \\in \\mathbb {R}\\textnormal { subject to }\\alpha (v^{(\\ell )}_1 + v^{(\\ell )}_2) + \\beta v^{(r)}_k = 0$ From the second statement of Lemma REF , for $B_L \\in T_L$ , a non-zero vector $v$ satisfying $B_L^T v = 0$ has $v_a = (-1)^{d(a,b)} v_b$ relative to the encoded graph $G_L$ .", "We have $B^{\\prime \\prime }_{[L]} \\in T_L$ .", "Vertices 1 and 2 are connected by an even path in the relevant graph, which establishes $v^{(\\ell )}_1 = v^{(\\ell )}_2$ , so in particular $(v^{(\\ell )}_1 + v^{(\\ell )}_2) \\ne 0$ .", "Thus, the null space of $B$ is precisely: $t (v^{(r)}_k v^{(\\ell )} - (v^{(\\ell )}_1 + v^{(\\ell )}_2) v^{(r)}) , t \\in \\mathbb {R}$ Observe that $(v^{(r)}_k v^{(\\ell )} - (v^{(\\ell )}_1 + v^{(\\ell )}_2) v^{(r)}) \\in \\mathbb {R}^k$ has all non-zero entries because the coefficients $v^{(r)}_k$ and $(v^{(\\ell )}_1 + v^{(\\ell )}_2)$ are non-zero, and vectors $v^{(\\ell )}$ and $v^{(r)}$ have disjoint support.", "Thus $B \\in \\mathcal {S}_{2k - 1, k}$ .", "Claim 14 $\\mathcal {S}_{2k - 1, k} \\subset \\mathcal {T}_k^+$ .", "Suppose $B \\in \\mathcal {S}_{2k - 1, k}$ .", "Since $B$ is a minimal dependency, none of its columns have a single 1.", "Further, $B$ must have at least $k - 1$ non-zero columns since $B$ must have rank $k - 1$ .", "Hence $B$ has $k-2$ columns with two 1's and one column with three 1s.", "Without loss of generality we suppose $B$ 's first column is the one with 3 1's.", "Again define $B^{\\prime \\prime }$ as $B$ omitting its first column.", "Because $B$ 's non-zero columns are linearly independent, $B^{\\prime \\prime }$ has rank $k-2$ .", "We show that $B^{\\prime \\prime }$ encodes two disjoint trees by proving that $B^{\\prime \\prime }$ must encode a graph with no more than two connected components.", "Suppose for sake of contradiction that $B^{\\prime \\prime }$ encodes a graph $G^{\\prime \\prime }$ having connected components $X,Y,Z$ , where $X,Y,Z$ are disjoint non-empty subsets of $[1,\\ldots ,k]$ .", "Let $v \\in \\mathbb {R}^k$ be an all-non-zero vector such that $B^T v = 0$ .", "For a vertex set $S \\subset [1,\\ldots ,k]$ write $v_S$ for the vector with $(v_S)_i = v_i$ for $i \\in S$ , $(v_S)_i = 0$ otherwise.", "So, $v = v_X + v_Y + v_Z$ , and $v_X, v_Y, v_Z$ have non-zero entries in disjoint locations.", "We have $B^T (v_X + v_Y + v_Z) = 0 \\in \\mathbb {R}^M$ , where $m$ is the number of columns in $B$ , so we have $(B^{\\prime \\prime })^T (v_X + v_Y + v_Z) = 0 \\in \\mathbb {R}^{m-1}$ .", "But because $X,Y,Z$ are disjoint in $G^{\\prime \\prime }$ , the non-zero entries of $(B^{\\prime \\prime })^T v_X$ , $(B^{\\prime \\prime })^T v_Y)$ , $(B^{\\prime \\prime })^T v_Z$ must be pairwise disjoint.", "Thus, $(B^{\\prime \\prime })^T (v_X + v_Y + v_Z) = 0 \\rightarrow (B^{\\prime \\prime })^T v_X = (B^{\\prime \\prime })^T v_Y =(B^{\\prime \\prime })^T v_Z = 0$ .", "Because $v_X, v_Y, v_Z$ are linearly independent, the nullity of $(B^{\\prime \\prime })^T$ is at least 3, so $\\mathrm {rank}(B^{\\prime \\prime }) = \\mathrm {rank}((B^{\\prime \\prime })^T) \\le k-3$ .", "This contradicts our result that $\\mathrm {rank}(B^{\\prime \\prime }) = k-2$ .", "Thus, $G^{\\prime \\prime }$ has at most two connected components.", "Because $B^{\\prime \\prime }$ has $k-2$ non-zero columns, $G^{\\prime \\prime }$ has $k-2$ edges.", "Since $G^{\\prime \\prime }$ has $k$ vertices and no more than two connected components, it must be two disjoint trees.", "We return to the first column of $B$ .", "Without loss of generality let vertices $[1,\\ldots , L]$ correspond to one of the disjoint trees in $G^{\\prime \\prime }$ , and $[L+1,\\ldots ,k]$ the others, with $1 \\le L \\le k-1$ .", "We want to reason about $a,b,c$ , the three coordinates with $B_{a1} = B_{b1} = B_{c1} = 1$ .", "Applying Lemma REF again to the two disjoint trees gives that there is a vector $v^{(\\ell )} \\in \\mathbb {R}^k$ non-zero on exactly the coordinates $1,2,\\ldots ,L$ with $(B^{\\prime \\prime })^T v^{(\\ell )} = 0$ , and a vector $v^{(r)} \\in \\mathbb {R}^k$ non-zero on exactly the coordinates $L+1,\\ldots ,k$ with $(B^{\\prime \\prime })^T v^{(\\ell )} = 0$ .", "We can rule out that all of $a,b,c \\in [1,\\ldots , L]$ : Suppose for sake of contradiction this was the case, then observe that $u = v^{(r)}$ would satisfy $(B^T u)_1 = 1 v^{(r)}_a + 1 v^{(r)}_b + 1 v^{(r)}_c = 0 + 0 + 0 = 0$ , and $(B^{\\prime \\prime })^T v^{(r)} = 0$ , so $B^T u = 0$ : $u$ is non-zero but not all of its entries are non-zero, which contradicts $B \\in \\mathcal {S}_{2k - 2, k}$ .", "Symmetrically, we cannot have all of $a,b,c \\in [L+1,\\ldots , k]$ .", "Then without loss of generality we have $a,b \\in [1,\\ldots ,L], c \\in [L+1,\\ldots ,k]$ .", "Write $G^{\\prime \\prime }_1$ for the subgraph of $G^{\\prime \\prime }$ induced by vertices $[1,\\ldots ,L]$ : this is a tree containing $a$ and $b$ .", "Suppose for sake of contradiction that $a$ and $b$ are connected by an odd length path.", "Then $v^{(\\ell )}_a = -v^{(\\ell )}_b$ .", "Then let $u = 1 v^{(\\ell )}$ : $(B^{\\prime \\prime })^T u = 0$ as before, and $(B^T u)_1 = 1 v^{(\\ell )}_a + 1 v^{(\\ell )}_b + 1 v^{(\\ell )}_c = - v^{(\\ell )}_b + v^{(\\ell )}_b + 0 = 0$ .", "Again we have non-zero $u$ with not all entries 0 satisfying $B^T u = 0$ – a contradiction.", "Thus we must have that $a,b$ are connected by a path of even length in $G^{\\prime \\prime }_1$ .", "It follows that $B$ is the vertex-hyperedge incidence matrix of a forest with two trees connected by a 3-hyperedge where the two vertices of the 3-hyperedge in the same tree are connected by an even length path.", "Thus $B \\in \\mathcal {T}_k^+$ .", "Lemma 29 $\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ .", "We prove this by showing both inclusions in the following two claims.", "Claim 15 $\\mathcal {T}_k^C \\subset \\mathcal {S}_{2k, k}^{\\prime }$ Suppose $B \\in \\mathcal {T}_k^C$ .", "Then $B$ has $k$ columns with two 1's each.", "Without loss of generality we may let column 1 of $B$ be an edge in the cycle; then let $B^{\\prime }$ be $B$ with the first column set to zero.", "Then $B^{\\prime } \\in \\mathcal {T}_k$ , so $B^{\\prime } \\in \\mathcal {S}_{2k - 2, k}$ by Lemma REF .", "Hence there is a unique (up to constant multiple) non-zero $v$ with $(B^{\\prime })^T v = 0$ , and $v$ has all non-zero entries.", "Write the first column of $B$ as $e_x + e_y$ , for $1 \\le x < y \\le k$ .", "We claim that $e_x + e_y$ is in the span of $B^{\\prime }$ 's columns.", "By definition, the edge $(x,y)$ is part of an even length cycle in the graph encoded by $B$ .", "We encode this cycle.", "For odd $n \\ge 1$ , there are ordered pairs $(a_i, b_i), 1 \\le i \\le n$ such that each sum of basis vectors $e_{a_i} + e_{b_i}$ is a column of $B^{\\prime }$ , $b_i = a_{i+1}$ for $1 \\le n-1$ , and $a_1 = x$ and $b_n = y$ .", "Then observe that $(e_{a_1} + e_{b_1}) - (e_{a_2} + e_{b_2}) + \\ldots + (-1)^{n+1} (e_{a_n} + e_{b_n}) = e_{a_1} + e_{b_n} = e_x + e_y.$ Thus column 1 of $B$ is in the span of the rest of the columns of $B$ .", "Thus $B^T v = 0$ as well, and because $B^{\\prime }$ has rank $k-1$ , $B$ must have rank $k-1$ as well.", "So $B \\in \\mathcal {S}_{2k, k}$ .", "Moreover, $B \\in \\mathcal {S}_{2k, k}^{\\prime }$ because it has $k$ non-zero columns each with two 1's.", "Claim 16 $\\mathcal {S}_{2k, k}^{\\prime } \\subset \\mathcal {T}_k^C$ .", "Let $B \\in \\mathcal {S}_{2k, k}^{\\prime }$ .", "Because $B$ has rank $k-1$ , and $k$ non-zero columns, there is a subset of $k-1$ columns of $B$ with rank $k-1$ .", "Then without loss of generality assume the first column of $B$ is in the span of the other columns, and let $B^{\\prime }$ be $B$ , but with the first column set to 0.", "There is a unique (up to constant multiple) non-zero vector $v$ such that $B^T v = 0$ , and $v$ has all non-zero entries.", "Then by construction $B^{\\prime T} v = 0$ .", "Further because $B^{\\prime }$ has rank $k-1$ , so does $B^{\\prime T}$ , and the rank of the nullspace of $B^{\\prime T}$ is 1.", "Thus $v$ must be the unique (up to constant multiple) non-zero vector satisfying $B^{\\prime T} v = 0$ .", "So $B^{\\prime } \\in \\mathcal {S}_{2k - 2, k}$ .", "Thus $B^{\\prime } \\in \\mathcal {T}_{k}$ .", "Write the first column of $B$ as $e_x + e_y$ , $1 \\le x < y \\le k$ .", "To show $B \\in \\mathcal {T}_k^C$ , we must show that $x$ and $y$ are connected by an odd length path in the tree encoded by $B^{\\prime }$ .", "We know that $e_x + e_y$ is in the span of the columns of $B^{\\prime }$ , so we can write $e_x + e_y = \\sum _{i=2}^{m} \\alpha _i (e_{a_i} + e_{b_i}),$ where $m$ is the width of $B^{\\prime }$ , and for each $i$ where $\\alpha \\ne 0$ and the column $B_i$ is non-zero, we have $B_i = e_{a_i} + e_{b_i}$ .", "Let $S = \\lbrace i : \\alpha _i \\ne 0 \\wedge B_i \\ne 0\\rbrace $ .", "If some $1 \\le j \\le k$ appears in only one pair in $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , it must be the case that $\\sum _{i=1}^{m} \\alpha _i (e_{a_i} + e_{b_i})$ has non-zero $e_j$ coefficient.", "So $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , is a collection of edges of a tree such that at most two vertices ($x$ and $y$ ) belong to only one edge.", "Then $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , must be a path, with $x$ and $y$ at the end points.", "Without loss of generality let $a_2 = x, b_{|S| + 1} = y$ and $a_i = b_{i+1}$ for $2 \\le i \\le |S|$ .", "For $2 \\le i \\le |S|$ , because $e_{b_i} \\perp e_x + e_y$ , we must have that $\\alpha _{i+1} = -\\alpha _i$ so that the sum will cancel in coordinate $b_i$ .", "It follows that $e_x + e_y = \\sum _{i=2}^{|S| + 1} (-1)^i \\alpha _2 (e_{a_i} + e_{b_i}) = \\alpha _2 e_x + (-1)^{|S| + 1} \\alpha _2 e_y.$ It is clear we must choose $\\alpha _2 = 1$ and $|S|$ must be odd.", "Thus $e_x + e_y$ fulfills the conditions of the additional edge forming an even cycle in $\\mathcal {T}_k^C$ .", "We thus have $B \\in \\mathcal {T}_k^C$ .", "We now prove Lemma REF , which we restate here.", "Lemma 30 Suppose we have two sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ where $A_S \\in \\mathcal {M}_{|S|}$ and $A_T \\in \\mathcal {M}_{|T|}$ .", "Let $\\ell $ be the number of non-zero entries in $A_{S \\cup T}$ .", "Then there are at least $\\max \\left(|S \\cup T| - 1, \\frac{\\ell }{2}\\right)$ non-zero entries in $A_{S \\cup T}$ that are not the first (top) non-zero entry in their column.", "Let $R := S \\cup T$ .", "By Observation REF , since $A_S$ and $A_T$ are minimal dependencies, there cannot be a column in $A_S$ or $A_T$ with a single one.", "Hence no column of $A_R$ has a single one.", "It follows immediately that there are at least $\\frac{\\ell }{2}$ non-zero entries that are not the first non-zero entry in their column in $A_R$ .", "Next we show that there are at least $|R| - 1$ non-zero entries in $A_R$ .", "Let $G = (R, E)$ be the hypergraph given by the vertex-hyperedge incidence matrix $A_R$ .", "We claim that $G$ must be connected.", "Indeed, each of the hypergraphs given by the incidence matrix $A_S$ and $A_T$ are connected on their own.", "Since $S \\cap T \\ne \\emptyset $ , the hypergraph given by $A_R$ must be connected.", "Let $F$ be an arbitrary spanning tree of $G$ whose edges are contained in the hyperedges of $G$ .", "Let $v \\in R$ be an arbitrary node in $G$ , which we call the root.", "For each $u \\in R \\setminus \\lbrace v\\rbrace $ , consider the entry $(u, e(u))$ of $A$ , where $e(u)$ is the index of the hyperedge containing the edge from $u$ towards $v$ in $F$ .", "Note that $A_{u, e(u)} = 1$ , and further, each hyperedge $e$ in $G$ contains at least one node $x(e)$ (which can be chosen arbitrarily) for which $e(x(e)) \\ne e$ .", "Consider $x(e)$ to be the “first\" non-zero entry in the column $e$ .", "It follows that the entries $\\lbrace (u, e(u))\\rbrace _{u \\in R \\setminus \\lbrace x\\rbrace }$ are not the “first\" in their column.", "We show this argument pictorially in Figure REF , where we draw an arrow from each $u$ to $e(u)$ , and we circle the entry $A_{u, e(u)}$ .", "Figure: Illustration of the proof that there are at least |R|-1|R| - 1 non-zero entries that are not first in their column.", "Each hyperedge contains at least one vertex that is not pointed at, otherwise, the arrows would create a cycle.", "Proof of Lemmas REF , REF and REF We prove Lemmas REF and REF .", "Lemma REF is a corollary of Lemma REF .", "* * We prove this by linearly combining the dependencies.", "Without loss of generality, let $S = [k]$ for some k. For all $i \\in [k]$ , since $A_i \\in H_i$ , there exists some $x^{(i)}$ such that $x^{(i)}_i \\ne 0$ and $Ax^{(i)} = 0$ .", "Observe that for each $i \\in [k]$ , we have $\\mathrm {supp}(x^{(i)}) \\subset S$ - otherwise it would imply that some column $A_j$ for $j \\notin S$ is spanned by $H_j$ .", "Choose random coefficients $c_i$ for $i \\in [k]$ from any continuous distribution and let $y = \\sum _i {c_i x^{(i)}}$ .", "Then with probability 1, $y$ is non-zero on all $i \\in S$ .", "* For the first part, let $v$ be such that $A^Tv = e_i$ .", "Then for any $w$ with $i \\in \\mathrm {supp}(w)$ , we have $w^TA^Tv = w_i \\ne 0$ , so it is impossible that $Aw = 0$ .", "For the converse, suppose $A_i \\notin H_i$ .", "Then there must exist some $w_i $ such that $\\langle {{w_i }, A_i}\\rangle \\ne 0$ , but ${w_i } \\perp A_j$ for all $j \\ne i$ .", "However, this implies that $A{w_i } = \\langle {{w_i }, A_i}\\rangle e_i$ , so $e_i \\in \\mathrm {Span}(A^T)$ , which is a contradiction.", "Proofs of Lemmas  and , and on bounds related to Binomial distributions * $\\begin{split}\\Pr \\left[\\mathrm {Bin}(n, p) \\ge t\\right] &\\le \\sum _{i = t}^{n}\\binom{n}{i}p^i \\\\&\\le \\sum _{i = t}^{n}\\binom{n}{t}p^i\\prod _{j = t + 1}^i\\left(\\frac{n - j + 1}{j}\\right) \\\\&\\le \\sum _{i = t}^{n}\\binom{n}{t}p^i\\prod _{j = t + 1}^i\\left(\\frac{n}{t}\\right) \\\\&= \\sum _{i = t}^{n}\\binom{n}{t}p^t\\left(\\frac{pn}{t}\\right)^{i - t} \\\\&\\le \\binom{n}{t}p^t\\sum _{j = 0}^{\\infty }\\left(\\frac{pn}{t}\\right)^{j} \\\\&= \\frac{\\binom{n}{t}p^t}{1 - \\frac{pn}{t}}.\\end{split}$ For $t \\ge 2np$ , plugging in Sterling's formula yileds the lemma.", "* We break down this sum as follows.", "$\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le \\sum _{\\ell = j}^{2j}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ]\\\\&\\qquad + \\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\qquad + \\sum _{\\ell \\ge n/3}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ]\\end{split}$ We bound the first term in the following claim.", "Claim 17 $\\sum _{\\ell = j}^{2j}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk}.$ $\\begin{split}\\sum _{\\ell = j}^{2j}&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 2j\\Pr \\left[\\mathrm {Bin}\\left(2j, \\frac{2j + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\& \\le 4j\\left(\\frac{e2j(2j + k)}{\\gamma nj}\\right)^j\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\&\\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j].\\end{split}$ Here the first inequality follows from the fact that the summand is highest for $\\ell = 2j$ , the second inequality follows from the tail bound in Lemma  in Section , and then third inequality follows from the fact that $j \\le k + 1 \\le 2k$ .", "Now $\\begin{split}\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] &\\le \\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\&\\le \\binom{\\gamma nk}{2j}\\left(\\frac{d}{n}\\right)^{2j}\\left(1 - \\frac{d}{n}\\right)^{\\gamma nk - 2j} \\\\&\\le \\left(\\frac{e\\gamma nk}{2j}\\right)^{2j}\\left(\\frac{d}{n - d}\\right)^{2j}\\left(1 - \\frac{d}{n}\\right)^{\\gamma nk} \\\\&\\le \\left(\\frac{e\\gamma dk}{j}\\right)^{2j}e^{-\\gamma dk}\\\\&\\le \\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk}\\end{split}$ Combining this with Equation REF yields the claim.", "We bound the second term in Equation REF in the following claim.", "Claim 18 $\\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}.$ .", "For $\\ell \\le n/3$ , using Lemma  and Sterling's forumla, we have $\\begin{split}&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 2\\left(\\frac{e\\ell (\\ell + k)}{\\gamma n\\frac{\\ell }{2}}\\right)^{\\lceil {\\ell /2}\\rceil }\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&= 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&= 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\binom{\\gamma nk}{\\ell }\\left(\\frac{d}{n}\\right)^\\ell \\left(1 - \\frac{d}{n}\\right)^{\\gamma nk - \\ell } \\\\&\\le 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{e\\gamma nk}{\\ell }\\right)^{\\ell }\\left(\\frac{d}{n - d}\\right)^\\ell \\left(1 - \\frac{d}{n}\\right)^{\\gamma nk} \\\\&\\le 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{e\\gamma nk}{\\ell }\\right)^{\\ell }\\left(\\frac{d}{n - d}\\right)^\\ell e^{-\\gamma dk} \\\\&\\le 2\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } e^{-\\gamma dk}.\\end{split}$ We do casework on the parity of $\\ell $ .", "Let $\\ell = 2a + b$ , where $b \\in \\lbrace 0, 1\\rbrace $ .", "If $b = 1$ , then $\\begin{split}\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } &= \\left(\\frac{4e(2a+1)}{\\gamma n}\\right)^{a + 1}\\left(\\frac{2e\\gamma dk}{2a + 1}\\right)^{2a + 1} \\\\&= \\left(\\frac{16e^3\\gamma ^2d^2k^2(2a+1)}{\\gamma n(2a + 1)^2}\\right)^a\\left(\\frac{8e^2\\gamma dk(2a+1)}{\\gamma n(2a + 1)}\\right) \\\\&\\le \\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\left(\\frac{8e^2dk}{n}\\right)\\end{split}$ Now since the maximum over $x$ of $f(x) = \\left(\\frac{y}{x}\\right)^x$ is achieved at $x = y/e$ , and above this value of $x$ , the $f(x)$ is decreasing, since $j \\ge k - 1 \\ge \\frac{e8e^3\\gamma d^2k^2}{n}$ , we have for all $a \\ge j$ , $\\begin{split}\\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\left(\\frac{8e^2dk}{n}\\right) &\\le \\left(\\frac{8e^3\\gamma d^2k^2}{nj}\\right)^a\\left(\\frac{8e^2dk}{n}\\right)\\\\&\\le \\left(\\frac{8e^3\\gamma d^2k^2}{n(k - 1)}\\right)^a\\left(\\frac{8e^2dk}{n}\\right) \\\\&\\le \\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{a + 1}.\\end{split}$ If $b = 0$ , then $\\begin{split}\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } &= \\left(\\frac{8ea}{\\gamma n}\\right)^{a}\\left(\\frac{2e\\gamma dk}{2a}\\right)^{2a} \\\\&= \\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\end{split}$ By the same reasoning as before, we have for all $a \\ge j + 1$ , $\\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a \\le \\left(\\frac{8e^3\\gamma d^2k^2}{n(j + 1)}\\right)^{a} \\le \\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{a}.$ Combining these two cases back into Equation REF , we have for all $\\ell \\ge 2j + 1$ , $\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 2e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{\\lceil {\\frac{\\ell }{2}}\\rceil }.$ Summing over all $\\ell \\ge 2j + 1$ , we have $\\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}.$ .", "Finally, we bound the third term in Equation REF in the following claim.", "Claim 19 $\\sum _{\\ell \\ge n/3}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le $ It suffices to bound the probability $\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\ge n/3].$ Again employing Lemma , we have $\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\ge n/3] \\le 2\\left(\\frac{3e\\gamma dk}{n}\\right)^{n/3}.$ Combining claims REF , REF and REF , we have $\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk} \\\\&\\qquad + 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}\\\\&\\qquad + 2\\left(\\frac{3e\\gamma dk}{n}\\right)^{n/3}.\\end{split}$ It is easy to check that this sum is dominated by the first term, and hence for some universal constant $c_{\\ref {masterlemma:small}}$ , $\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 16k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk} \\\\&\\le \\left(\\frac{k}{n}\\right)^je^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}.\\end{split}$ This proves the lemma.", "* Because $n/2 < n-k-1$ , we may bound the probability by: $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\frac{n}{2}, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].\\end{split}$ We use the Chernoff bound $\\Pr [X \\le (1 - \\delta )\\mu ] \\le e^{-\\mu \\delta ^2/2}$ for $\\mu = \\mathbb {E}[X]$ , plugging in $\\mu = \\frac{n}{2}\\left(1 - \\frac{1}{\\sqrt{kd/n}}\\right)$ and $\\delta = (1 - \\frac{1}{\\sqrt{kd/n}})^{-1} \\left(1 - \\frac{1}{\\sqrt{kd/n}} - \\frac{2k}{n}\\right)$ .", "This gives $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\frac{n}{2}, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right]\\\\&\\le e^{-\\frac{n}{2}\\left(1 - \\frac{1}{\\sqrt{kd/n}}\\right)\\left(1 - \\frac{1}{\\sqrt{kd/n}} - \\frac{2k}{n}\\right)^2}\\\\&\\le e^{-n\\epsilon },\\end{split}$ where $\\epsilon \\ge 1/36$ .", "To achieve this value of $\\epsilon $ , we plugged in $k < \\frac{n}{12}$ in the last inequality.", "Now we compute the sum over $k$ : $\\sum _{k = \\frac{2n}{d}}^{\\frac{n}{C}}\\binom{n}{k}e^{-\\epsilon n} \\le n \\binom{n}{n/C} e^{-\\epsilon n} \\le n (eC)^{n/C} e^{-\\epsilon n} = e^{n(\\frac{\\log n}{n} + \\frac{1 + \\log C}{C} - \\epsilon )}$ which for constant $C$ large enough, is $e^{-\\Theta (n)}$ .", "Proof of Claims REF and REF in the ABC Small Case In the following Appendix, we will prove the claims stated in the proof of Lemma $\\ref {lemma:abc_small}$ .", "Before proving these claims, we will need the following additional lemma: The following bound gives a general approximation for the probability mass function of a HyperGeometric Distribution.", "It also provides a second bound under the additional assumption that the number of draws in our distribution is not too large relative to the population size.", "Lemma 31 Let $\\mathcal {X}\\sim \\text{HyperGeom}(A,B,n)$ .", "Furthermore, let us define $(1-q)=\\frac{B}{A}$ .", "Then $\\Pr [\\mathcal {X}=k]\\le \\binom{n}{k}(1-q)^k\\left(q+\\frac{k}{A-n}\\right)^{n-k}.$ Furthermore, assuming $n\\le \\frac{3}{2}qA$ and $q\\le 1/2$ , we have: $\\begin{split}\\Pr [\\mathcal {X}=k]&\\le \\binom{n}{k}(1-q)^k q^{n-k}\\left(e^{\\left({{6ek}}\\right)}\\right)\\\\&\\le \\left(\\frac{en}{k}\\right)^k(1-q)^k q^{n-k}\\left(e^{\\left({{6ek}}\\right)}\\right) \\\\ &= \\left(\\right)^k\\end{split}$ Recall that, by definition of the hypergeometric distribution, we have: $\\Pr [\\mathcal {X}=k]=\\frac{\\binom{B}{k}\\binom{A - B}{n-k}}{\\binom{A}{n}}$ We can expand out the binomial terms into their factorial representations to see: $\\Pr [\\mathcal {X}=k]&=\\frac{\\frac{B!}{(B-k)!k!", "}\\cdot \\frac{(A - B)!", "}{(A - B-n+k)!(n-k)!}}{\\frac{A!}{(A-n)!n!}}\\\\&=\\binom{n}{k}\\left(\\frac{B!}{(B-k)!", "}\\cdot \\frac{(A-B)!}{(A-B-n+k)!", "}\\cdot \\frac{(A-n)!}{A!", "}\\right)\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}(B-k+i)\\prod _{i=1}^{n-k}(A-B-n+k+i)\\prod _{i=1}^n \\frac{1}{A-n+i}\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\frac{A-B-n+k+i}{A-n+i}\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\left(1-\\frac{B}{A-n+i}+\\frac{k}{A-n+i}\\right) \\\\&\\le \\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\left(1-\\frac{B}{A}+\\frac{k}{A-n}\\right)$ By definition of the hypergeometric distribution, $k\\le B$ .", "Thus, it follows $k-i<B<A$ for all $i\\in [k]$ .", "This implies: $\\frac{B-k+i}{A-k+i}\\le \\frac{B}{A}=(1-q)$ This gives us the first inequality of the lemma: $P[\\mathcal {X}=k]\\le \\binom{n}{k}\\prod _{i=1}^k (1-q) \\prod _{i=1}^{n-k} \\left(q+\\frac{k}{A-n}\\right)=\\binom{n}{k} (1-q)^k \\left(q+\\frac{k}{A-n}\\right)^{n-k}.$ We will now proceed under the assumption that $n\\le \\frac{3}{2}qA$ to achieve the second bound.", "We write: $\\left(q+\\frac{k}{A-n}\\right)^{n-k}&\\le \\left(q+\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{n-k}\\\\&=\\sum _{i=0}^{n-k} \\binom{n-k}{i}q^{n-k-i}\\left(\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k} \\left(\\frac{\\frac{3}{2}eqA}{i}\\right)^i\\left(\\frac{1}{q}\\right)^i\\left(\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k}\\left(\\frac{\\frac{3}{2}ek}{i(1-\\frac{3}{2}q)}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k} \\left(\\frac{1}{i!", "}\\right)\\left(\\frac{\\frac{3}{2}ek}{1-\\frac{3}{2}q}\\right)^{i}\\\\&\\le q^{n-k}\\left(e^{\\left(\\frac{\\frac{3}{2}ek}{1-\\frac{3}{2}q}\\right)}\\right)\\\\&\\le q^{n-k}\\left(e^{\\left({6ek}\\right)}\\right)$ Applying this to the first inequality gives: $P[\\mathcal {X}=k]\\le \\binom{n}{k}(1-q)^kq^{n-k}\\left(e^{\\left(6ek\\right)}\\right)$ as claimed.", "We are now ready to prove claims REF and REF .", "* By expanding the binomial distribution, we have, $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1-p)N}\\right)=j\\right] &\\le \\sum _{i=0}^j\\binom{\\ell }{i}\\left(\\frac{\\ell }{\\gamma (1 - p)N}\\right)^{i}\\left(1-\\frac{\\ell }{\\gamma (1-p)N}\\right)^{\\ell -i}\\\\&\\le 2^{\\ell } \\left(\\frac{\\ell }{\\gamma (1-p)N}\\right)^{\\ell -j}.\\end{split}$ Using the fact that $K\\le \\frac{3}{2}N$ , we employ the second bound of Lemma REF to find: $&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1 - p)N}\\right)=j\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]\\\\&\\le \\left(\\binom{\\gamma dK}{\\ell }(1-p)^{\\ell }p^{\\gamma dK-\\ell }e^{6e\\ell }\\right) 2^\\ell \\left(\\frac{\\ell }{c(1-p)N}\\right)^{\\ell -j}\\\\&\\le \\binom{\\gamma dK}{2K}p^{\\gamma dK-2K}e^{12eK} 2^{2K} \\left(\\frac{2K}{\\gamma N}\\right)^{\\ell -j}\\\\&\\le \\left(\\frac{9e^{12K+2}c^2d^2}{4}\\right)^{K}p^{\\gamma dK-2K} \\left(\\frac{K}{N}\\right)^{\\ell -j}\\\\&\\le \\left( p^{\\gamma d+c\\log (\\gamma d)} \\right)^K\\left(\\frac{K}{N}\\right)^{\\ell -j}$ for some universal constant $c$ .", "Next we show the second statement in the claim.", "Since for sufficiently large $\\gamma d$ , the mean of HyperGeom$(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)$ is greater than $4K$ , the term $\\Pr [\\mathrm {Bin}(\\ell , \\frac{K}{N})\\ge K-1]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]$ is maximized (over $\\ell \\in [1, 4K]$ ) at $\\ell = 4K$ .", "By Equation REF we have we have $\\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{K}{N}\\right)\\ge K-1\\right]&\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ] \\\\&\\le 4K\\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=4K]2^{4K} \\left(\\frac{K}{\\gamma (1 - p)N}\\right)^{K-1}\\\\&\\le 4K\\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=4K]2^{5K} \\left(\\frac{K}{N}\\right)^{K-1}$ Using the fact that $K\\le \\frac{3}{2}N$ , we employ the second bound of Lemma REF to find: $& \\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{4K}{\\gamma (1-p)N}\\right)\\ge K-1\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]\\\\&\\le 4K\\left(\\binom{\\gamma dK}{4K}p^{\\gamma dK-4K}e^{24eK}\\right)2^{5K}\\left(\\frac{4K}{N}\\right)^{K-1}\\\\&\\le 4K\\left(\\frac{e\\gamma d}{4}\\right)^{4K}p^{\\gamma dK-4K}e^{24eK}2^{5K}\\left(\\frac{K}{N}\\right)^{K-1}\\\\&\\le \\left(p^{\\gamma d-c^{\\prime }\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{K-1}.$ for some constant $c^{\\prime }$ .", "The claim follows from choosing $c_{\\ref {claim:abc1}}$ larger than $c$ and $c^{\\prime }$ .", "We now prove Claim REF .", "* We write: $&\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk} \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\sum _{i=0}^{\\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor }\\binom{\\ell }{i}\\left(\\frac{\\ell }{\\gamma n}\\right)^{\\ell -i}\\left(1-\\frac{\\ell }{\\gamma (1-p)n}\\right)^\\ell \\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]3^\\ell \\left(\\frac{\\ell }{\\gamma (1-p)n}\\right)^{\\ell /2}$ Applying the first bound of Lemma REF , we find: $&\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\binom{\\gamma dk}{\\ell }(1-p)^k\\left(p+\\frac{\\ell }{\\gamma d(n-k)}\\right)^{\\gamma dk-\\ell }\\right)3^\\ell \\left(\\frac{\\ell }{\\gamma (1-p)n}\\right)^{\\ell /2}\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\frac{1}{2}+\\frac{k}{(n-k)}\\right)^{\\gamma dk}\\left(\\frac{e\\gamma dk}{\\ell }\\right)^\\ell 6^\\ell \\left(\\frac{\\ell }{\\gamma n}\\right)^{\\ell /2}\\\\&\\le (0.625)^{-\\gamma dk} \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{\\ell n}}\\right)^{\\ell }\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}\\max _{\\ell \\in \\lbrace 2k+1,...,\\gamma dk\\rbrace }\\left(\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{\\ell n}}\\right)^{\\ell }.$ To show the function in the expression above is maximized at $2k+1$ when $k\\le \\frac{n}{18e\\gamma d^2}$ , we consider the derivative.", "Taking $C=\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{n}}$ , we write: $\\frac{d}{d\\ell }\\left[ \\left(\\frac{C}{\\sqrt{\\ell }}\\right)^\\ell \\right] &=\\frac{d}{d\\ell }\\left[ e^{\\ell \\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\right]\\\\&=e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\frac{d}{d\\ell }\\left[\\ell \\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right) \\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell \\frac{d}{d\\ell }\\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)\\right]\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell \\frac{1}{\\left(\\frac{C}{\\sqrt{\\ell }}\\right)}\\frac{d}{d\\ell }\\left[\\left(\\frac{C}{\\sqrt{\\ell }}\\right)\\right]\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell ^{1.5}\\frac{d}{d\\ell }\\ell ^{-0.5}\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)-\\frac{1}{2}\\right]\\\\&= \\frac{1}{2}e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C^2}{\\ell }\\right)-1\\right]\\\\&= \\frac{1}{2}e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C^2}{e\\ell }\\right)\\right].$ Thus, we see that the derivative is zero when $\\ell =\\frac{C^2}{e}=\\frac{36e\\gamma d^2k^2}{n}$ and strictly negative afterwards.", "For $k\\le \\frac{n}{18e\\gamma d^2}$ , this maximum occurs before $\\ell =2k+1$ .", "As the function is monotonically decreasing after the maximum, we conclude that $\\ell =2k+1$ is where the function is maximized on the interval $[2k+1,\\gamma dk]$ for $k\\le \\frac{n}{18e\\gamma d^2}$ .", "We continue: $&\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}\\left(\\frac{36\\gamma e^2d^2k^2}{(2k+1)n}\\right)^{k+1/2}\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}(18\\gamma e^2d^2)^{k+1/2}\\left(\\frac{k}{n}\\right)^{k+1/2}\\\\&\\le e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}$ for sufficiently large $\\gamma d$ .", "Lower bounds on distance to $\\mathbb {1}$ via number of all-zero rows In this appendix, we prove lower bounds on the number of zero rows in a random matrix $A$ .", "The following observation shows that this is sufficient to give a lower bound on the distance between $\\mathbb {1}$ and the span of $A$ .", "Observation 3 For any matrix $A$ , if there is a subset of zero rows $S$ , then for any $w$ , $|A_S w - \\mathbb {1}_S| = |S|$ thus $|A w - \\mathbb {1}| \\ge |S|$ .", "Lemma 32 Let $A \\sim \\text{BGC}(n, \\gamma , d)$ , then with probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1-o(1))e^{-\\gamma d} n$ .", "let $I_i$ be the indicator variable that row $i$ is all zero.", "$\\Pr [I_i = 1] = (1 - \\frac{d}{n})^{\\gamma n} = (1-o(1)) e^{-\\gamma d}$ .", "By a Chernoff bound, $\\Pr \\left[\\sum _i I_i \\le (1-n^{-1/3})(1-o(1))e^{-\\gamma d}n\\right] \\le e^{-(1-o(1))e^{-\\gamma d}n (n^{-2/3})/2} = o(1)$ Thus with probability $1-o(1)$ , $A$ has at least $(1-n^{-1/3})(1-o(1))e^{-\\gamma d} n = (1-o(1)) e^{-\\gamma d} n$ zero rows.", "The following lemma generically shows when we can apply the second moment method, which we will use to bound the number of all-zero rows for symmetric Bernoulli matrices and for the ABC.", "Lemma 33 Let $S_N = \\lbrace I^{(N)}_1,\\ldots , I^{(N)}_N\\rbrace $ be a family of sets of $\\lbrace 0,1\\rbrace $ -random variables such $\\mathbb {E}[I^{(N)}_i] = \\Theta (1)$ for all $i$ , and $\\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(1/N)$ for all $i \\ne j$ .", "Let $X^{(N)} = \\sum _{i=1}^{N} I^{(N)}_i$ .", "Then with probability $1-o(1)$ , $X^{(N)} \\ge (1-o(1)) \\mathbb {E}[X^{(N)}],$ where all big-O notation is in terms of $N \\rightarrow \\infty $ .", "$\\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(1/N)$ for $i \\ne j$ , and $\\text{Var}(I^{(N)}_i) = O(1)$ because $I^{(N)}_{i}$ is bounded, so $\\text{Var}(X^{(N)}) = \\sum _{i=1}^{n} \\text{Var}(I^{(N)}_i) + \\sum _{1 \\le i \\ne j \\le N} \\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(N)$ .", "By Chebyshev's inequality, $\\Pr \\left[|X^{(N)} - \\mathbb {E}[X^{(N)}]| > N^{1/3} \\sqrt{\\text{Var}(X^{(N)})}\\right] < N^{-1/3}$ Because $\\text{Var}(X^{(N)}) = O(N)$ , $N^{1/3} \\sqrt{\\text{Var}(X^{(N)})} = o(N)$ .", "Thus, we have with probability $1-o(1)$ that $X^{(N)} \\ge \\mathbb {E}[X^{(N)}] - o(N)$ .", "Because $\\mathbb {E}[X^{(N)}] = \\sum _{i=1}^{N} \\mathbb {E}[I^{(N)}_i] = \\Theta (N)$ , we equivalently assert that with probability $1-o(1)$ , $X^{(N)} \\ge (1-o(1))\\mathbb {E}[X^{(N)}]$ .", "Lemma 34 Let $A \\sim \\textnormal {SB}(n, d)$ .", "With probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1 - o(1))e^{-d}n$ .", "Define $I_i$ as the indicator variable of the event that row $i$ is all zero.", "We have $\\mathbb {E}[I_i] = (1-d/n)^{n}$ for all $i$ , but the $I_i$ are not independent.", "Let $X = \\sum _{i=1}^{N} I_i$ .", "We compute $\\text{Var}(X)$ .", "$\\text{Var}(X) = \\sum _{i,j=1}^{n} \\text{Cov}(I_i, I_j) = \\sum _{i,j=1}^{n} \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j]$ Note that $\\mathbb {E}[I_i I_j]$ is the probability of the event that both row $i$ and $j$ are 0.", "When $i \\ne j$ , rows $i$ and $j$ have identical entry $A_{ij} = A_{ji}$ , but all other entries are independent.", "Thus the probability of both rows $i,j$ having all zero entries is $(1-d/n)^{2n-1}$ .", "Thus, $\\text{Cov}(I_i, I_j) &= (1-d/n)^{2n-1} - (1-d/n)^{2n} \\\\&= \\left(\\frac{1}{1-d/n} - 1\\right) (1-d/n)^{2n} \\\\&= \\frac{d/n}{1-d/n} (1-d/n)^{2n} = O(1/n)$ Then from Lemma REF , with probability $1-o(1)$ , we have $\\mathbb {E}[X] \\ge (1-o(1)) n (1-d/n)^n = (1-o(1)) n e^{-d}$ .", "Lemma 35 Let $A \\sim \\textnormal {ABC}(n, \\gamma , d)$ .", "With probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1 - o(1))p^{\\gamma d}n$ .", "Define $I_i$ as the indicator variable of the event that row $i$ is all zero.", "Because we are interested in upper bounding the variance, we want to upper bound $\\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j]$ .", "A row $i$ of $A \\sim \\text{ABC}_p(n,\\gamma ,d)$ is zero only if all of the $\\gamma d$ 1's in row $i$ of $A_0$ fell into the $p\\gamma n$ last columns.", "In the setting of the configuration model, this means all of row $i$ 's $\\gamma d$ half edges are matched to the $p(\\gamma dn)$ half edges corresponding to the dropped columns.", "So $\\Pr [E_i] = \\frac{ \\binom{p(\\gamma dn)}{\\gamma d)} }{\\binom{\\gamma dn}{\\gamma d} }$ Let $j \\ne i$ .", "Observe that $\\mathbb {E}[I_j | I_i]$ can be computed similarly, but the conditioning implies that there are already $cd$ half edges from row $i$ assigned all to the $p(\\gamma dn)$ dropped column half-edges.", "$\\Pr [E_j | E_i] = \\frac{ \\binom{p(\\gamma dn) - cd}{cd)} }{\\binom{\\gamma dn - cd}{cd} } $ Observe that if two pairs of functions $a_i(n),b_i(n), i=1,2$ satisfy $a_i(n), b_i(n) = O(1)$ and $a_i(n) - b_i(n) = O(1/n)$ , then $a_1(n) a_2(n) = (b_1(n) + O(1/n))(b_2(n) + O(1/n)) = b_1(n) b_2(n) + (b_1(n) + b_2(n) + O(1))O(1/n) \\rightarrow a_1(n)a_2(n) - b_1(n)b_2(n) = O(1/n)$ .", "This can be applied iteratively to any constant number of functions.", "Observe that $\\Pr [E_i] - \\Pr [E_j | E_i] = \\prod _{\\ell =0}^{\\gamma d-1} \\frac{p(\\gamma dn)-\\ell }{\\gamma dn-\\ell } - \\prod _{\\ell =0}^{\\gamma d-1} \\frac{p(\\gamma dn)-\\gamma d-\\ell }{\\gamma dn-\\gamma d-\\ell }$ Note that the terms corresponding to a fixed $\\ell $ in each product have difference that is $O(1/n)$ : $\\frac{p(\\gamma dn) - \\ell }{\\gamma dn - \\ell } - \\frac{p(\\gamma dn) - cd - \\ell }{\\gamma dn - cd - \\ell } = \\frac{O(n)}{(\\gamma dn - \\ell )(\\gamma dn - cd - \\ell )} = O(1/n).$ It follows from the observation (and that $\\gamma d$ is a constant) that $\\Pr [E_i] - \\Pr [E_j | E_i] = O(1/n)$ .", "It follows that $\\text{Cov}(I_i, I_j) = \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j] = \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i]^2 = (\\Pr [E_i]) (\\Pr [E_i] - \\Pr [E_j | E_i]) = O(1/n)$ Then by Lemma REF , we have with probability $1-o(1)$ that $X \\ge (1-o(1))E[X] = (1-o(1)) p^{\\gamma d}n$ .", "Implication of Theorem REF on exact rank of random matrices.", "Define the 2-core of a matrix $A$ to be the matrix remaining after repeating the following peeling process: If there is a row with zero or one non-zero entries, remove that row, and remove the column corresponding to the position of the non-zero entry (if any).", "Note that this 2-core corresponds to viewing $A$ as the vertex-hyperedge incidence matrix of a hypergraph.", "Let $\\mathbf {n^*}$ and $\\mathbf {m^*}$ be the number of rows (vertices) and columns (hyperedges) respectively in the 2-core.", "We prove the following corollary to Theorem REF .", "Corollary 1 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ .", "With probability $1 - o(1)$ , $\\textnormal {rank}(A) = {\\mathbf {n^*}} + n - {\\mathbf {m^*}}$ Let $D$ be the set of rows involved in linear dependencies: $D = \\bigcup _{x: x^TA = 0}{\\mathrm {supp}(x)}.$ Let $I := [n] \\setminus D$ be the remaining rows.", "Let $P_R$ be the set of rows removed during the peeling process, and let $P_C$ be the set of columns removed.", "Let $A^{\\prime } \\in \\mathbb {R}^{{\\mathbf {n^*}} \\times {\\mathbf {m^*}}}$ be the 2-core matrix which remains after the peeling process.", "The following claim follows immediately from the definition of the peeling process: Claim 20 The row-span of $A_{P_R}$ is $\\mathbb {R}^{P_C}$ , which has rank $n - \\mathbf {m^*}$ .", "Then it suffices to show that following claim holds conditioned on the event in Theorem REF holding.", "Claim 21 $A^{\\prime }$ has full row rank, which is rank $\\mathbf {n^*}$ .", "First we show that conditioned on the event in Theorem REF holding, we have $D \\subseteq P_R$ .", "Indeed, conditioned on this event holding, we have $D = \\lbrace i \\in [n]: \\exists S \\ni i: A_S \\in \\mathcal {T}_{|S|}\\rbrace .$ Consider any set $S$ for which $A_S \\in \\mathcal {T}_{|S|}$ .", "By the definition of $\\mathcal {T}_{|S|}$ , $A_S$ is the vertex-edge incidence matrix of a tree.", "Hence the peeling process defined above to create the 2-core will necessarily remove all the rows in $S$ .", "Now to prove the claim, suppose for contradiction that there existed a linear dependency among the rows of $A^{\\prime }$ .", "Then since the row span of $A_{P_R}$ is $\\mathbb {R}^{P_C}$ , it must be the case that there is a linear dependency in $A$ which contains theses rows of $A^{\\prime }$ .", "That is, if $x^TA^{\\prime } = 0$ , let $y := x^TA_{[n] \\setminus P_R}$ such that $\\mathrm {supp}(y) \\in P_C$ .", "Then we can find $x^{\\prime }$ such that $x^{\\prime T}A_{P_R} = -y$ , and hence combining $x$ and $x^{\\prime }$ yields a vector $\\mathbf {x}$ such that $\\mathbf {x}^TA = 0$ .", "However, this is a contradiction, because it means some row in $[n] \\setminus P_R$ must be involved in a linear dependency — and hence a minimal linear dependency — contradicting the fact that $D \\subseteq P_R$ .", "Proof of Lemma  REF In this section we prove the following anticoncentration lemma.", "* Let $r_i$ be an independent Bernoulli random variable with parameter $2p$ and let $s_i$ be an independent Bernoulli random variable with parameter $1/2$ .", "It follows that $\\Pr [r_is_i=1]=\\Pr [z_i=1]$ .", "Thus, we have: $\\Pr (v^T z =c) &= \\Pr \\left(\\sum _{i=1}^n v_i r_is_i=c\\right)\\\\&\\le \\Pr \\left(\\sum _{i=1}^n v_ir_is_i=c \\Big | \\sum _{i=1}^n r_i\\ge mp\\right) + \\Pr \\left(\\sum _{i=1}^n r_i\\le mp\\right)$ We employ a result of Erdős [11] to note: $\\Pr \\left(\\sum _{i=1}^n v_ir_is_i=c \\Big | \\sum _{i=1}^n r_i\\ge mp\\right)\\le \\left(\\frac{1}{\\sqrt{\\pi /2}}\\right)\\frac{1}{\\sqrt{mp}}$ Finally, a Chernoff bound gives us: $\\Pr (\\sum _{i=1}^n r_i\\le mp) \\le \\left(2^p\\left(\\frac{1-2p}{1-p}\\right)^{1-p}\\right)^m\\le e^{(\\ln (2)-1)mp}$ Combining these two expressions gives the desired result." ], [ "Notation and Formal Set-up", "In this work, we consider the following three ensembles of random matrices.", "We refer to two of these ensembles as “codes\" to follow the gradient coding literature.", "Rectangular Bernoulli Matrix (Bernoulli Gradient Code) Let $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ denote a random matrix in $\\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ where each entry of $A$ is Bernoulli with parameter $d/n$ .", "We will consider this ensemble for any $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ .", "Symmetric Bernoulli Matrix Let $A \\sim \\textnormal {SB}(n, d)$ denote a random symmetrix matrix in $\\lbrace 0, 1\\rbrace ^{n \\times n}$ whose upper diagonal entries are i.i.d.", "Bernoulli random variables with parameter $d/n$ , and whose diagonal is 0.", "We will consider this ensemble of matrices for $d = \\omega (1)$ .", "Augmented Biregular Code The Augmented Biregular Code (ABC) is based on the adjacency matrix of a random biregular graph generated from the configuration model.", "Formally, consider the following process to generate a random matrix $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ from the distribution $\\textnormal {ABC}(n, \\gamma , d)$ , for $\\gamma > 1$ .", "Create $n$ row-nodes and $\\gamma n$ column-nodes and associate to each row-node $\\gamma d$ half-edges and to each column node $d$ half-edges.", "Create a multigraph $G$ by choosing a uniformly random pairing of the $\\gamma dn$ half-edges from the row-nodes to the $\\gamma dn$ half-edges from the column-nodes.", "Given this bipartite graph, we will take $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ to be the matrix where $(A_0)_{ij} = 1$ iff there is at least one edge from node $i$ to $j$ .", "We will study the ensemble of random matrices obtained by removing a random $p$ fraction of columns from this ABC matrix $A_0$ .", "Formally, we call this ensemble $\\textnormal {ABC}_p(n, \\gamma , d)$ such that $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ is an $n \\times \\gamma n(1-p)$ matrix formed by deleting $\\gamma n p$ random columns from an ABC matrix $A_0 \\sim \\textnormal {ABC}(n, \\gamma , d)$ , which has dimensions $n \\times \\gamma n$ .", "For a positive integer $n$ , we use $[n]$ to denote the set $\\lbrace 1, \\ldots , n\\rbrace $ .", "All big-O notation denotes limiting behaviour as $n \\rightarrow \\infty $ .", "We use $\\mathrm {Span}(A)$ to mean the span of the columns of a matrix $A$ .", "For a vector $v \\in \\mathbb {R}^n$ and a set $S \\subset [n]$ , let $v_S$ denote the vector $v$ restricted to entries indexed by elements of $S$ .", "Similarly, for a matrix $A \\in \\mathbb {R}^{n \\times m}$ and a set $S \\subset [n]$ , let $A_S$ denote the matrix $A$ restricted to the rows in the set $S$ .", "For $j \\le m$ , let $A^{:j}$ denote the matrix restricted to the first $j$ columns of $A$ .", "Let $A_j$ denote the $j$ th column of $j$ .", "For a symmetric matrix $A \\in \\mathbb {R}^{n \\times n}$ , for $j \\in [n]$ , let $A^{(j)}$ denote the $n - 1 \\times n - 1$ matrix equal to $A$ with its $j$ -th row and column removed.", "We refer to a matrix $A$ as the vertex-edge incidence matrix of a graph $G$ if each there is a bijection between edges of $G$ and columns of $A$ that maps an edge $(i, j)$ to a column with ones in locations $i$ and $j$ and zeros elsewhere.", "Let $e_i$ denote the $i$ th canonical basis vector with a 1 in position $i$ .", "We use the notation $D_{KL}(p||q)$ to refer to the KL divergence of two Bernoulli random variables with parameters $p$ and $q$ respectively." ], [ "Statement of Results", "In this section, we state our main results highlighted in the contributions.", "We begin with a few definitions, which we summarize in Table REF .", "Definition 1 A matrix in $B \\in \\mathbb {R}^{k \\times m}$ is a minimal linear dependency if it satisfies the following two properties: There exists $x \\in \\mathbb {R}^k$ such that $x^TB = 0$ and $\\mathrm {supp}(x) = [k]$ .", "$B$ has rank $k-1$ .", "Let $\\mathcal {M}_k \\subset \\bigcup _{m \\ge 1} \\mathbb {R}^{k \\times m}$ denote matrices of height $k$ which are minimal dependencies.", "We will distinguish among these matrices three particular types of linear dependencies, which are illustrated in Figure REF .", "Figure: The three structures of minimal dependencies that occur with constant probability in the random matrices in Propositions , and for k=5k = 5.", "(a) An element of 𝒯 k \\mathcal {T}_k.", "(b) An element of 𝒯 k + \\mathcal {T}_k^+.", "(c) An element of 𝒯 k C \\mathcal {T}_k^C.Definition 2 (Tree dependency) Define $\\mathcal {T}_k$ as the set of matrices $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with exactly $2k - 2$ non-zero entries such that the non-zero columns of $B$ form the vertex-edge incidence matrix of a tree on $k$ vertices.", "Definition 3 (Two-forest dependency) Let $\\mathcal {T}_k^+$ be the set of matrices $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with exactly $2k - 1$ non-zero entries satisfying the following: $B$ has $k-1$ non-zero columns: $k-2$ columns supported on 2 entries and one column supported on 3 entries.", "The submatrix of $B$ restricted to the columns of support 2 is the vertex-edge incidence matrix of a forest $F$ with two connected components $F_1,F_2$ .", "The column of support 3 contains 1's at rows $a,b,c$ where $a,b \\in F_i$ are connected by an even-length path, and $c \\in F_j$ for $\\lbrace i, j\\rbrace = \\lbrace 1, 2\\rbrace $ .", "Definition 4 (Tree-with-added-edge dependency) Define $\\mathcal {T}_k^C$ as the set of $B\\in \\bigcup _m \\lbrace 0,1\\rbrace ^{k\\times m}$ with $2k$ non-zero entries such that the non-zero columns of $B$ form the vertex-edge incidence matrix of a tree on $k$ vertices, with an added edge between two vertices in the tree of odd distance from each other.", "The additional edge may create a multi-edge in the this graph.", "Table: Notation in this work" ], [ "Characterization of Linear Dependencies", "Our main technique in addressing Question REF is a new characterization of the linear dependencies that occur among the rows of a random matrix $A$ .", "The following three theorems describe our main results: [Characterization BGC]theoremcharbgc There exists a universal constant $c$ such that for any $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ , for $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ , with probability $1 - o(1)$ : All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^{+} \\cup \\mathcal {T}_k^C$ .", "The number of rows involved in a linear dependency of $A$ is at most $ne^{-\\gamma d + c\\log (\\gamma d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le ne^{-\\gamma d + c\\log (\\gamma d)}.$ [Characterization Symmetric Bernoulli]theoremcharsquare Let $A \\sim \\textnormal {SB}(n, d)$ , where $d = \\omega (1)$ .", "Then with probability $1 - o(1)$ , All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k$ .", "The number of rows involved in a linear dependency of $A$ is at most $ne^{-d + o(d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le ne^{- d + o(d)}.$ [Characterization ABC]theoremcharabc There exist universal constants $c$ and $\\gamma _0$ such that for any constant $p < 1/2$ , $\\gamma > \\gamma _0$ , and $d \\ge d_0(\\gamma )$ , for $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ , with probability $1 - o(1)$ : All minimal dependencies of $k$ rows of $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^{+} \\cup \\mathcal {T}_k^C$ .", "The number of rows involved in a linear dependency of $A$ is at most $np^{\\gamma d - c\\log (\\gamma d)}$ , that is $\\left|\\bigcup _{x : x^TA = 0} \\mathrm {supp}(x) \\right| \\le np^{\\gamma d - c\\log (\\gamma d)}.$ Remark 1 With high probability, one can show that under the conditions of Theorem REF , $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ does have minimal dependencies in $\\mathcal {T}_k$ for $k \\lesssim \\frac{\\log (n)}{\\gamma d}$ .", "We expect the same to hold for the symmetric matrix and for the ABC.", "For constant $d$ , with constant probability, $A \\sim \\text{BGC}(n, \\gamma , d)$ has minimal dependencies in $\\mathcal {T}_k^+$ and $\\mathcal {T}_k^C$ for $k$ sufficiently small, including $k = 5$ .", "We expect the same to hold for the ABC.", "Remark 2 While our initial motivation was Question REF , we believe these characterization theorems may have implications for understanding the exact rank of a sparse random matrices.", "For instance, in Corollary REF in the appendix, we show for $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ , with probability $1 - o(1)$ , the rank of $A$ is exactly equal to the graph-theoretic 2-core rank bound." ], [ "Progress on Question ", "The following three bound the distance from $\\mathbb {1}$ to the column span of the random matrices we study.", "Theorem 1 (BGC Distance) Let $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "For $\\gamma > 1$ and $d \\ge d_0(\\gamma )$ , there exists a constant $c$ such that with probability $1 - o(1)$ , $(1 - o(1))e^{-\\gamma d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le e^{-\\gamma d + c\\log (\\gamma d)}.$ Theorem 2 (Square Bernoulli Distance) Let $A \\sim \\textnormal {SB}(n, d)$ .", "For $d = \\omega (1)$ , with probability $1 - o(1)$ , $(1 - o(1))e^{-d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le e^{-d + o(d)}.$ Theorem 3 (ABC Distance) Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ .", "For any $p < 1/2$ , there exists constants $c$ , $\\gamma _0$ and $d_0$ such that for $\\gamma \\ge \\gamma _0$ and $d \\ge d_0$ , with probability $1 - o(1)$ , $(1 - o(1))p^{\\gamma d} \\le \\frac{1}{n}\\min _{w}|A w - \\mathbb {1}|_2^2 \\le p^{\\gamma d + c\\log (\\gamma d)}.$ Remark 3 (Distances to arbitrary vectors: good news) These results can be extended beyond $\\mathbb {1}$ to unit vectors $v \\in \\mathbb {R}^n$ whose mass is well-distributed among the coordinates of $v$ .", "Formally, suppose $v$ is a distance of at least $\\rho $ from any $\\delta n$ -sparse unit vector, that is, $v$ is $(\\delta , \\rho )$ -incompressible.", "Then under the conditions of Theorem REF , with probability $1 - o(1)$ , $\\frac{\\rho ^4\\delta }{8}e^{-d} \\le \\min _{w}|A w - v|_2^2 \\le \\max (n|v|_{\\infty }^2e^{-d + o(d)}, 1)$ Similar results hold for $A \\sim \\textnormal {BGC}$ or $A \\sim \\textnormal {ABC}_p$ , with the exponent of $e$ begin the same exponent as in the respective theorems.", "Remark 4 (Distances to arbitrary vectors: bad news) For arbitrary vectors $v$ , we cannot hope to prove similar high probability bounds as for $\\mathbb {1}$ .", "For instance, if $v$ is a $O(1)$ -sparse unit vector, then under the conditions of Theorems REF , REF , or REF , with constant probability, the distance from $v$ to the span of $A$ is 0.", "Similarly, under the conditions of Theorems REF or REF , with constant probability (that may depend on $d$ ), the distance from $v$ to the span of $A$ is 1.", "The statements in these remarks are evident from the proofs of the theorems above, described in Section REF ." ], [ "Gradient Coding Results", "We are able to use Theorem REF on the distance between $\\mathbb {1}$ and $A \\sim \\textnormal {ABC}_p$ to analyze the expected decoding error of an assignment matrix based on the ABC ensemble.", "Since most results in the gradient coding literature concern a square assignment matrices where $m = n$ , we design an $n \\times n$ assignment matrix $B \\sim \\textnormal {ABC}_{\\text{stacked}}(n, \\gamma , d)$ by stacking together $\\gamma $ copies of a rectangular matrices $A_0 \\sim \\textnormal {ABC}(n/\\gamma , \\gamma , d/\\gamma )$ for an appropriate choice of $\\gamma $ .", "Full details of the construction are given in Section .", "We prove the following theorem about the assignment matrix $B$ : []theoremabcstacked Let $c,\\gamma _0,d_0$ be the universal constants from Theorem $\\ref {abc_random}$ .", "Choose any $\\gamma ,d\\in \\mathbb {Z}^+$ such that $\\gamma \\ge \\gamma _0$ , $\\gamma \\mid d$ and $\\frac{d}{\\gamma }\\ge d_0$ .", "For any sufficiently large $n$ divisible by $\\gamma $ , let $B \\sim \\textnormal {ABC}_{\\text{stacked}}(n, \\gamma , d)$ .", "Then with constant probability over the choice of $B$ : $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)} \\le p^{d - c\\log (d)}+o(1),$ and $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\left( \\textnormal {err}(B, S)\\right) \\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1).$" ], [ "Overview of Proofs", "In the next subsection, we give an overview of how we bound the distance from $\\mathbb {1}$ to the span of $A$ to yield Theorems REF , REF , and REF .", "In the following subsection, we give an overview of our proof of the characterization results." ], [ "Bounding the Distance from $\\mathbb {1}$ to the Span of {{formula:a63f25b4-1732-47bb-966c-f8dc57db21b8}}", "Given a matrix $A$ , we will partition its rows into two sets: $D$ , the set of all rows involved in a linear dependency, and $[n] \\setminus D$ .", "Formally, $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x).$ We will use the following lemma to bound the distance from a vector $v$ to the column span of $A$ .", "Lemma 1 Let $A \\in \\mathbb {R}^{n \\times m}$ , and let $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x)$ be the set of rows which are involved in a linear dependency.", "Then for any $v \\in \\mathbb {R}^n$ , we have $\\min _{w \\in \\mathbb {R}^m}|Aw - v|_2^2 \\le |v_D|_2^2.$ This lemma follows from applying the following lemma, which we prove in Appendix .", "lemmabasisone Let $A \\in \\mathbb {R}^{n \\times m}$ , and let $D = \\bigcup _{x : x^TA = 0} \\mathrm {supp}(x)$ .", "Then for any $i \\notin D$ we have $e_i \\in \\mathrm {Span}(A)$ .", "To see how Lemma REF follows from Lemma REF , for any $v$ , by Lemma REF , the vector $v^{\\prime } := \\sum _{i \\in [n] \\setminus D}v_ie_i \\in \\mathrm {Span}(A)$ .", "Hence $\\min _w |Aw - v|_2^2 \\le |v - v^{\\prime }|_2^2 = |v_D|_2^2,$ establishing Lemma REF .", "For $v = \\mathbb {1}$ , we have $\\min _w |Aw - \\mathbb {1}|_2^2 \\le |D|.$ Plugging in the bound on $|D|$ given in the characterization theorems from Section REF yields the upper bounds in the distance theorems in Section REF .", "The lower bounds on the distance are given by counting the number of of all-zero rows in $A$ .", "Notice that the squared distance between $\\mathbb {1}$ and the span of $A$ is at least the number of all-zero rows in $A$ .", "We formally prove these lower bounds using standard concentration tools in Section ." ], [ "Overview of Proof of Characterization", "Our proof of each characterization result is divided into either two or three main cases: a “small\" case, a “large\" case, and sometimes a “medium\" case.", "The small case proves that for some constant $c_{\\text{s}}$ , for $k \\lesssim \\frac{n}{d^{c_{\\text{s}}}}$ , with high probability, all minimal dependencies of $k$ rows in $A$ are in $\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "The large case proves that for a second constant $c_{\\ell }$ , for $k \\gtrsim \\frac{n}{d^{c_{\\ell }}}$ , with high probability there are no minimal dependencies of $k$ rows in $A$ .", "The exact constants $c_{\\text{s}}$ and $c_{\\ell }$ depend on the particular ensemble of random matrices.", "We require a medium case when $c_{\\text{s}} > c_{\\ell }$ , in order to account for all $k$ .", "While our proofs are organized by the specific ensemble of random matrices, we give here a short overview of the techniques in the small and large cases, as they are similar among all three ensembles we study.", "The main idea in proving this characterization of dependencies in the small and medium cases comes from the following two trivial observations.", "Observation 1 Let $B \\in \\lbrace 0, 1\\rbrace ^{k \\times m}$ be a matrix.", "Then $B$ cannot be a minimal dependency if some column of $B$ contains exactly one 1.", "Recall that our matrices are $0/1$ valued, and thus, if there is only one row with value 1 at index $j$ , that row is linearly independent to all other rows in our subset.", "Hence, no such dependency exists.", "Observation 2 Let $B \\in \\lbrace 0, 1\\rbrace ^{k \\times m}$ be a matrix.", "If $B$ has fewer than $2k-2$ entries that are 1, then $B$ is not a minimal dependency.", "The requirement that $B$ is rank $k-1$ means that at least $k-1$ columns must have at least one 1 in them and it follows from the previous observation that for a minimal dependency to exist, each of these columns must have at least two ones in them.", "Thus, we need at least $2k-2$ entries which are 1 in $B$ before a minimal dependency can exist." ], [ "Small Case", "The goal of the small case is to prove that with high probability all small minimal row dependencies ($k \\lesssim n/d^{c_s}$ ) are contained in $\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "We begin by selecting an arbitrary set $S$ of $k$ rows, which induces a submatrix $A_S$ .", "Then, by conditioning on $L$ , which is the number of 1s which appear in $A_S$ , we consider a random process derived from the distribution of $A$ which places these $L$ 1s in the submatrix one by one.", "Due to Observation REF , we only must consider the case when $L \\ge 2k - 2$ .", "By Observation 1, $A_S$ does not have a minimal dependency if there exists a column with exactly one 1 in $A_S$ .", "To lower bound the probability of this event, we consider a random walk which increases by 1 every time our random process places a 1 in an already occupied column and stays constant otherwise.", "As long as the value of this random walk is less than $L/2$ at the time we have placed of the $L$ 1s in the submatrix, we know that $A_S$ does not have a minimal dependency.", "For the case of symmetric matrices, we modify this argument slightly by coupling with a random walk which also increases every time a 1 is placed in the “symmetric\" portion of $A_S$ , that is, a column of $A_S$ indexed by an element of $S$ ." ], [ "Large Case", "The goal of the large case is to show that with high probability, for $k \\gtrsim n/d^{c_\\ell }$ , there are no minimal dependencies of $k$ rows.", "Our main tool in ruling out large dependencies is the following set of anti-concentration results, most of which are standard in the literature.", "Roughly speaking, these results state that the dot product of a random vector and a deterministic vector will not concentrate on any one value with too large probability.", "Each ensemble of random matrices we study requires an anti-concentation lemma tailored to the distribution of a column vector from that matrix.", "In the BGC matrix, we use the following version of the Littlewood-Offord theorem for sparse random vectors.", "[c.f.", "[10] Lemma 8.2]lemmalosparse Let $v \\in \\mathbb {R}^n$ be a deterministic vector with support at least $m$ .", "Let $z \\in \\mathbb {R}^n$ be the random vector with i.i.d.", "Bernoulli entries with parameter $p \\le 1/2$ .", "Then for any fixed $c$ , $\\Pr \\left[v^Tz=c\\right] \\le \\left(\\frac{1}{\\sqrt{\\pi /2}}\\right)\\frac{1}{\\sqrt{mp}}+\\left(e^{(\\ln (2)-1)mp}\\right).$ In particular, for $mp\\ge 9$ , we have: $\\Pr \\left[ v^Tz=c\\right] \\le \\frac{1}{\\sqrt{mp}}.$ In the symmetric matrix case, we will additionally use a quadratic Littlewood-Offord result, originally due to [8].", "We state a version for sparse random vectors from [10].", "Lemma 2 (c.f.", "[10] Lemma 8.4) Let $M \\in \\mathbb {R}^{n \\times n}$ be a deterministic matrix with a least $m$ non-zero entries in each of $m$ distinct columns of $M$ .", "Let $z \\in \\mathbb {R}^n$ be the random vector with i.i.d.", "Bernoulli entries with parameter $p \\le 1/2$ .", "Then for any fixed $c$ , $\\Pr \\left(z^TMz = c\\right) = O\\left(\\frac{1}{(mp)^{1/4}}\\right).$ For the ABC matrix, in which each column of $A$ is close to $d$ -regular, we will use the following weaker anti-concentration lemma, which we prove in Section .", "Lemma 3 (Anti-concentration for Sparse Regular Vectors) Let $v \\in \\mathbb {R}^n$ be an arbitrary vector whose most common entry is $a$ .", "Then for any $d \\le \\sqrt{\\frac{n}{2}}$ , if $z \\in \\lbrace 0, 1\\rbrace ^n$ is sampled uniformly from the set of vectors with exactly $d$ 1s, we have: $\\Pr \\left[v^Tz = c\\right]\\le 1/2+\\frac{d^2}{n}.$ for all $c \\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "Our general strategy for the rectangular BGC and ABC matrices is as follows.", "Fix a set $S \\subset [n]$ of size $k$ .", "We consider the random process $A^{:1}_S, A^{:2}_S, \\cdots A^{:m}_S = A_S$ , where we add columns of $A_S$ one at a time.", "(Recall that $A^{:1}_S$ is the submatrix of $A$ given by restricting to the rows and $S$ and the first $i$ columns.)", "At each step $i$ , we keep track of the left kernel of $A^{:i}$ .", "Since our goal is to show that the left kernel of $A_S$ contains no vectors with support size $k$ , we leverage the anti-concentration results above as follows: if the kernel of $A^{:i}_S$ contains a vector $v$ with support $k$ , then it is likely that the next column added, $(A_S)_{i + 1}$ , will not be orthogonal to $v$ .", "After adding enough columns, we show that with high probability, we “knock out” all candidate kernel vectors with large support.", "While this approach is relatively straightforward for the BGC matrix where the columns are independent, we must be more careful for the ABC since the columns are not independent.", "In this case, for each column added, we consider the pairing of the $d$ half-edges of the corresponding column-node, and we show that for at least half of the columns, these pairings are “sufficiently random\".", "We use a similar approach to rule out some large dependencies ($n/d^{c_\\ell } < k < \\Theta (n)$ ) for the symmetric Bernoulli matrices.", "However, this approach breaks down for when $k$ becomes close to $n$ since the matrix is square.", "For instance, in the extreme case when $k = n$ , we would need to “knock-out\" a kernel vector at every single step of adding columns to $A$ .", "This certainly won't occur with high probability.", "Our strategy for ruling out dependencies on the order of $\\Theta (n)$ rows is inspired by a combination of the approaches in [12] and [8].", "Using Markov's law, we show that if there exists a kernel vector of $A$ with large support, then $A$ must contain many columns $A_i$ which are in the span of the remaining columns $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "We bound this probability that $A_i \\in \\mathrm {Span}(\\lbrace A_j\\rbrace _{j \\ne i})$ by conditioning on $A^{(i)}$ , the matrix formed by removing the $i$ th row and column of $A$ , and leveraging the randomness of $A_i$ .", "We consider two main cases: one in which $A^{(i)}$ has a kernel vector with large support, and one in which it doesn't.", "If $A^{(i)}$ has a kernel vector $v$ with large support (on the order of $\\Theta (n)$ ), we use Lemma REF to show that with probability $1 - O(1/\\sqrt{d})$ , $A_i$ is orthogonal to $v$ , and hence $A_i$ is not in the span of the remaining columns $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "If $A^{(i)}$ has no kernel vectors with large support, we are able to construct a dense “pseudoinverse” $B$ for $A^{(i)}$ , for which $A_i^TBA_i \\ne 0$ implies that $A_i$ is not in the span of $\\lbrace A_j\\rbrace _{j \\ne i}$ .", "Lemma REF guarantees this occurs with probability $1 - O(1/\\@root 4 \\of {d})$ .", "Notice that unlike the results for the BGC and ABC which hold with probability that decays at least polynomially in $n$ , our results for square matrices hold with probability that decays polynomially in $d$ .", "For this reason, our results only hold with high probability if $d$ goes to infinity with $n$ ." ], [ "Medium Case", "We sometimes need an extra medium case to rule out dependencies of $k$ rows where $n/d^{c_s} < k < n/d^{c_\\ell }$ .", "This is easily accomplished using Observation REF by showing the existence of a column with a single 1." ], [ "Useful Lemmas", "We gather here a few definitions and lemmas that we use in our proofs for multiple of the random matrix ensembles we study.", "Their proofs are technical, so we defer them to the appendix.", "The first lemma allow us to show that the linear dependencies we encounter with constant probability have the graph-structures depicted in Figure REF .", "Definition 5 Define $\\mathcal {S}_{L, k}$ to be union over all integers $m$ of the set of matrices $B\\in \\lbrace 0,1\\rbrace ^{k\\times m}$ such that $B$ forms a minimal dependency and $B$ has exactly $L$ entries which are 1.", "Definition 6 Define $\\mathcal {S}_{L, k}^{\\prime } \\subset \\mathcal {S}_{L, k}$ to be the subset of matrices in $\\mathcal {S}_{L, k}$ which have exactly $k$ non-zero columns.", "The following lemma relates these sets to the sets $\\mathcal {T}_k, \\mathcal {T}_k^+$ and $\\mathcal {T}_k^C$ introduced earlier in Definitions REF , REF , and REF .", "[Classification of Dependencies]lemmalemmaclassification We have the following three equivalences: $\\mathcal {S}_{2k-2, k} = \\mathcal {T}_k$ .", "$\\mathcal {S}_{2k-1, k} = \\mathcal {T}_k^+$ .", "$\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ .", "We will use the next lemma to bound the correlation between the existence of linear dependencies in intersecting submatrices $A_S$ and $A_T$ .", "Lemma 4 Suppose we have two sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ where $A_S \\in \\mathcal {M}_{|S|}$ and $A_T \\in \\mathcal {M}_{|T|}$ .", "Let $\\ell $ be the number of non-zero entries in $A_{S \\cup T}$ .", "Then there are at least $\\max \\left(|S \\cup T| - 1, \\frac{\\ell }{2}\\right)$ non-zero entries in $A_{S \\cup T}$ that are not the first (top) non-zero entry in their column.", "Lemmas  and REF are proved in Appendix .", "Several of our proofs use the following tail bound on Binomial distributions with a small parameter.", "[Tail Bound on Binomial]lemmataillemma If $t \\ge 2np$ , then $\\Pr \\left[\\mathrm {Bin}(n, p) \\ge t\\right] \\le 2\\left(\\frac{enp}{t}\\right)^t.$ Several of our proofs in the small case will use the following black-box calculation to bound the probability of encountering minimal dependencies.", "[Small Case Binomial Calculation]lemmamastersmall For constants $\\gamma , d > 0$ , for $k \\le \\frac{n}{8e^4\\gamma d^2}$ , there exists a constant $c_{\\ref {masterlemma:small}}$ such that for any $j \\in \\lbrace k - 1, k, k + 1\\rbrace $ and $\\gamma \\ge 1/2$ , we have $\\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^j.$ Similarly, several of our large cases will use the following black-box calculation.", "[Large Case Binomial Calculation]lemmalargegeneral There exists constants $C$ , $d_0$ , and $c_{\\ref {large_general}}$ , such that for all $d \\ge d_0$ , for any positive integer $\\frac{4n}{d} \\le k \\le n/C$ , we have $\\sum _{k = \\frac{2n}{d}}^{n/C}\\binom{n}{k}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\le e^{-c_{\\ref {large_general}}n}$ Lemmas , and are proved in Appendix ." ], [ "Bernoulli Gradient Codes", "In this section, we prove Theorem REF on the characterization of linear dependencies in matrices $A \\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "As discussed in the overview, we will divide the proof of this result into small, medium, and large cases." ], [ "Small Case", "The goal of this section will be to prove the following lemma.", "Recall that $\\mathcal {M}_k$ is the set of matrices with $k$ rows that are minimal dependencies.", "Lemma 5 (BGC Small Case) Let $A\\sim \\textnormal {BGC}(n,\\gamma ,d)$ .", "Let $S \\subset [n]$ be a set of size $k$ for $k \\in [1,\\frac{n}{8e^4\\gamma d^2}]$ .", "There exist universal constants $c_{\\ref {masterlemma:small}}, d_0$ such that if $d > d_0$ , then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_{k}^+ \\cup \\mathcal {T}_{k}^C \\right)] =\\left(e^{-\\gamma d + c_{\\ref {masterlemma:small}}\\log (d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k+1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k]\\le \\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k-1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^+]\\le \\left(\\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{k}{n}\\right)\\right)^k$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^C]\\le \\left(\\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{k}{n}\\right)\\right)^k$ .", "Further if $S, T \\subset [n]$ with $S \\cap T \\ne \\emptyset $ and $j := |S \\cup T| \\le \\frac{n}{8e^4\\gamma d^2}$ , then $\\Pr [A_S \\in \\mathcal {M}_{|S|} \\wedge A_T \\in \\mathcal {M}_{|T|}] \\le \\left( \\left(e^{-\\gamma d+c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\right)\\left(\\frac{j}{n}\\right)\\right)^{j-1}.$ [Proof of Lemma REF ] Let $A\\sim BGC(n,\\gamma ,d)$ .", "For a row subset $S$ where $|S|=k$ , let $E_S$ denote the event that the submatrix $A_S$ induced by the row subset $S$ does not contain a column with exactly one 1.", "Let $L_S\\sim \\mathrm {Bin}(\\gamma k n,d/n)$ denote the number of 1s in $A_S$ .", "By Observation $\\ref {observation1}$ , we know the following: $\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]\\le \\Pr \\left[E_S|L_S=\\ell \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]$ By Observation $\\ref {observation2}$ , we already know the expression on the left hand-side is 0 when $L_S<2k-2$ .", "Thus, we only need to address the case where $L_S\\ge 2k-2$ .", "Recall that each entry in our matrix is an independent Bernoulli random variable.", "Thus, after conditioning on the event that $L_S=\\ell $ for some $\\ell \\in [2k-2, \\gamma k n]$ , the $\\ell $ ones in $A_S$ are distributed uniformly at random throughout the matrix $A_S$ .", "We can arbitrarily enumerate the 1s from 1 to $\\ell $ and consider the random process that places each 1 into $A_S$ sequentially starting from an $k \\times \\gamma n$ all zeros matrix.", "On each step of this process, we can query whether a column with exactly one 1 has been created.", "If such a column is created, we will call this step good.", "All other steps are bad.", "As we add exactly $\\ell $ ones into the matrix, and there are $\\gamma n$ columns, we conclude that the probability of each step being a good event is at least $1-\\frac{\\ell }{\\gamma n}$ .", "As each bad step can remove at most one column with exactly one 1 in it, it is clear that if more than half of the steps are good events, then our resulting submatrix $A_S$ must contain a column with exactly one 1.", "Let $\\lbrace X_i\\rbrace _{i = 1}^{\\ell }$ be the random process which counts the number of bad events that have occurred after $i$ ones have been added to the matrix.", "Thus we have: $\\Pr \\left[E_S|L_S=\\ell \\right]\\le \\Pr \\left[X_\\ell \\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right].$ Since the probability of $X_i$ increases at each step is at most $\\frac{\\ell }{\\gamma n}<\\frac{\\ell +k}{\\gamma n}$ , we can define $Y^{(\\ell )}\\sim \\mathrm {Bin}\\left(\\ell , \\frac{\\ell +k}{\\gamma n}\\right)$ , such that $\\Pr \\left[E_S|L_S=\\ell \\right]\\le \\Pr \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right].$ This implies that $\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]\\le \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]$ We now employ Lemma , which we restate here for convenience.", "* Using this Lemma , we conclude the following results.", "For the second statement in the lemma, we have $\\Pr \\left[A_S\\in \\mathcal {S}_{2k-2,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k-1\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k-1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma k n, d/n) = \\ell ]\\\\&\\le e^{-\\gamma d k + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k-1}.$ Similarly for the third and fourth statements, we have $\\Pr \\left[A_S\\in \\mathcal {S}_{2k-1,k}\\cup \\mathcal {S}_{2k,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k}.$ For the first statement in the lemma, observe that by Lemma , we have $\\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_{k}^+ \\cup \\mathcal {T}_{k}^C\\right) = \\left(\\bigcup _{\\ell \\ge 2k + 1} {\\mathcal {S}_{\\ell , k}}\\right) \\cup \\left(\\mathcal {S}_{2k, k} \\setminus \\mathcal {S}_{2k, k}^{\\prime }\\right).$ Now $\\sum _{\\ell =2k+1}^{\\gamma nk}\\Pr \\left[A_S\\in \\mathcal {S}_{\\ell ,k}\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge \\left\\lceil \\frac{\\ell }{2}\\right\\rceil \\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k+1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k+1}.$ Finally, we note that if $A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }$ , then there must be at least one column with three ones.", "Hence, the number of bad events cannot be exactly $k$ as an equal number of good and bad events means that each column has of $A_S$ has exactly 2 ones in it.", "Thus, we find: $\\Pr \\left[A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }\\right]&\\le \\Pr \\left[Y^{(\\ell )}\\ge k+1\\right]\\cdot \\Pr \\left[L_S=\\ell \\right]\\\\&\\le \\sum _{\\ell \\ge 1}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(k+1, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma n k, d/n) = \\ell ]\\\\&\\le e^{-\\gamma dk + c_{\\ref {masterlemma:small}}\\log (\\gamma d)}\\left(\\frac{k}{n}\\right)^{k+1}.$ Combining these two equations yields the first statement in the lemma.", "For the final setting, where we have two sets $S$ with $S \\cap T \\ne \\emptyset $ , by Lemma REF , it must be the case that if $A_S$ and $A_T$ are minimal dependencies, then $A_{S \\cup T}$ contains at least $\\max \\left(\\frac{\\ell }{2}, |S \\cup T| - 1\\right)$ ones that are not the first in their column, where $\\ell $ is the total number of ones in $A_{S \\cup T}$ .", "Plugging in Lemma  again proves that $\\Pr [A_S \\in \\mathcal {M}_{|S|}, A_T \\in \\mathcal {M}_{|T|}] \\le \\left(\\frac{|S \\cup T|}{n}\\right)^{|S \\cup T| - 1}e^{-\\gamma d|S \\cup T| + c_{\\ref {masterlemma:small}}|S \\cup T|\\log (\\gamma d)}.$ Finally, the lemma follows by using the classification lemma, Lemma , to replace $\\mathcal {S}_{2k - 2, k}$ with $\\mathcal {T}_k$ , $\\mathcal {S}_{2k - 1, k}$ with $\\mathcal {T}_k^+$ , and $\\mathcal {S}_{2k, k}^{\\prime }$ with $\\mathcal {T}_k^C$ ." ], [ "Medium Case", "To fill the small gap between where our small case ends and our large case begins, our proof requires the following lemma which shows that, with high probability, medium-sized minimal dependencies do not occur in our matrix $A$ .", "Unlike in the small case, tree-like dependency structures no longer occur with significant probability.", "Thus, it is sufficient to explicitly calculate the probability that $A_S$ has no columns with exactly 1 one without conditioning on the number of ones in $A_S$ .", "However, one should observe that this strategy will begin to fail as the expected number of 1s in each column begins to exceed 1.", "Thus, a more careful bound on the existence of a minimal dependency will be needed when $k$ is greater than approximately $n/d$ .", "This case is covered in the next subsection.", "Lemma 6 (BGC Medium Case) Let $A\\sim \\textnormal {BGC}(n,\\gamma ,d)$ .", "For any constants $\\alpha ,\\beta \\in \\mathbb {R}^+$ , there exists $d(\\alpha ,\\beta )$ , such that for any $d > d(\\alpha ,\\beta )$ , $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d]\\right] = o(1).$ We will take a union bound over all possible sets $S$ of size $k \\in [\\alpha n/d^2,\\beta n/d]$ of the probability that $A_S$ is a minimal dependency.", "For a row subset $S$ , let $E_S$ be the event that the matrix $A_S$ has a no columns with exactly one 1.", "Then by Observation REF , $\\Pr \\left[\\exists x: x^TA = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d] \\right]\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d} \\binom{n}{k}\\Pr \\left[E_{[k]}\\right].$ For sufficiently large $n$ , it follows: $\\Pr [E_{[k]}]&= \\left(1-k\\left(\\frac{d}{n}\\right)\\left(1-\\frac{d}{n}\\right)^{k-1}\\right)^{\\gamma n}\\\\&\\le \\left(1-k\\left(\\frac{d}{n}\\right)e^{-2dk/n}\\right)^{\\gamma n}\\\\&\\le e^{-\\gamma dk\\left(e^{-2dk/n}\\right)}.$ So we find: $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [\\alpha n/d^2,\\beta n/d]\\right] &\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d} \\binom{n}{k}\\Pr \\left[E_{[k]}\\right]\\\\&\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d}\\left(\\frac{en}{k}\\right)^ke^{-\\gamma dk\\left(e^{-2dk/n}\\right)}\\\\&\\le \\sum _{k=\\alpha n/d^2}^{\\beta n/d}\\left(e^{-\\gamma d\\left(e^{-2dk/n}\\right)+\\ln (en/k)}\\right)^k\\\\&\\le n\\left(e^{-\\gamma d\\left(e^{-2\\beta }\\right)+\\ln (ed^2/\\alpha )}\\right)^k\\\\&=o(1).$ for sufficiently large $d$ depending only on $\\alpha $ and $\\beta $ ." ], [ "Large Case", "The goal of the this section is to prove the following lemma: Lemma 7 (BGC Large Case) Let $A\\sim \\textnormal {BGC}(n, \\gamma , d)$ .", "For any $\\gamma > 1$ , there exists $d_0$ such that for all $d > d_0$ we have, $\\Pr \\left[\\exists x: A^Tx = 0, |\\mathrm {supp}(x)| \\in [9n/d, n]\\right] = o(1).$ Our main tool in proving this Lemma is the following following Littlewood-Offord Theorem due to Costello and Vu [10].", "However, their paper only states the result up to an implied constant.", "As our theorems rely on this implied constant, we reprove this result in Appendix using the same strategy as Costello and Vu, but with an explicit constant and sharper lower order terms.", "* We now prove Lemma REF .", "[Proof of Lemma REF ] We union bound over all $k \\in [9n/d, n]$ and all sets $S$ of size $k$ of the probability that $A_S$ is a minimal dependency.", "Fix a set $S$ of size $k$ .", "We will consider the random process where we generate the $\\gamma n$ independent columns $(A_S)_i$ for $i = 1\\ldots \\gamma n$ one at a time.", "For $i \\le \\gamma n$ , let $\\mathcal {N}_i \\in \\mathbb {R}^k$ be the nullspace of the first $i$ columns drawn, and let $\\mathcal {D}_i \\subset \\mathcal {N}_i$ be the span of the set of vectors in $\\mathcal {N}_i$ which have no zeros.", "Let $R_i$ be the dimension of $\\mathcal {D}_i$ .", "If $R_i > 0$ , then we can choose an arbitrary vector $v$ in $\\mathcal {D}_i$ with support $k$ , and by Lemma REF , with probability at least $1 - \\frac{1}{\\sqrt{kd/n}}$ , the $(i + 1)$ th column drawn is not orthogonal to $v$ .", "In this case $R_{i + 1} = R_i - 1$ .", "If at any point $R_i$ becomes 0, then this means there can be no dependency involving all the rows.", "It follows that since $R_0 = k$ , we have $\\Pr [A_S \\in \\mathcal {M}_k] \\le \\Pr \\left[R_{\\gamma n} \\ne 0\\right] \\le \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].$ Let $C$ be the universal constant in Lemma , which we restate for the reader's convenience: * We first handle when $k \\le n/C$ .", "Assuming $d$ is larger than the universal constant in Lemma , we can apply Lemma to compute the following union bound: $\\begin{split}\\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [9n/d, n/C]\\right] &\\le \\sum _{k = 9 n/d}^{n/C} \\Pr \\left[A_{[k]} \\in \\mathcal {M}_{|S|}\\right] \\\\&\\le \\sum _{k=9 n/d}^{n/C} \\binom{n}{k} \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\\\& \\le \\sum _{k=9 n/d}^{n/C} \\binom{n}{k} \\Pr \\left[\\mathrm {Bin}\\left(n-k-1, 1-\\frac{1}{\\sqrt{kd/n}}\\right) \\le k\\right] \\\\&\\le e^{-c_{\\ref {large_general}} n} = o(1).\\end{split}$ Now we handle the case where $k \\ge n/C$ .", "Choose $d$ large enough, dependent on $\\gamma $ and $C$ to satisfy $\\left(1 - \\frac{2}{\\sqrt{d/C}}\\right)\\gamma > 1.$ This implies that for $\\frac{n}{C} \\le k \\le n$ , we have $\\gamma n - k \\ge \\frac{2\\gamma n }{\\sqrt{(d/n)k}}.$ .", "The inequality in this form allows us to use the the Binomial Tail bound in Lemma , which establishes: $\\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{(d/n)k}}\\right) \\le k\\right] &= \\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, \\frac{1}{\\sqrt{(d/n)k}}\\right) \\ge \\gamma n - k\\right] \\\\&\\le 2 \\left(\\frac{e \\gamma n}{\\sqrt{(d/n) k} (\\gamma n - k)}\\right)^{\\gamma n- k} \\\\&\\le 2 \\left(\\frac{e \\gamma n}{\\sqrt{d/C} (\\gamma n - k)}\\right)^{\\gamma n-k} \\\\&\\le 2 \\left(\\frac{e\\gamma }{\\sqrt{d/C} (\\gamma - 1)}\\right)^{(\\gamma -1)n}.", "\\\\$ By choosing $d$ large enough depending on $\\gamma $ and the universal constant $C$ , we may get $(\\frac{e\\gamma }{\\sqrt{d/C} (\\gamma -1)})^{\\gamma -1} < 1/3$ , so $\\Pr \\left[\\mathrm {Bin}\\left(\\gamma n, 1-\\frac{1}{\\sqrt{(d/n) k}}\\right) \\le k\\right] \\le (1/3)^n.$ This overcomes the union bound over at most $2^n$ sets: $ \\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [n/C, n]\\right] \\le 2^n (1/3)^n = o(1).", "$ Thus, we may combine Equations REF and REF to obtain our result $\\Pr \\left[\\exists S: A_S \\in \\mathcal {M}_{|S|} \\wedge |S| \\in [9n/d, n]\\right] \\le o(1).$" ], [ "Proof of Theorem ", "We are now ready to prove Theorem REF , which we restate here for the reader's convenience.", "* [Proof of Theorem REF ] It follows immediately by combining Lemmas REF , REF (with $\\alpha = \\frac{1}{8e^4\\gamma }, \\beta = 9$ ), and Lemma REF that with probability $1 - o(1)$ , there are no sets $S$ where $A_S$ is a minimal dependencies $A_S$ , but $A_S$ is not in $\\cup _{k \\le \\log (n)}\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ .", "We bound the number of rows involved in minimal linear dependencies using the second moment method.", "For $S \\subset [n]$ , let $X_S$ be the indicator random variable of the event that $A_S$ is a minimal dependency.", "Then with probability $1 - o(1)$ , the number of rows involved in minimal dependencies is at most $X := \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|X_S.$ First we consider the expectation of $X$ .", "Combining the probability that $A_S$ is in $\\mathcal {T}_k$ , $\\mathcal {T}_{k}^+$ , or $\\mathcal {T}_k^C$ , we have from Lemma REF that for any $S$ with $|S| = k$ , for the constant $c_{\\ref {masterlemma:small}}$ from that lemma, $\\mathbb {E}[X_S] \\le \\left(\\frac{k}{n}\\right)^{k - 1}e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}$ Hence $\\begin{split}\\mathbb {E}[X] &= \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|\\mathbb {E}[X_S] \\\\&\\le \\sum _{k \\le n}\\binom{n}{k}k\\left(\\frac{k}{n}\\right)^{k - 1}e^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)} \\\\&\\le \\sum _{k \\le n}ne^ke^{-\\gamma d k + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)} \\\\& \\le \\sum _{k \\le n}ne^{-\\gamma dk + (c_{\\ref {masterlemma:small}} + 1)k\\log (\\gamma d)}\\\\& \\le ne^{-\\gamma d + c_1\\log (\\gamma d)}\\end{split}$ for some constant $c_1$ .", "Here the final inequality is given by the sum of geometric series.", "Next we bound the variance of $X$ .", "Let $S, T \\subset [n]$ with $|S|, |T| \\le \\log (n)$ .", "If $S \\cap T = \\emptyset $ , by the independence of $A_S$ and $A_T$ , we have $\\mathbb {E}[X_SX_T] = \\mathbb {E}[X_S]\\mathbb {E}[X_T].$ Alternatively, if $S \\cap T \\ne \\emptyset $ , then with $R := S \\cup T$ , by Lemma REF , we have $\\mathbb {E}[X_SX_T] \\le \\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}.$ Hence we can compute $\\begin{split}\\text{Var}(X) &= \\mathbb {E}[X^2] - \\mathbb {E}[X]^2 \\\\&= \\sum _{S: |S| \\le \\log (n)}\\sum _{T: |T| \\le \\log (n)} |S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&= \\sum _{S}\\sum _{T : T \\cap S = \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) + \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\mathbb {E}[X_SX_T] \\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}\\sum _{S, T \\subset R}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}\\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}2^{2|R|}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + c_{\\ref {masterlemma:small}}|R|\\log (\\gamma d)}\\\\&\\le \\sum _{R \\subset [n]: |R| \\ge 1}\\left(\\frac{|R|}{n}\\right)^{|R| - 1}e^{-\\gamma d|R| + (c_{\\ref {masterlemma:small}} + 1)|R|\\log (\\gamma d)}\\\\&\\le \\sum _{k \\ge 1}\\binom{n}{k}\\left(\\frac{k}{n}\\right)^{k- 1}e^{-\\gamma dk + (c_{\\ref {masterlemma:small}} + 1)k\\log (\\gamma d)} \\\\&\\le ne^{-\\gamma d + c_2\\log (\\gamma d)}\\end{split}$ for some constant $c_2$ .", "Using Markov's law, it follows that $\\begin{split}\\Pr [X \\ge \\mathbb {E}[X] + t] &\\le \\Pr [(X - \\mathbb {E}[X] )^2 \\ge t^2] \\le \\frac{\\text{Var}(X)}{t^2}.\\end{split}$ Plugging in $t = ne^{-\\gamma d}$ , we obtain that $\\begin{split}\\Pr [X \\ge ne^{-\\gamma d + c_1\\log (\\gamma d)} + ne^{-\\gamma d}] &\\le \\frac{ne^{-\\gamma d + c_2\\log (\\gamma d)}}{\\left(ne^{-\\gamma d}\\right)^2} \\le \\frac{1}{ne^{-\\gamma d - c_2\\log (\\gamma d)}}.\\end{split}$ The theorem follows by choosing $c \\ge c_1 + \\log (2)$ ." ], [ "Symmetric Bernoulli Matrices", "In this section, we prove Theorem REF on the characterization of minimal linear dependencies in symmetric Bernoulli matrices.", "As before, we break down our proof into small, medium, and large cases.", "The small and medium cases are similar to the last section on the BGC.", "However, Unlike for the BGC, however, the large case is not self contained: showing that there are no kernel vectors with large support relies on understanding the image of $Ax$ over vectors $x$ of small support (see Lemma REF below).", "This will require understanding small and medium dependencies in $A$ and in the matrix $A$ with one column removed, which is why our small case and medium case lemmas contain additional statements to this affect." ], [ "Small Case", "Our main goal in the small case is to prove the following lemma.", "Lemma 8 (Symmetric Small Case) There exists a universal constant $c_{\\ref {lemma:small}}$ such that the following holds.", "Let $A\\sim \\textnormal {SB}(n, d)$ for any $d \\ge 1$ .", "Let $S \\subset [n]$ be any set of size $k\\in [1,\\frac{n}{8e^4d^2}]$ .", "Then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] = e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^k.$ $\\Pr [A_S \\in \\mathcal {T}_{k}]\\le e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1}.$ The same result applies if $A$ has one column removed, i.e.", "$A = B^{:n - 1}$ , where $B \\sim \\textnormal {SB}(n, d)$ .", "We introduce some notation to prove this, pictured in Figure REF .", "Fix a set $S$ of $k$ rows and consider the submatrix $A_S$ induced by these rows.", "Let $E_{Sym}$ be the set of entries of $A_S$ whose columns are indexed by values in $S$ .", "Let $E_{SymAD}$ be subset of entries in $E_{Sym}$ that are above the diagonal of $A$ , and hence mutually independent.", "Let $E_{Asym}$ be the set of entries who columns are not in $S$ , and finally, let $E = E_{Asym} \\cup E_{SymAD}$ be the full set of mutually independent entries that determine $A_S$ .", "Formally: $\\begin{split}&E_{Sym} := \\lbrace (i, j): i, j \\in S \\rbrace \\\\&E_{SymAD} := \\lbrace (i, j): i, j \\in S, i < j \\rbrace \\\\&E_{Asym} := \\lbrace (i, j): i \\in S, j \\notin S \\rbrace \\\\&E := E_{SymAD} \\cup E_{Asym}\\end{split}$ Figure: Regions of A S A_S in a symmetric matrix.We will couple the process of putting non-zero entries in these rows with a random walk that counts the number of times a non-zero entry is inserted in $E_{Asym}$ in a column that already contains a non-zero entry or into $E_{SymAD}$ .", "We condition on $L_S$ , the number of non-zero entries in $E$ .", "Note that $L_S \\sim \\mathrm {Bin}(|E|, d/n)$ .", "Conditioned on $L_S = \\ell $ , the process of choosing random entries in $A_S$ is equivalent to randomly choosing $\\ell $ locations in $E$ for these non-zero entries without replacement.", "(Note that this is true even if we are considering a matrix with the last column removed, even though $E_{SymAD}$ may include some entries which are not repeated below the diagonal if the index of the last column is in $S$ ).", "Let $(X_{i})_{i \\in \\ell }$ be the random walk that increases by 1 if the $i$ th random location chosen is in $E_{SymAD}$ or if the $i$ th random location is in $E_{Asym}$ and is not the first non-zero entry placed in its column.", "Otherwise, let $X_i = X_{i - 1}$ .", "The following claims say that $X_{\\ell }$ must be large for $A_S$ to be a minimal dependency not in $\\mathcal {T}_k$ .", "Claim 1 If $X_{L_S} < \\max \\left(k, \\frac{L_S}{2}\\right)$ and there are at least $2k - 1$ non-zero entries total in the rows, then there is a column (whose index is not in $S$ ) with a single non-zero entry.", "Let $P$ be the number of non-zeros entries in $E_{Asym}$ , and let $M$ be the number of non-zero entries in $E_{SymAD}$ such that $M + P = L_S$ and $2M + P \\ge 2k - 1$ .", "Let $Y:= X_{L_S} - M$ be the number of non-zero entries in $E_{Asym}$ which are not the first in their column.", "Now the number of columns not in $S$ which have exactly one non-zero entry is at least $ (P - Y) - Y = P - 2X_{L_S} + 2M \\ge \\max (2k - 1, L_S) - 2X_{L_S} \\ge 1.$ The proof of the following claim is nearly identical.", "Claim 2 If $X_{L_S} < \\max \\left(k - 1, \\frac{L_S}{2}\\right)$ and there are at least $2k - 2$ non-zero entries total in the rows, then there is a column (whose index is not in $S$ ) with a single non-zero entry.", "Recall from Observations REF that any minimal dependency $A_S$ must have at least $2k - 2$ non-zero entries total.", "Further, any minimal dependency not in $\\mathcal {T}_k$ must have at least $2k - 1$ non-zero entries.", "Hence by Claim REF , $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] \\le \\Pr [X_{L_S} < \\max \\left(k, L_S/2\\right)]$ Further by Claim REF , $\\Pr [A_S \\in \\mathcal {T}_k] \\le \\Pr [X_{L_S} < \\max \\left(k - 1, L_S/2\\right)].$ To bound the probability that $X_{L_S}$ is large, we couple $X_i$ with a random walk $(Y_i)_{i \\in L_S}$ which increases by 1 with probability $\\frac{k(k + L_S)}{|E|}$ and otherwise stays constant.", "Observe that $Y_i$ stochastically dominates $X_i$ , because there are at most $k + L_S$ columns — and hence $k(k + L_S)$ locations in $E$ — in which placing a non-zero entry will increase $X_i$ .", "Then conditioned on $L_S = \\ell $ , for any $j$ , we have $\\begin{split}\\Pr \\left[X_{\\ell } \\ge j\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{k + \\ell }{n - k}\\right) \\ge j\\right].\\end{split}$ We now use Lemma , which we restate here, to sum this probability over all values of $L_S$ .", "* We employ this lemma with $\\gamma = \\frac{|E|}{nk} \\ge 1 - k/n$ and $j = \\max \\left(k, \\ell /2\\right)$ , to achieve $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\mathcal {T}_k] \\le e^{-dk + (c_{\\ref {masterlemma:small}}+1)klog(d)}\\left(\\frac{k}{n}\\right)^k.$ Further, setting $j = \\max \\left(k - 1, L_S/2\\right)$ , we achieve $\\Pr [A_S \\in \\mathcal {T}_k] \\le e^{-dk + (c_{\\ref {masterlemma:small}}+1)k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1}.$ Putting $c_{\\ref {lemma:small}} = c_{\\ref {masterlemma:small}} + 1$ proves the lemma." ], [ "Medium Case", "Our main goal of this section is to prove the following lemma.", "Lemma 9 (Symmetric Medium Case) Let $A \\sim \\textnormal {SB}(n, d)$ with $d$ at least some universal constant $d_0$ .", "Then $\\Pr \\left[\\exists x: A^Tx = 0, \\frac{n}{8e^4d^2} \\le \\mathrm {supp}(x) < \\frac{9n}{d}\\right] = o(1).$ The same result applies if $A$ has one column removed, ie.", "$A = B^{:n - 1}$ where $B \\sim \\text{SB}(n, d)$ .", "We will take a union bound over all possible sets $S$ of size $k \\in \\left[\\frac{n}{8e^4d^2}, \\frac{9n}{d}\\right]$ of the probability that $A_S$ is a minimal dependency.", "By Observation REF , it suffices to show that with probability $1 - o(1)$ , for all such sets $S$ , there is a column in $A_S$ with exactly one 1.", "Since the columns $(A_S)_i$ for $ i \\in [n] \\setminus S \\setminus \\lbrace n\\rbrace $ are mutually independent, we have $\\Pr [A_S \\in \\mathcal {M}_{|S|}] \\le \\left(1 - k\\frac{d}{n}\\left(1 - \\frac{d}{n}\\right)^{k - 1}\\right)^{n - k - 1}$ The calculation in Lemma REF gives the result that $\\sum _{k = \\frac{n}{8e^4d^2}}^{\\frac{9n}{d}}\\binom{n}{k} \\Pr [A_S \\in \\mathcal {M}_{k} \\text{ where $|S| = k$}] = o(1).$" ], [ "Large Case", "The first lemma in our large case rules out with high probability minimal dependencies of $k$ rows for $\\frac{9n}{d} \\le k < \\frac{n}{C}$ for some constant $C$ .", "The proof is similar to the large case for the BGC matrix.", "Lemma 10 (Symmetric Large Case 1) Let $A \\sim \\textnormal {SB}(n, d)$ .", "There exist constants $d_0$ and $C$ such that for all $d > d_0$ , $\\Pr \\left[\\exists x: A^Tx = 0, \\frac{9n}{d} \\le \\mathrm {supp}(x) < \\frac{n}{C}\\right] = o(1).$ The same result applies if $A$ has one column removed, ie.", "$A = B^{:n - 1}$ where $B \\sim \\text{SB}(n, d)$ .", "We union bound over all $k \\in [9n/d, n]$ and all sets $S$ of size $k$ of the probability that $A_S$ is a minimal dependency.", "Fix a set $S$ of size $k$ .", "We will consider the random process where we generate the $n - k - 1$ columns $(A_S)_i$ for $i \\in [n - 1] \\setminus S$ one at a time.", "Note that these columns are all mutually independent since they do not include columns indexed by $S$ .", "Further, they do not include the last column of $A$ .", "Consider the following process, where we draw these independent columns one at a time.", "For $i \\le n - k - 1$ , let $\\mathcal {N}_i \\in \\mathbb {R}^k$ be the nullspace of the first $i$ columns drawn, and let $\\mathcal {D}_i \\subset \\mathcal {N}_i$ be the span of the set of vectors in $\\mathcal {N}_i$ which have no zeros.", "Let $R_i$ be the dimension of $\\mathcal {D}_i$ .", "If $R_i > 0$ , then we can choose an arbitrary vector $v$ in $\\mathcal {D}_i$ with support $k$ , and by Lemma REF , with probability at least $1 - \\frac{1}{\\sqrt{kd/n}}$ , the $(i + 1)$ th column drawn is not orthogonal to $v$ .", "In this case $R_{i + 1} = R_i - 1$ .", "If at any point $R_i$ becomes 0, then this means there can be no dependency involving all the rows.", "It follows that since $R_0 = k$ , we have $\\Pr [R_{n - k - 1} \\ne 0] \\le \\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].$ Next we take a union bound over all $k$ in the desired range and all $\\binom{n}{k}$ sets $S$ of size $k$ .", "Employing the calculation from Lemma , for universal constants $c_{\\ref {large_general}}$ , $C$ , and $d_0$ , for $d \\ge d_0$ , we have $\\Pr [\\exists x: Ax = 0, 9n/d \\le \\mathrm {supp}(x) \\le n/C] \\le e^{-c_{\\ref {large_general}}n}.$ This concludes the proof of the lemma.", "The remainder of the large case is based off of ideas from the works of Ferber, Kwan, and Sauermann [12](see Lemma 2.1), and Costello, Tau, and Vu [8](see proof of Lemma 2.8).", "We will use the following two linear-algebraic lemmas, proved in Appendix .", "lemmanullspace Let $A$ be a matrix with columns $A_i$ for $i \\in [n]$ .", "Let $H_i$ be the space spanned by the column vectors $A_1, A_2, \\cdots A_{i - 1}, A_{i + 1}, \\cdots A_n$ .", "Let $S$ be the set of all $i$ such that $A_i \\in H_i$ .", "Then there exists some $y$ with $\\text{supp}(y) = S$ such that $Ay = 0$ .", "lemmabasis With the terminology of the previous lemma, $e_i \\in \\mathrm {Span}(A^T)$ , if and only if $A_i \\notin H_i$ .", "The following lemma is our main tool in ruling out large dependencies and proving Theorem REF .", "It breaks down the probability that there is a large linear dependency consisting of more than $t$ rows into the sum of the probabilities of several other events.", "Two of these probabilities (lines 1 and 5 of the right hand side of Equation REF in the lemma) we can show to be small via anti-concentration lemmas.", "Two of the probabilities (lines 2 and 4 of the right hand side of Equation REF in the lemma), we can show to be small using lemmas we proved in the small case.", "This is unlike the previous large case lemma, where weren't concerned with any small structures that might exist in $A$ .", "Lemma 11 (Symmetric Large Case Main) Let $A \\sim \\text{SB}(n, d)$ .", "Let $A^{(i)}$ denote the submatrix of $A$ given by the $i$ th row and column removed.", "Let $A_i$ denote the $i$ th column of $A$ , and let $A_i^\\prime $ be this vector with the $i$ th entry removed.", "For any $u , r, s, t \\in [n]$ with $r < s$ , $\\begin{split}\\Pr \\left[\\exists x: Ax = 0, \\mathrm {supp}(x) > t\\right] &\\le \\frac{n}{t}\\max _{x \\in \\mathbb {R}^{n - 1}: \\mathrm {supp}(x) \\ge s}\\Pr \\left[x^TA^{\\prime }_n = 0\\right]\\\\&\\quad +\\frac{n}{t}\\Pr \\left[\\exists x: A^{(n)}x = 0, r < |\\mathrm {supp}(x)| < s \\right]\\\\&\\quad + \\frac{n}{t} \\frac{dr}{n}\\\\&\\quad + \\frac{n}{t}\\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right]\\\\&\\quad + \\frac{n}{t}\\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[A^{\\prime T}_nXA^{\\prime }_n = 0\\right],\\end{split}$ where $\\mathcal {T}^{m}_{p, q}$ denotes the set of matrices in $\\mathbb {R}^{m \\times m}$ with some set of $p$ columns that each have at least $q$ non-zero entries.", "For $i \\in [n]$ , let $H_i$ be the space spanned by the column vectors $A_1, A_2, \\cdots A_{i - 1}, A_{i + 1}, \\cdots A_n$ .", "Then $Ax = 0$ for some $x$ implies that for all $i \\in \\text{supp}(x)$ , $A_i \\in H_i$ .", "Let $\\mathcal {E}_i$ denote the event that $A_i \\in H_i$ .", "Let $X_i$ be the indicator of this event, and let $X = \\sum _i X_i$ .", "Then by Markov's inequality and the exchangeability of the columns, $\\Pr \\left[\\exists x : Ax = 0, |\\mathrm {supp}(x)| \\ge t \\right] = \\Pr \\left[X \\ge t\\right] \\le \\frac{\\mathbb {E}[X]}{t} = \\frac{n\\Pr [\\mathcal {E}_n]}{t}.$ We will break down the probability $\\Pr [\\mathcal {E}_n]$ into several cases, depending on the size of the support of vectors in the kernel of $A^{(n)}$ .", "Let $S \\subset [n - 1]$ be the set of all $i$ such that $e_i \\in \\mathrm {Span}(A^{(n)})$ , such that by Lemma REF and Lemma REF , $k:= \\max (\\mathrm {supp}(x): A^{(n)}x = 0) = n - 1 - |S|.$ .", "Case 1: $A^{(n)}$ has a kernel vector $x$ with large support, that is, $k \\ge s$ .", "Case 2: $A^{(n)}$ has a kernel vector $x$ with medium support, that is $r < k < s$ .", "Case 3: $A^{(n)}$ does not have any kernel vectors with large or medium support vectors in its kernel, that is, $k \\le r$ .", "We can expand $\\begin{split}\\Pr [\\mathcal {E}_n] &= \\Pr [\\mathcal {E}_n| k \\ge s]\\Pr [k \\ge s]\\\\&\\qquad + \\Pr [\\mathcal {E}_n|r < k < s]\\Pr [r < k < s]\\\\&\\qquad + \\Pr [\\mathcal {E}_n| k \\le r]\\Pr [k \\le r].\\end{split}$ For simplicity, define $\\textbf {a} := A_n^{\\prime }$ to be the first $n - 1$ entries of the column $A_n$ .", "To evaluate the probability of the first case, we condition on $A^{(n)}$ and let $x$ be any vector of support at least $s$ in the kernel of $A^{(n)}$ .", "Observe that $\\mathcal {E}_n$ cannot hold if $x^T\\textbf {a}$ is non-zero.", "Indeed, if $x^T\\textbf {a} \\ne 0$ , then let $x^{\\prime } = (x_1, x_2,\\ldots ,x_{n-1},0)/(x^T\\textbf {a})$ such that $Ax^{\\prime } = e_n$ .", "Then by Lemma REF , $A_n \\notin H_n$ and hence $\\mathcal {E}_n$ does not occur.", "Since $\\textbf {a}$ is independent from $x$ , we have $\\Pr [\\mathcal {E}_n| k \\ge s]\\Pr [k \\ge s] \\le \\max _{x: \\mathrm {supp}(x) \\ge s}\\Pr [x^T\\textbf {a} = 0].$ Combined with Equation REF , the contribution from this case yields the first term in the right hand side of Equation REF .", "For the second case, we bound $\\Pr [\\mathcal {E}_n|r < k < s]\\Pr [r < k < s] \\le \\Pr [r < k < s] \\le \\Pr \\left[\\exists x: A^{(n)}x = 0, r < |\\mathrm {supp}(x)| < s \\right].$ Combined with Equation REF , the contribution from this case yields the second term in the right hand side of Equation REF .", "The third case will lead to the final three terms in the right hand side of Equation REF .", "In this case, we will show conditions under which we can algebraically construct a vector $v$ such that $Av = e_n$ .", "This will imply by Lemma REF that $A_n \\notin H_n$ .", "Recall that $S \\subset [n - 1]$ is the set of all $i$ such that $e_i \\in \\mathrm {Span}(A^{(n)})$ .", "For $i \\in S$ , let $w_i $ be any vector such that $A^{(n)}{w_i } = e_i$ .", "We next construct a sort of “pseudoinverse\" matrix $B \\in \\mathbb {R}^{n - 1 \\times n - 1}$ as follows: For $i \\in S$ , define $B_{ij}$ to be the $i$ th entry of $w_i $ .", "That is, for $i \\in S$ , the $i$ th column of $B$ is $w_i $ .", "Define all other entries of $B$ to be zero.", "The following claim shows a condition for $\\mathcal {E}_n$ not holding.", "Claim 3 If $\\mathrm {supp}(\\textbf {a}) \\subset S$ and $\\textbf {a}^TB\\textbf {a} \\ne 0$ , then $e_n \\in \\mathrm {Span}(A)$ .", "Let ${w^{\\prime }} := B\\textbf {a} = \\sum _{i \\in S}{\\textbf {a}_iw_i }$ such that $A^{(n)}{w^{\\prime }} = \\sum _{i \\in S}{\\textbf {a}_ie_i}.$ Hence if $\\mathrm {supp}(\\textbf {a}) \\subset S$ , $A^{(n)}{w^{\\prime }} = \\textbf {a}.$ In this case, define $w \\in \\mathbb {R}^n$ to be the vector with $w^{\\prime }$ in the first $n - 1$ entries and $-1$ in the final entry.", "Then the first $n - 1$ entries of $Aw$ are 0, and the last entry is ${\\bf {a}}^T w^{\\prime } = {\\bf {a}}^T B \\bf {a}$ .", "Evidently, if ${\\bf {a}}^T B {\\bf {a}} \\ne 0$ , then $\\frac{Aw}{{\\bf {a}}^T B {\\bf {a}}} = e_n,$ so $e_n \\in \\mathrm {Span}(A)$ .", "By definition, in the third case, we have $|S| \\ge n - 1 - r$ .", "Hence by Claim REF , $\\begin{split}\\Pr [\\mathcal {E}_n \\wedge k \\le r]&\\le \\Pr \\left[\\mathrm {supp}(\\textbf {a}) \\lnot \\subset S \\wedge |S| \\ge n - 1 - r\\right]\\\\&\\quad + \\Pr [\\textbf {a}^TB\\textbf {a} = 0 \\wedge |S| \\ge n - 1 - r].\\end{split}$ Notice that $S$ is a function of $A^{(n)}$ and so $\\textbf {a}$ is independent from $S$ .", "It is easy to check that for any set $S$ of size at least $n - 1 - r$ , $\\Pr \\left[\\mathrm {supp}(\\textbf {a}) \\lnot \\subset S\\right] \\le 1 - \\left(1 - \\frac{d}{n}\\right)^r \\le \\frac{dr}{n}.$ We will break up the second term in Equation REF by conditioning on whether the support of $B$ has many entries or not, and using the independence of $\\textbf {a}$ from $B$ : $\\Pr [\\textbf {a}^TB\\textbf {a} = 0] \\le \\Pr \\left[B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s} \\wedge |S| \\ge n - 1 - r\\right] + \\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[\\textbf {a}^TX\\textbf {a} = 0\\right]$ To further bound the first probability on the right hand side, observe that if $|S| \\ge n - 1 - r$ and $B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}$ , there must exist at least $u$ different $i \\in S$ such that $\\mathrm {supp}(w_i ) \\le s$ .", "So $\\begin{split}\\Pr \\left[B \\notin \\mathcal {T}^{n - 1}_{n - 1 - r - u, s} \\wedge |S| \\ge n - 1 - r\\right] &\\le \\Pr \\left[|\\lbrace i: \\exists x \\ne 0: A^{(n)}x = e_i, |\\mathrm {supp}(x)| < s\\rbrace | \\ge u \\right] \\\\ &\\le \\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right],\\end{split}$ where the last inequality follows by Markov's inequality.", "Combining this with Equations REF , REF , and REF yields $\\Pr [\\mathcal {E}_n \\wedge k \\le r] \\le \\frac{dr}{n} + \\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[\\textbf {a}^TX\\textbf {a} = 0\\right] + \\frac{n}{u}\\Pr \\left[\\exists x \\ne 0: A^{(n)}x = e_1, |\\mathrm {supp}(x)| < s\\right].$ Plugging this and Equations REF and REF into Equation REF and finally Equation REF yields the lemma.", "We instantiate Lemma REF with $t = \\frac{n}{C}$ , $s = \\frac{n}{C}$ , $r = \\frac{n}{d\\log (d)}$ , $u = \\frac{n}{2}$ , where $C$ is the constant from Lemma REF , to obtain the following lemma.", "Lemma 12 Let $A \\sim \\textnormal {SB}(n, d)$ for $d = \\omega (1)$ .", "With $C$ equal to the constant from Lemma REF , $\\begin{split}\\Pr [\\exists x: Ax = 0, \\mathrm {supp}(x) > n/C] &\\le C\\Pr [\\exists x: A^{(n)}x = 0, \\frac{n}{d\\log (d)} \\le |\\mathrm {supp}(x)| \\le n/C] \\\\&\\qquad + 2C\\Pr [\\exists x: A^{(n)}x = e_1, |\\mathrm {supp}(x)| \\le n/C] \\\\&\\qquad + o(1).\\end{split}$ This follows immediately from plugging in these values of $t, s, r$ and $u$ into Lemma REF and applying the anti-concentration results in Lemmas REF and REF to the first and last terms.", "Indeed, Lemmas REF shows that $\\frac{n}{t}\\max _{x \\in \\mathbb {R}^{n - 1}: \\mathrm {supp}(x) \\ge s}\\Pr \\left[x^TA^{\\prime }_n = 0\\right] \\le \\frac{n}{t}\\frac{1}{\\sqrt{sd/n}} = o(1).$ Lemma REF shows that $\\frac{n}{t}\\max _{X \\in \\mathcal {T}^{n - 1}_{n - 1 - r - u, s}}\\Pr \\left[A^{\\prime T}_nXA^{\\prime }_n = 0\\right] \\le O\\left(\\frac{1}{\\@root 4 \\of {\\min (s, n - 1 - r - u)d/n}}\\right) = o(1).$ .", "The third term is at most $O(1/\\log (d))$ which is also $o(1)$ ." ], [ "Proof of Theorem ", "We are now ready to put the results of the small, medium, and large cases together to prove Theorem REF .", "For the convenience of the reader, we restate this theorem: * We will need the following two lemmas to show that the first two terms in the right hand size of Lemma REF are $o(1)$ .", "Lemma 13 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Then $\\Pr \\left[\\exists x: Ax = 0, \\frac{n}{d\\log (d)} < |\\mathrm {supp}(x)| < \\frac{n}{C} \\right] = o(1).$ This is immediate from the medium and first large case Lemmas REF and REF and the fact that $d = \\omega (1)$ , which implies that $\\frac{n}{d\\log (d)} \\ge \\frac{n}{8e^4d^2}$ .", "Lemma 14 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Then $\\Pr \\left[\\exists x: Ax = e_1, |\\mathrm {supp}(x)| < \\frac{n}{C} \\right] = o(1)$ We reduce Lemma REF to Lemma REF , which are better suited to prove with our medium and small case lemmas.", "Lemma 15 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ and let $C$ be as in Lemma REF .", "Let $K = \\lbrace x: Ax = 0, |\\mathrm {supp}(x)| \\le \\frac{n}{C}\\rbrace $ .", "Then $\\Pr \\left[\\left|\\bigcup _{x \\in K}{\\mathrm {supp}(x)}\\right| \\ge \\frac{n}{d\\log (d)} \\right] = o(1).$ Similarly, suppose $A^{\\prime }$ is $A$ with the first row removed.", "Let $K^{\\prime } = \\lbrace x: A^{\\prime }x = 0, |\\mathrm {supp}(x)| \\le \\frac{n}{C}\\rbrace $ .", "Then $\\Pr \\left[1 \\in \\bigcup _{x \\in K^{\\prime }}{\\mathrm {supp}(x)} \\right] = o(1).$ We first prove Lemma REF from Lemma REF : [Proof of Lemma REF ] First we consider vectors $x$ with $1 \\notin \\mathrm {supp}(x)$ .", "Applying the first part of Lemma REF to $A^{(1)}$ , with probability $1 - o(1)$ , there exists some set $T \\subset [n] \\setminus \\lbrace 1\\rbrace $ with $|T| \\le \\frac{n}{d\\log (d)}$ such that $\\mathrm {supp}(x) \\subset T$ for all $x$ satisfying that $A^{(1)}x = 0$ and $\\mathrm {supp}(x) \\le \\frac{n}{C}$ .", "With probability $1 - o(1)$ , $\\mathrm {supp}(A_1) \\cap T = \\emptyset $ , so for all vectors $x$ with support in $[n] \\setminus \\lbrace 1\\rbrace $ and of size less than $\\frac{n}{C}$ , we do not have $Ax = e_1$ .", "Next we consider vectors $x$ with $1 \\in \\mathrm {supp}(x)$ .", "If such an $x$ exists, that is, $Ax = e_1$ and $1 \\in \\mathrm {supp}(x)$ , then it must be the case that $A^{\\prime }x = 0$ , where $A^{\\prime }$ is $A$ with the first row removed.", "By the second part of Lemma REF , the probability that such an $x$ exists is $o(1)$ .", "[Proof of Lemma REF ] Let $\\mathcal {L}_2$ be the event that $\\exists x: Ax = 0, \\frac{n}{d\\log (d)} < |\\mathrm {supp}(x)| < \\frac{n}{C}.$ Recall that by Lemma REF , this event occurs with probability $o(1)$ .", "To prove the first part, we will show that conditioned on $\\mathcal {L}_2$ not occurring, we have $\\left|\\bigcup _{x \\in K}{\\mathrm {supp}(x)}\\right| < \\frac{n}{d\\log (d)}$ .", "Index the set $K$ as follows: $K = \\lbrace x^{(1)}, \\cdots , x^{(|K|)}\\rbrace $ .", "For $i = 1$ to $|K|$ , let $y^{(i)} = \\sum _{j = 1}^i{c_i x^{(i)}}$ , where $c_i$ is chosen uniformly from the interval $[0, 1]$ .", "It follows that with probability 1, for all $i \\le |K|$ , $\\mathrm {supp}(y^{(i)}) = \\bigcup _{j \\le i}{\\mathrm {supp}(x^{(i)})}.$ If $\\mathcal {L}_2$ does not occur, then $|\\mathrm {supp}(x^{(i)})| \\le \\frac{n}{d\\log (d)}$ for all $i$ , so we have that $|\\mathrm {supp}(y^{(i + 1)})| \\le |\\mathrm {supp}(y^{(i)})| + \\frac{n}{d\\log (d)}$ .", "It follows that if $|\\mathrm {supp}(y^{(|K|)})| \\ge \\frac{n}{d\\log (d)}$ , then there exists some $i \\le |K|$ such that $|\\mathrm {supp}(y^{(i)})| \\in [\\frac{n}{d\\log (d)}, \\frac{2n}{d\\log (d)}]$ .", "However, this would imply that $\\mathcal {L}_2$ holds, which is a contradiction.", "For the second part, by Lemmas REF , REF , and REF , with probability $1 - o(1)$ , the only possible minimal dependencies in the columns of $A$ must have a total of between $2(k - 1)$ and $2k$ non-zero entries.", "(Note that we are applying the results of these lemmas to columns instead of rows).", "There are $\\binom{n}{k-1}$ possible sets of $k$ columns which include the first column.", "By Lemma REF , the probability of a dependency occurring in one of those sets of columns is at most $ \\left(\\frac{k}{n}\\right)^{k-1}e^{-kd + c_{\\ref {lemma:small}}k\\log (d)}.$ Hence by a union bound, with probability at most $ \\binom{n}{k - 1}\\left(\\frac{k}{n}\\right)^{k-1}e^{-kd + c_{\\ref {lemma:small}}k\\log (d)} = o(1),$ there are no small dependencies involving the first column.", "We restate Theorem REF for the reader's convenience before proving it.", "* [Proof of Theorem REF ] Combining Lemma REF with the Lemmas REF and REF presented above, with probability $1 - o(1)$ , for any $k \\ge \\frac{n}{d\\log (d)}$ , there are no minimal dependencies in $A$ of $k$ rows.", "Further applying Lemma REF , we see that with probability $1 - o(1)$ , all minimal dependencies of of size $k$ must be in $\\mathcal {T}_k$ .", "This proves the first part of the theorem.", "It remains to bound with high probability the number of rows involved in minimal dependencies in $\\mathcal {T}_k$ .", "For $S \\subset [n]$ , let $X_S$ be the indicator variable that $A_S \\in \\mathcal {T}_{|S|}$ .", "Then the total number of rows in minimal dependencies is at most $X = \\sum _{S \\subset [n]}|S|X_S.$ By Lemma REF , for the universal constant $c_{\\ref {lemma:small}}$ , we have $\\begin{split}\\mathbb {E}[X] &= \\sum _{k \\le \\frac{n}{8e^4d^2}}\\binom{n}{k}{k}e^{-dk + c_{\\ref {lemma:small}}k\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1} \\\\&\\le \\sum _{k \\le \\frac{n}{8e^4d^2}}\\left(\\frac{en}{k}\\right)^ke^{-dk + c_{\\ref {lemma:small}}\\log (d)}\\left(\\frac{k}{n}\\right)^{k - 1} \\\\&= \\sum _{k \\le \\frac{n}{8e^4d^2}}ne^ke^{-dk + c_{\\ref {lemma:small}}k\\log (d)} \\\\&= \\sum _{k \\le \\frac{n}{8e^4d^2}}ne^{-dk + c_{\\ref {lemma:small}}k\\log (d) + k} \\\\&\\le n\\frac{e^{-d + 1 + c_{\\ref {lemma:small}}\\log (d)}}{1 - e^{-d + 1 + c_{\\ref {lemma:small}}\\log (d)}}\\\\&\\le ne^{-d + c\\log (d)}\\end{split}$ for some constant $c$ .", "By Markov's law, we have $\\Pr \\left[X \\ge ne^{-d + (c + 1)\\log (d)} \\le \\frac{\\mathbb {E}[X]}{ne^{-d + (c + 1)\\log (d)}}\\right] \\le \\frac{1}{d}.$ Since $d = \\omega (1)$ , this proves the theorem." ], [ "Augmented Biregular Codes", "In this section, we prove Theorem REF on the characterization of minimal linear dependencies in matrices from $\\textnormal {ABC}_p$ .", "Our proof is broken down into a small and large case." ], [ "The ABC distribution via the Configuration Model", "Recall that we use the following process to generate a random matrix $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ from the distribution $\\textnormal {ABC}(n, \\gamma , d)$ : Create $n$ row-nodes and $\\gamma n$ column-nodes and associate to each row-node $\\gamma d$ half-edges and to each column node $d$ half-edges.", "Create a multi-graph $G$ by choosing a uniformly random pairing of the $\\gamma d n$ half-edges from the-row nodes to the $\\gamma d n$ half-edges from the columns-nodes.", "Given this bipartite graph, we will take $A_0 \\in \\lbrace 0, 1\\rbrace ^{n \\times \\gamma n}$ to be the matrix where $(A_0)_{ij} = 1$ iff there is at least one edge from node $i$ to $j$ .", "Because we will study the resulting matrix $A_0$ via the half-edge pairing process, in our proofs we will sometimes consider the permutation $\\rho \\in \\mathcal {S}_{\\gamma d n}$ which defines the random pairing of half-edges.", "From $\\rho $ , we can can construct an additional matrix $H_0 = H_0(\\rho )$ , which we call the row half-edge occupancy matrix .", "Let $P \\in \\lbrace 0, 1\\rbrace ^{\\gamma d n \\times \\gamma d n}$ be the permutation matrix of $\\rho $ .", "From $P$ , construct $H_0 \\in \\lbrace 0, 1\\rbrace ^{\\gamma d n \\times \\gamma n}$ by summing adjacent columns of $P$ corresponding to half-edges of the same column nodes.", "Symbolically, $(H_0)_{ij} = \\sum _{k=(j-1)d + 1}^{jd} P_{ik}.$ Note that $H_0$ has $\\gamma n$ columns; column $i$ represents column-node $i$ in the configuration model.", "$H_0$ has $\\gamma d n$ rows: rows $(j-1)\\gamma d + 1$ through $j\\gamma d$ represent the $\\gamma d$ half edges of row-node $j$ , for $1 \\le j \\le n$ .", "Each column contains exactly $d$ 1's, corresponding to that column-node's half edges.", "We may write $A_0 = A_0(\\rho )$ in terms of the random matrix $H_0$ .", "Because $(A_0)_{ij} = 1$ iff there is at least one edge from vertex $i$ to $j$ , we have $(A_0)_{ij} = \\mathbb {1}(\\exists k, d(i-1) < k \\le di, H_{kj} = 1 ).$ When generating a matrix $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ , we can without loss of generality generate $A_0 \\sim \\textnormal {ABC}(n, \\gamma , d)$ and take $A := A_0^{:\\gamma n(1 - p)}$ to be the matrix containing the first $\\gamma n(1 - p)$ columns of $A_0$ .", "We define $H := H_0^{:\\gamma n(1 - p)}$ to be the first $\\gamma n(1-p)$ columns of the row half-edge occupancy matrix associated with $A_0$ .", "For a set $S \\subset [n]$ with $|S| = k$ , let $H(S) \\in \\mathbb {R}^{\\gamma d k \\times \\gamma n(1-p)}$ be the matrix $H$ restricted to the $\\gamma d k$ rows corresponding to half-edges of the row-nodes in $S$ ." ], [ "Small Case", "The goal of this section will be to prove the following lemma: Lemma 16 Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ for $\\gamma \\ge 1$ .", "Let $S\\subset [n]$ be any set of size $k\\in [1,\\frac{n}{18e\\gamma d^2}]$ .", "There exists universal constants $c_{\\ref {lemma:abc_small}},d_0$ such that if $d>d_0$ , then: $\\Pr [A_S \\in \\mathcal {M}_k \\setminus \\left(\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C \\right)] = O\\left(e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\right)$ .", "$\\Pr [A_S \\in \\mathcal {T}_k]\\le \\left(p^{\\gamma d+ c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k-1}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^+]\\le \\left(p^{\\gamma d+c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k}$ .", "$\\Pr [A_S \\in \\mathcal {T}_k^C]\\le \\left(p^{\\gamma d+a_0\\log (\\gamma d)}\\right)^k\\left(\\frac{k}{n}\\right)^{k}$ .", "Further if $S, T \\subset [n]$ with $S \\cap T \\ne \\emptyset $ .", "Let $|S| = k$ , $|T| = j$ , and $R := S \\cup T$ , then $\\Pr [A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C \\wedge A_T \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C] \\le \\left( \\left(p^{\\gamma d+c_{\\ref {lemma:abc_small}}\\log (\\gamma d)}\\right)\\left(\\frac{|R|}{n}\\right)\\right)^{|R|-1}.$ To prove Lemma REF , we shall condition on $L_S$ , the number of 1s in $H(S)$ .", "We observe that $L_S\\sim \\text{HyperGeom}(\\gamma d n, \\gamma d(1-p)n, \\gamma dk)$ .", "Note that $L_S$ is greater than or equal to the number of entries which are 1 in the submatrix $A_S$ , with equality if and only if there does not exist a row node in $S$ which pairs multiple half-edges to the same column-node in $[\\gamma n (1 - p)]$ .", "We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Let $E_S$ denote the event that there are no columns in $H(S)$ with exactly 1 one.", "It follows from Observation REF that $\\Pr [A_S\\in \\mathcal {S}_{\\ell ,k}]\\le \\Pr [E_S |L_S=\\ell ]\\cdot \\Pr [L_S=\\ell ]$ Our general strategy to bound $\\Pr [E_S|L_S = \\ell ]$ relies on the following claim.", "Claim 4 Let $X$ be the number of non-zero columns in $H(S)$ , or equivalently, the number of 1s in $H(S)$ which are the first (top) 1 in their column.", "If $E_S$ occurs, then $X \\le \\lfloor {L_S/2}\\rfloor $ .", "If $X > \\lfloor {L_S/2}\\rfloor > L_S/2$ , then there are strictly less than $L_S/2$ ones in $H(S)$ which are not the first 1 in their column.", "By the pigeonhole principle, this means there must be at least one column with a “top\" 1 but no 1s below it, ie.", "this column has exactly one 1.", "We will bound the probability that $X$ is small by considering a random walk which counts the number of non-zero columns as half-edges are paired one at time.", "We formalize this random walk as follows.", "Conditioned on $L_S$ , the matrix $H(S) \\in \\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ is distributed like a uniformly random matrix on the set of all matrices in $\\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ with exactly $L_S$ ones.", "We construct $H(S)$ via the following random process $M^0, M^1, \\ldots , M^{L_S} = H(S)$ on matrices in $\\lbrace 0, 1\\rbrace ^{\\gamma d k \\times \\gamma n(1-p)}$ , in which the $L_S$ half-edges represented in $H(S)$ are paired one at a time.", "Formally, at each step $i$ , we construct $M^i$ from $M^{i - 1}$ by placing a 1 in a uniformly random location in $M^{i - 1}$ without a 1.", "For $i \\in 0, 1, \\ldots L_S$ , let $X_i$ equal the number of non-zero columns in $M^i$ , such that $X_{L_S} = X$ .", "When placing the $i$ -th 1, there are at most $i-1$ non-zero columns so far.", "In particular, since $i \\le L_S$ , there are at most $L_S$ non-zero columns throughout this process.", "Because $H(S)$ has $\\gamma (1-p)n$ columns, we thus have the bound $\\Pr [X_i - X_{i - 1} = 1] \\ge 1 - \\frac{L_S}{\\gamma (1-p)n}$ .", "It follows that the random variable $X$ first-order stochastically dominates the random variable $\\sum _{i=1}^{L_S} \\textnormal {Bernoulli}(1 - \\frac{L_S}{\\gamma (1-p)n})$ .", "So in particular, we have: $\\Pr \\left[X\\le \\lfloor L_S/2\\rfloor \\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(L_S, 1 - \\frac{L_S}{\\gamma (1-p)n}\\right) \\le \\lfloor L_S/2\\rfloor \\right],$ which implies $\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}]&\\le \\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1 - \\frac{\\ell }{\\gamma (1-p)n}\\right) \\le \\lfloor \\ell /2\\rfloor \\right]\\cdot \\Pr [ L_S=\\ell ].$ We will use the following claim proved in Appendix : claimclaimabc Let $p<1/2$ .", "Let $K\\le \\frac{3}{2}pN$ .", "There exists constants $c_{\\ref {claim:abc1}}$ , and $d_0$ such that for all $\\gamma >1$ and $d>d_0$ , the following two bounds hold.", "For $\\ell \\in \\lbrace 2K-2,2K-1,2K\\rbrace $ , and $j \\le \\lfloor \\ell /2\\rfloor $ , we have: $\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1-p)N}\\right)\\le j\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma d N,\\gamma d(1-p)N,\\gamma d K)=\\ell ]\\le \\left(p^{\\gamma d - c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{\\ell -j}.$ Further, $\\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{4K}{\\gamma (1 - p)N}\\right)\\ge K-1\\right]\\cdot \\Pr \\left[\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell \\right]\\le \\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{K-1}.$ Since $L_S \\sim \\textnormal {HyperGeom}(\\gamma d n, \\gamma d(1-p)n, \\gamma d k)$ , it must be the case that $L_S \\ge \\gamma d k - p\\gamma dn.$ Hence if $L_S \\in \\lbrace 2k-2,2k-1,2k\\rbrace $ , then necessarily $k \\le \\frac{p\\gamma d n}{\\gamma d - 2} \\le \\frac{3}{2}pn,$ for $\\gamma d \\ge 6$ , which is guaranteed if $d \\ge 6$ .", "Thus we can apply the first bound in Claim REF with $K = k$ and $N = n$ to study $\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}] \\le \\Pr \\left[Y\\le \\lfloor \\ell /2\\rfloor ]\\cdot \\Pr [ L_S=\\ell \\right]$ , yielding: $\\Pr [A_S\\in \\mathcal {S}_{2k-2,k} ]&=\\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k-1}.\\\\\\Pr [A_S\\in \\mathcal {S}_{2k-1,k} ]&=\\left(p^{\\gamma d- c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k}.\\\\\\Pr [A_S\\in \\mathcal {S}_{2k,k}^{\\prime }]\\le \\Pr [A_S\\in \\mathcal {S}_{2k,k} ]&=\\left(p^{\\gamma d-c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k}.$ We use the equivalences of $\\mathcal {S}_{2k - 2, k} = \\mathcal {T}_k$ , $\\mathcal {S}_{2k - 1, k} = \\mathcal {T}_k^+$ , and $\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ from Lemma  to yield statements 2, 3, and 4 in this lemma.", "Next we obtain a bound on the event that $A_S \\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)$ .", "Observe that $A_S \\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C) \\Rightarrow A_S \\in \\left(\\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }\\right) \\cup \\bigcup _{\\ell \\ge 2k + 1}\\mathcal {S}_{\\ell , k}.$ If $A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }$ , there must be a column with at least three 1s.", "So by a similar argument to Claim REF , it must be the case that $X \\le k - 1$ .", "Hence $\\Pr [A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k,k}^{\\prime }]&\\le \\Pr \\left[\\mathrm {Bin}\\left(2k, 1 - \\frac{2k}{\\gamma (1-p)n}\\right) \\le k-1\\right]\\cdot \\Pr [L_S=2k]$ Another application of the Claim REF gives us: $\\Pr [A_S\\in \\mathcal {S}_{2k,k}\\setminus \\mathcal {S}_{2k, k}^{\\prime }]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k+1}.$ It follows that $\\Pr [A_S\\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right) \\left(\\frac{k}{n}\\right)^{k+1}+\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr [A_S\\in \\mathcal {S}_{\\ell , k}]\\\\&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right) \\left(\\frac{k}{n}\\right)^{k+1}+\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1 - \\frac{\\ell }{\\gamma (1-p)n}\\right) \\le \\left\\lfloor {\\frac{\\ell }{2}}\\right\\rfloor \\right]\\cdot \\Pr [L_S = \\ell ]$ To bound the final term, we use the following claim, which we prove in Appendix .", "claimClaimabc Let $p<1/2$ .", "There exists $\\gamma _0$ and $d_0$ such that for all $\\gamma >\\gamma _0$ and $d>d_0$ , if $k\\le \\frac{n}{18e\\gamma d^2}$ , we have: $\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr \\left[\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor \\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma d n,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\le e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}$ This gives us the desired result for sufficiently large $\\gamma d$ : $\\Pr [A_S\\in \\mathcal {M}_k\\backslash (\\mathcal {T}_{k}\\cup \\mathcal {T}_{k}^+\\cup \\mathcal {T}_{k}^C)]&\\le \\left(p^{\\gamma d+c_{\\ref {claim:abc1}}\\log (\\gamma d)}\\right)^k \\left(\\frac{k}{n}\\right)^{k+1}+ e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\\\&\\le O\\left(e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}\\right).$ This yields the first statement in the lemma.", "Finally we prove the final statement about pairs of sets $S$ and $T$ of size $k$ and $j$ respectively.", "Recall that $R = S \\cup T$ .", "Since $A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ , and $A_S \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C$ , $A_S$ contains at most $2k$ non-zero entries and $A_T$ contains at most $2j$ non-zero entries.", "Hence $A_R$ contains at most $2(k + j) \\le 4|R|$ non-zero entries.", "By Lemma REF , $A_{R}$ must contain at least $|R| - 1$ non-zero entries that are not the first in their column.", "By the same argument as before, conditioned on $L_R = \\ell $ , the probability that $A_S \\in \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ , and $A_S \\in \\mathcal {T}_j \\cup \\mathcal {T}_j^+ \\cup \\mathcal {T}_j^C$ is at most $\\Pr [\\mathrm {Bin}(\\ell , \\frac{4|R|}{\\gamma (1 - p)n}) \\ge |R| - 1]$ .", "We check that the conditions of Claim REF hold with $K =|R|$ and $N = n$ .", "Since we only consider the case when $L_R \\le 4|R|$ , we must have $K = |R| \\le \\frac{p\\gamma d n}{\\gamma d - 4} = \\frac{p\\gamma d N}{\\gamma d - 4} \\le \\frac{3}{2}pN$ for $d \\ge 12$ .", "Hence employing Claim REF yields the lemma for $d_0 \\ge 12$ and $c_{\\ref {lemma:abc_small}} = c_{\\ref {claim:abc1}}$ ." ], [ "Large Case", "The main goal of the large case is to prove the following lemma.", "Lemma 17 (ABC Large Case) Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ for some constant $\\gamma \\ge 16$ .", "Then there exists a constant $d_0(\\gamma )$ such that for $d \\ge d_0$ , $\\Pr \\left[\\exists x: x^TA = 0, \\mathrm {supp}(x) \\ge \\frac{n}{18e\\gamma d^2}\\right] \\le o(1).$ Our main tool in proving this Lemma is an anti-concentration Lemma that is based on the sparse Littlewood-Offord Theorem from [9].", "While typically such anti-concentration lemmas concern the dot product of a deterministic vector and a random vector with independent entries, we derive a weaker result which concerns the dot product of a deterministic vector with a vector in which there are a fixed number of non-zero entries whose positions are random.", "Lemma 18 (Anti-concentration for Sparse Regular Vectors (Same as Lemma REF )) Let $v\\in \\mathbb {R}^N$ be an arbitrary vector whose most common entry is $a$ .", "Then for any $d \\le \\sqrt{\\frac{N}{2}}$ , if $x \\in \\lbrace 0, 1\\rbrace ^N$ is sampled uniformly from the set of vectors with exactly $d$ 1s, we have: $\\Pr \\left[x\\cdot v=w\\right]\\le 1/2+\\frac{d^2}{N}.$ for all $w\\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "Remark 5 As $d \\rightarrow \\infty $ , the anti-concentration probability approaches the smaller value of $1/e$ , though $1/2$ is tight for $d = 2$ .", "For $i \\in \\lbrace 1,...,d\\rbrace $ , let $j_i \\sim \\text{Uniform}([N])$ , and let $X_i := v_{j_i}$ .", "Then $\\Pr [x\\cdot v=w] &= \\Pr \\left[\\sum _{i=1}^d X_i =w\\ | \\text{All $j_i$ are unique}\\right] \\\\&\\le \\frac{\\Pr \\left[\\sum _{i=1}^d X_i =w\\right]}{\\Pr \\left[\\text{All $j_i$ are unique}\\right]}.$ Claim 5 $\\Pr \\left[\\text{All $j_i$ are unique}\\right] \\ge 1 - \\frac{d^2}{N}.$ By a union bound, $\\Pr \\left[\\exists i \\ne \\ell : j_i = j_{\\ell } \\right] &\\le \\binom{d}{2}\\cdot \\Pr \\left[j_1 = j_2\\right]\\\\&=\\binom{d}{2}\\left(\\frac{1}{N}\\right)\\\\&\\le \\frac{d^2}{N}.$ Next we show by induction on $d$ that $\\Pr \\left[\\sum _{i=1}^d X_i = w\\right] \\le 1/2$ for all $w\\in \\mathbb {R}\\backslash \\lbrace da\\rbrace $ .", "For $d=1$ , we note that the chosen element $w$ cannot be the most common element $a$ .", "Thus, $w$ is at worst equally as common as the most common element $a$ .", "This implies that the number of times $w$ appears in $v$ is at most $\\frac{N}{2}$ , hence $\\Pr [X_1 = w]\\le \\frac{1}{2}$ .", "Now assume that $\\Pr \\left[\\sum _{i=1}^{d-1} X_i = u\\right]\\le 1/2$ holds for all $u\\ne (d-1)a$ .", "For $w\\ne da$ , we write: $\\Pr \\left[\\sum _{i=1}^{d} X_i = w\\right] &=\\sum _{x\\in \\text{Supp($X_d$)}} \\Pr \\left[X_d = x\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1} X_i = w-x\\right]\\\\&=\\Pr \\left[X_d=a\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1}X_i=w-a\\right] +\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot \\Pr \\left[\\sum _{i=1}^{d-1} X_i = w-x\\right]$ Let $p_x := \\Pr \\left[\\sum _{i=1}^{d-1}X_i=w-x\\right]$ such that by the induction hypothesis $p_a \\le 1/2$ as $w-a=(d-1)a$ if and only if $w=da$ .", "Thus, we conclude: $\\Pr \\left[\\sum _{i=1}^{d} X_i = w\\right] &\\le \\max _{p_a \\le 1/2, \\sum _{x}p_x \\le 1}\\left(\\Pr \\left[X_d=a\\right]\\cdot p_a + \\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot p_x\\right) \\\\&\\le \\max _{\\sum _{x \\ne a}p_x \\le 1/2}\\left(\\Pr \\left[X_d=a\\right]\\cdot \\left(1/2\\right) + \\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} \\Pr \\left[X_d = x\\right]\\cdot p_x\\right) \\\\&\\le \\max _{\\sum _{x \\ne a}p_x \\le 1/2}\\left(\\Pr \\left[X_d=a\\right]\\cdot (1/2) + \\left(\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}} P(X_d = x)\\right)\\cdot \\left(\\sum _{x\\in \\text{Supp($X_d$)$\\backslash \\lbrace a\\rbrace $}}p_x\\right)\\right)\\\\&= \\Pr \\left[X_d=a\\right]\\cdot (1/2)+(1-\\Pr [X_d=a])\\cdot (1/2)\\\\&= 1/2.$ Here the second inequality follows from the fact that the optimum over the $p_x$ is achieved by putting the maximum possible mass on $p_a$ , that is, $p_a = 1/2$ .", "Returning to Equation REF , we have for all $w \\ne ad$ and $d \\le \\sqrt{\\frac{N}{2}}$ , $\\Pr [x\\cdot v=w] &\\le \\frac{\\Pr \\left[\\sum _{i=1}^d X_i =w\\right]}{\\Pr \\left[\\text{All $j_i$ are unique}\\right]} \\le \\frac{1/2}{1 - \\frac{d^2}{N}} \\le 1/2 + \\frac{d^2}{N},$ where the final inequality follows from the fact that $\\frac{1}{1 - x} \\le 1 + 2x$ for $0 \\le x \\le 1/2$ .", "To prove Lemma REF , we take a similar approach to the large case for the BGC.", "For a fixed set $S$ of size $k$ , we consider the stochastic process $A_S^{:1}, A_S^{:2}, \\ldots $ in which we add the columns of $A_S$ one by one.", "We define the space: $D(A_S^{:j})=\\mathrm {Span}\\left(\\lbrace v\\in (\\mathbb {R}\\backslash \\lbrace 0\\rbrace )^k: v A_S^{:j} =0 \\rbrace \\right),$ and let $R_j := \\text{Rank}(D(A_S^{:j})).$ If $R_{\\gamma n(1-p)} =0$ , then $A_S$ is not a minimal dependency.", "Each time we add a new column, $R_j$ either stays constant or decreases by at least 1 (note that $R_j$ can decrease by more than 1: if a new column $j$ has exactly one non-zero entry, then $R_j = R_{j+1} = \\ldots = 0$ no matter what $R_{j-1}$ was).", "We will use the following lemma to show for $j \\le \\gamma n/2$ , each column we add is close to random.", "Then, using Lemma REF , we show that with decent probability, $R_j$ decreases.", "We can then apply a Chernoff Bound to show that $R_{\\gamma n/2} = 0$ with high probability.", "More formally, we consider the process of constructing $H \\in \\lbrace 0,1\\rbrace ^{\\gamma d n\\times \\gamma n(1-p)}$ (the row half-edge occupancy matrix) one column at a time by pairing the $d$ half-edges from each column-node at each step to a random set of $d$ unpaired row-half-edges.", "Define $e(S, j) := \\gamma d k - |H^{:j}(S)|_1$ , that is the number of unpaired half-edges among the $k$ row-nodes in $S$ after the first $j$ column-nodes have randomly paired their half edges.", "The following lemma uses standard concentration bounds to show that for the first $\\gamma n/2$ columns, there are many unpaired half-edges out of row-nodes in $S$ .", "Lemma 19 Let $A \\sim \\textnormal {ABC}_p(n, \\gamma , d)$ .", "Let $\\Omega $ be the event in which for any set $S$ with $|S| \\ge \\frac{n}{18e\\gamma d^2}$ and for any $j\\in [1,\\frac{\\gamma }{2}n]$ , we have $\\frac{e(S,j)}{\\gamma dn-dj}\\ge \\frac{k}{2n}$ .", "For $\\gamma \\ge 2$ , there exists a constant $d_0(\\gamma )$ such that for $d \\ge d_0$ , we have $\\Pr [\\Omega ] \\ge 1 - \\gamma ne^{-n/d^3}$ .", "For $d$ larger than some constant $d_0(\\gamma )$ , we have $\\frac{n}{d^3} \\le \\frac{n}{18e\\gamma d^2}$ , hence it suffices to prove the result for all $k\\in [n/d^3,n)$ .", "Fix $k$ and let us define $\\eta _k$ to be the ratio $k/n$ .", "Choose a subset $S$ of size $k$ .", "Furthermore, fix $j\\le \\frac{\\gamma }{2}n$ .", "We proceed by applying the following tail bound on Hypergeometric distributions to $e(S,j)$ which is clear from [6].", "Lemma 20 (Hypergeometric Tail Bound from [6]) Let $X$ be given by a hypergeometric distribution with parameters $A$ , $B$ , $c$ .", "Then for all $t < \\frac{B}{A}$ , $\\Pr \\left[X \\le tc \\right] \\le e^{-cD_{KL}(t||B/A)}.$ Similarly, letting $Y$ be given by a hypergeometric distribution with parameters $A$ , $A - B$ , and $c$ , for $t > \\frac{B}{A}$ , we have $\\Pr \\left[X \\ge tc \\right] = \\Pr [Y \\le (1 - t)c] \\le e^{-cD_{KL}(1 - t||1 - B/A)} = e^{-cD_{KL}(t||B/A)}.$ Using this bound, we have $P[e(S,j)\\le ({\\eta _k-t})(\\gamma d n-dj)]&\\le e^{-D_{KL}\\left(\\eta _k-t||\\eta _k\\right)(\\gamma d n-dj)}\\\\&\\le e^{-D_{KL}\\left(\\eta _k-t||\\eta _k\\right)\\frac{\\gamma dn}{2}}$ for all $t\\in (0,\\eta _k)$ .", "The following claims expands this KL-divergences for $t = \\eta _k/2$ .", "Claim 6 $\\begin{split}D_{KL}\\left(\\eta _k-\\eta _k/2||\\eta _k\\right) \\ge \\frac{\\eta _k}{12}.\\end{split}$ $\\begin{split}D_{KL}\\left(\\eta _k/2||\\eta _k\\right) &= \\frac{\\eta _k}{2}\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\ln \\left(\\frac{1 - \\eta _k/2}{1 - \\eta _k}\\right) \\\\&\\ge \\frac{\\eta _k}{2}\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(\\frac{\\eta _k}{2} + \\frac{3\\eta _k^2}{8}\\right) \\\\&= \\frac{\\eta _k}{2}\\left(\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(1 + \\frac{3\\eta _k}{4}\\right)\\right) \\\\& \\ge \\frac{\\eta _k}{12}.\\end{split}$ Here the first inequality follows by using the Taylor expansion for $\\ln (1 - x)$ and the final inequality follows by noting that the quadratic $\\ln (1/2) + \\left(1 - \\frac{\\eta _k}{2}\\right)\\left(1 + \\frac{3\\eta _k}{4}\\right)$ achieves its minimum over $\\eta _k \\in [0, 1]$ at $\\eta _k \\in \\lbrace 0, 1\\rbrace $ .", "Union bounding over all $\\binom{n}{k}$ sets $S$ of size $k$ , we have $\\Pr \\left[\\exists S, k := |S| \\in [n/d^3, n] : e(S,j)\\le \\frac{\\eta _k}{2}(\\gamma dn-dj)\\right]&\\le \\sum _{k=n/d^3}^n \\binom{n}{k}e^{-\\frac{\\eta _k\\gamma dn}{24}}\\\\&\\le \\sum _{k=n/d^3}^n \\left(\\frac{en}{k}\\right)^ke^{-\\frac{k\\gamma d}{24}} \\\\&= \\sum _{k=n/d^3}^n e^{-k\\left(\\frac{\\gamma d}{24}- \\ln (1/\\eta _k) - 1\\right)} \\\\&\\le \\sum _{k=n/d^3}^n e^{-k\\left(\\frac{\\gamma d}{24}- \\ln (d^3) - 1\\right)} \\\\&\\le \\sum _{k=n/d^3}^n e^{-k} \\\\& \\le 2e^{-n/d^3}.$ We take a union bound over $j \\in [1, \\gamma n/2]$ to achieve $\\Pr \\left[\\exists S, k := |S| \\in [n/d^3,n] : e(S,j)\\le \\frac{\\eta _k}{2}(\\gamma dn-dj)\\right] \\le \\gamma ne^{-n/d^3}.$ Conditioned on $\\Omega $ , we can use our anti-concentration bound, Lemma REF , to show that for $j \\le \\gamma n/2$ , $R_j$ often decreases.", "Lemma 21 For $j \\le \\gamma n/2$ , conditioned on $\\Omega $ , if $R_j \\ge 1$ , then with probability at least $\\mu _k := \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k},$ we have $R_{j} \\le R_{j-1} - 1.$ Suppose $R_{j - 1} \\ge 1$ , and let $v$ be any vector in $(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )^k$ such that $A_S^{:j - 1}v=0$ .", "Let $v^{\\prime } \\in \\mathbb {R}^{\\gamma dk}$ be the vector which repeats each coordinate of $v$ $\\gamma d$ times: that is, $v^{\\prime }_i = v_{\\lceil \\frac{i}{\\gamma d}\\rceil }$ .", "Let $h_j \\in \\lbrace 0, 1\\rbrace ^{\\gamma dn}$ be $j$ th column of $H$ , which has exactly $d$ ones indicating the half-edges matched to the $d$ half-edges from the column-node $j$ .", "Let $h_j(S) \\in \\lbrace 0, 1\\rbrace ^{\\gamma d k}$ be the restriction of $h_j$ to the entries corresponding to half-edges from row-nodes in $S$ .", "Claim 7 If at most one half-edge is matched from column-node $j$ to a single row-node in $S$ and $h_j(S) \\cdot v^{\\prime } \\ne 0$ , then $R_{j} \\le R_{j-1} - 1$ .", "If at most one half-edge is matched from column-node $j$ to a single row-node in $S$ , then $v \\cdot (A_S)_j = \\sum _{i \\in S}v_iA_{ij} = \\sum _{i \\in S}v_i\\mathbb {1}(\\exists \\ell , \\gamma d(i-1) < \\ell \\le \\gamma di, H_{\\ell j} = 1 ) = \\sum _{i \\in S}v_i\\sum _{\\ell = \\gamma d(i - 1) + 1}^{\\gamma di} H_{\\ell j} = v^{\\prime } \\cdot h_j(S) \\ne 0.$ Hence $v \\notin D(A_S^{:j})$ , so the rank of $D(A_S^{:j})$ is strictly smaller than that of $D(A_S^{:j-1})$ .", "Claim 8 $\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | \\Omega ] \\le \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) + \\frac{d}{\\gamma k}.$ We condition on the number of non-zero entries in $h_j(S)$ , which we denote $s$ .", "Observe that conditioned on $s$ , the vector $h_j(S)$ is a uniformly random vector from the set of all vectors in $\\lbrace 0, 1\\rbrace ^{\\gamma d k}$ with $s$ 1s.", "This holds even when we conditioned on $\\Omega $ , because this event says nothing about which half-edges among the nodes in $S$ have been paired.", "By Lemma REF , since $v^{\\prime }$ contains no zeros (and hence its most common element is not zero), we have $\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | s \\ge 1, \\Omega ] \\le 1/2 + \\frac{d}{\\gamma k}.$ It remains to consider the probability that $s = 0$ , since in this case, we always have $h_j(S) \\cdot v^{\\prime } = 0$ .", "We know that $s$ is distributed like a hypergeometric random variable $\\text{HyperGeom}(\\gamma d n - dj, d, e(S, j))$ .", "Indeed, there are $d$ half-edges that are paired with the addition of the $(j + 1)$ -th column, there are $e(S, j)$ unpaired half-edges among the nodes in $S$ , and there are $\\gamma d n - dj$ total unpaired half-edges among the row nodes.", "Since we have conditioned on $\\Omega $ , we know that $\\frac{e(S,j)}{\\gamma d n-dj}\\ge \\frac{k}{2n}$ .", "Hence we can compute $\\Pr [s = 0 | \\Omega ] \\le \\left(1 - \\frac{k}{2n}\\right)^{d} \\le e^{-\\frac{dk}{2n}}.$ Combining Equations REF and REF , it follows that $\\begin{split}\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | \\Omega ] &= \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\Pr [h_j(S) \\cdot v^{\\prime } = 0 | s \\ge 1, \\Omega ]\\\\&\\le \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\left(1/2 + \\frac{d}{\\gamma k}\\right)\\\\&\\le \\Pr [s = 0| \\Omega ] + \\left(1 - \\Pr [s = 0| \\Omega ]\\right)\\left(1/2\\right) + \\frac{d}{\\gamma k} \\\\&= \\frac{1}{2}\\left(1 + \\Pr [s = 0| \\Omega ]\\right) + \\frac{d}{\\gamma k}\\\\&\\le \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) + \\frac{d}{\\gamma k}.\\end{split}$ Claim 9 The probability that more than one half-edge is matched from a column-node $j$ to a single row-node in $S$ is at most $\\frac{2\\gamma d^3}{k}$ .", "Conditioned on $\\Omega $ , we have at each step $j$ , there are at least $\\frac{k(\\gamma d n - dj)}{2n}$ open half-edges out of $S$ , and hence at least $\\frac{k(\\gamma d n - dj)}{2n\\gamma d} \\ge \\frac{k}{4}$ nodes in $S$ with at least one open half-edge.", "Since each node in $S$ has at most $\\gamma d$ open half-edges, each pair of half-edges from the $j$ th column collide with a row-node with probability at most $\\frac{\\gamma d}{k/4}$ .", "Hence by a union bound, the probability of collision is at most $\\binom{d}{2}\\left(\\frac{\\gamma d}{k/4}\\right) \\le \\frac{2\\gamma d^3}{k}.$ It follows from the previous two claims that the probability that $h_j(S) \\cdot v^{\\prime } \\ne 0$ and at most one half-edge is match from a column-node $j$ to a single row-node in $S$ is at least $\\begin{split}1 - \\Pr [\\textnormal {Event in Claim~\\ref {claim:double_match} occurs}] - \\Pr [h_j(S)\\cdot v^{\\prime } = 0 | \\Omega ]&\\ge 1 - \\frac{2\\gamma d^3}{k} - \\frac{d}{\\gamma k} - \\frac{1}{2}\\left(1 + e^{-\\frac{dk}{2n}}\\right) \\\\&= \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{2\\gamma d^3}{k} - \\frac{d}{\\gamma k} \\\\&\\ge \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k}.\\end{split}$ Using Claim REF , this proves the lemma.", "We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Recall that our goal is to show that with high probability, for all $k \\ge \\frac{n}{18e\\gamma d^2}$ , for all sets $S$ of size $k$ , we have $R_{\\gamma n/2} = 0$ .", "Throughout the rest of the proof, we assume that we have conditioned on $\\Omega $ , since $\\Pr [\\Omega ] = 1 - o(1)$ .", "For a fixed set $S$ of size $k$ , by Lemma REF , conditioned on each term being positive, the random process $R_1, R_2, \\cdots , R_{\\gamma n/2}$ is stochastically dominated by the random process $Y_1, Y_2, \\ldots ,Y_{\\gamma n/2}$ , where $Y_{i + 1} = Y_i - \\mathrm {Ber}(\\mu _k)$ and $\\mu _k = \\frac{1}{2}\\left(1 - e^{-\\frac{dk}{2n}}\\right) - \\frac{3\\gamma d^3}{k}.$ Hence $\\Pr [R_{\\gamma n/2} > 0] \\le \\Pr [Y_{\\gamma n/2} > 0].$ By a Chernoff Bound, since $Y_0 = k$ , for any $\\mu \\le \\mu _k$ , we have $\\begin{split}\\Pr [Y_{\\gamma n/2} > 0] &= \\Pr [Y_{0} - Y_{\\gamma n/2} < k]\\\\&\\le e^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k.\\end{split}$ Define $\\eta _k := k/n$ .", "Taking a union bound over all sets $S$ of size $k$ , conditioned on $\\Omega $ , the probability that at least one set $S$ has $R{\\gamma n/2} > 0$ is at most $\\begin{split}\\binom{n}{k}e^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k &\\le \\left(\\frac{en}{k}\\right)^ke^{-\\gamma n\\mu /2}\\left(\\frac{e\\gamma n\\mu }{2k}\\right)^k \\\\&= e^{-\\gamma n\\mu /2}\\left(\\frac{e^2\\gamma n^2\\mu }{2k^2}\\right)^k \\\\&= e^{-\\gamma n\\mu /2+ k\\log \\left(\\frac{e^2\\gamma n^2\\mu }{k^2}\\right)} \\\\&= e^{k\\left(-\\frac{\\gamma \\mu }{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu }{2\\eta _k^2}\\right)\\right)}\\end{split}$ We consider two cases: Case 1: $\\frac{n}{18e\\gamma d^2} \\le k \\le \\frac{5n}{d}$ .", "Case 2: $k \\ge \\frac{5n}{d}$ .", "In the first case, since $1 - e^{-x} \\ge x/2$ for $0 < x < 1$ , for $n$ large enough, we have $\\mu _k \\ge \\frac{1}{2}\\left(\\frac{d\\eta _k}{4}\\right) - \\frac{3\\gamma d^3}{k} \\ge \\frac{d\\eta _k}{10}.$ Hence for $\\mu = \\frac{d\\eta _k}{10}$ , we have $-\\frac{\\gamma \\mu }{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu }{2\\eta _k^2}\\right) \\le -\\frac{\\gamma d}{20} + \\log \\left(\\frac{e^2\\gamma d}{20\\eta _k^2}\\right) \\le -\\frac{\\gamma d}{20} + \\log \\left(\\frac{18^2e^3\\gamma ^3d^5}{20}\\right) \\le -0.25$ for $d$ larger than some constant $d_0$ .", "In the second case, we have $\\mu _k \\ge 0.45$ , and hence for $\\mu = 0.45$ and $\\gamma \\ge 16$ , we have $-\\frac{\\gamma \\mu _k}{2\\eta _k} + \\log \\left(\\frac{e^2\\gamma \\mu _k}{2\\eta _k^2}\\right) \\le \\max _{\\eta \\le 1} -\\frac{3.6}{\\eta } + \\log \\left(\\frac{3.6e^2}{\\eta ^2}\\right) \\le -0.25.$ Returning to Equation REF , conditioned on $\\Omega $ , the probability that there is a set $S$ of size $k$ for which $R_{\\gamma n/2} > 0$ is at most $e^{-k/4}$ .", "Summing over all $k \\ge \\frac{n}{168\\gamma d^2}$ yield a probability of failure among any $k$ of at most $\\frac{e^{-\\frac{n}{64e\\gamma d^2}}}{1 - e^{-1/4}} \\le 5e^{-\\frac{n}{64e\\gamma d^2}}.$ Unioning with the probability that $\\Omega $ doesn't occur from Lemma REF yields Lemma REF ." ], [ "Proof of Theorem ", "We are now ready to prove Theorem REF , which we restate here for the reader's convenience.", "* [Proof of Theorem REF ] By combining the result of the small case, Lemma REF , with the result of the large case, Lemma REF , we observe that $1 - o(1)$ , there are no minimal dependencies in $A$ that are not in $\\bigcup _{k \\le \\log (n)}\\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C .$ This proves the first statement in the theorem.", "Next we bound the size of $D$ , the set of rows involved in linear dependencies: $D := \\bigcup _{x: x^TA = 0}{\\mathrm {supp}(x)}.$ Define $\\tilde{\\mathcal {T}_k} := \\mathcal {T}_k \\cup \\mathcal {T}_k^+ \\cup \\mathcal {T}_k^C$ For a set $S \\subset [n]$ , let $X_S$ be the indicator of the event that $A_S \\in \\tilde{\\mathcal {T}_k}$ .", "Notice that with probability $1 - o(1)$ , $|D|$ is at most $X := \\sum _{S \\subset [n], |S| \\le \\log (n)}|S|X_S.$ We will bound $X$ with high probability via the second moment method with Lemma REF as our main tool.", "First we compute the expectation of $X$ .", "By Lemma REF , for some constant $c_{\\ref {lemma:abc_small}}$ , we have $\\begin{split}\\mathbb {E}[X] &= \\sum _{S: |S| \\le \\log (n)}{|S|\\Pr [A_S \\in \\tilde{\\mathcal {T}_k}]} \\\\&\\le \\sum _{k \\ge 1}k\\binom{n}{k}\\left(\\frac{k}{n}\\right)^{k - 1}p^{\\gamma dk + c_{\\ref {lemma:abc_small}}k\\log (\\gamma d)} \\\\&\\le n \\sum _{k \\ge 1}p^{\\gamma d k + (c_{\\ref {lemma:abc_small}} + 1)k\\log (\\gamma d)} \\\\&\\le n p^{\\gamma d - c_1\\log (\\gamma d)}\\end{split}$ for some constant $c_1$ .", "If $T, S \\subset [n]$ are such that $|S| \\le |T| \\le \\log (n)$ and $S \\cap T \\ne \\emptyset $ , then by Lemma REF , with $R := S \\cup T$ $\\mathbb {E}[X_SX_T] \\le \\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)}.$ In the next claim, we show that if $T \\cap S = \\emptyset $ , then the events $X_S$ and $X_T$ are almost uncorrelated.", "Claim 10 Let $T, S \\subset [n]$ and $|T|, |S| \\le \\log (n)$ .", "If $T \\cap S = \\emptyset $ , then $\\mathbb {E}[X_SX_T] \\le \\left(1 + \\frac{6\\gamma d\\min (|S|, |T|)}{n}\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T].$ [Proof of Claim REF ] Let $|S| = k$ and $|T| = j$ and without loss of generality assume $k \\le j$ .", "Let $\\rho $ be the permutation defining the mapping of half-edges used to generate $A$ .", "Let $\\mathcal {P}_{\\gamma dk, \\gamma dn}$ be the set of injections from $[\\gamma dk] $ to $[\\gamma dn]$ .", "For $\\pi \\in \\mathcal {P}_{\\gamma dk, \\gamma dn}$ , recall that $A(\\pi ) \\in \\lbrace 0, 1\\rbrace ^{k \\times \\gamma n(1-p)}$ is the first $k$ rows of the $\\textnormal {ABC}_p$ matrix which is generated from mapping half-edges according to $\\pi $ and then dropping the last $\\gamma pn$ columns.", "Let $\\mathcal {R}_k = \\lbrace \\pi \\in \\mathcal {P}_{\\gamma dk, \\gamma dn}: A(\\pi ) \\in \\tilde{\\mathcal {T}_k}\\rbrace .$ Since each element of $\\mathcal {P}_{\\gamma dk, \\gamma dn}$ is equally likely to be the restriction of $\\rho $ to $[\\gamma dk]$ , we have $\\mathbb {E}[X_S] = \\frac{|\\mathcal {R}_k|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}.$ Our goal is to compute $\\begin{split}\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T] &= \\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_S \\in \\mathcal {R}_k \\wedge \\rho |_T \\in \\mathcal {R}_j \\right] - \\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_S \\in \\mathcal {R}_k\\right]\\Pr _{\\rho \\sim \\mathcal {S}_{\\gamma dn}}\\left[\\rho |_T \\in \\mathcal {R}_j \\right] \\\\&= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\Pr \\left[\\rho |_S = \\pi _1 | \\rho |_T = \\pi _2\\right] - \\Pr \\left[\\rho |_S = \\pi _1\\right]\\right)\\left(\\Pr \\left[\\rho |_T = \\pi _2\\right]\\right).\\end{split}$ Here we have abused notation to interpret the restriction of $\\rho $ to a set $U$ as being an element of $\\mathcal {P}_{\\gamma d|U|, \\gamma dn}$ .", "If the images of $\\pi _1$ and $\\pi _2$ intersect, then $\\Pr _{\\rho }[\\rho |_S = \\pi _1 | \\rho |_S = \\pi _2] = 0.$ Otherwise, if the images do not intersect, we have $\\Pr _{\\rho }[\\rho |_S = \\pi _1 | \\rho |_S = \\pi _2] = \\frac{1}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|} = \\Pr [\\rho |_S = \\pi _1]\\frac{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|}$ It follows that $\\begin{split}\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T] &= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\Pr \\left[\\rho |_S = \\pi _1 | \\rho |_T = \\pi _2\\right] - \\Pr \\left[\\rho |_S = \\pi _1\\right]\\right)\\left(\\Pr \\left[\\rho |_T = \\pi _2\\right]\\right)\\\\&= \\sum _{\\pi _1 \\in \\mathcal {R}_k, \\pi _2 \\in \\mathcal {R}_j}\\left(\\frac{|\\mathcal {P}_{\\gamma dk, \\gamma dn}|}{|\\mathcal {P}_{\\gamma dk, \\gamma dn - \\gamma dk}|} - 1\\right)\\Pr \\left[\\rho |_T = \\pi _2\\right]\\left(\\Pr \\left[\\rho |_S = \\pi _1\\right]\\right) \\\\&= \\left(\\frac{(\\gamma dn)!", "(\\gamma dn - 2\\gamma dk)!", "}{(\\gamma dn - \\gamma dk)!", "(\\gamma dn - \\gamma dk)!", "}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&= \\left(\\left(\\prod _{i = 1}^{\\gamma dk}\\frac{\\gamma dn - i + 1}{\\gamma dn - \\gamma dk - i + 1}\\right)-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\left(1 + \\frac{2\\gamma dk}{\\gamma dn - \\gamma dk}\\right)^{\\gamma dk}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\left(1 + \\frac{3k}{n}\\right)^{\\gamma dk}-1\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T] \\\\&\\le \\left(\\frac{6\\gamma dk^2}{n}\\right)\\mathbb {E}[X_S]\\mathbb {E}[X_T].\\end{split}$ where the first inequality follows from the fact that $\\frac{1}{1 - x} \\le 1 + 2x$ for $x \\le \\frac{1}{4}$ .", "We can now bound the variance of $X$ : $\\begin{split}\\text{Var}(X) &= \\mathbb {E}[X^2] - \\mathbb {E}[X]^2 \\\\&= \\sum _{S: |S| \\le \\log (n)}\\sum _{T: |T| \\le \\log (n)}|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\sum _{S}\\sum _{T : T \\cap S = \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) + \\sum _{S}\\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\mathbb {E}[X_SX_T] - \\mathbb {E}[X_S]\\mathbb {E}[X_T]\\right) \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _S \\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\mathbb {E}[X_SX_T] \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _S \\sum _{T: T \\cap S \\ne \\emptyset }|S||T|\\left(\\frac{|S \\cup T|}{n}\\right)^{|S \\cup T| - 1}p^{\\gamma d|S \\cup T| - c_{\\ref {lemma:abc_small}}|S \\cup T|\\log (\\gamma d)}\\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _{R: 1 \\le |R| \\le 2\\log (n)} \\sum _{S, T \\subset R}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)}\\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + \\sum _{R: 1 \\le |R| \\le 2\\log (n)} 2^{2|R|}|R|^2\\left(\\frac{|R|}{n}\\right)^{|R| - 1}p^{\\gamma d|R| - c_{\\ref {lemma:abc_small}}|R|\\log (\\gamma d)} \\\\&\\le \\frac{6\\gamma d\\log (n)^2}{n}\\mathbb {E}[X]^2 + np^{\\gamma d - c_2\\log (\\gamma d)} \\\\&\\le 6\\gamma d\\log (n)^2np^{2\\gamma d - 2c_1\\log (\\gamma d)} + np^{\\gamma d - c_2\\log (\\gamma d)}.\\end{split}$ for some constant $c_2$ (the second to last line follows from a similar calculation as used when computing $\\mathbb {E}[X]$ ).", "It follows by Markov's law that for any $t$ , $\\Pr [X \\ge \\mathbb {E}[X] + t] \\le \\frac{\\text{Var}(X)}{t^2}$ Plugging in $t = \\left(np^{\\gamma d}\\right)^{3/4}$ , we have $\\begin{split}\\Pr [X \\ge \\mathbb {E}[X] + n^{3/4}p^{-\\gamma d}] &\\le \\frac{\\text{Var}(X)}{\\left(np^{\\gamma d}\\right)^{3/2}} \\\\&\\le \\frac{6\\gamma d\\log (n)^2np^{2\\gamma d - 2c_1\\log (\\gamma d)} + np^{\\gamma d - c_2\\log (\\gamma d)}}{\\left(np^{\\gamma d}\\right)^{3/2}} \\\\&\\le b^{-1/2}\\left(6\\gamma d\\log (n)^2p^{\\gamma d/2-2c_1\\log (\\gamma d)} + p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\right)\\\\&\\le n^{-1/2}p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\log (n)^2\\end{split}$ It follows that with probability at most $n^{-1/2}p^{-\\gamma d/2 - c_2\\log (\\gamma d)}\\log (n)^2 = o(1)$ , we have $X \\le np^{\\gamma d - c_1\\log (\\gamma d)} + \\sqrt{np^{\\gamma d}}\\log (n)^2p^{-c_2\\log (\\gamma d)} \\le np^{\\gamma d - c\\log (\\gamma d)},$ for some constant $c$ .", "This proves the theorem." ], [ "Applications to Gradient Coding", "In the following section we will address the motivating application of our work: the design of gradient codes with small decoding error.", "The literature on gradient codes largely focuses on the special case where the assignment matrix is a square.", "Hence, to align our results with this standard, we introduce the following stacked $ABC$ construction, which will us to apply our results on wide matrices to square $n\\times n$ assignment matrices.", "Definition 7 For $\\gamma ,d,n\\in \\mathbb {Z}^+$ such that $\\gamma \\mid d$ and $\\gamma \\mid n$ , we define the $\\gamma $ -stacked Augmented Biregular Code $B$ to be an $n\\times n$ matrix formed by sampling $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ and stacking $\\gamma $ identical copies of $A_0$ .", "We will denote the distribution of such matrices as $ABC_{\\gamma -stacked}(n,d)$ (see Figure REF ).", "Figure: The stacked ABCRemark 6 We note that these stacked designs can be viewed as a natural generalization of the Fractional Repetition Code (FRC) of [21].", "In particular, any $B\\sim \\textnormal {ABC}_{stacked}(N,d,d)$ is the $N\\times N$ FRC matrix with $d$ ones in each column, up to a permutation of the rows and columns.", "The key improvement is that our generalization allows the stacking variable $\\gamma $ — which increases the adversarial decoding error — to stay constant for arbitrarily large $d$ .", "Using these stacked ABC matrices, we will prove the following theorem: * While the first bound will be a direct corollary of Theorem $\\ref {abc_random}$ , the second bound relies on some external lemmas that will be introduced below.", "Thus, for the sake of clarity, we will split the proof of Theorem $\\ref {abc_stacked}$ into the following two lemmas.", "Lemma 22 Let $c,\\gamma _0,d_0$ be the universal constants from Theorem $\\ref {abc_random}$ .", "Choose any $\\gamma , d\\in \\mathbb {Z}^+$ such that $\\gamma geq \\gamma _0$ , $\\gamma \\mid d$ , and $\\frac{d}{\\gamma }\\ge d_0$ .", "For sufficiently large $N$ such that $\\gamma \\mid n$ , let $B\\sim \\textnormal {ABC}_{stacked}(n, \\gamma ,d)$ .", "Then with probability $1-o(1)$ over the choice of $B$ , we have that for any $p<\\frac{1}{2}$ : $\\frac{1}{n}\\mathbb {E}_{S\\sim \\binom{[n]}{pn}} \\textnormal {err}(B, S) \\le p^{d + c\\log (d)}+o(1)$ Lemma 23 Choose any $\\gamma ,d,N\\in \\mathbb {Z}^+$ such that $\\gamma \\mid d$ and $\\gamma \\mid n$ .", "Let $B\\sim \\textnormal {ABC}_{stacked}(n,\\gamma , d)$ .", "With constant probability, we have: $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\left( \\textnormal {err}(B, S)\\right) \\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1).$ Temporarily assuming these lemmas, we note that Theorem REF follows immediately.", "We will now prove Lemma $\\ref {stacked_prop1}$ .", "[Proof of Lemma REF ] Let $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ denote the ABC matrix which is stacked $\\gamma $ times to generate $B$ , then it follows that: $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}&= \\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|Bw - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|A_0w - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\mathbb {E}\\left[\\min _{w}|Aw - \\mathbb {1}|_2^2\\right]$ where $A\\sim \\textnormal {ABC}_p(n/\\gamma ,\\gamma ,d/\\gamma )$ .", "First observe trivially that $ \\frac{\\gamma }{n}\\min _{w}|Aw - \\mathbb {1}|_2^2$ is always at most 1.", "This can be seen by taking $w$ to be the vector of all zeros.", "By Theorem REF , there is a $1-o(1)$ chance that $A_0$ has the property that the following holds with probability $1-o(1)$ over the choice of $S$ : $\\frac{\\gamma }{n}\\min _{w}|Aw - \\mathbb {1}|_2^2\\le p^{d-c\\log (d)}+o(1).$ For $A_0$ where this holds, we can calculate the expectation as: $\\gamma \\mathbb {E}\\left[\\min _{w}|Aw - \\mathbb {1}|_2^2\\right]\\le (1-o(1))\\left(p^{\\gamma d-c\\log (\\gamma d)}+o(1)\\right)n+o(n).$ This gives us the desired result: $\\frac{1}{n}\\mathbb {E}_{S \\sim \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}\\le p^{d-c\\log (d)}+o(1)$ for a matrix $B\\sim \\textnormal {ABC}_{stacked}(n,\\gamma , d)$ with probability $1-o(1)$ .", "Lemma REF requires a bit more machinery.", "We are going employ the following lemma from [13] which allows us to bound the adversarial error of a gradient code as a function of the second largest singular value.", "Formally, we have the following lemma.", "Lemma 24 (Proposition 4.1 of [13]) Let $A\\in \\lbrace 0,1\\rbrace ^{N\\times M}$ be an assignment matrix such that each row has exactly $D$ ones.", "Let $\\sigma _2$ be the largest singular value of $A$ .", "Then for any set of stragglers $S$ such that $|S|=s$ , we have: $\\frac{1}{N}\\textnormal {err}(A,S)\\le \\frac{1}{N}\\left(\\frac{\\sigma _2}{D}\\right)^2\\frac{sM}{M-s}$ To calculate an upper bound on the second largest singular value of an $ABC$ matrix, we first need a result from [2] which states that with constant probability, the configuration model we use to generate our $ABC$ matrix has no rows which map to the same column node more than once ie., the bipartite graph produced is simple.", "Let $\\mathcal {G}(n, \\gamma n, d, \\gamma d)$ denote the uniform distribution on simple $(d, \\gamma d)$ -biregular bipartite graphs with $n$ left nodes and $\\gamma n$ right nodes.", "Lemma 25 ([2]) Let $A_0\\sim ABC(n,\\gamma ,d)$ , then the probability that $\\begin{pmatrix}0 & A_0\\\\A_0^T & 0\\end{pmatrix}$ is the adjacency matrix of a bipartite, biregular random graph $G$ is at least $\\varepsilon (d)>0$ .", "Furthermore, if we condition on this event occurring, then $G\\sim \\mathcal {G}(n,\\gamma n,d,\\gamma d)$ .", "The previous lemma allows us to apply the following result of [3] to bound the second largest singular value of these well behaved ABC matrices.", "Lemma 26 (Theorem 4 of [3]) Let $A=\\begin{pmatrix}0 & X\\\\X^T & 0\\end{pmatrix}$ be the adjacency matrix of a bipartite, biregular random graph $G\\sim \\mathcal {G}(n,\\gamma n,d,\\gamma d)$ .", "Then, with probability $1-o(1)$ , $A$ 's second largest eigenvalue $\\lambda _2$ satisfies $\\lambda _2 \\le \\sqrt{d_1-1}+\\sqrt{d_2-1}+o(1).$ We are now ready to prove Lemma REF .", "[Proof of Lemma REF ] Let $A_0\\sim \\textnormal {ABC}(n/\\gamma ,\\gamma ,d/\\gamma )$ denote the $ABC$ matrix which is stacked $\\gamma $ times to generate $B$ .", "It follows that: $\\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}{ \\textnormal {err}(B, S)}&= \\frac{1}{n}\\max _{S \\in \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|Bw - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}}\\min _{w: w_j = 0 \\: \\forall \\: j \\in S}|A_0w - \\mathbb {1}|_2^2\\\\&= \\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}} err(A_0,S)$ By Lemma $\\ref {stacked_lemma2}$ , with constant probability, $A^{\\prime }=\\begin{pmatrix}0 & A_0\\\\A_0^T & 0\\end{pmatrix}$ is the adjacency matrix of a bipartite, biregular random graph $G$ .", "Condition on this event occurring.", "Then $G$ is uniformly sampled from $\\mathcal {G}(n/\\gamma ,N,d/\\gamma ,d)$ .", "By Lemma $\\ref {stacked_lemma3}$ , with probability $1-o(1)$ , the second largest eigenvalue of $A^{\\prime }$ is $\\lambda _2\\le \\sqrt{d/\\gamma -1}+\\sqrt{d-1}+o(1)$ .", "Thus, the second largest singular value of $A_0$ is $\\sigma _2 \\le \\sqrt{d/\\gamma -1}+\\sqrt{d-1}+o(1)$ .", "Hence, by Lemma $\\ref {stacked_lemma1}$ , we have: $\\frac{\\gamma }{n}\\max _{S \\in \\binom{[n]}{pn}} err(A_0,S)&\\le \\frac{\\gamma }{n}\\left(\\frac{\\gamma \\sigma _2}{d}\\right)^2\\frac{pn^2}{(1-p)n}\\\\&\\le \\gamma ^3\\left(\\frac{2\\sqrt{d}+o(1)}{d}\\right)^2\\frac{p}{(1-p)}\\\\&\\le \\left(\\frac{4\\gamma ^3p}{d(1-p)}\\right)+o(1)\\\\&\\le \\left(\\frac{8\\gamma ^3p}{d}\\right)+o(1)$ as desired.", "Table: Comparison of Related Work.", "We have normalized the decoding error by 1/N1/N." ], [ "Acknowledgements", "The authors thank Mary Wootters for helpful comments on this manuscript." ], [ "Proof of Lemmas ", "Recall Lemma : * We break the proof of Lemma  into three lemmas, which we prove independently.", "Lemma 27 $\\mathcal {S}_{2k-2, k} = \\mathcal {T}_k$ .", "Further, for $B \\in \\mathcal {T}_k = \\mathcal {S}_{2k - 2, k}$ , there is a unique (up to constant multiple) non-zero vector $v$ satisfying $B^T v = 0$ , where $v_i = (-1)^{d_G(i,j)} v_j$ , and $d_G(i,j)$ is the path length from vertex $i$ to $j$ in the graph $G$ encoded by $B$ .", "We prove this by showing both inclusions in the following two claims.", "Claim 11 $\\mathcal {T}_k \\subset \\mathcal {S}_{2k-2, k}$ .", "Let $B \\in \\mathcal {T}_k$ .", "Since $B$ has $k-1$ non-zero rows, it has rank at most $k-1$ , and so the nullspace of $B^T$ has dimension at least 1.", "Let $v$ be any vector such that $B^Tv = 0$ .", "Suppose $(i,j)$ is an edge in the tree $G$ encoded by $\\mathcal {T}_k$ , then there is a column $C$ of $B$ such that $B$ has 1's exactly at coordinates $i$ and $j$ .", "We have $C \\cdot v = 0$ thus $v_j = -v_i$ .", "Applying this to paths of multiple edges gives that $v_i = (-1)^{d_G(i,j)} v_j$ where $d_G$ is the path length from $i$ to $j$ in $G$ (which is unique and well-defined because $G$ is a tree).", "By this formula, if any $v_i$ is non-zero, then all other entries of $v$ are non-zero, and are uniquely determined by $v_i$ .", "Thus $B^T$ (and $B$ ) have rank $k-1$ .", "We also know that $B$ has $k-1$ non-zero columns, each with 2 1's, so it has $2k-2$ non-zero entries.", "Thus $B \\in \\mathcal {S}_{2k - 2, k}$ .", "In addition, by the description of $v$ , this establishes the second statement of the lemma.", "Claim 12 $\\mathcal {S}_{2k - 2, k} \\subset \\mathcal {T}_k$ .", "Suppose $B \\in \\mathcal {S}_{2k - 2, k}$ .", "Because $B$ has rank $k-1$ it must have $\\ge k-1$ non-zero columns.", "Recall that by Observation REF , no column of $B$ can have exactly one 1.", "Thus, $B$ must have exactly $k-1$ non-zero columns, each with exactly 2 1's.", "Let $G$ be be the graph on vertices $[1,\\ldots ,k]$ encoded by incidence matrix given by the non-zero columns of $B$ .", "We claim $G$ is connected.", "Suppose otherwise, and without loss of generality let $[1,\\ldots ,L], [L+1,\\ldots ,k]$ be two disconnected components of $G$ for $1 \\le L \\le k-1$ .", "By definition of $\\mathcal {S}_{2k-2,k}$ there is non-zero $v \\in \\mathbb {R}^k$ with $B^T v = 0$ , and $v$ has all non-zero entries.", "Let $v^{(L)}, v^{(R)} \\in \\mathbb {R}^k$ with $v^{(L)} = (v_1,v_2,\\ldots ,v_L,0,0,\\ldots ,0)$ and $v^{(R)} = (0,0,\\ldots ,0,v_{L+1},v_{L+2},\\ldots ,v_k)$ .", "$B^T(v^{(L)} + v^{(R)}) = B^T v = 0$ .", "By the disconnectedness of $[1,\\ldots ,L]$ and $[L+1,\\ldots ,k]$ , the set of columns of $B$ having non-zero entries in rows $[1,\\ldots ,L]$ is disjoint from the set of columns of $B$ having non-zero entries in rows $[L+1,\\ldots , k]$ .", "Thus $B^T v^{(L)}$ and $B^T v^{(R)}$ are non-zero in disjoint coordinates.", "Thus $B^T(v^{(L)} + v^{(R)}) = 0 \\rightarrow B^T v^{(L)} = B^T v^{(R)} = 0$ .", "This implies the null space of $B^T$ has rank at least 2, which contradicts that $B$ has rank $k-1$ .", "Thus $G$ is a connected graph.", "Because $G$ is a connected graph on $k$ vertices with at most $k-1$ edges, $B$ must encode a tree, and all its $k-1$ non-zero columns encode distinct edges.", "Thus $B \\in \\mathcal {T}_k$ .", "Call a column of a $\\lbrace 0,1\\rbrace $ -matrix an $n$ -column if it has exactly $n$ non-zero entries.", "Lemma 28 $\\mathcal {T}_k^+ = \\mathcal {S}_{2k - 1, k}$ We prove this by showing both inclusions in the following two claims.", "Claim 13 $\\mathcal {T}_k^+ \\subset \\mathcal {S}_{2k - 1, k}$ .", "Suppose $B \\in \\mathcal {T}_k^+$ .", "$B$ is the incidence matrix of a forest with two trees connected by a 3-hyperedge; without loss of generality we may assume the first column of $B$ encodes this hyper-edge.", "Further, by relabeling rows appropriately, we may assume $B_{11} = B_{21} = B_{k1} = 1$ , with vertices 1 and 2 in the same tree and connected by an even path, and vertex $k$ in the other tree.", "Then we may further relabel rows such that rows $1,2,\\ldots ,L$ correspond to one tree, and $L+1,\\ldots ,k$ correspond to the other, for some $L$ with $2 \\le L \\le k-1$ .", "Let $B^{^{\\prime \\prime }}$ be $B$ without its first column.", "The entries of $B^{^{\\prime \\prime }}$ , restricted to rows $1,\\ldots ,L$ , define a tree in $T_L$ .", "So by the second statement of Lemma REF , there is a unique (up to constant multiple) non-zero vector $v_\\ell \\in \\mathbb {R}^L$ satisfying $(B^{^{\\prime \\prime }}_{[L]})^T v_\\ell = 0$ .", "Similarly, there is a unique (up to constant multiple) non-zero vector $v_r \\in \\mathbb {R}^{k-L}$ satisfying $(B^{^{\\prime \\prime }}_{[k] \\setminus [L]})^T v_r = 0$ .", "Because the non-zero columns of rows $1,\\ldots ,L$ are disjoint from the non-zero columns of rows $L+1,\\ldots ,k$ , the null space of $(B^{^{\\prime \\prime }})^T$ is the direct sum of the null spaces of $(B^{^{\\prime \\prime }}_{[L]})^T$ and $(B^{^{\\prime \\prime }}_{[k] \\setminus [L]})^T$ .", "Concretely, let $v^{(\\ell )}, v^{(r)} \\in \\mathbb {R}^k$ be defined $v^{(\\ell )} = v_\\ell || 0^{k-L}$ and $v^{(r)} = 0^L || v_r$ , where $||$ means concatenation.", "Then the null space of $B^{^{\\prime \\prime }}$ are precisely the vectors $\\alpha v^{(\\ell )} + \\beta v^{(r)}$ for $\\alpha ,\\beta \\in \\mathbb {R}$ .", "Recall $B$ is $B^{^{\\prime \\prime }}$ , with the additional first column with three 1's in $B_{11}, B_{21}$ , and $B_{k1}$ .", "Thus the null space of $B$ is: $\\alpha v^{(\\ell )} + \\beta v^{(r)}: \\alpha , \\beta \\in \\mathbb {R}\\textnormal { subject to }\\alpha (v^{(\\ell )}_1 + v^{(\\ell )}_2) + \\beta v^{(r)}_k = 0$ From the second statement of Lemma REF , for $B_L \\in T_L$ , a non-zero vector $v$ satisfying $B_L^T v = 0$ has $v_a = (-1)^{d(a,b)} v_b$ relative to the encoded graph $G_L$ .", "We have $B^{\\prime \\prime }_{[L]} \\in T_L$ .", "Vertices 1 and 2 are connected by an even path in the relevant graph, which establishes $v^{(\\ell )}_1 = v^{(\\ell )}_2$ , so in particular $(v^{(\\ell )}_1 + v^{(\\ell )}_2) \\ne 0$ .", "Thus, the null space of $B$ is precisely: $t (v^{(r)}_k v^{(\\ell )} - (v^{(\\ell )}_1 + v^{(\\ell )}_2) v^{(r)}) , t \\in \\mathbb {R}$ Observe that $(v^{(r)}_k v^{(\\ell )} - (v^{(\\ell )}_1 + v^{(\\ell )}_2) v^{(r)}) \\in \\mathbb {R}^k$ has all non-zero entries because the coefficients $v^{(r)}_k$ and $(v^{(\\ell )}_1 + v^{(\\ell )}_2)$ are non-zero, and vectors $v^{(\\ell )}$ and $v^{(r)}$ have disjoint support.", "Thus $B \\in \\mathcal {S}_{2k - 1, k}$ .", "Claim 14 $\\mathcal {S}_{2k - 1, k} \\subset \\mathcal {T}_k^+$ .", "Suppose $B \\in \\mathcal {S}_{2k - 1, k}$ .", "Since $B$ is a minimal dependency, none of its columns have a single 1.", "Further, $B$ must have at least $k - 1$ non-zero columns since $B$ must have rank $k - 1$ .", "Hence $B$ has $k-2$ columns with two 1's and one column with three 1s.", "Without loss of generality we suppose $B$ 's first column is the one with 3 1's.", "Again define $B^{\\prime \\prime }$ as $B$ omitting its first column.", "Because $B$ 's non-zero columns are linearly independent, $B^{\\prime \\prime }$ has rank $k-2$ .", "We show that $B^{\\prime \\prime }$ encodes two disjoint trees by proving that $B^{\\prime \\prime }$ must encode a graph with no more than two connected components.", "Suppose for sake of contradiction that $B^{\\prime \\prime }$ encodes a graph $G^{\\prime \\prime }$ having connected components $X,Y,Z$ , where $X,Y,Z$ are disjoint non-empty subsets of $[1,\\ldots ,k]$ .", "Let $v \\in \\mathbb {R}^k$ be an all-non-zero vector such that $B^T v = 0$ .", "For a vertex set $S \\subset [1,\\ldots ,k]$ write $v_S$ for the vector with $(v_S)_i = v_i$ for $i \\in S$ , $(v_S)_i = 0$ otherwise.", "So, $v = v_X + v_Y + v_Z$ , and $v_X, v_Y, v_Z$ have non-zero entries in disjoint locations.", "We have $B^T (v_X + v_Y + v_Z) = 0 \\in \\mathbb {R}^M$ , where $m$ is the number of columns in $B$ , so we have $(B^{\\prime \\prime })^T (v_X + v_Y + v_Z) = 0 \\in \\mathbb {R}^{m-1}$ .", "But because $X,Y,Z$ are disjoint in $G^{\\prime \\prime }$ , the non-zero entries of $(B^{\\prime \\prime })^T v_X$ , $(B^{\\prime \\prime })^T v_Y)$ , $(B^{\\prime \\prime })^T v_Z$ must be pairwise disjoint.", "Thus, $(B^{\\prime \\prime })^T (v_X + v_Y + v_Z) = 0 \\rightarrow (B^{\\prime \\prime })^T v_X = (B^{\\prime \\prime })^T v_Y =(B^{\\prime \\prime })^T v_Z = 0$ .", "Because $v_X, v_Y, v_Z$ are linearly independent, the nullity of $(B^{\\prime \\prime })^T$ is at least 3, so $\\mathrm {rank}(B^{\\prime \\prime }) = \\mathrm {rank}((B^{\\prime \\prime })^T) \\le k-3$ .", "This contradicts our result that $\\mathrm {rank}(B^{\\prime \\prime }) = k-2$ .", "Thus, $G^{\\prime \\prime }$ has at most two connected components.", "Because $B^{\\prime \\prime }$ has $k-2$ non-zero columns, $G^{\\prime \\prime }$ has $k-2$ edges.", "Since $G^{\\prime \\prime }$ has $k$ vertices and no more than two connected components, it must be two disjoint trees.", "We return to the first column of $B$ .", "Without loss of generality let vertices $[1,\\ldots , L]$ correspond to one of the disjoint trees in $G^{\\prime \\prime }$ , and $[L+1,\\ldots ,k]$ the others, with $1 \\le L \\le k-1$ .", "We want to reason about $a,b,c$ , the three coordinates with $B_{a1} = B_{b1} = B_{c1} = 1$ .", "Applying Lemma REF again to the two disjoint trees gives that there is a vector $v^{(\\ell )} \\in \\mathbb {R}^k$ non-zero on exactly the coordinates $1,2,\\ldots ,L$ with $(B^{\\prime \\prime })^T v^{(\\ell )} = 0$ , and a vector $v^{(r)} \\in \\mathbb {R}^k$ non-zero on exactly the coordinates $L+1,\\ldots ,k$ with $(B^{\\prime \\prime })^T v^{(\\ell )} = 0$ .", "We can rule out that all of $a,b,c \\in [1,\\ldots , L]$ : Suppose for sake of contradiction this was the case, then observe that $u = v^{(r)}$ would satisfy $(B^T u)_1 = 1 v^{(r)}_a + 1 v^{(r)}_b + 1 v^{(r)}_c = 0 + 0 + 0 = 0$ , and $(B^{\\prime \\prime })^T v^{(r)} = 0$ , so $B^T u = 0$ : $u$ is non-zero but not all of its entries are non-zero, which contradicts $B \\in \\mathcal {S}_{2k - 2, k}$ .", "Symmetrically, we cannot have all of $a,b,c \\in [L+1,\\ldots , k]$ .", "Then without loss of generality we have $a,b \\in [1,\\ldots ,L], c \\in [L+1,\\ldots ,k]$ .", "Write $G^{\\prime \\prime }_1$ for the subgraph of $G^{\\prime \\prime }$ induced by vertices $[1,\\ldots ,L]$ : this is a tree containing $a$ and $b$ .", "Suppose for sake of contradiction that $a$ and $b$ are connected by an odd length path.", "Then $v^{(\\ell )}_a = -v^{(\\ell )}_b$ .", "Then let $u = 1 v^{(\\ell )}$ : $(B^{\\prime \\prime })^T u = 0$ as before, and $(B^T u)_1 = 1 v^{(\\ell )}_a + 1 v^{(\\ell )}_b + 1 v^{(\\ell )}_c = - v^{(\\ell )}_b + v^{(\\ell )}_b + 0 = 0$ .", "Again we have non-zero $u$ with not all entries 0 satisfying $B^T u = 0$ – a contradiction.", "Thus we must have that $a,b$ are connected by a path of even length in $G^{\\prime \\prime }_1$ .", "It follows that $B$ is the vertex-hyperedge incidence matrix of a forest with two trees connected by a 3-hyperedge where the two vertices of the 3-hyperedge in the same tree are connected by an even length path.", "Thus $B \\in \\mathcal {T}_k^+$ .", "Lemma 29 $\\mathcal {S}_{2k, k}^{\\prime } = \\mathcal {T}_k^C$ .", "We prove this by showing both inclusions in the following two claims.", "Claim 15 $\\mathcal {T}_k^C \\subset \\mathcal {S}_{2k, k}^{\\prime }$ Suppose $B \\in \\mathcal {T}_k^C$ .", "Then $B$ has $k$ columns with two 1's each.", "Without loss of generality we may let column 1 of $B$ be an edge in the cycle; then let $B^{\\prime }$ be $B$ with the first column set to zero.", "Then $B^{\\prime } \\in \\mathcal {T}_k$ , so $B^{\\prime } \\in \\mathcal {S}_{2k - 2, k}$ by Lemma REF .", "Hence there is a unique (up to constant multiple) non-zero $v$ with $(B^{\\prime })^T v = 0$ , and $v$ has all non-zero entries.", "Write the first column of $B$ as $e_x + e_y$ , for $1 \\le x < y \\le k$ .", "We claim that $e_x + e_y$ is in the span of $B^{\\prime }$ 's columns.", "By definition, the edge $(x,y)$ is part of an even length cycle in the graph encoded by $B$ .", "We encode this cycle.", "For odd $n \\ge 1$ , there are ordered pairs $(a_i, b_i), 1 \\le i \\le n$ such that each sum of basis vectors $e_{a_i} + e_{b_i}$ is a column of $B^{\\prime }$ , $b_i = a_{i+1}$ for $1 \\le n-1$ , and $a_1 = x$ and $b_n = y$ .", "Then observe that $(e_{a_1} + e_{b_1}) - (e_{a_2} + e_{b_2}) + \\ldots + (-1)^{n+1} (e_{a_n} + e_{b_n}) = e_{a_1} + e_{b_n} = e_x + e_y.$ Thus column 1 of $B$ is in the span of the rest of the columns of $B$ .", "Thus $B^T v = 0$ as well, and because $B^{\\prime }$ has rank $k-1$ , $B$ must have rank $k-1$ as well.", "So $B \\in \\mathcal {S}_{2k, k}$ .", "Moreover, $B \\in \\mathcal {S}_{2k, k}^{\\prime }$ because it has $k$ non-zero columns each with two 1's.", "Claim 16 $\\mathcal {S}_{2k, k}^{\\prime } \\subset \\mathcal {T}_k^C$ .", "Let $B \\in \\mathcal {S}_{2k, k}^{\\prime }$ .", "Because $B$ has rank $k-1$ , and $k$ non-zero columns, there is a subset of $k-1$ columns of $B$ with rank $k-1$ .", "Then without loss of generality assume the first column of $B$ is in the span of the other columns, and let $B^{\\prime }$ be $B$ , but with the first column set to 0.", "There is a unique (up to constant multiple) non-zero vector $v$ such that $B^T v = 0$ , and $v$ has all non-zero entries.", "Then by construction $B^{\\prime T} v = 0$ .", "Further because $B^{\\prime }$ has rank $k-1$ , so does $B^{\\prime T}$ , and the rank of the nullspace of $B^{\\prime T}$ is 1.", "Thus $v$ must be the unique (up to constant multiple) non-zero vector satisfying $B^{\\prime T} v = 0$ .", "So $B^{\\prime } \\in \\mathcal {S}_{2k - 2, k}$ .", "Thus $B^{\\prime } \\in \\mathcal {T}_{k}$ .", "Write the first column of $B$ as $e_x + e_y$ , $1 \\le x < y \\le k$ .", "To show $B \\in \\mathcal {T}_k^C$ , we must show that $x$ and $y$ are connected by an odd length path in the tree encoded by $B^{\\prime }$ .", "We know that $e_x + e_y$ is in the span of the columns of $B^{\\prime }$ , so we can write $e_x + e_y = \\sum _{i=2}^{m} \\alpha _i (e_{a_i} + e_{b_i}),$ where $m$ is the width of $B^{\\prime }$ , and for each $i$ where $\\alpha \\ne 0$ and the column $B_i$ is non-zero, we have $B_i = e_{a_i} + e_{b_i}$ .", "Let $S = \\lbrace i : \\alpha _i \\ne 0 \\wedge B_i \\ne 0\\rbrace $ .", "If some $1 \\le j \\le k$ appears in only one pair in $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , it must be the case that $\\sum _{i=1}^{m} \\alpha _i (e_{a_i} + e_{b_i})$ has non-zero $e_j$ coefficient.", "So $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , is a collection of edges of a tree such that at most two vertices ($x$ and $y$ ) belong to only one edge.", "Then $\\lbrace (a_i,b_i)\\rbrace _{i \\in S}$ , must be a path, with $x$ and $y$ at the end points.", "Without loss of generality let $a_2 = x, b_{|S| + 1} = y$ and $a_i = b_{i+1}$ for $2 \\le i \\le |S|$ .", "For $2 \\le i \\le |S|$ , because $e_{b_i} \\perp e_x + e_y$ , we must have that $\\alpha _{i+1} = -\\alpha _i$ so that the sum will cancel in coordinate $b_i$ .", "It follows that $e_x + e_y = \\sum _{i=2}^{|S| + 1} (-1)^i \\alpha _2 (e_{a_i} + e_{b_i}) = \\alpha _2 e_x + (-1)^{|S| + 1} \\alpha _2 e_y.$ It is clear we must choose $\\alpha _2 = 1$ and $|S|$ must be odd.", "Thus $e_x + e_y$ fulfills the conditions of the additional edge forming an even cycle in $\\mathcal {T}_k^C$ .", "We thus have $B \\in \\mathcal {T}_k^C$ .", "We now prove Lemma REF , which we restate here.", "Lemma 30 Suppose we have two sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ where $A_S \\in \\mathcal {M}_{|S|}$ and $A_T \\in \\mathcal {M}_{|T|}$ .", "Let $\\ell $ be the number of non-zero entries in $A_{S \\cup T}$ .", "Then there are at least $\\max \\left(|S \\cup T| - 1, \\frac{\\ell }{2}\\right)$ non-zero entries in $A_{S \\cup T}$ that are not the first (top) non-zero entry in their column.", "Let $R := S \\cup T$ .", "By Observation REF , since $A_S$ and $A_T$ are minimal dependencies, there cannot be a column in $A_S$ or $A_T$ with a single one.", "Hence no column of $A_R$ has a single one.", "It follows immediately that there are at least $\\frac{\\ell }{2}$ non-zero entries that are not the first non-zero entry in their column in $A_R$ .", "Next we show that there are at least $|R| - 1$ non-zero entries in $A_R$ .", "Let $G = (R, E)$ be the hypergraph given by the vertex-hyperedge incidence matrix $A_R$ .", "We claim that $G$ must be connected.", "Indeed, each of the hypergraphs given by the incidence matrix $A_S$ and $A_T$ are connected on their own.", "Since $S \\cap T \\ne \\emptyset $ , the hypergraph given by $A_R$ must be connected.", "Let $F$ be an arbitrary spanning tree of $G$ whose edges are contained in the hyperedges of $G$ .", "Let $v \\in R$ be an arbitrary node in $G$ , which we call the root.", "For each $u \\in R \\setminus \\lbrace v\\rbrace $ , consider the entry $(u, e(u))$ of $A$ , where $e(u)$ is the index of the hyperedge containing the edge from $u$ towards $v$ in $F$ .", "Note that $A_{u, e(u)} = 1$ , and further, each hyperedge $e$ in $G$ contains at least one node $x(e)$ (which can be chosen arbitrarily) for which $e(x(e)) \\ne e$ .", "Consider $x(e)$ to be the “first\" non-zero entry in the column $e$ .", "It follows that the entries $\\lbrace (u, e(u))\\rbrace _{u \\in R \\setminus \\lbrace x\\rbrace }$ are not the “first\" in their column.", "We show this argument pictorially in Figure REF , where we draw an arrow from each $u$ to $e(u)$ , and we circle the entry $A_{u, e(u)}$ .", "Figure: Illustration of the proof that there are at least |R|-1|R| - 1 non-zero entries that are not first in their column.", "Each hyperedge contains at least one vertex that is not pointed at, otherwise, the arrows would create a cycle." ], [ "Proof of Lemmas ", "We prove Lemmas REF and REF .", "Lemma REF is a corollary of Lemma REF .", "* * We prove this by linearly combining the dependencies.", "Without loss of generality, let $S = [k]$ for some k. For all $i \\in [k]$ , since $A_i \\in H_i$ , there exists some $x^{(i)}$ such that $x^{(i)}_i \\ne 0$ and $Ax^{(i)} = 0$ .", "Observe that for each $i \\in [k]$ , we have $\\mathrm {supp}(x^{(i)}) \\subset S$ - otherwise it would imply that some column $A_j$ for $j \\notin S$ is spanned by $H_j$ .", "Choose random coefficients $c_i$ for $i \\in [k]$ from any continuous distribution and let $y = \\sum _i {c_i x^{(i)}}$ .", "Then with probability 1, $y$ is non-zero on all $i \\in S$ .", "* For the first part, let $v$ be such that $A^Tv = e_i$ .", "Then for any $w$ with $i \\in \\mathrm {supp}(w)$ , we have $w^TA^Tv = w_i \\ne 0$ , so it is impossible that $Aw = 0$ .", "For the converse, suppose $A_i \\notin H_i$ .", "Then there must exist some $w_i $ such that $\\langle {{w_i }, A_i}\\rangle \\ne 0$ , but ${w_i } \\perp A_j$ for all $j \\ne i$ .", "However, this implies that $A{w_i } = \\langle {{w_i }, A_i}\\rangle e_i$ , so $e_i \\in \\mathrm {Span}(A^T)$ , which is a contradiction." ], [ "Proofs of Lemmas ", "* $\\begin{split}\\Pr \\left[\\mathrm {Bin}(n, p) \\ge t\\right] &\\le \\sum _{i = t}^{n}\\binom{n}{i}p^i \\\\&\\le \\sum _{i = t}^{n}\\binom{n}{t}p^i\\prod _{j = t + 1}^i\\left(\\frac{n - j + 1}{j}\\right) \\\\&\\le \\sum _{i = t}^{n}\\binom{n}{t}p^i\\prod _{j = t + 1}^i\\left(\\frac{n}{t}\\right) \\\\&= \\sum _{i = t}^{n}\\binom{n}{t}p^t\\left(\\frac{pn}{t}\\right)^{i - t} \\\\&\\le \\binom{n}{t}p^t\\sum _{j = 0}^{\\infty }\\left(\\frac{pn}{t}\\right)^{j} \\\\&= \\frac{\\binom{n}{t}p^t}{1 - \\frac{pn}{t}}.\\end{split}$ For $t \\ge 2np$ , plugging in Sterling's formula yileds the lemma.", "* We break down this sum as follows.", "$\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le \\sum _{\\ell = j}^{2j}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ]\\\\&\\qquad + \\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\qquad + \\sum _{\\ell \\ge n/3}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ]\\end{split}$ We bound the first term in the following claim.", "Claim 17 $\\sum _{\\ell = j}^{2j}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk}.$ $\\begin{split}\\sum _{\\ell = j}^{2j}&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 2j\\Pr \\left[\\mathrm {Bin}\\left(2j, \\frac{2j + k}{\\gamma n}\\right) \\ge j\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\& \\le 4j\\left(\\frac{e2j(2j + k)}{\\gamma nj}\\right)^j\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\&\\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j].\\end{split}$ Here the first inequality follows from the fact that the summand is highest for $\\ell = 2j$ , the second inequality follows from the tail bound in Lemma  in Section , and then third inequality follows from the fact that $j \\le k + 1 \\le 2k$ .", "Now $\\begin{split}\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] &\\le \\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\le 2j] \\\\&\\le \\binom{\\gamma nk}{2j}\\left(\\frac{d}{n}\\right)^{2j}\\left(1 - \\frac{d}{n}\\right)^{\\gamma nk - 2j} \\\\&\\le \\left(\\frac{e\\gamma nk}{2j}\\right)^{2j}\\left(\\frac{d}{n - d}\\right)^{2j}\\left(1 - \\frac{d}{n}\\right)^{\\gamma nk} \\\\&\\le \\left(\\frac{e\\gamma dk}{j}\\right)^{2j}e^{-\\gamma dk}\\\\&\\le \\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk}\\end{split}$ Combining this with Equation REF yields the claim.", "We bound the second term in Equation REF in the following claim.", "Claim 18 $\\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}.$ .", "For $\\ell \\le n/3$ , using Lemma  and Sterling's forumla, we have $\\begin{split}&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 2\\left(\\frac{e\\ell (\\ell + k)}{\\gamma n\\frac{\\ell }{2}}\\right)^{\\lceil {\\ell /2}\\rceil }\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&= 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&= 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\binom{\\gamma nk}{\\ell }\\left(\\frac{d}{n}\\right)^\\ell \\left(1 - \\frac{d}{n}\\right)^{\\gamma nk - \\ell } \\\\&\\le 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{e\\gamma nk}{\\ell }\\right)^{\\ell }\\left(\\frac{d}{n - d}\\right)^\\ell \\left(1 - \\frac{d}{n}\\right)^{\\gamma nk} \\\\&\\le 2\\left(\\frac{2e(\\ell + k)}{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{e\\gamma nk}{\\ell }\\right)^{\\ell }\\left(\\frac{d}{n - d}\\right)^\\ell e^{-\\gamma dk} \\\\&\\le 2\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } e^{-\\gamma dk}.\\end{split}$ We do casework on the parity of $\\ell $ .", "Let $\\ell = 2a + b$ , where $b \\in \\lbrace 0, 1\\rbrace $ .", "If $b = 1$ , then $\\begin{split}\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } &= \\left(\\frac{4e(2a+1)}{\\gamma n}\\right)^{a + 1}\\left(\\frac{2e\\gamma dk}{2a + 1}\\right)^{2a + 1} \\\\&= \\left(\\frac{16e^3\\gamma ^2d^2k^2(2a+1)}{\\gamma n(2a + 1)^2}\\right)^a\\left(\\frac{8e^2\\gamma dk(2a+1)}{\\gamma n(2a + 1)}\\right) \\\\&\\le \\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\left(\\frac{8e^2dk}{n}\\right)\\end{split}$ Now since the maximum over $x$ of $f(x) = \\left(\\frac{y}{x}\\right)^x$ is achieved at $x = y/e$ , and above this value of $x$ , the $f(x)$ is decreasing, since $j \\ge k - 1 \\ge \\frac{e8e^3\\gamma d^2k^2}{n}$ , we have for all $a \\ge j$ , $\\begin{split}\\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\left(\\frac{8e^2dk}{n}\\right) &\\le \\left(\\frac{8e^3\\gamma d^2k^2}{nj}\\right)^a\\left(\\frac{8e^2dk}{n}\\right)\\\\&\\le \\left(\\frac{8e^3\\gamma d^2k^2}{n(k - 1)}\\right)^a\\left(\\frac{8e^2dk}{n}\\right) \\\\&\\le \\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{a + 1}.\\end{split}$ If $b = 0$ , then $\\begin{split}\\left(\\frac{4e\\ell }{\\gamma n}\\right)^{\\lceil {\\ell /2}\\rceil }\\left(\\frac{2e\\gamma dk}{\\ell }\\right)^{\\ell } &= \\left(\\frac{8ea}{\\gamma n}\\right)^{a}\\left(\\frac{2e\\gamma dk}{2a}\\right)^{2a} \\\\&= \\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a\\end{split}$ By the same reasoning as before, we have for all $a \\ge j + 1$ , $\\left(\\frac{8e^3\\gamma d^2k^2}{na}\\right)^a \\le \\left(\\frac{8e^3\\gamma d^2k^2}{n(j + 1)}\\right)^{a} \\le \\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{a}.$ Combining these two cases back into Equation REF , we have for all $\\ell \\ge 2j + 1$ , $\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 2e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{\\lceil {\\frac{\\ell }{2}}\\rceil }.$ Summing over all $\\ell \\ge 2j + 1$ , we have $\\sum _{\\ell \\ge 2j + 1}^ {n/3}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}.$ .", "Finally, we bound the third term in Equation REF in the following claim.", "Claim 19 $\\sum _{\\ell \\ge n/3}^{\\infty }\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\frac{\\ell }{2}\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\le $ It suffices to bound the probability $\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\ge n/3].$ Again employing Lemma , we have $\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) \\ge n/3] \\le 2\\left(\\frac{3e\\gamma dk}{n}\\right)^{n/3}.$ Combining claims REF , REF and REF , we have $\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 8k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk} \\\\&\\qquad + 4e^{-\\gamma dk}\\left(\\frac{8e^3\\gamma d^2k}{n}\\right)^{j + 1}\\\\&\\qquad + 2\\left(\\frac{3e\\gamma dk}{n}\\right)^{n/3}.\\end{split}$ It is easy to check that this sum is dominated by the first term, and hence for some universal constant $c_{\\ref {masterlemma:small}}$ , $\\begin{split}\\sum _{\\ell \\ge 1}^{\\infty }&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{\\ell + k}{\\gamma n}\\right) \\ge \\max \\left(j, \\frac{\\ell }{2}\\right)\\right]\\Pr [\\mathrm {Bin}(\\gamma nk, d/n) = \\ell ] \\\\&\\le 16k\\left(\\frac{20ek}{\\gamma n}\\right)^j\\left(2e\\gamma d\\right)^{4k}e^{-\\gamma dk} \\\\&\\le \\left(\\frac{k}{n}\\right)^je^{-\\gamma dk + c_{\\ref {masterlemma:small}}k\\log (\\gamma d)}.\\end{split}$ This proves the lemma.", "* Because $n/2 < n-k-1$ , we may bound the probability by: $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\frac{n}{2}, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right].\\end{split}$ We use the Chernoff bound $\\Pr [X \\le (1 - \\delta )\\mu ] \\le e^{-\\mu \\delta ^2/2}$ for $\\mu = \\mathbb {E}[X]$ , plugging in $\\mu = \\frac{n}{2}\\left(1 - \\frac{1}{\\sqrt{kd/n}}\\right)$ and $\\delta = (1 - \\frac{1}{\\sqrt{kd/n}})^{-1} \\left(1 - \\frac{1}{\\sqrt{kd/n}} - \\frac{2k}{n}\\right)$ .", "This gives $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(n - k - 1, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right] &\\le \\Pr \\left[\\mathrm {Bin}\\left(\\frac{n}{2}, 1 - \\frac{1}{\\sqrt{kd/n}}\\right) < k\\right]\\\\&\\le e^{-\\frac{n}{2}\\left(1 - \\frac{1}{\\sqrt{kd/n}}\\right)\\left(1 - \\frac{1}{\\sqrt{kd/n}} - \\frac{2k}{n}\\right)^2}\\\\&\\le e^{-n\\epsilon },\\end{split}$ where $\\epsilon \\ge 1/36$ .", "To achieve this value of $\\epsilon $ , we plugged in $k < \\frac{n}{12}$ in the last inequality.", "Now we compute the sum over $k$ : $\\sum _{k = \\frac{2n}{d}}^{\\frac{n}{C}}\\binom{n}{k}e^{-\\epsilon n} \\le n \\binom{n}{n/C} e^{-\\epsilon n} \\le n (eC)^{n/C} e^{-\\epsilon n} = e^{n(\\frac{\\log n}{n} + \\frac{1 + \\log C}{C} - \\epsilon )}$ which for constant $C$ large enough, is $e^{-\\Theta (n)}$ ." ], [ "Proof of Claims ", "In the following Appendix, we will prove the claims stated in the proof of Lemma $\\ref {lemma:abc_small}$ .", "Before proving these claims, we will need the following additional lemma: The following bound gives a general approximation for the probability mass function of a HyperGeometric Distribution.", "It also provides a second bound under the additional assumption that the number of draws in our distribution is not too large relative to the population size.", "Lemma 31 Let $\\mathcal {X}\\sim \\text{HyperGeom}(A,B,n)$ .", "Furthermore, let us define $(1-q)=\\frac{B}{A}$ .", "Then $\\Pr [\\mathcal {X}=k]\\le \\binom{n}{k}(1-q)^k\\left(q+\\frac{k}{A-n}\\right)^{n-k}.$ Furthermore, assuming $n\\le \\frac{3}{2}qA$ and $q\\le 1/2$ , we have: $\\begin{split}\\Pr [\\mathcal {X}=k]&\\le \\binom{n}{k}(1-q)^k q^{n-k}\\left(e^{\\left({{6ek}}\\right)}\\right)\\\\&\\le \\left(\\frac{en}{k}\\right)^k(1-q)^k q^{n-k}\\left(e^{\\left({{6ek}}\\right)}\\right) \\\\ &= \\left(\\right)^k\\end{split}$ Recall that, by definition of the hypergeometric distribution, we have: $\\Pr [\\mathcal {X}=k]=\\frac{\\binom{B}{k}\\binom{A - B}{n-k}}{\\binom{A}{n}}$ We can expand out the binomial terms into their factorial representations to see: $\\Pr [\\mathcal {X}=k]&=\\frac{\\frac{B!}{(B-k)!k!", "}\\cdot \\frac{(A - B)!", "}{(A - B-n+k)!(n-k)!}}{\\frac{A!}{(A-n)!n!}}\\\\&=\\binom{n}{k}\\left(\\frac{B!}{(B-k)!", "}\\cdot \\frac{(A-B)!}{(A-B-n+k)!", "}\\cdot \\frac{(A-n)!}{A!", "}\\right)\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}(B-k+i)\\prod _{i=1}^{n-k}(A-B-n+k+i)\\prod _{i=1}^n \\frac{1}{A-n+i}\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\frac{A-B-n+k+i}{A-n+i}\\\\&=\\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\left(1-\\frac{B}{A-n+i}+\\frac{k}{A-n+i}\\right) \\\\&\\le \\binom{n}{k} \\prod _{i=1}^{k}\\frac{B-k+i}{A-k+i}\\prod _{i=1}^{n-k}\\left(1-\\frac{B}{A}+\\frac{k}{A-n}\\right)$ By definition of the hypergeometric distribution, $k\\le B$ .", "Thus, it follows $k-i<B<A$ for all $i\\in [k]$ .", "This implies: $\\frac{B-k+i}{A-k+i}\\le \\frac{B}{A}=(1-q)$ This gives us the first inequality of the lemma: $P[\\mathcal {X}=k]\\le \\binom{n}{k}\\prod _{i=1}^k (1-q) \\prod _{i=1}^{n-k} \\left(q+\\frac{k}{A-n}\\right)=\\binom{n}{k} (1-q)^k \\left(q+\\frac{k}{A-n}\\right)^{n-k}.$ We will now proceed under the assumption that $n\\le \\frac{3}{2}qA$ to achieve the second bound.", "We write: $\\left(q+\\frac{k}{A-n}\\right)^{n-k}&\\le \\left(q+\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{n-k}\\\\&=\\sum _{i=0}^{n-k} \\binom{n-k}{i}q^{n-k-i}\\left(\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k} \\left(\\frac{\\frac{3}{2}eqA}{i}\\right)^i\\left(\\frac{1}{q}\\right)^i\\left(\\frac{k}{(1-\\frac{3}{2}q)A}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k}\\left(\\frac{\\frac{3}{2}ek}{i(1-\\frac{3}{2}q)}\\right)^{i}\\\\&\\le q^{n-k}\\sum _{i=0}^{n-k} \\left(\\frac{1}{i!", "}\\right)\\left(\\frac{\\frac{3}{2}ek}{1-\\frac{3}{2}q}\\right)^{i}\\\\&\\le q^{n-k}\\left(e^{\\left(\\frac{\\frac{3}{2}ek}{1-\\frac{3}{2}q}\\right)}\\right)\\\\&\\le q^{n-k}\\left(e^{\\left({6ek}\\right)}\\right)$ Applying this to the first inequality gives: $P[\\mathcal {X}=k]\\le \\binom{n}{k}(1-q)^kq^{n-k}\\left(e^{\\left(6ek\\right)}\\right)$ as claimed.", "We are now ready to prove claims REF and REF .", "* By expanding the binomial distribution, we have, $\\begin{split}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1-p)N}\\right)=j\\right] &\\le \\sum _{i=0}^j\\binom{\\ell }{i}\\left(\\frac{\\ell }{\\gamma (1 - p)N}\\right)^{i}\\left(1-\\frac{\\ell }{\\gamma (1-p)N}\\right)^{\\ell -i}\\\\&\\le 2^{\\ell } \\left(\\frac{\\ell }{\\gamma (1-p)N}\\right)^{\\ell -j}.\\end{split}$ Using the fact that $K\\le \\frac{3}{2}N$ , we employ the second bound of Lemma REF to find: $&\\Pr \\left[\\mathrm {Bin}\\left(\\ell , 1-\\frac{\\ell }{\\gamma (1 - p)N}\\right)=j\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]\\\\&\\le \\left(\\binom{\\gamma dK}{\\ell }(1-p)^{\\ell }p^{\\gamma dK-\\ell }e^{6e\\ell }\\right) 2^\\ell \\left(\\frac{\\ell }{c(1-p)N}\\right)^{\\ell -j}\\\\&\\le \\binom{\\gamma dK}{2K}p^{\\gamma dK-2K}e^{12eK} 2^{2K} \\left(\\frac{2K}{\\gamma N}\\right)^{\\ell -j}\\\\&\\le \\left(\\frac{9e^{12K+2}c^2d^2}{4}\\right)^{K}p^{\\gamma dK-2K} \\left(\\frac{K}{N}\\right)^{\\ell -j}\\\\&\\le \\left( p^{\\gamma d+c\\log (\\gamma d)} \\right)^K\\left(\\frac{K}{N}\\right)^{\\ell -j}$ for some universal constant $c$ .", "Next we show the second statement in the claim.", "Since for sufficiently large $\\gamma d$ , the mean of HyperGeom$(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)$ is greater than $4K$ , the term $\\Pr [\\mathrm {Bin}(\\ell , \\frac{K}{N})\\ge K-1]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]$ is maximized (over $\\ell \\in [1, 4K]$ ) at $\\ell = 4K$ .", "By Equation REF we have we have $\\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{K}{N}\\right)\\ge K-1\\right]&\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ] \\\\&\\le 4K\\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=4K]2^{4K} \\left(\\frac{K}{\\gamma (1 - p)N}\\right)^{K-1}\\\\&\\le 4K\\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=4K]2^{5K} \\left(\\frac{K}{N}\\right)^{K-1}$ Using the fact that $K\\le \\frac{3}{2}N$ , we employ the second bound of Lemma REF to find: $& \\sum _{\\ell =1}^{4K}\\Pr \\left[\\mathrm {Bin}\\left(\\ell , \\frac{4K}{\\gamma (1-p)N}\\right)\\ge K-1\\right]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dN,\\gamma d(1-p)N,\\gamma dK)=\\ell ]\\\\&\\le 4K\\left(\\binom{\\gamma dK}{4K}p^{\\gamma dK-4K}e^{24eK}\\right)2^{5K}\\left(\\frac{4K}{N}\\right)^{K-1}\\\\&\\le 4K\\left(\\frac{e\\gamma d}{4}\\right)^{4K}p^{\\gamma dK-4K}e^{24eK}2^{5K}\\left(\\frac{K}{N}\\right)^{K-1}\\\\&\\le \\left(p^{\\gamma d-c^{\\prime }\\log (\\gamma d)}\\right)^K \\left(\\frac{K}{N}\\right)^{K-1}.$ for some constant $c^{\\prime }$ .", "The claim follows from choosing $c_{\\ref {claim:abc1}}$ larger than $c$ and $c^{\\prime }$ .", "We now prove Claim REF .", "* We write: $&\\sum _{\\ell =2k+1}^{\\gamma d k}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk} \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\sum _{i=0}^{\\left\\lfloor \\frac{\\ell }{2}\\right\\rfloor }\\binom{\\ell }{i}\\left(\\frac{\\ell }{\\gamma n}\\right)^{\\ell -i}\\left(1-\\frac{\\ell }{\\gamma (1-p)n}\\right)^\\ell \\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]3^\\ell \\left(\\frac{\\ell }{\\gamma (1-p)n}\\right)^{\\ell /2}$ Applying the first bound of Lemma REF , we find: $&\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\binom{\\gamma dk}{\\ell }(1-p)^k\\left(p+\\frac{\\ell }{\\gamma d(n-k)}\\right)^{\\gamma dk-\\ell }\\right)3^\\ell \\left(\\frac{\\ell }{\\gamma (1-p)n}\\right)^{\\ell /2}\\\\&\\le \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\frac{1}{2}+\\frac{k}{(n-k)}\\right)^{\\gamma dk}\\left(\\frac{e\\gamma dk}{\\ell }\\right)^\\ell 6^\\ell \\left(\\frac{\\ell }{\\gamma n}\\right)^{\\ell /2}\\\\&\\le (0.625)^{-\\gamma dk} \\sum _{\\ell =2k+1}^{\\gamma dk}\\left(\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{\\ell n}}\\right)^{\\ell }\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}\\max _{\\ell \\in \\lbrace 2k+1,...,\\gamma dk\\rbrace }\\left(\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{\\ell n}}\\right)^{\\ell }.$ To show the function in the expression above is maximized at $2k+1$ when $k\\le \\frac{n}{18e\\gamma d^2}$ , we consider the derivative.", "Taking $C=\\frac{6edk\\sqrt{\\gamma }}{\\sqrt{n}}$ , we write: $\\frac{d}{d\\ell }\\left[ \\left(\\frac{C}{\\sqrt{\\ell }}\\right)^\\ell \\right] &=\\frac{d}{d\\ell }\\left[ e^{\\ell \\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\right]\\\\&=e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\frac{d}{d\\ell }\\left[\\ell \\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right) \\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell \\frac{d}{d\\ell }\\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)\\right]\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell \\frac{1}{\\left(\\frac{C}{\\sqrt{\\ell }}\\right)}\\frac{d}{d\\ell }\\left[\\left(\\frac{C}{\\sqrt{\\ell }}\\right)\\right]\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)+ \\ell ^{1.5}\\frac{d}{d\\ell }\\ell ^{-0.5}\\right]\\\\&= e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C}{\\sqrt{\\ell }}\\right)-\\frac{1}{2}\\right]\\\\&= \\frac{1}{2}e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C^2}{\\ell }\\right)-1\\right]\\\\&= \\frac{1}{2}e^{\\ell \\left(\\frac{C}{\\sqrt{\\ell }}\\right)} \\left[\\ln \\left(\\frac{C^2}{e\\ell }\\right)\\right].$ Thus, we see that the derivative is zero when $\\ell =\\frac{C^2}{e}=\\frac{36e\\gamma d^2k^2}{n}$ and strictly negative afterwards.", "For $k\\le \\frac{n}{18e\\gamma d^2}$ , this maximum occurs before $\\ell =2k+1$ .", "As the function is monotonically decreasing after the maximum, we conclude that $\\ell =2k+1$ is where the function is maximized on the interval $[2k+1,\\gamma dk]$ for $k\\le \\frac{n}{18e\\gamma d^2}$ .", "We continue: $&\\sum _{\\ell =2k+1}^{\\gamma dk}\\Pr [\\mathrm {Bin}(\\ell , 1-\\frac{\\ell }{\\gamma n})\\le \\lfloor \\frac{\\ell }{2}\\rfloor ]\\cdot \\Pr [\\textnormal {HyperGeom}(\\gamma dn,\\gamma d(1-p)n,\\gamma dk)=\\ell ]\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}\\left(\\frac{36\\gamma e^2d^2k^2}{(2k+1)n}\\right)^{k+1/2}\\\\&\\le (\\gamma dk) (0.625)^{-\\gamma dk}(18\\gamma e^2d^2)^{k+1/2}\\left(\\frac{k}{n}\\right)^{k+1/2}\\\\&\\le e^{-k}\\left(\\frac{k}{n}\\right)^{k+1/2}$ for sufficiently large $\\gamma d$ ." ], [ "Lower bounds on distance to $\\mathbb {1}$ via number of all-zero rows", "In this appendix, we prove lower bounds on the number of zero rows in a random matrix $A$ .", "The following observation shows that this is sufficient to give a lower bound on the distance between $\\mathbb {1}$ and the span of $A$ .", "Observation 3 For any matrix $A$ , if there is a subset of zero rows $S$ , then for any $w$ , $|A_S w - \\mathbb {1}_S| = |S|$ thus $|A w - \\mathbb {1}| \\ge |S|$ .", "Lemma 32 Let $A \\sim \\text{BGC}(n, \\gamma , d)$ , then with probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1-o(1))e^{-\\gamma d} n$ .", "let $I_i$ be the indicator variable that row $i$ is all zero.", "$\\Pr [I_i = 1] = (1 - \\frac{d}{n})^{\\gamma n} = (1-o(1)) e^{-\\gamma d}$ .", "By a Chernoff bound, $\\Pr \\left[\\sum _i I_i \\le (1-n^{-1/3})(1-o(1))e^{-\\gamma d}n\\right] \\le e^{-(1-o(1))e^{-\\gamma d}n (n^{-2/3})/2} = o(1)$ Thus with probability $1-o(1)$ , $A$ has at least $(1-n^{-1/3})(1-o(1))e^{-\\gamma d} n = (1-o(1)) e^{-\\gamma d} n$ zero rows.", "The following lemma generically shows when we can apply the second moment method, which we will use to bound the number of all-zero rows for symmetric Bernoulli matrices and for the ABC.", "Lemma 33 Let $S_N = \\lbrace I^{(N)}_1,\\ldots , I^{(N)}_N\\rbrace $ be a family of sets of $\\lbrace 0,1\\rbrace $ -random variables such $\\mathbb {E}[I^{(N)}_i] = \\Theta (1)$ for all $i$ , and $\\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(1/N)$ for all $i \\ne j$ .", "Let $X^{(N)} = \\sum _{i=1}^{N} I^{(N)}_i$ .", "Then with probability $1-o(1)$ , $X^{(N)} \\ge (1-o(1)) \\mathbb {E}[X^{(N)}],$ where all big-O notation is in terms of $N \\rightarrow \\infty $ .", "$\\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(1/N)$ for $i \\ne j$ , and $\\text{Var}(I^{(N)}_i) = O(1)$ because $I^{(N)}_{i}$ is bounded, so $\\text{Var}(X^{(N)}) = \\sum _{i=1}^{n} \\text{Var}(I^{(N)}_i) + \\sum _{1 \\le i \\ne j \\le N} \\text{Cov}(I^{(N)}_i, I^{(N)}_j) = O(N)$ .", "By Chebyshev's inequality, $\\Pr \\left[|X^{(N)} - \\mathbb {E}[X^{(N)}]| > N^{1/3} \\sqrt{\\text{Var}(X^{(N)})}\\right] < N^{-1/3}$ Because $\\text{Var}(X^{(N)}) = O(N)$ , $N^{1/3} \\sqrt{\\text{Var}(X^{(N)})} = o(N)$ .", "Thus, we have with probability $1-o(1)$ that $X^{(N)} \\ge \\mathbb {E}[X^{(N)}] - o(N)$ .", "Because $\\mathbb {E}[X^{(N)}] = \\sum _{i=1}^{N} \\mathbb {E}[I^{(N)}_i] = \\Theta (N)$ , we equivalently assert that with probability $1-o(1)$ , $X^{(N)} \\ge (1-o(1))\\mathbb {E}[X^{(N)}]$ .", "Lemma 34 Let $A \\sim \\textnormal {SB}(n, d)$ .", "With probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1 - o(1))e^{-d}n$ .", "Define $I_i$ as the indicator variable of the event that row $i$ is all zero.", "We have $\\mathbb {E}[I_i] = (1-d/n)^{n}$ for all $i$ , but the $I_i$ are not independent.", "Let $X = \\sum _{i=1}^{N} I_i$ .", "We compute $\\text{Var}(X)$ .", "$\\text{Var}(X) = \\sum _{i,j=1}^{n} \\text{Cov}(I_i, I_j) = \\sum _{i,j=1}^{n} \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j]$ Note that $\\mathbb {E}[I_i I_j]$ is the probability of the event that both row $i$ and $j$ are 0.", "When $i \\ne j$ , rows $i$ and $j$ have identical entry $A_{ij} = A_{ji}$ , but all other entries are independent.", "Thus the probability of both rows $i,j$ having all zero entries is $(1-d/n)^{2n-1}$ .", "Thus, $\\text{Cov}(I_i, I_j) &= (1-d/n)^{2n-1} - (1-d/n)^{2n} \\\\&= \\left(\\frac{1}{1-d/n} - 1\\right) (1-d/n)^{2n} \\\\&= \\frac{d/n}{1-d/n} (1-d/n)^{2n} = O(1/n)$ Then from Lemma REF , with probability $1-o(1)$ , we have $\\mathbb {E}[X] \\ge (1-o(1)) n (1-d/n)^n = (1-o(1)) n e^{-d}$ .", "Lemma 35 Let $A \\sim \\textnormal {ABC}(n, \\gamma , d)$ .", "With probability $1 - o(1)$ , the number of rows that are all zeros in $A$ is at least $(1 - o(1))p^{\\gamma d}n$ .", "Define $I_i$ as the indicator variable of the event that row $i$ is all zero.", "Because we are interested in upper bounding the variance, we want to upper bound $\\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j]$ .", "A row $i$ of $A \\sim \\text{ABC}_p(n,\\gamma ,d)$ is zero only if all of the $\\gamma d$ 1's in row $i$ of $A_0$ fell into the $p\\gamma n$ last columns.", "In the setting of the configuration model, this means all of row $i$ 's $\\gamma d$ half edges are matched to the $p(\\gamma dn)$ half edges corresponding to the dropped columns.", "So $\\Pr [E_i] = \\frac{ \\binom{p(\\gamma dn)}{\\gamma d)} }{\\binom{\\gamma dn}{\\gamma d} }$ Let $j \\ne i$ .", "Observe that $\\mathbb {E}[I_j | I_i]$ can be computed similarly, but the conditioning implies that there are already $cd$ half edges from row $i$ assigned all to the $p(\\gamma dn)$ dropped column half-edges.", "$\\Pr [E_j | E_i] = \\frac{ \\binom{p(\\gamma dn) - cd}{cd)} }{\\binom{\\gamma dn - cd}{cd} } $ Observe that if two pairs of functions $a_i(n),b_i(n), i=1,2$ satisfy $a_i(n), b_i(n) = O(1)$ and $a_i(n) - b_i(n) = O(1/n)$ , then $a_1(n) a_2(n) = (b_1(n) + O(1/n))(b_2(n) + O(1/n)) = b_1(n) b_2(n) + (b_1(n) + b_2(n) + O(1))O(1/n) \\rightarrow a_1(n)a_2(n) - b_1(n)b_2(n) = O(1/n)$ .", "This can be applied iteratively to any constant number of functions.", "Observe that $\\Pr [E_i] - \\Pr [E_j | E_i] = \\prod _{\\ell =0}^{\\gamma d-1} \\frac{p(\\gamma dn)-\\ell }{\\gamma dn-\\ell } - \\prod _{\\ell =0}^{\\gamma d-1} \\frac{p(\\gamma dn)-\\gamma d-\\ell }{\\gamma dn-\\gamma d-\\ell }$ Note that the terms corresponding to a fixed $\\ell $ in each product have difference that is $O(1/n)$ : $\\frac{p(\\gamma dn) - \\ell }{\\gamma dn - \\ell } - \\frac{p(\\gamma dn) - cd - \\ell }{\\gamma dn - cd - \\ell } = \\frac{O(n)}{(\\gamma dn - \\ell )(\\gamma dn - cd - \\ell )} = O(1/n).$ It follows from the observation (and that $\\gamma d$ is a constant) that $\\Pr [E_i] - \\Pr [E_j | E_i] = O(1/n)$ .", "It follows that $\\text{Cov}(I_i, I_j) = \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i] \\mathbb {E}[I_j] = \\mathbb {E}[I_i I_j] - \\mathbb {E}[I_i]^2 = (\\Pr [E_i]) (\\Pr [E_i] - \\Pr [E_j | E_i]) = O(1/n)$ Then by Lemma REF , we have with probability $1-o(1)$ that $X \\ge (1-o(1))E[X] = (1-o(1)) p^{\\gamma d}n$ ." ], [ "Implication of Theorem ", "Define the 2-core of a matrix $A$ to be the matrix remaining after repeating the following peeling process: If there is a row with zero or one non-zero entries, remove that row, and remove the column corresponding to the position of the non-zero entry (if any).", "Note that this 2-core corresponds to viewing $A$ as the vertex-hyperedge incidence matrix of a hypergraph.", "Let $\\mathbf {n^*}$ and $\\mathbf {m^*}$ be the number of rows (vertices) and columns (hyperedges) respectively in the 2-core.", "We prove the following corollary to Theorem REF .", "Corollary 1 Let $A \\sim \\textnormal {SB}(n, d)$ with $d = \\omega (1)$ .", "With probability $1 - o(1)$ , $\\textnormal {rank}(A) = {\\mathbf {n^*}} + n - {\\mathbf {m^*}}$ Let $D$ be the set of rows involved in linear dependencies: $D = \\bigcup _{x: x^TA = 0}{\\mathrm {supp}(x)}.$ Let $I := [n] \\setminus D$ be the remaining rows.", "Let $P_R$ be the set of rows removed during the peeling process, and let $P_C$ be the set of columns removed.", "Let $A^{\\prime } \\in \\mathbb {R}^{{\\mathbf {n^*}} \\times {\\mathbf {m^*}}}$ be the 2-core matrix which remains after the peeling process.", "The following claim follows immediately from the definition of the peeling process: Claim 20 The row-span of $A_{P_R}$ is $\\mathbb {R}^{P_C}$ , which has rank $n - \\mathbf {m^*}$ .", "Then it suffices to show that following claim holds conditioned on the event in Theorem REF holding.", "Claim 21 $A^{\\prime }$ has full row rank, which is rank $\\mathbf {n^*}$ .", "First we show that conditioned on the event in Theorem REF holding, we have $D \\subseteq P_R$ .", "Indeed, conditioned on this event holding, we have $D = \\lbrace i \\in [n]: \\exists S \\ni i: A_S \\in \\mathcal {T}_{|S|}\\rbrace .$ Consider any set $S$ for which $A_S \\in \\mathcal {T}_{|S|}$ .", "By the definition of $\\mathcal {T}_{|S|}$ , $A_S$ is the vertex-edge incidence matrix of a tree.", "Hence the peeling process defined above to create the 2-core will necessarily remove all the rows in $S$ .", "Now to prove the claim, suppose for contradiction that there existed a linear dependency among the rows of $A^{\\prime }$ .", "Then since the row span of $A_{P_R}$ is $\\mathbb {R}^{P_C}$ , it must be the case that there is a linear dependency in $A$ which contains theses rows of $A^{\\prime }$ .", "That is, if $x^TA^{\\prime } = 0$ , let $y := x^TA_{[n] \\setminus P_R}$ such that $\\mathrm {supp}(y) \\in P_C$ .", "Then we can find $x^{\\prime }$ such that $x^{\\prime T}A_{P_R} = -y$ , and hence combining $x$ and $x^{\\prime }$ yields a vector $\\mathbf {x}$ such that $\\mathbf {x}^TA = 0$ .", "However, this is a contradiction, because it means some row in $[n] \\setminus P_R$ must be involved in a linear dependency — and hence a minimal linear dependency — contradicting the fact that $D \\subseteq P_R$ ." ], [ "Proof of Lemma ", "In this section we prove the following anticoncentration lemma.", "* Let $r_i$ be an independent Bernoulli random variable with parameter $2p$ and let $s_i$ be an independent Bernoulli random variable with parameter $1/2$ .", "It follows that $\\Pr [r_is_i=1]=\\Pr [z_i=1]$ .", "Thus, we have: $\\Pr (v^T z =c) &= \\Pr \\left(\\sum _{i=1}^n v_i r_is_i=c\\right)\\\\&\\le \\Pr \\left(\\sum _{i=1}^n v_ir_is_i=c \\Big | \\sum _{i=1}^n r_i\\ge mp\\right) + \\Pr \\left(\\sum _{i=1}^n r_i\\le mp\\right)$ We employ a result of Erdős [11] to note: $\\Pr \\left(\\sum _{i=1}^n v_ir_is_i=c \\Big | \\sum _{i=1}^n r_i\\ge mp\\right)\\le \\left(\\frac{1}{\\sqrt{\\pi /2}}\\right)\\frac{1}{\\sqrt{mp}}$ Finally, a Chernoff bound gives us: $\\Pr (\\sum _{i=1}^n r_i\\le mp) \\le \\left(2^p\\left(\\frac{1-2p}{1-p}\\right)^{1-p}\\right)^m\\le e^{(\\ln (2)-1)mp}$ Combining these two expressions gives the desired result." ] ]
2105.11718
[ [ "Payment Does Not Imply Consensus (For Distributed Payment Systems)" ], [ "Abstract Decentralized payment systems such as Bitcoin have become massively popular in the last few years, yet there is still much to be done in understanding their formal properties.", "The vast majority of decentralized payment systems work by achieving consensus on the state of the network; a natural question to therefore ask is whether this consensus is necessary.", "In this paper, we formally define a model of payment systems, and present two main results.", "In Theorem 1, we show that even though there exists a single step black box reduction from Payment Systems to Byzantine Broadcast, there does not exist any black box reduction in the other direction which is significantly better than a trivial reduction.", "In Theorem 2, we show how to construct Payment Systems which only require a very small number of messages to be sent per transaction.", "In particular, global consensus about which transactions have occurred is not necessary for payments in this model.", "We then show a relation between the construction in Theorem 2 and the Lightning Network, relating the formal model constructions we have given to a practical algorithm proposed by the cryptocurrency community." ], [ "[block]1em" ], [ "[hang]1em" ], [ "[hang]1em [enumerate]label*=0.", "Decentralized payment systems such as Bitcoin have become massively popular in the last few years, yet there is still much to be done in understanding their formal properties.", "The vast majority of decentralized payment systems work by achieving consensus on the state of the network; a natural question to therefore ask is whether this consensus is necessary.", "In this paper, we formally define a model of payment systems, and present two main results.", "In Theorem REF , we show that even though there exists a single step black box reduction from Payment Systems to Byzantine Broadcast, there does not exist any black box reduction in the other direction which is significantly better than a trivial reduction.", "In Theorem REF , we show how to construct Payment Systems which only require a very small number of messages to be sent per transaction.", "In particular, global consensus about which transactions have occurred is not necessary for payments in this model.", "We then show a relation between the construction in Theorem REF and the Lightning Network, relating the formal model constructions we have given to a practical algorithm proposed by the cryptocurrency community." ], [ "Motivation", "Decentralized payment systems, commonly known as \"cryptocurrencies\", solve the following problem: provide a group of participants the ability to \"send\" and \"receive\" virtual money to each other, such that no small group of individuals can violate the security of the monetary system.", "For example, participants should not be able to spend a virtual $\\$100$ note twice, nor should they be able to \"steal\" money from other participants.", "Since the launch and successful operation of Bitcoin in 2009, we have seen a significant increase in interest, funding and research into understanding decentralized payment systems.", "For example, at its peak in December 2017, the market capitalization of Bitcoin alone had grown from nothing to over 300 billion USD in just nine years [46].", "Throughout this timeperiod, it is undeniable that research in distributed consensus algorithms pioneered by Lamport and others [2] in the 1980s has been significantly influential in shaping how both researchers and practitioners think about and design distributed payment systems.", "Algorithms for consensus aim to solve the problem of coordinating a group of participants to communicate in a way such that after some time, all participants agree on some fact, and there is no way for a small group of individuals to prevent this from happening.", "The connection to decentralized payment systems is that if everyone can use a consensus protocol to agree on how much money each person has spent, then we can ensure that no-one spends their $\\$100$ twice or steals money from someone else.", "To give a few examples of the extent of this influence: All of the top 10 cryptocurrenciesAs of March 2019 by market capitalization provide payment system functionality by reaching consensus on the current monetary state of the network.", "[46] The author makes no claim about the thousands of relatively unknown cyrotocurrencies which often do not have well understood security guarantees.", "Prominent researchers in this field introduce the distributed payment system problem as one in which it is crucial all participants agree on which transactions have occurred.", "[36].", "Current implementations of cryptocurrencies developed by the academic community are directly based on consensus solutions for Byzantine Agreement [42] or built on top of consensus protocols [23] which reach agreement on each transaction.", "Prominent members of the cryptocurrency community explicitly and repeatedly frame cryptocurrencies a solution for solving a global consensus problem about the monetary state of the network.", "The development of Bitcoin [21] is claimed to have been significantly influenced by known solutions and impossibility results for Byzantine Agreement.", "For example, it is claimed the author of Bitcoin (whose real identity is unknown) would introduce Bitcoin as a solution to the Byzantine Agreement problem on chat forums.", "Consequentially, the thousands of cryptocurrencies which are built on top of Bitcoin's Blockchain architecture are all derived from solving the consensus problem as well.", "The founder of Ethereum, which is considered one of the foundational cryptocurrencies within the cryptocurrency community, maintains a website at https://vitalik.ca/ where he gives his perspectives on designing distributed payment systems.", "The arguments he gives make heavy use of language and ideas drawn from distributed consensus.", "Based on the above, it seems that the following assumption is implicit in a significant part of the work and discussion in the cryptocurrency community: Assumption Distributed payment systems cannot exist without achieving regular global consensus about which payments have occurred.", "Understanding the truth of falsehood of this implicit assumption is of central importance: If we can show in which precise way the assumption is true, we will have a clearer understanding of the canonical structure of a decentralized payment system: any such implementation would need to use tools from consensus literature, and known impossibility results would apply.", "If proven false, we will have a better understanding of exactly where the equivalence between consensus and payment systems breaks down.", "Exploiting the point at which the two problems diverge may lead to new algorithms which break lower bound results inherent in any payment system implementation based on consensus of payments.", "The purpose of this thesis is to closely scrutinize this implicit assumption.", "By the end of this paper, we hope to convince the reader that the latter scenario is closer to the truth: while consensus certainly implies a payment system, the reverse implication is more nuanced.", "The two main results of this paper are as follows: In Theorem REF , we show that even though there exists a single step black box reduction from a certain model of payment systems to Byzantine Broadcast, there does not exist any black box reduction in the other direction which is significantly better than a trivial reduction.", "In Theorem REF , we show that in a certain trust model we can construct payment systems where in the best case (when all participants behave honestly), only $\\mathcal {O}(\\log _{d}(N))$ participants exchange messages per transaction.", "In particular, under certain conditions, transactions do not require global consensus to occur in the model we give." ], [ "Organization and Contributions", "We begin in part by giving a brief survey of the consensus literature and implementations of modern cryptocurrencies most relevant to the question at hand.", "The goal of this section is to (a) give context to the current understanding of consensus and cryptocurrencies, and (b) familiarize the reader with known results which will be used in subsequent parts.", "Results whose solutions provide useful intuition will be proved.", "We will also discuss how this thesis relates to prior work on trying to remove the need for frequent consensus in Bitcoin, by contrasting the Constructions in Part with the Lightning Network [31].", "The comparison between payment systems and consensus begins in part .", "We start by formally defining the minimal functionality any distributed payment system should satisfy in the fault tolerant model.", "This will lead to the definition of the \"Marker Problem\", a toy problem designed to encapsulate the key ideas of a distributed payment system.", "After formally defining a model for this problem, the first set of results we show towards resolving the central question of this thesis is that (section ) There exists a single step, black box reduction from the Marker Problem to Byzantine Broadcast.", "(Proposition REF ) There exists no black box reduction from Byzantine Broadcast to the Marker Problem which is significantly better than a trivial reduction.", "(Theorem REF ) Already, this suggests that in the particular model we have chosen, achieving consensus is not inherently required in a payment system.", "We continue by giving some concrete constructions of solutions to payment systems in the proposed model.", "In section REF we play closer attention to the best case message complexitythe number of messages sent when when all participants happen to behave honestly of payment systems, in an attempt to break the inherent lower bound of $\\Omega (N)$ messages per transaction in any consensus-based payment system which achieves consensus on transactionsSuch consensus based solutions form the backbone of all cryptocurrencies based on a blockchain construction., where $N$ is the number of participants: We show a lower bound of $\\Omega (Nf)$ on the \"total\" best case message complexity of payment systems in a weak adversarial model, where $f$ is the number of faults (Proposition REF ).", "We give solutions to the Marker Problem showing this lower bound is tight.", "We use the proof of Proposition REF to construct a solution to the Marker Problem \"cycle coin\", which has the curious locality property that certain payments require only $\\mathcal {O}(1)$ messages in the best case (Construction REF ).", "In part , we argue for reasonable trust extensions to the fault tolerant model which are natural for distributed payment systems.", "By building on the ideas in Construction REF , we then show that under this model, and under certain regularity assumptions about the distribution of transactions: There exist payment systems with best case message complexity $\\mathcal {O}(\\log _{d}(N))$ per transaction, where $d \\ge 4$ can be chosen to any function of $N$ if one is willing to assume all participants have initial income $\\Omega (d)$ (Theorem $\\ref {thrm:short-cyc}$ ) In a semi-egalitarian society where every process has initial income $\\Omega (N^{c})$ for $c>0$ , then the best case message complexity is $\\mathcal {O}(1)$.", "This Construction breaks the $\\Omega (N)$ best case message complexity lower bound inherent to any consensus based solution which achieves consensus on transactions.", "Using similar ideas, there exist simple and realistically plausible ways for participants to coordinate to further reduce the best case message complexity per transaction.", "We comment that the most valuable component of part is likely the conceptual idea of how to bootstrap multiple solutions of Construction REF in a certain trust model to create highly local transactions with small message complexity.", "The income regularity conditions are unnecessarily strong and not naturally motivated: there is likely significant room for improvement for developing more sophisticated randomized constructions with naturally motivated transaction distribution assumptions, and this is perhaps an interesting problem for future work.", "We conclude in part by summarizing the key ideas of this paper, commenting on the practical consequence of the results obtained in an idealized model, and proposing problems for future exploration.", "By the end, we hope that the reader considers the equivalence between distributed payment systems and consensus to be less trivial than initially believed." ], [ "Introduction", "There has been a substantial amount of research in designing fault tolerant systems for networked processes.", "The purpose of this subsection is to give a brief introduction and context to this area of research.", "We give a survey of different model settings in which this problem is considered, state some known results of this area, and give proofs of results we rely on in later chapters.", "One of the first formal definitions for consensus over a distributed network was given by Lamport et al.", "in the 1980s [2].", "The motivation given was similar to the following story: Imagine there are a collection of $N$ Byzantine Generals who are currently camping outside of enemy territory, and would like to decide on a plan to attack the enemy the following day.", "For simplicity, assume they can either decide to attack (denoted 1) or retreat (denoted 0).", "Most importantly, they need to make sure that they all agree on the same plan, or risk dividing the army in half and being defeated by the enemy.", "All this would be rather simple if the generals could sit together at a table and vote on a decision; however, our generals are rather shy and refuse to leave their tents.", "Each general $i$ will only communicate with another general via sending letters.", "To complicate matters further, it is known that up to $f<N$ of the generals are working in secret with the enemy, and will try to ruin any plans of the Byzantines.", "Each Byzantine has an opinion (either 0 or 1) of what the decision should be for the next day.", "We would like to come up with a strategy the generals can follow such that if all non-traitorous, honest Byzantines think the decision should be to attack (resp.", "retreat), then all honest Byzantines reach a consensus on this decision, even when the dishonest generals behave maliciously.", "Even if not all honest Byzantines have the same opinion, we still want to ensure that every honest Byzantine agrees on the same decision at the end (whichever that might be), to avoid half the army attacking and the other half retreating.", "To begin solving this problem, we need to formalize how the generals communicate and behave.", "We imagine a collection of processes/nodes $P_{1},\\dots ,P_{N}$ (i.e.", "generals) which function within a network.", "At each time step $t \\in \\mathbb {N}$ , each process may send and receive messages to other processes in the network.", "The reliability of the network to deliver these messages is either asynchronous, partially synchronous, or synchronous [39].", "While an asynchronous network may arbitrarily delay (but eventually deliver) a sent message, a synchronous network is guaranteed to reliably deliver every sent message by the next time period.", "Partially synchronous networks [13] model a region between these two extremes, where there are some (unknown to each process) guarantees on the delay of messages being delivered.", "This paper will focus on the synchronous network case.", "Formally, we have the following definition: Definition 4.1.1 A synchronous network consists of a collection of processes/nodes $P_1, \\dots , P_N$ .", "If you like, you can imagine each $P_n$ as being an algorithm running on node $n$ .", "At each time step $t \\in \\mathbb {N}$ , the following occurs for process $P_n$ , $n \\in [N]$ : $P_n$ receives all the messages that were sent to it from another processes at time $t-1$ .", "In particular, $P_n$ sees a list containing elements of the form $(m,n^{\\prime })$ , where $m$ is the sent message and $n^{\\prime }$ is the sender.", "Based on the contents of all received messages up until time $t$ , $P_n$ can send messages to any other process.", "Formally, $P_n$ is a deterministic function of all past received messages.", "If we were to stop here, we would be considering a model for the Byzantine Generals problem in the unauthenticated case.", "In this paper, we will primarily be concernedall communication will be assumed to be authenticated unless explicitly stated.", "with the authenticated case, where we give our generals some extra help: we imagine that each process $P_{n}$ can sign a message $m$ to produce the string $SIGN_{P_n}(m):=(m)_{P_n}$ .", "Such a signature is assumed to be unforgeable and tamperproof: no other process can produce a substring of the form $(m)_{P_n}$ unless they copied it from a message originally signed by $P_n$ .", "Such a formalism is designed to model real cryptographic signatures which have similar properties.", "Without loss of generality, we assume that in the authenticated model, all processes sign their messages before sending them.", "We imagine that at time $t=0$ , each process $P_n$ is given an initial value $v_n$ in some finite set $\\mathcal {V} \\supset \\lbrace 0,1\\rbrace $ , which is their opinion of how to attack the next day.", "We would like our processes to agree on some value at the end.", "Now, how do the enemy generals behave?", "What do the agreement strategies we construct have to protect against?", "Conceptually, it will be useful to imagine a single \"adversary\" which can corrupt honest generals to make them dishonest, and can coordinate the enemy generals against the honest Byzantine generals.", "Many different kinds of adversaries have been studied, depending on the kinds of applications considered.", "For example, fail-stop models include adversaries which can cause honest processes to terminate during network execution, and failure-omission models allow adversaries to cause some messages to be omitted from delivery.", "This paper will primarily be concerned with byzantine adversaries.", "Formally, an $f$ -Adversary is one which, at $t=0$ , can look at all the initial values $\\lbrace v_n\\rbrace _{n \\in [N]}$ , and knows the deterministic strategy all the processes will follow.", "It can then pick up to $f$ processes to corrupt, making them dishonest.", "From this point onward, the dishonest processes can behave arbitrarily, while the honest processes behave according to some specified strategy.", "While dishonest processes cannot forge the signature of any honest process in the authenticated model, each dishonest process is allowed to forge the signature of another dishonest process.", "Having formally described how both honest and dishonest generals behave, we now need to say what it means for them to reach agreement on a decision.", "There are two closely related problems which model this.", "Recall that each process $P_{n}$ is given an initial value $v_n$ at time $t=0$ .", "At some point in the future, $P_{n}$ decides on a value $d_n$ , the decision it will ultimately follow about the battle the next day.", "Our solution should consist of a collection of protocols, i.e.", "rules or deterministic functions, which each honest process $P_n$ follows protocol $p_n$ .", "Even though we have not mentioned any randomness, we will give a definition which allows some probability of failure in anticipation of a future model: Definition 4.1.2 (Byzantine Agreement): We say that a collection of protocols $\\lbrace p_i\\rbrace _{i \\in [N]}$ is a solution to the byzantine agreement problem in the presence of an $f-$ Adversary with error probability $\\epsilon $ if the following conditions hold with probability at least $1-\\epsilon $ : Consistency: For any two honest processes $P_i,P_j$ , we have $d_i=d_j$ .", "Validity: If $v_i=v$ for all honest processes, then $d_i=v$ for all honest processes.", "Termination: Each honest processes decides in finite time.", "If the set of initial values is $\\mathcal {V}=\\lbrace 0,1\\rbrace $ , we call this problem binary Byzantine Agreement.", "Thus, deterministic solutions to Byzantine Agreement, which are protocols which function within the deterministic model we have built, have error probability $\\epsilon =0$ .", "A closely related problem is Byzantine Broadcast: instead of each process $P_{n}$ receiving an initial value $v_n$ , only the specially selected process $P_{1}$ (the \"general leader\") receives an initial value $v_1$ .", "If the general leader is honest, then all honest generals should agree with the leader's decision.", "If the leader is dishonest, all generals should still agree on the same value.", "Definition 4.1.3 (Byzantine Broadcast): We say that a collection of protocols $\\lbrace p_i\\rbrace _{i \\in [N]}$ is a solution to the byzantine broadcast problem in the presence of an $f-$ Adversary with error probability $\\epsilon $ if the following conditions hold with probability at least $1-\\epsilon $ : Consistency: For any two honest processes $P_i,P_j$ , we have $d_i=d_j$ .", "Validity: If $P_1$ is honest, then $d_i=v_1$ for all honest processes.", "Termination: Each honest processes decides in finite time.", "If the initial set of values $\\mathcal {V}=\\lbrace 0,1\\rbrace $ , we call this problem Binary Byzantine broadcast.", "If we have a solution to these problems, it will be useful to quantify exactly how good the solution is.", "Towards this aim, we give the following definitions: Definition 4.1.4 Given any solution to a Byzantine problem, we define the following metrics: Message Complexity: The total number of messages sent across the network by all honest processors until consensus is reached (i.e.", "all processes decide).", "Note that we explicitly do not take message length into accountThis is because we will later solely focus on the number of messages sent in an attempt to differentiate consensus solutions from payment system solutions.", "The literature often also considers the number of bits per message.. Signature Complexity: The total number of signatures sent across the network by all honest processors during network operation.", "Note that multiple signatures can occur in a single message, and furthermore we count a signature multiple times if it is sent in multiple messages.", "If a process signs and sends a signed message, i.e.", "$((\"attack\")_{P_1})_{P_2}$ , this counts as multiple signatures.", "Round Complexity: The maximum number of time steps taken until consensus is reached.", "Before diving into the details, we briefly give an overview of some known results for these problems.", "After introducing and defining the Byzantine problems, [2] showed that for deterministic processes $\\lbrace P_n\\rbrace _{n \\in [N]}$ operating over a synchronous network in the unauthenticated case, the the Agreement and Broadcast problems are solvable if and only if $3f<N$ .", "In contrast, in the authenticated case, Byzantine Broadcast solvable for all $f \\le N$ [6], but Byzantine Agreement still only remains solvable iff $3f<N$ .", "In an asynchronous network, an important result from [10] showed that consensus is impossible even in the weak fail stop adversarial model allowing only a single process to arbitrarily terminate.", "It was shown in [13] that we can recover from this impossibility result and still reach consensus in a partially synchronous network.", "The first solutions to the Byzantine agreement problem required sending messages with a combined bit length which was exponential in $f$ , namely $\\mathcal {O}(N^{f+2})$ , and a round complexity of $f+1$ .", "It was later shown in [3] that this round length is optimal, and later solutions for Byzantine agreement were given which gave polynomial message complexity [6].", "In [11], a lower bound of $\\Omega (N(f+1))$ for the signature complexity in the authenticated model was given and shown to be tight.", "What about the difference in difficulty of Byzantine consensus when the set of possible initial values $\\mathcal {V}$ is large, verses the binary case $\\mathcal {V}=\\lbrace 0,1\\rbrace $ ?", "By giving a black box reduction from the multivalued case to the binary case, [9] showed that binary and multivalued Byzantine Agreement are essentially equivalent.", "We will therefore often think of the binary and multi-valued problems as being \"the same\".", "Following a categorization of the complexity of the Byzantine problems for deterministic solutions, researchers turned to randomness in an attempt to break these lower bounds.", "Many of these solutions were based on the idea of using randomness to create \"public coins\" which could be used to facilitate consensus [8].", "Recently, by making use of a shared random string, a random oracle and cryptographic signatures, [40] built on these ideas to give a solution BBA* to the agreement problem which runs in $\\mathcal {O}(1)$ rounds in expectation, has $\\mathcal {O}(N^2)$ message complexity in expectation, and tolerates $3f<N$ failures by a computationally bound adaptive adversary.", "Other randomized solutions have been given which tolerate up to $2f<N$ faults and also run in $\\mathcal {O}(1)$ rounds in expectation [20].", "From the lower bounds side, it has been shown that any randomized solution to the agreement problem has a probability of failing which decreases at best exponentially in the number of rounds in the non-adaptive fail-stop model [15], [22].", "The precise model in which a randomized solution to Byzantine Agreement/Broadcast is formulated relies on technical definitions of random oracles, digital signatures, and computationally bounded adversaries which are not central to the ideas of this paper.", "Despite this, randomized consensus algorithms are used in practical designs of modern cryptocurrencies, and the reductions we give in subsequent sections are fairly agnostic to whether we are in the deterministic or randomized model.", "To strike a balance, while still being able to give a comparison between payment systems and randomized solutions to consensus, we will simply state results which also apply to randomized consensus, and refer the interested reader to [40] for more details on what the model for randomized consensus looks like.", "We informally describe the differences between the randomized and deterministic models here, though these details will not be relevant in this paper: Protocols can now be randomized, i.e.", "they may flip random coins to decide what to do next.", "Processes have access to certain cryptographic tools, such as digital signatures through a public key interface, shared random oracles, and a public shared random string.", "The adversary can only run in polynomial time.", "At the beginning of each time step, it can view the entire network, and can choose some processes to behave dishonestly, up to a total of $f$ over all time.", "If a process is corrupted by an adversary at some time, we refer to it as dishonest (even before the time it has been corrupted).", "The adversary then directly controls each process.", "If a solution for a problem occurs in this model, we will call it a randomized solution.", "These solutions may have error probability $\\epsilon >0$ .", "Because the randomized setting only gives processes access to more primitives and requires the adversary to be computationally bounded, a solution in the deterministic model is automatically a solution in the randomized model." ], [ "Constructive Results", "We now present some specific solutions and relations between Byzantine Broadcast and Byzantine Agreement.", "Proposition 4.2.1 For any $f < N-1$ , there exists a deterministic solution to the Byzantine Broadcast problem which can tolerate up to $f$ corruptions.", "(From [6]) Recall that we let $SIGN_{P_i}(v)=(v)_{P_i}$ denote the string signed by $P_i$ .", "Likewise, we let $((v)_{P_{i_{1}}}\\dots )_{P_{i_k}}$ denote the string obtained by $P_{i_1}$ signing $v$ , and then $P_{i_2}$ signing the resulting message, and so forth.", "If $i_i=1$ (i.e.", "the first signature is from the broadcasting process $P_1$ ), and $P_{i_j} \\ne P_{i_w}$ for $j \\ne w$ (all the signature identities are distinct), we call a message of this form proper of length $k$ , and we can refer to $v$ as the value of the proper message.", "We also arbitrarily designate $f+1$ \"relay\" processes $P_{j_1},\\dots ,P_{j_{f+1}}$ , where none of these are the broadcaster $P_1$ .", "If $P_1$ is honest, it signs its initial value by computing $m=(v_1)_{P_1}$ , and sends this message to all processes.", "Now consider the following protocol for an honest process $P_n$ .", "Each processor keeps a list $L$ of values it has seen before.", "At the beginning of time step $i+1$ , $P_n$ lexicographical orders all its received messages during the previous time step.", "$P_n$ then iterates through each message $m$ in order and does the following: if $m$ is not proper, or is proper but is not of length $i$ , discard $m$ .", "if $value(m) \\in L$ , discard $m$ .", "if $|L| \\ge 2$ , discard $m$ .", "Otherwise, add $value(m)$ to $L$ .", "We say that $P_n$ extracts $value(v)$ at round $i+1$ .", "Sign $m$ to produce a proper string $m^{\\prime }=(m)_{P_n}$ of length $i+1$ .", "If $P_n$ is a relay processor, send a copy of $m^{\\prime }$ to every other processor.", "If $P_n$ is not a relay processor, and send a copy of $m^{\\prime }$ to every relay processor.", "At time $f+2$ , if $|L|\\ne 1$ , $P_n$ decides the value 0 (\"sender fault\").", "Otherwise $P_n$ decides the unique value in $L$ .", "We claim that such a construction solves the Byzantine Broadcast problem.", "First, suppose the broadcaster $P_1$ is honest.", "Then at step 1, all honest nodes $P_n$ extract value $v_1$ .", "Moreover, since $P_1$ never produces a signature of the form $(v^{\\prime })_{P_1}$ for $v^{\\prime } \\ne v_1$ , no proper message with a different value is ever sent across the network.", "Consequently, after $f+2$ steps, each $P_n$ has $|L|=1$ , and all honest processors decide $v_1$ .", "Next, we claim that all honest processors decide on the same value.", "In particular, we claim: suppose an honest process $P_n$ extracts value $v$ within $f+3$ steps.", "Then any other honest process $P_{n^{\\prime }}$ has either extracted $v$ or has extracted two distinct values.", "It then follows that either (a) all processes extract exactly 0 or 2 values, or (b) all processes extract the exact same value.", "In both cases, all honest processes agree on the same value after $f+3$ steps.", "To prove the claim, suppose that $P_{n}$ has extracted $v$ after $f+3$ steps, but some $P_{n^{\\prime }}$ has not.", "let $m$ be the proper message of length $i$ with value $v$ from which $P_{n}$ extracted $v$ .", "Choose $n$ such that $i$ is the smallest such integer (i.e.", "$i+1$ was the earliest round number in which $v$ was extracted by any honest process $P_{n}$ ).", "Then we must have $i<f+1$ .", "Otherwise, $m$ has been signed by at least one honest process at an earlier time (and hence $v$ was extracted at an earlier time), contradicting the choice of $i$ .", "Thus $P_{n}$ extracted $v$ by time $f+1$ .", "If $P_{n}$ is a relay process, it then transmits a proper message of length $i+1$ to every other honest process with value $v$ : hence, every honest process will either extract $v$ at round $i+1 \\le f+2$ , or it will not because it has already extracted two values.", "If $P_{n}$ is not a relay process, then since at least one relay process is honest, $P_{n}$ sends a proper message of length $i+1$ to some honest relay process $P_{n^{\\prime \\prime }}$ with value $v$ .", "There are now two cases: either $P_{n^{\\prime \\prime }}$ extracts $v$ in round $i+1$ , and we reduce to a previous case, concluding that all processes have either extracted $v$ or two values by time $i+2 \\le f+3$ .", "If $P_{n^{\\prime \\prime }}$ does not extract $v$ in round $i+1$ , then it has already extracted two values by round $i+1$ , and we reduce to the previous case again, where all processes will have extracted two values by round $i+2 \\le f+3$ .", "This completes the claim.", "Notice that in the construction above, the total number of messages sent by all honest processes is $\\mathcal {O}(Nf)$ .", "Each honest process, besides $P_1$ and the relay processors, sends at most two groups of messages to relay processors.", "Each relay processor sends at most 2 groups of messages to non-relay processors.", "This makes the total number of messages equal to $(\\textit {messages from process 1})+(\\textit {messages from relay processes})+(\\textit {messages from non-relay processes})$ $\\le N+2\\times (f+1)\\times N+2N\\times (f+1)=\\mathcal {O}(Nf)$ Since each message contains $\\mathcal {O}(f)$ signatures, the total number of signatures sent is $\\mathcal {O}(Nf^2)$ .", "Likewise, the number of time steps until a decision is made is $\\mathcal {O}(f)$ .", "These will be useful metrics to remember for two reasons: firstly it gives us a sense of how efficient our solution is.", "Secondly, there are known lower bounds of e.g.", "how many time steps are needed to achieve consensus in certain models, and we will make use of these in the future.", "While we will not use it, we also mention that there is a similar result for the Byzantine Agreement problem: Proposition 4.2.2 For any $f \\in \\mathbb {N}$ such that $3f<N$ , there exists a deterministic solution to the Byzantine Agreement problem which can tolerate up to $f$ corruptions.", "The message and signature complexity is polynomial in $N,f$ , and the round complexity is $\\mathcal {O}(f)$ .", "We comment again that, in contrast to the Broadcast problem, there is no solution for the Byzantine Agreement problem for any $f \\in \\mathbb {N}$ when $3f \\ge N$[2].", "For example, consider the case $3f=N,f=1$.", "For randomized Byzantine Agreement, we mention the following result: Proposition 4.2.3 (From [40]) For any $f \\in \\mathbb {N}$ such that $3f<N$ , and any $\\gamma \\ge 1$ , there exists a randomized solution to the Byzantine Agreement problem for $\\mathcal {V}=\\lbrace 0,1\\rbrace $ which can tolerate up to $f$ corruptions.", "The total number of messages sent and signatures made is $\\mathcal {O}(\\gamma N^2)$ , the number of time steps taken is $\\mathcal {O}(\\gamma )$ in expectation, and the probability of error is $2^{-\\Omega (\\gamma )}$ .", "We comment that there exist solutions tolerating $f$ faults with $2f<N$ and similar complexity given in [20].", "However, the construction given in REF uses ideas which are perhaps more directly relevant to the implementation of modern cryptocurrencies [42].", "We can strengthen proposition REF to handle the full Byzantine Agreement problem with arbitrary initial values using the following lemma: Lemma 4.2.1 There exists a black box reduction, using only two extra rounds and $\\mathcal {O}(N^2)$ extra messages and signatures, from multi-valued Byzantine Agreement tolerating $f<\\frac{N}{3}$ faults to Binary Byzantine Agreement tolerating $f$ faults, i.e.", "these two problems are essentially equivalent.", "(From [9]) Consider the following construction: At the first round, all honest processes send their initial value $v_i$ to every other process.", "We call an honest process perplexed if during this round, at least $\\frac{1}{2}(N-f)$ of the values it receives are different from its own (if it receives no value from a process, it assumes the default value 0 was sent); otherwise we say the honest process is content.", "At the second round, every honest perplexed process sends a message to every other process saying \"I am perplexed\".", "Now for each honest process $P_n$ , define two arrays $Value_{n}[i], Perplexed_{n}[i]$ .", "Set $Value_{n}[n]=v_n$ , and $Value_{n}[j]=$ the value process $P_j$ claimed to have during round 1.", "Likewise, set $Perplexed_{n}[n]=True$ if $P_n$ is perplexed, and $Perplexed_{n}[j]=True$ iff process $P_j$ claimed to be perplexed in round 2.", "Lastly, define $Alert_n=True$ if at least $N-2f$ of the elements of $Perplexed_{n}$ are $True$ , and $False$ otherwise.", "Lastly, have each honest process $P_{n}$ now run the Binary Byzantine Agreement protocol with initial value $Alert_{n}$ .", "Eventually all honest processes decide on a common value $Alert$ .", "If $Alert=True$ , then all honest processes decide on the default initial value 0.", "If $Alert=False$ , then $P_{n}$ decides as follows: $P_{n}$ initializes a list $L_{n}$ : for each $j$ such that $Perplexed_{n}[j]=False$ , $P_{n}$ adds $Value_{n}[j]$ to $L_{n}$ .", "$P_{n}$ then decides on the most frequently occurring value in $L_{n}$ .", "We claim this construction gives the required behavior: Termination: If the Binary Byzantine Agreement protocol terminates in $t$ steps, then this construction terminates in $t+2$ steps.", "Validity: Suppose all honest processes have the same initial value $v$ .", "Since $3f<N$ , each process receives at most $f+1<\\frac{1}{2}(N-f)$ distinct values at round 1, and so no honest process is perplexed.", "Thus at most $f<N-2f$ elements of $Perplexed_{n}$ are true, and $Alert_{n}=False$ for every honest process.", "By validity of the Binary Byzantine Agreement protocol, all honest processes agree on the value $Alert=False$ .", "Moreover, the value $v$ occurs at least $N-f>\\frac{N}{2}$ times in $L_{n}$ , so $P_{n}$ decides correctly.", "Consistency: If $Alert=True$ , all honest processes decide the same value 0.", "It remains to consider the case $Alert=False$ .", "Let $P_{n}$ be a content process with initial value $v_n$ , and let $v^{*}$ be a most frequently occurring value of the initial values of correct processes.", "Suppose $v_n \\ne v^{*}$ .", "Then $P_{n}$ receives at least $\\frac{1}{2}(N-f)$ values different from its own $v_i$ in step 1, contradicting that $P_{n}$ is content.", "It follows that $v^{*}$ is unique, and that the initial value of every content process equals $v^{*}$ .", "Since $Alert=False$ , at least $f+1$ honest processes are content; otherwise, at least $N-2f$ honest processes would be perplexed, and all correct processes would have $Alert_{n}=True$ at the end of the second round, contradicting the validity of the Binary Byzantine Agreement protocol.", "Thus each correct process $P_{n}$ has their list $L_{n}$ consisting of at least $f+1$ copies of $v^{*}$ from honest content processes, and at most $f$ other values from dishonest processes claiming to be content.", "It follows that $v^{*}$ is the unique majority value and all $P_{n}$ decide on the same value.", "Another way to simplify the number of definitions we have is to notice that a solution for Byzantine Agreement implies a solution of similar complexity for Byzantine Broadcast: Lemma 4.2.2 Suppose there exists a solution to the Byzantine Agreement problem which tolerates $f$ faults.", "Then there exists a solution to the Byzantine Broadcast problem which tolerates $f$ faults, takes 1 extra round, and sends $\\mathcal {O}(N)$ more messages and signatures than the original solution.", "In particular, we construct this solution via a black box reduction.", "Have processor 1 send a signed copy of its initial value $v_1$ to all processors.", "At time step 1, if an honest processor sees a single value $v_1$ signed and sent from $P_1$ , it takes this to be its value in the Byzantine Agreement game.", "Otherwise it chooses a default initial value $0 \\in \\mathcal {V}$ .", "Now all honest processors run the assumed solution for Byzantine Agreement, and eventually decide on a value.", "This construction uses one extra round and an additional $\\mathcal {O}(N)$ messages and signatures.", "If $P_1$ is honest, all honest processes start with the same initial value for Byzantine Agreement, and by assumption will all decide on value $v_1$ .", "If $P_1$ is dishonest, regardless of the initial values chosen by honest processors, because there are at most $f$ faults, all honest processors will come to a consensus on the same value by assumption of the correctness of the Byzantine Agreement solution.", "Corollary 4.2.1 For any $f \\in \\mathbb {N}$ such that $3f<N$ , and any $\\gamma \\ge 1$ , there exists a randomized solution to the Byzantine Broadcast problem which can tolerate up to $f$ corruptions.", "The total number of messages sent and signatures made is $\\mathcal {O}(\\gamma N^2)$ , the number of time steps taken is $\\mathcal {O}(\\gamma )$ , and the probability of error is $2^{-\\Omega (\\gamma )}$ .", "This follows from Propositions REF , REF and Lemma REF ." ], [ "Lowerbound Results", "Having given some positive constructions, we now survey some impossibility results for these problems.", "The first is the following: Proposition 4.3.1 Any deterministic solution to the Byzantine Generals problem tolerating $f$ faults requires at least $f+1$ time steps in the worst case.", "In particular, there exists a strategy the adversary can follow which forces the number of steps taken to be $f+1$ .", "We refer to [6] for a proof of this fact, which is based on a generalization of a similar theorem given in [3].", "Note that this is a lower bound result: not every possible network computation requires $f+1$ steps to reach consensus, but there is always some set of choices the adversary can make if it really wants to force $f+1$ steps to be made.", "For example, if the broadcaster is honest and sends the signed value $(0)_{P_1}$ to all processes, and each process $P_n$ happens to send $((0)_{P_1})_{P_n}$ to every other process in time step 2, then all processes can infer that agreement has been reached and terminate in 2 steps.", "By combining the round complexity lower bound (Proposition REF ) together with the black box reduction from Byzantine Broadcast to Byzantine Agreement (Lemma REF ), we get the immediate corollary: Corollary 4.3.1 Any deterministic solution to the Byzantine Agreement problem tolerating $f$ faults requires at least $f$ time steps in the worst cast.", "In particular, there exists a strategy the adversary can follow which forces the number of steps taken to be $f$ .", "A later paper [11] gives lower bounds for the message complexity and signature complexity of deterministic Byzantine Broadcast: Proposition 4.3.2 (From [11]) Any deterministic solution to the Byzantine Broadcast problem tolerating $f<N-1$ faults has signature complexity $N(f+1)/4$ , even when all processes behave honestly.", "Proposition REF is a strong result: it says that even when all processes behave correctly, $N(f+1)/4$ signatures are still exchanged by any protocol tolerating $f$ faults.", "We note again that we are assuming all sent messages are authenticated with a signature during the sending process.", "Consider two executions of the network: in the first execution $H$ , all processes are honest and the broadcaster $P_1$ has initial value $v_1=0$ .", "In the second execution $G$ , all processes are honest and the broadcaster $P_1$ has the initial value $v_1=1$ .", "Let $M_{H,a,b},M_{G,a,b}$ be the sets of messages with their associated send times, sent by process $a$ to process $b$ during histories $H,G$ respectively.", "Let $A(n)$ be the set of processes which either (a) received a signature from $n$ in histories $G$ or $H$ , or (b) sent their own signature to $n$ in histories $G$ or $H$ .", "Now if $\\forall n \\in [N]$ , we have $|A(n)| \\ge f+1$ , we are done, since one of history $G$ or $H$ involves sending at least $\\frac{1}{2}\\sum _{n \\in [N]} |A(n)|/2 \\ge N(f+1)/4$ signatures.", "Suppose for the sake of contradiction that $\\exists n \\in [N]$ such that $|A(n)|\\le f$ .", "We now define a new history $H^{\\prime }$ which proceeds as follows: we make all the processes in $A(n)$ faulty.", "During the execution of $H^{\\prime }$ , we make each process $P_{n^{\\prime }} \\in A(n)$ send the messages $M_{H,n^{\\prime },n}$ to $P_n$ at the appropriate times.", "Towards all other processes $n^{\\prime \\prime } \\ne n, n^{\\prime \\prime } \\notin A(n)$ , we make $P_{n^{\\prime }} \\in A(n)$ send the messages $M_{G,n^{\\prime },n^{\\prime \\prime }}$ to $P_{n^{\\prime \\prime }}$ at the appropriate times.", "We need to verify two properties of this construction.", "The first is that our construction is valid: the messages we require the dishonest processes to send do not violate the integrity of the signatures honest processes during the execution of history $H^{\\prime }$ .", "Note that $n^{\\prime } \\in A(n)$ only sends messages to $P_{n}$ which contain signatures from dishonest nodes $A(n)$ , and so the adversary is able to \"forge\" any signatures required for messages that need to be sent to $n$ .", "For $n^{\\prime \\prime } \\ne n, n^{\\prime \\prime } \\notin A(n)$ , note that $n^{\\prime \\prime }$ never receives a signature from $P_n$ .", "Thus for $n^{\\prime } \\in A(n)$ , all the signatures in a message $m \\in M_{G,n^{\\prime },n^{\\prime \\prime }}$ are either from signatures already received by $n^{\\prime }$ , or from signatures from processes in $A(n)$ (which can be forged).", "Lastly, note that the received messages of honest process $P_n$ look identical to those in history $H$ , so $P_n$ will decide 0.", "However, there is at least one honest process $n^{\\prime \\prime } \\ne n, n^{\\prime \\prime } \\notin A(n)$ whose received messages look identical to those in history $G$ , and so will decide 1, violating the consistency condition of Byzantine Broadcast.", "Again by Lemma REF , the analogous result holds for Byzantine Agreement.", "We also have a similar lower bound on the message complexity: Proposition 4.3.3 (From [11]) Any deterministic solution to the Byzantine Generals problem tolerating $f$ faults has message complexity at least $\\max ((N-1)/2,(1+\\frac{f}{2})^2)$ in the worst case.", "In particular, there exists a strategy the adversary can follow which forces the number of messages sent to be $\\max ((N-1)/2,(1+\\frac{f}{2})^2)$ .", "We comment that the adversarial strategy for proposition REF is rather weak: the adversary simply needs to ignore some of its received messages, and behave honestly otherwise." ], [ "Concatenating Protocols", "A common technique for building solutions to larger problems is to use protocols for smaller problems (Byzantine Generals, secret sharing) as building blocks, and we will frequently use this technique.", "For example, one may wish to have processes run two copies of Byzantine Broadcast in parallel or sequence, and then use the decisions from each consensus protocol and combine them in a particular way.", "However, naively combining protocols can lead to serious flaws in the concatenated protocol, and so is worth mentioning here briefly.", "For example, imagine simulating two \"copies\" of Byzantine Consensus in the authenticated setting, one after the other.", "In the first simulation, processes exchange authenticated messages and achieve consensus.", "In the second simulation, processes receive new input values and re-run the consensus protocol again.", "However, if the consensus protocol is blindly re-run, there is no longer a guarantee of consensus for the second roundFor example, see [18] for impossibility results in this direction; note that these results apply to stateless composition of protocols.", "The same paper shows that if we include the round number of a protocol in a message, we can arbitrarily compose solutions to Byzantine Agreement.", "This is because in the second simulation, dishonest processes can reuse the signatures of honest processes from the first simulation (which they would not have been able to acquire otherwise).", "However, this issue is easily overcome by including a nonce in all messages which uniquely identify which simulation the message belongs to.", "The next few statements are difficult to state formally in a way which captures their full generality without introducing substantial notation, even though the ideas are very simple.", "Instead we choose to make these claims as high level statements, where the proof will make clear exactly when it is valid to apply them.", "Definition 4.4.1 We say a protocol solution $\\lbrace p^{\\prime }_n\\rbrace _{n \\in [N]}$ simulates a collection of protocols solutions $\\lbrace (p_1)_{n}\\rbrace _{n \\in [N]},\\dots ,\\lbrace (p_K)_{n}\\rbrace _{n \\in [N]}$ with unique nonceswe assume these nonces have never been used before in the network execution $nonce_1,\\dots ,nonce_k$ if for all $n$ , $p^{\\prime }_n$ stipulates running copies of protocols $(p_1)_n,\\dots ,(p_K)_n$ with their associated nonces.", "Recall that in the authenticated setting, we assume all sent messages are signed as $m^{\\prime }=(m)_{P_{j}}$ .", "Formally, $p^{\\prime }$ behaves as follows: Whenever $(p^{\\prime })_n$ receives a message $m^{\\prime }$ , it checks to see that all signatures $(s)_{P_j}$ contained in $m^{\\prime }$ are of the form $s=m \\cdot nonce_i$ here $\\cdot $ denotes string concatenation with a unique symbol between the two concatenated strings for some fixed $i$ .", "We say such messages belong to simulation $i$ .", "If so, it passes the message $m^{\\prime }$ to the simulation $(p_i)_n$ .", "Otherwise it ignores the message.", "When the simulation $(p_i)_n$ receives message $m^{\\prime }$ , it pretends that it can't see any of the $\\cdot nonce_i$ components and behaves as usual.", "If $(p_i)_n$ wants to copy or sign the signatures of other processes and combine them in a new message, we implicitly assume that it includes the nonce identifier for the $i$ th simulation.", "Suppose protocol $(p_i)_n$ wants to send a message $m^{\\prime }$ on process $P_n$ .", "$m^{\\prime }$ consists of collections of signed messages where each signature $(s)_{P_j}$ is of the form $s=m \\cdot nonce_i$ for some fixed $i$ .", "$p^{\\prime }_n$ simply sends this message over the network.", "Note that the signature and message complexity of $\\lbrace p^{\\prime }_n\\rbrace _{n \\in [N]}$ is equal to the sums of the complexities of each simulation.", "The round complexity remains the same.", "Proposition 4.4.1 (Informal) Suppose $p_1,\\dots ,p_k$ are protocol solutions which individually succeed against an $f$ -Adversary in the deterministic setting.", "Then the simulation of these protocols combined, $p^{\\prime }$ , also succeeds (all of $p_1,\\dots ,p_K$ succeed together) against the $f$ -Adversary.", "In the randomized setting, if we combine the Byzantine Agreement protocols $p_1,\\dots ,p_K$ , where each $p_i$ has error $\\epsilon $ and is the algorithm in [40] used to prove proposition REF , then $p^{\\prime }$ also succeeds with error $K\\epsilon $ .", "(Informal:) We give the proof for the deterministic model.", "Suppose that the combined simulation did not succeed, so that without loss of generality, the particular simulation $(p_1)_n$ did not succeed running on honest process $P_n$ when the other simulations were run in conjunction.", "In particular, $(p_1)_n$ behaved (decided incorrectly, sent an incorrect message etc) in an unintended way at some time $t$ .", "For $j \\in [N]$ , let $R_{j,t},S_{j,t}$ be the set of all messages received/sent resp.", "by $P_j$ during time $t$ which belong to simulation 1, and let $D$ be the set of dishonest processes.", "By fixing any initial values to the problem in questione.g.", "the initial values $\\lbrace v_i\\rbrace _{n \\in [N]}$ in Byzantine Broadcast, for an honest process $P_j$ , $(p_1)_j$ 's actions only depend on $R_j$ .", "But now consider a new network execution $E$ where all honest processes only run protocol solution $\\lbrace (p_1)_{j}\\rbrace _{j \\in [N]}$ corresponding to simulation 1.", "Have the adversary choose the same set $D$ of dishonest processes, and mimic the execution of the 1st simulation in $E$ : Inductively, we claim that at step $t$ , the sent and received messages $S_{E,j,t},R_{E,j,t}$ of any process $P_j$ in $E$ at time $t$ is equal to those of $R_{j,t},S_{j,t}$ at time $t$ , modulo the identifier $nonce_i$ .", "The case $t=0$ is immediate.", "At the beginning of time $t$ , all processes $P_j$ in $E$ receive the same messages as in $R_{j,t}$ by induction, since these are just the sent messages of the prior round.", "We need to show all processes in $E$ also send the same messages at time $t$ .", "For honest processes, this follows immediately, because they are deterministic functions of their received messages.", "For a dishonest process $P_j \\in D$ during the execution of $E$ , we stipulate that it sends the same messages belonging to simulation 1 that were sent by $P_j$ at this time in the original simulation.", "We can do this only if $P_j$ is not forging any signatures of honest processes by sending a message $m^{\\prime } \\in S_{j,t}$ .", "But by construction, any signatures from honest processes appearing in $m^{\\prime }$ belong to simulation 1, and so must appear in $R_{j,t}$ (otherwise the adversary would not have been able to send $m^{\\prime }$ in the original simulation).", "By induction these signatures appear in $R_{E,j,t}$ (modulo $nonce_i$ ), and so the adversary can comply with this stipulation.", "It follows that all honest processes in $E$ receive the same messages as in the original simulation.", "Thus $P_n$ behaves incorrectly in $E$ , contradicting that $\\lbrace (p_1)_i\\rbrace _{i \\in [N]}$ succeeds individually." ], [ "Digital Money and Decentralization", "The idea of digital money has been previously studied by cryptographers, mainly with the concerns of privacy and security in mind.", "For example, [14] gives a construction of a protocol which would allow individuals to interact with a bank in an anonymous way: Alice will be able to spend money from her account without the bank being able to tell where she is spending it.", "However, if Alice ever tries to spend the same digital coin twice, then she ends up revealing her identity to the bank, and the bank can prove that Alice double spent a coin (and consequently follow with legal action).", "Similar constructions of this kind include Alice being able to prove, for example, whether the bank is being honest or stealing her money.", "These solutions make sense when two conditions are met: (a) when there is a specially designated individual, such as a bank, who can be relied on to behave in a certain way because of a regulatory environment, and (b) when there is a realistic threat of legal action if such behavior is not observed.", "But what if the \"bank\" Alice is using is an anonymous individual on the internet?", "Or even if the bank is a known start-up, what if it operates in a foreign country?", "Even if Alice can prove that the bank is cheating, whether Alice can reasonably follow up with punitive action is a non-trivial concern.", "One of the key problems decentralized payment systems solve is being able to co-ordinate a large number of individuals to form a payment system, even in the absence of a strong regulatory environment.", "This makes such a payment system highly robust and accessible to anyone with the minimal ability to send messages across the internet.There are a number of other security advantages which are often argued: for example, there is now no longer a central bank the government can use to change the money supply.", "Because we cannot rely on any fixed subset of individuals to behave in a particular way in this context, the notion of fault tolerance against any $f$ failures (i.e.", "bad behavior by any $f$ participants) is therefore certainly a necessary requirement for any decentralized payment system to have.", "Decentralized payment systems therefore traditionally focus on providing a protocol individuals can follow, so that even if any $f$ individuals behave badly, the payment system will still function correctly.", "In contrast to distributed consensus in formal models, our theoretical understanding of cryptocurrencies is still relatively underdeveloped and an active area of research.", "For example, a significant portion of current research is focused on just understanding and formalizing Bitcoin's [21] particular implementation of a distributed payment system [39], which operates via blockchain consensus.", "Other researchers are working on adapting known solutions to Byzantine Agreement to work over the internet as distributed payment systems [42], while some members of the cryptocurrency community attempt to more informally generalize the ideas behind the blockchain protocol to achieve greater transaction speeds via \"tangles\" [41].", "This research area is very new and constantly evolving.", "We will give a brief summary of the key model differences between a practical payment system which works on the internet, and a protocol which might operate in a fault tolerant model of the previous section.", "We will then briefly outline the high level idea of how Bitcoin facilitates a payment system.", "Since the vast majority of prominent cryptocurrencies operate on similar principles, this will be a faithful representation to keep in mind when thinking about current implementations of practical payment systems.", "The details of this representation are listed purely to give context, but are not needed for the rest of this paper.", "The key model differences between the internet and the \"fault tolerant\" model given in the previous section are as follows: All participants do not necessarily know each other (the \"communication graph\" is not fully connected); instead, processes only know of and can message a few neighbouring processes.", "They therefore communicate to others by \"gossiping\" to neighbouring processes.", "For example, if $P_{a}$ wants to send the message \"I pay $P_{b}$ $\\$1$ \", $P_{a}$ will send this message to its neighbours, and request that the message be inductively forwarded to their neighbours.", "Fault tolerance in different network communication typologies has been studied [2], but it is perhaps unclear how to model the connectivity of arbitrary participants on the internet in a robust way.", "Processes are usually allowed to be offline: for example, they can \"opt out\" of participating in a protocol at arbitrary times, and then rejoin later.", "Defining a notion of fault tolerance in networks where certain nodes can be \"online\" or \"offline\" has been worked on in [19] and [34].", "There are substantial financial incentives for processes to behave in non-trivial ways.", "Modeling the incentives of processes is actively being studied from a game theory perspective, particularly in the context of Bitcoin [43], [26].", "For example, a number of results have showed that Bitcoin is not incentive compatible, in the sense that even two thirds of the participants are honest, it can be more financially profitable for processes to behave dishonestly.", "[38], [32], [28], [29], [30] Despite these differences, Bitcoin is still an empirically successful algorithm at achieving distributed consensus between collections of anonymous individuals.", "At a high level, Bitcoin works as follows: at each round $i$ , all honest participants will reach agreement on a block of new transactions.", "This block is then appended to the list of transactions which have been agreed on previously; thus all honest participants have a consensus about who has paid whom and by how much.", "Thus when someone wants to make a new transaction, all honest participants can check the list of transactions they have already agreed on to verify that there is sufficient balance for the transaction to go through.", "Define a ledger $L_0$ , which consists of all transactions currently processed by the network at time $t=0$ .", "Practically, this might consist of a single entry $(pay,NULL,Alice,100)_{Null}$ , indicating that $Alice$ starts out with 100 bitcoins at $t=0$ .", "Now, we imagine a sequence of rounds $i=0,1,\\dots $ .", "During each round, any number of unknown participants may try to send and receive payments.", "At $i=0$ , Alice is the only one with a positive balance, so only she can make a payment.", "At round $i$ , a participant $Bob$ with non-zero balance might want to pay Alice 1 Bitcoin.", "He does this by gossiping the signed message $(pay,Bob,Alice,1,id)_{Bob}$Here id is a unique identifier to his neighbours, hoping that everyone will eventually receive this message.", "At the end of round $i$ , a random leader is elected from the set of all online participants.", "The ability to elect a random leader is one of the central ideas in being able to extend classical solutions for Byzantine Agreement to those which work when the participants are unknown.", "For now, let's take it on faith that at the end of round $i$ , all participants agree on a leader Charlie for round $i$ .", "Charlie, if he is honest, will look at all the transactions he has received through gossiping.", "He will then try to put them all together in an extension block $E_{i}$ , where $E_{i}$ contains the signed messages of all the payments in round $i$ .", "If he cannot include a particular transaction (because maybe Bob tried to pay Alice a bitcoin when he didn't have any balance), Charlie simply ignores this transaction.", "Finally, Charlie links $E_{i}$ with the ledger for the payments in all previous rounds $L_{i-1}$ , forming a chain $L_i=E_{i} \\rightarrow E_{i-1} \\rightarrow \\dots \\rightarrow L_0$ .", "Charlie then signs and publishes $L_i$ , and everyone agrees that everyone's balance at the beginning of round $i+1$ is as reflected by the payments listed in $L_i$ , provided the extension block $E_i$ is validall the signatures are correct, no-one has negative balance etc.", "We now briefly try to motivate why such a construction works, without getting tied down by details.", "Firstly, notice that regardless of how Charlie behaves, Charlie can never cause Bob to pay Alice an amount Bob did not intend to pay: this is because Bob needs to sign any payment before it can be included in an extension block.", "Thus, even if Charlie is dishonest, the most damage he can do is block all transactions by not including anyone's transaction in the next block.", "If we assume that $2f<N$ , then if we elect a random Charlie at each round, at least half of the time we will have an honest Charlie which will allow transactions to be appended to the chain.", "Thus we will always make some progress in processing transactions over time.", "The non-trivial part of Blockchain is electing a random leader: these details make the description just given slightly less clean.", "Bitcoin does this by allowing any process which would like to be the leader try to solve a random puzzle.", "As a concrete examplethere are less computationally intensive ways to achieve random leader election, imagine all processes have access to a random function $H$ ; if you can find a value $v$ such that the last $k$ digits of $H(v||E_{i} \\rightarrow L_{i-1})$ are all 0, then you can publish $v$ , together with an extension $E_i$ , to all processes to prove you are a leader for extending $L_{i-1}$ at round $i$ .", "processes who opt into this role, miners, are given incentives to do so by receiving a monetary reward in Bitcoin if they become the leader The assumption is that the only way to find such a $v$ is to try different random values, eventually finding such a $v$ after $2^{k}$ steps in expectation .", "The time at which the next leader finds such a $v$ is random.", "Moreover, there may be two distinct processes $P_{a},P_{b}$ , each with two distinct proposed extensions $E_{a},E_{b}$ , which both find valid values $v_{a},v_{b}$ for round $i$ .", "In this case, both extensions are accepted, and the network is currently uncertain whether $L_{a}=E_{a} \\rightarrow L_{i-1}$ or $L_{b}=E_{b} \\rightarrow L_{i-1}$ reflects the true balance.", "However, at round $i+1$ a new leader $P_{c}$ is elected.", "$P_{c}$ needs to choose which chain and extension they would like to propose; the value $v$ which $P_{c}$ finds is, with high probability, only valid for one of $L_{a},L_{b}$ , and so $P_{c}$ only extends one of these.", "We stipulate that honest leaders should only try to extend the longest chain, and moreover that the longest chain is the one which specifies the \"true\" balance of all participants.", "Under certain assumptions, one can show that after a few extensions, it is always clear whether a certain extension will continue to stay in the longest chain or will be forever rejected [24].", "If we can be sure $E_i$ is always in the longest chain, then we can consider all the transactions in $E_i$ has having been confirmed.", "In the particular example we have given, we are uniformally electing a leader proportional to how many times they choose to evaluate the function $H$ in an attempt to find a valid value $v$ , and we assume that no subset of participants utilizing a total of one half of the computational power of the network is all dishonest.", "This is like the condition $2f<N$ , where $N$ now represents the total computational power of the network.", "Presumably an adversary cannot maliciously coordinate so much computational powerThis has been shown to be a questionable assumption in practice, because large companies specialize in monopolizing computational power for mining bitcoin due to economies of scale.", "There are other choices for what $N$ can represent, for example the total money in the system.", "Then the condition $2f<N$ says that an adversary cannot coordinate more than $\\frac{1}{2}$ of the total wealth of the payment system to behave in an adversarial way., and when they cannot, Bitcoin is in some sense secure.", "But suppose that $f>\\frac{1}{2}N$ .", "Then the adversary can launch the notorious $51\\%$ attack: imagine Alice pays Bob $\\$1$ in block $E_1$ .", "After a few more blocks, where everyone behaves honestly, Bob sees the chain $E_5 \\rightarrow \\dots \\rightarrow E_1 \\rightarrow E_0$ .", "Since $E_1$ is so far down in the chain, ordinarily, if more than say two thirds of the computational power is honest, $E_1$ would stay in the longest chain forever with high probability.", "Bob sees that Alice's transaction is therefore confirmed, so he sends the physical goods Alice purchased.", "However, after receiving the goods from Bob, Alice uses her $51\\%$ computational power to produce a new block $E_1^{\\prime }$ with no transactions, and creates the chain $E_1^{\\prime }\\rightarrow E_0$ .", "Nothing has gone wrong yet; $E_5 \\rightarrow \\dots \\rightarrow E_1 \\rightarrow E_0$ is still the longest chain, so everyone agrees that $E_1$ (and hence Alice's payment to Bob) is confirmed, so Bob's balance is still $\\$1$ .", "But now Alice chooses to only extend the chain $E_1^{\\prime } \\rightarrow E_0$ .", "Because Alice has the majority of the computational power, after some amount of time she will be able to produce a long chain $E_k \\rightarrow \\dots \\rightarrow E_1^{\\prime } \\rightarrow E_0$ which is, with high probability, longer than any other chain that would have been produced by everyone else even if everyone else was only extending the chain $E_5 \\rightarrow \\dots \\rightarrow E_1 \\rightarrow E_0$ .", "Since the new longest chain no longer contains Alice's payment to Bob, Bob loses his $\\$1$ .", "Thus Bitcoin is not secure if $f>\\frac{N}{2}$ ." ], [ "Prior Work on Reducing Consensus", "A major drawback Bitcoin suffers from is the time it takes to create a new block extensionThis is a trade-off between security and efficiency: longer block times mean more stability, but slower transactions., and the large space required to store all transactions on a chain.", "As a result, Bitcoin can only handle on the order of ten transactions per second, compared to the tens of thousands per second achieved by modern credit card services.", "It is within this context that members of the cryptocurrency community have been trying to reduce the amount of information which needs to be agreed on through consensus in order for a transaction to occur: given the use of a consensus mechanism for transactions, how can we make it more efficient?", "Note that there's a nuanced difference between this question, and the one which starts by asking whether global consensus on transactions is needed at all.", "It is challenging to give a complete and accurate account of the efforts which have been pursued in this area.", "For example, there are over 2000 cryptocurrencies which are registered on CoinMarketCap aloneAs of March 2019..", "Many of these are slight variations of the blockchain protocol, tailored for a particular use case.", "Moreover, even when a purportedly novel solution to consensus is presented, it is usually done so in the following manner.", "A short whitepaper will be produced, sketching the software engineering details of how the protocol works.", "Sometimes there might be some discussion about various attacks against the protocol, and how they will not succeed.", "In rare cases, authors might heuristically argue that some statistical method will guarantee security by doing some calculations.", "But in almost all instances of practical releases of cryptocurrencies by the \"non-academic\" community, there are no proofs of security or correctness.", "Indeed, even the security of Bitcoin is a relatively open problem.", "It therefore makes it very difficult to assess which claims should be taken seriously: a software developer might publish a claim that they have a protocol with minimal use of consensus, but whether their protocol is provably secure in an adversarial model is another matter.", "Perhaps the most serious attempt to reduce the amount of consensus needed for transactions to occur is the lightning network [31].", "This is a solution designed to reduce the number of transactions which need to be globally published on the blockchain ledger.", "It is still a controversial solution within the blockchain communityFor example, Roger Ver and other prominent cryptocurrency figures are vocal skeptics [37], but it is currently being experimented with in a semi-live settingHowever, its launch has also repeatedly been delayed ever since its initial proposal in 2016.. By pursuing the question of consensus, we will be led to constructing a payment system in part which will turn out to have similar characteristics to the lightning network.", "We therefore include in the appendix a description of how the lightning network works, and how the constructions in Part relate to this prior work.", "This can be examined after reading Part , and we will draw the reader's attention to the appendix when this point comes." ], [ "Payment Systems", "Inspired by the task of trying to compare the consensus problem to the problem of sending payments over a distributed system, we now give a model and definition for a payment system.", "In particular, we would like our model to capture the minimal functionality any reasonable payment system should possess.", "Similar to the motivation of Byzantine Generals, we imagine $N$ participants who would like to decide on a set of rules such that, by only passing signed messages between them, they can create a system which allows people to send payments between each other.", "Such a system needs to be robust to the many $f$ individuals who would like to break the system due to financial incentives.", "Real currency has worked in the past as follows: individuals have unforgeable, or difficult to replicate, discrete physical objects (paper notes, shells, precious metals etc) which act as a marker that an individual has some value.", "These markers themselves need not have inherent value; what is important is the inductive belief that if Alice has such a marker, a second party Bob will accept the marker as payment.", "The only reason Bob accepts the marker as payment is because he too believes that he may find a third party Charlie who will accept the marker as payment from Bob, and so on.", "More formally, our definition of payment system should encapsulate the following notions Participants can hold some notion of \"marker\".", "If a participant holds a marker, this means that in the future, they can transfer the marker to someone else, and consequently lose the marker.", "If a second party receives a marker, they know who they received it from.", "Decentralized payment systems are concerned with providing these marking functionalities in the context where some (unspecified) participants may actively try to cheat.", "Physical currencies have the advantage that they cannot easily be replicated.", "In contrast, digital decentralized currencies are less obvious to implement because if one has a digital coin, it can easily be \"copied\".", "Concretely, if Alice has some protocol she can follow which involves interacting with some network participants $P$ and constitutes paying a coin to Bob, nothing stops her from repeating the same protocol with a disjoint set of participants $P^{\\prime }$ and paying the same coin to Charlie afterwards.", "Since $P,P^{\\prime }$ are distinct, they have no way of knowing that Alice spent her coin twice.", "Note however in our analogy, that if Alice gives Bob a shell as payment, there is no a priori need for Charlie to also agree that Alice gave Bob a shell (which would be the case if consensus about transactions is reached), unless perhaps Bob later chooses to pay Charlie.", "We now formalize these notions in the same fault tolerant model as Byzantine Consensus.", "Formally, we consider a set of processes $P_1,\\dots ,P_N$ which interact via point to point messages in the synchronous, authenticated setting of Definition REF .", "We break the time steps $t=0,\\dots $ into $K$ rounds, each round consisting of $T$ steps.", "Thus after $i$ rounds, $iT$ steps have passed.", "We imagine each process $P_n$ as starting with an initial balance of $v_{n,0} \\in \\mathbb {N}$ markers at the beginning of round 0, and we will notate $v_{n,i}$ to be the number of markers $P_n$ has at the beginning of round $i$ .", "In general, we will only be able to talk about $v_{n,i}$ for honest processes $P_{n} \\in \\mathcal {H}$ .", "At the beginning of round $i$ , each honest process $P_n$ receives an input $I_{n,i} \\in [N]$ .", "If $P_n$ is honest and $v_{n,i}=0$ , then we stipulate $I_{n,i}=n$ .", "Semantically, this means \"$P_{n}$ would like to send a coin to $I_{n,i}$ in round $i$ \".", "For the remaining $T$ steps of round $i$ , honest processors $P_{n}$ follow some protocol $p_{n}$ .", "We will let $\\mathcal {H}$ denote the set of all honest processes, and we assume that the adversary knows all future inputs of all processes.", "At the end of round $i$ , each honest process decides on A value $v_{n,i+1} \\in \\mathbb {N}, v_{n,i+1} \\ge 0$ , its balance at the end of round $i$ .", "A list of senders $S_{n,i} \\subset [N]\\cup \\lbrace \\perp \\rbrace $ which $P_{n}$ believes sent it coins in round $i$ , where we allow duplicate elements and $\\perp $ is a dummy sentinel value.", "Definition 6.0.1 We say that a collection of protocols $\\lbrace p_i\\rbrace _{i \\in [N]}$ is a solution to the payment system problem in the presence of an $f-$ Adversary with error probability $\\epsilon $ if, with probability at least $1-\\epsilon $ , for all rounds $i \\in [K]$ we have (safety S1, non-duplication) $\\sum _{P_n \\in \\mathcal {H}} v_{n,i}\\le \\sum _{P_n \\in \\mathcal {H}} v_{n,0}$ and $\\forall n \\in [N], v_{n,i} \\ge 0$ (safety S2, non impersonation) Suppose $P_{n_1},P_{n_2} \\in \\mathcal {H}$ , and $P_{n_1} \\in S_{n_2,i}$ .", "Then $I_{n_1,i}=n_2$ .", "(safety S3, self-consistency) Suppose $P_n \\in \\mathcal {H}$ .", "Let $\\delta =1$ if $v_{n,i}>0$ , 0 otherwise.", "Then $v_{n,i+1} = v_{n,i}+|S_{n,i}|-\\delta $ .", "(liveness L1) Suppose $P_{n_1} \\in \\mathcal {H}$ .", "Then $R_{n_1,i}\\subset \\lbrace P_{n_2} \\in \\mathcal {H}|I_{n_2,i}=n_1 \\wedge v_{n_2,i}>0\\rbrace $ These properties should hold over all input sequences over $K$ rounds, and over all initial value distributions $\\lbrace v_{n,0}\\rbrace _{n \\in [N]}$ .", "If at most one honest process $P_{n}$ receives an input $I_{n,i} \\ne n$ per round, we call this the single transaction per round model.", "Otherwise we refer to the multi-transaction per round model.", "The best way to get an intuition for these conditions is to consider the special case where $v_{1,0}=1$ , and $v_{n,0}=0$ for $n \\ne 1$ .", "In this case, after each round, call the (at most one) honest process which decides it has non-zero value the marked process.", "S1 simply says at most one honest process $P_{n}$ decides it is marked.", "S2 says that if $P_{n}$ thinks $P_{n^{\\prime }}$ was the marked process in the previous round, and $P_{n^{\\prime }}$ is honest, then $P_{n^{\\prime }}$ was indeed so.", "S3 and L1 together say that if $P_{n^{\\prime }}$ is marked and receives input $n$ , then $P_{n}$ will decide it is marked in the next round, and that the previous marked process was $P_{n^{\\prime }}$ .", "In this special case, we such a problem the marker game/marker problem.", "Note that this is necessarily a single transaction per round model.", "More formally, we have: Definition 6.0.2 (The Marker Problem) In the same setting as before, processes communicate for $K$ rounds and have to tolerate byzantine adversaries: If process $P_1$ is honest, it starts with a marker at the beginning of the first round.", "We say it is \"Marked\".", "Otherwise, the marked process is $\\perp $ (a dummy value indicating no marked process for this round).", "At the beginning of each round, the marked process $M \\in \\lbrace P_{n}\\rbrace _{n \\in [N]} \\cup \\lbrace \\perp \\rbrace $ is given an input $I_{i}=n \\in [N]$ .", "At the end of this round: (consistency) At most one honest process $P_n \\in \\mathcal {H}$ should decide that it has the marker and become the \"marked node\".", "(liveness) If $M,P_{n} \\in \\mathcal {H}$ , $I_{i}=n$ , then $P_{n}$ should decide that it now has the marker, and that the marked node in the previous round was $M$ .", "(non-impersonation) If $P_{i} \\in \\mathcal {H}$ becomes marked at the end of this round and $M=\\perp $ , then $P_{i}$ should decide that the previous marked node was $d \\in (\\lbrace P_{n}\\rbrace _{n \\in [N]}\\cup \\lbrace \\perp \\rbrace )\\setminus \\mathcal {H}$ .", "Similarly to the byzantine generals problem, we can define analogous metrics of solution complexity: Message Complexity Per Round: The total number of messages sent across the network by all honest processors per round.", "We let the amortized round complexity be this value divided by the number of honest processors which made a payment in the round in question.", "Signature Complexity Per Round: The total number of signatures sent across the network by all honest processors per round.", "Round Complexity: The number of time steps taken per round, i.e.", "$T$ .", "The first proposition we prove is that the marker game is as general as a full payment system: if we have a solution to the marker game, then we can construct one for a payment system as well with minimal overhead.", "It will therefore suffice to focus on understanding this simpler problem first.", "Proposition 6.0.1 Suppose a collection of protocols $\\lbrace p_n\\rbrace _{n \\in [N]}$ deterministically solve the marker game with round length $T$ for $K$ rounds.", "Then there exists a solution $\\lbrace p^{\\prime }_n\\rbrace _{n \\in [N]}$ which solves the payment problem with round length $T$ for $K$ rounds for any initial marker distribution $\\lbrace v_{n,0}\\rbrace _{n \\in [N]}$ .", "Let $V=\\sum _{n \\in [N]} v_{n,0}$ .", "The key idea is to have $p^{\\prime }_{n}$ simulate $V$ copies of the marker game solution.", "By permuting process labels, we can assume without loss of generality that for any $n \\in [N]$ , we have a solution to the marker game where $P_{n}$ starts as the marked process.", "For each $n \\in [N]$ , we obtain a group $g_n$ of $v_{n,0}$ copies of a solution to the marker game $\\lbrace p_{n^{\\prime }}\\rbrace _{n^{\\prime } \\in [N]}$ , where $P_n$ starts as the marked process.", "We define $p^{\\prime }_{n}$ to be the process which simulates the $V$ copies of the marker game for each group $g_{n^{\\prime }}, n^{\\prime } \\in [N]$ .", "By proposition REF , we know that we can assume each individual simulation has the same guarantees the marker game provides.", "$p^{\\prime }_{n}$ interacts with its simulations as follows: At each round $i$ , $p^{\\prime }_{n}$ decides that it has value equal to the number of times it is marked in each of its simulations, i.e.", "the number of marked simulations it has.", "If a marked simulation $p_{n^{\\prime }}$ of $p^{\\prime }_{n}$ decides that $s$ was the marked process in round $i$ , then $p^{\\prime }_{n}$ includes $s$ in $S_{n,i}$ .", "Now suppose $p^{\\prime }_{n}$ receives input $I_{n,i}$ in round $i$ .", "$p^{\\prime }_{n}$ does the following: First it tries to choose a marked simulation, otherwise it chooses no simulation (If $v_{n,i}>0$ , such a marked simulation can be chosen by definition).", "It then sets the input register of simulation $p_{n}$ to be $I_{n,i}$ , and continues to simulate all $V$ simulations for $T$ steps.", "This completes the description of $p^{\\prime }_{n}$ .", "For correctness, it suffices to check each condition: (safety S1, non-duplication) $\\sum _{P_n \\in \\mathcal {H}} v_{n,i}= \\textit { the total number of marked simulations across all processes } \\le V = \\sum _{P_n \\in \\mathcal {H}} v_{n,0}$ where the inequality follows by consistency of each simulation.", "$\\forall n \\in [N], v_{n,i} \\ge 0$ by construction.", "(safety S2, non impersonation) Suppose $P_{n_1},P_{n_2} \\in \\mathcal {H}$ , $P_{n_1} \\in S_{n_2,i}$ .", "This means $P_{n_1},P_{n_2}$ run each of their simulations honestly.", "Then some marked simulation $k \\in [V]$ of $P_{n_2}$ decided that $P_{n_1}$ sent its marker to $P_{n_2}$ in round $i$ .", "Suppose for the sake of contradiction that $I_{n_1,i} \\ne n_2$ .", "Then $P_{n_1}$ never set $n_2$ as input to simulation $k$ in round $i$ .", "If $P_{n_1}$ was marked in simulation $k$ at the beginning of round $i$ , then by liveness $I_{n_1,i} \\ne n_2$ is marked in the subsequent round, but this contradicts consistency of simulation $k$ at round $i+1$ .", "If $P_{n_1}$ was not marked in simulation $k$ at the beginning of round $i$ , then either non-impersonation (if there is no marked process in simulation $k$ at the beginning of round $i$ ) or liveness and consistency (if there is some marked process in simulation $k$ at round $i$ not equal to $P_{n_1}$ ) of the $k$ th simulation will give as the required contradiction.", "(safety S3, self-consistency) Suppose $P_n \\in \\mathcal {H}$ .", "$v_{n,i}$ is equal to the number of $P_{n}$ 's marked simulations at the beginning of round $i$ .", "By liveness and by construction, this is equal to $v_{n,i-1}$ plus the number of newly marked simulations in round $i-1$ , minus one if $P_{n}$ sends a marker in some simulation in round $i-1$ (which by construction occurs if $P_{n}$ has non-zero balance).", "(liveness L1) Suppose $P_{n_1},P_{n_2} \\in \\mathcal {H}$ .", "Then $I_{n_2,i}=n_1$ implies (by liveness) that by the end of the round, a marker game simulation belonging to $P_{n_1}$ decides $P_{n_2}$ was the previous marked process in this simulation, so $P_{n_2} \\in R_{n_1,i}$ by construction.", "One can apply a similar idea to show that a particular solution to the marker game implies one for a full payment system in the randomized model, with error probability at most $V\\epsilon $ (union bound).", "Note that in this construction, the round complexity is preserved.", "If in round $i$ this construction makes payments $\\lbrace I_{n,i}=r_n\\rbrace _{n \\in [N]}$ , then the associated message and signature complexities of this construction are equal to the sums of the corresponding complexities in each of the marker simulations which initiate these payments.", "Before moving to analyzing the marker problem in more detail, we preemptively comment on some concerns the reader may have about the definitions given Why do we only support sending discrete markers instead of general numerical values as payment?", "Would we not need to send $10^6$ markers just to make a single payment in some instances?", "Notice that traditional hard currency systems solve this problem by giving different discrete objects (i.e.", "notes) different values.", "By issuing a wide variety of different denominationsFor example, imagine a monetary system consisting of the notes $1,2,4,\\dots ,2^n$ .", "Then we can pay any integer value in the range $[0,2^{n+1}-1]$ using at nost $n$ notes, people are able to pay a wide range of values in cash with only a few notes/coins.", "In the construction of proposition REF , it is relatively simple to see how we might mark different simulations as corresponding to different values.", "Why do we require both the sender $P_{n_1}$ and the recipient $P_{n_2}$ to be honest for a payment to go through?", "We have two justifications for this: firstly, in a payment between two parties, it is generally implicit that both parties desire the payment to go through ($P_{n_1}$ wants to exchange the payment for goods, and $P_{n_2}$ would like to receive the payment).", "Thus each party can only harm their interests by behaving dishonestly.", "The second reason is semantics: one might be concerned by the notion that $P_{n_1}$ is unable to pay $P_{n_2}$ unless $P_{n_2}$ adheres to a strict set of rules, but this seems to undermine the ability of $P_{n_1}$ to be able to spend its money any way it chooses.", "Such a concern may be justified if we were dealing with hard currency backed by a central bank, where all merchants must e.g.", "accept paper currency as payment as decreed and enforced by rule of law.", "However, in the context of a decentralized payment system, where there is no central authority, there is no means by which to enforce that participants are forced to accept e.g.", "Bitcoin as a means of payment.", "We require $P_{n_2}$ to be honest in order for $P_{n_1}$ to be able to make a payment to $P_{n_2}$ , just as much as we require $P_{n_2}$ to believe that a cryptographic string of 0's and 1's is something it should provide goods to $P_{n_1}$ in exchange for.", "Nothing stops $P_{n_2}$ from simply turning off their laptop and deciding never to accept Bitcoin as a means of payment again.", "Why do we require non-impersonation?", "Why would there a priori be a problem with someone making a payment while claiming to be someone else?", "We give two reasons: firstly, this definition is easy to enforce: if $P_{n_1}$ wants to send a payment to $P_{n_2}$ , simply require that they first indicate this fact to $P_{n_2}$ by sending a signed statement of this intention.", "The second reason is that this condition gives us the convenience of being able to easily reduce unauthenticated communication to payment systemswe will say more on this in the next section.", "Why is the fault tolerant model the correct model of security for payment systems?", "This is an excellent question.", "Because crytocurrencies have been so strongly influenced by consensus work, the fault tolerant model is the trust model which has been traditionally assumed by the cryptocurrency community when analyzing the security of distributed payment systems.", "The primary reason we define payment systems in this model is therefore so that we can analyze the consensus assumption in the same context it is traditionally thought to hold in.", "Recall in section that we justified why such trust robustness is necessary in the context of the internet: when any particular subset of participants might be anonymous, or cannot reliably be legally challenged by law enforcement, we cannot rely on any particular subset of individuals to behave honestly.", "However, we never argued why such trust robustness is sufficient.", "For example, consider the following pathological example: For $P_1$ to send a marker to $P_N$ , each of $P_{2},\\dots ,P_{N-1}$ is given two choices, either (a) or (b).", "If at least one of these processes choose (a), then $P_1$ 's marker is sent through successfully.", "If all of $P_2,\\dots ,P_{N-1}$ choose (b), then the marker is randomly distributed to one of the participants $P_2,\\dots ,P_{N-1}$ .", "If we prescribe all honest processes must choose option (a), then the payment system just described is fault tolerant for any $N-3$ faults!", "Yet it is very unclear why we would expect any self-interested party to choose option (a) instead of option (b); said differently, it is unclear why honesty is the \"default\" behaviorIn contrast, \"honesty\" is a far more natural default state in the context of more traditional problems thought to be solved by consensus: For example, one common traditional motivation for Byzantine Agreement is that if you have a collection of processors in an airplane which communicate together, you would like to make sure that they still collectively reach a valid decision even when there are some hardware faults.", "In this context, it is clear why honesty is the default behavior – the processors were designed to be honest in the first place.. Thankfully, the payment systems we primarily consider will tolerate any $N$ faults, and will not suffer such pathological cases.", "Regardless, we encourage the reader to think about precisely what the fault tolerant model of trust means in any particular payment system." ], [ "Payment Systems in the Fault Tolerant Model", "In this section, we will study the relation between the marker problem and the byzantine consensus problem in the synchronous fault tolerant model.", "While this setting is less natural for real world payment systems, it has been well studied for Byzantine consensus, and provides a benchmark on which to compare the problems of consensus and payment." ], [ "Reductions", "We begin by analyzing reductions between the two problems.", "The first observation is that a solution to the Byzantine Broadcast problem immediately implies a solution for the marker problem through a black box reduction: in particular, Byzantine Broadcast is at least as hard as the marker problem.", "Proposition 7.1.1 There exists a black box reduction from the marker game to Byzantine Broadcast: Suppose $\\lbrace p_n\\rbrace _{n \\in [N]}$ is a solution which solves the Byzantine Broadcast problem with error $\\epsilon $ , tolerating $f$ faults, with round complexity $T$ , and message and signature complexities $mc,sc$ .", "Then for any $K \\in \\mathbb {N}$ , there exists a $K$ round solution to the marker problem with message and signature complexity $mc,sc$ per round, error $K\\epsilon $ , and where each round uses $T$ steps of communication.", "We concatenate together $K$ rounds of Byzantine Broadcast, where at each round the new marked process is the leader of the next round.", "The marked process $M$ of round $i$ broadcasts their input $I_{M,i}$ to the network at the beginning of the round, and all honest participants run Byzantine Broadcast with $M$ as the leader.", "At the end of the round, all honest processes decide that $I_{M,i}$ is the new marked process, and continue inductively.", "We initialize process 1 as being the marked in the first round.", "We need to check that such a construction satisfies the conditions of definition REF .", "Note by proposition REF , we can assume that with probability at least $1-K\\epsilon $ , all simulations of Byzantine Broadcast runs successfully.", "Conditioning on this event, we can assume that after the $i$ th round, all honest processes reach consensus on a new marked process for round $i+1$ .", "Let $M$ be the marked process in round $i$ , and let $I_{M,i}=n$ .", "consistency: All honest processes reach consensus on a single value at round $i$ , so at most one honest processes decides it is marked at round $i$ .", "liveness: if $M$ is honest, then all honest processes decide on the proposed value $n$ .", "If $n$ is honest, then $P_{n}$ decides $n$ at the end of round $i$ , so liveness is satisfied.", "Since $P_{n}$ is honest, it knows that $M$ was the previous marked process because it decided this in the previous round.", "non-impersonation: If $P_{n}$ is honest, then inductively $P_{n}$ decided on the leader/marked process of round $i$ at the end of round $i-1$ .", "If $M \\in \\mathcal {H}$ , then $n$ decides $M$ .", "If $M \\notin \\mathcal {H}$ , then $n$ decides $M \\notin \\mathcal {H}$ .", "This black box reduction gives two immediate corollaries: Corollary 7.1.1 For any $f<N, K \\in \\mathbb {N}$ , there exists a deterministic solution to the $K$ round marker game which tolerates up to $f$ faults, has message complexity $\\mathcal {O}(Nf)$ per round, signature complexity $\\mathcal {O}(Nf^2)$ per round, and $T=\\mathcal {O}(f)$ steps per round.", "This follows from known solutions to Byzantine Broadcast (proposition REF ) and the black box reduction (proposition REF ).", "Corollary 7.1.2 For any $f$ such that $3f<N$ and any $K,\\gamma \\in \\mathbb {N}$ , there exists a randomized solution to the $K$ round marker problem which tolerates up to $f$ faults, has message complexity $\\mathcal {O}(\\gamma \\log (K)N^2)$ per round, and $T=\\mathcal {O}(\\gamma \\log (K))$ steps per round, with error $2^{-\\Omega (\\gamma )}$ .", "By corollary REF , there exists a randomized solution to Byzantine Broadcast tolerating $f$ faults with signature and message complexity $\\mathcal {O}(\\gamma \\log (K)N^2)$ , running in $\\mathcal {O}(\\gamma \\log (K))$ steps, and with error probability $\\frac{1}{K}2^{\\Omega (\\gamma )}$ .", "We now apply proposition REF to concatenate the Byzantine Agreement protocols together.", "We comment that this black box reduction from payment systems to Byzantine Broadcast leads to a kind of \"global consensus\" about the state of the network at the end of each round, and is very reminiscent of the idea of a public ledger incorporated in cryptocurrencies which use blockchains [35].", "On the other hand, while a Byzantine Broadcast solution can easily solve the marker problem, it is less clear how a solution to the marker problem relates to Byzantine Broadcast.", "In particular, the safety and liveness assumptions of the marker problem are far more local.", "For example, even in $N-1$ payments, it does not follow that by the end of each round, honest processes will be in agreement with which process has been marked.", "Indeed, to get a sense of this locality, consider the following proposition: Proposition 7.1.2 There exists a deterministic solution to the marker problem which tolerates $f$ faults for $f<N$ , such that even when all nodes behave honestly, after $N-1$ rounds where nodes $1,\\dots ,N-1$ have each received a payment, node $N$ does not receive a single message during all rounds.", "We will prove proposition REF later in section REF .", "This proposition alone doesn't necessarily say anything: maybe there is some way we can combine solutions to the marker problem which efficiently solves Byzantine Broadcast, and shows us that payment systems are \"just as hard\" to construct as solutions to the Byzantine Broadcast problem.", "We have just given a black box reduction from the Marker Problem to Byzantine Broadcast; a natural question is therefore whether there exists a reduction the other way: Question 7.1.1 Does there exist a reduction from Byzantine Broadcast to the Marker Problem?", "To answer this question, we first need to formalize what we mean by a reduction.", "In particular, notice that we will allow ourselves even more power than is allowed by a black box reduction: our reduction will be able to look at the \"state\" of each process and its sent and received messages and possibly take advantage of the inner workings of a solution to the marker problem.", "If it is indeed true that every solution of a payment system essentially makes use of consensus, then by controlling the payment system and looking at its operation, we should be able to make all processes reach agreement on some value.", "We formalize this idea with the following definition: Definition 7.1.1 A reduction from Byzantine Broadcast to a solution $\\lbrace p_n\\rbrace _{n \\in [N]}$ of the Marker Game which tolerates $f$ faults is a protocol solution $\\lbrace p^{\\prime }_n\\rbrace _{n \\in [N]}$ to the Byzantine Broadcast problem tolerating $f$ faults, with the following exceptions Processes can no longer directly send messages across the network, but can only interact through access to $W$ uniquely identifiable simulations of the Marker Game solution $\\lbrace p_n\\rbrace _{n \\in [N]}$ (by permuting labels, we allow different simulations to correspond to different processes starting with the marker).", "Each reduction step $i$ corresponds to a single round execution of the marker problem solutions.", "At the beginning of reduction step $i$ (round $i$ in the marker problem), process $P_n$ can write to the input register $I_{n}$ of the $j\\in [W]$ th simulation of the marker game, and can view what the $j$ th simulation decides.", "At step $i$ , process $P_n$ can also look at the inner workings of simulation $w \\in [W]$ : for example, the memory state of the simulation, and all sent and received messages for $P_n$ indirectly made by running the $w$ th simulation.", "We call such a solution a strong reduction.", "If we remove condition REF and allow processes to send messages across the network as well, we call this a weak reduction.", "We say that Byzantine Broadcast is strongly/weakly reducible to the Marker Problem with fault tolerance $f$ if Byzantine Broadcast with fault tolerance $f$ is reducible to every solution $\\lbrace p_n\\rbrace _{n \\in [N]}$ of the Marker Problem tolerating $f$ faults.", "Again, we comment that if a black box reduction from Byzantine Broadcast to the Marker Problem exists, then such a reduction would show Byzantine Broadcast is strongly reducible to the Marker Problem.", "Thus this definition of reduction is weaker than that of a black box reduction.", "For strong reductions, we restrict processes to only be able to use the interface of the payment system to avoid the trivial case where processes simply ignore the payment system and implement Byzantine Broadcast by sending messages to each other.", "What should we be aiming for in such a reduction?", "Since the reduction from the Marker Problem to Byzantine Broadcast only used a single black box call to Byzantine Broadcast per round, and terminated immediately, at best we might hope that conversely there is a clever way to send around $\\mathcal {O}(1)$ coins in the network in $\\mathcal {O}(1)$ steps such that we force every honest process to essentially agree on something.", "Certainly, we should be able to do this with any payment system which works by reaching consensus at the end of every round.", "On the other extreme, focus on the easier problem of reducing just binary Byzantine agreement to the Marker Problem.", "We know there exist solutions to Byzantine Agreement in the unauthenticated setting (and without any help of a marker solution black box) which run in $\\mathcal {O}(f)$ steps and send $\\mathcal {O}(f^2)$ messages when $f=\\mathcal {O}(N)$ [4].", "These solutions imply the existence of naive reductions of similar complexity which make no use of the essential properties of a payment system whatsoever.", "Indeed, notice if we were to have access to a payment system which can send arbitrary valuesBy a previous footnote, certainly we can send an encoding of value $v \\in \\mathbb {N}$ with $\\mathcal {O}(\\log (v))$ markers., we can naively, via a payment system, simulate having access to a network which can send unauthenticated messages: if all processes start out with an arbitrarily large amount of money, an honest process $P_{n_1}$ can send an unauthenticated message $m$the only values we really need to send over the network for the solution of binary Byzantine Broadcast are the names of different processes and the values $0,1$ to $P_{n_2}$ by sending value $ENCODE(m)$ to $P_{n_2}$ , where $ENCODE(m)$ converts $m$ into an integer.", "In the next step, $P_{n_2}$ receives this value and determines that the sender was $P_{n_1}$ .", "It turns out that the answer to question REF is no; in particular, given simulation access to a payment system, in general we can't do much better at achieving consensus than we would have, had we not been given a payment system at all.", "Payment systems don't significantly help us solve consensus: Theorem 1 For any $f \\in \\mathbb {N}$ such that $3f<N$ , Byzantine Broadcast is not strongly reducible to the Marker Problem in fewer than $\\frac{f+1}{3}$ simulation steps.", "If $W$ copies of the Marker Problem are used in the reduction, then the number of steps of any reduction is at least $\\max (\\frac{f+1}{3},\\Omega (\\frac{N}{fW}))$ .", "Consequently, there also does not exist any black box reduction from Byzantine Broadcast to the Marker Problem in fewer than this many black box steps.", "For any $f \\in \\mathbb {N}$ such that $3f<N$ , Byzantine Broadcast is not weakly reducible to the Marker Problem in fewer than $\\frac{f+1}{3}$ simulation steps.", "If $W$ copies of the Marker Problem are used and $M$ messages are sent in the reduction, then the number of steps of any reduction is at least $\\max (\\frac{f+1}{3},\\Omega (\\frac{N-M}{fW}))$ .", "Consequently, there also does not exist any black box reduction, even when allowed to send additional authenticated messages across the network, from Byzantine Broadcast to the Marker Problem in fewer than this many black box steps.", "We note that since Byzantine Broadcast is reducible to Byzantine Agreement (proposition REF ), the analogous statements hold for Byzantine Agreement as well.", "We prove this theorem in two parts.", "First, we give a solution to the marker problem with round complexity 3, $\\mathcal {O}(f^2)$ signature complexity per round, and $\\mathcal {O}(f)$ message complexity per round.", "Construction 1 Arbitrarily designate $3f+1$ \"broadcast\" processes.", "(Step 1): At round $i$ , the marked process $M$ sends the signed message $m_{n}=$ \"I want to pay process $n$ ; here is a proof $p$ I am the marked process at round $i$ \" to each of the broadcast processes.", "(Step 2): If a broadcast process receives a valid message of the form $m_{n}$ , it signs and sends a message of the form $r_{n}=\"M$ pays $n$ at round $i\"$ to process $P_{n}$ .", "If it receives multiple valid messages of this form, it sends at most one message to a potential recipient during this round.", "(Step 3) If a process $P_{n}$ receives at least $2f+1$ signed messages of the form $r_{n}$ , then it decides it has the marker and $M$ was the sender.", "Otherwise it decides that it does not have the marker.", "The $2f+1$ signed messages constitute the \"proof\" that $P_{n}$ is the marked process at round $i+1$ .", "(During the first round, the \"proof\" that $P_1$ is the leader is just an empty string).", "Proposition 7.1.3 For any $K \\in \\mathbb {N}$ and any $f$ such that $3f<N$ , construction REF gives a solution to the marker problem which tolerates $f$ faults, has round complexity $T=3$ , and message and signature complexity per round $\\mathcal {O}(f),\\mathcal {O}(f^2)$ resp.", "Construction REF uses $T=3$ per round, and honest processes send at most $2\\times (3f+1)$ messages and $\\mathcal {O}((2f+1)(3f+1)+(3f+1))=\\mathcal {O}(f^2)$ signatures per round.", "We claim this gives a valid construction.", "To see this, first note that at any round $i$ , at most one process $P_{n}$ receives $2f+1$ messages of the form $r_n$ .", "Suppose not, and two distinct processes $P_{n},P_{n^{\\prime }}$ have this property.", "There are at least $f+1$ signers in common between the broadcasters who signed the messages for $P_{n}$ , and the broadcasters who signed the messages for $P_{n^{\\prime }}$ .", "At least one of these broadcasters is honest, contradicting that each honest broadcaster send at most one message during any particular round.", "Consistency: At most one honest process receives the required number of signatures to decide it is the marked node.", "Liveness: If $M,P_{n} \\in \\mathcal {H}$ , $M$ sends a message to all broadcast processes.", "At least $3f+1-f=2f+1$ broadcast processes forward a message to $P_{n}$ , and $P_{n}$ then decides it is the new marked process and that the previous marked process is $M$ .", "Because only $P_{n}$ has a proof it is the marked process, it can elect a new marked process in the next round if it behaves honestly.", "Non-impersonation: If $M=\\perp $ , then no honest process has a proof that it is marked at round $i$ .", "Consequentially, no honest $P_{n} \\in \\mathcal {H}$ which decides it is marked will decide that the previous marked process was in $\\mathcal {H}$ .", "We comment that solutions for the deterministic marker problem and randomized Byzantine Broadcast both have expected round length $T=\\mathcal {O}(1)$ .", "Construction REF for the marker problem in the randomized setting is an improvement of the $\\log (K)$ round complexity derived in corollary REF .", "We can now give a proof of Theorem REF : The key idea is to leverage the lower bound results from propositions REF , REF and REF to derive a contradiction.", "First consider the strong case, and suppose there is a reduction using fewer than $\\frac{f+1}{3}$ simulation steps.", "By proposition REF , this implies a solution to Byzantine Broadcast which simulates multiple copies of a marker game, where each simulation step corresponds to 3 network time steps.", "This solution tolerates $f$ faults and runs in fewer than $\\frac{f+1}{3}\\times 3=f+1$ steps, contradicting the lower bound in proposition REF .", "Now suppose that we only use $o(\\frac{N}{fW})$ rounds.", "By proposition REF , this implies a solution to Byzantine Broadcast with signature complexity $o(\\frac{N}{fW} \\times f^2 W)=o(Nf)$ , contradicting proposition REF (we note we could have derived the same bound by considering the message complexity).", "For the weak case, we can repeat the argument which considers the round complexity of Byzantine Broadcast to get that the number of simulation steps is at least $\\frac{f+1}{3}$ .", "Suppose for the sake of contradiction that the number of simulation steps is $o(\\frac{N-M}{fW})$ .", "Then by proposition REF , this implies a solution to Byzantine Broadcast with message complexity $o(N)$ , contradicting the lower bound of proposition REF ." ], [ "Best Case Message Complexity", "The result from the previous section tells us that, at least from the perspective of reductions in the model we have given, consensus and payment are fairly different problems.", "This gives us hope that we might be able to construct practical payment systems which behave fundamentally differently compared to any consensus based solution.", "With this goal in mind, the candidate property we choose to focus on for the rest of this paper is the best case message complexity of a payment system, which we define to be the number of messages sent per transaction in the event when all processes behave honestly.Note that the payment system is still robust to $f$ processes behaving dishonestly, but the number of messages sent may increase in this case.", "In particular, we will do our best, ignoring all other complexity considerations, to focus on building a payment system with good best case message complexity under reasonable conditions.", "We make this choice for the following reasons: While round complexity has strong lower bound results in the fault tolerant model, discrete time steps and round lower bounds don't realistically translate to protocols over the internet in practice.", "The signature complexity of a solution in the fault tolerant model does not necessarily translate to practice either: there are cryptographic techniques to \"compress\" chains of signatures in a single message into a short summary string.", "Notice that a message complexity of $\\Omega (N)$ per transaction in the best case is an innate property of any consensus based solution which reaches agreement on all transactions.", "This is because all participants need to agree on the updated state of the network after a transaction, and so need to all send/receive at least one message.", "By focusing on reducing the message complexity as much as possible, we will be forced to end up with solutions to the payment problem which are inherently local: for example, if only $\\mathcal {O}(1)$ messages are typically sent per transaction, this would intuitively need to rely on a method very different to global consensus of transactions.", "Best case complexity can be a realistic benchmark for practical payment systems.", "For example, suppose we only send large amounts of messages when processes are behaving dishonestly.", "In practice, if a single individual is causing excessive network stress and this can be detected, they can be ignored/removed from the network.", "Moreover, we can naturally build into a payment system financial costs or fees which are associated with how many messages are sent over the network.", "With this goal in mind, we begin by studying the best case complexity of payment systems.", "Recall from proposition REF that every solution to Byzantine Broadcast requires sending $N(f+1)/4$ signatures even when all processes behave honestly.", "The message complexity of the marker problem, and payment systems in general, is slightly more more nuanced.", "Recall we showed a construction for a payment system which always sent $\\Theta (f)$ messages per payment.", "It turns out that there are payment systems which might only send any number between 1 and $N$ messages per payment.", "Moreover, if we make certain assumptions about the distribution of income in a payment system, we can get away with sending significantly less than $\\Omega (N)$ messages per payment.", "We start by giving an analogous result for some metric of the message complexity of any deterministic payment system which closely mirrors that of proposition REF for Byzantine Broadcast: Proposition 7.2.1 For any $K=1$ round, $T$ step deterministic solution to the marker problem tolerating $f$ faults, let $z_{n}$ denote the number of messages sent by all processes after $T$ steps when the marked process $M=P_1$ receives input $I_{1}=n$ and all processes behave honestly.", "Then $\\sum _{n \\in [N]} z_{n} = \\Omega (Nf)$ We note that this result holds even in the case of a weak adversarial model, where the adversary is only able to simulate two copies of honest protocols (it does not take advantage of being able to see the entire network state).", "To prove this, first we will give a lemma, which in itself will be instructive in constructing a new solution to the marker problem.", "Let $X(n)$ be the set of processes which either send or receive a message during the first $T$ steps when $P_1$ receives input $I_{1}=n$ (so $z_n\\ge \\frac{1}{2}|X(n)|$ ), and all processes behave honestly.", "Lemma 7.2.1 Let $P_{n_1},P_{n_2} \\in \\lbrace P_{n}\\rbrace _{n \\in [N]}\\setminus \\lbrace P_{1}\\rbrace $ , and $D:= X(n_1) \\cap X(n_2)$ .", "Then either (i) $P_{n_1}$ is contained in $X(n_2)$ or $P_{n_2}$ is contained in $X(n_1)$ , or (ii) $|D|\\ge f-1$ .", "Suppose the claim were false: namely $P_{n_1},P_{n_2} \\notin X(n_1) \\cap X(n_2)$ , but $|D| \\le f-1$ .", "We now give a dishonest protocol each of the processes in $Z:=D\\cup \\lbrace P_1\\rbrace $ can follow ($|Z|\\le f$ ) which allows $P_{1}$ to pay both processes $P_{n_1}$ and $P_{n_2}$ its coin (i.e.", "\"double spend\"), resulting in both processes deciding they are the marked process for the next round, contradicting the consistency property of definition REF .", "The scenario is as follows: Figure: Lemma Have each node in $Z$ run two copies of the honest protocol (without nonces); the first copy corresponds to a simulation where $P_1$ pays $P_{n_1}$ , and the second simulation corresponds to a simulation where $P_{1}$ pays $P_{n_2}$ .", "When $P_{n_{a}},P_{n_{b}} \\in Z$ send messages to each other, they prepend a label $\\in \\lbrace 0,1\\rbrace $ to their messages to indicate which simulation copy the message belongs to.", "When they send messages to honest processes not in $Z$ , they omit this label.", "When they receive messages from processes not in $Z$ , they can infer which simulation the message belongs to because the sets $X(n_1)\\setminus Z, X(n_2)\\setminus Z$ are disjoint.", "To start off the simulation, we have $P_{1} \\in Z$ run two copies of honest protocols for $P_{1}$ , one where it \"imagines\" its input being $I(1)=n_1$ , and the other where it imagines $I(1)=n_2$ .", "Notice the following: after $T$ steps, the sent and received messages for all processes in $X(n_1)-Z$ are identical to those in the case where all processes are honest and $P_{1}$ has input $I(1)=n_1$ .", "Thus $P_{n_1}$ decides it is marked after $T$ steps.", "But the same argument tells us that $P_{n_2}$ decides it is marked as well, leading to a contradiction.", "Notice that in Construction REF , we always have $|D| \\ge 3f+1$ , and so this construction represents the \"one extreme\" of the condition in Lemma REF , namely we always have $|D|\\ge f-1$ .", "One might therefore wonder if it is possible to construct a solution which always satisfies the other extreme.", "Indeed, after finishing the proof of Proposition REF , we will construct a deterministic solution to a payment system which respects the \"other extreme\" of this condition, namely it will always be the case that either $P_{n_1}$ is contained in $X(n_2)$ or $P_{n_2}$ is contained in $X(n_1)$ .", "For now, we complete the proof: By lemma REF , $\\forall P_{n_1},P_{n_2} \\in \\lbrace P_{n}\\rbrace _{n \\in [N]} \\setminus \\lbrace P_1\\rbrace $ , either (i) $P_{n_1} \\in X(n_2)$ or $P_{n_2} \\in X(n_1)$ , or (ii) $|X(n_1)\\cap X(n_2)|\\ge f-1$ .", "Define $K_1=\\lbrace n \\in [N]\\setminus \\lbrace 1\\rbrace ||X(n)| \\ge f-1\\rbrace $ and its relative complement $K_2=[N]\\setminus (\\lbrace 1\\rbrace \\cup K_1)$ By condition (ii), it is easy to see that $\\sum _{n \\in K_2} |X(n)| \\ge \\frac{|K_2|(|K_2|-1)}{2}$ because for each unordered pair $(n_1,n_2)$ with $n_1,n_2 \\in K_2$ we have either $P_{n_1} \\in X(n_2)$ or $P_{n_2} \\in X(n_1)$ .", "Thus we have $2\\sum _{n \\in [N]\\setminus \\lbrace 1\\rbrace } z_n \\ge \\sum _{n \\in [N]\\setminus \\lbrace 1\\rbrace }|X(n)|$ $\\ge \\min _{|K_2| \\in \\mathbb {R}_{+}} \\frac{|K_2|(|K_2|-1)}{2}+|K_1|(f-1)$ $=\\min _{|K_2| \\in \\mathbb {R}_{+}} \\frac{|K_2|(|K_2|-1)}{2}+(N-1-|K_2|)(f-1)$ $=f(N-\\frac{1}{2})-N+\\frac{7}{8}-\\frac{1}{2}f^2=\\Omega (Nf)$ As a sanity check, note that Construction REF satisfies this lower bound and is tight: we have $\\sum _{n \\in [n]} z_n = N \\times 2 \\times (3f+1)=N(6f+2)$ .", "The intuition for Lemma REF is as follows: either the intersection $X(n_1)\\cap X(n_2)$ is $\\Omega (f)$ , so we can guarantee that there is some honest process within this intersection who will prevent $P_{1}$ from double spending.", "This is very similar in spirit to usual Byzantine Consensus reasoning.", "If this isn't the case, then the only other way we can guarantee $P_{1}$ can't double spend after paying $P_{n_1}$ is if $P_{n_2}$ itself saw $P_{1}$ spending its coin already, and hence $P_{n_2} \\in X(n_1)$ ; this is the point at which the reasoning for payment systems formally differs from that of Byzantine Consensus in the fault tolerant model, because we stipulate that $P_{n_2}$ must be honest if it is to have any protection from this double spendingwheres in Byzantine Consensus, we wouldn't be able to assume that this particular fixed process behaves honestly.. How might we construct a payment system where the property $P_{n_2} \\in X(n_1)$ or $P_{n_1} \\in X(n_2)$ always holds (in general, even when $P_1$ isn't the starting marked process)?", "One (and the only) way to do this is to connect all processes together in a directed cycle, and stipulate that payments can only move \"across\" the cycle.", "Since any two paths on a directed cycle originating from a common point always meet at some endpoint on one of the paths, this will give us the desired property.", "Using this idea, we now try to give another solution to the marker problem where the number of messages sent per transaction is highly variable – some transactions only require $O(1)$ messages in the best case, but only under highly unrealistic assumptions about the distribution of payments.", "In part , we will then use this construction as a building block to outline how we can achieve $\\mathcal {O}(\\log (N))$ messages per transaction in the best case under more reasonable assumptions about the distribution of payments.", "Figure: Illustration of a payment from process P 0 P_0 to process P 3 P_3 in construction .Construction 2 (Cycle Coin): For simplicity, we will first describe a construction which almost works, and requires all processes (regardless of whether they are faulty or not) respond in a particular way which is detectable.", "We will then describe how to remove this assumption.", "We number the $N$ processes from $0,\\dots ,N-1$ .", "Begin by connecting all processes into a cycle, with $P_{n}$ directionally connected and leading to $P_{n+1 \\text{ mod } N}$ .", "We let $P_{a,b}$ denote the path along this graph which connects $P_{a}$ to $P_{b}$ , with $P_{a}$ included and $P_{b}$ excluded.", "If $a=b$ , then we let $P_{a,b}$ be the empty path.", "Recall if we have values $v_i$ , we let $(\\dots ((v \\cdot v_1)_{P_{i_1}}\\cdot v_{2})_{P_{i_2}} \\dots \\cdot v_{i_k})_{P_{i_k}}$ denote the string produced when $v \\cdot v_1$ is signed by $P_{i_1}$ , the resulting string is concatenated with $v_2$ and then signed by $P_{i_2}$ and so on.", "Notionally, we rewrite this as $(v)[v_1 P_{i_1},\\dots ,v_k P_{i_k}]$ .", "We define a chain of length 0 ending at $P_0$ and extended by $P_0$ to be the string $(\"\")_{P_0}=()[P_0]=[P_0]$ .", "We define a chain of length $i$ ending at $P_{end}$ and extended by $P_{ext}$ to be a string of the form $(c)[P_{i_1},xP_{ext},P_{i_2},xP_{ext},P_{i_3},xP_{ext},\\dots , P_{i_k},yP_{ext}]$ , where $c$ is a chain of length $i-1$ ending at $P_{ext}$ , and $P_{ext}=P_{i_1},\\dots ,P_{i_k}$ are the processes along the ordered path $P_{P_{ext},P_{end}}$ .", "We let $(n_1,...,n_k)$ denote a chain $c$ of length $k$ constructed by a chain of length 1 ending at $P_{n_1}$ , extended by $P_{n_1}$ to a chain of length 2 ending at $P_{n_2}$ , and so forth, and finally ending at $P_{n_k}$ (note that there is a bijective correspondence between a chain $c$ and its representation $(n_1,...,n_k)$ ).", "As some concrete examples where $N=6$ : $(0)=[P_0,(yP_0)]$ $(1)=[P_0,(P_0,yP_0)]$ $(2)=[P_0,(P_0,xP_0,P_1,yP_0)]$ $(3)=[P_0,(P_0,xP_0,P_1,xP_0,P_2,yP_0)]$ $(1,3)=[P_0,(P_0,xP_0),(P_1,xP_1,P_2,yP_1)]$ The chain of length 0 corresponds to the empty tuple $()$ .", "Given a chain $c=(n_1,...,n_k)$ , we let $weight(c)$ be the length of the walk consisting of the paths $P_{0} \\rightarrow P_{n_1} \\rightarrow \\dots \\rightarrow P_{n_k}$ along the cycle (where we interpret $P_{a} \\rightarrow P_{a}$ as the empty path).", "A partially constructed chain $c^{\\prime }$ is a string of the form $c^{\\prime }=(n_1,\\dots ,P_{ext})[P_{i_1},xP_{ext},P_{i_2},xP_{ext},P_{i_3},xP_{ext},\\dots P_{i_k},xP_{ext}]$ with $P_{i_1}\\rightarrow \\dots \\rightarrow P_{i_k}$ being some cycle path starting from $P_{ext}$ .", "Given a partially constructed chain $c^{\\prime }$ , we let $weight(c^{\\prime })$ denote the weight of the (complete) chain produced by replacing the last $xP_{ext}$ with $yP_{ext}$ .", "The key property of the protocol is the following: an honest protocol $P_{n}$ will decide it is marked iff it receives a chain ending at $P_{n}$ , and it knows there does not exist a chain of greater weight.", "$P_{n}$ can be certain of this fact because if there were ever a chain of greater weight, $P_{n}$ would have had to sign it.", "Likewise, $P_{n}$ can always prevent chains of greater weight from being produced once it has decided it has been marked, by refusing to extend chains.", "Inductively, $P_{n}$ can then extend the chain of greatest weight to a new process $P_{end}$ as a form of payment.", "The details are as follows: At round 0, the marked process $P_{0}$ begins with the empty chain $()$ of length 0.", "At round $i=0,\\dots ,K-1$ , the marked process $M$ has a chain $c$ of length $i$ ending at $M$ : we say $c$ \"marks\" $M$ .", "It then does the following on receiving input $I_{M}=n$ : $M$ communicates with all processes on the path $P_{M,P_{n}}$ to create a chain $c^{\\prime \\prime }$ , where $c^{\\prime \\prime }$ is extended by $M$ , contains $c$ as a prefix, and ends at $P_{n}$ .", "If $P_{n} \\ne M$ , $M$ does this by beginning with the partial chain $c^{\\prime }=((c)_{M}\\cdot x)_{M}$ : for each process on the path $P_{M,P_{n}}$ in order (excluding $M$ ), $M$ messages process $P_{i_j}$ with query $c^{\\prime }$ and asks it to send back $(c^{\\prime })_{P_{i_j}}$ .", "$M$ then updates $c^{\\prime }=((c^{\\prime })_{P_{i_j}}\\cdot x)_{M}$ and moves to the next process in the path, or updates $c^{\\prime }=((c^{\\prime })_{P_{i_j}}\\cdot y)_{M}$ if $P_{i_j}$ is the last process on the path.", "If all processes on $P_{M,P_{n}}$ comply, this takes at most $\\mathcal {O}(|P_{M,P_{n}}|)$ messages.", "$M$ then sends the completely constructed chain $c^{\\prime \\prime }$ to process $P_{n}$ At round $i=0,\\dots ,K-1$ , an honest process $P_{n}$ reacts the following way to a request from $P_{a}$ to help extend a chain $c$ to a chain $c^{\\prime \\prime }$ .", "We assume $P_{n}$ is sent a partially constructed chain $c^{\\prime }$ which it is asked to append its signature to.", "Checks that $c$ is a chain of length $i$ , and that $c$ ends at $P_{a}$ , and that the form of $c^{\\prime }$ is valid.", "These properties are required for $P_{a}$ 's request to be valid, and these properties can be verified by any process.", "If $P_{n}$ has previously signed a partial chain $\\gamma $ with $weight(\\gamma )\\ge weight(c^{\\prime })$ , then $P_{n}$ refuses to extend $c^{\\prime }$ and replies with $(\\gamma )_{P_{n}}$ instead.", "If $P_{n}$ has received a chain $\\gamma $ ending at $P_{n}$ with $weight(\\gamma )\\ge weight(c^{\\prime })$ , then $P_{n}$ refuses to extend $c^{\\prime }$ and replies with $\\gamma $ instead.", "Otherwise $P_{n}$ responds to $P_{a}$ with $(c^{\\prime })_{P_{n}}$ .", "$P_{a}$ can then create a partial chain $((c^{\\prime })_{P_{n}} \\cdot x)_{P_{a}}$ or chain $((c^{\\prime })_{P_{n}} \\cdot y)_{P_{a}}$ with weight one greater than $c^{\\prime }$ .", "At round $i=0,\\dots ,K-1$ , an honest process $P_{n}$ reacts the following way when receiving a chain $c$ of length $i+1$ ending at $P_{n}$ : If $P_{n}$ has not signed a partial chain $\\gamma $ with $weight(\\gamma )=weight(c)$ , and $P_{n}$ has not previously received a chain $\\gamma $ ending at $P_{n}$ with $weight(\\gamma )=weight(c)$ , then $P_{n}$ decides it is the marked process for the next round, and that the marked process in the current round is the extender of $c$ .", "Proposition 7.2.2 Suppose that all processes, regardless of whether they are faulty, always produce some valid response when requested to append their signature to a valid partial chain in construction REF .", "That is, they always comply with an extension request, or publish a \"proof\" that they are not required to extend a particular request.Technically, we only require that the processes on the payment path $P_{M,P_{n}}$ behave in this way.", "Then construction REF is a solution to the marker problem which tolerates $f<N-1$ faults.", "If at round $i$ , $M$ sends the marker to $P_{n}$ , then the number of steps taken and number of messages sent until $P_{n}$ decides that it is the marked process for round $i+1$ is $\\mathcal {O}(|P_{M,P_{n}}|)$ .", "Moreover, $M$ has a proof it has paid $P_{n}$ in the following sense: If a third party Alice would like a proof $M$ paid $P_{n}$ at round $i$ , then $M$ sends to Alice the chain $c$ of length $i+1$ ending at $P_{n}$ it used to pay $P_{n}$ .", "Alice then sends $c$ to $P_{n}$ .", "After Alice sends $c$ to $P_{n}$ , if $P_{n}$ has not yet decided it is the marked process at round $i+1$ , then $P_{n}$ has a chain or partial chain $\\gamma $ with $weight(\\gamma )=weight(c)$ which does not equal $c$ , which proves $M$ did not pay $P_{n}$ or $M$ is dishonestIn the future, we will use the term dishonest to refer to both (a) $M$ not following the prescribed protocol, and (b) $M$ lying about paying $P_{b}$.", "In particular, $P_{n}$ cannot produce such a proof if $M$ is honest.", "We need to check the three conditions: Consistency: Suppose for the sake of contradiction that two distinct honest processes $P_{n},P_{n^{\\prime }}$ both decide in round $i$ they are marked for round $i+1$ .", "Then both processes possess corresponding chains $c_{n},c_{n^{\\prime }}$ of length $i+1$ which end at $P_{n},P_{n^{\\prime }}$ and are extended by $P_{ext},P_{ext^{\\prime }}$ resp.", "Without loss of generality, assume $weight(c_{n})>weight(c_{n^{\\prime }})$ .", "But then it must be the case that $P_{n^{\\prime }}$ signed a prefix chain/partial chain $\\gamma $ of $c_{n}$ with $weight(\\gamma )= weight(c_{n^{\\prime }})$ .", "By construction, $P_{n^{\\prime }}$ does not do this in round $r\\le i$ if it receives $c_{n^{\\prime }}$ before being requested to make this signature; thus $P_{n^{\\prime }}$ must have decided it was marked after signing $\\gamma $ , but this is a contradiction.", "Liveness: When $M$ decides it is marked by $c$ , there exist no other chains or partial chains of greater weight (otherwise $M$ would have had to sign them).", "By construction, this means when $M$ messages a process $P_j \\in P_{M,P_{n}}$ with partial chain $c^{\\prime }$ in an attempt to extend $c^{\\prime }$ , $P_{j}$ can only give the response $(c^{\\prime })_{P_j}$ .", "This is because $M$ is required to sign the end of every partial chain extended by $M$ , and so during this process the only partial chains of maximum weight that have been produced by the network are the partial chains $M$ explicitly creates, and all chains constructed up until this time have weight $\\le weight(c)$ (because $M$ never appended $y)_{M}$ to a partial chain with weight $\\ge weight(c)-1$ ).", "Thus after $M$ receives a response from the last process in the path, $M$ will have a string $\\gamma $ which it can sign to produce a completed chain $\\tilde{c}=(\\gamma \\cdot y)_{M}$ , where $\\tilde{c}$ is extended by $M$ , ends at $P_{n}$ , and contains $c$ as a prefix.", "$M$ then sends $\\tilde{c}$ to $P_{n}$ .", "By construction, $P_{n}$ will not have previously seen a chain of greater or equal weight ending at $P_{n}$ , nor will $P_{n}$ have signed a partial chain of weight $\\ge weight(\\tilde{c})$ .", "If $P_{n}$ is honest, then $P_{n}$ correctly decides that $M$ was the previously marked process and that $P_{n}$ is the next marked process.", "Non-impersonation: By the previous paragraph, the only way for an honest process $P_{n}$ to accept a payment via chain $c$ , and decide $c$ was extended by honest process $P_{a}$ , is if $P_{a}$ was indeed the extender and appended its signature to the end of $c$ .", "Note the following: if $M$ is honest and marked via chain $c$ during round $i$ , then all chains and partial chains of weight greater than $weight(c)$ produced by the network will have $c$ as a prefix: suppose not, and that $\\gamma $ is a chain/partial chain which does not contain $c$ as a prefix.", "Let $\\gamma ^{\\prime }$ be the prefix chain/partial chain of $\\gamma $ with $weight(\\gamma ^{\\prime })=weight(c)$ , $\\gamma ^{\\prime }\\ne c$ .", "Then $(\\gamma ^{\\prime })_{M}$ is a prefix of $\\gamma $ , but by construction if $M$ decides it is marked by chain $c$ in round $i$ , it will never sign and never has signed $\\gamma ^{\\prime }$ , contradicting that $\\gamma $ exists.", "With regards to the proof of payment, note that if $M \\in \\mathcal {H}$ is marked by chain $c$ , and did indeed pay $P_{n}$ with chain $c^{\\prime \\prime }$ extended by $M$ from $c$ , then all partial chains $\\gamma $ which do not contain $c$ as a prefix satisfy $weight(\\gamma )\\le weight(c)<weight(c^{\\prime \\prime })$ , and all chains $\\gamma $ which do not contain $c$ as a prefix satisfy $weight(\\gamma ) \\le weight(c)<weight(c^{\\prime \\prime })$ , so $P_{n}$ cannot prove $M$ is dishonest.", "Conversely, if chain $c^{\\prime \\prime }$ is not a valid chain which causes $P_{n}$ to accept a payment from $M$ in round $i$ , then $P_{n}$ has seen a chain/partial chain $\\gamma \\ne c^{\\prime \\prime }$ with $weight(\\gamma )= weight(c^{\\prime \\prime })$ .", "It remains to explain how we can deal with the issue that a dishonest process $P_{n}$ may refuse to respond altogether, instead of validly responding to an extension request.", "We can solve this problem by running Byzantine Broadcast as a subprocess.", "Suppose a process $P_{a}$ (honest or not) sends an extension request to process $P_{b}$ for the extension of a partial chain $c^{\\prime }$ , and $P_{b}$ does not respond validly within the correct time.", "We then allow $P_{a}$ to broadcast to all processes that it would like the network to decide on the response $P_{b}$ gives to query $c^{\\prime }$ .", "Importantly, note that we are not saying anything about deciding the guilt of $P_{b}$ (for example, it could be that $P_{a}$ simply behaved dishonestly and ignored $P_{b}$ 's valid response).", "Assuming all of this works, all honest processes then reach an agreement on some valid extension $\\gamma $ of $c^{\\prime }$ which is produced by $P_{b}$ , or the default null value $\\perp $ .", "If all processes agree on a valid extension $\\gamma $ , then in particular $P_{a}$ decides on a valid response $\\gamma $ from $P_{b}$ and the problem is resolved$\\gamma $ is necessarily a response from $P_{b}$ , because it ends with $P_{b}$ 's signature which cannot be forged.", "Note this case occurs if $P_{b}$ is honest.", "If $P_{a}$ instead decides on $\\perp $ ($P_{b}$ is dishonest), then we have all honest processes \"imagine\" that $P_{b}$ is deleted from the network, and $P_{b}$ 's signature is no longer required to construct chains.", "We need to be slightly careful about how we implement this idea.", "We give concise details below: Construction 3 (Proof of response for construction REF ) During round $i$ , the marked node $M$ tries to extend $c$ by asking for extensions along the path $P_{M,n}$ .", "We break this up into $N$ time periods (the maximum number of extensions needed) each of a predefined number of steps, where during the $k$ th time period: At step 1, $M$ sends an extension request $c^{\\prime }$ to $P_{j}$ , the $k$ th process in $P_{M,n}$ (this time period is empty if $k>|P_{M,n}|$ ) At step 1 in this time period, all honest processes each initialize $N^2$ instances of Byzantine Broadcast in parallel, the implementation of which is detailed in the proof of proposition REF .", "Each instance $(a,b) \\in [N]$ is a Byzantine Broadcast protocol for $P_{a}$ to issue a network query for $P_{b}$ about a query request $c^{\\prime }$ .", "The essential property of this particular implementation of Byzantine Broadcast is that no messages are sent if the broadcaster does not broadcast anything.", "At step 2, if $P_{j}$ does not respond, $M$ broadcasts $(c^{\\prime },P_{j})$ to the network.", "At step $z=\\mathcal {O}(f)$ , all the $N^2$ Byzantine Broadcast protocols have terminated.", "If $M$ is honest and broadcasts $(c^{\\prime },P_{j})$ , then all honest processes will decide this.", "Each honest process checks that the request $c^{\\prime }$ is valid, otherwise the request is ignored.", "For a further $\\mathcal {O}(f)$ steps, all honest processes run Byzantine Broadcast to reach agreement on any network queries of responsiveness for any process $P_{b}$ .", "If a response of $c^{\\prime }$ is requested for $P_{b}$ , then Byzantine Broadcast is run with $P_{b}$ as the leader, and $P_{b}$ 's broadcast is taken as its response.", "By the end of these steps, all honest processes agree which queries were made, and the responses to these queries.", "In total, each payment now takes $T=\\mathcal {O}(fN)$ steps to complete.", "However, the number of messages is still only $\\mathcal {O}(|P_{M,n}|)$ per round when all processes behave honestly, because no messages are sent by the Byzantine Broadcast protocols in this case.", "When processes behave dishonestly, the number of messages is $\\mathcal {O}(N \\times N^2 \\times Nf)=\\mathcal {O}(N^5)$ per round in the worst case.", "As a sanity check, note that construction REF satisfies the lower bound of proposition REF and is tight as well: when all processes behave honestly, $\\sum _{n \\in [N]} z_n = \\sum _{n \\in [N-1]} \\Omega (n)=\\Omega (Nf)$ since the construction tolerates $f=N-2$ faults.", "We now build on the ideas in this construction to give a fully fledged payment system which is robust to any $f<N-1$ failures.", "Moreover, if we have some control over the distribution of income and spending patterns, then when all processes behave honestly, the message complexity is at most $\\mathcal {O}(\\log (N))$ .", "This marks a distinct shift from constructions of payment systems which are based on consensus mechanisms which reach agreement on all transactions, where $\\Omega (N)$ messages per transaction are inherently required.", "The central idea is that if we could only \"hop around\" the parts of cycles which are large, then we could avoid the payment paths which require $\\Omega (N)$ messages." ], [ "Locality Through Trust", "We ended the last section by giving a simple construction of a payment system which has some degree of locality: in some transactions, only a few neighbouring processes are messaged in order for a transaction to occur.", "In this section, we will show how by expanding our model in a natural way, we can significantly bootstrap this locality to construct deterministic payment systems whose payments are highly local, and give best case message complexity significantly better than that of randomized consensus solutions, while still tolerating an arbitrary number of faults." ], [ "Trusted Anonymous Third Parties For Indirection", "The idea of having a trusted third party to mediate interactions between processes in a network has been used successfully to design a number of efficient protocolsfor example, secret sharing and anonymous messaging..", "However, we argue that there is a certain kind of trusted third party model which is particularly natural to payment systems.", "We give a story to motivate this kind of trust: Suppose Alice would like to pay Bob $\\$100$ for the comic books he gave her.", "However, unfortunately Bob's account is held with bank B, while Alice's account is held with bank A, and it is known that there is a very high transaction fee for transfers between these two banks.", "Luckily, Alice finds an advertisement on the internet from a fellow comic book enthusiast Charlie.", "Charlie's account is held with bank C, which happens to have very good transaction rates with both banks A and B. Alice hatches a plan: if she can ask Charlie to pay Bob on her behalf, then she could pay Charlie back the difference.", "With all the savings, she might even be willing to give Charlie an extra transaction fee, and then everyone (except the banks) would get a profit.", "However, Alice and Charlie have never met, and so Alice is cautious about trusting Charlie.", "To begin with, she sends Charlie $\\$1$ .", "Charlie, being the honest comic book enthusiast he is, happily sends across Alice's $\\$1$ to Bob.", "Alice then asks Bob to confirm the transaction went through.", "Being a little more confident now, Alice decides to send through $\\$2$ .", "Over time, Alice and Charlie might begin to develop a trusting relationship.", "If Charlie ever cheats and doesn't pay Bob on Alice's behalf, then Alice will know and stop using Charlie as an intermediary.", "Moreover, she will only lose at most the maximum amount she transacted through Charlie at any single time.", "Charlie probably doesn't want to cheat: he would lose a profitable revenue stream of future payments from Alice, and Alice might start ruining Charlie's reputation.", "In reality, we are more likely to have a free market situation: multiple agents like Charlie will try to build a business as efficient intermediaries between Alice and Bob.", "If an intermediary ever cheats, there will be many more intermediaries to happily take its place.", "The key properties of this scenario which make third party trust natural areWe note that this model of trust is also the de facto model under which Bitcoin has been operating under in certain contexts: participants will pay goods and service providers in digital currency on the belief that these participants will send them physical goods or provide services in return.", "Sometimes these providers can be completely anonymous, sometimes the goods themselves are legally questionable, and sometimes the service provides are foreign cryptocurrency startups with unclear legal regulation.", "There is therefore little opportunity for legal recourse if a trust assumption proves to be invalid.", "Alice can detect when Charlie is cheating, and get a guaranteed bound on her loss.", "Alice can stop using Charlie after Charlie cheats once.", "There are plausible financial incentives for there to exist many choices of good intermediaries Alice can turn to instead.", "Notice that it also makes sense to allow for multiple intermediaries, provided the number of intermediaries is small: Alice may ask Charlie to pay Bob.", "Instead of Charlie paying Bob directly, it may be cheaper for him to pay Bob via Sam, and so forth.", "We distill this discussion of trusted payment intermediaries into the following definition: Definition 8.1.1 In a trusted payment intermediary model of payment systems where process $P_{a}$ would like to send a coin to $P_{b}$ , we add the optional functionality of allowing process $P_{a}$ to form voluntary agreements with a subset of intermediary processes $\\lbrace P_{i_j}\\rbrace _{j \\in [Z]}$ with the semantics \"I, $P_{i_{j}}$ , promise to facilitate the payment of $P_{a}$ to $P_{b}$ \".", "We say the payment system in this model is valid if, when such such agreements are used to transact, they have the following properties: If $P_{a},P_{b},\\lbrace P_{i_j}\\rbrace _{j \\in [Z]}$ are honest, then the transaction from $P_{a}$ to $P_{b}$ is completed: $P_{b}$ decides $P_{a}$ send a coin to $P_{b}$ in the appropriate round.", "If $P_{a}$ is honest but $P_{b}$ claims it was not paid, then $P_{a}$ decides that at least one process in $\\lbrace P_{i_j}\\rbrace _{j \\in [Z]} \\cup \\lbrace P_{b}\\rbrace $ cheated.We only stipulate that $P_{a}$ can decide only one intermediary is honest for the following reason: it will be the case, as in the bank analogy, that only one intermediary is required to be dishonest for the transaction to fail: thus only one intermediary is really \"responsible for\" causing the transaction to fail, and it becomes technically messy to talk about multiple intermediaries being dishonest.", "Moreover, if multiple intermediaries have ill intentions towards $P_{a}$ , they could always collude so that only one of them needs to behave dishonestly at each round, while still preventing payment at each round.", "Of course, $P_{a}$ may then decide to use an entirely different set of intermediaries all together if a payment fails, and different intermediaries may form their own preferences about which other intermediaries are the most reliable to work with.", "If $P_{c} \\in \\lbrace P_{i_j}\\rbrace _{j \\in [Z]} \\cup \\lbrace P_{b}\\rbrace $ is honest, then $P_{a}$ does not decide that $P_{c}$ cheated.", "We give no guarantees that the payment will succeed or that $P_{a}$ will keep its coin if $P_{a}$ uses these agreements to transact and any of the parties behave dishonestly.", "From the previous section, we know the following payment system exists: Corollary 8.1.1 Given any complete directed cycle $C$ on the participant processes, there exists a payment system in the single transaction per round, deterministic model tolerating any $f<N-1$ faults which supports any initial coin distribution and has the following additional properties: If any process $P_{c}$ falsely claims to pay some other process $P_{b}$ in round $i$ , then $P_{b}$ has a proof to any third party $P_{a}$ in the sense of proposition REF that $P_{c}$ is dishonest.", "If a process $P_{a}$ pays a process $P_{b}$ in round $i$ , and all processes processes are honest, then the message complexity of this round is $\\mathcal {O}(|P_{P_{a},P_{b}}|)$ .", "This follows immediately from the cycle coin solution to the marker problem (construction REF ) and the reduction given from payment systems to the Marker Problem (proposition REF ), because the cycle coin construction sends no messages when a marker pays itself in a round.", "Using this building block, we are now ready to give the construction of a payment system which uses indirection between cycles to reduce message complexity.", "The key idea is the following: construct a payment system by stacking together many cycle payment systems consisting of cycles of different permutations.", "Now imagine that $P_{a}$ would like to pay $P_{b}$ .", "Since $P_{a}$ has non-zero balance, it has a coin in at least one of the subcycles $C$ which make up the payment system.", "Now $P_{a}$ has a choice: $P_{a}$ could pay $P_{b}$ using the subpayment system $C$ .", "But if $P_{a}$ is very far away from $P_{b}$ on the cycle, $P_{a}$ might be lucky by finding another process $P_{c}$ such that (a) $P_{a}$ is close to $P_{c}$ on cycle $C$ , and (b) $P_{c}$ is close to $P_{b}$ on some other cycle $C^{\\prime }$ .", "If $P_{c}$ has a coin on cycle $C^{\\prime }$ , then the following can happen: $P_{a}$ pays $P_{c}$ on cycle $C$ , and $P_{c}$ promises to ensure that a payment gets to $P_{b}$ .", "$P_{c}$ then pays $P_{b}$ on cycle $C^{\\prime }$ , and sends $P_{a}$ a proof of payment equal to the chain it used to pay $P_{b}$ .", "By \"hopping between cycles\" we are able to reduce the number of messages we need to send per transaction.", "In general, we might make multiple hops for a single payment.", "Proposition 8.1.1 Consider a collection of $K$ cycle payment systems of the form described in corollary REF , with associated cycles $C_{1},\\dots ,C_{K}$we will interchangably refer to payment systems through their associated cycles.", "Concatenate these payment systems together to form a new payment system $PS^{\\prime }$ as in the proof of proposition REF which tolerates $f<N-1$ faults, and let $\\mathcal {F} \\subset \\lbrace P_n\\rbrace _{n \\in [N]}$ be a set of \"trusted intermediary\" processes.", "At round $i$ , let $Value(C_{k},n)$ denote the value of process $P_{n}$ in the payment system associated with the simulation of payment system $C_{k}$ , and define the following graph $G_{i}$ on the vertex set $[K] \\times [N]$ and edge set $E$ : $((k,P_{a}),(k^{\\prime },P_{b})) \\in E \\textit { if } k=k^{\\prime } \\wedge (P_{a},P_{b}) \\in Edges(C_{k})$ (cycle step) or $((k,P_{a}),(k^{\\prime },P_{b})) \\in E \\textit { if } Value(C_{k^{\\prime }},b)>0 \\wedge a=b \\in \\mathcal {F}$ (cycle hop) Let $\\mathcal {W}=\\lbrace k \\in [K]|Value(C_{k},a)>0\\rbrace $ and $D_{a,b}=\\min _{k \\in \\mathcal {W},k^{\\prime } \\in [K]} dist_{G_{i}} ((k,P_{a}),(k^{\\prime },P_{b}))$ .", "Then in the trusted payment intermediary, single payment per round model, we can define the payment system $PS^{\\prime }$ to have the following property: At round $i$ when honest process $P_{a}$ makes a payment to honest process $P_{b}$ , if $P_{a}$ can find a path $P$ of length $L$ in $G_{i}$ by using $M$ messages, and all processes behave honestly, then the message complexity of a transaction from $P_{a}$ to $P_{b}$ is $\\mathcal {O}(M+L)$ .", "In particular, if $P_{a}$ knows $G_{i}$ at the start of round $i$ , then the message complexity is $\\mathcal {O}(D_{a,b})$ .", "Moreover, if $\\mathcal {O}(|P|)$ trusted intermediaries on the path $P$ behave honestly, then $P_{a}$ 's payment is guaranteed to go through to $P_{b}$ .", "We comment that it is a straightforward extension to consider a different set of trusted intermediates $\\mathcal {F}_{n}$ for each different process $P_{n}$ , however the core ideas are captured by this statement.", "Using the primitives we have developed, we describe one potential way for $P_{a}$ to send a payment to $P_{b}$ : Let the path from $P_{a}$ to $P_{b}$ be $P$ .", "$P$ consists of a number of cycle hops and cycle steps.", "Let $(P_{i_1},\\dots ,P_{i_Z})$ be the intermediate processes which facilitate the cycle hops in path $P$ (see figure REF ).", "We define a macro round which consists of a large number of micro rounds.", "Each macro round corresponds to a single round in $PS^{\\prime }$ .", "Each micro round corresponds to a single round in the simulated cycle payment systems.", "We might fix the number of micro rounds per macro round to be some upper bound on the diameter of the graph $G_{i}$ .", "At macro round $i$ , $P_{a}$ receives input $b \\in [N]$ during the execution of payment system $PS^{\\prime }$ .", "$P_{a}$ now sends $\\mathcal {O}(|P|)$ messages to the intermediate processes $P_{i_1},\\dots ,P_{i_Z}$ asking for a promise of the form \"I, $P_{i_{j}}$ , will pay $P_{i_{j+1}}$ in micro round $j+1$ , if I receive a payment from $P_{i_{j-1}}$ in round $j$ \".", "We notate $P_{i_{0}}:=P_{a}, P_{i_{Z+1}}:=P_{b}$ .", "At microstep $j=0,\\dots ,Z$ , $P_{i_{j}}$ pays a coin to $P_{i_{j+1}}$ using the path $P_{i_{j}} \\rightarrow P_{i_{j+1}}$ along the appropriate cycle $C_{k}$ determined by $P$ (cycle steps).", "Such an action is possible because (we assume without loss of generality) $P$ does not contain two cycle hops from the same process, and $Value(C_{k},P_{i_{j}})>0$ at the beginning of macro round $i$ .", "By corollary REF , all of these actions combined take $\\mathcal {O}(|P|)$ messages to perform assuming all processes behave honestly.", "At the end of all the microsteps, $P_{a}$ asks for proofs from all the intermediate processes that their payment obligations were satisfied.", "Conceptually, we have $P_{a}$ sign a message to $P_{b}$ at the beginning of macro round $i$ indicating that it intends to send a payment, and sends the signed promises of the intermediate processes $P_{i_{j}}$ to $P_{b}$ as well.", "If $P_{b}$ receives a payment from $P_{i_{Z}}$ , it then decides that $P_{a}$ paid $P_{b}$ in macro round $i$ .", "To see that this gives a valid payment system in the trusted payment intermediary, single transaction per round model, we need to check a few conditions.", "First note that $S1$ (non-duplication) in Definition REF always holds, because it holds for each simulated payment system for any number of faults $f<N-1$ .", "We assert that when no intermediaries are used, non-impersonation, self consistency and liveness in round $i$ hold by the construction given in the proof of Proposition REF .", "The new content we need to check is what happens when intermediaries are used in round $i$ .", "We will check non-impersonation still holds, liveness holds when the intermediaries are honest, and that in the case $P_{b}$ claims it is not paid, the conditions of definition of the trusted payment intermediary model (definition REF ) are met.", "non-impersonation: $P_{b}$ only decides that $P_{a}$ paid $P_{b}$ if it explicitly receives a signature from $P_{a}$ indicating it will pay $P_{b}$ at round $i$ .", "Therefore if $P_{a}$ is honest and does not pay $P_{b}$ at round $i$ , it cannot be impersonated.", "Liveness and proofs of honesty: If $P_{a},P_{b},\\lbrace P_{i_j}\\rbrace _{j \\in [Z]}$ are honest, then the payment goes through as described.", "If $P_{b}$ claims it did not receive a payment from intermediary $P_{i_{Z}}$ , then $P_{a}$ can iteratively go through each of the processes in the order $P_{b}=P_{i_{Z+1}},P_{i_{Z}},P_{i_{Z-1}},\\dots $ .", "Inductively, at step $j$ , if $P_{i_{j}}$ claims it was not paid by $P_{i_{j-1}}$ , then either $P_{i_{j-1}}$ claims it did pay $P_{i_{j}}$ and we have a proof of whether $P_{i_{j-1}}$ is being honest by corollary REF , or $P_{i_{j-1}}$ admits it did not pay $P_{i_{j}}$ , because it was not paid by $P_{i_{j-2}}$ .", "In this case, $P_{a}$ recursively moves down to step $j-1$ .", "At step $j$ , if $P_{j}$ is honest, then either $P_{j}$ is able to prove to $P_{a}$ that it did pay $P_{j+1}$ , or $P_{j-1}$ is unable to prove to $P_{a}$ that it paid $P_{j}$ .", "Thus $P_{a}$ will not decide that $P_{j}$ cheated.", "Since $P_{b}$ claims it was not paid and $P_{a}$ paid $P_{i_1}$ by assumption, $P_{a}$ will decide that at least one of $P_{b}\\cup \\lbrace P_{i_z}\\rbrace _{z \\in [Z]}$ cheated.", "Thus this protocol satisfies all of the conditions of Definition REF .", "We comment that in the above construction, it is not actually necessary for $P_{a}$ to know the entire path to $P_{b}$ in advance, and we can change the semantics of the intermediate trust assumptions as well: for example, $P_{a}$ could simply forward the payment to $P_{b}$ which it thinks is a good intermediate process for this transaction.", "$P_{b}$ can then sign the promise \"I promise to make sure a payment gets to $P_{b}$ by time $t$ \", and assume full responsibility and trust.", "Provided $P_{b}$ can figure out the remaining short path and find its own trusted intermediaries, the message complexity can still be made small.", "We will refer to payment systems of the form described in proposition REF consisting of cycles $C_1,\\dots C_K$ as a cycle payment system with cycles $C_1,\\dots ,C_K$Thus $C_1$ is a cycle payment system consisting of the cycle $C_1$ .. We now define an unnecessarily strong condition on the spending distribution in a payment system which allows us to state some simple constructions which give low message complexity: Definition 8.1.2 Consider a cycle payment system consisting of cycles $C_1,\\dots ,C_K$ .", "We say that the system is balanced in the single transaction per round model if, during the beginning of every round, we have that $Value(P_n,C_k)>0$ for all $n \\in [N], k \\in [K]$ .", "Theorem 2 Suppose for simplicity that $\\mathcal {F}=[N]$Even this extreme case is not entirely unrealistic: we might imagine that the default behavior of participants is to facilitate payments, at the benefit of gaining transaction fees and reputation for being a reliable facilitator.. Then there exists a deterministic cycle payment system consisting of $2K\\ge 4$ cycles $C_1,\\dots ,C_K$ and tolerating any $f<N-1$ faults in the trusted payment intermediary model, such that if the payment system is balanced, the best case message complexity is $\\mathcal {O}(\\log _{K}(N))$ per transaction.", "The number of trusted intermediaries per transaction is also $\\mathcal {O}(\\log _{K}(N))$ .", "Pick a random, undirected $r=2K\\ge 4$ regular graph on $N$ processes, $G_{N,r}$ .", "We use two well known properties about such graphs: If $r\\ge 4$ is even, then $G_{N,r}$ asymptotically almost surely has a complete Hamiltonian decomposition into edge disjoint Hamiltonian cycles $t_1,\\dots ,t_{\\frac{r}{2}}$ .", "[16] Almost every $r$ regular graph has diameter at most $d\\ge \\mathcal {O}\\left(\\frac{\\log (3rN\\log (N))}{\\log (r)}\\right)=\\mathcal {O}(\\log _{r}{N})$ .", "[5] Thus, for sufficiently large $N$ , we can pick $G_{N,2K}$ such that it has diameter $\\mathcal {O}(\\log _{K}{N})$ , and can be decomposed into undirected Hamiltonian cycles $t_1,\\dots ,t_K$ .", "For each undirected $t_i$ , choose two directed cycles $C_{2i},C_{2i+1}$ for each direction around the cycle $t_i$ .", "Now construct a cycle payment system $PS$ consisting of the cycles $C_{1},\\dots ,C_{2K}$ .", "Now we invoke proposition REF : consider the corresponding graph $G_{i}$ at round $i$ in the statement of the proposition.", "Because $PS$ is balanced, the cycle hop edges of $G_{i}$ do not change between rounds.", "Consequently, $G_{i}$ is fixed for all $i$ , and determined by the cycles $C_{1},\\dots ,C_{2K}$ ; thus we can assume each process $P_{a}$ can compute shortest paths in $G_{i}$ for any round $i$ without the need to send any messages.", "Moreover, we claim that $\\max _{a,b \\in [N]} D_{a,b}=\\mathcal {O}(\\log _{K}{N})$ , from which the claim follows by proposition REF .", "To see this, notice that for any $a,b \\in [N]$ , given a shortest path $P$ in $G_{N,2K}$ connecting $P_{a}$ to $P_{b}$ , we can find a corresponding path in $G_{i}$ with at most twice the number of edges: The path $P$ can be decomposed into a sequence of paths $l_1 \\rightarrow \\dots \\rightarrow l_z \\rightarrow \\dots \\rightarrow l_Z$ where each $l_z$ moves in a particular direction around some Hamiltonian cycle $t_{j_{z}}$ , and $t_{j_{z}}$ and $t_{j_{z+1}}$ are distinct cycles.", "For notational convenience, we will say that $l_{z}$ ends at the same vertex $l_{z+1}$ begins (so that adjacent paths share endpoints).", "By the choices of the cycles $C_{1},\\dots ,C_{2K}$ , we can follow each path $l_z$ in $G_{N,2K}$ with a path $l^{\\prime }_{z}$ of the same length in $G_{i}$ , by following the corresponding cycle (either $C_{2j_{z}}$ or $C_{2j_{z}+1}$ , depending on orientation) in $G_{i}$ : both paths begin and end at the same process.", "Moreover, by following a cycle hop between cycles and using the fact that the payment system is balanced, we can find an edge $e_{z}$ in $G_{i}$ so that the paths $l_{z} \\rightarrow l_{z+1}$ in $G_{N,2K}$ and $l^{\\prime }_{z} \\rightarrow e_{z} \\rightarrow l^{\\prime }_{z+1}$ both end at the same processes.", "Inductively, $P^{\\prime }=l^{\\prime }_{1} \\rightarrow e_{1} \\dots \\rightarrow e_{Z-1} \\rightarrow l^{\\prime }_{Z}$ has at most twice the path length of $l_{1} \\rightarrow \\dots \\rightarrow l_{Z}$ , and $P^{\\prime }$ starts at process $P_{a}$ and ends at process $P_{b}$ .", "It follows by another use of the well balanced condition that $D_{a,b} \\le |P^{\\prime }|$ .", "We emphasize that the statement of REF is, in an important measure, weaker than what can actually be achieved.", "Given a highly connected graph $G_{i}$ , the process $P_{a}$ initiating the payment may have reasonable flexibility in choosing which participants are required to be honest for the message complexity of the transaction to be low.", "In particular, $P_{a}$ can actively look for processes willing to facilitate the transaction when trying to find a short path through $G_{i}$ , rather than being at the mercy of an arbitrary subset of $\\mathcal {O}(\\log _{K}(N))$ participants to behave honestly.", "There are many different choices of cycles one can try to combine to get small graph diameter.", "Condition REF is overly strong because in practice, especially on random graphs, removing a few edges because we cannot make a cycle jump (say, because $Value(C_k,P_j)=0$ on a cycle $C_k$ ) will not significantly affect the graph diameter.", "$P_{a}$ might try a local search along a few different short paths before finding one which works.", "To illustrate this idea more concretely, we briefly sketch another construction using two cycles: while we don't give any analysis, we leave it to the reader to convince themselves that such a construction allows for small message complexity when the balance condition is satisfied, and that the construction is reasonably robust to removing some between-cycle edges.", "Construction 4 Start with the standard cycle on $N$ processes, where the edges have the orientation $n \\rightarrow n-1$ .", "Construct a sequence of paths recursively as follows: Start at process 0.", "Extend a directed edge to the median $m$ of $[0,N]$ .", "Now recursively (a): continue extending this path to the median of $[m,N]$ , and (b) start a new path at $m-1$ , extending it to the median of $[0,m-1]$ , and so on.", "At the end of this process, we will have a number of disjoint directed paths.", "Join them all together to get the second cycle.", "The picture looks like this: Figure: Recursively constructed \"binary search\" cycles for N=29N=29.", "We end this section with a final comment: One may wonder to what extent 3rd party trust is really necessary for this construction.", "For example, if $P_{c}$ promises $P_{a}$ it will pay $P_{b}$ in round $i$ but does not, can we not use a similar idea like the one in construction REF (proof of response) to have the network come to a consensus about whether $P_{c}$ cheated?", "The key issue is making sure that if we catch $P_{c}$ cheating, $P_{c}$ still has some non-zero value it can be forced to pay back to compensate $P_{a}$ .", "Note that a solution of low best case message complexity which doesn't rely on trusted intermediaries doesn't necessarily violate the message complexity lower bound of $\\Omega (Nf)$ for payment systems in the best case (Proposition REF ), because this was proved for the Marker Problem where the distribution of income is centered only on $P_1$ , but our low message constructions rely on the distribution of income being sufficiently spread out with respect to the network topology.", "Such an idea may therefore work, but the solution constructed in the fault tolerant model may also make heavy use of timing assumptions which would not realistically translate to use over the internet.", "In practice, one would need to make $P_{c}$ put money in escrow for a certain amount of time while it was behaving as an intermediary, and this may require some kind of global consensus to achieve.", "The idea of using peer to peer payment indirection, putting money in escrow, and using consensus to resolve peer to peer indirection disputes is very similar in spirit to the Lightning Network [31].", "We find it interesting that by considering a corner case of Lemma REF in an ideal model of payment systems, we have been led down a road which ended with ideas similar to those being experimented with in real world payment systems.", "We now turn the reader's attention to the appendix, which gives a high level overview of the lightning network and its relation to Part ." ], [ "Trusted Third Parties for Coordinating Payment Cancellation", "In this last subsection, we sketch a simple idea which can also be used to reduce the message complexity in a third party trusted model.", "Imagine a case where there is a trusted third party which is responsible for coordinating information about payments between processes.", "If the third party is dishonest, the only consequence is that the message complexity per transaction increases (but no security guarantees are violated).", "In practice, we imagine a market of such \"information intermediaries\" which compete to offer services which give the best information/reduction in message complexity.", "The idea is fairly simple: consider at round $i$ , in the multiple payments per round model, of processes making $Q$ payments $\\lbrace P_{a_i} \\rightarrow P_{b_i}\\rbrace _{i \\in [Q]}$ on a cycle payment system consisting of a single cycle $C$ .", "If $P_{a_i}$ is much closer to $P_{b_j}$ on $C$ than $P_{a_i}$ is to $P_{b_i}$ , and symmetrically, then the central coordinator can pair $P_{a_i},P_{a_j}$ together so that $P_{a_i}$ pays $P_{b_j}$ on behalf of $P_{a_j}$ and $P_{a_j}$ pays $P_{b_i}$ on behalf of $P_{a_i}$ under the same trust model of Definition REF .", "In general, multiple processes may cooperate together to collectively \"cancel\" their payment paths, and be required to collectively trust each other.", "In practice, we imagine a system where a central coordinator will have clients which routinely join and coordinate with other processes in this group.", "The membership of a process will be conditional on it never cheating in the group, and the coordinator may offer insurance against such behavior.", "Better coordinators will receive more clients, and well-behaved clients will be able to join coordinators with larger client pools, leading to reduced message complexity per transaction.", "How should the coordinator pair multiple transactions together?", "Given a collection of payers/sources $\\lbrace P_{a_i}\\rbrace _{i \\in [Q]}$ and recipients/sinks $\\lbrace P_{b_i}\\rbrace _{i \\in [Q]}$ , define the cost of the pairing $P_{a_i}$ with $P_{b_j}$ , $c_{i,j}$ to be the length of the directed cycle on $C$ from $P_{a_i}$ to $P_{b_j}$ , which is proportional to the message complexity associated with such a payment.", "By repairing difference sources/sinks, we would like to minimize the sums of these costs.", "Consider the algorithm GREEDY, which at each iteration $i \\in [Q]$ , picks an arbitrary source/sink, and pairs it with a sink/source not already paired which has the smallest cost associated with its pairing.", "Proposition 8.2.1 GREEDY produces a pairing with minimal total cost.", "Induct on the number of sources $Q$ , with $Q=1$ being immediate.", "Now suppose we have a collection $Q$ of sources $\\lbrace P_{a_i}\\rbrace _{i \\in [Q]}$ and sinks $\\lbrace P_{b_i}\\rbrace _{i \\in [Q]}$ which have not already been paired.", "Let $S$ be a pairing which minimizes the total cost.", "Without loss of generality, pick an arbitrary source $P_{a_i}$ which is paired to $P_{b_j}$ in solution $S$ .", "If $P_{b_j}$ is already a sink which minimizes the path length $|P_{P_{a_i},P_{b_j}}|$ , we remove the pairing $P_{a_i},P_{b_j}$ from the source and sink list and are done by induction.", "Otherwise, there is some sink $P_{b_{j^{\\prime }}}$ with $|P_{P_{a_i},P_{b_{j^{\\prime }}}}|<|P_{P_{a_i},P_{b_j}}|$ .", "Let $P_{a_{i^{\\prime }}}$ be the source connected to $P_{b_{j^{\\prime }}}$ in solution $S$ .", "Construct a new solution $S^{\\prime }$ which repairs $P_{a_{i}}$ with $P_{b_{j^{\\prime }}}$ and $P_{a_{i^{\\prime }}}$ with $P_{b_{j}}$ , hence having the property that $P_{a_{i}}$ is paired with a nearest sink.", "If we can show $S^{\\prime }$ does not have greater total cost than $S$ , then we will be done by induction.", "Since $|P_{P_{a_i},P_{b_{j^{\\prime }}}}|<|P_{P_{a_i},P_{b_j}}|$ , it must be that $P_{b_{j^{\\prime }}}$ lies on the path $P_{P_{a_i},P_{b_j}}$ and occurs strictly before $P_{b_j}$ .", "Consider three cases: (i), $P_{a_{j^{\\prime }}}$ lies on the path $P_{P_{a_i},P_{b_j}}$ and in addition (a) occurs on the path $P_{a}$ to $P_{b_{j^{\\prime }}}$ ($S^{\\prime }$ has the same cost as $S$ ), (b) occurs on the path $P_{b_{j^{\\prime }}}$ to $P_{b}$ ($S^{\\prime }$ has cost less than or equal to $S$ ), (ii) $P_{a_{j^{\\prime }}}$ does not lie on the path $P_{P_{a_i},P_{b_j}}$ ($S^{\\prime }$ has the same cost as $S$ ).", "Figure: Cases for GREEDY.While it is not done so here, it might be interesting to explore simple conditions on the distribution of payments in the multiple transaction per round model which cause GREEDY to give low cost solutions.", "Proposition 8.2.2 In the multiple transaction per round, trusted payment intermediary model, there exists a deterministic payment system tolerating any $f<N-1$ faults with best case message complexity in round $i$ of $|GREEDY(\\lbrace P_{a_i}\\rbrace _{i \\in [Q]},\\lbrace P_{b_i}\\rbrace _{i \\in [Q]})|/Q$ , where $\\lbrace P_{a_i}\\rbrace _{i \\in [Q]}$ are the processes which make payments to $\\lbrace P_{b_i}\\rbrace _{i \\in [Q]}$ in round $i$ ." ], [ "Conclusions", "This thesis began with the aim of interrogating the following assumption: Assumption Distributed payment systems cannot exist without achieving regular global consensus about which payments have occurred.", "By the end of this thesis, we were able to get a clearer idea of the validity of this assumption in the following ways:" ], [ "In the formal fault tolerance model, payment systems are weaker than consensus:", "We showed in proposition REF that there is a single step back box reduction from payment systems to Byzantine Broadcast, showing that if we can solve the consensus problem, then we can implement a payment system.", "On the other hand, by using known lower bound results for Byzantine Broadcast and constructing a low round complexity solution to the Marker Problem, we were able to show in Theorem REF that there exists no black box, or even \"see-through box\" reduction from Byzantine Broadcast to payment systems which is significantly better than trivially using a payment system as an unauthenticated messaging channel." ], [ "Under a reasonable trust model, we need not achieve regular global consensus in order to facilitate payments:", "In section REF , Construction REF , we showed how to construct simple solutions to the Marker Problem which did not require global consensus to transact when processors are well behaved.", "In contrast, we also showed a tight lower bound of $\\Omega (Nf)$ for the best case message complexity of the Marker Problem in Proposition REF , indicating that if we wanted to do much better than Construction REF , we would either need to change the model of our problem, or make assumptions about the distribution of income.", "In part , we extended our model to allow trusted intermediaries.", "We showed that there exist highly fault tolerant payment systems in this model which facilitate inherently local transactions in the best case.", "In Theorem REF we collected these ideas to show that under certain transaction distribution assumptions, we can tolerate any $f<N-1$ faults with best case message complexity $\\mathcal {O}(\\log _{d}(N))$ per transaction and $\\mathcal {O}(\\log _{d}(N))$ trusted intermediaries per transaction.", "Major cryptocurrencies such as Bitcoin do not tolerate more than $f$ faults for $2f<N$ , and require achieving consensus at every round about which transactions have occurred (in our model, which is not the same model as the one a real-world cryptocurrency operates under, this would require sending $\\Omega (N)$ messages per transaction in the best case).", "This result implies that in the model we have chosen, the assumption that payment systems require regular global consensus about which transactions have occurred is not necessarily true: at the least, this degree of consensus is not needed in the best case.", "Despite these results, it is also important to point out that all of the constructions given were analyzed in models which are different than the ones in which practical cryptocurrencies operate under (these differences are detailed in section part ).", "While we hope the ideas given in these formal models lead to useful translations in more practical settings, such translations are not always obvious.", "We state the gaps in this understanding in the form of posing new problems:" ], [ "Part ", "Question 1 Does there exist a reduction from the $K$ round marker problem to the 2 round Marker Problem?", "What about the 1 Round Marker Problem?", "(Definition REF ).", "Notice that the heavy requirement that \"all processes behave honestly\" in order to get low message complexity in Construction REF , and hence the construction in Theorem REF , is because the dispute resolution mechanism which forces processes to extend chains uses Byzantine Broadcast as a black box.", "If anyone requests the network to decide on a response for process $P_{n}$ , then the message complexity is dominated by the number of messages required to run Byzantine Broadcast.", "If we could therefore force processes to be responsive without this consensus mechanism, then we could significantly lower the message complexity even without best case behavior.", "Question 2 Is there a way to force, or strongly incentivize processes to be responsive in Construction REF without a global consensus mechanism?For example, if there is a financial penalty associated with being unresponsive, then we can have a bound on the message complexity in terms of how much an adversary needs to spend to increase the number of messages." ], [ "Part ", "Question 3 By using tools for leader election in Randomized Byzantine Agreement, can the ideas presented in part be extended to the realistic case where the $N$ participants are unknown and may be online or offline?", "Question 4 Can we find realistic, more well motivated conditions on the distribution of transactions which guarantee we can always efficiently find a short payment path in Proposition REF ?", "Question 5 Given a transaction distribution, can we always construct a cycle payment system with the guarantee that there exists (with high probability) a short payment path as in Proposition REF ?" ], [ "The Lightning Network and Part ", "We will briefly describe the core ideas behind the lightning network, and then move to explaining how Part relates to these ideas.", "The lightning network is designed to operate as a second layer protocol, on top of Bitcoin.", "There does not yet seem to be any formal analysis of its performance, although there has been some empirical analysis [44], [45].", "The idea is the following: suppose Alice and Bob regularly make small payments to each other.", "Instead of making all of these small transactions on the blockchain, Alice can create a payment channel with Bob.", "This involves Alice \"paying\" 1 Bitcoin into a new payment channel with Bob by putting a transaction on the blockchain, effectively putting her 1 Bitcoin in escrow.", "Alice's payment channel is effectively a local ledger between Alice and Bob, where Alice has balance 1 and Bob has balance 0.", "Now when Alice wants to pay Bob 0.5 Bitcoin, she sends it through her payment channel by updating the local ledger.", "Alice and Bob both sign the update, and Bob is \"paid\" through the payment channel.", "Alice and Bob now both have 0.5 Bitcoin on the local payment channel.", "If Bob wants to now pay Alice, he can send back the 0.5 Bitcoin on the payment channel by signing an updated copy of the local ledger.", "All of this happens without communicating to the blockchain, besides the initial setup of the payment channel.", "Now, if Bob has 0.5 Bitcoin in the payment channel with Alice, but would like to pay Charlie with this value, Bob dissolves the payment channel by publishing it to the blockchain.", "In particular, Bob publishes the most updated version of the local payment channel to the Blockchain.", "There's a chance Bob can cheat by publishing an outdated version of the payment channel which doesn't contain his payment to Alice (the blockchain cannot tell the difference, because the payment channel only involved communication between Alice and Bob).", "This problem is solved by putting a timelock on how quickly Bob can dissolve the channel: when he tries to do this, Alice has a few days to publish a \"more recent\" version of the payment channel to prove Bob is cheatingSince the more recent version contains Bob's signature in it, this proves that Bob intentionally published an outdated ledger..", "If Bob is cheating and Alice does this, Alice gets all the Bitcoin in the payment channel.", "Otherwise the payment channel is dissolved, and Alice's 1 Bitcoin which she originally deposited in escrow to create the payment channel is now split between Alice and Bob on the global blockchain, according to the local ledger of the payment channel.", "The reason the lightning network is interesting is the following: again, suppose that Bob wanted to pay Charlie.", "There is another way for Bob to do this without dissolving his payment channel with Alice.", "In particular, if Alice has a local payment channel with Charlie in which Alice has a positive balance of 0.5 Bitcoin, then Bob can make Alice sign the promise \"If Bob pays me 0.5 Bitcoin in the channel between Alice and Bob, I'll pay Charlie 0.5 Bitcoin in the channel between Alice and Charlie\".", "If this happens, then Bob pays Alice and Alice pays Charlie, meaning that Bob effectively pays Charlie.", "None of this required any messages on the blockchain.", "The main constraint of this construction is only the capacity of the payment channelsThis constraint also seems to be a major empirical limitation on the ability of the lightning network to facilitate transactions.", "(for example, if Alice and Bob's payment channel started with 1 Bitcoin, Alice can only be an intermediary for Bob for a value of at most 1 Bitcoin), and the topology of the connections of these channels.", "If Alice cheats and does not pay Charlie, then Bob publishes Alice's promise and proof of her cheating on the blockchain, and Alice loses the balance in her local payment channel.", "Of course, this idea of paying someone through an intermediary can be done inductively: the lightning network imagines that everyone might form connections and process the majority of small transactions through this network, leaving only large transactions for the blockchain.", "The lightning network's idea of using payment indirection between 2 party payment channels is very similar to Part 's idea to use payment indirection to hop over long cycles.", "Note however that the lightning network and Part operate in very different trust and network models: The lightning network is designed to be practical over the internet: participants might be temporarily offlineBut they cannot be offline for too long, because then Alice might not be able to catch Bob cheating in time if he publishes an outdated copy of the local payment channel to the blockchain., and there are less strong timing assumptions than in a synchronous network.", "The lightning network does not trust payment intermediaries (if an intermediary is dishonest, this can be proved and published to the blockchain), while Part is motivated in the context of having many intermediaries which can be trusted to behave well due to market incentives.", "Despite these model differences, we believe there is an intuitive way to view the relation between the constructions in Part and the lightning network: For each local payment channel of one unit of value between $P_n,P_{n^{\\prime }}$ in the lightning network, \"deposit\" this value into a cycle-coin construction where the only two participants in the cycle are $P_n,P_{n^{\\prime }}$ .", "Local two party payment channels in the lightning network correspond to two-member cycles: if Bob would like to pay Alice in the two-cycle, he signs the current longest chain, which currently ends at Bob, and sends it to Alice.", "Notice that in Construction REF , Alice's signature is not required to extend the chain in the special case of a 2-cycle: only the sender needs to sign the chain.", "Each payment which moves between two payment channels in the lightning network corresponds to a \"cycle hop\" between 2-cycles in the network.", "For Bob to dissolve a local payment channel and redeem a unit of value, he publishes the longest chain (ending at Bob) currently created by the 2-cycle which proves Bob is the marked process in this cycle.", "Alice has finite time to refute this proof by publishing a chain of greater weight.", "Figure: The lightning network as a collection of 2-cycles connected by cycle-hops.", "The solid arrows correspond to 2-cycle payment channels, and the dashed lines correspond to cycle hops between 2-cycles.In general, the lightning network corresponds to the network topology of having a large number of 2-cycles of Construction REF connected together by cycle hops.", "The lightning network chooses to reach consensus on whether participants are honest in these cycle hops, which is why it has stronger security guarantees every time a payment moves through a cycle hop.", "We justify this correspondence as follows: notice that two party payment systems and 2-cycles have the same security guarantees and mechanism of enforcement.", "Bob only accepts an updated ledger from Alice if it is an extended ledger from the one they agreed to most recently.", "Likewise, when redeeming value, Bob proves that his ledger is the most recent by allowing Alice the opportunity to present a counter example of a ledger which is longer (more recent) which contains Bob's signature, and Alice has a finite time in which to do this.", "In a 2-cycle of Construction REF , Bob only accepts payment from Alice if Alice signs a chain with greater weight than any chain Bob has seen before.", "Likewise, when redeeming value, Bob can prove that he has the marker in the local payment channel if he publishes a chain ending at Bob, and Alice cannot produce a greater weight chain.", "This proof of value is identical to the \"proof of valid response\" in Construction REF , where participants prove that a cycle extension request is invalid by publishing a chain of greater weight.", "Notice however that in the special case of a 2-cycle construction, there is never a need to ask another participant to extend a cycle if you are the marked process.", "Thus network consensus is only needed when wanting to prove one particular process is marked in the 2-cycle, but consensus is not needed for in-cycle payments when the marker moves between Alice and Bob, i.e.", "2-cycle payments do not require use of the global blockchain.", "In both a 2-cycle and a payment channel in the lightning network, both Alice and Bob only have finite time to broadcast this a proof or counter example to the networkFor 2-cycles, these need to be broadcast within the current round., so both the lightning network and Construction REF use a time locking mechanism.", "The cycle hops in both systems correspond to using individual participants to act as intermediaries between different payment channels/2-cycles.", "Thus, when we formally studied the constructions in Part , we were studying the properties of a structure which is closely related to that of the lightning network, the primary difference being that we used $N$ -cycles instead of 2-cycles.", "We hope that the formalism developed in parts , are therefore useful in setting up a way to formally understand the security and efficiency of systems like the lightning network." ] ]
2105.11821
[ [ "Nonparametric classes for identification in random coefficients models\n when regressors have limited variation" ], [ "Abstract This paper studies point identification of the distribution of the coefficients in some random coefficients models with exogenous regressors when their support is a proper subset, possibly discrete but countable.", "We exhibit trade-offs between restrictions on the distribution of the random coefficients and the support of the regressors.", "We consider linear models including those with nonlinear transforms of a baseline regressor, with an infinite number of regressors and deconvolution, the binary choice model, and panel data models such as single-index panel data models and an extension of the Kotlarski lemma." ], [ "Introduction", "Random coefficients models are used to incorporate multiple sources of unobserved heterogeneity in modelling various economic behaviors.", "In these models, the researcher can be interested in more than the average of the vector of random coefficients and even might want to recover its distribution (see, e.g., [6], [19], [24], [32], [40]).", "Imposing parametric assumptions on the law of the random coefficients is a widely used approach in the analysis of random coefficients models (see, e.g., [5], [57]) but can seriously drive the results (see [9], [28]).", "Economic theory rarely motivates such restrictions.", "When the coefficients are random with a law in a nonparametric class and independent from the regressors, identification often requires the regressors to have a support which is the whole space.", "However, in applications regressors may only have limited variation.", "The main contribution of this paper is to provide identification of the distribution of the coefficients in some random coefficients models with regressors independent from the random coefficients, namely exogenous ones, when their support is a proper subset, possibly discrete but countable.", "Estimation results along those lines for the linear model are available in [21].", "We provide a general identification strategy which is then used to study linear models including those with nonlinear transforms of a baseline regressor, with an infinite number of regressors and deconvolution, the binary choice model, and panel data models such as single-index panel data models and an extension of the Kotlarski lemma.", "We show in these models that the support of the regressors can be discrete if we maintain integrability assumptions on the distribution of the coefficients.", "Using several examples, we illustrate the trade-off between the variation the regressors and the restrictions on the nonparametric class of distributions of random coefficients we consider.", "The paper is organized as follows.", "Section introduces the notations and the nonparametric identification strategy.", "In Section , we provide the main identification results for the various linear random coefficients models.", "Section provides results on the random coefficients binary model and Section on some panel data models with random coefficients.", "The appendix provides complements on the tools used for identification and the proofs." ], [ "Preliminaries", "Bold letters are used for vectors, capital letters for indeterminates of polynomials or random variables/vectors.", "For a real number $r$ , $r_c$ is the vector, the dimension of which will be clear from the text, where each entry is $r$ .", "The notations ${\\mathbb {N}}$ and ${\\mathbb {N}}_0$ are used for the positive and nonnegative integers, for $p\\in {\\mathbb {N}}$ $[p]=\\lbrace 1,\\dots ,p\\rbrace $ , ${\\mathbb {R}}_+=\\lbrace x\\in {\\mathbb {R}}:\\ x>0\\rbrace $ , $\\mathbb {C}(X)$ denotes the set of rational functions, and ${\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\left\\lbrace \\cdot \\right\\rbrace $ the indicator function.", "$\\mathbb {K}[Z_1,\\dots ,Z_p]$ is the ring of polynomials of $p$ variables with coefficients in the ring $\\mathbb {K}$ .", "Let $S \\subseteq {\\mathbb {R}}^p$ , $C(S)$ and $C^{\\infty }(S)$ be the continuous and infinitely differentiable functions at every point in $S$ with values in $\\mathbb {C}$ .", "For $\\mathcal {S}\\subseteq {\\mathbb {C}}^{p}$ , $\\mathcal {A}(\\mathcal {S})$ and $\\mathcal {H}^{\\infty }(\\mathcal {S})$ are the analytic and bounded analytic functions on $\\mathcal {S}$ .", "Quasi-analytic classes of functions on ${\\mathbb {R}}^{p}$ are vector spaces of complex valued functions in $C^{\\infty }({\\mathbb {R}}^{p})$ characterized by the knowledge of their derivatives at ${0}_c$ .", "Unlike real analytic functions, the Taylor series of such functions do not need to converge.", "The notation $|\\cdot |_q$ for $q\\in [1,\\infty ]$ stands for the $\\ell _q$ norm of a vector with components in ${\\mathbb {C}}$ while $L^{q}(S,\\mu )$ for $q\\in [1,\\infty ]$ are the $q$ integrable functions on $S$ with respect to the measure $\\mu $ and the norms $\\Vert \\cdot \\Vert _{L^{q}(S,\\mu )}$ .", "When $\\mu $ is the Lebesgue measure we drop it from the notation.", "$\\mathbb {S}^p$ is the unit sphere in $\\mathbb {R}^{p+1}$ .", "$(e_j)_{j=1}^{p}$ is the canonical basis of ${\\mathbb {R}}^{p}$ .", "For $\\beta \\in {\\mathbb {C}}^{p}$ , $k\\in {\\mathbb {N}}_0$ , and $m\\in \\mathbb {N}_0^{p}$ , denote by $m!=\\prod _{j=1}^{p}m_j!$ , $|m|=\\sum _{j=1}^{p}m_j$ , $\\beta ^{m}=\\prod _{j=1}^{p}\\beta _j^{m_j}$ , and $|\\beta |^{m}=\\prod _{j=1}^{p}|\\beta _j|^{m_j}$ .", "$GL({\\mathbb {R}}^{p})$ are the invertible $p\\times p$ matrices with real coefficients.", "$\\mathfrak {M}_c(\\Omega )$ , $\\mathfrak {M}(\\Omega )$ , and $\\mathfrak {M}_1(\\Omega )$ are the sets of complex, nonnegative, and probability measures on a Borel measurable set $\\Omega $ .", "When $\\Omega \\subset {\\mathbb {R}}^p$ is closed, these are measures over ${\\mathbb {R}}^p$ with support in $\\Omega $ .", "For $\\mu \\in \\mathfrak {M}_c({\\mathbb {R}}^{p})$ and $m\\in {\\mathbb {N}}_0^{p}$ , $|\\mu |$ is its total variation, $s_{\\mu }(m)=\\int _{{\\mathbb {R}}^{p}}x^{m}d\\mu (x)$ the moments, and by $s_{|\\cdot |,|\\mu |}(m)=\\int _{{\\mathbb {R}}^{p}}|x^{m}|d|\\mu |(x)$ the absolute moments.", "Define $\\mathfrak {M}_c^{*}\\left({\\mathbb {R}}^{p}\\right)&=\\left\\lbrace \\mu \\in \\mathfrak {M}_c\\left({\\mathbb {R}}^{p}\\right):\\ \\forall m \\in {\\mathbb {N}}, \\ \\sum _{j=1}^p s_{|\\cdot |,|\\mu |}(2me_j)<\\infty \\right\\rbrace ,\\\\\\mathfrak {M}^{*}\\left({\\mathbb {R}}^{p}\\right)&=\\mathfrak {M}_c^{*}\\left({\\mathbb {R}}^{p}\\right)\\cap \\mathfrak {M}\\left({\\mathbb {R}}^{p}\\right),\\ \\mathfrak {M}_1^{*}\\left({\\mathbb {R}}^{p}\\right)=\\mathfrak {M}^{*}\\left({\\mathbb {R}}^{p}\\right)\\cap \\mathfrak {M}_1\\left({\\mathbb {R}}^{p}\\right).$ Let $\\mathcal {M}$ be $\\mathfrak {M}_c^*\\left(\\Omega \\right)$ or $\\mathfrak {M}^*\\left(\\Omega \\right)$ .", "A measure $\\mu $ is said to be determinate in $\\mathcal {M}$ if $\\mu \\in \\mathcal {M}$ and if $\\mu $ is uniquely determined in $\\mathcal {M}$ by $\\lbrace s_{\\mu }(m)\\rbrace _{m\\in {\\mathbb {N}}_0^{p}}$ .", "The Fourier transform of $\\mu \\in \\mathfrak {M}_c({\\mathbb {R}}^{d})$ (resp.", "$f$ in $L^q(\\mathbb {R}^{d})$ for $q=1,2$ ) is $\\mathcal {F}\\left[\\mu \\right]: \\ x \\mapsto \\int _{\\mathbb {R}^{d}}e^{ib^{\\top }x}d\\mu (b)$ (resp.", "$\\mathcal {F}\\left[f\\right]$ ).", "For a random vector $X$ , $\\mathbb {P}_{X}$ is its law, $F_{X}$ its CDF, $f_{X}$ its density, $\\varphi _{X}=\\mathcal {F}\\left[\\mathbb {P}_{X}\\right]$ , and $\\mathbb {S}_{X}$ its support.", "$ {\\mathbb {E}} _{\\mathbb {P}}\\left[ \\cdot \\right] $ is the expectation under $\\mathbb {P}$ , $\\otimes $ the product of measures, and $\\mathbb {P}_{Y|X}(\\cdot |x)$ for $x\\in \\mathbb {S}_{X}$ the conditional distribution of $Y$ given $X=x$ .", "This paper considers models of the form $Y = v(X,\\Gamma ),\\ X\\perp \\!\\!\\!", "\\perp \\Gamma ,$ where $v$ is a known measurable vector valued function, $Y$ is the vector of outcomes, $\\Gamma \\in \\Gamma $ and $X\\in \\mathcal {X}$ are vectors of unobserved and observed factors, and $\\Gamma $ and $\\mathcal {X}$ are the Euclidian space or the sphere.", "Everything in this paper holds if we impose independence given $Z$ , where $Z$ is a random vector from which we could have observations simultaneously with those of $Y$ and $X$ or which could be identifiable from a model for $X$ obtained by a control function approach.", "The independence means that $X$ is exogenous, it yields $\\forall x\\in \\mathbb {S}_{X},\\ \\mathbb {P}_{Y|X}(\\cdot |x)=\\mathbb {P}_{v(x,\\Gamma )}.$ In order to study identification, we proceed as if $(\\mathbb {P}_{Y|X=x})_{x\\in \\mathbb {S}_{X}}$ were known and denote the true law by $\\mathbb {P}_{\\Gamma }^*$ .", "We maintain restrictions $\\mathcal {R}$ on the primitives $\\mathbb {P}_{X}\\otimes \\mathbb {P}_{\\Gamma }$ which involve two conditions: (i) $V\\subseteq \\mathbb {S}_{X}$ and (ii) $\\mathbb {P}_{\\Gamma }\\in \\mathcal {P}$ , for well chosen sets $V$ and $\\mathcal {P}$ .", "We use the notation $(\\mathbb {S}_{ X },\\mathbb {P}_{\\Gamma })\\in \\mathcal {R}$ .", "Random coefficients models involve inner products of subvectors of $\\Gamma $ and $X$ , hence indices with random coefficients, and possibly additional idiosyncratic errors.", "The linear random coefficients model can be written as $Y = \\alpha + \\beta ^{\\top }X,\\ \\Gamma =(\\alpha , \\beta ^{\\top })^{\\top }\\perp \\!\\!\\!", "\\perp X.$ It is a baseline component of more complex random coefficient models.", "For example, [23] shows how identification of a discrete choice model with random coefficients and a special regressor is amenable to identification of (REF ) while without special regressor requires a treatment like in Section .", "Obtaining identification of $\\mathbb {P}_{\\Gamma }^*$ in these non separable models with multiple unobservables without restricting $\\mathcal {P}$ usually requires that $\\mathbb {S}_{X}=\\mathcal {X}$ .", "For model (REF ) this can be shown by the Cramer-Wold Theorem.", "$\\mathbb {S}_{X}=\\mathcal {X}$ is too demanding for a dataset.", "However, identification can hold when $\\mathbb {S}_{X}$ is a proper subset but $\\mathcal {P}$ is a restricted nonparametric class.", "Because random coefficient models involve indices, we can study identification when $\\mathbb {S}_{X}$ is replaced by a convenient invertible affine transformation.", "So when the results in this paper depend on the order of the regressors, the order is irrelevant.", "Indeed, for all $\\underline{x}\\in {\\mathbb {R}}^p$ and $M\\in GL({\\mathbb {R}}^p)$ , $\\alpha +\\beta ^{\\top }X=\\alpha +\\beta ^{\\top }\\underline{x}+\\beta ^{\\top }M^{-1}M\\left(X-\\underline{x}\\right)$ and there is a one to one relation between $\\mathbb {P}_{\\alpha +\\beta ^{\\top }\\underline{x},\\left(M^{-1}\\right)^{\\top }\\beta }$ and $\\mathbb {P}_{\\alpha ,\\beta }$ .", "Let $ \\mathcal {T}^{v}_{\\mathbb {S}_{X}}$ map $\\mathbb {P}_{\\Gamma }$ to the collection of direct image measures $\\left(\\mathbb {P}_{v(x,\\Gamma )}\\right)_{ x \\in \\mathbb {S}_{X}}$ .", "$\\mathbb {P}_{\\Gamma }^*$ is identified under $\\mathcal {R}$ if $\\mathcal {T}^{v}_{\\mathbb {S}_{X}}$ is injective.", "The proofs that we give often rely on the diagram $\\begin{tikzcd}[column sep=3.5pc]\\mathbb {P}_{\\Gamma }\\in \\mathcal {P} \\quad {r}{\\mathcal {T}_{\\mathbb {S}_{X}}^{v}} \\quad {d}{\\mathcal {G}\\text{ (injective)}}& \\left(\\mathbb {P}_{v(x,\\Gamma )}\\right)_{ x \\in \\mathbb {S}_{X}} {d}{\\Pi } \\\\\\mathcal {G}\\left[\\mathbb {P}_{\\Gamma }\\right]\\in \\mathfrak {F}(S)\\quad {r}{\\text{restriction to }U} &\\quad \\mathcal {G}\\left[\\mathbb {P}_{\\Gamma }\\right]_{|U}\\end{tikzcd}$ The choice of $\\Pi $ depends on $v$ .", "By restricting $\\mathcal {P}$ , $\\mathcal {G}[\\mathbb {P}_{\\Gamma }]$ belongs to a class of functions $\\mathfrak {F}(S)$ which also contains all differences between functions in $\\mathcal {G}[\\mathbb {P}_{\\Gamma }]$ (e.g., $\\mathfrak {F}(S)$ could be a vector space), $\\mathcal {G}$ is injective, and $U$ is a set of uniqueness of $\\mathfrak {F}(S)$ .", "Definition 1 $U\\subseteq S$ is a set of uniqueness for a vector space of functions $\\mathfrak {F}(S)$ on $S$ if every function of $\\mathfrak {F}(S)$ which is zero on $U$ is identically zero on $S$ ." ], [ "The linear random coefficients model", "The model is (REF ).", "It is a natural specification to account for heterogenous effects.", "It can be viewed as more general than the quantile regression.", "Indeed, when the conditional quantiles are strictly increasing, the quantile regression defines the same data generating process as a linear random coefficients model where the coefficients are functions of a scalar uniform distribution.", "The unobserved scalar uniform variable is a ranking variable.", "(REF ) allows for the coefficients to be a function of a multidimensional vector, possibly infinite.", "[23] allows for extensive generalizations of (REF ) including ones involving nonlinear transforms of a baseline regressor.", "This paper considers different specifications.", "In Section REF , (REF ) is an approximation of the nonlinear model $\\boxed{\\quad Y=g(X,\\Theta ),\\ \\Theta \\perp \\!\\!\\!", "\\perp X,}$ where $g$ is unknown, $\\Theta $ has arbitrary, possibly infinite, dimension.", "Indeed, when $\\mathbb {S}_{X}$ is compact and almost surely in $\\theta \\in \\mathbb {S}_{\\Theta }$ , $x \\mapsto g(x,\\theta )\\in L^2(\\mathbb {S}_{X})$ , $(f_{j})_{j\\in {\\mathbb {Z}}}$ is a Riesz basis and $f_0=1$ , then $g(x,\\theta )= \\gamma _{0}(\\theta )+\\sum _{j\\in {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace }\\gamma _{j}(\\theta )f_{j}(x).$ Recall that a Riesz basis of a separable Hilbert space is the image of an orthonormal basis by a bounded invertible operator and that the coefficients $\\gamma _{j}(\\theta )$ are the inner products of $x\\mapsto g(x,\\theta )$ with a uniquely defined biorthogonal system (see sections 1.7 and 1.8 in [60]).", "An approximation of (REF ) is thus $Y=\\gamma _{0}(\\Theta )+\\sum _{j\\in J}\\gamma _{j}(\\Theta )f_{j}(X),\\ \\Theta \\perp \\!\\!\\!", "\\perp X,$ where $J\\subset {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace $ is finite.", "Equation (REF ) can be used for extrapolation of the conditional distribution of $Y$ given $X=x$ when the functions $f_{j}$ are analytic (see also [20]).", "Hence, it is possible to extrapolate not only conditional expectation functions but also any conditional quantiles even under a nonparametric specification with multiple unobservables entering in a non-additively separable way.", "In (REF ) or its approximation (REF ), the researcher can also be interested in the law of the random coefficients to eventually obtain the law of the elasticity $\\partial _x g(x,\\Theta )x/g(x,\\Theta )$ or of the marginal effects $\\partial _x g(x,\\Theta )$ assuming they exist and $x$ belongs to the interior of $\\mathbb {S}_{X}$ .", "Remark 1 A variation of (REF ) consists in taking $J={\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace $ but there exists $j_0\\in {\\mathbb {N}}$ such that $\\gamma _j(\\Theta )$ for $|j|>j_0$ are deterministic and $\\sum _{j\\in {\\mathbb {Z}}} \\mathbb {E}\\left[\\gamma _j(\\Theta )^2\\right] < \\infty $ .", "${\\mathbb {E}} \\left[\\gamma _j(\\theta ) \\right]$ are identified because $ {\\mathbb {E}} \\left[\\gamma _j(\\theta ) \\right] = {\\mathbb {E}} \\left[ Y g_j(X)/f_{X}(X) \\right]$ , where $(g_j)_{j\\in {\\mathbb {Z}}}$ is the biorthogonal system associated to $(f_j)_{j\\in {\\mathbb {Z}}}$ .", "So this model is amenable to (REF ) with an a priori unknown set $J$ .", "However, by taking $j_0$ to be the smallest $k$ such that $j_0=\\text{argmin}_{k\\in {\\mathbb {N}}}\\inf _{\\mathbb {P}_{\\Gamma }}\\int _{{\\mathbb {R}}}\\mathbb {E}\\left[\\left(\\mathbb {P}\\left(Y\\le y\\left|X\\right.\\right)- \\mathbb {P}\\left(\\sum _{|j|\\le k}\\Gamma _{j}f_j(X)\\le y\\right)\\right)^2\\right]dy.$ Finally, we also consider in this section infinite dimensional linear random coefficients models.", "There, the random coefficients belongs to $\\ell _2({\\mathbb {N}}_0)$ a.s.", "It is denoted by $\\lbrace \\Gamma _m\\rbrace $ and is such that $\\Gamma _0=\\alpha $ and, for all $m\\in {\\mathbb {N}}$ , $\\Gamma _m=\\beta _m$ .", "The regressors $\\lbrace X_m\\rbrace $ belong to $\\ell _2({\\mathbb {N}})$ a.s. and the model can be written $Y = \\alpha + \\sum _{m=1}^{\\infty }\\beta _m X_m,\\ \\lbrace \\Gamma _m\\rbrace \\perp \\!\\!\\!", "\\perp \\lbrace X_m\\rbrace .$ It can be related to the random coefficients functional linear regression (see [59]) $ Y = \\int _{-1}^1 \\pi (s) X(s) ds + U,$ where a.s. $X\\in L^2(-1,1)$ , $ \\pi = \\sum _{m=0}^{\\infty } \\Pi _{m} \\nu _m\\in L^2(-1,1)$ and $X_m : = \\int _{-1}^1\\nu _k(s)X(s) ds$ belong respectively to $\\ell _2({\\mathbb {N}}_0)$ and $\\ell _2({\\mathbb {N}})$ , and $(\\nu _m)_{m\\in {\\mathbb {N}}_0}\\in L^2(-1,1)^{{\\mathbb {N}}_0}$ .", "Similar expressions are used for model (REF ).", "In this section, we consider different identifying restrictions which we denote by $\\mathcal {R}_{L,j}$ , for $j\\in [7]$ and are defined below.", "The main theorem of this section is the following.", "Theorem 1 $\\mathbb {P}_{\\Gamma }^*$ in (REF ) is identified under either of $\\mathcal {R}_{L,j}$ for $j\\in \\lbrace 1,2,3,6,7\\rbrace $ .", "$\\mathbb {P}_{\\lbrace \\Gamma _m\\rbrace }^*$ in (REF ) is identified under either of $\\mathcal {R}_{L,j}$ for $j\\in \\lbrace 4,5\\rbrace $ .", "Moreover, $\\mathbb {P}_{\\Gamma }^*$ in (REF ) is not identified if $\\mathcal {R}_{L,3}$ (REF ) holds but not $\\mathcal {R}_{L,3}$ (REF ).", "The proofs rely on diagram (REF ) with $\\mathcal {G} = \\mathcal {F}$ (sometimes extended to an open set of ${\\mathbb {C}}^{p+1}$ ), $U = \\lbrace (t,t x ), \\ (t, x ) \\in {\\mathbb {R}}\\times \\mathbb {S}_{X}\\rbrace $ , and $ \\Pi : \\ \\left(\\mathbb {P}_{(1,x^{\\top })\\Gamma }\\right)_{ x \\in \\mathbb {S}_{X}} \\mapsto \\left((t, x ) \\in {\\mathbb {R}}\\times \\mathbb {S}_{ X } \\mapsto \\int _{{\\mathbb {R}}^{p+1}} e^{it(1,x^{\\top })\\gamma }d\\mathbb {P}_{\\Gamma }(\\gamma )\\right),$ with the relevant modifications when dealing with sequences and model (REF ).", "We denote by $\\mathcal {P}(\\Omega )$ (resp.", "$\\mathcal {P}_c(\\Omega )$ ) the set of (resp.", "complex) measures with support included in $\\Omega $ which are determinate in $\\mathfrak {M}^*(\\Omega )$ (resp.", "$\\mathfrak {M}_c^*(\\Omega )$ ).", "Appendix REF reviews criteria for determinacy and classical examples are given in Remark REF .", "The vector space spanned by $\\mathcal {F}[\\mathcal {P}(\\Omega )]$ (resp.", "$\\mathcal {F}[\\mathcal {P}_c(\\Omega )]$ ) is a quasi-analytic class of functions on ${\\mathbb {R}}^{p}$ .", "It is known that sets with a nonempty interior and, when $p=1$ , sets which contain a bounded sequence of distinct points are sets of uniqueness.", "For analytic classes this is the Weierstrass theorem (see, e.g., Theorem 15.11 in [47]) and for quasi-analytic classes this follows by the arguments in the proof of Lemma 4.8 in [1] for the quasi-analytic class under consideration.", "We now give details on the restrictions." ], [ "Regressors with finite support and independence of the marginals of $\\mathbb {P}_{\\Gamma }$ , deconvolution", "This is a challenging situation in terms of limited variation of the regressors.", "We present restrictions in the spirit of [2].", "Let $\\Omega _0\\subseteq \\mathbb {R}$ and, for all $k\\in [p]$ , $\\Omega _k\\subseteq \\mathbb {R}$ be given closed sets.", "These sets account for possible prior information on the support of the marginals of $\\mathbb {P}_{\\Gamma }$ .", "When such an information is not available, we take the set to be ${\\mathbb {R}}$ .", "We consider either of the two following restrictions.", "Restriction 1 $\\left\\lbrace 0_c,x(1),\\dots ,x(p)\\right\\rbrace \\subseteq \\mathbb {S}_{X}$ and $(x(1),\\dots ,x(p))$ is an upper triangular matrix with nonzero diagonal elements; $\\mathcal {P}=\\lbrace \\mathbb {P}_{\\Gamma }=\\mathbb {P}_{\\alpha }\\otimes \\bigotimes _{k=1}^p\\mathbb {P}_{\\beta _k}:\\ \\forall k\\in [p],\\ \\mathbb {P}_{\\beta _k}\\in \\mathcal {P}(\\Omega _k)\\rbrace $ .", "Restriction 2 $\\mathcal {R}_{L,1}$ (REF ) holds ; $\\mathcal {P}=\\lbrace \\mathbb {P}_{\\Gamma }=\\mathbb {P}_{\\alpha }\\otimes \\bigotimes _{k=1}^p\\mathbb {P}_{\\beta _k}:\\ \\mathbb {P}_{\\alpha }\\in \\mathcal {P}(\\Omega _0),\\ \\forall k\\in [p-1],\\ \\mathbb {P}_{\\beta _k}\\in \\mathcal {P}(\\Omega _k)\\rbrace $ .", "The problem of deconvolution with two samples, one of the error and one of the sum of the signal and the error, fits model (REF ) with $p=1$ and $\\mathbb {S}_{X}=\\lbrace 0,1\\rbrace $ is a particular case (see [23]).", "$\\mathcal {R}_{L,2}$ (REF ) is in the spirit of the assumption used in [10].", "Under $\\mathcal {R}_{L,1}$ (REF ), $\\varphi _{\\alpha }$ can have zeros on an open set at the expense of a stronger assumption on $\\mathbb {P}_{\\beta }$ .", "There are classical examples of characteristic functions with compact support (e.g., $t\\mapsto (1-|t|^{r}){\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace |t|\\le 1\\rbrace $ for $0<r\\le 1$ ).", "Remark 1 in [36] relies on analyticity.", "[41] considers the estimation of compactly supported densities of $\\beta $ which implies $\\mathcal {R}_{L,1}$ (REF ) and $\\varphi _{\\beta }$ is analytic.", "Now on, we do not assume mutual independence of the random coefficients.", "This is important if $\\Gamma $ is of the form $\\Gamma (\\Theta )$ , where $\\Theta $ is a deep heterogeneity parameter $\\Theta $ as in (REF )." ], [ "$\\mathbb {S}_{X}$ is not in the zeros of a nonzero polynomial and nonlinear model", "Let $\\Omega \\subseteq \\mathbb {R}^{p+1}$ be a closed set and the researcher knows that $\\mathbb {S}_{\\Gamma }\\subseteq C$ .", "Restriction 3 $\\mathbb {S}_{ X }$ is not a subset of the zeros of a nonzero element of ${\\mathbb {R}}[Z_1,\\dots ,Z_{p}]$ ; $\\mathcal {P}$ is the set of measures which are determinate in $\\mathfrak {M}^*(\\Omega )$ .", "Lemma 2 in [40] considers the case where $\\Omega ={\\mathbb {R}}^{p+1}$ and assumes $\\mathbb {S}_{X}$ contains an open ball which implies $\\mathcal {R}_{L,3}$ (REF ) (see also [3]).", "$\\mathcal {R}_{L,3}$ (REF ) is much weaker than assuming that $\\mathbb {S}_{\\Gamma }$ is compact as in [3].", "By the usual properties of the Fourier transform, the vector space spanned by $\\mathcal {F}[\\mathcal {P}]$ is a quasi-analytic class.", "Because $\\mathcal {A}({\\mathbb {R}}^{p+1})$ is a small subset of $C^{\\infty }({\\mathbb {R}}^{p+1})$ (see Appendix REF ), it is important in the analysis of the paper to use larger classes and allow for Fourier transforms to not be analytic.", "The next examples satisfy $\\mathcal {R}_{L,3}$ (REF ) (i.e.", "are sets of uniqueness of $\\mathbb {R}_d[Z_1,\\dots ,Z_{p}]$ ).", "Example 1 $p=1$ and $|\\mathbb {S}_{X}|=\\infty $ ; $p\\ge 2$ and $U_{p}\\subseteq \\mathbb {S}_{X}$ , where $U_{p}$ is defined recursively via $U_{1}$ , such that $|U_{1}|=\\infty $ and, for all $j=2,\\dots ,p$ , $U_{j}=\\bigcup _{u\\in {U_{j-1}}}\\lbrace (u^{\\top },v)^{\\top },\\ v\\in {V}_{j}(u)\\rbrace $ , where $|{V}_{j}(u)|=\\infty $ .", "This is a set of uniqueness because $P\\in \\mathbb {R}_d[Z_1,\\dots ,Z_{p}]$ can be written as $P(Z_1,\\dots ,Z_p)=\\sum _{k=0}^dQ_k(Z_1,\\dots ,Z_{p-1})Z_p^k$ , where $Q_k\\in \\mathbb {R}_d[Z_1,\\dots ,Z_{p-1}]$ .", "Examples of sets $U_{p}$ are $\\prod _{k=1}^{p}V_k \\subseteq \\mathbb {S}_{ X }$ , where, for all $k\\in [p]$ , $|V_k|=\\infty $ , the infinite fan $\\lbrace x\\in {\\mathbb {R}}^2:\\ \\exists n\\in {\\mathbb {N}}, \\ x_2=nx_1\\rbrace $ , and infinite staircase $\\lbrace x\\in {\\mathbb {R}}^2:\\ x_2=\\left\\lceil {x_1}\\right\\rceil \\rbrace $ (see [7]).", "A consequence of the second statement in Theorem REF is that $\\mathbb {P}_{\\Gamma }^*$ in (REF ) when $X=(X\\ X^2)^{\\top }$ under $\\mathcal {R}_{L,3}$ (REF ) holds, even if $\\mathbb {S}_X={\\mathbb {R}}$ .", "Indeed, $\\mathbb {S}_{X}$ is included in the set of zeros of $Q(Z_1,Z_2) = Z_1^2 - Z_2$ .", "Example REF shows that we can handle other nonlinear transformations of a baseline variable which can have discrete support.", "It applies to model (REF ).", "Example 2 $\\lbrace (f_1(u),\\dots ,f_p(u))^{\\top },\\ u\\in U_q \\rbrace \\subseteq \\mathbb {S}_{X}$ , where $q\\in {\\mathbb {N}}$ , $\\mathcal {B}(\\mathcal {S})$ is a vector space of functions on $\\mathcal {S}\\subseteq {\\mathbb {C}}^q$ which is an algebra with respect to multiplication, the set of functions $\\lbrace f_j, j\\in [p]\\rbrace $ of $\\mathcal {B}(\\mathcal {S})$ is such that if $P(f_1,\\dots ,f_p)=0$ , with $P\\in {\\mathbb {R}}[Z_1,\\dots ,Z_p]$ , then $P=0$ , and $U_q$ a set of uniqueness of $\\mathcal {B}(\\mathcal {S})$ (see Section REF for classes of analytic and quasi-analytic functions).", "Because $\\mathcal {B}(\\mathcal {S})$ is an algebra with respect to multiplication, $P(f_1,\\dots ,f_p)\\in \\mathcal {B}(\\mathcal {S})$ .", "Because $U_q$ is a set of uniqueness of $\\mathcal {B}(\\mathcal {S})$ , if $P(f_1,\\dots ,f_p)(x)=0$ for all $x\\in U_q$ then $P(f_1,\\dots ,f_p)=0$ , so $P=0$ .", "Hence, $\\mathcal {R}_{L,3}$ (REF ) holds.", "Here, $p$ can be large and $U_q$ discrete.", "The last condition in Example REF means that $\\lbrace f_j, j\\in [p]\\rbrace $ is algebraically independent over ${\\mathbb {R}}$ .", "Clearly, algebraic independence over $k$ implies algebraic independence over ${\\mathbb {R}}$ if ${\\mathbb {R}}$ is a subfield of $k$ .", "We now give useful examples to apply Example REF .", "If $f_1(z)=z_1,\\dots ,f_{p-1}(z)=z_q$ , $f_p$ is a real or complex analytic function of $z_1,\\dots ,z_{p-1}$ and $\\lbrace f_j, j\\in [p]\\rbrace $ is a set of algebraically independent over ${\\mathbb {C}}$ , $f_p$ is said to be transcendental in ${\\mathbb {C}}$ .", "If $f$ is a transcendental function in ${\\mathbb {C}}$ , then so is $1/f$ and its inverse on a domain where it is invertible.", "If $f$ is meromorphic in ${\\mathbb {C}}$ then it is a transcendental function in ${\\mathbb {C}}$ iff it is not a rational function (e.g., $\\exp (z)/z$ , $\\sin (z)/(z-1)^2$ , $\\cosh (z)$ , $\\Gamma (z)$ , $\\zeta (z)$ ).", "Classical examples of transcendental functions include $z^{\\pi }$ , $\\exp (z)$ , $z^z$ , $z^{1/z}$ , and $\\log (z)$ .", "If $q\\le p-1$ and $f_1(x)=x_1,\\dots ,f_q(x)=x_q$ , then $\\lbrace f_j, j\\in [p]\\rbrace $ is an algebraically independent set over ${\\mathbb {C}}$ iff $\\lbrace f_j, j\\in \\lbrace q+1,\\dots ,p\\rbrace \\rbrace $ is algebraically independent over ${\\mathbb {C}}(Z_1,\\dots ,Z_q)$ .", "For example, $\\lbrace f_j, j\\in [p-1]\\rbrace $ are algebraically independent over ${\\mathbb {C}}(Z)$ if: $f_j(z)=e^{\\lambda _jz}$ , where $(\\lambda _j)_{j=1}^p$ are such that, if $(b_j)_{j\\in J} \\in \\mathbb {N}_0^{p}$ is such that $ \\sum _{j=1}^p b_j \\lambda _j = 0$ , then, for all $j=1,\\dots ,p$ , $b_j=0$ (see proof of Proposition REF ).", "$f_j=e^{\\varphi _j}$ and $\\varphi _j\\in \\mathcal {A}_0(\\mathcal {S})$ , where $\\mathcal {A}_0(\\mathcal {S})\\subseteq \\mathcal {A}(\\mathcal {S})$ and $\\mathcal {S}\\subseteq {\\mathbb {C}}$ is a simply connected open subset which does not contain 0 but a set of uniqueness of $\\mathcal {A}_0(\\mathcal {S})$ , such that, for $j=2,\\dots ,p$ , $\\lim _{x\\rightarrow \\infty }|\\varphi _j(x)/\\varphi _{j-1}(x)|=\\infty $ and $\\lim _{n\\rightarrow \\infty }|\\varphi _1(x)/\\log (x)|=\\infty $ .", "Thus, $\\left\\lbrace \\Psi ,e^{\\varphi _1\\circ \\Psi },\\dots ,e^{\\varphi _p\\circ \\Psi }\\right\\rbrace $ is algebraically independent over ${\\mathbb {C}}$ on a connected open subset $\\mathcal {S}_1\\subseteq {\\mathbb {C}}$ if $\\Psi :\\ \\mathcal {S}_1\\rightarrow \\mathcal {S}$ is analytic.", "A way to achieve the condition in (REF ) above is to rely on a transcendental number $r$ over ${\\mathbb {Z}}$ (e.g., $e$ or $\\pi $ ).", "It means that, for all $P\\in {\\mathbb {Z}}[Z]$ , $P(r)=0$ implies that $P=0$ .", "The following result shows that we can work with the specification (REF ) with functions from a Riesz basis which are algebraically independent.", "Proposition 1 Let $T\\in {\\mathbb {R}}_+$ , $\\lambda _j = j +1/(5r^{|j|})$ and $f_j(z)=e^{i \\pi \\lambda _j z/T }$ , for all $j\\in {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace $ and $r\\in (1,\\infty )$ is transcendental over ${\\mathbb {Z}}$ , and $f_0=1$ .", "We have: $\\lbrace f_j:\\ j\\in {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace \\rbrace $ is an algebraically independent family of functions over ${\\mathbb {C}}(X)$ ; $\\left(f_j\\right)_{j\\in {\\mathbb {Z}}}$ is a Riesz basis of $L^2(-T,T)$ .", "(P6.REF ) implies $\\lbrace f_j:\\ j\\in {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace \\rbrace $ is an algebraically independent family of functions over ${\\mathbb {R}}$ which is the condition in Example REF .", "Here $q=1$ and $U_q$ should be such that $U_q\\subseteq [-T,T]$ .", "Consider model (REF ).", "$(\\lbrace e_{j,m}\\rbrace )_{j\\in {\\mathbb {N}}}$ is now the canonical basis of $\\ell _2({\\mathbb {N}})$ .", "Let $G$ be a vector subspace of $\\ell _2({\\mathbb {N}})$ and $\\mu $ a probability measure on $\\ell _2({\\mathbb {N}})$ , then we denote by $\\Pi _{G*}\\mu $ the projection of $\\mu $ onto $G$ .", "Restriction 4 There exists $n_0\\in {\\mathbb {N}}$ and $\\lbrace \\underline{x}_m\\rbrace \\in \\ell _2({\\mathbb {N}})$ such that, for all $n\\ge n_0$ , $ \\lbrace (x_1,\\dots ,x_n) : \\lbrace x_m\\rbrace \\in \\mathbb {S}_{\\lbrace X_m\\rbrace },\\ \\forall j> n, x_j = \\underline{x}_j\\rbrace $ is not a subset of the zeros of a nonzero element of ${\\mathbb {R}}[Z_1,\\dots ,Z_{n}]$ ; $\\mathcal {P}$ is the set of measures whose projections onto $G_{n+1} := \\text{Span}(\\lbrace e_{1,m}\\rbrace ,\\dots , \\lbrace e_{n+1,m}\\rbrace )$ are determinate in $\\mathfrak {M}^*({\\mathbb {R}}^{n+1})$ .", "$\\mathcal {R}_{L,4}$ (REF ) allows discrete support of the regressors.", "Like in Section REF , less restrictive conditions on $\\mathbb {S}_{\\lbrace X_m\\rbrace }$ can be obtained assuming independence between marginals of $\\mathbb {P}_{\\lbrace \\Gamma _m\\rbrace }$ .", "An alternative restriction is: Restriction 5 There exists a Gaussian measure $\\mu $ on $\\ell _2({\\mathbb {N}})$ such that $\\lbrace \\lbrace u_m\\rbrace \\in {\\mathbb {R}}^{{\\mathbb {N}}_0}:\\ \\forall n\\in {\\mathbb {N}},\\ u_n=u_0x_n,\\ \\lbrace x_m\\rbrace \\in \\mathbb {S}_{\\lbrace X_m\\rbrace }\\rbrace $ has positive $\\mu $ -measure; $\\mathcal {P}$ is the set of measures such that, for all $n\\in {\\mathbb {N}}$ and $\\lbrace f_m\\rbrace \\in \\ell _2({\\mathbb {N}})\\setminus \\lbrace \\lbrace 0_m\\rbrace \\rbrace $ , the projections onto $G_{n+1} := \\text{Span}(\\lbrace e_{1,m}\\rbrace ,\\dots , \\lbrace e_{n,m}\\rbrace , \\lbrace f_m\\rbrace )$ are determinate in $\\mathfrak {M}^*({\\mathbb {R}}^{n+1})$ ." ], [ "Without restrictions on $\\alpha $", "$\\mathcal {R}_{L,3}$ (REF ) places restrictions on $\\mathbb {P}_{\\alpha }$ which we entirely remove.", "This is important to model income or wealth and allow $\\alpha $ to have, for example, a Pareto distribution which does not even belong to $\\mathfrak {M}^*({\\mathbb {R}})$ .", "We can replace the role of $\\alpha $ by $\\beta _k$ for $k\\in [p]$ or a combination of the coefficients if, starting from (REF ), we form, for example, $\\frac{Y}{X_k-\\underline{x}_k}=\\beta _k+\\frac{\\alpha }{X_k-\\underline{x}_k}+\\sum _{j\\ne k}\\beta _j\\frac{X_j-\\underline{x}_j}{X_k-\\underline{x}_k}$ and there exists $\\underline{x}_k\\notin \\mathbb {S}_{X_k}$ .", "If the support of a subvector of dimension $k$ of $X$ is ${\\mathbb {R}}^k$ then we can similarly handle situations where there are no restrictions on the joint distribution of the corresponding coefficients and $\\alpha $ .", "In this section, $\\Omega \\subseteq {\\mathbb {R}}^p$ is a closed set and it can be known that $\\mathbb {S}_{\\beta }\\subseteq \\Omega $ .", "A key idea is to rely on the partial Fourier transform.", "For example, $\\mathcal {F}[\\mathbb {P}_{\\Gamma }]$ can be analyzed as the collection of functions $\\mathcal {F}[\\mathbb {P}_{\\Gamma }](t,\\cdot )$ for $t\\in {\\mathbb {R}}$ and $\\forall t\\in {\\mathbb {R}},\\ \\mathcal {F}\\left[\\mathbb {P}_{\\Gamma }\\right](t,\\star )=\\mathcal {F}\\left[\\mathbb {P}_{\\beta ,t}\\right](\\star ),\\ \\text{where}\\ \\mathbb {P}_{\\beta ,t}=\\int _{{\\mathbb {R}}}e^{ita}d\\mathbb {P}_{\\Gamma }(a,\\cdot ).$ Because the vector space spanned by $\\mathcal {F}[\\mathcal {P}_c(\\Omega )]$ is a quasi-analytic class of functions on ${\\mathbb {R}}^{p}$ and by the continuity of the Fourier transform at 0, $\\mathbb {P}_{\\Gamma }^*$ in (REF ) is identified under $\\mathcal {R}_{L,6}$ below.", "Restriction 6 $\\mathbb {S}_{ X }$ has a nonempty interior or, when $p=1$ , $\\mathbb {S}_{X}$ contains a bounded sequence of distinct points; $\\mathcal {P}$ restricts $\\mathbb {P}_{\\Gamma }$ so that $\\mathbb {P}_{\\beta ,t}\\in \\mathcal {P}_c(\\Omega )$ for all $t\\ne 0$ .", "$\\mathcal {R}_{L,6}$ (REF ) implies that $t\\mathbb {S}_{ X }$ is a set of uniqueness of quasi-analytic classes for all $t\\ne 0$ .", "In order to have quantitative statements on $\\mathbb {S}_{X}$ yielding sets of uniqueness and explicit conditions so that $\\mathcal {R}_{L,6}$ (REF ) holds, we restrict our attention to the case where there are sets of points $V_{k}\\subseteq {\\mathbb {R}}$ for $k\\in [p]$ such that $\\prod _{k=1}^pV_{k} \\subset \\mathbb {S}_{ X }$ .", "The following restriction involves pairs $(C_k,\\mathcal {P}_k)$ for $k\\in [p]$ .", "$\\mathcal {P}_k$ are classes of probabilities of the form $\\mathcal {P}_k=\\left\\lbrace \\mathbb {P}_{\\beta _k}\\in \\mathfrak {M}_1^*(\\Omega _k): \\forall h\\in \\mathcal {H}_k,\\ \\mathbb {E}\\left[h\\left(\\beta _k\\right)\\right]\\le M_k(h)\\right\\rbrace ,$ where $\\Omega _k\\subseteq {\\mathbb {R}}$ is a closed set, which can be taken not equal to ${\\mathbb {R}}$ if the researcher knows $\\mathbb {S}_{\\beta _k}\\subseteq \\Omega _k$ , and $\\mathcal {H}_k$ can consist of one or a sequence of nonnegative measurable functions.", "The parameters of these classes are $\\Omega _k$ , $\\mathcal {H}_k$ , and $M_k$ .", "The proof relies on the related class $\\mathcal {P}_{c,k}=\\left\\lbrace \\mu \\in \\mathfrak {M}_c^*(\\Omega _k): \\forall h\\in \\mathcal {H}_k,\\ \\int _{{\\mathbb {R}}}h(z)d|\\mu |(z)\\le 2M_k(h)\\right\\rbrace .$ The classes $C_k$ are such that $ \\mathcal {F}[\\mathcal {P}_{c,k}]\\subseteq C_k\\subseteq C^{\\infty }({\\mathbb {R}})$ .", "Restriction 7 $\\mathbb {S}_{X}$ and $\\mathcal {P}$ are such that, for all $k\\in [p]$ , $tV_k$ is a set of uniqueness of $C_k$ for all $t\\ne 0$ ; $\\mathbb {P}_{\\beta _k}\\in \\mathcal {P}_k$ .", "Let us give examples of pairs $(V_k,\\mathcal {P}_k)_{k=1}^p$ and of corresponding sets of uniqueness.", "Some examples use log-convex sequences $(M_m)_{m\\in {\\mathbb {N}}_0}$ which are sequences of nonnegative numbers such that, for all $m\\in {\\mathbb {N}}$ , $M_m^2\\le M_{m-1}M_{m+1}$ and we also assume $M_0=1$ .", "Define also the iterated logarithms by $\\log ^{*0}(t) = t$ and for $j\\ge 1$ by $\\log ^{*j}(t) = \\log (\\log ^{*(j-1)} (t))$ provided $t$ is large enough for the quantity to be defined and $\\exp ^{*n}$ is the reciprocal function.", "For $k\\in [p]$ , ${\\mathbb {E}} \\left[e^{r \\left| \\beta _k\\right| }\\right] \\le m(r) $ for all $r\\in {\\mathbb {R}}_+$ , where $ \\lim _{r\\rightarrow \\infty }m(r) = \\infty $ and $V_{k}$ satisfies $\\forall t>0,\\ \\exists \\alpha >1:\\ \\overline{\\lim }_{r\\rightarrow \\infty }\\frac{\\log (\\alpha )}{\\log (m(\\alpha t r))}\\left|V_k\\cap \\left((-r,r)\\setminus \\lbrace 0\\rbrace \\right)\\right|>1.$ For $k\\in [p]$ , there exists $(b,c)\\in {\\mathbb {R}}_+^2$ such that ${\\mathbb {E}} \\left[ \\left| \\beta _k\\right| ^m\\right] \\le c b^m m!", "$ for all $m\\in {\\mathbb {N}}_0$ and $V_k$ satisfies $\\overline{\\lim }_{r\\rightarrow \\infty }\\frac{\\log \\left(\\left|V_k\\cap (-r,r)\\right|\\right)}{r}=\\infty .$ For $k\\in [p]$ , $\\mathbb {S}_{\\beta _k}\\subseteq {\\mathbb {R}}_+$ , there exist $(b,c)\\in {\\mathbb {R}}_+^2$ and a log-convex sequence $\\lbrace M_m\\rbrace $ such that $M_{0}=1$ , ${\\mathbb {E}} \\left[ \\left| \\beta _k\\right| ^m\\right] \\le c b^m M_{m} $ for all $m\\in {\\mathbb {N}}_0$ , $ \\sum _{m\\in {\\mathbb {N}}}\\frac{1}{M_{m}^{1/(2m)}}=\\infty ,$ and $V_{k}$ contains a bounded sequence of distinct points; For $k\\in [p]$ , there exist $(b,c)\\in {\\mathbb {R}}_+^2$ and a log-convex sequence $\\lbrace M_m\\rbrace $ such that $M_{0}=1$ , ${\\mathbb {E}} \\left[ \\left| \\beta _k\\right| ^m\\right] \\le c b^m M_{m} $ for all $m\\in {\\mathbb {N}}_0$ , and either $M_{m}= \\nu (m)m!$ , where $\\nu $ satisfies, for all $n\\in {\\mathbb {N}}$ , $\\nu (m)=1$ for $0\\le m<\\exp ^{*n}(1)$ and else $\\nu (m)=(\\log (m)\\log ^{*2}(m)\\cdot \\log ^{*n}(m))^m$ , and $V_k$ satisfies $\\overline{\\lim }_{r\\rightarrow \\infty }\\frac{\\log ^{*n+1}\\left(\\left|V_k\\cap (-r,r)\\right|\\right)}{r}=\\infty ;$ or $ \\lbrace M_{m}\\rbrace $ satisfies $\\sum _{m\\in {\\mathbb {N}}}\\frac{1}{M_{2m}^{1/(2m)}}=\\infty $ and $V_{k}$ contains a bounded sequence of distinct points; In (E.REF ), a sufficient condition for (REF ) when $\\mathbb {S}_{\\beta _k}\\subseteq [-\\rho ,\\rho ]$ , hence $m(r)=e^{\\rho r}$ , is $\\overline{\\lim }_{r\\rightarrow \\infty }\\frac{1}{r}\\left|V_k\\cap \\left((-r,r)\\setminus \\lbrace 0\\rbrace \\right)\\right|=\\infty .$ In (E.REF ), $C_{k}$ consists of the functions $f$ such that there exists $\\rho >0$ such that $f$ can be extended uniquely to $\\mathcal {H}^{\\infty }\\left(\\lbrace z\\in {\\mathbb {C}}:\\ |\\mathrm {Im}(z)|<\\rho \\rbrace \\right)$ (see Theorem 19.9 in [47]).", "In (E.REF ), and (E.REF ), $C_{k}$ are quasi-analytic classes which are detailed in Appendix REF .", "In (E.REF ), using Proposition REF (REF ) with $\\lbrace M_m\\rbrace =\\lbrace m!\\rbrace $ , $C_{k}$ is an analytic class and the set of uniqueness is given in (2) in Section 1 of [31].", "In particular, Proposition REF shows that $ \\mathcal {F}[\\mathcal {P}_{c,k}]\\subseteq C_k$ .", "Based on Theorem 4b in [31], Example 2 in Appendix REF gives more general forms of $\\nu $ in the definition of $\\lbrace M_m\\rbrace $ in (E.REF ) and associated sets $V_k$ .", "The relation between condition (REF ) (resp.", "(REF )) and the determinacy of measures in $\\mathfrak {M}^*({\\mathbb {R}})$ (resp.", "$\\mathfrak {M}^*({\\mathbb {R}}_+)$ ) is explicited in Section REF .", "Remark 2 The Student's $t$ with $0<\\nu <\\infty $ degrees of freedom, the generalized gamma $GG(a,b,p)$ for $0<a<1/2$ of density $ab^px^{ap-1}\\exp (-bx^a)/\\Gamma (p)$ on ${\\mathbb {R}}^+$ , any positive power of the lognormal, the law of $N^{2n+1}$ for all $n\\in {\\mathbb {N}}$ , $|N|^r$ for $r>4$ , $X^m$ for all $m\\in {\\mathbb {N}}\\setminus \\lbrace 1,2\\rbrace $ , $Y^r$ for all $|r|>2$ , where $N$ is a Gaussian, $X$ a Laplace, gamma or logistic, and $Y$ an inverse Gaussian random variables, respectively do not satisfy these conditions.", "However, $|N|^r$ for all $0<r\\le 4$ , $X^m$ for $m=1,2$ , $Y^r$ for all $-2\\le r\\le 2$ , $GG(a,b,p)$ for all $a\\ge 1/2$ (thus the $\\chi ^2$ with any degrees of freedom) satisfy these conditions, hence are determinate in the space of measures with the appropriate support restriction (i.e., $\\mathfrak {M}^*({\\mathbb {R}})$ or $\\mathfrak {M}^*({\\mathbb {R}}_+)$ ) (see [54], [44], [35] for more examples)." ], [ "The random coefficients binary choice model", "The random coefficients binary choice model takes the form $Y={\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace \\alpha +\\beta ^{\\top }X\\ge 0\\rbrace , \\ (\\alpha , \\beta ) \\perp X .$ The binary variable $Y$ is 1 when an individual chooses a good, treatment, or an action.", "In the context of treatment effects, $Y=1$ when an individual chooses treatment.", "[58] shows the monotonicity of [34] is equivalent to a selection equation with an additively separable latent index with a single unobservable.", "Hence monotonicity can be related to rank invariance (see [13]) similarly to the fact that, in Section , a linear random coefficients models where the random coefficients are functions of a scalar unobservable can be related to the quantile regression.", "It is well known that monotonicity is sometimes too strong and it is exemplified in [34].", "[29] call (REF ) the benchmark nonseparable, nonmonotonic model of treatment choice.", "Because the scale of the index is not identified, various normalizations can be used.", "A simple one used in consists in assuming one coefficient is 1.", "This requires that in the original scale the coefficient has a (strict) sign a.s. Based on this and under sufficient variation of the corresponding (special) regressor [23] show that identification corresponds to identification of the model of Section .", "It was mentioned that this idea applies to all sorts of models involving as constitutive element random coefficients indices lying in certain rectangles (e.g., choice models with multiple alternatives or entry games).", "[23] also considers the case where the special regressor and the remaining ones have limited variation.", "The scaling of [24] in a preliminary version [23] was removed and overlooked as not important.", "However, assuming that one coefficient is 1 is unnecessary and too restrictive because it still imposes some form of monotonicity.", "Some elements can be found in [22] in the context of selection equations and missing data in sample surveys.", "Assuming that $\\mathbb {P}(|(\\alpha ,\\beta ^{\\top })^{\\top }|_2=0)=0$ , (REF ) can be equivalently written as $Y={\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace \\Gamma ^{\\top }S\\ge 0\\rbrace , \\ \\Gamma \\perp S ,$ where $\\Gamma =(\\alpha ,\\beta ^{\\top })^{\\top }/|(\\alpha ,\\beta ^{\\top })^{\\top }|_2$ and $S=(1,X^{\\top })^{\\top }/|(1,X^{\\top })^{\\top }|_2$ .", "Clearly $|(1,X^{\\top })^{\\top }|_2\\ge 1$ and the support of $S$ is a closed subset of the hemisphere $H^+=\\lbrace s\\in \\mathbb {S}^p:\\ s_1\\ge 0\\rbrace $ .", "We consider identification of the density $f_{\\Gamma }^*$ of $\\mathbb {P}_{\\Gamma }^*$ with respect to $\\sigma $ , which is the surface measure on $\\mathbb {S}^p$ .", "In this section, we consider the following restriction of the class $\\mathcal {P}$ $\\mathcal {P}_{BC}=\\left\\lbrace \\mathbb {P}_{\\Gamma }\\in \\mathfrak {M}_1(\\mathbb {S}^p):\\ d\\mathbb {P}_{\\Gamma }=f_{\\Gamma }d\\sigma ,\\ f_{\\Gamma }(u)f_{\\Gamma }(-u)=0 \\ \\mathrm {for\\ a.e.", "}\\ u\\in \\mathbb {S}^p\\right\\rbrace .$ It is shown in [25] that $\\mathbb {P}_{\\Gamma }^*$ is identified under the restriction $\\mathbb {S}_{X}={\\mathbb {R}}^{p}$ and $\\mathcal {P} =\\mathcal {P}_{BC}$ .", "It is assumed in [24] that the support of $\\Gamma $ lies in an (unknown) hemisphere, namely, that there exists $n$ in $\\mathbb {S}^p$ such that $\\mathbb {P}(n^\\top \\Gamma \\ge 0)=1$ .", "This assumption first appeared in [33] and is just a sufficient but unnecessary assumption, similar to assuming that one coefficient is 1.", "Indeed, if $n\\in H^+$ , then we have $\\mathbb {P}(Y=1|S=n)=1$ , else $\\mathbb {P}(Y=1|S=-n)=0$ .", "This means that there exist limits of values of the regressors such that in the limit everyone chooses $Y=1$ or in the limit everyone chooses $Y=0$ .", "It is stronger than $f_{\\Gamma }(u)f_{\\Gamma }(-u)=0$ for a.e.", "$u\\ \\mathrm {in}\\ \\mathbb {S}^p$ which does not imply “unselected samples\".", "Consider now the case where the support of $X$ is a proper subset of ${\\mathbb {R}}^p$ or equivalently the support of $S$ is a proper subset of $H^+$ .", "The hemispherical transform is defined, for $s\\in \\mathbb {S}^p$ and $f\\in L^1\\left(\\mathbb {S}^p\\right)$ , by $\\mathcal {T}f(s)=\\int _{\\mathbb {S}^p}{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace u^{\\top }s\\ge 0\\rbrace f(u)d\\sigma (u)-1/2$ .", "Using the restrictions $\\mathcal {P}_{BC}$ , we have $\\forall s\\in \\mathbb {S}_{S},\\ \\mathcal {T}f_{\\Gamma }(s)+\\frac{1}{2}=\\mathbb {E}[Y|S=s].$ $V$ -quasi-analytic classes of functions on $\\mathbb {S}^p$ are vector spaces of functions on $\\mathbb {S}^p$ characterized by $\\left(\\Delta ^m f(x)\\right)_{m\\in {\\mathbb {N}}_0}$ for all $x\\in V$ .", "In this section, we consider two different identifying restrictions which we denote by $\\mathcal {R}_{BC,j}$ for $j\\in [2]$ .", "The main theorem of this section is the following Theorem 2 $\\mathbb {P}_{\\Gamma }^*$ in (REF ) is identified under either of $\\mathcal {R}_{BC,j}$ for $j\\in \\lbrace 1,2\\rbrace $ .", "Similarly to Section , the proofs rely on the diagram (REF ) with $\\mathcal {G}=\\mathcal {T}$ , $U = \\lbrace s = (1, x ^{\\top })/\\left|(1, x ^{\\top })\\right|_2, \\ x \\in \\mathbb {S}_{ X } \\rbrace $ , and $\\Pi : \\ \\left( \\mathbb {P}_{{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace \\Gamma ^{\\top }s\\ge 0\\rbrace } \\right)_{ s \\in \\mathbb {S}_{ S }} \\mapsto \\left( s \\in \\mathbb {S}^p \\mapsto \\int _{\\mathbb {S}^p}{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace u^{\\top }s\\ge 0\\rbrace f(u)d\\sigma (u)-\\frac{1}{2} \\right).$ In the rest of this section, we consider $\\mathcal {P}\\subseteq \\mathcal {P}_{BC}$ such that, for given $V\\subseteq \\mathbb {S}^p$ , the vector space spanned by $\\mathcal {T}\\left[f_{\\Gamma }\\right]$ for all $\\mathbb {P}_{\\Gamma }\\in \\mathcal {P}$ is a $V$ -quasi-analytic class of functions on $ \\mathbb {S}^p$ .", "We now give details on the restrictions.", "For functions on $\\mathbb {S}^p$ , the Laplacian $\\Delta $ has eigenspaces $H_{m,p}$ , eigenvalues $\\zeta _{m,p}=-m(m+p-1)$ , and $Q_{m,p}f(\\cdot )=\\int _{\\mathbb {S}^p}q_{m,p}(\\cdot ,y)f(y)d\\sigma (y)$ is the orthogonal projection of $f$ onto $H^{m,p}$ for all $m\\in {\\mathbb {N}}_0$ .", "Restriction 8 There exists $0<\\epsilon <1$ such that $\\mathbb {S}_{X}$ is a set of uniqueness of $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}^p:\\ \\left|\\mathrm {Im}(z)\\right|_{2}<\\epsilon \\right\\rbrace \\right)$ ; $\\mathcal {P}$ is such that $\\overline{\\lim }_{m\\rightarrow \\infty }\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}^{1/m}<1/(1+2\\epsilon )$ .", "$\\mathcal {R}_{BC,1}$ (REF ) is a sufficient condition for: ${\\mathbb {E}} \\left[Y | X = \\cdot \\right]$ belongs to $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}^p:\\ \\left|\\mathrm {Im}(z)\\right|_{2}<\\epsilon \\right\\rbrace \\right)$ .", "Clearly, a set of uniqueness of $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}^p:\\ \\left|\\mathrm {Im}(z)\\right|_{2}<\\epsilon \\right\\rbrace \\right)$ is a set of uniqueness of the superset $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}^p:\\ \\left|\\mathrm {Im}(z)\\right|_{\\infty }<\\epsilon \\right\\rbrace \\right)$ .", "Hence, a sufficient condition for $\\mathcal {R}_{BC,1}$ (REF ) is that $U_{p} \\subseteq \\mathbb {S}_{X}$ where for all $j=2,\\dots ,p$ , $U_{j}=\\bigcup _{u\\in {U_{j-1}}}\\left\\lbrace (u^{\\top },v)^{\\top },\\ v\\in {V}_{j}(u)\\right\\rbrace $ , where ${V}_{j}(u)$ and $U_{1}$ are sets of uniqueness of $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}:\\ \\left|\\mathrm {Im}(z)\\right|<\\epsilon \\right\\rbrace \\right)$ .", "A particular case is a product of sets of uniqueness of $\\mathcal {H}^{\\infty }\\left(\\left\\lbrace z\\in {\\mathbb {C}}:\\ \\left|\\mathrm {Im}(z)\\right|<\\epsilon \\right\\rbrace \\right)$ .", "We give examples in Section REF .", "The odd part $f^-$ of $f\\in L^q\\left(\\mathbb {S}^p\\right)$ is the limit in $L^q\\left(\\mathbb {S}^p\\right)$ of $f_n^-(x)=(f_n(x)-f_n(-x))/2$ , where $\\left(f_n\\right)_{n\\in {\\mathbb {N}}_0}\\in \\left(C\\left(\\mathbb {S}^p\\right)\\right)^{{\\mathbb {N}}_0}$ converges to $f$ in $L^q\\left(\\mathbb {S}^p\\right)$ .", "We make use of $C^{\\infty }_{{\\rm odd}}(\\mathbb {S}^p)$ , the restriction to odd functions of $C^{\\infty }(\\mathbb {S}^p)$ , and $C(\\lbrace M_m\\rbrace )=\\lbrace f\\in C^{\\infty }(\\mathbb {S}^p):\\ \\exists c,b\\in {\\mathbb {R}}_+:\\ \\forall m\\in {\\mathbb {N}}_0,\\ \\left\\Vert \\Delta ^m f^-\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\le cb^mM_m\\rbrace .$ Restriction 9 $\\lbrace M_m\\rbrace $ satisfies (REF ) and There exists $U\\subseteq \\mathbb {S}_{S}$ a set of uniqueness of harmonic homogenous polynomials of odd degree in ${\\mathbb {R}}[Z_1,\\dots ,Z_{p+1}]$ and, for all $f\\in C^{\\infty }_{{\\rm odd}}\\left(\\mathbb {S}^p\\right)\\cap \\mathcal {T}[C(\\lbrace M_m\\rbrace )]$ , if, for all $u\\in \\mathbb {S}_{S}$ , $f(u)=0$ , then, for all $u\\in U$ and $m\\in {\\mathbb {N}}_0$ , $\\Delta ^m f (u)=0$ ; $\\mathcal {P}$ is such that $ f_{\\Gamma } \\in C(\\lbrace M_m\\rbrace )$ .", "$U$ is a set of uniqueness of homogeneous polynomials in $\\mathbb {R}[Z_1,\\dots ,Z_{p+1}]$ if there exists $A\\in GL(\\mathbb {{\\mathbb {R}}}^{p+1})$ such that $\\lbrace (1, u ^{\\top })^{\\top }: u\\in \\widetilde{U}\\rbrace \\subseteq AU$ and $\\widetilde{U}$ is a set of uniqueness of $\\mathbb {R}[Z_1,\\dots ,Z_p]$ .", "If $\\mathbb {S}_{X}$ and thus $\\mathbb {S}_{S}$ contains an open set, then clearly $\\mathcal {R}_{BC,2}$ (REF ) is satisfied." ], [ "Extension of the Kotlarski lemma", "Consider the equation $Y_t= \\delta + \\epsilon _t,\\ t\\in \\lbrace 1,2\\rbrace ,$ where the vector of unobserved heterogeneity is $\\Gamma :=(\\epsilon _1,\\epsilon _2,\\delta )$ .", "Restriction 10 Define the restriction $\\mathcal {R}_K$ on $\\mathcal {P}$ by $\\left\\lbrace \\mathbb {P}_{\\Gamma }=\\mathbb {P}_{\\epsilon _1}\\otimes \\mathbb {P}_{\\epsilon _2}\\otimes \\mathbb {P}_{\\delta },\\ \\mathbb {P}_{\\epsilon _1}\\in \\mathcal {P}_1(\\Omega _1)\\right\\rbrace $ , where $\\Omega _1 \\subseteq {\\mathbb {R}}$ is a closed set and $\\mathcal {P}_1(\\Omega _1)$ is the set of measures which are determinate in $\\mathfrak {M}^*(\\Omega _1)$ and such that ${\\mathbb {E}} _{\\mathbb {P}_{\\epsilon _1}}[\\epsilon _1]=0$ Theorem 3 $\\mathbb {P}_{\\Gamma }^*$ in (REF ) is identified under $\\mathcal {R}_K$ .", "Lemma 1 in [36] assumes all characteristic functions do not vanish and in Remark 1 it is written that this can be extended to the case where all characteristic functions are analytic.", "Theorem REF shows that these assumptions are too strong and identification can be achieved when none of the characteristic functions are analytic and $\\epsilon _2$ and $\\delta $ might not have finite first absolute moments.", "[18] present a similar result under alternative assumptions, but assuming $\\varphi _{\\epsilon _1}$ is analytic." ], [ "Linear panel data model where regressors are monomials of a baseline scalar regressor", "Consider the equation $Y_t=\\alpha + \\sum _{j=1}^{T}\\beta _{j} X_t^j +\\epsilon _t, \\ t=1,\\dots ,T,$ where $X_t$ is a scalar regressor and denote by $\\Gamma = (\\alpha , \\beta ^{\\top })^{\\top }$ .", "For each $t$ , we have $\\mathbb {S}_{X_t,\\dots ,X_t^T}\\subseteq \\left\\lbrace {u}\\in {\\mathbb {R}}^p:\\ {u}_2={u}_1^2,\\hdots ,{u}_T={u}_1^T\\right\\rbrace ,$ hence the restriction $\\mathcal {R}_{L,3}$ does not hold.", "We show in this section that the availability of $T$ periods allows nonparametric identification.", "Remark 3 Note that (REF ) could be generalized to $Y_t=\\alpha + P(X_t) + \\epsilon _t$ , for all $t=1,\\dots ,T$ , where $P(X_t) = \\sum _{j=1}^{T}\\beta _{j} X_t^{\\theta (j)}$ and $\\theta \\in {\\mathbb {N}}_0^{{\\mathbb {N}}}$ and increasing.", "However, using our identification strategy would yield to consider the so-called generalized Vandermonde matrices, whose theoretical properties, in particular the ones of their inverse, are not yet well known.", "Restriction 11 $\\mathcal {X}_1 =\\lbrace x\\in \\mathbb {S}_{X}: x_1=\\dots =x_T\\rbrace \\ne \\emptyset $ .", "$ \\mathbb {P}_{\\Gamma } = \\bigotimes _{j=1}^T \\mathbb {P}_{\\epsilon _j} \\otimes \\mathbb {P}_{\\alpha ,\\beta }$ , $\\mathbb {P}_{\\epsilon _1}\\in \\mathcal {P}_1(\\Omega _1)$ , where $\\Omega _1 \\subseteq {\\mathbb {R}}$ is a closed set and $\\mathcal {P}_1(\\Omega _1)$ is the set of measures which are determinate in $\\mathfrak {M}^*(\\Omega _1)$ and such that ${\\mathbb {E}} _{\\mathbb {P}_{\\epsilon _1}}[\\epsilon _1]=0$ ; Using an extension of Theorem REF to $T$ periods, see proof of Theorem REF , $\\mathbb {P}_{\\epsilon }^*$ is identified under $\\mathcal {R}_{LP,0}$ .", "Note that the restriction ${\\mathcal {R}_{LP,0}}$ (REF ) is weaker than assuming that the covariates are centered $ 0 _c\\in \\mathbb {S}_{X}$ .", "The restriction ${\\mathcal {R}_{LP,0}}$ (REF ) is also maintained in [14], where they focus on the marginals of $\\Gamma $ without imposing our baseline independence assumption but considering only the individuals whose $ X $ belong to $\\mathcal {X}_1$ , which are the stayers.", "Restriction 12 For all $t\\in [T]$ , $\\mathbb {S}_{ X _t}$ contains a bounded sequence of distinct points; $\\mathbb {P}_{ \\beta _T} \\in \\mathcal {P}_T(\\Omega _T)$ , where $\\Omega _T \\subseteq {\\mathbb {R}}$ is a closed set and $\\mathcal {P}_T(\\Omega _T)$ is the set of measures which are determinate in $\\mathfrak {M}^*(\\Omega _T)$ .", "Theorem 4 $\\mathbb {P}^*_{\\Gamma , \\epsilon } $ in (REF ) is identified under $\\mathcal {R}_{LP,0}$ and $\\mathcal {R}_{LP}$ .", "Like Theorem REF under $\\mathcal {R}_{L,6}$ or $\\mathcal {R}_{L,7}$ , Theorem REF makes no assumptions on $\\mathbb {P}^*_{\\alpha , \\beta _1,\\dots , \\beta _{T-1}}$ ." ], [ "A single-index panel data model with two periods.", "Consider the equation $Y_t=f(\\Gamma ^{\\top }X_t) + \\eta _t,\\ t=1,2,\\ f\\text{ is increasing.", "}$ Restriction 13 Either of $\\mathcal {R}_{BC,j}$ for $j\\in \\lbrace 1,2\\rbrace $ holds and $\\lbrace (x_1,x_2)\\in \\mathbb {S}_{X_1,X_2}: x_1=x_2\\rbrace \\ne \\emptyset $ ; $ \\mathbb {P}_{\\Gamma , \\eta } = \\mathbb {P}_{\\Gamma } \\otimes \\mathbb {P}_{\\eta _1} \\otimes \\mathbb {P}_{\\eta _2} $ , $\\mathbb {P}_{ \\eta _1} \\in \\mathcal {P}_1(\\Omega _1)$ , where $\\Omega _1 \\subseteq {\\mathbb {R}}$ is a closed set and $\\mathcal {P}_1(\\Omega _1)$ is the set of measures which are determinate in $\\mathfrak {M}^*(\\Omega _1)$ , ${\\mathbb {E}} _{\\mathbb {P}_{\\eta _1}}\\left[\\eta _1\\right]=0$ ; $\\left|\\Gamma \\right|_2=1$ .", "Theorem 5 $\\mathbb {P}_{\\Gamma , \\eta }^*$ in (REF ) is identified under $\\mathcal {R}_{SI}$ .", "Theorem REF relies on both theorems REF and REF .", "Restriction ${R_{SI}}$ (REF ) means that there exist “stayers\" in the population, for which the value of the covariate stays the same accross periods, which is a mild assumption." ], [ "For a differentiable function $f$ of real variables, $f^{(m)}$ denotes $\\prod _{j=1}^{p}\\frac{\\partial ^{m_j}}{\\partial x _j^{m_j}}f$ and $\\mathrm {supp}(f)$ its support.", "Define, for $\\mu ,\\nu \\in \\mathfrak {M}_c^*\\left(\\Omega \\right)$ , where $\\Omega $ is a closed set of ${\\mathbb {R}}^p$ , the equivalence $\\mu \\sim \\nu $ if $s_{\\mu }(m)=s_{\\nu }(m)$ for all $m\\in {\\mathbb {N}}_0^{p}$ .", "${\\rm Aff}(\\mathbb {R}^p)$ for the group of invertible affine transformations of $\\mathbb {R}^p$ .", "Denote by $S^{\\prime }\\left(\\mathbb {S}^p\\right)$ the dual of $C^{\\infty }\\left(\\mathbb {S}^p\\right)$ (see [24])." ], [ "Analytic and quasi-analytic classes of Fourier transforms of complex measures", "Given $b\\in {\\mathbb {R}}_+$ , and $\\lbrace M_{m}\\rbrace \\in ((0,\\infty ]^p)^{{\\mathbb {N}}_0}$ where $\\lbrace M_{m,k}\\rbrace $ for all $k\\in [p]$ are log-convex sequences with $ M _{0,k}=1$ , let us introduce the class of functions $C^{\\lbrace M_{m}\\rbrace }(b)=\\left\\lbrace f\\in C^{\\infty }({\\mathbb {R}}^p):\\ \\exists c:\\ \\forall m\\in \\mathbb {N}_0^{p},\\ \\left\\Vert f^{(m)}\\right\\Vert _{L^{\\infty }({\\mathbb {R}}^p)}\\le cb^{|m|}\\prod _{k=1}^{p}M_{k,m_k}\\right\\rbrace .$ It consists of complex valued functions and is a vector space.", "We denote by $C^{\\lbrace M _{m}\\rbrace }=\\bigcup _{b\\in {\\mathbb {R}}_+}C^{\\lbrace M _{m}\\rbrace }(b)$ .", "Like $\\mathcal {A}(S)$ , $C^{\\lbrace M _{m}\\rbrace }$ is an algebra with respect to multiplication(see [47]).", "From Theorem 19.9 in [47], $C^{\\lbrace m!\\rbrace }$ consists of the functions $f$ such that there exists $\\rho >0$ such that $f$ can be extended uniquely to $\\mathcal {H}^{\\infty }\\left(\\lbrace z\\in {\\mathbb {C}}:\\ |\\mathrm {Im}(z)|<\\rho \\rbrace \\right)$ .", "Note that if $f\\in C^{\\lbrace M_{m}\\rbrace }(b)$ and $\\underline{x}\\in {\\mathbb {R}}^p$ , then the function obtained by fixing all variables but the $k^{\\text{th}}$ to those of $\\underline{x}$ defines a function of $C^{\\lbrace M_{k,{m}_k}\\rbrace }(b)$ .", "Thus, products of sets of uniqueness of the spaces of functions of a single variable are sets of uniqueness of $C^{\\lbrace M_m\\rbrace }(b)$ .", "By the Hölder inequality, if $\\Vert \\cdot \\Vert $ is a norm and $\\mu \\in \\mathfrak {M}^*\\left({\\mathbb {R}}^{p}\\right)$ , $\\left(\\int _{{\\mathbb {R}}^{p}}\\Vert x\\Vert ^md\\mu (x)\\right)_{m\\in {\\mathbb {N}}_0}$ is log-convex and $M_0=\\mu \\left({\\mathbb {R}}^{p}\\right)$ .", "Denote by $M(x)$ the trace function of $\\lbrace M_m\\rbrace $ , where, for all $m\\in {\\mathbb {N}}$ , $M_m^c=\\exp \\left(\\sup _{x\\ge 0}(mx-M(x))\\right)$ and $M(x)=\\sup _{m\\in {\\mathbb {N}}}(mx-\\log (M_m))$ (see [39]).", "The sequences $\\lbrace 1_m\\rbrace $ and $\\lbrace M^c_m\\rbrace $ are log-convex.", "If $S={\\mathbb {R}}$ and $\\underline{\\lim }_{m\\rightarrow \\infty }M_m^{1/m}=0$ then $C^{\\lbrace M_m\\rbrace }(S)=C^{\\lbrace 0_m\\rbrace }(S)$ and if $0<\\underline{\\lim }_{m\\rightarrow \\infty }M_m^{1/m}<\\infty $ then $C^{\\lbrace M_m\\rbrace }(S)=C^{\\lbrace 1_m\\rbrace }(S)$ , else, if the terms in the sequence are positive, $C^{\\lbrace M_m\\rbrace }(S)=C^{\\lbrace M^c_m\\rbrace }(S)$ .", "Let us also introduce certain subsets of $\\mathcal {P}_c(\\Omega )$ of the form $\\mathcal {P}_c^{\\lbrace M _{m}\\rbrace }(\\Omega )=\\left\\lbrace \\mu \\in \\mathfrak {M}_c^*\\left(\\Omega \\right): \\ \\exists c,b : \\forall m\\in {\\mathbb {N}}_0, k\\in [p], \\ s_{|\\cdot |,|\\mu |}(me_k) \\le c b^m M _{k,m} \\right\\rbrace .$ Proposition REF shows that $\\mathcal {F}\\left[\\mathcal {P}_{c} \\right] \\subseteq C$ for the classes $\\mathcal {P}_{c}$ and $C$ used in the examples (E.REF ), (E.REF ) and (E.REF ) related to the restriction $\\mathcal {R}_{L,7}$ .", "Proposition A.1 Let $\\Omega \\subseteq {\\mathbb {R}}$ be a closed set, $\\mu \\in \\mathfrak {M}_c^*(\\Omega )$ , $f = \\mathcal {F}[\\mu ]$ , and $\\lbrace M_m\\rbrace \\in (0,\\infty ]^{{\\mathbb {N}}_0}$ be log-convex sequence.", "If there exists an increasing function $m$ such that, for all $r\\in {\\mathbb {R}}_+$ , $\\int _{{\\mathbb {R}}} e^{r|x|}d|\\mu |(x)\\le 2 m(r)$ , then $f$ belongs to $\\mathcal {A}_0({\\mathbb {C}})=\\left\\lbrace f\\in \\mathcal {A}({\\mathbb {C}}):\\ \\forall r\\ge 0,\\ \\max _{z:|z|=r}|f(z)|\\le 2 m(r)\\right\\rbrace ;$ Let $\\mu \\in \\mathcal {P}_c^{\\lbrace M_{m}\\rbrace }({\\mathbb {R}})$ , where $\\lbrace M_m\\rbrace $ satisfies (REF ) then $f \\in C^{\\lbrace M_m\\rbrace }$ , which is a quasi-analytic class.", "Let $\\mu \\in \\mathcal {P}_c^{\\lbrace M_{m}\\rbrace }({\\mathbb {R}}_+)$ , where $\\lbrace M_m\\rbrace $ satisfies (REF ) then $f \\in C^{\\lbrace M_m\\rbrace }$ and $C^{\\lbrace M_m\\rbrace }\\bigcap \\left\\lbrace \\mathcal {F}[\\mu ],\\ \\mu \\in \\mathfrak {M}_c^*({\\mathbb {R}}_+)\\right\\rbrace $ is a quasi-analytic class.", "In all the cases of Proposition REF , $f$ is characterized by $\\left(f^{m}(0)\\right)_{m\\in {\\mathbb {N}}_0}$ , hence $\\mu $ is determinate in $\\mathfrak {M}^*_c({\\mathbb {R}})$ by injectivity of the Fourier transform.", "Consider (REF ).", "Use that there exists a complex Borel function $g$ with $|g|=1$ such that $d\\mu =gd|\\mu |$ and rewrite all integrals as integrals with respect to $|\\mu |$ .", "Let $r\\in {\\mathbb {R}}_+$ and $S_r =\\lbrace z \\in \\mathbb {C}: \\ |z|\\le r\\rbrace $ .", "For all $x\\in {\\rm supp}(\\mu )$ , $z\\in S_r\\mapsto e^{izx}$ is holomorphic and, for all $z\\in S_r$ , $x\\in {\\mathbb {R}}\\mapsto e^{izx}$ is $|\\mu |$ -integrable.", "For all compact $K\\subset S_r$ , for all $z\\in \\ K$ , we have, for all $x\\in {\\rm supp}(\\mu )$ , $|e^{izx}|\\le e^{r|x|}$ which is $|\\mu |-$ integrable.", "The rest follows by the same argument as those of the proof of theorem p91 of [56] for complex variables.", "Consider (REF ).", "Because $\\mu \\in \\mathfrak {M}_c^*({\\mathbb {R}})$ , we have $f\\in C^{\\infty }({\\mathbb {R}})$ but also clearly $f\\in C^{\\lbrace M_m\\rbrace }$ because, for all $m\\in {\\mathbb {N}}_0$ and $x\\in {\\mathbb {R}}$ , $\\left|f^{(m)}(x)\\right|\\le s_{ \\left| \\cdot \\right| , \\left| \\mu \\right| }(m)$ .", "The Denjoy-Carleman theorem (see Theorem 19.11 in [47]) yields that $C^{\\lbrace M_m\\rbrace }$ is quasi-analytic.", "Moreover, using that for all $m\\in {\\mathbb {N}}_0$ , $f^{(m)}(0)=i^{m}s_{\\mu }(m)$ , $\\mu $ is determinate in $\\mathfrak {M}_c^*({\\mathbb {R}})$ .", "Consider (REF ), we adapt the proof of Theorem 4.1 in [12].", "Define the measure $\\mu _1$ on ${\\mathbb {R}}$ by $ d\\mu _1 (t) =d\\mu (t^2)$ .", "Thus, we have, for all $m=2n$ with $n\\in {\\mathbb {N}}_0$ , $ s_{ \\left| \\cdot \\right| , \\left| \\mu _1\\right| }(m) = 2 s_{ \\left| \\cdot \\right| , \\left| \\mu \\right| }(n)$ hence $\\sum _{m\\in {\\mathbb {N}}}1/( s_{ \\left| \\cdot \\right| , \\left| \\mu _1\\right| }(2m))^{1/(2m)} = \\sum _{m\\in {\\mathbb {N}}}1/(2s_{ \\left| \\cdot \\right| , \\left| \\mu \\right| }(m))^{1/(2m)} $ and $\\sum _{m\\in {\\mathbb {N}}}1/(s_{ \\left| \\cdot \\right| , \\left| \\mu \\right| }(m))^{1/(2m)} \\ge \\sum _{m\\in {\\mathbb {N}}}1/M_{m}^{1/(2m)} = \\infty $ .", "Hence, applying (REF ) to $\\mu _1$ , $ \\mu $ is determinate in $\\mathfrak {M}_c^*({\\mathbb {R}}_+)$ .", "Because we have, for all $ (m,x)\\in {\\mathbb {N}}_0\\times {\\mathbb {R}}$ , $\\left|\\mathcal {F}^{(m)}[\\mu ](x) \\right| & = \\frac{1 }{2} \\left|\\mathcal {F}^{(2m)}[\\mu _1](x) \\right| \\le 2^{2m-1} s_{ \\left| \\cdot \\right| , \\left| \\mu _1\\right| }(2m)\\le 2^{2m} s_{ \\left| \\cdot \\right| , \\left| \\mu \\right| }(m),$ $\\mathcal {F}[\\mu ] \\in C^{\\lbrace M_m\\rbrace }$ , hence $\\mathcal {F}^{(m)}[\\mu ](0)=i^{m}s_{\\mu }(m)$ yields that $C^{\\lbrace M_m\\rbrace }\\bigcap \\left\\lbrace \\mathcal {F}[\\mu ],\\ \\mu \\in \\mathfrak {M}_c^*({\\mathbb {R}}_+)\\right\\rbrace $ is quasi-analytic." ], [ "Criteria for determinacy", "When $\\mu \\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ , the determinacy of $\\mu $ can be assessed by checking the Cramér condition: $ \\exists \\rho >0:\\ \\forall t:\\ |t|_2\\le \\rho ,\\ \\int _{{\\mathbb {R}}^{p+1}}e^{t^{\\top }x}d\\mu (x)<\\infty .$ When $\\mu \\in \\mathfrak {M}_1^*({\\mathbb {R}}^{p+1})$ , this means that the moment generating function exists for small $|t|_2$ .", "Using that $\\int _{{\\mathbb {R}}^{p+1}}\\cosh (|{t^{\\top }x}|)d\\mu (x)=\\int _{{\\mathbb {R}}^{p+1}}\\cosh ({t^{\\top }x})d\\mu (x)<\\infty $ , (REF ) is equivalent to: $ \\exists \\rho >0:\\ \\forall t:\\ |t|_2\\le \\rho ,\\ \\int _{{\\mathbb {R}}^{p+1}}e^{|t^{\\top }x|}d\\mu (x)<\\infty .$ The Cramér condition rules out laws with heavy tails and holds for example for multivariate normals and skew-normals (see [35] and references therein).", "For a random variable $X$ such that $\\mathbb {S}_{X}\\subseteq {\\mathbb {R}}_+$ , it is possible to use the weaker Hardy condition ${\\mathbb {E}} [e^{t\\sqrt{X}}]<\\infty $ for small $|t|$ to assess determinacy (see [26], [27], [55]).", "We now give a simple extension of Theorem 2 in [38] to $\\mathfrak {M}^*({\\mathbb {R}})$ .", "Proposition A.2 Let $\\mu \\in \\mathfrak {M}^*({\\mathbb {R}})$ with ${\\rm supp}(\\mu )={\\mathbb {R}}_+$ , then $\\overline{\\lim }_{m\\rightarrow \\infty }s_{\\mu }(m+1)/(m^2s_{\\mu }(m))<\\infty $ implies the following equivalent statements: $\\exists t>0:$ $\\int _{{\\mathbb {R}}_+}e^{t\\sqrt{x}}d\\mu (x)<\\infty $ ; $\\overline{\\lim }_{m\\rightarrow \\infty }s_{\\mu }(m)^{1/(2m)}/m<\\infty $ ; $\\overline{\\lim }_{m\\rightarrow \\infty }\\left(s_{\\mu }(m)\\right)^{1/m}/(2m)!<\\infty $ ; which imply $\\sum _{m\\in {\\mathbb {N}}}\\frac{1}{s_{\\mu }(m)^{\\frac{1}{m}}}=\\infty ,$ hence that $\\mu $ is determinate.", "(REF ) is the Carleman condition of the Stieltjes moment problem.", "Corollary A.1 (Theorem 1 in [38]) Let $\\mu \\in \\mathfrak {M}^*({\\mathbb {R}})$ , then $\\overline{\\lim }_{m\\rightarrow \\infty }s_{\\mu }(2(m+1))/(m^2s_{\\mu }(2m))<\\infty $ implies the following equivalent statements: $\\exists t\\in {\\mathbb {R}}:$ $\\int _{{\\mathbb {R}}_+}e^{tx}d\\mu (x)<\\infty $ ; $\\exists t>0:$ $\\int _{{\\mathbb {R}}_+}e^{t|x|}d\\mu (x)<\\infty $ ; $\\overline{\\lim }_{m\\rightarrow \\infty }s_{\\mu }(2m)^{1/(2m)}/2m<\\infty $ ; $\\overline{\\lim }_{m\\rightarrow \\infty }\\left(s_{\\mu }(2m)/(2m)!\\right)^{1/m}<\\infty $ ; $\\overline{\\lim }_{m\\rightarrow \\infty }\\left(s_{|\\cdot |,\\mu }(m)/m\\right)^{1/m}<\\infty $ ; which imply $\\sum _{m\\in {\\mathbb {N}}}\\frac{1}{s_{\\mu }(2m)^{\\frac{1}{m}}}=\\infty ,$ hence that $\\mu $ is determinate.", "(REF ) is the Carleman condition of the Hamburger moment problem.", "Corollary REF (see also Corollary REF below) allows the $\\overline{\\lim }$ in (C.A.1.REF ) and (C.A.1.REF ) to be taken over the subsequence of even integers.", "(REF ) when the support is known to lie in a half line is weaker than (REF ) and nonbinding when ${\\rm supp}(\\mu )$ is compact.", "Neither are necessary for determinacy (see [30] p93, [53] p113).", "Integral criteria for $\\mu \\in \\mathfrak {M}_1^*({\\mathbb {R}})$ with a density $f$ can be easier to check than (REF ) and (REF ).", "Due to Theorem 3 in [46], they can be used on marginals to assess determinacy of multivariate measures.", "We have the following results from [37]: When $f$ is positive and the Krein condition $\\int _{{\\mathbb {R}}}-\\log (f(x))/(1+x^2)dx<\\infty $ holds, $\\mu $ is not determinate; while, if $f$ is also even, differentiable, and there exists $x_0>0$ such that, for $x\\ge x_0>0$ and $x\\mapsto f(x)$ decreases to 0, $x\\mapsto -xf^{\\prime }(x)/f(x)$ increases to infinity (so-called Lin conditions), and $\\int _{{\\mathbb {R}}}-\\log (f(x))/(1+x^2)dx=\\infty $ then (REF ) holds.", "When ${\\rm supp}(\\mu )\\subseteq {\\mathbb {R}}^+$ , $f(x)$ is positive on $[x_0,\\infty )$ for $x_0>0$ , and the Krein condition $\\int _{x_0}^{\\infty }-\\log (f(x^2))/(1+x^2)dx<\\infty $ holds, $\\mu $ is not determinate (see [44] and [45] for weaker conditions on the interval where $f$ is positive); while, if $f$ is positive and differentiable on ${\\mathbb {R}}_+$ , and, for $x\\ge x_0$ , $x\\mapsto f(x)$ decreases to 0 and $x\\mapsto -xf^{\\prime }(x)/f(x)$ increases to infinity (so-called Lin conditions), and $\\int _{{\\mathbb {R}}_+}-\\log (f(x^2))/(1+x^2)dx=\\infty $ , then (REF ) holds.", "The Krein condition is not necessary for $\\mu \\in \\mathfrak {M}_1^*({\\mathbb {R}})$ to be indeterminate (see [53] p114) in the Hamburger and Stieltjes cases.", "Theorem 3 in [46] and Proposition REF imply the following results in higher dimensions.", "Corollary A.2 Let $q\\in \\lbrace 0,1,\\dots ,p+1\\rbrace $ , $\\mu _1,\\mu _2\\in \\mathfrak {M}_c^*({\\mathbb {R}}^{p+1})$ such that for known $\\mathfrak {B}\\in GL({\\mathbb {R}}^{p+1})$ ${\\rm supp}(\\mathfrak {B}_*\\mu _1)={\\rm supp}(\\mathfrak {B}_*\\mu _2)={\\mathbb {R}}^q\\times {\\mathbb {R}}_+^{p+1-q}$ and $\\mu _1\\sim \\mu _2$ .", "We have $\\mu _1=\\mu _2$ if for $\\mathfrak {A}\\in {\\rm Aff}({\\mathbb {R}}^{p+1})$ leaving invariant $\\mathrm {span}\\lbrace e(q+1),\\dots ,e(p+1)\\rbrace $ $&\\min _{j=1,\\dots ,q}\\sum _{m\\in {\\mathbb {N}}_0}\\frac{1}{\\left(s_{|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _1|}(2me(j))+s_{|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _2|}(2me(j))\\right)^{\\frac{1}{2m}}}=\\infty ;\\\\&\\min _{j=q+1,\\dots ,p+1}\\sum _{m\\in {\\mathbb {N}}_0}\\frac{1}{\\left(s_{|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _1|}(me(j))+s_{|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _2|}(me(j))\\right)^{\\frac{1}{2m}}}=\\infty ;$ or, when $\\mu _1,\\mu _2\\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ , $&\\min _{j=1,\\dots ,q}\\sum _{m\\in {\\mathbb {N}}_0}\\frac{1}{s_{(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _1}(2me(j))^{\\frac{1}{2m}}}=\\infty ;\\\\&\\min _{j=q+1,\\dots ,p+1}\\sum _{m\\in {\\mathbb {N}}_0}\\frac{1}{s_{(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _1}(me(j))^{\\frac{1}{2m}}}=\\infty ;$ where, in both cases, if $q=0$ , we remove the first minimum, while, if $q=p$ , we remove the second minimum.", "Proof of Corollary REF .", "When $q=p+1$ , the statement is obtained by applying the Denjoy-Carleman theorem to $\\mathcal {F}[(\\mathfrak {A}\\circ \\mathfrak {B})_*(\\mu _1-\\mu _2)]$ (see Theorem 3.2 in [12] for complex measures and else Theorem 2.3 in [16]).", "When $q<p$ , we apply this result to $(\\mathfrak {C}\\circ \\mathfrak {B})_*\\mu _j$ for $j=1,2$ , where $\\mathfrak {C}((x_1,\\dots ,x_{p+1})^{\\top })=(x_1,\\dots ,x_q,\\sqrt{x_{q+1}},\\dots ,\\sqrt{x_{p+1}})$ and proceed like in the proof of Theorem 4.1 in [12].$\\square $ The Denjoy-Carleman theorem applied to $\\mathcal {F}\\left[(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu \\right]$ where $\\mu \\in \\mathfrak {M}_c\\left({\\mathbb {R}}^{p+1}\\right)$ only requires assumptions on marginals, hence the conditions in Corollary REF only involve these marginals, this is in line with Theorem 3 in [46].", "[43] (see also Theorem 5 in [4]) states this result for elements of $\\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ when $q=p+1$ without allowing for affine transformations.", "Due to Proposition 3.10 in [17], it is possible that the sum of two functions belonging to classes defined by two different $p+1$ -tuples of sequences which both satisfy the sufficient condition of Denjoy-Carleman theorem do not belong to such a class.", "Rather than (REF )-(), also called Carleman conditions, [52] assumes the stronger condition $\\sum _{m\\in {\\mathbb {N}}_0}\\frac{1}{\\left(\\int _{{\\mathbb {R}}^{p+1}}\\sum _{j=1}^{p+1}x_j^{2m}d\\mu (x)\\right)^{\\frac{1}{2m}}}=\\infty .$ Corollary A.3 Let $q\\in \\lbrace 0,1,\\dots ,p+1\\rbrace $ , $\\mu _1,\\mu _2\\in \\mathfrak {M}_c^*({\\mathbb {R}}^{p+1})$ such that for known $\\mathfrak {B}\\in GL({\\mathbb {R}}^{p+1})$ ${\\rm supp}(\\mathfrak {B}_*\\mu _1)={\\rm supp}(\\mathfrak {B}_*\\mu _2)={\\mathbb {R}}^q\\times {\\mathbb {R}}_+^{p+1-q}$ and $\\mu _1\\sim \\mu _2$ .", "$\\mu _1=\\mu _2$ if there exist functions $W_j$ for $j=1,\\dots ,p+1$ from ${\\mathbb {R}}$ to $[0,\\infty ]$ such that: $\\min \\left\\lbrace \\min _{j=1,\\dots ,q}\\sum _{m\\in {\\mathbb {N}}}\\frac{1}{\\left\\Vert x\\mapsto \\frac{x^{2m}}{W_j\\left(\\sqrt{x}\\right)}\\right\\Vert _{L^{\\infty }({\\mathbb {R}})}^{\\frac{1}{2m}}},\\min _{j=q+1,\\dots ,p+1}\\sum _{m\\in {\\mathbb {N}}}\\frac{1}{\\left\\Vert x\\mapsto \\frac{x^{2m}}{W_j\\left(\\sqrt{x}\\right)}\\right\\Vert _{L^{\\infty }({\\mathbb {R}}_+)}^{\\frac{1}{2m}}}\\right\\rbrace =\\infty ;$ where, if $q=0$ , we remove the first minimum, while, if $q=p$ , we remove the second minimum; and $\\mathfrak {A}\\in {\\rm Aff}({\\mathbb {R}}^{p+1})$ leaving invariant $\\mathrm {span}\\lbrace e(q+1),\\dots ,e(p+1)\\rbrace $ such that: $&\\sum _{j=1,\\dots ,p+1}\\int _{{\\mathbb {R}}}W_j(x_j)d\\left(\\Pi [e(j)]_*\\left(|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _1|+|(\\mathfrak {A}\\circ \\mathfrak {B})_*\\mu _2|\\right)\\right)(x)<\\infty ;\\\\&\\mathrm {or}\\ \\sum _{j=1,\\dots ,p+1}\\int _{{\\mathbb {R}}}W_j(x_j)d\\left(\\left(\\Pi [e(j)]\\circ \\mathfrak {A}\\circ \\mathfrak {B}\\right)_*\\mu _1\\right)(x)<\\infty \\ \\mathrm {if}\\ \\mu _1,\\mu _2\\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1}).$ Moreover, $\\mu _1=\\mu _2$ if there exist functions $W_j$ for $j=1,\\dots ,p+1$ from ${\\mathbb {R}}$ to $[c_j,\\infty ]$ , where $c_j>0$ , satisfying (REF ), $\\mathfrak {A}\\in {\\rm Aff}({\\mathbb {R}}^{p+1})$ leaving invariant $\\mathrm {span}\\lbrace e(q+1),\\dots ,e(p+1)\\rbrace $ , and a function $W$ from ${\\mathbb {R}}^{p+1}$ to $[0,\\infty ]$ such that, for $|x|_2$ large enough, $W(\\mathfrak {A}\\circ \\mathfrak {B}(x))\\ge \\prod _{j=1}^{p+1}W_j(x_j)$ , with $\\int _{{\\mathbb {R}}^{p+1}}W(\\mathfrak {A}\\circ \\mathfrak {B}(x))d(|\\mu _1|+|\\mu _2|)(x)<\\infty $ or $\\int _{{\\mathbb {R}}^{p+1}}W(\\mathfrak {A}\\circ \\mathfrak {B}(x))d\\mu _1(x)<\\infty $ if $\\mu _1,\\mu _2\\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ .", "Proof of Corollary REF .", "Let us prove the moreover part of the statement.", "The first part is obtained in a similar manner.", "Assuming that $W_j$ are bounded from below by $c_j>0$ for $j=1,\\dots ,p+1$ , we use that for $j\\in \\lbrace 1,\\dots ,p+1\\rbrace $ , $m\\in {\\mathbb {N}}$ , and $\\mu ,\\nu \\in \\mathfrak {M}_c^*({\\mathbb {R}}^{p+1})$ , we have: $s_{|\\mu |}(me(j))+s_{|\\nu |}(me(j))&\\le \\left\\Vert x^{m}/W_j(x)\\right\\Vert _{L^{\\infty }({\\mathbb {R}})}\\int _{{\\mathbb {R}}^{p+1}}W_j(x_j)d\\left(|\\mu |+|\\nu |\\right)(x)\\\\&\\le \\frac{1}{\\prod _{\\begin{array}{c}k=1,\\dots ,p+1\\\\k\\ne j\\end{array}}c_k}\\left\\Vert x^{m}W_j(x)^{-1}\\right\\Vert _{L^{\\infty }({\\mathbb {R}})}\\int _{{\\mathbb {R}}^{p+1}}W(x)d\\left(|\\mu |+|\\nu |\\right)(x).$ We use the same argument for $\\mu _1\\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ .", "We conclude using Corollary REF .$\\square $ The first sufficient condition in Corollary REF is the less demanding.", "Due to Theorem 3.14 in [17], if $W$ is a function from ${\\mathbb {R}}$ to $[c,\\infty ]$ for $c>0$ , the next statements are equivalent: $\\sum _{m\\in {\\mathbb {N}}}1/\\left\\Vert x\\mapsto x^{2m}/W(x)\\right\\Vert _{L^{\\infty }({\\mathbb {R}})}^{1/(2m)}=\\infty $ ; There exists a function $\\tilde{W}$ with values in $[0,\\infty )$ and $R>0$ such that, for $|x|>R$ , $W(x)\\ge \\tilde{W}(x)$ and $\\tilde{W}(x)=\\tilde{W}(-x)<\\infty $ , $x\\mapsto \\log \\left(\\tilde{W}(e^x)\\right)$ is convex on $(\\log (R),\\infty )$ and $\\int _R^{\\infty }\\log \\tilde{W}(s)/(1+s^2)ds=\\infty $ ; There exist $\\epsilon >0$ , and a nondecreasing nonnegative function $\\rho \\in C^{\\infty }([0,\\infty ))$ equal to 0 on $[0,\\epsilon ]$ , such that, for all $x\\in {\\mathbb {R}}$ , $W(x)\\ge \\exp \\left(\\int _0^{|x|}(\\rho (s)/s) ds\\right)$ and $\\int _0^{\\infty }\\rho (s)/(1+s^2)ds=\\infty $ .", "For example, if there exist $R\\in {\\mathbb {R}}$ , $(a_j)_{j\\in {\\mathbb {N}}_0}\\in (0,\\infty )^{{\\mathbb {N}}_0}$ , and $(p_j)_{j\\in {\\mathbb {N}}_0}\\in (-\\infty ,1]^{{\\mathbb {N}}_0}$ equal to 0 for $j$ large enough, such that $W(x)\\ge \\exp \\left(\\frac{x^2}{\\prod _{j=0}^{\\infty }\\log _j^{p_j}(a_j|x|)}\\right){\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\left\\lbrace |x|\\ge R\\right\\rbrace \\quad \\left(\\mathrm {\\emph {e.g.", "},}\\ W(x)=\\exp {\\left(\\frac{|x|}{a_0\\log (a_1|x|)}\\right)}{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\left\\lbrace |x|\\ge R\\right\\rbrace \\right)$ then $W$ satisfies (QAW.REF ).", "Replacing $x$ by $\\sqrt{x}$ and using that the above lower bound is even, we obtain that $\\sum _{m\\in {\\mathbb {N}}}1/\\left\\Vert x\\mapsto x^{2m}/W(\\sqrt{x})\\right\\Vert _{L^{\\infty }({\\mathbb {R}}_+)}^{1/(2m)}=\\infty $ if there exist $(C,R)\\in {\\mathbb {R}}^2$ , $(a_j)_{j\\in {\\mathbb {N}}_0}\\in (0,\\infty )^{{\\mathbb {N}}_0}$ , and $(p_j)_{j\\in {\\mathbb {N}}_0}\\in (-\\infty ,1]^{{\\mathbb {N}}_0}$ equal to 0 for $j$ large enough, such that $W(x)\\ge C\\exp \\left(\\frac{x}{\\prod _{j=0}^{\\infty }\\log _j^{p_j}(a_j\\sqrt{x})}\\right){\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\left\\lbrace x\\ge R\\right\\rbrace \\quad \\left(\\mathrm {\\emph {e.g.", "},}\\ W(x)=\\exp {\\left(\\frac{\\sqrt{x}}{a_0\\log (a_1x)}\\right)}{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\left\\lbrace x\\ge R\\right\\rbrace \\right).$ Because $\\sum _{m\\in {\\mathbb {N}}}1/\\left\\Vert x\\mapsto x^{2m}/W(x)\\right\\Vert _{L^{\\infty }({\\mathbb {R}})}^{1/(2m)}=\\infty $ implies $\\sum _{m\\in {\\mathbb {N}}}1/\\left\\Vert x\\mapsto (x^{\\top }f(j))^{2m}/W(|x|_2)\\right\\Vert _{L^{\\infty }({\\mathbb {R}}^q)}^{1/(2m)}=\\infty $ for all bases $(f(j))_{j=1}^q$ of ${\\mathbb {R}}^q$ and similarly for ${\\mathbb {R}}_+^{p+1-q}$ , assuming that $\\mathfrak {B}$ is identity, we can replace (REF ) by the stronger condition (not involving $\\mathfrak {A}$ ) $W(x)\\ge \\widetilde{W}\\left(|(x_1,\\dots ,x_q)^{\\top }|_2\\right)\\widetilde{W}\\left(\\sqrt{|(x_{q+1},\\dots ,x_{p+1})^{\\top }|_2}\\right),$ where $\\widetilde{W}$ is a function from ${\\mathbb {R}}$ to $[c,\\infty ]$ for $c>0$ which satisfies the equivalent (QAW.REF )-(QAW.REF ).", "The function $1/W$ is called a quasi-analytic weight in [16], [17].", "The approach using quasi-analytic weights is closely related to the Krein and Lin conditions.", "Assume that $\\mathfrak {A}$ and $\\mathfrak {B}$ are the identity, $q=p+1$ (the remaining cases can be treated similarly), and $\\mu \\in \\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ has marginals $\\Pi [e(j)]_*\\mu $ with densities $f_j(x_j)$ for $j=1,\\dots ,p+1$ which are even and positive and such that $-x_jf_j^{\\prime }(x_j)/f_j(x_j)-\\alpha _j\\log (\\log (x_j))$ with $\\alpha _j>1$ is increasing for $|x_j|\\ge x_{0j}\\ge 0$ .", "Then, for $j=1,\\dots ,p+1$ , $W_j(x_j)=1/\\max (f_j(x_j)\\log (|x_j|)^{\\alpha _j},1)$ satisfies (QAW.REF ) and $\\int _{{\\mathbb {R}}}W_j(x_j)f_j(x_j)dx_j<\\infty $ , hence the conclusions of Corollary REF hold.", "The advantage of the approach using quasi-analytic weights is that, unlike the Krein and Lin conditions, it is not required that a density exists." ], [ "Complements on the sets of uniqueness", "Examples of sets of uniqueness for analytic classes are given in the examples (E.REF ) and (E.REF ).", "We only give here more details about (E.REF ).", "Let $m \\in {\\mathbb {R}}_+^{[0,\\infty )}$ , $\\lim _{r\\rightarrow \\infty }m(r)= \\infty $ , and $\\mathcal {A}_0({\\mathbb {C}})=\\left\\lbrace f\\in \\mathcal {A}({\\mathbb {C}}):\\ \\exists C>0:\\ \\forall r\\ge 0,\\ \\max _{z:|z|=r}|f(z)|\\le C m(r)\\right\\rbrace .$ $U$ is a set of uniqueness of such a class $\\mathcal {A}_0({\\mathbb {C}})$ if, for $\\epsilon >0$ , we have $\\exists \\alpha >1:\\ \\overline{\\lim }_{r\\rightarrow \\infty }\\frac{\\log (\\alpha )}{\\log (m(\\alpha r))}\\left|U\\cap \\left((-r,r)\\setminus \\lbrace 0\\rbrace \\right)\\right|>1.$ Hence, $t U$ is a set of uniqueness for all $t\\ne 0$ if $\\forall t>0,\\ \\exists \\alpha >1:\\ \\overline{\\lim }_{r\\rightarrow \\infty }\\frac{\\log (\\alpha )}{\\log (m(\\alpha t r))}\\left|U\\cap \\left((-r,r)\\setminus \\lbrace 0\\rbrace \\right)\\right|>1.$ A sufficient condition when $m(r)=e^{\\rho r}$ , which occurs when $f$ is the Fourier transform of the difference of two complex measures supported in $[-\\rho ,\\rho ]$ , is $\\overline{\\lim }_{r\\rightarrow \\infty }\\frac{1}{r}\\left|U\\cap \\left((-r,r)\\setminus \\lbrace 0\\rbrace \\right)\\right|=\\infty .$ Let us explain the above characterization.", "If $f\\in \\mathcal {A}_0(\\mathbb {C})$ , there exists $k\\in {\\mathbb {N}}$ such that $f(z)=z^kg(z)$ and $g(0)\\ne 0$ .", "Jensen's formula applied to $g$ (see 15.18 in [47]), with $ \\max _{z:|z|=r}|g(z)|= \\max _{z:|z|=r}|f(z)|/z^k \\le Cm(r)/r^k $ , ensures $ \\left| g(0)\\right| \\prod _{k=1}^{n_g(\\alpha r)}\\dfrac{\\alpha r}{|\\omega _k|} \\le \\frac{C m(\\alpha r) }{\\alpha ^k r^k},$ where $\\omega _1,\\dots ,\\omega _{n_g(\\alpha r)}$ are the zeros of $g$ in $B(0,\\alpha r)$ ranked according to their multiplicity and $n_g(\\alpha r)$ is the number of zeros of $g$ with multiplicity in $B(0,\\alpha r)$ .", "Thus, we have $ \\frac{C m(\\alpha r) }{\\alpha ^k r^k} \\ge \\left| g(0)\\right| \\prod _{k=1}^{n_g( r)}\\dfrac{\\alpha r}{|\\omega _k|} \\ge \\left| g(0)\\right| \\alpha ^{n_g(r)},$ hence $n_g( r) \\le \\log (C m(\\alpha r)(\\alpha r)^{-k}/ \\left| g(0)\\right| )/\\log (\\alpha )$ .", "We have $ \\frac{n_g( r) \\log (\\alpha ) }{\\log (m(\\alpha r))} \\le 1 + \\frac{\\log (C (\\alpha r)^{-k}/ \\left| g(0)\\right| ) }{\\log (m(\\alpha r))} $ then $ \\overline{\\lim }_{r\\rightarrow \\infty }n_g( r) \\log (\\alpha )/\\log (m(\\alpha r)) \\le 1$ .", "This yields that $U$ satisfying (REF ) is a set of uniqueness of $\\mathcal {A}_0({\\mathbb {C}})$ .", "The following sets $U$ are sets of uniqueness of the following quasi-analytic and $V$ -quasi-analytic classes and complement (E.REF ).", "Example 1.", "$V $ is a subset of the interior of $U$ , which is nonempty.", "Indeed, a function which is zero on $U$ has all its partial derivatives or Laplacians (for functions on $S\\subseteq \\mathbb {S}^p$ ) equal to zero at every point in $V$ .", "Example 2.", "(Theorem 4b in [31]) Let $M_m= \\nu (m)^mm!$ , where $x\\in [0,\\infty )\\mapsto \\nu (x)$ is increasing and continuously differentiable such that $\\nu (0)=1$ and $\\lim _{x\\rightarrow \\infty }\\frac{x\\nu ^{\\prime }(x)}{\\nu (x)}=0.$ $U$ is a set of uniqueness of $C^{\\lbrace M_m\\rbrace }(b)$ (which contains non-analytic functions) if $\\overline{\\lim }_{r\\rightarrow \\infty }\\frac{1}{r}\\int _1^{\\left|U\\cap (-r,r)\\right|}M(\\log (u))\\frac{du}{u^2}> \\frac{\\pi b}{2},$ where $M(x)$ is the trace function of $\\lbrace M_m\\rbrace $ defined by $M(x)=\\sup _{m\\in {\\mathbb {N}}}(mx-\\log (M_m))$ (see [39])." ], [ "$\\mathcal {A}(S)$ is a small subset of {{formula:665ef31f-70eb-43e7-b822-3835bc74904d}}", "Proposition REF can be found in [50], it is of the same spirit as the statement that the complement of nowhere analytic functions in $C^{\\infty }(S)$ is meager with empty interior, which we detail below.", "$C^{\\lbrace M_m\\rbrace }(S)$ is defined like $C^{\\lbrace M_m\\rbrace }$ but we replacing ${\\mathbb {R}}^p$ by an open set $S$ .", "Proposition A.3 If condition (REF ) fails, then $C^{\\lbrace M_m\\rbrace }(S)$ contains a vector space of dimension $2^{\\aleph _0}$ such that for all nonzero element $f$ there does not exists $x\\in S$ , a neighborhood $\\mathcal {N}_x$ of $x$ , and a sequence $\\lbrace M_{x,m}\\rbrace $ such that $f$ belongs to a quasi-analytic class $C^{\\lbrace M_{x,m}\\rbrace }(\\mathcal {N}_x)$ .", "Assume, for simplicity, that $S\\subset {\\mathbb {R}}^p$ is compact and as usual $C^{\\infty }(S)$ is equipped with the distance $d(f,g)=\\sum _{m\\in \\mathbb {N}_0^{p}}\\min \\left(1/2_c^{m},\\left\\Vert f^{(m)}-g^{(m)}\\right\\Vert _{L^{\\infty }(S)}\\right)$ .", "By the Cauchy-Hadamard theorem (see [51]), $f\\in C^{\\infty }(S)$ is analytic at $\\underline{x}$ if and only if there exist $b,c\\in \\mathbb {N}$ such that $f$ belongs to $T(\\underline{x},b,c)=\\left\\lbrace f\\in C^{\\infty }(S):\\ \\forall m\\in \\mathbb {N}_0^{p},\\ \\left|f^{(m)}(\\underline{x})\\right|\\le cb^{|m|}m!\\right\\rbrace ,$ thus $\\mathcal {A}(S)=\\bigcup _{\\underline{x}\\in \\mathbb {Q}^{p}\\cap S,b,c\\in \\mathbb {N}}T(\\underline{x},b,c)$ .", "This is a countable union of closed sets, hence closed.", "Moreover, $T(\\underline{x},b,c)$ has empty interior.", "Indeed (see, [49]), given $f$ in $T(\\underline{x},b,c)$ , for all $\\epsilon >0$ , $m\\in {\\mathbb {N}}$ such that $\\sum _{j=m}^{\\infty }2^{-j}<\\epsilon $ , $c<(\\epsilon b^{m}/(2m!", "))^{1/(2m)}$ , the functions $ f_{\\epsilon }: \\ x\\in S \\mapsto f(x)+\\epsilon \\cos \\left(c(x_{1}-\\underline{x}_{1})\\right)/(2c^m)$ are such that $d(f_{\\epsilon },f)<\\epsilon $ and $\\left|f_{\\epsilon }^{(2m)}(\\underline{x})-f^{(2m)}(\\underline{x})\\right|>b^{2m}(2m)!$ , hence $f_{\\epsilon }\\notin T(\\underline{x},b,c)$ .", "Due to Baire's theorem, the meager set $\\mathcal {A}(S)$ of the complete metric space $C^{\\infty }(S)$ has an empty interior.", "With the arguments in [11], the complement of nowhere analytic functions in $C^{\\infty }(S)$ , hence containing $\\mathcal {A}(S)$ , can be shown to be meager with empty interior." ], [ "Proofs", "Assume that $\\mathbb {P}_{\\Gamma }$ and $\\mathbb {P}_{\\Gamma }^*$ both give rise to the same collection $\\left(\\mathbb {P}_{\\left(1, x ^{\\top }\\right)\\Gamma }\\right)_{ x \\in \\mathbb {S}_{X}}$ .", "By (REF ), we have for all $x\\in \\mathbb {S}_{X}$ and $t$ in ${\\mathbb {R}}$ , $\\varphi _{\\alpha }(t)\\prod _{j=1}^p\\varphi _{\\beta _j}(tx_j)=\\varphi _{\\alpha }^*(t)\\prod _{j=1}^p\\varphi _{\\beta _j}^*(tx_j).$ Because the value at 0 of a characteristic function is 1, taking $x=0_c$ , yields $\\varphi _{\\alpha }=\\varphi _{\\alpha }^*$ .", "Because $\\varphi _{\\alpha }^*$ is continuous at 0, for all $x\\in \\mathbb {S}_{X}$ and $t\\in (-t_0,t_0)$ for $t_0$ small enough, $\\prod _{j=1}^p\\varphi _{\\beta _j}(tx_j)=\\prod _{j=1}^p\\varphi _{\\beta _j}^*(tx_j).$ Taking $x=x(1)$ yields, for all $t\\in (-t_0,t_0)$ , $\\varphi _{\\beta _1}(x(1)_1t)=\\varphi _{\\beta _1}^*(x(1)_1t)$ , hence $\\varphi _{\\beta _1}(t)=\\varphi _{\\beta _1}^*(t)$ for all $t\\in (-t_0/x(1)_1,t_0/x(1)_1)$ .", "Hence $\\mathbb {P}_{\\beta _1}$ and $\\mathbb {P}_{\\beta _1}^*$ have same moments, so $\\mathbb {P}_{\\beta _1}=\\mathbb {P}_{\\beta _1}^*$ .", "We conclude by iterating this procedure.", "By (REF ), (REF ) holds for all $x\\in \\mathbb {S}_{X}$ and $t$ in the complement of the zeros of $\\varphi _{\\alpha }^*$ .", "Because the complement of the set of zeros of $\\varphi _{\\alpha }^*$ is dense and $t\\mapsto \\prod _{j=1}^p\\varphi _{\\beta _j}(tx_j)$ and $t\\mapsto \\prod _{j=1}^p\\varphi _{\\beta _j}^*(tx_j)$ are continuous, (REF ) holds for all $x\\in \\mathbb {S}_{X}$ and $t\\in {\\mathbb {R}}$ .", "Taking $x=x(1)$ yields, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{\\beta _1}(x(1)_1t)=\\varphi _{\\beta _1}^*(x(1)_1t)$ , so $\\varphi _{\\beta _1}=\\varphi _{\\beta _1}^*$ .", "So, for all $x\\in \\mathbb {S}_{X}$ and $t\\in {\\mathbb {R}}$ , $\\prod _{j=2}^p\\varphi _{\\beta _j}(tx_j)=\\prod _{j=2}^p\\varphi _{\\beta _j}^*(tx_j)$ , and we conclude by iteration.", "Take $S = {\\mathbb {R}}^{p+1}$ and $\\mathfrak {F}(S)$ the vector space spanned by $\\mathcal {F}[\\mathcal {P}]$ .", "An element of $\\mathfrak {F}(S)$ is of the form $\\mathcal {F}[\\mu -\\nu ]$ , where $\\mu ,\\nu \\in \\mathcal {P}$ .", "For all $k\\in {\\mathbb {N}}_0$ and $ x \\in \\mathbb {S}_{ X }$ , $(-i)^k\\left.", "\\partial _t^{(k)}\\mathcal {F}\\left[ \\mu -\\nu \\right](t, t x )\\right|_{t=0} &= \\int _{{\\mathbb {R}}^{p+1}} \\left(\\gamma _1 + \\sum _{j=1}^{p}\\gamma _{j+1} x _j\\right)^k d(\\mu -\\nu )(\\gamma )= P_k( x ), $ where $P_k$ are the polynomials with real coefficients $P_k(Z_1,\\dots ,Z_{p}) = \\sum _{ j \\in {\\mathbb {N}}_0^{p+1}: \\ | j |_1 = k} \\left( \\begin{array}{c}k\\\\ j _1,\\dots , j _{p+1}\\end{array} \\right) \\left(\\int _{{\\mathbb {R}}^{p+1}} \\gamma ^{ j } d(\\mu -\\nu )(\\gamma )\\right)\\ Z_1^{ j _2}\\dots Z_{p}^{ j _{p+1}}.$ So, by (i), $\\mathcal {F}\\left[\\mu -\\nu \\right]=0$ on $U = \\lbrace t(1, x ): t\\in {\\mathbb {R}}, x \\in \\mathbb {S}_{ X }\\rbrace $ implies, for all $m\\in \\mathbb {N}_0^{p+1}$ , $s_{\\mu }(m)=s_{\\nu }(m)$ , hence $\\mu =\\nu $ .", "To prove the second statement, assume there exists $P\\in {\\mathbb {R}}[Z_1,\\dots ,Z_p]$ nonzero of degree $k\\in {\\mathbb {N}}$ , such that, $\\forall x\\in \\mathbb {S}_{X}$ , $P(x)=0$ .", "Let $Q$ be the corresponding homogeneous polynomial in ${\\mathbb {R}}[Z_1,\\dots ,Z_{p+1}]$ of degree $k$ such that, $\\forall x\\in \\mathbb {S}_{X}$ , $Q(1,x)=P(x)$ .", "Let $\\mathbb {P}_{\\Gamma }^*$ of support in $[-1,1]^{p+1}$ and density with respect to the Lebesgue measure bounded from below by $c>0$ , and the infinitely differentiable function $q(\\gamma ) = \\prod _{j=1}^{p+1} \\gamma _j \\exp (-1/(\\gamma _j^2-1)) {\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace \\gamma _j\\in [-1,1]\\rbrace $ .", "Then, $\\mathbb {P}_{\\Gamma } = \\mathbb {P}_{\\Gamma }^* + c h /\\Vert h\\Vert _{\\infty } d\\gamma $ , where $h = Q(\\partial _1, \\dots , \\partial _{p+1}) q $ is a probability.", "By properties of the Fourier transform of measures in $\\mathfrak {M}^*({\\mathbb {R}}^{p+1})$ and homogeneity, we conclude from $\\mathcal {F}\\left[\\mathbb {P}_{\\Gamma } \\right](t,tx) = \\mathcal {F}\\left[\\mathbb {P}_{\\Gamma }^* \\right](t,tx) + t^k Q(1,x)\\mathcal {F}\\left[h\\right](t,tx).$ Start by proving (P6.REF ).", "Take $J\\subseteq {\\mathbb {Z}}\\setminus \\lbrace 0\\rbrace $ finite , $J_s\\subseteq J$ the positive indices $j$ in $J$ such that $-j\\in J$ .", "Let $(b_j)_{j\\in J} \\in \\mathbb {N}_0^{|J|}$ such that $ \\sum _{j\\in J} b_j \\lambda _j = 0$ .", "We have $\\sum _{j\\in J} b_j \\lambda _j =\\sum _{j\\in J_s} \\frac{b_j+b_{-j}}{4r^{j}} + \\sum _{j\\in J\\setminus (J_s\\cup -J_s)}\\frac{b_j}{4r^{|j|}} + \\sum _{j\\in J} j b_j,$ hence, $P(r)=0$ , where $j_0 = \\max _{j\\in J} \\left| j\\right| $ and $P(X)=\\sum _{j\\in J_s} (b_j+b_{-j})X^{j_0-j}+\\sum _{j\\in J\\setminus J_s}b_jX^{j_0-|j|} + 4X^{j_0}\\sum _{j\\in J} j b_j \\in \\mathbb {Z}[X].$ Because $r$ is transcendental over $\\mathbb {Z}$ and $0\\notin J$ , for all $j\\in J\\setminus (J_s\\cup -J_s)$ , $b_j =0$ , and, for all $j\\in J_s$ , $b_j+b_{-j}=0$ thus $b_j=b_{-j}=0$ because $b_j,b_{-j}\\in {\\mathbb {N}}_0$ .", "Hence, for all $j\\in J$ , $b_j=0$ .", "Now, take $P\\in {\\mathbb {C}}(X)[X_1,\\dots ,X_p]$ , where $p=|J|$ , which is zero when evaluated at $(f_j)_{j\\in J}$ .", "Hence, we have $\\sum _k c_k(z)\\exp (z\\sum _{j\\in J}b_{k,j}\\lambda _j)=0$ , where the sum over $k$ is finite, all $b_{k,j}$ belong to ${\\mathbb {N}}_0$ , and $c_k$ are rational functions.", "All exponentials in this sum are distinct by the above computations and we conclude that all $c_k$ s are zero by taking limits in ${\\mathbb {C}}$ .", "This yields the result.", "(P6.REF ) follows from Kadec's $1/4$ -Theorem (see Theorem 14 page 42 in [60]) because $ \\sup _{j\\in {\\mathbb {Z}}}|\\lambda _j - j|<1/4$ .", "Denote by $ <,>_{\\ell _2({\\mathbb {N}})}$ the inner product in $\\ell _2({\\mathbb {N}})$ .", "Start by considering $\\mathcal {R}_{L,4}$ .", "By a similar argument as the one involving (REF ), we can consider without loss of generality $\\lbrace \\underline{x}_m\\rbrace = \\lbrace 0_m\\rbrace $ in $\\mathcal {R}_{L,4}$ (REF ).", "Take $S = \\ell _2({\\mathbb {N}})$ and $\\mathfrak {F}(S)$ the vector space spanned by $\\mathcal {F}[\\mathcal {P}]$ .", "An element of $\\mathfrak {F}(S)$ is of the form $\\mathcal {F}[\\mu -\\nu ]$ , where $\\mu ,\\nu \\in \\mathcal {P}$ .", "Let $n\\in {\\mathbb {N}}$ .", "We have, for all $\\lbrace z_m\\rbrace \\in G_{n+1}$ and $t\\in {\\mathbb {R}}$ , $ \\int _{\\ell _2({\\mathbb {N}})} e^{it <\\lbrace \\gamma _m\\rbrace ,\\lbrace z_m\\rbrace >_{\\ell _2({\\mathbb {N}})}} d(\\mu -\\nu )(\\lbrace \\gamma _m\\rbrace )= \\int _{ G_{n+1}} e^{it <\\lbrace \\gamma _m\\rbrace ,\\lbrace z_m\\rbrace >_{\\ell _2({\\mathbb {N}})}} d \\Pi _{ G_{n+1}*}(\\mu -\\nu )(\\lbrace \\gamma _m\\rbrace ) $ .", "We can now proceed like in the proof of Theorem REF under $\\mathcal {R}_{L,3}$ and obtain $\\Pi _{G_{n+1}*}\\mu =\\Pi _{G_{n+1}*}\\nu $ hence $ \\mathcal {F}\\left[ \\Pi _{G_{n+1}*}\\mu - \\Pi _{G_{n+1}*}\\nu \\right]=0 $ .", "Using that $\\cup _{n\\in {\\mathbb {N}}} G_{n+1}$ is dense in $\\ell _2({\\mathbb {N}})$ and the continuity of the Fourier transform we conclude $ \\mu = \\nu $ .", "Identification under $\\mathcal {R}_{L,5}$ is corollary of Theorem 4.1 in [15].", "The proof relies on a weaker form of diagram (REF ).", "Take $\\mathbb {P}_{\\Gamma }\\in \\mathcal {P}$ and $U_{l} = \\prod _{k=1}^l V_{k}$ for all $l\\in [p]$ .", "For $\\beta \\in {\\mathbb {C}}^p$ and $j\\in [p]$ , denote by $\\beta _{[j]}=(\\beta _1,\\dots ,\\beta _j)^{\\top }$ .", "The function $\\mathcal {F}[\\mathbb {P}_{\\Gamma }-\\mathbb {P}_{\\Gamma }^*]$ is 0 on $U$ , so for $t\\ne 0$ and $u\\in U_{p-1}$ , $g_{p,t,u}: z\\in {\\mathbb {R}}\\mapsto \\mathcal {F}[\\mathbb {P}_{\\Gamma }-\\mathbb {P}_{\\Gamma }^*](t,t u ,z)=\\mathcal {F}[\\mu _{p,t,u}(\\cdot )](z)$ is 0 on $tV_{p}$ , where $ \\mu _{p,t,u}(\\cdot ) =\\int _{{\\mathbb {R}}^{p}}e^{it\\left(a+b_{[p-1]}^{\\top }u\\right)}d(\\mathbb {P}_{\\Gamma }-\\mathbb {P}_{\\Gamma }^*)\\left(a,b_{[p-1]},\\cdot \\right).$ For $h$ nonnegative and measurable, we have $\\int _{{\\mathbb {R}}}h\\left(z\\right)d|\\mu _{p,t,u}|(z)\\le \\mathbb {E}_{\\mathbb {P}_{\\Gamma }}\\left[h\\left(\\beta _p\\right)\\right]+\\mathbb {E}_{\\mathbb {P}_{\\Gamma }^*}\\left[h\\left(\\beta _p\\right)\\right]\\le 2 M_k(h)$ (see Theorem 6.2 in [47]), so $g_{p,t,u}\\in C_p$ .", "Because $tV_{p}$ is a set of uniqueness of $C_p$ , $g_{p,t,u}$ is 0 on ${\\mathbb {R}}$ .", "Take now $(u,x)\\in U_{p-2}\\times {\\mathbb {R}}$ and $g_{p-1,t,u,x}: z\\in {\\mathbb {R}}\\mapsto \\mathcal {F}[\\mathbb {P}_{\\Gamma }-\\mathbb {P}_{\\Gamma }^*](t,t u ,z,x)=\\mathcal {F}[\\mu _{p-1,t,u,x}(\\cdot )](z)$ is 0 on $tV_{p-1}$ , where $\\mu _{p-1,t,u,x}(\\cdot )=\\int _{{\\mathbb {R}}^{p}}e^{it\\left(a+b_{[p-2]}^{\\top }u_{[p-2]}+b_{p}x\\right)}d\\mu \\left(a,b_{[p-2]},\\cdot ,b_{p}\\right)$ Because $g_{p-1,t,u,x}\\in C_{p-1}$ and $tV_{p-1}$ is set of uniqueness, $g_{p-1,t,u,x}$ is identically 0.", "We conclude by iterating this argument.", "Take $0<\\epsilon \\le 1/2$ .", "Assume that $\\mathbb {P}_{ \\Gamma }$ and $\\mathbb {P}_{ \\Gamma }^*$ both give rise to the same collection $ ({\\mathbb {E}} \\left[{\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace \\Gamma ^{\\top } s \\ge 0\\rbrace | S = s \\right])_{ s \\in \\mathbb {S}_{ S }} $ .", "Recall that, from [24], $\\mathcal {T}f(s)=\\mathcal {T}f^-(s)=\\left(\\mathcal {T}f\\right)^-(s).$ One has in $S^{\\prime }\\left(\\mathbb {S}^p\\right)$ , for all $f\\in L^q\\left(\\mathbb {S}^p\\right)$ , $\\mathcal {T} f=\\sum _{m\\in {\\mathbb {N}}_0}\\lambda _{2m+1,p}Q_{2m+1,p}f,$ where $\\lambda (2m+1,p)=(-1)^m2\\pi ^{p/2}1\\cdot 3\\cdots (2m-1)/(\\Gamma (p/2)p(p+2)\\cdots (p+2m))$ for all $m\\in {\\mathbb {N}}_0$ .", "Hence, we have, for all $x\\in \\mathbb {S}_{X}$ , $\\mathbb {E}[Y|X=x]=\\sum _{m\\in {\\mathbb {N}}_0}\\lambda _{2m+1,p}Q_{2m+1,p}f_{\\Gamma }\\left(\\frac{(1,x)}{\\sqrt{1+|x|_2^2}}\\right)+\\frac{1}{2}$ and similarly for $f_{\\Gamma }^*$ .", "Denote the dot product and Lie norm by $z^2=\\sum _{k=1}^pz_k^2$ and $L(z)=\\sqrt{|z|_2^2+\\sqrt{|z|_2^4-|z^2|^2}}$ respectively.", "Denote also by $G(z)=\\sum _{m\\in {\\mathbb {N}}_0}\\lambda _{2m+1,p}Q_{2m+1,p}f_{\\Gamma }(z) +1/2$ for all $z\\in {\\mathbb {C}}^{p+1}$ (resp.", "$G^*$ for $f_{\\Gamma }^*$ ) and $F(z)=(f_1(z),\\dots ,f_{p+1}(z))^{\\top }$ for all $z\\in {\\mathbb {C}}^{p}$ , where: $f_1(z)=\\frac{1}{\\sqrt{1+z^2}},\\ f_2(z)=\\frac{z_1}{\\sqrt{1+z^2}},\\ \\dots \\ ,\\ f_{p+1}(z)=\\frac{z_p}{\\sqrt{1+z^2}}.$ We have $\\mathbb {E}[Y|X=x]=G\\circ F(x)=G^*\\circ F(x)$ and we prove that $G\\circ F, G^*\\circ F\\in \\mathcal {A}({\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p)$ , hence $(G-G^*)\\circ F\\in \\mathcal {A}({\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p)$ .", "For this, we check the conditions of Theorem 1.2.3 in [48].", "First, we have $F\\in \\mathcal {A}({\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p)$ .", "Indeed, for all $z\\in {\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p$ , $1+z^2\\in ({\\mathbb {C}}\\setminus (-\\infty ,0])^{p+1}$ .", "Now, for all $z\\in {\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p$ , we have $L(F(z))^2=\\frac{L(1,z)^2}{\\left|\\sqrt{1+z^2}\\right|^2}\\le \\frac{1+\\left(|\\mathrm {Re}(z)|_2+|\\mathrm {Im}(z)|_2\\right)^2}{1+|\\mathrm {Re}(z)|_2^2-|\\mathrm {Im}(z)|_2^2}\\le 1+2\\epsilon .$ Moreover, using Lemma 3.23 in [42] for the first display and using the Young inequalities (see [24]) for the third display, we have, for all $z\\in {\\mathbb {C}}^{p+1}$ such that $L(z)\\le \\sqrt{1+2\\epsilon }$ , $\\left|\\lambda _{2m+1,p}Q_{2m+1,p}f_{\\Gamma }(z)\\right|&\\le L(z)^{2m+1}\\left\\Vert \\lambda _{2m+1,p}Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{\\infty }\\left(\\mathbb {S}^p\\right)}\\\\&\\le L(z)^{2m+1}\\left\\Vert \\mathcal {T}\\left[Q_{2m+1,p}f_{\\Gamma }\\right]\\right\\Vert _{L^{\\infty }\\left(\\mathbb {S}^p\\right)}\\\\&\\le L(z)^{2m+1}\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\\\& \\le \\left(1+2\\epsilon \\right)^{m+1/2}\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}.$ Now, because $\\overline{\\lim }_{m\\rightarrow \\infty }\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}^{1/m}\\le q/(1+2\\epsilon )$ with $q<1$ , there exists $m_0\\in {\\mathbb {N}}$ such that, for all $m\\ge m_0$ , $\\left(1+2\\epsilon \\right)^{m+1/2}\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\le \\sqrt{1+2\\epsilon }$ .", "Hence, there exists $C_{\\epsilon }<\\infty $ such that, for all $m\\in {\\mathbb {N}}_0$ , $\\left(1+2\\epsilon \\right)^{m+1/2}\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\le C_{\\epsilon }$ .", "As a result, for all $z\\in {\\mathbb {C}}^{p+1}$ such that $L(z)<\\sqrt{1+2\\epsilon }$ , we have $\\sup _{m\\in {\\mathbb {N}}_0}\\left|\\lambda _{2m+1,p}Q_{2m+1,p}f_{\\Gamma }(z)\\right|<\\infty $ .", "Using the fact that $\\overline{z}\\mapsto Q_{2m+1,p}f_{\\Gamma }(\\overline{z})$ are homogeneous harmonic polynomials and Theorem 1.5.6 in [48] yields $G\\circ F\\in \\mathcal {A}({\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p)$ .", "Moreover, for all $z\\in {\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p$ , we have $\\left|G\\circ F(z)\\right|&\\le \\sqrt{1+2\\epsilon }\\sum _{m\\in {\\mathbb {N}}_0}\\left(\\left(1+2\\epsilon \\right)\\left\\Vert Q_{2m+1,p}f_{\\Gamma }\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}^{1/m}\\right)^m + \\frac{1 }{2},$ and the upper bound is a convergent series using $\\mathcal {R}_{BC,1} \\ \\text{(ii)}$ .", "Hence, $G\\circ F\\in \\mathcal {A}( {\\mathbb {R}}^p+i\\epsilon \\mathbb {B}_{{\\mathbb {R}}}^p)$ is bounded and similarly for $G^*\\circ F$ , which yields the result.", "Assume that $\\mathbb {P}_{ \\Gamma }$ and $\\mathbb {P}_{ \\Gamma }^*$ both give rise to the same collection $ ({\\mathbb {E}} \\left[ \\Gamma ^{\\top } s \\ge 0\\rbrace | S = s \\right])_{ s \\in \\mathbb {S}_{ S }} $ .", "Due to (REF ), $\\mathcal {T}$ and $\\Delta $ commute.", "For $r\\ge 0$ , $q\\in [1,\\infty ]$ , and $f\\in \\ L^q\\left(\\mathbb {S}^p\\right)$ , $\\Delta ^k f$ exists in $S^{\\prime }\\left(\\mathbb {S}^p\\right)$ and is such that $\\Delta ^r f=\\sum _{m=0}^{\\infty }\\zeta _{m,p}^{r}Q_{m,p}f$ .", "When the sum converges in $L^q\\left(\\mathbb {S}^p\\right)$ , $f$ is said to belong to $W_{q}^{2r}\\left(\\mathbb {S}^p\\right)$ .", "This Sobolev space is equipped with $\\left\\Vert f\\right\\Vert _{W_{q}^{2r}\\left(\\mathbb {S}^p\\right)}=\\left\\Vert f\\right\\Vert _{L^q\\left(\\mathbb {S}^p\\right)} +\\left\\Vert \\Delta ^rf\\right\\Vert _{L^q\\left(\\mathbb {S}^p\\right)}$ .", "Using that $\\Delta $ commutes with $\\mathcal {T}$ , (REF ), the Young inequalities (see [24]), and $\\mathcal {R}_{BC,2}$ (REF ), yield that, for all $m\\in {\\mathbb {N}}_0$ , $\\mathcal {T}f\\in W_{\\infty }^m\\left(\\mathbb {S}^p\\right)$ and $\\left\\Vert \\Delta ^m \\mathcal {T}f\\right\\Vert _{L^{\\infty }\\left(\\mathbb {S}^p\\right)}\\le \\left\\Vert \\Delta ^mf^-\\right\\Vert _{L^1\\left(\\mathbb {S}^p\\right)}.$ Denote, for $T\\in {\\mathbb {N}}$ , by $\\mathcal {K}_Tf(\\cdot )=\\int _{\\mathbb {S}^p}k_T(\\cdot ,s)f(s)d\\sigma (s)$ , where $k_T(x,y)=\\sum _{l=0}^{T}\\psi \\left(l/T\\right)q_{l,p}(x,y)$ with $\\psi \\in C^{\\infty }([0,\\infty ))$ , nonnegative, nonincreasing, such that $\\psi (x)=1$ if $x\\in [0,1]$ , $0\\le \\psi (x)\\le 1$ if $x\\in [1,2]$ , and $\\psi (x)=0$ if $x\\ge 2$ , $\\Delta $ also commutes with $\\mathcal {K}_T$ .", "Now, using that $\\Delta $ commutes with $\\mathcal {T}$ and $\\mathcal {K}_T$ (first and third displays) and the Young inequalities (first display), we obtain, for all $m\\in {\\mathbb {N}}_0$ and $k\\in {\\mathbb {N}}$ , $\\left\\Vert \\Delta ^m\\mathcal {T}f-\\mathcal {T}\\mathcal {K}_T\\Delta ^m f\\right\\Vert _{\\infty }&\\le \\left\\Vert \\Delta ^mf^--\\Delta ^m\\mathcal {K}_Tf^-\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\\\&=\\left\\Vert \\Delta ^mf^--\\mathcal {K}_T\\Delta ^mf^-\\right\\Vert _{L^{1}\\left(\\mathbb {S}^p\\right)}\\\\&\\le CT^{-2k}\\left\\Vert \\Delta ^mf^-\\right\\Vert _{W_1^{2k}\\left(\\mathbb {S}^p\\right)}\\\\&\\le CT^{-2k}\\left(\\left\\Vert f^-\\right\\Vert _{W_1^{2m}\\left(\\mathbb {S}^p\\right)}+\\left\\Vert f^-\\right\\Vert _{W_1^{2m+2k}\\left(\\mathbb {S}^p\\right)}\\right),$ where the third display follows from Proposition A.2 in [24].", "Using $\\mathcal {T}\\mathcal {K}_T\\Delta ^m f=\\sum _{l=0}^{\\lfloor (T-1)/2\\rfloor }\\psi \\left(\\frac{2l+1}{T}\\right)\\lambda _{2l+1,p}Q_{2l+1,p}\\Delta ^m f,$ that $q_{2l+1,p}(x,y)$ is a polynomial in $x^{\\top }y$ , and that $\\Delta ^m f\\in L^1\\left(\\mathbb {S}^p\\right)$ , we obtain that $\\mathcal {T}\\mathcal {K}_T\\Delta ^m f$ is odd and continuous and conclude from (REF ) and $\\mathcal {R}_{BC,2}$ (REF ) that $\\mathcal {T}f\\in C^{\\infty }_{{\\rm odd}}(\\mathbb {S}^p)$ .", "$\\mathcal {R}_{BC,2}$ (REF ) and (REF ) yield $\\left\\Vert \\Delta ^m \\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)\\right\\Vert _{L^{\\infty }\\left(\\mathbb {S}^p\\right)}\\le \\widetilde{M}_m,$ where $ \\widetilde{M}_m=\\max (c_{f_{\\Gamma }}+c_{f_{\\Gamma }^*},1)\\max (b_{f_{\\Gamma }},b_{f_{\\Gamma }^*})^m M_m$ is such that $\\sum _{m\\in {\\mathbb {N}}}\\widetilde{M}_m^{1/m}\\ge \\max \\left(c_{f_{\\Gamma }}+c_{f_{\\Gamma }^*},1\\right)\\max \\left(b_{f_{\\Gamma }},b_{f_{\\Gamma }^*}\\right)\\sum _{m\\in {\\mathbb {N}}}M_m^{1/m}=\\infty $ because $\\lbrace M_m\\rbrace $ satisfies (REF ).", "Also, (REF ), the fact that $\\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)\\in C^{\\infty }_{{\\rm odd}}\\left(\\mathbb {S}^p\\right)$ , and $\\mathcal {R}_{BC,2}$ (REF ) imply that, for all $u\\in U$ and $m\\in {\\mathbb {N}}_0$ , $\\Delta ^m\\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)(u)=0$ .", "From the proof of Theorem 10 in [8] (paragraph after Lemma 5) we obtain, for all $u\\in U$ and $l\\in {\\mathbb {N}}_0$ , $ Q_{2l+1,p}\\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)(u)=0$ .", "Hence, by $\\mathcal {R}_{BC,2}$ (REF ) and the fact that $Q_{2l+1,p}\\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)$ is the restriction to $\\mathbb {S}^p$ of a harmonic homogeneous polynomial of degree $2l+1$ , we obtain, for all $s\\in \\mathbb {S}^p$ and $l\\in {\\mathbb {N}}_0$ , $Q_{2l+1,p}\\mathcal {T}\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)(s)=0$ .", "Thus $\\left(f_{\\Gamma }-f_{\\Gamma }^*\\right)^-=0$ and the odd parts of $f_{\\Gamma }$ and $f_{\\Gamma }^*$ coincide in $L^{1}\\left(\\mathbb {S}^p\\right)$ .", "By the argument in [25], using that for $f=f_{\\Gamma }$ or $f=f_{\\Gamma }^*$ , for a.e.", "$u\\in \\mathbb {S}^p$ $f(u)\\ge 0$ , the definition of $f^{-}$ , and $f_{\\Gamma }(u)f_{\\Gamma }(-u)=0$ for a.e.", "$u\\in \\mathbb {S}^p$ , we obtain that, for a.e.", "$u\\in \\mathbb {S}^p$ , $f(u)=2 f^{-}(u){\\rm {\\large 1}\\hspace{-2.3pt}{\\large l}}\\lbrace f^{-}(u)>0\\rbrace $ .", "Thus, there is a one-to-one mapping between $f$ and $f^-$ .", "This allows to conclude.", "Assume that $P=\\mathbb {P}_{\\Gamma }$ and $P=\\mathbb {P}_{\\Gamma }^*$ both give rise to the same pair $\\left(\\mathbb {P}_{\\Gamma _1 + \\Gamma _3}, \\mathbb {P}_{\\Gamma _2 + \\Gamma _3}\\right)$ .", "For $t_0>0$ small enough, on $(-t_0,t_0)$ , $\\varphi _{\\delta }$ , $\\varphi _{\\delta }^*$ , $\\varphi _{\\epsilon _1}$ , $\\varphi _{\\epsilon _1}^*$ , $\\varphi _{\\epsilon _2}$ , $\\varphi _{\\epsilon _2}^*$ do not vanish, hence there exist nonvanishing continuous functions $p_{\\delta }$ , $p_{\\epsilon _1}$ , and $p_{\\epsilon _2}$ such that $\\varphi _{\\delta }=p_{\\delta }\\varphi _{\\delta }^*$ , $\\varphi _{\\epsilon _1}=p_{\\epsilon _1}\\varphi _{\\epsilon _1}^*$ , $\\varphi _{\\epsilon _2}=p_{\\epsilon _2}\\varphi _{\\epsilon _2}^*$ .", "By the restriction $\\mathcal {R}_K$ , we have $p_{\\epsilon _1}\\in C^{\\infty }(-t_0,t_0)$ .", "We have $p_{\\delta }(t_1+t_2)p_{\\epsilon _1}(t_1)p_{\\epsilon _2}(t_2)=1\\quad \\text{for all}\\ -t_0< t_1,t_2,t_1+t_2< t_0,$ hence $p_{\\delta }(t)p_{\\epsilon _1}(t)=1\\quad \\text{and}\\quad p_{\\delta }(t)p_{\\epsilon _2}(t)=1\\quad \\text{for all}\\ -t_0< t< t_0.$ Injecting (REF ) into (REF ) we obtain, for all $-t_0< t_1,t_2,t_1+t_2< t_0$ , $p_{\\delta }(t_1+t_2)=p_{\\delta }(t_1)p_{\\delta }(t_2) $ , which, using again (REF ), yields $p_{\\epsilon _1}(t_1+t_2)=p_{\\epsilon _1}(t_1)p_{\\epsilon _1}(t_2)$ .", "Hence, we have $p_{\\epsilon _1}^{\\prime }(t_1+t_2)=p_{\\epsilon _1}^{\\prime }(t_1)p_{\\epsilon _1}(t_2)$ , which at $t_1=0$ and $t_2=t$ yields $p_{\\epsilon _1}^{\\prime }(t)=p_{\\epsilon _1}^{\\prime }(0)p_{\\epsilon _1}(t)\\quad \\text{for all} \\ -t_0< t< t_0,$ where $p_{\\epsilon _1}^{\\prime }(0)=\\varphi _{\\epsilon _1}^{\\prime }(0)-\\left(\\varphi _{\\epsilon _1}^*\\right)^{\\prime }(0)=i({\\mathbb {E}} _{\\mathbb {P}_{\\epsilon _1}}[\\epsilon _1]-{\\mathbb {E}} _{\\mathbb {P}_{\\epsilon _1}^*}[\\epsilon _1]) \\in i{\\mathbb {R}}$ which we denote by $p_{\\epsilon _1}^{\\prime }(0):=ib$ .", "Thus, we obtain, for all $t_0< t< t_0$ , $p_{\\epsilon _1}(t)=\\exp (ibt)$ .", "This yields that $t\\mapsto \\varphi _{\\epsilon _1}(t)-\\exp (ibt)\\varphi _{\\epsilon _1}^*(t)$ is 0 on $(-t_0,t_0)$ .", "Thus, using $\\mathcal {R}_K$ , we have $\\varphi ^{(1)}_{\\epsilon _1}(0)=ib +(\\varphi ^*_{\\epsilon _1})^{(1)}(0)$ then using that $\\mathbb {P}_{\\epsilon _1}$ and $\\mathbb {P}_{\\epsilon _1}^*$ are both mean 0 yields $b=0$ .", "Thus, $\\mathbb {P}_{\\epsilon _1}$ and $\\mathbb {P}_{\\epsilon _1}^*$ have the same moments so $\\mathbb {P}_{\\epsilon _1} =\\mathbb {P}_{\\epsilon _1}^*$ by $\\mathcal {R}_K$ .", "Now, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{Y_1}(t)=\\varphi _{\\delta }(t)\\varphi _{\\epsilon _1}^*(t)=\\varphi _{\\delta }^*(t)\\varphi _{\\epsilon _1}^*(t)$ , hence, because the zeros of $\\varphi _{\\epsilon _1}^*$ are isolated (see Lemma 4.8 in [1]) and $\\varphi _{\\delta }$ and $\\varphi _{\\delta }^*$ are continuous, we obtain $\\varphi _{\\delta }=\\varphi _{\\delta }^*$ .", "Similarly, because, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{Y_2-Y_1}(t)=\\varphi _{\\epsilon _2}^*(t)\\varphi _{\\epsilon _1}^*(-t)=\\varphi _{\\epsilon _2}(t)\\varphi _{\\epsilon _1}^*(-t)$ , the zeros of $\\varphi _{\\epsilon _1}^*$ are isolated and $\\varphi _{\\epsilon _2}$ and $\\varphi _{\\epsilon _2}^*$ are continuous, we obtain $\\varphi _{\\epsilon _2}=\\varphi _{\\epsilon _2}^*$ , hence the result.", "We use $\\mathcal {X}_0= \\lbrace x \\in {\\mathbb {R}}^T:\\ \\prod _{j=1}^T x _j\\ne 0,\\ \\prod _{m\\ne j} ( x _m - x _j)\\ne 0\\rbrace $ and, for all $( v , x )\\in {\\mathbb {R}}^T\\times \\mathcal {X}_0$ , $\\Theta :\\ ( v , x ) \\mapsto \\sum _{k=1}^T v _k \\left(\\sum _{j=1}^T b_{jk}( x ) x_j^{T}\\right),$ $\\left\\lbrace \\begin{array}{rll}b_{jk}( x ) = & \\displaystyle \\dfrac{(-1)^{k+1}}{\\displaystyle \\prod _{\\underset{m\\ne j}{1\\le m \\le T}} ( x _m - x _j)} \\sum _{\\underset{i_1,\\dots ,i_{T-k}\\ne j}{1\\le i_1 < \\dots < i_{T-k}\\le T}} x _{i_1} x _{i_2} \\dots x _{i_{T-k}} & \\text{for all} \\ k \\ne T \\\\b_{jk}( x ) = & \\dfrac{(-1)^{T+1}}{\\displaystyle \\prod _{\\underset{m\\ne j}{1\\le m \\le T}} ( x _m - x _j)} & \\text{for } \\ k = T .\\end{array} \\right.$ Assume that $\\mathbb {P}_{\\Gamma }$ and $\\mathbb {P}_{\\Gamma }^*$ both give rise to the same collections $\\left(\\mathbb {P}_{\\Gamma _1 + \\sum _{j=1}^T \\Gamma _{j+1} x ^j_t + \\epsilon _t}\\right)_{ x _t \\in \\mathbb {S}_{ X _t}, t=1,\\dots , T}$ .", "Using ${\\mathcal {R}_{LP,0}}$ (REF ), there exists $r>0$ such that $r 1 _c \\in \\mathcal {X}_1 $ and conditioning on $X = r 1 _c$ and using $\\delta =\\alpha +\\sum _{k=1}^T\\beta _k r^k$ yield, for all $ t \\in {\\mathbb {R}}^T$ , ${\\mathbb {E}} \\left[ e^{i t ^{\\top } Y } | X = r 1 _c \\right]= \\varphi _{\\delta }\\left(\\sum _{j=1}^{T} t _j\\right)\\prod _{j=1}^T\\varphi _{ \\epsilon _j}( t _j)$ , hence $ \\varphi _{\\delta }\\left(\\sum _{j=1}^{T} t _j\\right)\\prod _{j=1}^T\\varphi _{ \\epsilon _j}( t _j) = \\varphi _{\\delta }^*\\left(\\sum _{j=1}^{T} t _j\\right)\\prod _{j=1}^T\\varphi ^*_{ \\epsilon _j}( t _j).$ Then, following the same steps as in the proof of Theorem REF yields $\\mathbb {P}_{\\epsilon _1}=\\mathbb {P}_{\\epsilon _1}^*$ .", "Now, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{\\delta }(t)\\varphi _{\\epsilon _1}^*(t)=\\varphi _{\\delta }^*(t)\\varphi _{\\epsilon _1}^*(t),$ hence, because the zeros of $\\varphi _{\\epsilon _1}^*$ are isolated (see Lemma 4.8 in [1]) and $\\varphi _{\\delta }$ and $\\varphi _{\\delta }^*$ are continuous, we obtain $\\varphi _{\\delta }=\\varphi _{\\delta }^*$ .", "Similarly, because, for all $t\\in {\\mathbb {R}}$ and $j=2,\\dots , T$ , ${\\mathbb {E}} \\left[e^{it(Y_j-Y_{1})}| ( X _{1}, X _{j} )= (r,r)\\right]=\\varphi _{\\epsilon _j}^*(t)\\varphi _{\\epsilon _1}^*(-t)$ , we have $\\varphi _{\\epsilon _j}^*(t)\\varphi _{\\epsilon _1}^*(-t)=\\varphi _{\\epsilon _j}(t)\\varphi _{\\epsilon _1}^*(-t).$ Thus, using that the zeros of $\\varphi _{\\epsilon _1}^*$ are isolated and $\\varphi _{\\epsilon _j}$ and $\\varphi _{\\epsilon _j}^*$ are continuous for all $j=2,\\dots , T$ , we obtain $\\varphi _{\\epsilon _j}=\\varphi _{\\epsilon _j}^*$ for all $j=2,\\dots , T$ , hence, for all $t\\in {\\mathbb {R}}^{T}$ and $x\\in \\mathbb {S}_{X}$ , $\\varphi _{\\alpha , \\beta }\\left(\\sum _{j=1}^{T} t _j, \\sum _{j=1}^{T} t _j x _j, \\dots , \\sum _{j=1}^{T} t _j x _j^{T} \\right) = \\varphi _{\\alpha , \\beta }^*\\left(\\sum _{j=1}^{T} t _j, \\sum _{j=1}^{T} t _j x _j, \\dots , \\sum _{j=1}^{T} t _j x _j^{T} \\right).$ Then, for all $x\\in \\mathcal {X}_0$ , we use a change of variable that relates $t\\in {\\mathbb {R}}^{T}$ to $v\\in {\\mathbb {R}}^{T}$ such that $(\\sum _{j=1}^{T} t _j, \\sum _{j=1}^{T} t _j x _j, \\dots , \\sum _{j=1}^{T-1} t _j x _j^{T-1} ) = v $ .", "This change of variable allows to choose the values of the $T-1$ first variables in (REF ) independently from each other.", "This can be written as $ t = \\left(V^{-1}( x )\\right)^{\\top } v $ , where $ V( x )= \\left( \\begin{array}{ccccc}1 & x _1 & x _1^2 & \\dots & x _1^{T-1} \\\\\\colon & & & \\dots & \\colon \\\\1 & x _T& x _T^2 & \\dots & x _T^{T-1}\\end{array} \\right)$ is the Vandermonde matrix.", "We use $D( x )$ , the diagonal matrix which entries are the coordinates of $ x $ .", "It is a classical result that, for all $x\\in \\mathcal {X}_0$ , $D( x )$ and $V( x )$ are invertible hence $\\widetilde{V}( x ) = (D( x ) V( x ))^{\\top } $ is invertible.", "Then, for all $ v \\in {\\mathbb {R}}^T$ and $ x \\in \\mathcal {X}_0$ , we can express $ t $ as a function of $ v $ and $ x $ , and obtain for the last variable in $\\varphi _{\\alpha , \\beta }^*\\left(\\sum _{j=1}^{T} t _j, \\sum _{j=1}^{T} t _j x _j, \\dots , \\sum _{j=1}^{T} t _j x _j^{T} \\right),$ that $\\sum _{j=1}^{T} t _j x _j^T& = t ^{\\top } x ^T = \\left(\\left(V( x )^{-1}\\right)^{\\top } v \\right)^{\\top } x ^{T} =\\left( D( x )\\widetilde{V}( x )^{-1} v \\right)^{\\top } x ^{T}=\\Theta ( v , x ).$ Using (REF ), this yields, for all $ v \\in {\\mathbb {R}}^{T}$ and $x\\in \\mathcal {X}_0$ , $\\varphi _{\\alpha , \\beta }\\left( v , \\Theta ( v , x ) \\right) = \\varphi _{\\alpha , \\beta }^*\\left( v , \\Theta ( v , x ) \\right).$ Then, using that the vector space spanned by $\\mathcal {F}[\\mathcal {P}_c(\\Omega _T)]$ is a quasi-analytic class of functions on ${\\mathbb {R}}$ , we obtain that, for all $ v \\in {\\mathbb {R}}^{T}$ , $ z \\in {\\mathbb {R}}\\mapsto (\\varphi _{\\alpha ,\\beta } - \\varphi ^*_{\\alpha ,\\beta })( v ,z )$ belongs to a quasi-analytic class.", "Finally, for all $ v \\in {\\mathbb {R}}^{T}\\setminus \\mathcal {V}$ where $\\mathcal {V}$ is such that ${\\mathbb {R}}^{T}\\setminus \\mathcal {V}$ is dense in ${\\mathbb {R}}^{T}$ , $\\mathcal {R}_{LP}$ (REF ) and that $ x \\mapsto \\Theta ( v , x )$ is continuous on $\\mathbb {S}_{X} \\cap \\mathcal {X}_0$ yield that $U_{T, v }=\\left\\lbrace \\Theta ( v , u ),\\ \\forall u \\in \\mathbb {S}_{X} \\cap \\mathcal {X}_0\\right\\rbrace $ contains a bounded sequence of distinct points.", "We obtain that, for all $ v \\in {\\mathbb {R}}^{T}\\setminus \\mathcal {V}$ and $ z \\in {\\mathbb {R}}$ , $ \\varphi _{\\alpha ,\\beta }( v ,z ) = \\varphi ^*_{\\alpha ,\\beta }( v ,z )$ hence $\\mathbb {P}_{\\alpha ,\\beta } = \\mathbb {P}_{\\alpha ,\\beta }^*$ by continuity for all $ v \\in {\\mathbb {R}}^{T}$ .", "Let $0< \\epsilon < 1$ .", "Assume that $\\mathbb {P}_{\\Gamma }$ and $\\mathbb {P}_{\\Gamma }^*$ both give rise to the same collections $\\left(\\mathbb {P}_{f(\\Gamma ^{\\top } x _t) + \\eta _t}\\right)_{ x _t \\in \\mathbb {S}_{ X _t}, t=1,2}$ .", "Denote by $\\mathcal {X}_1=\\lbrace (x_1,x_2)\\in \\mathbb {S}_{X_1,X_2}: x_1=x_2\\rbrace $ .", "Using $\\mathcal {R}_{SI}$ (REF ), there exists $r >0$ such that $r ( 1 _c, 1 _c) \\in \\mathcal {X}_1$ , hence, for all $ t \\in {\\mathbb {R}}^{2}$ , using $\\delta : = f(\\Gamma ^{\\top } (r 1 _c))$ , we have $\\varphi _{\\delta }( t _1 + t _2)\\varphi _{ \\eta _1}( t _1)\\varphi _{ \\eta _2}( t _2)= \\varphi ^*_{\\delta }( t _1 + t _2)\\varphi ^*_{ \\eta _1}( t _1)\\varphi ^*_{ \\eta _2}( t _2).$ Then, following the same steps as in the proof of Theorem REF yields $ \\varphi _{ \\eta _1} = \\varphi _{ \\eta _1}^*$ .", "Now, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{Y_1| X _1}(t|r 1 _c)=\\varphi _{\\delta }(t)\\varphi _{\\eta _1}^*(t)=\\varphi _{\\delta }^*(t)\\varphi _{\\eta _1}^*(t)$ , hence, because the zeros of $\\varphi _{\\eta _1}^*$ are isolated (see Lemma 4.8 in [1]) and $\\varphi _{\\delta }$ and $\\varphi _{\\delta }^*$ are continuous, we obtain $\\varphi _{\\delta }=\\varphi _{\\delta }^*$ .", "Similarly, because, for all $t\\in {\\mathbb {R}}$ , $\\varphi _{Y_2-Y_{1}| X _1, X _2}(t |(r 1 _c,r 1 _c))=\\varphi _{\\eta _2}^*(t)\\varphi _{\\eta _1}^*(-t)=\\varphi _{\\eta _2}(t)\\varphi _{\\eta _1}^*(-t)$ , the zeros of $\\varphi _{\\eta _1}^*$ are isolated and $\\varphi _{\\eta _2}$ and $\\varphi _{\\eta _2}^*$ are continuous, we obtain $\\varphi _{\\eta _2}=\\varphi _{\\eta _2}^*$ .", "Thus, we obtain, for all $ t \\in {\\mathbb {R}}^{2}$ and $(x_1,x_2)\\in \\mathbb {S}_{ X _1, X _2}$ , $\\varphi _{f(\\Gamma ^{\\top }x_1),f(\\Gamma ^{\\top }x_2)}( t )=\\varphi ^*_{f(\\Gamma ^{\\top }x_1),f(\\Gamma ^{\\top }x_2)}( t ).$ This amount to study identification in $Z_t=f(\\Gamma ^{\\top }X_t)$ , $t=1,2$ and $Z_2\\ge Z_1$ is equivalent to $\\Gamma ^{\\top }\\left(X_{2}-X_{1}\\right)\\ge 0$ .", "This is the binary choice model, hence Theorem REF yields the result." ] ]
2105.11720
[ [ "Thermodynamics and phase transitions of black holes in contact with a\n gravitating heat bath" ], [ "Abstract We study the thermodynamics of a shell of self-gravitating radiation, bounded by two spherical surfaces.", "This system provides a consistent model for a gravitating thermal reservoir for different solutions to vacuum Einstein equations in the shell's interior.", "The latter include black holes and flat space, hence, this model allows for the study of black hole phase transitions.", "Following the analysis of arXiv:1103.3898 , we show that the inclusion of appropriate entropy terms to the spacetime boundaries (including the Bekenstein-Hawking entropy for black hole horizons) leads to a consistent thermodynamic description.", "The system is characterized by four phases, two black hole phases distinguished by the size of the horizon, a flat space phase and one phase that describes naked singularities.", "We undertake a detailed analysis of black-hole phase transitions, the non-concave entropy function, the properties of temperature at infinity, and system's heat capacity." ], [ "Introduction", "Ever since Bekenstein's proposal of black hole entropy [1] and Hawking's derivation of black hole radiation [2] , black holes are understood as thermodynamic objects.", "According to the generalized second law of thermodynamics (GSL)[3], black hole entropy adds up with the entropy of matter, and their sum is a non-decreasing function of time.", "The GSL implies that there exists a larger thermodynamic space that contains black holes and self-gravitating systems, i.e., systems of ordinary matter in which the gravitational self-interaction contributes significantly to their thermodynamic properties.", "Therefore, we expect the presence of phase transitions between black holes and self-gravitating systems.", "In this work, we undertake the analysis of such phase transitions in a simple system, that consists of a shell of self-gravitating radiation.", "The geometry inside the shell is either of Minkowski space, or of a black hole or of a singular solution.", "The different interior geometries correspond to different thermodynamic phases for the system, while the shell acts as a thermal bath for these phases.", "We analyze the thermodynamic properties of the system and the characteristics of the associated phase transitions." ], [ "Motivation", "There is substantial work on black-hole to black-hole phase transitions, first on Kerr-Newman black holes [4], [5], [6], and later on asymptotically anti-de Sitter (AdS) black holes [7], [8], [9], [10].", "Phase transitions in self-gravitating systems have also been extensively studied, see, for example, [11], [12], [13], [14].", "However, there is much less work on phase transitions between black holes and self-gravitating systems.", "The most well known case is the Hawking-Page phase transition between black holes and radiation [15], albeit in asymptotically AdS spacetimes.", "In asymptotically flat spacetimes, phase transitions have been studied by comparing the entropy of a Schwarzschild black hole in a box with the entropy of (non-gravitating) radiation in the box [4], [5], [16], [17], [18].", "Backreaction from the Hawking radiation can also be included in the thermodynamical description [19].", "Certainly, understanding phase transitions between black holes and self-gravitating systems is by itself important.", "Furthermore, progress in this direction could provide significant insights to quantum gravity theories [20].", "For example, consider a self-gravitating system that involves conserved quantities like particle numbers.", "These quantities disappear in the black hole case, by virtue of the no-hair theorem.", "A thermodynamic study of the formation of a black hole phase can provide novel information about the relation between black hole hair and quantum effects, the latter being incorporated in black hole entropy.", "Thermodynamic processes that cross from the black hole phase to ordinary matter may be relevant to discussions of black hole information loss.", "In the longer term, an analysis of critical exponents, or of non-equilibrium properties in such phase transitions could constrain candidate quantum theories of gravity.", "A different context for this work is the long-standing effort to understand the physics of non-extensive thermodynamic systems in equilibrium.", "Non-extensivity arises whenever the range of interactions of the system is larger than the size of the system: this is possible, either with short-range forces in small systems [21] or with long-range forces [22], [23], including gravity [24], [25].", "Non-extensive systems exhibit novel thermodynamic properties.", "They are spatially inhomogeneous even in equilibrium, their micro-canonical and canonical ensembles are inequivalent [26], their entropy function is not be concave (hence, heat capacities may be negative).", "The model examined here is novel and informative for non-extensive thermodynamics.", "It involves full General Relativity (rather than Newtonian approximations), it brings together black holes and self-gravitating system.", "Crucially, it can be treated semi-analytically, thereby, providing a workable testing ground for understanding the structure of gravitational thermodynamics." ], [ "Background", "The natural place to start an analysis of black hole phase transitions is the simplest self-gravitating system in General Relativity, namely, static, spherically symmetric solutions to Einstein's equations.", "These are described by the Tolman-Oppenheimer-Volkoff (TOV) equation.", "One might expect that for a given type of matter, there may exist a region of the parameter space that corresponds to black hole horizons coexisting with matter.", "This turns out to not be the case: if matter satisfies the dominant energy conditions, then no horizons are encountered when integrating the TOV equation from the boundary inwards [27], [28].", "In fact, if the equation of state is thermodynamically consistent, there are only two types of solutions, regular ones and singular ones [28].", "The former are everywhere locally Minkowskian, and they describe ordinary compact stars.", "The latter contain a strongly repulsive naked singularity at the center, where the geometry is locally isomorphic to negative-mass Schwarzschild spacetime.", "The situation is different when we consider a shell rather than a ball of matter.", "Suppose we place an interior boundary at $r = r_0$ so that the geometry for $r < r_0$ is a solution to the vacuum Einstein equations.", "Then, the associated thermodynamic state space contains regular solutions, singular solutions and also black hole solutions.", "Hence, we can undertake a thermodynamic analysis that includes black hole phase transitions.", "In this paper, we study a shell of radiation, bounded between two reflecting and non-thermally conducting surfaces at $r = r_0$ and at $r = R$ .", "The simplicity of the equation of state, implies that some properties of the solutions can be evaluated semi-analytically.", "For past studies of self-gravitating radiation, see, Refs.", "[29], [30], [31], [32], [33], [34].", "The radiation shell considered here defines a concrete model of a thermal reservoir in contact with a black hole.", "This model is an improvement of existing ones—see, for example, [18]—which employ the rather extreme idealization of switching off the gravitational interaction of the reservoir.", "Such an approximation is justified in extensive systems: when a system is brought into contact with a reservoir, it may acquire the reservoir's temperature or pressure, but its constitutive equations remain unchanged.", "This is due to the fact that short range forces affect only a small region around the interface of the system with the reservoir.", "In contrast, when long-range forces are involved, the reservoir acts directly on the whole of the system, and it can affect its constitutive equation.", "Our analysis employs the formal structure of equilibrium thermodynamics, as described by Callen [35].", "We mentioned earlier that there exists as yet no set of thermodynamic axioms applicable to non-extensive systems.", "In principle, some axiomatic approaches to thermodynamics [36], [37], [38] can be generalized for non-extensive systems; for work in this direction, see Ref.", "[39].", "Still, a definitive axiomatic formulation of thermodynamics for gravitating systems does not yet exist.", "We note the work of Martinez in adapting Callen's axioms to gravitational systems in Ref.", "[40], and Ref.", "[41] for formulating the three laws of thermodynamics in this context.", "The key point in Callen's formulation of thermodynamics is that an isolated thermodynamic system is described in terms of a set of macroscopic constraints that determine its thermodynamic state space $Q$ .", "The values of any variable that is not fixed by the constraints correspond to global maxima of the entropy function, subject to the constraints.", "This statement is the Maximum Entropy Principle (MEP).", "Eventually, all thermodynamic information is contained in the fundamental thermodynamic function, i.e., the entropy function $S: Q \\rightarrow R^+$ .", "In the present system, the thermodynamic state space $Q$ is three dimensional, it is determined by the shell radii $r_0$ and $R$ and by the Arnowitt-Deser-Misner (ADM) mass $M$ ." ], [ "Results", "Our results are the following.", "$1.$ The space $Z$ of solutions to the TOV equation is larger than the thermodynamic state space $Q$ .", "This means that the equilibrium values of the additional degrees of freedom must determined the MEP.", "We show that if the entropy function involves only contributions from radiation, the entropy function is unbounded and the MEP cannot be implemented.", "The system would then have no consistent thermodynamic description.", "This is an analogue of the gravothermal catastrophe [43], [44] that appears in non-relativistic self-gravitating systems.", "$2.$ To restore thermodynamic consistency, we must add entropy terms associated to the internal boundaries of the system, i.e., the horizons and the singularities that appear in the interior region.", "For horizons, we use the Bekenstein-Hawking entropy.", "For the repulsive singularity, the associated entropy is obtained by Wald's Noether charge for spacetime boundaries [42], modulo a multiplicative constant.", "The latter is determined uniquely by the requirement of thermodynamic consistency, i.e., that the MEP can be implemented.", "The resulting expression for the boundary entropy is consistent with a previous result of Ref.", "[32] for a ball of self-gravitating radiation.", "$3.$ After the implementation of the MEP, we construct the entropy function on the thermodynamic state space $Q$ .", "$Q$ splits into four regions, corresponding to four distinct phases.", "Phase F corresponds to a shell in locally Minkowski spacetime, phase $B_I$ describes a large black hole solution with little radiation in the shell, phase $B_{II}$ describes a small black hole coexisting with radiation, and phase S corresponds to singular solutions.", "$4.$ We find that the phase transitions between the F phase and the $B_I$ and $B_{II}$ phases are first-order.", "The latent heat in all transitions from the F phase to the black hole phases is negative, i.e., a heat must be removed from a self-gravitating system in order to form a black hole.", "We explain that this occurs because black holes constitute a higher-energy phase but not a higher-temperature phase.", "The transition between the $B_I$ and the $B_{II}$ phases is continuous, and so is the transition between the F and S phases.", "There is also one triple point for the F, $B_I$ and $B_{II}$ phases.", "$5.$ There is no coexistence curve between the black hole phases and the $S$ phase.", "This could change by a modification of the Bekenstein-Hawking expression for black hole entropy at very small masses.", "$6.$ We analyse the behavior of other thermodynamic observables, focusing on the temperature at infinity, on the heat capacity.", "The heat capacity of the system can become negative, however, in the context of non-extensive thermodynamics, this is not necessarily a sign of instability [45].", "The plan of this paper is the following.", "In Sec.", "2. we analyse the geometry of the shell of gravitating radiation, and all types of solutions.", "In Sec.", "3, we show that the inclusion of entropy contributions from boundary terms allows for a thermodynamically consistent description of the system.", "In Sec.", "4, we implement the MEP, we identify the four phases that characterize the system, and we analyze phase transitions and other thermodynamic properties.", "In Sec.", "5, we summarize and discuss our results.", "The Appendix contains a proof that the entropy of radiation in the box has no global maxima." ], [ "Constitutive equations", "The system under study is a spherical shell of self-gravitating radiation in thermal equilibrium.", "The thermodynamic equations for radiation are $\\rho = 3P = b T^4, \\qquad s=\\frac{4}{3}b^{1/4}\\rho ^{3/4}$ where $\\rho $ is the energy density, $P$ is the pressure, $T$ is the temperature, $s$ is the entropy density and $b=\\pi ^2/15$ in geometrized units ($c=G=\\mathalpha {\\usebox {}}=k_B=1$ ).", "The spacetime metric is static and spherically symmetric, $ds^2 = -L(r)^2 dt^2 + \\frac{dr^2}{1-\\frac{2m(r)}{r}} + r^2 (d\\theta ^2 + \\sin ^2\\theta d\\phi ^2)$ where $L(r)$ is the lapse function, $m(r)$ is the mass function and $(t,r,\\theta ,\\phi )$ is the usual coordinate system.", "The radiation is contained between two reflecting spherical boundaries, an external boundary at $r = R$ , and an internal one at $r = r_0 < R$ .", "For $r> R$ , the solution is Schwarzschild with ADM mass $M$ , i.e., $L(r ) = \\sqrt{1-\\frac{2M}{r}}, \\qquad \\qquad m(r ) = M.$ For $r \\in [r_0, R]$ , the geometry is determined by the TOV equation for radiation, $\\frac{dm}{dr}=4\\pi r^2\\rho , \\qquad \\qquad \\frac{d\\rho }{dr} = -\\frac{4\\rho }{r^2}\\frac{(m+\\frac{4}{3}\\pi r^3 \\rho )}{1-\\frac{2m}{r}}.$ For fixed ADM mass $M$ , we solve the TOV equation from the boundary $r = R$ inwards.", "To this end, we must specify the density $\\rho _R$ at $R$ (or equivalently the temperature $T_R$ at $r = R$ ), in addition to the mass $m(R)=M$ .", "The temperature at the boundary $T_R$ is related to the temperature at infinity $T_{\\infty }$ by Tolman's law, $LT = T_{\\infty }$ , $T_\\infty = T_R \\sqrt{1-\\frac{2M}{R}}.$ Tolman's law also determines the lapse function for $r \\in [r_0, R]$ , $L(r) = \\sqrt{1-\\frac{2M}{R}}\\bigg (\\frac{\\rho _R}{\\rho (r)}\\bigg )^{1/4}$ Let $m_0 := m(r_0)$ and $\\rho _0 := \\rho (r_0)$ .", "The solution for $r < r_0$ is Schwarzschild, since there is no matter present.", "In particular, $L(r) = \\kappa \\sqrt{1-\\frac{2m_0}{r}} \\qquad \\qquad m(r) = m_0,$ where $\\kappa $ is a scaling constant for the time coordinate $t$ .", "By continuity at $r = r_0$ , $\\kappa = \\sqrt{\\frac{1- 2M/R}{1- 2m_0/r_0}}\\bigg (\\frac{\\rho _R}{\\rho _0}\\bigg )^{1/4}$ If $m_0 > 0$ , there is a Schwarzschild horizon at $r = 2m_0$ The integration of the TOV equations from the boundary inwards never encounters a horizon, so $2m_{0} < r_{0}$ always.", ".", "If $m_0 = 0$ , the geometry for $r < r_0$ is Minkowskian.", "If $m_0 <0$ , the geometry for $r < r_0$ is Schwarzschild with negative mass, i.e., there is a naked curvature singularity at the center." ], [ "Solution curves", "Here, we consider the solution curves obtained when solving the TOV equations from the boundary inwards.", "A solution of Eqs.", "(REF ) is uniquely specified by the boundary data $(R, M, T_R)$ .", "It is convenient to introduce the variables, $\\xi := \\log \\frac{r}{R} \\qquad u:=\\frac{2m(r)}{r} \\qquad v:=4\\pi r^2 \\rho (r),$ which allow us to write the TOV equations as $\\frac{du}{d\\xi } = 2v-u, \\qquad \\qquad \\frac{dv}{d\\xi }= \\frac{2v(1-2u-\\frac{2}{3}v)}{1-u}.$ Eqs.", "(REF ) are simpler than Eqs.", "(REF ), because they define an autonomous two-dimensional dynamical system.", "They need only two positive numbers $(u_R, v_R)$ for boundary data, the outer boundary corresponding to $\\xi = 0$ .", "The center corresponds to $\\xi \\rightarrow - \\infty $ .", "This simplification is possible only in linear equations of state, because they introduce no scale into the TOV equations.", "Hence, the solutions are invariant under the transformation $r \\rightarrow \\lambda r, m \\rightarrow \\lambda m $ , and $\\rho \\rightarrow \\lambda ^{-2}\\rho $ , for any $\\lambda > 0$ .", "In order to describe the behavior of the solution curves, we identify two straight lines on the $u-v$ plane (Fig.REF ): $\\varepsilon _1: 2v-u=0$ , at which all solution curves satisfy $du/d\\xi = 0$ .", "$\\varepsilon _2: 1-2u-\\frac{2}{3}v=0$ , at which all solution curves satisfy $dv/d\\xi = 0$ .", "The two curves intersect at the point $K = (\\frac{3}{7}, \\frac{3}{14})$ .", "$K$ and $O = (0, 0)$ are the equilibrium points of the dynamical system (REF ).", "Figure: Solution curves to the Tolman-Oppenheimer-Volkoff equation for radiation.A typical solution curve $C: u=u(\\xi ),\\, v=v(\\xi )$ satisfies the following properties [28].", "There is a point $r_1 <R$ , such that $u(r) = 0$ .", "Furthermore, $u(r) < 0$ for all $r < r_1$ .", "There is a point $r_2 < r_1$ , such that $\\frac{dP}{dr}(r_2) = 0$ .", "Furthermore, $\\frac{dP}{dr}> 0$ , for $r < r_2$ .", "$\\lim _{\\xi \\rightarrow - \\infty } (u,v) = (-\\infty , 0) $ , $\\lim _{\\xi \\rightarrow \\infty } (u,v) = K $ .", "The only non-trivial exception is the curve $\\Gamma $ of regular solutions, i.e., solutions that satisfy $m(0)=0$ .", "On $\\Gamma $ , $\\lim _{\\xi \\rightarrow - \\infty } (u,v) = O$ .", "The variables $v$ and $u$ attain maximum values on $\\Gamma $ at the points $P$ and $Q$ respectively, where $(u_P, v_P) \\approx (0.3861, 0.3416)$ and $(u_Q, v_Q) \\approx (0.4926, 0.2463)$ .", "The following solution curves are degenerate cases.", "(i) The points $K$ and $O = (0, 0)$ are equilibrium points, so each defines a distinct solution curve.", "(ii) A point on the $u$ axis ($v_R = 0, u_R \\ne 0$ ), evolves with $u(\\xi ) = u_R e^{-\\xi },\\, v(\\xi ) = 0$ , and it encounters an even horizon.", "This corresponds to a Schwarzschild black hole without matter." ], [ "Shell configurations", "The solution curves uniquely determine the spacetime geometry associated to a shell of radiation—for a past treatment of this system, see, Ref.", "[34].", "We select $R, M, T_R$ , and we follow the solution curve until we encounter $r_0$ .", "The segment of the solution curve between $r_0$ and $R$ determines the shell's geometry, for $r > R$ , and $r < r_0$ the geometry is Schwarzschild with mass $M$ and $m_0$ respectively.", "Hence, the space $Z$ of equilibrium configurations for a shell of self-gravitating radiation is four-dimensional.", "It can be described by the coordinates $(R, r_0, M, T_R)$ or equivalently by the coordinates $(R, \\xi _0, u_R, v_R)$ , where $\\xi _0 = \\log (r_0/R)$ .", "The solutions classes fall into three types, depending on the value of $m_0 = m(r_0)$ .", "$m_0 = 0$ : Type F (flat) solution for $r < r_0$ .", "$m_0 > 0$ : Type B (black-hole), it contains a Schwarzschild horizon at $r < r_0$ .", "$m_0 < 0$ : Type S (naked singularity), it contains the negative-mass Schwarzschild singularity $r < r_0$ .", "F-type solutions form a set of measure zero in $Z$ that acts as a the boundary between the subset of B-type and S-type solutions.", "The curve of the F-type solutions on the $u_R-v_R$ plane (the F-curve for brevity ) depends only on $\\xi _0$ , because of the scaling symmetry.", "In Fig.", "REF , the F-curve is plotted for different $\\xi _0$ .", "B-type solutions lie in the region between the F-curve and the $u_R$ axis; the remainder corresponds to S-type solutions.", "As $\\xi _0$ decreases, so does the area of the $B$ phase.", "At $\\xi \\rightarrow - \\infty $ , B-type solutions disappear and the F-curve coincides with the line $\\Gamma $ of Fig.1.", "Figure: The F-curve for different values of ξ 0 \\xi _0, namely.", "Type B configurations lie in the region between the R-curve and the horizontal axis.", "Outside this region there exist only Type S configurations.", "As ξ 0 \\xi _0 decreases the F-curve shrinks to the solution curve Γ\\Gamma .For fixed $\\xi _0$ , the F-curve has two distinctive points.", "The point $u_{max}(\\xi _0)$ that corresponds to the maximum value of $u_R$ .", "The final point $u_f(\\xi _0) \\ne 0$ , where the curve intersects the horizontal axis ($v_R = 0$ ).", "From Fig.REF , we see that both $u_{max}(\\xi _0)$ and $u_f(\\xi _0)$ decrease with decreasing $\\xi _0$ —see, Fig.", "(REF ).", "As $\\xi _0 \\rightarrow -\\infty $ , $u_{max} \\rightarrow u_Q$ , i.e., it corresponds to the Oppenheimer-Volkoff limit for a sphere of radiation [29].", "In the same limit, $u_{f}$ vanishes as $e^{-\\xi _0}$ .", "The dependence of $u_{max}$ and of $u_f$ on $\\xi _0$ is plotted in Fig.", "(REF ).", "Figure: Plot of u max u_{max} against ξ 0 \\xi _0.", "Notice that lim ξ 0 →-∞ u max =u Q \\lim _{\\xi _0\\rightarrow -\\infty }u_{max} = u_Q.From Fig.", "REF , we can analyse how the three types of solution are distributed in the one-dimensional submanifolds (fibers) $V_{(u_R, \\xi _0, R)}$ of constant $(u_R, \\xi _0, R)$ .", "For given $\\xi _0$ , each fiber corresponds to a line $u_R =$ constant in the plots of REF .", "We characterize the fibers in terms of their intersection with the R-curve.", "There are three types of behavior, by which we characterize the fibers as being of type I, II, and III.", "Fibers of type I are defined by $u_R \\le u_f(\\xi _0)$ .", "The line $u_R =$ constant intersects the F-curve only once, at some point $v_1$ .", "For $v_R < v_1$ , the fiber contains solutions of type B, and for $v_R> v_1$ , it contains solutions of type S. Fibers of type II are defined by $u_f(\\xi _0) < u_R < u_{\\max }(\\xi _0)$ .", "The line $u_R =$ constant intersects the F-curve at least twice.", "Hence, these fibers involve two solutions of type F, at $v_R = v_1$ and $v_R = v_2$ , such that all solutions with $v \\notin [v_1, v_2]$ are of type S. In the interval $(v_1, v_2)$ the solutions are either of type B, or there exist alternating regions of type B and type S solutions.", "The latter is the case for large negative values of $\\xi _0$ where the R-curve develops a spiraling shape, hence, it intersects the line $u_R =$ constant more than twice.", "Fibers of type III are defined by $u_R > u_{\\max }(\\xi _0)$ .", "All points in these fibers correspond to solutions of type S." ], [ "The fundamental representation", "We proceed with a study of the thermodynamics of the shell system.", "We will be working with the fundamental representation (i.e., the entropy representation) of thermodynamic systems, in which the relevant thermodynamic potential is the entropy $S$ , expressed as a function of the total energy $M$ and the constraints in the extension of the system, namely the area of the bounding surfaces, as expressed in the variables $r_0$ and $R$ .", "In extensive systems, the choice of representation does not affect physical predictions.", "Extensivity together with the second law of thermodynamics imply that $S$ is a concave function of the extensive variables [35].", "The Legendre transform for concave functions fully preserves their information: it is an involution, i.e., the double Legendre transform of a functions $S$ returns the original function $S$ .", "Since the different representations are connected by a Legendre transform, they all contain the same information about the system.", "In a non-extensive system, the entropy $S$ needs not be a concave function.", "Typically, convex insertions appear, i.e., regions of the fundamental thermodynamic space where $S$ is convex.", "The double application of the Legendre transform does not return the original function $S$ but its concave hull [46]The Maxwell construction employed in the study of first-order phase transitions is a well known example of taking the concave hull of a non-concave entropy function.", "However, the Maxwell construction is physically meaningful only in extensive systems, where concavity of the entropy is a physical necessity..", "In this sense, the Legendre transform does not preserve all thermodynamically relevant information.", "As a result, the other thermodynamic representations are not equivalent to the fundamental one.", "At the microscopic level, this difference is manifested in the inequivalence between the microcanonical and the canonical ensembles.", "The equivalence of these ensembles in ordinary statistical mechanics follows from the requirement that the size of the system is much larger than the range of the force between the constituents.", "Then, the entropy obtained from the microcanonical distribution is concave [47].", "In presence of long-range forces, the microcanonical entropy is generically non-concave, leading to the inequivalence of ensembles.", "A convex insertion in some region of the fundamental space implies the presence of a first-order phase transition [48], [49].", "Hence, when studying an isolated gravitational system, it is necessary to use the fundamental representation, as any other representation will misrepresent the physicsIn Ref.", "[41], it was shown that the natural representation for a self-gravitating system is a free entropy representation, based upon the thermodynamic potential $\\Omega $ (the free entropy) that is obtained as a Legendre transform of the entropy with respect to the number $N$ of particles in a system.", "For radiation, the number of particles is not conserved, and $\\Omega $ coincides with the entropy function..", "Typically, the other representations (Gibbs, Helmholtz, enthalpy) are obtained by coupling the system to an external reservoir.", "We mentioned previously that idealized couplings to a reservoir make no sense in a gravitating system, because the reservoir can affect even the constitutive equations of the system.", "The shell of radiation considered here can be viewed as a gravitating reservoir for the system in the interior region.", "However, radiation is strongly affected and it strongly affects the interior region, so that a split between a system and a reservoir makes no sense.", "In gravitational systems, we must treat system and reservoir as a single isolated system, and for this reason, we must always work with the fundamental representation." ], [ "The maximum-entropy principle", "First, we consider the shell system with gravity switched off.", "In the entropy representation, the entropy $S$ is a function of $R, r_0$ and $M$ , $S(R, r_0, M) = \\frac{4}{3}b^{1/4} V^{1/4}(R, r_0) M^{3/4},$ where $V(R, r_0) = \\frac{4\\pi }{3}(R^3-r_0^3)$ is the volume of the shell.", "The thermodynamic state space $Q = \\lbrace (R, r_0, M)\\rbrace $ is three-dimensional.", "The thermodynamic state space remains the same when gravity is switched on.", "Since the volume is a variable in a gravitational system, the dependence of $S$ on $R$ and $r_0$ is non trivial.", "The problem now is that the set $Z$ of solutions to the TOV equation is four dimensional.", "Hence, the independent thermodynamic variables do not fix uniquely the solution.", "In compact stars ($r_0=0$ ), this problem is usually addressed by an additional assumption, namely, regularity at the center $m(0)=0$ .", "Regular solutions form a set of measure zero in the set of all solutions.", "Almost all solutions have $m(0)<0$ , and there are no solutions with $m(0) > 0$ .", "Solutions with $m(0) < 0$ have a curvature singularity at the center.", "However, this singularity causes no problems with causality and predictability: the spacetime has no inextensible geodesics, it is bounded-acceleration complete, and it is conformal to a globally hyperbolic spacetime with boundary [28].", "The problem with the regularity condition is that it does not cover the whole thermodynamic state space, as there are no solutions with $m(0) = 0$ , if $M> M_{OV}$ , where $M_{OV}$ is the Oppenheimer-Volkoff limit.", "In Ref.", "[32], it was also argued that there is no good reason to a priori exclude singular solutions from all considerations, since they would appear in the sum of geometries of a quantum theory of gravity, at least as virtual solutions.", "An important benefit of using a shell as our thermodynamic system is that it demonstrates unambiguously the inadequacy of the regularity condition for selecting equilibrium configurations.", "Even if one wants to a priori exclude solutions of type S—presumably because they involve a naked singularity—, there is no justification in excluding solutions of type B.", "Hence, the regularity condition cannot identify an equilibrium configuration for the radiation shell.", "We will select the equilibrium configurations by employing the maximum entropy principle.", "The MEP asserts that the values assumed by any parameters in absence of constraints are those that maximize the entropy over the manifold of constrained states [35].", "In the entropy representation, the total mass $M$ and the shell radii $R$ and $r_0$ are assumed to be constrained.", "As shown in Sec.", "2.3, the manifold $Z$ is foliated by surfaces of constant $(M, R, r_0)$ , or equivalently, of constant $(R, u_R, \\xi _0) $ .", "Each fiber $V_{(R, u_R, \\xi _0)}$ of the foliation is parameterized by $v_R$ .", "We can construct an entropy function $S_Z$ on $Z$ , $S_Z(R, u_R, \\xi _0, v_R)$ .", "The MEP asserts that the equilibrium state for $M, R, r_0$ is obtained by maximizing the entropy functional along the associated fiber, $S_{eq}(M, R, r_0) = \\mbox{max}_{v_R} S_Z(R, u_R, \\xi _0, v_R).", "$ If there is no global maximum of $S_Z$ on a fiber, the MEP fails to apply.", "This is the case in some gravitating systems, known as the gravothermal catastrophe [43].", "In what follows, we will show that if $S_Z(R, u_R, \\xi _0, v_R)$ involves only a contribution $ S_{rad} $ from radiation, the shell system is not thermodynamically consistent.", "In contrast, an appropriate gravity contribution $S_{gr}$ makes the system consistent.", "The gravitational entropy $S_{gr}$ is a Noether charge associated to the spacetime boundaries in the region $r < r_0$ .", "For B-type solutions, $S_{gr}$ is the Bekenstein-Hawking entropy.", "For the S-type solutions, the working expression for $S_{gr}$ is the same with the one identified in Ref.", "[32] From a thermodynamic point of view, the need of a term $S_{gr}$ from $r < r_0$ is obvious.", "If we view the radiation shell as a thermal reservoir in contact with the interior region, then it is necessary to include a contribution from the interior region, otherwise a black hole would be thermodynamically indistinguishable from flat space.", "Since the interior solution is vacuum, the only contribution to entropy can be of gravitational origin." ], [ "Radiation entropy", "The radiation entropy $S_{rad}$ of a solution to the TOV equation is given by $S_{rad} = 4 \\pi \\int _{r_0}^R \\frac{dr r^2 s}{\\sqrt{1- \\frac{2m(r)}{r}}}= \\frac{4}{3} (4\\pi b)^{1/4} \\int _{r_0}^R dr \\frac{r^{1/2} v^{3/4}}{\\sqrt{1-u}}.$ Solutions to the TOV equation satisfy [29] $ \\frac{r^{1/2}v^{3/4}}{\\sqrt{1-u}} = \\frac{d}{dr}\\bigg ( \\frac{v + \\frac{3}{2}u}{6 v^{1/4}\\sqrt{1-u}} r^{3/2} \\bigg ),$ from which we obtain $S_{rad}(R,\\xi _0,u_R,v_R) = \\frac{2}{9}(4\\pi b )^{1/4}\\left(\\frac{v_R+\\frac{3}{2}u_R}{v_R^{1/4}\\sqrt{1-u_R}} -\\frac{v_0+\\frac{3}{2}u_0}{v_0^{1/4}\\sqrt{1-u_0}}e^{3\\xi _0/2} \\right)R^{3/2}, $ where $u_0=u(\\xi _0)$ and $v_0 = v(\\xi _0)$ .", "The simple dependence of $S_{rad}$ on $R$ is due to the scaling symmetry.", "The key point here is that $S_{rad}$ has no global maximum.", "The reason is that it diverges at infinity.", "In particular, we found that For all $R, \\xi _0, u_R$ , $\\lim _{v_R \\rightarrow \\infty }S_{rad} = \\infty $ .", "For all $R$ , and for $\\xi _0 < \\log u_R$ , $\\lim _{v_R \\rightarrow \\infty } S_{rad} = \\infty $ .", "These limits can easily be seen in numerical calculation.", "Fig.", "REF shows the results of such a calculation, $S_{rad}/R^{3/2}$ is plotted as function of $v_R$ for fixed $u_R$ and $\\xi _0$ .", "We also obtained an analytic proof of the entropy limits above, which is detailed in the Appendix A.", "The proof is rather long, as it requires an explicit construction of the solution curves for $v_R >> 1$ and for $v_R << 1$ .", "Still, it is instructive as the same methods can be employed in the study of other solutions to the TOV equation.", "To conclude, the MEP cannot be applied in the shell system if $S_{rad}$ is the only contribution to the system's entropy.There is no global maximum of entropy in the fibers of constant $(R, u_R, \\xi _0)$ .", "Figure: S rad /R 3/2 S_{rad}/R^{3/2} as a function of v R v_R for fixed ξ 0 =-1\\xi _0 = -1 and for different values of u R u_R.", "(a) u R =0.01u_R=0.01 (b) u R =0.36u_R=0.36 (c) u R =0.367u_R=0.367 (d) u R =0.368u_R=0.368 (e) u R =0.37u_R=0.37 (f) u R =0.6u_R=0.6." ], [ "Entropy from spacetime boundaries", "Ever since Bekenstein's and Hawking's work on black hole thermodynamics, we know that entropy can be meaningfully assigned to gravitational degrees of freedom.", "Wald showed that black hole entropy can be expressed in terms of the Noether charge $Q(\\xi )$ of spacetime diffeomorphisms [42], as $S = \\frac{Q(\\xi )}{T_{\\infty }},$ The Noether charge $Q(\\xi )$ is defined in terms of the time-like Killing vector $\\xi = \\frac{\\partial }{\\partial t}$ , normalized so that $\\xi ^{\\mu }\\xi _{\\mu }=-1$ at infinity, and evaluated on the horizon, viewed as a boundary of the surfaces of constant $t$ : $Q(\\xi ) = \\frac{\\lambda }{4 \\pi } \\oint _{\\partial \\Sigma } d\\sigma _{\\mu \\nu } \\nabla ^{\\mu } \\xi ^{\\nu },$ where $\\lambda $ is an arbitrary multiplicative constant.", "For positive-mass Schwarzschild spacetime, $Q(\\xi ) = 2 \\lambda M$ , when evaluated at the horizon.", "Since $T_{\\infty } = 8 \\pi M$ , the Bekenstein-Hawking entropy $S_{BH} = 4\\pi M^2$ is obtained for $\\lambda = \\frac{1}{4}$ .", "We will use the Bekenstein-Hawking entropy for the entropy of the horizon that appears in solutions of type $B$ : $S_{grav} = 4 \\pi m_0^2$ .", "For type S solutions, the singularity at $r = 0$ defines a timelike boundary [28].", "For this boundary, $Q(\\xi ) = 2 \\lambda m_0\\kappa ,$ suggesting an entropy associated to singularity equal to $S_{sing} = \\frac{2 \\lambda m_0\\kappa }{ T_{\\infty }} = \\lambda (4\\pi b)^{1/4} \\frac{u_0}{v_0^{1/4}\\sqrt{1-u_0}}e^{3\\xi _0/2} R^{3/2}.$ The only way we have found to specify $\\lambda $ is through the requirement of thermodynamic consistency.", "In [32], it was shown that the only way to implement the MEP for a sphere of self-gravitating radiation is by assigning entropy to the singularity with $\\lambda = 2$ .", "The same holds in the shell system studied here.", "The only value of $\\lambda $ that provides a consistent implementation of the MEP is $\\lambda = 2$ .", "This is best seen in Fig.", "(REF ).There we define $S_{sing}$ for $\\lambda = 2$ , and consider candidate entropy functions $S_Z := S_{rad}+\\alpha S_{sigma}$ or different values of $\\alpha $ .", "Only the value $\\alpha = 1$ leads to a thermodynamically consistent function $S_Z$ : for $\\alpha < 1$ , $S_Z$ is not bounded from belowAn entropy function that is not bounded from below cannot be consistent with the statistical interpretation of entropy, where entropy is proportional to the logarithm of the number of microstates.", "If there is a lower bound, we can always add a constant to the entropy function and render it positive., and for $\\alpha > 1$ , $S_Z$ has no global maximum on fibers.", "Figure: Plot of the candidate entropy function (S rad +α·S sing )R -3/2 (S_{rad}+\\alpha \\cdot S_{sing})R^{-3/2} against v R v_R for (u R ,ξ 0 )=(0.44,-4)(u_R, \\xi _0)=(0.44,-4) and different values of α\\alpha (a)α=0\\alpha =0 , (b)α=0.9 \\alpha =0.9, (c) α=1\\alpha =1, (d) α=1.08\\alpha =1.08 and (e) α=2\\alpha =2.", "Any α<1\\alpha < 1, the entropy function is unbounded from below, and for α>1\\alpha > 1, it has not global maximum.", "Only the case α=1\\alpha = 1 corresponds to a physically admissible entropy function.In what follows, we will take $S_{sing}$ with $\\lambda = 2$ as the gravitational contribution to the total entropy in the S phase.", "$S_{sing}$ is negative, and vanishes for $m_0=0$ , thereby enhancing the stability of $R$ -type solutionsWe have tested this method to systems described by different equations of state, (e.g., fermionic matter) and we have found that the value $\\lambda =2$ is the only one that gives a consistent MEP.", "At the moment, we lack a fundamental explanation of this fact.. We found numerically that all F-type solutions at a fiber of constant $(u_R, R, \\xi _0)$ correspond to local maxima of $S_{rad}+S_{sing}$, with respect to $v_R$ .", "S-type maxima are only possible in fibers with no F-type solutions, i.e., for $u_R > u_c(\\xi _0)$ .", "This behavior is demonstrated in Fig.", "REF .", "This result is identical with the one of Ref.", "[32] for balls of radiation, and it also persists for other equations of state.", "Figure: Left: We plot S rad +S sing R 3/2 \\frac{S_{rad}+S_{sing}}{R^{3/2}} as a function of v R v_R for ξ 0 =-4\\xi _0=-4 and for different values of u R u_R: (a) u R =0.44u_R=0.44 (b) u R =0.47u_R=0.47, and (c) u R =0.50u_R=0.50.", "Case (a) has four F-type solutions, which correspond to the local maxima of the entropy function.", "Case (b) has two F-type solutions, again corresponding to local maxima of entropy.Case (c) has no F-type solution, the entropy maximum corresponds to a S-type solution.Right: S rad +S sing R 3/2 \\frac{S_{rad}+S_{sing}}{R^{3/2}} (blue) and u 0 u_0 (orange) are plotted as function of v R v_R for ξ 0 =-4\\xi _0=-4 and u R =0.44u_R=0.44.", "This plot demonstrates clearly the one-to-one correspondence between maxima of entropy and F-type solutions (u 0 =0u_0 = 0).We conclude that the entropy function $S_Z(R, \\xi _0, u_R, v_R)$ is $S_Z = \\left\\lbrace \\begin{array}{cc} S_{rad} + S_{BH}, & u_0 \\ge 0 \\\\ S_{rad} +S_{sing},& u_0 < 0 \\end{array} \\right..$ We note that both $S_{rad}(R, \\xi _0, u_R, v_R) $ and $S_{sing}(R, \\xi _0, u_R, v_R) $ can be expressed as $f(\\xi _0, u_R, v_R)R^{3/2}$ , i.e., they scale with $R^{3/2}$ .", "In contrast, $S_{BH} = 4\\pi m_0^2 = \\pi (u_0 e^{\\xi _0})^2 R^2$ scales with $R^2$ .", "The black hole contribution breaks the scaling invariance of the entropy that originates from the scale independence of the equation of state." ], [ "The four phases of the system", "Next, we implement the MEP on each fiber of constant $(R, u_R, \\xi _0)$ .", "There are three different scenarios, one for each type of fiber, see, Sec.", "2.3.", "Type I fibers: If $u_R < u_f(\\xi _0)$ , then the fiber contains a single solution of type F, say at $v_R = v_1$ , that is a local maximum.", "Solutions for $v_R > v_1$ are of type S, and they have all smaller entropy than the F-type solution.", "There is no local maximum of entropy for S-type solutions.", "For $v_R < v_1$ , solutions are of type B.", "The maximum value of entropy for B-type solutions occurs typically at very small $v_R$ , often numerically indistinguishable from $v_R = 0$ , i.e., a black hole with very little radiation in the shell.", "We will refer to this type of black hole, as a solution of type $B_I$ .", "Hence, the entropy along a type I fiber contains one local entropy maximum $S_F$ of type F, and one local maximum $S_{B_I}$ of type $B_I$ .", "The global maximum is the highest of the two local maxima.", "Typical plots of the entropy along a fiber are given in Fig.", "REF .", "The behavior is characteristic of a first-order phase transition.", "As the location of the fiber changes, so does the relative height of the two maxima.", "If $S_{B_I} > S_R$ , the equilibrium phase is of type $B_I$ ; if $S_F > S_{B_I}$ the equilibrium phase is of type F. The submanifold of $Q$ where $S_{B_I} = S_F$ is the coexistence curve for the B-F transition.", "Figure: Entropy maxima and phase transition in fibers of type II.Type II fibers: Type II fibers are characterized by several local maxima of type F and at least one local maximum of type B.", "The main difference from fiber I is that the B-type local maxima lie at an intermediate value between two F-type maxima—see, Fig.", "REF .", "In these B-type solutions a significant fraction of mass is in the form of radiation in the shell, and the black hole horizon is often very small.", "We will denote these solutions as $B_{II}$ t Figure: Entropy maxima and phase transition in fibers of type IIIIType III fibers: Type III fibers contain only S type solutions.", "Obviously, the entropy-maximizing solutions are of type S. To summarize, the implementation of the MEP demonstrates that the thermodynamic state space $Q$ splits into four components, which corresponds to phases of types $R$ , $B_I$ , $B_{II}$ and $S$ .", "The results of this analysis are summarized in Table REF .", "Table: The three types of fiber and the four phases." ], [ "Phase diagrams and coexistence curves", "In Fig.", "(REF ), we show how the submanifolds of constant $R$ are partitioned into the four phases, for different values of $R$ .", "We remark the following.", "1.", "The phases are separated by four coexistence curves.", "Two coexistence curves coincide with the submanifolds that separate fibers of different types.", "In particular, the surface $u_R = u_{max}(\\xi _0)$ separates between fibers of type I and fibers of type II, and the surface $u_R = u_f(\\xi _0)$ separates between fibers of type II and fibers of type III.", "Phase transitions across these surfaces are continuous, because the value of $v_R$ at maximum entropy is continuous.", "Hence, the temperature at infinity $T_{\\infty }$ is also continuous.", "The former surface describes the F-S phase transition, and the second surface describes the $B_I$ -$B_{II}$ phase transition.", "The F-$B_I$ and the F-$B_{II}$ transitions occur within the same fiber.", "As explained earlier, $v_R$ is discontinuous along the phase transition, so the transitions are of first order.", "There is one triple point (actually a curve in the thermodynamic state space $Q$ ) for the F-$B_I$ -$B_{II}$ phases.", "Both $u_R$ and $\\xi _0$ on the triple point decrease with $R$ .", "2.", "There is no coexistence curve between the S- and either of the $B_I$ and the $B_{II}$ phases.", "The S phase has only a coexistence curve with the R phase.", "This leads to a rather `bizarre' behavior, of a thin strip of F-phase being intermediate between the $B_{II}$ and the S phase.", "However, this is mathematically necessary, since one cannot go from positive values of $m_0$ to negative values of $m_0$ without crossing the surface $m_0 = 0$ .", "In absence of this strip, the $F$ phase is always at lower energy than the black hole phases, in accordance with the Page-Hawking phase transition [15] or the heuristic discussions of black hole formation in a box [4], [17].", "The strip of F-phase may be removed, if the Bekenstein-Hawking expression for entropy changes at small masses.", "A small-mass black hole emits more energy in Hawking radiation, and in presence of the box the black hole would have to coexist with its Hawking radiation.", "If the Hawking radiation contributes negatively to the energy, then it would be possible to have $m_o < 0$ even in presence of the horizon.", "It would then be possible to pass from the S phase to the B phase without crossing from the R phase.", "However, such a modification is not only conjecturalNote that the usual logarithmic corrections to the Bekenstein-Hawking entropy—see, for example Ref.", "[50]—apply in the regime of large masses, and they are not relevant to this problem.", "A treatment of a black hole in a box with backreaction from its Hawking radiation [41] leads to a correction linear with respect to mass, but again this works only for black holes of sufficiently large mass., but, at the current state of knowledge, it can only be implemented by inserting by hand a phenomenological parameter in the Bekenstein-Hawking formula.", "3.", "In linear scale for $u_R$ , the F-phase can only be distinguished in the thin strip interpolating between the $S$ and $B_{II}$ phases.", "We need a logarithmic scale for $u_R$ in order to see the intuitively obvious result that the F-phase dominates at small masses.", "The B-phases dominate at high $R$ and they are suppressed at small $R$ .", "This is obvious since the horizon contribution to the entropy grows faster than any other contribution with the scale of the system.", "However, the B-phases disappear as $\\xi _0 \\rightarrow -\\infty $ for all $R$ , reflecting the fact that there are no horizons in a ball of self-gravitating radiation.", "4.", "$B_I$ solutions have smaller ADM mass than $B_{II}$ solutions for the same $\\xi _0$ and $R$ .", "However, the area of the horizon (determined by $m_0$ ) in $B_I$ is typically larger, because a large part of the mass of $B_{II}$ consists of radiation in the shell.", "Two different black hole phases, one with little surrounding radiation and a large horizon , and one with a small horizon also appear in the heuristic analysis of a black hole coexisting with radiation in a box [4].", "The physical systems under consideration are different, but the main distinction is that in the model of [4], the large black hole phase is unstable, while here the phase $B_I$ is stable." ], [ "The entropy function in equilibrium", "After the implementation of the MEP, we evaluate the entropy function $S_{eq}(R, u_R, \\xi _0)$ on $Q$ , using Eq.", "(REF ).", "Characteristic plots of $S_{eq}$ as a function of $u_R$ , for fixed $\\xi _0$ and $R$ are given in Fig.", "REF .", "Figure: Plot of S eq /R 3/2 S_{eq}/R^{3/2} as a function of u R u_R for R=10 38 R=10^{38} and (a) ξ 0 =-1 \\xi _0=-1, (b) ξ 0 =-2\\xi _0=-2, (c) ξ 0 =-5\\xi _0=-5 .", "For ξ 0 =-1\\xi _0 = -1 auxiliary graphs zoom in specific ranges of u R u_R where the transitions F- B I B_I, B I -B II B_I - B_{II} and B II B_{II}-F-S take place, respectively.We see that the entropy function is, in general, a non-concave function of $u_R$ , and hence, of $M$ .", "By construction, $S$ is continuous across phase transitions.", "We note that the entropy is not an increasing function of $u_R$ , and that it is bounded from above for fixed $R$ and $\\xi _0$ .", "This maximum satisfies Bekenstein's bound [51], [52], $S< 2\\pi M R$ , or equivalently $\\frac{S}{u_R R^2} < \\pi ,$ for all values of $R$ that can be reasonably be considered as macroscopic.", "We emphasize that it is the consistent implementation of the MEP, through the inclusion of the term $S_{sing}$ , that makes this system satisfy Bekenstein's bound." ], [ "Temperature and heat capacity", "The maximization of the MEP also allows us to identify $v_R$ as a function on $Q$ for the equilibrium solutions.", "Then, the temperature at infinity is $T_{\\infty }(R, \\xi _0, u_R) = \\sqrt{1-u_R} \\left(\\frac{b v_R(R, \\xi _0, u_R)}{4 \\pi R^2}\\right)^{-1/4}$ The temperature $T_R$ cannot be identified with a partial derivative of the entropy function.", "This is only possible for solutions with a simply connected boundary [41], which includes regular solutions to the TOV equation [29].", "Plots of $T_R$ as a function of $u_R$ , for fixed $\\xi _0$ and $R$ are given in Fig.", "REF .", "The temperature $T_R$ is not an increasing function of the total energy $M$ for fixed $R$ and $r_0$ , and it has finite jumps at first-order phase transitions.", "Figure: Plot of T R T_R as a function of u R u_R for fixed R=10 38 R=10^{38} and different values of ξ 0 \\xi _0:(a) ξ 0 =-1 \\xi _0=-1, (b) ξ 0 =-2\\xi _0=-2, (c) ξ 0 =-5\\xi _0=-5 .", "The insets zoom near the phase transition points for curve (a) and demonstrate a finite discontinuity for T R T_R.The natural definition of heat capacity $C$ when the boundaries of the system are kept fixed (the analogue of heat capacity at constant volume) is [31] $C:=\\left(\\frac{\\partial M}{\\partial T_{\\infty }}\\right)_{R, r_0} = \\frac{R}{2} \\left(\\frac{\\partial T_{\\infty }}{\\partial u_R}\\right)_{R, \\xi _0}^{-1}.$ Clearly, the heat capacity is negative in any region of the thermodynamic state space $Q$ where $T_R$ decreases with $u_R$ —see, Fig.", "REF .", "At the local maxima of $T_R$ , $\\left(\\frac{\\partial T_{\\infty }}{\\partial u_R}\\right)_{R, \\xi _0} = 0$ , hence, $C$ diverges and changes sign.", "At the points of the first-order phase transition, $C$ exhibits discontinuities.", "Similar to the ball of self-gravitating radiation that was studied in Ref.", "[31], the present system is also characterized by alternating regions of positive and negative heat capacities throughout the thermodynamic state space." ], [ "Latent heat", "Since the transitions F-$B_I$ and $B_{II}$ -F are first-order, they involve latent heat.", "In extensive systems, the latent heat $L$ is identified as the difference $\\Delta H$ of the enthalpy between the two phases, while the Gibbs free energy is constant.", "Hence, $L = \\Delta H = \\Delta (G +TS) = \\Delta (TS)$ .", "In the fundamental representation, $S$ is constant along the transition, hence, $L = S \\Delta T$ .", "We employ the analogue of this formula in our system, i.e., we consider the quantity $L = S \\Delta T_{\\infty } $ as a candidate for the latent heat in the first order transitions between flat space and the black hole phases.", "This choice is based primarily on the basis of analogy with extensive thermodynamics.", "However, it appears plausible that this is the amount of heat (REF ) we must give the system on the coexistence curve—together with work $\\Delta W = - L$ , so that $\\Delta U = 0$ —for the phase change to occur.", "Figure: The reduced latent heat L/ML/M as a function for ξ 0 \\xi _0 and for constant RR: (a) F→B I F\\rightarrow B_{I} transition, (b)B II →FB_{II} \\rightarrow F transition.In Fig.", "(REF ), we plot the reduced latent heat $L/M$ as a function of $\\xi _0$ , for fixed $R$ .", "We note the following.", "1.", "The latent heat is negative for both the $F\\rightarrow B_{I}$ and the $F\\rightarrow B_{II}$ transitions.", "This means that the black hole phases are always low temperature phases compared to flat space, one needs to `boil' a black hole in order to remove the horizon.", "This result may appear surprising.", "Except for the the narrow strip of F phase before the S phase, the F phase has lower internal energy than the B phases.", "However, the temperature of the F phase is higher.", "This is due to the fact that the temperature $T_R$ is not a monotonous function of the energy.", "To the best of our knowledge, the fact that black holes have lower temperature than self-gravitating systems along the coexistence curve has not been noted.", "It is easy to see that this must be the case.", "Consider, for example, a simplified analysis of black hole phase transitions in the vein of [4].", "A box of radius $R$ contains either a black hole of mass $M$ or (non-gravitating) radiation with the same mass $M$ .", "In the former case, the entropy is given by the Bekenstein-Hawking expression $S_1 = 4 \\pi M^2$ , in the latter by $S_2 = \\frac{4 \\pi }{3} b^{1/4} M^{3/4} R^{3/4}$ , where $b$ is the constant in Eq.", "(REF ).", "The associated temperatures are $T_1 = (8\\pi M)^{-1}$ and $T_2 = \\frac{1}{\\pi } b^{-1/4} M^{1/4} R^{-3/4}$ .", "A black hole is entropically favored for $M > M_c$ , where the critical mass $M_c = \\left(\\frac{b}{81}\\right)^{1/5} R^{3/5}$ .", "The temperature of the black hole phase at $M_c$ is $T_1(M_c) = \\frac{1}{8 \\pi } \\left(\\frac{81}{b}\\right)^{1/5} R^{-3/5}$ / The temperature of the radiation phase is $T_2(M_c) = \\frac{1}{\\pi (81)^{\\frac{1}{20}}} b^{-1/5} R^{-3/5}$ .", "We compute $\\Delta T = T_1(M_c) - T_2(M_c)$ , $\\Delta T = -\\frac{5}{24 \\pi R^{3/5}} \\left(\\frac{81}{b}\\right)^{1/5} < 0,$ i.e., we verify that the black hole has lower temperature than radiation along the coexistence curve.", "It is important to understand the role of the negative latent heat in a non-equilibrium setting of gravitational collapse.", "A plausible conjecture is that it refers to the amount of radiation (electromagnetic or gravitational) that must be emitted before the system settles as a black hole.", "To test this conjecture, we must undertake an analysis of gravitational collapse in the context of non-equilibrium thermodynamics, while keeping track of all heat currents throughout the collapse.", "2.", "The reduced latent heat is almost constant for a large range of values of $\\xi _0$ .", "It vanishes as $\\xi _0 \\rightarrow - \\infty $ , well outside the range of the plots in Fig.", "(REF ), because the black hole phases disappear at this limit.", "3.", "The latent heat is much smaller in transitions involving the $B_{II}$ phase than in transitions with the $B_I$ phase.", "This is because the horizon in the $B_{II}$ phase is much smaller.", "In contrast, the latent heat with respect to the $B_I$ phase is a substantial fraction of the total mass." ], [ "Conclusions", "We analysed the thermodynamics of a shell of gravitating radiation surrounding a solution to vacuum Einstein's equation, which may either correspond to flat space, a black hole or a repulsive singularity.", "The shell can be interpreted as a gravitating heat reservoir.", "However, the presence of long range forces necessitates an analysis of the total system that consists of the shell and its interior.", "We showed that the only way to obtain a consistent thermodynamic description of the system is by assigning a specific expression for entropy to the naked singularity, thus, confirming the proposal of Ref.", "[32] in a more complex set-up.", "The result is a concrete model for describing phase transitions between black holes and self-gravitating systems that is fully compatible with the rules of thermodynamics.", "Such models are important for black hole thermodynamics and quantum gravity, but also for expounding the mathematical and physical structure of non-extensive gravitating systems.", "The methods of this paper can be straightforwardly generalized, for example, to different equations of state, rotating systems and other shell geometries.", "The self-gravitating shell of radiation can also be used as a generic thermal reservoir in studies of system-reservoir thermodynamics in self-gravitating systems.", "Furthermore, our results strongly suggest the importance of a thermodynamic analysis to gravitational collapse.", "The non-equilibrium evolution of a self-gravitating shell is perhaps the simplest model for analyzing the interplay between horizon formation and thermodynamics in a gravitating system.", "We expect that the solutions studied in this paper will correspond to asymptotic states of a non-equilibrium analysis." ], [ "Acknowledgements", "D.K.", "acknowledges financial support from the “Andreas Mentzelopoulos Foundation”." ], [ "The asymptotic behavior of radiation entropy", "In this section, we prove the asymptotic behavior of the radiation entropy $S_{rad}(u_R, v_R, R, \\xi _0)$ of (REF ), as $v_R\\rightarrow \\infty $ and as $v_R \\rightarrow 0$ , the other parameters being fixed.", "In particular, we show that (i) $S_{rad}\\rightarrow \\infty $ as $v_R \\rightarrow \\infty $ , for all $u_R, R, \\xi _0$ , (ii) $S_{rad}\\rightarrow \\infty $ as $v_R \\rightarrow 0$ , for $\\xi _0 < \\log u_R$ , and (iii)$S_{rad}\\rightarrow 0$ as $v_R \\rightarrow 0$ , for $\\xi _0 >\\log u_R$ ." ], [ "Main formulas", "The function $S_{rad}$ can be expressed as $S_{rad} = \\frac{2}{9}(4\\pi b )^{1/4} [\\sigma (0) - \\sigma (\\xi _0)]R^{3/2},$ where $\\sigma (\\xi ) := \\frac{u(\\xi )+\\frac{3}{2}v(\\xi )}{v(\\xi )^{1/4}\\sqrt{1-u(\\xi )}}e^{3\\xi /2}.$ The functions $u(\\xi ), v(\\xi )$ are solutions to the differential equations $u^{\\prime } = 2v-u, \\qquad \\qquad v^{\\prime } = \\frac{2v(1-2u-\\frac{2}{3}v)}{1-u}, $ for $\\xi < 0$ and with boundary conditions $u(0) = u_R$ and $v(0) = v_R$ .", "Using Eqs.", "(REF ), we find $\\frac{d\\sigma }{d\\xi } = \\frac{6 v^{3/4}e^{3\\xi /2}}{\\sqrt{1-u}} \\ge 0.", "$ Hence, $\\sigma (0) - \\sigma (\\xi ) \\ge 0 $ , for all $\\xi \\le 0$ .", "It is convenient to introduce the variable $\\epsilon := 1 - u$ , bringing Eqs.", "(REF ) into the form $\\epsilon ^{\\prime } = 1-\\epsilon - 2 v, \\qquad \\qquad v^{\\prime } =\\frac{2v(2\\epsilon -1-\\frac{2}{3}v)}{\\epsilon }.", "$ An important special case is the regime where either $v >> 1$ or $\\epsilon >> 1$ .", "In this regime, Eqs.", "(REF ) become $\\epsilon ^{\\prime } = -\\epsilon - 2 v, \\qquad \\qquad v^{\\prime } =\\frac{2v(2\\epsilon -\\frac{2}{3}v)}{\\epsilon }, $ and they admit exact solution $\\frac{\\epsilon }{v} = \\alpha _1 e^{-5\\xi } - \\frac{2}{15}, \\qquad \\epsilon (\\xi ) = \\alpha _2 e^{-\\xi }\\left(\\frac{15}{2} \\alpha _1 - e^{5\\xi }\\right)^3, $ where $\\alpha _1$ and $\\alpha _2$ are integration constants.", "For these solutions, Eq.", "(REF ) becomes $\\frac{d\\sigma }{d\\xi } = 6\\left(\\frac{15}{2}\\right)^{3/4} \\alpha _2^{1/4} e^{5 \\xi }.", "$" ], [ "Case $v_R \\rightarrow \\infty $", "Consider the case of $v_R >> 1$ .", "Eq.", "(REF ) applies in the vicinity of $\\xi = 0$ .", "As shown in [28], in all singular solutions, $v$ increases with decreasing $\\xi $ up to a point $\\xi _2$ , where $v^{\\prime }(\\xi _2) = 0$ , and then it decreases to zero as $\\xi \\rightarrow -\\infty $ .", "However, at $\\xi = \\xi _2$ , $v \\simeq 3 \\epsilon $ , hence, $\\epsilon (\\xi _2) >> 1$ .", "For $\\xi < \\xi _2$ , $\\epsilon $ keeps increasing with decreasing $\\xi $ .", "Hence, Eq.", "(REF ) applies to all $\\xi \\le 0$ , and we can evaluate the integration constants in Eq.", "(REF ) by the values of $\\epsilon $ and $v$ at $\\xi = 0$ .", "Then, we obtain $\\alpha _1 = \\frac{2}{15} + \\frac{\\epsilon _R}{v_R}, \\qquad \\alpha _2 = \\bigg (\\frac{2}{15}\\bigg )^3\\frac{v_R^3}{\\epsilon _R^2}.$ Hence, Eq.", "(REF ) gives $\\frac{d\\sigma }{d\\xi } = \\frac{v_R^{3/4}}{\\epsilon _R^{1/2}} e^{5\\xi }.$ Thus, we obtain $\\sigma (0) - \\sigma (\\xi ) = \\frac{6}{5}\\frac{v_R^{3/4}}{\\sqrt{1-u_R}}(1-e^{5\\xi _0}),$ to conclude that $\\lim _{v_R \\rightarrow \\infty }[\\sigma (0) - \\sigma (\\xi )] = \\infty $ ." ], [ "Case $v_R \\rightarrow 0$ and {{formula:f068efb9-34b4-4ce7-a6f3-267c5bd6d1ed}}", "Consider a solution with $v_R << u_R < 1$ .", "Integrating from $\\xi = 0$ , $u$ increases with decreasing $\\xi $ ; $v$ initially decreases with decreasing $\\xi $ and then increases again.", "The condition $v<< u$ remains valid for an interval $(\\xi _r, 0)$ , in which $u^{\\prime } = - u, \\qquad v^{\\prime } = \\frac{2v (1-2u)}{1-u}.", "$ These equations admit solutions $u(\\xi ) = u_R e^{-\\xi }, \\qquad v = \\frac{v_R u_R^2 (1 - u_R)^2}{u^2 (1-u)^2} $ For sufficiently small $v_R$ , Eq.", "(REF ) applies up to a point where $\\epsilon = 1 - u <<1$ .", "For $\\epsilon << 1$ (but not necessarily $v<< 1$ ), Eqs.", "(REF ) become $\\epsilon ^{\\prime } = 1 - 2v, \\qquad v^{\\prime } = -\\frac{2v (1 +\\frac{2}{3}v)}{\\epsilon }, $ Hence, we obtain $\\frac{d \\epsilon }{dv} = -\\frac{\\epsilon (1-2v)}{2v (1 +\\frac{2}{3}v)} $ Eq.", "(REF ) has solutions of the form $\\frac{v}{(v+\\frac{3}{2})^4} = \\frac{a}{\\epsilon ^2}, $ for some constant $a$ .", "The minimum value $\\epsilon _*$ of $\\epsilon $ occurs at $\\xi = \\xi _*$ , such that $\\epsilon ^{\\prime }(\\xi _*) = 0 $ , or equivalently $v(\\xi _*) = \\frac{1}{2}$ .", "By Eq.", "(REF ), $a = \\epsilon _*^2/32$ .", "Then, Eq.", "(REF ) becomes $\\frac{32v}{(v+\\frac{3}{2})^4} = \\left(\\frac{\\epsilon _*}{\\epsilon }\\right)^2.", "$ Eqs.", "(REF ) and (REF ) must approximately coincide in some open set of $\\xi $ where $v<< 1$ .", "This is only possible if $\\epsilon _* = \\frac{16}{9} u_R (1 - u_R) \\sqrt{2 v_R}.", "$ Eqs.", "(REF ) and (REF ) imply that $(v^{-1/2} + \\frac{3}{2} v^{-3/2}) v^{\\prime } = - \\frac{16 \\sqrt{2}}{3 \\epsilon _*}.$ Integrating from some reference point $\\xi = \\xi _r$ with $v(\\xi _r) = v_r$ , we find $2(\\sqrt{v(\\xi )} - \\sqrt{v_r}) - 3 \\left(\\frac{1}{\\sqrt{v(\\xi )}} - \\frac{1}{\\sqrt{v_r}}\\right) = - \\frac{16 \\sqrt{2}}{3 \\epsilon _*} (\\xi - \\xi _r) $ Using Eq.", "(REF ) for a choice of the reference point $\\xi = \\xi _r$ lying in the domain of validity of Eq.", "(REF ), $\\xi = \\log u_R + \\frac{3 \\epsilon _*}{16 \\sqrt{2}} \\left( \\frac{3}{\\sqrt{v(\\xi )}} - 2 \\sqrt{v(\\xi )} \\right).", "$ Setting $\\xi = \\xi _*$ in Eq.", "(REF ), we obtain $\\xi _* = \\log u_R + \\frac{3 \\epsilon _*}{8}.", "$ The key point in this analysis is that $\\epsilon _*\\rightarrow 0$ for $v_R \\rightarrow 0$ .", "By Eq.", "(REF ), the smallest value for $\\xi _0 < \\xi _*$ is $\\log u_R$ .", "For any $\\xi _0 < \\log u_R$ , we can choose sufficiently small $v_R$ , so that $\\frac{\\xi _0 - \\log u_R}{\\epsilon _*} >> 1$ , which by Eq.", "(REF ) implies that $v(\\xi _0) << 1$ .", "Hence, for sufficiently small $v_R$ , $\\xi _0$ lies always in the domain of validity of Eq.", "(REF ).", "Then, Eq.", "(REF ) becomes $\\frac{d\\sigma }{d\\xi } = \\frac{6 v_R^{3/4}u_R^{3/2}(1-u_R)^{3/2}}{(1-u_Re^{-\\xi })^2}e^{3\\xi }.$ We integrate to obtain $\\sigma (0) - \\sigma (\\xi _0) = 6 v_R^{3/4}u_R^{3/2}(1-u_R)^{3/2} F(\\xi _0),$ where $F(\\xi _0) = \\int _0^{\\xi _0} \\frac{d\\xi e^{3\\xi }}{(1 - u_Re^{-\\xi })^2}$ is a smooth function of $\\xi _0 \\in (\\log u_R, 0)$ .", "We conclude that $\\lim _{v_R \\rightarrow 0}[\\sigma (0) - \\sigma (\\xi )] = 0$ ." ], [ "Case $v_R \\rightarrow 0$ and {{formula:68399bef-5ab9-4e6e-a472-f15c6ea38efb}}", "As shown in Ref.", "[28], the solution $\\xi < \\xi _*$ is characterized by a point $\\xi _1$ , such that $u(\\xi _1) = 0$ , or $\\epsilon (\\xi _1) = 1$ .", "By continuity, there exists an interval $(\\bar{\\xi }, \\xi _*)$ where $ \\xi _* >\\bar{\\xi } > \\xi _1 \\epsilon (\\xi ) $ remains smaller than any arbitrary value $1 > \\bar{\\epsilon } > \\epsilon _*$ .", "Since $\\epsilon _* \\sim \\sqrt{v_R}$ , we can choose $\\bar{\\epsilon }$ to be proportional to $v_R^a$ , for $a < \\frac{1}{2}$ .", "It is convenient to choose $\\bar{\\epsilon } = \\frac{u_R(1-u_R)}{9\\sqrt{2}} v_R^{1/4},$ so that $\\bar{v} = v(\\bar{\\xi }) = v_R^{-1/6}$ .", "Eq.", "(REF ) for $\\xi _r = \\xi _*$ becomes $( 2 \\sqrt{v(\\xi )} - \\frac{3}{\\sqrt{v(\\xi )}} - 2 \\sqrt{2} ) = \\frac{16}{\\sqrt{2}\\epsilon _*}(\\log u_R - \\xi ).", "$ For fixed $\\xi < \\log u_R$ , we can choose $v_R$ so that the right hand side of Eq.", "(REF ) becomes very large.", "Since $v > \\frac{1}{2}$ for $\\xi > \\xi _*$ , in this limit $v>> 1$ , hence, $v(\\xi ) = \\frac{32}{\\epsilon _*^2} (\\log u_R - \\xi )^2.$ In this regime, Eq.", "(REF ) implies that $\\epsilon = \\epsilon _* \\frac{v^{3/2}}{4\\sqrt{2}}$ .", "Then, Eq.", "(REF ) becomes $\\frac{d\\sigma }{d\\xi } = 2^{1/4} \\frac{12 }{\\sqrt{\\epsilon _*}}.$ Hence, integrating from $\\log u_R$ to $\\xi _0 \\in (\\log u_R, \\bar{\\xi })$ , we find $\\sigma (\\log u_R) - \\sigma (\\xi _0) = 2^{1/4} \\frac{12 }{\\sqrt{\\epsilon _*}} (\\log u_R - \\xi _0).$ We see that $\\lim _{v_R \\rightarrow 0}\\sigma (\\log u_R) - \\sigma (\\xi _0) = \\infty $ .", "Since $\\sigma (0) - \\sigma (\\xi _0) > \\sigma (\\log u_R) - \\sigma (\\xi _0) $ , we conclude that for $\\xi _0 \\in (\\bar{\\xi }, \\log u_R)$ , $\\lim _{v_R \\rightarrow 0}\\sigma (0) - \\sigma (\\xi _0) = \\infty $ .", "Finally, we examine the case of $\\xi < \\bar{\\xi }$ .", "Since $\\bar{v}\\rightarrow \\infty $ for $v_R \\rightarrow 0$ , the solution from $\\bar{\\xi }$ to the center is of the type that has been studied in Sec.", "A.2.", "Hence, for any $\\xi _0 < \\bar{\\xi }$ , $\\lim _{v_R \\rightarrow 0}[\\sigma (\\bar{\\xi }) - \\sigma (\\xi _0)] = \\infty $ .", "Since $\\sigma (0) - \\sigma (\\xi _0) > \\sigma (\\log u_R) - \\sigma (\\xi _0) $ , we conclude that for $\\xi _0 \\in (-\\infty , \\bar{\\xi }]$ , $\\lim _{v_R \\rightarrow 0}[\\sigma (0) - \\sigma (\\xi _0)] = \\infty $ ." ] ]
2105.11817
[ [ "Complete Requirements-based Testing with Finite State Machines" ], [ "Abstract In this paper, new contributions to requirements-based testing with deterministic finite state machines are presented.", "Elementary requirements are specified as triples consisting of a state in the reference model, an input, and the expected reaction of the system under test defined by a set of admissible outputs, allowing for different implementation variants.", "Composite requirements are specified as collections of elementary ones.", "Two requirements-driven test generation strategies are introduced, and their fault coverage guarantees are proven.", "The first is exhaustive in the sense that it produces test suites guaranteeing requirements satisfaction if the test suite is passed.", "If the test suite execution fails for a given implementation, however, this does not imply that the requirement has been violated.", "Instead, the failure may indicate an arbitrary violation of I/O-equivalence, which could be unrelated to the requirement under test.", "The second strategy is complete in the sense that it produces test suites guaranteeing requirements satisfaction if and only if the suite is passed.", "Complexity considerations indicate that for practical application, the first strategy should be preferred to the second.", "Typical application scenarios for this approach are safety-critical systems, where safety requirements should be tested with maximal thoroughness, while user requirements might be checked with lesser effort, using conventional testing heuristics." ], [ "Background: Requirements-driven Model-based Testing", "In model-based black-box testing of embedded control systems, test suites can be generated from models with two alternative objectives in mind.", "(1) The test suite could aim at uncovering conformance violations; typical conformance relations are interface language equivalence or refinement.", "It is assumed that the model captures the complete expected behaviour of the system under test (SUT).", "(2) Alternatively, test suites can be constructed to uncover violations of specific requirements in an implementation behaviour.", "To this end, the requirement has to be specified in addition to the model, using, for example, temporal logic or test scenario specifications.", "Another option is to use modelling languages like SysML [21] allowing to relate requirements to behavioural or structural model elements, making it often unnecessary to add temporal logic or scenario specifications.", "Some requirements-driven test approaches advocate test generation from temporal logic formulas alone [39], so that a behavioural model becomes unnecessary.", "This, however, has the disadvantage that formulas can only refer to interface variables, making the formulas quite complex to specify.", "With a behavioural model at hand, formulas can also refer to internal state variables, facilitating the expression of requirements [23].", "Objective (1) typically applies to protocol testing or any other domain where the model is sufficiently small to induce conformance test suites of acceptable size [20].", "Objective (2) applies to domains where models are too large and too complex to perform conformance tests with acceptable size and sufficient test strength to uncover conformance violations.", "Moreover, development standards for safety-critical control systems typically require that testing should be requirements-driven, so that a conformance testing approach that does not relate test cases to requirements would be inadmissible [30], [6].", "Finally, requirements-driven testing is the preferred approach to systematic regression testing: after changes to the implementation, the requirements affected by these changes are identified, and it is tested whether the modified SUT conforms to these requirements.", "This allows to avoid re-testing the whole test suite, which may be too time consuming, especially in HW/SW integration testing or system testing, where test executions need to be performed in physical time and cannot be sped up by using faster processors or performing them in parallel on many CPU cores, as is possible for software tests." ], [ "Main Contributions", "In this article, two novel model-based requirements-driven test strategies are introduced, and their fault coverage guarantees are proven.", "Models are represented as deterministic finite state machines (DFSMs).", "We are aware of the fact that FSMs are not well-suited for modelling control systems with complex and large data types for interfaces and internal state variables.", "We have shown, however, that such systems may be abstracted to FSMs after having calculated state and input equivalence classes on the complex model which is interpreted, for example, as a Kripke Structure.", "Test suites with guaranteed fault coverage calculated for the FSM abstraction give rise to equivalence class tests with likewise guaranteed fault coverage for the complex system [13], [16].", "These considerations motivate the study of testing theories for finite state machines.", "We restrict ourselves to deterministic systems, since determinism is always required in the context of safety-critical control systems.", "Elementary requirements are represented as triples $R(q,x,Z)$ , where $q$ is a state in the reference model, $x$ is an input to the DFSM, and $Z$ is a subset of the machine's output alphabet, representing the admissible outputs that may be produced by the SUT after having processed any input sequence leading the reference model into state $q$ .", "A composite requirement is specified as a combination of elementary ones.", "It is easy to see that implementations whose true behaviour can be represented by a DFSM which is language-equivalent to that of the reference DFSM automatically fulfil all specified requirements.", "The two strategies have the following characteristics.", "The first strategy is exhaustive in the sense that it provides test suites that imply requirements satisfaction when passed.", "When failed, the SUT is guaranteed to violate language equivalence, but it does not necessarily violate the composite requirement that is being tested.", "This approach is called exhaustive testing of (composite) requirements.", "The second strategy is complete in the sense that it provides test suites that are passed by the SUT if and only if it conforms to the specified requirement.", "When a test suite generated according to this strategy is failed, it is guaranteed that the SUT violates the requirement.", "On the other hand, these test suites do not uncover any violations of language conformance, if these violations are unrelated to the requirement.", "We call this approach complete testing of (composite) requirements.", "Observe that the terms `exhaustive' and `complete' have been adopted from conformance testing, where a test suite is called `sound' if it is passed by all conforming implementations, `exhaustive' if non-conforming implementations will always fail at least one test case, and `complete' if the suite is sound and exhaustive [35].", "The test suite sizes depend on the requirement and on the difference between the maximal number $m$ of states assumed for the SUT and the (known) number of $n$ of states in the minimised reference model.", "For the first strategy, a detailed evaluation shows that test effort reductions between 20% and 60% can be achieved in comparison to test suites generated to prove language equivalence.", "We explain why the second strategy is of significant theoretical value, but usually results in larger test suites and is of lesser practical value than the first.", "The work presented here generalises previous publications [14], [15], where requirements referred to outputs of different criticality only, but could not be linked to states and inputs of the reference model." ], [ "FSM Library", "The test suites generated for the evaluation of the test strategies presented in this article have been calculated using the fsmlib-cpp library, an open source project programmed in C++.", "The library contains fundamental algorithms for processing Mealy Machine FSMs and a variety of model-based test generation algorithms.", "Download, contents, and installation of the library is explained in the lecture notes [24] which are also publicly available." ], [ "Overview", "In Section , basic definitions about finite state machines that are needed for the elaboration of results are presented.", "The notion of elementary and composite requirements is introduced in Section .", "The soundness of the concept is justified by proving that language equivalence can be alternatively expressed by composite requirements.", "In Section , we present two DFSM abstractions that are induced by our requirements notion and needed for the construction of complete test suites.", "In Section , a pass-relation for requirements-driven tests and the first main theorem about complete test suites implying requirements satisfaction are presented.", "The application of this main theorem to practical requirements-driven testing is illustrated and evaluated by means of several experiments in Section .", "In Section , the second main theorem yielding complete test suites to be passed by the SUT if and only if it fulfils the requirements is introduced and proven.", "The two test strategies are compared with respect to their complexity and to their practical value in Section .", "Section  discusses related work, and Section  presents a conclusion.", "In Appendix , it is explained how the test suites described in examples throughout the article can be re-generated using the fsmlib-cpp." ], [ "Basic Definitions", "A finite state machine (FSM) is a tuple $M = (Q,\\underline{q}, \\Sigma _I, \\Sigma _O, h)$ with finite state space $Q$ , initial state $\\underline{q}\\in Q$ , finite input and output alphabets $\\Sigma _I, \\Sigma _O$ , and transition relation $h\\subseteq Q\\times \\Sigma _I \\times \\Sigma _O \\times Q$ .", "The specified inputs of a state $q\\in Q$ denote the set $in(q)$ of input elements defined by $in(q) = \\lbrace x\\in \\Sigma _I~|~\\exists y\\in \\Sigma _O,q^{\\prime }\\in Q: (q,x,y,q^{\\prime })\\in h \\rbrace $ .", "The language $L(q)$ of a state $q\\in Q$ is the set of all finite sequences $\\tau = (x_1,y_1)\\dots (x_k,y_k)\\in (\\Sigma _I\\times \\Sigma _O)^*$ for which states $q_1,\\dots ,q_k$ satisfying $(q,x_1,y_1,q_1)\\in h,\\dots ,(q_{k-1},x_k,y_k,q_k)\\in h$ can be found.", "The empty sequence $\\varepsilon $ is also an element of $L(q)$ .", "Sequence $\\tau $ is called an I/O-trace of $q$ , and its length is denoted by $|\\tau | = k$ .", "Sequence $\\overline{x} = x_1\\dots x_k$ is called an input trace, and $\\overline{y} = y_1\\dots y_k$ an output trace.", "The language $L(M)$ of the FSM $M$ is the language $L(\\underline{q})$ of its initial state.", "We also use the alternative notations $\\overline{x} / \\overline{y}$ for $\\tau $ and $x_i/y_i$ for $(x_i,y_i)$ in $\\tau $ .", "Trace segments of some trace $\\tau $ from $p^{th}$ element to $q^{th}$ element are denoted by $\\tau ^{[p..q]}$ .", "If $p < q$ as, for example, in $\\tau ^{[1..0]}$ , this denotes the empty trace $\\varepsilon $ .", "If $L(M) = L(M^{\\prime })$ , the two machines are called I/O-equivalent or language equivalent; if $L(M^{\\prime })\\subseteq L(M)$ , the former is a reduction of the latter.", "An FSM is deterministic, and the machine is called a DFSM, if and only if a pair $(q,x)\\in Q\\times \\Sigma _I$ is associated with at most one transition element $(q,x,y,q^{\\prime })\\in h$ .", "The FSM is called completely specified if and only if every pair $(q,x)\\in Q\\times \\Sigma _I$ is associated with at least one transition element $(q,x,y,q^{\\prime })\\in h$ .", "For completely specified DFSMs, every pair $(q,x)\\in Q\\times \\Sigma _I$ is associated with exactly one transition element $(q,x,y,q^{\\prime })\\in h$ .", "In this case, the transition relation $h$ can be represented by a transition function $\\delta : Q\\times \\Sigma _I \\rightarrow Q$ and an output function $\\omega : Q\\times \\Sigma _I\\rightarrow \\Sigma _O$ , such that $(q,x,y,q^{\\prime })\\in h$ if and only if $(q,x)\\in Q\\times \\Sigma _I$ and $(q,x,y,q^{\\prime }) = (q,x,\\omega (q,x),\\delta (q,x))$ .", "Transition functions and output functions can be extended to sequences of inputs in a natural way by setting $\\delta & : & Q\\times \\Sigma _I^*\\rightarrow Q\\\\\\delta (q,\\varepsilon ) & = & q\\\\\\delta (q,x.\\overline{x}) & = & \\delta (\\delta (q,x),\\overline{x})$ $\\omega & : & Q\\times \\Sigma _I^*\\rightarrow \\Sigma _O^*\\\\\\omega (q,\\varepsilon ) & = & \\varepsilon \\\\\\omega (q,x.\\overline{x}) & = & \\omega (q,x).\\omega (\\delta (q,x),\\overline{x})$ An FSM is observable if and only if state, input and output uniquely determine the target state, that is, if the transition relation fulfils $\\forall (q,x,y,q^{\\prime }), (q,x,y,q^{\\prime \\prime }) \\in h: q^{\\prime } = q^{\\prime \\prime }.$ DFSMs are automatically observable, and each nondeterministic non-observable FSM can be transformed to a language-equivalent observable one [18].", "The prime machine of a DFSM $M$ is a DFSM $M^{\\prime }$ with minimal number of states, such that $M$ and $M^{\\prime }$ have the same language.", "$M^{\\prime }$ is uniquely determined up to isomorphism.", "The same holds for observable nondeterministic FSMs [34].", "For a DFSM $M$ and a state $q\\in Q$ , an input trace $\\overline{x}$ of an I/O-trace $\\tau \\in L(q), \\tau =\\overline{x}/\\overline{y}$ , leads to a uniquely determined target state $q_k = \\delta (q,\\overline{x})$ , since there is exactly one state sequence $q_1,\\dots ,q_k$ and output trace $\\overline{y}$ satisfying (REF ).", "We use notation $q\\text{-\\underline{after}-}\\overline{x} = q_k$ .", "Note that this is a partially defined operator, if the DFSM is not completely specified.", "We extend its domain with the empty input trace $\\varepsilon $ by setting $q\\text{-\\underline{after}-}\\varepsilon = q$ .", "We extend the $\\text{-\\underline{after}-}$ operator to set-valued right-hand side operands by setting $q\\text{-\\underline{after}-}X = \\lbrace q\\text{-\\underline{after}-}\\tau ~|~\\tau \\in X\\rbrace $ for $X\\subseteq \\Sigma ^*$ .", "A state cover of a DFSM $M$ is a set $V \\subseteq \\Sigma _I^*$ of input traces such that for each $q\\in Q$ , there exists a $v\\in V$ satisfying $\\underline{q}\\text{-\\underline{after}-}v = q$ .", "In this article, it is always required that $\\varepsilon \\in V$ , because the empty trace reaches the initial state, i.e.", "$\\underline{q}\\text{-\\underline{after}-}\\varepsilon = \\underline{q}$ .", "The following well-known lemma shows how a state cover can be constructed in a very basic (though not optimal) way.", "It will be used in the proof of the main theorem.", "Let $M=(Q,\\underline{q},\\Sigma _I,\\Sigma _O,h)$ be an FSM over input alphabet $\\Sigma _I$ and output alphabet $\\Sigma _O$ .", "Let $V\\subseteq \\Sigma _I^*$ be a finite set of input traces containing the empty trace $\\varepsilon $ .", "Then either $\\underline{q}\\text{-\\underline{after}-}V$ contains all reachable states, i.e., $\\underline{q}\\text{-\\underline{after}-}V = \\underline{q}\\text{-\\underline{after}-}\\Sigma _I^*$ , or $\\underline{q}\\text{-\\underline{after}-}(V\\cup V.\\Sigma _I)$ contains at least one additional state which is not contained in $\\underline{q}\\text{-\\underline{after}-}V$ , that is, $\\underline{q}\\text{-\\underline{after}-}V\\subsetneq \\underline{q}\\text{-\\underline{after}-}(V\\cup V.\\Sigma _I)$ .", "See, for example, [24].", "proof" ], [ "Requirements and Acceptable Deviations – Motivation", "In this section, elementary requirements are introduced as assertions stating that specific transitions of the reference model need to be performed correctly, whenever the SUT resides in a state corresponding to the transition's source state in the reference model.", "While, just as in the case of conformance testing, the correct behaviour is specified by the reference model, an acceptable deviation is also specified for each requirement.", "Such a deviation consists of a set of outputs that are considered as “harmless” for the specified state and input pair, though the correct output specified in the reference model is still the preferred result.", "Consider a speed monitor in a train that automatically triggers the service brakes when the train exceeds the admissible velocity.", "If the emergency brakes are triggered instead of the service brakes, this would be considered as an acceptable deviation: the safety goal to slow down the train will be reached even better than with the service brakes.", "Only the passenger comfort is reduced, since the emergency brakes are much harder than the normal ones.", "On the other hand, if the brakes are released instead of triggering them, this would lead to a safety violation and therefore be an unacceptable deviation.example The reason to introduce acceptable deviations is that the test effort can be reduced in a considerable way if (a) only critical requirements are tested with complete strategies, instead of using a complete conformance suite, and (b) we abstain from insisting on the exact behaviour specified in the reference model, as long as the resulting deviations are still acceptable." ], [ "Elementary Requirements, Acceptable Deviations, and Satisfaction", "Formalising this intuitive concept, let $M=(Q, \\underline{q}, \\Sigma _I, \\Sigma _O, h)$ be a reference model, represented by a deterministic, completely specified prime FSM with $|Q|=n\\ge 2$ .", "Let $\\omega _M:Q\\times \\Sigma _I\\rightarrow \\Sigma _O$ denote the output function of $M$ .", "An elementary requirement is denoted by $R(q,x,Z)$ , where $q\\in Q$ , $x\\in \\Sigma _I$ , and $Z\\subset \\Sigma _O$ , such that $Z \\ne \\Sigma _O$ and $\\omega _M(q,x)\\in Z$ .", "Output $\\omega _M(q,x)$ is called the expected output, and the outputs $y \\in Z-\\lbrace \\omega _M(q,x)\\rbrace $ are called acceptable deviations.", "The set $Z$ is required to be a true subset of the output alphabet, because it would not make sense to specify a requirement where any output $y\\in \\Sigma _O$ is considered as acceptable.", "An elementary requirement $R(q,x,Z)$ is fulfilled with acceptable deviations (written $S\\models R(q,x,Z)$ ) by some SUT $S = (S,\\underline{s},\\Sigma _I, \\Sigma _O, h_s)$ , if, after any input trace $\\overline{x}\\in \\Sigma _I^*$ leading to $q$ in the reference model, the input $x$ given to $S$ results in an output $z\\in Z$ .", "To formalise this notion, define $\\Pi (q) = \\lbrace \\overline{x}\\in \\Sigma _I^*~|~\\underline{q}\\text{-\\underline{after}-}\\overline{x} = q \\rbrace ;$ this is the (possibly infinite) set of all input traces reaching $q$ in reference model $M$ , when starting from the initial state $\\underline{q}$ .", "Then specify $S \\models R(q,x,Z)\\ \\text{if and only if}\\ \\forall \\pi \\in \\Pi (q): \\omega _S(\\underline{s}\\text{-\\underline{after}-}\\pi ,x) \\in Z.$" ], [ "Composite Requirements for DFSMs", "A composite requirement $R$ is written as a conjunction $R\\equiv R(q_1,x_1,Z_1)\\wedge \\dots \\wedge R(q_k,x_k,Z_k)$ of elementary requirements and interpreted in the natural way as $S\\models R \\ \\text{if and only if}\\ (S\\models R(q_1,x_1,Z_1)) \\wedge \\dots \\wedge (S\\models R(q_k,x_k,Z_k)).$ This means that the SUT $S$ fulfils each of the elementary requirements involved.", "As a running example, we will consider a reference model represented by the state machine $M$ shown in Fig.", "REF .", "The initial state $q_0$ is marked by a double circle.", "On $M$ , we specify the composite requirement $R \\equiv R(q_0,a,\\lbrace 0,1\\rbrace ) \\wedge R(q_1,b,\\lbrace 0,2\\rbrace ) \\wedge R(q_2,a,\\lbrace 0,1\\rbrace ).$ Consider an implementation with behaviour as modelled by the DFSM $S$ in Fig.", "REF .", "The implementation is not language equivalent to $M$ : for example, $\\omega _M(q_0,a.a.a) = 1.0.0$ , but $\\omega _S(s_0,a.a.a) = 1.0.1$ .", "However, the following example traces show that $S$ might still fulfil $R$ with acceptable deviationThis will be established more formally in the consecutive sections, where a complete test strategy for checking the implementation's compliance to composite requirements is elaborated.", ": (a) For sample input trace $b.b\\in \\Pi (q_0)$ , we observe that $\\omega _S(s_0\\text{-\\underline{after}-}b.b,a) = \\omega _S(s_0,a) = 1 = \\omega _M(q_0,a) = \\omega _M(q_0\\text{-\\underline{after}-}b.b,a)$ , so this result conforms to $R(q_0,a,\\lbrace 0,1\\rbrace )$ without any deviation.", "(b) For $a.b\\in \\Pi (q_1)$ , we get $\\omega _S(s_0\\text{-\\underline{after}-}a.b,b) = \\omega _S(s_1,b) = 0 = \\omega _M(q_1,b) = \\omega _M(q_0\\text{-\\underline{after}-}a.b,b)$ , so this result also conforms to the applicable elementary requirement $R(q_1,b,\\lbrace 0,2\\rbrace )$ without any deviation.", "(c) For $a.a\\in \\Pi (q_2)$ , $\\omega _S(s_0\\text{-\\underline{after}-}a.a,a) = \\omega _S(s_0,a) = 1$ , whereas $ \\omega _M(q_0\\text{-\\underline{after}-}a.a,a) = \\omega _M(q_2,a) = 0$ .", "But this deviation is still acceptable, since the applicable elementary requirement $R(q_2,a,\\lbrace 0,1\\rbrace )$ accepts both outputs 0 and 1.", "$\\Box $ Figure: NO_CAPTION$M$ .", "Figure: NO_CAPTION$S$ ." ], [ "Characterisation of Language Equivalence by Composite Requirements.", "A “very large” composite requirement can be defined by stating that the SUT should conform to every transition of the reference model, without leaving any alternatives for the expected output.", "The following theorem states that this requirement exactly captures language equivalence.", "Let $M=(Q,\\underline{q}, \\Sigma _I, \\Sigma _O,h_M)$ and $S=(S,\\underline{s},\\Sigma _I,\\Sigma _O,h_S)$ be deterministic, completely specified prime machines over the same alphabet with output functions $\\omega _M:Q\\times \\Sigma _I\\rightarrow {\\Sigma _O}$ and $\\omega _S:Q\\times \\Sigma _I\\rightarrow {\\Sigma _O}$ , respectively.", "Define composite requirement $R_{eq}=\\bigwedge _{(q,x)\\in Q\\times \\Sigma _I}R(q,x, \\lbrace \\omega _M(q,x)\\rbrace ).$ Then $L(S)=L(M)\\quad \\text{if and only if}\\quad S\\models R_{eq}.$ It is obvious that $L(S)=L(M)$ implies $S\\models R_{eq}$ , since $M\\models R_{eq}$ by construction of $R_{eq}$ .", "Therefore, we only need to show that $S\\models R_{eq}$ implies $L(S)=L(M)$ .", "Suppose $L(S)\\ne L(M)$ .", "Then there exists a shortest input sequence $\\overline{x}\\in \\Sigma _I^*$ such that $\\omega _S(\\underline{s},\\overline{x})\\ne \\omega _M(\\underline{q},\\overline{x})$ , where $\\omega _S, \\omega _M$ are the natural extension of the output functions to input sequences introduced in Section .", "Since $\\varepsilon \\in L(M)\\cap L(S)$ , $|\\overline{x}|\\ge 1$ and $\\overline{x}$ can be written as $\\overline{x}=\\pi .x$ , for some $\\pi \\in \\Sigma _I^*$ and $x\\in \\Sigma _I$ .", "Then we have $\\omega _S(\\underline{s},\\pi )=\\omega _M(\\underline{q},\\pi )\\wedge \\omega _S(\\underline{s}\\text{-\\underline{after}-}\\pi ,x)\\ne \\omega _M(\\underline{q}\\text{-\\underline{after}-}\\pi , x)$ .", "Hence $S\\lnot \\models R(\\underline{q}\\text{-\\underline{after}-}\\pi , x, \\lbrace \\omega _M(\\underline{q}\\text{-\\underline{after}-}\\pi , x)\\rbrace )$ , and this elementary requirement is a conjunct of the composite requirement $R_{eq}$ .", "Therefore, $S\\lnot \\models R_{eq}$ follows.", "proof Given a reference model $M$ and associated elementary requirements $R(q_i,x_i,Z_i)$ with $i=1,\\dots ,k$ as introduced above, we will now introduce two abstractions $M \\rightarrow M_1 \\rightarrow M_2$ that are needed to create test suites allowing to verify that an SUT $S$ fulfils all $R(q_i,x_i,Z_i)$ without having to test for the stronger property language equivalence.", "$M_1$ abstracts from concrete $M$ -outputs by using sets of output events $Z_i\\subset \\Sigma _O$ where requirement $R(q_i,x_i,Z_i)$ is involved, and using the whole output alphabet $\\Sigma _O$ as don't care symbol to specify outputs unrelated to any requirement.", "$M_2$ is the prime machine of $M_1$ .", "Note that $M_1$ may be no longer minimal, since some $M$ -states may not be distinguishable anymore due to output abstraction." ], [ "Construction of $M_1$", "Let $*=\\Sigma _O$ denote the don't care symbol specifying that “any output is allowed” in certain situations.", "Define $\\Sigma _O^{\\prime }=\\lbrace *, Z_1, \\dots , Z_k\\rbrace $ as the new output alphabet of a completely specified, deterministic abstraction $M_1$ with state space $Q$ , initial state $\\underline{q}$ , input alphabet $\\Sigma _I$ , and output and transition functions specified as follows.", "The transition function $\\delta _{M_1}$ of $M_1$ coincides with that of $M$ , that is, $\\delta _{M_1} = \\delta _M$ .", "The output function $\\omega _{M_1}$ of $M_1$ is defined by $\\omega _{M_1}(q_i,x_i) & = Z_i &\\ \\text{for}\\ i\\in \\lbrace 1,\\dots ,k\\rbrace \\\\\\omega _{M_1}(q,x) & = * &\\ \\text{for}\\ (q,x)\\in Q\\times \\Sigma _I\\setminus \\lbrace (q_1,x_1),\\dots ,(q_k,x_k) \\rbrace $ By construction, $M_1$ abstracts all outputs related to elementary requirements $R(q_i,x_i,Z_i)$ to $Z_i$ and all outputs that are unrelated to any requirement to $*$ .", "Being unrelated to any requirement means that the (state,input)-pair $(q,x)$ occurring in the original transition $(q, x, y, q^{\\prime })\\in h$ differs from all $(q_i,x_i)$ used in the specification of some requirement $R(q_i,x_i,Z_i)$ ." ], [ "Construction of $M_2$", "Let $M_2$ be the prime machine of $M_1$ .", "If $\\lbrace q_1,\\dots ,q_k\\rbrace =Q$ and $(x_i,Z_i)=(x_j,Z_j)$ , for all $i,j=1,\\dots , k$ , then $M_2$ contains only one state, otherwise, $M_2$ contains at least two states.", "Denote the states in $M_2$ by $[q]$ , which is the equivalence class of state $q$ in $M_1$ : $[q]=\\lbrace q^{\\prime }\\in Q~|~L_{M_1}(q^{\\prime })= L_{M_1}(q)\\rbrace $ The transition relation of $M_2$ is denoted by $h_2=\\lbrace ([q],x,y,[q^{\\prime }])~|~(q, x, y, q^{\\prime })\\in h_1\\rbrace $ .", "We have $[q\\text{-\\underline{after}-}\\overline{x}]=[q]\\text{-\\underline{after}-}\\overline{x}$ , for any $q\\in Q$ and $\\overline{x}\\in \\Sigma _I^*$ , where the transition \"$\\text{-\\underline{after}-}$ \" from the left side is according to $h$ and $h_1$ , since $M_1$ contains the same transitions of $M$ up to the outputs.", "The \"$\\text{-\\underline{after}-}$ \" from the right side is according to $h_2$ .", "Let $n^{\\prime }=|Q^{\\prime }|\\ge 1$ .", "For any $q_i,q_j\\in \\lbrace q_1,\\dots ,q_k\\rbrace $ , $[q_i]=[q_j]$ implies $x_i/Z_i=x_j/Z_j$ .", "Let $q\\in Q$ and suppose $[q]=[q_i]$ for some $i=1,\\dots , k$ .", "Then $(q,x_i,Z_i)\\in \\lbrace (q_1,x_1,Z_1),\\dots , (q_k,x_k,Z_k)\\rbrace $ , because otherwise the output of $(q, x_i)$ in $h_1$ is $*$ and $*\\ne Z_i$ , contradicting the fact that $q$ and $q_i$ are equivalent.", "Hence, from $[q]=[q_i]$ follows $q=q_j$ for some $j=1,\\dots , k$ .", "Define $[\\Pi (q_i)]=\\bigcup _{j\\in \\lbrace \\ell \\in \\lbrace 1,\\dots ,k\\rbrace ~|~[q_\\ell ]=[q_i]\\rbrace }\\Pi (q_j).$ This is the set of all input traces leading to states $q_j$ in $M$ that are equivalent to $q_i$ in $M_1$ .", "For the reference model $M$ introduced in Example REF and the composite requirement $R$ specified there, the abstracted machine $M_1$ is depicted in Fig.", "REF .", "Its minimised machine $M_2$ is shown in Fig.", "REF .", "$\\Box $ Figure: NO_CAPTION$M_1$ of reference model $M$ from Example REF .", "Figure: NO_CAPTION$M_1$ ." ], [ "Test Cases, Test Suites, and Pass Criteria", "A test suite $\\text{TS}$ is a set of input traces $\\overline{x}\\in \\Sigma _I$ ; the latter are called test cases.", "The expected result associated with test case $\\overline{x}$ is the output trace $\\omega _M(\\underline{q},\\overline{x})$ calculated from the reference model.", "An implementation $S=(S,\\underline{s},\\Sigma _I,\\Sigma _O,h_S)$ passes test case $\\overline{x}$ , if and only if the associated outputs observed when running $\\overline{x}$ against $S$ conform to the expected results.", "More formally, $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\overline{x} \\ \\ \\text{if and only if}\\ \\ \\omega _S(\\underline{s},\\overline{x}) = \\omega _M(\\underline{q},\\overline{x}).$ Implementation $S$ passes the test suite $\\text{TS}$ if and only if $S$ passes all test cases, that is, $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}\\ \\ \\text{if and only if}\\ \\ \\forall \\overline{x}:\\text{TS}: S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\overline{x}.$ We say that a test suite $\\text{TS}$ is exhaustive with respect to composite requirement $R = R(q_1,x_1,Z_1)\\wedge \\dots \\wedge R(q_k,x_k,Z_k)$ if and only if $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}\\Rightarrow S\\models R.$ In the remainder of this section, we will construct a finite test suite $\\text{TS}_\\Rightarrow $ , whose size depends on the reference model $M$ , the (composite) requirement, and the assumed maximal number $m$ of states in the implementation DFSM $S$ .", "We will show that $\\text{TS}_\\Rightarrow $ is exhaustive with respect to composite requirement $R = R(q_1,x_1,Z_1)\\wedge \\dots \\wedge R(q_k,x_k,Z_k)$ for the fault domain of completely specified, deterministic FSMs with at most $m$ states.", "Obviously, if $S$ is I/O-equivalent to $M$ , then $S$ satisfies the composite requirement $R$ .", "Hence, any SUT $S$ , with $S\\lnot \\models R$ will fail any complete test suite for fault model $F=(M,\\sim , D)$ , where $S\\in D$ and $\\sim $ is the language equivalence conformance relation.", "Therefore, any complete test suite for fault model $F=(M,\\sim , D)$ with $S\\in D$ satisfies the above two conditions.", "Our objective is to introduce a test strategy leading to fewer test cases than the well-known strategies for generating test suites checking language equivalence.", "It should be emphasised that the expected results associated with each test case reject any deviation of the SUT from the outputs expected according to the reference model.", "Therefore, if $S$ is in a state corresponding to some state $q_i$ in $M$ and reacts to input $x_i$ with some output $y$ which is in $Z_i$ but which differs from the expected value $\\omega _M(q,x_i)$ , the test execution will fail.", "It will be illustrated below, however, that the introduction of admissible deviation $y\\in Z_i-\\lbrace \\omega _M(q,x_i) \\rbrace $ may lead to fewer test cases.", "This reduction comes at the cost that some violations of expected results may be overlooked; but it can still be guaranteed that these violations are always admissible deviations.", "For transitions that are not linked to any elementary requirement $R(q_i,x_i,Z_i)$ , no guarantees are made whatsoever.", "Moreover, note that according to condition (REF ), failing the test suite does not necessarily imply that $R$ is violated: the test suite may also fail because an erroneous output is uncovered which is unrelated to any elementary requirement in $R$ .", "This conforms to our understanding of a “reasonably designed” test oracle.", "Any detected deviation in comparison to the reference model will lead to the test case to fail; it is just not guaranteed that all errors that are unrelated to $R$ will be uncovered." ], [ "Test Suite Construction", "Let $V$ , $\\varepsilon \\in V$ , be a state cover of $M$ .", "We define three auxiliary sets $A, B, C$ containing pairs of input traces.", "$A&=V\\times V\\\\B&=V\\times V.\\bigcup _{i=1}^{m-n+1} \\Sigma _I^{i}\\\\C&=\\lbrace (\\alpha ,\\beta )~|~ \\alpha \\in \\text{pref}(\\beta ), \\alpha , \\beta \\in V.\\bigcup _{i=1}^{m-n+1} \\Sigma _I^{i}\\rbrace $ For any $\\alpha ,\\beta \\in A\\cup B\\cup C$ , define $\\Delta _M(\\alpha ,\\beta )&=\\lbrace \\gamma \\in \\Sigma _I^*~|~\\omega _M(\\underline{q}\\text{-\\underline{after}-}\\alpha ,\\gamma ) \\ne \\omega _M(\\underline{q}\\text{-\\underline{after}-}\\beta ,\\gamma )\\rbrace \\\\\\Delta _{M_1}(\\alpha ,\\beta )&=\\lbrace \\gamma \\in \\Sigma _I^*~|~\\omega _{M_1}(\\underline{q}\\text{-\\underline{after}-}\\alpha ,\\gamma ) \\ne \\omega _{M_1}(\\underline{q}\\text{-\\underline{after}-}\\beta ,\\gamma )\\rbrace \\\\$ The set $\\Delta _M(\\alpha ,\\beta )$ contains all input traces distinguishing the states $q\\text{-\\underline{after}-}\\alpha $ and $q\\text{-\\underline{after}-}\\beta $ in $M$ by yielding different output traces when applied to these states.", "Set $\\Delta _{M_1}(\\alpha ,\\beta )$ contains all input traces distinguishing the states $q\\text{-\\underline{after}-}\\alpha $ and $q\\text{-\\underline{after}-}\\beta $ in $M_1$ .", "Note that $\\Delta _M(\\alpha ,\\beta )$ and $\\Delta _{M_1}(\\alpha ,\\beta )$ may be empty, since $\\underline{q}\\text{-\\underline{after}-}\\alpha $ and $\\underline{q}\\text{-\\underline{after}-}\\beta $ are not necessarily distinguishable in $M$ or $M_1$ , respectively.", "For an arbitrary set $P\\subseteq \\Sigma _I^*\\times \\Sigma _I^*$ of pairs of input traces, define $P(M)\\subseteq P$ and $P(M_1)\\subseteq P$ by $(\\alpha ,\\beta )\\in P(M)&\\Leftrightarrow (\\alpha ,\\beta )\\in P\\wedge \\Delta _M(\\alpha ,\\beta )\\ne \\varnothing \\\\(\\alpha ,\\beta )\\in P(M_1)&\\Leftrightarrow (\\alpha ,\\beta )\\in P\\wedge \\Delta _{M_1}(\\alpha ,\\beta )\\ne \\varnothing $ We will apply this notation to the sets $A, B, C$ defined above: $A(M)$ , for example, is the subset of all trace pairs $(\\alpha ,\\beta )$ from $A$ whose target states $\\underline{q}\\text{-\\underline{after}-}\\alpha $ and $\\underline{q}\\text{-\\underline{after}-}\\beta $ , respectively, are distinguishable in $M$ ." ], [ "Main Theorem on Exhaustive Test Suites", "The following main theorem shows that the criteria (REF ) and () suffice to guarantee that the test suite $\\text{TS}_\\Rightarrow $ is exhaustive.", "Let $m\\ge n$ be a positive integer.", "Let $S=(S, \\underline{s}, \\Sigma _I,\\Sigma _O, h_s)$ be a minimal DFSM with $|S|\\le m$ .", "Let $\\text{TS}_\\Rightarrow \\subseteq \\Sigma _I^*$ be any test suite satisfying $&V.\\bigcup _{i=0}^{m-n+1}\\Sigma _I^i\\subseteq \\text{TS}_\\Rightarrow ,\\, \\text{and}\\\\&\\forall (\\alpha ,\\beta )\\in A(M)\\cup B(M_1)\\cup C(M_1): \\exists \\gamma \\in \\Delta _M(\\alpha ,\\beta ): \\alpha .\\gamma , \\beta .\\gamma \\in \\text{TS}_\\Rightarrow $ Then $\\text{TS}_\\Rightarrow $ is exhaustive for composite requirement $R=R(q_1,x_1,Z_1)\\wedge \\dots \\wedge R(q_k,x_k,Z_k)$ , that is, $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}_\\Rightarrow \\Rightarrow S\\models R.$ Suppose $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}_\\Rightarrow $ .", "We first show that $V.\\bigcup _{i=0}^{m-n} \\Sigma _I^{i}$ is a state cover of $S$ .", "Since $V$ is a state cover of $M$ and $M$ is a prime machine, there are input traces $\\alpha .\\gamma , \\beta .\\gamma $ in $\\text{TS}_\\Rightarrow $ for each pair of the $n$ states in $Q$ , such that $\\alpha ,\\beta \\in V$ and states $\\underline{q}\\text{-\\underline{after}-}\\alpha $ and $\\underline{q}\\text{-\\underline{after}-}\\beta $ are distinguished by $\\gamma $ in $M$ .", "Since $S$ passes these test cases, this also distinguishes $n$ states in $S$ .", "Since $S$ has at most $m$ states, Lemma  can be applied to conclude that $V.\\bigcup _{i=0}^{m-n} \\Sigma _I^{i}$ reaches all states of $S$ .", "Suppose $S\\,\\lnot \\models \\, R$ .", "Then there is some $t\\in \\lbrace 1,\\dots , k\\rbrace $ with $S\\lnot \\models R(q_t,x_t,Z_t)$ .", "Hence there exists $\\pi \\in \\Pi (q_t)$ with $\\omega _M(\\underline{q}\\text{-\\underline{after}-}\\pi ,x_t)\\in Z_t$ , but $\\omega _S(\\underline{q}\\text{-\\underline{after}-}\\pi ,x_t)\\notin Z_t$ .", "Lifting these observations to the abstraction $M_2$ introduced in Section REF , this induces the existence of input traces $\\pi _2\\in [\\Pi (q_t)]$ such that $\\omega _{M_2}([\\underline{q}]\\text{-\\underline{after}-}\\pi _2,x_t) = Z_t$ and $\\omega _S(\\underline{s}\\text{-\\underline{after}-}\\pi _2,x_t)\\notin Z_t$ .", "Since $\\varepsilon \\in V$ , each $\\pi _2\\in [ \\Pi (q_t)]$ can be structured as $\\pi _2 = v.\\tau $ with $v\\in V$ and $\\tau \\in \\Sigma _I^*$ .", "Let $\\tau $ be a shortest sequence such that $\\exists v\\in V: v.\\tau \\in [ \\Pi (q_t)] \\wedge \\omega _S(\\underline{s}\\text{-\\underline{after}-}v.\\tau ,x_t)\\notin Z_t.$ Since $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}_\\Rightarrow $ and $V.\\bigcup _{i=0}^{m-n+1} \\Sigma _I^{i}\\subseteq \\text{TS}_\\Rightarrow $ , we have $|\\tau |\\ge m-n+1$ , because otherwise, the input trace $v.\\tau .x_t$ would have been tested, and the test suite would have failed.", "Let $\\tau _i=\\tau ^{[1..i]}, i\\le m-n+1$ .", "Then $v.\\tau _i\\ne v.\\tau _j$ for all $i\\ne j$ .", "Let $V=\\lbrace v_1,\\dots , v_n\\rbrace $ .", "Then $v_i\\ne v_j$ for $i\\ne j$ , since $V$ reaches $n$ different states in $M$ .", "Suppose $v_i=v.\\tau _j$ for some $i,j$ .", "Let $\\iota $ be the suffix of $\\tau _j$ with $\\tau _j.\\iota =\\tau $ .", "Then $v_i.\\iota =v.\\tau _j.\\iota =v.\\tau \\in [ \\Pi (q_t)]$ , and $|\\iota |<|\\tau |$ , a contradiction to the assumption that $\\tau \\in \\Sigma _I^*$ is a shortest sequence satisfying condition (REF ).", "As a consequence, the set $U = \\lbrace v_1,\\dots , v_n, v.\\tau _1, \\dots , v.\\tau _{m-n+1}\\rbrace $ contains $m+1$ elements.", "Observe that $U\\subseteq \\text{TS}_\\Rightarrow $ , since all elements start with a $v\\in V$ and – if extended by some $\\tau _i$ – are followed by an input trace of length less or equal to $m-n+1$ .", "Since $S$ contains only $m$ states, there exist $\\alpha \\ne \\beta \\in U$ reaching the same state in $S$ , that is, $\\underline{s}\\text{-\\underline{after}-}\\alpha =\\underline{s}\\text{-\\underline{after}-}\\beta $ .", "Assume that $\\underline{q}\\text{-\\underline{after}-}\\alpha , \\underline{q}\\text{-\\underline{after}-}\\beta $ are not equivalent in $M_1$ .", "This would imply $\\lbrace \\alpha ,\\beta \\rbrace \\subseteq A(M) \\cup B(M_1)\\cup C(M_1)$ and the existence of a distinguishing trace $\\gamma \\in \\Delta _M(\\alpha ,\\beta )$ such that $\\lbrace \\alpha .\\gamma , \\beta .\\gamma \\rbrace \\subseteq \\text{TS}_\\Rightarrow $ (see ()).", "This would lead to a failed test execution, since $\\gamma $ cannot distinguish $\\underline{s}\\text{-\\underline{after}-}\\alpha =\\underline{s}\\text{-\\underline{after}-}\\beta $ , but would lead to different outputs when applied to $\\underline{q}\\text{-\\underline{after}-}\\alpha $ and $\\underline{q}\\text{-\\underline{after}-}\\beta $ .", "This contradiction implies that $\\underline{q}\\text{-\\underline{after}-}\\alpha , \\underline{q}\\text{-\\underline{after}-}\\beta $ are equivalent in $M_1$ .", "As a consequence, $\\lbrace \\alpha , \\beta \\rbrace $ cannot be contained in $V$ , since pairs from $V$ always reach distinguishable states in $M$ .", "Without loss of generality, we therefore assume that one of the cases (a) $\\alpha =v_i\\in V, \\beta =v.\\tau _j$ , or (b) $\\alpha =v.\\tau _i, \\beta =v.\\tau _j, i<j$ applies.", "Let $\\iota $ be the suffix of $\\tau _j$ with $\\tau _j.\\iota =\\tau $ (in the case $j=m-n+1=|\\tau |, \\iota =\\varepsilon $ ).", "Then $\\beta .\\iota =v.\\tau $ .", "Since $\\underline{q}\\text{-\\underline{after}-}\\alpha , \\underline{q}\\text{-\\underline{after}-}\\beta $ are equivalent states in $M_1$ , we obtain that $\\underline{q}\\text{-\\underline{after}-}\\beta .\\iota $ and $\\underline{q}\\text{-\\underline{after}-}\\alpha .\\iota $ are equivalent states in $M_1$ , hence $\\alpha .\\iota \\in [ \\Pi (q_t)]$ .", "Since $\\underline{s}\\text{-\\underline{after}-}\\beta =\\underline{s}\\text{-\\underline{after}-}\\alpha $ in $S$ , we conclude that $\\underline{s}\\text{-\\underline{after}-}\\beta .\\iota =\\underline{s}\\text{-\\underline{after}-}\\alpha .\\iota $ and $ \\omega _S(\\underline{s}\\text{-\\underline{after}-}\\beta .\\iota ,x_t)=\\omega _S(\\underline{s}\\text{-\\underline{after}-}\\alpha .\\iota , x_t)\\notin Z_t$ .", "In the case $\\alpha =v_i\\in V$ , we have $|\\iota |<|\\tau |$ .", "For $\\alpha =v.\\tau _i$ , we have $\\alpha .\\iota =v.\\tau _i.\\iota $ and $|\\tau _i.\\iota |<|\\tau |$ .", "Both cases contradict the assumption that $\\tau \\in \\Sigma _I^*$ is a shortest sequence satisfying condition (REF ).", "This contradiction implies that $S\\ \\underline{\\text{\\bf pass}}_\\Rightarrow \\ \\text{TS}_\\Rightarrow $ and $S \\lnot \\models R$ cannot both be true and completes the proof of the theorem.", "proof Algorithms for creating test suites according to formulas (REF ) and () have been described in the original work [8] and in [15].", "Consider again reference model $M$ and the (in black box testing practise unknown) implementation model $S$ from Example REF .", "$M$ is minimal and has $n=3$ states.", "Under the hypothesis that the minimised model of the true implementation behaviour also has $m=3$ states, the following test suite with 4 test cases has been calculated according to the rules (REF ) and ().", "$a.a.b, a.b.b, b.a.b, b.b.a$ Applying this test suite to the implementation model $S$ results in I/O-traces $a.a.b/1.0.2,a.b.b/1.2.0,b.a.b/2.0.0,b.b.a/2.0.1$ These executions conform to the reference model $M$ , so the implementation passes the suite.", "This shows that $S \\models R$ holds.", "Applying the complete H-method to $M$ and hypothesis $m=n=3$ results in 5 test cases.", "These comprise the ones calculated according to (REF ) and (), adding a fifth test case $a.a.a$ .", "This new test case reveals the fact that $S$ is not language equivalent to $M$ , since $S$ produces $a.a.a/1.0.1$ where $a.a.a/1.0.0$ had been expected according to reference model $M$ .", "In Appendix , it is explained how these test suites can be automatically generated using the library fsmlib-cpp mentioned in the introduction.", "$\\Box $" ], [ "Fasten Seatbelt and Return-to-Seat Sign Control", "The following experiment is a (slightly simplified) real-world example concerning safety-related and uncritical indications in an aircraft cabinThe application used in this experiment has been originally published in [15].", "The application description has been reproduced here in its original form, in order to make this article sufficiently self-contained.. Table: State-transition table of DFSM specifying the control of FSB signs and RTS signs in an aircraft cabin.Table: Explanation of states 𝐬 0 \\mathbf {s_0},..., 𝐬 23 \\mathbf {s_{23}} in Table .A cabin controller in a modern aircraft switches the fasten seat belt (FSB) signs located above the passenger seats in the cabin and the return to seat (RTS) signs located in the lavatories according to the rules modelled in the DFSM shown in Table REF .", "Note that this DFSM is already minimal.", "As inputs, the cabin controller reads the actual position of the fasten seat belts switch in the cockpit, which has the position f0 (OFF), f1 (ON), and f2 (AUTO).", "Further inputs come from the cabin pressure control system which indicates “cabin pressure low” by event d1 and “cabin pressure ok” by d0.", "This controller also indicates “excessive altitude” by e1 or “altitude in admissible range” by e0.", "Another sub-component of the cabin controller determines whether the so-called AUTO condition is true (event a1) or false (a0).", "The cabin controller switches the fasten seat belt signs and return to seat signs on and off, depending on the actual input change and its current internal state.", "As long as the cabin pressure and the cruising altitude are ok (after initialisation of the cabin controller or if last events from the cabin pressure controller were d0, e0), the status of the FSB and RTS signs is determined by the cockpit switch and the AUTO condition: if the switch is in the ON position, both FSB and RTS signs are switched on (output 11 in Table REF ).", "Turning the switch into the OFF position switches the signs off.", "If the switch is in the AUTO position, both FSB and RTS signs are switched on if the AUTO condition becomes true with event a1, and they are switched off again after event a0.", "The AUTO condition may depend on the status of landing gears, slats, flaps, and oil pressure, these details are abstracted to a1, a0 in our example.", "As soon as a loss of pressure occurs in the cabin (event d1) or an excessive altitude is reached, the FSB signs must be switched on and remain in this state, regardless of the actual state of the cockpit switch and the AUTO condition.", "The RTS signs, however, need to be switched off, because passengers should not be encouraged to leave the lavatories in a low pressure or excessive altitude situation.", "After the cabin pressure and the altitude are back in the admissible range, the FSB and RTS signs shall automatically resume their state as determined by the “normal” inputs from cockpit switch and AUTO condition.", "Table REF facilitates the interpretation of the DFSM model shown in Table REF : for every DFSM state $\\mathbf {s_0}$ ,...,$\\mathbf {s_{23}}$ , the associated status of the cockpit switch, cabin decompression, excessive altitude, and AUTO condition, as well as the last output made when entering the state is displayed." ], [ "Complete Test Suites Checking Model Equivalence Computed by the H-Method", "Applying the H-Method [8] as implemented in the fsmlib-cppsee Appendix  for instructions how to use the test case generation program which is part of the library, the number of test cases needed to test an implementation to establish language equivalence with full fault coverage is shown in Table REF , column H. Recall that a test case is a sequence of inputs.", "When executing a test case against an implementation, the expected results are determined As the size of the test suite depends on the potential number $m$ of implementation states minus the number $n$ of states in the minimised model, test suites are calculated for $m-n = 0,1,2$ .", "Table: Comparison of the numbers of test cases needed to prove I/O-equivalence (H-Method) and to prove requirements satisfaction using exhaustive testing for requirements 𝐑 1 \\mathbf {R}_1 and 𝐑 2 \\mathbf {R}_2 specified in this section." ], [ "Requirement $\\mathbf {R}_1$ : Safety-relevant outputs on Decompression", "As a first requirement, we consider $\\mathbf {R}_1$ .", "Whenever cabin decompression occurs, the FSB signs shall be set to 1, and the RTS signs to 0.", "In the DFSM model shown in Table REF , this requirement is reflected by column $\\mathbf {d1}$ : regardless of the current state, output (FSB,RTS)=(1,0) is assigned on occurrence of input $\\mathbf {d1}$ .", "Encoding $\\mathbf {R}_1$ in our requirements specification formalism results in the following representation.", "$\\mathbf {R}_1 \\equiv \\bigwedge _{i=0}^{23} R(s_i,\\mathbf {d1},\\lbrace 10 \\rbrace )$ Obviously, the specification of $\\mathbf {R}_1$ is independent of the current state, and there are no alternative outputs that may be accepted in exchange for the output (FSB,RTS)=(1,0) expected according to the reference model.", "Creating an exhaustive test suite for $\\mathbf {R}_1$ according to Theorem REF results in the number of test cases shown in column $\\mathbf {R}_1$ of Table REF .", "As listed in column $\\mathbf {\\Delta _1^\\%}$ of this table, the test case reductions achieved in comparison to the H-based test suite vary with $m-n$ and are in range $56\\%$ — $63\\%$ .", "The test of this requirement could be advisable, for example, during regression testing after the sign controller's code has been modified with regard to the sub-function responsible for decompression handling.", "Instead of running the complete H-test suite, it would suffice to perform the significantly smaller test suite for $\\mathbf {R}_1$ which also guarantees full fault coverage as far as $\\mathbf {R}_1$ is concerned." ], [ "Requirement $\\mathbf {R}_2$ : All Safety-relevant outputs and Sign Activation by Manual Switch", "As second requirement, we consider a composite requirement related to all safety-critical events in combination to normal behaviour reactions (FSB,RTS)=(1,1) when setting the cockpit switch to position 1.", "$\\mathbf {R}_2$ .", "Whenever cabin decompression or excessive altitude occurs, the FSB signs shall be set to 1, and the RTS signs to 0.", "The signs stay activated until both decompression and execessive altitude are no longer present.", "In absence of cabin decompression and excessive altitude, both FSB and RTS signs shall be switched on when setting the cockpit switch to position 1.", "Formalising this requirement leads to $\\mathbf {R}_2 & \\equiv & \\bigwedge _{i=0}^{23} \\big ( R(s_i,\\mathbf {d1},\\lbrace 10 \\rbrace )\\wedge R(s_i,\\mathbf {e1},\\lbrace 10 \\rbrace ) \\big ) \\wedge {}\\\\& & \\bigwedge _{i\\in \\lbrace 3,\\dots ,11,15,\\dots ,23 \\rbrace }\\big ( R(s_i,\\mathbf {f0},\\lbrace 10 \\rbrace ) \\wedge R(s_i,\\mathbf {f1},\\lbrace 10 \\rbrace ) \\wedge R(s_i,\\mathbf {f2},\\lbrace 10 \\rbrace ) \\big ) \\wedge {}\\\\& & \\bigwedge _{i\\in \\lbrace 3,\\dots ,11,15,\\dots ,23 \\rbrace }\\big ( R(s_i,\\mathbf {a1},\\lbrace 10 \\rbrace ) \\wedge R(s_i,\\mathbf {a0},\\lbrace 10 \\rbrace ) \\big ) \\wedge {}\\\\& & \\bigwedge _{i\\in \\lbrace 6,\\dots ,11,18,\\dots ,23 \\rbrace } R(s_i,\\mathbf {d0},\\lbrace 10 \\rbrace ) \\wedge {}\\\\& & \\bigwedge _{i\\in \\lbrace 3,4,5,9,10,11,15,16,17,21,22,23 \\rbrace } R(s_i,\\mathbf {e0},\\lbrace 10 \\rbrace )\\\\& & \\bigwedge _{i\\in \\lbrace 0,1,2,12,13,14 \\rbrace } R(s_i,\\mathbf {f1},\\lbrace 11 \\rbrace )$ Creating an exhaustive test suite for $\\mathbf {R}_2$ according to Theorem REF results in the number of test cases shown in column $\\mathbf {R}_2$ of Table REF .", "As listed in column $\\mathbf {\\Delta _2^\\%}$ of this table, the test case reductions achieved in comparison to the H-based test suite are in range $23\\%$ — $35\\%$ .", "Testing this requirement would be advisable, for example, after the initial development of the sign controller, assuming that there would not be enough time to perform all tests derived by the H-method.", "Requirement $\\mathbf {R}_2$ leads to a smaller test suite, but still covers all safety-relevant reactions of the implementation and the most important user requirement.", "Then it could be justified to test the functionality related to the AUTO condition with less effort, since, as a fall back option for the pilot, signs could always be switched on manually when the AUTO mode is not properly functioning." ], [ "Complete Testing of Composite Requirements", "Suppose that DFSMs $M$ (reference model) and $S$ (implementation) are completely specified and consider the composite requirement $R=\\bigwedge _{i=1}^k R(q_i,x_i,Z_i)$ specified on $M$ .", "In this section, we are going to construct finite test suites $\\text{TS}_\\Leftrightarrow $ depending on $M$ and $R$ and a new pass relation $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ such that an implementation $S$ passes $\\text{TS}_\\Leftrightarrow $ if and only if it conforms to the requirements $R$ .", "In the terminology introduced in Section REF , these test suites are called complete." ], [ "A Nondeterministic Alternative to $M_1$", "Let $M_1$ be the FSM abstraction induced by $M$ and $R$ as described in Section .", "Recall that $M_1$ is deterministic with output alphabet $\\lbrace *, Z_1,\\dots , Z_k\\rbrace $ , where $*$ is short for $\\Sigma _O$ .", "The DFSM $M_1$ induces an alternative abstraction $M_1^{\\prime }$ of $M$ , which is nondeterministic and has output alphabet $\\Sigma _O$ .", "This nondeterministic FSM is specified by $M_1^{\\prime }=(Q, \\underline{q}, \\Sigma _I, \\Sigma _O, h_1^{\\prime })$ , where $(q,x,y,q^{\\prime })\\in h_1^{\\prime }\\Leftrightarrow \\big (\\delta _{M_1}(q,x) = q^{\\prime } \\wedge y \\in \\omega _{M_1}(q,x)\\big ).$ Given $(q,x)\\in \\Sigma _I\\times \\Sigma _I$ such that $\\omega _{M_1}(q,x) = *$ , the nondeterministic machine $M_1^{\\prime }$ possesses the transitions $(q,x,y,\\delta _{M_1}(q,x))$ with arbitrary $y\\in \\Sigma _O$ .", "For $i\\in \\lbrace 1,\\dots ,k \\rbrace $ , recall that $\\omega _{M_1}(q_i,x_i) = Z_i$ .", "For these cases, $M_1^{\\prime }$ reacts by nondeterministically taking one of the transitions $(q_i,x_i,y,\\delta _{M_1}(q,x))$ with $y\\in Z_i$ .", "When representing the abstraction $M_1$ from Fig.", "REF as a nondeterministic FSM, this results in $M_1^{\\prime }$ as depicted in Fig.", "REF .", "$\\Box $ Figure: Nondeterministic FSM M 1 ' M_1^{\\prime } created from M 1 M_1 displayed in Fig.", ".From the construction rules specified above and from the illustration in Example REF and Fig.", "REF , it is immediately clear that $M_1^{\\prime }$ is observable and completely specified.", "Also note that by construction, $M_1^{\\prime }$ may be interpreted as the “most nondeterministic FSM which still satisfies requirement $R$ ”: for (state,input)-pairs that are unrelated to $R$ , any output from $\\Sigma _O$ can occur.", "For pairs $(q_i,x_i)$ related to elementary requirements $R(q_i,x_i,Z_i)$ , any output from $Z_i$ can be nondeterministically selected, so that $R(q_i,x_i,Z_i)$ is never violated.", "Moreover, $M_1^{\\prime }$ is only nondeterministic with respect to the outputs produced in a given state $q$ for a given input $x$ , whereas the target state is uniquely determined by $q$ and $x$ .", "This means that the transition functions $\\delta _{M_1}, \\delta _{M_1^{\\prime }}: Q\\times \\Sigma _I \\rightarrow Q$ of $M_1$ and $M_1^{\\prime }$ , respectively, coincide in the sense that $\\forall (q,x)\\in Q\\times \\Sigma _I: \\delta _{M_1}(q,x) = \\delta _{M_1^{\\prime }}(q,x).$ As a consequence, the expressions $\\underline{q}\\text{-\\underline{after}-}\\overline{x},\\ \\overline{x}\\in \\Sigma _I^*,$ always result in the same uniquely determined target state, regardless of whether they are evaluated in $M_1$ or $M_1^{\\prime }$ .Recall that for general nondeterministic FSMs, $\\underline{q}\\text{-\\underline{after}-}\\overline{x}$ specifies a set of possible target states, since their transition relation $h$ may allow for different target states being reached for a given pre-state and input.", "Finally, note that, since $M_1^{\\prime }$ is nondeterministic, its output function is set-valued, $\\omega _{M_1^{\\prime }} : Q\\times \\Sigma _I^* \\rightarrow \\mathbb {P}(\\Sigma _O^*)$ .", "It is easy to see, however, that $\\omega _{M_1^{\\prime }}(q,x) = \\omega _{M_1}(q,x)$ for all states $q$ and inputs $x$ : FSM $M_1^{\\prime }$ has transitions $q \\xrightarrow{} \\delta _{M_1}(q,x)$ for every $y\\in \\omega _{M_1}(q,x)$ .", "The following theorem presents an important insight into the relationship between requirements satisfaction and reduction (i.e.", "language inclusion): an implementation machine $S$ satisfies the requirement $R$ , if and only if its language is contained in the language of the abstraction $M_1^{\\prime }$ constructed above.", "$S\\models R\\Leftrightarrow L(S)\\subseteq L(M_1^{\\prime })$ Recall that $\\Pi (q)=\\lbrace \\overline{x}\\in \\Sigma _I^*~|~\\underline{q}\\text{-\\underline{after}-}\\overline{x}=q\\rbrace $ .", "Also recall that by definition, $S\\models R\\Leftrightarrow \\forall i\\in \\lbrace 1,\\dots ,k\\rbrace , \\pi \\in \\Pi (q_i): \\omega _S(\\underline{s}\\text{-\\underline{after}-}\\pi ,x_i)\\in Z_i$ .", "Suppose that $L(S)\\subseteq L(M_1^{\\prime })$ .", "Since $S$ is completely specified by assumption, there exists a unique I/O-trace $\\overline{x}.x_i/\\overline{y}.y \\in L(S)$ for any input sequence $\\overline{x}.x_i$ with $\\overline{x} \\in \\Pi (q_i)$ and $i\\in \\lbrace 1,\\dots ,k\\rbrace $ .", "Since we assume that $S$ is a reduction of $M_1^{\\prime }$ , this I/O-trace must also be a trace of $M_1^{\\prime }$ .", "By construction of $M_1^{\\prime }$ , this means that $y\\in Z_i$ , so $S$ fulfils requirement $R(q_i,x_i,Z_i)$ .", "Since this argument was independent of $i\\in \\lbrace 1,\\dots ,k\\rbrace $ , $S\\models R$ follows.", "Now suppose that $L(S)\\lnot \\subseteq L(M_1^{\\prime })$ .", "Then there exists an I/O-trace $\\overline{x}.x/\\overline{y}.y \\in L(S)$ such that $\\overline{x}/\\overline{y} \\in L(M_1^{\\prime })$ , but $\\overline{x}.x/\\overline{y}.y \\notin L(M_1^{\\prime })$ .", "Suppose that $(\\underline{q}\\text{-\\underline{after}-}\\overline{x},x) \\ne (q_i,x_i)$ for all $i\\in \\lbrace 1,\\dots , k\\rbrace $ , where the $\\text{-\\underline{after}-}$ operator is evaluated in $M_1^{\\prime }$ .", "Then, by construction of $M_1^{\\prime }$ , $\\overline{x}.x/\\overline{y}.y \\in L(M_1^{\\prime })$ for all $y\\in \\Sigma _O$ , so this is a contradiction to the assumption $\\overline{x}.x/\\overline{y}.y \\notin L(M_1^{\\prime })$ .", "Thus the assumption $\\overline{x}.x/\\overline{y}.y \\notin L(M_1^{\\prime })$ implies the existence of an $i\\in \\lbrace 1,\\dots , k\\rbrace $ such that $(\\underline{q}\\text{-\\underline{after}-}\\overline{x},x) = (q_i,x_i)$ .", "As a consequence, $\\overline{x}.x_i/\\overline{y}.y \\notin L(M_1^{\\prime })$ and $\\overline{x}\\in \\Pi (q_i)$ .", "By construction of $M_1^{\\prime }$ , this means that $y\\notin Z_i$ .", "This implies that $S$ violates requirement $R(q_i,x_i,Z_i)$ , so $S\\lnot \\models R$ , and this completes the proof.", "$\\Box $" ], [ "The pass criterion $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $", "Let $\\overline{x}\\in \\Sigma _I^*$ and $\\text{TS}_\\Leftrightarrow \\subseteq \\Sigma _i^*$ .", "We define a new pass criterion for test cases by $S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\overline{x} \\equiv \\big ( \\omega _S(\\underline{s}, \\overline{x})\\in \\omega _{M_1^{\\prime }}(\\underline{q}, \\overline{x})\\big ),$ and extend this to test suites by $S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{TS}_\\Leftrightarrow \\equiv \\big (\\forall \\overline{x}\\in \\text{TS}_\\Leftrightarrow : S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\overline{x}\\big ).$ Intuitively speaking, a test case represented by some input trace $\\overline{x}$ is passed by the implementation $S$ if and only if the resulting I/O-trace $\\overline{x}/\\omega _S(\\underline{s}, \\overline{x})$ performed by $S$ is contained in the language of $M_1^{\\prime }$ .", "Using the respective output functions, this is expressed here by stating that the output trace generated by $S$ on input trace $\\overline{x}$ in an element of the set of output traces possible in $M_1^{\\prime }$ for this test case.", "Therefore, $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ is just the well-known pass criterion for reduction testing of $S$ against $M_1^{\\prime }$ , that is, for checking whether $L(S)\\subseteq L(M_1^{\\prime })$ holds.", "Figure: Implementation SS from Example .At first glance, one might ask whether it is possible to apply the test suites specified in Theorem REF just with the new pass criterion $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ , in order to create complete test suites for the given requirement.", "This, however, is not true: the following example shows that the suites from Theorem REF are no longer exhaustive, when applied with $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ .", "Consider again reference model $M$ and requirement $R \\equiv R(q_0,a,\\lbrace 0,1\\rbrace ) \\wedge R(q_1,b,\\lbrace 0,2\\rbrace ) \\wedge R(q_2,a,\\lbrace 0,1\\rbrace ).$ from Example REF .", "From Example REF , we know that $a.a.b, a.b.b, b.a.b, b.b.a$ is an exhaustive test suite for pass criterion $\\underline{\\text{\\bf pass}}_\\Rightarrow $ generated according to rules (REF ) and () from Theorem REF .", "Now consider another implementation $S$ as shown in Fig.", "REF .", "Applying the four test cases above to $S$ results in I/O-traces $a.a.b/1.1.1,\\ a.b.b/1.1.0,\\ b.a.b/1.1.0,\\ b.b.a/1.0.1$ It is easy to see that $S$ passes the four test cases when applying pass criterion $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ (just check the observed outputs against $M_1$ from Fig.", "REF ).", "However, $S$ does not satisfy requirement $R(q_1,b,\\lbrace 0,2\\rbrace )$ , and, equivalently, $S$ is not a reduction of $M_1^{\\prime }$ shown in Fig.", "REF : input trace $b.a.a.b$ applied to $S$ results in $b.a.a.b/1.1.0.1,$ but $\\underline{q}\\text{-\\underline{after}-}b.a.a = q_1$ in $M$ , and $R(q_1,b,\\lbrace 0,2\\rbrace )$ only allows 0 or 2 as output when $b$ is applied in state $q_1$ .", "In contrast to that, $S$ outputs 1 when given input $b$ in state $s_0\\text{-\\underline{after}-}b.a.a = s_0$ .", "Expressed in an equivalent way, $b.a.a.b/1.1.0.1\\notin L(M_1^{\\prime })$ , for $M_1^{\\prime }$ shown in Fig.", "REF .", "example" ], [ "Main Theorem on Complete Test Suites", "The following theorem shows that any complete test suite for reduction testing against $M_1^{\\prime }$ can be reduced in a specific way that still guarantees requirements satisfaction if and only if the resulting suite is passed.", "Let $\\text{TS}$ be any complete reduction test suite guaranteeing $L(S)\\subseteq L(M_1^{\\prime })$ if and only if $S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\text{TS}$ for all $S \\in {\\cal D}$ .", "Define $\\overline{\\Pi }=\\bigcup _{i=1}^{k}\\Pi (q_i).\\lbrace x_i\\rbrace .$ Then any test suite $\\text{TS}_\\Leftrightarrow $ satisfying $\\text{pref}(TS) \\cap \\overline{\\Pi }\\subseteq \\text{TS}_\\Leftrightarrow $ is complete for testing $R$ , that is, $S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{TS}_\\Leftrightarrow \\Leftrightarrow S\\models R$ holds for all $S\\in {\\cal D}$ , that is, $\\text{TS}_\\Leftrightarrow $ is a complete suite for testing composite requirement $R$ .", "Let $\\overline{x}\\in \\Sigma _I^*$ be any nonempty input sequence and $\\overline{y} = \\omega _S(\\underline{s}, \\overline{x})$ .", "Let $\\ell = |\\overline{x}|=|\\overline{y}|$ .", "We prove the following derivation for an arbitrary test case $\\overline{x}$ of the complete reduction test suite $\\text{TS}$ and its pass criterion $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ .", "$& & S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\overline{x}\\\\& \\Leftrightarrow & \\forall j\\in \\lbrace 1,\\dots ,\\ell \\rbrace :\\overline{y}(j)\\in \\omega _{M_1^{\\prime }}(\\underline{q}\\text{-\\underline{after}-}\\overline{x}^{[1..j-1]}, \\overline{x}(j))\\\\& \\Leftrightarrow & \\forall j \\in \\lbrace 1,\\dots ,\\ell \\rbrace , \\big (\\underline{q}\\text{-\\underline{after}-}\\overline{x}^{[1..j-1]},\\overline{x}(j)\\big )\\in \\lbrace (q_i,x_i)~|~i=1,\\dots ,k\\rbrace : \\nonumber \\\\ & &\\overline{y}(j)\\in \\omega _{M_1^{\\prime }}(\\underline{q}\\text{-\\underline{after}-}\\overline{x}^{[1..j-1]}, \\overline{x}(j))\\\\& \\Leftrightarrow & S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{pref}(\\overline{x})\\cap \\overline{\\Pi }$ The equivalence (REF ) $\\Leftrightarrow $ () follows from re-writing the original definition of $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ with explicit indexes for input and output events.", "Implication () $\\Rightarrow $ () is trivial, since we restrict the $\\overline{y}(j)$ under consideration to those that are outputs obtained when applying requirements-related inputs $x_i$ in state $q_i$ .", "Conversely, () $\\Leftarrow $ () follows from the fact that for $\\big (\\underline{q}\\text{-\\underline{after}-}\\overline{x}^{[1..j-1]},\\overline{x}(j)\\big )\\notin \\lbrace (q_i,x_i)~|~i=1,\\dots ,k\\rbrace $ , we have $\\omega _{M_1^{\\prime }}(\\underline{q}\\text{-\\underline{after}-}\\overline{x}^{[1..j-1]}, \\overline{x}(j)) = \\Sigma _O$ , so any output is acceptable.", "Finally, equivalence () $\\Leftrightarrow $ () follows from the fact that () is just the definition of $S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{pref}(\\overline{x})\\cap \\overline{\\Pi }$ with explicit indexes.", "With the derivation above, we have shown that in order to prove $L(S) \\subseteq L(M_1^{\\prime })$ , it suffices to check just the test cases $\\text{pref}(\\overline{x})\\cap \\overline{\\Pi }$ for all $\\overline{x}$ in the original complete test suite $\\text{TS}$ .", "This proves $L(S) \\subseteq L(M_1^{\\prime }) \\Leftrightarrow S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{TS}_\\Leftrightarrow .$ Now Theorem REF can be applied to conclude that $S\\models R\\Leftrightarrow S\\ \\underline{\\text{\\bf pass}}_\\Leftrightarrow \\ \\text{TS}_\\Leftrightarrow $ , and this completes the proof.", "$\\Box $ Applying the adaptive state counting algorithm from [10] to reference model $M_1^{\\prime }$ from Example REF and implementation model $S$ from Example REF , a test suite $\\text{TS}$ with 39 test cases of maximal length is obtained to test for reduction of $S$ against $M_1^{\\prime }$ with $n = 2$ for the number of states in the minimised observable FSM associated with $M_1^{\\prime }$ , and $m = 3$ for the number of states in the minimised version of $S$ (which is identical to $S$ ).", "Restricting this complete reduction test suite to the test cases that are also contained in $\\text{pref}(TS) \\cap \\overline{\\Pi }$ , where $\\overline{\\Pi }$ is specified according to Theorem REF for reference model $M$ and requirement $R$ specified in Example REF , results in 16 test cases $\\begin{array}{llll}a.a.a & a.b.b.b.b & b.a.b.b.b.a & b.b.b.a.b.a \\\\a.a.b.a.b & b.a.a.b.a.a & b.b.a.a & b.b.b.b.a\\\\a.b.a.b.a & b.a.b.a.a & b.b.a.b.a.b & b.b.b.b.b.b\\\\a.b.b.a & b.a.b.a.b.b & b.b.a.b.b & a.a.b.b\\end{array}$ It is easy to see that $S$ passes this test suite when applying pass criterion $\\underline{\\text{\\bf pass}}_\\Leftrightarrow $ and reference model $M_1^{\\prime }$ .", "Moreover, the decrease of test cases in comparison to the full reduction test suite is significant.", "However, observing that the exhaustive test suite for checking requirements satisfaction according to Theorem REF needs only 4 test cases (see Example REF ), shows at least for this example that complete requirements testing needs far more test cases than exhaustive requirements testing.", "This observation will be discussed in more detail in the next section.", "example" ], [ "Exhaustive requirements testing.", "Consider first the maximal length $\\text{tcl}_{max}^{exh}$ of test cases for exhaustive requirements testing according to Theorem REF .", "From the test suite specification in (REF ) and (), we conclude that $\\text{tcl}_{max}^{exh} & \\le & \\text{`maximal length of traces in $V$'} + {} \\\\& & m-n+1 + {} \\nonumber \\\\& & \\text{`maximal length of distinguishing traces $\\gamma $'} \\nonumber $ When using state covers $V$ with minimal-length input traces, the latter are bounded by $n-1$ , where $n$ is the number of states in the prime machine of the reference model.", "Also, minimal-length distinguishing traces are bounded by $n - 1$ .", "This gives us an upper bound $\\text{tcl}_{max}^{exh} \\le n + m - 1$ for the maximal test case length.", "For Example REF , the state cover has traces of maximal length 1, and the minimal-length distinguishing traces have length 1.", "Moreover, $m=n=3$ .", "Applying Formula (REF ) results in $\\text{tcl}_{max}^{exh} = 3$ , and this is confirmed by the test traces calculated in Example REF that are all of length 3." ], [ "Complete requirements testing.", "It is well known that testing for language equivalence requires shorter test cases than testing for reduction.", "This is discussed, for example, in the lecture notes [24]: there, it is shown that for certain reference models, any complete reduction test suite needs test cases of maximal length $\\text{tcl}_{max}^{cmp} = m\\cdot n,$ where $m$ is the maximal number of SUT states, and $n$ the number of states in the minimised observable reference model [24].", "This is reflected by Example REF , where $m = 3$ and $n = 2$ (number of states in the prime machine of $M_1^{\\prime }$ ), and the longest test case has $m\\cdot n = 6$ inputs.", "When calculating upper bounds for the number of test cases needed in an exhaustive or complete test suite, test cases that are prefixes of others can be removed from the suite: if an input traces $\\overline{x}_1$ reveals an error in the implementation, then this error will also be revealed by any longer input traces $\\overline{x}_1.\\overline{x}_2$ which has $\\overline{x}_1$ as prefix.", "The bounds presented here take this observation into account." ], [ "Exhaustive requirements testing.", "The maximal number $\\text{tc}_{max}^{exh}$ of test cases for exhaustive requirements testing coincides with the maximal number of test cases needed for language equivalence testing, since, when choosing requirement $R_{eq}$ specified in Theorem REF , this characterises language equivalence.", "As a consequence, the estimate $\\text{tc}_{max}^{exh}$ for exhaustive requirements testing according to Theorem REF is that of the H-method.", "As pointed out by the inventors of the H-Method in [8], the upper bound depends on the implementation technique for the method in a critical way.", "Based on experiments made in [8] and on our implementation in the fsmlib-cpp, the H-Method usually requires significantly fewer test cases than the well-known W-Method, for which the upper bound $n^2\\cdot |\\Sigma _I|^{m-n+1}$ is well-known [38], [7].", "As a consequence, it is safe to assume that test suites created by the H-Method fulfil $\\text{tc}_{max}^{exh} \\le n^2\\cdot |\\Sigma _I|^{m-n+1}$" ], [ "Complete requirements testing.", "For complete requirements testing according to Theorem REF and Theorem REF , the maximal number $\\text{tc}_{max}^{cmp}$ of test cases required is the maximal number required for reduction testing, where the reduction test suite is generated from the nondeterministic abstraction $M_1^{\\prime }$ created from the original reference model $M$ and the requirement $R$ as described in Section REF .", "We assume that the minimised equivalent of $M_1^{\\prime }$ has $n$ states.", "Any complete reduction testing strategy can be used for this purpose, and from the resulting test suites $\\text{TS}$ , all input traces outside $\\text{pref}(\\text{TS})\\cap \\overline{\\Pi }$ can be removed, as shown in Theorem REF .", "A very basic strategy derived from an investigation of product automata shows that the set $\\Sigma _I^{nm}$ of all input sequences of length $m\\cdot n$ is a complete reduction test suite (see, for example, the lecture notes [24]).", "For most practical examples, (adaptive) state counting methods as published in [28], [29], [10] need significantly fewer test cases that $|\\Sigma _I^{nm}|$ .", "In the general case, however, they may perform even worse than this bound derived from product automata.", "Fortunately, Theorem REF deals with deterministic implementations only, and the specific structure of the nondeterministic reference models $M_1^{\\prime }$ introduced in Section REF guarantees that every state is deterministically reachable in the sense that we can calculate state covers as in the deterministic case, where every input trace is guaranteed to lead to the specified target state – only the outputs accompanying this input trace may vary nondeterministically.", "As a consequence, there exists a deterministic state cover $V$ of $M_1^{\\prime }$ with $n$ elements whose traces have bounded length less or equal to $n-1$ .", "A standard argument from state counting methods (see references above) now implies that test suites of the form $V.\\bigcup _{i=0}^{mn-n + 1} \\Sigma _I^i$ are complete for checking language inclusion.", "Observing that $|V| = n$ and prefixes of other test cases can be removed from a test suite without impairing its completeness properties, this results in the upper bound $\\text{tc}_{max}^{cmp} = n\\cdot |\\Sigma _I|^{mn - n + 1},$ for the number of test cases needed to check whether the SUT is a reduction of $M_1^{\\prime }$ ." ], [ "Discussion of Bounds for the Number of Test Cases", "Comparing the dominating values in formulas (REF ) and (REF ), we find that the exhaustive strategy is only exponential in the difference $m-n$ , whereas the complete strategy is exponential in the product $m\\cdot (n-1)$ .", "This confirms the observation from Example REF that complete requirements testing needs considerably more test cases that the exhaustive strategy.", "From a practical perspective, it will be useful in most situations to learn additional errors about violations of language equivalence, even if they do not represent violations of requirements.", "The disadvantage of having to debug whether a failed test case points to a requirements violation or “only” to a general violation of language equivalence seems of lesser importance to us than the fact that the exhaustive strategy needs fewer test cases." ], [ "Related Work", "The use of formal specifications, in particular, reference models with formal behavioural semantics, has a long tradition in testing [11], [27], [1].", "Among the numerous formal approaches, complete test strategies have received special attention, because they guarantee full fault coverage with respect to a reference model and a conformance relation under certain well-defined hypotheses concerning the potential faults of the system under test (SUT).", "Complete strategies are of particular interest in the domain of safety-critical systems, where a justification of the test case selection is required in order to obtain certification credit.", "These strategies have been comprehensively investigated in the context of conformance testing.", "Typical conformance relations were I/O-language equivalence or language containment [38], [7], [8], [8], [32], [10], [28], [13], refinement relations for process algebras and related formalisms [4], [5], and the well-known ioco-relation [36].", "In the context of hybrid systems, new conformance relations have been proposed, for example, in [2].", "Theorem REF states that any complete reduction test suite can be modified to yield a (usually smaller) complete suite for requirements testing.", "Typical complete reduction testing strategies are based on a state counting method; these are essential for testing reduction relations between nondeterministic reference models and (deterministic or nondeterministic) SUTs.", "State counting has been explained in [10], [28].", "The fact that reduction (in our case, the reduction of a model abstraction $M_1^{\\prime }$ presented in Section REF ) preserves requirements satisfaction is a very general insight which holds for different modelling formalisms and requirements specification methods.", "We refer here the well-known fact that LTL specifications can be checked by a maximally nondeterministic Buchi automaton and all refinements thereof fulfil the formula as well [3].", "The Unified Theories of Programming [12] investigate both requirements satisfaction and refinement on a more general logical level and provide the insight that refinement is strongly related to logical implication.", "Therefore, requirements satisfaction is preserved by refinement as a simple logical consequence.", "While conformance testing is preferred in the field of protocol verification [20], [37], other application areas follow the property-driven approach, where it has to be established that the SUT implements a collection of requirements in a correct way [19], [9], [17], [33], [23].", "Again, models reflecting the requirements under consideration may be used.", "Alternatively, implicit specifications in linear temporal logic can be constructed to identify the I/O-traces fulfilling a requirement; the underlying theory has been elaborated, for example, in [3] and [31].", "Since logic specifications referring to the interfaces of the SUT alone can become very complex, it is often advocated to combine models with temporal logic specifications, so that the latter may also refer to internal states of the model, thereby simplifying the formulae [22], [23].", "When applied to complex real-world systems, complete FSM-based testing methods usually require very large test suites.", "This problem has been mitigated in recent years by abstraction techniques based on equivalence classes and symbolic state machines, see, for example, [16], [26]: the original FSM-based test generation algorithms can be applied to abstractions of more complex models, such as extended finite state machines or UML state machines.", "Examples presented in [25] show that these abstraction techniques allow for complete testing of quite complex control systems with feasible effort." ], [ "Conclusion", "In this paper, a notion for specifying elementary and composite requirements in deterministic finite state machine models has been defined.", "Two black-box testing strategies proving or disproving that an implementation satisfies these requirements have been presented.", "The first is exhaustive in the sense that every implementation violating the requirement will fail at least one test case generated according to the strategy.", "Failing a test case implies that the SUT is not language equivalent to the reference model, but may not necessarily mean that the implementation violates the requirement.", "The second is complete in the sense that it is exhaustive and guarantees that failing a test case alway implies that the system under test violates the requirement.", "The exhaustiveness and completeness properties, respectively, of the test suites generated according to these strategies hold under the assumption that the implementation has no more than $m \\ge n$ states, where $n$ is the known number of states in the reference model.", "The implementation of the first strategy is based on the H-method.", "Using a real-world application, it is demonstrated that the first strategy frequently requires significantly less test cases than a complete method establishing language equivalence by means of the original H-method.", "Therefore, the new method is well-suited for testing a selection of critical requirements with guaranteed fault coverage, while less critical requirements can be tested in the conventional way, using heuristics for test case generation.", "The implementation of the second strategy is based on a state counting method which has been originally used to test for language inclusion.", "While the second strategy leads to smaller test suites in comparison to complete methods showing language inclusion, it usually results in more test cases than needed for establishing language equivalence.", "Consequently, it is of more theoretical interest, at least if a model can be constructed that is equivalent to the desired behaviour of the implementation.", "The authors would like to thank Robert Sachtleben for pointing out important details about the complexity of state counting methods." ], [ "Tool Support and Resources", "The test suites presented in the examples above have been generated using the open source library fsmlib-cpp.", "The library is downloaded, compiled, and some executables are created according to the instructions given in [24].", "After that, an executable fsm-generator is available which allows for test suite generation according to various strategies without having to write own main programs accessing the library classes and their methods.", "Reference model $M$ and SUT model $S$ from Example REF can be found in the fsmlib-cpp installation, directory resources/complete-requirements-based-testing/Example-2-3-4 as files M.csv and S.csv, respectively (deterministic FSMs can be specified in CSV format as explained in [24]).", "To re-generate the H-method test suite checking language equivalence as described in Example REF , change into the Example-2-3-4-directory and type command <path-to-executable>/fsm-generator -h -a 0 M.csv Option -a specifies the maximal number of additional states allowed in the SUT, so this generation creates an H-test suite from $M$ which is complete for language equivalence testing under the assumption that $S$ does not have more states than $M$ .", "For generating the exhaustive requirements test suite from $M$ and $R$ according to Example REF , the abstraction $M_1$ has to be manually created from $M$ and $R$ as specified in Example REF .", "This DFSM is also stored in the Example-2-3-4 directory as file M1.csv.", "The exhaustive requirements test suite is now generated by command <path-to-executable>/fsm-generator -s -h -a 0 M.csv M1.csv The -s parameter specifies requirements-based testing, and in such a case, a second FSM specification file (here: M1.csv) is expected as parameter.", "This call to the generator creates exactly the test suite with the 4 test cases shown in Example REF for requirements-based testing according to Theorem REF .", "For generating the test suites related to the Fasten-Seat-Belt and Return-to-Seat-Sign controller described in Section , change into directory resources/complete-requirements-based-testing/Section-6 The reference model described in Section  is stored in this directory as FSBRTSX.csv (note that the csv-file uses other state names than the $s_i$ shown in Table REF ).", "The DFSM abstraction created from requirement $\\mathbf {R}_1$ is contained in file FSBRTSX-ABS-R1.csv.", "To re-create the test suites for different values of $m-n$ as specified in Table REF , column $\\mathbf {R}_1$ , use commands <path-to-executable>/fsm-generator -s -h -a 0 FSBRTSX.csv FSBRTSX-ABS-R1.csv <path-to-executable>/fsm-generator -s -h -a 1 FSBRTSX.csv FSBRTSX-ABS-R1.csv <path-to-executable>/fsm-generator -s -h -a 2 FSBRTSX.csv FSBRTSX-ABS-R1.csv For requirement $\\mathbf {R}_2$ specified in Section , use abstraction file file FSBRTSX-ABS-R2.csv and commands that are equivalent to the ones shown above.", "To create the complete test suites for language equivalence with the H-Method, use commands <path-to-executable>/fsm-generator -h -a 0 FSBRTSX.csv <path-to-executable>/fsm-generator -h -a 1 FSBRTSX.csv <path-to-executable>/fsm-generator -h -a 2 FSBRTSX.csv" ] ]
2105.11786
[ [ "Detectable Gravitational Wave Signals from Affleck-Dine Baryogenesis" ], [ "Abstract In Affleck-Dine baryogenesis, the observed baryon asymmetry of the Universe is generated through the evolution of the vacuum expectation value (VEV) of a scalar condensate.", "This scalar condensate generically fragments into non-topological solitons (Q-balls).", "If they are sufficiently long-lived, they lead to an early matter domination epoch, which enhances the primordial gravitational wave signal for modes that enter the horizon during this epoch.", "The sudden decay of the Q-balls into fermions results in a rapid transition from matter to radiation domination, producing a sharp peak in the gravitational wave power spectrum.", "Avoiding the problem of gravitino over-abundance favours scenarios where the peak frequency of the resonance is within the range of the Einstein Telescope and/or Decigo.", "Therefore, we show this scenario provides an observable signal, providing a mechanism to test Affleck-Dine baryogenesis." ], [ "Introduction", "The slight asymmetry between matter and anti-matter is one of the cornerstone puzzles of modern particle cosmology, as the Standard Model fails to provide an explanation [1], [2], [3].", "An elegant paradigm for explaining the slight asymmetry is the Affleck-Dine mechanism [4], [5], [3], [6].", "Supersymmetric theories generically have flat directions [7], [5], which have non-zero baryon or lepton number.", "During inflation, a scalar condensate generically develops in these directions, whose non-zero vacuum expectation value (VEV) spontaneously breaks C and CP.", "At the end of inflation, a baryon and/or lepton asymmetry is generated as the VEV coherently evolves and the condensate fragments [8].", "These resulting clumps may be long-lived non-topological solitons (Q-balls) [9], [10], [11], [12], carrying either lepton or baryon number [13].", "This global charge is transferred to Standard Model particles when the Q-balls decay.", "However, the Affleck-Dine mechanism is generically a high-scale phenomenon, making it difficult to confirm observationally.", "In this paper, we argue that a broad class of Affleck-Dine models significantly enhance the primordial gravitational wave power spectrum.", "This provides a novel mechanism to test or constrain these models.", "Generically, the Q-balls produced through the fragmentation of the Affleck-Dine condensate are large and long-lived.", "Consequently, they may evolve as non-relativistic matter, and eventually come to dominate the energy density of the Universe.", "If the Q-balls decay rapidly, there is a sudden change in the equation of state for the Universe.", "This results in rapidly oscillating scalar perturbations, which enhances the primordial gravitational wave spectrum from inflation.", "This is analogous to the so-called poltergeist mechanism, in which the sudden decay of black holes also enhances the gravitational wave spectrum [14].", "This is in contrast to the case where gravitational waves are produced during the fragmentation of the condensate, as it is typical that the condensate is a small fraction of the initial total energy [15], [16].", "Our proposed test is also potentially complimentary to tests that consider predictions on the ratio of scalar to isocurvature perturbations in D-term inflation [17] and the backreaction of the Affleck-Dine potential onto the inflaton potential which can conflict with cosmic microwave background constraints [18].", "In this letter, we first argue that Affleck-Dine scenarios generically have this epoch of early matter domination, and secondly, the Q-ball decay rate is sufficiently fast to enhance the gravitational wave spectrum.", "In particular, the sudden transition avoids the suppression that occurs in a gradual transition like Moduli decay [19].", "The conditions for a fast transition are easily satisfied.", "Analytical arguments and simulations show that the Q-ball mass distribution is sharply peaked [8], [20], [21], [22].", "Secondly, the charge quanta inside the Q-balls decay to fermions.", "The decay rate is suppressed by a surface area to volume factor, as explained below.", "The decay rate then accelerates as the Q-ball decays, similar to black hole decay.", "Furthermore, avoiding an overabundance of gravitinos results in a gravitational wave spectrum that is at sufficiently low frequencies to be observed.", "Finally, although in this work we make no statistical claims, we present a variety of points in parameter space where the Q-balls are sufficiently long lived to dominate the energy density and produce a detectable gravitational wave signal.", "Thus, if such a signal is observed, we can narrow the cause down to two known scenarios - an early period of Q-ball domination, which is likely a consequence of Affleck-Dine baryogenesis, or an early period of light primordial black hole domination [14]." ], [ "Q-ball Induced Early Matter Domination", "During inflation, the field $\\Phi $ acquires a vacuum expectation value when averaged over super-horizon scales [23], [24], [4], [25], [26].", "At the end of inflation, it relaxes towards its equilibrium value as the field fragments to form Q-balls [8], [3].", "During the relaxation process, a charge excess is produced as the field VEV follows a curving path in field space, which is biased either clockwise or counter-clockwise by the small CP-violating operator.", "However, as a higher dimensional operator, it will also be sensitive to the initial post-inflationary VEV, which is subject to random fluctuations during inflation.", "Consequently, some Hubble patches will have an excess of $Q$ charge while other have an excess of $\\bar{Q}$ charge.", "Therefore, there are symmetric and asymmetric components to the initial Q-ball density.", "After fragmentation, most of the condensate's initial energy is contained in Q-balls rather than individual particles, particularly if the couplings between the scalar field and fermions is small [8], [20].", "If the asymmetric component is small (as is expected due to the smallness of the observed baryon asymmetry), the symmetric component must then be large.", "We parameterize the asymmetric component of the Q-ball energy density $r = \\frac{n_{\\bar{Q}}- n_{Q}}{n_{\\bar{Q}}+n_Q} \\ ,$ and we expect $r$ to be within an order of magnitude of the baryon asymmetry.", "(This can also be understood as a consequence of a highly elliptical orbit during relaxation, which simulations connect to a large symmetric component [22].)", "In the thin wall regime, the vacuum expectation value inside the Q-ball can be found by minimizing $V(\\Phi ^2) \\slash \\Phi ^2$ where $\\Phi $ parameterizes the flat direction in the Affleck-Dine potential.", "The energy per unit charge of a single Q-ball is given by $\\omega = \\sqrt{ \\dfrac{2V(v)}{ v^2}},$ where $v$ is the VEV inside the Q-ball.", "(We discuss specific potentials in Appendix .)", "The total initial energy in Q-balls after fragmentation is then $\\rho _Q = Q_0 \\omega n_0$ where $n_0\\sim N_Q H_0^3$ is the initial number density of the Q-balls and $Q_0$ is their initial charge.", "Simulations suggest $N_Q \\sim 1000$ for gravity-mediated SUSY scenarios and $N_Q \\sim \\mathcal {O}(1)$ for most gauge-mediated scenarios, if higher dimensional operators are negligible.", "However, in this scenario the resulting Q-balls are in the thick wall (as opposed to thin wall) regime [21].", "Although we focus on thin wall Q-balls in this work, we note that scaling arguments suggest thick wall Q-balls are longer lived and thus also can induce early matter dominated epochs (see Appendix ).", "It is straightforward to derive a condition for the initial charge of the Q-balls in terms of the initial baryon asymmetry $Y_{B0}$ , $r$ , and the reheating temperature $T_0$ , $Q_0 = \\frac{3 Y_{B0} M_{\\rm Pl}^3}{800 \\sqrt{5} \\pi ^{5/2} g_\\ast r T_0 ^3}.$ The initial Q-balls produced after fragmentation are typically quite large; in our benchmark scenarios, the initial charges are above $10^{29}$ .", "Consequently, they will travel at non-relativistic speeds in the post-inflationary plasma.", "Then, if the Q-balls live long enough, they dominate the energy density of the universe.", "We can approximate the temperature of matter-radiation equality in the limit where Q-ball decay is negligible as $T_{\\rm eq} \\sim \\frac{4 Y _{B0} \\omega }{3 r} \\ .$ Although long-lived, the Q-balls produced by the fragmentation of the Affleck-Dine condensate are not absolutely stable; indeed, they cannot be since their conserved charge must be transferred to Standard Model particles.", "The sfermions carrying the charge can decay to a quark (or lepton) and neutralino or chargino.", "Expressions for the relevant coupling can be found in Ref.", "[27], although we will parameterize the vertex in terms of an effective Yukawa coupling $y_{\\rm eff}$ .", "This decay happens only at the surface of the Q-ball, for one of two reasons.", "First, if the VEV of the squark or slepton fields inside the Q-ball (which carry the charge) is significantly larger then the energy per unit charge $\\omega $ , then the large induced fermion masses can forbid the decay inside the Q-ball.", "The induced masses of the Standard Model fermions have magnitude $gv$ , where $g= g_3$ for quarks if the Q-ball is made of squarks and $g = g_2$ for leptons if the Q-balls is made of sleptons [28].", "Therefore, if $gv \\gtrsim \\omega $ , then the decay occurs only at the surface of the Q-ball, where the VEV drops to zero.", "Alternatively, if the decay is not forbidden, then decays in the interior of the Q-ball rapidly fill up the Fermi sea.", "Thereafter, the Q-ball quanta decay only at the surface as long as the diffusion time, $t_D \\sim 3 R^2/\\lambda $ , is sufficiently long.", "The mean free path is $\\lambda \\sim 1/{\\sigma _{\\psi \\phi } n}$ , where number density $n = 3 Q_0/ (4 \\pi R^3)$ refers to the density of scalar quanta inside the Q-ball.", "The diffusion time is shortest for the highest momentum, which is $\\sim \\omega \\slash 2$ when the decay is energetically forbidden.", "The diffusion time can then be approximated using the scattering cross section $\\sigma _{\\psi \\phi } \\sim g^4_i \\slash (\\omega \\slash 2)^2$ , where $g_i \\in (g_Y,g_2,g_3)$ depending on the Standard Model fermion and sfermion involved.", "For the benchmark points presented below, decays inside the Q-ball are suppressed for the first reason.", "Regardless of the reason, the Q-ball evaporation rate is suppressed by the ratio of the surface area to the volume.", "Therefore, the decay rate accelerates as the Q-ball shrinks, resulting in an effectively instantaneous decay, similar to black holes [29], [14].", "The charge depletion per unit time per unit area of a Q ball obeys the equation [30] $\\frac{dQ}{dt\\, dA} = \\frac{y_{\\rm eff} v \\omega ^2}{64 \\pi }$ where $v$ is the field value of the condensate and $y_{\\rm eff}$ is the effective Yukawa coupling, accounting for mixing angles.", "In the thin wall limit, $R = \\left( \\dfrac{3Q}{4 \\pi \\omega v^2} \\right)^{1 \\slash 3}, $ which gives $\\Gamma _{\\rm Q-ball} = \\frac{y_{\\rm eff} v \\omega ^2}{16} \\left( \\frac{3Q}{4 \\pi \\omega v^2} \\right)^{2 \\slash 3} \\ .$ Q-ball decay becomes rapid when $\\Gamma _{\\rm Q-ball} \\slash HQ \\sim 1$ .", "For the Q-balls to decay after dominating the energy density, we must therefore require $\\Gamma _{\\rm Q-ball} \\slash HQ \\big |_{T=T_{\\rm eq}} \\ll 1 $ .", "Approximating the left side at $Q = Q_0$ , we find the condition $\\frac{0.178 y_{\\rm eff} r^{7/3}T_0}{ Y_{B0} ^{7/3} \\omega ^{2/3} ( g_\\ast v)^{1/3}}\\left(\\frac{1000}{N_Q} \\right)^{1/3} \\ll 1 \\ .$ The large symmetric component $r \\sim Y_{B0}$ is vital due to the $Y_{B0}^{- 7 \\slash 3}$ factor, which would otherwise make this condition difficult to satisfy.", "As expected, this prefers small Yukawa couplings, which result in long-lived Q-balls.", "We emphasize that for our numerical analysis, we solved the differential equation $dQ \\slash dt = -\\Gamma _{\\rm Q-Ball}(Q,T)$ .", "We also note that the initial baryon asymmetry will be larger than it is today because the decay of the Q-ball dilutes the charge asymmetry.", "The final asymmetry is given by $Y_{B} = Y_{B0} \\left( 1+ \\frac{4Y_{B0}}{3 r} T_{\\rm dec} \\right)^{-3/4},$ where $T_{\\rm dec}$ is the temperature at which the Q-balls decay and $Y_B=8.59\\times 10^{-11}$ as given by Planck [31].", "Because the Q-ball mass fraction is sharply peaked at a single value and the decay is effectively instantaneous, the scale factor and Hubble approximately obey step function solutions $\\frac{a(\\eta )}{a(\\eta _R) } = \\left\\lbrace \\begin{array}{cc} \\left( \\frac{\\eta }{\\eta _R} \\right)^2 \\\\ 2 \\frac{\\eta }{\\eta _R} - 1 \\end{array} \\right.", "\\ , \\quad H(\\eta ) = \\left\\lbrace \\begin{array}{cc}\\frac{2}{\\eta } & (\\eta \\le \\eta _R) \\\\\\frac{1}{\\eta - \\eta _R/2} & (\\eta > \\eta _R)\\end{array} \\right.$ where $\\eta $ is the conformal time; $\\eta _R$ is specifically the conformal time at which radiation domination recommences." ], [ "Gravitational waves", "We assume inflation generates a primordial scalar power spectrum of the form ${\\cal P} _\\zeta (k) = \\Theta (k_{\\rm inf} - k) A_s \\left( \\frac{k}{k_\\ast } \\right)^{n_s-1} $ for some cutoff scale $k_{\\rm inf}$ with $n_s$ being the spectral tilt, $k_\\ast $ being the pivot scale and $A_s$ being the amplitude at the pivot scale.", "We take typical values of $A_s=2.1\\times 10^{-9} $ , $n_s =0.97$ , $k_\\ast = 0.05 \\ {\\rm Mpc} ^{-1} $  [31].", "Scalar perturbations grow with the scale factor during any matter domination epoch, including the Q-ball dominated epoch mentioned above, which can in turn induce gravitational waves [29], [19].", "Our analysis of the induced gravitational wave signal follows [19] and therefore we similarly work within the conformal Newtonian gauge and assume Gaussian curvature perturbations.", "If matter domination is sufficiently long, then perturbations at small scales can enter the non-linear regime where a linear analysis is insufficient.", "Such non-linearities become important at scales $k_{\\rm NL} \\sim 470/\\eta _R$ , where $\\eta _R$ is the conformal time at which the Q-ball-caused matter domination era abruptly ended.", "In this work, we neglect the non-linear regime and therefore we restrict ourselves to points in parameter space at which the maximum comoving mode enhanced by early matter domination satisfies $k_{\\rm max} \\lesssim 470/\\eta _R$ .", "We note that there may still be detectable gravitational wave signals in the parameter space where this is not satisfied, although we leave the analysis of the non-linear regime to future work.", "Using the step function approximations given above, the power spectrum of gravitational waves at conformal time $\\eta $ is [32] $\\Omega _{\\rm GW} (\\eta , k) = \\frac{1}{24} \\left( \\frac{k}{a (\\eta )H(\\eta )} \\right)^2 \\overline{{\\cal P } _h (\\eta ,k)}$ where the time averaged power spectrum of the induced gravitational waves is related to the scalar (curvature) power spectrum as $\\overline{{\\cal P } _h (\\eta ,k)} = 4 \\int _0 ^\\infty dv && \\int _{|1-v|}^{1+v} du \\left( \\frac{4 v^2-(1+v^2 - u ^2)^2}{4 v u} \\right) ^2 \\nonumber \\\\ && \\times \\overline{I^2(u,v,k,\\eta ,\\eta _R} {\\cal P} _\\zeta (uk) {\\cal P} _\\zeta (vk) \\ .$ In the above the time dependence of the gravitational waves is $I(u,v,k,\\eta ,\\eta _R) = \\int _0 ^{k \\eta } d (\\overline{k \\eta }) \\frac{a (\\bar{\\eta })}{a(\\eta )} k G_k (\\eta , \\bar{\\eta } ) f(u,v, \\overline{k \\eta } , k \\eta _R)$ where the Greens function is the solution of the equation $\\frac{\\partial ^2 G(\\eta , \\bar{\\eta })}{\\partial \\eta ^2} + \\left( k^2 - \\frac{1}{a}\\frac{\\partial ^2 a}{\\partial \\eta ^2} \\right)G(\\eta , \\bar{\\eta }) = \\delta (\\eta - \\bar{\\eta } )$ and the source function has the form $f(u,v, \\overline{k \\eta } , k \\eta _R) &=& \\frac{3}{25(1+w)} \\left( 2(5+3 w) \\Phi (u \\overline{k \\eta } ) \\Phi (v \\overline{ k \\eta }) \\right.", "\\nonumber \\\\&& \\left.", "+ 4 H^{-1} \\frac{\\partial }{\\partial \\eta } \\left( \\Phi (u \\overline{k \\eta } ) \\Phi (v \\overline{ k \\eta }) \\right) \\right.", "\\nonumber \\\\ && \\left.", "+4 H^{-2} \\frac{\\partial }{\\partial \\eta } \\Phi (u \\overline{k \\eta } )\\frac{\\partial }{\\partial \\eta } \\Phi (v \\overline{k \\eta } ) \\right) \\ .$ In these equations $w$ is the equation of state parameter and $\\Phi $ is the transfer function of the gravitational potential, which obeys the evolution equation [33] $\\frac{\\partial ^2 \\Phi }{\\partial \\eta ^2} + 3(1+w) H \\frac{\\partial \\Phi }{\\partial \\eta } + w k^2 \\Phi = 0 \\ .$ For a sufficiently quick transition from matter to radiation domination, we can use the analytic formulae for the gravitational wave power spectrum in Ref.", "[29] which we give in the supplementary material.", "This rapid transition is necessary to produce the sharp peak through the “poltergeist” mechanism [14].", "During the early matter domination epoch, density perturbations in non-relativistic Q-ball modes grow and form overdensities.", "The Q-ball decay, which is rapid as compared to the Hubble time, converts these overdensities into relativistically moving sound waves, which serve as sources of gravitational waves.", "Gravitational waves exhibit a rapidly growing resonance mode which is amplified by interactions with a sound wave comoving at a certain angle [34], [29], [14].", "This resonance results in a dramatic enhancement at a certain frequency, as can be seen in our Fig.", "REF .", "It is important that the transition to radiation domination is rapid, because otherwise the overdensities dissolve gradually and do not result in any relativistically moving modes." ], [ "Results", "We present the gravitational wave signal for three sample points in parameter space in Fig.", "REF .", "These were chosen to have Yukawa couplings similar in size to those in the Standard Model; the precise values of the parameters are given in Table REF .", "To retain generality, we specify the VEV $v$ and energy density per charge $\\omega $ of the Q-balls, instead of specializing to a particular potential.", "Gravity and gauge-mediation models which produce Q-balls with these properties are discussed in Appendix .", "Calculated quantities, such as the equality and decay temperatures, for these benchmark points are given in Table REF .", "We note that since $\\omega $ is within one order of magnitude of $T_0$ , the temperature at which Q-balls are produced, it is self-consistent to neglect finite temperature corrections to $\\omega $ , which are induced via loop corrections.", "The observable range is controlled by the proposed frequency sensitivity of future gravitational wave detectors with high temperature probed by higher frequencies.", "At present the highest frequency gravitational wave detectors with enough sensitivity are the Cosmic Explorer [42] and the Einstein Telescope [44] although higher frequency proposals are a promising work in progress [45].", "We see that DECIGO has particularly good coverage of our expected signal.", "There is a modest trend for points with smaller Yukawa couplings to decay later and therefore to have lower frequency peaks.", "For the signal to be observable, the Q-balls must decay when the temperature falls in the range $20 \\ {\\rm GeV} < T_{\\rm dec} < 2 \\times 10^7 \\ {\\rm GeV} $ .", "The upper bound is frequently satisfied even for large reheating temperatures, although a low reheating temperature is often preferred to avoid overproduction of gravitinos, though the exact bound on the reheating temperature depends on the mass of the gravitino [46], [47], [48], [49].", "We have imposed $T_R<10^7$ GeV for all benchmarks.", "Table: Parameters used in our three benchmark points in Fig. .", "In addition to the Q-ball parameters ω\\omega and vv, Y B Y_B is the initial charge asymmetry after fragmentation which occurs at temperature T 0 T_0, N Q N_Q is the average initial number of Q-balls per Hubble volume after fragmentation, and r∼Y B r \\sim Y_B is the ratio of the asymmetric component.", "Note that the Yukawa couplings are equal to that of the Standard Model bottom quark, up quark, and electron in the top, middle, and bottom rows.", "Additionally, we have taken g * =106g_*=106 in our analysis.Table: Calculated quantities for the three benchmark points in .", "T eq T_{\\rm eq} is the temperature of Q-ball-radiation equality and T dec T_{\\rm dec} is the temperature of Q-ball decay." ], [ "Conclusions", "As a high-scale phenomenon, it is difficult to observationally confirm Affleck-Dine baryogenesis.", "We have shown here that broad class of Affleck-Dine models produce a detectable gravitational wave signal within the range of the Einstein Telescope and/or Decigo.", "Such signals are a consequence of the sudden end of an early matter-domination epoch, which occurs if the Q-balls from the fragmentation of the Affleck-Dine condensate are sufficiently long-lived.", "A low reheating temperature, motivated by the gravitino problem, ensures a signal within the observable frequency range, but we find that this is not a requirement.", "Thus, if a signal is indeed observed, we can narrow the cause down to two known scenarios- an early period of Q-ball domination, which is a natural outcome of Affleck-Dine baryogenesis, or an early period of light primordial black hole domination [14]." ], [ "Acknowledgments", "The works of AK and GW were supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.", "A.K.", "was supported the U.S. Department of Energy (DOE) grant No.", "DE-SC0009937 and by Japan Society for the Promotion of Science (JSPS) KAKENHI grant No.", "JP20H05853.", "We thank Kazunori Kohri for useful calculations." ], [ "Explicit Potentials", "To keep the discussion as general as possible, we have avoided specifying a potential.", "In this appendix, we discuss Q-balls in both gauge-mediated and gravity-mediated scenarios.", "In the gauge-mediated supersymmetric scenario the potential is $V(\\Phi ) = m^4 \\log \\left( 1+ \\frac{|\\Phi |^2}{m^2} \\right) + \\frac{1}{\\Lambda ^2} |\\Phi |^6,$ plus a small CP-violating term.", "When the second term is negligible, the resulting Q-balls are thick wall Q-ball [21], for which the analysis presented here is inapplicable.", "However, scaling arguments favor a Q-ball domination epoch.", "Fragmentation tends to produce one large Q-ball in each Hubble volume [20], resulting in a sharply peaked mass distribution at large masses, which tend to be long-lived.", "Furthermore, the radius now scales as $Q^{1 \\slash 4}$ , and after accounting to the scalings of the VEV and energy per unit charge with $Q$ , we expect the Q-ball decay rate to scale as $Q^{1 \\slash 4}$ .", "This is suppressed compared to the thin-wall rate, and therefore, the Q-balls will tend to be longer-lived.", "$\\Gamma _{\\rm Q-ball} \\slash HQ$ then scales as $Q^{-3 \\slash 4}$ , which increases as the charge decreases, leading to the rapid matter-to-radiation transition.", "We plan to address this scenario more fully in future work.", "When the second term in (REF ) is not negligible, then the resulting Q-balls are in the thin wall regime, although since the energy per charge $\\omega $ is independent of the charge $Q$ , the equilibrium state is not one Q-ball per Hubble volume.", "Alternatively, one can consider gravity-mediated SUSY breaking, in which case the Affleck-Dine condensate has the potential $V(\\Phi ) = m^2 |\\Phi |^2 \\left( 1 + K \\log \\left( \\dfrac{|\\Phi |^2}{m^2} \\right)\\right) + \\dfrac{1}{\\Lambda ^2} |\\Phi |^6,$ where $\\Lambda $ is an effective scale for the higher-dimensional operator, $m$ is the mass of the scalar field, and $K \\approx -0.01$ to $-0.1$ is a one-loop correction.", "Since $K<0$ , Q-balls can be formed, and in the thin wall limit, the VEV inside the Q-ball is given by [11] $v \\approx \\left( \\Lambda M_{\\rm Pl} \\sqrt{ \\dfrac{|K|}{2}} \\right)^{1 \\slash 2},$ from which an expression for $\\omega $ can be found.", "In Table REF we show one choice of parameters for the gauge-mediated potential (left) and gravity-mediated potential (right) for each benchmark set of parameters discussed in the text.", "That is, in each row the Q-balls produced have the same VEV $v$ and energy-per-unit-charge $\\omega $ as in corresponding row in Table REF .", "For the gauge-mediated scenario, we have ensured that the sixth order term is relevant so that we are in the thin-wall regime.", "We see that in all cases the scale of the effective operator, $\\Lambda $ , is well above the reheating temperature.", "Table: Parameters for the potentials () and () which produce Q-balls corresponding to our three benchmark points.", "The first row corresponds to our first benchmark in Table , the second row corresponds the second benchmark, and the third corresponds to the third.", "For the gravity-mediated scenario, we have fixed k=-0.05k=-0.05." ], [ "Supplementary material", "We here outline the gravitational wave spectrum from an instantaneous transition from matter to radiation domination, following Ref [29].", "We define the following functions: $& s_0 (x,x_{\\rm max})= \\left( \\Theta \\left[ \\frac{2 x_{\\rm max}}{1+\\sqrt{3}}-x \\right] \\right.", "+ \\left.", "\\left(2 \\frac{x_{\\rm max}}{x} - \\sqrt{3} \\right) \\right.", "\\nonumber \\\\ & \\left.", "\\Theta \\left[ x- 2 \\frac{x_{\\rm max}}{1+ \\sqrt{3}} \\right] \\Theta \\left[ 2 \\frac{x_{\\rm max}}{\\sqrt{3}} -x \\right] \\right) \\times \\Theta \\left[ 2 \\frac{x_{\\rm max}}{ \\sqrt{3}} - x \\right] ,$ along with $S_i (x) &= \\int _0 ^x dz \\frac{\\sin z}{z} , \\qquad C_i (x) = - \\int _x ^\\infty dz \\frac{\\cos z}{z} \\ .$ The gravitational wave spectrum involves a resonant contribution, $\\Omega _{\\rm res}$ , an infrared contribution, $\\Omega _{\\rm IR}$ , and a non-resonant UV contribution, $\\Omega _{\\rm UV}$ , such that the total spectrum is given by $& \\Omega _{\\rm GW}(x,x_{\\rm max}) = \\Omega _{\\rm res} + \\nonumber \\\\&3 A_s^2 x_{\\rm max}^8 \\frac{4 C_i[\\frac{x}{2}]^2+(\\pi - 2 S_i [\\frac{x}{2}])^2}{2^{17+2 n_s}625(3+2n_s)} \\left( \\frac{2 x_{\\rm max}}{x} -1 \\right) ^{2n_s} \\nonumber \\\\ & \\times \\left( \\frac{x}{x_\\ast } \\right) ^{2 (n_s -1)} \\left( \\Omega _{\\rm IR} \\Theta [x_{\\rm max}-x] + \\Omega _{\\rm UV} \\Theta \\left[ x-x_{\\rm max} \\right] \\right), \\nonumber \\\\$ where we define $X \\equiv x \\slash x_{\\rm max}$ to write $& \\Omega _{\\rm IR}= \\frac{1}{(2+n_s)(3+n_s)(4+n_s)(5+2n_s)(7+2n_s)} \\nonumber \\\\&\\times \\left( 1536-6144 X + (7168-1920n_s-256 n_s^2) X^2 \\right.", "\\nonumber \\\\& \\left.", "+ (5760 n_s +768 n_s^2)X^3 \\right.", "\\nonumber \\\\ & \\left.", "+(1328 n_s +3056 n_s^2 + 832 n_s^3 +64 n_s^4) X^4 \\right.", "\\nonumber \\\\& - \\left.", "(7168 +12256 n_s +7392 n_s^2 +1664 n_s^3 +128 n_s^4)X^5 \\right.", "\\nonumber \\\\& + \\left.", "(7392 + 10992 n_s +5784 n_s^2 + 1248 n_s^3 +96 n_s^4) X^6 \\right.", "\\nonumber \\\\& - \\left.", "(2784+3904 n_s +1960 n_s^2 +416 n_s^3 + 32 n_s^4) X^7 \\right.", "\\nonumber \\\\& + \\left.", "(370 + 503 n_s +247 n_s^2 + 52 n_s^3 + 4 n_s^4) X^8 \\right.", "\\nonumber \\\\& - \\left.", "256 \\left(1- X\\right)^6 \\left[(6+ 6(2+n_s) X \\right.", "\\right.", "\\nonumber \\\\& \\left.", "\\left.", "+(2+n_s)(5+2n_s)X^2\\right]\\left(1-\\frac{X}{2- X}\\right)^{2 n_s} \\right),$ and $& \\Omega _{\\rm UV} = 2 \\left( 2 - X\\right)^4 \\Gamma [4+2n_s]\\nonumber \\\\& \\times \\left( \\dfrac{X^4}{\\Gamma [5+2n_s]} - \\frac{4 X^2 \\left(2-X \\right)^2}{\\Gamma [7+2n_s]} + \\frac{24 \\left( 2-X\\right)^4}{\\Gamma [9+2n_s]} \\right),$ $\\Omega _{\\rm res} &= \\frac{2.3 \\sqrt{3} 3^{n_s}}{625 \\times 2^{13+2 n_s}} x^7 X^{2(n_s-1)} A_s^2 s_0(x,x_{\\rm max}) \\nonumber \\\\ & \\times \\left( 4 _2F_1 [\\frac{1}{2} , 1-n_s , \\frac{3}{2},\\frac{s_0(x,x_{\\rm max})^2}{3}] \\right.", "\\nonumber \\\\ &- \\left.", "3 _2F_1 [\\frac{1}{2} , -n_s , \\frac{3}{2},\\frac{s_0(x,x_{\\rm max})^2}{3}] \\right.", "\\nonumber \\\\ & \\left.", "- s_0(x,x_{\\rm max})^2 {}_2F_1 [\\frac{3}{2} , -n_s , \\frac{5}{2},\\frac{s_0(x,x_{\\rm max})^2}{3}] \\right) \\nonumber \\\\$ where $ {}_2 F_1$ is the hypergeometric function.", "To convert to a frequency spectrum, simply take $x_\\ast = 1/k_\\ast \\eta _r$ and $ &\\Omega _{\\rm GW} (f) h^2 = 0.39 h^2 \\Omega _r \\nonumber \\\\ &\\quad \\times \\Omega _{\\rm GW} \\left[4.1 \\times 10^{-24} \\left(\\frac{\\eta _r}{\\rm GeV^{-1}}\\right) f,\\frac{T_{\\rm eq}}{T_{\\rm dec}} \\right].", "$" ] ]
2105.11655
[ [ "ST-HOI: A Spatial-Temporal Baseline for Human-Object Interaction\n Detection in Videos" ], [ "Abstract Detecting human-object interactions (HOI) is an important step toward a comprehensive visual understanding of machines.", "While detecting non-temporal HOIs (e.g., sitting on a chair) from static images is feasible, it is unlikely even for humans to guess temporal-related HOIs (e.g., opening/closing a door) from a single video frame, where the neighboring frames play an essential role.", "However, conventional HOI methods operating on only static images have been used to predict temporal-related interactions, which is essentially guessing without temporal contexts and may lead to sub-optimal performance.", "In this paper, we bridge this gap by detecting video-based HOIs with explicit temporal information.", "We first show that a naive temporal-aware variant of a common action detection baseline does not work on video-based HOIs due to a feature-inconsistency issue.", "We then propose a simple yet effective architecture named Spatial-Temporal HOI Detection (ST-HOI) utilizing temporal information such as human and object trajectories, correctly-localized visual features, and spatial-temporal masking pose features.", "We construct a new video HOI benchmark dubbed VidHOI where our proposed approach serves as a solid baseline." ], [ "Introduction", "Thanks to the rapid development of deep learning [10], [16], machines are already surpassing or approaching human level performance in language tasks [44], acoustic tasks [46], and vision tasks (e.g., image classification [15] and visual place recognition [4]).", "Researchers thus started to focus on how to replicate these successes to other semantically higher-level vision tasks (e.g., visual relationship detection [28], [5]) and vision-language tasks (e.g., image captioning [38] and visual question answering [1]) so that machines learn not just to recognize the objects but to understand their relationships and the contexts.", "Especially, human-object interaction (HOI) [33], [3], [8], [29], [24], [40], [39], [13], [23], [41], [25], [37], [48] aiming to detect actions and spatial relations among humans and salient objects in images/videos has attracted increasing attention, as we sometimes task machines to understand human behaviors, e.g., pedestrian detection [47] and unmanned store systems [26].", "Although there are abundant studies that have achieved success in detecting HOI in static images, the fact that few of them [18], [29], [35] consider temporal information (i.e., neighboring frames before/after the target frame) when performed on video data means they are actually “guessing” temporal-related HOIs with only naive co-occurrence statistics.", "While conventional image-based HOI methods (e.g., the baseline model in [39]) can be used for inference on videos, they treat input frames as independent and identically distributed (i.i.d.)", "data and make independent predictions for neighboring frames.", "However, video data are sequential and structured by nature and thus are not i.i.d.", "What is worse is that, without temporal context these methods are unable to differentiate (especially, opposite) temporal interactions, such as push versus pull a human and open versus close a door.", "As shown in Figure REF (a), given a video segment, traditional HOI models operate on a single frame at a time and make predictions based on 2D-CNN (e.g., [16]) visual features.", "These models by nature could not distinguish interactions between two people such as push, pull, lean on and chase, which are visually similar in static images.", "A possible reason causing video-based HOI underexplored is the lack of a suitable video-based benchmark and a feasible setting.", "To bridge this gap, we first construct a video HOI benchmark from VidOR [30], dubbed VidHOI , where we follow the common protocol in video and HOI tasks to use a keyframe-centered strategy.", "With VidHOI, we urge the use of video data and propose VideoHOI as – in both training and inference – performing HOI detection with videos.", "Spatial-Temporal Action Detection (STAD) is another task bearing a resemblance to VideoHOI by requiring to localize the human and detect the actions being performed in videos.", "Note that STAD does not consider the objects that a human is interacting with.", "STAD is usually tackled by first using a 3D-CNN [36], [2] as the backbone to encode temporal information into feature maps.", "This is followed by RoI pooling with object proposals to obtain actor features, which are then classified by linear layers.", "Essentially, this approach is similar to a common HOI baseline illustrated in Figure REF (a) and differs only in the use of 3D backbones and the absence of interacting objects.", "Based on conventional HOI and STAD methods, a naive yet intuitive idea arises: can we enjoy the best of both worlds, by replacing 2D backbones with 3D ones and exploiting visual features of interacting objects?", "This idea, however, did not work straightforwardly in our preliminary experiment, where we replaced the backbone in the 2D baseline [39] with the 3D one (e.g., SlowFast [7]) to perform VideoHOI.", "The relative change of performance after replacing the backbone is presented in the left most entry in Figure REF (a) with a blue bar.", "In VideoHOI experiment, the 3D baseline provides only a limited relative improvement ($\\sim $ 2%), which is far from satisfactory considering the additional temporal context.", "In fact, this phenomenon has also been observed in two existing works under similar settings [11], [20], where both experiments in STAD and another video task Spatial-Temporal Scene Graph Generation (STSGG) present an even worse, counter-intuitive result: replacing the backbone is actually harmful (also presented as blue bars in Figure REF (a)).", "We probed the underlying reason by analyzing the architecture of these 3D baselines and found that, surprisingly, temporal pooling together with RoI pooling does not work reasonably.", "As illustrated in Figure REF (b), temporal pooling followed by RoI pooling, which is a common practice in conventional STAD methods, is equivalent to cropping features of the same region across the whole video segment without considering the way objects move.", "It is not unusual for moving humans and objects in neighboring frames to be absent from its location in the target keyframe.", "Temporal-and-RoI-pooling features at the same location could be getting erroneous features such as other humans/objects or meaningless background.", "Dealing with this inconsistency, we propose to recover the missing spatial-temporal information in VideoHOI by considering human and object trajectories.", "The performance change of this temporal-augmented 3D baseline on VideoHOI is represented by the tangerine bar in Figure REF (a), where it achieves $\\sim $ 23% improvement, in sharp contrast to $\\sim $ 2% of the original 3D baseline.", "This experiment reveals the importance of incorporating the \"correctly-localized\" temporal information.", "Keeping the aforementioned ideas in mind, in this paper we propose Spatial-Temporal baseline for Human-Object Interaction detection in videos, or ST-HOI, which makes accurate HOI prediction with instance-wise spatial-temporal features based on trajectories.", "As illustrated in Figure REF (b), three kinds of such features are exploited in ST-HOI: (a) trajectory features (moving bounding boxes; shown as the red arrow), (b) correctly-localized visual features (shown as the yellow arrow), and (c) spatial-temporal actor poses (shown as the green arrow).", "The contribution of our work is three-fold.", "First, we are among the first to identify the feature inconsistency issue existing in the naive 3D models which we address with simple yet “correct” spatial-temporal feature pooling.", "Second, we propose a spatial-temporal model which utilizes correctly-localized visual features, per-frame box coordinates and a novel, temporal-aware masking pose module to effectively detect video-based HOIs.", "Third, we establish the keyframe-based VidHOI benchmark to motivate research in detecting spatial-temporal aware interactions and hopefully inspire VideoHOI approaches utilizing the multi-modality data, i.e., video frames, texts (semantic object/relation labels) and audios.", "HOI Detection aims to reason over interactions between humans (actors) and target objects.", "HOI is closely related to visual relationship detection [28], [5] and scene graph generation [45], in which the subject in (subject-predicate-object) are not restricted to a human.", "HOI in static images has been intensively studied recently [33], [3], [8], [29], [24], [40], [39], [13], [23], [41], [25], [37], [48].", "Most of the existing methods can be divided into two categories by the order of human-object pair proposal and interaction classification.", "The first group [8], [3], [24], [40], [39], [13], [37], [23] performs human-object pair generation followed by interaction classification, while the second group [9], [29], [41], [25] first predicts the most probable interactions performed by a person followed by associating them with the most-likely objects.", "Our ST-HOI belongs to the first group as we establish a temporal model based on trajectories (continuous object proposals).", "In contrast to the popularity of image-based HOI, there are only a few of studies in VideoHOI [22], [18], [29], [35] and, to the best of our knowledge, all of which conducted experiments on CAD-120 [21] dataset.", "In CAD-120, the interactions are defined by merely 10 high-level activities (e.g., making cereal or microwaving food) in 120 RGB-D videos.", "This setting is not favorable to real-life scenarios where machines may be asked to understand more fine-grained actions.", "Moreover, previous methods [22], [18], [29] adopted pre-computed hand-crafted features such as SIFT [27] which have been outperformed by deep neural networks, and ground truth features including 3D poses and depth information from RGB-D videos which are unlikely to be available in real life scenarios.", "While [35] adopted a ResNet [16] as their backbone, their method is inefficient by requiring $M \\times N$ computation for extracting $M$ humans' and $N$ objects' features.", "Different from these existing methods, we evaluate on a larger and more diversified video HOI benchmark dubbed VidHOI, which includes annotations of 50 predicates on thousands of videos.", "We then propose a spatial-temporal HOI baseline that operates on RGB videos and does not utilize any additional information." ], [ "Spatial-Temporal Action Detection (STAD)", "STAD aims to localize actors and detect the associated actions (without considering interacting objects).", "One of the most popular benchmark for STAD is AVA [11], where the annotation is done at a sampling frequency of 1 Hz and the performance is measured by framewise mean AP.", "We followed this annotation and evaluation style when constructing VidHOI, where we converted the original labels into the same format.", "As explained in section , a standard approach to STAD [36], [2] is extracting spatial-temporal feature maps with a 3D-CNN followed by RoI pooling to crop human features, which are then classified by linear layers.", "As shown in Figure REF (a), a naive modification that incorporates RoI-pooled human/object features does not work for VideoHOI.", "In contrast, our ST-HOI tackles VideoHOI by incorporating multiple temporal features including trajectories, correctly-localized visual features and spatial-temporal masking pose features.", "Figure: An illustration of the two proposed spatial-temporal features.", "(a) In contrast to performing RoI pooling followed by temporal pooling like , , we adopt a reverse approach to first frame-wise RoI-pool instance feature maps using trajectories, which are then averaged pool along the time axis to get correctly-localized visual features.", "(b) With NN object trajectories (including MM human), for each frame we utilize a trained human pose prediction model (e.g., ) to generate 2D actor pose feature and extract a dual spatial mask for all M×(N-1)M \\times (N-1) valid pair.The pose feature and the mask are concatenated and down-sampled, followed by two 3D convolution layers and spatial-temporal pooling to generate the masking pose features." ], [ "Spatial-Temporal Scene Graph Generation", "Spatial-Temporal Scene Graph Generation (STSGG) [20] aims to generate symbolic graphs representing pairwise visual relationships in video frames.", "A new benchmark, Action Genome, is also proposed in [20] to facilitate researches in STSGG.", "Ji et al.", "[20] dealt with STSGG by combining off-the-shelf scene graph generation models with long-term feature bank [43] on top of a 2D- or 3D-CNN, where they found that the 3D-CNN actually undermines the performance.", "While observing similar results in VidHOI (Figure REF (a)), we go one step further to find out the underlying reason is that RoI features across frames were erroneously pooled.", "We correct this by utilizing object trajectories and applying Tube-of-Interest (ToI) pooling on generated trajectories to obtain correctly-localized position information and feature maps throughout video segments." ], [ "Overview", "We follow STAD approaches [36], [2], [7] to detect VideoHOI in a keyframe-centric strategy.", "Denote $V$ as a video which has $T$ keyframes with sampling frequency of 1 Hz as $\\lbrace I_t\\rbrace , t=\\lbrace 1,...,T\\rbrace $ , and denote $C$ as the number of pre-defined interaction classes.", "Given $N$ instance trajectories including $M$ human trajectories ($M \\le N$ ) in a video segment centered at the target frame, for human $m \\in \\lbrace 1,...,M\\rbrace $ and object $n \\in \\lbrace 1,...,N\\rbrace $ in keyframe $I_t$ , we aim to detect pairwise human-object interactions $r_t = \\lbrace 0,1\\rbrace ^C$ , where each entry $r_{t,c}, c \\in \\lbrace 1,...,C\\rbrace $ means whether the interaction $c$ exists or not.", "Refer to Figure REF (b) for an illustration of our ST-HOI.", "Our model takes in a video segment (sequence of $T$ frames) centered at $I_t$ and utilizes a 3D-CNN as the backbone to extract spatial-temporal feature maps of the whole segment.", "To rectify the mismatch caused by temporal-RoI pooling, based on $N$ object (including human) trajectories $\\lbrace j_i\\rbrace , i=\\lbrace 1,..,N\\rbrace , j_i \\in \\mathbb {R}^{T \\times 4}$ we generate temporal-aware features including correctly-localized features and spatial-temporal masking pose features.", "These features together with trajectories are concatenated and classified by linear layers.", "Note that we aim at a simple but effective temporal-aware baseline to VideoHOI so that we do not utilize tricks in STAD such as non-local block [42] or long-term feature bank [43], and in image-based HOI like interactiveness [24], though we note that these may be used to boost the performance." ], [ "Correctly-localized Visual Features", "We have discussed in previous sections on inappropriately pooled RoI features.", "We propose to tackle this issue by reversing the order of temporal pooling and RoI-pooling.", "This approach has recently been proposed in [17] and named as tube-of-interest pooling (ToIPool).", "Refer to Figure REF (a) for an illustration.", "Denote $v \\in \\mathbb {R}^{d \\times T \\times H \\times W}$ as the output of the penultimate layer of our 3D-CNN backbone, and denote $v_t \\in \\mathbb {R}^{d \\times H \\times W}$ as the $t$ -th feature map along the time axis.", "Recall that we have $N$ trajectories centered at a keyframe.", "Following the conventional way, we also exploit visual context when predicting an interaction, which is done by utilizing the union bounding box feature of a human and an object.", "For example, the sky between human and kite could help to infer the correct interaction fly.", "Recall that $j_i$ represents the trajectory of object $i$ , where we further denote $j_{i,t}$ as the 2D bounding box at time $t$ .", "The spatial-temporal instance features $\\lbrace \\bar{v}_i\\rbrace $ are then obtained using ToIPool with RoIAlign [14] by $\\bar{v}_i = \\frac{1}{T} \\sum _{t=1}^{T} \\text{RoIAlign}(v_{t}, j_{i,t}),$ where $\\bar{v}_i \\in \\mathbb {R}^{d \\times h \\times w}$ and $h$ and $w$ means height and width of the pooled feature maps, respectively.", "$\\bar{v}_i$ is flattened before concatenating with other features." ], [ "Spatial-Temporal Masking Pose Features", "Human poses have been widely utilized in image-based HOI methods [24], [13], [39] to exploit characteristic actor pose to infer some special actions.", "In addition, some existing works [40], [39] found that spatial information can be used to identify interactions.", "For instance, for human-ride-horse one can imagine the actor's skeleton as legs widely open (on horse sides), and the bounding box center of human is usually on top of that of horse.", "However, none of the existing works consider this mechanism in temporal domain: when riding a horse the human should be moving with horse as a whole.", "We argue that this temporality is an important property and should be utilized as well.", "The spatial-temporal masking pose module is presented at Figure REF (b).", "Given $M$ human trajectories, we first generate $M$ spatial-temporal pose features with a trained human pose prediction model.", "On frame $t$ , the predicted human pose $h_{i,t} \\in \\mathbb {R}^{17 \\times 2}, i=\\lbrace 1,..,M\\rbrace , t=\\lbrace 1,..,T\\rbrace $ is defined as 17 joint points mapped to the original image.", "We transform $h_{i,t}$ into a skeleton on a binary mask with $f_h: \\lbrace h_{i,t}\\rbrace \\in \\mathbb {R}^{17 \\times 2} \\rightarrow \\lbrace \\bar{h}_{i,t}\\rbrace \\in \\mathbb {R}^{1 \\times H \\times W}$ , by connecting the joints using lines, where each line has a distinct value $x \\in [0, 1]$ .", "This helps the model to recognize and differentiate different poses.", "For each of $M \\times (N-1)$ valid human-object pairs on frame $t$ , we also generate two spatial masks $s_{i,t} \\in \\mathbb {R}^{2 \\times H \\times W}, i=\\lbrace 1,...,M \\times (N-1)\\rbrace $ corresponding to human and object respectively, where the values inside of each bounding box are ones and outsides are zeroed-out.", "These masks enable our model to predict HOI with reference to important spatial information.", "For each pair, we concatenate the skeleton mask $\\bar{h}_{i,t}$ and spatial masks $s_{i,t}$ along the first dimension to get the initial spatial masking pose feature $p_{i,t} \\in \\mathbb {R}^{3 \\times H \\times W}$ : $p_{i,t} = [s_{i,t}; \\bar{h}_{i,t}].$ We then down-sample $\\lbrace p_{i, t}\\rbrace $ , feed into two 3D convolutional layers with spatial and temporal pooling, and flatten to obtain the final spatial-temporal masking pose feature $\\lbrace \\bar{p}_{i,t}\\rbrace $ .", "Table: A comparison of our benchmark VidHOI with existing STAD (AVA ), image-based (HICO-DET and V-COCO ) and video-based (CAD-120 and Action Genome ) HOI datasets.", "VidHOI is the only dataset that provides temporal information from video clips and complete multi-person and interacting-object annotations.VidHOI also provides the most annotated keyframes and defines the most HOI categories in the existing video datasets.†\\dagger Two less categories as we combine adult, child and baby into a single category, person." ], [ "Prediction", "We fuse the aforementioned features, including correctly-localized visual features $\\bar{v}$ , spatial-temporal masking pose features $p$ , and instance trajectories $j$ by concatenating them along the last axis $v_{\\text{so}} = [\\bar{v}_s; \\bar{v}_u; \\bar{v}_o; j_s; j_o; \\bar{p}_{so}], \\\\$ where we slightly abuse the notation to denote the subscriptions $s$ as the subject, $o$ as the object and $u$ as their union region.", "$v_{\\text{so}}$ is then fed into two linear layers with the final output size being the number of interaction classes in the dataset.", "Since VideoHOI is essentially a multi-label learning task, we train the model with per-class binary cross entropy loss.", "During inference, we follow the heuristics in image-based HOI [3] to sort all the possible pairs by their softmax scores and evaluate on only top 100 predictions." ], [ "Dataset and Performance Metric", "While we have discussed in section REF about the problem of lacking a suitable VideoHOI dataset by analyzing CAD-120 [21], we further explain why Action Genome [20] is also not a feasible choice here.", "First, the authors acknowledged that the dataset is still incomplete and contains incorrect labels [19].", "Second, Action Genome is produced by annotating Charades [32], which is originally designed for activity classification where each clip contains only one \"actor\" performing predefined tasks; should any other people show up, there are neither any bounding box nor interaction label about them.", "Finally, the videos are purposedly-generated by volunteers, which are rather unnatural.", "In contrast, VidHOI are based on VidOR [30] which is densely annotated with all humans and predefined objects showing up in each frame.", "VidOR is also more challenging as the videos are non-volunteering user-generated and thus jittery at times.", "A comparison of VidHOI and the existing STAD and HOI datasets is presented in Table REF .", "VidOR is originally collected for video visual relationship detection where the evaluation is trajectory-based.", "The volumetric Interaction Over Union (vIOU) between a trajectory and a ground truth needs to be over 0.5 before considering its relationship prediction; however, how to obtain accurate trajectories with correct start- and end-timestamp remains challenging [34], [31].", "We notice that some image-based HOI datasets (e.g., HICO-DET [3] and V-COCO [12]) as well as STAD datasets (e.g., AVA [11]) are using a keyframe-centered evaluation strategy, which bypasses the aforementioned issue.", "We thus adopt the same and follow AVA to sample keyframes at a 1 FPS frequency, where the annotations on the keyframe at timestamp $t$ are assumed to be fixed for $t \\pm 0.5$ s. In detail, we first filter out those keyframes without presenting at least one valid human-object pair, followed by transforming the labels from video clip-based to keyframe-based to align with common HOI metrics (i.e., frame mAP).", "We follow the original VidOR split in [30] to divide VidHOI into a training set comprising 193,911 keyframes in 6,366 videos and a validation setThe VidOR testing set is not available publicly.", "with 22,808 keyframes in 756 videos.", "As shown in Figure REF , there are 50 relation classes including actions (e.g., push, pull, lift, etc.)", "and spatial relations (e.g., next to, behind, etc.).", "While half (25) of the predicate classes are temporal-related, they account for merely $\\sim $ 5% of the dataset.", "Figure: Predicate distribution of the VidHOI benchmark shows that most of the predicates are non-temporal-related.Figure: Performance comparison in predicate-wise AP (pAP).The performance boost after adding trajectory features is observed for most of the predicates.Interestingly, both spatial (e.g., next to, behind) and temporal (e.g., towards, away) predicates benefit from the temporal-aware features.Predicates sorted by the number of occurrence.Models in Oracle mode.Following the evaluation metric in HICO-DET, we adopt mean Average Precision (mAP), where a true positive HOI needs to meet three below criteria: (a) both the predicted human and object bounding boxes have to overlap with the ground truth boxes with IOU over 0.5, (b) the predicted target category need to be matched and (c) the predicted interaction is correct.", "Over 50 predicates, we follow HICO-DET to define HOI categories as 557 triplets on which we compute mean AP.", "By defining HOI categories with triplets we can bypass the polysemy problem [48], i.e., the same predicate word can represent very different meaning when pairing with distinct objects, e.g., person-fly-kite and person-fly-airplane.", "We report the mean AP over three categories: (a) Full: all 557 categories are evaluated, (b) Rare: 315 categories with less than 25 instances in the dataset, and (c) Non-rare: 242 categories with more than or equal to 25 instances in the dataset.", "We also examine the models in two evaluation modes: Oracle models are trained and tested with ground truth trajectories, while models in Detection mode are tested with predicted trajectories." ], [ "Implementation Details", "We adopt Resnet-50 [16] as our 2D backbone for the preliminary experiments, and utilize Resnet-50-based SlowFast [7] as our 3D backbone for all the other experiments.", "SlowFast contains the Slow and Fast pathways, which correspond to the texture details and the temporal information, respectively, by sampling video frames in different frequencies.", "For a 64-frame segment centered at the keyframe, $T=32$ frames are alternately sampled to feed into the Slow pathway; only $T/\\alpha $ frames are fed into the Fast pathway, where $\\alpha = 8$ in our experiments.", "We use FastPose [6] to predict human poses and adopt the predicted trajectories generated by a cascaded model of video object detection, temporal NMS and tracking algorithm [34].", "Like object detection is to 2D HOI detection, trajectory generation is an essential module but not a main focus of this work.", "If a bounding box is not available in neighboring frames (i.e., the trajectory is shorter than $T$ or not continuous throughout the segment), we fill it with the whole-image as a box.", "We train all models from scratch for 20 epochs with the initial learning rate $1 \\times 10^{-2}$ , where we use step decay learning rate to reduce the learning rate by $10\\times $ at the 10th and 15th epoch.", "We optimize our models using synchronized SGD with momentum of 0.9 and weight decay of $10^{-7}$ .", "We train each 3D video model with eight NVIDIA Tesla V100 GPUs with batch size being 128 (i.e., 16 examples per GPU), except for the full model where we set batch size as 112 due to the memory restriction.", "We train the 2D model with a single V100 with batch size being 128.", "During training, following the strategy in SlowFast we randomly scale the shorter side of the video to a value in $[256, 320]$ pixels, followed by random horizontal flipping and random cropping into $224 \\times 224$ pixels.", "During inference, we only resize the shorter side of the video segment to 224 pixels." ], [ "Quantitative Results", "Since we aim to deal with a) the lack of temporal-aware features in 2D HOI methods, b) the feature inconsistency issue in common 3D HOI methods and c) the lack of a VideoHOI benchmark, we mainly compare with the 2D model [39] and its naive 3D variant on VidHOI to understand if our ST-HOI addresses these issues effectively.", "The performance comparison between our full ST-HOI model (Ours-T+V+P) and baselines (2D model, 3D model) are presented in Table REF , in which we also present ablation studies on our different features (modules) inlcuding trajectory features (T), correctly-localized visual features (V) and spatial-temporal masking pose features (P).", "Table REF shows that 3D model only has a marginal improvement compared to 2D model (overall $\\sim $ 2%) under all settings in both evaluation modes.", "In contrast, adding trajectory features (Ours-T) leads to a much larger 23% improvement in Oracle mode or 15% in Detection mode, showing the importance of correct spatial-temporal information.", "We also find that by adding additional temporal-aware features (i.e., V and P) increasingly higher mAPs are attained, and our full model (Ours-T+V+P) reports the best mAPs in Oracle mode, achieving the highest $\\sim $ 25% relative improvement.", "We notice that the performance of Ours-T+V is close to that of Ours-T under Oracle setting, which is possibly because the ground truth trajectories (T) have provided enough “correctly-localized” information so that the correct features do not help much.", "We also note that the performance of Ours-T+P is slightly higher than that of Ours-T+V+P under Detection mode, which is assumably due to the same, aforementioned reason and the inferior performance resulting from the predicted trajectories.", "The overall performance gap between Detection and Oracle models is significant, indicating the room for improvement in trajectory generation.", "Another interesting observation is that Full mAPs are very close to Rare mAPs, especially under Oracle mode, showing that the long-tail effect over HOIs is strong (but common and natural).", "To understand the effect of temporal features on individual predicates, we compare with predicate-wise AP (pAP) shown in Figure REF .", "We observe that, again, under most of circumstances naively replacing 2D backbones with 3D ones does not help video HOI detection.", "Both temporal predicates (e.g., towards, away, pull) and spatial (e.g., next_to, behind, beneath) predicates benefit from the additional temporal-aware features in ST-HOI.", "These findings verify our main idea about the essential use of trajectories and trajectory-based features.", "In addition, each additional features do not seem to contribute equally for different predicates.", "For instance, we see that while Ours-T+V+P performs the best on some predicates (e.g., behind and beneath), our sub-models achieve the highest mAP on other predicates (e.g., watch and ride).", "This is assumedly because predicate-wise performance is heavily subject to the number of examples, where major predicates have 10-10000 times more examples than minor ones (as shown in Figure REF ).", "Figure: Results (in predicate-wise AP) of the baselines and our full model w.r.t.", "top frequent temporal predicates.Table: Results of temporal-related and spatial (non-temporal) related triplet mAP.", "T%/S% means relative temporal/spatial mAP change compared to 2D model .Since the majority of HOI examples are spatial-related ($\\sim $ 95%, as shown in Figure REF ), the results above might not be suitable for demonstrating the temporal modeling ability of our proposed model.", "We thus focus on the performance on only temporal-related predicates in Figure REF , which demonstrates that Ours-T+V+P greatly outperforms the baselines regarding the top frequent temporal predicates.", "Table REF presents the triplet mAPs of spatial- or temporal-only predicates, showing Ours-T significantly improves the 2D model on temporal-only mAP by relative +73.9%, in sharp contrast to -7.1% of the 3D model in Oracle mode.", "Similar to our observation with Table REF , Ours-T performs on par with Ours-T+V+P for temporal-only predicates; however, it falls short of spatial-only predicates, showing that spatial/pose information is still essential for detecting spatial predicates.", "Overall, these results demonstrate the outstanding spatial-temporal modeling ability of our approach.", "Figure: Performance comparison (in AP) of some temporal-related HOIs in VidHOI validation set.Compared to 2D model, 3D model only shows limited improvement for the presented examples, while our ST-HOI variants provide huge performance boost.Models are in Oracle mode.Figure: Examples of video HOIs predicted by the 2D model and our ST-HOI, both in Oracle mode.Each consists of five consecutive keyframes sampled in 1 Hz, where an entry in tables denotes whether a predicate between the subject (human; a green box) and the object (also human in both cases; a red box) is detected correctly (True Positive) or not (False Positive or False Negative).Compared to the 2D baseline, our model predicts more accurate temporal HOIs (e.g., hold_hand_of in T 4 T_4 and T 5 T_5 of the upper example and lift in T 1 T_1 of the lower example).ST-HOI also produces less false positives in both examples.We also compare the performance with respect to some HOI triplets in Figure REF .", "Similar to the results on predicate-wise mAP, we also observe the large gap between naive 2D/3D models and our models with the temporal features.", "ST-HOI variants are more accurate in predicting especially temporal-aware HOIs (hug/lean_on-person and push/pull-baby_walker).", "We also see in some examples that Ours-T+V+P does not perform the best among all the variants, e.g., lean_on-person), which is similar to the phenomenon we observed in Figure REF ." ], [ "Qualitative Results", "To understand the effectiveness of our proposed method, we visualize two video HOI examples of VidHOI predicted by the 2D model [39] and Ours-T+V+P (both in Oracle mode) in Figure REF .", "Each (upper and lower) example is a 5-second video segment (i.e., five keyframes) with a HOI prediction table where each entry means either True Positive (TP), False Positive (FP), False Negative (FN) or True Negative (TN) for both models.", "The upper example shows that, compared to the 2D model, Ours-T+V+P makes more accurate HOI detection by successfully predicting hold_hand_of at $T_4$ and $T_5$ .", "Moreover, Ours-T+V+P is able to predict interactions that requires temporal information, such as lift at $T_1$ in the lower example.", "However, we can see that there is still room for improvement for Ours-T+V+P in the same example, where lift is not detected in the following $T_2$ to $T_4$ frames.", "Overall, our model produces less false positives throughout the dataset, which in turn contributes to its higher mAP and pAP." ], [ "Conclusion", "In this paper, we addressed the inability of conventional HOI approaches to recognize temporal-aware interactions by re-focusing on neighboring video frames.", "We discussed the lack of a suitable setting and dataset for studying video-based HOI detection.", "We also identified a feature-inconsistency problem in a common video action detection baseline which arises from its improper order of RoI feature pooling and temporal pooling.", "To deal with the first issue, we established a new video HOI benchmark dubbed VidHOI and introduced a keyframe-centered detection strategy.", "We then proposed a spatial-temporal baseline ST-HOI which exploits trajectory-based temporal features including correctly-localized visual features, spatial-temporal masking pose features and trajectory features, solving the second problem.", "With quantitative and qualitative experiments on VidHOI, we showed that our model provides a huge performance boost compared to both the 2D and 3D baselines and is effective in differentiating temporal-related interactions.", "We expect that the proposed baseline and the dataset would serve as a solid starting point for the relatively underexplored VideoHOI task.", "Based on our baseline, we also hope to motivate further VideoHOI works to design advanced models with the multi-modal data including video frames, semantic object/relation labels and audios." ], [ "Acknowledgment", "This research is supported by Singapore Ministry of Education Academic Research Fund Tier 1 under MOE's official grant number T1 251RES2029." ] ]
2105.11731
[ [ "Dynamics of Uniaxial-to-Biaxial Nematics Switching in Suspensions of\n Hard Cuboids" ], [ "Abstract Field-induced reorientation of colloidal particles is especially relevant to manipulate the optical properties of a nanomaterial for target applications.", "We have recently shown that surprisingly feeble external stimuli are able to transform uniaxial nematic liquid crystals (LCs) of cuboidal particles into biaxial nematic LCs.", "In the light of these results, here we apply an external field that forces the reorientation of colloidal cuboids in nematic LCs and sparks a uniaxial-to-biaxial texture switching.", "By Dynamic Monte Carlo simulation, we investigate the unsteady-state reorientation dynamics at the particle scale when the field is applied (uniaxial-to-biaxial switching) and then removed (biaxial-to-uniaxial switching).", "We detect a strong correlation between the response time, being the time taken for the system to reorient, and particle anisotropy, which spans from rod-like to plate-like geometries.", "Interestingly, self-dual shaped cuboids, theoretically considered as the most suitable to promote phase biaxiality for being exactly in between prolate and oblate particles, exhibit surprisingly slow response times, especially if compared to prolate cuboids." ], [ "Introduction", "Colloids are biphasic systems comprising particles homogeneously dispersed in a medium.", "In colloidal suspensions, the dispersed phase consists of solid particles, while the continuous phase is a liquid.", "In particular, the dispersed particles should have at least in one direction a dimension roughly between 1 nm and 1 $\\mu $ m, so that gravitational and thermal forces compensate each other [1].", "This balance allows the dispersed particles to remain suspended and to diffuse randomly via Brownian motion, named after the Scottish botanist Robert Brown who, in 1827, described the persistent and casual jumpy moves of organelles suspended in water [2].", "If the suspended particles are anisotropic, under certain conditions, they can self-organise into liquid-crystalline phases.", "Liquid crystals (LCs) are mesophases that flow like liquids but, exhibit a significant degree of internal ordering like crystals.", "A common LC morphology is the uniaxial nematic ($\\rm N_{U}$ ) phase where particles have one axis pointing collectively in the same direction, but their centres of mass are randomly distributed.", "This merely orientational ordering allows nematic LCs to exhibit optical birefringence while maintaining mechanical fluidity.", "Currently, LCs are deployed in a multitude of optical technology, including (but not limited to) commercial displays [3], displays for virtual augmented reality [4] and even smart windows with memory displays [5].", "The roll-out of these high-tech products are coupled alongside further advances to understand and control the morphology of LCs [6] as well as techniques to optimise how they are manufactured [7], highlighting the relevance of LCs in today's research landscape.", "Very recently, there has been reignited interest in the biaxial nematic phase ($\\rm N_{B}$ ) and its potential to be incorporated into display technology.", "In contrast to the $\\rm N_{U}$ phase, the $\\rm N_{B}$ phase possesses two optical axes due to the alignment of the three directors, making it very appealing for the design of nanomaterials with novel optical properties [8].", "Equally important, the $\\rm N_{B}$ phase is also foreseen to realise faster switching through its minor axis switching mode, an aspect that could potentially improve refresh rates in displays [9], [10], [11].", "Despite these promising features, the existence of stable molecular $\\rm N_{B}$ phases is still an ongoing debate within the LC community.", "While a biaxial geometry is indeed necessary to observe $\\rm N_{B}$ phases, it has been shown that this requirement might not be sufficient as the $\\rm N_B$ phase tends to be metastable with respect to other phases, including $\\rm N_U$ and smectic (Sm) LCs [12].", "This is for instance the case of colloidal cuboids, which can only form $\\rm N_B$ phases at sufficiently large size dispersity [13], [14], extreme anisotropy [15], in the presence of depletants [16] or upon application of an external field [17], [18].", "Unless at least one of these conditions are met, systems of monodisperse or bidisperse cuboids cannot form $\\rm N_B$ phases [19], [20].", "For the monodisperse case, simulation results showed excellent qualitative agreement with sedimentation experiments of highly uniform colloidal cuboids synthesised by Yang and co-workers, which also preclude the existence of stable $\\rm N_{B}$ phases at equilibrium [21].", "Research efforts have also been made to ascertain, by theory, simulation and experiments, the existence of $\\rm N_B$ phases in systems of other biaxial particles [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] or mixtures of uniaxial particles [32].", "Despite such a widespread interest in mapping the phase behaviour of colloidal suspensions of biaxial particles, including cuboids, the study of their dynamics is still at an embryonic stage, especially for the difficulty of finding suitable interaction potentials that could describe exotic geometries and still be framed within a simulation technique.", "Exotic shapes are commonly described by hard-core potentials, but these cannot be directly employed in Brownian dynamics (BD) or Molecular Dynamics (MD) simulations.", "Nevertheless, it is only by assessing the dynamics that one will be able to draw relevant conclusions on the potential use of nematic or other LC phases in specific applications.", "With this in mind, over the last years, our group has developed a stochastic method that can qualitatively and quantitatively mimic the Brownian motion of colloids as obtained by BD simulations [33], [34], [35], [36].", "This method, referred to as Dynamic Monte Carlo (DMC), has become an established simulation technique not only for the study of the dynamics of biaxial particles, such as cuboids and curved rods [37], [38], but also for the dynamics of uniaxial particles, like rods, for which soft potentials are indeed available [39], [40].", "With regards to the equilibrium dynamics of cuboids, we found that the system long-time relaxation dramatically depends on particle anisotropy, being slower at the self-dual shape, the geometry that would preferentially stabilise biaxial nematics [38].", "By definition, the self-dual shape is an intermediate geometry between prolate and oblate, where length ($L$ ), width ($W$ ) and thickness ($T$ ) are such that $W = \\sqrt{LT}$ .", "Our simulations also confirmed the occurrence of a Fickian and Gaussian dynamics at both short and long times, thus providing an alternative picture to the claimed universality of Fickian yet non-Gaussian dynamics in soft-matter systems [41].", "For its potential impact in nanotechnology, equally intriguing is the out-of-equilibrium dynamics of cuboids, especially because it can spark phase switching and new material properties.", "In general, the reorientation dynamics of biaxial particles induced by an external stimulus has received very limited attention.", "At the molecular scale, Lee and co-workers studied the reorientation dynamics of $\\rm N_{B}$ phases of bent-core mesogens and measured primary and secondary axis switching, finding the latter either 3 or 100 times faster than the former depending on the mesogen [9].", "Although this study was met with some scepticism [42], Zannoni and co-workers later on performed MD simulations on $\\rm N_{B}$ phases of biaxial Gay-Berne ellipsoids and confirmed that the rotation of minor axes is indeed faster, although only up to one order of magnitude, than the rotation of the main axis, both in the bulk [10] and under confinement [11].", "Following our recent findings on the field-induced stability of the $\\rm N_B$ phase [18], here we explore the field-induced dynamics of switching from uniaxial to biaxial nematics of colloidal cuboids, with special interest in the particle reorientation dynamics and associated response time.", "More specifically, we are interested to study the kinetics of reorientation of LCs transitioning between two different nematic textures, namely an $\\rm N_{U}$ $\\rightarrow $ $\\rm N_{B}$ transition under an external field, and an $\\rm N_{B}$ $\\rightarrow $ $\\rm N_{U}$ relaxation when the field is switched off, and estimate the associated response times.", "To gain an insight into the impact of particle anisotropy on the dynamics of phase switching, we consider monodisperse systems of prolate, oblate and self-dual shaped cuboids.", "Their ability of reorienting under the effect of an external field is assessed by employing a DMC algorithm specifically designed to track the dynamics of out-of-equilibrium colloidal systems [35].", "This paper is organised as follows.", "We first introduce the methodology to simulate our systems and characterise the dynamics.", "We then discuss the results of our simulations by analysing the effect of particle anisotropy on the out-of-equilibrium dynamics in $\\rm N_{U}$ $\\rightarrow $ $\\rm N_{B}$ and $\\rm N_{B}$ $\\rightarrow $ $\\rm N_{U}$ switching before finally drawing our conclusions." ], [ "Model and Simulation Methodology", "We modelled monodisperse colloidal cuboids as hard board-like particles (HBPs) constrained in a cubic box with periodic boundaries.", "The behaviour of hard-core systems is basically determined by the packing fraction, which is given by: $\\eta \\equiv \\frac{Nv_{o}}{V}$ where $N$ is the number of particles, $v_{o}$ the volume of an individual HBP and $V$ the volume of the simulation box.", "The particle thickness, $T$ , is set as the system unit length.", "Consequently, particle length and width are given in units of $T$ , and read $L^* \\equiv L/T$ and $W^* \\equiv W/T$ , respectively.", "In particular, $L^*=12$ for all systems studied, while $W^*$ assumed values between 1 (rod-like HBPs) and 12 (plate-like HBPs) and included $W^*=\\sqrt{L^*}$ at the self-dual shape.", "Similar to our previous work [18], we apply an external field that promotes alignment of the particle intermediate axis, defined by: $U_{\\rm ext} = -\\frac{\\varepsilon _{f}}{2} \\Big [3 \\cdot (\\rm \\hat{\\textbf {x}}_{i} \\cdot \\rm \\hat{\\textbf {e}})^{2} -1 \\Big ]$ where $\\varepsilon _f$ is the field strength, $\\rm \\hat{\\textbf {x}}$ is the unit vector associated with the width of particle $i$ , while $\\rm \\hat{\\textbf {e}}$ is the field direction.", "We have set the reduced field strength, $\\varepsilon ^*_f \\equiv \\varepsilon _f \\beta =3$ , with $\\beta $ the inverse temperature, which provides a measure of the relative strength of the field applied with respect to thermal energy.", "The unit vectors $\\rm \\hat{\\textbf {y}}$ and $\\rm \\hat{\\textbf {z}}$ are associated with thickness and length, respectively.", "Similar to our previous work, we remind that this external field model does not intend to mimic a real electric or magnetic field, but rather, we focus on its effect to reorient particles.", "The orientation of the particle unit vectors before and after application of the field is schematically displayed in Fig.", "REF .", "Figure: Reorientation of a prolate (a) and oblate (b) HBP due to external field 𝐞 ^\\rm \\hat{\\textbf {e}} coupled to the particle intermediate axis 𝐱 ^\\rm \\hat{\\textbf {x}}.", "The N U \\rm N_{U} phases of self-dual shaped particles in this work always have a director along 𝐧 ^\\hat{\\textbf {n}}, and therefore rotate around 𝐳 ^\\hat{\\textbf {z}} (similar to rod-like particles) when an external field is applied.The focus of this work is on the reorientation dynamics of HBPs in an external field.", "We first performed standard Monte Carlo (MC) simulations in the canonical ensemble in a cubic box containing $N = 2000$ HBPs to equilibrate $\\rm N_{U}^{+}$ and $\\rm N_{U}^{-}$ phases at $\\eta = 0.34$ , where $\\rm N_{U}^{+}$ and $\\rm N_{U}^{-}$ refer, respectively to prolate and oblate nematic LCs.", "At this packing fraction, the $\\rm N_{U}$ phases (either prolate or oblate) are stable across all anisotropies [19].", "Each MC cycle consists of $N$ attempts to displace and/or rotate HBPs, which are accepted if no overlaps are detected.", "To determine the occurrence of overlaps between pairs of HBPs, we implemented the separating axes theorem by Gottschalk et al.", "[43], adapted by John and Escobedo to study tetragonal parallelepipeds [44], [45].", "To quantify the system long-range orientational order, we calculated the nematic order parameter and director associated to each particle axis.", "To this end, we performed the diagonalisation of a second-rank symmetric tensor of the form: $\\textbf {Q}^{\\lambda \\lambda } = \\frac{1}{2N} \\Biggl \\langle \\sum _{i=1}^{N} \\big (3\\hat{\\lambda }_{i} \\cdot \\hat{\\lambda }_{i} - \\textbf {I}\\big ) \\Biggr \\rangle $ where $\\hat{\\lambda }_i$ = $\\hat{\\textbf {x}}$ , $\\hat{\\textbf {y}}$ , $\\hat{\\textbf {z}}$ are unit vectors aligned with W, T and L respectively, while I is the identity tensor.", "The diagonalisation of $\\textbf {Q}^{\\lambda \\lambda }$ results in three eigenvalues ($S_{2,W}$ , $S_{2,T}$ , $S_{2,L}$ ) and their corresponding eigenvectors ($\\rm \\hat{\\textbf {m}}$ , $\\rm \\hat{\\textbf {l}}$ , $\\rm \\hat{\\textbf {n}}$ ).", "The nematic director of an $\\rm N_U$ phase is the eigenvector with the largest eigenvalue.", "For instance, if the largest positive eigenvalue is $S_{2,L}$ , then the nematic director is $\\hat{\\textbf {n}}$ , indicating high degree of orientational order along the $\\hat{\\textbf {z}}$ axis of the particles.", "The biaxial order parameters can also be evaluated using the same symmetric tensor.", "For example, the biaxial order parameter that quantifies the fluctuations of particles' axes $\\hat{\\textbf {x}}$ and $\\hat{\\textbf {y}}$ respectively along the directors $\\hat{\\textbf {m}}$ and $\\rm \\hat{\\textbf {l}}$ reads [19] $B_{2,L} = \\frac{1}{3} \\; (\\hat{\\bf m} \\cdot \\textbf {Q}^{xx} \\cdot \\hat{\\bf m} + \\hat{\\bf l} \\cdot \\textbf {Q}^{yy} \\cdot \\hat{\\bf l} - \\hat{\\bf m} \\cdot \\textbf {Q}^{yy} \\cdot \\hat{\\bf m} - \\hat{\\bf l} \\cdot \\textbf {Q}^{xx} \\cdot \\hat{\\bf l})$ The values of $B_{2,W}$ and $B_{2,T}$ can be calculated using similar expressions.", "Following the definition introduced in our former work, a phase is considered to be biaxial if $B_{2} \\ge 0.35$ , although weak biaxial phases can already be observed for $B_{2} \\ge 0.20$ [18], [14].", "We monitor the evolution of uniaxial and biaxial order parameters until they have plateaued and fluctuate in a bounded range.", "The equilibrated configurations are then used for external field application in DMC simulations.", "To study the dynamics, we performed DMC simulations in the canonical ensemble.", "Because our goal is producing realistic time trajectories, unphysical moves, such as cluster moves, swaps, jumps and changes in box dimension (which would result in centres of mass rescaling) are not implemented.", "The position of the particle centre of mass is updated by decoupling the displacement $\\delta \\textbf {r}_{i}$ into three contributions, with $\\delta \\textbf {r}_{i} = X_{W}\\hat{\\textbf {x}} + X_{T}\\hat{\\textbf {y}} + X_{L}\\hat{\\textbf {z}}$ .", "Rotational moves are performed by three consecutive reorientations around $\\hat{z}$ , $\\hat{x}$ and $\\hat{y}$ , with maximum rotations of $Y_{L}$ , $Y_{W}$ and $Y_{T}$ , respectively.", "The extent of particle displacement and rotation are picked from uniform distributions that depend on the particle translational, $D^{tra}_{\\alpha ,i}$ , and rotational, $D^{rot}_{\\alpha ,i}$ , diffusion coefficients at infinite dilution, with $\\alpha ={L,W,T}$ .", "Maximum displacements and rotations are given by: ${X_{\\alpha }} \\le \\sqrt{2D^{tra}_{\\alpha ,i}\\delta t_{MC}}$ ${Y_{\\alpha }} \\le \\sqrt{2D^{rot}_{\\alpha ,i}\\delta t_{MC}}$ where $\\delta t_{MC}$ is the DMC timescale for one cycle, and is set to $\\delta t_{MC} = 10^{-2} \\tau $ for all simulations, with $\\tau $ the time unit.", "The coefficients $D^{tra}_{\\alpha ,i}$ and $D^{rot}_{\\alpha ,i}$ have been estimated by using the open-source software HYDRO++ [46], [47].", "The interested reader is referred to our previous work [38] for the specific values of these translational and rotational diffusivities in units of $D_{o} \\equiv T^{2}\\tau ^{-1}$ and $D_{r} \\equiv $ rad$^{2}\\tau ^{-1}$ respectively.", "For a monodisperse out-of-equilibrium system, the Brownian dynamics timescale, $\\delta t_{BD}$ , can be obtained by rescaling the MC time scale as follows $\\delta t_{BD} = \\frac{\\mathcal {A}_{c}}{3} \\delta t_{MC}$ where $\\mathcal {A}_{c}$ is the time-dependent acceptance rate calculated at the $c$ th MC cycle over the transitory unsteady state [35].", "We determine $\\mathcal {A}_{c}$ by performing an MC cycle at a fixed $\\delta t_{MC}$ and integrating the above equation numerically: $t_{\\text{BD}}(\\mathcal {C}_{\\text{MC}}) = \\delta t_{\\text{MC}} \\sum _{c=0}^{\\mathcal {C}_{\\text{MC}}} \\frac{\\mathcal {A}_{c}}{3}$ where $t_{\\rm BD}(\\mathcal {C}_{MC})$ is the Brownian time after $\\rm \\mathcal {C}_{MC}$ MC cycles.", "It should be noted that the DMC method does not consider solvent mediated hydrodynamic interactions, which are expected to become especially relevant at strong external fields or large packing fractions.", "Figure: (a) Uniaxial and biaxial order parameters of a system with HBPs of W * =1.0W^{*} = 1.0 undergoing equilibration with an external field with ε f * =3\\varepsilon ^*_f = 3.", "The field is switched on at t/τ≈280t/\\tau \\approx 280 and switched off at t/τ≈550t/\\tau \\approx 550.", "The shaded orange area corresponds to t ON t_{\\rm ON}, while the green shaded area corresponds to t OFF t_{\\rm OFF}.", "(b) Response times (t ON t_{\\rm ON} and t OFF t_{\\rm OFF}) as a function of W * W^{*} with ε f * =3\\varepsilon ^*_f = 3.", "The dashed vertical line in (b) represents the self-dual shape that separates the prolate and oblate geometries.To characterise the dynamics, we estimated (i) the response times, (ii) the mean square angular displacement (MSAD), and (iii) the angular self-part of the van-Hove function (s-VHF).", "We refer to the field-on ($t_{\\rm ON}$ ) and field-off ($t_{\\rm OFF}$ ) response times as the time taken for the biaxial order parameter to reach, respectively, 95$\\%$ (field-on) and 105$\\%$ (field-off) of its equilibrium value.", "In particular, when a field is applied to an $\\rm N_{U}^{+}$ phase, $t_{\\rm ON}$ is the time taken for $B_{2,L}$ to reach 95$\\%$ of its equilibrium value in the field-on steady state.", "A schematic illustration of how we performed this evaluation is reported in Fig.", "REF (a).", "Both sets of response times have been calculated from an average over 50 independent trajectories per system.", "Approximately 2$\\%$ of these trajectories have given response times that were very different from those generally observed.", "Since these anomalies tend to distort averages and give misleadingly large error bars, we have considered them as outliers and excluded them from the average.", "To this end, we employed the Modified Z-score method, a multiple outlier rejection technique to identify statistical anomalies [48].", "In particular, the Modified Z-score, $M_{j}$ , is given by the expression: $M_{j} = \\frac{0.6745 \\times \\big (t_{j} - \\bar{t}\\big )}{MAD}$ where $t_{j}$ is the response time of trajectory $j$ , $\\bar{t}$ is the median response time of the 50 trajectories and $MAD$ stands for median absolute deviation.", "An observable is considered an outlier only if $M_{j} > 3.5$ [48].", "The MSAD provides the ensemble average of the particle angular displacements over time.", "To compute the MSAD, we employ a definition of an unbounded MSAD akin to the translational mean square displacement.", "To this end, we first introduce the definition of a rotational displacement vector which takes the form [49], [50]: $ \\overrightarrow{\\varphi }(t) = \\int _{0}^{t} \\delta \\overrightarrow{\\varphi }(t^{\\prime }) dt^{\\prime }$ where $ \\delta \\overrightarrow{\\varphi }(t^{\\prime })$ is a vector with direction ${\\hat{\\lambda }_{i}}(t^{\\prime }) \\times \\hat{\\lambda }_{i}(t^{\\prime }+dt^{\\prime })$ and magnitude $| \\delta \\overrightarrow{\\varphi }(t^{\\prime })|= \\cos ^{-1}[{\\hat{\\lambda }_{i}}(t^{\\prime }) \\cdot \\hat{\\lambda }_{i}(t^{\\prime }+dt^{\\prime })]$ .", "From this, we can define the MSAD, which is mathematically expressed as: $\\langle \\varphi ^{2} (t)\\rangle = \\frac{1}{N} \\Bigg \\langle \\sum \\limits _{i=1}^N |{\\overrightarrow{\\varphi }}_{i}(t) - {\\overrightarrow{\\varphi }}_{i}(0)|^{2} \\Bigg \\rangle $ where $\\overrightarrow{\\varphi }_{i}$ is the rotational displacement vector of particle $i$ defined in Eq.", "REF .", "Angular brackets denote average over different trajectories.", "Finally, the so-defined rotational displacements are employed to compute the angular s-VHF [49], [50]: $G(\\varphi ,t) = \\frac{1}{N} \\Biggl \\langle \\sum \\limits _{i=1}^N \\delta (\\varphi - |\\overrightarrow{\\varphi }_{i}(t+t_{0}) - \\overrightarrow{\\varphi }_{i}(t_{0})|) \\Biggr \\rangle $ where the symbol $\\delta $ is the Dirac delta function.", "Basically, $G(\\varphi ,t)$ provides the probability distribution of angular displacements of particles within a time $t+t_{0}$ given their position at time $t_{0}$ ." ], [ "Results", "Upon application of a sufficiently strong external field, an $\\rm N_U$ phase can be transformed into an $\\rm N_{B}$ phase [18].", "This transformation is not permanent, and, when the field is removed, uniaxiality is restored.", "The time taken by the particles to reorient along the field director measures the system's ability of switching to a more ordered configuration.", "Vice-versa, when the field is removed, the particles are left free to rotate and the system recovers its original uniaxial state.", "A schematic illustration of both transitory states is given in Fig.", "REF .", "We have measured the response time associated to both $\\rm N_{U}$ $\\rightarrow $ $\\rm N_{B}$ and $\\rm N_{B}$ $\\rightarrow $ $\\rm N_{U}$ transitions upon application of the field $U_{\\rm ext}$ with $\\varepsilon ^*_f=3$ .", "The effect of changing field intensity between $\\varepsilon ^*_f=1.5$ and 3 on the $\\rm N_{U}$ $\\rightarrow $ $\\rm N_{B}$ response time has also been assessed and is available, for the interested reader, in Appendix A.", "The resulting response times, $t_{\\rm ON}$ and $t_{\\rm OFF}$ , are reported in Fig REF (b).", "Since the formation of a field-induced $\\rm N_{B}$ phase is dependent on the alignment of the particle intermediate axis $\\hat{x}$ with the external field, the discussion that follows is relative to this axis, unless otherwise stated.", "To start with, we notice that $t_{\\rm ON} < t_{\\rm OFF}$ across the complete set of anisotropies (see Fig.", "REF (b)).", "In other words, at a given particle width, the $\\rm N_{U}$ $\\rightarrow $ $\\rm N_{B}$ switching is faster than the $\\rm N_{B}$ $\\rightarrow $ $\\rm N_{U}$ switching.", "To understand the origin of this behaviour, we calculated the MSADs of our systems and compared the field-on and field-off profiles for each anisotropy.", "The MSAD of systems with $W^{*} = 2.5$ and 6 are shown, respectively, in the top and bottom frames of Fig.", "REF .", "At very short times, the field-on and field-off MSADs are very similar to each other, with the former becoming larger immediately after and up to relatively long time scales.", "Over this period of time, field-induced rotation is faster than free rotation.", "However, on time scales comparable to $t_{\\rm ON}$ , a crossover between the two MSADs is observed.", "On these time scales and beyond, free rotation grows significantly much faster with time than field-induced rotation.", "We therefore conclude that the presence of the external field accelerates the system orientational dynamics by forcing the reorientation of the particle $\\hat{x}$ axis along the field director.", "As more and more HBPs are oriented, the field-on MSAD grows less and less with time and would eventually saturate to a plateau if the field strength was sufficiently large to offset and overcome thermal forces.", "By contrast, the field-off MSAD practically shows the same behaviour with time over the full time scale, as expected in free rotational diffusion.", "As for the effect of anisotropy on the response time, we first discuss the case of the field-induced uniaxial-to-biaxial transitory state.", "Figure: MSAD in field-on and field-off scenarios of a system of HBPs with reduced width (a) W * =2.5W^{*} = 2.5 and (b) W * =6W^{*} = 6.", "The field-on simulations apply an external field of strength ε f * =3\\varepsilon ^*_{f} = 3.", "The dashed vertical lines indicate t ON t_{\\rm ON} of each systems (t OFF t_{\\rm OFF} is out of scale and not shown).", "The insets in (a) and (b) show the MSAD at shorter timescales.Figure: MSAD of the x ^\\hat{x}, y ^\\hat{y} and z ^\\hat{z} axes of HBPs for systems undergoing N U →N B \\rm N_{U} \\rightarrow N_{B} transition with ε f * =3\\varepsilon ^*_f = 3 for (a) prolate; (b) self-dual and (c) oblate HBPs, and N B →N U \\rm N_{B} \\rightarrow N_{U} transition when the field is switched off for a (d) prolate; (e) self-dual and (f) oblate HBPs.", "The dashed vertical lines in each figure represent the t ON t_{\\rm ON} for (a)-(c) and t OFF t_{\\rm OFF} for (d)-(f).In Fig.", "REF (b), $t_{\\rm ON}$ increases with $W^*$ , implying that the reorientation is slower for oblate than for prolate particles.", "More specifically, for rod-like HBPs ($W^{*} = 1$ ), we observe a rapid switching with $t_{\\rm ON}/\\tau \\approx 41$ , whereas for plate-like HBPs ($W^{*} = 12$ ), it is significantly slower, with $t_{\\rm ON}/\\tau \\approx 4200$ .", "Consequently, making HBPs more oblate leads to a slower field-induced $\\rm N_{U} \\rightarrow N_{B}$ transition.", "To confirm these preliminary tendencies, we compare the MSADs of the field-on regimes of each anisotropy along the three axes.", "The top frames of Fig.", "REF display the field-on MSADs of systems containing HBPs with $W^{*} = 1$ , 3.46 and 12.", "We notice that the MSAD of the particle axis oriented as the nematic director of the original $\\rm N_U$ phase is the smallest across all the geometries.", "More specifically, the MSAD of rod-like particles in Fig.", "REF (a) exhibits a strong rotational coupling between $\\hat{x}$ and $\\hat{y}$ particle axes, while $\\hat{z}$ is practically unaffected by the application of the field.", "Such a strong angular correlation between $\\hat{x}$ and $\\hat{y}$ , with $\\langle \\varphi ^2_W \\rangle = \\langle \\varphi ^2_T \\rangle $ over time, is due to the square cross-sectional area of this specific set of HBPs, where $W=T$ .", "For similar reasons, plate-like HBPs with $W=L$ exhibit strong rotational correlations between their axes $\\hat{x}$ and $\\hat{z}$ , with $\\langle \\varphi ^2_W \\rangle = \\langle \\varphi ^2_L \\rangle $ (see Fig.", "REF (c)), while $\\langle \\varphi ^2_T \\rangle $ , slightly increasing over time for mere thermal fluctuations, remains very small, practically insensible to the external field.", "In systems of self-dual shaped HBPs ($W^{*} = 3.46$ ), we observe that the MSADs of $W$ and $T$ are initially coupled, but then diverge over time.", "This behaviour is observed for all anisotropies that are not perfectly rod-like or plate-like and agrees very well with the tendencies reported in our recent work on the equilibrium dynamics of HBPs [38].", "When analysing the field-on MSADs of the particle axes perpendicular to the original nematic director, we also notice an initially linear, rather steep dependence on time, followed by an intermediate non-linear behaviour and subsequently by a second linear regime at times comparable to $t_{\\rm ON}$ .", "Such a long-time linear regime suggests that HBPs' angular displacements are gradually reducing, due to the system approaching a new equilibrium state.", "Under these conditions, further rotations of the particle intermediate axis $\\hat{x}$ , which is already aligned with the field, are suppressed, and only small angular fluctuations are detected.", "At much larger field strengths, with thermal fluctuations completely inhibited, we expect this second linear regime to plateau at long times.", "In agreement with Fig.", "2(b), we also observe that systems with rod-like HBPs ($W^{*} = 1$ ) only take $t/\\tau \\approx 17$ to reach $<\\varphi ^{2}_{W}> = 0.6$ rad$^2$ , whereas systems with self-dual shaped ($W^{*} = 3.46$ ) or plate-like ($W^{*} = 12$ ) HBPs take, respectively, $t/\\tau \\approx 250$ and $t/\\tau \\approx 1600$ to achieve the same MSAD value.", "This suggests that prolate HBPs tend to reorient significantly faster when an external field is applied, leading to a relatively rapid equilibration.", "Figure: (a) Angular s-VHFs of (a) field-on reorientation at time t/τ=35t/\\tau = 35 and (b) field-off reorientation at time t/τ=100t/\\tau = 100.To gain a better insight into the dynamics of reorientation during this first transitory unsteady state, we calculated the s-VHFs of all anisotropies at $t/\\tau = 35$ , corresponding to a time when all field-on cases are still undergoing equilibration.", "As the MSAD for field-off scenarios are linear, we arbitrarily picked $t/\\tau = 100$ to show the field-off s-VHFs.", "The s-VHFs shown in Fig.", "REF refer to the intermediate axis and have been normalised such that $\\int _{0}^{\\infty } 4\\pi \\varphi ^{2}G(\\varphi ,t) d\\varphi = 1$ .", "The first evident conclusion, confirming the results discussed so far, is that prolate HBPs rotate faster than oblate HBPs.", "This can be appreciated in Fig.", "REF (a) by pinpointing the location of the peak of $G(\\varphi _W,t)$ , which indicates the most probable rotation achieved by particles of a given geometry at $t/\\tau = 35$ .", "In particular, the peak of $G(\\varphi _{W=T},t)$ and $G(\\varphi _{W=L},t)$ suggests that rod-like and plate-like particles have rotated, respectively, by $\\varphi _{W} \\approx 1.4$ rad and $\\varphi _{W} \\approx 0.2$ rad.", "By increasing particle width from $W^*=1$ to 12, the peak of the angular s-VHFs gradually displaces towards lower rotations.", "Not only does the particle anisotropy determine the location of the peak of these distributions at a given time, but also their broadness.", "In other words, the angular s-VHFs provide relevant information on the most probable rotation performed by HBPs and on the existence of HBPs that rotate faster or slower than the average.", "In particular, the presence of fast- and slow-responsive HBPs is evinced by the tails of the distributions in Fig.", "REF (a), especially broad at $W^*=1$ and then narrower and narrower up to $W^*=12$ .", "Therefore, rod-like HBPs rotate relatively fast, but heterogeneously (broad $G(\\varphi _W,t)$ peaked at large distances), whereas plate-like HBPs are significantly slower, but rotate much more homogeneously (narrow $G(\\varphi _W,t)$ peaked at short distances).", "Figure: (a) Angular s-VHFs at various times, expressed as percentage of t OFF t_{\\rm OFF}, in systems of HBPs with (a) W * =3.46W^{*} = 3.46 and (b) W * =4W^{*} = 4.When the field is switched off, the system recovers its original uniaxial symmetry with the particles free to reorient under the mere effect of thermal fluctuations.", "The time $t_{\\rm OFF}$ taken by this field-off reorientation to re-establish the $\\rm N_U$ phase is again much shorter in nematics of prolate HBPs (see Fig.", "REF (b)).", "In particular, $t_{\\rm OFF}$ shows a tendency to increase with particle width up to $W^{*} = 4$ , where $t_{\\rm OFF}/\\tau \\approx 15 \\cdot 10^3$ .", "When HBPs acquire a modest oblate geometry, such as from $W^{*} = 6$ to $W^{*} = 8$ , $t_{\\rm OFF}$ drastically decreases to around $t_{\\rm OFF}/\\tau \\approx 6 \\cdot 10^3$ , before increasing again at $W^{*} = 12$ .", "We also notice that the free reorientation at $W^{*} = 3.46$ and 4 is particularly slower than that of other anisotropies and deserves an explanation.", "The MSADs in the field-off scenarios, shown in Fig.", "REF (d)-(f), exhibit a linear profile throughout the simulation due to the absence of an external field and decrease upon increasing $W^{*}$ .", "This tendency is also detected in the field-off s-VHFs of Fig.", "REF (b), where, similarly to the field-on case, the angular displacement decreases at increasing particle width.", "These elements would suggest a scenario where $t_{\\rm OFF}$ increases and free rotation becomes slower upon increasing $W^{*}$ .", "Because $t_{\\rm OFF}$ is peaked for approximately self-dual shaped particles and then decreases (see Fig.", "REF (b)), there must be an additional element contributing to the field-free reorientation from the $\\rm N_B$ to the $\\rm N_U$ phase.", "We believe that this element is related to the ability of self-dual shaped HBPs of retaining phase biaxiality when the field is switched off.", "In our recent work on the field-induced phase behaviour of HBPs, we found that the self-dual shape requires a surprisingly weak external field, compared to prolate and oblate geometries, to spark an $\\rm N_{U} \\rightarrow N_{B}$ transition [18].", "In particular, the minimum field strength to stabilise $\\rm N_{B}$ LCs was found to be $\\varepsilon ^*_f=0.1$ and 0.25 at $W^*=3.46$ and 4, respectively, and then increasing to $\\varepsilon ^*_f=0.5$ at $W^*=3$ and to $\\varepsilon ^*_f=1$ at $W^*=6$ .", "Therefore, we believe that the field free $\\rm N_{U} \\rightarrow N_{B}$ transition at or very close to the self-dual shape is affected by a metastability of the $\\rm N_{B}$ phase in off-field case.", "The underlying metastability allows the system to retain the induced biaxiality over a longer time when compared with nematics of oblate of prolate HBPs.", "In Fig.", "REF , we show the evolution of the s-VHFs of $W^{*} = 3.46$ and $W^{*} = 4$ at different times up to $t_{\\rm OFF}$ .", "At short to intermediate time scales, after having switched the field off, these s-VHFs exhibit a double peak that suggests the presence of two populations of HBPs.", "Because these two populations rotate at sufficiently different rates, we can label them as slow and fast.", "The first peak survives over a relatively long period of time, between 0.2$t_{\\rm OFF}$ and 0.7$t_{\\rm OFF}$ , turning gradually into a shoulder that disappears at longer times.", "These peaks and subsequent shoulders are especially pronounced in the case of $W^{*} = 4$ (Fig.", "REF (b)), explaining why $t_{\\rm OFF}$ at $W^{*} = 4$ is significantly slower than $t_{\\rm OFF}$ at $W^{*} = 3.46$ .", "Double peaks and shoulders are not observed at other anisotropies or in field-on transitions (not shown here), indicating that these tendencies are especially relevant only in the field-off relaxation of self-dual shaped particles.", "In addition to their propensity towards biaxiality retention, HBPs with $W^*=4$ rotate more slowly than perfectly self-dual shaped particles, as shown in Fig.", "REF (b) and in agreement with the tendencies observed in $\\rm N_U$ phase in the absence of external fields [38].", "The resulting large value of $t_{\\rm OFF}$ is therefore determined by the interplay between the particle's ability to rotate and the system's tendency of retaining phase biaxiality.", "This interplay explains the non-monotonic trend of $t_{\\rm OFF}$ with particle shape in Fig.", "REF (b)) and provides, along with $t_{\\rm ON}$ , a useful guideline to select the most suitable particle anisotropy for the design of field-responsive nanomaterials." ], [ "Conclusions", "In summary, by Dynamic Monte Carlo simulation, we studied the field-induced dynamics in uniaxial nematic LCs of colloidal HBPs.", "By forcing the particles to reorient around the nematic director, the external field induces an $\\rm N_{U} \\rightarrow N_{B}$ phase transition that takes the system to a new steady state.", "When the field is switched off, the biaxiality is gradually lost and the $\\rm N_U$ phase is restored.", "The time taken for the system to reorient, also referred to as response time, strongly depends on the particle anisotropy.", "The response times in $\\rm N_{U} \\rightarrow N_{B}$ and $\\rm N_{B} \\rightarrow N_{U}$ switching were calculated and compared across all anisotropies studied.", "Despite being the optimal shape to promote phase biaxiality, the switching dynamics of self-dual shape HBPs is less satisfactory compared to prolate HBPs.", "In particular, rod-like HBPs with $W^{*} = 1$ exhibit the fastest reorientation times in both the field-on and field-off cases.", "The analysis of MSADs and s-VHFs show that the response time is a result of a trade-off between particle rotational diffusion and phase biaxiality retention, being both determined by shape anisotropy.", "Prolate HBPs were found to rotate faster than self-dual shaped or oblate HBPs, allowing rapid phase switching between the two nematic phases.", "Systems of HBPs with geometry equal or very close to the self-dual shape exhibit a particularly slow field-free reorientation, most likely due to relatively low field strength required to transform $\\rm N_U$ into $\\rm N_B$ phases and favouring the former in absence of an external field [18].", "The ability of retaining biaxiality over a longer period of time is corroborated by the existence of a double peak in the angular s-VHFs of $\\rm N_{B} \\rightarrow N_{U}$ transition at short-to-intermediate time scales.", "This double peak suggests the existence of two populations of (quasi) self-dual shaped HBPs whose reorientation is not uniform and delays the system relaxation.", "While prolate HBPs are especially field-responsive and exhibit a rapid field-free reorientation, when one analyses the distribution of their angular displacements over time, this appears to be very broad, with particles exhibiting a very heterogeneous ability of rotating.", "By contrast, oblate HBPs, while significantly less responsive, are characterised by a very narrow distribution of angular displacements.", "All these elements offer a fundamental understanding of the impact of shape anisotropy on the dynamics of uniaxial-to-biaxial switching and a guidance to formulate nanomaterials with specific switching dynamics for target applications." ], [ "Acknowledgements", "EMR would like to thank the Malaysian Government Agency Majlis Amanah Rakyat for funding his PhD at the University of Manchester.", "AP and LT acknowledges the financial support from the Leverhulme Trust Research Project Grant RPG-2018-415.", "AC acknowledge the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER for funding (project PGC2018-097151-B-I00).", "EMR, LT, DC and AP acknowledge the assistance given by IT Services and the use of Computational Shared Facility at the University of Manchester.", "Finally, we thank Gerardo Campos-Villalobos (Utrecht University) for his assistance in generating the snapshots in Fig.", "REF ." ], [ "Data Availability", "The data that support the findings of this study are available from the corresponding author upon reasonable request." ], [ "Effect of Field Strength", "In this appendix, we briefly discuss the effect of altering field intensity on the field-on response times, $t_{\\rm ON}$ of HBPs.", "Here, we report the response times for the field-on case for all anisotropies studied at field strengths from $\\varepsilon ^*_f = 1.5$ to $\\varepsilon ^*_f = 3$ .", "These field intensities result in the formation of strong $\\rm N_{B}$ phases with $B_{2} \\ge 0.35$ [18].", "The results are shown in Fig.", "REF .", "Figure: Changes in field-on response time, t ON t_{\\rm ON} as a function of ε f * \\varepsilon ^*_f across different anisotropies.At constant $\\varepsilon ^*_{f}$ , we observe that $t_{\\rm ON}$ generally increases with $W^{*}$ and this increment is significant.", "For instance, at $\\varepsilon ^*_{f} = 2$ , the response time increases by two orders of magnitude from $t_{\\rm ON} \\approx 57$ at $W^{*} = 1$ to $ t_{\\rm ON} \\approx 5700$ at $W^{*} = 12$ .", "We conclude that prolate particles tend to rotate faster than oblate particles, regardless the field strength.", "Upon increasing field strength, the reorientation becomes faster as suggested by the gradual decrease of $t_{\\rm ON}$ with $\\varepsilon ^*_{f}$ .", "In addition, we note that the statistical errors in $t_{\\rm ON}$ decrease with $\\varepsilon ^*_{f}$ , most likely due to a stronger suppression of rotational fluctuations.", "This tendency is consistent with the works by Zannoni and co-workers [10].", "For rod-like HBPs ($W^{*} = 1$ ), increasing $\\varepsilon _{f}$ does not significantly affect $ t_{\\rm ON}$ , probably because the reorientation capability of these HBPs is very close to its saturation value." ], [ "Response Times", "In Table I, we report $t_{\\rm ON}$ for $\\varepsilon ^*_{f}=1.5$ , 2, 2.5 and 3, and $t_{\\rm OFF}$ for $\\varepsilon ^*_{f} = 3$ ." ] ]
2105.11725
[ [ "Multi-Task Learning of Generation and Classification for Emotion-Aware\n Dialogue Response Generation" ], [ "Abstract For a computer to naturally interact with a human, it needs to be human-like.", "In this paper, we propose a neural response generation model with multi-task learning of generation and classification, focusing on emotion.", "Our model based on BART (Lewis et al., 2020), a pre-trained transformer encoder-decoder model, is trained to generate responses and recognize emotions simultaneously.", "Furthermore, we weight the losses for the tasks to control the update of parameters.", "Automatic evaluations and crowdsourced manual evaluations show that the proposed model makes generated responses more emotionally aware." ], [ "Introduction", "The performance of machine translation and summarization has been approaching a near-human level in virtue of pre-trained encoder-decoder models, such as BART [5] and T5 [14].", "The same technology has been applied to dialogue systems, which are now expected to be put to practical use.", "To interact naturally with a human, the computer needs to be human-like.", "Several methods have been proposed to build such dialogue systems.", "They include a system interacting based on knowledge and common sense [2] and that interacting by considering one's own and the other's personality [21].", "In particular, we focus on the viewpoint of emotion as targeted in [15].", "In this paper, we propose a multi-task learning method for building a dialogue system that takes the speaker's emotions into account.", "Also, we focus on the hierarchy of emotions [4] and simultaneously train multiple emotion recognition tasks with different granularity.", "Our multi-task learning model is not expected to share complementary information among similar tasks as previous work [9], and we do not aim at improving the accuracy of emotion recognition.", "Instead, we focus on generating emotion-aware responses.", "Also, concerned that the ratio of emotion recognition in multi-task learning is too large, we explore further quality improvement by weighting each loss.", "We build a model based on BART [5], a pre-trained Transformer [17] model, to implement multi-task learning of response generation and emotion recognition.", "Experiments are performed using a dialogue corpus without context.", "The effectiveness of the proposed method in generating responses is confirmed by automatic and manual evaluations.", "Multi-task learning of response generation and emotion recognition makes generated responses more emotionally aware of utterances.", "The improvement is not only on the emotional aspect but also on the quality of fluency, informativeness, and relevance.", "We also found that controlling the parameters by weighting the losses improved the performance of the model.", "Figure: The architecture of our model, based on BART .It contains one LM head and several CLS heads, which solve generation and classification, respectively.", "In our experiments, three CLS heads are used for the emotion recognition tasks with different granularity." ], [ "Related Work", "One of the previous studies on emotion-based response generation is the Emotional Chatting Machine (ECM) [23].", "ECM is used together with an emotion classifier to generate a response based on a given emotion.", "EmpTransfo [20] is a similar model to ours.", "Given an utterance, a model based on GPT [13] learns an emotion and an action simultaneously in addition to a response, which improves the quality of generated responses.", "These models focus on the emotion of a response so that they do not generate a response based on the emotion of an utterance.", "[10] incorporate an emotion encoder into a hierarchical seq2seq architecture, enabling a system to understand the emotional context on a user.", "TG-EACM [18], the successor of EACM [19], is a model that considers not only the emotion in an utterance but also the emotion that a response should have.", "The model learns a distribution to infer both the emotion of the utterance and the response from a given utterance.", "CARE [22] uses some commonsense to generate a response with both rationality and emotion.", "Through latent concepts obtained from an emotionally aware knowledge graph, predicted responses can be emotional and rational.", "Actually, the above models require separate units or special architecture for understanding emotion in a dialogue.", "In contrast, our proposed model achieves that with a single structure, inherited from Transformer [17] and BART [5].", "In other words, our model does not need an extra unit.", "Therefore, the proposed method consequently reduces the redundancy of Transformer parameters [3] and realizes more efficient understanding of emotion to generate a response." ], [ "Overview", "Our model learns response generation as a generation task and emotion recognition as a classification task.", "By learning response generation and emotion recognition simultaneously through multi-task learning, it is possible to generate a response by considering the emotion of a given utterance.", "Multi-task learning often involves several similar tasks because they can share information and thus the performance of each task can be improved.", "However, the purpose of our multi-task learning method is to improve the quality of response generation, not to improve the performance of emotion recognition.", "This is different from general multi-task learning.", "Our model is based on BART [5].", "Its architecture is shown in Figure REF .", "The model has several output layers, or heads, for the tasks to be trained, which include an LM head for generating words in response generation and CLS heads for solving classification tasks.", "Given a sentence, the CLS head predicts its label such as positive or negative.", "One CLS head is set for each classification task.", "The input/output format of each task is the same as that in BART.", "In the generation task, we put an utterance and a right-shifted response into the encoder and decoder, respectively.", "In the classification task, we put an utterance and a right-shifted utterance into the encoder and decoder, respectively.", "Following the learning algorithm of MT-DNN [9], each task that the model learns is selected for each mini-batch.", "A different loss is calculated for each task, and the parameters are updated for each mini-batch." ], [ "Losses of Generation and Classification Tasks", "Let ${x} = (x_1, \\ldots , x_M)$ be the given utterance and ${\\theta }$ be the parameters of the model.", "Our model is trained by updating ${\\theta }$ based on the loss for each task." ], [ "Generation", "The response to ${x}$ is defined as ${y} = (y_1, \\ldots , y_N)$ .", "The model infers an appropriate ${y}$ from ${x}$ .", "The generation loss $\\mathcal {L}_\\mathrm {gen}$ is calculated as the negative log-likelihood loss.", "$\\mathcal {L}_\\mathrm {gen} = -\\sum _{j=1}^N \\log p (y_j | {x}, y_1, \\ldots , y_{j-1}; {\\theta })$" ], [ "Classification", "If the correct label of ${x}$ is $c$ , the model infers $c$ from ${x}$ .", "The negative log-likelihood loss is also used for the classification loss $\\mathcal {L}_\\mathrm {cls}$ .", "$\\mathcal {L}_\\mathrm {cls} = -\\log p (c | {x}; {\\theta })$" ], [ "Loss Weighting", "Although the proposed multi-task learning model learns the generation and classification tasks simultaneously, there is a possibility that the ratio of learning for the classification task is too large.", "When solving a general classification task, the end of learning is often determined by the convergence of the loss in the validation data.", "On the other hand, the target of our model is a generation task, and the number of epochs required for generation is larger than that of the classification task.", "Therefore, we consider weighting the loss functions.", "While the weight for response generation is fixed at 1, the weight for emotion recognition is varied between 0 and 1.", "This makes the contribution of the classification task reduced in updating the parameters.", "Table: The statistics of the datasets for our experiments, where TEC stands for Twitter Emotion Corpus.Because TEC and CrowdFlower have no split of train, validation, and test, we split them into three at 8:1:1.Table: Evaluation results of our models by multi-task learning.", "R stands for response generation, and E•\\bullet is emotion recognition with •\\bullet labels.", "Emo, flu, info, and relv are the four aspects for the manual evaluation by crowdsourcing." ], [ "Datasets", "We train a model with three tasks of emotion recognition in addition to response generation using multi-task learning.", "Each emotion recognition task is a classification task with 6, 2, and 12 labels, and we call them emotion recognition, coarse-grained emotion recognition, and fine-grained emotion recognition, respectively.", "The datasets for such emotion recognition were selected according to [1].", "The numbers of instances are summarized in Table REF ." ], [ "Response Generation", "DailyDialog [7] is used for response generation.", "The dataset is a multi-turn dialogue corpus, and we obtain pairs of an utterance and a response by extracting two turns at a time.", "Each utterance in the corpus has an emotion label, but we do not use these labels in the experiment.", "This is because almost all of the emotion labels are other, which is not suitable for our method." ], [ "Emotion Recognition", "For the core emotion recognition dataset, we use the Twitter Emotion Corpus [11].", "It was constructed based on Twitter hashtags and consists of six labels: {anger, disgust, fear, joy, sadness, surprise}.", "Because there is no distinction between train, validation, and test in the dataset, 80% of the total samples is assigned to train, and the remaining 10% each is assigned to validation and test." ], [ "Coarse-Grained Emotion Recognition", "For coarse-grained emotion recognition, we use SST-2 [16].", "This is a dataset of movie comments labeled with {positive, negative}.", "To maintain a balance with the number of instances for the other emotion recognition tasks, we reduce the number of instances for training to 25%." ], [ "Fine-Grained Emotion Recognition", "For fine-grained emotion recognition, we use the emotionally-tagged corpus provided by CrowdFlower.The original link is no longer available.", "An alternative is https://data.world/crowdflower/sentiment-analysis-in-text.", "We exclude the label empty and adopt this corpus for a classification task with 12 labels: {anger, boredom, enthusiasm, fun, happiness, hate, love, neutral, relief, sadness, surprise, worry}.", "As with the Twitter Emotion Corpus, this corpus does not have a split of train, validation, and test, and thus the whole data is divided into 8:1:1.", "Furthermore, for the same reason as in SST-2, only 50% of the total data is used." ], [ "Training", "The hyperparameters are set based on BART [5] and the Fairseq example.https://github.com/pytorch/fairseq/blob/master/examples/bart/README.summarization.md.", "The learning rate is set to 3e-5, and the parameters are optimized by Adam with weight decay.", "For response generation, we apply label smoothing of 0.1 to the negative log-likelihood loss.", "The number of input and output tokens is set to 64, and training is performed for 64 epochs.", "We use beam search with 5 beams to select words and eliminate cases where there are more than three repeated $n$ -grams.", "Training and generation are performed on NVIDIA Tesla V100.", "Figure: An example of the manual evaluation by crowdsourcing on Amazon Mechanical Turk.Workers are supposed to answer such questions by rating the given dialogue on a five-point scale." ], [ "Evaluation Metrics", "We evaluate the trained models automatically and manually." ], [ "Automatic Evaluation", "First, we evaluate how much the output responses are related to the correct response using Bleu [12].", "Second, we evaluate whether the output responses are lexically diverse using distinct [6].", "For distinct, distinct-1 and distinct-2 are calculated, which focus on unigrams and bigrams, respectively.", "We also compare the average number of words in output responses, which is based on the assumption that the longer a response is, the less common it is.", "The large average number indicates that generated responses tend to be not dull." ], [ "Manual Evaluation", "Actually, the lack of correlation between automatic and manual evaluation [8] has been indicated especially in regards to generation tasks.", "Thus, we perform manual evaluation by crowdsourcing, where Amazon Mechanical Turk is used as the platform.", "We use four metrics mainly following [15]: emotion, fluency, informativeness, and relevance.", "Each of the questions asks whether the generated response takes into account the emotion of the utterance, whether the generated response is syntactically correct, whether a generated response provides some information for the utterance, and whether the content of the response is appropriately related to the utterance.", "A total of 100 randomly selected responses for the test data are asked to rate the above four metrics on a five-point scale.", "US residents are designated as workers, and seven workers are requested for each metric of each sample.", "The final score is obtained as the average of the values obtained from the seven workers.", "An example of the questions asked to the workers is shown in Figure REF ." ], [ "Multi-Task Learning", "The evaluation results are shown in Table REF .", "The response generation is denoted by R, and the emotion recognition for the Twitter Emotion Corpus, SST-2, and CrowdFlower datasets is denoted by E6, E2, and E12, respectively.", "In terms of automatic evaluation, R+E6+E2 and R+E6+E12 maximized the distinct and Bleu, respectively.", "In the proposed multi-task learning model, therefore, emotion recognition of different granularity is effective in relevance and diversity.", "For manual evaluation, all models that include emotion recognition outperformed the model with only response generation.", "Moreover, R+E6 scores were particularly high for all four metrics.", "The proposed multi-task learning model not only makes the generated responses more emotionally aware but can also improve the quality of other metrics, such as fluency and informativeness.", "Several examples of responses generated by the obtained model are shown in Table REF .", "We compare the given utterances and their responses of R and R+E6.", "We can see that R+E6 generated more emotion-sensitive sentences, such as “Yeah, yeah, I know” and “good idea.” In addition, we show the results of emotion recognition in Table REF , which is especially on a six-label classification task.", "We calculate accuracy and F1-score as metrics for evaluation.", "The result shows that, on emotion recognition, increasing the number of tasks to train does not necessarily lead to improvement of the scores.", "We can see that models with training of emotion recognition on fine-grained labels tend to outperform the other models.", "However, the goal of our model is not improvement of classification but that of generation, so that those score variation is not essential in this work." ], [ "Loss Weighting", "The evaluation results for different loss weighting are shown in Table REF .", "The weight for the loss of E$\\bullet $ is denoted as $\\lambda _{\\mathrm {E}\\bullet }$ .", "In automatic evaluation, we can see the improvement of the scores by weighting, especially in the model with E12.", "On the other hand, the manual evaluation shows that weighting improves some scores, with the case (.5, .5, 0) producing the highest score.", "Therefore, weighting each loss can improve the quality of generated responses, and in the condition of our experiment, it is most effective to reduce the weights of E6 and E2 by half.", "Table: Evaluation results for differed loss.", "λ E• \\lambda _{\\mathrm {E}\\bullet } indicates the weight for the loss of E•\\bullet , and the metrics are the same as those of Table .", "The weight for the response generation loss (λ R \\lambda _\\mathrm {R}) is fixed at 1 throughout the experiments.", "Note that (1, 0, 0) is equivalent to R+E6 in Table ." ], [ "Conclusion", "We worked on improving the quality of neural network-based response generation.", "Focusing on the aspect of emotion, we proposed a multi-task learning response generation model that includes the tasks of generation and classification.", "Through automatic and manual evaluations, we confirmed that the proposed model improved several metrics of performance.", "Moreover, we further improved the quality of the model by weighting losses.", "As a result, we found that such weighting improved several scores and the balance of parameter updates was also an important factor.", "This paper focused on the emotion of the dialogue and generated responses that take into account the emotion of an utterance.", "On the other hand, we did not focus on the emotion of a response, which is a subject for our future work.", "We plan to work on estimating the emotions that a response should have and generating a response based on a specified emotion.", "In the experiments of this paper, we omitted the context of a dialogue.", "However, it is also necessary to consider past utterances and their effects on emotions for generating responses, which is also an issue to be addressed in the future." ], [ "Acknowledgements", "This work was supported by JSPS KAKENHI Grant Number JP18H03286." ] ]
2105.11696
[ [ "Fat Tails and Black Swans: Exact Results for Multiplicative Processes\n with Resets" ], [ "Abstract We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails -- ubiquitous in the statistics of social, economic, and ecological systems.", "Our main goal is to provide a series of exact results on the dynamics and asymptotic behaviour of increasingly complex versions of a basic multiplicative process with resets, including discrete and continuous-time variants and several degrees of randomness in the parameters that control the process.", "In particular, we show how the power-law distributions are built up as time elapses, how their moments behave with time, and how their stationary profiles become quantitatively determined by those parameters.", "Our discussion emphasizes the connection with financial systems, but these stochastic processes are also expected to be fruitful in modeling a wide variety of social and biological phenomena." ], [ "Introduction", "“At first glance, the facts of human life do not seem to subordinate themselves, as do the phenomena of nature, to certain general laws.", "Statistics show, however, that this is just a gradual difference due to the more complicated nature of human relations, and that man only has to learn to read the statistics correctly in order to extract more general conclusions; and it is not uncommon to find interesting and strange laws.” The quote above opens a report written in 1913 by Felix Auerbach,[1] where he discussed the skewed abundance distribution of cities of various sizes in Germany, in what is likely the first published document on the topic.", "Beyond the identification of an uncommon statistical pattern, Auerbach's paper reveals a visionary intuition on the existence of general laws underlying the statistical properties of social phenomena.", "Three decades later, in his book “Human Behavior and The principle of least effort,”[4] George Kingsley Zipf compiled multiple data with the aim of demonstrating regularities and subsequently discussing plausible, simple underlying principles.", "The distribution of city sizes and the frequency of words in written texts —an observation that goes back to 1916[2]— are undoubtedly the two best-known examples, but Zipf also analyzed the length of intervals between repetitions in a Mozart concerto, the number of retail stores of like kind, the number of passengers travelling by airway, the personal income in different countries —first described by Vilfredo Pareto in 1896[10]— and the number of composers of chamber music as a function of their year of birth, among many others.", "He ordered each set of events by decreasing size, $S(k)$ , assigning rank $k=1$ to the largest event, $k=2$ to the second largest, and so on.", "Though the relationship $S(k) \\propto k^{-\\alpha }$ , with $\\alpha \\approx 1$ , was pervasively found (and became known as Zipf's law), there were also many cases of data characterized by different values of $\\alpha $ , and even by other functional relations.", "Ever since, research on the mechanisms underlying emergent properties of collective human behaviour has been pursued with much effort and also remarkable success.", "[11] Quantities whose frequency distributions are given by power-law functions have always awaken especial interest in the study of a variety of social systems, as it has the search for simple processes behind this kind of distribution.", "[12], [13] Pure stochastic multiplicative processes (SMPs) yield lognormal distributions for the relevant variables,[14] and can produce bona fide power laws when acting in conjunction with additional mechanisms.", "Actually, SMPs appear at the core of many successful explanatory models for power laws, such as Yule's birth-death process to explain the distribution of taxa,[3] or Simon's model to derive the abundance of words in written texts.", "[5] These early models rationalize power-law distributions in other systems as well, as in city sizes[15] or growth of business firms.", "[16] Furthermore, suitable variations of Simon's model reproduce the observed abundance of family names,[17], [18] while it yields distributions that fit remarkably well the actual frequency of words in written texts[19] and the usage of notes in musical compositions.", "[20] Stochastic processes with multiplicative noise plus reinjection are characterized by distributions with power-law tails.", "[21], [22] This kind of mechanism has applications, for instance, in population dynamics and investment portfolio growth.", "[23] A variant of reinjection that sets a minimum value for the dynamical variable [24] can be interpreted in an economic context as a subsidy that keeps individuals above a critical poverty line.", "SMPs with conservation of the total population, which is a generalization of Zeldovich's intermittency model,[25] has been used to explain Zipf's law for cities.", "[26] Introduction of reset events in SMPs renders power laws with exponents which depend on the reset probability and on the distribution of growth rates,[7] a situation that can account for the power-law tail of the personal income distribution.", "[27] Though reset events were introduced in the generic context of SMPs yielding power laws,[7] this mechanism has been subsequently applied to a broad variety of problems.", "[8], [9], [28], [29] SMPs belong to a class of processes that can exhibit non-self-averaging effects.", "In physics, this property was first described for spin glasses, where the fact that different realizations of the process visit different areas of the phase space, even in the thermodynamic limit, yields different observable quantities (e.g., moments of the distribution of visited states) for each of the system replicas.", "[30] Later, it was shown that non-self-averaging behavior was present in various systems,[31], [32] such as sums of power-law distributed random variables, branching processes, and one-dimensional random walks with return to the origin —a simple case of stochastic additive processes with resets.", "[33] A major consequence of the lack of self-averaging is that quantitative properties of the system estimated through averages over time or realizations can be highly unreliable and deeply differ from the actual asymptotic properties of the stochastic process.", "Interestingly, the same phenomenology has been observed in economics and finance, where its relevance is difficult to overstate.", "[34] In these contexts, multiplicative processes are implicit in Gibrat's law,[35] (or “law or proportionate effect”), which states that the rate of growth of a business firm is independent of its absolute size.", "Though, as a pure SMP, Gibrat's law implies a lognormal distribution of the relevant variable, the ample evidence of power-law distributed quantities in economics and finance,[34] suggests that additional mechanisms must be at play.", "Indeed, since sustained exponential growth is unfeasible in reality, Gibrat's law cannot hold at all times.", "Financial crises and market crashes act as “control mechanisms” in the form of catastrophic, punctuated interruptions of the idealized multiplicative growth.", "Actually, examples of such events abound: price indices suffered severe drops in the crises of 2008 and 2011[36] (see Fig.", "REF ); in the 1980s, some banks lost more money than they ever made in their history;[37] financial bubbles more than often cause market crashes, like the well-known Wall Street crash of 1929 and many others.", "The abrupt loss of gains in a time much shorter than that required to accumulate them is commonplace in finance, and resets appear as a qualitatively suitable mechanism to mimic such dynamics.", "Nassim Taleb's Black Swan Theory[38] explains the prominent role of such “unexpected” catastrophic events —as rare black swans among ordinary birds— not only in finance, but also in many other contexts, such as history, technology, and science.", "These hardly predictable, large-magnitude occurrences have vastly stronger effects than regular episodes, due to psychological biases and a generalized poor understanding of the role of probability in social phenomena, related to the subjective notions of luck and fate.", "[6] Such disproportionate consequences are a direct corollary of the unavoidable shortness of historical records, which tends to magnify the exceptionality of those events.", "Figure: Monthly price indices for three major commodities between years 2001 and 2012 (with 2002-2004 average =100=100).", "Sustained multiplicative growth might precede severe drops to low values.", "Among the factors put forward a posteriori to explain such events, one finds weather, increasing demand for meat, use of biofuels, and variations in currency exchange rates.", "However, on the basis of a deterministic model, it has been shown that the two sharp peaks around 2008 and 2011 are specifically due to investor speculation.", "Public-domain data from the Food and Agriculture Organization (United Nations, www.fao.org).In this contribution, we revisit SMPs with resets with the aim of improving our understanding of the dynamics of such processes.", "We derive a number of exact results that allow for precise comparison with quantities obtained from dynamical simulations and clarify the limitations of time-series or realization averages to estimate the moments of the stochastic process —and therefore to achieve statistical predictions of future events based on knowledge of the past.", "Sections , , and discuss three models of increasing complexity and present exact results for their dynamics.", "Their applicability in a financial context is further discussed in Section , which closes this paper by highlighting other possible applications and future extensions of SMPs with resets." ], [ "Uniform multiplication and reset", "The simplest version of the class of multiplicative processes that we consider here is a Markov chain for a variable $x_t$ , evolving in discrete time $t=0,1,\\dots $ , with $x_t>0$ for all $t$ and $x_0=1$ .", "At each time step, $x_t$ is either multiplied by a constant positive factor $\\mu \\ne 1$ or it is reset to its initial value.", "The two instances occur with probabilities $1-r$ and $r$ , respectively ($0<r<1$ ).", "Namely, $ x_{t+1}=\\left\\lbrace \\begin{array}{ll}\\mu x_t & \\mbox{with probability $1-r$}, \\\\1 & \\mbox{with probability $r$}.\\end{array}\\right.$ Through the change of variables $x^{\\prime }= | \\ln x /\\ln \\mu |$ , this stochastic process is equivalent to the so-called Sisyphus random walk, for which several of the results discussed in this section have been obtained in previous work.", "[39] For the sake of concreteness, we focus on the choice $\\mu >1$ , which implies that $x_t \\ge 1$ for all $t$ .", "Power-law tails in the probability distribution for large $x$ , in fact, develop for $\\mu >1$ only.", "However, results can be straightforwardly extended to the case $\\mu <1$ , by exploiting the symmetry of the problem under the transformation $\\mu \\rightarrow \\mu ^{-1}$ , $x_t\\rightarrow x_t^{-1}$ .", "In this case, the probability exhibits power-law behavior for $x\\rightarrow 0$ .", "Figure REF shows the time dependence of $x_t$ along two realizations of process (REF ), with $\\mu =1.1$ and different values of the reset probability $r$ .", "The evolution consists of a succession of “bursts” of exponential growth induced by the multiplicative process, each of them terminated by a reset event.", "For $r=0.08$ , the less frequent reset events occasionally allow for rather high bursts, as compared with those obtained for $r=0.12$ in the time span of the simulation.", "Figure: Two typical realizations of the stochastic process of Eq.", "(), for μ=1.1\\mu =1.1 and two values of the reset probability: r=0.08r=0.08 (dark curve) and r=0.12r=0.12 (light-shaded curve).The stochastic process (REF ) can be readily dealt with, by noting that the variable $x_t$ can only adopt the values $1,\\mu ,\\mu ^2, \\dots $ or, generally, $ \\mu ^m$ , for $m=0,1,2,\\dots $ .", "From Eq.", "(REF ) it immediately follows that the probability $p_t^{(m)}$ that the variable equals $\\mu ^m$ satisfies the Chapman-Kolmogorov equation $p_{t+1}^{(m)} = (1-r) p_t^{(m-1)} (1-\\delta _{m0}) + r\\delta _{m0},$ where $\\delta _{ij}$ is Kronecker's delta.", "The solution reads $ p_t^{(m)}=\\left\\lbrace \\begin{array}{ll}r(1-r)^m & \\mbox{for $m<t$}, \\\\p_0^{(m-t)} (1-r)^t & \\mbox{for $m\\ge t$},\\end{array}\\right.$ in terms of a generic initial condition $p_0^{(m)}$ .", "For $x_0=1$ , we have $p_0^{(m)}=\\delta _{m0}$ , and $ p_t^{(m)}=\\left\\lbrace \\begin{array}{ll}r(1-r)^m & \\mbox{for $m<t$}, \\\\(1-r)^m & \\mbox{for $m=t$}, \\\\0 & \\mbox{for $m>t$},\\end{array}\\right.$ cf. Eq.", "(7) in Ref. 39.", "In Eqs.", "(REF ) and (REF ), all the information about the initial condition is accounted for in the range $m \\ge t$ so that, as time elapses, it becomes progressively relegated to exponentially higher values of $x$ .", "The long-time stationary behavior of the probability $p_t^{(m)}$ is built up from small values of $m$ , given by the first line in both equations: $ p_\\infty ^{(m)} = r (1-r)^m,$ cf. Eq.", "(11) in Ref. 39.", "The moment $\\langle x^\\gamma \\rangle _t =\\sum _m \\mu ^{\\gamma m} p_t^{(m)}$ can be exactly calculated for any order $\\gamma $ at all times.", "It reads $ \\langle x^\\gamma \\rangle _t =\\frac{r }{1-(1-r) \\mu ^\\gamma } + \\left[\\langle x^\\gamma \\rangle _0 - \\frac{r}{1-(1-r) \\mu ^\\gamma } \\right] (1-r)^t \\mu ^{\\gamma t} ,$ with $\\langle x^\\gamma \\rangle _0=1$ for $x_0=1$ .", "In contrast with the probability $p_t^{(m)}$ which, for asymptotically long times, always attains the stationary form given by Eq.", "(REF ), the moment $\\langle x^\\gamma \\rangle _t$ converges to a finite value, namely $\\langle x^\\gamma \\rangle _\\infty =\\frac{r }{1-(1-r) \\mu ^\\gamma },$ only if $(1-r) \\mu ^\\gamma <1$ .", "The convergence is exponential in time, within a typical time scale $ |\\ln [ (1-r) \\mu ^\\gamma ] |^{-1}$ .", "On the contrary, if $(1-r) \\mu ^\\gamma \\ge 1$ , $\\langle x^\\gamma \\rangle _t$ diverges exponentially with time.", "This convergence or divergence of the moment $ \\langle x^\\gamma \\rangle _t$ depending on whether the order $\\gamma $ is respectively lower or larger than the critical value $\\gamma _{\\rm c} = | \\ln (1-r) /\\ln \\mu |$ is compatible with a probability distribution (for a continuous variable $x$ ) decaying as the power law $x^{-1-\\gamma _{\\rm c}}$ .", "In fact, starting from the stationary distribution of Eq.", "(REF ), extending the index $m = \\ln x/\\ln \\mu $ to the real positive domain, and changing variables from $m$ to $x$ , we get the stationary (“fat-tailed”) power-law probability distribution $ f_\\infty (x) = \\frac{r}{\\ln \\mu } x^{-1-|\\ln (1-r)/\\ln \\mu |}.$" ], [ "Dissecting the process: properties of individual realizations", "In certain applications, it is necessary to “dissect” each possible realization of the stochastic process, determining the probability of each different way in which the system can evolve.", "Specifically, for Eq.", "(REF ), we first ask what is the probability that, up to a given time $\\tau $ , the evolution consists of a succession of exactly $k$ bursts of multiplicative growth ($k=1,\\dots ,\\tau +1$ , including one-step bursts between two contiguous reset events).", "Equivalently, we ask for the probability that exactly $k-1$ resets occur somewhere between times $t=1$ and $\\tau $ , both inclusive.", "For $k=1$ , the probability that no reset has occurred is $(1-r)^\\tau $ .", "In turn, each one of the $\\tau $ options for one reset ($k=2$ ) has probability $(1-r)^{\\tau -1} r$ .", "Generally, the probability that the evolution up to time $\\tau $ includes $k-1$ resets, i.e.", "$k$ bursts, is $\\rho _\\tau ^{(k)}= \\left( \\begin{array}{c} \\tau \\\\ k-1 \\end{array} \\right)(1-r)^{\\tau -k+1} r^{k-1},$ for $0\\le k\\le \\tau $ .", "Now, consider the set of realizations where, up to time $\\tau $ , the evolution consists of exactly $k$ bursts.", "How many of all these bursts have a duration of $m+1$ steps ($m=0,1,\\dots ,\\tau $ ), i.e.", "at how many times does $x_t$ reach the value $\\mu ^m$ just before the burst ends (either by a reset event or because time $\\tau $ has been reached)?", "This number can be calculated by considering the so-called compositions of $\\tau $ into $k$ parts,[40] and turns out to be $K_\\tau ^{(k,m)} =\\left\\lbrace \\begin{array}{ll}1 & \\mbox{for $k=1$ and $m=\\tau $}, \\\\ \\\\k \\left( \\begin{array}{c} \\tau -m-1 \\\\ k-2 \\end{array} \\right) &\\mbox{for $1<k\\le \\tau -m+1$ }, \\\\ \\\\0 & \\mbox{otherwise}.\\end{array}\\right.$ Taking into account that, if $x_t$ reaches the value $\\mu ^m$ during a burst, all the lower values $1,\\mu ,\\mu ^2,\\dots , \\mu ^{m-1}$ have also been previously attained, the total number of times that $x_t=\\mu ^m$ has been attained at any moment $t$ up to time $\\tau $ , in all the realizations with exactly $k$ bursts, can be calculated from $K_\\tau ^{(k,m)}$ , yielding $M_\\tau ^{(k,m)} =\\left\\lbrace \\begin{array}{ll}k \\left( \\begin{array}{c} \\tau -m \\\\ k-1 \\end{array} \\right) &\\mbox{for $1\\le k \\le \\tau -m+1$ }, \\\\ \\\\0 & \\mbox{otherwise}.\\end{array}\\right.$ Finally, if the system has evolved up to time $\\tau $ , the probability that the value of $x_t$ is $\\mu ^m$ at a randomly chosen time $t\\le \\tau $ reads $ \\sum _{k=1}^{\\tau +1} \\rho _\\tau ^{(k)} \\frac{M_t^{(k,m)}}{\\sum _{m^{\\prime }=0}^{\\tau } M_t^{(k,m^{\\prime })} } = \\frac{1+r (\\tau -m)}{1+\\tau } (1-r)^m,$ which, up to the normalization factor $(1+\\tau )^{-1}$ , corresponds to Eq.", "(39) in Ref. 39.", "Note carefully the difference between Eqs.", "(REF ) and (REF ).", "While the former gives the probability that a given value of $x$ is attained after $t$ evolution steps, the latter is the probability constructed by recording the frequency of all the values of $x$ along the whole evolution until time $\\tau $ .", "For $t=\\tau $ , the two expressions coincide in the limit $\\tau \\rightarrow \\infty $ only; cf. Eq.", "(REF )." ], [ "Remarks on self-averaging and ergodicity", "It is well known that systems involving stochastic variables with fat-tailed distributions exhibit peculiar statistical features, directly associated with the fact that some of the leading distribution moments —such as the mean value and/or the variance— are not finite.", "[14], [22] Although, for any finite time $t$ , the moments for the variable $x_t$ in the stochastic process (REF ) are all finite, as given by Eq.", "(REF ), similar features manifest themselves as time elapses and a fat-tailed asymptotic distribution progressively builds up.", "Lack of self-averaging —namely, the failure of a scaled sum of stochastic variables to converge to a well-defined value— is intuitive for distributions with divergent mean value.", "This phenomenon has been recognized as a typical feature of random multiplicative processes,[41] and has been widely discussed for a variety of systems of physical interest,[31], [32] including Markov processes with resets.", "[33] Self-averaging also fails when the mean value is finite but the variance diverges, a situation relevant to the theory of financial revenues and related problems in economics.", "[38], [42] The fat-tailed asymptotic distribution of process (REF ) has finite mean value $\\langle x \\rangle $ and divergent variance $\\sigma ^2 = \\langle x^2 \\rangle - \\langle x \\rangle ^2$ for $1<\\gamma _{\\rm c} = | \\ln (1-r) /\\ln \\mu | <2$ .", "The upper panel of Fig.", "REF shows $x_t$ , up to $t=10^4$ , along a single realization of the process with $\\mu =1.1$ and $r=0.1$ , for which $\\gamma _{\\rm c} \\approx 1.1$ .", "The light-shaded curve is the prediction of Eq.", "(REF ) for the mean value (i.e., with $\\gamma =1$ ), which approaches $\\langle x \\rangle _\\infty =10$ for long times.", "The large fluctuations of $x_t$ around its expected average are apparent in the main plot and in the inset, which shows a detail for intermediate times.", "Figure: Upper panel: x t x_t as a function of time, along a single realization of the stochastic process () for μ=1.1\\mu =1.1 and r=0.1r=0.1 (dark curve).", "The light-shaded curve stands for the analytical prediction for 〈x〉 t \\langle x\\rangle _t, Eq. ().", "The inset shows a close-up for intermediate times.", "Middle panel: Cumulative average of x t x_t (dark curve; see main text) for the same realization as in the upper panel, and the corresponding analytical prediction, Eq.", "() (light-shaded curve).", "Lower panel: As in the middle panel, averaging over 10 6 10^6 realizations of the stochastic process.", "The inset shows the result for short times.The middle panel of Fig.", "REF shows the accumulated average of the stochastic variable, defined as $\\bar{x}_t = (t+1)^{-1} \\sum _{\\tau =0}^t x_\\tau $ , for the same realization as in the upper panel.", "The light-shaded curve stands for the corresponding analytical value, ${ \\langle \\bar{x} \\rangle }_t = && \\frac{r }{1-(1-r) \\mu } \\nonumber \\\\ && + \\left[1 - \\frac{r}{1-(1-r)\\mu } \\right] \\frac{1-(1-r)^{t+1} \\mu ^{t+1}}{(t+1)[1-(1-r) \\mu ]}, $ derived from Eq.", "(REF ).", "Comparison of the two curves puts in evidence, in particular, the role of large values of $x_t$ in building up the cumulative average.", "Their contribution, in fact, punctuates with abrupt upward jumps the otherwise decaying evolution of $\\bar{x}_t$ .", "It is clear, however, that —although the typical convergence time for the mean value, $|\\ln [(1-r)\\mu ]|^{-1} \\approx 100$ , has long passed— $\\bar{x}_t$ is nowhere close to $\\langle x \\rangle _\\infty $ .", "This non-self-averaging effect remains even if an additional average is performed over many realizations of the stochastic process —as implicit in Eqs.", "(REF ) and (REF ).", "In the lower panel of Fig.", "REF we plot $\\bar{x}_t$ averaged over $10^6$ realizations.", "While, as shown in the inset, the coincidence between numerical and analytical results is very good for small $t$ , a sustained sizable discrepancy —which decreases only very slowly as time elapses— persists for long times.", "Somehow disappointingly, there is no formal theory to describe the behavior of the mean value of a stochastic variable —calculated as the average of successive random draws from a prescribed distribution— when its variance is infinite.", "Heuristic approaches based on the estimation of bootstrap distributions or the use of surrogate tail-trimmed variables, however, have been advanced in applications to finance.", "[43] A different viewpoint is provided by the extreme-value theory, which focuses on the statistics of the rare events where the variable exhibits severe deviations from the mean value.", "[41], [44] In our case, they correspond to the high bursts in the upper panel of Fig.", "REF , which cause the sharp steps in the middle panel.", "For the stochastic process (REF ) we can ask what is the mean waiting time $w_X$ until $x_t$ reaches, for the first time, a given (large) value $X =\\mu ^M$ .", "This event happens at the $M$ -th step of a sufficiently long burst, assuming that all the preceding bursts have been shorter than $M$ steps.", "The corresponding waiting time is the total duration of all the preceding bursts plus $M$ additional steps.", "Its mean value turns out to be $w_X = \\frac{1}{r} \\left[ \\frac{1}{(1-r)^M}-1 \\right] \\approx \\frac{1}{r} X^{\\gamma _{\\rm c}},$ where the approximation holds for large $X$ , i.e.", "for large $M$ ; cf. Eq.", "(18) in Ref. 39.", "On the other hand, the mean number $d_X$ of random draws from the stationary distribution (REF ) until the value $X=\\mu ^M$ is obtained, assuming that all the preceding draws have produced lower values, can be immediately found: $d_X = \\frac{r}{(1-r)^M} = r X^{\\gamma _{\\rm c}}.$ Note that the waiting time $w_X$ is larger than $d_X$ by a factor $r^{-2}$ ($=100$ in the realizations of Fig.", "REF ).", "This difference can be viewed as a form of “transient non-ergodicity” in our Markov chain.", "Along the stochastic process, in fact, observation of $x_t=X$ requires waiting the total duration of a succession of shorter bursts where $x_t$ adopts one or more times each value lower than $X$ .", "Instead, when drawing the stochastic variable directly from the distribution, the occurrence of $X$ is independent of the preceding draws.", "Evaluation of the distribution from the time evolution of $x_t$ , consequently, can be much slower than from random realizations of the stochastic variable." ], [ "The non-uniform case", "Although the stochastic process (REF ) already captures the key mechanisms that lead to the generation of a power-law distribution for the variable $x_t$ , it is useful to introduce a couple of generalizations that relax some assumptions implicit in the formulation of the above multiplicative process but, at the same time, preserve most of its analytical tractability.", "In particular, we now admit that the multiplicative coefficient $\\mu $ and the value of $x_t$ after each reset event vary randomly with time.", "Namely, we consider the Markov chain[7] $ x_{t+1}=\\left\\lbrace \\begin{array}{ll}\\mu _t x_t & \\mbox{with probability $1-r$}, \\\\s_t & \\mbox{with probability $r$},\\end{array}\\right.$ where, at each time step, $\\mu _t>0$ and $s_t>0$ are drawn from distributions $P(\\mu )$ and $F(s)$ , respectively.", "In this variant, in contrast with Eq.", "(REF ), $x_t$ generally adopts continuous positive values, and its probability is described by a distribution $f_t(x)$ .", "The initial value $x_0$ is chosen from a prescribed distribution $f_0(x)$ .", "The previous case is reobtained taking delta-like profiles for $P(\\mu )$ , $F(s)$ , and $f_0(x)$ .", "In order to avoid hindering our presentation with the discussion of special pathological situations, we assume that the moments of the distributions $P(\\mu )$ and $F(s)$ , $ \\langle \\mu ^\\gamma \\rangle =\\int _0^\\infty \\mu ^\\gamma P (\\mu ) d\\mu , \\ \\ \\ \\ \\ \\langle s^\\gamma \\rangle =\\int _0^\\infty s^\\gamma F (s) ds,$ are finite for any order $\\gamma \\in (-\\infty , \\infty )$ .", "This is the case if $P(\\mu )$ and $F(s)$ drop rapidly enough to zero for both $\\mu , s\\rightarrow 0$ and $\\mu , s \\rightarrow \\infty $ .", "In typical applications, in particular, the reset values $s_t$ are restricted to some finite domain, so that the support of the distribution $F(s)$ is a bounded interval, and the finiteness of $\\langle s^\\gamma \\rangle $ is guaranteed for all $\\gamma $ .", "The results that we obtain below, however, do not always require that $\\langle \\mu ^\\gamma \\rangle $ and $\\langle s^\\gamma \\rangle $ are finite.", "In order to obtain the solution of the stochastic process (REF ), it is convenient to consider the evolution of the logarithmic variable $y_t = \\ln x_t$ : $ y_{t+1}=\\left\\lbrace \\begin{array}{ll}y_t+ \\nu _y & \\mbox{with probability $1-r$}, \\\\u_t & \\mbox{with probability $r$},\\end{array}\\right.$ where $\\nu _t =\\ln \\mu _t$ and $u_t = \\ln s_t$ .", "The stochastic variables $\\nu _t$ and $u_t$ are respectively drawn from distributions $Q(\\nu )$ and $G(u )$ , with $Q(\\nu ) d\\nu =P(\\mu ) d\\mu $ and $G(u) du =F(s) ds$ .", "In other words, $Q(\\nu ) =P(\\exp \\nu ) \\exp \\nu $ and $G(u) = F(\\exp u) \\exp u$ .", "The same transformation yields for the initial distribution $g_0(y)$ in terms of $f_0(x)$ .", "The Markov chain for $y_t$ , Eq.", "(REF ), is an additive (generally, drift plus diffusion) stochastic process with resets.", "The Chapman-Kolmogorov equation for the probability distribution of $x_t$ , $f_t(x)$ , is $ f_{t+1} (x)= (1-r) \\int _0^\\infty P(\\mu ) \\mu ^{-1} f_t \\left( \\mu ^{-1} x \\right) d\\mu + r F(x) .$ Correspondingly, the probability distribution for $y_t$ , $g_t (y)$ , satisfies $ g_{t+1} (y)= (1-r) \\int _{-\\infty }^\\infty Q(\\nu ) g_t (y-\\nu ) d\\nu + r G(y) .$ In the Fourier representation, which we define as $\\hat{g}_t (\\eta ) = \\int _{-\\infty }^\\infty g_t (y) \\exp (-2\\pi i \\eta y ) d y$ and analogous expressions for all the other functions of $y$ , the solution to Eq.", "(REF ) reads $\\hat{g}_t(\\eta ) && = \\frac{r\\hat{G}(\\eta )}{1-(1-r)\\hat{Q}(\\eta )} \\nonumber \\\\&& +\\left[ \\hat{g}_0 (\\eta )-\\frac{r\\hat{G}(\\eta )}{1-(1-r)\\hat{Q}(\\eta )} \\right](1-r)^t\\hat{Q}(\\eta )^t.$ Since, up to a multiplicative constant in its variable, $\\hat{Q} (\\eta )$ is the characteristic function of the variable $\\nu $ , it satisfies $|\\hat{Q} (\\eta )| \\le 1$ for all $\\eta $ .", "Therefore, $\\hat{g}_t(\\eta )$ converges to the stationary solution $ \\hat{g}_\\infty (\\eta ) =\\frac{r\\hat{G}(\\eta )}{1-(1-r) \\hat{Q}(\\eta )}$ for $t\\rightarrow \\infty $ .", "Except for the term including the initial condition, the solution $ \\hat{g}_t (\\eta )$ in Eq.", "(REF ) is directly proportional to $\\hat{G} (\\eta )$ , the Fourier transform of the distribution of the stochastic variable immediately after each reset event —see second line of Eq.", "(REF ).", "The same happens with the stationary distribution $\\hat{g}_\\infty (\\eta )$ in Eq.", "(REF ).", "This proportionality reveals that $g_t(y)$ and, correspondingly, $f_t(x)$ are given —up to a term involving their initial values— by a linear superposition of contributions coming from each possible value of the variable after resets, which act as mutually independent “starts” for the ensuing multiplicative process.", "Upon Fourier antitransforming, in fact, both $g_t(y)$ and $f_t(x)$ would be respectively given by convolutions of $G(u)$ and $F(s)$ with distributions representing the contribution of each $u$ and $s$ .", "The effect of having admitted that after resets the variable can adopt different values is, therefore, rather straightforward.", "By Fourier antitransforming Eq.", "(REF ), it is in principle possible to find the solution $g_t(y)$ to equation (REF ).", "In turn, using the identity $g_t(y) dy = f_t(x) dx$ , we would obtain the solution to equation (REF ), $f_t (x) =x^{-1} g_t(\\ln x) $ .", "For generic forms of $Q(\\nu )$ and $G(u)$ , however, this calculation can seldom be explicitly performed.", "On the other hand, it is straightforward to exactly find the moments of the distribution $f_t (x)$ , $\\langle x^\\gamma \\rangle _t =\\int _0^\\infty x^\\gamma f_t (x) dx,$ for any order $\\gamma $ , by noting that $\\langle x^\\gamma \\rangle _t= \\hat{g}_t (i\\gamma /2\\pi )$ .", "Evaluating Eq.", "(REF ) at $\\eta =i\\gamma /2\\pi $ , in fact, we get $\\langle x^\\gamma \\rangle _t && =\\frac{r\\langle s^\\gamma \\rangle }{1-(1-r)\\langle \\mu ^\\gamma \\rangle }\\nonumber \\\\ &&+ \\left[\\langle x^\\gamma \\rangle _0 - \\frac{r\\langle s^\\gamma \\rangle }{1-(1-r)\\langle \\mu ^\\gamma \\rangle } \\right] (1-r)^t \\langle \\mu ^\\gamma \\rangle ^t, $ which generalizes Eq.", "(REF ).", "Note that, except for the term involving the initial condition, $\\langle x^\\gamma \\rangle _t$ is proportional to the corresponding moment of the distribution of values after resets, $F(s)$ .", "Much as for the stochastic process (REF ), while the distribution for $x_t$ always attains a well-defined asymptotic form, given by Eq.", "(REF ) in the Fourier representation for the logarithmic variable $y_t$ , the moment $\\langle x^\\gamma \\rangle _t$ converges to a finite value, $\\langle x^\\gamma \\rangle _\\infty =\\frac{r \\langle s^\\gamma \\rangle }{1-(1-r) \\langle \\mu ^\\gamma \\rangle },$ only if $(1-r) \\langle \\mu ^\\gamma \\rangle <1$ .", "It is interesting to note that —while, in the uniform case considered in Section , convergence or divergence of $ \\langle x^\\gamma \\rangle _t$ depends on whether $\\gamma $ is respectively lower or larger than a critical value $\\gamma _{\\rm c}$ — in the present case it can happen that the moment converges for $\\gamma $ inside a finite interval ($\\gamma _-,\\gamma _+$ ), with $\\gamma _-<0<\\gamma _+$ , and diverges elsewhere.", "This requires, in particular, that the distribution $P(\\mu )$ allows for values of $\\mu $ at both sides of $\\mu =1$ .", "Take, for instance, the two-delta distribution $P(\\mu ) = a \\delta (\\mu -\\mu _0)+(1-a) \\delta (\\mu -\\mu _0^{-1})$ with $0<a<1$ and $\\mu _0\\ne 1$ .", "The interval of convergence for the order $\\gamma $ is limited by $\\gamma _\\pm = \\frac{1}{\\ln \\mu _0} \\ln \\frac{1\\pm \\sqrt{1-4a(1-a)(1-r)^2}}{2a(1-r)} ,$ where, without generality loss, we have assumed $\\mu _0>1$ .", "Figure REF shows the interval ends $\\gamma _\\pm $ as functions of $a$ for three values of $r$ .", "Figure: The ends γ ± \\gamma _\\pm of the interval of convergence of the moment 〈x γ 〉 t \\langle x^\\gamma \\rangle _t for the multiplicative process with resets (), with μ\\mu drawn from the distribution P(μ)=aδ(μ-μ 0 )+(1-a)δ(μ-μ 0 -1 )P(\\mu ) = a \\delta (\\mu -\\mu _0)+(1-a) \\delta (\\mu -\\mu _0^{-1}) and μ 0 >1\\mu _0>1.", "The values of γ ± \\gamma _\\pm are scaled by lnμ 0 \\ln \\mu _0, and plotted as functions of aa for three values of the reset probability rr.The long-time convergence or divergence of $ \\langle x^\\gamma \\rangle _t$ for $\\gamma $ respectively inside and outside the interval$(\\gamma _-,\\gamma _+)$ is compatible with a distribution $f_\\infty (x)$ which behaves as $x^{-1-\\gamma _+}$ for $x\\rightarrow \\infty $ (in agreement with the results of Section ), and as $x^{-1-\\gamma _-}$ for $x\\rightarrow 0$ .", "Finding its complete form, however, would require to antitransform Eq.", "(REF )." ], [ "State-dependent multiplication and reset frequency with continuous time", "Coming back to the uniform case considered in Section , a continuous-time description can be straightforwardly introduced by first assigning a duration $\\Delta t$ to each evolution step.", "Writing the multiplicative coefficient in the first line of Eq.", "(REF ) as $\\mu = 1+\\lambda \\Delta t$ and taking the limit $\\Delta t\\rightarrow 0$ , the purely multiplicative part of the stochastic process for $x(t)$ becomes a linear equation, $\\dot{x} = \\lambda x$ , whose solution grows or decays exponentially, depending on the sign of $\\lambda $ .", "This exponential evolution is punctuated by the reset events, which now occur with frequency (probability per unit time) $q$ , related to the reset probability $r$ of Eq.", "(REF ) through $r=q \\Delta t$ .", "As in Section , we assume here that the stochastic variable is reset to $x=1$ after each event.", "The Chapman-Kolmogorov equation for the probability distribution $f(x,t)$ reads now $ \\partial _t f + \\lambda \\partial _x ( x f) = -q f(x,t) + q\\delta (x-1) \\int _0^\\infty f(x^{\\prime },t) dx^{\\prime }.$ Its interpretation as a continuity equation is transparent.", "The left-hand side describes probability drift, driven by multiplication, towards larger or smaller values of $x$ , depending on $\\lambda $ being positive or negative, respectively.", "The loss and gain terms in the right-hand side, meanwhile, stand for probability sinks and sources associated with reset events.", "It can be readily seen that Eq.", "(REF ) preserves the distribution norm $\\int _0^\\infty f(x,t) dx=1$ at all times, so that the integral in the last term is, in reality, a constant (see, however, the generalization in Eq.", "(REF ) below).", "Moreover, if $\\lambda >0$ and the support of the initial condition $f_0(x)$ is included in the interval $[1,\\infty )$ or, conversely, if $\\lambda < 0$ and the support of $f_0(x)$ is included in $(0,1]$ , the distribution $f(x,t)$ will only adopt non-zero values inside either interval at all times.", "To simplify the presentation, and in agreement with our discussion in Section , we assume that $\\lambda >0$ and restrict the analysis to the interval $[1,\\infty )$ .", "Under these conditions, Eq.", "(REF ) can be fully solved by treating the delta-like gain term in the right-hand side as a boundary condition at $x=1$ .", "In fact, assuming that the solution jumps from $f(x,t)=0$ for $x<1$ to a finite value $f(1^+,t)$ just above the boundary, the delta-like term must be balanced by the $x$ -derivative in the left-hand side, so that $f(1^+,t)= q/\\lambda $ .", "Taking this condition into account, the solution is $f(x,t) = && \\exp [-(\\lambda +q)t] f_0[x \\exp (-\\lambda t)] \\Theta [x\\exp (-\\lambda t)-1] \\nonumber \\\\ && +\\frac{q}{\\lambda }x^{-1-q/\\lambda } \\Theta [1-x\\exp (-\\lambda t)] , $ where $\\Theta (x)$ is Heaviside's step function.", "The distribution $f(x,t)$ is neatly divided into two contributions, corresponding to the two terms in the right-hand side of Eq.", "(REF ).", "For $x>\\exp (\\lambda t)$ , in the first term, we have the contribution of the initial condition, which shifts towards increasingly larger values of $x$ and, at the same time, is exponentially damped and stretched.", "For $x<\\exp (\\lambda t)$ , in the second term, the asymptotic power-law distribution is established.", "For $t\\rightarrow \\infty $ , this second contribution spans the whole domain of the variable $x$ , yielding $ f_\\infty (x) = \\frac{q}{\\lambda } x^{-1-q/\\lambda },$ to be compared with Eq.", "(REF ).", "As an illustration, Fig.", "REF shows, in log-log scale, four snapshots of $f(x,t)$ as a function of $x$ , evolving from an initial condition $f_0(x) = \\exp (1-x)$ , with $x\\in [1,\\infty )$ and $\\lambda =q=1$ .", "The exponential cutoff originating in $f_0(x)$ shifts to the right as time elapses, while the asymptotic distribution $f_\\infty (x)=x^{-2}$ builds up from the left.", "Figure: The probability distribution f(x,t)f(x,t), Eq.", "(), as a function of xx for t=0t=0, 2, 5 and ∞\\infty , starting from an initial condition f 0 (x)=exp(1-x)f_0(x)= \\exp (1-x), with λ=q=1\\lambda =q=1.", "Dashed vertical lines stand at the boundary x=exp(λt)x=\\exp (\\lambda t), which separates the contributions of the initial condition and the asymptotic long-time distribution, f ∞ (x)=x -2 f_\\infty (x)=x^{-2}.Equation (REF ) can be immediately generalized to the case where both the coefficient $\\lambda $ and the reset frequency $q$ depend on the current value of the stochastic variable $x$ .", "This extension reads $ \\partial _t f + \\partial _x [ \\lambda (x) f] = -q(x) f(x,t) + \\delta (x-1) \\Phi (t) ,$ with $ \\Phi (t) = \\int _0^\\infty q(x^{\\prime }) f(x^{\\prime },t) dx^{\\prime }.$ Note that the factor $x$ in the $x$ -derivative of Eq.", "(REF ) has been absorbed by $\\lambda (x)$ .", "As in the homogeneous case above, we focus on the problem restricted to the interval $x\\in [1,\\infty )$ , with $\\lambda (x)>0$ for all $x$ .", "Treating the last term in the right-hand side of Eq.", "(REF ) as a boundary condition at $x=1$ , we find $ f(1^+,t) = \\frac{\\Phi (t)}{\\lambda (1)}.$ In practice, this condition requires to resort to a self-consistent calculation, where $f(x,t)$ is first solved for arbitrary $\\Phi (t)$ , in such a way that Eq.", "(REF ) is satisfied, and then $\\Phi (t)$ is found from Eq.", "(REF ).", "Whether this calculation can be explicitly performed depends on the functional form of $\\lambda (x)$ and $q(x)$ .", "On the other hand, assuming that $f(x,t)$ tends to a well-defined limit for long times, the asymptotic distribution can be readily written as $ f_\\infty (x) = \\frac{\\Phi _\\infty }{\\lambda (x)} \\exp \\left[-\\int _1^x \\frac{q(x^{\\prime })}{\\lambda (x^{\\prime })} dx^{\\prime } \\right] ,$ with $\\Phi _\\infty $ the asymptotic value of $\\Phi (t)$ for $t\\rightarrow \\infty $ .", "Equation (REF ) makes it clear that $\\Phi _\\infty $ is essentially fixed by the normalization of $f_\\infty (x)$ .", "In fact, it can be easily verified that Eqs.", "(REF ) to (REF ) are mutually consistent if the distribution is normalized to unity.", "Such normalization, however, requires that the integral in the exponential of Eq.", "(REF ) diverges as $x\\rightarrow \\infty $ , namely, $ \\int _1^\\infty \\frac{q(x^{\\prime })}{\\lambda (x^{\\prime })} dx^{\\prime } = \\infty .$ This condition is verified when the reset frequency increases and/or the multiplicative coefficient decreases sufficiently fast as $x$ grows.", "It expresses the fact that there is no probability “leaking” towards large values of $x$ due to excessively weakened resets and/or strengthened multiplication.", "From Eq.", "(REF ), moreover, it is apparent that admitting state-dependent multiplication and reset frequency can significantly widen the class of stationary distributions covered by the model, well beyond power-law decaying functions.", "As an illustration, consider the case of constant reset frequency $q$ , with a multiplicative coefficient with algebraic dependence on $x$ , namely, $\\lambda (x)= \\lambda _0 x^{1-\\alpha }$ .", "Condition (REF ) is fulfilled if $\\alpha \\ge 0$ .", "For $\\alpha >0$ , the resulting distribution is $ f_\\infty (x) = \\frac{q }{\\lambda _0 x^{1-\\alpha }} \\exp \\left[- \\frac{q(x^\\alpha -1)}{\\lambda _0 \\alpha }\\right],$ which reduces to Eq.", "(REF ) for $\\alpha \\rightarrow 0$ .", "This form of $f_\\infty (x)$ , shown in Fig.", "REF for $\\lambda _0=q=1$ and some values of the exponent $\\alpha $ , is closely related to the Weibull and the stretched exponential distribution, which play a key role in the quantitative description of several socioeconomic systems.", "The present model, thus, provides a unified mechanism for the occurrence of a variety of distributions relevant to this kind of problems.", "Figure: The probability distribution of Eq.", "() for λ 0 =q=1\\lambda _0=q=1 and four values of the exponent α\\alpha .", "For α=0\\alpha =0, it reduces to the power-law distribution f ∞ (x)=x -2 f_\\infty (x) =x^{-2}." ], [ "Discussion and conclusion", "Stochastic multiplicative processes (SMPs) with reset events, in their several variants, are characterized by fat-tailed, often power-law, distributions of the relevant variable.", "These processes show punctuated behaviour, where periods of growth are terminated by sudden shifts to a previous state.", "High-order moments of the probability distribution diverge in most variants of the general model.", "These features are characteristic of, among others, financial processes, a context where higher-order moments are relevant.", "Kurtosis, for instance, is typically used as a measure of financial risk: divergence of the fourth moment (even with finite variance and skewness) yields unreliable predictions if evaluated using finite data sets.", "[38], [42] An additional difficulty in the numerical estimation of moments —and, therefore, of risk— is due to the very slow convergence of estimated values to the exact values of the process, even if the associated moments are finite.", "This “slow law of large numbers” is caused by the large weight of rare events (black swans), which take a lot of data to show up, and prevent a proper estimation of the moments of such processes through the moments of a sample.", "[42] The interpretation of SMPs with resets in the context of finance is quite straightforward.", "Regardless of whether the model is implemented in discrete or continuous time, the relevant variable, either $x_t$ or $x(t)$ , can be understood as the gain accumulated at a given time $t$ , while its dynamics are controlled by two parameters, one characterizing the momentary gain ($\\mu $ or $\\lambda $ ), and another one quantifying the risk of the investment ($r$ or $q$ ).", "The uniform model analyzed in Section would therefore describe an ensemble of gamblers devoid of strategy and democratically dependent on luck.", "Still, they experience dissimilar fortunes and could be, a posteriori, evaluated as poor, mediocre, or excellent investors.", "It has been suggested that the search for causality in lucky realizations of random processes, together with the elimination of unlucky series from samples, might lead to highly misleading interpretations of the actual causal mechanisms of data, a phenomenon of particular relevance in finance known as “survivorship bias.” [45] The simple model with constant risk and momentary gain clarifies other aspects of the process as well.", "The exponent of the power law depends on both parameters: the higher the frequency and magnitude of resets, the smaller the exponent (in absolute value).", "That is, black swans will have larger impact if the momentary gain increases, and also if the risk is high, as it could have been guessed.", "Furthermore, the estimation of the waiting time to observe an event of size $X$ shows that the typical, additional waiting time until an event of, say, twice that size occurs is $2^{\\gamma _c}$ time steps longer.", "If $\\gamma _c$ is unknown, or if its estimation is affected by large errors, predictions of the expected time until the next black swan and of its magnitude can be highly inaccurate, at best.", "The non-uniform model in Section extends those results by considering a broad class of probability distributions for momentary gains and reset values, while the risk is kept fixed.", "Momentary gain is now time-dependent and can be larger or smaller than one (in the latter case, it becomes a momentary loss), this generalization leading to two-sided power laws and to moments converging within a finite interval of $\\gamma $ values.", "The qualitative properties of this model are fully comparable to those of the uniform case, though additional situations can be now embraced.", "The accumulated gain can increase or decrease in random amounts at each time step, and the distribution of reset values $F(s)$ could take into account agents with different strategies regarding the fraction of their gains put at stake.", "Prudent investors might save part of their previous gains to avoid too severe drops.", "The details of this strategy directly affect the moments of the accumulated gain distribution, though the power-law function persists.", "Finally, in Section , we have studied a variant where the momentary gain and risk can depend on the accumulated gain.", "In this case, the model has been formulated in continuous time for the sake of analytical tractability, but can be interpreted along the same lines as discrete-time models.", "This last case incorporates the possibility that agents, perhaps content with the gains accumulated so far, develop cautious strategies by investing in products with lower momentary gain but which are less risk-prone.", "Interestingly, this non-greedy strategy can transform the power-law distribution into different fat-tailed distributions, where the effects of black swans might be strongly suppressed.", "A relevant situation that we have not explicitly explored in this work is the plausible relationship between risk and gain.", "In practice, it comes to reason that financial gain cannot be maximized while simultaneously minimizing risk.", "For example, if the momentary gain in the uniform model becomes a linear function of risk, $\\mu = 1+\\kappa r$ ($\\kappa >0$ ), then $\\gamma _c \\approx 1/\\kappa $ for small risk.", "Under the constraint of mutual dependency, strategies decrease the number of their degrees of freedom, and similar restrictions would hold if the relationship between gain and risk affects their statistical distribution (note that, implicitly, Eq.", "(REF ) establishes a form of weak constraint linking the two parameters).", "Still, the implementation of realistic constraints in scenarios where, furthermore, those distributions depend on the accumulated gain appears as an interesting avenue to explore.", "In a different context, it has been shown that a reset probability that depends on time can improve the efficiency of search processes.", "[46] Another extension with potential applicability is the consideration of trends in, e.g., the minimum reset value.", "A variety of situations might be subject to such trends, as Fig.", "REF illustrates for food price indices.", "The trend might be positive (inflation), negative (deflation), or be itself subject to large variations, as it happens with hyperinflation followed by currency devaluation.", "Finance offer multiple situations that can potentially be modelled through SMPs with resets and, occasionally, additional mechanisms tailored to specific scenarios.", "Possible applications are however broader and extend beyond the many cases already described in this work.", "[47] In population dynamics, also dominated by multiplicative (demographic) growth processes, the fast, local extinction of a population is often followed by “reinjection” in the form of a small number of migrating individuals.", "This situation could describe parasitic infection bursts in metapopulations [48] and explain the persistence of populations that would otherwise become extinct.", "[49] Resets could be also rephrased as any process that finishes the multiplicative growth, since the properties described do not depend on whether the reset is repeatedly applied to realizations that have a continuity in time or to many different realizations that are independently “born,” and then terminated at the time of resetting.", "An example can be found in models for cascade fracture with stopping events.", "[50] Applications of SMPs to such areas as material sciences and population dynamics would require their generalization to spatially extended systems, considering coupled stochastic processes occurring at neighbor sites.", "Given the ubiquity of multiplicative processes possibly amended by a variety of mechanisms —resets, among many others— the profusion of power-law (or fat-tailed) distributed quantities found in natural and social sciences no longer comes as a surprising fact.", "Such distributions entail non-trivial dynamical properties, as those characterized in this work, that severely limit our ability to predict future outcomes of the process.", "Despite the advances of over a century of research on this topic, further applications and deeper analytical approaches are yet to come to improve our understanding of the mechanisms generating fat-tailed distributions and our control of black-swan-like events." ], [ "Acknowledgements", "S. M. is supported by grant FIS2017-89773-P (MINECO/FEDER, E. U.", ")." ] ]
2105.11679
[ [ "Millimeter-Wave Beamforming with Continuous Coverage for Mobile\n Interactive Virtual Reality" ], [ "Abstract Contemporary Virtual Reality (VR) setups commonly consist of a Head-Mounted Display (HMD) tethered to a content-generating server.", "\"Cutting the wire\" in such setups and going truly wireless will require a wireless network capable of delivering enormous amounts of video data at an extremely low latency.", "Higher frequencies, such as the millimeter-wave (mmWave) band, can support these requirements.", "Due to high attenuation and path loss in the mmWave frequencies, beamforming is essential.", "For VR setups, beamforming must adapt in real-time to the user's head rotations, but can rely on the HMD's built-in sensors providing accurate orientation estimates.", "In this work, we present coVRage, a beamforming solution tailored for VR HMDs.", "Based on past and current head orientations, the HMD predicts how the Angle of Arrival (AoA) from the access point will change in the near future, and covers this AoA trajectory with a dynamically shaped beam, synthesized using sub-arrays.", "We show that this solution can cover such trajectories with consistently high gain, unlike regular single-beam solutions." ], [ "Introduction", "A wide variety of VR applications have been investigated over the years, in fields including education, medicine and manufacturing [1], [2], [3].", "Such applications require a reliable high-throughput and low-latency connection to an external device providing VR content [4].", "This may be live video recorded elsewhere, such as for remote collaboration or meetings, or 3D graphics generated on a PC or edge cloud, such as for gaming applications.", "The recently introduced Oculus Quest 2 HMD is capable of generating content on-device and thereby working without any connection to other devices.", "As this restricts the device from running many such connected and computationally intensive applications, it also offers the option to tether it to a PC.", "This setup, along with most others currently on the market, relies on a wired connection for content delivery.", "While this easily meets reliability, latency and throughput requirements, it limits the user's range of movement, hindering true immersion.", "To achieve truly wireless connected HMD, mmWave networking, in the 30 to 300 band, is most often considered, as lower frequencies cannot meet the VR requirements [4].", "Solutions often rely on the existing IEEE 802.11ad and IEEE 802.11ay Wi-Fi standards for mmWave [5] or on 5G NR's mmWave capabilities [6].", "The main challenges in building such a system stem from mmWave's inherently high path loss and attenuation.", "To achieve sufficiently high signal strength at the HMD, the transmitter and the HMD must both focus their energy towards each other, in a process called beamforming.", "mmWave transceivers usually implement beamforming using phased antenna arrays, consisting of many separate antenna elements [7].", "The path lengths of the signal from each element will differ slightly in a given direction, meaning the different signals are generally not phase-aligned.", "By carefully shifting the phase of each element, a beamforming algorithm ensures signals towards an intended receiver are phase-aligned and therefore at maximum amplitude.", "As this phenomenon also applies to signals received at phased arrays, beamforming should also occur when in receive mode, focusing towards the transmitter.", "While basic beamforming consists of a single beam in one direction, we exploit a more advanced approach using a variable number of sub-beams.", "By subdividing the array into sub-arrays providing sub-beams, the combined beam can cover a dynamically shaped area.", "Such flexible coverage is highly advantageous for beamforming on an HMD.", "An angular beam misalignment of a few degrees can have a significant impact on SNR [8], and a human head can reach an instantaneous angular velocity of hundreds of degrees per second [9], [10], [11].", "As such, a flexibly shaped beam, stretched in the direction of a head rotation, can provide an HMD with consistently high receive gain, essential for uninterrupted low-latency video delivery.", "To form such a beam proactively, head rotations must be accurately predicted.", "Fortunately, HMD are, by design, equipped with orientation estimation capabilities.", "Current and historical estimations enable the design of reasonably accurate predictors of future orientations and rotations.", "In this paper, we present coVRage, a novel beamforming method for HMD, supporting uninterrupted connectivity during rapid head movements.", "This is, to the best of our knowledge, the first HMD-focused beamforming method offering proactive AoA trajectory coverage through sub-arrays.", "Using simulation, we demonstrate that coVRage provides a stable gain in a single-user VR scenario.", "The remainder of this paper is organised as follows.", "In Section , we provide background and related work on sub-arrays, mmWave VR and head rotation prediction.", "Section  investigates how phased arrays may be placed within an HMD, along with an appropriate system model.", "Section  outlines how to represent 3D orientations.", "Next, Section  presents coVRage, and in Section  we evaluate how well it performs in simulation.", "Finally, Section  concludes this paper.", "To form beams of flexible size and shape, sub-arrays are crucial.", "Therefore, we provide an overview of sub-arrays and the related literature.", "A sub-array may be either localized, with all elements adjacent, or interleaved, with elements spread across the entire array, as illustrated in Fig.", "REF .", "The sub-array configuration can be supported at a hardware level, by having multiple RF chains, allowing each sub-array to send a different signal.", "This includes hybrid arrays, with one chain per sub-array, and digital arrays, with one chain per element [12].", "When only one RF chain is available for all elements, the array is called analog.", "Several works present design decisions for hybrid and digital arrays for localized [13], [14], [15], [16] or interleaved [12], [17] sub-array antennas.", "Zhang et al.", "compare the two in terms of performance and feasibility [18].", "For beamforming with hybrid arrays, many approaches have been proposed.", "These may either form a single main lobe [19], [20], or provide simultaneous coverage for multiple users [21], [22], [23], [24].", "The hybrid phased array has also been used to design hierarchical codebooks, facilitating a binary-search approach to beamforming with gradually narrowing beams [25].", "Physical sub-arrays, based on the array's design, can be further subdivided into logical sub-arrays.", "This allows for more flexible hierarchical codebook design [26].", "Such codebooks can also be designed with logical sub-arrays only, which only requires an analog array [27], [28], [29].", "Multi-user coverage with logical sub-arrays has also been investigated, both by assigning a sub-array per user [30], or by synthesizing one large beam of flexible shape, covering all recipients [31].", "Our algorithm extends this final approach to cover the upcoming trajectory of one peer, rather than the current locations of several peers." ], [ "Wireless VR", "Several works have considered mmWave for cutting the cord in VR.", "In the MoVR solution, a ceiling-mounted relay assists the AP at the edge of the playing field [8].", "The HMD's built-in location and orientation tracking is used to steer transmit and receive beams directly at peers.", "Zhong et al.", "present a programmable mmWave wireless solution using COTS hardware and investigate rendering-based optimizations [32].", "Other works further investigate such optimizations [33], [34].", "Elbamby et al.", "outline the challenges of mmWave VR [4].", "Na et al.", "measure attainable VR throughput with COTS IEEE 802.11ad hardware [5].", "The IEEE 802.11ad standard was shown to be a good fit for interactive VR, with its channel access settings having a significant impact on the attainable datarate [35].", "Kim, Lee and Lee propose a dynamic power control algorithm for energy-efficient VR delivery over IEEE 802.11ad [36].", "Several proposed designs incorporate falling back to legacy Wi-Fi to cover mmWave signal loss [37], [6].", "In case of pre-recorded VR content, frames can be sent proactively over mmWave using predicted future viewing directions [38].", "Pose information-assisted networks leverage location and orientation measurements from on-device sensors, such as in HMD, for beam selection as well as AP selection, focused on spatial sharing between clients [39].", "Finally, OScan proposes fast 3D beam steering for mobile clients such as HMD, using UV-coordinates [9].", "Of these works, only OScan considers HMD-side beamforming, but it does not support proactively covering upcoming AoA.", "As such, our work is complementary to most of the aforementioned works.", "Figure: Localized and interleaved sub-arrays in a Uniform Rectangular Array" ], [ "Head Rotation Prediction", "Several approaches of varying complexity have been considered for head rotation prediction.", "A variety of works has shown the effectiveness of classical approaches such as autoregression and Kalman filters for head rotation estimation and prediction [40], [41], [42], [43].", "The more recent field of viewport prediction essentially solves the same problem [44], [45].", "Recent work uses deep learning to further improve the results [46], and may use video content as additional inputs [47].", "While the different approaches are difficult to compare directly due to varying prediction horizons and datasets, most approaches provide predictions amply accurate for our application.", "Several of the above approaches achieve an average error under a third of that of a baseline predictor which outputs the latest known orientation as prediction.", "In this section, we describe the environment coVRage is expected to operate in, provide array design guidelines based on this environment, and outline an appropriate system model." ], [ "Expected Environment", "CoVRage considers a VR setup where a ceiling-mounted mmWave AP serves an HMD-wearing user on the ground.", "The user can freely rotate their head.", "Within the time span of a single rotation, the user's location is expected to remain static (the location intuitively changes more slowly than the rotation).", "The AP is assumed to run some beamforming algorithm enabling it to always perfectly focus its beam at the HMD.", "The HMD can estimate its own orientation with high accuracy, and can accurately predict its orientation in the near future [48], [49].", "Given this orientation, the HMD is able to derive the direction towards the AP.", "The HMD is equipped with a mmWave phased array.", "The goal of coVRage is then to tune the receive beam of the HMD such that the received signal strength is consistently high while the HMD rotates towards the predicted orientation.", "CoVRage achieves this by synthesizing a beam covering the entire (shortest) trajectory between the current and predicted orientation.", "The prediction horizon should be large enough to encompass a single fast head movement, e.g., 200." ], [ "Antenna Array Design", "The antenna array for the HMD should be designed with the expected environment outlined above in mind.", "We provide some guidelines, then present a specific design.", "First of all, we eliminate hybrid and digital arrays.", "While their many RF chains would offer more flexible beamforming, their power consumption and cost are prohibitive for a battery-powered consumer device [28].", "We therefore opt for an analog array.", "A next trade-off to consider is between the number of elements in the array, and the spacing between these elements.", "For an $N$ -element ULA, the attainable beamwidth in radians is $b_\\alpha = \\frac{0.886 \\lambda }{Nd\\cos \\alpha }$ at a steering angle $\\alpha $ ($\\alpha =0$ being broadside), with an inter-element spacing of $d$ .", "Such a ULA, with all elements on one line, will however not suffice, as it can only beamform with one degree of freedom [50].", "As coVRage requires 3D beamforming, with both azimuth and elevation of the beam controllable, a URA is needed.", "For a URA of size $N=N_xN_y$ aimed at $(\\phi ,\\theta )$ , the azimuthal and elevational beamwidths are calculated separately, replacing $N$ and $\\alpha $ in (REF ) with either $N_x$ and $\\phi $ or with $N_y$ and $\\theta $ .", "The beamwidth equation implies that, for a fixed physical area, adding more elements within said area will not tighten the beamwidth.", "As such, an inter-element spacing of $d=0.5\\lambda $ is often used throughout the industry, as a tighter spacing leads to unwieldily wide beams, while wider spacing is known to create grating lobes; undesired side lobes with a directional gain as high as the main lobe's.", "This rule of thumb, however, no longer applies when using interleaved sub-arrays.", "With $M_i$ interleaved sub-arrays in a URA, the inter-element spacing within the sub-array is is $\\sqrt{M_i}d$ , as illustrated by Fig.", "REF .", "As such, the physical inter-element spacing should be chosen with a specific $M_i$ in mind.", "Whenever the sub-beams that these $M_i$ interleaved sub-arrays can create are unable to cover a full trajectory, they should be further subdivided into sub-sub-arrays, which would be localized within the sub-array.", "For the remainder of this paper, we consider a specific instantiation of the phased array within the HMD.", "Measuring many modern HMD showed that a square URA of length 4 is feasible.", "We will use the 60 band, as this unlicensed band is free to use, and already widely used for mmWave Wi-Fi.", "Then, we use $M_i = 4$ interleaved sub-arrays, meaning the inter-element spacing becomes $d = 0.25\\lambda =$ 0.125.", "At this configuration, creating sub-sub-arrays would lead to rather large beams, meaning this is mainly a feasible option for higher frequencies.", "At 300, often considered the upper limit of mmWave, a sub-sub-array could consist of $40\\times 40$ elements, having a beamwidth of only 2.54." ], [ "System Model", "CoVRage is a receiver-side beamforming method, which assumes a LoS path always exists and ignores reflected pathsWith the indoor ceiling-to-floor transmissions we consider, LoS is unlikely to be broken.", "First-order reflections are most likely via walls, and their power is assumed to be negligible as long as the user is not right next to the wall and grating lobes are avoided.", "Redirected walking [51] can keep mobile users away from walls.. As such, we opt for a simple system model [36], [52], [53], calculating the received power as $P_R = P_T + G_T - PL(d) + G_R $ where $P_T$ and $P_R$ are input and received power in , $G_T$ and $G_R$ are transmitter and receiver gain in and $PL(d)$ is the path loss over $d$ meters in .", "Transmitter-side beamforming is assumed to be perfectIf pose information is forwarded from HMD to AP, beamforming at the (static) AP is considerably simpler than at the (rotating) HMD, and therefore considered to be solved for the purpose of our channel model., so the transmitter's EIRP is constantly at the maximum legally allowed strength (30 in Europe), and Figure: At an AoA φ\\phi , the path shortens by dsinφd\\sin \\phi for every next element for a ULA.", "With a URA and AoA (φ,θ)(\\phi ,\\theta ), this becomes dsinφcosθd\\sin \\phi \\cos \\theta in the xx-direction and dsinθd\\sin \\theta in the yy-direction.$EIRP = P_T + G_T$ Using the well-known log-distance path loss model, we approximate the path loss as $PL(d) = PL_{fs}(d_0) + 10n \\log _{10}\\left(\\frac{d}{d_0}\\right)$ where $d$ is the transmitter-receiver distance in meters, $d_0$ is some reference distance, $n$ is the path loss exponent and $PL_{fs}(d_0)$ is the Friis free-space path loss over $d_0$ : $PL_{fs}(d_0) = 20\\log _{10}\\left(\\frac{4\\pi d_0}{\\lambda }\\right)$ where $\\lambda $ is the wavelength.", "The path loss exponent is estimated as 2 for an indoor LoS mmWave setting [54], so given a wavelength of 0.005 for 60, and using $d_{0}={1}{}$ , the model simplifies to approximately $PL(d) = 68 + 20\\log _{10}(d)$ To determine the receiver gain, we first determine the phase shift between antenna elements.", "For a URA of size $N_xN_y$ , using element $A_{0,0}$ as reference element, the phase shift becomes [50] $\\delta _{x,y}(\\phi ,\\theta ) = e^{j 2 \\pi d \\lambda ^{-1} (-x\\sin \\phi \\cos \\theta - y\\sin \\theta )}$ for element $A_{x,y}$ with an AoA of azimuth $\\phi $ and elevation $\\theta $ , as illustrated in Fig.", "REF .", "Then, the phase shifters of the receive array are configured with AWV $\\mathbf {w}$ with $N_xN_y$ complex elements each with magnitude 1, such that the received signal is modified with coefficient $ C_R(\\phi ,\\theta ) = \\sum _{x=0}^{N_x-1}\\sum _{y=0}^{N_y-1} [\\mathbf {w}]_{x, y} \\delta _{x,y}(\\phi ,\\theta )$ such that the final directional receive gain in for some AWV and AoA is $G_R(\\phi ,\\theta ) = 10\\log _{10}(|C_R(\\phi ,\\theta )|^2) $ where $(\\phi ,\\theta )$ may be omitted for brevity when they represent the AoA.", "To beamform the receiver towards a specific direction, its gain must be maximized.", "For this, the weight elements $[\\mathbf {w}]_{x, y}$ of weight $\\mathbf {w}$ must be set to: $[\\mathbf {w}]_{x, y} = \\frac{1}{\\delta _{x,y}} = e^{j 2 \\pi d \\lambda ^{-1} (x\\sin \\phi \\cos \\theta + y\\sin \\theta )}$" ], [ "Orientations and directions", "Several methods of representing orientations and directions in 3D space have seen common use over the years [55], [56], [57].", "Each has its own advantages and disadvantages, meaning no single most useful representation exists, and care must be taken to select the most appropriate representation for an application.", "These representations may vary in interpretability, compactness, uniqueness, numerical stability, computational efficiency, ease of combination/subdivision and susceptibility to gimbal lock.", "Different graphical VR engines supply user orientations in different representations, and throughout coVRage several representations are deliberately used to exploit their advantages." ], [ "Representations", "An easily interpretable representation is that of the Euler angles.", "In this system, an orientation is described by three chained rotations around the three axes of the coordinate system, where this coordinate system rotates along with the body.", "As 3D rotations are not commutative, the order of orientations must be properly defined.", "The separate rotations are often referred to as yaw, pitch and roll, assigned the variables $\\phi $ , $\\theta $ and $\\psi $ respectively.", "This is easily converted from an orientation to a direction; by simply omitting the final rotation, a direction in 3D space is represented compactly.", "In this interpretation, the two remaining rotations are frequently called the azimuth and elevation.", "In graphical engines, rotations are often represented by unit quaternions.", "Quaternions, first covered in the mid 19th century, are an extension of complex numbers, containing three imaginary units $i$ , $j$ and $k$ , all equal to $-1$ when squared, rather than just the one.", "In this paper, we represent the quaternion $w+xi+yj+zk$ as the vector $\\mathbf {q} = [w, x, y, z]^{\\mathrm {T}}$ .", "The set of unit quaternions (i.e., of norm 1) is a double-cover of the 3D rotation group, meaning that for each rotation in 3D space, exactly two unit quaternion representations exist ($\\mathbf {q}$ and $-\\mathbf {q}$ , as negating both the magnitude and axis of a rotation results in the same rotation).", "Quaternions are mathematically convenient; they are numerically stable, do not suffer from gimbal lock and are computationally efficient.", "Furthermore, quaternions are easily combined by simply multiplying them using the Hamilton product.", "When representing a vector $\\vec{v}$ as quaternion $\\mathbf {v} = [0, \\vec{v}_x, \\vec{v}_y, \\vec{v}_z]^{\\mathrm {T}}$ , the product $\\mathbf {v}^{\\prime } = \\mathbf {q}\\mathbf {v}\\mathbf {q}^{\\mathrm {*}}$ , where $\\mathbf {q}^{\\mathrm {*}}$ is the complex conjugate, represents $\\vec{v}$ rotated by $\\mathbf {q}$ .", "Interpolation and extrapolation are also simple: $\\mathbf {q}^a$ maintains the rotational axis but multiplies the magnitude by $a$ .", "As a final representation, we consider uv-coordinates [58], [9].", "$(u,v)$ , consisting of only two real variables, only has enough degrees of freedom to represent directions in 3D, similar to the azimuth-elevation representation.", "UV-coordinates however exist in sine-space, meaning $\\lbrace (u,v) | u,v \\in [-1,1]\\rbrace $ covers a hemisphere whose center $(0,0)$ is equivalent to azimuth and elevation 0.", "Why these coordinates are commonly used for beamforming is outlined in Section REF ." ], [ "Conversions", "As different components within the beamforming system presented in this paper require different representations of orientations and directions, we often need to convert between them.", "The following conversions are used for the remainder of the paper." ], [ "Quaternions to Euler angles", "To convert a quaternion $\\mathbf {q} = [w, x, y, z]^{\\mathrm {T}}$ to Euler angles $(\\phi , \\theta , \\psi )$ , calculate [55]: $\\begin{aligned}\\phi &= \\arctan \\frac{2(wx+yz)}{1 - 2(x^2+y^2)}\\\\\\theta &= \\arcsin (2(wy+xz))\\\\\\psi &= \\arctan \\frac{2(wz+xy)}{1 - 2(y^2+z^2)}\\end{aligned}$ where the arctangent must be implemented using the atan2 function, returning a result in $[-\\pi ,\\pi ]$ ." ], [ "Euler angles to UV-coordinates", "For this conversion, first convert the orientation to a direction, by simply discarding the roll $\\psi $ .", "Then, the UV-coordinates are [9] $\\begin{aligned}u=&\\cos \\theta \\sin \\phi \\\\v=&\\sin \\theta \\end{aligned}$ Note that this definition differs from the one commonly used for the similar UV-mapping in graphical engines, which covers the full sphere." ], [ "UV-coordinates to Euler angles", "In the opposite direction, $\\phi $ and $\\theta $ can be recovered as $\\begin{aligned}\\phi &= \\arctan \\frac{u}{\\sqrt{1 - u^2 - v^2}}\\\\\\theta &= \\arcsin v\\end{aligned}$ again using atan2 in the implementation.", "This clearly shows that not every $(u,v)$ is a valid coordinate.", "If $u^2 + v^2 > 1$ , the azimuth is no longer a real number, meaning such coordinates are invalid." ], [ "CoVRage", "In this section, we provide a step-by-step explanation of how coVRage works, along with a brief analysis of its computational efficiency." ], [ "The Algorithm", "CoVRage must convert measured current and predicted future HMD orientations to a set of phase shifts for the phased array in the HMD.", "We decompose this process into three distinct steps.", "First, we determine how the AP appears to move relative to the HMD, the reference point.", "Specifically, we determine the direction of the AP at the start and end of the rotation between HMD orientations, and the shortest trajectory between these directions, in UV-space.", "Next, we determine a set of beams that covers this trajectory, achievable by the phased array.", "Finally, we minimize the destructive interference between the sub-arrays on the trajectory, to avoid having \"blind\" spots along the trajectory.", "Figure: Interpolated path between (10,10)(10,10) and (85,50)(85,50), performed with different representations.", "Only Slerp shows the actual shortest path.To present trajectory generation, we borrow some terminology from 3D graphics.", "All objects in 3D space are located relative to the world coordinate system, which is attached to the HMD.", "During a head rotation, this world coordinate system rotates.", "It is simple to see that this is equivalent to applying the inverse rotation, around the world coordinate system, to all other objects in 3D space.", "In quaternion terms, the HMD rotates from orientation $\\mathbf {q}_1$ to $\\mathbf {q}_2$ by rotation $\\mathbf {q}_2\\mathbf {q}_1^{\\mathrm {*}}$ , meaning the AP will appear to perform the rotation $(\\mathbf {q}_2\\mathbf {q}_1^{\\mathrm {*}})^{\\mathrm {*}}= \\mathbf {q}_1\\mathbf {q}_2^{\\mathrm {*}}$ around the user.", "To translate rotations to absolute directions, the AP direction at one point must be known.", "This can be hard-coded, or measured using existing AP sensing approaches [39].", "As the HMD is only expected to provide the start and end of the expected rotation within some brief time-frame, coVRage is responsible for generating the path of the AP direction during the rotation, between those two points.", "The representation of the orientation depends on the used framework.", "OpenVR provides rotation matrices, Unreal uses Euler angles and Unity gives quaternions.", "The goal of this step was to determine the AP trajectory in UV-space, so some conversion is definitely required.", "Furthermore, determining the shortest trajectory between two orientations (i.e., a single rotation, known to exist from Euler's rotation theorem) is not straightforward with UV-coordinates.", "To generate this UV-space trajectory, we will need to first generate it in another representation, sample some points from it, and convert those to UV-coordinates.", "More directly, we need to interpolate between the points.", "In the 3D graphics world, it is widely known that naive interpolation does not work well with rotation matrices and Euler angles [59], as this does not generate orientations on the shortest trajectory between the reference orientations.", "Quaternions, on the other hand, are known to be a perfect fit for interpolation.", "Given two quaternions $\\mathbf {q}$ and $\\mathbf {p}$ , the quaternion $\\mathbf {p}\\mathbf {q}^{\\mathrm {*}}$ represents the rotation from the orientation represented by $\\mathbf {q}$ to that represented by $\\mathbf {p}$ .", "The set of quaternions $(\\mathbf {p}\\mathbf {q}^{\\mathrm {*}})^a$ for $a \\in [0,1]$ covers exactly all intermediate orientations achieved during said rotation.", "This algorithm is known as Slerp and widely used in 3D graphics [59].", "The resulting quaternions are easily converted to UV-coordinates using (REF ) and (REF ).", "As trajectories have only a modest curve in UV-space, this approximation is very close.", "Fig.", "REF shows interpolations performed with quaternions, Euler angles and UV-coordinates.", "As only the quaternion-based interpolation provides the shortest path, this is used in coVRage." ], [ "Sub-Beamforming", "Once the AP trajectory as seen from the HMD is determined, the algorithm needs to synthesize a beam covering it.", "As the beam will consist of a variable number of sub-beams from sub-arrays, the number of beams, and, as an effect, their width, must first be determined.", "Here, the choice for UV-coordinates becomes clear.", "Remember from (REF ) that the beamwidth depends on the angular distance from broadside ($\\alpha =0$ ).", "In UV-coordinates however, the beamwidth is nearly invariant to the beam's direction [58].", "As such, the beamwidth in UV-space can be approximated by the constant $b_{uv} = \\frac{0.886 \\lambda }{Nd}$ Figure: 9 beams, equally spaced in Euler angle-space, all appear as near-perfect circles in UV-space.with an error always under 2, highest near the edges of the hemisphere.", "As shown in Fig.", "REF , a rectangular sub-array's beam anywhere in UV-space is as such accurately represented by a circle of constant radius, eliminating the need for complicated, time-consuming beam shape calculations.", "The problem of trajectory coverage with sub-beams is essentially reduced to covering a curve using circles.", "The first substep here is to determine how many beams are needed to cover the entire trajectory, noting that more beams means fewer elements per beam and therefore wider beams.", "Estimate the trajectory length $l_t$ as the sum of distances between adjacent UV-space trajectory points.", "We will aim the first sub-beam towards the current direction, then divide the remaining beams along the trajectory such that each point lies within at least one sub-beam's beamwidth.", "When each interleaved sub-array has a beamwidth of $w_i$ (0.111 for $16\\times 16$ ), the required number of sub-arrays $M_s$ is $M_s = \\left\\lceil \\frac{l_t + 0.5w_i}{w_i} \\right\\rceil $ as only half the beamwidth of the first sub-beam, aimed at the first point, covers the trajectory.", "Experimentation showed that aiming the first sub-beam such that the first point is at the edge of its beamwidth provided insufficient coverage at that first point.", "If $M_s$ exceeds the available number of interleaved sub-arrays $M_i$ (4 for the 4 $\\times $ 4 array), each must be further subdivided into localized sub-sub-arrays.", "For each subdivision, each sub-beam's width doubles and the number of sub-beams quadruples, meaning the required number of subdivisions is the minimal value of $s$ for which $l_t + 2^{(s-1)} w_i <= 4^sM_i\\,2^sw_i$ As the coverable trajectory length at 60 is already over 3 for $s=1$ , this is mainly of practical use with higher frequencies.", "Another possibility is that fewer than the available number of interleaved sub-arrays are needed.", "Some approaches choose to simply deactivate unneeded sub-arrays [28], which requires hardware support.", "We instead reinforce sub-beams by steering multiple sub-arrays in the same direction.", "When only one sub-array is needed, all are aimed in the same direction, effectively eliminating the sub-arraying mechanism entirely.", "With two sub-beams required, diagonally located pairs of sub-arrays steer towards the same direction.", "Finally, when three of the four are needed, the first sub-beam, closest to the current AP direction, is formed by two diagonally located sub-arrays.", "Once the number of beams is determined, aiming these is relatively straightforward.", "CoVRage iterates through the available sample points on the trajectory curve and determines for each point if a beam should be aimed towards it.", "This is determined by checking if a sub-beam focused at the point would cover all previously considered points not yet covered by a previous sub-beam.", "As long as this is the case, no candidate sub-beams are locked in.", "However, once a candidate sub-beam could no longer cover all as of yet uncovered previous points, the candidate sub-beam at the previous point is selected.", "To avoid coverage gaps between two adjacent sample points, we may require that a sub-beam also covers the most recently considered point already covered by the previously selected sub-beam.", "As such, two consecutive sub-beams will overlap at (at least) one sample point.", "With this algorithm, a sub-beam covering the final part of the trajectory may not be found.", "If this occurs, we extrapolate the trajectory and continue the algorithm until all original sample points are covered.", "The current implementation uses a simple linear extrapolation using the final two sample points.", "Using this set of sub-beams, phase shift weights for sub-beam syncing can be calculated.", "Experiments showed that the impact of how sub-beams are mapped to sub-arrays is negligible.", "Figure: The coVRage algorithmFigure: Directional receive gain using coVRage for two different trajectories.", "Gains under 30 shown as 30.", "Final two images show the effect of disabling features of coVRage.", "Green and purple crosses indicate sub-beams' focus points and overlap points, respectively.To calculate the sub-AWV $\\mathbf {w}_i$ of the $i$ -th sub-array, (REF ) still applies, with $x$ and $y$ being the element indices within the sub-array.", "To construct the full AWV, we first introduce helper functions $f_{i}(x,y)$ and $f_{c}(x,y)$ , which map array-wide element coordinates $(x,y)$ to the index of the sub-array said element is assigned to, and to the coordinates within that sub-array, respectively.", "The elements of the full AWV are then $[\\mathbf {w}]_{x,y} = \\frac{[\\mathbf {a}]_{f_{i}(x,y)}}{[\\mathbf {w}_{f_{i}(x,y)}]_{f_{c}(x,y)}}$ where $\\mathbf {a}$ contains sub-array-level phase shifts, detailed in the following subsection.", "The first two functions in Algorithm REF summarize this step." ], [ "Sub-Beam Syncing", "Once the sub-array layout is determined and each sub-array is aimed properly, the remaining step is to synchronize the sub-beams, eliminating destructive interference between sub-arrays along the trajectory.", "As global optimisation at this level is challenging and expensive, we apply a heuristic inspired by previous work on analog sub-arrays [31].", "Specifically, we minimize destructive interference between adjacent sub-beams where it is expected to be the most impactful.", "In this case, this is the point along the trajectory equidistant from the two sub-beams.", "Sub-beam selection in Section REF was carefully designed to ensure (at least) one sample point of overlap between adjacent beams' coverages.", "The algorithm iterates through all adjacent sub-beam pairs, determines the phase difference between the two sub-beams at the overlapping point, and applies a uniform additional phase shift to all elements of the second sub-beam, making the two sub-beams phase-aligned at the overlapping sample point.", "To determine the phase difference of sub-beams $i$ and $k$ (where $k=i+1$ ) at point $(u_m,v_m)$ , first convert this point to Euler angles $(\\phi _m,\\theta _m)$ using (REF ).", "Then determine $C^i_R(\\phi _m,\\theta _m)$ and $C^k_R(\\phi _m,\\theta _m)$ by applying (REF ) with the elements of only sub-array $i$ or $k$ .", "Then set the sub-array-level phase shift such that it undoes this phase difference at sub-array $k$ : $[\\mathbf {a}]_k = e ^ {j (\\angle C^i_R(\\phi _m,\\theta _m) - \\angle C^k_R(\\phi _m,\\theta _m))}$ where $\\angle $ denotes the angle (i.e., the phase) of the complex value.", "For the first sub-beam, there is no phase shift: $[\\mathbf {a}]_0 = 1$ .", "The third function in Algorithm REF summarizes this step." ], [ "Computational Complexity", "As coVRage is designed to run in real-time on an HMD, it must be computationally efficient.", "The entire procedure consists of closed-form expressions.", "The first and third functions in Algorithm REF are of complexity $O(\\log t_l)$ with $t_l$ the trajectory length.", "Considering the limitations of human head movement, $O(1)$ also approximates their complexity.", "The second function is $O(|P|)$ with $P$ the sampled points on the trajectory.", "If required, the sampling rate can be reduced to meet any beamforming deadlines.", "Any calculated sub-beam direction differs from the optimal direction by at most one sampling interval." ], [ "Evaluation", "In this section, we simulate coVRage to evaluate how well it performs in the envisioned scenario.", "First, we assess its performance in its trajectory-covering goal.", "Then, we analyse the performance within the VR application, assessing the impact on attainable datarate using mmWave Wi-Fi.", "To evaluate coVRage, we simulate the 4 60 array, and select two AP trajectories requiring all 4 interleaved sub-arrays to be fully covered.", "Fig.", "REF shows the directional receive gain, calculated using (REF ) with both Euler angles and UV-coordinates.", "For clarity, all gains are raised to at least 30, and only half the hemisphere is shown.", "This clearly shows that the gain along the entire trajectory is consistently high.", "Some deviation from the predicted trajectory is also inherently supported with this beamwidth, without losing excessive energy far away from the trajectory.", "This provides coVRage with some inherent robustness to prediction errors that may occur with contemporary prediction methods.", "In trajectory B, extrapolation provided the final sub-beam direction.", "Next, Fig.", "REF and REF illustrate the advantage of some coVRage design decisions.", "In Fig.", "REF , sub-beam syncing is disabled, instead using arbitrary, implementation-dependent sub-array-level phase shifts.", "Overall, the gain is lower, with coverage near the midpoints being especially poor.", "This indicates that sub-beam syncing is essential to the proper working of the algorithm.", "In Fig.", "REF , the first sub-beam is not placed at the first trajectory point, but rather at the farthest point whose beam still covers the first point.", "As there is no sub-beam syncing aimed at optimising gain at this first point, its gain decreases by 7 compared to having a sub-beam pointed directly at it.", "As this is the actual AP direction at the time of beamforming, high coverage for this point is arguably the most important.", "Next, we evaluate how coVRage compares to steering only a single beam in one specific direction.", "We consider three possible directions: (1) towards the current AP direction, (2) at the farthest trajectory point still covering the current position and (3) halfway along the trajectory.", "As the single beam uses the full array, with only half the inter-element spacing of the sub-arrays, this beam will be twice as wide as any sub-beam.", "Using the two trajectories from Fig.", "REF , we measure the directional receive gain along the entire trajectory using coVRage and the three single-beam approaches.", "As Fig.", "REF and REF show, the algorithm's coverage of the trajectory is very consistent.", "In trajectory A, the gain range is 4.75, largely due to a decrease at the the end of the trajectory.", "With trajectory B, the final sub-beam is aimed beyond the final trajectory point, meaning coverage remains very stable throughout, with a range of only 1.4.", "Higher coverage at the end of the trajectory could be enforced by requiring a final sub-beam beyond the trajectory.", "Figure: Directional receive gain and noise penalty across the trajectory, using coVRage and single-beam solutions.The single-beam approaches, as expected, manage to outperform coVRage at and around their steering direction.", "Away from that direction, however, gain reduces quickly, in contrast to coVRage.", "In analyzing the impact on SNR, both the receive gain in the AoA direction and the maximum receive gain are of importance.", "The former determines the intensity of the intended signal, while the latter influences that of the noise, assuming it is isotropic.", "Therefore, we quantify the approximate impact of gain fluctuations throughout the trajectory using a penalty $N(\\phi _{AoA}, \\theta _{AoA}) = \\max _{\\phi ,\\theta }(G_R(\\phi ,\\theta )) - G_R(\\phi _{AoA},\\theta _{AoA})$ for some AoA.", "This decibel-scale term can be subtracted from the SNR directly, and therefore represents the SNR loss caused by high directional gain away from the AoA (but possibly elsewhere along the trajectory).", "As long as the maximum directional gain lies along the trajectory, the gain fluctuation along the trajectory also sets an upper bound to this noise penalty.", "Fig.", "REF and REF show the penalty for the two trajectories under consideration.", "To assess this penalty's impact on performance, we rely on the IEEE 802.11ad standard's minimum received signal intensity for each MCS [36], [60].", "From the highest to lowest non-control MCS, offering 4620Mbps and 385Mbps throughput respectively, the required intensity drops by 15.", "As such, when using any single-beam solution, even if the maximum SNR is an exceptionally high 25 above the requirement for maximum MCS, it will drop so low along the trajectory that the datarate reverts to a control-level 27.5Mbps or connectivity is even lost altogether.", "Either halts delivery of video content to the HMD and is extremely disruptive to the user experience.", "Ignoring the coverage reduction at the end of trajectory A, a maximum SNR of just 1.5 above the required SNR of the highest MCS will suffice to maintain the maximum datarate throughout the trajectory.", "This was previously shown to be sufficient to serve multiple 4K HMD at 120 and a transmission latency under 1 [35].", "Hence, our solution can, in contrast to single-beam solutions, support truly wireless contemporary immersive VR setups." ], [ "Conclusion", "In this paper, we presented coVRage, the first beamforming algorithm designed specifically for HMD-side beamforming with mobile VR, where uninterrupted reception even during fast head rotations is crucial for maintaining user experience.", "Using the HMD's built-in orientation detection capabilities, a predictor can estimate how the AoA of incoming wireless video data will change in the near future.", "By subdividing the phased array into sub-arrays and aiming each sub-array's beam at a different point along the predicted trajectory, coVRage is able to guarantee uninterrupted coverage along the full trajectory, at a very stable signal strength.", "Simulations using a simple channel model show that coVRage can design beams with a signal strength variation of only a few decibels.", "A single-beam solution is shown to instead vary by tens of decibels, enough to decimate the attainable throughput, therefore causing a substantial negative impact on, or even fully impairing the end-user’s experience.", "In future work, we will further investigate capabilities with different array configurations and frequencies.", "Furthermore, we will quantify the impact of prediction errors and of residual destructive interference between sub-beams, and if needed harden the algorithm against this.", "Finally, we will combine coVRage with specific trajectory predictors and AP-side beamforming to evaluate the performance of an end-to-end system." ], [ "Acknowledgment", "The work of Jakob Struye was supported by the Research Foundation - Flanders (FWO): PhD Fellowship 1SB0719N.", "The work of Filip Lemic was supported by the EU Marie Skłodowska- Curie Actions Individual Fellowships (MSCA-IF) project Scalable Localization-enabled In-body Terahertz Nanonetwork (SCaLeITN), grant nr.", "893760.", "In addition, this work received support from the University of Antwerp's University Research Fund (BOF)." ] ]
2105.11793
[ [ "The perturbed prox-preconditioned spider algorithm: non-asymptotic\n convergence bounds" ], [ "Abstract A novel algorithm named Perturbed Prox-Preconditioned SPIDER (3P-SPIDER) is introduced.", "It is a stochastic variancereduced proximal-gradient type algorithm built on Stochastic Path Integral Differential EstimatoR (SPIDER), an algorithm known to achieve near-optimal first-order oracle inequality for nonconvex and nonsmooth optimization.", "Compared to the vanilla prox-SPIDER, 3P-SPIDER uses preconditioned gradient estimators.", "Preconditioning can either be applied \"explicitly\" to a gradient estimator or be introduced \"implicitly\" as in applications to the EM algorithm.", "3P-SPIDER also assumes that the preconditioned gradients may (possibly) be not known in closed analytical form and therefore must be approximated which adds an additional degree of perturbation.", "Studying the convergence in expectation, we show that 3P-SPIDER achieves a near-optimal oracle inequality O(n^(1/2) /epsilon) where n is the number of observations and epsilon the target precision even when the gradient is estimated by Monte Carlo methods.", "We illustrate the algorithm on an application to the minimization of a penalized empirical loss." ], [ "Introduction", "Consider the following composite, nonconvex, and possibly nonsmooth optimization problem $\\operatorname{Argmin}_{s \\in \\mathcal {S}} \\lbrace \\operatorname{W}(s) + g(s) \\rbrace \\;,$ where $\\mathcal {S}$ is a closed convex subset of $\\mathbb {R}^q$ , $\\operatorname{W}:\\mathcal {V} \\rightarrow \\mathbb {R}$ is a smooth function defined on a neighborhood $\\mathcal {V}$ of $\\mathcal {S}$ and $g: \\mathcal {S}\\rightarrow \\left(-\\infty , + \\infty \\right]$ is a proper lower semi-continuous convex function (with an easy to compute proximal term).", "This paper addresses the case when $\\operatorname{W}$ has a finite-sum structure $\\operatorname{W}(s) = \\frac{1}{n} \\sum _{i=1}^n \\operatorname{W}_i(s) \\;,$ and the optimization problem (REF ) is solved by a preconditioned-gradient based algorithm.", "Optimization problems (REF )-(REF ) often arise in machine learning.", "In such case, $n$ is the number of examples which is typically very large, $\\operatorname{W}_i$ is the loss function associated to example $\\# i$ , and $g$ is a non-smooth regularization term, e.g.", "$\\ell _1$ norm, Elastic net, etc.", "The preconditioning setting may naturally arise \"implicitly\", for example in the stochastic finite-sum version of the Expectation-Maximization (EM) algorithm in the exponential family, which was the main motivation for this work; see sec:logistic and the companion paper [1].", "Minimization problems (REF ) and (REF ) cover a broad range of applications in machine learning, statistics, and signal processing; see [2].", "State-of-the art methods to solve these problems rely on stochastic optimization approaches [3], [4].", "In the nonconvex case, while numerical algorithms for solving the noncomposite setting (i.e.", "$g=0$ ), are well-developed and have received significant attention [5], [6], methods for composite optimization remain scarce [7], [8].", "The authors in [7] proposes and studies a non-composite finite-sum problem using SVRG estimator from [9].", "This method is extended to the composite setting by applying the proximal operator of $g$ as in the proximal-gradient scheme (see [10], [11], [12] for literature review on the proximal-gradient algorithm).", "This technique is based on gradients and does not use preconditioning.", "This scheme has been later improved with SPIDER, where the gradient control variate is sequentially updated to improve the estimation accuracy: SPIDER is known to achieve near optimal oracle complexity in nonconvex optimization [13], [14], [8].", "This paper analyzes the 3P-SPIDER algorithm designed to solve $s \\in \\mathcal {S}: \\qquad 0 \\in \\nabla W(s) + \\partial g(s) \\;,$ by combining (i) a variance-reduced preconditioned-gradient algorithm designed for the finite sum setting, and (ii) a proximal step to take into account the (non smooth) regularizer $g$ .", "Furthermore 3P-SPIDER covers the case when the preconditioned gradient - of the form $n^{-1} \\sum _{i=1}^n \\mathsf {h}_i(s)$ , see AREF below - is not computable in a closed-form and is approximated: both the cases of an approximation of the full sum over $n$ terms by a sum over a random subsample, and an approximation of the functions $\\mathsf {h}_i$ 's are considered.", "3P-SPIDER was introduced in [1] for a specific application to large scale learning solved by a Expectation Maximization-based optimization method.", "The main contribution of this paper is to provide explicit control of the convergence in expectation of 3P-SPIDER and deduce complexity bounds in terms of the design parameters of this algorithm.", "A comparison to the state of the art gradient-based methods in terms of complexity bounds, is also provided in the case the quantities $\\mathsf {h}_i$ are expectations and are approximated by a Monte Carlo integration: it is established that the number of Monte Carlo samples can be chosen so that 3P-SPIDER reaches the oracle complexity bounds corresponding to the case where the $\\mathsf {h}_i(s)$ 's are known in closed form.", "Notations.", "$\\mathbb {R}_+^\\star $ and ${\\mathbb {N}}^\\star $ denote respectively (resp.)", "the positive real line and the positive integers.", "For $n \\in {\\mathbb {N}}^\\star $ , set $[n]^\\star \\stackrel{\\mathrm {def}}{=}\\lbrace 1, \\cdots , n\\rbrace $ and $[n] \\stackrel{\\mathrm {def}}{=}\\lbrace 0, \\cdots , n\\rbrace $ .", "For $x \\in \\mathbb {R}$ , $\\lceil x \\rceil $ is the nearest integer greater than or equal to $x$ .", "Vectors are column-vectors; for $a,b$ in $\\mathbb {R}^\\ell $ , $\\left\\langle a,b\\right\\rangle $ denotes the Euclidean scalar product, and $\\Vert a\\Vert $ the associated norm.", "For a matrix $A$ , we denote by $A^T$ and $A^{-1}$ resp.", "its transpose and its inverse.", "$\\mathrm {I}_d$ is the $d \\times d$ identity matrix.", "The random variables are defined on a probability space $(\\Omega , \\mathcal {A},\\mathbb {P})$ ; $\\mathbb {E}$ denotes the associated expectation.", "For random variables $U$ and a sigma-field $\\mathcal {F}$ , $\\mathbb {E}[U \\vert \\mathcal {F}]$ is the conditional expectation of $U$ given $\\mathcal {F}$ .", "For a smooth function $f$ , $\\nabla f$ is the gradient of $f$ .", "For a proper lower semi-continuous convex function $g$ and $x$ in its (assumed) non-empty domain, $\\partial g(x)$ is the subdifferential of $g$ at $x$ ." ], [ "The 3P-SPIDER algorithm", "The optimization problem at hand is the problem (REF ) in the case when A 1 $\\mathcal {S}$ is a closed convex subset of $\\mathbb {R}^q$ .", "$\\operatorname{W}: \\mathcal {V}\\rightarrow \\mathbb {R}$ is a continuously differentiable function on $\\mathcal {V}$ , an open neighborhood of $\\mathcal {S}$ .", "Its gradient $\\nabla \\operatorname{W}$ is globally Lipschitz-continuous on $\\mathcal {S}$ with Lipschitz constant $L_{\\dot{\\operatorname{W}}}$ .", "$g: \\mathcal {S}\\rightarrow \\left(-\\infty , + \\infty \\right]$ is a proper lower semi-continuous convex function.", "We consider a gradient approach for solving (REF ) and allow the use of a preconditioning matrix $B(s)$ which may depend on the current value of the parameter $s$ .", "We assume that A 2 For any $s \\in \\mathcal {S}$ , $B(s)$ is a $q \\times q$ positive definite matrix and there exist $0 < v_{\\min } \\le v_{\\mathrm {max}}$ such that for any $s \\in \\mathcal {S}$ , the spectrum of $B(s)$ is in $\\left[v_{\\min },v_{\\mathrm {max}}\\right]$ .", "For all $i \\in [n]^\\star $ , there exists a globally Lipschitz function $\\mathsf {h}_i: \\mathcal {S}\\rightarrow \\mathbb {R}^q$ , with constant $L_i$ , such that $- B^{-1}(s) \\, \\nabla \\operatorname{W}(s) = \\frac{1}{n} \\sum _{i=1}^n \\mathsf {h}_i(s)\\;.$ We introduce a weighted proximal operator: for a $q \\times q$ positive definite matrix $B$ , define for any $s \\in \\mathbb {R}^q$ and $\\gamma >0$ , $\\mathrm {Prox}_{B, \\gamma g}(s) \\stackrel{\\mathrm {def}}{=}\\operatorname{Argmin}_{s^{\\prime } \\in \\mathcal {S}} \\left\\lbrace \\gamma g(s^{\\prime })+ \\frac{1}{2} (s^{\\prime }-s)^T B (s^{\\prime }-s) \\right\\rbrace \\;.$ Set $\\mathsf {h}(s) \\stackrel{\\mathrm {def}}{=}n^{-1} \\sum _{i=1}^n \\mathsf {h}_i(s)$ .", "Under AREF and AREF , for any $s,s^{\\prime } \\in \\mathcal {S}$ and $\\gamma > 0$ , $\\mathrm {Prox}_{B(s), \\gamma g}(s^{\\prime })$ exists and is unique and, since $s^{\\prime }= \\mathrm {Prox}_{B(s),\\gamma g}(s + \\gamma \\mathsf {h}(s))$ if and only if $0 \\in \\partial g(s) + B(s) (s^{\\prime }-s- \\gamma \\mathsf {h}(s))$ , we obtain $\\lbrace s \\in \\mathcal {S}: \\mathrm {Prox}_{B(s),\\gamma g}(s + \\gamma \\mathsf {h}(s)) =s \\rbrace \\\\ = \\lbrace s\\in \\mathcal {S}: 0 \\in \\nabla \\operatorname{W}(s) + \\partial g(s)\\rbrace \\;.", "$ This property implies that the solutions of (REF ) are the roots of the function $s \\mapsto \\mathrm {Prox}_{B(s), \\gamma g}(s+ \\gamma \\mathsf {h}(s))-s$ restricted to $\\mathcal {S}$ , whatever $\\gamma >0$ .", "For any minibatch $\\mathcal {B}$ of size $\\mathsf {b}$ , sampled at random from $[n^\\star ]$ - with or without replacement, we have (see e.g.", "[15]) $\\mathsf {h}(s) = \\frac{1}{\\mathsf {b}} \\mathbb {E}\\left[ \\sum _{i \\in \\mathcal {B}} \\mathsf {h}_i(s)\\right] \\;,$ thus implying that in the finite-sum setting, the preconditioned gradient $-B^{-1}(s) \\nabla \\operatorname{W}(s)$ can be approximated by a sum with $\\mathsf {b}$ terms where the indexes of summation $i$ are sampled randomly: $\\mathsf {b}^{-1} \\sum _{i \\in \\mathcal {B}} \\mathsf {h}_i(s)$ .", "Therefore, a natural extension of the Proximal-Gradient algorithm to the finite-sum setting would define a sequence $\\lbrace \\widehat{S}_k, k \\ge 0 \\rbrace $ by $\\widehat{S}_{k+1} = \\mathrm {Prox}_{B(\\widehat{S}_k), \\gamma _{k+1} g} \\left( \\widehat{S}_k +\\frac{\\gamma _{k+1}}{\\mathsf {b}} \\sum _{i \\in \\mathcal {B}_{k+1}} \\mathsf {h}_i(\\widehat{S}_k)\\right)$ where $\\lbrace \\gamma _k, k \\ge 0\\rbrace $ is a positive deterministic sequence and $\\lbrace \\mathcal {B}_{k+1}, k \\ge 0\\rbrace $ is a sequence of minibatches of size $\\mathsf {b}$ sampled at random from $[n]^\\star $ .", "3P-SPIDER reduces the variance of this stochastic perturbation by the construction of a control variate, which is defined as an approximation of $\\mathsf {h}(\\widehat{S}_k)$ correlated with the random variable $\\mathsf {b}^{-1}\\sum _{i \\in \\mathcal {B}_{k+1}} \\mathsf {h}_i(\\widehat{S}_k)$ .", "This control variate is refreshed at each so-called outer loop, indexed by $t$ in Algorithm ; and then evolves along the inner loops, indexed by $k$ .", "Finally, 3P SPIDER allows approximations on the computation of $\\mathsf {h}_i(\\widehat{S}_{t,k})$ , approximations denoted by $\\widehat{\\mathsf {h}}_i^{t,k}$ .", "The algorithm is given in Algorithm .", "At the start of each outer loop $\\# t$ , the control variate $\\mathsf {S}_{t,0}$ is initialized (see Lines  and ) in order to approximate $\\mathsf {h}(\\widehat{S}_{t,-1})$ ; a natural idea is to choose $\\mathcal {E}_t=0$ .", "Nevertheless, the computational cost is important since it involves a sum over $n$ terms, and this full sum can be substituted with a sum having $\\mathsf {b}^{\\prime } \\ll n$ indices defined by a minibatch $\\mathcal {B}_{t,0}^{\\prime }$ sampled at random from $[n]^\\star $ ; in that case, $\\mathcal {E}_t \\ne 0$ .", "The control variate is modified at each inner loop $\\# k$ (see Line ): since $\\mathsf {S}_{t,0} \\approx \\mathsf {h}(\\widehat{S}_{t,-1})$ , we have $\\mathsf {S}_{t,k+1}\\approx \\mathsf {h}(\\widehat{S}_{t,k})$ upon noting that $\\mathsf {S}_{t,k+1} -\\mathsf {S}_{t,k} \\approx \\mathsf {h}(\\widehat{S}_{t,k}) - \\mathsf {h}(\\widehat{S}_{t,k-1})$ .", "The key property is the choice of the same minibatch $\\mathcal {B}_{t,k+1}$ when approximating $ \\mathsf {h}(\\widehat{S}_{t,k})$ and $ \\mathsf {h}(\\widehat{S}_{t,k-1})$ : the correlation of these quantities is the essence of the control variate mechanism.", "[htbp] The Perturbed Prox-Preconditioned SPIDER (3P-SPIDER) algorithm.", "$k_\\mathrm {out}, k_\\mathrm {in}\\in {\\mathbb {N}}^\\star $ ; $\\widehat{S}_\\mathrm {init}\\in \\mathcal {S}$ ; $\\gamma _{t,0} \\ge 0$ , $\\gamma _{t,k} >0$ for $t \\in [k_\\mathrm {out}]^\\star $ , $k \\in [k_\\mathrm {in}]^\\star $ .", "The 3P-SPIDER sequence $\\lbrace \\widehat{S}_{t,k}, t \\in [k_\\mathrm {in}]^\\star , k \\in [k_\\mathrm {in}]\\rbrace $ $\\widehat{S}_{1,0}= \\widehat{S}_{1,-1} = \\widehat{S}_\\mathrm {init}$ $\\mathsf {S}_{1,0} = n^{-1} \\sum _{i=1}^n\\widehat{\\mathsf {h}}_i^{1,-1}+ \\mathcal {E}_1$ $t=1,\\cdots , k_\\mathrm {out}$ $k=0,\\ldots ,k_\\mathrm {in}-1$ Sample a mini batch $\\mathcal {B}_{t,k+1}$ of size $\\mathsf {b}$ in $[n]^\\star $ $\\mathsf {S}_{t,k+1} = \\mathsf {S}_{t,k} + \\mathsf {b}^{-1} \\sum _{i \\in \\mathcal {B}_{t,k+1}} \\left( \\widehat{\\mathsf {h}}_i^{t,k} - \\widehat{\\mathsf {h}}_i^{t,k-1}\\right) $ $\\widehat{S}_{t,k+1/2} =\\widehat{S}_{t,k} + \\gamma _{t,k+1} \\mathsf {S}_{t,k+1}$ $\\widehat{S}_{t,k+1} =\\mathrm {Prox}_{B(\\widehat{S}_{t,k}), \\gamma _{t,k+1} g}\\left( \\widehat{S}_{t,k+1/2}\\right)$ $\\widehat{S}_{t+1,-1} =\\widehat{S}_{t,k_\\mathrm {in}}$ $\\mathsf {S}_{t+1,0}= n^{-1} \\sum _{i=1}^n\\widehat{\\mathsf {h}}_i^{t+1,-1} + \\mathcal {E}_{t+1}$ $\\widehat{S}_{t+1,-1/2} = \\widehat{S}_{t+1,-1} + \\gamma _{t+1,0} \\mathsf {S}_{t+1,0} $ $\\widehat{S}_{t+1,0} = \\mathrm {Prox}_{B(\\widehat{S}_{t+1,-1}),\\gamma _{t+1,0}}(\\widehat{S}_{t+1,-1/2})$ The SPIDER algorithm ([13], [14], [8]) corresponds to the case $g=0$ , $\\mathcal {S}= \\mathbb {R}^q$ , $B(s) = \\mathrm {I}_q$ and $\\widehat{\\mathsf {h}}^{t,k}_i = \\mathsf {h}_i(\\widehat{S}_{t,k})$ for any $i \\in [n]^\\star $ , $t\\in [k_\\mathrm {out}]^\\star $ and $k \\in [k_\\mathrm {in}-1]$ .", "In the case $B(s)= \\mathrm {I}_q$ , 3P-SPIDER is a perturbed proximal-gradient algorithm (see e.g.", "[16]); the convergence analysis below addresses the non convex case ($\\operatorname{W}$ is not assumed to be convex).", "In the case $g=0$ and $\\mathcal {S}= \\mathbb {R}^q$ , 3P-SPIDER is a Stochastic Approximation algorithm designed to find the roots of the preconditioned gradient $s \\mapsto \\mathsf {h}(s) = -B^{-1}(s)\\nabla \\operatorname{W}(s)$ .", "Applied with $g= 0$ or $g= \\chi _{\\mathcal {K}}$ - the characteristic function of a closed convex set $\\mathcal {K}$ , 3P-SPIDER is a variance reduced incremental EM algorithm (see [17], [1], see also section  for an application to the minimization of a penalized empirical loss)." ], [ "Convergence in expectation and complexity bounds", "For ease of exposition (see [18] for the general case), it is assumed hereafter that A 3 $\\gamma _{t+1,0} =0$ and $\\mathcal {E}_{t+1} =0$ for any $t \\in [k_\\mathrm {out}]^\\star $ .", "The intractable functions $s \\mapsto \\mathsf {h}_i(s)$ are defined as an integral with respect to (w.r.t.)", "a distribution $\\pi _{i,s}$ $\\mathsf {h}_i(s) \\stackrel{\\mathrm {def}}{=}\\int _\\mathsf {Z}\\mathcal {H}_i(z) \\, \\pi _{i,s}(\\mathrm {d}z) \\;.$ They are approximated by a Monte Carlo sum: $\\mathsf {h}_i(\\widehat{S}_{t,k-\\ell }) \\approx \\widehat{\\mathsf {h}}_i^{t,k-\\ell } \\stackrel{\\mathrm {def}}{=}\\frac{1}{m_{t,k+1}} \\sum _{r=1}^{m_{t,k+1}}\\mathcal {H}_i(Z_{r}^{i,t,k-\\ell }), \\ell \\in \\lbrace 0,1\\rbrace $ where conditionally to the past of the algorithm $\\mathcal {F}_{t,k}$ , the random variables $\\lbrace Z_r^{i,t,k-\\ell }, r \\ge 0 \\rbrace $ are independent and identically distributed (i.i.d.)", "with distribution $\\pi _{i,\\widehat{S}_{t,k-\\ell }}$ .", "Section  provides an example of this setting.", "More precisely, $\\mathcal {F}_{t,k}$ is the filtration associated to the history of the algorithm up to the outer loop $\\# t$ and the inner loop $\\# k$ , $\\mathcal {F}_{t,-1} & \\stackrel{\\mathrm {def}}{=}\\mathcal {F}_{t-1, k_\\mathrm {in}} \\;, \\quad \\mathcal {F}_{t,0} \\stackrel{\\mathrm {def}}{=}\\mathcal {F}_{t,-1} \\bigvee \\sigma (\\widehat{\\mathsf {h}}_i^{t,-1}, i \\in [n]^\\star ) \\;,\\\\ \\mathcal {F}_{t,k} &\\stackrel{\\mathrm {def}}{=}\\mathcal {F}_{t,k-1} \\bigvee \\sigma \\left( \\mathcal {B}_{t,k};\\widehat{\\mathsf {h}}_i^{t,k-1}, \\widehat{\\mathsf {h}}_i^{t,k-2}, i \\in \\mathcal {B}_{t,k} \\right) \\;.$ For any $t \\in [k_\\mathrm {out}]^\\star $ and $k \\in [k_\\mathrm {in}-1]$ , define $\\error _{t,k+1} \\stackrel{\\mathrm {def}}{=}\\frac{1}{\\mathsf {b}} \\sum _{i \\in \\mathcal {B}_{t,k+1}}\\left( \\widehat{\\mathsf {h}}^{t,k}_i - \\widehat{\\mathsf {h}}^{t,k-1}_i - \\mathsf {h}_i(\\widehat{S}_{t,k}) +\\mathsf {h}_i(\\widehat{S}_{t,k-1}) \\right) \\;;$ $\\error _{t,k+1}$ corresponds to the errors when approximating the quantities $\\mathsf {h}_i(\\widehat{S}_{t,k-\\ell })$ for $\\ell \\in \\lbrace 0,1\\rbrace $ at outer loop $\\# t$ and inner loop $\\# (k+1)$ .", "From standard computations on i.i.d.", "random variables (the randomness being the selection of the mini-batch $\\mathcal {B}_{t,k+1}$ ), we have $& \\mathbb {E}\\left[ \\error _{t,k+1} \\vert \\mathcal {F}_{t,k} \\right] = 0\\;, \\\\ & \\mathbb {E}\\left[ \\Vert \\error _{t,k+1} -\\mathbb {E}\\left[ \\error _{t,k+1} \\vert \\mathcal {F}_{t,k} \\right] \\Vert ^2 \\vert \\mathcal {F}_{t,k}\\right] \\le \\frac{C_v}{ \\mathsf {b}\\, m_{t,k+1}}\\;, $ where $C_v \\stackrel{\\mathrm {def}}{=}2 \\sup _{s \\in \\mathcal {S}} n^{-1} \\sum _{i=1}^n \\int _\\mathsf {Z}\\Vert \\mathcal {H}_i(z) - \\mathsf {h}_i(s) \\Vert ^2 \\pi _{i,s}(\\mathrm {d}z) \\;.$ The result (REF ) claims that the errors $\\eta _{t,k+1}$ are unbiased, while () is the control of its conditional variance - which is a decreasing function of the batch size and the number of points in the Monte Carlo sum.", "In (), the control is uniform with respect to the past (the right hand side is deterministic while the left hand side is random): this assumption can be difficult to check when the optimization problem is not constrained in order to ensure that the points $\\widehat{S}_{t,k}$ remain in a bounded subset of $\\mathbb {R}^q$ .", "Theorem REF provides an upper bound on a control in expectation of the convergence of the algorithm.", "First, it controls the difference of two successive values $\\widehat{S}_{t,k} - \\widehat{S}_{t,k-1}$ ; then it controls the quantity $\\Delta _{t,k} \\stackrel{\\mathrm {def}}{=}\\frac{\\Vert \\mathrm {Prox}_{B(\\widehat{S}_{t,k}), \\gamma _{t,k} g} \\hspace{-2.84544pt}\\left(\\widehat{S}_{t,k-1} + \\gamma _{t,k} \\mathsf {h}(\\widehat{S}_{t,k-1}) \\right) \\hspace{-2.84544pt}- \\widehat{S}_{t,k-1} \\Vert ^2}{\\gamma ^2_{t,k}}$ which is a kind of distance to the set of the solutions to (REF ) (see (REF )).", "When $B(s) = \\mathrm {I}_d$ and $g=0$ , $\\Delta _{t,k} = \\Vert \\mathsf {h}(\\widehat{S}_{t,k})\\Vert ^2 = \\Vert \\nabla \\operatorname{W}(\\widehat{S}_{t,k})\\Vert ^2$ .", "The proof of Theorem REF is given in [18].", "Theorem 1 Assume AREF , AREF and AREF .", "For any $t \\in [k_\\mathrm {out}]^\\star $ and $k \\in [k_\\mathrm {in}-1]$ , set $\\gamma _{t,k} = \\gamma _\\star \\stackrel{\\mathrm {def}}{=}\\frac{v_{\\min }}{L_{\\dot{\\operatorname{W}}} + 2Lv_\\mathrm {max}\\sqrt{k_\\mathrm {in}} / \\sqrt{\\mathsf {b}}}.$ Denote by $(\\tau , K)$ a uniform random variable on $[k_\\mathrm {out}]^\\star \\times [k_\\mathrm {in}]$ , independent of the path $\\lbrace \\widehat{S}_{t,k}, t \\in [k_\\mathrm {out}]^\\star , k \\in [k_\\mathrm {in}] \\rbrace $ .", "Then, $& \\frac{v_{\\min }^2}{2 \\left( L_{\\dot{\\operatorname{W}}} + 2 L v_{\\mathrm {max}} \\sqrt{k_\\mathrm {in}}/\\sqrt{\\mathsf {b}} \\right)} \\mathbb {E}\\left[\\frac{\\Vert \\widehat{S}_{\\tau ,K} -\\widehat{S}_{\\tau ,K-1}\\Vert ^2}{\\gamma _{\\tau ,K}^2} \\right] \\\\& \\le \\frac{1}{k_\\mathrm {out}(1+k_\\mathrm {in})}\\left( \\operatorname{W}(\\widehat{S}_\\mathrm {init}) + g(\\widehat{S}_\\mathrm {init}) -\\min (\\operatorname{W}+ g) \\right) \\\\ & + C_v \\, \\frac{v_\\mathrm {max}}{2 L}\\frac{1}{\\sqrt{k_\\mathrm {in}\\mathsf {b}}} \\mathbb {E}\\left[ \\frac{k_\\mathrm {in}-K}{m_{\\tau ,K+1}} \\right] \\;.$ In addition $&\\left( \\frac{2}{v_{\\min }} \\left\\lbrace L_{\\dot{\\operatorname{W}}} + 2L v_\\mathrm {max}\\sqrt{\\frac{k_\\mathrm {in}}{\\mathsf {b}} } \\right\\rbrace + L \\sqrt{\\frac{k_\\mathrm {in}}{\\mathsf {b}}} \\right)^{-1} \\mathbb {E}\\left[ \\Delta _{\\tau ,K} \\right] \\\\ & \\le \\left\\lbrace \\frac{L_{\\dot{\\operatorname{W}}}}{L v_{\\min }} + 2 \\frac{v_\\mathrm {max}}{v_{\\min }}\\sqrt{\\frac{k_\\mathrm {in}}{\\mathsf {b}} }\\right\\rbrace ^{-1} \\hspace{-5.69046pt}\\left(\\frac{1}{L}+ \\left(\\frac{v_\\mathrm {max}}{v_{\\min }}\\right)^2\\gamma _\\star \\sqrt{\\frac{k_\\mathrm {in}}{\\mathsf {b}}} \\right) \\cdots \\\\ & \\qquad \\qquad \\times \\mathbb {E}\\left[ \\frac{\\Vert \\widehat{S}_{\\tau ,K} -\\widehat{S}_{\\tau ,K-1}\\Vert ^2}{\\gamma ^2_{\\tau ,K}} \\right] \\\\ & \\qquad +\\left(\\frac{v_\\mathrm {max}}{v_{\\min }}\\right)^2 \\frac{C_v}{L}\\frac{1}{\\sqrt{\\mathsf {b}k_\\mathrm {in}}} \\mathbb {E}\\left[\\frac{k_\\mathrm {in}-K}{m_{\\tau ,K}} \\right] \\;.$ In Theorem REF , the expectations are w.r.t.", "the stochastic path of the algorithm $\\lbrace \\widehat{S}_{t,k}\\rbrace $ and to the randomness of the times $(\\tau , K)$ .", "This theorem provides a control of the errors when the algorithm is stopped at some random time $(\\tau , K)$ .", "Such a control is classical in the non-convex case to show non-asymptotic convergence of stochastic gradient methods to a stationary point [19]: it consists in fixing a maximal number of iterations (here set to $k_\\mathrm {out}\\times k_\\mathrm {in}$ ) of the algorithm, and draw at random, prior the run of the algorithm, a stopping time $(\\tau , K)$ .", "Let us discuss the complexity bounds in the case $m_{t,k} = m$ for any $t \\in [k_\\mathrm {out}]^\\star $ and $k \\in [k_\\mathrm {in}]^\\star $ .", "The total number of proximal calls is equal to $ \\mathcal {N}_{P} \\stackrel{\\mathrm {def}}{=}k_\\mathrm {out}( k_\\mathrm {in}+1)$ .", "The total number of approximations of the functions $\\mathsf {h}_i$ is equal to $\\mathcal {N}_{A} \\stackrel{\\mathrm {def}}{=}k_\\mathrm {out}(n+ 2 \\mathsf {b}k_\\mathrm {in})$ .", "The total number of Monte Carlo draws is $\\mathcal {N}_{MC} \\stackrel{\\mathrm {def}}{=}m\\mathcal {N}_A$ .", "Let us fix $\\epsilon >0$ .", "Among the values $\\mathcal {K}(n,\\epsilon )$ of the positive integers $(k_\\mathrm {out}, k_\\mathrm {in},\\mathsf {b}, m) \\in {\\mathbb {N}}^4$ which guarantee that $\\epsilon $ -stationarity is reached i.e.", "$\\mathbb {E}\\left[ \\Delta _{\\tau ,K} \\right] \\le \\epsilon \\;,$ the complexity bounds in terms of proximal calls are defined as $\\mathcal {K}_{P} \\stackrel{\\mathrm {def}}{=}\\min _{\\mathcal {K}(n,\\epsilon )} \\mathcal {N}_P$ ; similarly, we define the complexity in terms of $\\mathsf {h}_i^{\\prime }s$ approximations, and in terms of Monte Carlo draws.", "By choosing $\\mathsf {b}= k_\\mathrm {in}=\\sqrt{n}$ , $k_\\mathrm {out}= 1/(\\sqrt{n} \\epsilon )$ and $m=1/ \\epsilon $ , it is easily seen from Theorem REF that $\\mathbb {E}\\left[ \\Delta _{\\tau ,K} \\right] = O(\\epsilon )$ and $\\mathcal {K}_P =O(\\epsilon ^{-1}) \\;, \\quad \\mathcal {K}_{A} =O(\\sqrt{n} \\epsilon ^{-1}) \\;, \\quad \\mathcal {K}_{MC} = O(\\sqrt{n}\\epsilon ^{-2})\\;.$ When $B(s) =\\mathrm {I}_q$ , $g=0$ and the gradient functions $\\mathsf {h}_i$ 's can be computed exactly, the state of the art complexity of variance-reduced gradient algorithm in terms of total number of computations of these gradient functions is $O(\\sqrt{n} \\epsilon ^{-1})$ [8].", "This bound is also reached by the variance reduced incremental EM named SPIDER-EM, which corresponds to the case $g=0$ , $B(s) \\ne \\mathrm {I}_d$ , and the preconditioned gradient functions $\\mathsf {h}_i$ 's are explicit (see [15]).", "Our complexity $\\mathcal {K}_A$ reaches this optimal value: in that sense, 3P-SPIDER is optimal since the bound $O(\\sqrt{n} \\epsilon ^{-1})$ is reached despite the introduction of a proximal operator and the approximations of the preconditioned gradient functions $\\mathsf {h}_i$ '.", "To reach this optimal bound, the Monte Carlo complexity is $O(\\sqrt{n} \\epsilon ^{-2})$ ." ], [ "Application: Inference in the Logistic Regression Model", "We illustrate the convergence of 3P-SPIDER applied to inference in the following logistic regression model.", "Given $n$ covariate vectors $\\lbrace X_i, i \\in [n]^\\star \\rbrace $ in $\\mathbb {R}^d$ and $\\theta \\in \\mathbb {R}^d$ , the $\\lbrace -1, 1\\rbrace $ -valued binary observations $\\lbrace Y_i, i \\in [n]^\\star \\rbrace $ are assumed independent with success probability $\\mathbb {P}_\\theta (Y_i = 1)$ equal to $\\frac{1}{\\sigma ^d \\sqrt{2 \\pi }^d}\\int _{\\mathbb {R}^d} \\left(1 + \\exp (- \\left\\langle X_i,z_i\\right\\rangle ) \\right)^{-1} \\,\\exp \\left( - \\frac{\\Vert z_i - \\theta \\Vert ^2}{2 \\sigma ^2}\\right) \\mathrm {d}z_i \\;.$ This model corresponds to a predictor $Z_i$ for each individual $\\# i$ and these predictors $Z_j, j \\in [n]^\\star $ , are independent with distribution $\\mathcal {N}_d(\\theta , \\sigma ^2 \\mathrm {I}_d)$ .", "It is assumed that $\\sigma ^2$ is known; the unknown parameter $\\theta $ is learnt by minimization of the penalized normalized negative log-likelihood $\\theta \\mapsto F(\\theta )$ , with penalty term $\\mathrm {pen}(\\theta ) \\stackrel{\\mathrm {def}}{=}\\tau \\Vert \\theta \\Vert ^2$ for some $\\tau >0$ .", "$F$ is equal to (see [1]) $\\theta \\mapsto -\\frac{1}{n} \\sum _{i=1}^n \\log \\int _\\mathbb {R}\\left( 1 +\\exp \\left(-Y_i \\Vert X_i\\Vert z \\right) \\right)^{-1} \\exp \\left(\\left\\langle \\mathsf {s}_i(z),\\theta \\right\\rangle \\right) \\\\ \\times \\, \\exp ( - z^2/(2\\sigma ^2))\\mathrm {d}z + \\mathsf {R}(\\theta )$ where $\\mathsf {s}_i(z) \\stackrel{\\mathrm {def}}{=}z X_i / (\\Vert X_i\\Vert \\sigma ^2)$ , $\\mathsf {R}(\\theta ) \\stackrel{\\mathrm {def}}{=}(1/2) \\theta ^T \\Omega ^{-1} \\theta $ and $\\Omega \\stackrel{\\mathrm {def}}{=}\\left(\\frac{1}{\\sigma ^2 n} \\sum _{i=1}^n \\frac{X_iX_i^T}{\\Vert X_i\\Vert ^2} + 2 \\tau \\mathrm {I}_d \\right)^{-1}$ .", "The minimization of this criterion by a EM algorithm can be solved equivalently in the expectation space in order to minimize $\\operatorname{W}: s \\mapsto F(\\Omega s)$ (see e.g.", "[20], [1]).", "In that case, EM finds the roots on $\\mathbb {R}^d$ of $s \\mapsto \\mathsf {h}(s) \\stackrel{\\mathrm {def}}{=}\\frac{1}{n} \\sum _{i=1}^n \\int _\\mathbb {R}\\mathsf {s}_i(z)p_i(z;\\Omega s) \\mathrm {d}z -s \\;;$ $z \\mapsto p_i(z;\\theta )$ is the probability density function proportional to $z \\mapsto \\left( 1 + \\exp \\left(-Y_i \\Vert X_i\\Vert z \\right) \\right)^{-1}\\exp \\left( \\left\\langle \\mathsf {s}_i(z),\\theta \\right\\rangle - z^2/(2\\sigma ^2) \\right).$ We have $\\nabla \\operatorname{W}(s) \\stackrel{\\mathrm {def}}{=}- \\Omega \\mathsf {h}(s)$ for any $s \\in \\mathbb {R}^d$ (see [1]).", "Furthermore, upon noting that $\\mathbb {P}_\\theta (Y_i = y_i) \\le 1$ , it can be shown that the minima of $F$ are in the set $\\lbrace \\theta \\in \\mathbb {R}^d: \\tau \\Vert \\theta \\Vert ^2 \\le \\ln 4 \\rbrace $ , which implies that EM in the expectation space will find the roots of $\\mathsf {h}$ in $\\mathcal {K} \\stackrel{\\mathrm {def}}{=}\\lbrace s \\in \\mathbb {R}^d: s^T \\Omega s \\le \\ln 4 /(\\tau \\lambda _{\\min }) \\rbrace $ where $\\lambda _{\\min }$ is the positive minimal eigenvalue of $\\Omega $ .", "Therefore, we set $g$ equal to the characteristic function of $\\mathcal {K}$ ; with such a definition of $\\mathcal {K}$ , the associated weighted proximal is explicit.", "The data set is built from the MNIST data set, as described in [1]: $n = 24 \\, 989$ and $d=51$ .", "SPIDER-EM is run with $\\sigma ^2 = 0.1$ , $\\tau = 1$ , $k_\\mathrm {out}= 20$ , $k_\\mathrm {in}= \\lceil \\sqrt{n}/10 \\rceil =16$ , $\\mathsf {b}= \\lceil 10 \\sqrt{n}\\rceil $ , $\\mathcal {E}_t= 0$ , $\\gamma _{t,0} = 0$ , $\\gamma _{t,k} = 0.1$ and $m_{t,k} = 2 \\lceil \\sqrt{n} \\rceil $ until the outer loop $\\# 9$ and then $m_{t,k} = 10 \\lceil \\sqrt{n} \\rceil $ .", "On Figure REF (a), the 51 components of the sequence $\\lbrace \\widehat{S}_{t,k_\\mathrm {in}}, t \\in [k_\\mathrm {out}]^\\star \\rbrace $ are displayed vs the index of the outer loop $t$ .", "The convergence can be observed.", "On Figure REF (b), we display the quantiles $0.25$ , $0.5$ and $0.75$ of the squared norm $\\Vert \\widehat{S}_{t,k}\\Vert ^2$ as a function of the cumulated number of inner loops; these quantiles are computed over 25 independent runs of 3P-SPIDER.", "Here again, the convergence and the stability of the path over the independent runs can be observed.", "Finally, Figures REF (c,d) display the quantiles $0.25$ and $0.75$ of $ \\widehat{\\Delta }_{t,k} \\stackrel{\\mathrm {def}}{=}\\Vert \\widehat{S}_{t,k} -\\widehat{S}_{t,k-1}\\Vert ^2 / \\gamma _{t,k}^2$ as a function of the cumulated number of inner loops; these quantiles are estimated over 25 independent runs of 3P-SPIDER.", "We observe the gain when increasing the number of Monte Carlo points in order to reduce the fluctuations of the approximations of the $\\mathsf {h}_i$ 's; see [1] for a detailed study of the design parameters of 3P-SPIDER.", "To illustrate the benefit of the variance-reduction step in 3P-SPIDER, we also run Prox-Online-EM with $\\mathsf {b}= \\lceil 10 \\sqrt{n} \\rceil $ and $m_{t,k} = 2 \\lceil \\sqrt{n} \\rceil $ .", "Prox-Online-EM corresponds to Online-EM combined with a proximal step i.e.", "a proximal-preconditioned gradient algorithm.", "The quantiles $0.25$ and $0.75$ of $\\Vert \\widehat{S}_{t} - \\widehat{S}_{t-1}\\Vert ^2 / \\gamma _{t}^2$ are displayed on Figures REF (c,d) as a function of the number of iterations $t$ .", "It illustrates that 3P-SPIDER, as a proximal variance-reduced preconditioned gradient method, clearly improves on Online-EM.", "Figure: [(a) top left] Sequence {S ^ t,k in ,t∈[k out ] ☆ }\\lbrace \\widehat{S}_{t,k_\\mathrm {in}}, t \\in [k_\\mathrm {out}]^\\star \\rbrace [(b) top right] Quantiles of ∥S ^ t,k ∥ 2 \\Vert \\widehat{S}_{t,k}\\Vert ^2[(c) bottom left] Quantile 0.250.25 of Δ ^ t,k \\widehat{\\Delta }_{t,k} [(d)bottom right] Quantile 0.750.75 of Δ ^ t,k \\widehat{\\Delta }_{t,k}" ] ]
2105.11733
[ [ "Inconsistency of the data on the $K_1(1270) \\to \\pi K^*_0(1430)$ decay\n width" ], [ "Abstract We show, using the same Lagrangian for the $K_1(1270) \\to \\pi K^*_0(1430)$ and $K^*_0(1430) \\to K_1(1270) \\pi$ decays, that the present PDG data on the partial decay width of $K_1(1270) \\to \\pi K^*_0(1430)$ implies a width for $K^*_0(1430) \\to K_1(1270) \\pi$ decay which is about ten times larger than the total $K^*_0(1430)$ width.", "A discussion on this inconsistency is done, stressing its relationship to the existence of two $K_1(1270)$ states obtained with the chiral unitary theory, which are not considered in the experimental analyses of $K\\pi\\pi$ data." ], [ "ACKNOWLEDGEMENT", "This work is partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No.", "FIS2017-84038-C2-1-P B and by Generalitat Valenciana under contract PROMETEO/2020/023.", "This project has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No.", "824093 for the “STRONG-2020\" project.", "This work is also partly supported by the National Natural Science Foundation of China under Grants No.", "11975083 and No.", "11947413." ] ]
2105.11768
[ [ "Inhomogeneous Superconducting State Probed by $^{125}$Te NMR on UTe$_2" ], [ "Abstract UTe$_2$ is a recently discovered promising candidate for a spin-triplet superconductor.", "In contrast to conventional spin-singlet superconductivity, spin-triplet superconductivity possesses spin and angular momentum degrees of freedom.", "To detect these degrees of freedom and obtain the solid evidence of spin-triplet superconductivity in UTe$_2$, we performed $^{125}$Te-NMR measurement.", "We previously reported that the shoulder signal appears in NMR spectra below the superconducting (SC) transition temperature $T_{\\rm c}$ in $H \\parallel b$, and that a slight decrease in the Knight shift along the $b$ and $c$ axes ($K_b$ and $K_c$, respectively) below $T_{\\rm c}$ at a low magnetic field $H$.", "To clarify the origin of the shoulder signal and the trace of the decrease in $K_b$, we compared the $^{125}$Te-NMR spectra obtained when $H~\\parallel~b$ and $H~\\parallel~c$ and measured the $^{125}$Te-NMR spectra for $H~\\parallel~b$ up to 14.5~T.", "The intensity of the shoulder signal observed for $H~\\parallel~b$ has a maximum at $\\sim 6$~T and vanishes above 10~T, although the superconductivity is confirmed by the $\\chi_{\\rm AC}$ measurements, which can survive up to 14.5~T (maximum $H$ in the present measurement).", "Moreover, the decrease in $K_b$ in the SC state starts to be small around 7~T and almost zero at 12.5~T.", "This indicates that the SC spin state gradually changes with the application of $H$.", "Meanwhile, in $H~\\parallel~c$, unexpected broadening without the shoulder signals was observed below $T_{\\rm c}$ at 1~T, and this broadening was quickly suppressed with increasing $H$.", "We construct the $H$--$T$ phase diagram for $H~\\parallel~b$ and $H~\\parallel~c$ based on the NMR measurements and discuss possible SC states with the theoretical consideration.", "We suggest that the inhomogeneous SC state characterized by the broadening of the NMR spectrum originates from the spin degrees of freedom." ], [ "Introduction", "Superconductors exhibit two types of spin states in general: spin-singlet ($S$ = 0) and spin-triplet ($S$ = 1).", "For most superconductors, an $s$ -wave pairing with a spin-singlet state and zero momentum results in a superconducting (SC) state with no degrees of freedom.", "In contrast, the spin degrees of freedom remain in the spin-triplet state, and they possibly cause unusual behaviors.", "This includes the magnetic field ($H$ )-boosted superconductivity, multiple SC phases, and rotation of the ${d}$ vector, which is an SC order parameter for the spin-triplet pairing[5], [1].", "However, there are few materials in which the spin-triplet state is considered to be realized.", "In addition, the experimental observations for these phenomena are quite limited.", "Some of the most promising candidates for spin-triplet superconductors are uranium (U)-based ferromagnetic (FM) superconductors UGe$_2$ , URhGe, and UCoGe[2], [3], [4], in which the ferromagnetism coexists with the superconductivity.", "For these superconductors, many experimental and theoretical studies have suggested that spin-triplet superconductivity is induced by the Ising-type FM spin fluctuations[5].", "One of the most convincing results is the $H$ -boosted superconductivity caused by the development of the FM spin fluctuations, which has been observed in the vicinity of the $H$ -induced quantum critical point for the U-based FM superconductors URhGe[6] and UCoGe[7].", "Therefore, research on the spin-triplet pairing state has been conducted for these superconductors.", "Recently, the superconductivity of UTe$_2$ was discovered[8]; since then, UTe$_2$ has attracted significant attention because of its unique SC properties.", "UTe$_2$ has the orthorhombic crystal structure with the space group $Immm$ (#71, $D_{2h}^{25}$ ).", "Its SC upper critical fields ($H_{\\rm c2}$ ) for all three crystalline axes greatly exceed the Pauli limiting field[8], [9]; particularly for $H \\parallel b$ , the SC transition temperature $T_{\\rm c}$ has an anomalous upturn at $\\mu _0 H~=~15$  T, which is similar to the $H$ -boosted behavior[10].", "This $H_{\\rm c2}$ behavior cannot be understood by the scenario in the spin-singlet superconductors, and it is reminiscent of the FM superconductors.", "In addition to $H_{\\rm c2}$ , the Ising-type anisotropy of the static magnetic susceptibility[8], [9], and the presence of the moderate Ising-type fluctuations[11] are similar to those observed in FM superconductors[12], [4], [13], [14], [15], [16], [17].", "However, UTe$_2$ does not show any magnetic order at the ambient pressure[8].", "Therefore, it is suggested that UTe$_2$ is located in the vicinity of the FM quantum critical point and that the spin-triplet superconductivity mediated by the Ising-type FM spin fluctuations is realized in UTe$_2$ .", "Furthermore, multiple SC phases have been reported in the pressure ($P$ )–temperature ($T$ ) phase diagram for UTe$_2$[18], [19], [20].", "Another SC phase is induced by applying pressure, and the evolution to more complex multiple SC phases in $H~\\parallel ~a$ under pressure was reported[21].", "These experimental results strongly suggest that UTe$_2$ is a spin-triplet superconductor with degenerate multiple SC order parameters.", "To investigate the SC properties of UTe$_2$ from a microscopic point of view, we previously performed $^{125}$ Te-NMR measurements at the ambient pressure[22], [23].", "This is because the NMR measurement is sensitive to the changes in the local spin susceptibility at a nuclear site even in the SC state.", "When the ${d}$ vector is pinned to one of the crystalline axis, Knight shift proportional to the spin susceptibility decreases in the SC state in ${H} \\parallel {d}$ , but remains unchanged in ${H} \\perp {d}$ .", "Thus, the magnetic properties of the spin-triplet SC order parameters can be investigated.", "During our previous measurements[23], we focused on the reduction in the spin susceptibility in the SC state and measured the NMR Knight shift along the $b$ and $c$ axes ($K_b$ and $K_c$ , respectively).", "$K_b$ and $K_c$ decrease slightly at 1 T, which indicates that the ${d}$ vector has the $b$ and $c$ components.", "This result limits the SC pairing symmetry in UTe$_2$ to $A_{u}$ or $B_{3u}$ in odd-parity states within the irreducible representations of $D_{2h}$ .", "We also reported the anisotropic response of the Knight shift reduction against $H$ .", "The reduction in $K_b$ is nearly constant of $H$ up to 6.5 T; however, the reduction in $K_c$ decreases with increasing $H$ .", "In addition, $K_c$ does not change even below $T_{\\rm c}$ at 5.5 T. This anisotropic response of the spin susceptibility is considered to be related to the SC pairing symmetry.", "Although we have performed intensive research thus far, the following issues remain unresolved.", "First, the origin of additional NMR signals in the SC state for $H~\\parallel ~b$ is unclear.", "As previously reported[23], in addition to the shift of the main peak, additional shoulder signals appear in the SC-state NMR spectrum in $H~\\parallel ~b$ .", "Although the presence of additional signals suggests the appearance of another ordered state inside the SC phase, we cannot clarify the origin and characteristics from the previous experiments.", "Second, it must be clarified whether ${d}$ -vector rotation occurs with a high $H$ along the $b$ axis.", "The reduction in $K_b$ in the SC state implies that the SC state is unstable with a high $H$ , which is due to the competition between the SC-condensation and the Zeeman energies.", "In such a case, ${d}$ -vector rotation that is related to a change in the SC pairing symmetry is expected at a critical magnetic field $H_{\\rm pin}$ .", "We estimated the $\\mu _0H_{\\rm pin} \\sim $ 13 T from the decrease in $K_b$ , as reported in our previous paper[23].", "To solve these issues, in the present study, we performed precise NMR spectrum measurements up to 6.5 T in $H~\\parallel ~b$ and $H~\\parallel ~c$ , and a high-$H$ NMR measurement up to 14.5 T in $H~\\parallel ~b$ .", "We found that the additional NMR signals appear well below $T_{\\rm c}$ and the area fraction for those at 0.1 K against the main peak reaches a maximum at $\\sim $ 6 T, and it becomes zero above 10 T. Meanwhile, an unexpected large broadening of the NMR spectrum was observed in the SC state for $H~\\parallel ~c$ , and it is quickly suppressed by $H$ .", "Furthermore, the reduction in $K_b$ starts to decrease at $\\sim $ 7 T and it becomes zero at $\\sim $ 12.5 T. This indicates that the $b$ components of the ${d}$ vector disappear above 12.5 T. Considering all these results, we discuss the possible SC states based on the phase diagram." ], [ "Experimentals", "The single-crystal UTe$_2$ , which is the same sample used in our previous study[23], was grown using the chemical transport method with iodine as a transport agent[24].", "Natural uranium and 99.9% $^{125}$ Te-enriched metals were used as the starting materials for the present sample.", "There are two crystallographically inequivalent Te sites, 4$j$ and 4$h$ , with point symmetries $mm$ 2 and $m$ 2$m$ , respectively.", "We denote these sites as Te(1) and Te(2) according to the previous reports[23].", "The measurements were performed using the single crystal with a size of $3.5 \\times 0.7 \\times 1.4$ mm$^3$ .", "$T_{\\rm c}$ was determined by the AC susceptibility that was measured by recording the resonance frequency of the NMR-tank circuit during cooling.", "The frequency-swept $^{125}$ Te ($I$  = 1/2, gyromagnetic ratio $^{125}\\gamma _n/2\\pi $  = 13.454 MHz/T)-NMR spectrum was obtained by the integration of the Fourier transform (FT) for a spin-echo signal that was observed after a radio frequency (RF) pulse sequence with a 3 kHz step for a fixed $H$ .", "The typical RF pulse length is 12 $\\mu $ s. The NMR spectra are plotted against $K~\\equiv ~(f-f_0)/f_0$ .", "Here, $f$ is the NMR frequency, and $f_0$ is the reference frequency determined as $f_0~=~(\\gamma _n/2 \\pi )\\mu _0H$ .", "The magnetic field was calibrated using $^{63}$ Cu ($^{63}\\gamma _n/2\\pi $ =11.285 MHz/T) and $^{65}$ Cu ($^{65}\\gamma _n/2\\pi $ =12.089 MHz/T) NMR signals from the NMR coil.", "In the present measurement, the errors of the NMR intensity and the linewidth are determined from the background noise amplitude and the resolution of FT signals, respectively.", "All the NMR spectra in the SC state were recorded with the field-cooling process.", "Low-temperature NMR measurements down to 80 mK were performed using a $^3$ He-$^4$ He dilution refrigerator, and the single-crystalline sample was immersed into the mixture to reduce the heating effect.", "In the low-$T$ NMR measurements, we reduced the power of the RF-pulses for the observation of the spin-echo signal as small as possible.", "Due to this effort, we detected the broadening of the NMR spectra below $T_{\\rm c}$ originating from SC diamagnetism in all measurements.", "We performed the measurements using two setups.", "In the first setup (Setup A), we used the SC split magnet, which generates a horizontal field up to 6.5 T with a single-axis rotator to apply $H$ exactly parallel to the $b$ axis or $c$ axis with the $a$ axis as the rotation axis.", "This is the same setup as in the previous measurements[23].", "The sample alignment and angle dependence of the NMR spectrum are reported in the supplemental materials in the previous report[23].", "In the second setup (Setup B), we used the SC solenoid magnet, which generates a vertical field up to 14.5 T to apply a higher $H$ along the $b$ axis.", "In Setup B, because the sample cannot be rotated in $H$ , a misalignment with $H$ occurred, which is evaluated to be $3 \\sim 13$ degrees tilted from the $b$ axis.", "The evaluation of the misalignment is discussed in a later section." ], [ "Setup A: measurements with a rotator ", "First, we summarized the experimental results in Setup A.", "Figure REF (a) shows the $^{125}$ Te-NMR spectrum at several temperatures measured at 5.5 T in $H~\\parallel ~b$ .", "The peak shown by the arrows shifts to a lower $K$ direction below $T_{\\rm c}$ .", "This indicates a decrease in the spin susceptibility by forming Cooper pairs.", "In addition, we found that additional signals appear at low temperatures.", "Hereafter, we refer to the peak of the NMR spectrum and the additional signal that is observed at low temperatures as the “main peak” and “shoulder signal\", respectively.", "To investigate the origin of the shoulder signal appearing in the SC state upon cooling, the temperature dependence of the NMR intensity at $K~=~$ 5.25% [shown by the dotted line in Fig.", "REF (a)], which is normalized by the main peak intensity, is plotted in Fig.", "REF (b).", "We can define the characteristic temperature $T_x$ , at which the intensity at $K~=~$ 5.25% starts to increase.", "$T_x$ shown by the solid arrows in Fig.", "REF (b) is $\\sim 1$  K and is almost independent of $H$ in the measured $H$ region, which is different from $T_{\\rm c}$ .", "These results indicate the presence of another phase below $T_x$ .", "The difference between $T_{\\rm c}$ and $T_x$ is $\\sim $  0.6 K at 1 T, which is significantly larger than the separation in the double transition that is observed by a specific heat at zero fields ($\\sim $  0.2 K)[20], [25].", "It is noted that the shoulder signal has a characteristic structure.", "There are two peaks at the lower $K$ side and a broad peak that tails to a higher $K$ than that of the main peak, as shown in Fig.", "REF (c).", "This indicates that the spin susceptibility is inhomogeneously distributed in the low-$T$ phase.", "The area fraction of the shoulder signal at 0.1 K against the main peak increases with increasing $H$ up to 5.5 T, although the main peak still remains.", "Since we require higher-$H$ NMR measurements than 6.5 T to clarify how the area fraction of the shoulder signal changes against $H$ , the measurement was performed using Setup B as described in section 3.2.", "We also performed $^{125}$ Te-NMR measurements in $H~\\parallel ~c$ by Setup A.", "Figure REF (a) shows the $^{125}$ Te-NMR spectrum against $K$ at several temperatures measured at 1 T in $H~\\parallel ~c$ .", "Although a single symmetric peak was observed in the normal state, this peak consists of two NMR signals arising from the Te(1) and Te(2) sites because of the closeness of the Knight shift at the two sites when $H$ is parallel to the $c$ axis[11], [23].", "The symmetric NMR spectrum becomes broader in the SC state without showing any shoulder peak structure observed in $H~\\parallel ~b$ , and it slightly shifts to a lower $K$ direction, as we previously reported [23].", "The temperature dependence of the full width at the half maximum (FWHM) of the NMR spectrum, from which the normal-state value is subtracted ($\\Delta $ FWHM), in the frequency units at 1, 3, and 5.5 T are shown in Fig.", "REF (b).", "Although the $\\Delta $ FWHM for all the $H$ values increases below $T_{\\rm c}$ , the $\\Delta $ FWHM at the lowest $T$ becomes smaller with increasing $H$ , as shown in the inset of Fig.", "REF (b).", "It is noted that the broadening at the lowest $T$ is one order of magnitude larger than the broadening expected from the conventional SC diamagnetic effect.", "The broadening of the NMR spectrum by the conventional SC diamagnetic effect is of the order of the SC lower critical field $\\mu _0H_{\\rm c1} \\sim 1.2$  mT[23].", "This gives the upper limit of the $^{125}$ Te-NMR spectrum broadening $\\sim $ 16 kHz.", "Therefore, it is concluded that the broadening of the $^{125}$ Te-NMR in $H~\\parallel ~c$ is not simply due to the conventional SC diamagnetic effect.", "Instead, it originates from the unconventional SC properties in UTe$_2$ .", "The unexpected large broadening of the NMR spectrum in the SC state suggests that the spin susceptibility is inhomogeneously distributed also in $H \\parallel c$ .", "It is also noteworthy that the magnetic field dependence of the broadening of the NMR spectrum is different between $H~\\parallel ~b$ and $H~\\parallel ~c$ .", "The NMR spectrum at the lowest temperature, which is shown against $K$ , quickly becomes narrower with an increase in $H$ when $H~\\parallel ~c$ , as shown in the inset of Fig.", "REF (b) and Fig.", "REF (c).", "Although the magnetic field at which the broadening behavior disappears is ambiguous in $H \\parallel c$ , it is roughly estimated as $\\sim $  3 T above which the $\\Delta K$ is almost independent of $H$ as shown in the inset of Fig.", "REF (b).", "Here, $\\Delta K$ is the $\\Delta $ FWHM at the lowest temperature divided by the measured frequency.", "In contrast, the broadening behavior becomes more obvious with an increase in $H$ when $H~\\parallel ~b$ , as shown in Fig.", "REF (c).", "This indicates that the spin-susceptibility distributed signal, observed in the low fields, is suppressed in $H \\parallel c$ , but it seems to be enhanced in $H \\parallel b$ .", "Figure: (Color online) (a) TT variation of 125 ^{125}Te-NMR spectrum against KK measured at 1 T along the cc axis.The dashed line denotes the Knight shift in the normal state.", "(b) Temperature dependence of the FWHM for the NMR spectrum from which the normal-state value is subtracted (Δ\\Delta FWHM) at μ 0 H=1,3,\\mu _0H = 1, 3, and 5.5 T.The arrows show T c T_{\\rm c}s.The inset shows the magnetic field dependence of the Δ\\Delta FWHM at the lowest temperature divided by the measured frequency (ΔK\\Delta K).", "(c) 125 ^{125}Te-NMR spectrum at the lowest temperature at μ 0 H=1,3,\\mu _0H =1, 3, and 5.5 T along the cc axis.For comparison purposes, the normal-state NMR spectrum at 5.5 T is also shown." ], [ "Setup B: high-$H$ measurements in {{formula:af61c8cc-3c6f-4aa2-971f-31118f00d9d3}} ", "To investigate the origin of the shoulder signal and the SC properties for a high $H$ , we performed the NMR measurements using Setup B.", "As described in Section 2, there is a small sample misalignment in Setup B because the field direction cannot be controlled.", "Figure REF (a) shows the $^{125}$ Te-NMR spectrum in Setup B along with those in Setup A at several field angles from the $b$ to $c$ axis at 1 T and 1.65 K. The NMR spectrum of Setup B almost overlaps with the NMR spectrum at 12$^\\circ $ in Setup A.", "From the comparison between the Knight shift in Setup B and the angular dependence of the Knight shift in Setup A, we estimated that the tilted angle is 12.5$^\\circ $ from the $b$ axis to the $c$ axis when we assume that the sample misalignment originates from only the $b$ –$c$ rotation as shown in the inset of Fig.REF  (a).", "We can also estimate that the tilted angle is $\\sim $  3$^\\circ $ from the $b$ axis to the $a$ axis because there is a possibility of sample misalignment with the $b$ –$a$ rotation.", "Here, we used the Knight shift along the $a$ axis $K_a$  $\\sim $  40% that was obtained by extrapolating the temperature dependence of $K_a$ from 20 K to 1.6 K. It has been reported that the NMR signal for $H~\\parallel ~a$ disappears below 20 K due to the divergence of $1/T_2$[11]; thus, $K_a$ at low temperatures could not be measured.", "To check the effect of the misalignment on the superconductivity, we measured the magnetic field dependence of $T_{\\rm c}$ in Setup B from the $T$ -scan of $\\chi _{\\rm AC}$ , as shown in the inset of Fig.", "REF (b).", "$T_{\\rm c}$ was determined by the intersection point of the slope of the SC diamagnetism and the normal state.", "The obtained magnetic field dependence of $T_{\\rm c}$ is plotted in Fig.", "REF (b) with that measured in Setup A, where $H$ is exactly parallel to the $b$ axis.", "Although $T_{\\rm c} (H)$ is slightly reduced from that in $H~\\parallel ~b$ , the shape of the phase diagram is almost the same up to 14.5 T. This indicates that the effect of the misalignment is small for the superconductivity.", "Figure: (Color online) (a) 125 ^{125}Te-NMR spectrum in Setup B together with those in Setup A at several field angles from the bb to cc axis at 1 T and 1.65 K.(Inset)Angular dependence of the Knight shift in Setup A (squares) together with the Knight shift in Setup B (dashed line).", "(b)The HH–TT phase diagram for H∥bH \\parallel b in Setup A and for H∥∼bH \\parallel \\sim b in Setup B.The broken curves represent H c2 H_{\\rm c2} from the reference.", "(Inset)Temperature dependence of χ AC \\chi _{\\rm AC} at several HH in Setup B.T c T_{\\rm c}s, which are represented by the arrows, were determined by the intersection point of the slopes of the SC diamagnetism and the normal state.Next, we show the magnetic field dependence of the magnetic properties in the normal state.", "Figure REF (a) shows the $^{125}$ Te-NMR spectrum at several $H$ values in the normal state.", "A peak in the NMR spectrum gradually shifts to a higher $K$ direction with increasing $H$ .", "The Knight shift in the normal state, $K_{\\rm normal}$ is almost independent of $T$ below 2 K, as shown in the inset of Fig.", "REF (a).", "Because $K_{\\rm normal}$ is related to the Pauli paramagnetism, the increase in $K_{\\rm normal}$ against $H$ can be interpreted as an increase in the density of the states, which is consistent with the previous bulk measurements[26], [27].", "The magnetic field dependence of $K_{\\rm normal}$ and the Sommerfeld coefficient $\\gamma $[26] is presented in Fig.", "REF (b).", "The observed increase in $K_{\\rm normal}$ was scaled with $\\gamma $ .", "Figure: (Color online) (a) 125 ^{125}Te-NMR spectrum at several HH in the normal state.The dashed line is a guide for the eye.", "(Inset)Temperature dependence of K normal K_{\\rm normal} at 14 T.(b) Magnetic field dependence of K normal K_{\\rm normal} and the Sommerfeld coefficient γ\\gamma .The shoulder signal in the SC state displays peculiar behavior.", "Figure REF (a) shows the $^{125}$ Te-NMR spectrum measured at $\\sim 0.1$  K up to 5.5 T in Setup B.", "As previously mentioned and illustrated later, the Knight shift in the normal and SC states depend on $H$ .", "Thus, to simply compare the NMR line shape, in which the NMR spectra are normalized with the maximum peak, it is plotted against $K-K_{\\rm peak}$ , where $K_{\\rm peak}$ is the Knight shift at the maximum peak of the NMR spectrum.", "Although the peaks were located at slightly different positions due to the misalignment from the $b$ axis, the shoulder signal observed in Setup A was also observed in Setup B.", "Therefore, we could investigate the $H$ evolution of the shoulder signal in the high $H$ region.", "Figure: (Color online) 125 ^{125}Te-NMR spectrum at ∼\\sim 0.1 K in (a) μ 0 H≤\\mu _0 H \\le 5.5 T and (b) μ 0 H≥5.5\\mu _0 H \\ge 5.5 T in Setup B.The dashed line denotes the position of K-K peak =-0.2K - K_{\\rm peak}~=~-0.2%.", "(Inset) Magnetic field dependence of the NMR intensity at K-K peak =-0.2K - K_{\\rm peak}~=~-0.2% measured at 0.1 K.Figure REF (b) shows the $^{125}$ Te-NMR spectrum that was measured at $\\sim 0.1$  K above 5.5 T. This is plotted in the same manner as Fig.", "REF (a).", "As illustrated, the intensity of the shoulder signal decreases with increasing $H$ , and it almost vanishes at approximately 10 T. We plotted the magnetic field dependence of the intensity at $K-K_{\\rm peak}~=~-0.2$ %, as shown by the dotted line in Fig.", "REF (b) at $\\sim 0.1$  K in the inset of Fig.", "REF (b).", "From the figure, we can determine the critical field $\\mu _0 H_x$ where the shoulder signal disappears to be 10 T. Figure: (Color online) Magnetic field dependence of the change in the spin component of the Knight shift in the SC state, ΔK spin \\Delta K_{\\rm spin} in Setup A and B.The dotted line indicates ΔK spin \\Delta K_{\\rm spin} = 0.The dashed lines are a guide for the eye.", "(Inset) 125 ^{125}Te-NMR spectrum at ∼0.1\\sim 0.1 K (SC state) measured at 8, 10.5, and 12.5 T in Setup B.For comparison purposes, the normal-state NMR spectrum is also shown.In addition to $H_{x}$ , we also investigated the critical field at which the decrease in $K_b$ becomes zero.", "In the inset of Fig.", "REF , the NMR spectrum at $\\sim 0.1$  K (SC state) is compared with the normal-state NMR spectrum at 8, 10.5, and 12.5 T. While the NMR spectrum at 8 T and 10.5 T shifts to a lower $K$ direction in the SC state, no spectrum shift was observed between the normal and SC states at 12.5 T. For the quantitative discussion, we evaluated the change in the spin component of the Knight shift in the SC state, $\\Delta K_{\\rm spin}$ , which is described as $\\Delta K_{\\rm spin} = K(T \\sim 0.1 \\, {\\rm K}) - K_{\\rm normal} - \\Delta K_{\\rm dia},$ and we plotted the magnetic field dependence of $\\Delta K_{\\rm spin}$ in Fig.", "REF .", "Here, $K(T)$ is the Knight shift at $T$ , and $\\Delta K_{\\rm dia}$ is the contribution from the conventional SC diamagnetism.", "$\\Delta K_{\\rm dia}$ was estimated from the formula suggested by de Gennes[28], which is expressed as $\\Delta K_{{\\rm dia}} = - \\frac{H_{\\rm c1}}{H} \\frac{\\ln \\left( \\frac{\\beta d}{\\sqrt{e} \\xi } \\right) }{\\ln {\\kappa }}.$ As shown in Fig.", "REF , the obtained $\\Delta K_{\\rm spin}$ is almost independent of $H$ up to 7 T, which is consistent with the previous results[23].", "Meanwhile, the absolute value of $\\Delta K_{\\rm spin}$ decreases gradually above 7 T and becomes almost zero above $\\sim $ 12.5 T. We comment that the intensity of the shoulder signal starts to decrease at around 7 T, indicating that the SC state might be gradually changed from this field.", "It is noted that the superconductivity survives at least up to 14.5 T because the SC transition was confirmed by the $\\chi _{\\rm AC}$ measurement and the broadening of the NMR spectrum.", "The zero value of $\\Delta K_{\\rm spin}$ implies that the ${d}$ vector has no $b$ component above 12.5 T." ], [ "Discussion", "Here, we discuss how the ${d}$ vector changes against $H$ along the $b$ axis.", "In a previous report[23], we pointed out that the low-$H$ SC state is unstable; moreover, it should change in the high-$H$ region in $H~\\parallel ~b$ because $K_b$ decreases in the SC state, and the competition between the SC-condensation and Zeeman energies is expected.", "We estimated the critical magnetic field $\\mu _0H_{\\rm pin}$ to be $\\sim $ 13 T from the decrease in $K_b$ , which is consistent with the observed critical field in which the decrease in $K_b$ becomes zero.", "The observed magnetic field dependence of $\\Delta K_{\\rm spin}$ is gradual, not drastic.", "One possibility is that the SC pairing symmetry below and above 12.5 T is the same representation in $H~\\parallel b$ .", "Therefore, the change in the SC state is a crossover rather than a first-order phase transition, which gives strong constraints for the SC pairing symmetry.", "Next, we discuss the low-$T$ SC state, in which the sharp signal arising from the high-$T$ SC phase and the broad spin-susceptibility-distributed signal (the shoulder signal in $H \\parallel b$ and the unexpected broad signal in $H \\parallel c$ ) are observed.", "One of the characteristic features of the low-$T$ SC state is that the broad signal exists only below $T_c$ ($H$ ), indicating that this signal is closely related to the superconductivity.", "The other is the anisotropic field response of the spectrum broadening.", "The NMR spectrum at the lowest temperature quickly becomes narrower with an increase in $H$ when $H~\\parallel ~c$ , although the broadening behavior becomes more obvious with an increase in $H$ and disappears at 10 T when $H~\\parallel ~b$ .", "These results exclude the possibility of the magnetic origin for the spectrum broadening.", "Therefore, one possible scenario to interpret the origin of the broad signal is that other SC states with different SC pairing symmetries from that in the high-$T$ SC phase are realized in the specific regions of the sample.", "($i.e.$ the broad signal arises from the low-$T$ SC phase.)", "Since the NMR spectrum is extremely broad, a spatially distributed order parameter or multi-component order parameter generating the internal field are expected for the low-$T$ SC phase.", "As only the high-$T$ SC phase exists between $T_c$ and $T_x$ , the low-$T$ SC phase is expected to grow to replace a part of high-$T$ SC phase.", "In general, such coexistence is induced by disorders or impurities.", "However, the spectrum in the normal state is very sharp as shown in Figs.", "REF (a) and (c), and there are no indications of the presence of impurities or disorders.", "Therefore, it is considered that the high-$T$ and low-$T$ SC phases are highly degenerated, and the ground state can change with a slight difference in the local situation.", "Figure: (Color online) Summarized HH–TT phase diagram for (a) H∥bH~\\parallel ~b and (b) H∥cH~\\parallel ~c.The diamonds, triangles and circles represent H c2 H_{\\rm c2} determined by χ AC \\chi _{\\rm AC} in this study.The broken curves represent H c2 H_{\\rm c2} from reference.The stars represent T x T_{x} and H x H_{x} as described in the text.The SC phase can be distinguished into four regions based on the NMR results.Region boundaries are indicated by dotted lines.Each region in the phase diagrams is numbered as I∼\\sim IV.We now summarize the $H$ –$T$ phase diagram for $H~\\parallel ~b$ and $H~\\parallel ~c$ in UTe$_2$ in Figs.", "REF (a) and (b), respectively.", "Based on the present results, we can distinguish the SC phase into four regions, and number each region in the phase diagrams as I$\\sim $ IV.", "In region I, only the main peak was observed, and $K_b$ and $K_c$ decrease in the SC state.", "This corresponds to the high-$T$ SC phase and indicates the single-phase SC state with the nonzero $b$ and $c$ components in the ${d}$ vector (${d_b}$ and ${d_c}$ , respectively).", "Considering the $D_{2h}$ point group symmetry, the $A_{u}$ or $B_{3u}$ state is a candidate for SC pairing symmetry, which is consistent with the theoretical suggestion[29], [30], [31].", "In region II, the shoulder signal appears in addition to the main peak in $H~\\parallel ~b$ .", "The region II corresponds to the low-$T$ SC state where the high-$T$ SC phase and the low-$T$ SC phase coexist as discussed above.", "The determination of the boundary of the region II for $H \\parallel c$ is ambiguous.", "We consider that region II in $H \\parallel c$ appears also below 1 K from the constraint that the transition between region I and region II occurs at the same temperature at the zero field.", "In region III, $K_b$ does not decrease in the SC state.", "This indicates that the SC state with ${d} \\perp {b}$ is realized.", "Meanwhile, $K_c$ does not decrease in the SC state in region IV, indicating that the SC state with ${d} \\perp {c}$ .", "Because the change in $\\Delta K_{\\rm spin}$ against $H$ is gradual in $H \\parallel b$ and $H \\parallel c$ , it seems that the transitions between region I and III and between region I and IV are crossovers.", "Since no phase transitions were detected between the different regions with other experimental techniques, a further experimental confirmation is desired.", "Finally, we discuss a possible SC pairing symmetry in each region that is based on the present experimental results and the theoretical considerations.", "The Table REF represents the character tables for point groups (a) $D_{2h}$ , (b) $C^b_{2h}$ , and (c) $C^c_{2h}$[29].", "Here, the $D_{2h}$ symmetry is reduced to $C_{2h}^{b}$ in $H~\\parallel ~b$ or $C_{2h}^{c}$ in $H~\\parallel ~c$ .", "The SC pairing symmetry is classified into the one of the irreducible representations listed on the tables, and the ${d}$ vector can be represented by the corresponding basis function.", "Table: Character tables for point groups (a) D 2h D_{2h}, (b) C 2h b C^b_{2h}, and (c) C 2h c C^c_{2h}.", "Only odd-parity irreducible representations (IR) are listed.The notation x,y,zx, y, z in the reference correspond to a,b,ca, b, c in the table.As mentioned above, the plausible SC pairing symmetry in region I is $A_u$ or $B_{3u}$ .", "In region III(IV), the $K_b$ ($K_c$ ) does not decrease and thus, the $B_{2u}(B_{1u})$ state without $d_b$ ($d_c$ ) component is expected.", "From the crossover behavior of the boundary between region I and region III(IV), the SC pairing symmetry of region I should be the same irreducible representation as that of region III(IV) under $H \\parallel b$ ($H \\parallel c$ ).", "As shown in Table REF , the $A_u$ state which is in the same representation as $B_{2u}$ state in $C_{2h}^b$ and as $B_{1u}$ state in $C_{2h}^c$ is more suitable than the $B_{3u}$ state for the SC pairing symmetry of region I.", "To determine the SC pairing symmetry at region I, the Knight shift measurement in $H \\parallel a$ is crucial.", "A recent theoretical calculation that is based on a periodic Anderson model reveals that the $A_{u}$ and $B_{3u}$ states are almost degenerate and the dominant component of the ${d}$ -vector for the $A_{u}$ ($B_{3u}$ ) state is ${d_b}$ (${d_c}$ )[30].", "This is consistent with our experimental results.", "According to the calculation, it seems that the degeneracy of the $A_{u}$ and $B_{3u}$ states results in the coexistence of the high-$T$ and low-$T$ SC phases expected in region II.", "In region II, the broad signal is quickly suppressed against $H$ along the $c$ -axis, indicating that the low-$T$ SC phase includes $B_{3u}$ component with a large ${d_c}$ .", "We suggest that the $\\alpha A_u + \\beta B_{3u}$ state, whose complex coefficients $\\alpha , \\beta $ are changed depending on the local situations, can be consistent with the characteristic NMR spectrum observed in region II.", "For multi-component SC state, the time-reversal symmetry-breaking SC state originating from a chiral or a nonunitary state is expected.", "Although our measurements were performed under fields, our results are possibly related to the recent results with the polar Kerr effect[25] and the scanning tunneling microscopy measurements[32].", "We also point out that the spin-susceptibility-distributed signal in region II can be also explained by the properties of the superconductivity with the spin degree of freedom.", "It might be possible that the texture of ${d}$ -vector induces such a spatially modulated SC state in spin-triplet superconductors." ], [ "Conclusion", "We performed $^{125}$ Te-NMR measurements on a single-crystal UTe$_2$ to investigate the SC properties from a microscopic point of view.", "A shoulder signal appeared inside the SC phase when $H \\parallel b$ .", "The area fraction of the shoulder signal shows a maximum at $\\sim 6$  T and vanishes above 10 T, although the superconductivity confirmed by the $\\chi _{\\rm AC}$ measurements survives to at least 14.5 T. In contrast, an unexpected broadening without the shoulder signal was observed when $H~\\parallel ~c$ at a temperature below $T_{\\rm c}$ .", "This broadening is quickly suppressed by increasing $H$ .", "In addition, the decrease in $K_b$ in the SC state starts to be small at around 7 T and almost zero at 12.5 T. This indicates that the SC spin state gradually changes with the application of $H$ .", "Based on the experimental results, we constructed phase diagrams for both field directions.", "The SC phase can be distinguished into four regions based on the character of the SC spin state.", "In particular, an inhomogeneous low-$T$ SC state, where the high-$T$ and low-$T$ SC phases coexist, is expected to be realized in the low-$T$ and low-$H$ regions.", "The low-$T$ SC phase originates from the presence of the spin degrees of freedom.", "This finding places strong constraints on the SC pairing symmetry of UTe$_2$ , and it also opens up new possibilities for the spin-triplet SC states.", "The authors would like to thank M. Manago, J. Ishizuka, Y. Yanase, S. Fujimoto, Y. Maeno, and S. Yonezawa for their valuable discussions.", "This work was supported by the Kyoto University LTM Center, Grants-in-Aid for Scientific Research (Grant Nos.", "JP15H05745, JP17K14339, JP19K03726, JP16KK0106, JP19K14657, JP19H04696, JP20H00130, and JP20KK0061, and a Grant-in-Aid for JSPS Research Fellow (Grant No.", "JP20J11939) from JSPS." ] ]
2105.11823
[ [ "PAS-MEF: Multi-exposure image fusion based on principal component\n analysis, adaptive well-exposedness and saliency map" ], [ "Abstract High dynamic range (HDR) imaging enables to immortalize natural scenes similar to the way that they are perceived by human observers.", "With regular low dynamic range (LDR) capture/display devices, significant details may not be preserved in images due to the huge dynamic range of natural scenes.", "To minimize the information loss and produce high quality HDR-like images for LDR screens, this study proposes an efficient multi-exposure fusion (MEF) approach with a simple yet effective weight extraction method relying on principal component analysis, adaptive well-exposedness and saliency maps.", "These weight maps are later refined through a guided filter and the fusion is carried out by employing a pyramidal decomposition.", "Experimental comparisons with existing techniques demonstrate that the proposed method produces very strong statistical and visual results." ], [ "Introduction", "High dynamic range (HDR) technology aims at producing high quality images similar to human perception.", "However, the dynamic range gap between high contrast scenes and low dynamic range (LDR) capture/display devices causes information loss in highlights and shadows [1].", "In order to minimize distortions and detail loss, i.e., to capture and display high quality images, there are three main approaches: (i) using HDR compatible capture and display devices, (ii) employing tone-mapping operators to map HDR onto LDR displays, and (iii) using multi-exposure LDR fusion (MEF) to create HDR-like content for LDR screens [2].", "The user-grade technology manufacturers generally prefer to use MEF to obtain high quality LDR images, since MEF has significantly lower cost than hardware-based solutions; and with MEF, it is also possible to avoid tone-mapping related problems such as low-subjective contrast and color saturation [3], [4].", "MEF mainly aims at keeping the most informative parts of each exposure image via extracting weight maps, and then it blends them into a single HDR-like image [1].", "There are several MEF studies present in the literature.", "In the milestone study of Mertens et al.", "[1], a weight map extraction scheme is proposed which is based on contrast, saturation and well-exposedness.", "The fusion of the input stack is carried out by taking the Gaussian pyramid of weight maps and the Laplacian pyramid of exposures, which is inspired from Burt and Kolczynski [5].", "In a recent study of Lee et al.", "[6], a weight map is formed by employing an adapted version of well-exposedness of Mertens and a second map is characterized via the gradient information in each exposure.", "Finally, a pyramidal image decomposition is employed to carry out the fusion process.", "In Li and Zhang [7] (Li18), convolutional neural networks (CNNs) are employed for MEF.", "The first layer of a pretrained classification network is used for feature extraction.", "Then, local visibility and temporal consistency maps are extracted and adopted for the weighted fusion operation.", "In Liu and Leung [8], a method is proposed for both MEF and decolorization.", "In this study, the local gradient information of each exposure is extracted and provided to a CNNs model.", "The proposed algorithm produces satisfying results, however it can operate on stacks consisting of three exposures only.", "In Hayat and Imran [9], the weight map characterization scheme of Li and Kang [10] is modified with the dense-SIFT descriptor, and guided filtering is used to eliminate discontinuities and noise in these maps.", "Finally, image fusion is conducted via a pyramid decomposition approach.", "In the study of Li et al.", "[11] (Li20), the method of Ma et al.", "[12] is investigated and improved by forming adopted weight maps via signal strength, signal structure and mean intensities.", "Recursive downsampling and processing are included into the model and halo effects are significantly reduced.", "In a recent study of Ulucan et al.", "[2], a MEF algorithm is developed which is based on linear embeddings of images and watershed masking.", "Weights maps are first characterized via linear embeddings of exposure image patch spaces, while preserving local geometry of the sampled manifold structure.", "These weight maps are then refined via watershed masking to highlight most informative parts of each exposure in the input stack.", "Lastly, the fusion process is performed through weighted averaging.", "As it can be deduced from above studies, each specific algorithm generally differs in the way of extraction and/or characterization of weight maps.", "Therefore, new weight map extraction methods will enlighten the path leading to a general map formation framework.", "To this end, in this paper, a novel MEF method is proposed to fuse static exposure stacks.", "The weight map extraction algorithm relies on principal component analysis (PCA), adaptive well-exposedness and saliency map features.", "These maps are later refined by a guided filter, and then exposure images are fused via a pyramidal decomposition.", "The proposed method is compared with well-known MEF algorithms and it demonstrates very strong outputs both statistically and visually.", "To the best of available knowledge, this is the first time that PCA is used for extracting weights in MEF.", "Furthermore, this is the first study that modifies the well-exposedness (i.e., indicating brightness) feature to be fully adaptive, while existing MEF algorithms employ a fixed parameter and/or constant for well-exposedness.", "Finally, it is worth mentioning here that saliency maps are used in this study to mimic the human visual system (HVS) and give larger weights to the “best” parts of images as in the primary visual cortex.", "This paper is organized as follows.", "The proposed MEF technique is detailed in Sec. .", "Experimental results are presented and discussed in Sec. .", "Lastly, a brief summary and possible future directions for this study are given in Sec.", "." ], [ "The Proposed MEF Method: PAS-MEF", "A simple flowchart of the proposed method is given in Fig.", "REF .", "Given the input exposures, there are three branches for extracting PCA, adaptive well-exposedness and saliency maps.", "These maps are later combined to obtain final fusion weights in order to output a fused image via a pyramidal decomposition.", "Figure: A flowchart of the proposed MEF method.Weight extraction via PCA.", "PCA performs an orthogonal transformation to obtain linearly uncorrelated variables from possibly correlated data [13].", "The correlated data is projected onto the PCA space by taking advantage of the eigenvectors of the covariance matrix.", "The first principal component is in the direction of the highest variance (first eigenvector), while the second principal component lies in the subspace perpendicular to the first one and the next principal components are computed similarly.", "The representations of data in the PCA space can be called as scores.", "PCA has already been used in image fusion but, to the best of available knowledge, it has not been employed in MEF studies [14].", "Therefore, it is investigated in this study first by vectorizing $N$ number of gray-scale versions of exposure images $I_n, n=1\\dots N,$ into column vectors of the size $rc \\times 1$ where $r$ and $c$ represent the number of rows and columns of each image, respectively.", "The obtained $N$ column vectors are then stacked into the columns of an $rc \\times N$ data matrix, in which there are $rc$ observations with $N$ variables each, for calculating the scores of observations via PCA.", "Subsequently, each variable-score vector is linearly normalized to have a range between $[0~1]$ , then reshaped back to an $r\\times c$ matrix followed by a simple smoothing Gaussian filter.", "Finally, $N$ number of PCA weight maps ($P_n$ ) are obtained with a sum-to-one normalization applied at each spatial position.", "As an example, the extracted PCA maps of the Venice stack are demonstrated in Fig.", "REF .", "Weight extraction via adaptive well-exposedness.", "The well-exposedness feature is initially introduced by Mertens.", "For a given exposure image $I_n$ , it is extracted for each red-green-blue channel separately using a Gaussian curve as $exp\\left(-( I_n-0.5)^2/2\\sigma ^2\\right)$ where $\\sigma = 0.2$ .", "This weight aims at keeping pixel intensities which are not too close to 0 (under-exposed) or 1 (over-exposed), hence it favors pixels in well-exposed regions with intensity values close to $0.5$ .", "However, this exposedness feature sometimes can not sufficiently preserve bright regions of short-exposure images, as well as dark regions of long-exposure images [6].", "To overcome this problem, Lee proposed an adaptive well-exposedness based on the mean of pixel intensities of exposure images.", "In this adaptive form, the constant values of $0.5$ and $\\sigma $ of Mertens are replaced with some functions of the mean of pixel intensities of $I_n$ and its neighboring exposure images $I_{n-1}$ and $I_{n+1}$ .", "However, there still exists a fixed constant parameter to calculate the $\\sigma $ value.", "In this study, a fully adaptive weight calculation is proposed for the well-exposedness scheme in order to allocate large weights for dark regions when the image is a long-exposure, as well as for bright regions when the image is a short-exposure.", "All computations are carried out on the luminance channel $Y_n$ of $I_n$ as given in Eqn.", "(REF ), $\\footnotesize \\centering A_n = exp \\left( -\\frac{\\left( Y_n-(1-\\mu _{Y_n}) \\right)^2}{2\\sigma _{Y_n}^2} \\right)$ where $\\mu _{Y_n}$ and $\\sigma _{Y_n}$ represent the mean of pixel intensities and the standard deviation of $Y_n$ , respectively.", "This finally leads to an adaptive algorithm since the Gaussian curve controlling parameters are extracted via self statistical information of each single exposure, and larger weights are given to the “best” luminance intensities of the input.", "The obtained adaptive well-exposedness weights ($A_n$ ) of the Venice stack are illustrated in Fig.", "REF .", "Figure: Weights for Venice.", "(Top-to-bottom) PCA maps, adaptive well-exposedness weights, saliency maps, final fusion weights.Weight extraction via saliency map.", "When processed in the HVS, the attention that objects gather depends on the task at hand and stimulus-driven factors such as prominent colors [15].", "In the literature, several computational models are proposed to mimic the HVS; thus to highlight salient regions, increase the visual appeal and quality of images.", "In this study, saliency maps are used for assigning larger weights to regions which are more attractive to human observers.", "Since the design of a salient region detection algorithm is out of context, the technique introduced by Hou et al.", "[16] is integrated into the proposed MEF method.", "This saliency algorithm is based on a descriptor called image signature, which is the sign of the Discrete Cosine Transform (DCT) coefficients.", "Briefly, the DCT is first computed for each red-green-blue channel separately.", "Then, image reconstruction is carried out by calculating the inverse DCT of the sign of the DCT coefficients.", "The saliency maps ($S_n$ ) are finally obtained from the reconstructed image.", "For detailed information, the reader may refer to [16].", "The saliency maps of the Venice stack are shown in Fig.", "REF .", "Weight refinement and fusion.", "After all three weight maps are characterized, they are combined to form a single refined map for each exposure in the input stack given in Eqn.", "(REF ) as follows, $W_n = \\texttt {GuidFilt}(P_n \\times A_n \\times S_n),~n=1\\dots N,$ where GuidFilt is an edge-aware (edge-preserving) smoothing filter called guided filter [17], which is generally used to eliminate possible discontinuities and noise in weight maps, e.g., [9].", "All these maps $W_n$ are finally normalized to satisfy a sum-to-one constraint at each spatial position to form the final weights for fusion.", "The obtained final fusion weights of the Venice stack are given in Fig.", "REF .", "A pyramidal decomposition is applied to blend the input stack in order to further avoid artifacts such as halo effects at sharp texture and color changeovers [5], [1].", "In detail, the Laplacian pyramid ($L$ ) is employed to decompose each input exposure into $\\ell $ -levels of distinct resolutions, while the Gaussian pyramid ($G$ ) to carry out a similar operation for final fusion weights.", "The blending operation is applied at each pyramidal level, and as a result a fused Laplacian pyramid is obtained for the fused image given in Eqn.", "(REF ) as follows, $L\\lbrace F^\\ell \\rbrace = \\sum _{n=1}^N G\\lbrace W_n^\\ell \\rbrace \\times L\\lbrace I_n^\\ell \\rbrace $ where the fused pyramid $L\\lbrace F^\\ell \\rbrace $ is finally collapsed to acquire the final fused image $F$ .", "Figure: Laurenziana: (Left) Ulucan (0.9890.989), (right) Proposed (0.9910.991)." ], [ "Experimental Results", "The proposed MEF algorithm (PAS-MEF) is compared to Mertens [1], Lee [6], Li18 [7], Liu [8], Hayat [9], Ulucan [2] and Li20 [11] over 13 image stacks obtained from datasets in [18], [19], [20].", "All experiments are performed on an AMD Ryzen(TM) 5 3600x CPU @ $3.80$ GHz 6-core 16GB RAM machine using MATLAB R2020a.", "All competing algorithms are employed with their default settings, including the proposed method, without any optimization.", "A statistical performance analysis is performed through the multi-scale structural similarity framework for MEF, i.e., MEF-SSIM [19].", "MEF-SSIM is a perceptual quality assessment metric, which takes both the global luminance consistency and the local structure preservation into account to produce statistical results in the range $[0~1]$ .", "A score closer to 1 indicates a better perceptual quality.", "The obtained statistical scores of all algorithms are reported in Table REF , in which the bottom three rows present the average accuracy (avg), standard deviation (std) and average execution time (run-time) of each method.", "It can be clearly observed that PAS-MEF produces highly competitive results (i.e., being in the second best place) and surpasses six of the competing state-of-the-art MEF approaches on average.", "In addition, these statistical results indicate that the standard deviation of PAS-MEF scores is the smallest (together with Li20) and the computational complexity is very conceivable.", "Table: MEF-SSIM scores for each exposure stack used in experiments.", "The highest scores are in boldface.Figure: Tower: (Left) Ulucan (0.9820.982), (right) Proposed (0.9830.983).A side-by-side visual comparison of the fusion outputs of PAS-MEF and Ulucan is given for the Laurenziana stack in Fig.", "REF .", "In this particular example, PAS-MEF reaches the highest MEF-SSIM score when compared to other competing techniques.", "It can be clearly seen that the sky has a more natural color in the output of PAS-MEF.", "Furthermore, the details on the rooftops are better preserved in the result of this technique, while the tree in the middle has more vivid colors in Ulucan.", "Another visual comparison for the Tower stack is demonstrated in Fig.", "REF .", "The grass and flowers on the foreground are much better recovered in the proposed PAS-MEF, while clouds are better preserved and the tower has vivid colors in Ulucan.", "In Fig.", "REF , the fusion results are presented for the Mask stack.", "PAS-MEF produces the highest MEF-SSIM score (together with Hayat) for this exposure sequence.", "When compared to Li20 which has lower brightness and less information in several parts in the fused image, the proposed method clearly preserves the details of the building and the mask.", "On the other side, the sky has a more plausible color in Li20.", "In Fig.", "REF , the fusion outputs of PAS-MEF and Hayat are compared for the Kluki stack.", "Although PAS-MEF has its lowest MEF-SSIM score for this exposure sequence among other stacks in the dataset, the sky has more well-settled colors in blue regions, and the rooftop of the house and the grass have more vivid colors when compared to Hayat, whose MEF-SSIM is slightly higher.", "It can be concluded from Table REF that PAS-MEF reaches its highest MEF-SSIM score for the Chinese Garden stack and some visual fusion results are presented in Fig.", "REF .", "When compared to Lee, the sky region is more plausible in the proposed technique and overall a natural-looking image is obtained while avoiding any artifacts.", "In addition, it can be observed from Fig.", "REF that both PAS-MEF and Lee output natural-looking images for Venice.", "However, the sky and buildings on the left contain more vivid colors in PAS-MEF, which led to a significantly higher MEF-SSIM score than Lee.", "Further fusion examples are illustrated in Fig.", "REF and Fig.", "REF for Lighthouse and OldHouse, respectively.", "The rooftop of the light house, rocks and trees in the background have more striking colors in PAS-MEF, while a darker output is generated by Li20.", "Next in Fig.", "REF , Li20 contains very bright regions on the old house.", "The building has more well-settled colors in PAS-MEF which has the highest MEF-SSIM score when compared to other methods.", "Figure: Mask: (Left) Li20 (0.9910.991), (right) Proposed (0.9920.992).Figure: Kluki: (Left) Hayat (0.9800.980), (right) Proposed (0.9790.979).Figure: Chinese Garden: (Left) Lee (0.9900.990), (right) Proposed (0.9930.993).Figure: Venice: (Left) Lee (0.9720.972), (right) Proposed (0.9800.980).Figure: Lighthouse: (Left) Li20 (0.9780.978), (right) Proposed (0.9820.982).Figure: OldHouse: (Left) Li20 (0.9900.990), (right) Proposed (0.9920.992)." ], [ "Conclusion", "MEF is commonly used for obtaining HDR-like high quality images and numerous studies are present in this field.", "In general, the existing methods differ in the weight map characterization process.", "In this study, a novel weight extraction method is introduced which is based on PCA, adaptive well-exposedness and saliency maps.", "The obtained weights are refined via a guided filter and then image fusion is carried out through a pyramidal decomposition.", "The proposed algorithm presents very strong results both statistically and visually, and it outperforms several state-of-the-art MEF techniques.", "It is worth noting here that, to the best of available knowledge, this is the first study which incorporates PCA and fully adaptive well-exposedness into the MEF problem.", "As a future direction, the proposed algorithm will be optimized for increasing its statistical and visual performance, and for further reducing the run-time complexity.", "Moreover, PAS-MEF will be extended to fuse dynamic scenes in the image stack." ] ]
2105.11809
[ [ "Combinatoire des Sous-Groupes de Congruence du Groupe Modulaire II" ], [ "Abstract In this paper, we study combinatorics of congruence subgroups of the modular group.", "More precisely, we consider the matrix equation that naturally arises in the theory of Coxeter friezes and investigate its irreducible solutions.", "We give new properties for minimal monomial solutions.", "Furthermore, we introduce the notion of minimal dynomial solutions and study their irreducibility." ], [ "Introduction", "L'une des propriétés les plus intéressantes du groupe modulaire $SL_{2}(\\mathbb {Z})=\\left\\lbrace \\begin{pmatrix}a & b \\\\c & d\\end{pmatrix}\\;\\vert \\;a,b,c,d \\in \\mathbb {Z},\\;ad-bc=1\\right\\rbrace $ est l'existence de parties génératrices à deux éléments.", "On peut, en particulier, prendre les deux matrices suivantes (voir par exemple [1]) : $T=\\begin{pmatrix}1 & 1 \\\\[2pt]0 & 1\\end{pmatrix}, S=\\begin{pmatrix}0 & -1 \\\\[2pt]1 & 0\\end{pmatrix}.$ Á partir de ce choix, on peut montrer que pour toute matrice $A$ de $SL_{2}(\\mathbb {Z})$ il existe un entier strictement positif $n$ et des entiers strictement positifs $a_{1},\\ldots ,a_{n}$ tels que $A=T^{a_{n}}ST^{a_{n-1}}S\\cdots T^{a_{1}}S=\\begin{pmatrix}a_{n} & -1 \\\\[4pt]1 & 0\\end{pmatrix}\\begin{pmatrix}a_{n-1} & -1 \\\\[4pt]1 & 0\\end{pmatrix}\\cdots \\begin{pmatrix}a_{1} & -1 \\\\[4pt]1 & 0\\end{pmatrix}.$ On utilisera la notation $M_{n}(a_{1},\\ldots ,a_{n})$ pour désigner la matrice $\\begin{pmatrix}a_{n} & -1 \\\\[4pt]1 & 0\\end{pmatrix}\\begin{pmatrix}a_{n-1} & -1 \\\\[4pt]1 & 0\\end{pmatrix}\\cdots \\begin{pmatrix}a_{1} & -1 \\\\[4pt]1 & 0\\end{pmatrix}.$ Ces matrices interviennent également dans la théorie des frises de Coxeter (voir par exemple [6], [8] pour la définition des frises de Coxeter).", "En effet, les solutions de l'équation $M_{n}(a_{1},\\ldots ,a_{n})=-Id$ permettent de construire des frises de Coxeter, et, à partir d'une telle frise, on peut obtenir une solution de cette équation (voir [2] et [8] proposition 2.4).", "Les frises de Coxeter possèdent par ailleurs des connections avec de nombreux autres domaines mathématiques (voir par exemple [15]).", "Ceci amène naturellement à l'étude de l'équation généralisée suivante : $M_{n}(a_1,\\ldots ,a_n)=\\pm Id.$ V.Ovsienko (voir [18] Théorèmes 1 et 2) a résolu celle-ci sur $\\mathbb {N}^{*}$ et donné une description combinatoire des solutions en terme de découpages de polygones (généralisant par ailleurs un théorème antérieur dû à Conway et Coxeter, voir [5], [6], [10]).", "On dispose également des solutions de cette équation sur $\\mathbb {N}$ (voir [7] Théorème 3.1), sur $\\mathbb {Z}$ (voir [7] Théorème 3.2) et sur $\\mathbb {Z}[\\alpha ]$ avec $\\alpha $ un nombre complexe transcendant (voir [14] Théorème 2.7).", "On peut aussi étudier l'équation (REF ) en remplaçant $\\pm Id$ par $\\pm M$ avec $M$ une matrice du groupe modulaire, notamment pour $M=S$ et $M=T$ (voir [12]).", "On va s'intéresser ici aux cas des anneaux $\\mathbb {Z}/N\\mathbb {Z}$ , c'est-à-dire à la résolution sur $\\mathbb {Z}/N\\mathbb {Z}$ de l'équation : $M_{n}(a_1,\\ldots ,a_n)=\\pm Id.\\qquad \\mathrm {(E_{N})}$ On dira, en particulier, qu'une solution de (REF ) est de taille $n$ si cette solution est un $n$ -uplet d'éléments de $\\mathbb {Z}/N\\mathbb {Z}$ .", "L'étude de l'équation ci-dessus permet notamment de chercher toutes les écritures des éléments des sous-groupes de congruence ci-dessous : $\\hat{\\Gamma }(N)=\\lbrace A \\in SL_{2}(\\mathbb {Z})~{\\rm tel~que}~A= \\pm Id~( {\\rm mod}~N)\\rbrace $ sous la forme $M_{n}(a_1,\\ldots ,a_n)$ avec les $a_{i}$ des entiers strictement positifs.", "L'équation (REF ) a déjà été étudiée dans des travaux précédents (voir [11], [13]).", "Pour mener à bien cette étude on avait utilisé une notion de solutions irréductibles à partir de laquelle on peut construire l'ensemble des solutions (voir section suivante).", "Ceci nous avait permis de résoudre complètement (REF ) pour $N \\le 6$ (voir [13] section 4).", "On avait également obtenu des résultats généraux d'irréductibilité en définissant notamment la notion de solutions monomiales minimales (voir [13] section 3.3 et la section suivante).", "D'autres éléments pouvant être reliés aux cas des anneaux $\\mathbb {Z}/N\\mathbb {Z}$ , avec $N$ premier, peuvent également être trouvés dans [16].", "L'objectif ici est de poursuivre cette étude en obtenant des résultats sur la taille et l'irréductibilité des solutions monomiales minimales (voir section ) et d'obtenir des résultats d'irréductibilité pour d'autres solutions (voir section )." ], [ "Définitions et résultats principaux", "L'objectif de cette section est de rappeler les définitions introduites notamment dans [7] et [13] utiles à l'étude de l'équation (REF ) et d'énoncer les résultats principaux de ce texte.", "Sauf mention contraire, $N$ désigne un entier naturel supérieur à 2 et si $a \\in \\mathbb {Z}$ on note $\\overline{a}=a+N\\mathbb {Z}$ .", "Définition 2.1 ([19], définition 1.8) Soient $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\in (\\mathbb {Z}/N \\mathbb {Z})^{n}$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{m}}) \\in (\\mathbb {Z}/N \\mathbb {Z})^{m}$ .", "On définit l'opération ci-dessous : $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\oplus (\\overline{b_{1}},\\ldots ,\\overline{b_{m}})= (\\overline{a_{1}+b_{m}},\\overline{a_{2}},\\ldots ,\\overline{a_{n-1}},\\overline{a_{n}+b_{1}},\\overline{b_{2}},\\ldots ,\\overline{b_{m-1}}).$ Le $(n+m-2)$ -uplet obtenu est appelé la somme de $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}})$ avec $(\\overline{b_{1}},\\ldots ,\\overline{b_{m}})$ .", "Exemples Voici quelques exemples de sommes : $(\\overline{1},\\overline{2},\\overline{3}) \\oplus (\\overline{4},\\overline{1},\\overline{3},\\overline{2})= (\\overline{3},\\overline{2},\\overline{7},\\overline{1},\\overline{3})$  ; $(\\overline{4},\\overline{0},\\overline{1},\\overline{2}) \\oplus (\\overline{-1},\\overline{0},\\overline{1}) = (\\overline{5},\\overline{0},\\overline{1},\\overline{1},\\overline{0})$  ; $n \\ge 2$ , $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\oplus (\\overline{0},\\overline{0}) = (\\overline{0},\\overline{0}) \\oplus (\\overline{a_{1}},\\ldots ,\\overline{a_{n}})=(\\overline{a_{1}},\\ldots ,\\overline{a_{n}})$ .", "En particulier, si $(\\overline{b_{1}},\\ldots ,\\overline{b_{m}})$ est une solution de (REF ) alors la somme $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\oplus (\\overline{b_{1}},\\ldots ,\\overline{b_{m}})$ est solution de (REF ) si et seulement si $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}})$ est solution de (REF ) (voir [7], [19] et [13] proposition 3.7).", "En revanche, l'opération $\\oplus $ n'est pas commutative (voir [19] exemple 2.1).", "Définition 2.2 ([19], définition 1.5) Soient $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\in (\\mathbb {Z}/N \\mathbb {Z})^{n}$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{n}}) \\in (\\mathbb {Z}/N \\mathbb {Z})^{n}$ .", "On dit que $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\sim (\\overline{b_{1}},\\ldots ,\\overline{b_{n}})$ si $(\\overline{b_{1}},\\ldots ,\\overline{b_{n}})$ est obtenu par permutation circulaire de $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}})$ ou de $(\\overline{a_{n}},\\ldots ,\\overline{a_{1}})$ .", "On montre facilement que $\\sim $ est une relation d'équivalence sur les $n$ -uplets d'éléments de $\\mathbb {Z}/N \\mathbb {Z}$ (voir [19], lemme 1.7).", "De plus, si $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}}) \\sim (\\overline{b_{1}},\\ldots ,\\overline{b_{n}})$ alors $(\\overline{a_{1}},\\ldots ,\\overline{a_{n}})$ est solution de (REF ) si et seulement si $(\\overline{b_{1}},\\ldots ,\\overline{b_{n}})$ est solution de (REF ) (voir [7] proposition 2.6).", "Définition 2.3 ([7], définition 2.9) Une solution $(\\overline{c_{1}},\\ldots ,\\overline{c_{n}})$ avec $n \\ge 3$ de (REF ) est dite réductible s'il existe une solution de (REF ) $(\\overline{b_{1}},\\ldots ,\\overline{b_{l}})$ et un $m$ -uplet d'éléments de $\\mathbb {Z}/N \\mathbb {Z}$ $(\\overline{a_{1}},\\ldots ,\\overline{a_{m}})$ tels que $(\\overline{c_{1}},\\ldots ,\\overline{c_{n}}) \\sim (\\overline{a_{1}},\\ldots ,\\overline{a_{m}}) \\oplus (\\overline{b_{1}},\\ldots ,\\overline{b_{l}})$  ; $m \\ge 3$ et $l \\ge 3$ .", "Une solution est dite irréductible si elle n'est pas réductible.", "Remarque On ne considère pas $(\\overline{0},\\overline{0})$ comme une solution irréductible de (REF ).", "L'un de nos objectifs principaux est de trouver des solutions irréductibles de l'équation (REF ).", "En particulier, on a introduit dans [13] la notion de solutions monomiales rappelée ci-dessous : Définition 2.4 i) Soient $n \\in \\mathbb {N}^{*}$ et $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "On appelle solution $(n,\\overline{k})$ -monomiale un $n$ -uplet d'éléments de $\\mathbb {Z}/ N \\mathbb {Z}$ constitué uniquement de $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ et solution de (REF ).", "ii) On appelle solution monomiale une solution pour laquelle il existe $m \\in \\mathbb {N}^{*}$ et $\\overline{l} \\in \\mathbb {Z}/N\\mathbb {Z}$ tels qu'elle est $(m,\\overline{l})$ -monomiale.", "iii) On appelle solution $\\overline{k}$ -monomiale minimale une solution $(n,\\overline{k})$ -monomiale avec $n$ le plus petit entier pour lequel il existe une solution $(n,\\overline{k})$ -monomiale.", "iv) On appelle solution monomiale minimale une solution $\\overline{k}$ -monomiale minimale pour un $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "On connaît déjà un certain nombre de propriétés d'irréductibilité pour ces solutions.", "Celles-ci seront évoquées dans la section suivante où on démontrera également le résultat ci-dessous : Théorème 2.5 Soit $N$ un entier pair supérieur à 4.", "On a deux cas : Si $4 \\mid N$ alors la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale de (REF ) est de taille 4 ; Si $4 \\nmid N$ alors la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale de (REF ) est de taille 6.", "De plus, celle-ci est irréductible dans les deux cas.", "Pour obtenir de nouveaux résultats d'irréductibilité on définit une nouvelle classe de solutions de (REF ).", "Définition 2.6 i) Soient $n \\in \\mathbb {N}^{*}$ pair et $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "On appelle solution $(n,\\overline{k})$ -dynomiale une solution de taille $n$ de (REF ) de la forme $(\\overline{k},\\overline{-k},\\ldots ,\\overline{k},\\overline{-k})$ .", "ii) On appelle solution dynomiale une solution pour laquelle il existe $m \\in \\mathbb {N}^{*}$ et $\\overline{l} \\in \\mathbb {Z}/N\\mathbb {Z}$ tels qu'elle est $(m,\\overline{l})$ -dynomiale.", "iii) On appelle solution $\\overline{k}$ -dynomiale minimale une solution $(n,\\overline{k})$ -dynomiale avec $n$ le plus petit entier pour lequel il existe une solution $(n,\\overline{k})$ -dynomiale.", "iv) On appelle solution dynomiale minimale une solution $\\overline{k}$ -dynomiale minimale pour un $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "Remarque Les propriétés suivantes des solutions dynomiales sont immédiates : Une solution dynomiale est toujours de taille paire.", "Une solution $(n,\\overline{k})$ -dynomiale est équivalente à une solution $(n,\\overline{-k})$ -dynomiale.", "Si $\\overline{2k}=\\overline{0}$ alors une solution $(n,\\overline{k})$ -dynomiale est une solution $(n,\\overline{k})$ -monomiale.", "On dispose pour cette classe de solutions du résultat d'irréductibilité suivant démontré dans la section  : Théorème 2.7 Soient $N$ un nombre premier supérieur à 5 et $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "On suppose que les deux conditions suivantes sont vérifiées : $\\overline{k} \\ne \\overline{0}$  ; $\\overline{k}^{2}+\\overline{8}$ n'est pas un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ .", "La solution $\\overline{k}$ -dynomiale minimale de (REF ) est irréductible.", "Ce théorème permet d'obtenir plusieurs résultats d'irréductibilité intéressants exposés dans la section REF .", "On montre notamment dans celle-ci que la solution $\\overline{2}$ -dynomiale minimale de $(E_{N})$ est irréductible lorsque $N$ est un nombre premier supérieur à 5 vérifiant $N \\lnot \\equiv \\pm 1 [12]$ ." ], [ "Propriétés des solutions monomiales minimales", "Comme nous venons de l'évoquer, les solutions monomiales minimales possèdent un certain nombre de propriétés intéressantes (voir [13] section 3.3).", "On dispose notamment des deux résultats d'irréductibilité énoncés ci-dessous : Théorème 3.1 Soit $N$ un entier naturel supérieur à 2. i) ([13], Théorème 3.16) Si $N$ est premier alors les solutions monomiales minimales de (REF ) différentes de $(\\overline{0},\\overline{0})$ sont irréductibles.", "ii) ([13], Théorème 2.6) Si $N \\ge 3$ , $(\\overline{2},\\ldots ,\\overline{2}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{N}$ est une solution monomiale minimale irréductible de (REF ).", "L'objectif de cette section est d'approfondir l'étude de ces solutions en obtenant notamment des éléments sur leur taille et de nouveaux résultats d'irréductibilité.", "Dans cette partie, $N$ est un entier naturel supérieur ou égal à 2." ], [ "Taille des solutions monomiales minimales", "L'un des problèmes soulevés lors de l'étude de ces solutions (voir [13] problème 1) était d'avoir des informations sur la taille des solutions monomiales minimales.", "Notre objectif ici est de fournir des éléments de réponse à ce problème.", "Par le théorème REF , on sait que la solution $\\overline{2}$ -monomiale minimale de (REF ) est de taille $N$ .", "L'étude des solutions de (REF ) pour les petites valeurs de $n$ permet également de répondre précisément à notre question pour certaines valeurs de $\\overline{k}$ .", "Proposition 3.2 ([13], section 3.1) i) (REF ) n'a pas de solution de taille 1. ii) $(\\overline{0},\\overline{0})$ est l'unique solution de (REF ) de taille 2. iii) $(\\overline{1},\\overline{1},\\overline{1})$ et $(\\overline{-1},\\overline{-1},\\overline{-1})$ sont les seules solutions de (REF ) de taille 3. iv) Les solutions de (REF ) de taille 4 sont de la forme $(\\overline{-a},\\overline{b},\\overline{a},\\overline{-b})$ avec $\\overline{ab}=\\overline{0}$ et $(\\overline{a},\\overline{b},\\overline{a},\\overline{b})$ avec $\\overline{ab}=\\overline{2}$ .", "On en déduit que la solution $\\overline{0}$ -monomiale minimale est de taille 2 et que les solutions $\\pm \\overline{1}$ -monomiales minimales sont de taille 3.", "De plus, pour tout $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ , la solution $\\overline{k}$ -monomiale minimale et la solution $\\overline{-k}$ -monomiale minimale ont la même taille.", "Un simple calcul permet de montrer que la solution $\\overline{3}$ -monomiale minimale de $(E_{6})$ est de taille 6 et que la solution $\\overline{3}$ -monomiale minimale de $(E_{7})$ est de taille 4.", "Tout ceci nous permet de connaître précisément les tailles des solutions monomiales minimales pour $N \\in \\lbrace 2, 3, 4, 5, 6, 7\\rbrace $ .", "Pour le cas général on dispose du théorème ci-dessous qui donne une majoration de la taille dans le cas général : Théorème 3.3 ([3], page 216) Soit $N$ un entier naturel supérieur à 2.", "L'ordre des éléments de $SL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ est inférieur à $3N$ .", "Dans le cas où $N$ est premier, on peut avoir des informations plus précises sur la taille des solutions monomiales minimales.", "La preuve qui suit est une adaptation à notre situation de la preuve du cas $N$ premier du théorème précédent fournie dans [3].", "Théorème 3.4 Soient $N$ un nombre premier et $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "On a deux cas : si $\\overline{k}=\\pm \\overline{2}$ alors la solution $\\overline{k}$ -monomiale minimale est de taille $N$  ; si $\\overline{k} \\ne \\pm \\overline{2}$ alors la taille de la solution $\\overline{k}$ -monomiale minimale divise $\\frac{N-1}{2}$ ou $\\frac{N+1}{2}$ .", "On montre, par récurrence, que pour tout $n$ dans $\\mathbb {N}^{*}$ on a $M_{n}(2,\\ldots ,2)=\\begin{pmatrix}n+1 & -n \\\\n & -n+1\\end{pmatrix}.$ Donc, $M_{N}(\\overline{2},\\ldots ,\\overline{2})=\\begin{pmatrix}\\overline{N+1} & \\overline{-N} \\\\\\overline{N} & \\overline{-N+1}\\end{pmatrix}=Id$ ce qui implique que la taille de la solution $\\overline{2}$ -monomiale minimale de (REF ) divise $N$ .", "De plus, (REF ) n'a pas de solution de taille 1 et $N$ est premier.", "Donc, la taille de la solution $\\overline{2}$ -monomiale minimale de (REF ) est égale à $N$ .", "Celle de la solution $\\overline{-2}$ -monomiale minimale de (REF ) est par conséquent aussi égale à $N$ .", "On suppose maintenant $\\overline{k} \\ne \\pm \\overline{2}$ .", "Le polynôme caractéristique de $M_{1}(\\overline{k})$ est $\\chi (X)=X^{2}-\\overline{k}X+\\overline{1}$ .", "Ce polynôme a pour discriminant $\\Delta =\\overline{k}^{2}-\\overline{4}=(\\overline{k}-\\overline{2})(\\overline{k}+\\overline{2}) \\ne \\overline{0}$ .", "On a donc deux cas : $\\Delta $ est un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ .", "Dans ca cas, $M_{1}(\\overline{k})$ a deux valeurs propres distinctes et donc est diagonalisable dans $\\mathbb {Z}/N\\mathbb {Z}$ .", "Notons $\\overline{a}$ et $\\overline{b}$ ses valeurs propres.", "Il existe $P \\in GL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ tel que $M_{1}(\\overline{k})=P\\begin{pmatrix}\\overline{a} & \\overline{0} \\\\[2pt]\\overline{0} & \\overline{b}\\end{pmatrix}P^{-1}.$ De plus, $\\mathbb {Z}/N\\mathbb {Z}-\\lbrace \\overline{0}\\rbrace $ est un groupe de cardinal $N-1$ .", "Ainsi, $\\overline{a}^{N-1}=\\overline{1}$ ($\\overline{a} \\ne \\overline{0}$ puisque $\\overline{ab}=\\overline{1}$ ).", "Donc, $(\\overline{a}^{\\frac{N-1}{2}}-\\overline{1})(\\overline{a}^{\\frac{N-1}{2}}+\\overline{1})=\\overline{0}$ .", "On en déduit que $\\overline{a}^{\\frac{N-1}{2}}=\\pm \\overline{1}$ (puisque $\\mathbb {Z}/N\\mathbb {Z}$ est intègre).", "De même, $\\overline{b}^{\\frac{N-1}{2}}=\\pm \\overline{1}$ .", "Or, $\\overline{ab}=\\overline{1}$ donc $\\overline{a}^{\\frac{N-1}{2}}=\\overline{b}^{\\frac{N-1}{2}}=\\pm \\overline{1}$ .", "Il en découle $M_{1}(\\overline{k})^{\\frac{N-1}{2}}=P\\begin{pmatrix}\\overline{a}^{\\frac{N-1}{2}} & \\overline{0} \\\\[2pt]\\overline{0} & \\overline{b}^{\\frac{N-1}{2}}\\end{pmatrix}P^{-1}=\\pm Id.$ Par conséquent, l'ordre de $M_{1}(\\overline{k})$ dans $PSL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ , c'est-à-dire la taille de la solution $\\overline{k}$ -monomiale minimale de (REF ), divise $\\frac{N-1}{2}$ .", "$\\Delta $ n'est pas un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ .", "Soit $K$ un corps de décomposition de $\\chi $ (voir [9] Théorème V.18).", "$\\chi $ a deux racines distinctes dans $K$ .", "Notons les $x$ et $y$ .", "On a $(X-x^{N})(X-y^{N}) &=& X^{2}-(x^{N}+y^{N})X+x^{N}y^{N} \\\\&=& X^{2}-(x+y)^{N}X+x^{N}y^{N}~({\\rm morphisme~de~Frobenius}) \\\\&=& X^{2}-(x+y)^{N}X+(xy)^{N}~({\\rm commutativit\\acute{e}~de}~K) \\\\&=& X^{2}-\\overline{k}^{N}X+\\overline{1}~(x~{\\rm et}~y~{\\rm racines~de}~\\chi ) \\\\&=& X^{2}-\\overline{k}X+\\overline{1}.", "\\\\$ Donc, $x^{N}$ et $y^{N}$ sont des racines de $\\chi $ .", "De plus, $x^{N} \\ne x$ .", "En effet, supposons par l'absurde que $x^{N}=x$ .", "Dans ce cas, $x^{N-1}=\\overline{1}$ ($x \\ne \\overline{0}$ car $xy=\\overline{1}$ ).", "Or, le polynôme $Q(X)=X^{N-1}-\\overline{1}$ a au plus $N-1$ racines dans $K$ et les éléments non nuls de $\\mathbb {Z}/N\\mathbb {Z}$ sont des racines.", "Donc, les seules racines de $Q$ sont les éléments non nuls de $\\mathbb {Z}/N\\mathbb {Z}$ .", "On a donc, $x \\in \\mathbb {Z}/N\\mathbb {Z}$ .", "Ceci est absurde puisque $\\chi $ n'a pas de racine dans ce corps.", "Ainsi, $x^{N}=y$ et donc $x^{N+1}=xy=\\overline{1}$ .", "En procédant comme dans le cas précédent, on obtient $\\overline{x}^{\\frac{N+1}{2}}=\\overline{y}^{\\frac{N+1}{2}}=\\pm \\overline{1}$ ce qui implique que la taille de la solution $\\overline{k}$ -monomiale minimale de (REF ) divise $\\frac{N+1}{2}.$ On donne en annexe (voir annexe ) les valeurs des tailles des solutions monomiales minimales pour les nombres premiers compris entre 11 et 47.", "Remarque Il existe des cas où la taille des solutions monomiales minimales est égale à $\\frac{N-1}{2}$ ou $\\frac{N+1}{2}$ (voir annexe ).", "On peut également obtenir la taille précise de certaines solutions monomiales minimales lorsque $N$ et $\\overline{k}$ vérifient certaines propriétés (voir sous-partie suivante)." ], [ "Preuve du théorème ", "On suppose que $N$ est un entier pair supérieur à 4. i) Si $4 \\mid N$ , alors $\\overline{\\frac{N}{2}}\\overline{\\frac{N}{2}}=\\overline{\\frac{N^{2}}{4}}=\\overline{N\\frac{N}{4}}=\\overline{0}$ et donc $(\\overline{-\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{-\\frac{N}{2}})=(\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}})$ est solution de (REF ) (voir proposition REF ).", "La taille de la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale divise 4 (puisque c'est l'ordre de $M_{1}(\\overline{\\frac{N}{2}})$ dans le groupe $PSL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ ).", "(REF ) n'a pas de solution de taille 1.", "Comme $\\overline{\\frac{N}{2}} \\ne \\overline{0}$ , $(\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}})$ n'est pas solution et donc la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale est de taille 4.", "Comme, $\\overline{\\frac{N}{2}} \\ne \\overline{\\pm 1}$ , la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale est irréductible.", "En effet, une solution réductible de taille 4 est la somme de deux solutions de taille 3 et donc contient nécessairement $\\pm \\overline{1}$ .", "ii) Si $4 \\nmid N$ , on note $K=\\frac{N}{2}$ .", "En particulier, on a $K$ impair et $\\overline{2K}=\\overline{0}$ .", "On a $M_{6}(\\overline{K},\\overline{K},\\overline{K},\\overline{K},\\overline{K},\\overline{K}) = \\begin{pmatrix}\\overline{K^{6}-5K^{4}+6K^{2}-1} & \\overline{-K^{5}+4K^{3}-3K} \\\\\\overline{K^{5}-4K^{3}+3K} & \\overline{-K^{4}+3K^{2}-1}\\end{pmatrix}.$ Or, $\\overline{K^{6}-5K^{4}+6K^{2}} &=& \\overline{K^{6}-K^{4}-4K^{4}+3\\times 2K^{2}} \\\\&=& \\overline{K^{6}-K^{4}} \\\\&=& \\overline{K^{4}(-1+K^{2})}.", "\\\\$ De plus, $K^{2}$ est impair (produit de deux entiers impairs) donc $(-1+K^{2})$ est pair.", "Ainsi, il existe un entier $j$ tel que $(-1+K^{2})=2j$ .", "Donc, $\\overline{K^{6}-5K^{4}+6K^{2}}=\\overline{2jK^{4}}=\\overline{jK^{3}(2K)}=\\overline{0}.$ De même, on a $\\overline{-K^{5}+4K^{3}-3K} &=& \\overline{-K^{5}+2K(2K^{2})-K-2K} \\\\&=& \\overline{-K^{5}-K} \\\\&=& \\overline{-K(1+K^{4})}.", "\\\\$ Or, $K^{4}$ est impair (produit de quatre entiers impairs) donc $1+K^{4}$ est pair.", "Il existe un entier $j^{\\prime }$ tel que $1+K^{4}=2j^{\\prime }$ .", "Ainsi, $\\overline{-K^{5}+4K^{3}-3K}=\\overline{-K(1+K^{4})}=\\overline{-j^{\\prime }N}=\\overline{0}.$ On procède de façon analogue pour $\\overline{-K^{4}+3K^{2}}$ .", "On a $\\overline{-K^{4}+3K^{2}}=\\overline{K^{2}(1-K^{2})}$ .", "Or, $K^{2}$ est impair (produit de deux entiers impairs) donc $(1-K^{2})$ est pair.", "Il existe un entier $j^{\\prime \\prime }$ tel que $(1-K^{2})=2j^{\\prime \\prime }$ .", "Donc, $\\overline{-K^{4}+3K^{2}}=\\overline{K^{2}(1-K^{2})}=\\overline{2j^{\\prime \\prime }K^{2}}=\\overline{Nj^{\\prime \\prime }K}=\\overline{0}.$ Ainsi, $M_{6}(\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}})=-Id$ .", "Il en découle que la taille de la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale divise 6 (puisque c'est l'ordre de $M_{1}(\\overline{\\frac{N}{2}})$ dans le groupe $PSL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ ) c'est-à-dire que celle-ci est égale à 1, 2, 3 ou 6.", "(REF ) n'a pas de solution de taille 1.", "$\\overline{\\frac{N}{2}} \\ne \\overline{0}$ , donc, $(\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}})$ n'est pas solution.", "$\\overline{\\frac{N}{2}} \\ne \\overline{\\pm 1}$ , donc, $(\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}})$ n'est pas solution.", "On en déduit que la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale de (REF ) est de taille 6.", "Si celle-ci est réductible alors elle est la somme de deux solutions de taille 4 (puisqu'elle ne contient pas $\\pm \\overline{1}$ ).", "Dans ce cas, (REF ) a une solution de la forme $(\\overline{a},\\overline{\\frac{N}{2}},\\overline{\\frac{N}{2}},\\overline{a})$ avec $\\overline{a} \\ne \\overline{\\frac{N}{2}}$ (sinon la solution minimale serait de taille 4).", "Donc, on a $\\overline{\\frac{N^{2}}{4}}=\\overline{0}$ et $\\overline{a}=\\overline{-\\frac{N}{2}}$ .", "Ainsi, Il existe un entier $l$ tel que $K^{2}=2lK$ .", "Donc, $K^{2}$ est pair ce qui implique $K$ pair.", "Ce qui est absurde.", "Donc, la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale est irréductible de taille 6.", "$\\Box $ Remarque Si $N=2$ , alors la solution $\\overline{\\frac{N}{2}}$ -monomiale minimale est la solution $\\overline{1}$ -monomiale minimale qui est irréductible de taille 3." ], [ "Généralisations partielles", "On cherche à généraliser le théorème REF pour des diviseurs de $N$ différents de 2.", "On commence par le résultat suivant : Proposition 3.5 Si $l^{2} \\mid N$ avec $l \\ge 2$ alors $(\\overline{\\frac{N}{l}},\\ldots ,\\overline{\\frac{N}{l}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l}$ est solution de (REF ).", "On a $M_{2l}(\\overline{\\frac{N}{l}},\\ldots ,\\overline{\\frac{N}{l}}) &=& (\\begin{pmatrix}\\overline{\\frac{N}{l}} & \\overline{-1} \\\\\\overline{1} & \\overline{0}\\end{pmatrix}\\begin{pmatrix}\\overline{\\frac{N}{l}} & \\overline{-1} \\\\\\overline{1} & \\overline{0}\\end{pmatrix})^{l} \\\\&=& \\begin{pmatrix}\\overline{\\frac{N^{2}}{l^{2}}-1} & \\overline{-\\frac{N}{l}} \\\\\\overline{\\frac{N}{l}} & \\overline{-1}\\end{pmatrix}^{l} \\\\&=& \\begin{pmatrix}\\overline{N\\frac{N}{l^{2}}-1} & \\overline{-\\frac{N}{l}} \\\\\\overline{\\frac{N}{l}} & \\overline{-1}\\end{pmatrix}^{l}~{\\rm car}~l^{2} \\mid N \\\\&=& \\begin{pmatrix}\\overline{-1} & \\overline{-\\frac{N}{l}} \\\\\\overline{\\frac{N}{l}} & \\overline{-1}\\end{pmatrix}^{l} \\\\&=& \\overline{(-Id+\\frac{N}{l}S)^{l}} \\\\&=& \\overline{ \\sum _{k=0}^{l} \\binom{l}{k} (-1)^{l-k}(\\frac{N}{l}S)^{k}}~({\\rm bin\\hat{o}me~de~Newton}) \\\\&=& \\overline{ (-1)^{l}\\binom{l}{0}Id+(-1)^{l-1}\\binom{l}{1}\\frac{N}{l}S+\\sum _{k=2}^{l} \\binom{l}{k} (-1)^{l-k}\\frac{N^{k}}{l^{k}}S^{k}} \\\\&=& \\overline{ (-1)^{l}Id+(-1)^{l-1}NS+\\sum _{k=2}^{l} \\binom{l}{k} (-1)^{l-k}N\\frac{N}{l^{2}}\\frac{N^{k-2}}{l^{k-2}}S^{k}} \\\\&=& \\overline{(-1)}^{l}Id.", "\\\\$ L'avant dernière égalité est valide car $\\frac{N}{l^{2}}$ et $\\frac{N^{k-2}}{l^{k-2}}$ sont des entiers puisque $l^{2} \\mid N$ et $l^{k-2} \\mid N^{k-2}$ .", "Remarque Si $l=1$ alors $(\\overline{\\frac{N}{l}},\\ldots ,\\overline{\\frac{N}{l}})=(\\overline{0},\\overline{0}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2}$ est aussi solution de (REF ).", "Le résultat ci-dessous résout la question de l'irréductibilité de ces solutions.", "Proposition 3.6 Soit $l^{2} \\mid N$ avec $l \\ge 2$ .", "$(\\overline{\\frac{N}{l}},\\ldots ,\\overline{\\frac{N}{l}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l}$ est une solution irréductible de (REF ) si et seulement si $l=2$ .", "Si $l=2$ alors la solution est irréductible (voir théorème REF ).", "Si $l \\ge 3$ alors la solution n'est pas irréductible car $(\\overline{-\\frac{N}{l}},\\overline{\\frac{N}{l}},\\overline{\\frac{N}{l}},\\overline{-\\frac{N}{l}})$ est solution de (REF ) puisque $\\overline{\\frac{N^{2}}{l^{2}}}=\\overline{N\\frac{N}{l^{2}}}=\\overline{0}$ (voir proposition REF ) et $(\\overline{\\frac{N}{l}},\\ldots ,\\overline{\\frac{N}{l}})=(\\overline{2\\frac{N}{l}},\\overline{l},\\ldots ,\\overline{l},\\overline{2\\frac{N}{l}}) \\oplus (\\overline{-\\frac{N}{l}},\\overline{\\frac{N}{l}},\\overline{\\frac{N}{l}},\\overline{-\\frac{N}{l}}).$ $(\\overline{2\\frac{N}{l}},\\overline{l},\\ldots ,\\overline{l},\\overline{2\\frac{N}{l}})$ est de taille $2l-2 \\ge 4 >3$ .", "Remarque Ces propositions généralisent les propriétés 3.10 et 3.11 de [13].", "Avant de continuer, on a besoin des résultats suivants qui permettent d'obtenir une expression de $M_{n}(a_{1},\\ldots ,a_{n})$  : Soit $x$ un réel, on note $E[x]$ la partie entière de $x$ .", "On pose $K_{-1}=0$ , $K_{0}=1$ et on note pour $i \\ge 1$ $K_i(a_{1},\\ldots ,a_{i})=\\left|\\begin{array}{cccccc}a_1&1&&&\\\\[4pt]1&a_{2}&1&&\\\\[4pt]&\\ddots &\\ddots &\\!\\!\\ddots &\\\\[4pt]&&1&a_{i-1}&\\!\\!\\!\\!\\!1\\\\[4pt]&&&\\!\\!\\!\\!\\!1&\\!\\!\\!\\!a_{i}\\end{array}\\right|.$ $K_{i}(a_{1},\\ldots ,a_{i})$ est le continuant de $a_{1},\\ldots ,a_{i}$ .", "On dispose de l'égalité suivante (voir [4], [17]) : $M_{n}(a_{1},\\ldots ,a_{n})=\\begin{pmatrix}K_{n}(a_{1},\\ldots ,a_{n}) & -K_{n-1}(a_{2},\\ldots ,a_{n}) \\\\K_{n-1}(a_{1},\\ldots ,a_{n-1}) & -K_{n-2}(a_{2},\\ldots ,a_{n-1})\\end{pmatrix}.$ De plus, on dispose de l'expression classique ci-dessous : Lemme 3.7 Soit $n \\ge 0$ , $K_{n}(x,\\ldots ,x)=\\sum _{k=0}^{E[\\frac{n}{2}]} (-1)^{k}\\binom{n-k}{k}x^{n-2k}$ .", "Cela se prouve par récurrence sur $n$ .", "En effet, la formule est vraie pour $n=0$ et pour $n=1$ .", "Supposons qu'il existe un entier positif $n$ tel que la formule est vraie pour $n$ et $n-1$ .", "On suppose $n$ pair.", "En développant le déterminant définissant $K_{n+1}(x,\\ldots ,x)$ suivant la première colonne, on obtient : $K_{n+1}(x,\\ldots ,x) &=& xK_{n}(x,\\ldots ,x)-K_{n-1}(x,\\ldots ,x) \\\\&=& \\sum _{k=0}^{E[\\frac{n}{2}]} (-1)^{k}\\binom{n-k}{k}x^{n+1-2k}-\\sum _{k=0}^{E[\\frac{n-1}{2}]} (-1)^{k}\\binom{n-1-k}{k}x^{n-1-2k} \\\\&=& \\sum _{k=0}^{\\frac{n}{2}} (-1)^{k}\\binom{n-k}{k}x^{n+1-2k}-\\sum _{k=0}^{\\frac{n}{2}-1} (-1)^{k}\\binom{n-1-k}{k}x^{n-1-2k}~({\\rm car}~n~{\\rm est~pair}) \\\\&=& \\sum _{k=0}^{\\frac{n}{2}} (-1)^{k}\\binom{n-k}{k}x^{n+1-2k}-\\sum _{l=1}^{\\frac{n}{2}} (-1)^{l-1}\\binom{n-l}{l-1}x^{n+1-2l} \\\\ &=& \\sum _{k=0}^{\\frac{n}{2}} (-1)^{k}\\binom{n-k}{k}x^{n+1-2k}+\\sum _{l=1}^{\\frac{n}{2}} (-1)^{l}\\binom{n-l}{l-1}x^{n+1-2l} \\\\&=& x^{n+1}+\\sum _{k=1}^{\\frac{n}{2}} (-1)^{k}(\\binom{n-k}{k}+\\binom{n-k}{k-1})x^{n+1-2k} \\\\&=& x^{n+1}+\\sum _{k=1}^{\\frac{n}{2}} (-1)^{k}\\binom{n+1-k}{k}x^{n+1-2k}~({\\rm triangle~de~Pascal}) \\\\&=& \\sum _{k=0}^{\\frac{n}{2}} (-1)^{k}\\binom{n+1-k}{k}x^{n+1-2k}.", "\\\\$ On procède de façon analogue si $n$ est impair.", "Cela prouve la formule par récurrence.", "Proposition 3.8 Si $p^{2} \\mid N$ avec $p$ premier alors $(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2p}$ est une solution monomiale minimale de (REF ).", "Si $p=2$ alors le résultat est vrai (voir théorème REF ).", "On suppose maintenant $p \\ge 3$ .", "Par ce qui précède, $(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2p}$ est une solution monomiale de (REF ).", "Ainsi, la taille de la solution $\\overline{\\frac{N}{p}}$ -monomiale minimale divise $2p$ .", "Donc, celle-ci est égale à 1, 2, $p$ ou $2p$ .", "(REF ) n'a pas de solution de taille 1, et $\\overline{\\frac{N}{p}} \\ne \\overline{0}$ (sinon $N \\mid \\frac{N}{p}$ et donc $\\frac{N}{p} \\ge N$ ) donc la solution $\\overline{\\frac{N}{p}}$ -monomiale minimale n'est pas de taille 2.", "Supposons par l'absurde que la solution $\\overline{\\frac{N}{p}}$ -monomiale minimale est de taille $p$ .", "Il existe $\\epsilon $ dans $\\lbrace \\pm 1\\rbrace $ tel que $\\overline{\\epsilon }Id=M_{p}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}})=\\begin{pmatrix}K_{p}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}}) & -K_{p-1}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}}) \\\\K_{p-1}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}}) & -K_{p-2}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}})\\end{pmatrix}.$ Donc, $K_{p-1}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}})=\\overline{0}$ .", "Notons $K=K_{p-1}(\\overline{\\frac{N}{p}},\\ldots ,\\overline{\\frac{N}{p}})$ .", "On a, par le lemme précédent, $K &=& \\overline{\\sum _{k=0}^{E[\\frac{p-1}{2}]} (-1)^{k}\\binom{p-1-k}{k}(\\frac{N}{p})^{p-1-2k}} \\\\&=& \\overline{\\sum _{k=0}^{\\frac{p-1}{2}} (-1)^{k}\\binom{p-1-k}{k}(\\frac{N}{p})^{p-1-2k}}~({\\rm car}~p-1~{\\rm est~pair}) \\\\&=& \\overline{\\sum _{k=0}^{\\frac{p-1}{2}-1} (-1)^{k}\\binom{p-1-k}{k}\\frac{N^{p-1-2k}}{p^{p-1-2k}}+ (-1)^{\\frac{p-1}{2}}\\binom{\\frac{p-1}{2}}{\\frac{p-1}{2}}} \\\\&=& \\overline{(-1)^{\\frac{p-1}{2}} + \\sum _{k=0}^{\\frac{p-1}{2}-1} (-1)^{k}\\binom{p-1-k}{k}N\\frac{N}{p^{2}}\\frac{N^{p-1-2k-2}}{p^{p-1-2k-2}}}~(p-1-2k-2 \\ge 0~{\\rm et}~p^{2} \\mid N) \\\\&=& \\overline{(-1)^{\\frac{p-1}{2}}} \\\\& \\ne & \\overline{0}.", "\\\\$ Ceci est absurde.", "Donc, la solution $\\overline{\\frac{N}{p}}$ -monomiale minimale est de taille $2p$ ." ], [ "Réductibilité dans le cas $N=l^{n}$", "On se place dans le cas où $N=l^{n}$ avec $n$ et $l$ supérieurs à 2.", "On sait que le $2l^{n-1}$ -uplet d'éléments de $\\mathbb {Z}/N\\mathbb {Z}$ est une solution de (REF ) (voir [13] proposition 3.14).", "Cependant, la question de l’irréductibilité potentielle de cette solution reste ouverte.", "On se propose ici de répondre à cette dernière en démontrant le résultat suivant : Théorème 3.9 Si $N=l^{n}$ avec $l,n \\ge 2$ alors $(\\overline{l},\\ldots ,\\overline{l}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}}$ est une solution irréductible de (REF ) si et seulement si $l=2$ .", "Pour cela on a besoin de plusieurs résultats intermédiaires.", "On commence par le résultat classique suivant : Lemme 3.10 Soient $n \\in \\mathbb {N}^{*}$ et $k \\in [\\![1;n]\\!", "]$ , $\\frac{n}{{\\rm pgcd}(n,k)}~{\\rm divise}~\\binom{n}{k}$ .", "${n \\atopwithdelims ()k}=\\frac{n}{k}{n-1 \\atopwithdelims ()k-1}=\\frac{\\frac{n}{{\\rm pgcd}(n,k)}}{\\frac{k}{{\\rm pgcd}(n,k)}}{n-1 \\atopwithdelims ()k-1}.$ Donc, comme ${n \\atopwithdelims ()k} \\in \\mathbb {N}^{*}$ , on a $\\frac{k}{{\\rm pgcd}(n,k)}~{\\rm divise}~\\frac{n}{{\\rm pgcd}(n,k)}{n-1 \\atopwithdelims ()k-1}$ .", "Comme $\\frac{k}{{\\rm pgcd}(n,k)}$ et $\\frac{n}{{\\rm pgcd}(n,k)}$ sont premiers entre eux, on a, par le lemme de Gauss, $\\frac{k}{{\\rm pgcd}(n,k)}~{\\rm divise}~{n-1 \\atopwithdelims ()k-1}$ .", "Donc, $\\frac{n}{{\\rm pgcd}(n,k)}~{\\rm divise}~{n \\atopwithdelims ()k}$ .", "Lemme 3.11 Soient $(n,l) \\in (\\mathbb {N}^{*})^{2}$ , $n \\ge 3$ , $l \\ge 2$ et $j \\in [\\![2;n-1]\\!", "]$ .", "On a $l^{n-j}~{\\rm divise}~\\binom{2l^{n-2}}{j}$ .", "Si $j=2$ alors $\\binom{2l^{n-2}}{j}=\\frac{2l^{n-2}(2l^{n-2}-1)}{2}=l^{n-2}(2l^{n-2}-1)$ et donc le résultat est vrai.", "On peut ainsi supposer $n \\ge 4$ et $j \\ge 3$ .", "On a par le lemme précédent $\\frac{2l^{n-2}}{{\\rm pgcd}(2l^{n-2},j)}~{\\rm divise}~\\binom{2l^{n-2}}{j}.$ Notons $l=p_{1}^{\\alpha _{1}} \\ldots p_{r}^{\\alpha _{r}}$ la décomposition de $l$ en facteurs premiers.", "On a deux cas : $l$ est impair.", "Dans ce cas, pour tout $i$ appartenant à $[\\![1;r]\\!", "]$ , $p_{i} \\ne 2$ .", "$\\exists (\\beta _{1},\\ldots ,\\beta _{r},a) \\in \\mathbb {N}^{r+1}$ tels que ${\\rm pgcd}(2l^{n-2},j)=2^{a}p_{1}^{\\beta _{1}} \\ldots p_{r}^{\\beta _{r}}$ .", "Si $j$ est pair alors $a=1$ et si $j$ est impair alors $a=0$ .", "Montrons que pour tout $i$ dans $[\\![1;r]\\!", "]$ , $\\beta _{i} \\le \\alpha _{i}(j-2)$ .", "Supposons par l'absurde qu'il existe $i$ dans $[\\![1;r]\\!", "]$ tel que $\\beta _{i} > \\alpha _{i}(j-2)$ .", "Par récurrence, on montre que si $j \\ge 3$ on a $p_{i}^{j-2} \\ge j$ (car $p_{i} > 2$ ).", "On a $p_{i}^{\\beta _{i}}> p_{i}^{\\alpha _{i}(j-2)} \\ge p_{i}^{j-2} \\ge j.$ Donc, ${\\rm pgcd}(2l^{n-2},j) >j$ ce qui est absurde.", "Ainsi, pour tout $i$ appartenant à $[\\![1;r]\\!", "]$ , $\\beta _{i} \\le \\alpha _{i}(j-2)$ .", "De plus, on a $a=1$ si $j$ est pair et $a=0$ si $j$ est impair.", "On en déduit que $l^{n-j}$ divise $\\frac{2l^{n-2}}{{\\rm pgcd}(2l^{n-2},j)}$ .", "Donc, $l^{n-j}$ divise $\\binom{2l^{n-2}}{j}$ .", "$l$ est pair.", "Dans ce cas, on peut supposer $p_{1}=2$ , et donc $p_{j} >2$ pour $j$ dans $[\\![2;r]\\!", "]$ .", "$\\exists (\\beta _{1},\\ldots ,\\beta _{r}) \\in \\mathbb {N}^{r}$ tels que ${\\rm pgcd}(2l^{n-2},j)=p_{1}^{\\beta _{1}} \\ldots p_{r}^{\\beta _{r}}$ .", "On montre, en procédant comme dans le premier cas, que pour tout $i$ dans $[\\![2;r]\\!", "]$ , $\\beta _{i} \\le \\alpha _{i}(j-2)$ .", "Montrons que $\\beta _{1} \\le \\alpha _{1}(j-2)+1$ .", "Si $\\beta _{1} > \\alpha _{1}(j-2)+1$ alors $p_{1}^{\\beta _{1}} > p_{1}^{\\alpha _{1}(j-2)+1} \\ge p_{1}^{j-1} \\ge j.$ Donc, ${\\rm pgcd}(2l^{n-2},j) >j$ ce qui est absurde.", "On en déduit que pour tout $i$ dans $[\\![2;r]\\!", "]$ , $\\beta _{i} \\le \\alpha _{i}(j-2)$ et $\\beta _{1} \\le \\alpha _{1}(j-2)+1$ .", "Ainsi, $l^{n-j}$ divise $\\frac{2l^{n-2}}{{\\rm pgcd}(2l^{n-2},j)}$ .", "Donc, $l^{n-j}~{\\rm divise}~\\binom{2l^{n-2}}{j}$ .", "Donc, le résultat est vrai.", "Lemme 3.12 Si $N=l^{n}$ avec $l>2$ et $n \\ge 3$ alors $(\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}-4l^{n-2}+2}$ est une solution de (REF ).", "On a $2l^{n-1}-4l^{n-2}+2=2l^{n-2}(l-2)+2 \\ge 2l^{n-2}+2 \\ge 2l+2 \\ge 8$ .", "$(\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}}) \\sim (\\overline{2l^{n-1}},\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l})$ .", "Donc, $(\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}})$ est solution de (REF ) si et seulement si $(\\overline{2l^{n-1}},\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l})$ l'est aussi.", "On a $M=M_{2l^{n-1}-4l^{n-2}+2}(\\overline{2l^{n-1}},\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l})=M_{2l^{n-1}-4l^{n-2}}(\\overline{l},\\ldots ,\\overline{l})M_{2}(\\overline{2l^{n-1}},\\overline{2l^{n-1}}).$ $M_{2l^{n-1}-4l^{n-2}}(\\overline{l},\\ldots ,\\overline{l}) &=& M_{2l^{n-1}}(\\overline{l},\\ldots ,\\overline{l})M_{4l^{n-2}}(\\overline{l},\\ldots ,\\overline{l})^{-1} \\\\&=& M_{4l^{n-2}}(\\overline{l},\\ldots ,\\overline{l})^{-1}~\\\\&=& (\\begin{pmatrix}\\overline{0} & \\overline{1} \\\\\\overline{-1} & \\overline{l}\\end{pmatrix})^{4l^{n-2}}\\\\&=& (\\begin{pmatrix}\\overline{0} & \\overline{1} \\\\\\overline{-1} & \\overline{l}\\end{pmatrix}^{2})^{2l^{n-2}} \\\\&=& \\begin{pmatrix}\\overline{-1} & \\overline{l} \\\\\\overline{-l} & \\overline{-1+l^{2}}\\end{pmatrix}^{2l^{n-2}} \\\\&=& \\overline{(-Id+l\\begin{pmatrix}0 & 1 \\\\-1 & l\\end{pmatrix})^{2l^{n-2}}} \\\\&=& \\overline{ \\sum _{k=0}^{2l^{n-2}} \\binom{2l^{n-2}}{k} (-1)^{2l^{n-2}-k}l^{k}\\begin{pmatrix}0 & 1 \\\\-1 & l\\end{pmatrix}^{k}}~{\\rm (bin\\hat{o}me~de~Newton)} \\\\&=& \\overline{ \\sum _{k=0}^{n-1} \\binom{2l^{n-2}}{k} (-1)^{2l^{n-2}-k}l^{k}\\begin{pmatrix}0 & 1 \\\\-1 & l\\end{pmatrix}^{k}} \\\\&=& \\overline{(-1)^{2l^{n-2}}Id+(-1)^{2l^{n-2}-1}(2l^{n-2})l\\begin{pmatrix}0 & 1 \\\\-1 & l\\end{pmatrix}}.", "\\\\$ La dernière égalité est valide car, par le lemme précédent, $l^{n-k}$ divise $\\binom{2l^{n-2}}{k}$ pour $2 \\le k \\le n-1$ .", "De plus, $M_{2}(\\overline{2l^{n-1}},\\overline{2l^{n-1}})=\\begin{pmatrix}\\overline{-1} & \\overline{-2l^{n-1}} \\\\\\overline{2l^{n-1}} & \\overline{-1}\\end{pmatrix}.$ Donc, on a $M &=& M_{2l^{n-1}-4l^{n-2}}(\\overline{l},\\ldots ,\\overline{l})M_{2}(\\overline{2l^{n-1}},\\overline{2l^{n-1}})\\\\&=& (\\overline{(-1)^{2l^{n-2}}Id}+\\overline{(-1)^{2l^{n-2}-1}(2l^{n-2})l}\\begin{pmatrix}\\overline{0} & \\overline{1} \\\\\\overline{-1} & \\overline{l}\\end{pmatrix})\\begin{pmatrix}\\overline{-1} & \\overline{-2l^{n-1}} \\\\\\overline{2l^{n-1}} & \\overline{-1}\\end{pmatrix} \\\\&=& \\overline{(-1)^{2l^{n-2}}}\\begin{pmatrix}\\overline{-1} & \\overline{-2l^{n-1}} \\\\\\overline{2l^{n-1}} & \\overline{-1}\\end{pmatrix}+\\overline{(-1)^{2l^{n-2}-1}(2l^{n-1})}\\begin{pmatrix}\\overline{2l^{n-1}} & \\overline{-1} \\\\\\overline{1} & \\overline{2l^{n-1}-l}\\end{pmatrix}\\\\&=& \\overline{(-1)^{2l^{n-2}}}(\\begin{pmatrix}\\overline{-1} & \\overline{-2l^{n-1}} \\\\\\overline{2l^{n-1}} & \\overline{-1}\\end{pmatrix}-\\begin{pmatrix}\\overline{0} & \\overline{-2l^{n-1}} \\\\\\overline{2l^{n-1}} & \\overline{0}\\end{pmatrix}) \\\\&=& \\overline{(-1)^{2l^{n-2}-1}Id}.$ On peut maintenant démontrer le résultat principal de la section.", "Soit $N=l^{n}$ avec $l \\ge 2$ et $n \\ge 2$ .", "$(\\overline{l},\\ldots ,\\overline{l}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}}$ est solution de (REF ) (voir [13] proposition 3.14).", "Si $n=2$ alors le résultat est vrai (voir proposition REF ).", "Supposons $n \\ge 3$ .", "Si $l=2$ alors la solution est irréductible (voir théorème REF ).", "Supposons $l>2$ .", "On a $(\\overline{l},\\ldots ,\\overline{l})=(\\overline{l-2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{l-2l^{n-1}}) \\oplus (\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}})$ avec $(\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}-4l^{n-2}+2}$ et $(\\overline{l-2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{l-2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{4l^{n-2}}$ .", "De plus, par le lemme précédent, $(\\overline{2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}-4l^{n-2}+2}$ est une solution de (REF ) de taille supérieure à 3.", "$(\\overline{l-2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{l-2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{4l^{n-2}}$ est de taille supérieure à 4.", "Donc, $(\\overline{l},\\ldots ,\\overline{l}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{2l^{n-1}}$ est une solution réductible de (REF ).", "Remarque En particulier, $(\\overline{l-2l^{n-1}},\\overline{l},\\ldots ,\\overline{l},\\overline{l-2l^{n-1}}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{4l^{n-2}}$ est aussi une solution de (REF )." ], [ "Solution dynomiales-minimales", "On s'intéresse dans cette section au concept de solutions dynomiales-minimales défini dans la section en démontrant notamment le théorème REF ." ], [ "Résultats préliminaires", "Pour effectuer la preuve du théorème REF , on a besoin de plusieurs résultats intermédiaires.", "On commence par le lemme ci-dessous : Lemme 4.1 Soit $n \\in \\mathbb {N} \\cup \\lbrace -1\\rbrace $ , $K_{n}(x_{1},\\ldots ,x_{n})=(-1)^{n}K_{n}(-x_{1},\\ldots ,-x_{n})$ .", "On raisonne par récurrence sur $n$ .", "$K_{0}=1$ et $K_{-1}=0$ donc le résultat est vrai pour $n=-1$ et pour $n=0$ .", "Supposons qu'il existe $n \\in \\mathbb {N}$ tel que la formule est vraie au rang $n$ et $n-1$ .", "On a : $K_{n+1}(x_{1},\\ldots ,x_{n+1}) &=& x_{1}K_{n}(x_{2},\\ldots ,x_{n+1})-K_{n-1}(x_{3},\\ldots ,x_{n+1}) \\\\&=& (-1)^{n}x_{1}K_{n}(-x_{2},\\ldots ,-x_{n+1})-(-1)^{n-1}K_{n-1}(-x_{3},\\ldots ,-x_{n+1}) \\\\&=& (-1)^{n-1}(-x_{1}K_{n}(-x_{2},\\ldots ,-x_{n+1})-K_{n-1}(-x_{3},\\ldots ,-x_{n+1})) \\\\&=& (-1)^{n+1}(-x_{1}K_{n}(-x_{2},\\ldots ,-x_{n+1})-K_{n-1}(-x_{3},\\ldots ,-x_{n+1})) \\\\&=& (-1)^{n+1}K_{n+1}(-x_{1},\\ldots ,-x_{n+1}).", "\\\\$ Par récurrence, le résultat est vrai.", "On a également besoin du résultat suivant qui est l'analogue de la proposition 3.15 de [13] pour les solutions dynomiales.", "Lemme 4.2 Soient $n \\in \\mathbb {N}^{*}$ , $n \\ge 4$ et $(\\overline{a},\\overline{b},\\overline{k}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{3}$ .", "Soit $\\alpha \\in \\lbrace \\pm 1\\rbrace $ .", "i) Si $(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{n}$ est solution de (REF ) alors $\\overline{a}=\\overline{-b}$ et on a $\\overline{0}=\\overline{a}(\\overline{\\alpha k}+\\overline{a}).$ ii)Si $(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{n}$ est solution de (REF ) alors $\\overline{a}=\\overline{b}$ et on a $\\overline{2}=\\overline{a}(\\overline{\\alpha k}+\\overline{a}).$ i) $n$ est pair.", "Comme $(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b})$ est solution de (REF ), il existe $\\epsilon $ dans $\\lbrace -1,1\\rbrace $ tel que $\\overline{\\epsilon } Id &=& M_{n}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b}) \\\\&=& \\begin{pmatrix}K_{n}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b}) & -K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b}) \\\\K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) & -K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})\\end{pmatrix}.\\\\$ Ainsi, $K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})=-K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b})=\\overline{0}$ et $K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})=-\\overline{\\epsilon }.$ Or, $K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})&=& \\overline{a}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&-& K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{-\\epsilon a}-K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}).", "\\\\$ Donc, comme $\\overline{\\epsilon }^{2}=\\overline{1}$ , on a $\\overline{a}=\\overline{-\\epsilon }K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}).$ De même, on a $K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b})&=& \\overline{b}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&-& K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&=& \\overline{-\\epsilon b}-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}).", "\\\\$ Il en découle $\\overline{b} &=& \\overline{-\\epsilon }K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&=& \\overline{(-\\epsilon )}\\overline{(-1)^{n-3}}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{-a}~{\\rm car}~n~{\\rm est~pair}.\\\\$ De plus, on dispose des égalités ci-dessous : $\\overline{-\\epsilon } &=& K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{\\alpha k}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})-K_{n-4}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})\\\\$ et $M_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})$ $=\\begin{pmatrix}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) & -K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) & -K_{n-4}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})\\end{pmatrix}$ $=\\begin{pmatrix}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) & K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) & -K_{n-4}(-\\overline{\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})\\end{pmatrix} \\in SL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ .", "On en déduit l'égalité suivante : $-K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})K_{n-4}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})^{2}=\\overline{1}.$ Or, comme $K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})=\\overline{-\\epsilon }$ , on a $\\overline{\\epsilon }K_{n-4}(-\\overline{\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})^{2}=\\overline{1},$ c'est-à-dire $K_{n-4}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k}) &=& K_{n-4}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})~{\\rm car}~n-4~{\\rm est~pair}\\\\&=& \\overline{\\epsilon }(\\overline{1}+K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})^{2}).\\\\$ Ainsi, on a $\\overline{-\\epsilon } &=& \\overline{\\alpha k}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})-K_{n-4}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& -\\overline{\\alpha k}K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})-\\overline{\\epsilon }(\\overline{1}+K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})^{2}) \\\\&=& -\\overline{\\alpha k}K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})-\\overline{\\epsilon }-\\overline{\\epsilon }K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})^{2} \\\\&=& \\overline{-\\alpha \\epsilon k a}-\\overline{\\epsilon }-\\overline{\\epsilon a^{2}}.\\\\$ Donc, $\\overline{0}=\\overline{-\\alpha \\epsilon k a}-\\overline{\\epsilon a^{2}}=-\\overline{\\epsilon }\\overline{a}(\\overline{\\alpha k}+\\overline{a})$ et donc $\\overline{0}=\\overline{a}(\\overline{\\alpha k}+\\overline{a})$ .", "ii) $n$ est impair.", "Comme $(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b})$ est solution de (REF ), il existe $\\epsilon $ dans $\\lbrace -1,1\\rbrace $ tel que $\\overline{\\epsilon } Id &=& M_{n}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b}) \\\\&=& \\begin{pmatrix}K_{n}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b}) & -K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b}) \\\\K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) & -K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})\\end{pmatrix}.\\\\$ Donc, $K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})=-K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b})=\\overline{0}$ et $K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})=-\\overline{\\epsilon }.$ Or, $K_{n-1}(\\overline{a},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})&=& \\overline{a}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&-& K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&=& \\overline{-\\epsilon a}-K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}).", "\\\\$ Ainsi, comme $\\overline{\\epsilon }^{2}=\\overline{1}$ , on a $\\overline{a}=\\overline{-\\epsilon }K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}).$ De même, on a $K_{n-1}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{b})&=& \\overline{b}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&-& K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{-\\epsilon b}-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}).", "\\\\$ Donc, $\\overline{b} &=& \\overline{-\\epsilon }K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{(-\\epsilon )}\\overline{(-1)^{n-3}}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&=& \\overline{(-\\epsilon )}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})~{\\rm car}~n-3~{\\rm pair}\\\\&=& \\overline{a}.", "\\\\$ De plus, on a $\\overline{-\\epsilon } &=& K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\&=& \\overline{\\alpha k}K_{n-3}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})-K_{n-4}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\$ et $M_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})$ $=\\begin{pmatrix}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) & -K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k}) \\\\K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) & -K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})\\end{pmatrix}$ $=\\begin{pmatrix}K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k}) & -K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) & -K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})\\end{pmatrix} \\in SL_{2}(\\mathbb {Z}/N\\mathbb {Z})$ .", "Ainsi, $-K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})+K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})^{2}=\\overline{1}.$ Or, comme $K_{n-2}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{\\alpha k})=\\overline{-\\epsilon }$ , on a $\\overline{\\epsilon }K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})+K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})^{2}=\\overline{1},$ c'est-à-dire $K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})=\\overline{\\epsilon }(\\overline{1}-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})^{2}).$ Donc, on a $\\overline{-\\epsilon } &=& \\overline{\\alpha k}K_{n-3}(\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k})+K_{n-4}(\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\\\&=& \\overline{\\alpha k}K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})+\\overline{\\epsilon }(\\overline{1}-K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})^{2}) \\\\&=& \\overline{\\alpha k}K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})+\\overline{\\epsilon }-\\overline{\\epsilon }K_{n-3}(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})^{2} \\\\&=& \\overline{-\\alpha \\epsilon k a}+\\overline{\\epsilon }-\\overline{\\epsilon a^{2}}.\\\\$ Donc, $\\overline{-2\\epsilon }=\\overline{-\\alpha \\epsilon k a}-\\overline{\\epsilon a^{2}}=-\\overline{\\epsilon }\\overline{a}(\\overline{\\alpha k}+\\overline{a})$ et donc $\\overline{2}=\\overline{a}(\\overline{\\alpha k}+\\overline{a})$ .", "Remarque Dans le cas où $n$ est pair, il est possible que $\\overline{a} \\ne \\overline{0}$ et $\\overline{a} \\ne \\overline{k}$ .", "Par exemple, si on pose $N=14$ , le 18-uplet $(\\overline{7},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{3},\\overline{11},\\overline{7})$ est solution de (REF )." ], [ "Preuve du théorème d'irréductibilité", "On peut maintenant démontrer le résultat principal.", "Soient $\\overline{k} \\in \\mathbb {Z}/N\\mathbb {Z}$ , $\\overline{k} \\ne \\overline{0}$ , et $n \\in \\mathbb {N}^{*}$ tels que le $n$ -uplet $(\\overline{k},\\overline{-k},\\ldots ,\\overline{k},\\overline{-k})$ d'éléments de $\\mathbb {Z}/N\\mathbb {Z}$ est une solution dynomiale minimale de (REF ).", "On suppose par l'absurde que cette solution peut s'écrire comme une somme de deux solutions non triviales.", "$\\overline{k} \\ne \\pm \\overline{1}$ car $(\\pm \\overline{1})^{2}+\\overline{8}=\\overline{9}=\\overline{3}^{2}$ .", "Donc, si $n=4$ , $(\\overline{k},\\overline{-k},\\ldots ,\\overline{k},\\overline{-k})$ est irréductible, puisque les solutions réductibles de (REF ) de taille 4 contiennent toujours $\\pm \\overline{1}$ .", "On suppose maintenant $n \\ge 6$ .", "Il existe $(\\overline{a_{1}},\\ldots ,\\overline{a_{l}})$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{l^{\\prime }}})$ solutions de (REF ) différentes de $(\\overline{0},\\overline{0})$ avec $l+l^{\\prime }=n+2$ et $l,l^{\\prime } \\ge 3$ tels que $(\\overline{k},\\overline{-k},\\ldots ,\\overline{k},\\overline{-k}) \\sim (\\overline{b_{1}+a_{l}},\\overline{b_{2}},\\ldots ,\\overline{b_{l^{\\prime }-1}},\\overline{b_{l^{\\prime }}+a_{1}},\\overline{a_{2}},\\ldots ,\\overline{a_{l-1}}).$ De plus, $\\overline{k} \\notin \\lbrace \\overline{0},\\overline{-1},\\overline{1}\\rbrace $ donc $l,l^{\\prime }>3$ .", "Il existe $\\alpha $ dans $\\lbrace \\pm 1\\rbrace $   tel que $(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k})=(\\overline{b_{1}+a_{l}},\\overline{b_{2}},\\ldots ,\\overline{b_{l^{\\prime }-1}},\\overline{b_{l^{\\prime }}+a_{1}},\\overline{a_{2}},\\ldots ,\\overline{a_{l-1}}).$ On a deux cas : Cas 1 : Si $l$ est pair alors $l^{\\prime }=n+2-l$ est pair.", "On a donc $(\\overline{a_{1}},\\ldots ,\\overline{a_{l}})=(\\overline{a_{1}},\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{a_{l}}).$ Comme $(\\overline{a_{1}},\\ldots ,\\overline{a_{l}})$ est solution de (REF ), on a, par le lemme REF , $\\overline{a_{1}}=\\overline{a}=\\overline{-a_{l}}$ et $\\overline{0}=\\overline{a}(\\overline{a}+\\overline{\\alpha k})$ .", "Comme $N$ est premier, $\\mathbb {Z}/N\\mathbb {Z}$ est intègre et donc l'équation $\\overline{0}=\\overline{a}(\\overline{a}+\\overline{\\alpha k})$ a pour solutions $\\overline{a}=\\overline{0}$ et $\\overline{a}=\\overline{ -\\alpha k}$ .", "Si $\\overline{a}=\\overline{0}$ alors $(\\overline{a_{2}},\\ldots ,\\overline{a_{l-1}})=(\\overline{\\alpha k},\\overline{-\\alpha k},\\ldots ,\\overline{\\alpha k},\\overline{-\\alpha k}) \\in (\\mathbb {Z}/N\\mathbb {Z})^{l-2}$ est encore solution de (REF ) ce qui contredit la minimalité de la solution (si $\\alpha =-1$ alors $(\\overline{-a_{2}},\\ldots ,\\overline{-a_{l-1}})=(\\overline{k},\\overline{-k},\\ldots ,\\overline{ k},\\overline{-k})$ est encore solution).", "Donc, $\\overline{a}=\\overline{-\\alpha k}$ et par minimalité de la solution on a $l \\ge n$ ce qui implique $l^{\\prime } \\le 2$ .", "Donc, $l^{\\prime }=2$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{l^{\\prime }}})=(\\overline{0},\\overline{0})$ ce qui est absurde.", "Cas 2 : Si $l$ est impair alors $l^{\\prime }=n+2-l$ est impair.", "On a donc $(\\overline{a_{1}},\\ldots ,\\overline{a_{l}})=(\\overline{a_{1}},\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{a_{l}})$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{l^{\\prime }}})=(\\overline{b_{1}},\\overline{-\\alpha k},\\overline{\\alpha k},\\ldots ,\\overline{-\\alpha k},\\overline{\\alpha k},\\overline{-\\alpha k},\\overline{b_{l^{\\prime }}}).$ Comme $(\\overline{a_{1}},\\ldots ,\\overline{a_{l}})$ et $(\\overline{b_{1}},\\ldots ,\\overline{b_{l^{\\prime }}})$ sont solutions de (REF ), on a par le lemme REF  : [label=$\\circ $ ] $\\overline{a_{1}}=\\overline{a}=\\overline{a_{l}}$ et $\\overline{2}=\\overline{a}(\\overline{a}-\\overline{\\alpha k})$  ; $\\overline{b_{1}}=\\overline{b}=\\overline{b_{l^{\\prime }}}$ et $\\overline{2}=\\overline{b}(\\overline{b}-\\overline{\\alpha k})$ .", "$\\overline{a}$ et $\\overline{b}$ sont des racines de $P(X)=X(X-\\overline{\\alpha k})-\\overline{2}=X^{2}-\\overline{\\alpha k}X-\\overline{2}$ .", "Comme $N$ est premier différent de 2, $\\mathbb {Z}/N\\mathbb {Z}$ est un corps de caractéristique différente de 2 et le discriminant de $P$ est $\\Delta =(-\\overline{\\alpha k})^{2}-4\\times \\overline{-2}=\\overline{k}^{2}+\\overline{8}$ .", "Comme $\\overline{k}^{2}+\\overline{8}$ n'est pas un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ , $P$ n'a pas de racine dans $\\mathbb {Z}/N\\mathbb {Z}$ ce qui est absurde.", "Ainsi, on arrive à une absurdité dans les deux cas et donc $(\\overline{k},\\overline{-k},\\ldots ,\\overline{k},\\overline{-k})$ est irréductible.", "Remarques i) Il existe des solutions dynomiales minimales réductibles.", "Par exemple, posons $N=11$ .", "Une solution $\\overline{2}$ -dynomiale minimale (qui est de taille 12) est réductible.", "En effet, $(\\overline{6},\\overline{9},\\overline{2},\\overline{9},\\overline{6})$ est solution de $(E_{11})$ et $(\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9}) = (\\overline{7},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{2},\\overline{9},\\overline{7}) \\oplus (\\overline{6},\\overline{9},\\overline{2},\\overline{9},\\overline{6}).$ Cet exemple montre que, contrairement aux solutions monomiales minimales, il existe des solutions dynomiales minimales réductibles même si $N$ est premier.", "De même, contrairement aux solutions monomiales minimales, il existe des solutions $\\overline{2}$ -dynomiales minimales réductibles.", "ii) La condition $\\overline{k}^{2}+\\overline{8}$ n'est pas un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ n'est pas une condition nécessaire.", "Par exemple, $(\\overline{6},\\overline{-6},\\overline{6},\\overline{-6})$ est une solution dynomiale minimale irréductible pour $N=19$ et $\\overline{6}^{2}+\\overline{8}=\\overline{36}+\\overline{8}=\\overline{6}=\\overline{5}^{2}$ ." ], [ "Applications", "La condition $\\overline{k}^{2}+\\overline{8}$ n'est pas un carré dans $\\mathbb {Z}/N\\mathbb {Z}$ peut être vérifiée à l'aide la loi de réciprocité quadratique de Gauss (puisque $N$ est premier).", "Si $p$ est un nombre premier impair et si $a$ est un entier premier avec $p$ , on note $\\left(\\dfrac{a}{p}\\right)$ le symbole de Legendre c'est-à-dire : $\\left(\\dfrac{a}{p}\\right)=\\left\\lbrace \\begin{array}{ll}1 & \\mbox{si } \\overline{a}~{\\rm est~un~carr\\acute{e}~dans}~\\mathbb {Z}/p\\mathbb {Z}; \\\\-1 & \\mbox{sinon }.\\end{array}\\right.", "\\\\ $ Le symbole de Legendre vérifie les propriétés suivantes : Lemme 4.3 ([9], proposition XII.20) Soient $p$ un nombre premier impair et $a$ et $b$ deux entiers premiers avec $p$ .", "On a : i) (critère d'Euler) $\\left(\\dfrac{a}{p}\\right) \\equiv a^{\\frac{p-1}{2}}~[p]$  ; ii) (multiplicativité) $\\left(\\dfrac{ab}{p}\\right)=\\left(\\dfrac{a}{p}\\right)\\left(\\dfrac{b}{p}\\right)$  ; iii) $\\left(\\dfrac{-1}{p}\\right)=(-1)^{\\frac{p-1}{2}}$ .", "Théorème 4.4 (Loi de réciprocité quadratique de Gauss ; [9], Théorème XII.25 ) Soient $p$ et $q$ deux nombres premiers impairs distincts.", "On a $\\left(\\dfrac{p}{q}\\right)\\left(\\dfrac{q}{p}\\right)= (-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}.$ Par exemple, si $\\overline{k}=\\overline{3}$ , on a $\\overline{k^{2}+8}=\\overline{17}$ .", "Comme 17 est premier, on peut utiliser la loi de réciprocité quadratique de Gauss pour savoir si $\\overline{k^{2}+8}$ est un carré modulo $N$ (avec $N$ un entier premier supérieur à 5).", "On a alors par exemple : Proposition 4.5 La solution $\\overline{3}$ -dynomiale minimale de $(E_{97})$ est irréductible.", "97 est premier et $\\overline{3} \\notin \\lbrace \\overline{0},\\overline{1},\\overline{-1}\\rbrace $ .", "De plus, $97=5\\times 17+12$ et les carrés modulo 17 sont : $\\lbrace \\overline{0},\\overline{1},\\overline{4},\\overline{9},\\overline{-1},\\overline{8},\\overline{2},\\overline{15},\\overline{13}\\rbrace .$ Donc, d'après la loi de réciprocité quadratique de Gauss, on a : $\\left(\\dfrac{17}{97}\\right)=\\left(\\dfrac{97}{17}\\right)(-1)^{\\frac{17-1}{2}\\frac{97-1}{2}}=\\left(\\dfrac{12}{17}\\right)(-1)^{8 \\times 48}=-1.$ Donc, $\\overline{3^{2}+8}$ n'est pas un carré modulo 97.", "Donc, par le théorème REF , la solution $\\overline{3}$ -dynomiale minimale de $(E_{97})$ est irréductible.", "On peut aussi s'intéresser aux cas des solutions $\\overline{2}$ -dynomiales minimales de (REF ) où $N$ est un entier premier supérieur à 5.", "On commence par le résultat intermédiaire ci-dessous : Lemme 4.6 Soit $p$ un nombre premier.", "$\\overline{3}~{\\rm est~un~carr\\acute{e}~dans}~\\mathbb {Z}/p\\mathbb {Z} \\Longleftrightarrow \\left\\lbrace \\begin{array}{ll}p=2; \\\\p=3; \\\\p \\equiv \\pm 1~[12].\\end{array}\\right.", "\\\\ $ Si $p=2$ alors $\\overline{3}=\\overline{1}=\\overline{1}^{2}$ et si $p=3$ alors $\\overline{3}=\\overline{0}=\\overline{0}^{2}.$ On suppose maintenant $p$ supérieur à 5.", "D'après la loi de réciprocité quadratique de Gauss on a $\\left(\\dfrac{3}{p}\\right)=\\left(\\dfrac{p}{3}\\right)(-1)^{\\frac{p-1}{2}\\frac{3-1}{2}}=\\left(\\dfrac{p}{3}\\right)(-1)^{\\frac{p-1}{2}}.$ Comme $p$ est premier, $p \\equiv 1~[3]$ ou $p \\equiv -1~[3]$ .", "De plus, on a $\\left(\\dfrac{1}{3}\\right)=1$ et $\\left(\\dfrac{-1}{3}\\right)=-1$ .", "On distingue les deux cas : Si $p=3k+1$ avec $k$ un entier naturel pair $\\left(\\dfrac{3}{p}\\right)=1 \\Longleftrightarrow \\left(\\dfrac{1}{3}\\right)(-1)^{\\frac{3k}{2}}=1 \\Longleftrightarrow (-1)^{\\frac{3k}{2}}=1 \\Longleftrightarrow 4~{\\rm divise}~k \\Longleftrightarrow p \\equiv 1~[12].$ Si $p=3k-1$ avec $k$ un entier naturel pair $\\left(\\dfrac{3}{p}\\right)=1 \\Longleftrightarrow \\left(\\dfrac{-1}{3}\\right)(-1)^{\\frac{3k-2}{2}}=1 \\Longleftrightarrow (-1)^{\\frac{3k}{2}}=1 \\Longleftrightarrow 4~{\\rm divise}~k \\Longleftrightarrow p \\equiv -1~[12].$ Proposition 4.7 Si $N$ est un entier premier supérieur à 5 tel que $N \\lnot \\equiv \\pm 1[12]$ alors la solution $\\overline{2}$ -dynomiale minimale de (REF ) est irréductible.", "Par le lemme précédent, $\\overline{3}$ est un carré modulo $N$ si et seulement si $p \\equiv \\pm 1 [12]$ .", "$N$ est premier.", "De plus, on a $\\overline{2^{2}+8}=\\overline{12}$ et donc par multiplicativité du symbole de Legendre : $\\left(\\dfrac{12}{N}\\right)=\\left(\\dfrac{2^{2} \\times 3}{N}\\right)=\\left(\\dfrac{2^{2}}{N}\\right)\\left(\\dfrac{3}{N}\\right)=\\left(\\dfrac{2}{N}\\right)^{2}\\left(\\dfrac{3}{N}\\right)=\\left(\\dfrac{3}{N}\\right)=-1.$ Donc, par le théorème REF , la solution $\\overline{2}$ -dynomiale minimale de (REF ) est irréductible.", "Remarque Par le théorème faible de la progression arithmétique de Dirichlet (voir [9] proposition VII.13), il existe une infinité de nombres premiers supérieurs à 5 congrus à 1 modulo 12.", "La condition de la proposition précédente n'est pas nécessaire.", "Par exemple, si $N=59 \\equiv -1~[12]$ alors la solution $\\overline{2}$ -dynomiale minimale de (REF ) est irréductible.", "En effet, celle-ci est de taille 20 et la seule façon de la réduire est de trouver une solution de (REF ) de la forme $(\\overline{a},\\overline{-2},\\overline{2},\\overline{-2},\\ldots ,\\overline{2},\\overline{-2},\\overline{a})$ de taille inférieure à 19 (car les éléments donnés dans le cas 1 de la preuve du théorème REF sont toujours valides).", "Par le lemme REF , les seules valeurs de $\\overline{a}$ possibles sont $\\overline{12}$ et $\\overline{-10}$ .", "Or, en calculant toutes les possibilités on ne trouve aucune solution de (REF ), ce qui implique l'irréductibilité de la solution $\\overline{2}$ -dynomiale minimale de $(E_{59})$ .", "Ceci nous amène au problème ouvert suivant : Problème Soit $N$ un nombre premier.", "Trouver des conditions nécessaires et suffisantes sur $N$ pour l'irréductibilité de la solution $\\overline{2}$ -dynomiale minimale de (REF ).", "On donne en annexe les éléments concernant la réductibilité ou l'irréductibilité des solutions $\\overline{2}$ -dynomiales minimales de (REF ) pour les nombres premiers congrus à $\\pm 1$ modulo 12 et inférieurs à 500.", "Remerciements.", "Je remercie Valentin Ovsienko pour son aide précieuse." ], [ "Éléments sur la réductibilité des solutions $\\overline{2}$ -dynomiales minimales de (", " Table: NO_CAPTION" ] ]
2105.11719
[ [ "The Hipparcos-Gaia Catalog of Accelerations: Gaia EDR3 Edition" ], [ "Abstract We present a cross-calibration of Hipparcos and Gaia EDR3 intended to identify astrometrically accelerating stars and to fit orbits to stars with faint, massive companions.", "The resulting catalog, the EDR3 edition of the Hipparcos-Gaia Catalog of Accelerations (HGCA), provides three proper motions with calibrated uncertainties on the EDR3 reference frame: the Hipparcos proper motion, the Gaia EDR3 proper motion, and the long-term proper motion given by the difference in position between Hipparcos and Gaia EDR3.", "Our approach is similar to that for the Gaia DR2 edition of the HGCA, but offers a factor of ~3 improvement in precision thanks to the longer time baseline and improved data processing of Gaia EDR3.", "We again find that a 60/40 mixture of the two Hipparcos reductions outperforms either reduction individually, and we find strong evidence for locally variable frame rotations between all pairs of proper motion measurements.", "The substantial global frame rotation seen in DR2 proper motions has been removed in EDR3.", "We also correct for color- and magnitude-dependent frame rotations at a level of up to ~50 $\\mu$as/yr in Gaia EDR3.", "We calibrate the Gaia EDR3 uncertainties using a sample of radial velocity standard stars without binary companions; we find an error inflation factor (a ratio of total to formal uncertainty) of 1.37.", "This is substantially lower than the position dependent factor of ~1.7 found for Gaia DR2 and reflects the improved data processing in EDR3.", "While the catalog should be used with caution, its proper motion residuals provide a powerful tool to measure the masses and orbits of faint, massive companions to nearby stars." ], [ "Introduction", "The Early Third Data Release of the Gaia astrometry mission [16] has now released position, parallax, and proper motion measurements for more than 1 billion stars [21].", "EDR3 represents a substantial improvement over the Second Data Release [15], [20], with proper motions improving by a magnitude-dependent factor of two to four.", "Gaia EDR3 has already improved distance estimates to anchor the cosmic distance ladder [29], [30] and detected proper motions of nearby dwarf galaxies [25].", "It has also, for the first time, provided a direct detection of the Solar acceleration in the Galactic potential [17].", "Gaia presently only performs single-star fits (generally with five astrometric parameters); accelerating and non-single stars will be treated in future data releases.https://www.cosmos.esa.int/web/gaia/release However, for bright stars, the Hipparcos mission thirty years ago provides a precise and independently measured position.", "This fact was used to construct the first Gaia proper motion catalog, the Tycho-Gaia Astrometric Solution [26], [22].", "The long-term proper motion given by the position difference between the Hipparcos and Gaia epochs, similar to TGAS, provides a precise measurement that may be compared to Gaia's proper motions.", "This permits an external calibration of Gaia, and allows the catalog to be used to find and study astrometrically accelerating systems even before the release of higher-order astrometric fits by the Gaia team itself.", "As released, however, Gaia DR2's proper motions are not suitable for finding accelerating stars or for fitting orbits.", "A cross-calibration is required first to enforce a common reference frame and to ensure Gaussian residuals with the appropriate variance [5].", "Gaia EDR3 provides positions and proper motions based on 34 months of data rather than the 22 months used for DR2.", "The additional data, combined with better control of systematics, result in formal proper motion uncertainties nearly a factor of 4 lower than DR2 for bright stars.", "The Gaia Collaboration has performed extensive verification of the EDR3 catalog, but this has focused on stars fainter than $G \\approx 10$ [19], [12].", "The published verification analyses use a combination of binary stars, cluster measurements, and measurements of quasars and stars in the Large Magellanic Cloud.", "In these cases, the parallax and/or proper motions are known, or can be compared between stars known to have the same values to well within Gaia's uncertainty.", "The goal of this paper is to cross-calibrate Hipparcos specifically to Gaia EDR3, and to extend the cross-calibration and catalog verification of Gaia EDR3 to the brightest stars present in EDR3 ($G \\approx 4$ ).", "Gaia EDR3 represents a substantial increase in precision relative to DR2, especially in its proper motion measurements.", "With an updated cross-calibration, this improved precision will translate directly into better sensitivity to astrometric acceleration [18], [14], and better masses and orbits of faint companions to nearby stars [6], [7], [4], [24], [9], [36], [3].", "We structure the paper as follows.", "Section reviews our cross-match between Hipparcos and Gaia EDR3.", "Section summarizes the cross-calibration results of Brandt2018 with Gaia DR2, and describes how the improvements of EDR3 affect our approach.", "Section describes our construction of Hipparcos astrometric parameters and reference epochs from the two Hipparcos reductions.", "Section describes our local cross-calibration of the reference frames.", "In Section , we construct a reference sample and derive a spatially uniform inflation factor for the Gaia EDR3 uncertainties.", "Section summarizes the results of the cross-calibration and shows the final improvements relative to Gaia DR2.", "We describe the construction and structure of the catalog in Section .", "We conclude with Section ." ], [ "Cross-Matching ", "The first step in a cross-calibration of Hipparcos and Gaia EDR3 is a cross-match of sources in the two catalogs.", "The Gaia archive does provide such a cross-match, but it is very incomplete.", "Out of nearly 118,000 Hipparcos stars the cross-matches in the archive provide matches for just 100,000.", "This reflects quality checks that we wish to relax: we want a more complete cross-match that keeps stars that accelerate between the catalogs.", "Brandt2018 cross-match Hipparcos and Gaia based on a lenient positional criterion and subsequently matching colors and magnitudes.", "We take a simpler approach here: we propagate the Gaia EDR3 proper motions to the Hipparcos catalog epoch of 1991.25 and search for positional matches in Hipparcos within $1^{\\prime \\prime }$ using the conesearch function on the Gaia archive.", "We restrict the search to stars brighter than $G=13.7$ to limit the computational demand.", "Our $1^{\\prime \\prime }$ cross-match returns 116,224 results.", "A few hundred Hipparcos stars have multiple matches in Gaia EDR3.", "For these stars we choose the brightest Gaia EDR3 source.", "This leaves 115,346 matches, with one Gaia star for each Hipparcos star.", "Several thousand Hipparcos stars, including many extremely bright stars with $G \\lesssim 3$ , do not have matches with five-parameter solutions in Gaia EDR3.", "These stars saturate Gaia too heavily to recover reliable astrometry even with the most aggressive gating [21].", "We take the cross-matched sample of 115,346 stars as the starting point for the rest of our analysis." ], [ "Early DR3 vs. DR2", "In this section we summarize the main results from cross-calibrating Gaia DR2 to Hipparcos.", "We then review the differences between DR2 and EDR3 that most significantly impact a cross-calibration.", "Figure: Comparison of Gaia DR2 and EDR3 precisions.", "Left: the median uncertainties on DR2 proper motions (blue dot-dashed line), on long-term proper motions (orange dashed line), and their quadrature sum (green solid line) compared the median absolute difference between the DR2 and long-term proper motions (black line).", "The green curve represents the formal sensitivity of Hipparcos +Gaia DR2 to astrometric acceleration.Right: the same uncertainties and residuals but for Gaia EDR3 rather than DR2.", "The formal uncertainties on proper motion have fallen by a magnitude-dependent factor of ∼\\sim 3; the median astrometric residuals have also fallen accordingly.", "While the long-term proper motion was the most precise measurement for nearly all Hipparcos stars in DR2, the Gaia EDR3 proper motions are now the most precise measurements for about half of Hipparcos stars." ], [ "The ", "The DR2 version of the HGCA Brandt2018 represents a cross-calibration of three proper motions: the Hipparcos proper motions, the Gaia DR2 proper motions, and the position difference between Hipparcos and Gaia divided by the difference in epoch between the two surveys.", "This long-term proper motion is the most precise measurement for nearly 98% of stars after calibrating the Gaia DR2 uncertainties.", "This permits a clean separation of Hipparcos and Gaia proper motions: both can separately be calibrated to the more precise long-term proper motions.", "The uncertainties and systematics in Hipparcos and Gaia DR2 are, as a result, relatively straightforward to disentangle.", "Brandt2018 find, with $\\sim $ 150$\\sigma $ significance, that a 60/40 weighted average of the [11] and [34] Hipparcos reductions outperforms either on its own.", "They also find that the Hipparcos proper motions display small-scale structure that can be removed.", "Doing so improves the agreement of the Hipparcos proper motions with the long-term proper motions in a cross-validation set at high significance.", "Finally, Brandt2018 find that an additional 0.2 mas yr$^{-1}$ has to be added in quadrature to the proper motion uncertainties to achieve a statistical agreement with the long-term proper motions.", "The calibrated Hipparcos proper motion uncertainties are comparable to the reported uncertainties in the 1997 catalog [11] and larger than the uncertainties in [34], especially for bright stars.", "For Gaia DR2, Brandt2018 recover the frame rotation seen by [20] with modest additional small-scale variations.", "They determine a spatially variable error inflation (ratio of true to formal uncertainties) averaging 1.7.", "This error inflation varies spatially, primarily with ecliptic latitude and hence with the average number of observations according to Gaia's scanning pattern.", "Finally, Brandt2018 find that the stars in Gaia with the largest uncertainties (predominantly the very bright stars) have underestimated uncertainties even after error inflation." ], [ "Changes with EDR3", "Gaia EDR3 has implemented a number of improvements over DR2.", "The most important for the HGCA is a factor of nearly 4 improvement in proper motion uncertainties for bright stars [21], [12].", "A factor of $(34\\,{\\rm months}/22\\,{\\rm months})^{3/2} \\approx 2$ is attributable to the longer observational baseline; the rest is due to improved data processing and control of systematics [12].", "Gaia EDR3 has also removed most of the frame rotation that affected the bright star reference frame of DR2, and it has corrected the DOF bug [21].", "Figure REF shows the improvements from DR2 (on the left) to EDR3 (on the right).", "In this and subsequent figures, we show proper motions in right ascension and declination together.", "All of our quantitative analyses include the separate uncertainties and covariance of the two components of proper motion.", "From DR2 to EDR3, the formal proper motion uncertainties have fallen by a magnitude-dependent factor of three or four, while the median residuals between the long-term proper motion and the Gaia proper motion (corrected for frame rotation in DR2's case) have fallen by a factor of two to three.", "The long-term proper motions adopt the 60/40 mix of the two Hipparcos reductions found by Brandt2018.", "In DR2, the long-term proper motion uncertainties were dominated by Gaia errors for very bright magnitudes ($G \\lesssim 5$ ).", "They are now dominated by Hipparcos uncertainties at all magnitudes.", "In the DR2 edition of the HGCA, the median precision on the long-term proper motion was a factor of $\\sim $ 4 better than the Gaia proper motions.", "Structure and systematics in the proper motion residuals were overwhelmingly due to the rotation of Gaia's bright star reference frame.", "The Gaia EDR3 and long-term proper motions are comparably precise.", "Their difference is now equally sensitive to systematic rotations in the Gaia reference frame and positional systematics in the Hipparcos reference frame.", "Despite the difficulty in interpreting structures and systematics in the proper motion residuals, we follow the same approach to cross-calibrate the catalogs.", "Figure: Residuals between the Gaia EDR3 proper motions and the long-term proper motions, normalized by their combined uncertainty.", "We compute the normalized residuals separately for right ascension and declination and include both in the histograms.", "The residuals are much narrower than for DR2 (c.f.", "Figure 1 of Brandt2018) but still show a significant magnitude dependence.", "At the brighter magnitudes where Gaia EDR3 proper motion uncertainties dominate the error budget, the distributions are substantially wider than a unit Gaussian and reflect underestimated uncertainties.Figure REF shows the distribution of $z$ -scores: the residuals between the Gaia EDR3 and the long-term proper motions, divided by their standard errors.", "In this and subsequent figures, we compute $z$ -scores in right ascension and declination separately and then plot them together.", "The uncertainties on both are Gaussian even when the two components of proper motion are covariant.", "The standard errors are the quadrature sums of the formal uncertainties in the EDR3 proper motions and the positional uncertainties in EDR3 and the merged Hipparcos catalog, scaled by the 24.75 year baseline between the catalogs.", "The distributions deviate from a unit Gaussian.", "The width of the distribution increases at bright magnitudes where Gaia EDR3 proper motions dominate the error budget.", "This suggests that the EDR3 formal uncertainties underestimate the true uncertainties.", "This underestimate is expected [23] and also seen by the Gaia team [21], [12].", "The distributions of Figure REF also have tails well in excess of those of a Gaussian; these are likely from stars that are astrometrically accelerating.", "Figure REF compares favorably to Figure 1 of Brandt2018, the equivalent plot for Gaia DR2.", "The systematics from frame rotation are mostly gone and the formal uncertainties appear to be closer to the true uncertainties.", "A cross-calibration of Hipparcos and Gaia remains necessary, but the catalogs as published are in much closer agreement than they were for Gaia DR2." ], [ "Hipparcos Positions, Proper Motions, and Epochs", "The cross-calibration of Hipparcos in the HGCA Brandt2018 relies on Gaia positions, proper motions, and parallaxes.", "While all are more precise in EDR3, Gaia astrometry was already far more precise than the Hipparcos values for most stars in DR2.", "Our approach and results to the Hipparcos proper motions are essentially the same here as in Brandt2018.", "This applies especially to the linear combination of the two Hipparcos reductions used by Brandt2018 and the use of Gaia parallaxes to refine the other Hipparcos astrometric parameters.", "We summarize the calibration steps for the Hipparcos catalogs here.", "All of these steps closely match steps taken for the DR2 edition of the HGCA Brandt2018." ], [ "Use of the Gaia EDR3 parallaxes", "We use the Gaia EDR3 parallax measurements and uncertainties to refine the other Hipparcos astrometric parameters in the same way as Brandt2018.", "We refer to Section 4 of that paper for details.", "For Gaia DR2, Brandt2018 found an improvement of $\\sim $ 1% in the agreement between Hipparcos and Gaia proper motions after incorporating Gaia parallax measurements to refine the other Hipparcos astrometric parameters.", "We do not apply an error inflation to the Gaia EDR3 parallaxes for this purpose.", "[10] found that the EDR3 parallax uncertainties were underestimated by $\\lesssim $ 30% for stars with well-behaved fits.", "We find that the EDR3 parallaxes offer the same $\\sim $ 1% improvement to the Hipparcos proper motions as the DR2 parallaxes did.", "The DR2 parallaxes were precise enough relative to the Hipparcos measurements that EDR3 provides a negligible gain.", "The parallaxes also improve the Hipparcos positions: the mean absolute deviation between the long-term proper motions (whose uncertainties are dominated by Hipparcos) and the Gaia EDR3 proper motions fall by about 0.4% in right ascension and 0.2% in declination when incorporating Gaia parallaxes into the Hipparcos solution.", "Finally, we assess whether correcting for the parallax bias according to the prescriptions derived by [19] can further improve Hipparcos astrometry.", "These bias corrections are functions of source magnitude, color, and ecliptic latitude.", "We find that applying these corrections to EDR3 parallaxes offers a tiny improvement in the agreement of Hipparcos proper motions with long-term proper motions.", "However, it slightly degrades the agreement between long-term proper motions and Gaia EDR3 proper motions.", "As a result, we do not apply any bias corrections to the EDR3 parallaxes.", "The effects of applying a bias correction to the EDR3 parallaxes are at least two orders of magnitude smaller than the improvement from using the EDR3 parallaxes directly.", "Our analysis says little about the accuracy of the [19] bias corrections at bright magnitudes.", "The biases are much smaller than the Hipparcos parallax uncertainties, especially around $G \\sim 12$ where the corrections change abruptly with magnitude [19].", "Further, we can only measure the agreement between Gaia parallaxes and true parallaxes in the Hipparcos frame, which could be subject to their own biases and systematics." ], [ "The Astrometric Reference Epoch", "Our next step is to propagate positions to the central epochs of the Hipparcos observations (the epochs that minimize the positional uncertainties), exactly as was done by Brandt2018.", "The central epochs generally differ in right ascension and declination, and they also differ from the catalog epoch of 1991.25.", "This step results in a smaller uncertainty on position and removes covariance between position and proper motion.", "It also provides a more accurate approximation to the true epoch at which the proper motion is effectively measured.", "We apply this exact same approach to Gaia EDR3 using its covariance matrix.", "The perturbation to the reference epoch in right ascension is given by $\\delta t_{\\alpha *} = - \\frac{{\\rm Cov}(\\alpha *, \\mu _{\\alpha *})}{\\sigma ^2[\\mu _{\\alpha *}]}$ where ${\\rm Cov}(\\alpha *, \\mu _{\\alpha *})$ is the covariance between position and proper motion in right ascension ($\\alpha * \\equiv \\alpha \\cos \\delta $ ) and $\\sigma ^2[\\mu _{\\alpha *}]$ is the variance in proper motion.", "We compute Equation (REF ) and its counterpart for declination separately and propagate the positions to $t_{\\rm catalog} + \\delta t_{\\alpha *}$ and $t_{\\rm catalog} + \\delta t_{\\delta }$ .", "We then update the covariance matrices for the astrometric parameters as described in Brandt2018.", "With the astrometric reference epoch defined as above, we compute the long-term proper motions as, e.g., $\\mu _{HG, \\alpha *} = \\left( \\frac{\\alpha _{\\it Gaia} - \\alpha _{\\it Hip}}{t_{\\it Gaia,\\alpha } - t_{\\it Hip, \\alpha }} \\right) \\cos \\left[ \\frac{\\delta _{\\it Gaia} + \\delta _{\\it Hip}}{2} \\right].$ Once calibrated relative to the Hipparcos and Gaia EDR3 reference frames, this serves as one of the three proper motions supplied by the catalog." ], [ "The composite Hipparcos catalog", "Brandt2018 adopted a 60/40 mixture of the two Hipparcos reductions for both position and proper motion, with an additional uncertainty of 0.2 mas yr$^{-1}$ added in quadrature with the Hipparcos proper motions.", "The relative weights of the [11] and [34] reductions and the additional proper motion uncertainty were driven almost entirely by the Gaia DR2 position measurements and the Hipparcos proper motions.", "While the Gaia positions have improved with EDR3, they were already so precise with DR2 that the additional precision has no effect on the main results of Brandt2018.", "We perform one more check on the Hipparcos catalog mixture.", "The Gaia EDR3 proper motions are now comparable in precision to the long-term proper motions.", "We therefore compute the Hipparcos weights that give the best agreement between the Gaia EDR3 proper motions and the long-term proper motions.", "This is sensitive to the Hipparcos positions, but independent of the Hipparcos proper motions.", "We again find that a 60/40 mix is optimal, in agreement with the optimal mix found by Brandt2018 for the Hipparcos proper motions.", "We adopt this 60/40 linear combination of the two Hipparcos catalogs, and an error inflation of 0.2 mas yr$^{-1}$ added in quadrature, from Brandt2018." ], [ "A Common Reference Frame", "Our next step is to place the Hipparcos proper motions, the Gaia EDR3 proper motions, and the long-term proper motions on a common reference frame.", "Our approach is the same as that in Brandt2018, and we refer to that paper for full details.", "We provide a brief summary of the approach here before discussing our results.", "Brandt2018 first fit for global rotations between the three reference frames defined by the Hipparcos proper motions, the Gaia DR2 proper motions, and the long-term proper motions, but found that locally variable rotations were statistically preferred.", "We therefore begin with locally variable calibrations here.", "Our approach is to divide the sky into small tiles with approximately equal numbers of stars, fit for cross-calibration parameters in each one, and then produce a smoothed map for the correction using a Gaussian process regression with a Matérn covariance function.", "We do not know, a priori, how much structure there is in the residuals between the three reference frames.", "We therefore tile the sky at a range of resolutions in order to find the one that provides the best cross-calibration.", "We wish to use only stars that have constant proper motion to calibrate the reference frames.", "We therefore discard stars for which any proper motion is inconsistent at $>10\\sigma $ between Hipparcos, Gaia, and/or the long-term proper motion.", "This step removes about 28,000 stars (most of which are physically accelerating), leaving just over 87,000 to match the reference frames.", "Ever-finer tilings of the sky also carry a risk of overfitting, so we hold back 10% of the stars as a cross-validation data set.", "We also hold back our radial velocity reference stars (described in Section REF ) to avoid biasing our eventual calibration of the Gaia EDR3 proper motion uncertainties.", "Given a small patch of sky and a small number of stars in that patch, we fit for the frame rotation that maximizes the likelihood of a Gaussian mixture model ${\\cal L} &= \\prod _{{\\rm stars}~i} \\Bigg \\lbrace \\frac{g}{2\\pi \\sqrt{\\det {\\bf C}_i}} \\exp \\left[ -\\frac{\\chi _i^2}{2} \\right] \\nonumber \\\\&\\qquad \\qquad + \\frac{1 - g}{2\\pi \\sigma ^2} \\exp \\left[ -\\left( \\frac{\\left( \\Delta \\mu _{\\alpha *,i}\\right)^2 + \\left(\\Delta \\mu _\\delta ,i \\right)^2}{2\\sigma ^2} \\right)\\right] \\Bigg \\rbrace $ with $\\chi _i^2 =\\begin{bmatrix}\\Delta \\mu _{\\alpha *,i} & \\Delta \\mu _{\\delta ,i}\\end{bmatrix}{\\bf C}_i^{-1}\\begin{bmatrix}\\Delta \\mu _{\\alpha *,i} \\\\\\Delta \\mu _{\\delta ,i}\\end{bmatrix}.$ In Equation (REF ), $\\sigma $ is the width of a distribution to capture outliers, and $g$ , the prior for a star not to be an outlier, is set to $g=1/2$ ; both $g$ and $\\sigma $ are the same for all stars.", "For Hipparcos proper motions, we use the covariance matrix ${\\bf C}$ given by the weighted sum of the covariance matrix from each Hipparcos reduction, with a weight of 0.6 applied to the [34] reduction and a weight of 0.4 to the [11] reduction.These coefficients would be $0.4^2$ and $0.6^2$ if the two Hipparcos reductions were independent.", "However, they are based on the same observations.", "The improved agreement with long-term proper motions is much less than the factor of $\\sim \\sqrt{2}$ that would be expected for truly independent data sets.", "We then add an additional 0.2 mas yr$^{-1}$ in quadrature along the diagonal Brandt2018.", "We do not add any additional uncertainty to the positions.", "In a cross-calibration, an additive uncertainty to the Hipparcos positions is fully degenerate with an additive uncertainty to the Gaia EDR3 proper motions.", "We defer that analysis to Section .", "We fit for a frame rotation appropriate to all stars in each patch of sky.", "The frame rotation on an axis passing through this tile is unconstrained, so we set it to zero and constrain only the other two components.", "We maximize the likelihood given by Equation (REF ).", "Finally, we use the cross-validation sample to constrain the hyperparameters of a Gaussian Process Regression and to determine the optimal tiling of the sky." ], [ "Nonlinear Motion", "A star moving uniformly through space will apparently accelerate when its motion is projected onto spherical coordinates.", "We take each star's instantaneous position, proper motion, parallax, and radial velocity in Gaia EDR3 and convert it into a three-dimensional position and space velocity.", "For stars without radial velocities in EDR3, we use radial velocities taken from the Extended Hipparcos compilation [1].", "About 80% of Hipparcos stars have radial velocities from one source or the other.", "We assume zero radial velocity when computing three-dimensional velocities for the remaining 20% of stars.", "We propagate three-dimensional positions backwards using each star's three-dimensional velocity.", "We propagate by the time difference between Hipparcos and Gaia EDR3; this time varies from star to star and between right ascension and declination (Section REF ).", "We then measure the difference between the correctly propagated position $\\delta x$ and the linearly propagated position $\\delta x_{\\rm lin} = \\mu \\delta t$ divided by this time.", "We apply this nonlinearity correction to the difference between the long-term proper motion and the Gaia proper motion.", "We apply the same correction between the Hipparcos proper motion and the long-term proper motion; this is equivalent to applying twice the correction to the difference between the Hipparcos and the Gaia proper motions.", "We apply our computed nonlinearity corrections before calibrating the reference frames between Hipparcos, Gaia EDR3, and the long-term proper motions.", "The corrections are small for most stars but reach 16 mas yr$^{-1}$ (in declination) for Barnard's Star.", "For all but fifteen stars, the nonlinearity corrections in both right ascension and declination are less than 1 mas yr$^{-1}$ ." ], [ "The Hipparcos Proper Motions", "The Hipparcos data themselves have not changed since Gaia DR2 (apart from tiny changes in the Gaia parallax used to refine Hipparcos astrometry).", "As a result, our cross-calibration between the Hipparcos proper motions and the long-term proper motions is nearly identical to the results of Brandt2018.", "It does differ very slightly because we use a different set of stars to fit the reference frame and for cross-validation: Brandt2018 used stars with Hipparcos IDs ending in zero, while we use stars with a Gaia EDR3 random index ending in zero.", "Figure REF shows the local corrections to the Hipparcos proper motions to bring them into agreement with the long-term proper motions.", "These plots are very similar to Figure 6 of Brandt2018.", "Similarly as for the DR2 edition of the HGCA, these corrections to the Hipparcos proper motion frame increase the likelihood of the 8828 stars used for cross-validation by about $e^{340}$ over a constant frame rotation, and by about $e^{550}$ over no frame rotation at all.", "For the 87,386 stars consistent at $10\\sigma $ with constant proper motion, the correction of Figure REF increases the likelihood by a factor of about $e^{5000}$ over a constant frame rotation.", "This is somewhat more than the tenth power of $e^{340}$ , suggesting a modest degree of overfitting just as for the DR2 edition of the HGCA." ], [ "The Gaia Proper Motions", "We follow the same approach for Gaia.", "We use the same Gaussian mixture model as for DR2, but decrease the width of the broader, outlier Gaussian from 1 mas yr$^{-1}$ to 0.5 mas yr$^{-1}$ given Gaia's much improved precision.", "Brandt2018 used a coarser tiling of the sky than for Hipparcos proper motions, with more stars in each tile, to fit for the local frame rotation.", "Here, we fit for frame rotations between Gaia EDR3 and the long-term proper motions in 2578 tiles, with an average of 45 stars per tile, compared to 4586 tiles (with 25 stars per tile) for Hipparcos.", "Accounting for accelerating and cross-validation stars, we fit about 30 and 17 stars per tile for Gaia EDR3 and Hipparcos, respectively.", "The likelihood improvement for the cross-validation set of EDR3 proper motions is about $e^{290}$ over a constant frame rotation, and $e^{350}$ over no frame rotation; the corresponding improvements for the entire sample are about $e^{4900}$ and $e^{5400}$ .", "This again indicates only a modest degree of overfitting.", "Increasing the number of tiles (decreasing the number of stars per tile) offers a negligible improvement in the cross-validation sample at the cost of substantial overfitting; decreasing the number of tiles by 40% incurs a penalty of $\\sim \\!e^{20}$ in the likelihood of the cross-validation stars.", "The top panels of Figure REF show the resulting corrections to the proper motions as a function of position in equatorial coordinates.", "Figure: Local corrections to the difference between Gaia EDR3 proper motions and long-term proper motions.", "The top panels show the corrections independent of color and magnitude.", "The lower panels show corrections Ψ\\Psi that depend on color and magnitude; Equations () and () give the corrections for an individual star.", "The maps are Hammer projections in equatorial coordinates with α=δ=0\\alpha = \\delta = 0 at the center; east is left and north is up.", "[12] find that the Gaia EDR3 reference frame does show some dependence on magnitude and color.", "We test this possibility by using a coarser division of the sky and dividing our sample (again excluding the cross-validation set) by color and by magnitude.", "We use 920 tiles for this step, with $\\sim $ 100 stars per tile.", "We split along the median color, $\\nu _{\\rm eff} \\approx 1.57~\\mu $ m$^{-1}$ , and at $G = 7.1$ (slightly below the median $G \\approx 8$ ).", "We choose $G=7.1$ as it turns out to provide the best agreement with the data, and it is close to the median $G$ magnitude.", "It is also approximately the magnitude that separates the use of Gaia's lowest and its second-lowest gatings (gates being used to limit exposures and avoid saturation).", "We divide the sample sequentially.", "We first split by color, separately fitting the redder half and the bluer half of stars.", "We then split by magnitude and again separately fit the brighter and the fainter stars.", "Our Gaussian process regression will then operate on the difference between the correction for red and blue stars, or between bright and faint stars.", "We next choose the functional forms that we assume for the dependence of an individual star's correction on $\\nu _{\\rm eff}$ and $G$ magnitude.", "We assume that the correction will be proportional to $\\nu _{\\rm eff}$ relative to the catalog median normalized by its median absolute deviation, capping this value at $\\pm 3$ based on our results in the cross-validation data set.", "For magnitude, we instead choose a smoothed version of a Heaviside step function.", "This is inspired by the shift in gating and based on improved results over a linear relationship.", "We adopt the functional form $f(G) = 1 - \\frac{2}{1 + \\exp \\left(5(G - 7.1) \\right)} .$ The factor of 5 in the exponent allows for a smooth transition over about a magnitude.", "It performs slightly better in the cross-validation set than a step function and avoids a discontinuity.", "Variations in the reference frame as a function of color and magnitude are shown in Figure REF , and are highly significant.", "A color correction to the Gaia proper motions improves the likelihood of the cross-validation sample by a factor of $e^{168}$ , equivalent to about $18\\sigma $ .", "The improvement to the likelihood from the magnitude correction is lower at $e^{82}$ , but still significant at about $13\\sigma $ .", "Including both corrections improves the likelihood by $e^{257}$ .", "This is nearly equal to the product of the two improvements and indicates that the corrections may be regarded as independent.", "Intriguingly, the correction maps for color and magnitude look almost identical.", "Combining the two correction maps before smoothing with a Gaussian process regression slightly improves the likelihood of the cross-validation sample; it also reduces the potential for overfitting.", "We therefore compute and use a single set of maps to correct for frame rotation as a function of both color and magnitude.", "These maps are the ones that we use to compute likelihood ratios and that we show in Figure REF .", "The actual correction applied to an individual star is the value in the map multiplied by a factor $f(G, \\nu _{\\rm eff}) = 1 - \\frac{2}{1 + \\exp \\left(5(G - 7.1) \\right)} + \\frac{\\nu _{\\rm eff} - {\\rm M}( \\nu _{\\rm eff})}{{\\rm M}(|\\nu _{\\rm eff} - {\\rm M}(\\nu _{\\rm eff})|)}$ where ${\\rm M}(\\nu _{\\rm eff})$ denotes the median value of $\\nu _{\\rm eff}$ ; the denominator is the median absolute deviation.", "The color and magnitude-dependent corrections for a given star are the product of Equation (REF ) and the maps $\\Psi (\\alpha ,\\delta )$ shown in the lower panels of Figure REF .", "The total cross-calibration term between the Gaia EDR3 and long-term proper motions is then $\\Delta \\mu = {\\rm Base} + f(G, \\nu _{\\rm eff}) \\times \\Psi .$ For the full set of stars (about 10 times the cross-validation sample size), the likelihood improvements are nearly equal to the improvements in the cross-validation set raised to the tenth power.", "For color, magnitude, and color+magnitude corrections, they are approximately $e^{1530}$ , $e^{850}$ , and $e^{2390}$ , respectively.", "These indicate an almost complete lack of overfitting and reflect the smoothness of the correction maps shown in Figure REF .", "Most of the frame rotation that was present for the bright stars in Gaia DR2 was removed during the astrometric processing [21] and is not present in the maps shown in Figure REF .", "Those rotations were easily the most apparent parts of their DR2 counterparts (Figure 6 of Brandt2018).", "The maps still retain considerable structure, though the typical magnitude of this structure is a factor of $\\sim $ 20 smaller than for the Hipparcos proper motions.", "Both Hipparcos (when dividing the positions by 25 years) and Gaia EDR3 have similar uncertainties, and both contribute to the structure seen.", "The magnitude and color-dependent effects are likely attributable to the Gaia proper motions; [12] reported color- and magnitude-dependent frame rotations of up to 0.1 mas yr$^{-1}$ .", "The structure seen in the top panels of Figure REF probably has a component attributable to the Hipparcos positions and one attributable to the Gaia EDR3 proper motions." ], [ "The Gaia Uncertainties", "Brandt2018 calibrated the Gaia DR2 uncertainties using the same Gaussian mixture model they used to calibrate the proper motions.", "This succeeded for two reasons.", "First, the long-term proper motions were considerably more precise than the DR2 proper motions; the Hipparcos positional uncertainties were relatively unimportant.", "Second, the stars with real astrometric accelerations contributed little to the core of the distribution of residuals.", "In other words, the core of the distribution shown in Figure 9 of Brandt2018 appeared to be relatively uncontaminated by the distribution of real accelerators.", "Brandt2018 found strong evidence that the ratio of actual uncertainties to formal uncertainties in Gaia DR2—the error inflation factor—was a function of position on the sky.", "The analysis leading to this was possible only because of the low contamination of significant astrometric accelerators in the calibration sample.", "The same analysis on Gaia EDR3 finds no evidence of spatial variation in the error inflation factor.", "We speculate that the origin of this factor in DR2 may be associated with the DOF bug, which required an ad-hoc scaling of uncertainties.", "This ad-hoc scaling was a function of magnitude but not of the number of transits used, or of ecliptic latitude.", "It was through ecliptic latitude that the dependence was most visible in DR2 (see Figure 7 of Brandt2018).", "Our Gaussian mixture model with a constant variance for the background distribution turns out to be ill-suited to inferring the error inflation factor directly from the bulk of the data.", "Figure REF shows that real accelerators comprise a non-negligible fraction of stars even in the core of the distribution, with $z$ -scores near zero.", "In the following section we construct a calibration data set with almost no intrinsic astrometric acceleration suitable for calibrating the Gaia uncertainties.", "Throughout this section, we neglect the fact that a few percent of the stars have six-parameter astrometric solutions (where the color is fit together with the position, parallax, and proper motion).", "We have far too few stars to characterize any possible difference in the uncertainty calibration for these sources relative to the bulk of the stars with five-parameter fits." ], [ "A Set of Reference Stars", "To calibrate the uncertainties on the long-term proper motions and the Gaia EDR3 proper motions, we seek a sample of stars that are not accelerating within the precision of the measurements.", "For this purpose we turn to long-term radial velocity monitoring with precision spectrographs.", "We use the radial velocity time series as published by HIRES [8] and recalibrated by [31]; the uniformly calibrated HARPS database [32]; and the older but long-baseline Lick sample [13].", "We take all Hipparcos targets in each of our three radial velocity data sets and fit a linear trend, adding a jitter to enforce a reduced $\\chi ^2$ of unity.", "This subsumes any real companions, like planets, into a jitter term and/or a trend.", "This is not a problem because we are searching for stars with low jitter and no trend.", "Once we have determined the jitter, a simple linear fit returns the standard error of the trend, the radial velocity acceleration.", "For stars that were observed by more than one spectrograph, we combine the measurements by each spectrograph using inverse-variance weighted means.", "We next convert our radial velocity accelerations and their uncertainties to angular-equivalent units by multiplying by the Gaia EDR3 parallaxes.", "We then convert these values into proper motion units by multiplying by 12.375 years.", "This is half the time baseline between the Hipparcos and Gaia EDR3 catalog epochs, i.e., the effective time between the long-term proper motion and the EDR3 proper motion.", "The resulting precision is directly comparable to the precision of the difference between the long-term proper motion and the Gaia EDR3 proper motion.", "We seek to define a sample of stars showing no astrometric acceleration.", "Each star in our sample of candidate proper motion standard stars must meet five criteria: It must not have a common parallax companion in Gaia EDR3 within 10,000 AU in projection; Its radial velocity baseline must be at least five years on a single spectrograph; Its radial velocity jitter must be $<$ 10 m s$^{-1}$ for HIRES and HARPS, or $<$ 15 m s$^{-1}$ for Lick, to identify well-measured stars without planetary companions; Its radial velocity trend must be consistent with zero at $1\\sigma $ ; and Its radial velocity trend precision must be better than the precision of the proper motion difference.", "For the last step, we take the Gaia EDR3 formal uncertainties and add these in quadrature with the formal uncertainties of the Hipparcos positions divided by 24.75 years.", "We combine the published variances of the two Hipparcos reductions using a 60/40 mixture but apply no further error inflation.", "We then average the proper motion precisions in right ascension and declination to get a single value for each star.", "Figure: zz-scores for the set of radial velocity benchmark stars we use to calibrate Gaia EDR3 uncertainties.", "The thin blue line shows the best-fit trend divided by its standard error.", "The thicker orange line normalizes the best-fit trend by the combined uncertainty from the Gaia EDR3 and long-term proper motions, divided by 12.375 years and converted to physical units.", "The thickest green line adds the error inflation for Gaia EDR3 proper motions that we derive in Section .", "We use the average of the uncertainties in right ascension and declination for each star.Our procedure results in a set of 374 astrometric standard stars.", "Figure REF shows their $z$ -scores, the ratio of the best-fit radial velocity trend normalized to three different uncertainties.", "First, we normalize to the standard error of the radial velocity trend (after using jitter to enforce a reduced $\\chi ^2$ of unity).", "Second, we normalize to the formal uncertainty in the proper motion difference after converting units and averaging the precisions in right ascension and declination.", "Finally, we normalize to the calibrated uncertainties.", "These latter two distributions, shown with progressively thicker lines in Figure REF , are much narrower than unit Gaussians: the limits on line-of-sight accelerations are significantly smaller than Gaia EDR3's precision for tangential accelerations.", "Given our crude approach to modeling radial velocity jitter, we do not expect the distribution of $z$ -scores to be accurately Gaussian, and we make no inference on the true distribution of radial velocity accelerations using this data set.", "Some of our radial velocity standard stars could have low levels of real astrometric acceleration, e.g., from widely separated planets or brown dwarfs, but the purity of this sample will be far higher than for the catalog at large.", "In the next section we will derive an error inflation by applying a slightly revised Gaussian mixture model to the radial velocity standards." ], [ "Calibration", "We use the Gaussian mixture model given in Equation (REF ) to fit for two error inflation terms: an additive term (in quadrature) with the Hipparcos positional uncertainties, and a multiplicative term for the Gaia EDR3 uncertainties: $\\sigma ^2 = a^2 \\sigma ^2[\\mu _{G}] + \\frac{0.6 \\sigma ^2[x_{H2}] + 0.4 \\sigma ^2[x_{H1}] + a^2 \\sigma ^2[x_{G}] + b^2}{(t_G - t_H)^2},$ using $x$ to denote position in either right ascension or declination and the subscript to denote the catalog ($H1$  = [11], $H2$  = [34], $G$  = Gaia EDR3).", "The additive term could be identified with Hipparcos positions and/or with Gaia EDR3 proper motions, while the multiplicative term is clearly associated with Gaia EDR3.", "The covariance terms between right ascension and declination retain the factor $a$ but omit the term $b$ .", "A possible multiplicative error inflation on Hipparcos was ruled out by Brandt2018, and we see no evidence of one in the residuals that we derive below.", "We make two modifications to our fitting approach for Equation (REF ).", "First, we take our outlier distribution to be a Gaussian with a covariance matrix 4 times that given by Equation (REF ) (equivalent to 2 times the standard deviation) rather than one with a static width.", "This better accommodates stars with widely varying precision.", "Second, we fit for the probability $g$ that a star is not an outlier, i.e., that it is a good astrometric reference.", "We also tried this alternative outlier distribution in the analysis of Section , but found that it performed slightly worse under those circumstances.", "Figure: Proper motion residuals for our sample of 374 radial velocity standards without (left) and with (right) applying calibrations to the Gaia EDR3 astrometry.", "The histograms include proper motions in both right ascension and declination.", "The calibrated proper motion residuals (right panel) follow a Gaussian with unit variance.", "The deviations from the unit Gaussian are consistent with Poisson statistics.Our approach returns a best-fit $a = 1.37$ , $b = 0$ , and $g=1$ : all of the stars may be fit acceptably well by Equation (REF ) and do not require an outlier distribution.", "Figure REF shows the $z$ -scores of the astrometry of the radial velocity standards before (left) and after (right) applying this calibration to the uncertainties: the final distribution is accurately Gaussian with unit variance.", "Our result for $a$ is smaller than the error inflation found by Brandt2018 for Gaia DR2, and confirms EDR3's better control of systematic uncertainties [12], [21].", "The value of $b=0$  mas may be compared to the value of 0.2 mas yr$^{-1}$ found by Brandt2018 for proper motion uncertainties in Hipparcos.", "It suggests that no error inflation is needed on the Hipparcos positions and may, in part, be due to the use of Hipparcos positions to establish Gaia EDR3's bright star reference frame [21].", "We perform several consistency checks on our results.", "Modest variations in the sample size and inclusion criteria have a minor effect on our inferred error inflation factor $a$ .", "We describe these variations, and the effects on our results, below.", "First, we vary the width of the outlier distribution in Equation (REF ).", "Changing its variance by a factor from two to sixteen (1.4 to 4 in standard deviation) does not affect the results.", "As the outlier distribution becomes narrower still, it begins to merge with the central distribution and slightly decreases our inferred error inflation to $a \\approx 1.30$ However, this brings a negligible benefit to the likelihood: about $e^{0.2}$ or $\\Delta \\ln {\\cal L} = 0.2$ .", "As the outlier distribution merges with the central distribution the best-fit $g$ decreases; $g$ becomes effectively an additional free parameter.", "The data themselves do not demand this extra flexibility.", "For any relative width of the outlier distribution, we always infer an error inflation factor for Gaia EDR3 proper motions of at least 1.30.", "We also perform consistency checks using bootstrap resampling on our set of radial velocity standard stars, and by varying the criteria for inclusion in this set.", "Bootstrap resampling gives a mean value of $a=1.35$ with a standard deviation of 0.08, and a mean value of $b=0.08$ with a standard deviation of 0.11.", "The values of $a$ and $b$ are covariant: if we hold $b$ fixed at zero, the mean and standard deviation of $a$ become 1.37 and 0.07, respectively.", "Loosening the criteria for inclusion in our radial velocity standard sample slightly increases our inferred error inflation and makes it more sensitive to the outlier distribution, likely due to the inclusion of astrometric accelerators.", "Increasing our stringency for radial velocity precision to require $0.7\\sigma $ consistency with zero trend and a trend precision twice that of the astrometric precision yields just 142 reference stars, but this sample gives consistent best-fit values of $a=1.34$ , $g=1$ , and $b=0$ .", "Varying the sample of standard stars and the width of the outlier distribution change our inferred error inflation factor by only a few percent.", "We adopt $a=1.37$ at $b=0$ as our final values for the cross-calibration.", "Though we adopt $b=0$ , it is difficult to distinguish a small value of $b$ from a small change in $a$ given our limited sample size: part of our inferred error inflation might be properly ascribed to the Hipparcos positions.", "Some of the Hipparcos systematic uncertainties will also be shared by the Gaia EDR3 proper motions given the use of Hipparcos positions to establish the bright star reference frame [21].", "Any systematics shared by the two catalogs will cancel out in a cross-calibration.", "If we adopt an additional uncertainty of 0.2 mas in Hipparcos (c.f.", "the additional proper motion uncertainty inferred by Brandt2018), this would bring $a$ close to 1.3.", "Adopting $a=1.30$ and $b=0.3$  mas rather than $a=1.37$ and $b=0$ produces distributions of residuals that are indistinguishable by eye from those shown in Figure REF .", "Continued radial velocity monitoring will ultimately produce better and larger sets of standard stars and enable a better characterization of Gaia's true astrometric uncertainties." ], [ "Results", "The results of the cross-calibration are three proper motions on a common reference frame with calibrated uncertainties.", "For nearly all stars, the two most precise proper motions are the Gaia EDR3 proper motion and the long-term proper motion.", "Figure REF shows the residuals between these two proper motions, normalized by their uncertainties, before and after the calibrations we apply here.", "We overplot the sum of two Gaussians as a reference to guide the eye; it is not intended to rigorously fit the data.", "The left panel of Figure REF shows the increasing width of the distribution of $z$ -scores (normalized residuals) with stellar brightness.", "This reflects the fact that the formal Gaia EDR3 proper motion uncertainties underestimate the true uncertainties.", "For faint stars, the Hipparcos positional uncertainties dominate the error budget.", "The middle panel shows the effects of the calibrated Gaia proper motion uncertainties.", "These bring the $z$ -scores at different magnitudes into much better agreement with one another and with a unit Gaussian in the core.", "The right panel shows the additional effect of the proper motion calibrations shown in Figure REF .", "While the improvement does reflect some overfitting, our cross-validation sample shows that this overfitting is modest and that most of the improvement is real.", "Most of the stars shown in Figure REF are not accelerating within the uncertainties of Gaia EDR3.", "However, a substantial minority do appear to show astrometric acceleration.", "The proper motion difference has two components (one each for right ascension and declination), so the relevant $\\chi ^2$ distribution has two degrees of freedom.", "Overall, about 35,000 stars, or 30% of the sample, have a $\\chi ^2$ value for the difference between their Gaia EDR3 and long-term proper motions of at least 11.8.", "This threshold corresponds to a 0.3% false alarm rate for Gaussian uncertainties.", "Most of these stars have residuals too large for them to appear on Figure REF .", "Many more stars will show significant astrometric acceleration in future Gaia data releases.", "Figure: Left: median calibrated uncertainties as a function of magnitude, c.f.", "Figure .", "Right: precision improvement of EDR3 over DR2 in the calibrated Gaia proper motions, and in the astrometric acceleration (the difference between the long-term proper motions and the more precise of the Gaia or Hipparcos proper motions).", "The calibrated Gaia EDR3 proper motions represent a factor of ∼\\sim 4 improvement over their DR2 counterparts.", "This improvement is probably underestimated for very bright stars (G≲6G \\lesssim 6); these showed evidence that their proper motion uncertainties remained underestimated even after error inflation (Figure 9 of Brandt2018).", "The required error inflation for Gaia proper motions has fallen from a spatially dependent value averaging 1.7 in DR2 to a spatially uniform value of 1.37 in EDR3.", "The EDR3 proper motions are now more precise than the long term proper motions for about half of all Hipparcos stars, and are more precise than Hipparcos proper motions even for very bright stars.", "Overall, EDR3 offers a factor of ∼\\sim 3 improvement in sensitivity to astrometric acceleration compared to DR2.", "This factor is lower for G≲5G \\lesssim 5, where Hipparcos proper motions were more precise than DR2 proper motions, and at faint magnitudes where Hipparcos positional uncertainties dominate the astrometric acceleration error budget.Figure REF shows the final, calibrated uncertainties and residuals of the catalog as a function of magnitude, and the improvement over the DR2 edition of the HGCA.", "The calibrated proper motion uncertainties of Gaia EDR3 are now more precise than the long-term proper motions for about 40% of Hipparcos stars.", "The typical precision of the proper motion difference is about 50 $\\mu $ as yr$^{-1}$ between magnitudes 6 and 9.", "At a fiducial distance of 50 pc and taking the timescale of this discrepancy to be 12.375 years, this translates to an acceleration precision of about 1 m s$^{-1}$  yr$^{-1}$ .", "The precision is even better for closer stars.", "It represents an improvement by a factor of $\\sim $ 3 over the precision offered by the Gaia DR2 HGCA." ], [ "Use and Caveats", "The cross-calibration we provide assumes that the model sky paths used by Hipparcos and Gaia EDR3 are good approximations to the actual motions of the stars.", "This is appropriate for single stars and for stars with much fainter companions on long-period orbits.", "For these stars the acceleration between Hipparcos and Gaia will be large compared to the acceleration within either Hipparcos or Gaia.", "Our cross-calibrated proper motions will be unreliable for stars with companions of comparable brightness.", "In this case, Hipparcos, Gaia, or both will measure something closer to the photocenter motion of the system.", "Hipparcos provides seven and nine-parameter fits to stars that showed astrometric acceleration within Hipparcos.", "The Gaia EDR3 proper motions for these systems will likely be unreliable until the treatment of non-single stars in DR3.", "The cross-calibration should be taken with caution for extremely high proper motion stars.", "As an example, Barnard's Star has a $\\chi ^2$ between the Gaia EDR3 and long-term proper motions of about 80, corresponding to more than $8\\sigma $ .", "However, its discrepancy in proper motion in declination (0.56 mas yr$^{-1}$ ) is less than 4% of the 16 mas yr$^{-1}$ correction for nonlinear motion.", "The proper motion residual thus represents the apparent acceleration over about 6 months (4% of 12.375 years).", "Uncertainty in the true central epoch at which the proper motion is measured could account for much of this difference.", "When astrometric accelerations are measured at very high significance, uncertainties in the effective epoch at which proper motions are measured can be an important contribution to uncertainties in orbit fitting.", "This situation will be resolved when future Gaia data releases include the epoch astrometry.", "Finally, we caution the user against making inferences using the distribution of $\\chi ^2$ residuals.", "The catalog is intended to identify astrometrically accelerating stars for follow-up and to fit orbits to these systems.", "The residuals will reflect a mixture of astrometric acceleration due to faint companions, photocenter motion for stars with bright companions, short-period orbits for which the Gaia sky paths are not yet good approximations, and possible residual systematics in the catalog.", "However, for stars with faint, long-period companions, the combination of Hipparcos and Gaia provides a powerful tool for identifying systems, constraining orbits, and measuring masses." ], [ "Construction and Structure of the Catalog", "The catalog contains three proper motions, three covariance matrices, the epochs at which these are measured, and cross-calibration terms.", "The structure of the catalog is identical to that of the HGCA Brandt2018, with one additional entry.", "We provide the $\\chi ^2$ value (denoted chisq in the table) between the two most precise proper motion measurements.", "These are almost always Gaia EDR3 and the long-term proper motions.", "The $\\chi ^2$ values are not intended for population-level analyses but to select likely accelerators for follow-up by radial velocity or direct imaging surveys.", "The computed values account for covariance between right ascension and declination, and have two degrees of freedom.", "A false positive rate of 0.3%, equivalent to three Gaussian sigma, translates to $\\chi ^2 \\approx 11.8$ .", "The catalog provides the calibration terms between the various proper motions.", "This includes cross-calibration offsets between the Hipparcos, Gaia EDR3, and the long-term proper motion reference frames.", "It also includes the corrections for nonlinear apparent motion on the celestial sphere.", "These corrections can be undone to recover the Hipparcos or the long-term proper motion in its own reference frame.", "They are identical in name and form to their equivalents in the DR2 edition of the HGCA Brandt2018.", "The Hipparcos proper motions in the Gaia EDR3 reference frame are given by, e.g., $\\mu _{\\alpha *,H} = f \\mu _{\\alpha *,{H2}} + (1 - f)\\mu _{\\alpha *,{H1}} + \\xi _{\\alpha *,H} + 2 \\gamma _{\\alpha *}$ where $\\mu _{H1}$ and $\\mu _{H2}$ refer to the [11] and [34] reductions of Hipparcos, respectively, and $f = 0.6$ is the weight given to the Hipparcos re-reduction (Section ).", "The term $2\\gamma $ is the correction for projecting linear motion onto the celestial sphere, and $\\xi $ is the cross-calibration term between Hipparcos and Gaia EDR3.", "We compute $\\xi $ in practice as the sum of the calibration terms between Hipparcos and the long-term proper motion, and between the long-term proper motion and Gaia EDR3 proper motion.", "The long-term proper motions in the Gaia EDR3 frame are given by $\\mu _{\\alpha *,HG} = \\frac{f \\alpha _{H2} + (1 - f)\\alpha _{H1} - \\alpha _G}{t_G - (f t_{H2} + (1 - f)t_{H1})}\\cos \\delta + \\xi _{\\alpha *,HG} + \\gamma _{\\alpha *}$ where $\\alpha *$ refers to right ascension times cosine declination, $t$ refers to the astrometric epoch in a given catalog, and $\\xi $ , $\\gamma $ , and all subscripts have the same meanings as in Equation (REF ).", "lcrrrrrrrrrrr 0pt The Hipparcos–Gaia Catalog of Accelerations: Hipparcos Proper Motions Hipparcos Gaia EDR3 $\\mu _{\\alpha *,H}$ a $\\sigma _{\\alpha *,H}$ $\\mu _{\\delta ,H}$ a $\\sigma _{\\delta ,H}$ Corr $t_{\\alpha *,H}$ $t_{\\delta ,H}$ $\\xi _{\\alpha *,H}$ $\\xi _\\delta ,H$ $2\\gamma _{\\alpha *}$ $2\\gamma _{\\delta }$ Number Source ID 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 2cyear 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 1 2738327528519591936 $-$ 4.87 1.31 $-$ 1.02 0.80 0.35 1991.55 1991.28 $-$ 0.03 0.47 0.00 0.00 2 2341871673090078592 183.28 1.41 $-$ 1.27 0.78 0.14 1991.48 1991.42 $-$ 0.38 0.04 0.00 0.00 3 2881742980523997824 4.65 0.49 $-$ 2.97 0.41 0.17 1990.85 1991.05 0.02 0.25 0.00 0.00 4 4973386040722654336 62.85 0.60 0.55 0.57 $-$ 0.16 1991.01 1991.18 0.14 0.32 0.00 0.00 5 2305974989264598272 1.87 0.63 8.64 0.70 0.09 1991.10 1991.48 $-$ 0.15 0.07 0.00 0.00 6 2740326852975975040 224.05 5.77 $-$ 14.07 3.19 0.25 1991.34 1991.26 $-$ 0.02 0.29 0.00 0.00 7 2846308881856186240 $-$ 207.91 1.09 $-$ 200.99 0.78 0.41 1991.29 1991.23 0.08 0.21 0.00 0.00 8 2853169937491828608 19.04 1.32 $-$ 6.35 0.77 0.05 1991.57 1991.45 $-$ 0.06 0.19 0.00 0.00 9 2880160886370458368 $-$ 6.82 1.05 8.72 0.64 0.11 1991.26 1991.20 $-$ 0.16 0.25 0.00 0.00 10 4976500987226833024 42.09 0.96 40.81 0.80 $-$ 0.11 1991.23 1991.41 0.12 0.29 0.00 0.00 aValues include all local and nonlinearity corrections (e.g.", "$\\xi $ and $\\gamma $ , see Equation (REF )).", "lcrrrrrrrrr 0pt The Hipparcos–Gaia Catalog of Accelerations: Gaia EDR3–Hipparcos Scaled Position Differences Hipparcos Gaia EDR3 $\\mu _{\\alpha *,HG}$ a $\\sigma _{\\alpha *,HG}$ $\\mu _{\\delta ,HG}$ a $\\sigma _{\\delta ,HG}$ Corr $\\xi _{\\alpha *,HG}$ $\\xi _{\\delta ,HG}$ $\\gamma _{\\alpha *}$ $\\gamma _{\\delta }$ Number Source ID 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 1 2738327528519591936 $-$ 5.832 0.051 $-$ 5.093 0.029 0.37 $-$ 0.001 $-$ 0.004 0.000 0.000 2 2341871673090078592 181.510 0.048 $-$ 0.440 0.029 0.12 $-$ 0.012 0.008 0.000 0.001 3 2881742980523997824 5.762 0.014 $-$ 2.474 0.012 0.08 0.003 0.000 0.000 0.000 4 4973386040722654336 61.972 0.017 1.307 0.021 $-$ 0.32 $-$ 0.023 0.014 0.000 0.000 5 2305974989264598272 0.986 0.022 8.744 0.022 0.07 $-$ 0.015 $-$ 0.012 0.000 0.000 6 2740326852975975040 223.167 0.174 $-$ 11.586 0.096 0.37 0.003 $-$ 0.015 0.000 0.000 7 2846308881856186240 $-$ 211.040 0.039 $-$ 197.017 0.031 0.33 0.038 $-$ 0.005 0.001 0.000 8 2853169937491828608 18.713 0.048 $-$ 6.635 0.031 0.02 0.008 $-$ 0.025 0.000 0.000 9 2880160886370458368 $-$ 6.043 0.034 9.244 0.021 0.01 0.000 $-$ 0.009 0.000 0.000 10 4976500987226833024 42.321 0.028 40.833 0.028 $-$ 0.11 $-$ 0.022 0.013 0.000 0.000 aValues include all local and nonlinearity corrections (e.g.", "$\\xi $ and $\\gamma $ , see Equation (REF )).", "lcrrrrrrrrrr 0pt The Hipparcos–Gaia Catalog of Accelerations: Gaia EDR3 Proper Motions Hipparcos Gaia EDR3 $\\mu _{\\alpha *,G}$ a $\\sigma _{\\alpha *,G}$ $\\mu _{\\delta ,G}$ a $\\sigma _{\\delta ,G}$ Corra $t_{\\alpha *,G}$ $t_{\\delta ,G}$ $\\chi ^2[\\Delta \\mu ]$ Number Source ID 2cmas yr$^{-1}$ 2cmas yr$^{-1}$ 2cyear ($N_{\\rm d.o.f.}", "= 2$ ) 1 2738327528519591936 $-$ 0.360 0.079 $-$ 5.053 0.030 0.11 2015.93 2015.53 3515.5 2 2341871673090078592 179.805 0.783 $-$ 1.041 0.579 0.07 2015.82 2015.51 5.5 3 2881742980523997824 5.761 0.038 $-$ 2.406 0.032 0.19 2016.61 2016.14 4.2 4 4973386040722654336 61.965 0.019 1.302 0.025 $-$ 0.33 2015.97 2016.02 0.2 5 2305974989264598272 1.022 0.028 8.733 0.026 0.03 2016.38 2015.88 1.1 6 2740326852975975040 223.197 0.040 $-$ 11.498 0.026 0.23 2015.95 2015.43 0.8 7 2846308881856186240 $-$ 206.509 0.039 $-$ 196.098 0.017 $-$ 0.10 2015.62 2015.34 6859.7 8 2853169937491828608 18.746 0.094 $-$ 6.472 0.049 $-$ 0.22 2016.07 2015.82 8.6 9 2880160886370458368 $-$ 6.071 0.028 9.296 0.021 $-$ 0.15 2015.94 2015.85 3.3 10 4976500987226833024 42.298 0.014 40.841 0.018 $-$ 0.38 2016.01 2016.10 0.6 aValues are identical to those published in Gaia EDR3 [21].", "Tables , , and list the first ten rows of the Gaia EDR3 edition of the HGCA.", "Table is very similar to Table 2 of Brandt2018.", "The proper motions are systematically offset from their values in the DR2 edition of the HGCA because they are calibrated to the EDR3 reference frame rather than the rotating DR2 reference frame.", "Table likewise closely matches Table 3 in Brandt2018.", "Table shows considerable improvements in sensitivity compared to the DR2 proper motions (Table 4 of Brandt2018).", "Two of the first ten Hipparcos stars, HIP 1 and HIP 7, are accelerating at high significance.", "The acceleration in right ascension for HIP 1, for example, is significant at nearly $60\\sigma $ .", "The $\\chi ^2$ values in the last column show that the other eight stars are not accelerating at an equivalent significance of $3\\sigma $ ($\\chi ^2 = 11.8$ ).", "HIP 8 has $\\chi ^2 = 8.6$ , which is about $2.5\\sigma $ significant.", "The catalog format and data fields are identical to the DR2 edition of the HGCA apart from the $\\chi ^2$ field that we have added for convenience.", "Table lists the field names and their definitions.", "It is almost identical to Table 5 of Brandt2018." ], [ "Conclusions", "In this paper we have derived a cross-calibration of Hipparcos and Gaia EDR3 suitable for identifying astrometrically accelerating systems and for fitting orbits.", "The catalog contains just over 115,000 stars, of which about 30% are inconsistent with constant proper motion at 99.7% (equivalent to $3\\sigma $ in a Gaussian distribution).", "Our approach to the cross-calibration matches that of Brandt2018, and our results for Hipparcos proper motions carry over largely unchanged.", "This consists of a 60/40 mixture of the [34] and [11] reductions with an additional uncertainty of $0.2$  mas yr$^{-1}$ added in quadrature with the proper motion uncertainties, and a locally variable frame rotation of up to $\\sim $ 1 mas yr$^{-1}$ .", "We have confirmed the 60/40 mix of the two Hipparcos reductions using only Hipparcos positions.", "The Gaia Collaboration removed the global frame rotation from EDR3 using the Hipparcos positional measurements.", "As a result, the global frame rotation that was a dominant aspect of the cross-calibration for DR2 is gone.", "However, we find strong evidence for a locally variable cross-calibration, and for locally-variable frame rotation as a function of color and magnitude.", "The latter result is consistent with the findings of the Gaia Collaboration [12].", "The Gaia EDR3 proper motion uncertainties are more difficult to calibrate than their DR2 counterparts.", "For this purpose, we turn to long-term radial velocity surveys to construct a sample of astrometric reference stars.", "We find an error inflation factor, a ratio of total to formal uncertainties, of 1.37.", "Varying the details of the analysis and of the sample of radial velocity standard stars results in inferred inflation factors ranging from about 1.30 to just over 1.40.", "An error inflation of 1.37 is broadly consistent with the results for fainter stars [12] and is substantially lower than the spatially-variable factor of $\\sim $ 1.7 found by Brandt2018 for Gaia DR2.", "This reflects the improvements in data processing in EDR3, while the lack of spatial dependence may be related to the fixing of the DOF bug [20].", "The resulting catalog represents a factor of $\\sim $ 3 improvement in sensitivity to astrometric accelerations over the Gaia DR2 edition of the HGCA.", "The improvement varies with magnitude.", "It is lower for faint stars where the uncertainty is already dominated by the Hipparcos positional uncertainties, and it is lower for very bright stars where the Hipparcos proper motions were more precise than the Gaia DR2 proper motions.", "Hipparcos positional measurements will continue to provide valuable data points for bright stars even in future Gaia data releases that treat astrometrically accelerating stars internally.", "Orbital fits derived by our cross-calibration will also provide an independent check on the accelerations inferred by Gaia DR3.", "Software: astropy [2], [28], scipy [35], numpy [27], [33] T.D.B.", "thanks Daniel Michalik and G. Mirek Brandt for input on a draft of this paper, and an anonymous referee for corrections and suggestions for improvement.", "T.D.B.", "gratefully acknowledges support from National Aeronautics and Space Administration (NASA) under grant 80NSSC18K0439 and from the Heising-Simons Foundation (HSF) under grant 20190295.", "This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium).", "Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.", "lcr 0pt The Hipparcos–Gaia Catalog of Accelerations, EDR3 Edition: Description of Catalog Contents Parameter Name Units Description ${\\tt hip\\_id}$ Hipparcos identification number ${\\tt gaia\\_source\\_id}$ Gaia EDR3 source identification number ${\\tt gaia\\_ra}$ degrees Gaia EDR3 measured right ascension ${\\tt gaia\\_dec}$ degrees Gaia EDR3 measured declination ${\\tt radial\\_velocity}$ km s$^{-1}$ Measured radial velocity ${\\tt radial\\_velocity\\_error}$ km s$^{-1}$ Radial velocity standard error ${\\tt radial\\_velocity\\_source}$ Source of measured radial velocity, Gaia DR2 or XHIP [1] ${\\tt parallax\\_gaia}$ mas Gaia EDR3 parallax ${\\tt parallax\\_gaia\\_error}$ mas Gaia EDR3 parallax standard error ${\\tt pmra\\_gaia}$ mas yr$^{-1}$ Gaia EDR3 proper motion in right ascension, $d\\alpha /dt \\cos \\delta $ ${\\tt pmra\\_gaia\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmra\\_gaia}$ ${\\tt pmdec\\_gaia}$ mas yr$^{-1}$ Gaia EDR3 proper motion in declination ${\\tt pmdec\\_gaia\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmdec\\_gaia}$ ${\\tt pmra\\_pmdec\\_gaia}$ Correlation between ${\\tt pmra\\_gaia}$ and ${\\tt pmdec\\_gaia}$ ${\\tt pmra\\_hg}$ mas yr$^{-1}$ Calibrated proper motion in right ascension from the Hipparcos–Gaia positional difference ${\\tt pmra\\_hg\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmra\\_hg}$ ${\\tt pmdec\\_hg}$ mas yr$^{-1}$ Calibrated proper motion in declination from the Hipparcos–Gaia positional difference ${\\tt pmdec\\_hg\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmdec\\_hg}$ ${\\tt pmra\\_pmdec\\_hg}$ Correlation between ${\\tt pmra\\_hg}$ and ${\\tt pmdec\\_hg}$ ${\\tt pmra\\_hip}$ mas yr$^{-1}$ Calibrated proper motion in right ascension from the composite Hipparcos catalog ${\\tt pmra\\_hip\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmra\\_hip}$ ${\\tt pmdec\\_hip}$ mas yr$^{-1}$ Calibrated proper motion in declination from the composite Hipparcos catalog ${\\tt pmdec\\_hip\\_error}$ mas yr$^{-1}$ Calibrated uncertainty in ${\\tt pmdec\\_hip}$ ${\\tt pmra\\_pmdec\\_hip}$ Correlation between ${\\tt pmra\\_hip}$ and ${\\tt pmdec\\_hip}$ ${\\tt epoch\\_ra\\_gaia}$ year Central epoch of Gaia EDR3 right ascension measurement ${\\tt epoch\\_dec\\_gaia}$ year Central epoch of Gaia EDR3 declination measurement ${\\tt epoch\\_ra\\_hip}$ year Central epoch of Hipparcos right ascension measurement ${\\tt epoch\\_dec\\_hip}$ year Central epoch of Hipparcos declination measurement ${\\tt crosscal\\_pmra\\_hg}$ mas yr$^{-1}$ Difference in ${\\tt pmra\\_hg}$ from the catalog-computed value: $\\xi _{\\alpha *,HG}$ from Table ${\\tt crosscal\\_pmdec\\_hg}$ mas yr$^{-1}$ Difference in ${\\tt pmdec\\_hg}$ from the catalog-computed value: $\\xi _{\\delta ,HG}$ from Table ${\\tt crosscal\\_pmra\\_hip}$ mas yr$^{-1}$ Difference in ${\\tt pmra\\_hip}$ from the catalog-computed value: $\\xi _{\\alpha *,H}$ from Table ${\\tt crosscal\\_pmdec\\_hip}$ mas yr$^{-1}$ Difference in ${\\tt pmra\\_hip}$ from the catalog-computed value: $\\xi _{\\delta ,H}$ from Table ${\\tt nonlinear\\_dpmra}$ mas yr$^{-1}$ Correction to ${\\tt pmra\\_hg}$ from projecting linear motion onto the celestial sphere: $\\gamma _{\\alpha *}$ from Table ${\\tt nonlinear\\_dpmdec}$ mas yr$^{-1}$ Correction to ${\\tt pmdec\\_hg}$ from projecting linear motion onto the celestial sphere: $\\gamma _{\\delta }$ from Table ${\\tt chisq}$ $\\chi ^2$ value for a model of constant proper motion with 2 degrees of freedom" ] ]
2105.11662
[ [ "A Central Limit Theorem for Semidiscrete Wasserstein Distances" ], [ "Abstract We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $\\sqrt{n}\\{\\mathcal{T}_c(P_n,Q)-\\mathcal{W}_c(P,Q)\\}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported.", "We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability $Q$ and on the cost.", "Such results imply the central limit theorem for the $p$-Wassertein distance, for $p\\geq 1$.", "Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials." ], [ "Introduction", "A large number of problems in statistics or computer science require the comparison between histograms or, more generally, measures.", "Optimal transport has proven to be an important tool to compare probability measures since it enables to define a metric over the set of distributions which convey their geometric properties., see [27].", "Moreover, together with the convergence of the moments, it metrizes the weak convergence, see Chapter 7.1. in [28].", "It is nowadays used in a large variety of fields, moreover, in statistics and in particular in Machine learning, OT based methods have been developed to tackle problems in fairness as in [17], [13], [7], in domain adaptation ([24]), or transfer learning ([12]).", "Hence there is a growing need for theoretical results to support such applications and provide theoretical guarantees on the asymptotic distribution.", "In this framework, the optimal transport between distributions has to be estimated from a random sample of the empirical distributions.", "Hence some controls have to be added on the rate of convergence of the the empirical version of the optimal transport cost to its true value.", "Moreover determining the asymptotic behavior of the limit distribution enables to control precisely the deviations of the estimation errors and certify the accuracy of the estimation.", "This work focuses on the semi-discrete optimal transport, i.e.", "when one of both probabilities is supported on a discrete set.", "Such a problem has been studied in many contexts, including on resource allocation problem, points versus demand distribution, positions of sites such that the mean allocation cost is minimal ([15]), resolution of the incompressible Euler equation using Lagrangian methods ([10]), non-imaging optics; matching between a point cloud and a triangulated surface; seismic imaging( [20]), generation of blue noise distributions with applications for instance to low-level hardware implementation in printers( [4]), in astronomy ([19]).", "But in a more statistical point of view it can also be used to implement Goodness-of-fit-tests, in detecting deviations from a density map to have $P\\ne Q$ , by using the fluctuations of $\\mathcal {W}(P_n,Q)$ , see [15] and to the new transport based generalization of the distribution function, proposed by [5], when the probability is discrete.", "The most general formulation of the optimal transport problem considers $\\mathcal {X}, \\mathcal {Y}$ both Polish spaces.", "We use the notation $\\mathcal {P}(\\mathcal {X})$ (resp.", "$\\mathcal {P}(\\mathcal {Y})$ ) for the set of Borel probability measures on $\\mathcal {X}$ (resp.", "$\\mathcal {Y}$ ).", "The optimal transport problem between $P\\in \\mathcal {P}(\\mathcal {X})$ and $Q\\in \\mathcal {P}(\\mathcal {Y})$ for the cost $c:\\mathcal {X} \\times \\mathcal {Y}\\rightarrow [0,\\infty )$ is formulated as the solution of $\\mathcal {T}_c(P,Q):=\\inf _{\\gamma \\in \\Pi (P,Q)}\\int _{\\mathcal {X}\\times \\mathcal {Y}} c(\\textbf {x},\\textbf {y}) d \\pi (\\textbf {x}, \\textbf {y}),$ where $\\Pi (P,Q)$ is the set of probability measures $\\pi \\in \\mathcal {P}(\\mathcal {X}\\times \\mathcal {Y})$ such that $\\pi (A\\times \\mathcal {Y})=P(A)$ and $\\pi (\\mathcal {Y} \\times B)=Q(B)$ for all $A,B$ measurable sets.", "If $c$ is continuous and there exist two continuous functions $a\\in L^1(P)$ and $b\\in L^1(Q)$ such that $\\text{for all $(\\mathbf {x},\\mathbf {y})\\in \\operatorname{supp}(P)\\times \\operatorname{supp}(Q)$,}\\ \\ c(\\mathbf {x},\\mathbf {y})\\ge a(\\mathbf {x})+b(\\mathbf {y}),$ then Kantorovich problem (REF ) can be formulated in a dual form, as $\\mathcal {T}_c(P,Q)=\\sup _{(f,g)\\in \\Phi _c(P,Q)}\\int f(\\textbf {x}) dP(\\textbf {x})+\\int g(\\textbf {y}) dQ(\\textbf {y}),$ where $\\Phi _c(P,Q)=\\lbrace (f,g)\\in L_1(P)\\times L_1(Q): \\ f(\\textbf {x})+g(\\textbf {x})\\le c(\\textbf {x},\\textbf {y}) \\rbrace $ , see for instance Theorem 5.10 in [29].", "It is said that $\\psi \\in L_1(P)$ is an optimal transport potential from $P$ to $Q$ for the cost $c$ if there exists $\\varphi \\in L_1(Q)$ such that the pair $(\\psi , \\varphi )$ solves (REF ).", "We assume the observation of a sample $X_1, \\dots , X_n$ i.i.d.", "with law $P$ .", "The empirical measure $P_n=\\frac{1}{n}\\sum _{k=1}^n\\delta _{X_k}$ defines a random $\\mathcal {T}_c(P_n,Q)$ .", "Supposing that $P,Q\\in \\mathcal {P}(\\mathbb {R}^d)$ and both are absolutely continuous with respect to the Lebesgue measure, [9] proved that $E\\mathcal {W}_1(P_n,Q)$ converges to 0 with an order approximately of $\\frac{1}{n^{1/d}}$ .", "By using the triangular inequality, valid in Wassertein distances, we can not expect better order of convergence for the difference $\\lbrace \\mathcal {W}_1(P_n,Q)-\\mathcal {W}_1(P,Q)\\rbrace $ .", "But when it comes to the difference $\\sqrt{n}\\lbrace \\mathcal {T}_c(P_n,Q)-E\\mathcal {T}_c(P_n,Q)\\rbrace $ , the results of [6], [8] prove, using the Efron-Stein's inequality, that it follows a Gaussian asymptotic behavior.", "In the case of $P,Q$ being supported in a finite (resp.", "countable) set, in [25] (resp.", "[26]) proved that $\\sqrt{n}\\lbrace \\mathcal {W}_c(P_n,Q)-\\mathcal {W}_c(P_n,Q)\\rbrace $ has a weak limit $X$ .", "Such a limit is the supremun of a Gaussian process.", "They start by the identification of the space of distribution supported in a finite set $\\mathcal {X}=\\lbrace \\mathbf {x}_1, \\dots , \\mathbf {x}_m\\rbrace $ with $\\mathbb {R}^m$ , and then give a proof based on the directional Hadamard differentiability of the functional $(\\mathbf {p},\\mathbf {q})\\mapsto \\mathcal {T}_c(\\sum _{i=1}^m p_i\\delta _{\\mathbf {x}_i}, \\sum _{i=1}^m{q_i}\\delta _{\\mathbf {x}_i})$ .", "In this work we are concerned with the asymptotic behaviour of $\\sqrt{n}\\lbrace \\mathcal {T}_c(P_n,Q)-\\mathcal {T}_c(P_n,Q)\\rbrace $ when $P$ is finitely supported but not $Q$ .", "When the quadratic cost is involved and $Q$ is absolutely continuous with respect to the Lebesgue measure, with convex support, [8] proves that the limit is in fact Gaussian.", "Their approach is also based in some differentiability properties of the optimal transport problem.", "But one of their main arguments is that the optimal transport potential is unique which is no longer true in general costs and neither in general Polish spaces.", "We need to use weaker notions of derivative and follow similar arguments of those of [25] and [26] to derive our main result, Theorem REF .", "As pointed out before, the result of [25] establishes that, if $P$ and $Q$ are both supported in a finite set, then $\\sqrt{n}\\left(\\mathcal {T}_c(P_n,Q)- \\mathcal {T}_c(P,Q)\\right)\\overset{w}{\\longrightarrow }\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)} \\mathbb {G}(\\mathbf {z}), \\ \\ \\text{where} \\ \\ \\mathbb {G}(\\mathbf {z}):=\\sum _{i=1}^m z_i X_i$ and $(X_1,\\dots ,X_n)$ is a centered Gaussian vector and $\\operatorname{Opt}_c(P,Q)$ is the set of solutions of the dual problem (REF ), both described in section .", "Lately [26] extended the same result for probabilities supported in countable spaces.", "Our work covers and generalizes the previously mentioned result of [25] through Theorem REF .", "We use similar tools but we make a slightly different extension.", "First we define in (REF ) a functional $\\mathcal {M}$ countinously differentiable with respect to $\\mathbf {p}$ in the positive hyperoctant, and where the optimization is made in the whole space $\\mathbb {R}^m$ , see Lemma REF .", "Hence it is a unconstrained optimization problem, consequently we can apply Danskin's theorem (Theorem 4.13 in [2]) instead of its Lagrangian counterpart, i.e.", "Theorem 4.24 in [2]).", "Such a slightly modification of the functional allows to realize that the central limit theorem holds assuming only finite support on one of both probabilities, which yields the central limit theorem for the semidiscrete case.", "Summarizing, the main difference between the approach of [25] and the one we make in section  is the lack of assumptions on the probability $Q$ .", "Section 3 goes further and analyses the cases where the transport potential is unique up to additive constants.", "In such cases, under some assumption of regularity on the cost and on $Q$ , the limit is not a supremun anymore, but simply a centered Gaussian random variable.", "The last section studies the semidiscrete O.T.", "in manifolds and gives a CLT not for the transport cost but for the solutions of the dual problem (REF ).", "We underline that if both probabilities are continuous and the space is not one dimensional, we cannot expect such type of central limit for the potentials, since, as commented before, the expected value of the estimation of the transport cost converges with rate $O(n^{-\\frac{1}{d}})$ and no longer $o(n^{-\\frac{1}{2}})$ .", "When the two samples are discrete, even if such a rate is $o(n^{-\\frac{1}{2}})$ , the lack of uniqueness of the dual problem does not allow to prove such type of problems.", "In consequence, the semidiscrtete is the unique case where such results, for the potentials of the O.T.", "problem in general dimension, can be expected." ], [ "Central Limit Theorem", "The goal is to prove a Central Limit theorem for the optimal transport cost in general Polish spaces $\\mathcal {X}, \\mathcal {Y}$ for the semi-discrete problem.", "For that reason during the rest of the paper we will assume that $P$ is supported in a a finite set.", "Let $\\mathbb {X}=\\lbrace \\mathbf {x}_1, \\dots , \\mathbf {x}_m \\rbrace \\subset \\mathcal {X}$ be such that $\\mathbf {x}_i\\ne \\mathbf {x}_j$ , for $i\\ne j$ , and denote by $\\mathcal {P}(\\mathbb {X})$ the set of probabilities defined in such set, i.e.", "any $P\\in \\mathcal {P}(\\mathbb {X})$ can be written as $\\text{$P:=\\sum _{k=1}^mp_k\\delta _{\\mathbf {x}_k}$, where $p_i>0$, for all $i=1, \\dots , m$, and $\\sum _{k=1}^mp_k=1$.", "}$ In consequence $P$ is characterized by the vector $\\mathbf {p}=(p_1, \\dots , p_m)\\in \\mathbb {R}^m$ .", "The following result shows that for the previous defined $P$ , the optimal transport problem is equivalent to a optimization problem on the first coordinate of the following functional over $\\mathbb {R}^m\\times \\mathbb {R}^m$ defined as $\\begin{split}\\mathcal {M}:\\mathbb {R}^m\\times \\mathbb {R}^m &\\rightarrow \\mathbb {R}\\\\(\\mathbf {z},\\mathbf {p})&\\mapsto \\mathcal {M}(\\mathbf {z},\\mathbf {p}):= \\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}|p_k| z_k +\\int \\min _{i=1, \\dots , m} \\lbrace c(\\mathbf {y},\\mathbf {x}_i ) -z_i \\rbrace dQ(\\mathbf {y}),\\end{split}$ where $||\\mathbf {p} ||_1=\\sum _{k=1}^{m}|p_k|$ .", "The following result, which is a straightforward adaptation of Proposition 4.2 in [8], shows that for the previous defined $P$ , the optimal transport problem is equivalent to a optimization problem on the first coordinate of $\\mathcal {M}$ .", "Lemma 2.1 Let $P\\in \\mathcal {P}(\\mathbb {X})$ , $Q\\in \\mathcal {P}(\\mathcal {Y})$ and $c$ be a cost satisfying (REF ), then $\\mathcal {T}_c(P,Q)=\\sup _{\\mathbf {z}\\in \\mathbb {R}^m} \\mathcal {M}(\\mathbf {z}, \\mathbf {p}).$ Remark 2.2 Note that if $\\varphi $ denotes an optimal transport potential from $P$ to $Q$ for the cost $c$ , then $\\mathcal {T}_c(P,Q)=\\mathcal {M}((\\varphi (\\mathbf {x}_1),\\dots , \\varphi (\\mathbf {x}_m)), \\mathbf {p}),$ which makes the link between the optimal transport potentials and optimal values of (REF ) as $\\mathbf {z}=(\\varphi (\\mathbf {x}_1),\\dots , \\varphi (\\mathbf {x}_m))$ .", "Let $X_1, \\dots , X_n$ be a sequence of i.i.d.", "random variables with law $P$ , since $X_k\\in \\mathbb {X}$ for all $k=1,\\dots ,n$ , the empirical measure $P_n:=\\frac{1}{n}\\sum _{k=1}^n\\delta _{X_k}$ belongs also to $\\mathcal {P}(\\mathbb {X})$ .", "In consequence it can be written as $P_n:=\\sum _{k=1}^mp_k^n\\delta _{\\mathbf {x}_k}$ , where $p_1^n, \\dots , p_m^n$ are real random variables such that $p_i^n\\ge 0$ , for all $i=1, \\dots , m$ , and $\\sum _{k=1}^np_k^n=1$ .", "We want to study the weak limit of the sequence $\\left\\lbrace \\sqrt{n}\\left(\\mathcal {T}_c(P_n,Q)- \\mathcal {T}_c(P,Q)\\right)\\right\\rbrace _{n\\in \\mathbb {N}}.$ Consider a centered Gaussian vector, $(X_1,\\dots , X_m)$ with variance matrix where $\\Sigma (\\mathbf {p}):= \\begin{bmatrix}p_1(1-p_1) & -p_1p_2 & \\cdots & -p_1p_m\\\\-p_2p_1 & p_2(1-p_2) & \\cdots &-p_2p_m\\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\-p_mp_1 & \\cdots &p_mp_{m-1}&p_m(1-p_m)\\end{bmatrix}.$ and define the class of optimal values $\\operatorname{Opt}_c(P,Q):=\\left\\lbrace \\mathbf {z}\\in \\mathbb {R}^d: \\ \\mathcal {T}_c(P,Q)= \\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\right\\rbrace .$ The following result shows that the class of optimal values $ \\operatorname{Opt}_c(P,Q)$ is non-empty and the main theorem makes sense.", "Lemma 2.3 Let $P\\in \\mathcal {P}(\\mathbb {X})$ , $Q\\in \\mathcal {P}(\\mathcal {Y})$ and $c$ be such that (REF ) holds, then $ \\operatorname{Opt}_c(P,Q)\\ne \\lbrace \\emptyset \\rbrace $ .", "The proof of Lemma REF is postponed to the appendix and is based on Lemma REF , described below and whose proof can be found also in the appendix.", "The following theorem shows that $\\left\\lbrace \\sqrt{n}\\left(\\mathcal {T}_c(P_n,Q)- \\mathcal {T}_c(P,Q)\\right)\\right\\rbrace _{n\\in \\mathbb {N}}$ has a weak limit.", "Theorem 2.4 Let $P\\in \\mathcal {P}(\\mathbb {X})$ , $Q\\in \\mathcal {P}(\\mathcal {Y})$ and $c$ be such that it satisfies (REF ) and $\\int c( \\mathbf {y}, \\mathbf {x}_i)dQ(\\mathbf {y})<\\infty , \\ \\text{ for all $i=1,\\dots , m$}.$ Then $\\sqrt{n}\\left(\\mathcal {T}_c(P_n,Q)- \\mathcal {T}_c(P,Q)\\right)\\overset{w}{\\longrightarrow }\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)} \\sum _{i=1}^m z_i X_i,$ where $(X_1,\\dots ,X_n)\\sim \\mathcal {N}(\\mathbf {0}, \\Sigma (\\mathbf {p}))$ and $\\Sigma (\\mathbf {p}) $ is defined in (REF ).", "When $\\mathcal {X}$ and $\\mathcal {Y}$ are contained in the same Polish space $(\\mathcal {Z},d)$ , a particular cost that satisfies the assumptions of Theorem REF is the metric $d$ .", "Then the value${\\mathcal {T}_{d^p}(P,Q)}$ follows also the asymptotic behaviour described in Theorem REF .", "Using the classic delta-method, the following theorem establishes the asymptotic behavior of the well known $p$ -Wasserstsein distance $\\mathcal {W}_p(P,Q)=\\@root p \\of {\\mathcal {T}_{d^p}(P,Q)}$ .", "Corollary 2.5 Let $P\\in \\mathcal {P}(\\mathbb {X})$ and $Q\\in \\mathcal {P}(\\mathcal {Z}) $ be such that $\\int d( \\mathbf {y}, \\mathbf {x}_0)dQ(\\mathbf {y})<\\infty , \\ \\text{ for some $\\mathbf {x}_0\\in \\mathcal {X}$}.$ Then, for any $p\\ge 1$ , we have $\\sqrt{n}\\left(\\mathcal {W}_p(P_n,Q)- \\mathcal {W}_p(P,Q)\\right)\\overset{w}{\\longrightarrow }\\frac{1}{p\\left(\\mathcal {W}_p\\right)^{p-1}}\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_{d^p}(P,Q)} \\sum _{i=1}^m z_i X_i,$ where $(X_1,\\dots ,X_n)\\sim \\mathcal {N}(\\mathbf {0}, \\Sigma (\\mathbf {p}))$ and $\\Sigma (\\mathbf {p}) $ is defined in (REF ).", "[Proof of Theorem REF ] We have divided the proof in a sequence of lemmas, finally we will use the Delta-Method for Hadamard differentiable maps.", "Recall that a function $f:\\Theta \\rightarrow \\mathbb {R}$ , defined in an open set $\\Theta \\subset \\mathbb {R}^m$ , is said to be Hadamard directionally differentiable at $\\theta \\in \\Theta $ if there exists a a function $f^{\\prime }_{\\theta }:\\mathbb {R}^m\\rightarrow \\mathbb {R}$ such that $\\frac{f(\\theta +t_n\\mathbf {h}_n)-f(\\theta )}{t_n}\\xrightarrow[n\\rightarrow \\infty ]{} f_{\\theta }(\\mathbf {h}),\\ \\ \\text{for all sequences $t_n\\searrow 0$ and $\\mathbf {h}_n\\rightarrow \\mathbf {h}$.", "}$ Let $\\Theta \\subset \\mathbb {R}^m$ be an open set, $\\theta \\in \\Theta $ and $\\lbrace X_n\\rbrace _{n\\in \\mathbb {N}}$ be a sequence of random variables such that $X_n:\\Omega _n\\rightarrow \\Theta $ and $r_n(X_n-\\theta )\\stackrel{w}{\\longrightarrow } X$ for some sequence $r_n\\rightarrow +\\infty $ and some random element $X$ that takes values in $\\mathbb {R}^m$ .", "If $f:\\Theta \\rightarrow \\mathbb {R}$ is Hadamard differentiable at $\\theta $ with derivative $f^{\\prime }_{\\theta }(\\cdot ):\\mathbb {R}^m\\rightarrow \\mathbb {R}$ , then Theorem 1 in [23], so-called delta-method, states that $r_n(f(X_n)-f(\\theta ))\\stackrel{w}{\\longrightarrow } f^{\\prime }_{\\theta }(X)$ .", "Since $p_i>0$ for all $i=1,\\dots ,m$ , it is enough to consider the hyperoctant $\\mathcal {U}_+:=\\lbrace \\mathbf {p}\\in \\mathbb {R}^m:\\ p_i>0, \\ i=1, \\dots , m\\rbrace .$ The aim is to prove that the map $\\begin{split}\\Gamma :\\mathcal {U}_+&\\rightarrow \\mathbb {R}\\\\\\mathbf {p}&\\mapsto \\sup _{\\mathbf {z}\\in \\mathbb {R}^m} \\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\end{split}$ is Hadamard differentiable.", "With that in mind, we start by proving the following technical lemma.", "Lemma 2.6 Let $Q\\in \\mathcal {P}(\\mathcal {Y})$ be such that (REF ) holds for the cost $c$ which satisfies (REF ).", "Then the functional $\\mathcal {M}$ satisfies: it is continuous in $\\mathbb {R}^m\\times \\mathcal {U}_+$ , it is continuously differentiable in the second variable over the set $\\mathcal {U}_+$ , moreover $\\nabla _{\\mathbf {p}} \\mathcal {M}(\\mathbf {z},\\mathbf {p})=\\left(\\frac{z_i\\sum _{j=1}^m p_j-\\sum _{j=1}^m z_j p_j}{\\left(\\sum _{j=1}^m p_j\\right)^2} \\right)_{i=1,\\dots , m}, \\ \\ \\text{for all $\\mathbf {z}\\in \\mathbb {R}^m$,}$ for all $\\mathbf {p}_0\\in \\mathcal {U}_+$ there exists $\\alpha \\in \\mathbb {R}$ and $\\epsilon >0$ satisfying that the set $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p}):= \\left\\lbrace \\mathbf {z}\\in \\mathbb {R}^m: \\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\ge \\alpha , \\ z_1=0\\right\\rbrace $ is compact and non empty, for all $\\mathbf {p}_0\\in \\mathcal {U}_+$ such that $|\\mathbf {p}_0-\\mathbf {p}|<\\epsilon $ .", "Note that Lemma REF yields that the functional $\\mathcal {M}$ satisfies the hypothesis of Danskin's theorem, see for instance Theorem 4.13 in [2].", "In consequence $\\Gamma $ is directionally differentiable.", "By proving that $\\Gamma $ is locally Lipschitz in $\\mathcal {U}_+$ , we obtain the following result.", "Lemma 2.7 The map $\\Gamma $ defined in (REF ) is Hadamard differentiable, with derivative $\\Gamma ^{\\prime }_{\\mathbf {p}} (\\mathbf {q})=\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)}\\sum _{i=1}^m \\frac{z_i\\sum _{j=1}^m p_j-\\sum _{j=1}^m z_j p_j}{\\left(\\sum _{j=1}^m p_j\\right)^2}q_i, \\ \\ \\text{for all $\\mathbf {q}\\in \\mathbb {R}^m.$}$ Note that since $P=\\sum _{j=1}^m p_j\\delta _{\\mathbf {x}_j}$ is a probability, $\\sum _{j=1}^m p_j=1$ and (REF ) can be simplified: $\\Gamma ^{\\prime }_{\\mathbf {p}} (\\mathbf {q})=\\sum _{i=1}^m \\left(z_i-\\sum _{j=1}^m z_j p_j \\right)q_i, \\ \\ \\text{for all $\\mathbf {q}\\in \\mathbb {R}^m.$}$ Then recall that the multivariate central limit theorem yields that $\\sqrt{n}\\left( \\mathbf {p}_n- \\mathbf {p}\\right)\\xrightarrow{} \\mathbf {X}=(X_1,\\dots ,X_m)\\sim N\\left(\\mathbf {0}, \\Sigma (\\mathbf {p})\\right),$ where $\\Sigma (\\mathbf {p})$ is defined in (REF ).", "The aforementioned Delta-Method for Hadamard differentiable maps and (REF ) yields that $\\sqrt{n}\\left( \\Gamma (\\mathbf {p}_n)- \\Gamma (\\mathbf {p})\\right)\\stackrel{w}{\\longrightarrow }\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)}\\sum _{i=1}^m \\left(z_i-\\sum _{j=1}^m z_j p_j \\right) X_i.$ Since $\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)$ implies that $\\mathbf {z}-\\mathbf {1}\\sum _{j=1}^m z_j p_j\\in \\operatorname{Opt}_c(P,Q)$ , then we have that $\\sqrt{n}\\left( \\Gamma (\\mathbf {p}_n)- \\Gamma (\\mathbf {p})\\right)\\stackrel{w}{\\longrightarrow }\\sup _{\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)}\\sum _{i=1}^m z_i X_i,$ which concludes the proof.", "Remark 2.8 Note that, while the idea of proving a central limit theorem for the optimal transport cost through the Hadamard differentiability properties of a suitable functional is not new at all (the same was done by [25] and [26]), the functional used in other references only holds for finitely (or countably) supported $Q\\in \\mathcal {P}(\\mathbb {Y})$ , with $\\mathbb {Y}=\\lbrace y_1,\\dots , y_l\\rbrace $ .", "In [25] the dual problem deals with the functional.", "More precisely $\\mathcal {G}(\\mathbf {u},\\mathbf {v},\\mathbf {p}, \\mathbf {q}):=\\sum _{k=1}^{m}p_k u_k +\\sum _{k=1}^{l}q_k v_k .$ The optimization problem is formulated under the following constraints $\\sup _{\\mathbf {u},\\mathbf {v}} \\mathcal {G}(\\mathbf {u},\\mathbf {v},\\mathbf {p}, \\mathbf {q}), \\ \\ s.t.", "\\ \\ u_k+v_j\\le c(x_k,y_j), \\ \\ \\text{for $k\\in \\lbrace 1,\\dots , m \\rbrace $ and $j\\in \\lbrace 1,\\dots , l \\rbrace $.", "}$ With regards to [26], the approach is based on a direct analysis of the primal formulation which is also a constrained optimization problem.", "The definition of $\\mathcal {M}$ and Lemma REF allow to use Danskin's theorem instead of constrained versions such as Theorem 4.24 in [2]).", "Moreover the division by the term $|| \\mathbf {p}||_1$ in the definition of $\\mathcal {M}$ implies the Hadamard differentiablity not only for a fixed direction, but in a general sense.", "This is a consequence of the existence of finite solutions of $\\sup _{\\mathbf {z}\\in \\mathbb {R}^m} \\mathcal {M}(\\mathbf {z}, \\mathbf {p})$ , for every $\\mathbf {p}\\in \\mathcal {U}_+$ .", "Moreover, the point 3 in Lemma REF is no longer true without such regularization by $|| \\mathbf {p}||_1$ , and in consequence the argument does not hold." ], [ "Asymptotic Gaussian distribution optimal transport cost.", "The aim of this section is to show that in the case where $\\mathcal {X},\\mathcal {Y}\\subset \\mathbb {R}^d$ and under some regularity assumptions, enumerated as (A1)-(A3), the limit distribution is Gaussian.", "Let $Q\\in \\mathcal {P}(\\mathbb {R}^d)$ be a probability measure uniformly continuous with respect to the Lebesgue measure in $\\mathbb {R}^d$ .", "Assume that $c(\\mathbf {x},\\mathbf {y})=h(\\mathbf {x}-\\mathbf {y})$ where $h:\\mathbb {R}^d\\rightarrow [0, \\infty )$ is a non negative function satisfying: $h$ is strictly convex on $\\mathbb {R}^d$ .", "Given a radius $r\\in \\mathbb {R}^+$ and an angle $\\theta \\in (0,\\pi ) $ , there exists some $M:=M(r, \\theta )>0$ such that for all $|\\mathbf {p} |>M$ , one can find a cone $K(r, \\theta , \\mathbf {z},\\mathbf {p}):=\\left\\lbrace \\mathbf {x}\\in \\mathbb {R}^d : | \\mathbf {x}-\\mathbf {p}|| \\mathbf {z}|\\cos (\\theta /2)\\le \\left< \\mathbf {z},\\mathbf {x}-\\mathbf {p} \\right>\\le r| \\mathbf {z}| \\right\\rbrace ,$ with vertex at $\\mathbf {p}$ on which $h$ attains its maximum at $\\mathbf {p}$ .", "$\\lim _{|\\mathbf {x} | \\rightarrow 0}\\frac{h(\\mathbf {x})}{|\\mathbf {x} | }= \\infty $ .", "Under such assumptions, [11] shows the existence of an optimal transport map $T$ solving $\\mathcal {T}_c(P,Q):=\\inf _{T}\\int c(\\textbf {x},T(\\textbf {x})) d P(\\textbf {x}), \\ \\ \\text{and}\\ \\ T_{\\#}P=Q.$ The notation $T_{\\#}P$ represents the Push-Forward measure, it is the measure such that for each measurable set $A$ we have $T_{\\#}P(A):=P(T^{-1}(A))$ .", "The solution of (REF ) is called optimal transport map from $P$ to $Q$ .", "Moreover it is defined as the unique Borel function satisfying $T(\\mathbf {x})=\\mathbf {x}-\\nabla h^*(\\nabla \\varphi (\\mathbf {x})), \\ \\ \\text{where $\\varphi $ solves (\\ref {dual}).", "}$ Here $h^*$ denotes the convex conjugate of $h$ , see [22].", "Such uniqueness enabled [6] to deduce the uniqueness, under additive constants, of the solutions of $\\varphi $ .", "They assumed (A1)-(A3) to show that if two $c-$ concave functions have the same gradient almost everywhere for $\\ell _d$ in a connected open set, then both are equal, up to an additive constant.", "In consequence assuming that $h$ is differentiable, the support of $Q$ is connected and with Lebesgue negligible boundary $\\ell _d\\left(\\partial \\operatorname{supp}(Q)\\right)=0$ , the uniqueness, up to additive constants, of the solutions of (REF ) holds.", "The proof of the main theorem is consequence of Lemma REF , which proves that there exists an unique, up to an additive constant, $\\mathbf {z}\\in \\operatorname{Opt}(P,Q) $ .", "We use within this section the notation $\\mathbf {1}:=(1,\\dots ,1)$ .", "Lemma 3.1 Under the hypothesis of Theorem REF , if $\\tilde{\\mathbf {z}}$ , $\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)$ , then $\\tilde{\\mathbf {z}}=\\mathbf {z}-L\\mathbf {1}$ for some constant $L\\in \\mathbb {R}$ .", "The following theorem states, under the previous assumptions, that the limit distribution of Theorem REF is the centered Gaussian variable $\\sum _{i=1}^m z_i X_i$ , where $\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)$ .", "Note that Lemma REF implies that the random variable $\\sum _{i=1}^m z_i X_i$ follows the same distribution independently of the chosen $\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)$ , which is $\\mathcal {N}(0,\\sigma ^2(P,\\mathbf {z})),$ with $\\sigma ^2(P,\\mathbf {z})=\\operatorname{Var}( \\sum _{i=1}^m z_i X_i)\\ \\ \\text{and } \\ \\ (X_1,\\dots ,X_m)\\sim \\mathcal {N}(\\mathbf {0},\\Sigma (\\mathbf {p})),$ where $\\Sigma (\\mathbf {p})$ is defined in (REF ).", "Since for every $\\lambda \\in \\mathbb {R}$ , we have that $ \\sigma ^2(P,\\mathbf {z})= \\sigma ^2(P,\\mathbf {z}+\\lambda \\mathbf {1})$ , then the asymptotic variance obtained in the following theorem is well defined.", "Theorem 3.2 Let $P\\in \\mathcal {P}(\\mathbb {X})$ and $Q\\in \\mathcal {P}(\\mathbb {R}^d)$ be such that $Q\\ll \\ell _d$ and its support is connected with Lebesgue negligible boundary.", "If the cost $c$ satisfies (A1)-(A3) and $\\int c( \\mathbf {y}, \\mathbf {x}_i)dQ(\\mathbf {y})<\\infty , \\ \\text{ for all $i=1,\\dots , m$.", "}$ Then $\\sqrt{n}\\left(\\mathcal {T}_c(P_n,Q)- \\mathcal {T}_c(P,Q)\\right)\\stackrel{w}{\\longrightarrow }\\mathcal {N}(0,\\sigma ^2(P,\\mathbf {z})),$ with $\\sigma ^2(P,\\mathbf {z})$ defined in (REF ) for $\\mathbf {z}\\in \\operatorname{Opt}_c(P,Q)$ .", "Since, in particular, the potential costs $c_p=|\\cdot |^p$ , for $p>0$ , satisfy (A1)-(A3), then the following result follows immediately from Theorem REF and the Delta-Method for the function $t\\mapsto |t|^{\\frac{1}{p}}$ .", "Recall that, in the potential cost cases, $\\mathcal {T}_p(P,Q)$ denotes the optimal transport cost and $\\mathcal {W}_p(P,Q)=\\left(\\mathcal {T}_p(P,Q)\\right)^{\\frac{1}{p}}$ the $p$ -Wasserstein distance.", "Corollary 3.3 Let $P\\in \\mathcal {P}(\\mathbb {X})$ be as in (REF ) and $Q\\in \\mathcal {P}(\\mathbb {R}^d)$ be such that $Q\\ll \\ell _d$ , has finite moments of order $p$ and its support is connected with Lebesgue negligible boundary.", "Then, for every $p>1$ , we have that $\\sqrt{n}\\left(\\mathcal {T}_p(P_n,Q)- \\mathcal {T}_p(P,Q)\\right)\\xrightarrow{} \\mathcal {N}(0,\\sigma ^2(P,\\mathbf {z})),$ and $\\sqrt{n}\\left(\\mathcal {W}_p(P_n,Q)- \\mathcal {W}_p(P,Q)\\right)\\stackrel{w}{\\longrightarrow }\\mathcal {N}\\left(0,\\left(\\frac{1}{p\\left(\\mathcal {W}_p\\right)^{p-1}}\\right)^2\\sigma ^2(P,\\mathbf {z})\\right),$ with $\\sigma ^2(P,\\mathbf {z})$ defined in (REF ) for $\\mathbf {z}\\in \\operatorname{Opt}_{c_p}(P,Q)$ .", "[Proof of Theorem REF ] Note that Theorem REF states that $\\sqrt{n}\\left(\\mathcal {T}_p(P_n,Q)- \\mathcal {T}_p(P,Q)\\right)\\stackrel{w}{\\longrightarrow }\\sup _{z\\in \\operatorname{Opt}_{c}(P,Q)}\\sum _{i=1}^m z_i X_i,$ where $(X_1,\\dots ,X_n)\\sim \\mathcal {N}(\\mathbf {0}, \\Sigma (\\mathbf {p}))$ , $\\Sigma (\\mathbf {p}) $ is defined in (REF ) and $\\operatorname{Opt}_c(P,Q)$ in (REF ).", "Lemma REF shows that such class is in fact a singleton, up to additive constants.", "Note that Corollary REF is a particular case of Corollary REF in the cases where the optimal transport potential is unique—the hypothesis of Theorem REF hold—which is the reason why the case $p=1$ can not be considered.", "With regards to the other potential costs, $p>1$ , it is straightforward to see that the hypothesis (A1)-(A3) hold, see for instance [6] or [11]." ], [ "A central Limit theorem for the potentials.", "The aim of the section is to deduce a CLT for the potentials $(\\psi ,\\varphi )$ of the transport problem.", "Recall that it refers to all pairs solving (REF ).", "In the semidiscrete case the potentials are pairs formed by $\\mathbf {z}=(z_1,\\dots ,z_m)\\in \\operatorname{Opt}_c(P,Q)$ and $\\varphi (\\mathbf {y}):=\\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y},\\mathbf {x}_i ) -{z_i} \\rbrace $ Note that if $(\\psi ,\\varphi )$ solves (REF ) then $(\\psi +C,\\varphi -C)$ , for all constant $C$ .", "That makes necessary the study the properties of the following functional, defined in $\\langle \\mathbf {1}\\rangle ^{\\perp } $ which denotes the orthogonal complement of the vector space generated by $\\mathbf {1}=(1,\\dots , 1)$ , $\\mathcal {M}_{\\mathbf {p}}:\\langle \\mathbf {1}\\rangle ^{\\perp } &\\longrightarrow \\mathbb {R}\\\\\\mathbf {z}&\\mapsto \\mathcal {M}(\\mathbf {z},\\mathbf {p}),$ where $\\mathcal {M}(\\mathbf {z},\\mathbf {p})$ is defined in (REF ).", "The idea behind the proof of the main result of the section, Threoem REF , is to use classical results in $M$ -estimation.", "In consequence, the strictly positiveness of the Hessian of the previously defined $\\Gamma $ is required.", "Such study of the second derivative have already been addressed in [18], where they used a completely different notation, more related to the angle of mathematical analysis than the statistical.", "Consequently, this section changes slightly the notations of the previous ones.", "In order to make the readers work easier, we will try to be as consistent as possible with the notation adopted in the foregoing sections and, at the same time, to adapt it to the one proposed in [18].", "First we will assume that $\\mathcal {Y}$ is an open domain of a $d$ -dimensional Riemannian manifold endowed with the volume measure $\\mathcal {V}$ and metric $d$ .", "We will follow the point of view and notations presented in [18] .", "For further details we refer to this paper and references therein.", "Following the approach of [18], let's assume the following assumptions on the cost: $\\text{\\qquad \\mathrm {(Reg)}$c(\\mathbf {x}_i,\\cdot )\\in \\mathcal {C}^{1,1}(\\mathcal {Y})$, for all $i=1,\\dots , m$,}$ $\\text{\\qquad \\mathrm {(Twist)}$D_{\\mathbf {y}}c(\\mathbf {x}_i,\\mathbf {y}):\\mathcal {Y}\\rightarrow T^*_{\\mathbf {y}}(\\Omega )$ is injective as a function of $\\mathbf {y}$, for all $i=1,\\dots , m$},$ where $D_{\\mathbf {y}}c$ denotes the partial derivative of $c$ w.r.t.", "the second variable.", "For every $i\\in \\lbrace 1, \\dots , m\\rbrace $ there exists $\\Omega _i\\subset \\mathbb {R}^d$ open and convex set, and a $\\mathcal {C}^{1,1}$ diffeomorphism $\\exp ^c_{i}:\\Omega _i\\rightarrow \\Omega $ such that the functions $\\text{\\qquad \\mathrm {(QC)}$\\Omega _{i}\\ni \\mathbf {p}\\mapsto f_{i,j}(\\mathbf {p}):=c(\\mathbf {x}_i,\\exp ^c_{i}\\mathbf {p})-c(\\mathbf {x}_j,\\exp ^c_{i}\\mathbf {p})$ are quasi-convex for all $j=1,\\dots , m$.", "}$ Here quasi-convex, according to [18], means that for every $\\lambda \\in \\mathbb {R}^d$ the sets $f_{i,j}^{-1}([-\\infty , \\lambda ])$ are convex.", "Assumptions on the cost are not enough at all, it becomes necessary that the probability is supported in a $c$ -convex set $\\Omega $ , which means that $ (\\exp ^c_{i})^{-1}(\\Omega )$ is convex, for every $i=1,\\dots , m$ .", "Formally, let $\\mathcal {Y}\\subset \\mathcal {M}$ be a compact $c$ -convex set, $P\\in \\mathcal {P}(\\mathbb {X})$ be as in (REF ) and suppose that $\\text{\\qquad \\mathrm {(Hol)}$Q\\in \\mathcal {P}(\\mathcal {Y})$ satisfies $Q\\ll \\mathcal {V}$ with density $q\\in \\mathcal {C}^{o,\\alpha }(\\mathcal {Y})$}.$ Recall that $f\\in \\mathcal {C}^{0,\\alpha }(\\mathcal {Y})$ if there exist $C>0$ such that $||f(x)-f(y)||\\le Cd(x,y)^{\\alpha .}", "$ The last required assumption in [18] is that $Q$ satisfies a Poincaré Wirtinger inequality with constant $C_{PW}$ : a probability measure $Q$ supported in a compact set $\\mathcal {Y}\\subset \\mathcal {M}$ satisfies a Poincaré Wirtinger inequality with constant $C_{PW}$ if for every $f\\in \\mathcal {C}^1(\\mathcal {Y})$ we have that for $Y\\sim Q$ $wE(|f(Y)-E(f(Y))|)\\le C_{PW} E(|\\nabla f(Y)|).\\qquad \\mathrm {(PW)}$ In order to clarify the feasibility of such assumptions, we will provide some insights on them at the end of the section.", "[18] proved the following assertions.", "Under assumptions (REF ) and (REF ) the function $\\mathcal {M}(\\cdot , \\mathbf {p})$ is concave and differentiable with derivative $\\nabla _{\\mathbf {z}}\\mathcal {M}(\\mathbf {z}, \\mathbf {p})= (-Q(A_2(\\mathbf {z}))+p_2, \\dots , -Q(A_m(\\mathbf {z}))+p_m ),$ where $A_k(\\mathbf {z}):=\\lbrace \\mathbf {y} \\in \\mathbb {R}^d :\\ \\ c(\\mathbf {x}_k,\\mathbf {y} ) -z_k <c(\\mathbf {x}_i,\\mathbf {y} )-z_i, \\ \\ \\text{for all $i\\ne k$} \\rbrace .$ Under assumptions (REF ),(REF ) and (REF ), the function $\\mathcal {M}_{\\mathbf {p}}$ is twice continuously differentiable with Hessian matrix $D_{\\mathbf {z}}^2\\mathcal {M}(\\mathbf {z},{\\mathbf {p}})=\\left(\\frac{\\partial ^2 \\mathcal {M}_{\\mathbf {p}}}{\\partial z_i\\partial z_j}(\\mathbf {z})\\right)_{i,j=1,\\dots ,m}$ and partial derivatives $\\begin{split}\\frac{\\partial ^2 \\mathcal {M}}{\\partial z_i\\partial z_j}(\\mathbf {z},{\\mathbf {p}})&=\\int _{A_k(\\mathbf {z})\\cap A_k(\\mathbf {z}) } \\frac{1}{|\\nabla _{\\mathbf {y}} c(\\mathbf {x}_i,\\mathbf {y})-\\nabla _{\\mathbf {y}} c(\\mathbf {x}_j,\\mathbf {y})|}dQ(\\mathbf {y}), \\ \\ \\text{if $i\\ne j$,}\\\\\\frac{\\partial ^2 \\mathcal {M}}{\\partial ^2 z_i}(\\mathbf {z},{\\mathbf {p}})&=-\\sum _{j\\ne i}\\frac{\\partial ^2 \\mathcal {M}}{\\partial z_i\\partial z_j}(\\mathbf {z},{\\mathbf {p}}).\\end{split}$ Under assumptions (REF ),(REF ) and (REF ), and if $Q$ satisfies (REF ), defined below, then there exists a constant $C$ such that $\\text{$\\langle D^2_{\\mathbf {z}}\\mathcal {M}(\\mathbf {z},{\\mathbf {p}}) \\mathbf {v}, \\mathbf {v} \\rangle \\le -C \\epsilon ^3 |\\mathbf {v}|^2$, for all $\\mathbf {z}\\in \\mathcal {K}^{\\epsilon }$ and $\\mathbf {v}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }$,}$ where $ \\mathcal {K}^{\\epsilon }:=\\lbrace \\mathbf {z}\\in \\mathbb {R}^d: \\ Q(A_i(\\mathbf {z}))>\\epsilon , \\ \\text{ for all $i=1, \\dots , m$.}", "\\rbrace .$ These previous results imply immediately the following Lemma.", "Lemma 4.1 Let $\\mathcal {Y}\\subset \\mathcal {M}$ be a compact $c$ -convex set, $P\\in \\mathcal {P}(\\mathbb {X})$ and $Q\\in \\mathcal {P}(\\mathcal {Y})$ .", "Under assumptions (REF ), (REF ) and (REF ) on the cost $c$ and (REF ) and (REF ) on $Q$ , we have that the function $\\mathcal {M}_{\\mathbf {p}}$ is strictly concave and twice continuously differentiable, with $\\nabla \\mathcal {M}_p(\\mathbf {z})&=\\nabla _{\\mathbf {z}}\\mathcal {M}(\\mathbf {z}, \\mathbf {p})|_{\\langle \\mathbf {1}\\rangle ^{\\perp }}, \\\\D^2 \\mathcal {M}_p(\\mathbf {z})&=D^2_{\\mathbf {z}}\\mathcal {M}(\\mathbf {z}, \\mathbf {p})|_{\\langle \\mathbf {1}\\rangle ^{\\perp }}.$ Moreover if $\\bar{\\mathbf {z}}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }\\cap \\operatorname{Opt}_c(P,Q)$ , then there exists a constant $C$ such that $\\text{$\\langle D^2\\mathcal {M}_{\\mathbf {p}}(\\bar{\\mathbf {z}}) \\mathbf {v}, \\mathbf {v} \\rangle \\le -C \\inf _{i=1, \\dots , m}|p_i| ^3 |\\mathbf {v}|^2$, for all $\\mathbf {v}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }$.", "}$ Note that it only remains to prove that (REF ) holds.", "But it is a direct consequence of (REF ).", "Actually, since $\\bar{\\mathbf {z}}$ is the unique $\\mathbf {z}\\in \\operatorname{Opt}(P,Q)$ then $Q(A_k(\\bar{\\mathbf {z}}))=p_k$ for $k=1,\\dots ,m$ and we can conclude.", "Then we can formulate the main theorem of the section which yields a central limit theorem for the empirical estimation of the potentials.", "We follow classical arguments of $M$ -estimation by writing the function $\\mathcal {M}_{\\mathbf {p}}(\\mathbf {z})$ as $ E(g(X, \\mathbf {z}))$ with $X\\sim P$ and $g:\\mathbb {X}\\times \\langle \\mathbf {1}\\rangle ^{\\perp }\\rightarrow \\mathbb {R}$ defined by $g(\\mathbf {x}_k, \\mathbf {z})&:= z_k+\\int \\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y},\\mathbf {x}_i ) -z_i \\rbrace dQ(\\mathbf {y}),$ for each $\\mathbf {z}=(z_1,\\dots ,z_m)$ .", "The weak limit is a centered multivariate Gaussian distribution with covariate matrix, depending on the optimal $\\tilde{\\mathbf {z}}$ , defined by the bilinear map $\\Sigma (\\tilde{\\mathbf {z}})=\\left(D^2\\mathcal {M}_{\\mathbf {p}}(\\tilde{\\mathbf {z}} )\\right)^{-1}A\\left(D^2\\mathcal {M}_{\\mathbf {p}}(\\tilde{\\mathbf {z}} )\\right)^{-1},\\ \\text{where $A=\\sum _{i=1}^m p_i \\nabla _{\\tilde{\\mathbf {z}}}g(\\mathbf {x}_i, \\mathbf {z}) \\nabla _{\\tilde{\\mathbf {z}}}g(\\mathbf {x}_i, \\mathbf {z})^{t}.$}$ Theorem 4.2 Let $\\mathcal {Y}\\subset \\mathcal {M}$ be a compact $c$ -convex set, $P\\in \\mathcal {P}(\\mathbb {X})$ and $Q\\in \\mathcal {P}(\\mathcal {Y})$ .", "Under assumptions (REF ), (REF ) and (REF ) on the cost $c$ , and (REF ) and (REF ) on $Q$ , we have that $\\sqrt{n}\\left(\\hat{\\mathbf {z}}_n-\\tilde{\\mathbf {z}}\\right)\\stackrel{w}{\\longrightarrow }N(\\mathbf {0},\\Sigma (\\tilde{\\mathbf {z}})),$ where $\\tilde{\\mathbf {z}}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }\\cap \\operatorname{Opt}_c(P,Q)$ (resp.", "$\\hat{\\mathbf {z}}_n\\in \\langle \\mathbf {1}\\rangle ^{\\perp }\\cap \\operatorname{Opt}_c(P_n,Q)$ ), and $\\Sigma (\\tilde{\\mathbf {z}})$ is defined in (REF ).", "Now let $g:\\mathbb {X}\\times \\langle \\mathbf {1}\\rangle ^{\\perp }\\rightarrow \\mathbb {R}$ be defined in (REF ).", "It satisfies that if $X\\sim P$ then $ E(g(X, \\mathbf {z}))=\\mathcal {M}_{\\mathbf {p}}(\\mathbf {z})$ .", "Lemma REF implies in particular that: the function $\\mathbf {z}\\mapsto g(\\mathbf {x}_k, \\mathbf {z})$ is concave for every $\\mathbf {x}_k$ .", "There exists a unique $\\tilde{\\mathbf {z}}\\in \\arg \\sup _{\\mathbf {z}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }} E(g(X, \\mathbf {z})).$ The empirical potential is defined as $\\hat{\\mathbf {z}}_n\\in \\arg \\sup _{\\mathbf {z}\\in \\langle \\mathbf {1}\\rangle ^{\\perp }} P_n(g(X, \\mathbf {z})).$ The function $\\mathbf {z}\\mapsto E(g(X, \\mathbf {z}))$ is twice differentiable in $\\tilde{\\mathbf {z}}$ with strictly negative definite Hessian matrix $D^2\\mathcal {M}_{\\mathbf {p}}(\\tilde{\\mathbf {z}} )$ .", "For every $\\mathbf {z}\\in \\mathbb {R}^{m-1}$ we have that $|\\nabla _{\\mathbf {z}}g(\\mathbf {x}_k, \\mathbf {z})|\\le |(Q(A_2(\\mathbf {z}))+1, \\dots , Q(A_m(\\mathbf {z}))+1 )|\\le \\sqrt{2(m-1)}.$ Then all the assumptions of Corollary 2.2 in [16] are satisfied by the function $g$ .", "In consequence we have the limit $\\sqrt{n}\\left(\\hat{\\mathbf {z}}_n-\\tilde{\\mathbf {z}}\\right)\\xrightarrow{} N\\left(\\mathbf {0},\\left(D^2\\mathcal {M}_{\\mathbf {p}}(\\tilde{\\mathbf {z}} )\\right)^{-1}A\\left(D^2\\mathcal {M}_{\\mathbf {p}}(\\tilde{\\mathbf {z}} )\\right)^{-1}\\right),$ where $A=E\\left( \\nabla _{\\mathbf {z}}g(X, \\mathbf {z}) \\nabla _{\\mathbf {z}}g(X, \\mathbf {z})^{t}\\right).$ Note that computing $A$ we obtain the expression (REF ).", "For $\\tilde{\\mathbf {z}}$ defined as in Theorem REF , set $\\varphi (\\mathbf {y}):=\\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y},\\mathbf {x}_i ) -\\bar{z_i} \\rbrace $ and note that it is an optimal transport map from $Q$ to $P$ , set also the value $i(y)\\in \\lbrace 1,\\dots , m\\rbrace $ where the infumum of (REF ) is attained.", "As previously we can define their empirical counterparts $\\varphi _n(\\mathbf {y}):=\\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y},\\mathbf {x}_i ) -\\hat{z}_i \\rbrace ,$ which is an optimal transport map from $Q$ to $P_n$ , and $i_{n}(y)$ the index where the infimum of (REF ) is attained.", "Then we have that $\\sqrt{n} (\\hat{z}_{i_n(y)}-\\bar{z}_{i_n(y)}) \\le \\sqrt{n}(\\varphi _n(\\mathbf {y})-\\varphi (\\mathbf {y}))\\le \\sqrt{n} (\\hat{z}_{i(y)}-\\bar{z}_{i(y)}).$ We can take supremums over $\\mathbf {y}$ in both sides of (REF ) and derive that $\\sqrt{n}\\sup _{i=1, \\dots , m}( \\hat{z}_{i}-\\bar{z}_{i})=\\sqrt{n}\\sup _{\\mathbf {y}\\in \\mathcal {Y}}(\\varphi _n(\\mathbf {y})-\\varphi (\\mathbf {y})).$ By symmetry we have that $\\sqrt{n}\\sup _{i=1, \\dots , m}| \\hat{z}_{i}-\\bar{z}_{i}|=\\sqrt{n}\\sup _{\\mathbf {y}\\in \\mathcal {Y}}|\\varphi _n(\\mathbf {y})-\\varphi (\\mathbf {y})|,$ which implies the following corollary.", "Corollary 4.3 Under the hypothesis of Theorem REF , for $\\varphi $ and $\\varphi _n$ defined in () and (), we have that $\\sqrt{n}\\sup _{\\mathbf {y}\\in \\mathcal {Y}}|\\varphi _n(\\mathbf {y})-\\varphi (\\mathbf {y})|\\stackrel{w}{\\longrightarrow }\\sup _{i=1,\\dots ,m} |N_i|,$ where $(N_1,\\dots , N_m)\\sim N(\\mathbf {0},\\Sigma (\\tilde{\\mathbf {z}})).$ We will conclude by some comments on the assumptions made in this section.", "Note that if we consider $\\mathcal {M}=\\mathbb {R}^d$ and the quadratic cost, then (REF ), (REF ) and (REF ) are obviously satisfied, by taking the function $\\exp _j$ as the identity.", "Actually the map $\\mathbf {y}\\mapsto |\\mathbf {x}_j-\\mathbf {y} |^2$ is $\\mathcal {C}^{\\infty }(\\mathbb {R}^d)$ and $\\mathbf {y}-\\mathbf {x}_j$ is its derivative w.r.t.", "$\\mathbf {y}$ .", "Finally note that the function $ \\mathbb {R}^d\\ni \\mathbf {p}\\mapsto |\\mathbf {x}_i-\\mathbf {p}|^2-|\\mathbf {x}_j-\\mathbf {p}|^2=|\\mathbf {x}_i|^2-|\\mathbf {x}_j|^2+\\langle \\mathbf {x}_j-\\mathbf {x}_i,\\mathbf {p}\\rangle $ is linear in $\\mathbf {p}$ and consequently quasi-convex.", "Assumption (REF ) on the probability $Q$ has been widely studied in the literature for its implications in PDEs, see [1].", "They proved that (REF ) holds for a uniform distribution on a convex set $\\Omega $ .", "In [21], Lemma 1 claims that (REF ) is equivalent to the bound of $\\inf _{t\\in \\mathbb {R}} E(|f(Y)-t|)$ , for every $f\\in \\mathcal {C}^1(\\mathcal {Y})$ .", "Let $Y\\sim Q$ be such that there exists a $\\mathcal {C}^1(\\mathcal {Y})$ map $T$ satisfying the relation $T(U)=Y$ , where $U$ follows a uniform distribution on a compact convex set $A$ .", "Since $f\\circ T\\in \\mathcal {C}^1(A)$ , by the powerful result of [1], there exists $C_A>0$ such that $\\inf _{t\\in \\mathbb {R}} E(|f(Y)-t|)&= \\inf _{t\\in \\mathbb {R}} E(|f(T(U))-t|)\\le C_{A} E(|\\nabla f(T(U))|\\cdot ||T^{\\prime }(U)||_2)\\\\&\\le C_{A}\\sup _{\\mathbf {u}\\in A}||T^{\\prime }(\\mathbf {u})||_2E(|\\nabla f(T(U))|),$ where $|| T^{\\prime }(U)||_2$ denotes the matrix norm.", "We deduce that in such cases (REF ) holds.", "Note that the existence of such map relies on the well known existence of continuously differentiable optimal transport maps, which is treated by Caffarelli's theory.", "We refer to the most recent work [3] and references therein.", "However, as pointed out in [18], more general probabilities can satisfy that assumption such as radial functions on $\\mathbb {R}^d$ with density $\\frac{p(|\\mathbf {x}|)}{| \\mathbf {x}|^{d-1}}, \\ \\ \\text{for $|\\mathbf {x}|\\le R$, with $p=0$ in $[0,r] $ and concave in $[r,R]$.", "}$ Moreover the spherical uniform $\\mathbb {U}_d$ , used in [14] to generalize the distribution function to general dimension, where we first choose the radium uniformly and then, independently, we choose a point in the sphere $\\mathbf {S}_{d-1}$ , also satisfies (REF ).", "This can be proved by using previous reasoning with the function $T(\\mathbf {x})=\\mathbf {x}| \\mathbf {x}|^{d-1}$ , which is continuously differentiable.", "But note that such probability measure does not satisfy the regularity condition of continuous density over a convex set, in consequence some additional work should be done which is left as a future work.", "In the same way as the regularity of the transport can be derived in the continuous case by a careful treatment of the Monge-Ampére equation, see [5], we conjecture that Theorem REF could hold also in that discrete framework." ], [ "Appendix", "[Proof of Lemma REF ] Note that Lemma REF yields that there exist $\\alpha >0$ such that $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p}):= \\left\\lbrace \\mathbf {z}\\in \\mathbb {R}^m: \\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\ge \\alpha , \\ z_1=0\\right\\rbrace $ is non empty and compact.", "Lemma REF also claims that $\\mathcal {M}$ is continuous, then $\\sup _{\\mathbf {z}\\in \\mathbb {R}^m} \\mathcal {M}(\\mathbf {z}, \\mathbf {p})=\\sup _{\\mathbf {z}\\in \\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p})} \\mathcal {M}(\\mathbf {z}, \\mathbf {p}) $ whose supremun is attained in some $\\mathbf {z}\\in \\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p})$ by Weierstrass extreme value theorem.", "[Proof of Lemma REF ] To prove the first point we realize that $\\mathcal {M}=F_1+F_2$ , where $ F_1((\\mathbf {z},\\mathbf {p})):=\\int \\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y}-\\mathbf {x}_i ) -z_i \\rbrace dQ(\\mathbf {y})\\ \\ \\text{and} \\ \\ F_2((\\mathbf {z},\\mathbf {p})):= \\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}|p_k| z_k $ are both continuous functions in $\\mathbb {R}^m\\times \\mathcal {U}_+$ .", "An easy computation shows the point 2. of Lemma REF .", "For the part 3, we set $\\mathbf {p}_0\\in \\mathcal {U}_+$ and a sequence $\\lbrace \\mathbf {p}_n\\rbrace _{n\\in \\mathbb {N}}$ converging to $\\mathbf {p}_0$ then the measure $ P_n:=\\frac{1}{\\sum _{j=1}^m p_j^n}\\sum _{j=1}^m p_i^n\\delta _{\\mathbf {x}_j}$ converges weakly to $ P_0=\\frac{1}{\\sum _{j=1}^m p_j^0}\\sum _{j=1}^m p_i^0\\delta _{\\mathbf {x}_j}$ .", "In consequence $\\mathcal {T}_c(P_n,Q)\\rightarrow \\mathcal {T}_c(P,Q)=\\lambda >0$ , when $n\\rightarrow \\infty $ .", "This means that there exists some $\\epsilon >0$ such that $\\sup _{\\mathbf {z}\\in \\mathbb {R}^m}\\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\ge \\alpha :=\\lambda /2$ , for all $\\mathbf {p}$ such that $|\\mathbf {p}_0-\\mathbf {p}|<\\epsilon $ .", "Since $\\mathcal {U}_+$ is open, we can assume that $\\mathbb {B}_{2\\epsilon }(\\mathbf {p}_0)=\\lbrace \\mathbf {q}\\in \\mathbb {R}^m: |\\mathbf {p}_0-\\mathbf {q}|<2\\epsilon \\ \\rbrace \\subset \\mathcal {U}_+$ .", "This means that $1-\\epsilon >\\frac{p_j}{||\\mathbf {p}||_1}>\\epsilon , \\ \\ \\text{for all $\\mathbf {p}\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p}_0)$}$ Therefore the set $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p}):= \\left\\lbrace \\mathbf {z}\\in \\mathbb {R}^m: \\mathcal {M}(\\mathbf {z}, \\mathbf {p})\\ge \\alpha , \\ z_1=0\\right\\rbrace $ is not empty for every $\\mathbf {p}\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p}_0)$ .", "The only point remaining concerns the compactness of $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p})$ .", "Note that $&\\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}|p_k| z_k +\\int \\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {y}-\\mathbf {x}_i ) -z_i \\rbrace dQ(\\mathbf {y})\\\\&\\le \\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}p_k z_k +\\inf _{i=1, \\dots , m} \\lbrace \\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y}) -z_i \\rbrace \\\\&\\le \\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}p_k z_k +\\sup _{i=1, \\dots , m}\\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y}) -\\sup _{i=1, \\dots , m} z_i\\\\&=\\frac{1}{||\\mathbf {p}||_1}\\sum _{k=1}^{m}p_k \\left(z_k-\\sup _{i=1, \\dots , m} z_i\\right) +\\sup _{i=1, \\dots , m}\\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y}) \\\\&=\\frac{1}{||\\mathbf {p}||_1}\\sum _{k=2}^{m}p_k \\left(z_k-\\sup _{i=1, \\dots , m} z_i\\right) +\\sup _{i=1, \\dots , m}\\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y}) +\\frac{1}{||\\mathbf {p}||_1}p_1 \\left(-\\sup _{i=1, \\dots , m} z_i\\right)\\\\&\\le K -\\frac{p_1}{||\\mathbf {p}||_1} \\left(\\sup _{i=1, \\dots , m} z_i\\right),$ where $K=\\sup _{i=1, \\dots , m}\\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y})\\in \\mathbb {R}$ .", "From (REF ) we deduce that $\\sup _{k} z_k $ has to be bounded $\\sup _{k} z_k \\le M$ .", "The task now becomes to show that $\\inf _{k} z_k $ has to be bounded.", "Since $z_1=0$ then $\\mathcal {M}(\\mathbf {z},\\mathbf {p} )&\\le \\inf _j \\frac{p_j}{||\\mathbf {p}||_1} \\inf _{k} z_k+\\sup _j \\frac{p_j}{||\\mathbf {p}||_1} \\sup _{k} z_k+ \\sup _{i}\\int c(\\mathbf {y}-\\mathbf {x}_i )dQ(\\mathbf {y})-z_1\\\\&\\le M+K+\\inf _j \\frac{p_j}{||\\mathbf {p}||_1}\\inf _{k} z_k.$ Finally (REF ) implies that $\\inf _{k} z_k \\ge M^{\\prime }$ .", "In consequence the set $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p})$ is bounded, it is clear that also closed, hence compact.", "[Proof of Lemma REF ] Danskin's theorem (Theorem 4.13 in [2]) considers an optimization problem $G(u)=\\max _{x\\in X}g(x,u)\\ \\ s.t.", "\\ \\ x\\in \\Phi ,$ where $u\\in U$ and it is a Banach space with the norm $|\\cdot |$ , $X$ is a topological Hausdorff space, $\\Phi \\subset X$ is nonempty and closed and $g$ is continuous.", "If the function $g(x,\\cdot )$ is differentiable, $g$ and its derivative w.r.t.", "$u$ in the direction $u$ $D_u g(x,u)$ are countinous in $X\\times U$ and for all $u_0\\in U$ there exists $\\alpha \\in \\mathbb {R}$ and $\\epsilon >0$ satisfying that the set $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p}):= \\left\\lbrace x\\in \\Phi : f(x,u)\\ge \\alpha \\ \\right\\rbrace $ is compact and non empty, for all $u$ such that $|u-u_0|<\\epsilon $ .", "Then the function $G$ is Fréchet directionally differentiable in $u_0$ , which means that, for every $h\\in U $ , the limit $G^{\\prime }_{u_0}(h)= \\lim _{t\\downarrow 0}\\frac{G(u_0+th)-G(u_0)}{t} $ exists and satisfies that $G(u_0+h)=G(u_0)+G^{\\prime }_{u_0}(h)+o(|h|).", "$ Note that Lemma REF yields that the functional $\\mathcal {M}$ satisfies the assumptions of Danskin's theorem in $\\mathbb {R}^d\\times \\mathcal {U}_+$ , but $\\mathcal {U}_+$ is not a Banach space, consequently some additional work has to be done.", "Set $\\mathbf {p}\\in \\mathcal {U}_+$ and note that there exists some $\\epsilon >0$ such that $\\overline{\\mathbb {B}_{\\epsilon }(\\mathbf {p})}\\subset \\mathcal {U}_+$ .", "The function $\\mathcal {M}$ restricted to such set $\\mathcal {M}|_{\\mathbb {R}^m\\times \\overline{\\mathbb {B}_{\\epsilon }(\\mathbf {p})}}$ is a continuous function in $\\mathbb {R}^m\\times \\overline{\\mathbb {B}_{\\epsilon }(\\mathbf {p})}$ , then we can apply Theorem 1 in [30] to conclude that there exists a function $\\tilde{\\mathcal {M}}:\\mathbb {R}^m\\times \\mathbb {R}^m\\rightarrow \\mathbb {R}$ such that $\\tilde{\\mathcal {M}}=\\mathcal {M}$ in $\\mathbb {R}^m\\times \\overline{\\mathbb {B}_{\\epsilon }(\\mathbf {p})}$ , $\\tilde{\\mathcal {M}}$ is analytic in $\\mathbb {R}^m\\times \\mathbb {R}^m\\setminus \\left(\\mathbb {R}^m\\times \\overline{\\mathbb {B}_{\\epsilon }(\\mathbf {p})}\\right)$ .", "Note that the equality $\\Gamma (\\mathbf {p})= \\sup _{\\mathbf {z}\\in \\mathbb {R}^m} \\mathcal {M}(\\mathbf {z}=\\sup _{\\mathbf {z}\\in \\mathbb {R}^m}\\tilde{ \\mathcal {M}}(\\mathbf {z},\\mathbf {p})$ implies that we can apply Danskin's theorem to $\\tilde{\\mathcal {M}}$ and in consequence $\\Gamma $ is Fréchet directionally differentiable in $\\mathbf {p}.$ We claim that $\\Gamma $ is locally Lipschitz.", "To prove it recall that Lemma REF yields that there exist $\\mathbf {z}$ such that $\\mathcal {M}(\\mathbf {z},\\mathbf {p})=\\mathcal {T}_c(P,Q)$ and $z_1=0$ .", "Lemma REF claims also that there exists $\\alpha \\in \\mathbb {R}$ and $\\epsilon >0$ such $\\operatorname{lev}_{\\alpha }\\mathcal {M}(\\cdot , \\mathbf {p})$ is non empty and compact for all $\\mathbf {p}^{\\prime }\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p})$ .", "Then for all $\\mathbf {p}^{\\prime }\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p})$ and all $\\mathbf {z}^{\\prime }$ solving $\\mathcal {M}(\\mathbf {z}^{\\prime },\\mathbf {p}^{\\prime })=\\mathcal {T}_c(P^{\\prime },Q)$ and $z^{\\prime }_1=0$ , we have that $|z^{\\prime }_1|\\le M$ .", "Moreover such $\\epsilon $ can be chosen small enough in order to have $\\mathbb {B}_{2\\epsilon }(\\mathbf {p})\\subset \\mathcal {U}_+$ In consequence we have that $\\Gamma (\\mathbf {p})-\\Gamma (\\mathbf {p}^{\\prime })&\\le \\mathcal {M}(\\mathbf {z},\\mathbf {p})-\\mathcal {M}(\\mathbf {z},\\mathbf {p}^{\\prime })=\\sum _{j=1}^m \\left(\\frac{p_k}{|\\mathbf {p}|_1}-\\frac{p^{\\prime }_k}{|\\mathbf {p}^{\\prime }|_1}\\right)z_j\\\\&\\le M \\sum _{j=1}^m \\left|\\frac{p_k}{|\\mathbf {p}|_1}-\\frac{p^{\\prime }_k}{|\\mathbf {p}^{\\prime }|_1}\\right|=M|F(\\mathbf {p})-F(\\mathbf {p}^{\\prime })|_1,$ where $F(\\mathbf {p}):=\\left( \\frac{p_1}{|\\mathbf {p}|_1}, \\dots , \\frac{p_m}{|\\mathbf {p}|_1} \\right)$ .", "We remark that the reverse inequality holds exactly with the same arguments.", "Then it suffices to prove that $F$ is Lipchitz in $\\mathbb {B}_{\\epsilon }(\\mathbf {p})$ .", "It can be seen by computing for any $\\mathbf {p}^{\\prime }\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p})$ the derivative $F^{\\prime }(\\mathbf {p}^{\\prime })=\\frac{1}{\\left(\\sum _{i= 1}^m p^{\\prime }_i\\right)^2} \\begin{bmatrix}{-\\sum _{i\\ne 1}p^{\\prime }_i} & -1 & \\cdots & -1\\\\-1 & {-\\sum _{i\\ne 2}p^{\\prime }_i} & \\cdots &-1\\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\-1 & \\cdots & -1&{-\\sum _{i\\ne m}p^{\\prime }_i}\\\\\\end{bmatrix},$ and checking that the matrix norm $||F^{\\prime }(\\mathbf {p}^{\\prime }) ||_{1}\\le K:= \\frac{m-1+|\\mathbf {p}|_1+\\epsilon }{\\left(|\\mathbf {p}|_1-\\epsilon \\right)^2}$ .", "Note that this inequality holds since $\\epsilon $ is small enough to make $|\\mathbf {p}|_1-\\epsilon >0 $ .", "Therefore we have that $|\\Gamma (\\mathbf {p})-\\Gamma (\\mathbf {p}^{\\prime })|\\le MK| \\mathbf {p}-\\mathbf {p}^{\\prime }|_1\\le \\sqrt{m}MK| \\mathbf {p}-\\mathbf {p}^{\\prime }|, \\ \\ \\text{for all $\\mathbf {p}^{\\prime }\\in \\mathbb {B}_{\\epsilon }(\\mathbf {p})$}.$ Then $\\Gamma $ is locally Lipschitz and directionally differentiable in $\\mathcal {U}_+$ .", "By Proposition 2.49 in [2], we conclude that it is differentiable at $\\mathbf {p}$ in the Hadamard sense.", "[Proof of Lemma REF ] Set $\\tilde{\\mathbf {z}},\\mathbf {z}\\in \\operatorname{Opt}(P,Q)$ .", "Lemma REF implies that the $c$ -concave functions $\\psi ( \\mathbf {y}):=\\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {x}_i, \\mathbf {y}) -z_i \\rbrace $ and $\\tilde{\\psi }( \\mathbf {y}):=\\inf _{i=1, \\dots , m} \\lbrace c(\\mathbf {x}_i, \\mathbf {y}) -\\tilde{z}_i \\rbrace $ are both solutions of the dual optimal transport problem from $Q$ to $P$ .", "Moreover, from optimal transport theory, the map ${\\psi }^c$ (resp.", "$\\tilde{\\psi }^c$ ) satisfies that $\\varphi (\\mathbf {x}_j)=z_j$ (resp.", "$\\tilde{\\varphi }(\\mathbf {x}_j)=\\tilde{z}_j$ ).", "Hence Theorem 2.6 in [6] implies that $\\psi =\\tilde{\\psi }+L$ holds in the interior of the support of $Q$ .", "Since $\\tilde{z}_i:=\\tilde{\\psi }^c (\\mathbf {x}_i)=\\psi ^c(\\mathbf {x}_i)-L= z_i -L,$ we have that $\\tilde{\\mathbf {z}}=\\mathbf {z}-L$ for some constant $L\\in \\mathbb {R}$ ." ] ]
2105.11721
[ [ "Gravitational Waves in a Closed Spacetime via Deviation Equation" ], [ "Abstract Within the closed universe, we obtain the amplitude and frequency of gravitational waves in the terms of discrete wave numbers, wave propagation time, and cosmological constant using the deviation equation in the first-order perturbed metric.", "We demonstrate that the cosmological constant effect on GWs is only seen in the early universe.", "Also, by considering the time evolution of a gravitational wave in a closed spacetime, we investigate its effect on a circle of nearby massless particles, which will be compared with this case in the flat spacetime.", "Expanding the universe has effective damping on GWs; thus, we suggest it can be used as a tool to characterize the large-scale curvature of the universe" ], [ "Introduction", "The recent detection of gravitational waves (GWs) by LIGO-Virgo has opened a new window into the universe [1].", "It was a first hint of the great potential of GWs detection for the universe exploration and understanding.", "For the standard theoretical analysis, GWs rely on the split of the spacetime metrics $g_{\\mu \\nu } $ into small perturbation in addition to the background metric.", "In general relativity, the curvature of the background is the fundamental field, characterizing gravity, which is not required to be the Minkowski spacetime.", "While our universe appears to be spatially flat approximately at the accuracy of $0.5\\%$ , the data do not entirely rule out the case of $K=1 $ [2], [3], [4].", "Recently, the Planck legacy 2018 CMB power spectra provide statistically significant indications for a closed universe [5].", "The curvature tension is considered evidence for closed spacetime in[6].", "Also, a positive curvature is marginally suggested by the ages of the oldest stars[7], [8].", "In addition, Planck 2018 CMB lensing [9] and baryon acoustic oscillations [10], [11], [12] suggest a flat universe are quantitatively inconsistent at $ 2.5 $ to $ 3\\sigma $ with CMB data.", "Hence, by choosing the maximally extended de-Sitter metric (K = 1) as the unperturbed background, we studied the propagation of gravitational waves [13] and its damping by neutrinos in closed spacetime [14], [15].", "As before, many authors have considered the de Sitter background to study GWs; M.Shibata et al.", "investigated the dynamical evolution of axisymmetric gravitational waves in asymptotically de-Sitter spacetime [16] and found that, if the mass of gravitational waves is larger than the critical value, $ M_{crit,}=(3\\sqrt{\\Lambda })^{-1} $ the formation of black holes will be prevented, even in the presence of highly non-linear and localized gravitational waves.", "Within Bondi-Saches formalism, some properties of GWs about de Sitter background have been investigated[17].", "Also, there has been an analysis on the behavior of the gravitational (electromagnetic) waves with the simplicity of the conformal transformation around de-Sitter background [18], [19].", "The GW's propagation has been investigated in an asymptotically de-Sitter spacetime by expanding the perturbation around the Minkowski spacetime in the presence of the cosmological constant, which at once $ \\Lambda $ is an additional source ( de-Donder gauge) and after is a gauge ($ \\Lambda $ -gauge) and then concluded that the cosmological constant, $ \\Lambda $ impedes the detection of gravitational waves at the distance larger than $ L_{crit}\\approx r_{\\Lambda }^{2} $ with $r_{\\Lambda }=\\frac{1}{\\sqrt{\\Lambda }}$ [20].", "The effect of the cosmological constant on the gravitational waves as a background perturbation, corresponding to the curvature; $ \\eta _{\\mu \\nu } + h_{\\mu \\nu }^{\\Lambda }$ , has been investigated; the wave-like perturbations can absolutely have a detectable impact on pulsar timing arrays and these waves are modified both in the phase and the amplitude [21].", "The linearized gravitational waves in de Sitter spacetime are analyzed in the presence of a positive cosmological constant to obtain the quadrupole formula and the theory of gravitational radiation[22], [23], [24].", "Another use of the cosmological constant, $ \\Lambda $ is in the energy-momentum pseudo-tensor of the gravity and the result can be formulated in the form of the critical distance proportional to $r_{\\Lambda }=\\frac{1}{\\sqrt{\\Lambda }}$ , the frequency and the amplitude of GWs, as well as the distance of the source from the detector [25].", "As mentioned in [13], we found the propagation of the gravitational waves in the closed background which depends on the wave numbers only.", "Thus far, the frequency and amplitude of gravitational waves have not been achieved; its effect on the particle's ring in the de Sitter background has not been investigated.", "The current paper is organized as follows: In Section II, we review the geodesic deviation equation (GDE) in a closed spacetime and its lowest order solution will be found in the next section with its effect on the nearby massless particles.", "Also, we provide the amplitude and frequency of GWs.", "Then, in Section IV, using the general solution of gravitational waves equation in closed spacetime, we investigate the passing of these waves on the particles ring and study the presence of an effectively closed background.", "Section V contains a discussion on more significant effects, quantitative differences, or sensitivity of the detector.", "Finally, we compare the results with those of the flat case." ], [ " Geodesic Deviation Equation on Closed Spacetime ", "To study the physical effects due to gravitational waves, the motion of test particles in the presence of this wave is considered.", "Although these test particles are massless, in Newtonian McVittie's background, the propagation of gravitation waves has been investigated in the presence of point mass [26].", "Also, to obtain the coordinate independent measurement, the relative motion of the nearby particles is described by the geodesic deviation equation (GDE).", "Therefore, there are some nearby particles with four-velocity described by a single vector field $ U^{\\mu }(x)$ and a separation vector $ S^{\\mu } $ .", "The geodesic deviation equation is [27]: $\\dfrac{D^{2}}{D\\tau ^{2}}S^{\\mu }=R^{\\mu }_{~\\rho \\sigma \\nu }U^{\\rho }U^{\\sigma }S^{\\nu }$ as we work in the first order metric perturbation in $h_{\\mu \\nu }$ , e.g.", "$ g_{\\mu \\nu }=\\bar{g}_{\\mu \\nu }+ h_{\\mu \\nu }$ where $ \\bar{g}_{\\mu \\nu }$ is the metric of background.", "If we take test particles to be moving slowly, the four-velocity can be expressed as a unit vector in the time direction, plus corrections of order $h_{\\mu \\nu }$ and higher.", "The Reimann tensor already is in the first order, so the correction to $ U^{\\nu } $ may be ignored.", "Thus, we write: $U^{\\rho }=(1,0,0,0)$ Therefore, we need to calculate $ R^{\\mu }_{00\\nu } $ .", "In general, the background is flat (or Minkowski) spacetime that can be found in any textbook of General Relativity or Cosmology (e.g., in [27]).", "This paper considers the maximally extended de Sitter spacetime as a background metric, the components of which in the Cartesian coordinates are [28] $\\bar{g}_{00}=-1~~~~~ ,~~~~ \\bar{g}_{0i}=0 ~~~~ and~~~~ \\bar{g}_{ij}=a^{2}(t)\\tilde{g}_{ij}= a^{2}(t)(\\delta _{ij}+K\\dfrac{x^{i}x^{j}}{1-K x^{2}})$ where $i$ and $j$ run over the values 1, 2 and 3, and $ x^{0}\\equiv t $ is the time coordinate in our units, with the speed of light equal to unity.", "Also, $x^{i}$ are the quasi-Cartesian coordinates, $K$ is curvature constant, and $ a(t)=\\alpha \\cosh (\\dfrac{t}{\\alpha }) $ is the scale factor in de Sitter spacetime with $ \\alpha =\\sqrt{\\dfrac{3}{\\Lambda }} $ Note: We assume that the radius of de Sitter spacetime ($l$ ) is unity.", "So, what we consider as cosmological constant is $\\Lambda l^2$ , which is a dimensionless value.", "If we consider $l\\approx 10^{26} m$ as the radius of our observable universe, our dimensionless $\\Lambda $ is in the unit of $10^{-52} m^{-2}$ ..", "The Riemann tensor with the first order perturbation is $R^{\\rho }_{\\mu \\nu \\lambda }=\\bar{R}^{\\rho }_{\\mu \\nu \\lambda }+\\delta R^{\\rho }_{\\mu \\nu \\lambda }$ where $ \\bar{R}^{\\rho }_{\\mu \\nu \\lambda }$ is the Riemann tensor of the background.", "The $R^{i}_{~00j} $ is: $R^{i}_{~00j}=\\dfrac{\\ddot{a}}{a}\\delta ^{i}_{\\nu }+\\dfrac{1}{2}\\tilde{g}^{i k}\\ddot{D}_{k j}+\\dfrac{\\dot{a}}{a}\\tilde{g}^{ik}\\dot{D}_{kj}$ where $D_{ij}= a^{-2}(t) h_{ij}$ and it is the solution for gravitational waves equation in curved spacetime [13]: $\\nabla ^{2} D_{ij}-a^{2}\\ddot{D}_{ij}-3 a \\dot{a}\\dot{D}_{ij} -2KD_{ij}=0$ The last term is the presence of non-zero curved background.", "To be more specific and without losing the generality, we chose the solution of equation (REF ) as a move in the z-direction in closed spacetime ($K=1$ ), e.g.", "for $ + $ mode from the traceless and transverse conditions in the closed background, we have[13] $ D^{i}_{+j}=\\frac{D_{+}(z,\\tau )}{\\sqrt{1-X^2}}\\left(\\begin{array}{ccc}\\dfrac{1}{(1-x^{2}-z^{2})}& 0 &\\frac{xz}{(1-z^{2})(1-x^{2}-z^2)}\\cr 0& -\\dfrac{1}{1-y^{2}-z^{2}}& \\dfrac{-yz}{(1-z^{2})(1-y^{2}-z^{2})}\\cr \\dfrac{xz}{(1-z^{2})(1-x^{2}-z^{2})}&\\dfrac{-yz}{(1-z^{2})(1-y^{2}-z^{2})}&\\dfrac{z^{2}(x^{2}-y^{2})}{(1-z^{2})(1-y^{2}-z^{2})(1-y^{2}-z^{2})}\\end{array}\\right).$ where $ X^{2}=x^{2}+y^{2}+z^{2} $ and $ D_{+}(z,t) $ is: ${D}_{+}(z,\\tau )=\\frac{(1-z^2)}{1-n^2}(\\cos \\tau \\mp in\\sin \\tau )\\left\\lbrace \\begin{array}{c}exp[\\pm in(\\arccos z+\\tau )]\\cr exp[\\pm in (\\arccos z-\\tau )].\\end{array}\\right.$ where $ n $ is the wave number and is an integer which should be discrete (our attempt to find periodic waves requires the wave number to be integer.", "We will have the same result for $\\times $ mode; but, only its matrix will be different and ${D}_{+}(z,\\tau )={D}_{\\times }(z,\\tau )$ (For more details, please see [13]).", "It can be seen that the time propagation of GWs in the closed spacetime is not simply a plane wave.", "Also, it has all properties of the transverse-traceless condition (please see Appendix A).", "Therefore, the deviation equation in the closed background becomes [13] $ \\dfrac{d^{2}}{d\\tau ^{2}}S^{i}=\\frac{\\Lambda }{3}S^{i}+\\dfrac{\\dot{a}}{a}S^{j}\\frac{d}{d\\tau } D^{i}_{~j}+\\dfrac{1}{2}S^{j}\\dfrac{d^{2}}{d\\tau ^{2}}D^{i}_{~j}$ As is clear from the above equation, the relative acceleration between the two neighboring geodesics is proportional to the tensor perturbation, the cosmological constant, and the expansion parameter.", "The first term in Eq.", "(REF ) provides exponential expansion, which is a general characteristic of de Sitter spacetime.", "The $H(t)=\\dfrac{\\dot{a}(t)}{a(t)}$ is the Hubble parameter, expressing the effect of expanding the universe.", "By considering all terms on the right-hand side of Eq.", "(REF ), the acceleration of neighboring geodesics will be interpreted physically as a gravitational tidal force.", "However, in the next section, we will find the general solution of the deviation equation in closed spacetime and, then, investigate only the effect of closed gravitational waves on closely spaced particles." ], [ " General Solution of Deviation Equation and its Effect", "To find complete solution of deviation equation for slowly moving particle, or the lowest order solution, we could write Eq.", "(REF ) as: $\\Big (\\dfrac{d^{2}}{d\\tau ^{2}}-\\frac{\\Lambda }{3}\\Big )S^{i}=S^{j} L_\\tau D^{i}_{~j}$ where $L_\\tau =\\dfrac{\\dot{a}}{a}\\dfrac{d}{d\\tau } +\\dfrac{1}{2}\\dfrac{d^{2}}{d\\tau ^{2}}$ is a linear differential operator.", "Also, as mentioned, $ D_{+}(z,\\tau )= D_{\\times }(z,\\tau )$ , we consider the mixed tensor as $D^{i}_{~j}=\\delta ^{i}_{~j} D_{+, \\times }(z,\\tau )$ .", "Then, $\\dfrac{d^{2}}{d\\tau ^{2}}S^{i}=\\frac{\\Lambda }{3} f(\\tau ,\\vec{x},\\Lambda ) S^{i}$ where $f(\\tau ,\\vec{x},\\Lambda )$ is defined as follows: $f(\\tau , \\vec{x},\\Lambda )=1+\\dfrac{3}{\\Lambda } L_\\tau D_{+, \\times }(z,\\tau )$ For the wave traveling in the z-direction, only $ S^{1} $ and $ S^{2} $ will be affected.", "For slow-moving particles, we have $ \\tau =x^{0}=t $ .", "Therefore, to the lowest order for $ \" + \" $ modes, the solutions are: $S^{1}(t, n, \\Lambda )&=& \\lbrace 1+\\dfrac{\\Lambda }{3}f(t,n, \\Lambda ) \\rbrace S^{1}(0)\\nonumber \\\\S^{2}(t, n, \\Lambda )&=& \\lbrace 1- \\dfrac{\\Lambda }{3} f(t,n, \\Lambda )\\rbrace S^{2}(0)$ Also, for $ \" \\times \" $ modes, we can find: $S^{1}(t, n, \\Lambda )&=& S^{1}(0)+ \\dfrac{\\Lambda }{3}f(t,n, \\Lambda )S^{2}(0)\\nonumber \\\\S^{2}(t, n, \\Lambda )&=&S^{2}(0)+ \\dfrac{\\Lambda }{3} f(t,n, \\Lambda ) S^{1}(0)$ where $ f(t,n, \\Lambda ) $ by using $ D_{+,\\times }(0,t) $ will be: $f(t,n, \\Lambda )&=&1+\\dfrac{3}{2\\Lambda } e^{i n t} \\frac{\\left(3 n^2+1\\right) \\cos t+i n \\left(n^2+3\\right) \\sin t}{ \\left(n^2-1\\right)} \\nonumber \\\\&&~~~~~~~~~~~~~~~+ \\sqrt{\\frac{3}{\\Lambda }} e^{i n t} \\tanh ({\\sqrt{\\dfrac{\\Lambda }{3}}}t) \\frac{\\left(n^2+1\\right) \\sin t-2 i n \\cos t }{n^2-1}$ Note again that $ n$ is the wave number and should be a discrete number.", "In this case, the effect of gravitational waves on massless particles depends on $ n $ and $ \\Lambda $ .", "So, the movement of particles will be different from the flat case.", "By comparing (REF ) and (REF ) with Relations (7.114) and (7.115) in Ref.", "[27], we can define $ h(t,n,\\Lambda ) $ and $ \\omega (t,n,\\Lambda ) $ based on the real and imaginary parts of $f(t,n,\\Lambda ) $ as: $\\dfrac{\\Lambda }{3} f(t,n,\\Lambda )= h(t,n,\\Lambda ) e^{i \\omega (t,n,\\Lambda ) t}$ Figure: The plot of h(t,n,Λ) h(t,n,\\Lambda ) respect to n n on two different ranges of time t t .", "(a): In the short time (or λ≫L B \\lambda \\gg L_{B}), the amplitude has a minimum and increases with the large number of n n (solid line).", "For Λ \\Lambda greater than 0.001, one can see its effect and the increase in the amplitude (dashed line is for Λ=0.003 \\Lambda =0.003 ).", "(b): In the long-time propagation, or λ≪L B \\lambda \\ll L_{B}, the amplitude reduces to the constant value (solid line).", "Comparing (a) and (b) shows that, for a small value of n, amplitudes behave like each other and are independent of the time of propagation.where $ h(t,n,\\Lambda ) $ and $ \\omega (t,n,\\Lambda ) $ are the amplitude and frequency (or phase term) of gravitational waves in closed spacetime, respectively, as: $h(t,n,\\Lambda )&=&\\dfrac{2}{3(n^{2}-1)}\\lbrace [2 \\Lambda \\left(n^2-1\\right)-3 n (n^2+3) \\sin (t) \\sin (n t)+3 (3 n^2+1) \\cos (t) \\cos (n t) \\nonumber \\\\ &+& 2\\sqrt{3\\Lambda }\\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t) \\left((n^2+1) \\sin (t) \\cos (n t)+2 n \\cos (t) \\sin (n t)\\right)]^{2}\\nonumber \\\\&+&[3 n \\left(n^2+3\\right) \\sin (t) \\cos (n t) + 3 \\left(3 n^2+1\\right) \\cos (t) \\sin (n t)\\nonumber \\\\&+& 2\\sqrt{3\\Lambda } \\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t) \\left((n^2+1) \\sin (t) \\sin (n t)-2 n \\cos (t) \\cos (n t)\\right)]^{2}\\rbrace ^{1/2}$ and $\\omega (t,n,\\Lambda )&=&\\dfrac{1}{t} Arctan \\lbrace [3 n \\left(n^2+3\\right) \\sin (t) \\cos (n t)+3 \\left(3 n^2+1\\right) \\cos (t) \\sin (n t)\\nonumber \\\\&+& 2\\sqrt{3\\Lambda }\\left(3 n(n^2+3) \\sin (t) \\cos (n t)+3(3 n^2+1) \\cos (t) \\sin (n t) \\right) \\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t)] \\nonumber \\\\&\\times &[2 \\Lambda (n^2-1)-3 n (n^2+3) \\sin (t) \\sin (n t)+3 (3 n^2+1) \\cos (t) \\cos (n t) \\nonumber \\\\ &+&2\\sqrt{3\\Lambda } \\left( \\left(n^2+1\\right) \\sin (t) \\cos (n t)+2 n \\cos (t) \\sin (n t)\\right) \\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t)]^{-1}\\rbrace $ The amplitude of the gravitational wave depends on the time duration of the wave propagation.", "As usual, there are two widely separated spatio-temporal scales for the GWs[29], [30], see also [31], [32] for GWs effect on the second order perturbation of background metric and nonlinear effect such as GW turbulence [33].", "Let us define $L_{B}$ and $\\lambda $ as the length scale of variation of the background and the scale of the ripple, respectively.", "In a short time $ \\lambda \\gg L_{B}$ and in a long time $ \\lambda \\ll L_{B}$ which is so-called \" short wave approximation\".", "In the short time (or $ \\lambda \\gg L_{B}$ ), amplitude has a minimum value and will increase for the large values of $ n $ and the effect of $ \\Lambda $ is turned on from $ \\Lambda \\gtrsim 0.001 $ .", "However, for the long-time propagation or short-wave approximation, it goes to a finite amount for a large value of $ n $ , which is similar to the flat case.", "The effect of $ \\Lambda $ ’s can be inferred from $ \\Lambda \\gtrsim 0.0001 $ (Fig.REF ).", "Figure: The rate of h(t,n,Λ=0.0007) h (t,n,\\Lambda =0.0007) with respect to n n for the long-time propagation or short wave approximation.", "It is noteworthy that Λ \\Lambda induces the oscillating behavior to the amplitude.", "This effect is visible at the amplitude of greater than 0.0001.For the small value of $ n $ , its dynamic relation to $ n $ is independent of the time of wave propagation.", "The role of $ \\Lambda $ is different in each case: for the short time, $ \\Lambda $ ’s effect increases the amplitude (Fig.REF , (a) dashed line), while for the long time, the effect of $ \\Lambda $ makes an oscillatory behavior around the state of $ \\Lambda =0 $ (Fig.REF ).", "In the Planck result [3], the $ \\Lambda $ is $ (4.24\\pm 0.11) \\times 10^{-66} eV^{2}= (2.846\\pm 0.076 ) \\times 10^{-122} m_{Pl}^{2}\\approx 10^{-56} cm^{-2}$ in natural units, where $ m_{Pl} $ is the Planck mass.", "Therefore, to make cosmological constant effective for GW’s, it should be approximately 52 orders of magnitude larger than the present observable value.", "Here, $ \\rho _{\\Lambda } $ should be of order $ \\rho _{_{Electro-Weak}}\\sim 10^{23} g~cm^{-3} $ , where it is the stage of softly broken super-symmetric in the universe [34].", "For a small $ n $ , the frequency can be written in a series of time as: $w(t,n,\\Lambda )= n + \\dfrac{3 \\tan (t) }{t}+\\dfrac{2 \\Lambda \\left(-5 t-t \\sin ^2(t)+6 \\sin (t)+t \\cos ^2(t)+2 t \\cos (t)\\right)}{3 t (\\cos (2 t)+1)}\\,.$ In the long-time propagation, the second and third terms can be ignored.", "But, for a short time, its rate is similar to the previous one ( Fig.REF ), so the frequency is independent of the time propagation of the GWs.", "From $ \\Lambda \\ge 0.1 $ , it will affect the decrease in the frequency of gravitational waves.", "Figure: The rate of w(t,n,Λ) w (t,n,\\Lambda ) with respect to n n for the short time propagation; the effect of Λ \\Lambda appears when exceeding 0.1 0.1 (see dashed line).In the short-time propagation, the amplitude and frequency behaviors are very similar to each other although the amount of $ \\Lambda $ ’s effect on them is different (Figs.REF , .REF and .REF ).", "Now, we are looking for the effect of exact solutions on the nearby particles.", "So, Relations (REF ) and (REF ) become: $S^{1}(t, n, \\Lambda )&=& \\lbrace 1+\\dfrac{\\Lambda }{3}+\\dfrac{1}{2} e^{i n t} \\frac{\\left(3 n^2+1\\right) \\cos t+i n \\left(n^2+3\\right) \\sin t}{ \\left(n^2-1\\right)} \\nonumber \\\\&&~~~~~~~~~~~~~~~+ \\sqrt{\\frac{\\Lambda }{3 }} e^{i n t} \\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t) \\frac{\\left(n^2+1\\right) \\sin t-2 i n \\cos t }{n^2-1} \\rbrace S^{1}(0)\\nonumber \\\\S^{2}(t, n, \\Lambda )&=& \\lbrace 1-\\dfrac{\\Lambda }{3}-\\dfrac{1}{2} e^{i n t} \\frac{\\left(3 n^2+1\\right) \\cos t+i n \\left(n^2+3\\right) \\sin t}{ \\left(n^2-1\\right)} \\nonumber \\\\&&~~~~~~~~~~~~~~~- \\sqrt{\\frac{\\Lambda }{3 }} e^{i n t} \\tanh (\\sqrt{\\dfrac{\\Lambda }{3}}t) \\frac{\\left(n^2+1\\right) \\sin t-2 i n \\cos t}{n^2-1}\\rbrace S^{2}(0)$ Figure: The effect of the gravitational wave with the wave number n=2 in closed with Λ=0 \\Lambda =0 (dotted blue line), Λ=0.1 \\Lambda =0.1 (dotted-black), and flat spacetime (solid-yellow line) on the nearby particles; the effect of the cosmological constant is the oscillatory effect.By using numerical methods we compare our result with the flat case which it only examines the frequency and mode of the polarization.", "Obviously, $ n=1 $ is not the answer.", "For $ n=2 $ , moving the nearby particles is shown in Fig.REF .", "At lower $n $ 's, the difference between the flat and closed cases is quite evident with $ \\Lambda =0 $ (dotted-blue).", "If we assume $ \\Lambda =0.1 $ or larger, it has a very small change on the frequency of particles ring (Fig.", "REF , dotted-black).", "It can be predicted that, if the cosmological constant is introduced as a source term, only the lower frequency modes of GWs will be affected by the background and its detection can be used to analyze the Dark energy [20], [25].", "In comparison, if $ \\Lambda $ has the role of isotropic motion in the geodesic deviation equation, the ring of particles is deformed into an ellipse and their normal modes have $\\dfrac{\\pi }{4}$ relative to each other[35].", "For large $ n $ , expanding the universe destroys the oscillation of gravitational waves (Fig.REF ).", "These results were obtained for $ \\times $ mode.", "Thus, the effect of GWs in the closed background shows itself only in low wave numbers.", "In the following, we will show the effect of closed GWs on the nearby particles and the quantitative difference between the closed and flat GWs." ], [ " Effect of Closed Gravitational Wave on Nearby Particles ", "The Hubble parameter, $H(t)=\\dfrac{\\dot{a}(t)}{a(t)} $ , is the effect of the expanding universe; its impact on massive objects has been seen on the scale of galaxies and clusters [36].", "Yet, the source of GWs is located at the cosmologically large distances from the observer, e.g.", "$ z\\ge 0.1 $ , the effect of cosmological expansion on GWs are relevant and the GW150914 ($ z=0.09 $ ) is in this limit.", "In the wide range of the studies taken into account, the expansion of background [37], [38], [39], [40], [41], [42], [43].", "Since the test particles are very close to each other and move very slowly, only their relative motion has been taken into consideration.", "Therefore, neither the first term, nor the Hubble constant, in the Eq.", "(REF ), has any effect on this circle of nearby particles.", "On the other, the first and second terms of Eq.", "(REF ) are very smaller than the last term and we can ignore them.", "For slowly-moving particles, we have $ \\tau =x_{0}=t $ and the geodesic deviation equation becomes: Figure: The effect of the gravitational wave with the wave number n=20 in closed with Λ=0.1 \\Lambda =0.1 (dotted-black) and flat spacetime (solid-yellow line) on the nearby particles.", "The GWs will not fluctuate the massless particles.$\\dfrac{\\partial ^{2}}{\\partial t^{2}}S^{i}=\\dfrac{1}{2} S^{j}\\dfrac{\\partial ^{2}}{\\partial t^{2}} D^{i}_{~j}$ As mentioned, $D^{i}_{~j}$ is the solution for gravitational waves equation in curved space time [13].", "By using (REF ), the deviation equations for $ + $ mode will be: $\\frac{d^{2}}{d t^{2}}S^{1} &=&\\frac{1}{2}S^{1}\\frac{d^{2}}{d t^{2}} D^{1}_{~1}+\\frac{1}{2}S^{2}\\frac{d^{2}}{d t^{2}} D^{1}_{~2}+\\frac{1}{2}S^{3}\\frac{d^{2}}{d t^{2}} D^{1}_{~3} \\nonumber \\\\\\frac{d^{2}}{d t^{2}}S^{2} &=&\\frac{1}{2}S^{1}\\frac{d^{2}}{d t^{2}} D^{2}_{~1}+\\frac{1}{2}S^{2}\\frac{d^{2}}{d t^{2}} D^{2}_{~2}+\\frac{1}{2}S^{3}\\frac{d^{2}}{d t^{2}} D^{2}_{~3}\\nonumber \\\\\\frac{d^{2}}{d t^{2}}S^{3}&=&\\frac{1}{2}S^{1}\\frac{d^{2}}{d t^{2}} D^{3}_{~1}+\\frac{1}{2}S^{2}\\frac{d^{2}}{d t^{2}} D^{3}_{~2}+\\frac{1}{2}S^{3}\\frac{d^{2}}{d t^{2}} D^{3}_{~3}$ Assuming that the particles remain on the $ S_{1}-S_{2} $ plane, we will have $ S_{3}=0 $ and $D_{+,\\times }(z=0,t) =\\dfrac{\\cos (t) \\cos (nt) \\pm n\\sin (t) \\sin (nt)}{1- n^{2}}$ We see that the gravitational wave functions are also dependent on wave numbers.", "For $n=0$ , the solution of the gravitational waves equation is: $S_{1,+}(S_{2,+})=c_{1} MathieuC(0,1,\\frac{t}{2})\\pm c_{2}MathieuS(0,1,\\frac{t}{2})$ where in general Mathieu C(a, q, z) and Mathieu S(a, q, z) are even and odd solutions of the Mathieu differential equation, respectively [44].", "As seen in Fig.REF , this solution is not stable.", "Also, $n=1$ is a gauge mode, the solution of which is $x(t)=constant $ .", "As seen in Fig.", "REF , the solution is not an oscillation, so that it has no effect on the particles.", "For $ n=2 $ , there is no analytical solution; but, for mode $ n=3 $ , the solution of Eq.", "(REF ) can be written as the Heun C function[45] as: Figure: The solution for n=0 (redline) as a Mathieu function which is not stable.", "The solution n=1 (blue line) is a gauge mode which has no oscillation.$S_{1,+}(t)=(c_{1}+ c_{2}\\cos (t))e^{\\sqrt{2}\\cos ^{2}(t)}HeunC(2\\sqrt{2},\\frac{1}{2},\\frac{1}{2},\\frac{1}{4},0,\\cos ^{2}(t))\\nonumber \\\\S_{2,+}(t)=(c_{3}- c_{4}\\cos (t))e^{\\sqrt{2}\\cos ^{2}(t)}HeunC(2\\sqrt{2},\\frac{1}{2},\\frac{1}{2},\\frac{1}{4},0,\\cos ^{2}(t))$ where the HeunC function is the solution of the Heun Confluent equation and $c_{1}$ , $c_{2}$ , $c_{3}$ and $c_{4}$ are constant.", "For other modes, there are no analytical solution.", "Even though these equations have no exact analytical solutions for any values of $n $ , the most simple solutions are similar to (7.114) and (7.115) of ref [27], except that the time evolution of gravitational waves in closed spacetime is different from that of the flat case (i.e.", "$ D_{+,\\times }(z=0,t) $ has been used instead of $ \\exp (iwt) $ in ordinary GW's equation).", "Therefore, the terms of $ O(x^{3})$ and high orders are ignored and the solutions can be written in terms of the series expansion for any value of the wave number as: $S_{1,+} (n,t)&=& \\lbrace 1+\\sum _{i=2}^{\\infty } g_{i}(n)t^{i}\\rbrace S_{1}(0) \\nonumber \\\\S_{2,+} (n,t)&=&\\lbrace 1-\\sum _{i=2}^{\\infty } g_{i}(n)t^{i}\\rbrace S_{2}(0)$ where $ \\sum _{i=2}^{\\infty } g_{i}(n)t^{i}= f(t, n, \\Lambda =0) $ and $\\times $ mode gives: $S_{1,\\times }(n,t) &=& S_{1}(0)+ \\sum _{i=2}^{\\infty } g_{i}(n)t^{i}S_{2}(0) \\nonumber \\\\S_{2,\\times }(n,t) &=& S_{2}(0)+ \\sum _{i=2}^{\\infty } g_{i}(n)t^{i}S_{1}(0)$ Figure: Left: The solution of equation for n=2(black), n=3(dot-dashed) and n=4(dashed).", "As seen, there is difference between the lower mode solution of the closed background and the flat case.", "Right: The solution of equation for n=19(solid line) and n=20(dot-dashed) and higher.", "In the upper modes, the solutions will be the same as the flat case.The particles are located on the circle of radius $ S_{1}(0)=S_{2}(0)=1 $ before getting hit by GWs.", "As mentioned, in the $ n=0 $ , the solution is not stable and $ n=1 $ is a gauge mode which is not an oscillation.", "For the lower modes (Fig.", "REF Left), these solutions differ from the flat cases.", "Then, the amplitude of particles oscillation is different from that of the flat case, as we can see in Fig.REF and Fig.REF .", "The interesting point is that there is a new oscillation in every period; in other words, the particles oscillation in each period is different from that of others.", "When the mode number increases (for example, $n=18$ ,$n=19$ and $n=20$ or higher wave numbers), the solution in the closed background will be the same as the flat spacetime; these two cases cannot be distinguished ( Fig.", "REF Right and Fig.REF ).", "When the particles oscillate like the flat spacetime, this ensures the stability of the solutions." ], [ "Closed Background Effect", "Now, we consider the next order terms of $O(x^{3})$ , that is all the polarization matrix sentences of the (REF ).", "So, the solution of Eq.", "(REF ) with the condition of the remaining particles in $ S_{1}-S_{2} $ and $ S_{3}=0 $ for $ + $ mode becomes: $S_{1,+} &=& \\lbrace 1+\\sum _{i=2}^{\\infty } k_{i}(n)t^{i}\\rbrace S_{1}(0) \\nonumber \\\\S_{2,+} &=&\\lbrace 1-\\sum _{i=2}^{\\infty } k_{i}(n)t^{i}\\rbrace S_{2}(0)$ And the $\\times $ mode solutions is: Figure: The effect of the gravitational wave with the wave number n=2 in closed (dashed line ) and flat spacetime (solid line) on the nearby particles.", "In the next periods in closed spacetime, unlike the flat case, these shapes of oscillation cannot be seen and there are new shapes.$S_{1,\\times } &=& S_{1}(0)+ \\sum _{i=2}^{\\infty } h_{i}(n)t^{i}S_{2}(0) \\nonumber \\\\S_{2,\\times } &=& S_{2}(0)+ \\sum _{i=2}^{\\infty } v_{i}(n)t^{i}S_{1}(0)$ It can be seen that the solutions are different for each mode, e.g.", "$ k_{i}(n) $ is different from $ h_{i}(n) $ and $v_{i}(n)$ .", "There is also asymmetry in the solution of mode $ \\times $ , because of the difference of $ h_{i}(n) $ from $ v_{i}(n)$ .", "With these solutions, the particles start to oscillate at first.", "But, their oscillation will not remain stable later, i.e.", "the amplitude and frequency of particles will change with time in an inconsistent way for each wave number.", "By defining $ S_{\\times }=\\sqrt{S_{1,\\times }^{2}+S_{2,\\times }^{2}} $ , the movements of test particles will not be stable, as shown in Fig.", "REF , for lower wave numbers.", "In high wave numbers, the particles will not be oscillating.", "This is true for $+$ mode as well, i.e.", "the closed structure operates as an anomaly effect for harmonic oscillation.", "Figure: The effect of the gravitational wave with the wave number n=3 in closed (dashed line ) and flat spacetime (solid line) on the nearby particles.", "In the next periods in closed spacetime, the above shape will not be repeated." ], [ "Quantitative Difference between Closed and Flat GWs ", "For quantitative comparison of our results with the flat case, we should define: $\\Delta S^{i}(t)=S^{i}_{closed}(t)-S^{i}_{flat}(t)$ where $ S^{i}_{flat}(t)= \\dfrac{1}{2} h e^{i^{w_{_{GW}}t}}$ and $ h $ and $ w_{GW} $ are amplitude and frequency of GWs in flat spacetime, respectively.", "So, $\\Delta S^{i}(t)=[\\dfrac{\\Lambda }{3}f(t,n, \\Lambda )-\\dfrac{1}{2}h e^{iw_{GW}t}]\\delta ^{ij} S^{j}(0) ~~~~~,~~~~~i=1,2$ for $ \"+ \"$ mode $ i=j $ .", "This term, $ \\dfrac{\\Lambda }{3}f(t,n, \\Lambda )-\\dfrac{1}{2}h e^{iw_{GW}t} $ could separate the effect of two cases of flat and closed approaches.", "Therefore, $\\dfrac{\\Delta S^{i}(t)}{S^{i}(0)}&=&\\dfrac{\\Lambda }{3}f(t,n, \\Lambda )-\\dfrac{1}{2}h e^{iw_{GW}t}\\nonumber \\\\&\\cong &\\dfrac{\\Lambda }{3}+\\dfrac{3n^{2}+1}{2(n^{2}-1)} \\cos (t) \\cos (nt) - \\dfrac{n(n^{2}+1)}{2(n^{2}-1)} \\sin (t) \\sin (nt)-\\dfrac{1}{2} h(f) \\cos (w_{gw} t)\\nonumber \\\\$ Figure: The effect of the gravitational wave with the wave number n=20 in closed (dashed line ) and flat spacetime (solid line) on the nearby particles.", "In high wave numbers, the effects of closed and flat background are the same.for $ \"+ \"$ mode $ i=j $ .", "This term, $ \\dfrac{\\Lambda }{3}f(t,n, \\Lambda )-\\dfrac{1}{2}h e^{iw_{GW}t} $ could separate the effect of two cases of flat and closed approaches.", "Therefore, This relation can be interpreted as the sensitivity of detector for a land-based detector, such as LIGO or VIRGO.", "We can explore the differences between the two cases in the same interval time and duration of the gravitational wave passes.", "Here, $ t $ can be considered as the length of time, in which the detector is turned on.", "Therefore, for instance, considering the $ \\Lambda $ at the moment [3], amplitude spectral density, $ 10^{-23} < h(f)< 10^{-19} $ , and observed frequency, $ 10 (Hz) < f < 10^{4} (Hz) $ from the first detection of GWs, GW150914 [1], one can see that the amount of sensitivity depends on the $ \\Lambda $ and $ n $ .", "Figure: The plot of S × =S 1,× 2 +S 2,× 2 S_{\\times }=\\sqrt{S_{1,\\times }^{2}+S_{2,\\times }^{2}} for the nearby particles concerning the time in closed spacetime in the presence of the O(x 3 ) O(x^3) terms for lower and higher wave numbers.", "As seen in the next times, in lower wave numbers, both the frequency and amplitude will change momentarily and the system of particles will not remain stable.", "Also, in high wave numbers, the particles will not be oscillating." ], [ "Discussion and Conclusion", "So far, no study has been conducted on the frequency and amplitude of GWs as well as their effect on the nearby particles in closed de Sitter spacetime.", "The amplitude and frequency of gravitational waves are obtained by comparing the solutions of deviation equation in closed background, with its flat case independent of its $+ $ or $ \\times $ modes.", "The behavior of the amplitude relative to the wave number depends on the time duration of wave propagation, while the frequency is independent.", "The effect of the cosmological constant on GWs in closed spacetime is seen nearly after the epoch of Electro-Weak in the universe.", "Its effect on amplitude is weaker than on the frequency, because if $ \\Lambda \\ge 0.0001 $ , the amplitude is affected.", "The same happens for frequency when $ \\Lambda \\ge 0.1 $ .", "Also, the role of $ \\Lambda $ is different for them; it reduces the frequency, while its effect on the amplitude is oscillatory with the dissipation effect.", "The effect of this wave, which is on the large scale, can be used for the nearby particles in the small scale, because it is the solution of vacuum field equation of tensor perturbation and its traceless-transverse properties.", "Nevertheless, the effect on the nearby particles can only be seen at lower wave numbers, while in large numbers, the expansion of the universe will eliminate the oscillatory effect of GWs.", "The second approach corresponds to the flat case, but the time evolution of GWs in de Sitter spacetime differs from the Minkowski background.", "It has been concluded that the behavior of the low-frequency modes will be affected by the de-Sitter background.", "The differences between the two cases are also seen in the amplitude of the particle oscillations.", "Although the infrared modes are sensitive to the cosmological constant, it was found that the propagation of the gravitational waves would be observed at modes lower than the infrared ones when the wavelength of gravitational waves is smaller than the de Sitter radius of curvature and higher than the deviation.", "Therefore, the gravitational radiation can be used to test the intrinsic curvature of the large-scale spacetimes.", "The particle oscillation of flat spacetime, which is recovered within the closed spacetime for the large wave numbers, indicates the presence of stabilities in the solutions.", "Additionally, in lower wave numbers within closed spacetime, the particles’ oscillation is not harmonic and, rather, have small fluctuations in the back-point or at the maximum amplitudes.", "In the next order terms ($ O(x^{3}) $ and higher), caused by the closed structure, the particles start to oscillate; but after a short time, the ring of the test particles will become unstable.", "Hence, with the new approach, we can reconfirm that de Sitter background space operates as the cut-off oscillation.", "More interestingly, the only way to investigate the curvature of background spacetime by detecting gravitational waves is perhaps the direct study of GWs in lower wave numbers (very low frequencies) or at very high wavelengths.", "However, the precision of this detector must be in the order of $ \\Lambda $ -measurement at the moment in order to measure the distance between the two points, i.e.", "one under the influence of plane wave (flat wave) and the other under the influence of closed gravitational waves." ], [ "Traceless-transverse Gauge or TT Gauge", "The transverse-traceless gauge in curved spacetime is defined as: $D_{i0}=0~~~~~~~,~~~~~~~D^{i}_{~i}=K x^{j}x^{k}D_{jk}~~~~~~~,~~~~~~~\\partial _{i}D_{ik}=K x^{i}x^{j}\\partial _{i}D_{jk}-4K x^{l}D_{kl}$ In the TT gauge, GWs have an specially simple form.", "If a test mass is at rest at $\\tau =0$ , from the geodesic equation, we have $ \\dfrac{dx^{i}}{d\\tau }=0$ , the mass initially has been taken at rest when $ \\Gamma _{00}^{i}=0$ .", "By writing $ g_{\\mu \\nu }=\\bar{g}_{\\mu \\nu }+ h_{\\mu \\nu }$ and expanding to first order in $ h_{\\mu \\nu }$ , the Christoffel symbol $ \\Gamma _{00}^{i}$ ( from equations (3) and (11) in [13]) becomes: $\\Gamma _{00}^{i}&=& \\bar{\\Gamma }_{00}^{i}+\\delta \\Gamma _{00}^{i}\\nonumber \\\\&=&\\dfrac{\\dot{a}}{a}\\delta _{0i}+\\dfrac{1}{2a^{2}}[2\\dot{h}_{0i}-\\partial _{i}h_{00}-2 K x^{i}x^{j}\\partial _{0}h_{0j}+Kx^{i}x^{j}\\partial _{j}h_{00}]$ This quantity vanishes because both $h_{00}$ and $h_{0i}$ are set to zero.", "Therefore, if at time $\\tau =0$ , $ \\dfrac{dx^{i}}{d\\tau }$ and its derivative $ \\dfrac{d^{2}x^{i}}{d\\tau ^{2}}$ vanish; so, $ \\dfrac{dx^{i}}{d\\tau }$ remains zero at all times.", "In the TT gauge, particles were at rest before the arrival of the wave and, then, remained at rest even after the arrival of the wave.", "Therefore, in the closed background, the GWs have all properties of transverse-traceless gauge." ] ]
2105.11690
[ [ "Distortions Characterization for Dynamic Carrier Allocation in Ultra\n High-Throughput Satellites" ], [ "Abstract A novel analytical formula for the characterization of linear and nonlinear distortions in future ultra high-throughput communication payloads is proposed in this work.", "In this context, the carrier-to-interference ratio related to single-carrier and multicarrier signals is derived.", "Through the analysis of its behavior valuable insights are created, especially regarding the interaction between linear and nonlinear intersymbol interference.", "Furthermore, the principle of dynamic carrier allocation optimization is highlighted in a realistic scenario.", "Within the presented framework, it is proven that a significant gain can be achieved even with a limited number of carriers.", "Finally, a complexity and accuracy analysis emphasizes the practicality of the proposed approach." ], [ "Introduction", "Ultra high-throughput satellite (UHTS) systems development has been driven by the ever-increasing demand for high data rate satellite services.", "For the next generation of UHTS systems, it is foreseen that up to 2.9 GHz of downlink bandwidth per polarization will be achieved by shifting feeder links to the Q/V-band and exploiting the available Ka-band resources in the user links .", "This will help to meet the need of high data rate services for future beyond 5G communications (e.g., broadband internet services and delivery to the edge) .", "Furthermore, the allocation of payload resources (e.g., bandwidth, time, power, and coverage) will become increasingly flexible, especially through the evolution of digital processors.", "This will enable to adapt to different communication services and traffic demands.", "In particular, a dynamic carrier allocation (DCA) within the satellite will pave the way to a fully flexible connectivity between uplink and downlink beams.", "On one side, this strategy removes the need of a centralized gateway.", "One the other side, it prevents from increasing the processing power onboard the satellite, which is a major drawback of alternative strategies such as beam-hopping (physical layer) or flexible routing protocols (higher layer).", "The DCA is well known in the context of terrestrial communication systems, where high capacity gains can be achieved .", "However, it is still an uncharted territory in the context of flexible satellite payloads.", "Indeed, conventional approaches rely on static carrier allocation, which under-exploits the available frequency resources and does not allow any adaptation to the highly varying traffic demand.", "To enable the DCA in UHTS systems, two main phenomena need to be taken into account: the linear distortions and the nonlinear distortions.", "On the one hand, the analytical characterization and modeling of the nonlinearities entailed by the satellite high power amplifier (HPA) have been well covered in the literature , .", "On the other hand, the work of has recently enabled the analytical characterization of the linear distortions in UHTS systems, essentially entailed by the wideband output multiplexer (OMUX) filters.", "Therefore, it is currently possible to describe efficiently the main frequency-dependent components in the link budget and optimize the allocation of each user carrier.", "This key technology will enable to further increase the capacity of the next-generation UHTS systems.", "Thus, the original contribution of the present paper with respect to the state of the art is the following: A novel analytical characterization of both linear and nonlinear intersymbol interference is derived, enabling the study of their mutual interaction.", "The results are validated by simulations.", "A low-complexity analytical formula of the carrier-to-interference ratio (CIR) is derived in order to be exploited in the context of DCA optimization.", "The feasibility of near real-time DCA is emphasized in a practical scenario.", "Within this framework, the accuracy and complexity of the proposed CIR formula, along with the allocation gain are discussed.", "The remainder of this paper is structured as follows.", "Section II covers the UHTS system model.", "In Section III, the linear and nonlinear distortions are thoroughly analyzed and the total CIR formula is derived.", "Section IV contains a practical DCA scenario.", "The conclusions are presented in Section V. Figure: Block diagram of the future UHTS communication system." ], [ "UHTS System Model", "The UHTS system model is depicted in Fig.", "REF .", "The forward link of a geostationary UHTS is considered.", "This means that the uplink white Gaussian noise (WGN) can be neglected due to better link budget in the feeder link.", "The downlink WGN at the $r^{\\mathrm {th}}$ -user receiver terminal has a power of $\\sigma _r^2$ and is denoted $w_r(t)$ .", "Furthermore, the commonly used square-root raised cosine filter (SRRCF) $h_s(t)$ is considered as pulse shaping and matched filter.", "The roll-off factor is denoted $\\alpha $ .", "Thus, the single-carrier transmit signal ($N_T=1$ )The extension to multicarrier signals is performed in the next section.", "after pulse shaping, used as a basis for the later calculations, can be expressed as follows: $ x(t)=\\sum _{p=0}^{N_s-1} s_{p}\\cdot h_s(u-p T_s)\\ ,$ where the realizations $\\lbrace s_p\\rbrace _{0\\le p\\le N_s-1}$ are associated to the random variables $\\lbrace S_p\\rbrace _{0\\le p\\le N_s-1}$ considered uncorrelated.", "The number of symbols and the symbol duration are denoted $N_s$ and $T_s$ , respectively.", "The symbols are sent with a fixed average transmit power $\\mathrm {V}[S_p]=P$ , $\\forall p\\in \\lbrace 0;N_s-1\\rbrace $ .", "$E[\\mathcal {Z}]$ and $\\mathrm {V}[\\mathcal {Z}]$ are the expectation and variance of a complex variable $\\mathcal {Z}$ , respectively, where $\\mathrm {V}[\\mathcal {Z}]=E[|\\mathcal {Z}|^2]-|E[\\mathcal {Z}]|^2$ .", "For the sake of simplicity, it is assumed in the next section that the uplink gains and losses have been taken into account and that $P$ corresponds to the symbols power at the input of the HPA." ], [ "Distortions Characterization", "In this section, an analytical expression of the CIR is determined.", "Firstly, the nonlinear intersymbol interference (NL-ISI) and linear intersymbol interference (L-ISI) are studied for a single-carrier signal.", "Then, an extension to multicarrier signals and adjacent channel interference (ACI) is discussed." ], [ "Nonlinear Distortions", "The most widespread memoryless HPA model is the Saleh model .", "This model describes mathematically the HPA nonlinearities from an input signal $x(t)$ to an output signal $y(t)$ depending on $|x(t)|^2$ , the input signal instantaneous power at time $t$ .", "In this paper, the normalized parameters for a typical HPA are assumed as described in .", "The corresponding AM/AM and AM/PM characteristics are depicted in Fig.", "REF .", "Moreover, the input power $P$ is defined such that it corresponds to the input back-off (the input and output power at saturation are normalized).", "To improve the mathematical tractability, it is convenient to consider the truncated memoryless polynomial: $ y(t)=x(t)\\cdot \\sum _{k=0}^{N} \\gamma _{2k+1}(a) \\cdot |x(t)|^{2k}\\ ,$ where $\\lbrace \\gamma _{2k+1}(a)\\rbrace _{0\\le k\\le N}$ are the complex coefficients resulting from the Taylor approximation of the Saleh model when the instantaneous power $|x(t)|^2$ gets close to a constant $a$ .", "$N$ refers to to the number of odd order distortions to be considered.", "Without loss of generality, the case $N=1$ is considered.", "Thus, the relevant polynomial coefficients are: $ \\begin{aligned}[c]&\\gamma _1(a)=\\frac{\\alpha _{\\mathcal {I}} (1+2\\beta _{\\mathcal {I}} a)}{(1+\\beta _{\\mathcal {I}} a)^2}\\ ,\\\\&\\gamma _3(a)=-\\frac{\\alpha _{\\mathcal {I}} \\beta _{\\mathcal {I}}}{(1+\\beta _{\\mathcal {I}} a)^2}+j\\cdot \\frac{\\alpha _{\\mathcal {Q}}}{(1+\\beta _{\\mathcal {Q}} a)^2}\\ ,\\end{aligned}$ where $\\gamma _1(a)$ and $\\gamma _3(a)$ are the HPA linear gain and the third order complex coefficient, respectively, when the input back-off gets close to the value $a$ .", "Furthermore, $\\alpha _{\\mathcal {I}}=1.90947$ , $\\beta _{\\mathcal {I}}=1.07469$ , $\\alpha _{\\mathcal {Q}}=4.35023$ , and $\\beta _{\\mathcal {Q}}=2.33525$ are the normalized Saleh parameters.", "The dependency on the variables $a$ and $r$ will be omitted in the following for more compact notations.", "Figure: NO_CAPTIONGiven the typical complex coefficients, the received signal can be approximated as: $ z(q T_s) = \\gamma _1\\cdot z_1(q T_s) + \\gamma _3\\cdot z_3(q T_s) + (w*h_s)(q T_s)\\ ,$ where $ \\begin{aligned}[c]&z_1(q T_s)=\\sum _{k_1\\in \\mathbb {Z}} s_{q-k_1}\\cdot \\int _{\\mathbb {R}} h^\\ddagger (u) h_s(u+k_1 T_s) du\\ ,\\\\&z_3(q T_s)=\\sum _{k_1\\in \\mathbb {Z}} \\sum _{k_2\\in \\mathbb {Z}} \\sum _{k_3\\in \\mathbb {Z}} s_{q-k_1}s_{q-k_2}s^*_{q-k_3}\\cdot \\\\&\\int _{\\mathbb {R}} h^\\ddagger (u) h_s(u+k_1 T_s) h_s(u+k_2 T_s) h_s(u+k_3 T_s) du\\ ,\\end{aligned}$ are the received sampled signal components of first and third order, respectively.", "They are obtained after some mathematical manipulations similar to .", "The filter border effects at the beginning and at the end of the symbol sequence are also neglected.", "$h^\\ddagger (u)$ represents the post-HPA filters, which means that without linear distortions $h^\\ddagger (u)=h_s(u)$ ." ], [ "Linear Distortions", "Since the linear distortions caused by the OMUX filter after the HPA are dominant , the gain and group delay variations model will be based – without loss of generality – on the realistic re-scaled OMUX characteristic as illustrated in Fig.", "REF .", "A first-order approximation of the linear distortions has been proven to be very accurate (especially in a multicarrier configuration), while greatly lowering the implementation complexity .", "This so-called slope-based model relies on the gain slope $g$ $\\mathrm {(dB/MHz)}$ and group delay slope $d$ $\\mathrm {(ns/MHz)}$ .", "These slopes are determined by least-squares approximations over the considered carrier bandwidth.", "Thus, the post-HPA impulse response can be expressed as: $ h^\\ddagger (u)=\\int _{-\\infty }^{+\\infty } H_{\\mathrm {LD}}(f)\\sqrt{H_{\\mathrm {RCF}}(f)}e^{j2\\pi f u}df\\ ,$ where $ H_{\\mathrm {LD}}(f) = e^{\\xi g f - j\\pi d f^2}\\ ,$ is the first-order linear distortions transfer function and $\\xi =\\ln (10)/20$ .", "It is trivial to see that the frequency-dependent gain and group delay variations are affecting all kernels through $h^\\ddagger (u)$ .", "As discussed in , it is more convenient to normalize the expressions with respect to the symbol rate $R_s$ .", "Thus, the only two linear distortions variables are $x_g=g\\cdot R_s$ and $y_d=d\\cdot R_s^2$ and expressed in $\\mathrm {(dB/MHz)(Mbauds)}$ and $\\mathrm {(ns/MHz^2)(Mbauds)^2}$ ." ], [ "Carrier-to-Interference Ratio", "The analytical determination of the CIRSince the sampled filtered noise power results in a constant independent of the frequency, it can be omitted from the analysis and allocation optimization.", "can be viewed as the ability to separate the useful signal power (constructive) from the interference power (destructive) in the expression of $\\mathrm {V}[z(q T_s)]$ .", "The variance of the received sampled signal can be expressed as: $ \\begin{aligned}[c]\\mathrm {V}[z(q T_s)]=&|\\gamma _1|^2 \\cdot \\mathrm {V}[z_1(q T_s)] + |\\gamma _3|^2 \\cdot \\mathrm {V}[z_3(q T_s)] +\\\\&\\gamma _1 \\gamma _3^* \\cdot \\mathrm {Cov}[z_1(q T_s);z_3(q T_s)]+\\\\&\\gamma _3 \\gamma _1^* \\cdot \\mathrm {Cov}[z_3(q T_s);z_1(q T_s)]+\\\\&\\mathrm {V}[(w*h_s)(q T_s)]\\ .\\end{aligned}$ Thus, each frequency-dependent term can be divided into its useful and interfering part, such as: $ \\begin{aligned}[c]\\mathrm {V}[z_1(q T_s)]&=P\\cdot (\\kappa _{u,1,1} + \\kappa _{i,1,3}),\\\\\\mathrm {V}[z_3(q T_s)]&=P^3\\cdot (\\kappa _{u,3,3} + \\kappa _{i,3,3}),\\\\\\mathrm {Cov}[z_1(q T_s);z_3(q T_s)]&=P^2\\cdot (\\kappa _{u,1,3} + \\kappa _{i,1,3}),\\\\\\mathrm {Cov}[z_3(q T_s);z_1(q T_s)]&=P^2\\cdot (\\kappa _{u,3,1} + \\kappa _{i,3,1}),\\end{aligned}$ where, for instance, $ \\begin{aligned}[c]\\kappa _{u,1,1}=&\\left|\\int _{\\mathbb {R}} h^\\ddagger (u)h_s(u) du \\right|^2\\ , \\\\\\kappa _{i,1,1}=&\\sum _{k_1\\in \\mathbb {Z}\\backslash \\lbrace 0\\rbrace } \\left|\\int _{\\mathbb {R}} h^\\ddagger (u) h_s(u+k_1 T_s) du \\right|^2\\ ,\\end{aligned}$ are the useful and interfering part of the Volterra kernel power related to $z_1(qT_s)$ .", "As opposed to the first order, the third order of nonlinear distortions contains many terms to separate.", "The useful and interference terms start as follows: $ \\begin{aligned}[c]\\kappa _{u,3,3}=&\\sum _{k_2,k_3\\in \\mathbb {Z}} \\mathcal {H}(0,k_2,k_3)+\\cdots \\ , \\\\\\kappa _{i,3,3}=&\\sum _{\\begin{array}{c}k_1\\in \\mathbb {Z}\\backslash \\lbrace 0\\rbrace ;k_2,k_3\\in \\mathbb {Z}\\end{array}} \\mathcal {H}(k_1,k_2,k_3)+\\cdots \\ ,\\end{aligned}$ where $ \\begin{aligned}[c]&\\mathcal {H}(k_1,k_2,k_3)=\\\\&\\left|\\int _{\\mathbb {R}} h^\\ddagger (u)h_s(u+k_1 T_s) h_s(u+k_2 T_s) h_s(u+k_3 T_s) du \\right|^2\\ .\\nonumber \\end{aligned}$ Figure: Re-scaled DVB-S2x OMUX filter characteristic including an example of gain slope gg and group delay slope dd approximation with regards to a carrier 𝒞\\mathcal {C} of bandwidth 200 MHz 200\\ \\mathrm {MHz} and center frequency -150 MHz -150\\ \\mathrm {MHz}.Thus, the single-carrier CIR can be derived based on this decomposition and can be expressed as: $ \\left(\\frac{C_{\\mathrm {ISI}}}{I_{\\mathrm {ISI}}}\\right)=\\frac{P\\cdot c_1+ P^2\\cdot c_2+ P^3\\cdot c_3}{P\\cdot i_1+ P^2\\cdot i_2+ P^3\\cdot i_3}\\ ,$ where $ \\begin{split}c_1&=|\\gamma _1|^2\\cdot \\kappa _{u,1,1}\\ ,\\ \\qquad \\\\c_2&=\\gamma _1\\gamma _3^* \\cdot \\kappa _{u,1,3}\\ \\ \\qquad \\\\&+\\gamma _1^*\\gamma _3 \\cdot \\kappa _{u,3,1}\\ ,\\ \\qquad \\\\c_3&=|\\gamma _3|^2\\cdot \\kappa _{u,3,3}\\ ,\\ \\qquad \\\\\\end{split}\\begin{split}i_1&=|\\gamma _1|^2\\cdot \\kappa _{i,1,1}\\ ,\\\\i_2&=\\gamma _1\\gamma _3^* \\cdot \\kappa _{i,1,3}\\\\&+\\gamma _1^*\\gamma _3 \\cdot \\kappa _{i,3,1}\\ ,\\\\i_3&=|\\gamma _3|^2\\cdot \\kappa _{i,3,3}\\ .\\\\\\end{split} \\nonumber $ It is worth noting that with non-constant modulus symbol constellations, e.g.", "in the case 16APSK symbols, an additional factor depending on the ring ratio comes out of the variance terms in $P^2$ and $P^3$ .", "This factor is worsening the terms in $P^2$ and $P^3$ around the saturation.", "Figure REF illustrates the novel characterization of the CIR related to ISI as a function of the input power $P$ , the roll-off $\\alpha $ , and the amount of linear distortions represented by $x_g$ and $y_d$ .", "To best describe the interactions between L-ISI and NL-ISI, the limit cases are first discussed.", "On the one hand, only NL-ISI are present in the system when $x_g=y_d=0$ .", "This leads to $\\kappa _{u,1,1}=1$ and $\\kappa _{i,1,1}=\\kappa _{i,1,3}=\\kappa _{i,3,1}=0$ based on the properties of the SRRCF.", "In this case, it can be observed that the lower the roll-off, the lower the CIR.", "On the other hand, only L-ISI are present in the system when the HPA is operated in the linear region, i.e.", "$P\\rightarrow 0$ .", "This leads to $(C_{\\mathrm {ISI}}/I_{\\mathrm {ISI}})\\rightarrow \\kappa _{u,1,1}/\\kappa _{i,1,1}$ , which exactly corresponds to the closed-form solution derived in .", "Two examples of linear distortions are given to highlight the trade-off between L-ISI and NL-ISI.", "Both are considered in the case of a dominant gain slope, which happens for instance when a carrier is located at the edge of the wideband OMUX.", "When $x_g=10^0\\ \\mathrm {dB/MHz\\cdot Mbauds}$ , higher degree kernels are not affected by linear distortions and a transition region is observed around $P_{\\mathrm {dB}}=-9\\ \\mathrm {dB}$ where the lower degree kernels become dominant.", "It is only when the linear distortions are stronger, i.e.", "$\\kappa _{u,1,1}/\\kappa _{i,1,1}$ lying below $20\\ \\mathrm {dB}$ , that the higher degree kernels, here $\\kappa _{i,3,3}$ , start to have a noticeable impact near the saturation point.", "Finally, it is worth noting that the simulations of the system model confirm with high accuracy the theoretical formula.", "Figure: Single-carrier CIR analysis.", "The units of x g x_g and y d y_d are ( dB / MHz )·( Mbauds )\\mathrm {(dB/MHz)\\cdot (Mbauds)} and ( ns / MHz )·( Mbauds ) 2 \\mathrm {(ns/MHz)\\cdot (Mbauds)^2}, respectively.", "The Saleh model coefficients are approximated at a=0a=0 for N=1N=1 and the sums limitis L=10L=10.Kernels are simulated as partial sums where $k_i\\in \\lbrace -L,-L+1,\\hdots ,L\\rbrace $ ." ], [ "Extension to Multicarrier Scenario", "The extension to more than one carrier is now considered ($N_T>1$ ).", "Two new phenomena must be taken into consideration: the power imbalance between the carriersIn this context, $P$ designates the total signal input power.", "and the ACI caused by the carriers intermodulation products.", "On the one side, the useful part of the carrier power is dependent on the split of power between the carriers; it is denoted $C_{\\mathrm {ISI},\\lbrace i\\rbrace }$ , where $i$ represents the carrier index under consideration.", "This translates in an additional factor – function of the power split – arising in $c_1$ , $c_2$ and $c_3$ , which will not be the focus point of this paper.", "On the other side, an analytical expression of the ACI has been derived in .", "The variance of this expression is considered to compute $I_{\\mathrm {ACI},\\lbrace i,\\nu \\rbrace }$ , where $\\nu $ refers to the index of a carriers combination based on a list of possible combinations.", "In the frame of this paper, the ACI of order 1 and 3 are derived.", "For the sake of simplicity, the cross-terms between $I_{\\mathrm {ISI}}$ and $I_{\\mathrm {ACI},\\lbrace i,\\nu \\rbrace }$ will be neglected along with the constructive terms contained in the ACI.", "Finally, the final CIR taking into account ISI and ACI is expressed as: $ \\left(\\frac{C}{I}\\right)_{\\lbrace i,\\nu \\rbrace }\\approx \\left(\\frac{C_{\\mathrm {ISI},\\lbrace i\\rbrace }}{I_{\\mathrm {ISI}}+I_{\\mathrm {ACI},\\lbrace i,\\nu \\rbrace }}\\right)\\ .$ Thus, by taking into account the main useful and interfering contributions, this expression will allow low-complexity CIR calculations." ], [ "Scenario", "In this section, a carrier allocation scenario is considered to demonstrate the practicality of the proposed formula (REF ).", "A multicarrier signal composed of $N_c=3$ carriers $\\lbrace \\mathcal {C}_i\\rbrace _{1\\le i\\le N_c}$ is considered at the HPA input.", "The carriers characteristics are listed out in Table REF .", "Since the signal power at the HPA input is $P$ , a carrier with power split of 1/4 means that the input carrier power is $P/4$ .", "The wideband OMUX depicted in Fig.", "REF is considered as post-HPA linear distortions characteristic.", "Since practical multicarrier systems rely to some extent on HPA linearization or predistortion techniques with input back-off closer to saturation, an equivalent input power of $P_{\\mathrm {dB}}=-15\\ \\mathrm {dB}$ is chosen to translate these compensation mechanisms.", "This will also help to emphasize the contribution of each type of interference.", "Table: Parameters for Carrier Allocation Scenario" ], [ "Optimization Problem Formulation", "The center frequency of carrier $\\mathcal {C}_i$ corresponding to the $\\nu ^{\\mathrm {th}}$ -possible carrier combination is denoted $f_{0,i,\\nu }$ .", "We define the set of all possible carrier placements $\\mathcal {F}_{0}$ such that any $N_c-$ tuple $\\lbrace f_{0,i,\\nu }\\rbrace _{1\\le i\\le N_c}$ belonging to this set is a valid carrier allocation combination, where $\\nu $ stands for the index of the combination.", "For clarity purposes, the solution space of the carrier allocation problem has been reduced to a discrete set with cardinality $|\\mathcal {F}_{0}|=N_c!=6$ possibilities.", "Although different optimization approaches can be considered, this paper will focus on a user fairness approach, which aims at maximizing the sum capacity.", "As described in , this can be reduced to a max-min CIR optimization.", "The optimization problem can then be formulated as follows: $ &\\mathcal {P}:\\quad \\underset{1\\le \\nu \\le |\\mathcal {F}_{0}|}{\\mathrm {max}}\\left\\lbrace \\min \\limits _{\\begin{array}{c}1\\le i\\le N_c\\\\ f_{0,i,\\nu } \\in \\mathcal {F}_{0}\\end{array}}\\left\\lbrace \\left(\\frac{C}{I}\\right)_{{\\ \\ \\ \\lbrace i,\\nu \\rbrace }}\\ (f_{0,i,\\nu })\\right\\rbrace \\right\\rbrace $ where the goal is to maximize, by carrier allocation, the minimum CIR between the carriers given the set of possible carrier placements.", "The key enabler in making carrier allocation dynamic is the computation speed of the CIR.", "This analysis has been performed in in the context of linear distortions, where numerical integration and slope-based distortions approximation was proven to be a good balanced between accuracy and complexity.", "Therefore, this computation method has been extended here to nonlinear distortions effects.", "Figure: NO_CAPTION" ], [ "Results Discussion", "The carrier combination $\\mathcal {C}_1 \\mathcal {C}_2 \\mathcal {C}_3$ is depicted in Fig.", "REF .", "As illustrated, the total CIR can be decomposed into types of interference: L-ISI, NL-ISI and ACI.", "In this scenario, L-ISI is critical on the edge of the filter and essentially depend on the symbol rate with small variations due to the roll-off.", "Indeed, $\\mathcal {C}_3$ symbol rate is $1.5$ higher than the other carriers, therefore, it is subject to noticeable degradation in the order of $6\\ \\mathrm {dB}$ .", "On the other hand, the NL-ISI is critical in the center of the filter and essentially rely on the signal input power $P$ with small variations due to the power split between the carriers and the roll-off.", "As such, the carriers $\\mathcal {C}_2$ and $\\mathcal {C}_3$ have a higher input power, and consequently, are getting more gain due to the power robbing effect.", "However, lower roll-offs degrade more the CIR, which counterbalances the power robbing effect to some extent in this specific scenario.", "By performing an exhaustive search, the best carrier allocation is $\\mathcal {C}_2 \\mathcal {C}_3 \\mathcal {C}_1$ with a minimum CIR of $23.64\\ \\mathrm {dB}$ , whereas the worst combination is $\\mathcal {C}_3 \\mathcal {C}_2 \\mathcal {C}_1$ with a minimum CIR of $19.77\\ \\mathrm {dB}$ .", "This means that even with a low number of carriers it is possible to reach an allocation gain of $3.87\\ \\mathrm {dB}$ .", "The order of magnitude of the complexity per CIR computation is highlighted in Fig.", "REF .", "The curves denoted “SIM”, “TH1” and “TH2” represent the CIR computation by conventional transmission chain simulation, the theoretical CIR formula without pre-computation, and the theoretical CIR formula with pre-computed kernels, respectively.", "Indeed, the proposed formula provides a significant reduction in the implementation speed (by a factor $6\\cdot 10^2$ between “SIM” and “TH1”).", "Since the ISI are independent from the carriers combination, the majority of the kernels can be computed beforehand when knowing the gain and group delay distortions, carriers symbol rate, roll-off, and power distribution.", "This provides further complexity reduction (by a factor $7\\cdot 10^1$ between “TH1” and “TH2”).", "In this scenario, the minimum, mean and maximum accuracy of the proposed theoretical formula with respect to the simulations are $97.38\\%$ , $98.74\\%$ , and $99.98\\%$ , respectively.", "The CIR formula leads to the same optimal allocation, which emphasizes the superiority of the proposed approach.", "Figure: Conclusion" ] ]
2105.11803
[ [ "Resolution a la Kronheimer of $\\mathbb{C}^3/\\Gamma$ singularities and\n the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane\n solutions of supergravity" ], [ "Abstract We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence.", "We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions $Y$ of singularities $\\mathbb{C}^3/\\Gamma$ with $\\Gamma$ a finite subgroup of $SU(3)$.", "Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on $Y$, with particular attention to the case $\\Gamma=\\mathbb{Z}_4$.", "We conjecture that on the exceptional divisor the Kronheimer K\\\"ahler metric and the Ricci-flat one, that is locally flat at infinity, coincide.", "The conjecture is shown to be true in the case of the Ricci-flat metric on ${\\rm tot} K_{{\\mathbb WP}[112]}$ that we construct, which is a partial resolution of $\\mathbb{C}^3/\\mathbb{Z}_4$.", "For the full resolution we have $Y=\\operatorname{tot} K_{\\mathbb{F}_{2}}$, where $\\mathbb{F}_2$ is the second Hizebruch surface.", "We try to extend the proof of the conjecture to this case using the one-parameter K\\\"ahler metric on $\\mathbb{F}_2$ produced by the Kronheimer construction as initial datum in a Monge-Amp\\`{e}re (MA) equation.", "We exhibit three formulations of this MA equation, one in terms of the K\\\"ahler potential, the other two in terms of the symplectic potential; in all cases one can establish a series solution in powers of the fiber variable of the canonical bundle.", "The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, but uniquely determines all the subsequent terms as local functionals of the initial datum.", "While a formal proof is still missing, numerical and analytical results support the conjecture.", "As a by-product of our investigation we have identified some new properties of this type of MA equations that we believe to be so far unknown." ], [ "Introduction", "We report on the advances obtained on the following special aspect of the gauge/gravity correspondence, within the context of quiver gauge–theories [1], [2], [3], [4], [5]: the relevance of the generalized Kronheimer construction[6], [7] for the resolution of $\\mathbb {C}^3/\\Gamma $ singularities.", "In particular, after an introduction about D3-brane supergravity solutions, we consider, within this framework, the issues of the construction of a Ricci-flat metric on the smooth resolution $Y^\\Gamma $ of $\\mathbb {C}^3/\\Gamma $ .", "We begin with the general problem of establishing holographic dual pairs whose members are A) a gauge theory living on a D3-brane world volume, B) a classical D3-brane solution of type IIB supergravity in D=10 supergravity.", "Gauge theories based on quiver diagrams have been extensively studied in the literature [1], [2], [3], [4], [5] in connection with the problem of establishing holographic dual pairs as described above.", "Indeed the quiver diagram is a powerful tool which encodes the data of a Kähler quotient describing the geometry of the six directions transverse to the brane.", "The linear data of such a Kähler (or HyperKähler) quotient are the menu of the dual supersymmetric gauge theory, as they specify: the gauge group factors, the matter multiplets, the representation assignments of the latter with respect to the gauge group factors.", "The possibility of testing the holographic principle [8], [9], [10], [11], [12] and resorting to the supergravity side of the correspondence in order to perform, classically and in the bulk, quantum calculations that pertain to the boundary gauge theory is tightly connected with the quiver approach whenever the classical brane solution has a conformal point corresponding to a limiting geometry of the following type: $M_{D} \\, = \\, \\mathrm {AdS}_{p+2} \\times \\mathrm {SE}^{D-p-2}$ In the above equation by $\\mathrm {AdS}_{p+2}$ we have denoted anti de Sitter space in $p+2$ -dimensions while $\\mathrm {SE}^{D-p-2}$ stands for a Sasaki-Einstein manifold in $D-p-2$ -dimensions [13].", "Within the general scope of quivers a special subclass is that of McKay quivers that are group theoretically defined by the embedding of a finite discrete group $\\Gamma $ in an $n$ -dimensional complex unitary group $\\Gamma \\hookrightarrow \\mathrm {SU}(n)$ and are associated with the resolution of $\\mathbb {C}^n/\\Gamma $ quotient singularities by means of a Kronheimer-like construction [15], [16], [17].", "The case $n=2$ corresponds to the HyperKähler quotient construction of ALE-manifolds as the resolution of the $\\mathbb {C}^2/\\Gamma $ singularities, the discrete group $\\Gamma $ being a finite Kleinian subgroup of $\\mathrm {SU(2)}$ , as given by the ADE classificationFor a recent review of these matters see chapter 8 of [18]..", "The case $n=3$ was the target of many interesting and robust mathematical developments starting from the middle of the nineties up to the present day [19], [20], [21], [22], [23], [24], [25], [26], [27].", "The main and most intriguing result in this context, which corresponds to a generalization of the Kronheimer construction and of the McKay correspondence, is the group theoretical prediction of the cohomology groups $\\mathrm {H}^{(p,q)}\\left(Y^\\Gamma _{[3]}\\right)$ of the crepant smooth resolution $Y^\\Gamma _{[3]}$ of the quotient singularity $\\mathbb {C}^3/\\Gamma $ .", "Specifically, the main output of the generalized Kronheimer construction for the crepant resolution of a singularity $\\mathbb {C}^3/\\Gamma $ is a blowdown morphism: $\\mathrm {BD}\\, : \\quad Y^\\Gamma _{[3]} \\, \\longrightarrow \\,\\frac{\\mathbb {C}^3}{\\Gamma }$ where $Y^\\Gamma _{[3]}$ is a noncompact smooth three-fold with trivial canonical bundle.", "On such a complex three-fold a Ricci-flat Kähler metric $\\text{ds}^2_{\\mathrm {RFK}}(Y^\\Gamma _{[3]})\\, = \\,\\mathrm {\\mathbf {g}}^{\\mathrm {RFK}}_{\\alpha \\beta ^\\star } \\,dy^\\alpha \\otimes dy^{\\beta ^\\star }$ with asymptotically conical boundary conditions (Quasi-ALE) is guaranteed to exist (see e.g.", "[28], Thm.", "3.3), although it is not necessarily the one obtained by means of the Kähler quotient.", "According to the theorem proved by Ito-Reid [19], [22], [23] and based on the concept of age gradingFor a recent review of these matters within a general framework of applications to brane gauge theories see [6], [7]., the homology cycles of $Y_{[3]}^\\Gamma $ are all algebraic and its non vanishing cohomology groups are all even and of type $\\mathrm {H}^{(q,q)}$ .", "We actually have a correspondence between the cohomology classes of type $(q,q)$ and the discrete group conjugacy classes with age-grading $q$ , encoded in the statement: $\\mbox{dim} \\, \\mathrm {H}^{1,1}\\left(Y^\\Gamma _{[3]}\\right)& =& \\# \\, \\mbox{ of junior conjugacy classes in $\\Gamma $;}\\nonumber \\\\\\mbox{dim} \\, \\mathrm {H}^{2,2}\\left(Y^\\Gamma _{[3]}\\right)& =& \\# \\, \\mbox{ of senior conjugacy classes in $\\Gamma $;}\\nonumber \\\\&& \\mbox{all other cohomology groups are trivial}$ The absence of harmonic forms of type $(2,1)$ implies that the three-folds $Y^\\Gamma _{[3]}$ admit no infinitesimal deformations of their complex structure and is also a serious obstacle, as we discuss in section to the construction of supergravity D3-brane solutions based on $Y^\\Gamma _{[3]}$ that have transverse three-form fluxes.", "There is however a possible way out that is provided by the existence of mass-deformations.", "This is the main point of another line of investigation which we hope to report on soon.", "If the McKay quiver diagram has certain properties, the superpotential $\\mathcal {W}(\\Phi )$ on the gauge-theory side of the correspondence can be deformed by well defined mass-terms and, after (gaussian) integration of the massive fields, the McKay quiver is remodeled into a new non-McKay quiver associated with the Kähler or HyperKähler quotient description of smooth Kähler manifolds, like the resolved conifold, that admit harmonic $(2,1)$ -forms and sustain adequate D3-brane solutions.", "On the basis of the above remarks we can spell-out the scope of the present paper in the following way.", "The embedding (REF ) determines in a unique way a McKay quiver diagram which determines: the gauge group $\\mathcal {F}_\\Gamma $ , the matter field content $\\Phi ^I$ of the gauge theory, the representation assignments of all the matter fields $\\Phi ^I$ , the possible (mass)-deformations of the superpotential $\\mathcal {W}(\\phi )$ , the Ricci-flat metric on $Y$ can be inferred, by means of the Monge-Ampère equation, from the Kähler metric on the exceptional compact divisor (in those cases where it exists) in the resolution of $\\mathbb {C}^3/\\Gamma $ , which, on its turn, is determined by the McKay quiver through the Kronheimer construction.", "In relation with point 4) of the above list, to be discussed in a future paper, for the case $\\mathbb {C}^3/\\mathbb {Z}_4$ we anticipate the following.", "By means of gaussian integration we get a new quiver diagram that is not directly associated with a discrete group, yet it follows from the McKay quiver of $\\Gamma $ in a unique way.", "The group theoretical approach allows us to identify deformations of the superpotential and introduce new directions in the moduli space of the crepant resolution.", "In this sense, we go beyond the Ito-Reid theorem.", "Both physically and mathematically this is quite interesting and provides a new viewpoint on several results, some of them well known in the literature.", "Most of the latter are based on cyclic groups $\\Gamma $ and rely on the powerful weapons of toric geometry.", "Yet the generalized Kronheimer construction applies also to non abelian groups $\\Gamma \\subset \\mathrm {SU(3)}$ and so do the cohomological theorems proved by Ito-Reid, Ishii and Craw.", "Hence available mass-deformations are encoded also in the McKay quivers of non abelian groups $\\Gamma $ and one might explore the geometry of the transverse manifolds emerging in these cases.", "In relation with point 5) of the above list, fully treated in this paper for the same case $\\mathbb {C}^3/\\mathbb {Z}_4$ , we stress that, although the Kronheimer metric on $Y^\\Gamma _{[3]}$ is not Ricci-flat, yet its restriction to the exceptional divisor provides the appropriate starting point for an iterative solution of the Monge Ampère equation which determines the Ricci-flat metric.", "In view of the above considerations we can conclude that the McKay quiver diagram does indeed provide a determination of both sides of a D3-brane dual pair, the gauge theory side and the supergravity side.", "In this paper we focus on two paradigmatic examples, namely $\\mathbb {C}^3/\\mathbb {Z}_3$ (with $\\mathbb {Z}_3$ diagonally embedded in $\\mathrm {SU(3)}$ ) and $\\mathbb {C}^3/\\mathbb {Z}_4$ .", "The latter case was studied in depth in a recent publication [7].", "Relying on those results here we concentrate on the issue of the Ricci-flat Kähler metric.", "While in the case of HyperKähler quotients (yielding $\\mathcal {N}=2$ gauge theories and corresponding to the original Kronheimer construction of $\\mathbb {C}^2/\\Gamma $ resolutions) the Kronheimer metric is automatically Ricci-flat, in the case of Kähler quotients and of the generalized Kronheimer construction of $\\mathbb {C}^3/\\Gamma $ resolutions, the Kronheimer metric is not Ricci-flat and one needs to resort to different techniques in order to find a Ricci-flat metric on the same three-fold $Y^\\Gamma _{[3]}$ that is algebraically determined by the construction.", "The fascinating scenario that emerges from our combined analytical and numerical results is summarized in the following discussion.", "From the point of view of complex algebraic geometry the resolved variety $Y^\\Gamma _{[3]}$ is in many cases and, in particular in those here analyzed, the total space of a line-bundle over a compact complex two-fold, which coincides with the exceptional divisor $\\mathcal {ED}$ of the resolution of singularities: $Y^\\Gamma _{[3]} \\,\\stackrel{\\pi }{\\longrightarrow } \\,\\mathcal {ED}_{[2]} \\quad ; \\quad \\forall \\, p \\, \\in \\,\\mathcal {ED}_{[2]} \\quad : \\quad \\pi ^{-1}(p) \\, \\sim \\, \\mathbb {C}$ In the paradigmatic example, recently studied in [7], of the resolution à la Kronheimer of the $\\mathbb {C}^3/\\mathbb {Z}_4$ singularity, $\\mathcal {ED}$ is indeed the compact component of the exceptional divisor emerging from the blow-up of the singular point in the origin of $\\mathbb {C}^3$ and it happens to be the second Hirzebruch surface $\\mathbb {F}_2$ .", "Other cases are possible.", "Hirzebruch surfaces are $\\mathbb {P}^1$ bundles over $\\mathbb {P}^1$ , so that $\\mathcal {ED}_{[2]} \\,\\stackrel{\\widetilde{\\pi }}{\\longrightarrow } \\,\\mathbb {P}^1 \\quad ; \\quad \\forall \\, p \\, \\in \\,\\mathbb {P}^1 \\quad : \\quad \\widetilde{\\pi }^{-1}(p) \\, \\sim \\, \\mathbb {P}^1$ This double fibration is illustrated in a pictorial fashion in fig.REF .", "Given this hierarchical structure, the sought for Ricci-flat metric is constrained to possess the following continuous isometry group: $\\mathrm {G}_{iso} \\, = \\, \\mathrm {SU(2) \\times U(1)_{v} \\times U(1)_{w}}$ whose holomorphic algebraic action on the three coordinates $u,v,w$ is described later in eq.", "(REF ).", "The chosen isometry group implies that the sought for Ricci-flat metric is toric, as each of the three complex coordinates is acted on by an independent $\\mathrm {U(1)}$ -isometry.", "Furthermore the enhancement of one of the $\\mathrm {U(1)}$ 's to $\\mathrm {SU(2)}$ guarantees that either the Kähler potential $\\mathcal {K}$ in the standard complex formulation of Kähler geometry, or the symplectic potential $\\mathcal {G}$ , the Legendre transform of the former appearing in the available symplectic formalism [60], are functions only of two invariant real variables (see section and REF ).", "Assuming that we possess either one of these two real functions for the Ricci-flat metric For conventions see once again sections REF and .", ": $\\mathcal {K}_{\\text{Ricci-flat}}(\\varpi ,\\mathfrak {f} )\\quad \\text{or}\\quad \\mathcal {G}_{\\text{Ricci-flat}}(\\mathfrak {v},\\mathfrak {w})$ we can reduce the corresponding geometry to that of the exceptional divisor by setting a section of the $Y^\\Gamma _{[3]}$ bundle to zero as: $w \\, = \\, 0 \\quad \\Leftrightarrow \\quad \\mathfrak {f}=0 \\,\\, , \\,\\, \\mathfrak {w} \\, = \\, {\\textstyle \\frac{3}{2}}$ The fascinating scenario we have alluded to some lines above is encoded in the following: Figure: A conceptual picture of the resolvedthree-fold Y [3] Γ Y^\\Gamma _{[3]} displaying its double fribrationstructure.", "The orange sphere in the middle symbolizes the basemanifold of the bundle ℰ𝒟 [2] \\mathcal {ED}_{[2]}.", "A dense complexcoordinate patch for this ℙ 1 \\mathbb {P}^1 is named uu in the mainbody of the article.", "The blueish spheres around the orange onesymbolize the ℙ 1 \\mathbb {P}^1 fibers of ℰ𝒟 [2] \\mathcal {ED}_{[2]}.", "A densecomplex coordinate patch for these fibers is named vv in the mainbody of the article.", "Finally the greenish rays enveloping thedivisor ℰ𝒟 [2] \\mathcal {ED}_{[2]} symbolize the noncompact fibers of thebundle Y [3] Γ Y^\\Gamma _{[3]}.", "A dense coordinate patch for these fibersis named ww in the main body of the article.Conjecture 1.1 The Kronheimer Kähler metric $\\mathrm {ds}^2_{Kro}[Y^\\Gamma _{[3]}]$ on the line bundle (REF ) and the Ricci-flat one $\\mathrm {ds}^2_{Ricflat}[Y^\\Gamma _{[3]}]$ on the same manifold, that has the same isometries and is asymptotically locally flatMore precisely, this metric is Quasi-ALE in the sense of [28].", "are different, yet they coincide on the exceptional divisor $\\mathcal {ED}$ .", "The basic argument in favor of this conjecture is provided by an in depth analysis of a particular orthotoric metric that we construct in this paper and that is shown to describe the Ricci-flat metric on a degenerate limit of three-fold $Y^\\Gamma _{[3]}$ , as described in [7].", "This is a partial resolution of the $\\mathbb {C}^3/\\mathbb {Z}_4$ singularity and it occurs when the stability parameters (Fayet-Iliopolous parameters in the physics jargon) are restricted to be on the unique type III wallAccording to the terminology in [23], a wall in the space of stability parameters is of type III when it corresponds to a degeneration which contracts divisors to curves.", "In this case the noncompact component $\\mathbb {P}^1\\times \\mathbb {C}$ of the exceptional divisor shrinks to $\\mathbb {C}$ .", "appearing in the chamber structure associated with the generalized Kronheimer construction for this McKay quiver.", "From the algebraic geometry viewpoint, this variety $Y_{[3]}$ is the total space of the canonical bundle over the weighted projective space ${\\mathbb {W}P}[112]$ : $Y_{[3]}\\, = \\, {\\rm tot} K_{{\\mathbb {W}P}[112]}$ and its exceptional divisor is ${\\mathbb {W}P}[112]$ .", "We show that the Kähler metric induced on ${\\mathbb {W}P}[112]$ by our new Ricci-flat orthotoric metric is precisely identical with that obtained from the Kronheimer construction once reduced to the divisor.", "The various inspections of this known case within the framework of different formalisms and using different coordinate patches provided us with the means to make conjecture REF more robust.", "The main tool at our disposal is provided by the Monge-Ampère (MA) equation for Ricci-flatness of which we develop two versions, one in terms of the Kähler potential $\\mathcal {K}(\\varpi ,\\mathfrak {f} )$ (see section REF ) and one in terms of the symplectic potentialSee section for the definition of the real variables $\\mathfrak {v},\\mathfrak {w}$ .", "$\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ (see section REF ).", "In both cases we showed that the potential can be developed in power series of the invariant associated with the non compact fibers (either $\\mathfrak {f}$ or $\\mathfrak {w}-{\\textstyle \\frac{3}{2}}$ ) and that the MA equation imposes no restriction on the 0-th order potentials $\\mathcal {K}_0(\\varpi )$ or $\\mathcal {G}_0(\\mathfrak {v})$ , namely on the geometry chosen for the exceptional divisor.", "Rather, dealing carefully with the boundary conditions, we discovered that in both cases the MA equation completely determines all the other terms once $\\mathcal {K}_0(\\varpi )$ or $\\mathcal {G}_0(\\mathfrak {v})$ are given.", "Hence we can start with $\\mathcal {K}_0^{Kro}(\\varpi )$ or $\\mathcal {G}_0^{Kro}(\\mathfrak {v})$ as they are determined by the Kronheimer construction and going through the power series treatment of the MA equation we can construct a corresponding Ricci-flat metric.", "The only question which remains open is whether this Ricci-flat metric is asymptotically locally flat.", "In the case of ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ it is.", "This supports the conjecture.", "In order to transform the conjecture into a theorem one should first resum the series and study the metric at large distances.", "In this respect our study of the symplectic potential produced encouraging results.", "First of all we were able to construct an explicit form $\\mathcal {G}_{{\\mathbb {W}P}[112]}(\\mathfrak {v},\\mathfrak {w})$ of such potential for the orthotoric case.", "The function $\\mathcal {G}_{{\\mathbb {W}P}[112]}(\\mathfrak {v},\\mathfrak {w})$ , which is relatively simply written in terms of elementary transcendental functions, satisfies the MA equation and can be expanded in series of $(\\mathfrak {w}-{\\textstyle \\frac{3}{2}})$ .", "The remarkably similar behavior of the series truncations of the exact solution corresponding to ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ with the same truncations of the series determined by the MA equation for the smooth case ${\\rm tot} K_{\\mathbb {F}_{2}}$ suggests that also in the latter case there exists a summation of the series to some simple deformation of the function $\\mathcal {G}_{{\\mathbb {W}P}[112]}(\\mathfrak {v},\\mathfrak {w})$ .", "We postpone to future publications further attempts to sum the series solution and prove, if possible, our conjecture." ], [ "D3-brane supergravity solutions on resolved $\\mathbb {C}^3/\\Gamma $ \nsingularities", "An apparently general property of the $Y_{[3]}^\\Gamma $ manifolds that emerge from the crepant resolution construction, at least when $\\Gamma $ is abelian and cyclic is the following.", "The non-compact $Y_{[3]}^\\Gamma $ corresponds to the total space of some line-bundle over a complex two-dimensional compact base manifold $\\mathcal {M}_2$ : $Y_{[3]}^\\Gamma \\, \\stackrel{\\pi }{\\longrightarrow } \\,\\mathcal {M}_2$ According with this structure we name $u,v,w$ the three complex coordinates of $Y_{[3]}^\\Gamma $ , $u,v$ being the coordinates of the base manifold $\\mathcal {M}_2$ and $w$ being the coordinate spanning the fibers.", "We will use the same names also in more general cases even if the interpretation of $w$ as fiber coordinate will be lost.", "Hence we have: $y \\equiv y^\\alpha \\, = \\, \\left\\lbrace u,v,w\\right\\rbrace \\quad ;\\quad \\overline{y} \\equiv y^{\\overline{\\alpha }}\\, = \\,\\left\\lbrace \\overline{u},\\overline{v},\\overline{w}\\right\\rbrace $ An important observation which ought to be done right at the beginning is that other Kähler metrics $\\widehat{\\mathrm {\\mathbf {g}}}_{\\alpha \\beta ^\\star }$ do exist on the three-fold $Y_{[3]}$ that are not Ricci-flat, although the cohomology class of the associated Kähler form $\\widehat{\\mathbf {K}}$ can be the same as the cohomology class of $\\mathbf {K}_{\\mathrm {RFK}}$ .", "Within the framework of the generalized Kronheimer construction, among such Kähler (non-Ricci flat) metrics we have the one determined by the Kähler quotient according to the formula of Hithchin, Karlhede, Lindström and Roček [30].", "Indeed, as we show later in explicit examples, the Kähler metric: $\\text{ds} ^2_{\\mathrm {HKLR}}(Y_{[3]})\\, = \\,\\mathrm {\\mathbf {g}}^{\\mathrm {HKLR}}_{\\alpha \\beta ^\\star } \\, dy^\\alpha \\otimes dy^{\\beta ^\\star }$ which emerges from the mathematical Kähler quotient construction and which is naturally associated with $Y_{[3]}$ when this latter is interpreted as the space of classical vacua of the D3-brane gauge theory (set of extrema of the scalar potential), is generically non Ricci-flat.", "On the other hand on the supergravity side of the dual D3-brane pair we need the Ricci-flat metric in order to construct a bona-fide D3-brane solution of type IIB supergravity.", "In particular, calling $Y^\\Gamma _{[3]}$ the crepant resolution of the $\\mathbb {C}^3/\\Gamma $ singularity, admitting a Ricci-flat metric, we can construct a bona-fide D3 brane solution which is solely defined by a single real function $H$ on $Y^\\Gamma _{[3]}$ , that should be harmonic with respect to the Ricci-flat metric, namely: $\\Box _{\\mathbf {g}^{\\mathrm {RFK}}} \\, H \\, = \\, 0$ Indeed the function $H(\\mathbf {y})$ is necessary and sufficient to introduce a flux of the Ramond 5-form so as to produce the splitting of the 10-dimensional space into a 4-dimensional world volume plus a transverse 6-dimensional space that is identified with the three-fold $Y^\\Gamma _{[3]}$ .", "This is the very essence of the D3-picture.", "Yet there is another essential item that was pioneered in [31], [32], [33] namely the consistent addition of fluxes for the complex 3-forms $\\mathcal {H}_\\pm $ that appear in the field content of type IIB supergravity.", "These provide relevant new charges on both sides of the gauge/gravity correspondence.", "In [34], [35] such fluxes were constructed explicitely relying on a special kind of three-fold: $Y_{[3]} \\, = \\, Y_{[1+2]} \\, = \\, \\mathbb {C}\\times \\mathrm {ALE}_\\Gamma $ where $\\mathrm {ALE}_\\Gamma $ denotes one of the $\\mathrm {ALE}$ -manifolds constructed by Kronheimer [15], [16] as HyperKähler quotients resolving the singularity $\\mathbb {C}^2/\\Gamma $ with $\\Gamma \\subset \\mathrm {SU(2)}$ a finite Kleinian subgroup.", "As we explain in detail below, the essential geometrical feature of $Y_{[3]}$ , required to construct consistent fluxes of the complex 3-forms $\\mathcal {H}_\\pm $ , is that $Y_{[3]}$ should admit imaginary (anti)-self-dual, harmonic 3-forms $\\Omega ^{(2,1)}$ , which means: $\\star _{\\mathbf {g}^{\\mathrm {RFK}}} \\Omega ^{(2,1)} \\, = \\, \\pm \\, {\\rm i} \\, \\Omega ^{(2,1)}$ and simultaneously: $d\\Omega ^{(2,1)} \\, = \\, 0 \\quad \\Rightarrow \\quad d\\star _{\\mathbf {g}^{\\mathrm {RFK}}}\\Omega ^{(2,1)}\\, = \\, 0$ Since the Hodge-duality operator involves the use of a metric, we have been careful in specifying that (anti)-self-duality should occur with respect to the Ricci-flat metric that is the one used in the rest of the supergravity solution construction.", "The reason why the choice (REF ) of the three-fold allows the existence of harmonic anti-self dual 3-forms is easily understood recalling that the $\\mathrm {ALE}_\\Gamma $ -manifold obtained from the resolution of $\\mathbb {C}^2/\\Gamma $ has a compact support cohomology group of type $(1,1)$ of the following dimension: $\\mathrm {dim}\\,\\mathrm {H}^{(1,1)}_{comp}\\left(\\mathrm {ALE}_\\Gamma \\right) \\, = \\, r \\quad \\mbox{where}\\quad r \\, = \\, \\# \\,\\mbox{ of nontrivial conjugacy classes of$\\Gamma $}$ Naming $z \\in \\mathbb {C}$ the coordinate on the factor $\\mathbb {C}$ of the product (REF ) and $\\omega _I^{(1,1)}$ a basis of harmonic anti-self dual one-forms on $\\mathrm {ALE}_\\Gamma $ , the ansatz utilized in [34], [35] to construct the required $\\Omega ^{(2,1)}$ was the following: $\\Omega ^{(2,1)} \\, \\equiv \\, \\partial _z\\, \\mathfrak {f}^I (z) \\, dz \\, \\wedge \\, \\omega ^{(1,1)}_I$ where $\\mathfrak {f}^I (z)$ is a set of holomorphic functions of that variable.", "As it is well known $r$ is also the rank of the corresponding Lie Algebra in the ADE-classification of the corresponding Kleinian groups and the 2-forms $\\omega ^{(1,1)}_I$ can be chosen dual to a basis homology cycles $\\mathcal {C}_I$ spanning $H_{2}\\left(\\mathrm {ALE}_\\Gamma \\right)$ , namely we can set: $\\int _{\\mathcal {C}_I} \\,\\omega ^{(1,1)}_J \\, = \\, \\delta _{IJ}$ The form $\\Omega ^{(2,1)}$ is closed by construction: $d \\Omega ^{(2,1)} \\, = \\,0$ and it is also anti-selfdual with respect to the Ricci-flat metric: $\\text{ds}^2_{Y_{[1+2]}} \\, = \\, dz\\otimes d\\overline{z}\\, + \\, \\text{ds}^2_{\\mathrm {ALE}_\\Gamma }$ Hence the question whether we can construct sufficiently flexible D3-solutions of supergravity with both 5-form and 3-form fluxes depends on the nontriviality of the relevant cohomology group: $\\mbox{dim} \\, \\mathrm {H}^{(2,1)}\\left(Y_{[3]}\\right)\\, > \\, 0$ and on our ability to find harmonic (anti)-self dual representatives of its classes (typically not with compact support and hence non normalizable).", "At this level we find a serious difficulty.", "It seems therefore that we are not able to find the required $\\Omega ^{(2,1)}$ forms on $Y^\\Gamma _{[3]}$ and that no D3-brane supergravity solution with 3-form fluxes can be constructed dual to the gauge theory obtained from the Kronheimer construction dictated by $\\Gamma \\subset \\mathrm {SU(3)}$ .", "Fortunately, the sharp conclusion encoded in eq.", "(REF ) follows from a hidden mathematical assumption that, in physical jargon, amounts to a rigid universal choice of the holomorphic superpotential $\\mathcal {W}(\\Phi )$ .", "Under appropriate conditions that we plan to explain and which are detectable at the level of the McKay quiver diagram, the superpotential can be deformed (mass deformation) yielding a family of three-folds $Y^{\\Gamma ,\\mu }_{[3]}$ which flow, for limiting values of the parameter ($\\mu \\rightarrow \\mu _0$ ) to a three-fold $Y^{\\Gamma ,\\mu _0}_{[3]}$ admitting imaginary anti self-dual harmonic (2,1)-forms.", "Since the content and the interactions of the gauge theory are dictated by the McKay quiver of $\\Gamma $ and by its associated Kronheimer construction, we are entitled to see its mass deformed version and the exact D3-brane supergravity solution built on $Y^{\\Gamma ,\\mu _0}_{[3]}$ as dual to each other.", "This will be the object of a future work.", "Here we begin with an accurate mathematical summary of the construction of D3-brane solutions of type IIB supergravity using the geometric formulation of the latter within the rheonomy framework [36]." ], [ "Geometric formulation of Type IIB supergravity ", "In order to discuss conveniently the D3 brane solutions of type IIB that have as transverse space the crepant resolution of a $\\mathbb {C}^3/\\Gamma $ singularity, we have to recall the geometric Free Differential Algebra formulation of the chiral ten dimensional theory fixing with care all our conventions, which is not only a matter of notations but also of principles and geometrical insight.", "Indeed the formulation of type IIB supergravity as it appears in string theory textbooks [37], [38] is tailored for the comparison with superstring amplitudes and is quite appropriate to this goal.", "Yet, from the viewpoint of the general geometrical set up of supergravity theories this formulation is somewhat unwieldy.", "Specifically it neither makes the $\\mathrm {SU(1,1)/U(1)}$ coset structure of the theory manifest, nor does it relate the supersymmetry transformation rules to the underlying algebraic structure which, as in all other instances of supergravities, is a simple and well defined Free Differential algebra.", "The Free Differential Algebra of type IIB supergravity was singled out many years ago by Castellani in [39] and the geometric, manifestly $\\mathrm {SU(1,1)}$ –covariant formulation of the theory was constructed by Castellani and Pesando in [40].", "Their formulae and their transcription from a complex $\\mathrm {SU(1,1)}$ basis to a real $\\mathrm {SL(2,\\mathbb {R})}$ basis were summarized and thoroughly explained in a dedicated chapter of a book authored by one of us [41] which we refer the reader to." ], [ "The D3-brane solution with a $Y_{[3]}$ transverse\nmanifold", "In this section we discuss a D3-brane solution of type IIB supergravity in which, transverse to the brane world-manifold, we place a smooth non compact three-fold $ Y_{[3]}$ endowed with a Ricci-flat Kähler metric.", "The ansatz for the D3-brane solution is characterized by two kinds of flux; in addition to the usual RR 5-form flux, there is a non-trivial flux of the supergravity complex 3-form field strengths $ \\mathcal {H}_{\\pm }$ .", "We separate the ten coordinates of space-time into the following subsets: $x^M = \\left\\lbrace \\begin{array}{rcll}x^\\mu &:& \\mu =0,1,2,3& \\mbox{coordinates of the 3-brane world volume} \\\\y^\\tau &:& \\tau =4,5,6,7,8,9 & \\mbox{real coordinates of the $Y$variety} \\ \\end{array} \\right.$" ], [ "The D3 brane ansatz", "We make the following ansatz for the metricAs explained in appendix A, the conventions for the gamma matrices and the spinors are set with a mostly minus metric $d\\tau ^2$ .", "In the discussion of the solution, however, we use $ds^2= -d\\tau ^2$ for convenience.", "We hope this does not cause any confusion.", ": $\\text{ds}^2_{[10]}&=&H(y,\\overline{y})^{-\\frac{1}{2}}\\left(-\\eta _{\\mu \\nu }\\,dx^\\mu \\otimes dx^\\nu \\right)+H(y,\\overline{y})^{\\frac{1}{2}} \\, \\left(\\mathrm {\\mathbf {g}}^{\\mathrm {RFK}}_{\\alpha \\beta ^\\star } \\, dy^\\alpha \\otimes dy^{\\beta ^\\star }\\right) \\, \\nonumber \\\\\\text{ds}^2_{Y}&=&\\mathrm {\\mathbf {g}}_{\\alpha \\beta ^\\star }^{\\mathrm {RFK}} \\, dy^\\alpha \\otimes dy^{\\beta ^\\star }\\nonumber \\\\{\\rm det}(g_{[10]})&=&H(y,\\overline{y}){\\rm det}(\\mathrm {\\mathbf {g}^{\\mathrm {RFK}}})\\nonumber \\\\\\eta _{\\mu \\nu }&=&{\\rm diag}(+,-,-,-)$ where $\\mathrm {\\mathbf {g}}^{\\mathrm {RFK}}$ is the Kähler metric of the $Y_{[3]}$ manifold $\\mathrm {\\mathbf {g}}_{\\alpha \\overline{\\beta }}^{\\mathrm {RFK}} \\, = \\, \\partial _\\alpha \\,\\partial _{\\overline{\\beta }} \\,\\mathcal {K}^{\\mathrm {RFK}}\\left(y,\\overline{y}\\right)$ the real function $\\mathcal {K}^{\\mathrm {RFK}}\\left(y,\\overline{y}\\right)$ being a suitable Kähler potential." ], [ "Elaboration of the ansatz", "In terms of vielbein the ansatz (REF ) corresponds to $V^{A}= \\left\\lbrace \\begin{array}{rcll}V^a & = & H(y,\\overline{y})^{-1/4} \\, dx^a & a=0,1,2,3\\\\V^\\ell & = & H(y,\\overline{y})^{1/4} \\,\\mathbf {e}^\\ell & \\ell \\, = \\, 4,5,6,7,8,9\\end{array}\\right.$ where $\\mathbf {e}^\\ell $ are the vielbein 1-forms of the manifold $Y_{[3]}$ .", "The structure equations of the latter areThe hats over the spin connection and the Riemann tensor denote quantities computed without the warp factor.", ": $0& = & d \\, \\mathbf {e}^i - \\widehat{\\omega }^{ij} \\, \\wedge \\, \\mathbf {e}^k \\, \\eta _{jk}\\nonumber \\\\\\widehat{R}^{ij} & = & d \\widehat{\\omega }^{ij} -\\widehat{\\omega }^{ik}\\, \\wedge \\, \\widehat{\\omega }^{\\ell j} \\,\\eta _{k\\ell } = \\widehat{R}^{ij} _{\\phantom{ij}\\ell m } \\,\\mathbf {e}^\\ell \\,\\wedge \\, \\mathbf {e}^m $ The relevant property of the $Y$ metric that we use in solving Einstein equations is that it is Ricci-flat: $\\widehat{R}^{im}_{\\phantom{ij}\\ell m } = 0 $ What we need in order to derive our solution and discuss its supersymmetry properties is the explicit form of the spin connection for the full 10-dimensional metric (REF ) and the corresponding Ricci tensor.", "From the torsion equation one can uniquely determine the solution: $\\omega ^{ab} & = & 0 \\nonumber \\\\\\omega ^{a\\ell } & = & {\\textstyle \\frac{1}{4}} \\, H^{-3/2} \\, dx^a \\eta ^{\\ell k} \\,\\partial _k \\, H\\nonumber \\\\\\omega ^{\\ell m} & = & \\widehat{\\omega }^{\\ell m} + \\Delta \\omega ^{\\ell m} \\quad ; \\quad \\Delta \\omega ^{\\ell m} =- {\\textstyle \\frac{1}{2}} \\,H^{-1} \\, \\mathbf {e}^{[\\ell } \\, \\eta ^{m]k} \\, \\partial _k H$ Inserting this result into the definition of the curvature 2-form we obtainThe reader should be careful with the indices.", "Latin indices are always frame indices referring to the vielbein formalism.", "Furthermore we distinguish the 4 directions of the brane volume by using Latin letters from the beginning of the alphabet while the 6 transversal directions are denoted by Latin letters from the middle and the end of the alphabet.", "For the coordinate indices we utilize Greek letters and we do exactly the reverse.", "Early Greek letters $\\alpha ,\\beta ,\\gamma ,\\delta ,\\dots $ refer to the 6 transverse directions while Greek letters from the second half of the alphabet $\\mu ,\\nu ,\\rho ,\\sigma ,\\dots $ refer to the D3 brane world volume directions as it is customary in $D=4$ field theories.", ": $R^{a}_{b} & = & - \\frac{1}{8}\\, \\left[ H^{-3/2} \\Box _{\\mathbf {g}}\\, H - H^{-5/2} \\,\\partial _k H\\partial ^k H \\right] \\, \\delta ^a_b \\nonumber \\\\R^{a}_{\\ell } & = & 0\\nonumber \\\\R_\\ell ^m &=& \\frac{1}{8}H^{-3/2} \\Box _{\\mathbf {g}} H\\delta _\\ell ^m- \\frac{1}{8} H^{-5/2}\\partial _s H\\partial ^sH\\delta _\\ell ^m+\\frac{1}{4} H^{-5/2} \\partial _\\ell H\\partial ^m H$ where for any function $f\\left(y,\\overline{y}\\right)$ with support on $Y_{[3]}$ : $\\Box _{\\mathbf {g}} \\, f\\left(y,\\overline{y}\\right) \\, =\\,\\frac{1}{\\sqrt{\\mathrm {det}\\mathbf {g}}}\\, \\left( \\partial _\\alpha \\left(\\sqrt{\\mathrm {det}\\mathbf {g}}\\,\\,\\mathbf {g}^{\\alpha \\beta ^\\star } \\,\\partial _{\\beta ^\\star } \\,f \\right) \\right) $ denotes the action on it of the Laplace–Beltrami operator with respect to the metric (REF ) which is the Ricci-flat one: we have omitted the superscript $\\mathrm {RFK}$ just for simplicity.", "Indeed on the supergravity side of the correspondence we use only the Ricci-flat metric and there is no ambiguity." ], [ "Analysis of the field equations in geometrical terms", "The equations of motion for the scalar fields $\\varphi $ and $C_{[0]}$ and for the 3-form field strength $F^{NS}_{[3]}$ and $F^{RR}_{[3]}$ can be better analyzed using the complex notation.", "Defining, as we did above: ${\\mathcal {H}}_\\pm & = & \\pm 2 \\,e^{-\\varphi /2} F^{NS}_{[3]} + {\\rm i} 2 \\,e^{\\varphi /2} \\,F^{RR}_{[3]} \\\\P & =& {\\textstyle \\frac{1}{2}} \\, d\\varphi -{\\rm i} {\\textstyle \\frac{1}{2}} \\, e^\\varphi \\,F_{[1]}^{RR} $ eq.s (REF )-() can be respectively written as: $d(\\star P)- {\\rm i} e^{\\varphi } dC_{[0]}\\wedge \\star P + {\\textstyle \\frac{1}{16}}{\\mathcal {H}}_+ \\, \\wedge \\, \\star {\\mathcal {H}}_+=0 \\\\d \\star {\\mathcal {H}}_+ - \\frac{{\\rm i}}{2} e^{\\varphi } dC_{[0]}\\wedge \\, \\star {\\mathcal {H}}_+={\\rm i} \\, {F}_{[5]}^{RR}\\,\\wedge \\, {\\mathcal {H}}_+ - P \\wedge \\star {\\mathcal {H}}_- $ while the equation for the 5-form becomes: $d\\star F^{RR}_{[5]} = {\\rm i} \\, {\\textstyle \\frac{1}{8}} \\, {\\mathcal {H}}_+ \\wedge {\\mathcal {H}}_- $ Besides assuming the structure (REF ) we also assume that the two scalar fields, namely the dilaton $\\varphi $ and the Ramond-Ramond 0-form $C_{[0]}$ are constant and vanishing: $\\varphi =0 \\quad ; \\quad C_{[0]}=0$ As we shall see, this assumption simplifies considerably the equations of motion, although these two scalar fields can be easily restored [33]." ], [ "The three-forms", "The basic ansatz characterizing the solution and providing its interpretation as a D3-brane with three-form fluxes is described below.", "The ansatz for the complex three-forms of type IIB supergravity is given below and is inspired by what was done in [35], [34] in the case where $Y_{[3]}= \\mathbb {C}\\times \\mathrm {ALE}_\\Gamma $ : ${\\mathcal {H}}_+ \\, = \\, \\Omega ^{(2,1)} $ where $\\Omega ^{(2,1)}$ is localized on $Y_{[3]}$ and satisfies eq.s (REF -REF ) If we insert the ansätze (REF ,REF ) into the scalar field equation (REF ) we obtain: ${\\mathcal {H}}_+ \\, \\wedge \\, \\star _{10} {\\mathcal {H}}_+=0$ This equation is automatically satisfied by our ansatz for a very simple reason that we explain next.", "The form ${\\mathcal {H}}_+$ is by choice a three-form on $Y_{[3]}$ of type $(2,1)$ .", "Let $\\Theta ^{[3]}$ be any three-form that is localized on the transverse six-dimensional For the sake of the present calculation and the following ones where we have to calculate a Hodge dual, it is more convenient to utilize a set of 6 real coordinates $t^I$ ($I=1,\\dots ,6$ ) for the manifold $Y_{[3]}$ .", "Let $\\partial _I \\equiv \\frac{\\partial }{\\partial t^I}$ denote the standard partial derivatives with respect to such coordinates.", "manifold $Y_{[3]}$ : $\\Theta ^{[3]} \\, = \\, \\Theta _{IJK} \\, dt^I\\wedge dt^J \\wedge dt^K$ When we calculate the Hodge dual of $\\Theta ^{[3]}$ with respect to the 10-dimensional metric (REF ) we obtain a 7-form with the following structure: $\\star _{10}\\,\\Theta ^{[3]} \\, = \\, H^{-1} \\,\\mbox{Vol}_{\\mathbb {R}^{(1,3)}} \\, \\wedge \\, \\widetilde{\\Theta }^{[3]}$ where: $\\mbox{Vol}_{\\mathbb {R}^{(1,3)}} \\, = \\, {\\textstyle \\frac{1}{4!}}", "\\, dx^{\\mu } \\wedge dx^{\\nu }\\wedge dx^\\rho \\wedge dx^\\sigma \\, \\epsilon _{\\mu \\nu \\rho \\sigma }$ is the volume-form of the flat D3-brane and $\\widetilde{\\Theta }^{[3]} \\, \\equiv \\, \\star _{\\mathbf {g}} \\, \\Theta ^{[3]}$ is the dual of the three-form $\\Theta ^{[3]}$ with respect to the metric $\\mathbf {g}$ defined on $Y_{[3]}$ .", "Let us now specialize the three-form $\\Theta ^{[3]}$ to be of type $(2,1)$ : $\\Theta ^{[3]} \\, = \\, \\mathrm {Q}^{(2,1)}$ As shown in [31], [32], preservation of supersymmetry requires the complex three-form ${\\mathcal {H}}_+$ to obey the conditionIt also requires ${\\mathcal {H}}_+$ to be primitive.", "$\\star _{\\mathbf {g}} \\, \\mathrm {Q}^{(2,1)} \\, = - {\\rm i} \\,\\mathrm {Q}^{(2,1)}$ Hence: ${\\mathcal {H}}_+ \\wedge \\star _{10}\\, {\\mathcal {H}}_+ \\, = - {\\rm i} \\,\\mathrm {Q}^{(2,1)} \\, \\wedge \\,\\mathrm {Q}^{(2,1)}\\,\\wedge \\, H^{-1} \\mbox{Vol}_{\\mathbb {R}^{(1,3)}}\\, = \\, 0$" ], [ "The self-dual 5-form", "Next we consider the self-dual 5-form $F_{[5]}^{RR}$ which by definition must satisfy the following Bianchi identity: $d \\, F_{[5]}^{RR} = {\\rm i} \\, {\\textstyle \\frac{1}{8}} \\,{ \\mathcal {H}}_+\\, \\wedge \\, { \\mathcal {H}}_- $ Our ansatz for $F_{[5]}^{RR}$ is the following: $F_{[5]}^{RR} & = & \\alpha \\left( U + \\star _{10}\\, U \\right) \\\\U & = & d \\left( H^{-1} \\, \\mbox{Vol}_{\\mathbb {R}^{(1,3)}}\\right) $ where $\\alpha $ is a constant to be determined later.", "By construction $F_{[5]}^{RR}$ is self-dual and its equation of motion is trivially satisfied.", "What is not guaranteed is that also the Bianchi identity (REF ) is fulfilled.", "Imposing it, results into a differential equation for the function $H\\left(y,\\overline{y}\\right)$ .", "Let us see how this works.", "Starting from the ansatz () we obtain: $U &=& -\\frac{1}{4!}", "\\, \\epsilon _{\\mu \\nu \\rho \\sigma } \\,dx^\\mu \\wedge dx^\\nu \\wedge dx^\\rho \\wedge dx^\\sigma \\wedge \\frac{dH}{H^2} \\\\U_{\\mu \\nu \\rho \\sigma I} &=& -\\frac{1}{4!", "}\\epsilon _{\\mu \\nu \\rho \\sigma } \\, \\frac{\\partial _I H}{H^2} \\quad ;\\quad \\mbox{all other components vanish}$ Calculating the components of the dual form $\\star _{10}\\, U$ we find that they are non vanishing uniquely in the six transverse directions: $\\star _{10} U &=& \\widetilde{U}_{I_1\\dots I_5} \\,\\,dt^{I_1} \\wedge \\dots \\wedge dt^{I_5} \\nonumber \\\\\\widetilde{U}_{I_1\\dots I_5} &=& - \\, \\frac{\\sqrt{\\mbox{det} \\,g_{(10)}}}{5!}", "\\epsilon _{I_1\\dots I_5 J}\\,\\epsilon _{\\mu \\nu \\rho \\sigma } \\, g_{(10)}^{JK} \\, g_{10}^{\\mu \\mu ^\\prime }\\, g_{(10)}^{\\nu \\nu ^\\prime }\\, g_{(10)}^{\\rho \\rho ^\\prime }\\,g_{(10)}^{\\sigma \\sigma ^\\prime } \\, U_{\\mu ^\\prime \\nu ^\\prime \\rho ^\\prime \\sigma ^\\prime J}\\nonumber \\\\&=& \\frac{\\sqrt{\\mbox{det}\\mathbf {g}}}{5!}", "\\,\\epsilon _{I_1\\dots I_5 J} \\,\\mathbf {g}^{JK} \\, \\partial _K \\,H$ The essential point in the above calculation is that all powers of the function $H$ exactly cancel so that $\\star _{10} U$ is linear in the $H$ -derivatives Note that we use $\\mathbf {g}_{IJ}$ to denote the components of the Kähler metric (REF ) in the real coordinate basis $t^I$ .. Next using the same coordinate basis we obtain: $d \\, F_{[5]}^{RR}& = & \\alpha \\, d\\star U \\, =\\,\\alpha \\,\\underbrace{\\frac{1}{\\sqrt{\\mbox{det} \\, \\mathbf {g} }} \\, \\partial _I \\,\\left( \\sqrt{\\mbox{det} \\, \\mathbf {g} } \\, \\mathbf {g}^{IJ} \\, \\partial _JH\\right)}_{\\Box _{\\mathbf {g}} \\, H} \\, \\times \\, \\mbox{Vol}_{Y_{[3]}} \\nonumber \\\\&=& \\alpha \\,\\Box _{\\mathbf {g}} \\, H(y,\\overline{y}) \\, \\times \\,\\mbox{Vol}_{Y_{[3]}}$ where: $\\mbox{Vol}_{Y_{[3]}} & \\equiv &\\sqrt{\\mbox{det} \\, \\mathbf {g} } \\,\\frac{1}{6!}", "\\epsilon _{I_1 \\dots I_6} dt^{I_1}\\wedge \\dots \\wedge dt^{I_6} \\nonumber \\\\& = & \\sqrt{\\mbox{det} \\, \\mathbf {g}} \\,\\frac{1}{(3!", ")^2}\\,\\epsilon _{\\alpha \\beta \\gamma }\\,dy^\\alpha \\wedge dy^\\beta \\wedge dy^\\gamma \\, \\wedge \\epsilon _{\\overline{\\alpha }\\overline{\\beta }\\overline{\\gamma }}\\,d\\overline{y}^{\\overline{\\alpha }} \\wedge d\\overline{y}^{\\overline{\\beta }} \\wedge d\\overline{y}^{\\overline{\\gamma }}$ is the volume form of the transverse six-dimensional space.", "Once derived with the use of real coordinates, the relation (REF ) can be transcribed in terms of complex coordinates and the Laplace-Beltrami operator $\\Box _\\mathbf {g}$ can be written as in eq.", "(REF ).", "Let us now analyze the source terms provided by the three-forms.", "With our ansatz we obtain: ${\\textstyle \\frac{1}{8}} \\,{ \\mathcal {H}}_+ \\, \\wedge \\, {\\mathcal {H}}_- &= & \\mathbb {J}\\left(y,\\overline{y}\\right)\\, \\times \\,\\mbox{Vol}_{Y_{[3]}} \\nonumber \\\\\\mathbb {J}\\left(y,\\overline{y}\\right) & = & - \\, \\frac{1}{72\\, \\sqrt{\\mbox{det}\\, \\mathbf {g}}} \\, \\, \\,\\Omega _{\\alpha \\beta \\overline{\\eta }}\\,\\, \\overline{\\Omega }_{\\overline{\\delta }\\overline{\\theta }\\gamma } \\,\\, \\epsilon ^{\\alpha \\beta \\gamma } \\,\\,\\epsilon ^{\\overline{\\eta }\\overline{\\delta }\\overline{\\theta }}$ we conclude that the Bianchi identity (REF ) is satisfied by our ansatz if: $\\Box _\\mathbf {g}\\, H = - \\frac{1}{\\alpha }\\, \\mathbb {J}\\left(y,\\overline{y}\\right)$ This is the main differential equation to which the entire construction of the D3-brane solution can be reduced to.", "We are going to show that the parameter $\\alpha $ is determined by Einstein's equations and fixed to $\\alpha =1$ ." ], [ "The equations for the three–forms", "Let us consider next the field equation for the complex three-form, namely eq.", "().", "Since the two scalar fields are constant the $\\mathrm {SU(1,1)/O(2)}$ connection vanishes and we have: $d \\star {\\mathcal {H}}_+ = {\\rm i} \\, F^{RR}_{[5]}\\, \\wedge \\, {\\mathcal {H}}_+ $ Using our ansatz we immediately obtain: $d \\star {\\mathcal {H}}_+ = & = &- 2\\, {\\rm i} H^{-2}\\mathrm {}dH \\, \\wedge \\, \\widetilde{\\Omega }^{(2,1)}, \\wedge \\,\\Omega _{\\mathbb {R}^{1,3}}\\, +2 {\\rm i} \\, H^{-1} \\,d\\widetilde{\\Omega }^{(2,1)} \\wedge \\, \\Omega _{\\mathbb {R}^{1,3}}\\nonumber \\\\{\\rm i} \\, F^{RR}_{[5]}\\, \\wedge \\, {\\mathcal {H}}_+ & = & - 2\\,\\alpha {\\rm i} H^{-2} dH \\, \\wedge \\, \\Omega ^{(2,1)} \\, \\wedge \\,\\Omega _{\\mathbb {R}^{1,3}}\\, $ Hence if $\\alpha =1$ , the field equations for the three-form reduces to: $\\widetilde{\\Omega }^{(2,1)} \\, \\equiv \\, \\star _{\\mathbf {g}} {\\Omega }^{(2,1)} \\, = \\,-\\, {\\rm i}\\,{\\Omega }^{(2,1)} \\quad ;\\quad d\\star _{\\mathbf {g}} {\\Omega }^{(2,1)} \\, = \\,0 \\quad ;\\quad d {\\Omega }^{(2,1)} \\, = \\,0$ which are nothing else but eq.s (REF -REF ).", "In other words the solution of type IIB supergravity with three-form fluxes exists if and only if the transverse space admits closed and imaginary anti-self-dual forms $\\Omega ^{(2,1)}$ as we already statedBy construction a closed anti-self-dual form is also coclosed, namely it is harmonic..", "In order to show that also the Einstein's equation is satisfied by our ansatz we have to calculate the (trace subtracted) stress energy tensor of the five and three index field strengths.", "For this purpose we need the components of $F_{[5]}^{RR}$ .", "These are easily dealt with.", "Relying on the ansatz () and on eq.", "(REF ) for the vielbein we immediately get: $F_{A_1 \\dots A_5} =\\left\\lbrace \\begin{array}{ccc}F_{i abcd} & = & \\frac{1}{5!}", "\\, f_i \\, \\epsilon _{abcd} \\\\F_{ j_1\\dots i_5} & = & \\frac{1}{5!}", "\\epsilon _{i j_1\\dots j_5} \\, f^i \\\\\\mbox{otherwise} & = & 0 \\ \\end{array} \\right.$ where: $f_i = - \\alpha \\, H^{-5/4} \\, \\partial _i H$ Then by straightforward algebra we obtain: $T^{a}_{b}\\left[ F_{[5]}^{RR}\\right] & \\equiv & -75 \\, F^{a \\,\\cdot \\, \\cdot \\, \\cdot \\,\\cdot } \\,F_{b\\,\\cdot \\, \\cdot \\, \\cdot \\,\\cdot } = - \\frac{1}{8} \\, \\delta ^{a}_{b}\\, f_p \\, f^p \\nonumber \\\\& = & - \\alpha ^2 \\, \\frac{1}{8} \\, \\delta ^{a}_{b}\\, H^{-5/2} \\partial _p H\\, \\partial ^p H \\nonumber \\\\T^{i}_{j}\\left[ F_{[5]}^{RR}\\right] & \\equiv & -75 \\, F^{i \\,\\cdot \\, \\cdot \\, \\cdot \\,\\cdot } \\,F_{j\\,\\cdot \\, \\cdot \\, \\cdot \\,\\cdot } = \\frac{1}{4}\\, f^i \\, f_j \\, - \\, \\frac{1}{8} \\,\\delta ^i_j \\, f_p \\, f^p \\nonumber \\\\& = & \\alpha ^2 \\, \\left[ \\frac{1}{4}\\, H^{-5/4} \\partial ^i H \\, \\partial _j H \\, - \\, \\frac{1}{8}\\, \\delta ^i_j \\, H^{-5/4} \\partial ^p H \\, \\partial _p H \\right]$ Inserting eq.s (REF ) and (REF ) into Einstein's equations: $R^a_b & = & T^{a}_{b}\\left[F_{[5]}^{RR} \\right]\\nonumber \\\\R^i_j & = &T^{i}_{j}\\left[F_{[5]}^{RR}\\right]$ we see that they are satisfied, provided $\\alpha = 1$ and the master equation (REF ) is satisfied.", "This concludes our proof that an exact D3-brane solution with a $Y$ transverse space does indeed exist." ], [ "An example without mass deformations and no harmonic $\\Omega ^{(2,1)}$ :\n{{formula:13b91eaf-b838-404d-bd2c-87044b947fcc}}", "In [6] as a master example of the generalized Kronheimer construction of crepant resolutions the following case was considered: $Y_{[3]} \\, = \\, \\mathcal {O}_{\\mathbb {P}^2}(-3)\\, \\longrightarrow \\, \\frac{\\mathbb {C}^3}{\\mathbb {Z}_3}$ the action of the group $\\mathbb {Z}_3\\subset \\mathrm {SU(3)}$ on the three-complex coordinates $\\lbrace x,y,z\\rbrace $ being generated by the matrix: $\\mathfrak {g} \\, = \\, \\left(\\begin{array}{ccc}e^{\\frac{2 {\\rm i} \\pi }{3}} & 0 & 0 \\\\0 & e^{\\frac{2 {\\rm i} \\pi }{3}} & 0 \\\\0 & 0 & e^{\\frac{2 {\\rm i} \\pi }{3}} \\\\\\end{array}\\right)$ Following the steps of the construction one arrives at the following nine-dimensional flat Kähler manifold $\\mathcal {S}_{\\mathbb {Z}_3} \\, \\equiv \\,\\mbox{Hom}\\left(\\mathcal {Q}\\otimes R,R\\right)^{\\mathbb {Z}_3}$ where $\\mathcal {Q}$ is the three dimensional representation of $\\mathbb {Z}_3$ generated by $\\mathfrak {g}$ , while $R$ denotes the regular representation.", "The points of $\\mathcal {S}_{\\mathbb {Z}_3}$ are identified with the following triplet of matrices of $3\\times 3$ matrices: $A\\, = \\, \\left(\\begin{array}{ccc}0 & 0 & \\Phi ^A_{1,3} \\\\\\Phi ^A_{2,1} & 0 & 0 \\\\0 & \\Phi ^A_{3,2} & 0 \\\\\\end{array}\\right) \\quad ; \\quad B\\, = \\, \\left(\\begin{array}{ccc}0 & 0 & \\Phi ^B_{1,3} \\\\\\Phi ^B_{2,1} & 0 & 0 \\\\0 & \\Phi ^B_{3,2} & 0 \\\\\\end{array}\\right) \\quad ; \\quad C\\, = \\, \\left(\\begin{array}{ccc}0 & 0 & \\Phi ^C_{1,3} \\\\\\Phi ^C_{2,1} & 0 & 0 \\\\0 & \\Phi ^C_{3,2} & 0 \\\\\\end{array}\\right)$ The nine complex coordinates of $\\mathcal {S}_{\\mathbb {Z}_3}$ are the matrix entries $\\Phi ^{A,B,C}_{1,3}$ , $\\Phi ^{A,B,C}_{2,1}$ , $\\Phi ^{A,B,C}_{3,2}$ .", "With reference to the quiver diagram of fig.", "REF Figure: The quiver diagram of the diagonalembedding of the group ℤ 3 → SU (3)\\mathbb {Z}_3 \\rightarrow \\mathrm {SU(3)}which is dictated by the McKay matrix $\\mathcal {A}_{ij}$ appearing in the decomposition $\\mathrm {D}_i$ denote the irreducible representations of the group $\\Gamma = \\mathbb {Z}_3$ and each node of the quiver diagram corresponds to one of them.", "The number of lines going from node $i$ to node $j$ is equal to integer value of $\\mathcal {A}_{ij}$ .", "In each node $i$ we have a component $U_i(n_i\\times N)$ of the gauge group $\\mathcal {F}_\\Gamma $ where $n_i$ is the dimension of the irrep $D_i$ and $N$ is the number of D3-branes.", ": $\\mathcal {Q}\\otimes \\mathrm {D}_i \\, = \\, \\bigoplus _{j=1}^3 \\, \\mathcal {A}_{ij}\\,\\mathrm {D}_j$ the entries $\\Phi ^{A,B,C}_{1,3}, \\dots $ are interpreted as the complex scalar fields of as many Wess-Zumino multiplets in the bifundamental of the $\\mathrm {U_i(N)}$ groups mentioned in the lower suffix.", "In the case of a single brane (N=1) the quiver group $\\mathcal {G}_{\\mathbb {Z}_3}$ has the following structure: $\\mathcal {G}_{\\mathbb {Z}_3} \\, = \\, \\mathbb {C}^\\star \\otimes \\mathbb {C}^\\star \\, \\simeq \\, \\frac{\\mathbb {C}^\\star \\otimes \\mathbb {C}^\\star \\otimes \\mathbb {C}^\\star }{\\mathbb {C}^\\star _{central}}$ and its maximal compact subgroup $\\mathcal {F}_{\\mathbb {Z}_3}\\subset \\, \\mathcal {G}_{\\mathbb {Z}_3}$ is the following: $\\mathcal {F}_{\\mathbb {Z}_3} \\, = \\,\\mathrm { U(1)} \\otimes \\mathrm {U(1)} \\, \\simeq \\, \\frac{\\mathrm { U(1)} \\otimes \\mathrm {U(1)}\\otimes \\mathrm {U(1)}}{\\mathrm {U(1)}_{central}}$ The gauge group $\\mathcal {F}_{\\mathbb {Z}_3}$ and its complexification $\\mathcal {F}_{\\mathbb {Z}_3}$ are embedded into $\\mathrm {SL(3,\\mathbb {C})}$ by defining the following two generators: $\\mathbf {t}_1 \\, = \\,\\left(\\begin{array}{ccc}{\\rm i} & 0 & 0 \\\\0 & -{\\rm i} & 0 \\\\0 & 0 & 0 \\\\\\end{array}\\right) \\quad ; \\quad \\mathbf {t}_2 \\, = \\,\\left(\\begin{array}{ccc}0 & 0 & 0 \\\\0 & {\\rm i} & 0 \\\\0 & 0 & -{\\rm i} \\\\\\end{array}\\right)$ and setting: $\\mathcal {F}_{\\mathbb {Z}_3} \\, = \\, \\exp \\left[\\theta _1 \\, \\mathbf {t}_1+\\theta _2 \\, \\mathbf {t}_2\\right] \\quad \\theta _{1,2} \\in [0,2\\pi ]\\quad ; \\quad \\mathcal {G}_{\\mathbb {Z}_3} \\, = \\, \\exp \\left[w_1 \\, \\mathbf {t}_1+w_2 \\, \\mathbf {t}_2\\right] \\quad w_{1,2} \\in \\mathbb {C}$" ], [ "The HKLR Kähler potential", "The Kähler potential of the linear space $\\mathcal {S}_{\\mathbb {Z}_3}$ , which in the D3-brane gauge theory provides the kinetic terms of the nine scalar fields $\\Phi ^{A,B,C}_{1,2,3}$ is given by: $\\mathcal {K}_0\\left(\\Phi \\right)\\, = \\, \\mbox{Tr}\\left(A^\\dagger \\,A + B^\\dagger \\, B + C^\\dagger C \\right)$ where the three matrices $A,B,C$ are those of equation (REF ).", "According with the principles of the Kronheimer construction, the superpotential is given by $\\mathcal {W}\\left(\\Phi \\right)\\, = \\, \\text{const}\\times \\text{Tr}\\left( \\left[A,B\\right] \\, C\\right)$ .", "The final HKLR Kähler metric, whose determination requires two steps of physical significance: Reduction to the critical surface of the superpotential i.e.", "$\\partial _\\Phi \\,\\mathcal {W} \\, = \\, 0$ Reduction to the level surfaces of the gauge group moment maps by solving the algebraic moment map equations, was calculated in [6] according with the general theory there summarized, which is originally due to the authors of [30].", "The final form of HKLR Kähler potential is provided by: $\\mathcal {K}_{HKLR}(z,\\overline{z}, \\zeta ) & = &\\mathcal {K}_0 + \\zeta _I \\, \\mathfrak {C}^{IJ}\\, \\log \\left[ \\Upsilon _J^{\\alpha _{J,\\zeta }} \\right]\\nonumber \\\\& = & \\alpha \\Big \\lbrace \\left(2 \\zeta _1-\\zeta _2\\right) \\log \\left[\\Upsilon _1\\right]-\\left(\\zeta _1-2 \\zeta _2\\right) \\log \\left[\\Upsilon _2\\right]\\Big \\rbrace +\\frac{\\Sigma \\left(\\Upsilon _1^3+\\Upsilon _2^3+1\\right)}{\\Upsilon _1 \\Upsilon _2}$ where, as it was extensively discussed in [7], the coefficient $\\alpha $ might be adjusted, chamber by chamber, in chamber space, so as to make the periods of the tautological line bundles integer on the homology basis.", "Setting: $\\Sigma \\, \\equiv \\, |z _1|^2+|z _2|^2+|z _3|^2 \\quad ; \\quad \\Upsilon _{1,2} \\, = \\, \\Upsilon _{1,2} \\, = \\,\\Lambda _{1,2}\\left(\\Sigma ,\\zeta \\right)$ where $z_{1,2,3}$ are the three complex coordinates and $\\zeta \\, = \\, \\left\\lbrace \\zeta _1,\\zeta _2\\right\\rbrace $ the two Fayet-Iliopoulos parameters.", "Let us describe the explicit form of these functions.", "To this effect let us name $\\zeta _1=p$ , $\\zeta _2 = q$ , and let us introduce the following blocks: $\\mathfrak {A} & = & \\sqrt{p^6 \\left(\\left(2 p^3 q^3+9 p^2 q \\Sigma ^3+9 p q^2\\Sigma ^3+27 \\Sigma ^6\\right)^2-4 \\left(p^2 q^2+3 p\\Sigma ^3+3 q \\Sigma ^3\\right)^3\\right)}\\nonumber \\\\\\mathfrak {B} & = & 2 p^6 q^3+9 p^5 q \\Sigma ^3+9 p^4 q^2 \\Sigma ^3+27 p^3 \\Sigma ^6+\\mathfrak {A}$ then we have: $\\Lambda _{1}\\left(\\Sigma ,p,q\\right) &=& \\@root 3 \\of {\\frac{\\@root 3 \\of {2} p^4 q^2}{3 \\@root 3 \\of {\\mathfrak {B}}\\Sigma ^3}+\\frac{\\@root 3 \\of {2}p^3}{\\@root 3 \\of {\\mathfrak {B}}}+\\frac{\\@root 3 \\of {2} p^2q}{\\@root 3 \\of {\\mathfrak {B}}}+\\frac{\\@root 3 \\of {\\mathfrak {B}}}{3\\@root 3 \\of {2} \\Sigma ^3}+\\frac{p^2 q}{3 \\Sigma ^3}+1} \\\\\\Lambda _{2}\\left(\\Sigma ,p,q\\right) & = &{1\\over {18\\ 6^{2/3} \\mathfrak {B}^{2/3} p \\Sigma ^5}}\\left[\\frac{2^{2/3} \\mathfrak {B}^{2/3}+2\\@root 3 \\of {\\mathfrak {B}} \\left(p^2 q+3 \\Sigma ^3\\right)+2\\@root 3 \\of {2} p^2 \\left(p^2 q^2+3 p \\Sigma ^3+3 q \\Sigma ^3\\right)}{\\@root 3 \\of {\\mathfrak {B}} \\Sigma ^3}\\right]^{2/3}\\times \\nonumber \\\\&&\\left[6 \\mathfrak {B}^{2/3} p^2 \\Sigma ^3 (p-q)-\\@root 3 \\of {2}\\mathfrak {B}^{4/3}+2^{2/3} \\mathfrak {B} \\left(p^2 q+3 \\Sigma ^3\\right)+ \\right.", "\\nonumber \\\\ && \\left.2 \\@root 3 \\of {2 \\mathfrak {B}} p^2\\left(p^4 q^3+3 p^3 q \\Sigma ^3+6 p^2 q^2 \\Sigma ^3+9p \\Sigma ^6+9 q \\Sigma ^6\\right)\\right.\\nonumber \\\\&&\\left.\\qquad -2\\ 2^{2/3} p^4\\left(p^2 q^2+3 p \\Sigma ^3+3 q \\Sigma ^3\\right)^2\\right]$" ], [ "The issue of the Ricci-flat metric", "One main question is whether the metric arising from the Kähler quotient, which is encoded in eq.", "(REF ) is Ricci-flat.", "A Ricci-flat metric on the crepant resolution of the singularity $\\mathbb {C}^3/\\mathbb {Z}_3$ , namely on $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ , is known in explicit form from the work of CalabiSuch metrics were also re-discovered in the physics literature in [52].", "[42], yet it is not a priori obvious that the metric defined by the Kähler potential (REF ) is that one.", "The true answer is that it is not, as we show later on.", "Indeed we are able to construct directly the Kähler potential for the resolution of $\\mathbb {C}^n/\\mathbb {Z}_n$ , for any $n\\ge 2$ , in particular determining the unique Ricci-flat metric on $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ with the same isometries as the metric (REF ) and comparing the two we see that they are different.", "Here we stress that the metric defined by (REF ) obviously depends on the level parameters $\\zeta _1,\\zeta _2$ while the Ricci-flat one is unique up to an overall scale factor.", "This is an additional reason to understand a priori that (REF ) cannot be the Ricci flat metric.", "Actually Calabi in [42] found an easy form of the Kähler potential of a Ricci-flat metric on the canonical bundle of a Kähler-Einstein manifold, and that result applies to the cases of the canonical bundle of $\\mathbb {P}^2$ .", "However, in view of applications to cases where we shall consider the canonical bundles of manifolds which are not Kähler-Einstein, in the section we stick with our strategy of using the metric coming from the Kähler quotient as a starting point." ], [ "The Ricci-flat metric on $Y_{[3]}=\\mathcal {O}_{\\mathbb {P}^2}(-3)$", "As we have noticed above the HKLR Kähler metric defined by the Kähler potential (REF ) depends only on the variable $\\Sigma $ defined in eq.", "(REF ).", "It follows that the HKLR Kähler metric admits $\\mathrm {U(3)}$ as an isometry group, which is the hidden invariance of $\\Sigma $ .", "The already addressed question is whether the HKLR metric can be Ricci-flat.", "An almost immediate result is that a Ricci-flat Kähler metric depending only on the sum of the squared moduli of the complex coordinates is unique (up to a scale factor) and we can give a general formula for it.", "We can present the result in the form of a theorem.", "Theorem 3.1 Let $\\mathcal {M}_n$ be a non-compact $n$ -dimensional Kähler manifold admitting a dense open coordinate patch $z_i$ , $i=1,\\dots ,n$ which we can identify with the total space of the line bundle $\\mathcal {O}_{\\mathbb {P}^{n-1}}(-n)$ , the bundle structure being exposed by the coordinate transformation: $z_i\\,=\\, u_i \\, w^{{\\textstyle \\frac{1}{n}}} \\quad , \\quad (i=1,\\dots ,n-1) \\quad \\quad ; \\quad z_n\\, = \\, w^{{\\textstyle \\frac{1}{n}}}$ where $u_i$ is a set of inhomogenous coordinates for $\\mathbb {P}^{n-1}$ .", "The Kähler potential $\\mathcal {K}_n$ of a $\\mathrm {U(n)}$ isometric Kähler metric on $\\mathcal {M}_n$ must necessarily be a real function of the unique real variable $\\Sigma \\, = \\, \\sum _{i=1}^n |z_i|^2$ .", "If we require that metric should be Ricci-flat, the Kähler potential is uniquely defined and it is the following one: $\\mathcal {K}_n(\\Sigma )\\, = \\, k \\, + \\frac{\\left(\\Sigma ^n+\\ell ^n\\right)^{-\\frac{n-1}{n}}\\left((n-1) \\left(\\Sigma ^n+\\ell ^n\\right)-\\ell ^n\\left(\\Sigma ^{-n} \\ell ^n+1\\right)^{\\frac{n-1}{n}} \\,_2F_1\\left(\\frac{n-1}{n},\\frac{n-1}{n};\\frac{2n-1}{n};-\\ell ^n \\Sigma ^{-n}\\right)\\right)}{n-1}$ where $k$ is an irrelevant additive constant and $\\ell >0$ is a constant that can be reabsorbed by rescaling all the complex coordinates by a factor $\\ell $ , namely $z_i\\rightarrow \\ell \\widetilde{z}_i$ .", "Proof 3.1.1 The proof of the above statement is rather elementary.", "It suffices to recall that the Ricci tensor of any Kähler metric $\\mathbf {g}_{ij^\\star }\\, = \\, \\partial _i\\partial _{j^\\star }\\mathcal {K}(z,\\overline{z})$ can always be calculated as follows: $\\mathrm {Ric}_{ij^\\star }[\\mathbf {g}] \\, = \\,\\partial _i\\,\\partial _{j^\\star } \\, \\log \\left[\\mathrm {Det}\\left[\\mathbf {g}\\right]\\right]$ In order for the Ricci tensor to be zero it is necessary that $\\mathrm {Det}\\left[\\mathbf {g}\\right]$ be the square modulus of a holomorphic function $|F(z)|^2$ , on the other hand under the hypotheses of the theorem it is a real function of the real variable $\\Sigma $ .", "Hence it must be a constant.", "It follows that we have to impose the equation: $\\mathrm {Det}\\left[\\mathbf {g}\\right] \\, = \\, \\ell ^2 \\, = \\,\\mbox{const}$ Let $\\mathcal {K}(\\Sigma )$ be the sought for Kähler potential, calculating the Kähler metric and its determinant we find: $\\mathrm {Det}\\left[\\mathbf {g}\\right] \\, = \\, \\Sigma ^{n-1} \\mathcal {K}(\\Sigma )^{\\prime } \\left(\\Sigma ^2\\mathcal {K}(\\Sigma )^{\\prime \\prime }+\\Sigma \\, \\mathcal {K}(\\Sigma )^{\\prime } \\right)$ Inserting eq.", "(REF ) into eq.", "(REF ) we obtain a non linear differential equation for $\\mathcal {K}(\\Sigma )$ of which eq.", "(REF ) is the general integral.", "This proves the theorem.", "$\\diamondsuit $" ], [ "Particular cases", "It is interesting to analyze particular cases of the general formula (REF )." ], [ "The case $n=2$ yielding a Ricci flat metric on $\\mathcal {O}_{\\mathbb {P}^1}(-2)$ is the Eguchi-Hanson case namely the crepant resolution of the Kleinian singularity $\\mathbb {C}^2/\\mathbb {Z}_2$ .", "This is known to be a HyperKähler manifold and all HyperKähler metrics are Ricci-flat.", "Hence also the HKLR metric must be Ricci-flat and identical with the one defined by eq.", "(REF ).", "Actually we find: $\\mathcal {K}_2(\\Sigma )&=&\\left(\\Sigma ^2+\\ell ^2\\right)^{-\\frac{1}{2}}\\left( \\left(\\Sigma ^2+\\ell ^2\\right)-\\ell ^2\\left(\\Sigma ^{-2} \\ell ^2+1\\right)^{\\frac{1}{2}} \\,_2F_1\\left(\\frac{1}{2},\\frac{1}{2};\\frac{3}{2};-\\ell ^2 \\Sigma ^{-2}\\right)\\right)\\nonumber \\\\&=&\\sqrt{\\Sigma ^2+\\ell ^2}-\\ell \\log \\left(\\sqrt{\\Sigma ^2+\\ell ^2}+\\ell \\right)+\\ell \\log (\\Sigma ) \\, + \\, \\mbox{const}$ which follows from the identification of the hypergeometric function with combinations of elementary transcendental functions occurring for special values of its indices.", "The second transcription of the function is precisely the Kähler potential of the Eguchi-Hanson metric in its HKLR-form as it arises from the Kronheimer construction (see for instance [6])." ], [ "The next case is that of interest for the D3-brane solution.", "For $n=3$ , setting $\\ell =1$ , which we can always do by a rescaling of the coordinates, we find: $\\mathcal {K}_{Rflat}(\\Sigma ) &=& \\frac{2 \\left(\\Sigma ^3+1\\right)-\\left(\\frac{1}{\\Sigma ^3}+1\\right)^{2/3} \\,_2F_1\\left(\\frac{2}{3},\\frac{2}{3};\\frac{5}{3};-\\frac{1}{\\Sigma ^3}\\right)}{2 \\left(\\Sigma ^3+1\\right)^{2/3}}\\nonumber \\\\&=&\\frac{2 (\\Sigma ^3+1)-\\,_2F_1\\left(\\frac{2}{3},1;\\frac{5}{3};\\frac{1}{\\Sigma ^3+1}\\right)}{2 \\left(\\Sigma ^3+1\\right)^{2/3}}$ The second way of writing the Kähler potential follows from one of the standard Kummer relations among hypergeometric functions.", "There is a third transcription that also in this case allows to write it in terms of elementary transcendental functions.", "Before considering it we use eq.", "(REF ) to study the asymptotic behavior of the Kähler potential for large values of $\\Sigma $ .", "We obtain; $\\mathcal {K}_{Rflat}(\\Sigma ) \\, \\stackrel{\\Sigma \\rightarrow \\infty }{\\approx } \\, \\Sigma -\\frac{1 }{6\\Sigma ^2} +\\frac{1}{45 \\Sigma ^5} +\\mathcal {O}\\left(\\frac{1}{\\Sigma ^7}\\right)$ Eq.", "(REF ) shows that the Ricci-flat metric is asymptotically flat since the Kähler potential approaches that of $\\mathbb {C}^3$ .", "As anticipated, there is an alternative way of writing the Kähler potential (REF ) which is the following: $\\mathcal {K}_{Rflat}(\\Sigma ) &=&\\frac{\\pi }{2\\sqrt{3}}+ \\frac{1}{6} \\left(6 \\@root 3 \\of {\\Sigma ^3+1}+2 \\log \\left(\\@root 3 \\of {\\Sigma ^3+1}-1\\right)\\right.\\nonumber \\\\&&\\left.-\\log \\left(\\left(\\Sigma ^3+1\\right)^{2/3}+\\@root 3 \\of {\\Sigma ^3+1}+1\\right)-2 \\sqrt{3} \\tan ^{-1}\\left(\\frac{2\\@root 3 \\of {\\Sigma ^3+1}+1}{\\sqrt{3}}\\right)\\right)$ The identity of eq.", "(REF ) with eq.", "(REF ) can be worked with analytic manipulations that we omit.", "The representation (REF ) is particularly useful to explore the behavior of the Kähler potential at small values of $\\Sigma $ .", "We immediately find that: $\\mathcal {K}_{Rflat}(\\Sigma ) \\, \\stackrel{\\Sigma \\rightarrow 0}{\\approx } \\,\\log [\\Sigma ]+\\frac{\\pi }{2\\sqrt{3}}+\\mathcal {O}\\left(\\Sigma ^6\\right)$ The behavior of $\\mathcal {K}_{Rflat}(\\Sigma )$ is displayed in fig.REF .", "Figure: The plot of the Kähler potential𝒦 Rflat (Σ)\\mathcal {K}_{Rflat}(\\Sigma ) for the Ricci-flat metric on𝒪 ℙ 2 (-3)\\mathcal {O}_{\\mathbb {P}^2}(-3).", "The asymptotic flatness of themetric is evident from the plot.", "For large values of Σ\\Sigma itbecomes a straight line with angular coefficient 1." ], [ "The harmonic function in the case $Y_{[3]}=\\mathcal {O}_{\\mathbb {P}^2}(-3)$", "Let us now consider the equation for a harmonic function $H(z,\\overline{z})$ on the background of the Ricci-flat metric of $Y_{[3]}$ that we have derived in the previous sections.", "Once again we suppose that $H=H(\\Sigma )$ is a function only of the real variable $\\Sigma $ , viz.", "$R=\\sqrt{\\Sigma }$ .", "For the Ricci-flat metric the Laplacian equation takes the simplified form: $\\partial _i \\left(g^{ij^\\star } \\, \\partial _{j^\\star }H(\\Sigma )\\right) \\, = \\, 0$ , since the determinant of the metric is constant.", "Using the Kähler metric that follows from the Kähler potential $K_{Rflat}(\\Sigma )$ defined by eq.s (REF ),(REF ), we obtain a differential equation that upon the change of variable $\\Sigma =\\@root 3 \\of {r}$ takes the following form: $3 r (r+1) C^{\\prime \\prime }(r)+(5 r+3) C^{\\prime }(r)\\, = \\, 0$ The general integral eq.", "(REF ) is displayed below: $C(r)\\, = \\, \\kappa +\\lambda \\left(\\log \\left(1-\\@root 3 \\of {r+1}\\right)-\\frac{1}{2} \\log \\left((r+1)^{2/3}+\\@root 3 \\of {r+1}+1\\right)-\\sqrt{3} \\tan ^{-1}\\left(\\frac{2\\@root 3 \\of {r+1}+1}{\\sqrt{3}}\\right)\\right)$ $\\kappa ,\\lambda $ being the two integration constants.", "We fix these latter with boundary conditions.", "We argue in the following way: if the transverse space to the brane were the original $\\mathbb {C}^3/\\mathbb {Z}_3$ instead of the resolved variety $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ , then the harmonic function describing the D3-brane solution would be the following: $H_{orb}(R) \\, = \\, 1 + \\frac{1}{R^4} \\quad ; \\quad R \\, \\equiv \\,\\sqrt{\\Sigma } \\, = \\, \\sqrt{\\sum _{i}^2 |z_i|^2} \\, = \\, \\@root 6 \\of {r}$ The asymptotic identification for $R\\rightarrow \\infty $ of the Minkowski metric in ten dimension would be guaranteed, while at small values of $R$ we would find (via dimensional transmutation) the standard $\\mathrm {AdS_5}$ -metric times that of $\\mathbb {S}^5$ (see the following eq.s (REF ) and (REF )).", "In view of this, naming $R$ the square root of the variable $\\Sigma $ , we fix the coefficients $\\kappa ,\\lambda $ in the harmonic function $H_{res}(R)$ in such a way that for large values of $R$ it approaches the harmonic function pertaining to the orbifold case (REF ).", "The asymptotic expansion of the function: $H_{res}(R) \\, \\equiv \\, C(r^6) $ is the following one: $H_{res}(R)\\, \\stackrel{R\\rightarrow \\infty }{\\approx } \\, \\left(\\lambda -\\frac{\\pi \\kappa }{2 \\sqrt{3}}\\right)-\\frac{1}{2} \\kappa \\left(\\frac{1}{R}\\right)^4+O\\left(\\left(\\frac{1}{R}\\right)^5\\right)$ Hence the function $H_{res}(R)$ approximates the function $H_{orb}(R)$ if we set $ \\kappa \\, = \\, 2 \\, , \\, \\lambda =\\frac{\\pi }{\\sqrt{3}} $ .", "In this way we conclude that: $H_{res}(R) & = & \\frac{1}{3} \\left(2 \\log \\left(\\@root 3 \\of {R^6+1}-1\\right)-\\log \\left(\\left(R^6+1\\right)^{2/3}+\\@root 3 \\of {R^6+1}+1\\right)\\right.\\nonumber \\\\&&\\left.-2 \\sqrt{3} \\tan ^{-1}\\left(\\frac{2 \\@root 3 \\of {R^6+1}+1}{\\sqrt{3}}\\right)\\right)+\\frac{\\pi }{\\sqrt{3}}$ The overall behavior of the function $H_{res}(R)$ is displayed in fig.REF .", "Figure: The plot of the harmonic functionH res (R)H_{res}(R) for the Ricci-flat metric on𝒪 ℙ 2 (-3)\\mathcal {O}_{\\mathbb {P}^2}(-3)." ], [ "The asymptotic limits of the Ricci-flat metric for the D3-brane solution on $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ \n", "In the case of a standard D3-brane on $Y_{[3]}\\, = \\,\\mathbb {C}^3\\simeq \\mathbb {R}^6$ one writes the same ansatz as in eq.", "(REF ) and (REF -) where now the Kähler metric is $ \\mathbf {g}_{\\alpha \\beta ^\\star } \\, = \\,\\delta _{\\alpha \\beta ^\\star } $ Rewriting the complex coordinates in terms of polar coordinates $ z_1= e^{{\\rm i} \\varphi _1} R \\cos \\phi ,\\,z_2= e^{{\\rm i} \\varphi _2} R \\cos \\chi \\sin \\phi ,\\, z_3= e^{{\\rm i}\\varphi _3} R \\sin \\chi \\sin \\phi $ we obtain that: $\\text{ds}^2_{\\mathbb {C}^3} &\\equiv & \\sum _{i=1}^3 |dz_i|^2\\,= \\, dR^2+R^2 \\, \\text{ds}^2_{\\mathbb {S}^5}$ where: $\\text{ds}^2_{\\mathbb {S}^5} \\, = \\, d{\\varphi _1}^2 \\cos ^2 \\phi +\\sin ^2\\phi \\left(d{\\varphi _2}^2 \\cos ^2\\chi +d{\\varphi _3}^2 \\sin ^2\\chi +d\\chi ^2\\right)+d\\phi ^2$ is the $\\mathrm {SO(6)}$ -invariant metric of a 5-sphere in polar coordinates.", "In other words the Ricci-flat Kähler metric $ \\text{ds}^2_{\\mathbb {C}^3}$ (which is also Riemann-flat) is that of the metric cone on the Sasaki-Einstein metric of $\\mathbb {S}^5$ .", "At the same time the $\\mathrm {SO(6)}$ -invariant harmonic function on $\\mathbb {C}^3$ is given by the already quoted $H_{orb}(R)$ in (REF ), and the complete 10-dimensional metric of the D3-brane solution takes the form: $\\text{ds}^2_{10|orb}\\,=\\,\\frac{1}{\\sqrt{1+\\frac{1}{R^4}}}\\, \\text{ds}^2_{\\mathrm {Mink}_{1,3}}\\, + \\,\\sqrt{1+\\frac{1}{R^4}}\\, \\left(dR^2+R^2 \\,\\text{ds}^2_{\\mathbb {S}^5}\\right)$ For $R\\rightarrow \\infty $ the metric (REF ) approaches the flat Minkowski metric in $d=10$ , while for $R\\rightarrow 0$ it approaches the following metric: $\\text{ds}^2_{10|orb}\\, \\stackrel{R\\rightarrow 0}{\\approx } \\, \\underbrace{R^2\\,\\, \\text{ds}^2_{\\mathrm {Mink}_{1,3}} \\, + \\, \\frac{dR^2}{R^2}}_{\\mathrm {AdS}_5} \\, +\\,\\underbrace{ \\text{ds}^2_{\\mathbb {S}^5}}_{\\mathbb {S}^5}$ Let us now cosider the asymptotic behavior of the Ricci-flat metric on $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ .", "In order to obtain a precise comparison with the flat orbifold case the main technical point is provided by the transcription of the $\\mathbb {S}^5$ -metric in terms of coordinates well adapted to the Hopf fibration: $\\mathbb {S}^5 \\, \\stackrel{\\pi }{\\longrightarrow } \\, \\mathbb {P}^2\\quad ; \\quad \\forall \\; p \\in \\mathbb {P}^2 \\quad \\pi ^{-1}(p) \\sim \\mathbb {S}^1$ To this effect let $Y=\\left\\lbrace u,v\\right\\rbrace $ be a pair of complex coordinates for $\\mathbb {P}^2$ such that the standard Fubini-Study metric on this compact 2-fold is given by: $\\text{ds}^2_{\\mathbb {P}^2}\\, = \\, g^{\\mathbb {P}^2}_{ij^\\star }\\, dY^i \\, d\\overline{Y}^{j^\\star }\\, \\equiv \\,\\frac{ {dY}\\cdot d\\overline{Y}}{1+Y\\cdot \\overline{Y}}-\\frac{\\left(\\overline{Y}\\cdot d {Y}\\right)\\left(Y\\cdot d\\overline{Y}\\right)}{\\left(1+Y\\cdot \\overline{Y}\\right)^2}$ the corresponding Kähler 2-form being $\\mathbb {K}_{\\mathbb {P}^2}\\,= \\, \\frac{{\\rm i}}{2\\pi } \\,g^{\\mathbb {P}^2}_{ij^\\star }\\, dY^i \\,\\wedge \\,d\\overline{Y}^{j^\\star }$ .", "Introducing the one form: $ \\Omega \\, =\\,\\frac{{\\rm i} \\left({Y}\\cdot d\\overline{Y}-\\overline{Y} \\cdot dY\\right)}{2\\left(1+Y\\cdot \\overline{Y}\\right)}$ whose exterior derivative is the Kähler 2-form, $d\\Omega \\, = \\, 2\\pi \\,\\mathbb {K}_{\\mathbb {P}^2}$ , the metric of the five-sphere in terms of these variables is the following one: $\\text{ds}^2_{\\mathbb {S}^5} \\, = \\, \\text{ds}^2_{\\mathbb {P}^2} \\, + \\,\\left(d\\varphi + \\Omega \\right)^2$ where the range of the coordinate $\\varphi $ spanning the $\\mathbb {S}^1$ fiber is $\\varphi \\in \\left[0,2\\pi \\right]$ .", "In this way the flat metric on the metric cone on $\\mathbb {S}^5$ , namely (REF ) can be rewritten as follows: $\\text{ds}^2_{\\mathbb {C}^3} \\, = \\, dR^2 + R^2 \\, \\text{ds}^2_{\\mathbb {P}^2} \\,+ R^2 \\left(d\\varphi + \\Omega \\right)^2$" ], [ "Comparison of the Ricci-flat metric with the orbifold\nmetric", "In order to compare the exact Ricci-flat metric streaming from the Kähler potential (REF ) with the metric (REF ) it suffices to turn to toric coordinates $z_1 = u \\@root 3 \\of {w}\\, , \\quad z_2\\, = \\, v \\@root 3 \\of {w}\\, , \\quad z_3 \\, = \\, \\@root 3 \\of {w} \\quad ; \\quad \\Sigma \\, = \\,\\left(1+\\varpi \\right) \\, {\\mathfrak {f}}^{1/3} \\quad ; \\quad \\varpi =|u|^2+|v|^2 \\quad ; \\quad {\\mathfrak {f}} \\, = \\, |w|^2$ The toric coordinates $\\lbrace u,v\\rbrace \\equiv Y$ span the exceptional divisor $\\mathbb {P}^2$ while $w$ is the fiber coordinate in the bundle.", "Setting: $w \\, = \\, e^{{\\rm i}\\psi } |w| \\, = \\, e^{{\\rm i}\\psi } \\, \\left(\\frac{R^2}{1 + |u|^2 + |v|^2}\\right)^{{\\textstyle \\frac{3}{2}}}$ we obtain: $\\text{ds}^2_{Rflat} & = & h(R) dR^2 \\, + \\, f(R) \\, \\text{ds}^2_{\\mathbb {P}^2}\\, + \\, g(R) \\, \\left(d\\psi + 3 \\, \\Omega \\right)^2 \\nonumber \\\\f(R) & = &\\@root 3 \\of {R^6+1} \\quad ; \\quad h(R) \\, = \\, \\frac{R^4}{\\left(R^6+1\\right)^{2/3}} \\quad ;\\quad g(R) \\, = \\, \\frac{R^6}{9 \\left(R^6+1\\right)^{2/3}}\\quad $ From eq.", "(REF ) we derive the asymptotic form of the metric for large values of $R$ , namely: $\\text{ds}^2_{Rflat}\\, \\stackrel{R \\rightarrow \\infty }{\\approx }\\,{dR}^2+R^2 \\text{ds}_{\\mathbb {P}_2}^2+R^2 \\left(\\frac{\\text{d$\\psi $}}{3}+\\Omega \\right)^2$ The only difference between eq.", "(REF ) and eq.", "(REF ) is the range of the angular value $\\varphi \\, =\\,\\frac{\\psi }{3}$ .", "Because of the original definition of the angle $\\psi $ , the new angle $\\varphi \\in \\left[0,\\frac{2\\pi }{3}\\right]$ takes one third of the values.", "This means that the asymptotic metric cone is quotiened by $\\mathbb {Z}_3$ as it is natural since we resolved the singularity $\\mathbb {C}^3/\\mathbb {Z}_3$ ." ], [ "Reduction to the exceptional\ndivisor", "The other important limit of the Ricci-flat metric is its reduction to the exceptional divisor $\\mathcal {ED}$ .", "In the present case the only fixed point for the action of $\\Gamma =\\mathbb {Z}_3$ on $\\mathbb {C}^3$ is provided by the origin $z_{1,2,3}=0$ which, comparing with eq.", "(REF ), means $w=0 \\Rightarrow \\mathfrak {f}=0$ .", "This is the equation of the exceptional divisor which is created by the blowup of the unique singular point.", "In the basis of the complex toric coordinates $Y^i\\, \\equiv \\lbrace u,v,w\\rbrace $ , the Kähler metric derived from the Kähler potential (REF ) has the following appearance: $g_{ij^\\star }^{Rflat} \\, = \\, \\left(\\begin{array}{ccc}\\frac{v \\overline{v}+\\mathfrak {f} (\\varpi +1)^4+1}{(\\varpi +1)^2 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} & -\\frac{v \\overline{u}}{(\\varpi +1)^2 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} & \\frac{w (\\varpi +1)^2 \\overline{u}}{3 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} \\\\-\\frac{u \\overline{v}}{(\\varpi +1)^2 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} &\\frac{u \\overline{u}+\\mathfrak {f} (\\varpi +1)^4+1}{(\\varpi +1)^2 \\left(\\mathfrak {f}(\\varpi +1)^3+1\\right)^{2/3}} & \\frac{w (\\varpi +1)^2 \\overline{v}}{3 \\left(\\mathfrak {f}(\\varpi +1)^3+1\\right)^{2/3}} \\\\\\frac{u (\\varpi +1)^2 \\overline{w}}{3 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} &\\frac{v (\\varpi +1)^2 \\overline{w}}{3 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} &\\frac{(\\varpi +1)^3}{9 \\left(\\mathfrak {f} (\\varpi +1)^3+1\\right)^{2/3}} \\\\\\end{array}\\right)$ where the invariants $\\mathfrak {f},\\varpi $ are defined in equation (REF ).", "Hence the reduction of the metric to the exceptional divisor is obtained by setting $\\mathrm {d}w=\\mathrm {d}{\\overline{w}}=0$ in the line element $\\text{ds}^2_{Rflat}\\, \\equiv \\, g_{ij^\\star }^{Rflat} dY^i \\,\\,d\\overline{Y}^{j^\\star } $ and performing the limit $\\mathfrak {f}\\rightarrow 0$ on the result.", "We obtain: $\\text{ds}^2_{Rflat}\\, \\stackrel{\\mathcal {ED}}{\\longrightarrow } \\,\\text{ds}_{\\mathbb {P}^2}^2 \\, \\equiv \\,\\frac{\\mathrm {d}v\\left( \\mathrm {d}{\\overline{v}}+u\\,{\\overline{u}} \\mathrm {d}{\\overline{v}}-u{\\overline{v}}\\,\\mathrm {d}{\\overline{u}}\\right)+\\mathrm {d}u\\left( \\mathrm {d}{\\overline{u}}+ v\\,\\overline{v}\\mathrm {d}{\\overline{u}}-{\\overline{u}}\\,v\\,\\mathrm {d}{\\overline{v}}\\right)}{\\left(1+u \\overline{u}+v \\overline{v}\\right)^2}$ which is the standard Fubini-Study metric on $\\mathbb {P}^2$ obtained from the Kähler potential: $\\mathcal {K}_{\\mathbb {P}^2}^{FS}(\\varpi )\\, = \\, \\log \\left(1+\\varpi \\right)$ As we see, the metric on the exceptional divisor obtained from the Ricci-flat metric has no memory of the Fayet Iliopoulos (or stability parameters) $p,q$ which characterize instead the HKLR metric obtained from the Kronheimer construction.", "This is obvious since the Ricci-flat metric does not depend on $p,q$ .", "On the other hand the HKLR metric, that follows from the Kähler potential (REF ), strongly depends on the Fayet Iliopoulos parameters $\\zeta _1=p\\, , \\, \\zeta _2=q$ and one naturally expects that the reduction of $\\text{ds}^2_{HKLR}$ to the exceptional divisor will inherit such a dependence.", "Actually this is not the case since the entire dependence from $p,q$ of the HKLR Kähler potential, once reduced to $\\mathcal {ED}$ , is localized in an overall multiplicative constant and in an irrelevant additive constant.", "This matter of fact is conceptually very important in view of our conjecture that the Ricci-flat metric is completely determined, by means of the Monge-Ampère equation, from the Kähler metric on the exceptional divisor, as it is determined by the Kronheimer construction.", "In the present case where, up to a multiplicative constant, i.e.", "a homothety there is only one Ricci-flat metric on $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ with the prescribed isometries, our conjecture might be true only if the reduction of the HKLR metric to the exceptional divisor is unique and $p,q$ -independent, apart from overall rescalings.", "It is very much reassuring that this is precisely what actually happens." ], [ "The case $Y\\rightarrow \\mathbb {C}^3/\\mathbb {Z}_4$ and the general problem of determining a Ricci-flat metric", "The next case of interest to us at present is the resolution $Y\\rightarrow \\mathbb {C}^3/\\mathbb {Z}_4$ whose associated Kronheimer construction was studied in detail in [7].", "(A study of $ \\mathbb {C}^3/\\mathbb {Z}_4$ as a non-complete intersection affine variety in $\\mathbb {C}^9$ is presented in the Appendix.)", "The corresponding MacKay quiver is displayed in fig.REF .", "Differently from the case of the resolution $Y\\rightarrow \\mathbb {C}^3/\\mathbb {Z}_3$ studied in section , here the HKLR Kähler metric cannot be derived explicitly since the moment map equations form a system of algebraic equations of higher degree.", "Yet as it was explained in [7] one can work out the restriction of such metric to the compact component of the exceptional divisor which is the second Hirzebruch surface $\\mathbb {F}_2$ .", "Indeed it was shown that the quotient singularity $\\mathbb {C}^3/\\mathbb {Z}_4$ can be completely resolved by ${\\rm tot} K_{\\mathbb {F}_{2}}$ [7], that denotes the total space of the canonical bundle over the second Hirzebruch surface.", "Hence the main goal we would like to achieve is the construction of a Ricci-flat Kähler metric on ${\\rm tot} K_{\\mathbb {F}_{2}}$ which restricted to the base $\\mathbb {F}_2$ of the bundle hopefully coincides with Kähler metric on the same surface provided by the Kronheimer construction.", "Being a non-compact Calabi-Yau variety the existence of a Ricci-flat Kähler metric on ${\\rm tot} K_{\\mathbb {F}_{2}}$ is not implied by the classic Yau theorem, valid for smooth compact manifolds.", "To ask whether Ricci-flat metrics do exist, one has to specify boundary conditions.", "Figure: The quiver diagram describing theℂ 3 /ℤ 4 \\mathbb {C}^3/\\mathbb {Z}_4 singular quotient and codifying itsresolution via Kähler quotient à la Kronheimer.", "The samequiver diagram codifies the construction of the corresponding gaugetheory for a stack of D3-branes.", "Each node is associated with one ofthe 4 irreducible representations of ℤ 4 \\mathbb {Z}_4 and in each nodewe located one of the U i (1)\\mathrm {U_i(1)} groups with respect to whichwe perform the Kähler quotient.", "This is the case of oneD3-brane.", "For NN D3–branes, all gauge groups U i (1)\\mathrm {U_i(1)} arepromoted to U i (N)\\mathrm {U_i(N)}.We will be interested in metrics that, just as in the previous example, are asymptotically conical, namely of the formNote that without specifying the boundary conditions there can exist more than one Ricci-flat metric.", "Explicit examples of non-asymptotically conical Ricci-flat metrics in six real dimensions can be found in [43].", "$\\text{ds}^2 (Y) \\approx dR^2 + R^2 \\text{ds}^2 (X_5)$ for a suitable radial coordinate approaching $R\\rightarrow \\infty $ .", "Essentially by definition, $ \\text{ds}^2 (X_5)$ is a Sasaki-Einstein metric on a compact manifold (or orbifold) $X_5$ .", "Then we fix the boundary conditions for our metric by requiring that asymptotically it approaches the cone over $\\mathbb {S}^5/\\mathbb {Z}_4$ .", "With this boundary conditionThe results in [28] require some more precise estimate on the fall-off of the metric at infinity.", "the theorems in [28] imply the existence of a unique Ricci-flat Kähler metric in every Kähler class of the resolved variety $Y$ .", "Analogous existence results for isolated quotient singularities $\\mathbb {C}^m/\\Gamma $ were given in [44] and later extended in [45] and [46] for crepant resolutions of general isolated conical singularities.", "See also [47] for of applications of the general existence results in the toric context, including the resolution of the conical singularities on the $Y^{p,q}$ Sasaki-Einstein five-manifolds [48].", "The existence results are analogous to Yau's theorem in the compact case.", "In fact, recently there has been some renewed interest and activity in this area, with some new results concerning for example the existence of Sasaki-Einstein manifolds, outside the toric realm.", "These results are related to the idea of “stability”.", "For reference, recent work on this subject include [49], [50].", "For many purposes, knowledge of the existence of a metric, together with some of its key properties, can be sufficient for extracting interesting physical information.", "This is true also in the case of the AdS/CFT correspondence.", "However, if one is interested in constructing the metrics explicitly, namely write them down in some coordinate systems, then the existence theorems are not helpful, because they are not constructive (as far as we know).", "The classic examples of explicit Ricci-flat Kähler metrics in real dimension four include Eguchi-Hanson, Gibbons-Hawking, Taub-NUT, Atiyah-Hitchin.", "In real dimension six, for a long time the resolved and deformed metrics on the conifold singularity constructed by Candelas and de la Ossa [51] were the only (non trivial) known examples of explicit Ricci-flat Kähler metrics.", "The so-called “resolved conifold” metric is a metric on the total space of the vector bundle ${\\cal O}(-1)\\oplus {\\cal O}(-1)\\rightarrow \\mathbb {P}^1$ , the isometry group is $\\mathrm {SU(2)\\times SU(2)\\times U(1)}$ and asymptotically it approaches the cone over the Sasaki-Einstein manifold $\\mathrm {T^{1,1}}$ (with the same isometry).", "In other cases, different kind of resolutions exist, where instead of a $\\mathbb {P}^1$ one replaces the singularity with a compact four-dimensional manifold (or orbifold) ${\\cal M}_4$ .", "A general ansatz that yields explicit Ricci-flat Kähler metrics was constructed by Page and Pope (in any dimension) [52], but this is somewhat limited as it assumes that the metric induced on ${\\cal M}_4$ is Kähler-Einstein.", "Explicit Kähler-Einstein metrics on smooth four-dimensional manifolds are known only for ${\\cal M}_4 =\\mathbb {P}^2$ and ${\\cal M}_4 = \\mathbb {P}^1\\times \\mathbb {P}^1$ .", "The former leads to the construction of an explicit Ricci-flat Kähler metric on the total space of ${\\cal O}_{\\mathbb {P}^2}(-3)\\simeq \\mathrm {tot}K_{\\mathbb {P}^2}$ , which is the resolution of the quotient singularity $\\mathbb {C}^3/\\mathbb {Z}_3$ and was fully described in section (see also[42]).", "The latter leads to the construction of an explicit Ricci-flat Kähler metric on the total space of $\\mathrm {tot}K_{\\mathbb {P}^1\\times \\mathbb {P}^1}$ , which is the resolution of the conical singularity (conifold)$/\\mathbb {Z}_2$ .", "The corresponding Sasaki-Einstein manifolds at infinity are, respectively, $\\mathbb {S}^5/\\mathbb {Z}_3$ and $\\mathrm {T^{1,1}}/\\mathbb {Z}_2$ .", "For the case of $\\mathrm {tot}K_{\\mathbb {P}^1\\times \\mathbb {P}^1}$ , a generalisation was constructed [53], namely an explicit Ricci-flat Kähler metric that depends on the two independent Kähler classes parameters: this construction however uses the $\\mathrm {SU(2)\\times SU(2)\\times U(1)}$ symmetry and as a result the metric is co-homogeneity one, although it does not fit in the ansatz of [42] and [52].", "Recently, the ansatz of [42], [52] was used to produce explicit Ricci-flat Kähler metrics on the canonical bundle of generalised flag manifolds [54].", "Extensions that include the dependence on several Kähler class parameters have appeared in [55], [56]." ], [ "The Ricci-flat Kähler metric on ${\\rm tot} K_{\\mathbb {F}_{1}}$", "The metric that we shall present in the sequel has some distinctive features that are shared with an explicit Ricci-flat Kähler metric on ${\\rm tot} K_{\\mathbb {F}_1}$ , where $ \\mathbb {F}_1$ is the first Hirzebruch surface, i.e., the first del Pezzo surface $\\mathrm {dP}_1$ , constructed in [57].", "This metric is many ways “more complicated” than all the other metrics mentioned above.", "Let us summarise some of its salient properties: Asymptotically it approaches the cone over the Sasaki-Einstein manifoldIncidentally, $Y^{2,1}$ can also be viewed as circle bundle over $\\mathbb {F}_1$ .", "See section 5 of [58].", "$Y^{2,1}$ .", "The isometry group is $\\mathrm {SU(2)\\times U(1) \\times U(1)}$ .", "It is cohomogeneity two.", "In particular, there is a homogeneous base, given by a round $\\mathbb {P}^1$ , and then the metric depends non-trivially on two coordinates.", "It is toric, in that there is a $\\mathrm {U(1)^3 \\in SU(2)\\times U(1) \\times U(1)}$ subgroup of isometries that leaves invariant the Kähler form, and contains the torus of the toric three-fold ${\\rm tot} K_{\\mathbb {F}_1}$ .", "This group allows one to introduce three moment map coordinates and three angular coordinates (“action-angle” coordinates system).", "It also possesses an additional “hidden symmetry” corresponding to the existence of a so-called Hamiltonian two-form [14], that implies the existence of a coordinate system (called “orthotoric”) in which the metric components are all given in terms of functions of one variable.", "Imposing this extra symmetry however, comes at the price of loosing one of the two Kähler class parameters.", "Indeed it was later demonstrated in [29] that the two-parameter metric (that is known to exist thanks to the general theorems of [45], [46]) does not posses such Hamiltonian two-form.", "The metric induced on exceptional divisor ${\\cal M}_4= \\mathbb {F}_1$ is obviously Kähler, but it is not Einstein.", "Indeed, a Kähler-Einstein metric on $\\mathbb {F}_1$ does not exist.", "In [14] (further explored in detail in [59]) it was shown that this metric is part of a family of (in general only partialThis means that for general $p$ and $q$ the compact divisor ${\\cal M}_4$ has orbifold singularities [59], [14].", "This is because the metric ansatz is “too simple” to account for all the necessary Kähler class parameters; but completely resolved metrics are known to exist [47].", "For the special case $p=2$ and $q=1$ the metric is completely smooth.", "We also note that in [14] were constructed different types of partial resolutions, corresponding to various “chambers”.", "Moreover, the paper discusses the case of general dimension, while for our purposes we shall focus on the case of real dimension $d=6$ .)", "resolutions of the conical Ricci-flat metrics on the whole family of $Y^{p,q}$ Sasaki-Einstein manifolds.", "In [29] it is given a relation between the orthotoric coordinates and a set of complex coordinates that is well adapted to the complex structure of ${\\rm tot} K_{\\mathbb {F}_{1}}$ , with one complex coordinate on the non-compact fiber $\\mathbb {C}$ , one coordinate on the fiber $\\mathbb {P}^1$ in $\\mathbb {F}_1$ and one coordinate on the base $\\mathbb {P}^1$ in $\\mathbb {F}_1$ .", "A set of local complex coordinates explicitly related to the orthotoric coordinates was given in section 2.2 of [14].", "It would be interesting to work out the relation between these and the complex coordinates defined in [29].", "Since, similarly to $ \\mathbb {F}_1$ , also $ \\mathbb {F}_2$ does not admit a Kähler-Einstein metric, the Ricci-flat metric on ${\\rm tot} K_{\\mathbb {F}_{2}}$ cannot be found through the Calabi ansatz [42], [52].", "We expect the Ricci-flat metric on ${\\rm tot} K_{\\mathbb {F}_{2}}$ to share many features with that on ${\\rm tot} K_{\\mathbb {F}_{1}}$ , summarised above.", "One difference is that at infinity it must approach the cone over the Sasaki-Einstein orbifold $\\mathbb {S}^5/\\mathbb {Z}_4$ , as opposed to the cone over the Sasaki-Einstein manifold $Y^{2,1}$ .", "The Ricci-flat metric on ${\\rm tot} K_{\\mathbb {F}_{2}}$ will also be toric and moreover it should have again isometry group $\\mathrm {SU(2)\\times U(1)\\times U(1)}$ .", "This immediately implies that the metric should be co-homogeneity two and in practice it leads to PDE's in two variables.", "For example, one can write the Monge-Ampere equation for the Kähler potential as a PDE in two variables, or similarly the corresponding equation for the symplectic potential.", "Without further assumptions, these equations are unlikely to be solvable in closed form.", "A natural assumption to make is that the metric admits a Hamiltonian two-form, namely that it can be put in the orthotoric form.", "This is natural because the partial resolution of all the $Y^{p,q}$ singularities arise in this ansatz, with $p=2$ , $q=1$ giving the complete resolution above.", "Strictly speaking the $p>q>0$ should hold, however, it is known that by performing a scaling limit of the $Y^{p,q}$ Sasaki-Einstein metrics, one can recover the limiting cases $Y^{p,p}=\\mathbb {S}^5/\\mathbb {Z}_{2p}$ and $Y^{p,0}=\\mathrm {T^{1,1}}/\\mathbb {Z}_{p}$ , suggesting that the partial resolution metrics may also be extended to these regimes of parametersIn fact, in Appendix A of [14] the metric ansatz of [52] is recovered in a limit.." ], [ "A general set up for a metric ansatz with separation of variables", "In the sequel we begin by considering a metric on a 6-dimensional manifold $\\mathcal {M}_6$ which is Kähler and by construction admits SU(2)$\\times $ U(1)$\\times $ U(1) as an isometry group.", "This metric depends on two functions $\\Upsilon (s)$ and $P(t)$ of two real coordinates $s,t$ invariant with respect to the isometry group.", "The other coordinates are four angles, with ranges and periodicities specified according with the following summary table: $\\begin{array}{cccccc}s\\le -3\\, , & -\\tfrac{3}{2} \\le t\\le 0,& 0 \\le \\theta \\le \\pi , & \\phi \\in [0,2\\pi ]\\, , & \\tau \\in [0,2\\pi ]\\, , & \\chi \\in [0,\\frac{3}{2}\\pi ]\\\\\\end{array}$ The metric, which is defined by means of the following vielbein $E^1&=&\\frac{1}{2} \\sqrt{s t} {d\\theta }\\nonumber \\\\E^2&=&\\frac{1}{2} \\sqrt{s t} \\sin \\theta {d\\phi } \\nonumber \\\\E^3&=&\\frac{1}{2} \\,\\sqrt{\\frac{s-t}{3+s}}\\,\\Upsilon (s) d s\\nonumber \\\\E^4&=&\\frac{1}{2} \\,\\sqrt{t-s} P(t)\\, dt\\nonumber \\\\E^5&=&-\\frac{1}{\\sqrt{\\frac{s-t}{3+s}}\\,\\Upsilon (s)}\\,\\left[-\\frac{1}{2} t \\left({d\\tau }+(1-\\cos {\\theta })d\\phi -\\frac{2 {d\\chi }}{3}\\right)+{d\\chi }\\right]\\nonumber \\\\E^6&=&-\\frac{1}{\\sqrt{t-s} P(t)} \\left[-\\frac{1}{2} s\\left({d\\tau }+(1-\\cos {\\theta }) d{\\phi }-\\frac{2{d\\chi }}{3}\\right)+{d\\chi }\\right]$ is derived, by generalization, from the orthotoric metrics discussed inIn particular, see the line element (4.1) in [59], after correcting some typos in that expression.", "The relation to our coordinates is given by $t=y-1$ , $s=x-1$ .", "Moreover, we have $\\theta _\\mathrm {here}= \\theta _\\mathrm {there}$ , $\\phi _\\mathrm {here}= \\phi _\\mathrm {there}$ , as well as $\\chi _\\mathrm {here}= \\tau _\\mathrm {there}$ , $\\tau _\\mathrm {here}=2\\psi _\\mathrm {there}+\\tfrac{2}{3}\\tau _\\mathrm {there}$ .", "[59], [14] where the relation of latter with the metrics on Sasakian 5-manifolds $Y^{p,q}$ is also presented.", "Although in those references it was assumed that $p>q$ , presently we will consider setting $p=q=2$ and show that this yields an orthotoric metric that we shall identify as a Ricci-flat Kähler metric on ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ .", "The asymptotic metric corresponds to a cone over the limiting case $Y^{2,2}=\\mathbb {S}^5/\\mathbb {Z}_4$ of the Sasaki-Einstein manifolds $Y^{p,q}$ [48].", "The line-element: $\\text{ds}_{\\text{ort}}^2&=&\\sum _{i=1}^6\\left(E^i\\right){}^2\\\\&=&\\frac{1}{4} s \\, t \\,\\left({d\\theta }^2+\\sin ^2\\theta {d\\phi }^2 \\right)+\\frac{(s-t)\\Upsilon (s)^2}{4 (3+s)} \\, d s^2+\\frac{1}{4} (t-s) P(t)^2 \\, \\text{dt}^2\\nonumber \\\\&&+\\frac{(3+s)}{(s-t) \\Upsilon (s)^2}\\left[-\\frac{1}{2} t \\left(\\text{d$\\tau $}+(1-\\cos {\\theta }) {d\\phi }-\\frac{2 \\text{d$\\chi $}}{3}\\right)+{d\\chi }\\right]^2\\nonumber \\\\&&+\\frac{1}{(t-s)P(t)^2}\\left[-\\frac{1}{2} s \\left(\\text{d$\\tau $}+(1-\\cos {\\theta }){d\\phi }-\\frac{2 \\text{d$\\chi $}}{3}\\right)+{d\\chi }\\right]^2$ is Kählerian by construction since it admits the following closed Kähler 2-form: $\\mathbb {K}_{\\text{ort}}&=&E_1\\wedge E_2+E_3\\wedge E_5+E_4\\wedge E_6\\nonumber \\\\&=&\\frac{1}{2} \\left\\lbrace t \\left[-\\frac{1}{2} \\cos \\theta \\text{ds}\\wedge {d\\phi }+\\frac{1}{2} \\left(ds\\wedge {d\\tau }-\\frac{2}{3} ds\\wedge {d\\chi }\\right)+\\frac{1}{2} ds\\wedge {d\\phi }\\right] -ds\\wedge {d\\chi } \\right\\rbrace \\nonumber \\\\&&+\\frac{1}{2} \\left\\lbrace s \\left[-\\frac{1}{2} \\cos \\theta \\text{dt}\\wedge {d\\phi }+\\frac{1}{2} \\left(\\text{dt}\\wedge {d\\tau }-\\frac{2}{3} \\text{dt}\\wedge {d\\chi }\\right)+\\frac{1}{2} \\text{dt}\\wedge {d\\phi }\\right]-\\text{dt}\\wedge {d\\chi }\\right\\rbrace \\nonumber \\\\&&+\\frac{1}{4} s t \\sin {\\theta } {d\\theta }\\wedge {d\\phi }$ Indeed $\\mathbb {K}_{\\text{ort}}$ is closed by construction and it is a Kähler 2-form since we have: $\\mathbb {K}_{\\text{ort}}=\\frac{1}{2}\\sum _{i=1}^6 \\sum _{j=1}^6\\mathbb {J}_{\\text{ij}}E^i\\wedge E^j$ where: $\\mathbb {J}\\,=\\,\\left(\\begin{array}{cccccc}0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\\\end{array}\\right)\\quad ;\\quad \\mathbb {J}^2= - \\mathbf {1}$ is an antisymmetric tensor which squares to minus the identity, namely it is a frame-index complex structure tensor.", "It should be noted that the Kähler form in eq.", "(REF ) is independent from the two functions $\\Upsilon (s)$ and $P(t)$ , namely it is universal for an entire class of metrics." ], [ "The orthotoric metric on $K_{{\\mathbb {W}P}[112]}$", "Within the general scope of the above described setup we have that the metric () is Ricci-flat for the following choice of the two functions parameterizing the line-element: $\\Upsilon (s)\\,=\\,\\sqrt{\\frac{-s}{\\frac{2}{3}\\,s^2-s+3}}\\quad ;\\quad P(t)\\,=\\, \\frac{1}{\\sqrt{-\\frac{2}{3}t^2-t}}$ With the choice (REF ), from eq.", "() we obtain: $\\text{ds}_{{\\rm tot} K_{{\\mathbb {W}P}[112]}}^2&=&\\frac{1}{4} \\left\\lbrace \\frac{4\\left(\\frac{2 s^2}{3}+s+\\frac{9}{s}\\right)\\left[\\left(\\frac{t}{3}+1\\right) {d\\chi }-\\frac{1}{2}t [(1-\\cos {\\theta }){d\\phi } +{d\\tau }]\\right]^2}{t-s}+\\right.\\nonumber \\\\&&\\left.+\\frac{4 t (2 t+3) \\left(\\left[\\frac{s}{3}+1\\right){d\\chi }-\\frac{1}{2} s [(1-\\cos \\theta ){d\\phi } +{d\\tau }]\\right]^2}{3 (s-t)}\\right.\\nonumber \\\\&& \\left.+st \\left(\\sin ^2\\theta {d\\phi }^2+d\\theta ^2\\right)+\\frac{ds^2 (t-s)}{\\frac{2 s^2}{3}+s+\\frac{9}{s}}+\\frac{3dt^2 (s-t)}{t (2 t+3)}\\right\\rbrace $ The reason for the subscript ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ is that the Ricci flat metric (REF ) turns out to be defined over the total space of the canonical bundle of the (singular) projective space ${{\\mathbb {W}P}[112]}$ namely on ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ .", "It is a simple matter to verify that asymptotically, for $s\\rightarrow -\\infty $ , the metric (REF ) is indeed approximatively conical, and therefore Quasi-ALE [28].", "To see this, one can set $s=-\\tfrac{2}{3}R^2$ , so that $\\text{ds}_{{\\rm tot} K_{{\\mathbb {W}P}[112]}}^2 \\, \\stackrel{R \\rightarrow \\infty }{\\approx }\\, dR^2 + R^2 \\text{ds}^2_{X_5}$ at leading order in $R$ .", "Since the metric is Ricci-flat Kähler, and it takes the form of a cone over a five-dimensional space, it follows that locally the five-dimensional metric $\\text{ds}^2_{X_5}$ is a Sasaki-Einstein metric.", "In appendix we discuss the metric $\\text{ds}^2_{X_5}$ in more detail, showing that $X_5=\\mathbb {S}^5/\\mathbb {Z}_4$ , with a specific $\\mathbb {Z}_4$ action.", "As we will show in sect.", ", the metric induced by (REF ) on the exceptional divisor ${{\\mathbb {W}P}[112]}$ is the same as the one obtained on that space while resolving a $\\mathbb {C}^3/\\mathbb {Z}_4$ orbifold singularity by means of the Kronheimer construction localized on the unique type III wall $\\mathcal {W}_2$ displayed by its chamber structure (see sect.", "6.4 of [7])." ], [ "Integration of the complex structure and the complex\ncoordinates", "In their algebraic geometry description, the varieties of the type here considered are complex threefolds $\\mathcal {K}_3$ that are canonical bundles of some compact Kähler two-fold $\\mathcal {D}_2$ which, on its turn, is the total space of a line-bundle over $\\mathbb {P}_1$ : $\\mathcal {M}_6=\\mathcal {K}_3\\overset{\\pi }{\\longrightarrow }\\mathcal {D}_2 \\overset{\\widetilde{\\pi }}{\\longrightarrow } \\mathbb {P}_1$ This hierarchical structure implies a hierarchy in the complex coordinates that can be organized and named in the following way according with the nomenclature of [7]: $\\begin{array}{rclcrcl}u &=& \\text{coordinate on the }\\mathbb {P}_1\\text{ base of }\\mathcal {D}_2&;&v &=& \\text{coordinate on the fibers of }\\mathcal {D}_2\\\\w &=& \\text{coordinate on the fibers of }\\mathcal {K}_3&\\hbox{}&\\hbox{}\\\\\\end{array}$ This structure is reflected in the integration of the complex structure that can be deduced from the combination of the Kähler 2-form with the metric." ], [ "The path to the integration", "Indeed, having the metric and the Kähler form we can construct the complex structure tensor.", "Then we try to integrate the complex structure we have found.", "This is very important in order to organize the fibred structure of the manifold.", "First from eq.", "(REF ) one reads off the vielbein $E_{\\mu }^i$ defined as: $E^i &=& E_{\\mu }^i d x^{\\mu }\\quad ; \\quad x^{\\mu } \\,=\\,\\lbrace s,t,\\theta ,\\phi ,\\tau ,\\chi \\rbrace $ The $6\\times 6$ matrix $E_{\\mu }^i$ depends only on the $s, t$ variables and on the angle $\\theta $ (as we will see $\\theta $ can be traded for the coordinate $\\rho =\\tan \\frac{\\theta }{2}$ and in the symplectic formalism it is a moment variable).", "The true angular variables are the phases of the three complex coordinates namely $\\phi $ , $\\tau $ , $\\chi $ .", "As a next step one introduces the inverse vielbein which is just the matrix inverse of $E_{\\mu }^i$ according with the definition $E_{\\mu }^i E_j^{\\mu } =\\delta _j^i$ This enables us to write the differentials of the coordinates as linear combinations of the vielbein ${dx}^{\\mu }= E_j^{\\mu }E^i$ ." ], [ "Using the vielbein matrix and its inverse we can convert the frame indices of the complex structure tensor to coordinate ones and we get: $\\mathbb {JW}_\\mu {}^\\nu =\\left(\\begin{array}{cccccc}0 & 0 & 0 & 0 & -\\frac{1}{3} \\Upsilon (s)^2 & -\\frac{s \\Upsilon (s)^2}{6+2 s} \\\\0 & 0 & 0 & 0 & -\\frac{1}{3} (3+t) P(t)^2 & -\\frac{1}{2} t P(t)^2 \\\\0 & 0 & 0 & \\csc {\\theta } & -\\tan \\frac{\\theta }{2} & 0 \\\\-\\frac{2 (3+s) t \\sin ^2\\frac{\\theta }{2}}{(s-t) \\Upsilon (s)^2} & -\\frac{2 s \\sin ^2\\frac{\\theta }{2}}{(t-s)P(t)^2} & -\\sin {\\theta } & 0 & 0 & 0 \\\\-\\frac{(3+s) t}{(s-t) \\Upsilon (s)^2} & -\\frac{s}{(t-s) P(t)^2} & 0 & 0 & 0 & 0 \\\\\\frac{2 (3+s) (3+t)}{3 (s-t) \\Upsilon (s)^2} & \\frac{2 (3+s)}{3 (t-s) P(t)^2} & 0 & 0 & 0 & 0 \\\\\\end{array}\\right)$" ], [ "Integration of the autodifferentials", "The matrix $\\mathbb {JW}$ has three eigenvectors corresponding to the eigenvalue ${\\rm i}$ and three corresponding to the eigenvalue $-{\\rm i}$ (their complex conjugates).", "The three eigenvectors corresponding to ${\\rm i}$ are the rows of the following matrix $Y_{\\mu }^i=\\left(\\begin{array}{cccccc}\\frac{{\\rm i} s \\Upsilon (s)^2}{6+2 s} & \\frac{1}{2} {\\rm i} t P(t)^2 & 0 & 0 & 0 & 1 \\\\\\frac{1}{3} {\\rm i} \\Upsilon (s)^2 & \\frac{1}{3} {\\rm i} (3+t) P(t)^2 & {\\rm i} \\tan \\frac{\\theta }{2} & 0 & 1 & 0 \\\\0 & 0 & - {\\rm i} \\csc {\\theta } & 1 & 0 & 0 \\\\\\end{array}\\right)$ combining with the differentials $dY^i \\, =\\,Y_{\\mu }^i dx^{\\mu }$ we obtain three closed one-forms $d\\left(dY^i\\right)\\,=\\, 0$ that can be integrated to yield the three-complex variables $u,v$ and $w$ ." ], [ "is obtained from the integration of $dY^3$ : $u =\\exp {{\\rm i}\\int (-{\\rm i} \\csc {\\theta } d\\theta +d\\phi )}= \\tan \\frac{\\theta }{2} e^{{\\rm i}\\phi }$" ], [ "is obtained from the integration of $dY^2$ : $v =\\exp {\\rm i}\\int \\left(d\\tau +\\frac{1}{3} {\\rm i} (3+t)P(t)^2 \\,dt + {\\rm i} \\tan \\frac{\\theta }{2}\\,d\\theta +\\frac{1}{3} {\\rm i} \\Upsilon (s)^2 \\, ds \\right)=\\cos ^2\\frac{\\theta }{2} H(t) \\, \\Psi (s)e^{{\\rm i} \\tau }$ where we have introduced the following new functions of $t$ and $s$ : $H(t)=\\exp \\left(-\\frac{1}{3}\\int _{\\text{const}}^t(3+x)P(x)^2dx\\right)\\quad ;\\quad \\Psi (s)=\\exp \\left(-\\frac{1}{3}\\int _{-\\infty }^s \\Upsilon (x)^2 \\, dx \\right)$" ], [ "is obtained from the integration of $dY^1$ .", "Here we are not assisted by $\\mathrm {SU(2)}$ invariance to define the exact coefficient in front of the differential.", "We choose a coefficient that appears reasonable from the result and what we obtain is either the coordinate $w$ of other approaches or a power $w^a$ .", "In the sequel comparing with the construction from the iterative procedure we will see what is the correct identification of the power $a$ .", "At the beginning our educated guess suggests the use of a coefficient $4/3$ .", "So we set $w=\\exp {\\rm i}\\frac{4}{3} \\,\\int \\left( d\\chi +\\frac{1}{2} {\\rm i} \\, t \\,P(t)^2 dt\\, +\\, \\frac{{\\rm i} \\, s \\, \\Upsilon (s)^2}{6+2 s}\\, ds\\right) =\\Phi (s) K(t) e^{{\\rm i}\\frac{4}{3}\\chi }$ where we have introduced the new functions: $\\Phi (s)=\\exp \\left(-\\frac{2}{3}\\int _{-\\infty }^s\\frac{x}{x+3}\\Upsilon (x)^2 dx \\right)\\quad ;\\quad K(t)=\\exp \\left(-\\frac{2}{3}\\int _{\\text{const}}^t x\\,P(x)^2\\,dx\\right)$ One necessary property that must be possessed by the function $\\Phi (s)$ is: $\\Phi (-3) = 0$ which defines the exceptional divisor at $w = 0$ Notice that with the ranges of the coordinates that we specified in (REF ), we see that $u$ is a complex coordinate on a $\\mathbb {P}^1$ , while $\\nu $ and $w$ are complex coordinates on two copies of $\\mathbb {C}$ ." ], [ "AMSY symplectic formalism and transcription\nof the metric in this formalism", "According to the formalism introduced by Abreu [60] and developed by Martelli, Sparks and Yau [61], in the case of toric Kähler varieties of complex dimension $n$ , one can find moment maps $\\mu ^i$ and angular variables $\\Theta _i$ such that the Kähler 2-form takes the universal form: $\\mathbb {K}=\\sum _{i=1}^n d\\mu ^i\\wedge d\\Theta _i$ At the same time there exist a function $G(\\mu ^i)$ of the $n$ real moment variables, named the symplectic potential, such that the metric takes the following universal form: $\\text{ds}_{\\text{symp}}^2= G_{ij} d\\mu ^i d\\mu ^j+\\left(G^{-1}\\right){}^{ij }d\\Theta _id\\Theta _j$ where by definition: $G_{ij}\\equiv \\partial _{i,j}G\\text{}$ is the Hessian of the symplectic potential and $\\left(G^{-1}\\right){}^{ij}$ is the inverse of the Hessian matrix.", "In our case the three angular variables are $ \\Theta =\\lbrace \\phi ,\\tau ,\\chi \\rbrace $ and the Kähler form is given by $\\mathbb {K}$ as defined in eq.", "(REF ).", "Transforming the pseudo angle $\\theta $ to the variable $\\rho $ by setting $ \\theta \\,=\\, 2 \\arctan \\rho $ and implementing such change of variables in the Kähler form we obtain: $\\mathbb {K} &=& \\frac{1}{12} \\left[3 t ds\\wedge d\\tau +\\frac{6 t \\rho ^2 ds\\wedge d\\phi }{1+\\rho ^2}-2 (3+t) ds\\wedge d\\chi +3 s \\left(dt\\wedge d\\tau +\\frac{2 \\rho ^2 dt\\wedge d\\phi }{1+\\rho ^2}\\right)\\right.\\nonumber \\\\&&\\left.", "-2 (3+s) dt\\wedge d\\chi +\\frac{12 s t \\rho d\\rho \\wedge d\\phi }{\\left(1+\\rho ^2\\right)^2}\\right]$ which is compatible with eq.", "(REF ) if the coefficient of each of the three angular variables $\\tau $ , $\\chi $ , $\\phi $ is a closed differential that can be integrated to a single new moment coordinate function of the real coordinates $\\rho ,s,t$ .", "Hence we introduce the vector of moments: $\\mu =\\lbrace \\mathfrak {u},\\mathfrak {v},\\mathfrak {w}\\rbrace $ and the Kähler 2-form (REF ) can be rewritten as: $\\mathbb {K}=d\\mathfrak {u}\\wedge d\\phi +d\\mathfrak {v}\\wedge d\\tau +d\\mathfrak {w}\\wedge d\\chi $ provided we have defined the coordinate transformation: $\\mathfrak {u}\\,=\\,\\frac{s \\, t \\,\\rho ^2}{2+2 \\rho ^2}\\quad ; \\quad \\mathfrak {v}\\,=\\, \\frac{s\\, t}{4}\\quad ; \\quad \\mathfrak {w}=\\frac{1}{6} \\left(-3 t-s (3+t)\\right)$ The unique inverse transformation of the above coordinate change is the following one: $\\begin{array}{rclcrcl}\\rho &=& \\frac{\\sqrt{\\mathfrak {u}}}{\\sqrt{-\\mathfrak {u}+2 \\mathfrak {v}}}&;&t &=& \\frac{1}{6} \\left(-4 \\mathfrak {v}-6\\mathfrak {w}+\\sqrt{-144 \\mathfrak {v}+(4 \\mathfrak {v}+6\\mathfrak {w})^2}\\right)\\\\\\hbox{}&\\hbox{}&\\hbox{}&\\hbox{}& s &=& \\frac{1}{3} \\left(-2 \\mathfrak {v}-3\\mathfrak {w}-\\sqrt{4 (-9+\\mathfrak {v}) \\mathfrak {v}+12 \\mathfrak {v}\\mathfrak {w}+9 \\mathfrak {w}^2}\\right)\\\\\\end{array}$ The new real coordinates are named $\\mathfrak {u}$ ,$\\mathfrak {v}$ ,$\\mathfrak {w}$ with gothic letters since they are the symplectic counterparts of the complex coordinates $u,v,w$ yet, differently from the latter, we do not need the complex structure to find them and hence they are independent from the metric." ], [ "Transcription of the metric in the toric symplectic form", "At this point we try to rewrite the metric depending on the two functions: $M(s,t) \\equiv \\sqrt{\\frac{s-t}{3+s}}\\, \\Upsilon (s) \\quad ; \\quad \\Phi (s,t) \\equiv \\sqrt{t-s} \\, P(t)$ in the symplectic form (REF ).", "Setting: $M(s,t)\\,=\\, \\mathfrak {M}(\\mathfrak {v},\\mathfrak {w})\\equiv \\mathfrak {M}\\quad ;\\quad \\Phi (s,t)\\,=\\,\\mathfrak {F}(\\mathfrak {v},\\mathfrak {w})\\equiv \\mathfrak {F}\\quad ;\\quad \\Omega \\, \\equiv \\,4 (-9+\\mathfrak {v}) \\mathfrak {v}+12 \\mathfrak {v}\\mathfrak {w}+9 \\mathfrak {w}^2$ we easily derive that $ds^2_{ort}$ takes the form (REF ) with the following matrix $\\mathcal {G}_{ij}$ : $\\mathcal {G}_{11} &=& -\\frac{\\mathfrak {v}}{\\mathfrak {u}^2-2 \\mathfrak {u} \\mathfrak {v}}\\nonumber \\\\\\mathcal {G}_{12} &=& \\frac{1}{\\mathfrak {u}-2 \\mathfrak {v}}\\nonumber \\\\\\mathcal {G}_{13} &=& 0 \\nonumber \\\\\\mathcal {G}_{22} &=& \\frac{1}{9} \\left[-\\frac{9 \\mathfrak {u}}{\\mathfrak {u}\\mathfrak {v}-2 \\mathfrak {v}^2}+\\frac{\\mathfrak {F}^2 \\left(-2\\mathfrak {v}-3 \\mathfrak {w}+\\sqrt{\\Omega }+9\\right)^2}{\\Omega }+\\frac{\\mathfrak {M}^2 \\left(2 \\mathfrak {v}+3\\mathfrak {w}+\\sqrt{\\Omega }-9\\right)^2}{\\Omega }\\right] \\nonumber \\\\\\mathcal {G}_{23} &=& \\frac{1}{6 \\Omega }\\left\\lbrace \\mathfrak {F}^2 \\left[8 \\mathfrak {v}^2+\\mathfrak {v} \\left(24\\mathfrak {w}-4 \\sqrt{\\Omega }-54\\right)-3 (2 \\mathfrak {w}-3)\\left(\\sqrt{\\Omega }-3\\mathfrak {w}\\right)\\right] \\right.\\nonumber \\\\&&\\left.+\\mathfrak {M}^2 \\left[8 \\mathfrak {v}^2+\\mathfrak {v}\\left(24 \\mathfrak {w}+4 \\sqrt{\\Omega }-54\\right)+3 (2 \\mathfrak {w}-3) \\left(3 \\mathfrak {w}+\\sqrt{\\Omega }\\right)\\right] \\right\\rbrace \\nonumber \\\\\\mathcal {G}_{33} &=& \\frac{1}{16 \\Omega }\\left[ 4 \\mathfrak {F}^2 \\left(-2 \\mathfrak {v}-3\\mathfrak {w}+\\sqrt{\\Omega }\\right)^2+4 \\mathfrak {M}^2 \\left(2\\mathfrak {v}+3 \\mathfrak {w}+\\sqrt{\\Omega }\\right)^2\\right]$ It remains to be seen if we are able to retrieve the symplectic potential from which the above matrix is obtained through double derivatives.", "With some integrations and some educated guesses we find that the form (REF ) of the matrix can be reproduced if we write the symplectic potential as follows: $G(\\mathfrak {u},\\mathfrak {v},\\mathfrak {w}) \\, = \\,G_0(\\mathfrak {u},\\mathfrak {v})+\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ where $G_0(\\mathfrak {u},\\mathfrak {v})=\\frac{1}{2} \\left(-\\mathfrak {u}+\\mathfrak {u} \\log \\mathfrak {u}\\right)-\\mathfrak {v} \\log \\mathfrak {v}+\\left(-\\frac{\\mathfrak {u}}{2}+\\mathfrak {v}\\right)\\log (-\\mathfrak {u}+2 \\mathfrak {v})$ and where $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ is some function of the two fibre coordinates $\\mathfrak {v}$ ,$\\mathfrak {w}$ only.", "With this choice the matrix $G_{\\text{ij}}$ becomes: $G_{ij}=\\left(\\begin{array}{ccc}-\\frac{\\mathfrak {v}}{\\mathfrak {u}^2-2 \\mathfrak {u} \\mathfrak {v}} &\\frac{1}{\\mathfrak {u}-2 \\mathfrak {v}} & 0 \\\\\\frac{1}{\\mathfrak {u}-2 \\mathfrak {v}} &-\\frac{\\mathfrak {u}}{\\mathfrak {u} \\mathfrak {v}-2 \\mathfrak {v}^2}+\\mathcal {G}^{(2,0)}(\\mathfrak {v},\\mathfrak {w})& \\mathcal {G}^{(1,1)}(\\mathfrak {v},\\mathfrak {w}) \\\\0 & \\mathcal {G}^{(1,1)}(\\mathfrak {v},\\mathfrak {w}) &\\mathcal {G}^{(0,2)}(\\mathfrak {v},\\mathfrak {w}) \\\\\\end{array}\\right)$ and the full-fledged expression of the line element can be obtained by substitution.", "Comparing the obtained result with eq.", "(REF ) we easily see that the functions $\\mathfrak {M}(\\mathfrak {v}, \\mathfrak {w})=M(s,t)$ and $\\mathfrak {F}(\\mathfrak {v}, \\mathfrak {w}) = \\Phi (s,t)$ can be expressed in terms of the derivatives $\\mathcal {G}^{(2,0)}(\\mathfrak {v},\\mathfrak {w})$ , $\\mathcal {G}^{(0,2)}(\\mathfrak {v},\\mathfrak {w})$ , but in order to avoid other functions we get a second order differential constraint on the symplectic potential $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ that relates its mixed derivatives to $\\mathcal {G}^{(2,0)}(\\mathfrak {v},\\mathfrak {w})$ , $\\mathcal {G}^{(0,2)}(\\mathfrak {v},\\mathfrak {w})$ .", "This differential is expressed in a simpler way by means of the original coordinates $s,t$ .", "We shall presently derive it.", "We anticipate that its solution very strongly limits the possibilities so that it has to be discarded.", "In other words we have to accept a generic function $\\mathcal {G}(\\mathfrak {v}, \\mathfrak {w})$ and try to match it with the boundary conditions on the exceptional divisor." ], [ "Orthotoric separation of variables and the symplectic\npotential", "In order to compare the generic metric in symplectic formalism provided by the symplectic potential displayed in eq.s (REF ), (REF ) with the following two-function metricAt this level we do not require $M(s,t)$ and $\\Phi (s,t)$ to have the specific form of eq.", "(REF ): $\\text{ds}^2_{2fun} &=& \\frac{1}{4} s t \\left(d\\phi ^2 \\sin ^2\\theta +d\\theta ^2\\right)+\\frac{1}{M(s,t)^2}\\left[d\\chi -\\frac{1}{2} t \\left(d\\phi (1-\\cos (\\theta ))+d\\tau -\\frac{2 d\\chi }{3}\\right)\\right]^2\\nonumber \\\\&&+\\frac{1}{\\Phi (s,t)^2}{\\left[d\\chi -\\frac{1}{2} s \\left(d\\phi (1-\\cos {\\theta })+d\\tau -\\frac{2 d\\chi }{3}\\right)\\right]^2}+\\frac{1}{4} dt^2 \\Phi (s,t)^2+\\frac{1}{4} ds^2M(s,t)^2\\nonumber \\\\$ we make the following steps.", "First we regard the function $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ as a function only of $t$ and $s$ , as it is evident from the transformation rule (REF ), and we write: $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})\\, \\equiv \\, \\Gamma (t,s) $ .", "By means of the transformation (REF ) we can rewrite the generic metric (REF ) produced by the symplectic potential (REF -REF ) in terms of the variables $s,t$ , instead of $\\mathfrak {v},\\mathfrak {w}$ .", "The result coincides with $\\text{ds}^2_{2fun}$ as given in eq.", "(REF ) if the following conditions hold true: $&&\\frac{\\partial ^2}{\\partial s^2}\\Gamma (t,s) \\,=\\, \\frac{1}{4} \\,M(s,t)^2 \\quad ; \\quad \\frac{\\partial ^2}{\\partial t^2}\\Gamma (t,s) \\,=\\, \\frac{1}{4} \\, \\Phi (s,t)^2 \\nonumber \\\\&&\\frac{\\partial }{\\partial s}\\Gamma (t,s)\\, - \\,\\frac{\\partial }{\\partial t}\\Gamma (t,s)+(s-t)\\frac{\\partial ^2}{\\partial s\\,\\partial t}\\Gamma (t,s) \\,=\\, 0$ The first two equations in (REF ) just provide the identification of the two functions $M(s,t)$ and $\\Phi (s,t)$ in terms of second order derivatives of the symplectic potential.", "On the other hand the last equation of (REF ) is a very strong constraint on the function $\\Gamma (t,s)$ which severely restricts the available choices of $\\Gamma (t,s)$ ." ], [ "The symplectic potential of the Ricci-flat orthotoric metric\non ${\\rm tot} K_{{\\mathbb {W}P}[112]}$", "In the case of the canonical bundle ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ , whose Ricci–flat metric is given by eq.", "(REF ), eq.s (REF ) imply $\\Gamma ^{(0,2)}(t,s) & = & \\frac{(s-t) \\Upsilon (s)^2}{4(s+3)} \\, = \\, -\\frac{s (s-t)}{4 (s+3) \\left(\\frac{2s^2}{3}-s+3\\right)}\\quad \\Rightarrow \\quad \\Upsilon (s)^2 = \\frac{s }{\\left(\\frac{2s^2}{3}-s+3\\right)}\\nonumber \\\\\\quad \\Gamma ^{(2,0)}(t,s) & = &\\frac{1}{4} P(t)^2 (t-s)\\, =\\, \\frac{s-t}{4 \\left(\\frac{2 t^2}{3}+t\\right)} \\quad \\Rightarrow \\quad P(t)^2 \\, = \\, -\\frac{1}{\\frac{2}{3}\\, t\\,\\left(t+3\\right)}$ By means of two double integrations and modulo linear functions in $s,t$ (they are irrelevant for the metric) we determine the explicit form of the potential $\\Gamma (t,s)$ : $\\Gamma _{{{\\mathbb {W}P}[112]}} (t,s) &=& \\frac{1}{224}\\left\\lbrace -7\\left[(3 t-st+3s)\\log (2 s^2-3s+9)+2 (s+3) (t+3) \\log {(s+3)}-8 s t \\log {t}\\right.\\right.\\nonumber \\\\&&\\left.\\left.+2 (2 s+3) (2t+3) \\log (2 t+3)\\right]-6 \\sqrt{7} (s t+s+t-6) \\arctan \\frac{3-4 s}{3 \\sqrt{7}}\\right\\rbrace $ The function $\\Gamma _{{{\\mathbb {W}P}[112]}}(t,s)$ satisfies by construction the differential constraint encoded in the third of eq.s (REF ).", "Using the transformation rule (REF ) we can rewrite it as a function of the symplectic variables $\\mathfrak {v},\\mathfrak {w}$ .", "In this way we arrive at the following symplectic potential where we have used the liberty of adding linear functions of $\\mathfrak {v}$ or $\\mathfrak {w}$ to obtain the most convenient form of its reduction to the exceptional divisor, located at $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$ .", "The function $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}\\left(\\mathfrak {v},\\mathfrak {w}\\right) &=&\\frac{1}{224} \\left\\lbrace 7 \\left[6 (2 \\mathfrak {w}-3) \\log \\left(-\\sqrt{(2 \\mathfrak {v}+3 \\mathfrak {w})^2-36\\mathfrak {v}}-2 \\mathfrak {v}-3 \\mathfrak {w}+9\\right)\\right.\\right.\\nonumber \\\\&&\\left.\\left.+16\\mathfrak {v} \\log \\left(\\sqrt{(2 \\mathfrak {v}+3\\mathfrak {w})^2-36 \\mathfrak {v}}-2 \\mathfrak {v}-3\\mathfrak {w}\\right)^2\\right.\\right.\\nonumber \\\\&&\\left.\\left.-2 (8 \\mathfrak {v}-12\\mathfrak {w}+9) \\log \\left(\\sqrt{(2 \\mathfrak {v}+3\\mathfrak {w})^2-36 \\mathfrak {v}}-2 \\mathfrak {v}-3\\mathfrak {w}+\\frac{9}{2}\\right)\\right.\\right.\\nonumber \\\\&&\\left.\\left.+2 (4 \\mathfrak {v}+3\\mathfrak {w}) \\log \\left(\\frac{1}{567} \\left[4 \\sqrt{(2\\mathfrak {v}+3 \\mathfrak {w})^2-36 \\mathfrak {v}}+8\\mathfrak {v}+12 \\mathfrak {w}+9\\right]^2+1\\right)\\right]\\right.\\nonumber \\\\&&\\left.-4\\sqrt{7} (4 \\mathfrak {v}-3 (\\mathfrak {w}+3))\\, \\text{arctan}\\left(\\frac{4 \\sqrt{(2 \\mathfrak {v}+3 \\mathfrak {w})^2-36\\mathfrak {v}}+8 \\mathfrak {v}+12 \\mathfrak {w}+9}{9\\sqrt{7}}\\right)\\right.\\nonumber \\\\&&\\left.-(8 \\mathfrak {v}+9) \\log \\frac{34359738368}{823543}+2 \\sqrt{7} (8\\mathfrak {v}-27)\\, \\arctan \\frac{5}{\\sqrt{7}}\\right\\rbrace $ is expressed in terms of elementary transcendental functions, yet it has the remarkable property of satisfying the Monge-Ampère equation for Ricci-flatness, so that it may be called “the miraculous function\".", "On the exceptional divisor it reduces to $\\mathcal {D}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v})\\, \\equiv \\,\\mathcal {G}_{{{\\mathbb {W}P}[112]}}\\left(\\mathfrak {v},{\\textstyle \\frac{3}{2}}\\right)\\, = \\,\\frac{1}{16} \\left[ 8 \\mathfrak {v} \\log \\left(16\\mathfrak {v}^2\\right)+(9-8 \\mathfrak {v}) \\log \\left(\\frac{9}{2}-4 \\mathfrak {v}\\right)\\right].$" ], [ "Kähler metrics on Hirzebruch surfaces and their canonical bundles", "For the case of the canonical bundle on $\\mathbb {F}_2$ , which is the complete resolution of the $\\mathbb {C}^3/\\mathbb {Z}_4$ singularity, we have additional information that is relevant and inspiring for the general case.", "Let us summarize the main points.", "According to the results of [7] there is a well adapted system of complex coordinates that arise from the toric analysis of $\\mathbb {C}^3/\\mathbb {Z}_4$ and of its resolution.", "These coordinates are named as follows: $z_i \\, = \\, \\left\\lbrace u, \\, v, \\, w \\right\\rbrace $ and are defined on a dense open chart reaching all components of the exceptional divisor.", "Their interpretation was already anticipated in eq.", "(REF ) and it is the following.", "The coordinate $w$ spans the fibers in the canonical bundle $Y\\,\\stackrel{\\pi }{\\longrightarrow } \\, \\mathbb {F}_2 $ while $u,v$ span a dense open chart for the base manifold (i.e.", "the compact component $\\mathbb {F}_2$ of the exceptional divisor $\\mathcal {ED}$ ).", "In particular since $\\mathbb {F}_2$ is a $\\mathbb {P}^1$ bundle over $\\mathbb {P}^1$ , namely $ \\mathbb {F}_2\\stackrel{\\pi }{\\longrightarrow } \\, \\mathbb {P}^1$ , the coordinate $u$ is a standard Fubini-Study coordinate for the base $\\mathbb {P}^1$ while $v$ spans a dense open chart of the fibre $\\mathbb {P}^1$ .", "This set of coordinates can be used for any $\\mathbb {F}_n$ Hirzebruch surface with $n \\ge 1$ .", "The action of the isometry group (REF ) on these coordinates was described in [7] and it is as follows: $\\forall \\, \\mathbf {g} &=& \\left(\\begin{array}{cc}a & b \\\\c & d \\\\\\end{array}\\right) \\, \\in \\, \\mathrm {SU(2)}\\quad : \\quad \\mathbf {g}\\left(u,v,w\\right) \\, = \\,\\left(\\frac{a \\, u + b}{c\\, u +d}, \\quad v \\, \\left(c \\,u+d\\right)^{n},\\quad w\\right)\\nonumber \\\\\\forall \\, \\mathbf {g} &=& \\exp [{\\rm i}\\,\\theta _1] \\, \\in \\, \\mathrm {U(1)}_v\\quad \\quad : \\quad \\mathbf {g}\\left(u,v,w\\right) \\, = \\,\\left(u,\\quad \\exp [{\\rm i}\\,\\theta _1]\\, v \\,\\quad w\\right)\\nonumber \\\\\\forall \\, \\mathbf {g} &=& \\exp [{\\rm i}\\,\\theta _2] \\, \\in \\, \\mathrm {U(1)}_w\\quad \\quad : \\quad \\mathbf {g}\\left(u,v,w\\right) \\, = \\,\\left(u,\\quad \\, v,\\quad \\exp [{\\rm i}\\,\\theta _2]\\, w\\right) $ The above explicit action of the isometry group on the $u,v,w$ coordinates suggests the use of an invariant real combination $\\varpi _n \\, \\equiv \\, \\left(1+\\mid u\\mid ^2\\right)^n \\, \\mid v\\mid ^2$ and the assumption that the Kähler potential $\\mathcal {K}_{\\mathbb {F}_n}$ of the Kähler metric $\\mathbf {g}_{\\mathbb {F}_n}$ should be a function (up to trivial terms $\\mathrm {Re} f(z)$ ) only of $\\varpi _n$ : $\\mathcal {K}_{\\mathbb {F}_n}\\, = \\, G_n(\\varpi _n)$ The function $G_n(\\varpi _n)$ should also depend on two parameters (we name them $\\ell $ ,$\\alpha $ ) which are associated to the volumes of the two homology cycles of $\\mathbb {F}_n$ , respectively named $C_1$ and $C_2$ that also form a basis for the homology group of the total space $Y$ , namely the canonical bundle on $\\mathbb {F}_n$ .", "Indeed the homology of $Y$ coincides with the homology of the base manifold $\\mathbb {F}_n$ .", "Introducing the Kähler two form: $\\mathbf {K}_{\\mathbb {F}_n} \\, \\equiv \\, \\frac{{\\rm i} }{2\\pi } \\partial \\overline{\\partial } \\mathcal {K}_{\\mathbb {F}_n}$ we need to find: $&&\\int _{C_1}\\mathbf {K}_{\\mathbb {F}_n} \\,=\\, \\frac{9}{16} \\, \\alpha \\,\\ell \\quad ; \\quad \\int _{C_2}\\mathbf {K}_{\\mathbb {F}_n} \\,=\\, \\frac{9}{16} \\, (2 +\\alpha ) \\, \\ell $ where $\\ell $ is a dimensionful parameter providing the scale and $\\alpha $ is some dimensionless parameter parameterizing the ratio between the two volumes.", "The two toric cycles $C_{1,2}$ are respectively defined by the following two equations: $C_1 \\, \\Leftrightarrow \\, v=0 \\quad ; \\quad C_2 \\, \\Leftrightarrow \\, u=0$ As pointed out in [7], in addition to the above two properties of the Kähler form, if we consider the Ricci two-form of the Kähler metric on $\\mathbb {F}_n$ $\\mathbf {Ric}_{\\mathbb {F}_n} \\, = \\, \\frac{{\\rm i} }{2\\pi } \\partial \\overline{\\partial } \\log \\left[\\mbox{det}\\left(\\mathbf {g}^{\\mathbb {F}_n}\\right)\\right]\\quad ; \\quad \\mathbf {g}^{\\mathbb {F}_n}_{ij^\\star } \\, =\\, \\partial _i\\partial _{j^\\star }\\,\\mathbf {K}_{\\mathbb {F}_n}\\quad \\quad i=1,2 \\quad j^\\star = 1^\\star ,2^\\star $ we must find: $&&\\int _{C_1}\\mathbf {Ric}_{\\mathbb {F}_n} \\, =\\, 2-n \\quad ; \\quad \\int _{C_2}\\mathbf {Ric}_{\\mathbb {F}_n} \\, =\\, 2$ It appears that eq.s (REF -REF ) are strong constraints on the function $G_n(\\varpi _n)$ .", "It is interesting to see how they are realized in the metric on $\\mathbb {F}_2$ obtained from the Kronheimer construction.", "We will show this below." ], [ "The metric on $\\mathbb {F}_2$ induced by the Kronheimer\nconstruction", "In [7], relying on the Kronheimer construction, we have constructed an analytically defined Kähler metric on the total space of the canonical bundle of $\\mathbb {F}_2$ .", "The Kähler potential has only an implicit definition as the largest real root of a sextic equation.", "Yet its reduction to the compact exceptional divisor, which is indeed the 2nd Hirzebruch surface, is explicit and the Kähler potential of this metric can be exhibited in closed analytic form.", "We think that this information is very important for the comparison between the parameters of the Ricci-flat metric appearing in supergravity with those emerging in the Kronheimer construction that are the Fayet Iliopoulos parameters of the dual gauge theory.", "Following the chamber structure discussed in [7] we choose the chamber VI defined by the following inequalities on the three Fayet Iliopoulos parameters $\\zeta _{1,2,3}$ : $\\zeta _1-\\zeta _2-\\zeta _3<0 \\quad ;\\quad -\\zeta _1+\\zeta _2-\\zeta _3>0 \\quad ;\\quad -\\zeta _1-\\zeta _2+\\zeta _3<0$ and chamber VIII, defined instead by the following ones: $\\zeta _1-\\zeta _2-\\zeta _3<0 \\quad ;\\quad -\\zeta _1+\\zeta _2-\\zeta _3<0 \\quad ;\\quad -\\zeta _1-\\zeta _2+\\zeta _3<0$ Inside those two chambers we make the choice: $\\zeta _1\\, = \\, \\zeta _3 \\, = \\, r \\quad ; \\quad \\zeta _2\\, = \\, (2+\\alpha ) r \\quad ; \\quad \\quad r>0$ For $\\alpha >0$ we are in chamber VI, while for $\\alpha <0$ we are in chamber VIII.", "For $\\alpha =0$ we are instead on the wall where the non singular variety: $Y \\, \\equiv \\,{\\rm tot} K_{{\\mathbb {F}}_2}$ degenerates in $ Y_3 \\, \\equiv \\, {\\rm tot} K_{{\\mathbb {W}P}[112]}$ denoting by ${\\rm tot} K_{\\mathcal {M}}$ the total space of the canonical bundle of a Kähler manifold (or orbifold) $\\mathcal {M}$ .", "The solution of the moment map equations for the two independent moment maps reduced to the exceptional divisor by performing the limit $w\\rightarrow 0$ is the following one: $T_1=T_3 \\, =\\, {\\sqrt{\\frac{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi }{2(\\alpha +2) \\sqrt{{\\varpi }/{v \\overline{v}}}}}} \\, ; \\,T_2 \\, =\\, \\frac{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi ^2+8 \\varpi }+3\\alpha +\\varpi +4}{2 \\alpha ^2+6 \\alpha +4}$ The complete Kähler potential of the quotient is made of two addends, the pull-back on the constrained surface of the Kähler potential of the flat ambient metric plus the logarithmic term: $\\mathcal {K}_{quotient} \\, = \\, \\mathcal {K}_{0} \\, + \\, \\underbrace{\\zeta _I\\mathfrak {C}^{IJ} \\, \\log T_J}_{\\mathcal {K}_{log}} \\quad ; \\quad \\mathfrak {C}^{IJ}\\, = \\, \\left(\\begin{array}{ccc}2 & -1 & 0 \\\\-1 & 2 & -1 \\\\0 & -1 & 2 \\\\\\end{array}\\right)$ In the present case we explicitly find: $\\mathcal {K}_{0} \\, = \\,2 \\frac{\\alpha \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+2 \\varpi +1\\right)+\\sqrt{\\alpha ^2+6\\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha ^2+3 \\varpi }{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi }$ and $\\mathcal {K}_{log}& = & 2 (\\alpha +1) \\log \\frac{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi ^2+8 \\varpi }+3 \\alpha +\\varpi +4}{2 \\alpha ^2+6 \\alpha +4}-2 \\alpha \\log {\\sqrt{\\frac{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi }{2(\\alpha +2) \\sqrt{{\\varpi }/{v\\overline{v}}}}}}\\nonumber \\\\$ By explicit calculation we were able to verify that the Kähler potential of the quotient $\\mathcal {K}_{quotient}$ yields a metric satisfying all the constraints (REF -REF ).", "We show this in section REF ." ], [ "Reduction to the exceptional divisor", "In this section we consider the reduction to the exceptional divisor for a generic metric of the class described in section , emphasizing that the Kähler metric induced on the divisor is completely determined by the real function $P(t)$ of the real variable $t$ .", "We carefully consider what are the differential constraints on such a function required by the topology and complex structure of the second Hirzebruch surface $\\mathbb {F}_2$ showing that they are all met by the $P(t)$ function that one obtains by localizing the generalized Kronheimer construction of the $ \\mathbb {C}^3/\\mathbb {Z}_4$ singularity resolution on the exceptional divisor." ], [ "The reduction", "The reduction to the exceptional divisor is obtained in the Kähler form and in the metric by setting $s = -3$ .", "The Kähler form on the divisor is the following one $\\mathbb {K}_{\\mathcal {ED}}=\\frac{1}{12} \\left(-9 \\,t \\, \\sin {\\theta }{d\\theta }\\wedge {d\\phi }-9 \\,dt\\wedge {d\\tau }+9 \\, (\\cos {\\theta } -1)\\, dt\\wedge d\\phi \\right)$ while the metric is the following one: $\\text{ds}_{\\mathcal {ED}}^2=\\,-\\frac{3\\,t}{4}({d\\theta }^2 + \\sin ^2\\theta {d\\phi }^2)+\\frac{1}{4} (t+3) P(t)^2 {dt}^2 +\\frac{9 [{d\\tau }+(1-\\cos \\theta ){d\\phi } ]^2}{4 (t+3) P(t)^2}$ and it is completely determined by the function $P(t)$ .", "For the choice: $P(t)=\\left(-\\frac{2}{3}t^2-t\\right)^{-\\frac{1}{2}}$ it is the metric on the orbifold ${\\mathbb {W}P}[112]$ while for other choices of $P(t)$ , obtainable from the Kronheimer construction, $\\text{ds}_{\\mathcal {ED}}^2$ can indeed be a good Kähler metric on the second Hirzebruch surface $\\mathbb {F}_2$ .", "From eq.", "(REF ) specifying the Kähler 2-form of the exceptional divisor and eq.", "(REF ) providing its Kähler metric, we immediately work out also the complex structure tensor that has the following appearence: $\\mathbb {J}_{\\mathcal {ED}}\\, = \\,\\left(\\begin{array}{cccc}0 & 0 & 0 & -\\frac{1}{3} (t+3) P(t)^2 \\\\0 & 0 & \\csc \\theta & -\\tan {\\theta \\over 2} \\\\{6 \\sin ^2{\\theta \\over 2} \\over (t+3) P(t)^2} &-\\sin \\theta & 0 & 0 \\\\\\frac{3}{(t+3) P(t)^2} & 0 & 0 & 0 \\\\\\end{array}\\right)$" ], [ "Topology and the functions of the $t$ coordinate", "We have two important informations on the topology of $\\mathbb {F}_2$ , which provide an extremely selective test in order to know whether a certain metric is indeed defined on $\\mathbb {F}_2$ or on some different twofold, may be degenerate.", "The tests are related with the integrals of the Kähler 2-form $\\mathbb {K}$ and of the Ricci 2-form $\\mathbb {R}$ ic on the two toric curves $C_{1,2}$ respectively defined by the vanishing of either coordinate $(u,v)$ $C_1=\\lbrace v=0\\rbrace \\quad ; \\quad C_2=\\lbrace u=0\\rbrace $ Indeed, as we illustrated in section we must find $\\int _{C_1}\\mathbb {K}\\ne \\infty \\quad ;\\quad \\int _{C_2}\\mathbb {K}\\ne \\infty \\quad ; \\quad \\int _{C_1}\\mathbb {K}\\ne 0\\quad ;\\quad \\int _{C_2}\\mathbb {K}\\ne 0 \\quad ; \\quad \\int _{C_1}\\text{$\\mathbb {R}$ic}=0 \\quad ; \\quad \\int _{C_2}\\text{$\\mathbb {R}$ic}=2$ The explicit reduction of the Kähler form $\\mathbb {K}_{\\mathbb {F}_2}$ to the two cycles $C_1$ and $C_2$ is very simple when $\\mathbb {K}_{\\mathbb {F}_2}$ is written in the basis of the real coordinates ($t$ ,$\\theta $ ,$\\tau $ ,$\\phi $ ).", "Indeed in order to set $v=0$ we have just to look for the zeros of the above defined function $H(t)$ that depends by integration from $P(t)$ .", "Let us suppose that $H(-|t_{max}| )=0$ .", "We obtain the reduction of the Kähler form to the cycle $C_1$ by setting $t = - |t_{max}| $ = const $<$ 0, while we get the reduction to the cycle $C_2$ by setting $\\theta $ = 0.", "$\\mathbb {K}\\mid _{C_1} \\,=\\, \\frac{3}{4}\\,|t_{max}|\\, \\sin {\\theta }\\,{d\\theta }\\wedge {d\\phi } \\quad ; \\quad \\mathbb {K}\\mid _{C_2} \\,=\\, -\\frac{3}{4}{dt}\\wedge {d\\tau }$ Hence we see that in order to get $\\mathbb {F}_2$ as exceptional divisor we need two conditions, that are necessary, although not sufficient.", "$|t_{max}| $ $\\ne $ 0 the range of the coordinate $t$ must be finite [$-|t_{max}| $ ,$-|t_{min}|$ ] in order to get a finite size for the cycle $C_2$ If the zero of the function $H(t)$ is at t=0 we immediately know that there is a degeneration and this is indeed the case of ${\\mathbb {W}P}[112]$ .", "If we integrate the complex structure of the exceptional divisor displayed in eq.", "(REF ) with the same method we used for the whole 6-dimensional space, we find that the coordinate $u$ is exactly the same as in eq.", "(REF ), while for $v$ we find: $v \\, = \\, H(t) \\cos ^2\\frac{\\theta }{2}e^{{\\rm i} \\tau }$ Comparison with the result for $v$ in the entire space (eq.s (REF -REF )) tells us that the function $\\Psi (s)$ must be finite and non vanishing at $s = -3$ in order to have a consistent reduction to the divisor: $\\Psi (-3) = 1 \\quad ; \\quad \\Phi (-3) = 0$ The normalization $\\Psi (-3)$ = 1 can always be obtained by an irrelevant rescaling in the definition of $v$ if $-3$ is not a zero of $\\Psi (s)$ while it must be a zero of $\\Phi (s)$ ." ], [ "Interpretation of the function $H(t)$", "From the explicit integration of the complex structure we obtain a very important interpretation of the function $H(t)$ in relation with the complex Kähler geometry of the exceptional divisor.", "Since the Kähler metric on this two-fold has isometry $\\mathrm {SU(2)\\times U(1)}$ , $\\mathrm {SU(2)}$ acting on the $u$ variable by linear fractional transformation and on $v$ by multiplication with the $u$ -compensator $(c u+d)^2$ , as described in eq.s (REF ), the Kähler potential $\\mathcal {K}$ can be a function only of the invariant combination $\\varpi \\equiv \\varpi _2$ defined in eq.", "(REF ).", "Relying on the representation of $u$ and $v$ derived from the integration of the complex structure we easily obtain: $\\varpi \\, =\\, \\cos ^4\\frac{\\theta }{2}\\left(\\tan ^2\\frac{\\theta }{2}+1\\right)^2 H(t)^2 \\, = \\, H(t)^2$ It follows that: $t\\, = \\, H^{-1}(\\sqrt{\\varpi })$ where $H^{-1}$ denotes the inverse function.", "Since the range of $\\sqrt{\\varpi }$ is $[0,\\infty ]$ , it is necessary that the inverse function $H^{-1}$ maps the semi-infinite interval $[0,\\infty ]$ in a finite one $[-|t_{max}|,-|t_{min}|]$ defined by: $-|t_{max}| \\, = \\, \\lim _{\\varpi \\rightarrow 0}\\, H^{-1}(t) \\quad ; \\quad -|t_{min}| \\, = \\,\\lim _{\\varpi \\rightarrow \\infty }\\, H^{-1}(t)$" ], [ "Topological constraints on the function $P(t)$", "Given the above topology results characterizing the second Hirzebruch surface and considering the metric of the divisor as given in eq.", "(REF ) and its Kähler form (REF ) we immediately obtain the conditions on the function $P(t)$ .", "Indeed, while calculating the Ricci form we can specify integral differential conditions on $P(t)$ from the values of its periods mentioned above.", "We know the explicit form of the complex structure on the exceptional divisor that is obtained by reduction to $s = -3$ of the complex structure pertaining the full 6-dimensional manifold $\\mathcal {M}_6$ .", "The complex structure of the exceptional divisor was displayed in eq.", "(REF ).", "The Ricci form can be calculated by setting its antisymmetric components equal to $\\text{$\\mathbb {R}$ic}_{ij}$ = $J_i^{\\text{}k}$$R_{\\text{kj}}$ where $R_{kj}$ is the standard Ricci tensor.", "In this way we obtain the following general result that exclusively depends on the function $P(t)$ : $\\text{$\\mathbb {R}$ic}_{\\mathcal {ED}}\\,=\\,\\mathfrak {A}(t)\\sin {\\theta }\\,{d\\theta }\\wedge {d\\phi }+\\mathfrak {B}(t)\\, \\sin ^2\\frac{\\theta }{2}\\, {dt}\\wedge {d\\phi }\\,+\\, \\mathfrak {C}(t){dt}\\wedge {d\\tau }$ where $\\mathfrak {A}(t),\\mathfrak {B}(t),\\mathfrak {C}(t)$ are functions of the $t$ -variable expressed as rational functions of $P(t)$ and its first and second derivative with simple $t$ -dependent coefficient.", "We do not write them explicitly for shortness.", "Then the Ricci 2-form can be easily localized on the two cycles $C_1$ and $C_2$ , yielding: $\\text{$\\mathbb {R}$ic}|_{C_1} \\, =\\,\\mathfrak {A}(-|t_{max}|)\\,\\sin {\\theta }\\, d\\theta \\wedge d\\phi \\quad ;\\quad \\text{$\\mathbb {R}$ic}|_{C_2}\\, =\\,\\mathfrak {C}(t)\\, dt \\wedge d\\tau $ Hence, in order to realize the second Hirzebruch surface not only the range of $t$ must have finite extrema [$-|t_{max}| $ ,$-|t_{min}|$ ] but we should also have: $\\mathfrak {A}(-|t_{max}|)=0 \\quad ; \\quad \\int _{-|t_{min}|}^{-|t_{max}| } \\mathfrak {C}(t) \\, dt = 2$" ], [ "The relation between the function $P(t)$ and the\nKähler potential {{formula:47ac66bf-1783-4d5f-839f-cdd0e77668bd}} of the exceptional\ndivisor", "Our goal is that of determining a Ricci-flat metric on the canonical bundle ${\\rm tot} K_{\\mathbb {F}_2}$ , starting from a given bona fide Kähler metric on the second Hirzebruch surface, described in terms of the real variables $t$ , $\\theta $ , $\\tau $ , $\\phi $ .", "In the complex description, any Kähler metric is determined by a suitable Kähler potential; given the isometries and their realization on the chosen complex coordinates $u,v$ , the Kähler potential for the $\\mathbb {F}_2$ surface is a real function of the invariant combination $\\varpi $ defined in eq.", "(REF ) which we generically denote $\\mathcal {K}$ ($\\varpi $ ).", "Therefore it is important to determine the relation between the real variables and the standard complex ones at the same time with the relation between the Kähler potential $\\mathcal {K}$ ($\\varpi $ ) and the function $P(t)$ which determines the metric in the real variables.", "In this respect the essential point to be stressed is that the relation between the real variables and the complex ones is not universal and fixed once for all, rather it depends on the choice of the Kähler potential or viceversa of the function $P(t)$ .", "Hence it is convenient to introduce a name for the inverse function: $H^{-1} \\left(\\sqrt{\\varpi }\\right) = G_T(\\varpi )$ and find its differential relation with the Kähler potential which follows from a comparison between the metric as determined in complex Kähler geometry from $\\mathcal {K}$ ($\\varpi $ ) and as written in real variables.", "For convenience we rewrite the general real form of the metric on the exceptional divisor in the following more compact way $\\text{ds}_{\\mathcal {ED}}^2\\, = \\, -\\frac{3}{4} \\left(t\\underbrace{[{d\\theta }^2+{d\\phi }^2\\sin ^2\\theta ]}_{\\text{metric on $\\mathbb {P}^1$}}+ F(t)\\,{dt}^2\\,+\\frac{1}{F(t)}{\\underbrace{[{d\\tau }+(1-\\cos {\\theta }){d\\phi }]}_{\\text{connectionon the $\\mathrm {U(1)}$ bundle}}}^2\\right) \\quad ; \\quad F(t)\\,\\equiv \\, -\\frac{1}{3}(t+3) P(t)^2$ which clearly displays the fibred structure of the exceptional divisor.", "Next we convert the metric in eq.", "(REF ) using the substitution rule $t\\,=\\, G_T(\\varpi ),\\quad \\theta \\,=\\, 2\\arctan \\sqrt{u\\overline{u}}, \\quad \\tau \\, =\\, -\\frac{1}{2} {\\rm i}\\,\\log \\left(\\frac{v}{\\overline{v}}\\right),\\quad \\phi \\,=\\,-\\frac{1}{2} {\\rm i}\\log \\left(\\frac{u}{\\overline{u}}\\right)$ In this way we transform the metric (REF ) to the complex coordinates $u,v$ and we compare it with the generic metric obtained from a generic Kähler potential $\\mathcal {K}$ ($\\varpi $ ).", "We find that the two metrics coincide provided the following two conditions are satisfied: $t\\, \\equiv \\, G_T(\\varpi )\\,=\\, -\\frac{2}{3} \\varpi \\partial _{\\varpi }\\mathcal {K}(\\varpi )\\quad ; \\quad P(t)\\,= \\,\\pm \\frac{3}{2\\sqrt{-\\varpi \\left(3-\\frac{2}{3} \\varpi \\partial _{\\varpi }\\mathcal {K}(\\varpi )\\right) \\left(\\partial _{\\varpi }\\mathcal {K}(\\varpi )+\\varpi \\partial _{\\varpi ,\\varpi }\\mathcal {K}(\\varpi )\\right)}}$ Given the Kähler potential $\\mathcal {K}$ ($\\varpi $ ), which is supposed to depend also on a deformation parameter, the above equation (REF ) allows to rewrite the same metric in real coordinates, provided one is able to invert the first formula, namely, to find $\\varpi $ as a function of $t$ and of the deformation parameter $\\alpha $ ." ], [ "The Kronheimer Kähler potential for the\n$\\mathbb {F}_2$ surface and its associated {{formula:68bffc69-2522-4399-a62a-f529edfb2e72}} function", "From the Kronheimer construction of the $\\mathbb {C}^3/\\mathbb {Z}_4$ resolution reduced to the exceptional divisor we have the Kähler potential derived in section REF .", "The result obtained in eq.s (REF ,REF ,REF ) can be summarized writing the following general form of the Kähler potential: $\\mathcal {K}_{\\mathbb {F}_2}(\\varpi ,\\alpha )&=&-\\frac{9}{16}\\left\\lbrace -4 (\\alpha +1) \\log \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi ^2+8\\varpi }+3 \\alpha +\\varpi +4\\right)\\right.", "\\nonumber \\\\&&-\\frac{4 \\left[\\alpha \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+2 \\varpi +1\\right)+\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha ^2+3 \\varpi \\right]}{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi }\\nonumber \\\\&&\\left.", "+4 \\alpha \\log \\sqrt{\\frac{\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi }{\\sqrt{\\varpi }}}+8+16 \\log {2}\\right\\rbrace $ where for additional convenience we have changed the overall normalization of the metric multiplying by a $9/8$ factor, have disregarded the irrelevant addends proportional to $\\log [v\\,\\overline{v}]$ and have added a convenient constant addend.", "For $\\alpha $ =0 the surface described by the Kähler metric corresponding to the potential (REF ) degenerates into the singular ${\\mathbb {W}P}[112]$ while for other values of $\\alpha $ such that $0 < \\mid \\alpha \\mid < 1$ we have a metric on a smooth $\\mathbb {F}_2$ surface." ], [ "The degenerate case ${\\mathbb {W}P}[112]$", "It is interesting to see how we recover the degenerate case ${\\mathbb {W}P}[112]$ from the general case.", "Setting $\\alpha $ =0 we obtain: $\\mathcal {K}_{ {\\mathbb {W}P}[112]}(\\varpi )& =&\\frac{9}{4}\\left[\\frac{3 \\varpi +\\sqrt{\\varpi (\\varpi +8)}}{\\varpi +\\sqrt{\\varpi (\\varpi +8)}}+\\log \\left(\\varpi +\\sqrt{\\varpi (\\varpi +8)}\\right)-2-4 \\log {2}\\right]\\nonumber \\\\t&=&-\\frac{3 \\varpi ^2 \\left(\\varpi +\\sqrt{\\varpi (\\varpi +8)}+8\\right)}{\\sqrt{\\varpi (\\varpi +8)} \\left(\\varpi +\\sqrt{\\varpi (\\varpi +8)}\\right)^2} \\quad \\Rightarrow \\quad \\varpi =\\frac{8t^2}{3 (2 t+3)}$ This implies that the interval [0,$\\infty $ ] of $\\varpi $ is mapped into the interval [0,-3/2] and this suffices to guarantee that the cycle $C_1$ is contracted to zero as we have already explained.", "Finally for the function $P(t)$ , using the above general formulae we get: $P(t)=\\sqrt{\\frac{-3}{2 t^2+3t}}$" ], [ "The smooth $\\mathbb {F}_2$ case", "First we can verify that when $\\alpha $ is either $-1$ or $-2$ , the surface degenerates, as the metric depends only on the variable $u$ and no longer on $v$ .", "Using the formula (REF ) we can calculate $t$ and $P(t)$ .", "We find the following relatively complicated answer: $t&=& G_T(\\varpi )\\,=\\, \\frac{N_T}{D_T} \\nonumber \\\\N_T &=& -3 \\left\\lbrace \\alpha ^4+\\alpha \\varpi \\left[3 \\varpi \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+16\\right)+8 \\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+3 \\varpi ^2\\right]\\right.\\nonumber \\\\&&\\left.+4 \\varpi ^2 \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\varpi +8\\right)+\\alpha ^2 \\varpi \\left[6 \\left(\\sqrt{\\alpha ^2+6\\alpha \\varpi +\\varpi (\\varpi +8)}+2\\right)+19 \\varpi \\right]\\right.\\nonumber \\\\&&\\left.+\\alpha ^3 \\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+9 \\varpi \\right)\\right\\rbrace \\nonumber \\\\D_T &=& 4 \\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}\\left(\\sqrt{\\alpha ^2+6 \\alpha \\varpi +\\varpi (\\varpi +8)}+\\alpha +\\varpi \\right)^2$ The new function $G_T(\\varpi )$ maps the interval [0,$\\infty $ ] of $\\varpi $ into the interval $\\left[-\\frac{3 \\alpha }{8},-\\frac{3}{8} (4+3 \\alpha )\\right]$ so that the range of the negative variable $t$ is $t\\in \\left[-\\frac{3}{8}(4+3\\alpha ),-\\frac{3\\alpha }{8}\\right]$ and, as expected, the cycle $C_1$ does not shrink to zero unless $\\alpha $ = 0.", "Quite surprisingly the function $G_T(\\varpi )$ can be easily inverted and we find: $\\varpi \\equiv H(t,\\alpha )^2=\\frac{64 t^2-9 \\alpha ^2}{54 \\alpha +48 t+72}$ while for $P(t)$ we get: $P(t,\\alpha )\\,=\\,2 \\sqrt{\\frac{27 \\alpha ^2+432 \\alpha t+192 t(t+3)}{(t+3) (9 \\alpha +8 t+12) \\left(9 \\alpha ^2-64 t^2\\right)}}$ and we verify that $P(t,0 )=\\sqrt{\\frac{-3}{2 t^2+3t}}$ which is the correct result for the singular case ${\\mathbb {W}P}[112]$ .", "In terms of the function $F(t)$ parameterizing the metric (REF ) we have: $F(t,\\alpha ) \\, = \\, \\frac{4 \\left(27 \\alpha ^2+432\\alpha t+192 t (t+3)\\right)}{3 (9 \\alpha +8 t+12) \\left(64 t^2-9 \\alpha ^2\\right)} \\, = \\, \\frac{1}{2}\\,\\left(\\frac{1}{\\frac{3 \\alpha }{8}+t}-\\frac{1}{\\frac{3}{8} (3 \\alpha +4)+t}+\\frac{1}{t-\\frac{3\\alpha }{8}}\\right)$ The above structure of the function $F(t,\\alpha )$ is very much inspiring.", "As we see, it is just the sum of three simple poles that are alternatively simple poles of the $dt^2$ -coefficient and zeros of the coefficient of the $(d\\tau +(1-\\cos (\\theta ) d\\phi )^2$ -term.", "The range of the variable $t$ turns out to be the interval between two such poles where the sign of the function $F(t)$ is the correct one for in order for the metric (REF ) to have Euclidian signature.", "The three poles are: $t_1 \\, =\\, -\\frac{3 \\alpha }{8} \\quad ; \\quad t_2 \\, = \\, -\\frac{3}{8} (3 \\alpha +4) \\quad ; \\quad t_3 \\, =\\, \\frac{3 \\alpha }{8}$ We also see what is the mechanism of the degeneration producing the singular ${\\mathbb {W}P}[112]$ case: the two poles $t_1$ and $t_3$ come to coincide and the coincidence point is zero.", "This produces the vanishing of the $C_1$ -cycle as we explained above.", "Substituting the function $F(t,\\alpha )$ as given in eq.", "(REF ) into the metric we get a final form of a specific Kähler metric on the second Hirzebruch surface which follows from the Kronheimer construction.", "This metric provides the boundary condition for the Ricci-flat metric on the canonical bundle $ {\\rm tot} K_{\\mathbb {F}_2}$ which must reduce to it when setting $ds=0$ , $d\\chi =0$ and $s = - 3$ ." ], [ "As a matter of check we calculate the periods of the Kähler and Ricci 2-forms also in the real formalism, obtaining the following expected result which holds true for $0 < \\mid \\alpha \\mid < 1$ : $\\int _{C_1}\\mathbb {K} = \\frac{9 \\alpha }{16}\\quad ;\\quad \\int _{C_2}\\mathbb {K} = \\frac{9 (2+\\alpha )}{16}\\quad ;\\quad \\int _{C_1}\\text{$\\mathbb {R}$ic} = 0\\quad ;\\quad \\int _{C_2}\\text{$\\mathbb {R}$ic} = 2$ The above result for the Kähler form is immediate once the function $P(t)=P(t,\\alpha )$ is specified.", "It is instead interesting to see the subtle way in which the result for the Ricci form is obtained independently from the value of $\\alpha $ .", "Calculating the Ricci tensor of the metric in eq.", "(REF ) with the function $F(t,\\alpha )$ of eq.", "(REF ) we find the symmetric matrix $\\mathcal {R}ic$ which, multiplied by the transpose of the complex structure (REF ) with $P(t)$ as in eq.", "(REF ) produces the Ricci form $\\text{$\\mathbb {R}$ic}_{{\\mathcal {ED}}}$ with the structure displayed in eq.", "(REF ) and the following explicit expressions for the functions $\\mathfrak {A}(t)$ and $\\mathfrak {C}(t)$ .", "$\\mathfrak {A}(t) &=& \\frac{(8 t-3 \\alpha ) (3 \\alpha +8 t) \\left(27 \\alpha ^2 (3 \\alpha +4)+512 t^3+576 (3\\alpha +4) t^2+216 \\alpha ^2 t\\right)}{8 t \\left(9 \\alpha ^2+144 \\alpha t+64 t(t+3)\\right)^2} \\\\\\mathfrak {C}(t)&=& \\frac{d}{dt} \\,U(t) \\quad ; \\quad U(t) \\, = \\, \\frac{864 (\\alpha +1) (\\alpha +2) \\left(3 \\alpha ^2+8 (3 \\alpha +4) t\\right)}{\\left(9\\alpha ^2+144 \\alpha t+64 t (t+3)\\right)^2}-\\frac{3 (3 \\alpha +4)}{8t} $ We immediately see that $-|t_{max}|=-\\frac{3 \\alpha }{8}$ is a zero of $\\mathfrak {A}(t)$ so that $\\int _{C1}\\mathbb {R}\\text{ic}_{\\mathcal {ED}}\\, = \\, 0$ , while setting as we must $-|t_{min}|=-\\frac{3}{8} (3 \\alpha +4)$ we obtain: $U(-|t_{max})-U(-|t_{min})\\, = \\, 2 \\quad \\Rightarrow \\quad \\int _{C2}\\mathbb {R}\\text{ic}_{\\mathcal {ED}}\\, = \\, 0$" ], [ "The exceptional divisor in symplectic coordinates.", "Considering next the description of the 6-dimensional manifold $\\mathcal {M}_6$ in terms of symplectic coordinates $\\left\\lbrace \\mathfrak {u},\\mathfrak {v},\\mathfrak {w},\\phi ,\\tau ,\\chi \\right\\rbrace $ (see sect.)", "we easily find that the localization $s =-3$ of the exceptional divisor corresponds to $\\mathfrak {w}=\\frac{3}{2}$ , $d\\mathfrak {w}={d\\chi } =\\, 0$ .", "Hence defining $\\mathcal {D}(\\mathfrak {v})\\equiv \\mathcal {G}\\left(\\mathfrak {v},\\frac{3}{2}\\right),$ where $\\mathcal {G}\\left(\\mathfrak {v},\\mathfrak {w}\\right)$ is the variable part of the overall symplectic prepotential, we obtain that the Kähler metric on the exceptional divisor has also a description in terms of a symplectic potential given by $\\mathfrak {D}(\\mathfrak {u},\\mathfrak {v})=G_0(\\mathfrak {u},\\mathfrak {v})+\\mathcal {D}(\\mathfrak {v})=\\mathcal {D}(\\mathfrak {v})+\\left(\\mathfrak {v}-\\frac{\\mathfrak {u}}{2}\\right)\\log (2 \\mathfrak {v}-\\mathfrak {u})+\\frac{1}{2} (\\mathfrak {u} \\log \\mathfrak {u}-\\mathfrak {u})-\\mathfrak {v} \\log \\mathfrak {v}$ with moment and angular variables $ \\mu ^i=\\lbrace \\mathfrak {u},\\mathfrak {v}\\rbrace $ , $\\Theta _j=\\lbrace \\phi ,\\tau \\rbrace $ and line element as follows: $\\text{ds}_{\\mathcal {ED}}^2 \\, = \\,D_{ij}{d\\mu }^i{d\\mu }^j+ (D^{-1})^{ij}{d\\Theta }_i{d\\Theta }_j$ where the two matrices are: $&& D_{ij} \\,=\\, \\left(\\begin{array}{cc}-\\frac{\\mathfrak {v}}{\\mathfrak {u}^2-2 \\mathfrak {u} \\mathfrak {v}} &\\frac{1}{\\mathfrak {u}-2 \\mathfrak {v}} \\\\\\frac{1}{\\mathfrak {u}-2 \\mathfrak {v}} &{\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})}-\\frac{\\mathfrak {u}}{\\mathfrak {u}\\mathfrak {v}-2 \\mathfrak {v}^2} \\\\\\end{array}\\right)\\quad ; \\quad (D^{-1})^{ij}\\,=\\, \\left(\\begin{array}{cc}\\frac{\\mathfrak {u} \\left(\\mathfrak {v} (2 \\mathfrak {v}-\\mathfrak {u})\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})+\\mathfrak {u}\\right)}{\\mathfrak {v}^2\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})}& \\frac{\\mathfrak {u}}{\\mathfrak {v} \\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})} \\\\\\frac{\\mathfrak {u}}{\\mathfrak {v} \\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})} &\\frac{1}{\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})} \\\\\\end{array}\\right)$ Reduced to the exceptional divisor, the coordinate transformation (REF ) is very simple.", "We have: $\\mathfrak {u}=\\frac{3}{4}t(-1+\\cos \\theta )$ , $\\mathfrak {v}=-\\frac{3t}{4}$ .", "So if we declare that the function $\\mathcal {D}(\\mathfrak {v}) = \\Pi (t)$ , is a function of $t$ we obtain $\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v})=\\frac{16}{9}\\Pi ^{\\prime \\prime } (t)$ and replacing these transformation in (REF -REF ) we obtain that the line element in symplectic coordinates coincides with the line element of eq.", "(REF ) provided that: $\\mathcal {D}(\\mathfrak {v}) \\equiv \\Pi (t) \\quad ; \\quad P(t)^2=-\\frac{4 \\Pi ^{\\prime \\prime }(t)}{t+3} \\quad \\Rightarrow \\quad \\Pi ^{\\prime \\prime }(t)\\, = \\, -\\frac{3}{4} F(t)$ So the function $F(t)$ determining the Kähler geometry of the exceptional divisor, linked to its Kähler potential by eq.", "(REF ), is just $4/3$$\\times $ the second derivative of the non-fixed part of the symplectic potential." ], [ "Applying the above scheme to the Kähler metric on $\\mathbb {F}_2$ induced by the Kronheimer construction, namely utilizing in eq.", "(REF ) $F(t)=F(t,\\alpha )$ as given in eq.", "(REF ) we obtain the following differential equation: $\\mathcal {D}^{\\prime \\prime }(\\mathfrak {v};\\alpha )=16 \\left(\\frac{1}{27 \\alpha -32\\mathfrak {v}+36}+\\frac{1}{32 \\mathfrak {v}-9 \\alpha }+\\frac{1}{9\\alpha +32 \\mathfrak {v}}\\right)$ which, modulo linear functions implies $\\mathcal {D}(\\mathfrak {v},\\alpha )\\,=\\,\\mathcal {D}_{\\mathbb {F}_2}(\\mathfrak {v},\\alpha )$ where $\\mathcal {D}_{\\mathbb {F}_2}(\\mathfrak {v},\\alpha )& \\equiv & \\frac{1}{2} \\mathfrak {v} \\log \\left(1024 \\mathfrak {v}^2-81 \\alpha ^2\\right)+\\frac{1}{64} (27\\alpha -32 \\mathfrak {v}+36) \\log (27 \\alpha -32\\mathfrak {v}+36) \\nonumber \\\\ && + \\frac{9}{32} \\alpha \\,\\text{arctanh}\\left(\\frac{32 \\mathfrak {v}}{9 \\alpha }\\right)-\\frac{\\mathfrak {v}}{2}$ We also find: $\\mathcal {D}_{\\mathbb {F}_2}(\\mathfrak {v},0)& =&-\\frac{1}{16}(-9+8\\mathfrak {v})\\log \\left(3-\\frac{8}{3}\\mathfrak {v}\\right)+\\mathfrak {v} \\log \\mathfrak {v}\\quad \\text{modulo a linearfunction of}\\,\\, \\mathfrak {v}$ For $\\alpha $ = 0 this is the correct result for ${\\mathbb {W}P}[112]$ ." ], [ "The Monge-Ampère equation and its series expansion", "In this section we arrive at the core of the issue, i.e.", "the construction of Ricci-flat metrics on the spaces we are concerned with.", "The common general feature of these is that they are the total space of the canonical bundle of a complex two-dimensional compact Kähler manifold $\\mathcal {M}_4$ , the exceptional divisor when the total space is the full or partial resolution of a quotient singularity.", "In this interpretation the base of the canonical bundle is indeed the exceptional divisor produced by the blow up of an isolated singular point.", "The additional common structural feature of the Ricci-flat metrics we want to consider is, as we already stressed several times, the group of continuous isometries that they should possess, mentioned in equation (REF ).", "The action of $\\mathrm {G}_{iso}$ on the three complex coordinates $u,v,w$ that originate from the integration of the complex structure was displayed in eq.s (REF ).", "The presence of these isometries imposes very stringent constraints on the Kähler metric that are most efficiently handled at the level of the potential $\\mathfrak {P}$ from which the metric can be obtained by means of derivatives.", "The condition of Ricci-flatness of the metric is translated into a nonlinear differential equation to be satisfied by the potential $\\mathfrak {P}$ that we name the Monge-Ampère equation.", "As we have seen in the previous pages, there are three equivalent formulations of the Kähler geometry of the toric six-dimensional manifolds $\\mathcal {M}_6$ we are concerned with: A) The complex setup where the geometry is encoded in the Kähler potential $\\mathfrak {P}\\, = \\,\\mathcal {K}\\left(u,v,w,\\overline{u},\\overline{v},\\overline{w}\\right)$ B) The symplectic setup where the geometry is encoded in the symplectic potential $\\mathfrak {P}\\, =\\,\\mathcal {G}\\left(\\mathfrak {u},\\mathfrak {v},\\mathfrak {w}\\right)$ C) The hybrid setup where the geometry is encoded in the symplectic potential, but instead of the coordinates $v,w$ we use the coordinates $s$ ,$t$ related to them by the coordinate transformation (REF -REF ).", "Correspondingly there are, to begin with, two formulations of the Monge-Ampère equation, one for the Kähler potential, one for the symplectic potential.", "In both cases the constraints imposed by the chosen isometries reduce the effective potential to be a function of only two real variables so that the Monge-Ampère equation is a non linear partial differential equation in two variables.", "At this point the symplectic case still splits into two versions depending on whether we employ the pure symplectic variables or the hybrid ones $s, t$ .", "In all formulations, as we show below, the equation has the property that we can fix as boundary condition an arbitrarily chosen Kähler metric on the exceptional divisor." ], [ "The Monge-Ampère equation for the Kähler\npotential", "We begin with the Monge-Ampère equation written in terms of the Kähler potential.", "It follows from the chosen isometries that the Kähler potential $\\mathcal {K}$ must be a function only of the two invariants: $\\mathfrak {f} \\, \\equiv \\, \\mid w\\mid ^2 \\quad ; \\quad \\varpi \\, \\equiv \\,\\left( 1 +\\mid u \\mid ^2\\right)^2 \\, \\mid v \\mid ^2 \\quad \\text{or}\\quad \\mathcal {T} \\, \\equiv \\,4+\\varpi - \\sqrt{\\varpi \\,(\\varpi +8)}$ so that we can set: $\\mathcal {K} \\, = \\, G(\\varpi ,\\mathfrak {f}) \\quad \\text{or} \\quad \\mathcal {K} \\, = \\, G(\\mathcal {T},\\mathfrak {f})$ The use of the alternative combination $\\mathcal {T}$ simplifies the Kähler potential in certain cases.", "The Monge-Ampère equation in this setup is simply the statement that the determinant of the Kähler metric is constant.", "Indeed in the complex coordinate setup the hermitian Ricci tensor is obtained from the logarithm of the metric determinant in the same way as the Kähler metric is obtained from the Kähler potential: $\\text{Ric}_{ij^\\star }\\, = \\, \\frac{\\partial }{\\partial z^i} \\,\\frac{\\partial }{\\partial \\overline{z}^{j^\\star }} \\log \\left[\\text{det}\\mathbf {g}\\right] \\quad ; \\quad \\mathbf {g} \\, = \\, g_{ij^\\star }\\, = \\, \\frac{\\partial }{\\partial z^i} \\,\\frac{\\partial }{\\partial \\overline{z}^{j^\\star }}G(\\mathcal {T},\\mathfrak {f})\\quad ; \\quad z^i \\equiv \\lbrace u,v,w\\rbrace $ Hence if $\\text{det} \\, \\mathbf {g} \\, = \\, \\kappa $ where $\\kappa $ is a constant parameter, the Ricci tensor is necessarily zero and we have a Ricci-flat metric.", "The Monge-Ampère equation is obtained by replacing in eq.", "(REF ) the expression of $\\text{det} \\, \\mathbf {g} $ in terms of derivatives of the Kähler potential $G(\\mathcal {T},\\mathfrak {f})$ .", "Relying on the definition of the invariants provided in eq.", "(REF ) we obtain: $4 \\mathcal {T}^3 G^{(1,0)}(\\mathcal {T},\\mathfrak {f} )\\left\\lbrace G^{(0,1)}(\\mathcal {T},\\mathfrak {f} )\\left[\\left(\\mathcal {T}^2+8 \\mathcal {T}-16\\right) G^{(1,0)}(\\mathcal {T},\\mathfrak {f} )+\\mathcal {T}\\left(\\mathcal {T}^2-16\\right) G^{(2,0)}(\\mathcal {T},\\mathfrak {f} )\\right]\\right.\\nonumber \\\\\\left.+\\mathfrak {f}\\left[G^{(0,2)}(\\mathcal {T},\\mathfrak {f} ) \\left\\lbrace \\left(\\mathcal {T}^2+8\\mathcal {T}-16\\right)G^{(1,0)}(\\mathcal {T},\\mathfrak {f} )+\\mathcal {T}\\left(\\mathcal {T}^2-16\\right) G^{(2,0)}(\\mathcal {T},\\mathfrak {f})\\right\\rbrace \\right.\\right.\\nonumber \\\\\\left.\\left.-\\mathcal {T}\\left(\\mathcal {T}^2-16\\right) G^{(1,1)}(\\mathcal {T},\\mathfrak {f})^2\\right]\\right\\rbrace \\, = \\,-\\kappa (\\mathcal {T}+4)^4$ One can solve the Monge-Ampère equation in the above form by developing the Kähler potential in power series of $\\mathfrak {f}$ : $G(\\mathcal {T},\\mathfrak {f} )= G_0(\\mathcal {T})+\\sum _{n=1}^{\\infty }G_n(\\mathcal {T}) \\,\\mathfrak {f} ^n $ where $G_0(\\mathcal {T})$ is the Kähler potential of a convenient Kähler metric defined over the exceptional divisor.", "Indeed it is a property of the considered system that inserting (REF ) into the Monge-Ampère equation (REF ), the function $G_0(\\mathcal {T})$ corresponding to the Kähler potential of the Kähler metric on the exceptional divisor is undetermined, while all the other $G_n(\\mathcal {T})$ functions can be iteratively determined in terms of the previous $G_{k<n}(\\mathcal {T})$ .", "As we discussed before, it is quite remarkable that on the exceptional divisor located at $s=-3$ the Ricci-flat orthotoric metric (REF ) reduces precisely to the Kähler metric on ${\\mathbb {W}P}[112]$ , which was obtained in [7] from the Kronheimer construction while performing the partial resolution of the $\\mathbb {C}^3/\\mathbb {Z}_4$ singularity on a type III wall." ], [ "Recursive solution for the Kähler potential in the case\n${\\rm tot} K_{{\\mathbb {W}P}[112]}$", "In this section we present the recursive solution of the Monge-Ampère equation which was obtained by means of a dedicated MATHEMATICA code using as 0-th order Kähler potential the following one $G_0(\\mathcal {T})= 4\\log \\mathcal {T}+\\mathcal {T}$ which yields the Kronheimer Kähler metric on ${\\mathbb {W}P}[112]$ .", "The Kähler potential of the full metric on ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ can be expressed as follows $G(\\mathcal {T},\\mathfrak {f} )=4\\log \\mathcal {T} +\\mathcal {T}+\\frac{1}{2}\\left( \\sum _{k=1}^{\\infty } \\frac{1}{(k!", ")}\\frac{\\mathcal {P}_{2k-2}(\\mathcal {T})}{(2\\mathcal {T})^{2k}(\\mathcal {T}+4)^{2k-3}}(\\kappa \\mathfrak {f} )^k\\right)$ where the symbol $\\mathcal {P}_{2k-2}(\\mathcal {T})$ denotes a polynomyal of degree $2k-2$ in the variable $\\mathcal {T}$ .", "The remarkable feature is that the coefficients of the polynomials $\\mathcal {P}_{2k-2}(\\mathcal {T})$ are all integer numbers whose decomposition into prime factors involves prime numbers of increasing values.", "We show the first 6 of these intriguing polynomials $k = 1 ~ | ~ \\mathcal {P}_{0}(\\mathcal {T})= 2$ $k = 2 ~ | ~ \\mathcal {P}_{2}(\\mathcal {T})= 112+16 \\mathcal {T}+\\mathcal {T}^2$ $k = 3 ~ | ~ \\mathcal {P}_{4}(\\mathcal {T})= 2 \\left(10112+4000 \\mathcal {T}+408 \\mathcal {T}^2+30 \\mathcal {T}^3+\\mathcal {T}^4\\right)$ $k = 4 ~ | ~ \\mathcal {P}_{6}(\\mathcal {T})=6563840+4347392 \\mathcal {T}+925952 \\mathcal {T}^2+82624 \\mathcal {T}^3+7112 \\mathcal {T}^4+350 \\mathcal {T}^5+8 \\mathcal {T}^6 $ $k = 5 ~ | ~ \\mathcal {P}_{8}(\\mathcal {T})= 3128950784+2919825408\\mathcal {T}+987267072 \\mathcal {T}^2+150301696 \\mathcal {T}^3+13354240\\mathcal {T}^4+1313920 \\mathcal {T}^5\\\\{}\\qquad \\qquad \\qquad \\; +76064\\mathcal {T}^6+2812 \\mathcal {T}^7+49 \\mathcal {T}^8$ $k = 6 ~ | ~ \\mathcal {P}_{10}(\\mathcal {T})= 1980772122624+2387983728640\\mathcal {T}+1118459035648 \\mathcal {T}^2+256754671616\\mathcal {T}^3\\\\{}\\qquad \\qquad \\qquad ~~\\;\\;+32204621824 \\mathcal {T}^4+3128804864\\mathcal {T}^5+331169920 \\mathcal {T}^6+20666912 \\mathcal {T}^7+975904\\mathcal {T}^8+28886 \\mathcal {T}^9\\\\{}\\qquad \\qquad \\qquad ~~\\;+407 \\mathcal {T}^{10} $" ], [ "The Monge-Ampère equation of Ricci-flatness for the\nsymplectic potential", "According to [60], [61] the condition for Ricci-flatness can be written as a differential condition on the symplectic potential which is the following $\\text{Det}\\left[G_{ij}\\right]=\\text{const} \\times \\text{Exp}\\sum _{h=1}^n c^h\\partial _hG$ where $c^h$ are some constants.", "In the case of our general metric with isometry $\\mathrm {SU(2)\\times U(1)\\times U(1)}$ , the symplectic form of the Monge-Ampère equation simplifies since we have the particular form (REF ) of the matrix $G_{ij}$ .", "Indeed we find: $\\text{detHes} \\equiv \\text{Det}\\left[G_{ij}\\right]=\\frac{\\mathfrak {v}}{\\mathfrak {u}(\\mathfrak {u}-2 \\mathfrak {v})}\\left[\\mathcal {G}^{(1,1)}(\\mathfrak {v},\\mathfrak {w})^2-\\mathcal {G}^{(0,2)}(\\mathfrak {v}, \\mathfrak {w})\\mathcal {G}^{(2,0)}(\\mathfrak {v},\\mathfrak {w})\\right]$ This facilitates the study of the Ricci-flatness condition because the coefficients $c^{\\mathfrak {u}}$ and $c^{\\mathfrak {v}}$ are already fixed by the need to reproduce the $\\mathfrak {u}$ -dependence of $\\mathrm {detHes}$ .", "We easily find: $\\text{Exp}\\left[-2\\partial _{\\mathfrak {u}}G(\\mathfrak {u},\\mathfrak {v},\\mathfrak {w})\\, -2\\partial _{\\mathfrak {v}}G(\\mathfrak {u},\\mathfrak {v},\\mathfrak {w})\\,+\\,k\\,\\partial _{\\mathfrak {w}}G(\\mathfrak {u},\\mathfrak {v},\\mathfrak {w})\\right]\\, = \\, -\\frac{e^{k \\mathcal {G}^{(0,1)}(\\mathfrak {v},\\mathfrak {w})-2\\mathcal {G}^{(1,0)}(\\mathfrak {v},\\mathfrak {w})}\\mathfrak {v}^2}{\\mathfrak {u}^2-2 \\mathfrak {u} \\mathfrak {v}}$ Hence in the symplectic formalism the Monge-Ampère equation for Ricci flatness reduces to the following relation: $c e^{k \\mathcal {G}^{(0,1)}(\\mathfrak {v},\\mathfrak {w})-2\\mathcal {G}^{(1,0)}(\\mathfrak {v},\\mathfrak {w})}\\mathfrak {v}+\\mathcal {G}^{(1,1)}(\\mathfrak {v},\\mathfrak {w})^2-\\mathcal {G}^{(0,2)}(\\mathfrak {v},\\mathfrak {w})\\mathcal {G}^{(2,0)}(\\mathfrak {v},\\mathfrak {w}) = 0$ imposed solely on the function of two variables $\\mathcal {G}$ [$\\mathfrak {v}$ ,$\\mathfrak {w}$ ].", "We have explicitly verified that the function $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v},\\mathfrak {w})$ defined in equation (REF ), which corresponds to the orthotoric Ricci-flat metric on ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ satisfies eq.", "(REF ) with: $k =- \\frac{8}{3}\\quad ; \\quad c=\\frac{72 e^3}{7}$" ], [ "Discussion of the boundary condition", "As we show below, differently from the case of the Monge-Ampère equation for the Kähler potential in the symplectic case, there is a subtle issue concerning the choice of boundary condition to be imposed on the function while restricting it to the exceptional divisor.", "The important point is that at the level of the metric the limit $\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}} $ should reproduce the metric on the divisor derived from the potential $\\mathcal {D}(\\mathfrak {v})\\, = \\,\\mathcal {G}\\left(\\mathfrak {v},{\\textstyle \\frac{3}{2}}\\right)$ .", "There are only two ways to obtain this.", "If the symplectic potential $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ is holomorphic at $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$ and admits a Taylor series expansion in $\\mathfrak {w}-{\\textstyle \\frac{3}{2}}$ we are obliged to impose that $\\partial _w\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ be a constant at $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$ and this results in a recursive solution with coefficients that are rational functions of increasing order and can hardly define a convergent series.", "Furthermore the only known solution of the Monge Ampère equation, provided by the function (REF ) corresponding to the orthotoric metric on ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ has not this holomorphic behavior.", "Indeed $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v},\\mathfrak {w})$ provides a paradigmatic example of the other possible boundary condition which foresees a logarithmic singularity of the symplectic potential while approaching the exceptional divisor: $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})\\,\\stackrel{\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}}}{\\approx } \\,\\left(\\mathfrak {w}-{\\textstyle \\frac{3}{2}}\\right) \\, \\log \\left(\\mathfrak {w}-{\\textstyle \\frac{3}{2}}\\right) \\, + \\,\\mathcal {G}_0(\\mathfrak {v})+ \\mathcal {O}\\left(\\mathfrak {w}-{\\textstyle \\frac{3}{2}}\\right)$ In the sequel we show that with the second type of boundary condition we can reconstruct the known solution $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v},\\mathfrak {w})$ of equation (REF ) and also derive a series solution pertaining to the smooth ${\\mathbb {F}}_2$ case which displays the same general features as $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v},\\mathfrak {w})$ .", "Unfortunately, up to the present moment we can only give numerical evidences of the last statement.", "In view of what we explained above we skip the details concerning the first type of boundary condition (holomorphicity at $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$ and jump directly to the case of a logarithmic singularity at $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$ .", "Indeed, a logarithmic singularity is known to be the correct behaviour to ensures smoothness of the toric Kähler metrics near to divisors [62], [63], [60]." ], [ "The boundary condition with a logarithmic singularity\nat $\\mathfrak {w}={\\textstyle \\frac{3}{2}}$", "We implement the second type of boundary condition requiring that following two properties should be preserved: a) The symplectic potential $\\mathcal {G}(\\mathfrak {v},\\mathfrak {w})$ has a finite limit for $\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}}$ b) The limit for $\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}}$ of the bundle metric should be exactly the exceptional divisor metric (REF -REF ) Namely we must have: $\\lim _{\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}}} \\mathcal {G}(\\mathfrak {v},\\mathfrak {w}) \\, = \\,\\mathcal {D}(\\mathfrak {v}) \\quad ; \\quad \\lim _{\\mathfrak {w}\\rightarrow {\\textstyle \\frac{3}{2}}} \\mathrm {ds}^2_{symp} \\, = \\,\\mathrm {ds}^2_{{\\mathcal {ED}}}$ To discuss this alternative boundary condition it is convenient to use rescaled variables defined as follows $x=2\\mathfrak {v} \\quad ; \\quad y = 3\\mathfrak {w} \\quad ; \\quad y=\\frac{9}{2}+\\omega \\Rightarrow \\omega = 3\\, \\left(\\mathfrak {w} - {\\textstyle \\frac{3}{2}}\\right)$ In terms of such variables the Monge-Ampère equation (REF ) becomes $c\\, x\\, \\exp \\left[-8 \\,\\mathcal {G}^{(0,1)}(x,\\omega )-4\\,\\mathcal {G}^{(1,0)}(x,\\omega )\\right]-\\mathcal {G}^{(1,1)}(x,\\omega )^2+\\mathcal {G}^{(0,2)}(x,\\omega ) \\mathcal {G}^{(2,0)}(x,\\omega )\\,= \\, 0$ Instead of assuming that $\\mathcal {G}(x,\\omega )$ is holomorphic at $\\omega =0$ , we impose that it has a logarithmic singularity of the form $\\omega \\log \\omega $ .", "Indeed this is the unique alternative way in which the metric on the total space can reduce to the metric exceptional divisor in the limit $\\omega \\rightarrow 0$ .", "Furthermore this behavior for $\\mathfrak {w} \\rightarrow {\\textstyle \\frac{3}{2}}$ is precisely that displayed by the symplectic potential $\\mathcal {G}_{{{\\mathbb {W}P}[112]}}(\\mathfrak {v},\\mathfrak {w})$ explicitly written down in eq.", "(REF ).", "Hence we assume the following different series expansion which isolates a logarithmic singularity at $\\omega =0$ : $\\mathcal {G}(x,\\omega )\\, =\\, {\\textstyle \\frac{1}{8}} \\,\\omega \\, \\log [\\omega ]+ \\mathcal {G}_0(x) \\, + \\,\\sum _{k=1}^{\\infty } \\omega ^k \\mathcal {G}_k(x)$ The function $\\mathcal {G}_0(x)$ is free.", "All the functions $\\mathcal {G}_k(x)$ ($k\\ge $ 1) are determined in terms of $\\mathcal {G}_0(x)$ .", "For instance we have: $\\mathcal {G}_0(x) &=& \\mathcal {G}_0(x) \\nonumber \\\\\\mathcal {G}_1(x)&=& \\frac{1}{8} \\log \\left(-\\frac{e^{-4 \\mathcal {G}_0^{\\prime }(x)} x}{\\mathcal {G}_0^{\\prime \\prime }(x)}\\right)\\nonumber \\\\\\mathcal {G}_2(x) &=& \\frac{32 x^2 \\mathcal {G}_0^{\\prime \\prime }(x){}^3-12 x \\mathcal {G}_0^{\\prime \\prime }(x){}^2+16 x^2 \\mathcal {G}_0{}^{(3)}(x) \\mathcal {G}_0^{\\prime \\prime }(x)+2 \\mathcal {G}_0^{\\prime \\prime }(x)-2 x \\mathcal {G}_0{}^{(3)}(x)+x^2 \\mathcal {G}_0{}^{(4)}(x)}{256x^2 \\mathcal {G}_0^{\\prime \\prime }(x){}^2} \\nonumber \\\\\\mathcal {G}_3(x) &=& \\frac{1}{36864 x^4 \\mathcal {G}_0^{\\prime \\prime }(x){}^4}\\left(-48 x^2 \\left(16 \\mathcal {G}_0{}^{(3)}(x) x^2+7\\right) \\mathcal {G}_0^{\\prime \\prime }(x){}^4-48 x \\left(12 x^3 \\mathcal {G}_0{}^{(4)}(x)-5\\right)\\mathcal {G}_0^{\\prime \\prime }(x){}^3\\right.\\nonumber \\\\&&\\left.+4 \\left(144 \\mathcal {G}_0{}^{(3)}(x){}^2 x^4-18\\mathcal {G}_0{}^{(5)}(x) x^4+38 \\mathcal {G}_0{}^{(4)}(x) x^3+12\\mathcal {G}_0{}^{(3)}(x) x^2-11\\right)\\mathcal {G}_0^{\\prime \\prime }(x){}^2\\right.\\nonumber \\\\&&\\left.+2 x \\left(-152 x^2 \\mathcal {G}_0{}^{(3)}(x){}^2+\\left(72\\mathcal {G}_0{}^{(4)}(x) x^3+4\\right) \\mathcal {G}_0{}^{(3)}(x)\\right.\\right.\\nonumber \\\\&&\\left.\\left.-x \\left(\\mathcal {G}_0{}^{(6)}(x) x^2-6\\mathcal {G}_0{}^{(5)}(x) x+2 \\mathcal {G}_0{}^{(4)}(x)\\right)\\right)\\mathcal {G}_0^{\\prime \\prime }(x)+9 x^2 \\left(x \\mathcal {G}_0{}^{(4)}(x)-2\\mathcal {G}_0{}^{(3)}(x)\\right){}^2\\right)\\nonumber \\\\$" ], [ "Recursive solution of the symplectic Monge-Ampère equation in the case\nwhere the smooth ${\\mathbb {F}}_2$ surface is at the\nboundary", "Relying on the results of the previous subsection we consider the case where the symplectic potential at the boundary (i.e.", "on the exceptional divisor) is the one yielding the Kronheimer Kähler metric on ${\\mathbb {F}}_2$ .", "In terms of the $x$ variable and setting, $ \\Delta = {\\textstyle \\frac{9}{8}} \\alpha $ the function in eq.", "(REF ) can be rewritten as follows: $\\mathcal {G}_0(x,\\Delta ) \\equiv \\frac{1}{4} x \\log \\left(4 x^2-\\Delta ^2\\right)+\\frac{1}{8} \\,\\Delta \\log \\left(\\frac{\\Delta +2 x}{\\Delta -2 x}\\right)+ \\frac{1}{8}\\left(3 \\Delta -2x+\\frac{9}{2}\\right) \\log \\left(3 \\Delta -2x+\\frac{9}{2}\\right)$ from the formal solution discussed in the previous section we obtain: $\\mathcal {G}(x,\\omega ) = \\mathcal {G}_0(x,\\Delta ) +\\frac{1}{8} \\omega \\log \\left(\\frac{x (6 \\Delta -4 x+9)^2}{2 e\\left[\\Delta ^2+4 x^2-6 (2 \\Delta +3) x\\right]}\\right)+ \\frac{1}{8} \\, \\omega \\, \\log \\omega +\\sum _{k=1}^\\infty \\frac{N_{k+1}(x,\\Delta )}{D_{k+1}(x,\\Delta )} \\,\\omega ^{k+1}$ where $N_{k+1}(x,\\Delta )$ and $D_{k+1}(x,\\Delta )$ are polynomials whose degrees are as follows: $\\text{degree} \\left[N_{k+1}(x,\\Delta )\\right]\\, = \\, 6 \\, k \\quad ; \\quad \\text{degree} \\left[D_{k+1}(x,\\Delta )\\right]\\, = \\, 7 \\, k$ Hence the degree of the coefficient of $\\omega ^{k+1}$ in the series expansion is a rational function of $x$ of degree $-k$ , a feature that looks promising for convergence.", "By means of a dedicated MATHEMATICA code we can calculate the polynomials $N_{k+1}(x,\\Delta )$ , $D_{k+1}(x,\\Delta )$ to any desired order.", "For reason of typographical space we display here only the first terms up to order $k=2$ .", "$N_2\\text{(x) = }-3888x^4-864 x^5+128 x^6-5184 x^4 \\Delta -576 x^5 \\Delta +720 x^3 \\Delta ^2-1536 x^4 \\Delta ^2+480 x^3 \\Delta ^3-81 \\Delta ^4+162 x \\Delta ^4-120 x^2 \\Delta ^4-108 \\Delta ^5+108 x \\Delta ^5-36 \\Delta ^6$ $N_3\\text{(x) = }49152 x^{12}-811008 x^{11} (3+2 \\Delta )-891 \\Delta ^8 (3+2 \\Delta )^4+108 x \\Delta ^6 (3+2 \\Delta )^3\\left(810+1080 \\Delta +421 \\Delta ^2\\right)-36 x^2 \\Delta ^6 (3+2\\Delta )^2 \\left(23652+31536 \\Delta +10961 \\Delta ^2\\right)+384 x^6\\Delta ^4 \\left(31347+41796 \\Delta +12692 \\Delta ^2\\right)$ $+1024 x^{10} \\left(42039+56052 \\Delta +18412 \\Delta ^2\\right)-1536 x^7 \\Delta ^2 \\left(-9963-19926 \\Delta -8238 \\Delta ^2+412 \\Delta ^3\\right)$ $-1536 x^5 \\Delta ^4 \\left(15795+31590 \\Delta +19092\\Delta ^2+3368 \\Delta ^3\\right)$ $ -3072 x^9 \\left(28431+56862 \\Delta +41124 \\Delta ^2+10568 \\Delta ^3\\right)+576 x^4 \\Delta ^4 \\left(35721+95256 \\Delta \\right.$ $\\left.+83151 \\Delta ^2+26196 \\Delta ^3+1655 \\Delta ^4\\right)\\\\+256 x^8 \\left(-137781-367416 \\Delta -270540 \\Delta ^2-34128 \\Delta ^3+23272 \\Delta ^4\\right)$ $+864 x^3 \\Delta ^4 \\left(-6561-21870 \\Delta -17739 \\Delta ^2+3402 \\Delta ^3+8805 \\Delta ^4+2558 \\Delta ^5\\right)$ $D_2\\text{(x) = }128 x^2 (-9+4 x-6 \\Delta )\\left(-18 x+4 x^2-12 x \\Delta +\\Delta ^2\\right)^2$ $D_3\\text{(x) = }9216 x^4 (9-4 x+6 \\Delta )^2 \\left(4x^2+\\Delta ^2-6 x (3+2 \\Delta )\\right)^4$" ], [ "Numerical study in the case $\\Delta = {\\textstyle \\frac{3}{4}}$", "Since so far we have not been able to guess the sum of the series in terms of elementary or higher transcendental functions, to get some understanding of the solution we have resorted to a numerical study of the approximants to the solution obtained by truncating the series in eq.", "(REF ) to various orders performing the plots.", "The relevant thing is that for the special value $\\Delta = 0$ of the parameter we know the exact sum of the series.", "It is provided by the symplectic potential (REF ) which pertains to the case of ${\\rm tot} K_{{\\mathbb {W}P}[112]}$ .", "This fortunate occurrence enables us to compare the plot of the exact function with those of its approximants.", "This comparison, as we are going to see, turns out to be quite inspiring since it elucidates the meaning of certain oscillatory behaviors of the approximants that are completely analogous in the case $\\Delta =0$ , where we know the sum of the series and in the case $\\Delta > 0$ where the sum is unknown.", "In terms of the variables $x$ and $\\omega $ the symplectic potential of the orthotoric metric takes the following explicit expression: $\\mathcal {G}_{\\Delta =0}(x,\\omega ) &=&\\frac{1}{224} \\left\\lbrace 7 \\left[(4 \\omega +16 x) \\log \\left({9\\over 2} -\\sqrt{\\left(x+\\omega +\\frac{9}{2}\\right)^2-18 x}- x-\\omega \\right)\\right.\\right.\\nonumber \\\\&&\\left.\\left.-2 (4 x-4 \\omega -9) \\log \\left(\\sqrt{\\left(x+\\omega +\\frac{9}{2}\\right)^2-18 x}-x-\\omega \\right)\\right.\\right.\\nonumber \\\\&&\\left.\\left.+(4 x+2 \\omega +9) \\log \\left(\\frac{1}{567} \\left[4 \\sqrt{\\left(x+\\omega +\\frac{9}{2}\\right)^2-18 x}+4 x+4 \\omega +27\\right]^2+1\\right)\\right]\\right.\\nonumber \\\\&&\\left.", "-2 \\sqrt{7} (4 x-2 \\omega -27)\\arctan \\frac{4 \\sqrt{\\left(x+\\omega +\\frac{9}{2}\\right)^2-18 x}+4 x+4 \\omega +27}{9\\sqrt{7}}\\right.\\nonumber \\\\&&\\left.", "-(4 x+9) \\log \\frac{34359738368}{823543}+2 \\sqrt{7} (4 x-27)\\arctan \\frac{5}{\\sqrt{7}}\\right\\rbrace $ For comparison we choose the series solution in the case $\\Delta ={\\textstyle \\frac{3}{4}}$ .", "This value, corresponding to $\\alpha = {\\textstyle \\frac{2}{3}}$ , introduces various simplifications in the solution and, for no other good reason, provides a good reference point.", "In this case the symplectic potential takes the following appearance $G_{\\Delta = {\\textstyle \\frac{3}{4}}}(x,\\omega ) &=& \\frac{1}{32} \\left[8 x \\log \\left(4 x^2-\\frac{9}{16}\\right)+(27-8 x)\\log \\left(\\frac{27}{4}-2 x\\right)+6 \\, \\text{arctanh}\\frac{8x}{3}\\right]\\nonumber \\\\&&\\frac{1}{8} \\omega \\log (\\omega ) +\\frac{1}{8} \\omega \\log \\left(\\frac{2 (27-8 x)^2 x}{e [16 x (4x-27)+9]}\\right)+ \\sum _{k=1}^{\\infty } \\frac{\\widehat{N}_{k+1}(x)}{\\widehat{D}_{k+1}(x)} \\,\\omega ^{k+1}$ We omit the explicit presentation of the rational functions $\\frac{\\widehat{N}_{k+1}(x)}{\\widehat{D}_{k+1}(x)}$ that we have calculated by means of a computer programme up to order $k=10$ and higher.", "We rather present the plots of such approximants.", "Let us first consider the plot of the function $\\mathcal {G}_{\\Delta =0}(x,\\omega )$ displayed in fig.REF .", "Figure: Plots of the exact symplectic potential𝒢 Δ=0 (x,ω)\\mathcal {G}_{\\Delta =0}(x,\\omega ) for small values of the distanceω\\omega from the exceptional divisor (plot on the left) andextending to large values (plot on the right.", ")As we distinctly see from the picture, the exact function, namely the sum of the infinite series in $\\omega $ defines parametrically a perfectly smooth surface in three dimensions that however features a nontrivial structure provided by a sort of smooth bending along a line that starts approximately at $x={\\textstyle \\frac{9}{4}},\\omega = 0$ and goes up towards $x={\\textstyle \\frac{9}{2}}, \\omega =\\infty $ .", "The geometrically meaning of this bending is not entirely clear, yet one can guess that it corresponds to a transition region from a near divisor geometry to an asymptotic geometry that is that of a metric cone over the Sasakian orbifold $\\mathbb {S}^5/\\mathbb {Z}_4$ .", "It is now interesting to compare the behavior of the exact function with its approximants obtained truncating the series to various orders.", "Figure: Plots of the exact symplectic potential𝒢 Δ=0 (x,ω)\\mathcal {G}_{\\Delta =0}(x,\\omega ) compared to its approximants oforder 6 and 7 respectively: on the right for small values ofω\\omega , on the left extending to large values of ω\\omega .Let us now consider the plots displayed in fig.REF .", "In the plot on the right, the surface plotted in the middle is the sum of the series (i.e.", "the exact function), while the other two surfaces, respectively bending, one up, the other down, are two consecutive approximants (the first of even order, the second of odd order).", "As we clearly see, the series converges to the exact function and does it rapidly, in the region before the bending structure illustrated above.", "As we come close to such a line of bending the series no longer converges and its various truncations oscillate violently creating a peculiar canyon.", "Let us now compare this behavior of the case $\\Delta = 0$ with that of the series solution for $\\Delta ={\\textstyle \\frac{3}{4}}$ .", "To this effect let us consider the figure REF .", "Figure: Plots of approximants of even and oddorder of the function 𝒢 Δ=3 4 (x,ω)\\mathcal {G}_{\\Delta ={\\textstyle \\frac{3}{4}}}(x,\\omega ) thesum of whose series representation is unknown.", "As in the other casesthe plot on the right is for small values of ω\\omega and displaystwo consecutive approximants of order 7 and 8, respectively, whilethe plot on the left extends to large values of ω\\omega anddisplays several approximants.The structure of the plots of the truncated series are qualitatively the same in the case $\\Delta = {\\textstyle \\frac{3}{4}}$ , as they are in the case $\\Delta = 0$ .", "Furthermore, in a completely analogous way to the case $\\Delta =0$ , for small values of $\\omega $ and $x$ also the series representation of $G_{\\Delta = {\\textstyle \\frac{3}{4}}}(x,\\omega )$ converges rapidly to some well defined function while approaching the region of the bending it starts oscillating.", "Hence we are led to conclude that we should be able to retrieve an analytically defined solution of the Monge-Ampère equation for the symplectic potential which reduces to the Kronheimer metric on ${\\mathbb {F}}_2$ at $\\omega =0$ .", "It is a matter of finding some alternative way of summing the series by a smart change of variables or by means of some smart integral transform." ], [ "The Hybrid version of the Monge-Ampère equation", "The most promising setup to study the MA equation for the symplectic potential is the hybrid one.", "Working in the $s,t$ coordinates defined in eq.s (REF -REF ) and setting $\\mathcal {G}(v,w)\\,=\\,\\Gamma (t,s)$ , the equation (REF ) is transformed into the following one: $\\frac{1}{4} c e^{\\mathcal {C}} s t = \\frac{64\\mathcal {B}}{(s-t)^2}-\\frac{64 \\mathcal {A}}{(s-t)^4}$ where: $\\mathcal {C} &\\equiv & \\frac{8 (s+1) \\Gamma ^{(0,1)}(t,s)-8 (t+1) \\Gamma ^{(1,0)}(t,s)}{s-t}\\nonumber \\\\\\mathcal {B} &\\equiv & \\Gamma ^{(0,2)}(t,s) \\Gamma ^{(2,0)}(t,s) \\nonumber \\\\\\mathcal {A} &\\equiv & \\left(\\Gamma ^{(0,1)}(t,s)-\\Gamma ^{(1,0)}(t,s)+(s-t) \\Gamma ^{(1,1)}(t,s)\\right)^2$ It is an important observation that the term $\\mathcal {A}$ is the square of the constraint whose vanishing implies the orthotoric separation of the $s,t$ variables (see the last of eq.s (REF ).", "It is interesting to see how with this separation of variables, namely when $\\mathcal {A}=0$ , the differential equation (REF ) does indeed split in two equations, one for the $t$ variable, the other for the $s$ variable.", "On the other hand the equation for the $t$ variable implies the $\\mathbb {W}P[112]$ symplectic potential.", "The argument goes as follows.", "Generalizing the structure of the known solution for the case $\\mathbb {W}P[112]$ we introduce the following ansatz: $\\Gamma (s,t)\\, =\\, -\\frac{1}{3} (2 s+3) {\\Pi }(t)+\\mathcal {P}(s)+(s+3) \\left(\\mathcal {Q}(t)-\\frac{1}{16} (t+3) \\log (s+3)\\right)+s t \\text{Y}_1(s)+(s+t) \\text{Y}_2(s)$ where $\\Pi (t)$ is an unknown function of $t$ that we would like to identify with the symplectic potential of the exceptional divisor metric and $Y_{1,2}(s)$ are also two unknown functions of $s$ .", "On the other hand the other two functions entering the ansatz are integral differential functionals of $Y_{1,2}(s)$ and $\\Pi (t)$ , respectively : $\\mathcal {P}(s) &=& \\frac{1}{16} \\int \\left(-16 \\kappa _1-16 s^2 \\text{Y}_1^{\\prime }(s)-32 s \\text{Y}_2^{\\prime }(s)+s+3\\right)\\, ds \\\\\\mathcal {Q}(t) &=& \\frac{1}{3} (t+3) \\int \\frac{(2 t+3) \\Pi ^{\\prime }(t)-2 \\Pi (t)}{(t+3)^2} \\,dt+\\kappa _1 $ With these choices the term $\\mathcal {C}$ in eq.", "(REF ) splits into separate functions of different variables: $\\mathcal {C} &=& \\mathcal {U}(s)+T(t)\\nonumber \\\\T(t) &=& \\frac{8 \\left(3 (t+1) \\Pi ^{\\prime }(t)+2 (t+3) \\int \\frac{(2 t+3) \\Pi ^{\\prime }(t)-2\\Pi (t)}{(t+3)^2} \\, dt-4 \\Pi (t)\\right)}{3 (t+3)} \\nonumber \\\\\\mathcal {U}(s) &=& \\frac{1}{2} \\left(-16 s^2 \\text{Y}_1^{\\prime }(s)-16 s \\text{Y}_1^{\\prime }(s)-16 \\text{Y}_1(s)-16 s\\text{Y}_2^{\\prime }(s)-16 \\text{Y}_2^{\\prime }(s)+16 \\text{Y}_2(s)+s-2 \\log (s+3)+1\\right)\\nonumber \\\\$ On the other hand we find that $\\mathcal {A}\\, = \\,0$ while the $\\mathcal {B}$ -term factorises as follows: $\\mathcal {B} &=& \\mathcal {H}(t) J(s) \\\\\\mathcal {H}(t) &=& \\frac{4 \\Pi ^{\\prime \\prime }(t)}{t+3} \\\\J(s) &=& \\frac{16 s (s+3) \\text{Y}_1^{\\prime \\prime }(s)+32 (s+3) \\text{Y}_1^{\\prime }(s)+16 s \\text{Y}_2^{\\prime \\prime }(s)+48\\text{Y}_2^{\\prime \\prime }(s)-1}{s+3}$ In this way the solution of the MA equation reduces to the solution of two separate integral differential equations one in the $s$ variable, one in the $t$ -variable: $\\frac{\\lambda }{2}\\, t \\, \\exp [T(t)] \\, = \\,\\mathcal {H}(t) \\quad ; \\quad \\frac{\\mu }{2}\\, s \\, \\exp [\\mathcal {U}(s)] \\, = \\,J(s)$ We focus on the first in the variable $t$ .", "With rather simple manipulations it can be reduced to an ordinary differential equation of higher order, namely: $\\frac{8 (t+1) \\Pi ^{\\prime \\prime }(t)}{t+3}-\\frac{\\Pi ^{(3)}(t)}{\\Pi ^{\\prime \\prime }(t)}+\\frac{2t+3}{t^2+3 t} \\, = \\, 0$ which is a differential equation of the first order for the function $F(t)$ .", "Apart from an integration constant which is fixed by the topological constraints on the periods of Ricci form, the unique solution of eq.", "(REF ) is $F(t,0)$ corresponding to the geometry of ${\\mathbb {W}P}[112]$ .", "This shows that in order to impose a boundary function consistent with $\\alpha \\ne 0$ we need to modify the ansatz (REF ) in such a way as to introduce a certain $s,t$ -mixing." ], [ "Conclusions", "As we advocated in the introduction, the present paper is an illustration of the conjecture REF for which we have strong support from the fact that it is verified for the value $\\Delta =0$ of the parameter in the paradigmatic case of the $\\mathbb { C}^3/\\mathbb {Z}_4$ singularity resolution.", "Further numerical evidence emerges from the study of the power series solution of the Monge-Ampère equation in the symplectic potential formulation.", "This latter in its hybrid version seems to provide the most promising approach since different series expansions might be glued together to prolong the solution beyond the valleys of oscillations.", "Assuming that in due time our conjecture can be transformed into a proof, we would like to stress its relevance.", "According to our view point, Conjecture REF provides a precise mathematical relationship to realise the gauge/gravity correspondence in a proper way.", "The generalized Kronheimer construction fixes all the items of the gauge theory on the brane world-volume: field content, gauge group, flavor symmetries and interactions.", "As maintained by Conjecture REF , the same Kronheimer construction determines, via the Monge-Ampère equation, also the Ricci-flat Kähler metric to be used in the construction of the dual D3-brane solution of supergravity.", "If REF is proved we can say that, for the class of theories realised on D3 branes at $\\mathbb {C}^3/\\Gamma $ Calabi-Yau singularities, the McKay quiver determines uniquely both sides of the correspondence." ], [ "Acknowledgments", "We thank M. Graffeo and V. Peragine for help with the computations involved in Appendix A. M. B. would like to acknowledge illuminating discussions on quivers and dimers with D. Bufalini, S. Mancani, A. Antinucci and F. Riccioni." ], [ "The affine variety $\\mathbb {C}^3/{\\mathbb {Z}}_4$", "We study the quotient $\\mathbb {C}^3/{\\mathbb {Z}}_4$ as an affine variety, i.e., as a closed (in the Zariski topology) subset of an affine space $D$ cut by an ideal $I$ of the polynomial ring $x_1,\\dots ,x_D]$ .", "In particular, we show that $\\mathbb {C}^3/{\\mathbb {Z}}_4$ is not a (schematic) complete intersection." ], [ "We recall the notions of set-theoretic and schematic intersection.", "Definition A.1 Let $X \\subset D$ be an affine variety, and denote by $d$ its codimension in $D$ .", "$X$ is a set-theoretic complete intersection if it is cut by $d$ equations as a subset of $D$ .", "$X$ is a schematic complete intersection if it is cut by $d$ equations as an affine variety.", "In other terms, if $A(X)$ is the coordinate ring of $X$ (the ring of regular functions on $X$ ), then $A(X) = x_1,\\dots ,x_D]/I$ , where the minimal number of generators of the ideal $I$ is $d$ .", "It turns out that all quotients $3/\\Gamma $ , where $\\Gamma $ is a finite abelian subgroup of $\\operatorname{SL}_3($ , are set-theoretic complete intersections, and therefore so is the case for $\\mathbb {C}^3/{\\mathbb {Z}}_4$ .", "However, we shall not prove this fact here, and rather concentrate on proving that $\\mathbb {C}^3/{\\mathbb {Z}}_4$ is not a schematic complete intersection." ], [ "An affine toric variety $X$ can be expressed as $ X_\\sigma = \\operatorname{Specm}\\, S_\\sigma ] $ where $\\sigma \\subset N\\otimes \\widehat{R}$ is a strongly convex polyhedral cone, $N$ is a lattice, and $S_\\sigma $ is the semigroup $S_\\sigma =\\sigma ^\\vee \\cap M$ , with $ \\sigma ^\\vee $ the dual cone to $\\sigma $ ; $M$ is the dual of the lattice $N$ .", "Specm denotes the maximal spectrum, i.e., the set of maximal ideals of $S_\\sigma ]$ with the Zariski topology.", "Basically following [64], we delineate a procedure to find the equations for the affine toric variety $X$ .", "We remind that a Hilbert basis $\\mathcal {H}_\\sigma $ for the semigroup $S_\\sigma $ is a minimal set of generators for $S_\\sigma $ which contains the rational generators of the rays of $\\sigma ^\\vee $ .", "Define $D = \\# \\mathcal {H}_\\sigma $ .", "Then the elements of $\\mathcal {H}_\\sigma $ are related by $D-n$ relations, which generate an ideal $I_{\\sigma ,0}$ of $x_1,\\dots ,x_D]$ .", "Given two ideals $I$ , $J$ in a ring $R$ , the saturation of $I$ with respect to $J$ is defined as $I:J^\\infty = \\lbrace a\\in R \\, \\vert \\, a^NJ\\subset I \\ \\mbox{for} \\ N\\gg 0.\\rbrace $ Then one proves that the ideal $I_\\sigma $ of $X_\\sigma $ in $D$ is the saturation of $I_{\\sigma ,0}$ wih the respect to the ideal $K_\\sigma =(x_1\\cdots x_D) \\subset x_1,\\dots ,x_D].$ Remark A.1 If $I$ , $J$ are ideals in $x_1,\\dots ,x_D]$ , the affine variety corresponding to the ideal $I:J^\\infty $ is $ V(I:J^\\infty ) = \\overline{V(I)\\setminus V(J)}$ where the closure is taken in $D$ (in the Zariski topology), and for every ideal $L$ in $x_1,\\dots ,x_D]$ , $V(L)$ denotes the closed set in $D$ corresponding to $L$ ." ], [ "Now we check that $3/{\\mathbb {Z}}_4$ is not a schematic complete intersection, as noted in [65].", "Realizing $X$ as in equation (REF ) we can take for $\\sigma $ the cone with generators $(1,0,0)$ , $(-1,2,0)$ , $(0,-1,2)$ in the latttice $N={\\mathbb {Z}}^3$ .", "The dual cone $\\sigma ^\\vee $ has rational generators $(4,2,1),$ $ (0,2,1),$ $(0,0,1)$ in $M\\simeq {\\mathbb {Z}}^3$ .", "A Hilbert basis of $S_\\sigma $ is obtained by adding the lattice points $ (1,1,1), \\ (0,1,1), \\ (1,2,1), \\ (2,1,1), \\ (2,2,1), \\ (3,2,1) .$ Assigning variables $x_1,\\dots ,x_9$ to these lattice points we obtain that $I_{\\sigma ,0}$ is generated by the 6 equations $x_1x_8-x_9^2=0,\\quad x_2x_9^2-x_8^3=0, \\quad x_3x_9^2-x_7^2x_8 = 0, \\\\x_4x_9-x_7x_8=0, \\quad x_5x_9^2-x_7x_8^2=0, \\quad x_6x_9-x_8^2=0$ Saturating this ideal with respect to $K=(x_1\\cdots x_9)$ one sees that $I_\\sigma $ is generated by the 20 quadratic equations (the equation needed to cut $X$ from 9 with the correct schematic structure): $\\begin{array}{ccccc}x_8^2-x_6x_9; & x_7x_8-x_4x_9; & x_6x_8-x_2x_9; & x_4x_8-x_4x_5; & x_1x_8-x_9^2; \\\\x_6x_7-x_5x_9; & x_5x_7-x_3x_8; & x_4x_7-x_3x_9; & x_2x_7-x_5x_8 ; & x_6^2-x_2x_8 ; \\\\x_4x_6-x_5x_8; & x_1x_6-x_8x_9; & x_4x_5-x_3x_6; & x_1x_5-x_4x_9; & x_4^2-x_3x_8; \\\\x_2x_4-x_5x_6; & x_1-x_7x_9; & x_2x_3-x_5^2; & x_1x_3-x_7^2 ; &x_1x_2 - x_6x_9.\\end{array}$ These are a minimal set of generators.", "So $X$ is the intersection of 20 quadrics in 9.", "All these quadrics are singular along their intersection with a plane of codimension 3 (when their equation contains a square) or 4 (when their equation does not contain a square).", "The dimension of the singular locus is 6 and 5 respectively (not 5 and 4!)", "It may be interesting to see what variety does the ideal $I_{\\sigma ,0}$ describe.", "To this end one computes the primary decomposition of the ideal [66].", "This yields 5 ideals; one is radical, and coincides with $I_\\sigma $ , so that one component of the variety is $3/{\\mathbb {Z}}_4$ .", "The other ideals are generated by monomials, and correspond to (intersections of) coordinate planes of different dimensions, counted with multiplicities." ], [ "The orbifold $\\mathbb {S}^5/\\mathbb {Z}_4$", "Setting $s=-\\tfrac{2}{3}R^2$ with $r\\rightarrow \\infty $ in the metric (REF ), it is straightforward to verify that this takes the approximate form $\\text{ds}_{{\\rm tot} K_{{\\mathbb {W}P}[112]}}^2 \\stackrel{R\\rightarrow \\infty }{\\approx } dR^2 + R^2 \\text{ds}^2_{X_5}$ at leading order in $R$ .", "Since the metric is Ricci-flat Kähler, and it takes the form of a cone over a five-dimensional space, it follows that locally the five-dimensional metric $\\text{ds}^2_{X_5}$ is Sasaki-Einstein.", "Below we shall show that globally, this is precisely a Sasaki-Einstein metric on the orbifold $\\mathbb {S}^5/\\mathbb {Z}_4$ .", "In the coordinates used in the paper, the five-dimensional metric reads $\\text{ds}^2_{X_5}&=&- \\frac{t}{6} \\left(\\sin ^2\\theta {d\\phi }^2+d\\theta ^2\\right)-\\frac{dt^2}{2t (2 t+3)}\\left.-\\frac{2 t (2 t+3)}{9} \\left[ \\frac{d\\chi }{3}-\\frac{1}{2} [(1-\\cos \\theta ){d\\phi } +{d\\tau }]\\right]^2\\right.\\nonumber \\\\&& +\\frac{4}{9}\\left[\\left(\\frac{t}{3}+1\\right) {d\\chi }-\\frac{1}{2}t [(1-\\cos {\\theta }){d\\phi } +{d\\tau }]\\right]^2$ After introducing the new coordinate $\\sigma \\in [0,\\frac{\\pi }{2}]$ as $t = - \\frac{3}{2}\\sin ^2\\sigma $ it becomes $\\text{ds}^2_{X_5}&=& d\\sigma ^2 + \\frac{\\sin ^2\\sigma }{4} \\left(\\sin ^2\\theta {d\\phi }^2+d\\theta ^2\\right)+\\frac{\\sin ^2\\sigma \\cos ^2\\sigma }{4} \\left[ \\frac{2}{3}d\\chi - d\\tau -d\\phi + \\cos \\theta {d\\phi }\\right]^2\\nonumber \\\\&& +\\frac{1}{9}\\left[2 d\\chi -\\frac{3}{2}\\sin ^2\\sigma \\left( \\frac{2}{3}d\\chi - d\\tau -d\\phi + \\cos \\theta {d\\phi }\\right)\\right]^2$ and one can check that this is indeed locally a Sasaki-Einstein metric, where the first line is a Kähler-Einstein metric.", "In order to uncover the relation with the metric on the five-sphere $\\mathbb {S}^5$ , it is convenient to redefine the angular coordinates as $\\widetilde{\\phi }= \\phi \\, , \\qquad \\beta = \\tfrac{2}{3}\\chi - \\phi - \\tau \\, , \\qquad \\psi = 2\\chi \\, ,$ with inverse $\\phi = \\widetilde{\\phi }\\, , \\qquad \\tau = \\tfrac{1}{3}\\psi - \\phi - \\beta \\, , \\qquad \\chi = \\tfrac{1}{2}\\psi \\, ,$ where, after performing the change of coordinates, we can drop the tilde on $\\widetilde{\\phi }$ and simply continue to denote this as $\\phi $ .", "The metric then reads $\\text{ds}^2_{X_5}&=& d\\sigma ^2 + \\frac{\\sin ^2\\sigma }{4} \\left(\\sin ^2\\theta {d\\phi }^2+d\\theta ^2\\right)+\\frac{\\sin ^2\\sigma \\cos ^2\\sigma }{4} \\left( d\\beta + \\cos \\theta d\\phi \\right)^2\\nonumber \\\\&& +\\frac{1}{9}\\left[d\\psi -\\frac{3}{2}\\sin ^2\\sigma \\left( d\\beta + \\cos \\theta {d\\phi }\\right)\\right]^2$ It is well-known (and simple to verify) that taking $\\phi \\sim \\phi + 2\\pi $ and $\\beta \\sim \\beta +4\\pi $ , with $\\theta \\in [0,\\pi ]$ and $\\sigma \\in [0,\\tfrac{\\pi }{2}]$ , the first line is the standard Einstein metric on $\\mathbb {P}^2$ .", "Moreover, with $\\psi \\sim \\psi + 6\\pi $ , the five-dimensional metric is the round metric on $\\mathbb {S}^5$ , viewed as the total space of a circle bundle $\\mathbb {S}^5 \\overset{\\pi }{\\longrightarrow } \\mathbb {P}^2$ , normalised so to obey the equation $R_{ij}^{X_5} & = 4 g_{ij}^{X_5}$ On the other hand, we are not free to chose the ranges of the coordinates, but these are inherited from the ranges of the original coordinates $(\\phi ,\\tau ,\\chi )$ , fixed in (REF ).", "From the change of coordinates (REF ), it followsThe periodicities of $\\phi $ and $\\psi $ are obvious.", "The simplest way to determine the periodicity of $\\beta $ is by demanding that the total volume of the three-torus with coordinates $(\\phi ,\\chi ,\\tau )$ is preserved by the coordinate transformation (REF ).", "that we must enforce the following periodicities: $\\phi \\sim \\phi +2\\pi \\, , \\qquad \\beta \\sim \\beta + 2\\pi \\, , \\qquad \\psi \\sim \\psi +3\\pi \\, \\, ,$ thus suggesting that globally the space is an orbifold $\\mathbb {S}^5/\\mathbb {Z}_4$ .", "However, the precise form of the $\\mathbb {Z}_4$ action is not transparent from these considerations.", "Next, we will show that the $\\mathbb {Z}_4$ action is precisely the correct action inherited from the $\\mathbb {C}^3/\\mathbb {Z}_4$ orbifold singularity.", "We start with three standard complex coordinates $(z_1,z_2,z_3)$ on $\\mathbb {C}^3$ and consider the following change of coordinates $z_1 = R \\sin \\sigma \\cos \\tfrac{\\theta }{2} e^{{\\rm i}\\left(-\\frac{\\beta +\\phi }{2} + \\frac{\\psi }{3}\\right)}\\, , \\qquad z_2 = R \\sin \\sigma \\sin \\tfrac{\\theta }{2} e^{{\\rm i}\\left(\\frac{\\phi -\\beta }{2} +\\frac{\\psi }{3}\\right)}\\, , \\qquad z_3 = R \\cos \\sigma e^{{\\rm i}\\frac{\\psi }{3}}\\, ,$ where $|z_1|^2+|z_2|^2+|z_3|^2& =R^2$ It can be checked that the metric induced at $R=1$ , ${\\rm d s}^2_5 &= (|dz_1|^2+|dz_2|^2+|dz_3|^2) |_{R=1}$ coincides with (REF ), and more generally, the six-dimensional metric is the cone ${\\rm d s}^2_{\\rm cone} =dR^2+R^2 \\text{ds}^2_{X_5}$ .", "To see that these are good coordinates on $\\mathbb {C}^3$ we can also view it as $\\mathbb {C}^3\\simeq \\mathbb {R}^6 = \\mathbb {R}^2\\oplus \\mathbb {R}^2\\oplus \\mathbb {R}^2$ , by defining $z_1 = \\rho _1 e^{{\\rm i}\\varphi _1}\\, , \\qquad z_2 = \\rho _2 e^{{\\rm i}\\varphi _2}\\, , \\qquad z_3 = \\rho _3 e^{{\\rm i}\\varphi _3}$ so that the induced metric reads ${\\rm d s}^2_{\\rm cone} &= |dz_1|^2+|dz_2|^2+|dz_3|^2 =d\\rho _1^2 + \\rho _1^2 d\\varphi _1^2+d\\rho _2^2 + \\rho _2^2 d\\varphi _2^2+d\\rho _3^2 + \\rho _3^2 d\\varphi _3^2$ where for $\\mathbb {C}^3$ the ranges of the coordinates are now $\\rho _i\\in [0,+\\infty )$ and $\\varphi _i \\sim \\varphi _i + 2\\pi $ , for $i=1,2,3$ .", "Defining $y_i=\\tfrac{1}{2}\\rho _i^2$ , this gives the standard metric in symplectic-toric coordinates, with Kähler form $\\mathbb {K}_{\\mathbb {C}^3} = d y_1\\wedge d\\varphi _1+ d y_2\\wedge d\\varphi _2+d y_3\\wedge d\\varphi _3$ From this it is clear that the $U(1)^3$ torus action on $\\mathbb {C}^3$ $(z_1,z_2,z_3) \\rightarrow (\\lambda _1 z_1,\\lambda _2z_2,\\lambda _3 z_3)$ with $|\\lambda _i|=1$ , $\\lambda _i=e^{{\\rm i}c_i}$ descends on the $\\varphi _i$ coordinates to $(\\varphi _1,\\varphi _2,\\varphi _3) \\rightarrow (\\varphi _1+c_1,\\varphi _2+c_2,\\varphi _3+c_3)$ Notice that on $\\mathbb {C}^3$ the periodicities of the two sets of angular coordinates are consistent with the change of coordinates $\\varphi _1 = -\\frac{\\beta +\\phi }{2} + \\frac{\\psi }{3}\\, , \\qquad \\varphi _2 = \\frac{-\\beta +\\phi }{2} + \\frac{\\psi }{3}\\, \\, , \\qquad \\varphi _3 = \\frac{\\psi }{3}\\, ,$ with inverse $\\phi = -\\varphi _1+\\varphi _2 , \\qquad \\beta = -\\varphi _1-\\varphi _2 +2\\varphi _3\\, \\, , \\qquad \\psi = 3\\varphi _3\\, ,$ as we have $(2\\pi )^3 = \\int d\\varphi _1 d\\varphi _2 d\\varphi _3 = \\frac{1}{6} \\int d\\phi d\\beta d\\psi = \\frac{1}{6} (2\\pi )(4\\pi )(6\\pi )$ Let us now reformulate the standard orbifold action of a discrete group $\\Gamma \\in SU(3)$ on $\\mathbb {C}^3$ with the corresponding action on $\\mathbb {S}^5$ in the above $(\\phi , \\beta ,\\psi )$ coordinates.", "We will restrict to $\\Gamma =\\mathbb {Z}_n$ for simplicity.", "In the $(z_1,z_2,z_3)$ coordinates on $\\mathbb {C}^3$ , a $\\mathbb {Z}_n$ orbifold action is defined by the identification $(z_1,z_2,z_3)\\sim (\\omega _n^{a_1} z_1,\\omega _n^{a_2} z_2,\\omega _n^{a_3} z_3)$ where $\\omega _n$ is a $n$ -th root of unity.", "The requirement that $\\mathbb {Z}_n\\in SU(3)$ implies that $a_1+a_2+a_3=0\\quad \\mathrm {mod}\\quad n$ Using (REF ), the above orbifold action implies the following identification in the $\\varphi _i$ coordinates $(\\varphi _1,\\varphi _2,\\varphi _3) \\sim (\\varphi _1+a_1\\tfrac{2\\pi }{n},\\varphi _2+a_2\\tfrac{2\\pi }{n},\\varphi _3+a_3\\tfrac{2\\pi }{n})$ and, equivalently, the following identification in the $(\\phi , \\beta ,\\psi )$ coordinates $(\\phi , \\beta ,\\psi ) \\sim (\\phi +(-a_1+a_2)\\tfrac{2\\pi }{n},\\beta +(-a_1-a_2+2a_3)\\tfrac{2\\pi }{n},\\psi +3a_3\\tfrac{2\\pi }{n})$ The simplest example is the $\\mathbb {C}^3/\\mathbb {Z}_3$ orbifold, with $\\mathbb {Z}_3$ action on $\\mathbb {C}^3$ given by $ (z_1,z_2,z_3) \\sim ( e^ {{\\rm i}\\tfrac{2\\pi }{3}} z_1,e^ {{\\rm i}\\tfrac{2\\pi }{3}}z_2,e^ {{\\rm i}\\tfrac{2\\pi }{3}} z_3),$ which using (REF ) corresponds simply to $ \\psi \\sim \\psi + 2\\pi $ .", "In this case, the metric (REF ), taking $\\phi \\in [0,2\\pi ]$ , $\\beta \\in [0,4\\pi ]$ , is the metric on $\\mathbb {S}^5/\\mathbb {Z}_3$ .", "This space can also be viewed as the unit circle bundle inside $\\mathcal {O}_{\\mathbb {P}^2}(-3)$ , namely the total space of the canonical line bundle over $\\mathbb {P}^2$ .", "Let us now discuss our main example, the orbifold $\\mathbb {C}^3/\\mathbb {Z}_4$ .", "In the table below we summarise the action of the three non-trivial elements of $\\mathfrak {g} \\in \\mathbb {Z}_4$ , including the identifications both in the $(\\varphi _1,\\varphi _2,\\varphi _3)$ and the $(\\phi , \\beta ,\\psi )$ coordinates.", "$\\begin{array}{|c|c|c|c|}\\hline \\mathfrak {g} : (z_1,z_2,z_3) & \\lbrace a_1,a_2,a_3 \\rbrace & (\\varphi _1,\\varphi _2,\\varphi _3)\\sim & (\\phi , \\beta ,\\psi ) \\sim \\\\\\hline ({\\rm i},{\\rm i}, -1) & \\lbrace 1,1,2 \\rbrace & (\\varphi _1+\\tfrac{\\pi }{2},\\varphi _2+\\tfrac{\\pi }{2},\\varphi _3+\\pi ) & (\\phi , \\beta +\\pi ,\\psi +3\\pi )\\\\(-1,-1, 1) & \\lbrace 2,2,0 \\rbrace & (\\varphi _1+\\pi ,\\varphi _2+\\pi ,\\varphi _3) & (\\phi , \\beta +2\\pi ,\\psi )\\\\(-{\\rm i},-{\\rm i}, -1) & \\lbrace 3,3,2 \\rbrace & (\\varphi _1+\\tfrac{3\\pi }{4},\\varphi _2+\\tfrac{3\\pi }{4},\\varphi _3+\\pi )& (\\phi , \\beta +3\\pi ,\\psi +3\\pi )\\\\\\hline \\end{array}$ As we see, in either of these two sets of angular coordinates the identifications are not diagonal.", "In the coordinates $(\\phi , \\beta ,\\psi )$ the clearest identification is the action of (junior) element $\\lbrace 2,2,0 \\rbrace $ , which implies that the base space, with metric in the first line of (REF ), is $\\mathbb {P}^2/\\mathbb {Z}_2$ .", "The action of the (junior) element $\\lbrace 1,1,2 \\rbrace $ means that as $\\beta $ goes half way around its circle, the coordinate $\\psi $ goes once around the $\\psi $ -circle, with period $3\\pi $ .", "The action of the (senior) element $\\lbrace 3,3,2 \\rbrace $ is simply a consequence of the previous two.", "In order to clarify the orbifold action on $S^5$ , it is useful to adopt a set of angular coordinates in which the $\\mathbb {Z}_4$ action is diagonal.", "It is then simple to verify that this is achieved precisely by the original coordinates $(\\phi ,\\tau ,\\chi )$ defined in (REF ).", "We summarise this diagonal action in the table below, where for convenience we defined $\\gamma \\equiv \\tfrac{4}{3}\\chi $ .", "$\\begin{array}{|c|c|c|}\\hline \\mathfrak {g} : (z_1,z_2,z_3) & \\lbrace a_1,a_2,a_3 \\rbrace & (\\phi , \\tau ,\\gamma ) \\sim \\\\\\hline ({\\rm i},{\\rm i}, -1) & \\lbrace 1,1,2 \\rbrace & (\\phi , \\tau ,\\gamma +2\\pi )\\\\(-1,-1, 1) & \\lbrace 2,2,0 \\rbrace & (\\phi , \\tau +2\\pi ,\\gamma )\\\\(-{\\rm i},-{\\rm i}, -1) & \\lbrace 3,3,2 \\rbrace & (\\phi , \\tau +2\\pi ,\\gamma +2\\pi )\\\\\\hline \\end{array}$ This shows that the indeed, the $\\mathbb {Z}_4$ action on $\\mathbb {C}^3$ induces the correct $\\mathbb {Z}_4$ action on the asymptotic metric on $\\mathbb {S}^5$ .", "In order to further clarify the orbifold action on $\\mathbb {S}^5$ , it is convenient to rewrite the metric (REF ) in the form of a circle fibration over a base space, that turns out to be precisely $\\mathbb {W} P [112]$ .", "In particular, rearranging the terms in (REF ) we find $\\text{ds}^2_{X_5}& = \\widetilde{\\text{ds}}^2_{\\mathbb {W} P [112]}+ \\frac{1}{16} (1+3\\cos ^2\\sigma ) \\left[ d\\gamma + \\frac{2\\sin ^2\\sigma }{1+3\\cos ^2\\sigma } (d\\tau - \\cos \\theta d\\phi )\\right]^2$ whereInterestingly, precisely this metric was found in [67] as a limiting case of a more general one-parameter family of smooth metrics on $\\mathbb {F}_2$ , in the context of AdS$_5$ solutions of eleven-dimensional supergravity.", "See eq.", "(5.7) of this reference.", "$\\widetilde{\\text{ds}}^2_{\\mathbb {W} P [112]}&= d\\sigma ^2 + \\frac{1}{4}\\sin ^2\\sigma \\left(d\\theta ^2 + \\sin ^2\\theta d\\phi ^2 \\right) + \\frac{\\sin ^2 \\sigma \\cos ^2\\sigma }{1+3\\cos ^2\\sigma }\\left( d\\tau - \\cos \\theta d\\phi \\right)^2\\, ,$ which clearly displays the fact that $S^5/\\mathbb {Z}_4$ arises as the total space of a circle fibration over $\\mathbb {W} P [112]$ , equipped with the metric (REF ).", "We decorated this metric with a tilde to distinguish it from the different metric on $\\mathbb {W} P [112]$ , that we discuss in the main body of the paper, namely the metric (REF ) induced on the exceptional divisor by the Ricci-flat metric (REF ).", "Below we will rewrite the latter metric in different coordinates, to facilitate the comparison with the metric in (REF ).", "Let us discuss briefly how to see that the underlying (singular) variety to the metric defined in (REF ) is indeed $\\mathbb {W}P[112]$ .", "With the ranges of coordinates and periodicities $\\sigma \\in [0,\\frac{\\pi }{2}]$ , $\\theta \\in [0,\\pi ]$ , $\\phi \\in [0,2\\pi ]$ , $\\tau \\in [0,2\\pi ]$ we see that near to $\\sigma \\approx 0$ the metric develops an $\\mathbb {R}^4/\\mathbb {Z}_2$ singularity (it is a cone over the Lens space $\\mathbb {S}^3/\\mathbb {Z}_2$ ), while near to $\\sigma \\approx \\tfrac{\\pi }{2}$ , the space shrinks smoothly to $\\mathbb {S}^2 \\times \\mathbb {R}^2$ .", "Following a reasoning analogous to that in the main body of the paper, one can see that there exists only one non-trivial two-cycle $C_2\\quad \\Leftrightarrow \\quad \\lbrace \\theta = {\\rm constant}, ~\\phi = {\\rm constant}\\rbrace $ while the other two-cycle of $\\mathbb {F}_2$ , that would be defined by $C_1 \\Leftrightarrow \\lbrace t=t_{max}=-\\tfrac{3}{2}\\sin ^2\\sigma _{max}\\ne 0\\rbrace $ is shrunk to zero size in the above metricSince we have not established whether the above metric is Kähler, the simplest way to see this is probably to consider the explicit one-parameter familiy of metrics on $\\mathbb {F}_2$ desingularising (REF ), presented in [67].. From the metric (REF ) we now read off the connection one-form $\\widetilde{\\cal A} & \\equiv \\frac{2\\sin ^2\\sigma }{1+3\\cos ^2\\sigma } (d\\tau - \\cos \\theta d\\phi )$ whose associated first Chern class can be integrated on $C_2$ to give $\\frac{1}{2\\pi }\\int _{C_2} d \\widetilde{\\cal A} & = 2$ showing that indeed this is a connection on the unit circle bundle inside the canonical bundle of $\\mathbb {W} P [112]$ .", "We conclude this appendix by writing the metric on the exceptional divisor (REF ) induced by the Ricci-flat metric (REF ) in a form that makes more transparent the comparison with the above discussion.", "Using (REF ) we have that $\\text{ds}^2_{\\mathcal {ED}} &= \\frac{9}{4} \\left[ (1+\\cos ^2\\sigma ) d\\sigma ^2 +\\frac{1}{2}\\sin ^2\\sigma \\left(d\\theta ^2 + \\sin ^2\\theta d\\phi ^2 \\right) + \\frac{\\sin ^2 \\sigma \\cos ^2\\sigma }{1+\\cos ^2\\sigma }\\left( d\\tau - \\cos \\theta d\\phi \\right)^2\\right]$ from which the similarity with the metric (REF ) is apparent.", "For completeness, let us also display the behaviour of the orthotoric metric (REF ) near to the exceptional divisor.", "Setting $s=-3-\\rho ^2$ in (REF ), for $\\rho \\rightarrow 0$ we have $\\text{ds}_{{\\rm tot} K_{{\\mathbb {W}P}[112]}}^2& \\stackrel{\\rho \\rightarrow 0}{\\approx } \\text{ds}^2_{\\mathcal {ED}} + \\frac{3(1+\\cos ^2\\sigma ) }{8} \\left[ d\\rho ^2 + \\rho ^2 \\left[ d \\gamma + \\frac{2\\sin ^2\\sigma }{1+\\cos ^2\\sigma }(d\\tau - \\cos \\theta d\\phi )\\right]^2 \\right]$ where the angular variables $(\\phi ,\\tau ,\\gamma \\equiv \\tfrac{4}{3}\\chi )$ are precisely those defined in (REF ), which all have canonical $2\\pi $ -periodicities.", "This shows that the metric (REF ) is smooth in the neighborhood of the exceptional divisor $\\mathcal {ED}={\\mathbb {W}P}[112]$ , in particular locally it has the topology of ${\\mathbb {W}P}[112]\\times \\mathbb {C}^2$ .", "The connection one-form ${\\cal A} & \\equiv \\frac{2\\sin ^2\\sigma }{1+\\cos ^2\\sigma } (d\\tau - \\cos \\theta d\\phi )$ read off from (REF ) has first Chern class again given by $\\frac{1}{2\\pi }\\int _{C_2} d {\\cal A} & = 2$ as it should be.", "To summarise, in this appendix we have shown that the orbifold action of $\\mathbb {Z}_4$ on $\\mathbb {S}^5$ , induced by the $\\mathbb {C}^3/\\mathbb {Z}_4$ quotient, is not diagonal in the canonical coordinates where the Sasaki-Einstein metric on $\\mathbb {S}^5$ can be viewed as a $U(1)$ fibration over $\\mathbb {P}^2$ with its Kähler-Einstein metric.", "This action is diagonalised precisely by the coordinates $(\\phi ,\\tau ,\\tfrac{4}{3}\\chi )$ used in the main part of the paper, and adapting the metric to these coordinates, it takes the form of a $U(1)$ fibration over ${\\mathbb {W}P}[112]$ , with the non-Einstein metric (REF ).", "This is precisely the unit circle bundle in the canonical line bundle over ${\\mathbb {W}P}[112]$ .", "The metric on the exceptional divisor of the partial resolution, induced from the orthotoric Ricci-flat metric, is a similar, but manifestly different non-Einstein metric (REF ).", "Statement about conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest." ] ]
2105.11704
[ [ "Non-real zeros of derivatives" ], [ "Abstract A number of results are proved concerning non-real zeros of derivatives of real meromorphic functions.", "In particular, the paper supersedes the previous arxiv submission \"Non-real zeros of linear differential polynomials in real meromorphic functions\"." ], [ "Introduction", "This paper concerns non-real zeros of derivatives of real meromorphic functions in the plane, that is, meromorphic functions mapping $\\mathbb {R}$ into $\\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ .", "The case of real entire functions has seen extensive research [2], [3], [6], [13], [19], [20], [30], [33], motivated at least in part by the Wiman conjecture (proved in [3], [30], [33]) that if $f$ is a real entire function and $f$ and $f^{\\prime \\prime }$ have only real zeros, then $f$ belongs to the Laguerre-Pólya class consisting of locally uniform limits of real polynomials with real zeros.", "The following theorem combines results from [3], [25].", "Theorem 1.1 ([3], [25]) Let $f$ be a real meromorphic function of infinite order in the plane such that $f$ or $1/f$ has finitely many poles and non-real zeros.", "Then $f^{\\prime \\prime }/f^{\\prime }$ has infinitely many non-real zeros, that is, $f^{\\prime \\prime }$ has infinitely many non-real zeros which are not zeros of $f^{\\prime }$ .", "This result will be strengthened as follows.", "Theorem 1.2 Let $f$ be a real meromorphic function of infinite order in the plane, with finitely many non-real zeros and poles.", "Assume that $f = f_1/f_2$ , where the $f_j$ are real entire functions, with no common zeros, and that at least one of $f_1$ and $f_2$ has finite lower order.", "Then $f^{\\prime \\prime }/f^{\\prime }$ has infinitely many non-real zeros.", "Thoerem REF represents a fairly substantial improvement of Theorem REF , since under the hypotheses of the latter one of $f_1, f_2$ may be assumed to be a polynomial.", "The theorem is applicable, in particular, if $f$ is a real meromorphic function of infinite order in the plane, with finitely many non-real zeros and poles, for which the exponent of convergence of either the zeros or the poles of $f$ is finite.", "Theorem REF will be deduced from the next result: here, and subsequently, $H$ denotes the open upper half-plane $\\lbrace z \\in \\mathbb {C}: \\, {\\rm Im} \\, z > 0 \\rbrace $ .", "Theorem 1.3 Let $L$ be a real meromorphic function in the plane, with finitely many non-real poles, and assume that $L$ has a representation $L = h R_1 \\psi _1 + R_2 \\psi _2 ,$ in which: $h$ is a real transcendental entire function; the $R_j$ are real rational functions, with $R_1 \\lnot \\equiv 0$ ; each $\\psi _j$ satisfies either $\\psi _j \\equiv 1$ or $\\psi _j(H) \\subseteq H$ .", "Then $L + L^{\\prime }/L$ has infinitely many non-real zeros.", "To deduce Theorem REF from Theorem REF it will suffice to show that $L = f^{\\prime }/f$ has a representation (REF ): this is well known if $f$ is as in Theorem REF , in which case the Levin-Ostrovskii factorisation of $f^{\\prime }/f$ has $R_2 \\equiv 0$ [3], [25], [30] (see also Lemma REF ).", "The methods of Theorems REF and REF turn out also to be applicable to a strand initiated in [22], in which the second derivative $f^{\\prime \\prime }$ is replaced by $f^{\\prime \\prime } + \\omega f$ , with $\\omega \\in \\mathbb {R}\\setminus \\lbrace 0 \\rbrace $ .", "Here attention is necessarily restricted to the case $\\omega > 0$ , as illustrated by an example cited in [22]: for $a \\in \\mathbb {R}$ writing $f^{\\prime }(z)/f(z) = a + e^{-2az} $ makes $f$ , $1/f$ and $f^{\\prime \\prime }(z) - a^2 f(z) = e^{-4az} f(z)$ all zero-free.", "Theorem 1.4 Let $f$ be as in the hypotheses of Theorem REF , and let $\\omega $ be a positive real number.", "Then $f^{\\prime \\prime } + \\omega f$ has infinitely many non-real zeros.", "When $f$ is a real entire function of infinite order with finitely many non-real zeros, Theorem REF is not new [22], but the present proof is considerably simpler than that of [22] and the result substantially more general.", "For results on non-real zeros of $f^{\\prime \\prime }+\\omega f$ when $\\omega \\ge 0$ and $f$ has finite order, the reader is referred to [22], [24], [27] and [26].", "As with the proof of Theorem REF , Theorem REF will be deduced from a result involving functions of the form (REF ).", "Theorem 1.5 Let $L$ be as in the hypotheses of Theorem REF , and let $a, b$ be positive real numbers.", "Then $L^{\\prime } + a L^2+ b$ has infinitely many non-real zeros.", "The simple example $L(z)= \\tan z$ , $a = b = 1$ , shows that the requirement that $h$ is transcendental in (REF ) is not redundant in Theorem REF .", "The following property will play a pivotal role in the proofs of Theorems REF and REF .", "Definition 1.1 A transcendental meromorphic function $L$ in the plane has the UHWV property if there exist $\\tau , \\gamma $ with $1/2 < \\tau < \\gamma < 1$ , an unbounded subset $E_1 $ of $[1, + \\infty )$ and a function $N(r): E_1 \\rightarrow (1, + \\infty )$ satisfying the following: (A) $\\lim _{r \\rightarrow + \\infty , r \\in E_1} N(r) = + \\infty $ ; (B) for each $r \\in E_1$ there exists $z_0 = z_0(r)$ with $|z_0| = r, \\quad N(r)^{-\\tau } < \\arg z_0 < \\pi - N(r)^{-\\tau } ,$ such that, uniformly as $r \\rightarrow + \\infty $ in $E_1$ , $L (z) \\sim L(z_0) \\left( \\frac{z}{z_0} \\right)^{N(r)} \\quad \\hbox{and} \\quad N(r)^{1/2} = o \\left( \\log ^+ |L(z)| \\right)$ on $Q_r = \\left\\lbrace z \\in \\mathbb {C}: \\,\\left| \\log \\frac{z}{z_0} \\right| \\le N(r)^{-\\gamma } \\right\\rbrace .$ Here UHWV stands for ”upper half-plane Wiman-Valiron” and standard results from the Wiman-Valiron theory [11] imply that if $L$ is a real transcendental entire function then $L$ has the UHWV property, with $N(r)$ the central index (see Lemma REF ).", "Theorem 1.6 Let $L$ be a real transcendental meromorphic function in the plane with finitely many non-real poles and assume that $L$ has the UHWV property.", "Then $L+L^{\\prime }/L$ has infinitely many non-real zeros.", "To prove Theorem REF it will suffice to show that $L$ has the UHWV property and apply Theorem REF directly.", "It is not clear whether, under the hypotheses of Theorem REF , $L^{\\prime }+aL^2 + b$ automatically has non-real zeros when $a, b > 0$ , and the proof of Theorem REF will use the representation (REF ) alongside the UHWV property.", "The next focus of the present paper is the general problem of classifying all real meromorphic functions in the plane which, together with some of their derivatives, have only real zeros and poles [14], [16], [17].", "In this direction, the following conjecture was advanced in [14].", "Conjecture 1.1 ([14]) Let $f$ be a real transcendental meromorphic function in the plane with at least one pole, and assume that all zeros and poles of $f$ , $f^{\\prime }$ and $f^{\\prime \\prime }$ are real, and that all poles of $f$ are simple.", "Then $f$ satisfies $f(z) = C \\tan (az+b) + Dz + E, \\quad a, b, C, D, E \\in \\mathbb {R}.$ If $f$ is allowed multiple poles then there are further examples for which $f$ , $f^{\\prime }$ and $f^{\\prime \\prime }$ have only real zeros and poles [15].", "Results from [14], [18], [21], [23], [32] show that the conjecture is true if, in addition, $f^{\\prime }$ omits some finite value.", "Furthermore, Theorem REF and [24] together show that there are no functions $f$ satisfying the hypotheses of Conjecture REF such that either of the following holds: $f$ has infinite order and the zeros or poles of $f$ have finite exponent of convergence; $f$ has finite order and infinitely many poles but finitely many zeros.", "The conjecture was also proved in [28] for real transcendental meromorphic functions in the plane which map the open upper half-plane $H$ into itself.", "All zeros and poles of such functions are automatically real and simple and interlaced [29]: that is, between any two consecutive poles of $f$ there is a zero, and between consecutive zeros of $f$ lies a pole (this follows from a consideration of residues for $f$ and $1/f$ ).", "Theorem 1.7 Let $f$ be a real transcendental meromorphic function in the plane with infinitely many zeros and poles, all real, simple and interlaced.", "If $f^{\\prime \\prime }$ has finitely many non-real zeros then $f(z) = Az+ B + \\frac{R(z)e^{icz}-1}{A_1R(z)e^{icz} - \\overline{A_1} } ,$ where $A, B, c$ and $A_1$ are constants with $A, B, c$ real and $A_1 \\in H$ , while $R$ is a rational function with all its zeros in $H$ and $|R(x)| = 1$ for all real $x$ .", "If $f^{\\prime \\prime }$ has only real zeros then $f$ is as in (REF ).", "In particular, Conjecture REF is true under the additional hypothesis that $f$ has infinitely many zeros and poles, all simple and interlaced.", "Theorem REF will be deduced from [28] and the following result involving real meromorphic functions with real zeros and poles such that, with finitely many exceptions, all poles are simple and adjacent poles are separated by at least one zero.", "Theorem 1.8 Let $U$ be a real meromorphic function in the plane, all but finitely many of whose zeros are real.", "Assume further that $U$ has infinitely many poles $X$ , all but finitely many of which are real and simple and have a corresponding real zero $Y $ of $U$ with $X < Y$ and $U(x) \\ne \\infty $ on $(X, Y)$ .", "Then $U$ satisfies the following.", "(i) $U$ has a representation $U = S \\psi ,$ where $S$ is a real meromorphic function in the plane with finitely many poles and $\\psi (H) \\subseteq H$ .", "(ii) If $S$ has infinite order then $U^{\\prime \\prime }/U^{\\prime }$ has infinitely many non-real zeros.", "(iii) If $S = Re^P$ , with $R$ a real rational function and $P$ a non-constant polynomial, then $U^{(m)} $ has infinitely many non-real zeros, for each $m \\ge 2$ .", "Part (i) is not new, but its inclusion is convenient for the statement and proof of parts (ii) and (iii): the standard construction, which determines $\\psi $ up to a rational factor, is outlined in Lemma REF .", "If $S$ is transcendental with finitely many zeros and poles in (REF ), then either (ii) or (iii) is applicable, although Theorem REF says nothing about the case where $S$ has finite order and infinitely many zeros.", "Simple examples such as $\\cot z$ show that Theorem REF (iii) fails for $m=2$ if $P$ is constant: for an example not of the form (REF ) set $V(z) = z \\cot z, \\quad V^{\\prime \\prime }(z) = -2 \\operatorname{cosec}^2 z + 2z \\operatorname{cosec}^2 z \\cot z= 2 ( z - \\tan z ) \\operatorname{cosec}^2 z \\cot z .$ Since the iterates of $\\tan z$ converge to 0 on $\\mathbb {C}\\setminus \\mathbb {R}$ , all fixpoints of $\\tan z$ are real, and so are all zeros of $V^{\\prime \\prime }$ .", "This paper is organised as follows.", "After preliminary considerations in Sections and , Theorem REF is proved in Section .", "The UHWV property is discussed further in Section , the proofs of Theorems REF and REF appearing in Section REF .", "Next, Theorem REF is proved, and Theorem REF is deduced from it, in Section .", "Finally, Theorem REF is established in Section , and Theorem REF in Section ." ], [ "Preliminary lemmas", "Lemma 2.1 There exists a positive constant $c_0$ such that if $\\psi : H \\rightarrow H$ is analytic then, for $ r \\ge 1$ and $\\theta \\in (0,\\pi )$ , $\\frac{|\\psi (i)| \\sin \\theta }{5 r} < |\\psi (re^{i\\theta })| <\\frac{5r |\\psi (i)|}{\\sin \\theta }\\quad \\hbox{and} \\quad \\left| \\frac{ \\psi ^{\\prime }( re^{i \\theta } ) }{ \\psi ( re^{i \\theta } ) } \\right|\\le \\frac{c_0}{r \\sin \\theta } .\\quad $ Both of these estimates are standard: the first is essentially just Schwarz' lemma [29], while the second follows from Bloch's theorem applied to $\\log \\psi $ .", "$\\Box $ Lemma 2.2 ([5]) Let $\\Omega $ be a plane domain.", "Let $\\mathcal {L}$ be the family of all analytic functions $L$ on $\\Omega $ such that $\\Psi _2(L) + 1 = L^{\\prime } + L^2 + 1$ has no zeros on $\\Omega $ .", "Then $\\mathcal {L}$ is normal.", "Lemma REF is a special case of [5].", "$\\Box $ Lemma 2.3 ([7]) Let $1 < r < R < + \\infty $ and let the function $g$ be meromorphic in $|z| \\le R$ .", "Let $I(r)$ be a subset of $[0, 2 \\pi ]$ of Lebesgue measure $\\mu (r)$ .", "Then $\\frac{1}{2 \\pi } \\int _{I(r)}\\log ^+ | g(re^{i \\theta })| \\, d \\theta \\le \\frac{ 11 R \\mu (r)}{R - r} \\left( 1 + \\log ^+ \\frac{1}{\\mu (r)} \\right)T(R, g).$ Lemma 2.4 ([10]) Let $S(r)$ be an unbounded positive function on $[1, + \\infty )$ which is non-decreasing and continuous from the right.", "Let $A > 1, B > 1$ and $G = \\lbrace r \\ge 1 : S(Ar) \\ge B S(r) \\rbrace $ .", "Then the upper logarithmic density of $G$ satisfies $\\overline{{\\rm logdens }} \\, G = \\limsup _{r \\rightarrow \\infty } \\left( \\frac{1}{\\log r} \\, \\int _{[1, r] \\cap G} \\, \\frac{1}{t} \\, dt \\right)\\le \\left( \\frac{\\log A}{\\log B} \\right) \\limsup _{r \\rightarrow \\infty } \\frac{ \\log ^+ S(r)}{\\log r } .$ The next lemma proves Theorem REF (i).", "Lemma 2.5 Let $U$ be as in the hypotheses of Theorem REF (i).", "Then $U$ has a representation (REF ) in which $S$ is a real meromorphic function with finitely many poles in the plane and $\\psi (H) \\subseteq H$ .", "Proof.", "This is the standard Levin-Ostrovskii construction [3], [30].", "By assumption, all but finitely many poles of $U$ are real and simple, and all but finitely many of these may be labelled $x_k$ , in such a way that $x_k < x_{k+1}$ and there is a zero $y_k$ of $U$ with $x_k < y_k < x_{k+1}$ .", "If $|k|$ is large, say $|k| \\ge k_0$ , then $x_k$ and $x_{k+1}$ have the same sign and the product $\\psi (z) = \\prod _{|k| \\ge k_0} \\frac{1-z/y_k}{1-z/x_k} ,$ converges by the alternating series test, and maps $H$ into $ H$ because, for $z \\in H$ , $\\arg \\psi (z)= \\sum _{k \\in K} \\arg \\frac{y_k-z}{x_k-z} \\in (0, \\pi ).$ This proves Lemma REF , with $\\psi $ determined up to a rational factor.", "$\\Box $ Lemma 2.6 Let $m \\ge 2$ be an even integer and let $Q_m(y) = \\sum _{j=0}^{m} { m \\atopwithdelims ()j } (j+1)!", "\\, y^{j} .$ Then there exists $d_m > 0$ such that $Q_m(y) \\ge d_m \\max \\lbrace 1, \\, y^{m} \\rbrace $ for all $y \\in \\mathbb {R}$ .", "Proof.", "Since $Q_m(0) = 1$ it suffices to show that $Q_m(y) > 0$ for $y \\ne 0$ , this being obvious if $y > 0$ .", "For $y < 0$ write $x = 1/y < 0$ and, using Leibniz' rule and the fact that $m$ is even, $P(x) &=& e^{-x} x^{-2} = \\frac{1}{x^2} - \\frac{1}{x} + \\frac{1}{2!}", "- \\frac{x}{3!}", "+ \\ldots + \\frac{x^m}{(m+2)!}", "-\\frac{x^{m+1}}{(m+3)!}", "+ \\dots , \\\\0 &<&\\frac{(m+1)!", "}{x^{m+2}} - \\frac{m!", "}{x^{m+1}}+ \\frac{m!}{(m+2)!}", "- \\frac{(m+1)!}{(m+3)!}", "\\, x + \\dots \\\\&=& P^{(m)}(x)= \\sum _{j=0}^{m} { m \\atopwithdelims ()j } (-1)^{m-j} e^{-x} (-1)^j (j+1)!", "\\, x^{-2-j}\\\\&=& \\sum _{j=0}^{m} { m \\atopwithdelims ()j } e^{-x} (j+1)!", "\\, x^{-2-j} = x^{-2} e^{-x} Q_m(y) .$ $\\Box $" ], [ "Transcendental singularities of the inverse function", "Throughout this section let $G$ be a transcendental meromorphic function in the plane.", "Suppose first that $G(z) \\rightarrow a \\in \\mathbb {C}\\cup \\lbrace \\infty \\rbrace $ as $z \\rightarrow \\infty $ along a path $\\gamma $ ; then the inverse $G^{-1}$ is said to have a transcendental singularity over the asymptotic value $a$  [1], [31].", "If $a \\in \\mathbb {C}$ then for each $\\varepsilon > 0$ there exists a component $\\Omega = \\Omega ( a, \\varepsilon , G)$ of the set $\\lbrace z \\in \\mathbb {C}: |G(z) - a | < \\varepsilon \\rbrace $ such that $\\gamma \\setminus \\Omega $ is bounded, these components being called neighbourhoods of the singularity [1].", "Two paths $\\gamma , \\gamma ^{\\prime }$ on which $G(z) \\rightarrow a$ determine distinct singularities if the corresponding components $\\Omega ( a, \\varepsilon , G)$ , $\\Omega ^{\\prime }( a, \\varepsilon , G)$ are disjoint for some $ \\varepsilon > 0$ .", "The singularity is called direct [1] if $\\Omega ( a, \\varepsilon , G)$ , for some $\\varepsilon > 0$ , contains finitely many zeros of $G-a$ , and indirect otherwise.", "A transcendental singularity will be referred to as lying in an open set $D$ if $\\Omega (a, \\varepsilon , G) \\subseteq D$ for all sufficiently small positive $\\varepsilon $ .", "Transcendental singularities over $\\infty $ may be classified using $1/G$ .", "The following lemmas from [22], [27] link asymptotic values approached on paths in $H$ with the growth of the Tsuji characteristic $ \\mathfrak {T} (r, g) $ [3], [8], [34], which is defined for meromorphic functions $g$ on the closed upper half-plane $\\overline{H} =\\lbrace z \\in \\mathbb {C}: \\mathrm {Im} \\, z \\ge 0 \\rbrace $ .", "Lemma 3.1 ([27], Lemma 2.2) Let $L \\lnot \\equiv 0$ be a real meromorphic function in the plane such that $\\mathfrak {T}(r, L) = O( \\log r )$ as $r \\rightarrow \\infty $ , and define $F$ by $F(z) = z-1/L(z)$ .", "Assume that at least one of $L$ and $1/L$ has finitely many non-real poles.", "Then there exist finitely many $\\alpha \\in \\mathbb {C}$ such that $F(z)$ or $L(z)$ tends to $\\alpha $ as $z$ tends to infinity along a path in $\\mathbb {C}\\setminus \\mathbb {R}$ .", "Lemma 3.2 ([22], Lemma 2.4) Let $G$ be a meromorphic function in the plane such that $\\mathfrak {T}(r, G) = O( \\log r )$ as $r \\rightarrow \\infty $ .", "Then there is at most one direct transcendental singularity of $G^{-1}$ lying in $H$ .", "The following proposition is a stronger version of [22], with a simpler proof, whih will occupy the remainder of this section.", "Here $B(a, r)$ denotes the open ball of centre $a \\in \\mathbb {C}$ and radius $r > 0$ .", "Proposition 3.1 Suppose that $R \\in (0, + \\infty )$ and the transcendental meromorphic function $G$ has no asymptotic values $w$ with $0 < |w| < R < \\infty $ , and finitely many critical values $w$ with $|w| < R$ .", "Let $A$ be a component of the set $G^{-1}(B(0, R))$ .", "Then the number of zeros of $G$ in $A$ , counting multiplicities, plus the number of transcendental singularities of $G^{-1}$ over 0, lying in $A$ , exceeds by at most 1 the number of zeros of $G^{\\prime }$ in $A$ , again counting multiplicities.", "Proof.", "It may be assumed that there exists a component $A$ of $G^{-1}(B(0, R))$ which contains a finite number, $M$ say, of zeros of $G^{\\prime }$ , counting multiplicities, but also contains zeros $u_1, \\ldots , u_p$ of $G$ , repeated according to multiplicity, as well as $q$ pairwise disjoint neighbourhoods $\\Omega _j(0, s, G)$ of transcendental singularities of $G^{-1}$ over 0, where $s > 0$ is small and $M+1 \\le p + q < \\infty $ .", "It is not assumed at this stage that there are no other zeros of $G$ , nor other transcendental singularities of $G^{-1}$ over 0, lying in $A$ , nor even that the number of these is finite.", "Choose points $v_j \\in \\Omega _j(0, s, G)$ , for $j=1, \\ldots , q$ .", "Then $u_1, \\ldots , u_p, v_1 , \\ldots , v_q$ may be joined to each other by paths in $A$ and so all lie in a compact connected subset of $A$ on which $|G(z)| \\le S_1$ , and hence in a component $B \\subseteq A$ of $G^{-1}(B(0,S_2))$ , for some $S_1, S_2$ with $s < S_1 < S_2 < R$ .", "These observations show that it is enough to prove that $p+q \\le M+1$ when $G$ has no critical or asymptotic values $w$ with $|w| = R$ .", "Let $w_1, \\ldots , w_N$ be the critical values of $G$ with $0 < |w| < R$ .", "Join each $w_j$ to a point $w_j^*$ on $|w| = R$ by a straight line segment $\\lambda _j$ in the annulus $2s < |w| \\le R$ , in such a way that these $\\lambda _j$ are pairwise disjoint; if the $w_j$ have distinct arguments modulo $2 \\pi $ , the $\\lambda _j$ may be taken to be radial segments, while if repetition occurs the segments may be rotated slightly about $w_j$ .", "Let $E_0 = B(0, R)$ and, for $m=1, \\ldots , N$ , set $E_m = E_{m-1} \\setminus \\lambda _m =E_0 \\setminus \\left( \\bigcup _{j=1}^m \\lambda _j \\right) .$ Since $E_N \\setminus \\lbrace 0 \\rbrace $ contains no asymptotic or critical values of $G$ , a straightforward modification, almost identical to that in [22], of a standard argument from [31] shows that every component $C$ of $G^{-1}(E_N)$ is simply connected, and contains either no zeros of $G$ and one transcendental singularity of $G^{-1}$ over 0, or exactly one point at which $G(z) = 0$ , which may be a multiple zero.", "This is accomplished by deleting from the half-plane ${\\rm Re} \\, v < \\log R$ all pre-images of the $\\lambda _j$ under $e^v$ , and considering $\\phi (v) =G^{-1}(e^v)$ on the resulting simply connected domain $U_0$ , the two possible conclusions for $C$ corresponding to whether or not $\\phi $ is univalent on $U_0$ .", "To prove Proposition REF it now suffices to establish the following lemma.", "Lemma 3.3 If $m \\in \\lbrace 0, \\ldots , N \\rbrace $ and $C$ is a component of $G^{-1}(E_m)$ contained in $A$ , let $Z_m(C)$ be the number of zeros of $G$ in $C$ , counting multiplicities, plus the number of neighbourhoods $\\Omega _j(0, s, G)$ contained in $C$ , and let $Y_m (C)$ be the number of zeros of $G^{\\prime }$ in $C$ , again counting multiplicities.", "Then $Z_m(C) \\le 1 + Y_m(C) .$ Proof.", "The lemma will be proved by backwards induction, and (REF ) clearly holds when $m=N$ .", "Now suppose that $0 < m \\le N$ , and that (REF ) holds whenever $C$ is a component of $G^{-1}(E_m)$ contained in $A$ .", "Let $D$ be a component of $G^{-1}(E_{m-1})$ contained in $A$ ; the idea of the proof is to delete from $D$ pre-images of $\\lambda _m$ , thus leaving residual components of $G^{-1}(E_{m})$ , to each of which the induction hypothesis can be applied.", "Take all points $\\zeta _j$ in $D$ with $G(\\zeta _j) = w_{m}$ ; each pre-image of $\\lambda _m$ in $D$ contains at least one $\\zeta _j$ .", "If $\\zeta _j$ is not a critical point, continuation of $G^{-1}$ along $\\lambda _{m}$ gives a path $\\sigma _j$ from $\\zeta _j$ to $\\partial D$ .", "These paths $\\sigma _j$ are pairwise disjoint, because $G$ has no critical values on $\\lambda _{m}$ apart from $w_{m}$ itself.", "Delete all of these $\\sigma _j$ from $D$ ; the set $D^{\\prime }$ which is left is open and is still connected, because if a path joining two points in $D$ meets any of these $\\sigma _j$ , then it meets only finitely many of them, and may be diverted around each without leaving $D^{\\prime }$ .", "Next, consider the finitely many multiple $w_m$ -points of $G$ in $D^{\\prime }$ .", "Let $\\zeta _j \\in D^{\\prime }$ be a zero of $G - w_{m}$ of multiplicity $m_j + 1 \\ge 2$ .", "Then there are $m_j+1$ pre-images $\\tau _{j,k} \\subseteq D^{\\prime }$ of $\\lambda _{m}$ starting at $\\zeta _j$ and joining $\\zeta _j$ to $\\partial D$ .", "Here the $\\tau _{j,k}$ for a given $j$ are disjoint, apart from their common starting point $\\zeta _j$ , and those starting at distinct $\\zeta _j$ do not meet at all.", "Let $t$ be small and positive and let $T_j = \\bigcup _{k=1}^{m_j+1} \\tau _{j,k} $ ; then $U_j = B(\\zeta _j, t) \\setminus T_j$ has $m_j+1$ components, and every $ \\zeta \\in D^{\\prime } \\setminus T_j$ can be joined initially to $\\zeta _j$ by a path in $D^{\\prime }$ , and hence to a point in $U_j$ by a path in $D^{\\prime } \\setminus T_j$ .", "It follows that if the $T_j$ are deleted one at a time from $D^{\\prime }$ , each step increases the number of residual components by at most $m_j$ .", "Hence the number $r$ of components $C_j$ of $G^{-1}(E_m)$ contained in $D$ exceeds by at most 1 the number of zeros of $G^{\\prime }$ in $D$ which are also zeros of $G-w_{m}$ .", "It now follows from the induction hypothesis that $Z_{m-1}(D) \\le \\sum _{j=1}^r Z_m(C_j) \\le \\sum _{j=1}^r ( 1 + Y_m(C_j) ) = r + \\sum _{j=1}^r Y_m(C_j) \\le 1 + Y_{m-1}(D).$ $\\Box $" ], [ "Proof of Theorem ", "Let the function $L$ be as in the hypotheses of Theorem REF , and assume that $L + L^{\\prime }/L$ has finitely many non-real zeros.", "The proof follows quite closely the method of [27].", "Lemma 4.1 Define $F$ by $F(z) = z - \\frac{1}{L(z)}, \\quad F^{\\prime } = 1 + \\frac{L^{\\prime }}{L^2} .$ Then $F$ is transcendental, $F^{\\prime }$ has finitely many non-real zeros, and $L$ and $F$ satisfy $\\mathfrak {T}(r, L) + \\mathfrak {T}(r, F) = O( \\log r )$ as $r \\rightarrow \\infty $ .", "Moreover, there exist finitely many $\\alpha \\in \\mathbb {C}$ such that $F(z)$ or $L(z)$ tends to $\\alpha $ as $z$ tends to infinity along a path in $\\mathbb {C}\\setminus \\mathbb {R}$ .", "For real $K > 0$ let $H_K = \\lbrace z \\in H : |z| > K \\rbrace , \\quad W_K = \\lbrace z \\in H : F(z) \\in H_K \\rbrace .$ Then there exists a large positive real number $K$ such that $F$ has neither critical nor asymptotic values in $H_K $ , and $F$ maps each component of $W_K$ conformally onto $H_K$ .", "Proof.", "First, $F$ is transcendental because $L$ is.", "Since $L$ has finitely many non-real poles, while $L + L^{\\prime }/L$ has finitely many non-real zeros, the functions $Q = 1/L$ and $1- Q^{\\prime } = F^{\\prime } = Q(L+L^{\\prime }/L)$ have finitely many non-real zeros.", "Hence applying Hayman's alternative [9] to $Q$ as in [3], with the Tsuji characteristic replacing that of Nevanlinna, delivers (REF ), whereupon Lemma REF shows that there exist finitely many $\\alpha \\in \\mathbb {C}$ such that $F(z)$ or $L(z)$ tends to $\\alpha $ as $z$ tends to infinity along a path in $\\mathbb {C}\\setminus \\mathbb {R}$ , and the existence of $K$ follows.", "$\\Box $ Lemma 4.2 There exist $\\theta \\in (\\pi /4, 3\\pi /4 )$ and $N_0 \\in \\mathbb {N}$ with the following properties.", "First, $L$ has neither critical nor asymptotic values in $R^+ = \\lbrace r e^{i \\theta } : 0 < r < + \\infty \\rbrace $ , nor in $R^- = \\lbrace r e^{-i \\theta } : 0 < r < + \\infty \\rbrace $ .", "Next, define $x$ by $x \\sin \\theta = K$ , with $K$ as in Lemma REF .", "Then there exist at most $N_0 $ points $z$ lying on the circle $S(0, 2x )$ of centre 0 and radius $2x$ which satisfy $L(z) \\in R^+$ .", "Proof.", "Both assertions follow from Lemma REF and the fact that $L $ is real and transcendental.", "$\\Box $ Lemma 4.3 Let $D$ be a component of $W_K$ , let $a \\in \\partial D$ be a zero of $L$ , and let $\\rho $ be small and positive.", "Then $a$ is unique, and there exists at most one path lying in $D$ and tending to $a$ which is mapped by $L$ onto the arc $\\Omega _{\\theta ,\\rho } = \\lbrace t e^{i \\theta }: \\, 0 < t < \\rho \\rbrace $ .", "Proof.", "This is [27], but with $L_m$ in the notation of [27] replaced by $L$ , and rests on two facts: first, $F(z) \\sim -1/L(z)$ as $z \\rightarrow a$ ; second, since $F$ is univalent on $D$ , there is precisely one component of $ \\left\\lbrace z \\in \\mathbb {C}: \\, 1/2\\rho < |F(z)| < +\\infty , \\, \\pi /16 < \\arg F(z) < 15 \\pi /16 \\right\\rbrace $ in $D$ .", "$\\Box $ The next lemma is [27] and follows from Lemma REF .", "Lemma 4.4 ([27]) There exists a positive integer $N_1$ with the following property.", "Let $D$ be a component of $W_K$ .", "Then there exist at most $N_1$ components $\\Gamma $ of $\\partial D$ with $\\Gamma \\subseteq H$ .", "$\\Box $ The proof of Theorem REF will now be completed using a combination of ideas from [25], [27].", "Assume henceforth that $r \\in E_1$ is large, where $E_1$ is as in Definition REF .", "Lemma 4.5 The set $Q_r$ in (REF ) is contained in a component $D$ of $W_K$ .", "Proof.", "Let $z \\in Q_r$ .", "Since (REF ) and (REF ) give $1/L(z) = o( {\\rm Im} \\, z )$ , it follows from (REF ) that $|F(z)| > K$ and ${\\rm Im} \\, F(z) > 0$ .", "$\\Box $ Lemma 4.6 Let $N_2$ be a large positive integer.", "Then there exist $S > 0$ and pairwise distinct points $w_j$ , for $j=1, \\ldots , 4N_2$ , each of large modulus and satisfying $L (w_j) = Se^{i \\theta } \\in R^+$ , where $R^+$ is as in Lemma REF , and all lying in the same component $D$ of the set $W_K$ .", "Proof.", "Use (REF ) to write, on $Q_r$ , $\\zeta = \\log \\frac{z}{z_0}, \\quad g(z) = \\log L(z) = N(r) \\zeta + \\log L(z_0) + o(1).$ Since $|N(r) \\zeta | = N(r)^{1-\\gamma } $ on $ \\partial Q_r$ , Rouché's theorem implies that $g(Q_r)$ contains the closed disc of centre $ \\log L(z_0) $ and radius $N(r)^{(1-\\gamma )/2}$ .", "This gives $N_2$ distinct points $w_j \\in Q_r$ , all satisfying $L(w_j) = Se^{i \\theta }$ for some large positive $S$ , where $\\theta $ is as in Lemma REF , and hence $L(w_j) \\in R^+$ .", "$\\Box $ Lemma 4.7 For $j=1, \\ldots , 4N_2$ choose a component $\\sigma _j$ of $L^{-1}(R^+)$ with $w_j \\in \\sigma _j$ .", "Then the $\\sigma _j$ are pairwise disjoint and each is mapped univalently onto $R^+$ by $L$ .", "Moreover at least $2N_2$ of the $\\sigma _j$ are such that $\\sigma _j $ lies in $ H_{2x} \\cap D$ and has the following property: as $w \\rightarrow 0$ on $R^+$ the pre-image $z = L^{-1}(w) \\in \\sigma _j$ tends to infinity in $D$ .", "Proof.", "The first two assertions follow from the choice of $\\theta $ in Lemma REF .", "Since the $|w_j|$ and $N_2$ are large, Lemma REF implies that at least $3N_2$ of the $\\sigma _j$ lie in $H_{2x}$ : take $z $ on one of these $\\sigma _j$ .", "Because $L(z) \\in H$ , (REF ) gives $F(z) \\in H$ .", "If $|L(z)| \\ge 1/x$ then $|F(z)| > 2x - x > K$ while $|L(z)| = s < 1/x$ implies that $|F(z)| \\ge {\\rm Im} \\, F(z) \\ge \\frac{\\sin \\theta }{s} > x \\sin \\theta = K.$ Thus at least $3N_2$ of the $\\sigma _j$ lie in $W_K$ and so in $D$ .", "As $L(z) \\rightarrow 0$ on one of these $\\sigma _j$ , the pre-image $z$ tends either to infinity or to a zero $a \\in \\partial D$ of $L$ , the latter possible for at most one $\\sigma _j$ by Lemma REF .", "$\\Box $ Assume, after re-labelling if necessary, that for $j=1, \\ldots , 2N_2$ the path $\\sigma _j$ satisfies the conclusions of Lemma REF , and let $\\sigma _j^{\\prime }$ be the maximal subpath of $\\sigma _j$ on which $|L(z)| \\le S$ .", "These $\\sigma _j^{\\prime }$ can be extended to simple paths $\\tau _j$ in $D$ , these pairwise disjoint except for a common starting point $z^* \\in D$ .", "Since $N_2$ is large, Lemma REF gives at least $N_2$ pairwise disjoint domains $\\Omega _k $ , each bounded by two $\\tau _j$ , and so by two of the $\\sigma _j^{\\prime }$ and a bounded simple path $\\lambda _k \\subseteq D$ , such that the closure of $\\Omega _k$ lies in $ D$ .", "Because $F(z) \\ne \\infty $ on $D$ , there exists a small positive $r_k $ such that $\\hbox{for all $z \\in \\partial \\Omega _k$, either $\\arg L(z) = \\theta $ or $|L(z)| \\ge r_k$.", "}$ For each $\\Omega _k $ , Lemma REF delivers $P_k \\in (0, r_k)$ such that the circle $S(0, P_k)$ contains no critical values of $L$ and no $\\alpha \\in \\mathbb {C}$ such that $L(z) \\rightarrow \\alpha $ along a path tending to infinity in $H$ .", "Choose $u_k \\in \\partial \\Omega _k$ , lying on one of the $\\sigma _j^{\\prime }$ , with $L(u_k) = P_k e^{i \\theta }$ , and continue $z = L^{-1}(w)$ along $S(0, P_k)$ in the direction taking $z$ into $\\Omega _k$ .", "By (REF ) this leads to $v_k \\in \\Omega _k$ with $L(v_k) = P_k e^{-i \\theta }$ .", "Next, Lemma REF permits $ L^{-1}(w)$ to be continued along the half-ray $w = t e^{-i \\theta } $ , so that $t$ decreases and $z = L^{-1}(w)$ starts at $v_k$ and, by (REF ) again, remains in $\\Omega _k \\subseteq D$ .", "Since $L(z) \\ne 0$ on $D$ this gives a path tending to infinity in $\\Omega _k$ on which $L(z) \\rightarrow 0$ with $\\arg L(z) = - \\theta $ .", "Hence there exists an unbounded component $V_k$ of $\\lbrace z \\in \\mathbb {C}: {\\rm Im } \\, (1/L(z)) > 2/P_k \\rbrace $ , with $V_k \\cup \\partial V_k \\subseteq \\Omega _k \\subseteq D$ by (REF ) again, and the function $u_k(z) = {\\rm Im} \\, \\frac{1}{L(z)} - \\frac{2}{P_k} \\quad (z \\in V_k), \\quad u_k(z) = 0 \\quad (z \\notin V_k) ,$ is non-constant and subharmonic in the plane.", "There are at least $N_2$ of these $u_k$ , with disjoint supports, and $N_2$ is large.", "Thus the Phragmén-Lindelöf principle [12] gives, for at least one $k$ , a point $z \\in V_k \\subseteq D$ , with $|z| $ large and ${\\rm Im} \\, 1/L(z) > |z|^2 $ , and hence ${\\rm Im} \\, F(z) < 0$ by (REF ), which is plainly a contradiction.", "$\\Box $" ], [ "Sufficient conditions for the UHWV property", "The main focus of this section will be on proving that if $L$ is as in the hypotheses of Theorem REF then $L$ has the UHWV property in Definition REF .", "Let $h(z) = \\sum _{n=0}^\\infty \\alpha _n z^n$ be a transcendental entire function.", "Then for $r > 0$ the central index $N(r)$ of $h$ is the largest $n$ for which $|\\alpha _n| r^n = \\max \\lbrace | \\alpha _m | r^m : \\, m = 0, 1, 2, \\ldots \\rbrace $ , and $N(r)$ tends to infinity with $r$ [11].", "The following is a routine consequence of the Wiman-Valiron theory [11].", "Lemma 5.1 Let $h$ be a real transcendental entire function, denote by $N(r)$ the central index of $h$ and let $1/2 < \\tau < \\gamma < 1$ .", "Then there exists a set $E_0 \\subseteq [1, + \\infty )$ of finite logarithmic measure such that $\\lim _{r \\rightarrow + \\infty , r \\notin E_0} \\frac{N(r)}{\\left( \\log M(r, h) \\right)^2 } = 0 .$ Furthermore, for each $r \\notin E_0$ there exists $z_0 = z_0(r)$ satisfying (REF ), such that $|h(z_0)| \\sim M(r, h)$ and, uniformly as $r \\rightarrow + \\infty $ outside $E_0$ , $h (z) \\sim h(z_0) \\left( \\frac{z}{z_0} \\right)^{N(r)}\\quad \\hbox{and} \\quad \\log |h(z)| \\ge (1-o(1)) \\log M(r, h)$ on the set $Q_r$ in (REF ).", "Proof.", "Choose any $\\sigma , \\tau , \\gamma $ with $1/2 < \\sigma < \\tau < \\gamma < 1$ .", "By a standard result from the Wiman-Valiron theory [11], there exists a set $E_0 $ of finite logarithmic measure such that (REF ) holds; furthermore, if $r \\in [1, \\infty ) \\setminus E_0$ and $|z_1| = r$ , $|h(z_1)| \\sim M(r, h)= \\max \\lbrace |h(z)| : |z| = r \\rbrace $ , then $h (z) \\sim h(z_1) \\left( \\frac{z}{z_1} \\right)^{N(r)}\\quad \\hbox{for} \\quad \\left| \\log \\frac{z}{z_1} \\right| \\le N(r)^{-\\sigma } .$ Since $h$ is real, it may be assumed that ${\\rm Im} \\, z_1 \\ge 0$ for $r \\in [1, \\infty ) \\setminus E_0$ , and so there exists $z_0$ satisfying (REF ) and the first estimate of (REF ).", "Next, (REF ) and the fact that $0 < 1 - \\gamma < 1/2$ yield the second estimate of (REF ) via $\\log |h(z)| \\ge \\log M(r, h) - N(r)^{1-\\gamma } - o(1) \\ge (1-o(1)) \\log M(r, h) .$ $\\Box $ Lemma REF shows every real transcendental entire function has the UHWV property.", "The same is in fact true of any real transcendental meromorphic function in the plane for which the inverse function has a direct transcendental singularity over $\\infty $ : this can be proved identically, but using the version of Wiman-Valiron theory developed in [4] for functions with direct tracts.", "Lemma 5.2 Let $L$ be as in the hypotheses of Theorem REF .", "Denote by $N(r)$ the central index of $h$ and let $1/2 < \\tau < \\gamma < 1$ .", "Then there exists a set $E_0 \\subseteq [1, + \\infty )$ of finite logarithmic measure with the following property: for each $r \\in [1, + \\infty ) \\setminus E_0$ there exists $z_0 = z_0(r)$ satisfying (REF ) such that (REF ) holds on the set $Q_r$ in (REF ), uniformly as $r \\rightarrow + \\infty $ outside $E_0$ .", "In particular, $L$ has the UHWV property, with $E_1 = [1, + \\infty ) \\setminus E_0$ .", "Proof.", "Combining (REF ), (REF ) and (REF ) shows that, for large $r \\in [1, \\infty ) \\setminus E_0$ and $z \\in Q_r$ , $R_1(z) \\psi _1(z) &\\sim & R_1(z_0) \\psi _1( z_0), \\\\\\log \\frac{1}{ |R_1(z) \\psi _1(z)| } + \\log ^+ |R_2(z) \\psi _2(z)|&\\le & O( \\log r) + O( \\log N(r) ) \\le o( \\log M(r, h) ), \\\\\\log |L(z)| &\\ge & (1-o(1)) \\log M(r, h) ,$ from which (REF ) follows.", "$\\Box $ Lemma 5.3 Let $g = g_1/g_2 $ , where $g_1, g_2$ are real entire functions, with no common zeros.", "Assume that $g_2$ has finite lower order, but $g$ has infinite order.", "Then $L = g^{\\prime }/g$ satisfies the hypotheses of Theorem REF , and hence the conclusions of Lemma REF .", "Proof.", "The logarithmic derivative of each $g_j$ has a Levin-Ostrovskii factorisation [3], [30] $\\frac{g_j^{\\prime }}{g_j} = \\phi _j \\psi _j ,$ in which: $\\phi _j$ and $\\psi _j$ are real meromorphic functions; $\\phi _j$ has finitely many poles; if $g_j$ has finitely many zeros then $\\psi _j \\equiv 1$ ; if $g_j$ has infinitely many zeros then (REF ) is obtained by applying Lemma REF in conjunction with Rolle's theorem, in which case $\\psi _j(H) \\subseteq H$ .", "Since $g_2$ has finite lower order, (REF ) and the lemma of the logarithmic derivative give $T(r, \\phi _2) \\le m(r, \\phi _2) + O( \\log r) \\le m(r, 1/\\psi _2) + m(r, g_2^{\\prime }/g_2) + O( \\log r) = O( \\log r )$ on a sequence of $r$ tending to infinity, and so $\\phi _2$ is a rational function.", "Thus [3] implies that $g_2$ has finite order.", "Because $g$ has infinite order, so has $g_1$ , and applying [3] again shows that $\\phi _1 $ is transcendental.", "Hence $L $ has a representation (REF ) as required.", "$\\Box $" ], [ "Proof of Theorems ", "First, let $L$ be as in the hypotheses of Theorem REF : then $L$ has the UHWV property, by Lemma REF , whereupon Theorem REF implies that $L+L^{\\prime }/L$ has infinitely many non-real zeros.", "Next, if $f$ is as in the hypotheses of Theorem REF then $L = f^{\\prime }/f$ satisfies those of Theorem REF , by Lemma REF applied to $f$ or $1/f$ .", "$\\Box $" ], [ "Proof of Theorem ", "Let $L, \\phi , \\psi , a, b$ be as in the hypotheses of Theorem REF and suppose that $L^{\\prime } + a L^2 + b $ has finitely many non-real zeros.", "Writing $L(z) = \\alpha L_1( \\beta z)$ , where $\\alpha = \\sqrt{b/a}$ and $\\beta = \\sqrt{ab}$ , makes it possible to assume that $a=b=1$ .", "The following estimate for the Tsuji characteristic of $L$ was deduced in [22] from an argument of Tumura-Clunie type [9].", "Note that [22] was stated for the special case in which $L = f^{\\prime }/f$ , where $f$ is an entire function such that $f$ and $f^{\\prime \\prime }+f$ have finitely many non-real zeros, but the proof depends only on $L$ having finitely many non-real poles and $L^{\\prime }+L^2+1$ finitely many non-real zeros.", "Lemma 6.1 (Lemma 4.1, [22]) The Tsuji characteristic of $L$ satisfies $\\mathfrak {T}(r, L) = O( \\log r ) \\quad \\hbox{as $r \\rightarrow \\infty $}.$ $\\Box $ Lemma 6.2 The transcendental entire function $h$ in (REF ) has order at most 1.", "Proof.", "This is fairly standard.", "First, (REF ) and (REF ) imply that, as $r \\rightarrow + \\infty $ , $T(r, h) = m(r, h) \\le m(r, L) + O( \\log r ) .$ This implies in turn that, as $R \\rightarrow + \\infty $ , by (REF ) and an inequality of Levin and Ostrovskii [30] (see also [3] or [22]), $\\frac{T(R, h)}{2R^2} \\le \\int _R^\\infty \\frac{T(r, h)}{r^3} \\, dr \\le 2\\int _R^\\infty \\frac{\\mathfrak {T}(r, L)}{r^2 } \\, dr +O \\left( \\frac{\\log R}{R^2} \\right) = O \\left( \\frac{\\log R}{R} \\right) .$ $\\Box $ The proof in [22] made extensive use of the auxiliary function $F = (TL-1)/(L+T)$ , where $T(z) = \\tan z $ .", "For the present paper it turns out to be simpler to work with $G(z) = e^{2iz} \\left( \\frac{L(z)-i}{L(z)+i} \\right) = - \\left( \\frac{ F(z)-i}{F(z)+i} \\right) ,\\quad G^{\\prime }(z) = \\frac{ 2i e^{2iz} (L^{\\prime }(z)+L(z)^2 + 1)}{(L(z)+i)^2} .$ Then $|G(x)| = 1$ for $x \\in \\mathbb {R}$ , and $G^{\\prime }$ has finitely many zeros in $\\mathbb {C}\\setminus \\mathbb {R}$ , while $Y = \\lbrace z \\in H: \\, L(z) \\in H \\rbrace \\subseteq W = \\lbrace z \\in H: \\, |G(z)| < 1 \\rbrace .$ There now follows a sequence of lemmas which together show that $G$ has finitely many asymptotic values $\\alpha \\in \\mathbb {C}$ with $| \\alpha | \\ne 1$ , using a method which substantially simplifies the approach in [22].", "For $\\alpha \\in \\mathbb {C}$ , use (REF ) to define $s_\\alpha $ by $s_\\alpha (z) =\\frac{ G(z) - \\alpha }{e^{2iz} - G(z)}= \\frac{1}{2i} \\left( L(z)-i - \\alpha e^{-2iz} (L(z)+i) \\right) .$ Since $L$ has finitely many non-real poles, so has each $s_\\alpha $ .", "Lemma 6.3 Let $\\alpha , \\beta \\in \\mathbb {C}$ satisfy $\\alpha \\ne \\beta $ .", "Then there exists $c_1 > 0$ such that if $z \\in H$ and $|z|$ is large then $| s_\\alpha (z) | + | s_\\beta (z) | \\ge c_1$ .", "Proof.", "Assume that there exists a sequence $z_n \\rightarrow \\infty $ in $H$ such that $| s_\\alpha (z_n) | + | s_\\beta (z_n) | \\rightarrow 0$ .", "Since $|e^{2i z_n} | \\le 1$ in (REF ), it must be the case that $G(z_n) = O(1)$ , from which it follows that $G(z_n) \\rightarrow \\alpha $ and $G(z_n) \\rightarrow \\beta $ , which is impossible.", "$\\Box $ Lemma 6.4 Let $\\alpha , \\beta \\in \\mathbb {C}$ satisfy $\\alpha \\ne \\beta $ , and let $c_2 > 0$ .", "Then there exists $c_3 > 0$ such that if $z_n \\rightarrow \\infty $ in $H$ with $|e^{2iz_n} - \\alpha | \\ge c_2$ and $G(z_n) \\rightarrow \\alpha $ then $s_\\alpha (z_n) \\rightarrow 0$ and $| s_\\beta (z_n)| \\le c_3$ .", "Proof.", "This follows from (REF ) and the fact that $2 |e^{2iz_n} - G(z_n)| \\ge c_2$ for all large $n$ .", "$\\Box $ Lemma 6.5 Let $\\alpha _1, \\dots , \\alpha _N \\in \\mathbb {C}$ be pairwise distinct, with $| \\alpha _1 | \\ne 0, 1$ , and let $G(z) \\rightarrow \\alpha _1$ on a path $\\gamma $ tending to infinity in $H$ .", "Then there exists a path $\\lambda $ tending to infinity in $H$ on which $s_{\\alpha _1}(z) \\rightarrow 0$ and $s_{\\alpha _j}(z)$ is bounded for $j=2, \\ldots , N$ .", "Proof.", "Evidently there exists $q \\in \\mathbb {C}\\setminus \\mathbb {R}$ such that the solutions of $e^{2iz} = \\alpha _1 $ are $a_n = n \\pi + q $ , $n \\in \\mathbb {Z}$ .", "Let $\\varepsilon $ be small and positive.", "Then Lemma REF shows that $s_{\\alpha _1}(z)$ is small, and the remaining $s_{\\alpha _j}(z)$ are uniformly bounded, for all $z \\in \\gamma $ such that $|z|$ is large and $z$ lies outside the union of the discs $B(a_n, \\varepsilon )$ .", "It may therefore be assumed that $\\gamma $ meets the disc $B(a_n, \\varepsilon )$ for all $n$ in an unbounded set $E \\subseteq \\mathbb {Z}$ , since otherwise there is nothing further to prove.", "Then $0 < | \\alpha _1 | < 1$ and for each $n \\in E$ there exists a simple subpath $\\sigma _n$ of $\\gamma $ which lies in the annulus $ 2 \\varepsilon \\le |z-a_n| \\le 4 \\varepsilon $ and joins the two boundary circles.", "Lemma REF implies that $\\lim _{|n| \\rightarrow \\infty , n \\in E} \\tau _n = 0,\\quad \\tau _n = \\max \\lbrace |G(z) - \\alpha _1 | + |s_{\\alpha _1}(z)| : \\, z \\in \\sigma _n \\rbrace .$ Moreover, standard estimates [31] give a positive constant $C$ , independent of $n \\in E$ , such that the harmonic measure $\\omega (z, \\sigma _n, B(a_n, 4\\varepsilon ) \\setminus \\sigma _n)$ is at least $C$ for $|z-a_n| \\le \\varepsilon $ .", "Let $E_1$ be the set of $n \\in E$ such that $|n|$ is large and there exists $z_1$ in $B(a_n, 4 \\varepsilon )$ with $|L(z_1)| \\le 2$ .", "Since the functions $L_n(z) = L(a_n + z)$ , $n \\in E_1$ , satisfy $L_n \\ne \\infty $ and $L_n^{\\prime }+L_n^2+1 \\ne 0$ on $B(0, 8 \\varepsilon )$ , Lemma REF and (REF ) deliver $K_1, K_2 > 0$ , independent of $n$ , such that $|L(z)| \\le K_1$ and $| s_{\\alpha _j}(z)| \\le K_2$ for $z$ in $B(a_n, 4 \\varepsilon )$ , $n \\in E_1$ and $j=1, \\ldots , N$ .", "This makes $u_1(z) = \\log |s_{\\alpha _1}(z)/K_2|$ subharmonic and non-positive on $B(a_n, 4 \\varepsilon )$ , for $n \\in E_1$ , and a standard combination of (REF ) with the two constants theorem [31] yields, for $|z-a_n| \\le \\varepsilon $ , $u_1(z) \\le C \\log \\left( \\frac{\\tau _n}{K_2} \\right) , \\quad |s_{\\alpha _1}(z)| \\le K_2 \\left( \\frac{\\tau _n}{K_2} \\right)^C .$ Thus for $n \\in E_1$ and $z \\in \\gamma \\cap B(a_n, \\varepsilon )$ , the function $s_{\\alpha _1}(z)$ is small, by (REF ), while $| s_{\\alpha _j}(z)| \\le K_2$ for $j=2, \\ldots , N$ .", "It remains only to deal with the set $E_2$ of $n \\in E \\setminus E_1$ such that $|n|$ is large.", "These $n$ are such that $|L(z)| > 2$ for all $z$ in $B(a_n, 4 \\varepsilon )$ , and hence $|G(z)| \\le 3$ there, by (REF ).", "This time $u_2(z) = \\log |(G(z)- \\alpha _1)/4| $ is subharmonic and non-positive on $B(a_n, 4 \\varepsilon )$ , and combining (REF ) with the two constants theorem yields $\\displaystyle {|G(z)- \\alpha _1| \\le 4 \\left( \\frac{\\tau _n}{4} \\right)^C}$ for $|z-a_n| \\le \\varepsilon $ .", "Thus for $|z-a_n| = \\varepsilon $ , where $n \\in E \\setminus E_1$ and $|n|$ is large, (REF ) and Lemma REF imply that $s_{\\alpha _1}(z)$ is small, and the remaining $s_{\\alpha _j}(z)$ are uniformly bounded.", "The proof is now completed by replacing any part of $\\gamma $ which enters and leaves $B(a_n, \\varepsilon )$ , for $n \\in E_2$ , by an arc of the circle $| z- a_n| = \\varepsilon $ .", "$\\Box $ Lemma 6.6 The function $G$ has finitely many asymptotic values $\\alpha \\in \\mathbb {C}$ with $| \\alpha | \\ne 1$ .", "Proof.", "Since $|G(x)| = 1$ for $x \\in \\mathbb {R}$ it suffices to show that there do not exist pairwise distinct $\\alpha _1, \\alpha _2, \\alpha _3 \\in \\mathbb {C}$ , with $| \\alpha _j | \\ne 0, 1$ , such that $G(z) \\rightarrow \\alpha _j$ along a path $\\gamma _j$ tending to infinity in $H$ .", "Assume the contrary: then Lemma REF gives paths $\\lambda _1, \\lambda _2, \\lambda _3$ in $H$ such that $s_{\\alpha _j}(z)$ tends to 0, while the remaining $s_{\\alpha _k}(z)$ are bounded, as $z \\rightarrow \\infty $ on $\\lambda _j$ .", "Hence $Q(z) = s_{\\alpha _1}(z) s_{\\alpha _{2}}(z) s_{\\alpha _{3}}(z)$ tends to 0 on each $\\lambda _j$ .", "By Lemma REF , each intersection $\\lambda _j \\cap \\lambda _{j^{\\prime }}$ is bounded for $j \\ne j^{\\prime }$ .", "Choose a large $R \\in (0, \\infty )$ .", "It may be assumed that the $\\lambda _j$ start on $|z| = R$ and divide $\\lbrace z \\in H : \\, |z| > R \\rbrace $ into four disjoint unbounded domains $D_0, \\ldots , D_3$ , such that $\\lambda _j$ separates $D_{j-1}$ from $D_j$ for $j =1, 2, 3$ .", "Suppose first that, as $z$ tends to infinity in $D_1$ , the function $Q(z) $ is bounded, and so tends to 0 by the Phragmén-Lindelöf principle.", "Lemma REF implies that $|s_{\\alpha _2}(z) s_{\\alpha _3}(z)| > |s_{\\alpha _1}(z)| $ as $z \\rightarrow \\infty $ on $\\lambda _1$ , while $|s_{\\alpha _1}(z)| > |s_{\\alpha _2}(z) s_{\\alpha _3}(z)| $ as $z \\rightarrow \\infty $ on $\\lambda _2$ .", "Hence there exists $z \\in D_1$ , with $|z|$ arbitrarily large, $Q(z)$ small and $|s_{\\alpha _1}(z)| = |s_{\\alpha _2}(z) s_{\\alpha _3}(z)|$ .", "But this implies that $s_{\\alpha _1}(z)$ and at least one of $s_{\\alpha _2}(z), \\, s_{\\alpha _3}(z)$ are both small, which contradicts Lemma REF .", "It follows that $Q(z)$ is unbounded as $z$ tends to infinity in $D_1$ and, by the same argument, in $D_2$ also, so that $Q^{-1}$ has at least two direct singularities over $\\infty $ , lying in $H$ .", "Since $\\mathfrak {T}(r, Q) = O( \\log r )$ as $r \\rightarrow \\infty $ , by (REF ) and (REF ), this contradicts Lemma REF .", "$\\Box $ Lemma 6.7 If $a \\in \\mathbb {C}$ and $|a| \\ne 1$ , and if $G^{-1}$ has a transcendental singularity over $a $ , then the singularity is direct.", "Proof.", "This follows from Lemma REF , the fact that $G^{\\prime }$ has finitely many non-real zeros, and the standard classification of isolated singularities of the inverse function [31].", "$\\Box $ Lemma 6.8 If $G^{-1}$ has a transcendental singularity over $a \\in \\mathbb {C}$ , then $a = 0$ or $|a|= 1$ , and there exists $N_0 \\in \\mathbb {N}$ such that if $D$ is a component of the set $W$ in (REF ) then $G$ has at most $N_0$ zeros in $D$ , counting multiplicities.", "Moreover, $L^{-1}$ cannot have a direct transcendental singularity over $a \\in \\mathbb {C}\\setminus \\mathbb {R}$ , and $L$ has finitely many asymptotic values in $\\mathbb {C}\\setminus \\mathbb {R}$ .", "Finally, there exists $\\theta \\in (\\pi /4, 3 \\pi /4)$ such that $L$ has no critical or asymptotic values on the open half-line $R^+$ given by $w = i + t e^{i \\theta } $ , $0 < t < + \\infty $ .", "Proof.", "This is a modification of [22].", "To prove the first two assertions let $g$ be $G$ or $L$ and assume that $g^{-1}$ has a direct transcendental singularity over $a \\in \\mathbb {C}$ , with $|a| \\ne 0, 1$ if $g = G$ , and $a \\in \\mathbb {C}\\setminus \\mathbb {R}$ if $g = L$ .", "Since $|G(x)| = 1$ on $\\mathbb {R}$ and $L(\\mathbb {R}) \\subseteq \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ , it may be assumed that the singularity lies in $H$ .", "Let $\\delta _1, \\delta _2 $ be small and positive.", "Then there exists a component $D \\subseteq H$ of $\\lbrace z \\in \\mathbb {C}: |g(z) - a| < \\delta _1 \\rbrace $ such that $g(z) \\ne a$ on $D$ and $v(z) = \\log \\frac{\\delta _1}{|g(z) - a|} \\quad (z \\in D), \\quad v(z) = 0 \\quad (z \\in \\mathbb {C}\\setminus D),$ defines a non-constant subharmonic function on $\\mathbb {C}$ .", "Because $\\mathfrak {T}(r, g) = O( \\log r )$ as $r \\rightarrow \\infty $ by (REF ) and (REF ), a standard argument as in [22] shows that $v$ has order of growth at most 1.", "Let $N_1$ be a large positive integer.", "By Lemma REF , there exists a real polynomial $P_1$ , of degree at most $N_1-1$ , such that $h_1(z) = z^{-N_1}( h(z) - P_1(z))$ is entire and transcendental of order at most 1.", "Let $C$ be a component of the set $\\lbrace z \\in \\mathbb {C}: |h_1(z)| > 1 \\rbrace $ .", "If $z \\in C$ and $|z|$ is large, and $\\delta _2 < | \\arg z | < \\pi - \\delta _2$ , then combining (REF ), (REF ) and (REF ) shows that $|L(z)| = |L( \\bar{z})|$ is large; furthermore one of $|G(z)|$ and $|G(\\bar{z})|$ is small and the other is large.", "Thus neither $z$ nor $\\bar{z}$ lies in $D$ , and $z$ cannot lie in the reflection of $D$ across $\\mathbb {R}$ .", "For $s > 0$ denote by $\\theta _C (s), \\theta _D (s)$ the angular measure of the intersection of $C$ , respectively $D$ , with the circle $|z| = s$ , and let $\\theta _C^*(s) = + \\infty $ if the whole circle $|z| = s$ lies in $C$ , with $\\theta _C^*(s) = \\theta _C(s) $ otherwise.", "Then the Cauchy-Schwarz inequality and the fact that $\\delta _2$ is small give, for large $s$ , $\\theta _C(s) + 2 \\theta _D(s) \\le 2 \\pi + 8 \\delta _2 ,\\quad 9 \\le \\left( \\frac{1}{ \\theta _C^*(s)} + \\frac{2}{\\theta _D(s)} \\right) ( 2 \\pi + 8 \\delta _2 ).", "$ Now $v_1 = \\log ^+ |h_1|$ and $v_2 = v$ are both subharmonic of order at most 1, so set $B^*(r, v_j) = \\max \\lbrace v_j(z) : \\, |z| = r \\rbrace $ for $r > 0$ , fix a large positive $r_0$ and let $r \\rightarrow + \\infty $ .", "Then a standard application of Carleman's estimate for harmonic measure, exactly as in [22], leads to a contradiction via $\\left( \\frac{9}{2 + 8 \\delta _2 /\\pi } \\right) \\log r &\\le &\\int _{r_0}^r \\left( \\frac{\\pi }{s \\theta _C^*(s)} + \\frac{2\\pi }{s \\theta _D(s)} \\right) \\, ds \\\\&\\le &\\log B^*(2r, v_1) + 2\\log B^*(2r, v_2) + O(1) \\\\&\\le & (3+o(1)) \\log r .$ It now follows, in view of Lemma REF , that if $a \\in \\mathbb {C}$ is an asymptotic value of $G$ then $a = 0$ or $|a|= 1$ , and $N_0$ exists by Proposition REF and the fact that $G^{\\prime }$ has finitely many non-real zeros.", "Next, $L$ cannot have infinitely many asymptotic values $a \\in \\mathbb {C}\\setminus \\mathbb {R}$ because otherwise $L^{-1}$ would have at least two direct transcendental singularities over $\\infty $ , lying in $H$ , which by (REF ) contradicts Lemma REF .", "The existence of $\\theta $ follows at once.", "$\\Box $ Since $L$ satisfies the hypotheses of Theorem REF , which are the same as those of Theorem REF , Lemma REF may now be applied to $L$ .", "Lemma 6.9 Let $r \\notin E_0$ be large, with $E_0$ as in Lemma REF .", "Then the set $Q_r$ in (REF ) is contained in a component $D$ of the set $W$ in (REF ).", "Proof.", "(REF ), (REF ) and (REF ) give a positive constant $c$ such that, on $Q_r$ , ${\\rm Im } \\, z &\\ge & c N(r)^{-\\tau }, \\quad \\frac{1}{L(z)} = o \\left( N(r)^{-\\tau } \\right),\\\\G(z) &=& e^{2iz} \\cdot \\frac{1 -i/L(z)}{1+i/L(z)} = e^{2iz} ( 1 + \\varepsilon _1(z)), \\quad \\varepsilon _1(z) = o \\left( N(r)^{-\\tau } \\right),\\\\\\log |G(z)| &\\le & - 2c N(r)^{-\\tau } + o \\left( N(r)^{-\\tau } \\right) < 0 .$ $\\Box $ Lemma 6.10 Let $N_2$ be a large positive integer.", "Then for large enough $r$ as in Lemma REF there exist $S > 0$ and pairwise distinct $w_j \\in Q_r$ , for $j=1, \\ldots , N_2$ , such that $L(w_j) = i + S e^{i \\theta } \\in R^+$ , where $\\theta $ and $R^+$ are as in Lemma REF .", "For each $j \\in \\lbrace 1, \\ldots , N_2 \\rbrace $ , let $\\sigma _j$ be the component of $L^{-1}(R^+)$ with $w_j \\in \\sigma _j$ .", "Then the $\\sigma _j$ are pairwise disjoint and lie in the same component $D$ of $W$ as $Q_r$ , and each is mapped univalently onto $R^+$ by $L$ .", "Furthermore, at least one of the $\\sigma _j$ has the property that as $w \\rightarrow i$ on $R^+$ the pre-image $z = L^{-1}(w) \\in \\sigma _j$ tends to infinity in $D$ .", "Proof.", "Let $r$ be large and as in Lemma REF .", "The existence of $S$ and the $w_j$ is proved exactly as in Lemma REF , using the fact that $L(z)-i \\sim L(z)$ on $Q_r$ .", "The next three assertions follow from the fact that $L^{-1}$ has no singular values on $R^+$ , by the choice of $\\theta $ , and the inclusions $w_j \\in Q_r \\subseteq D$ and (REF ).", "Now, as $w \\rightarrow i$ on $R^+$ the pre-image $z = L^{-1}(w) \\in \\sigma _j$ lies in $D$ and tends either to a zero of $L-i$ , which by (REF ) is a zero of $G$ in $D$ of the same multiplicity, or to infinity.", "Because $N_2 $ is large, Lemma REF now implies that $z= L^{-1}(w)$ must tend to infinity for at least one $j$ .", "$\\Box $ Thus $L(z) $ tends to $i$ along a path $\\mu $ tending to infinity in the component $D$ of $W$ .", "This gives $t \\in (0, 1/2)$ and a neighbourhood $\\Omega (t)$ of a transcendental singularity of $L^{-1}$ over $i$ , such that $\\mu \\setminus \\Omega (t)$ is bounded.", "Moreover, $\\Omega (t)$ lies in $H$ , and so in $Y \\subseteq W$ , and hence in $D$ .", "By Lemma REF and (REF ), $G$ and $L-i$ have finitely many zeros in $D$ .", "But this implies that $L^{-1}$ has a direct transcendental singularity over $i$ , which contradicts Lemma REF , and thereby proves Theorem REF .", "$\\Box $" ], [ "Proof of Theorem ", "Assume that $f$ is as in the hypotheses.", "As in Section REF , $L = f^{\\prime }/f$ satisfies the identical hypotheses of Theorems REF and REF , and the latter implies that $L^{\\prime }+L^2 + \\omega = (f^{\\prime \\prime }+\\omega f)/f$ has infinitely many non-real zeros.", "$\\Box $" ], [ "Proof of Theorem\n", "Let $U$ be as in the hypotheses of Theorem REF .", "Part (i) was already proved in Lemma REF , with $\\psi $ as in (REF ).", "Next, assume that $S$ has infinite order in (REF ) and write $\\frac{U^{\\prime }}{U} = L + \\frac{\\psi ^{\\prime }}{\\psi } , \\quad L = \\frac{S^{\\prime }}{S} .$ Since $S$ has finitely many poles and non-real zeros, applying Lemma REF , with $g = S$ , shows that $L$ satisfies the hypotheses of Theorem REF and so has the UHWV property by Lemma REF .", "Then the UHWV property for $U^{\\prime }/U$ follows from the fact that (REF ), (REF ) and (REF ) give, for large $r \\in E_1$ and $z \\in Q_r$ , $\\left| \\frac{\\psi ^{\\prime }(z)}{\\psi (z)} \\right| = O\\left( \\frac{N(r)^\\tau }{r} \\right) = o( |L(z)| ), \\quad \\frac{U^{\\prime }(z)}{U(z)} \\sim L(z) .", "$ Since $U^{\\prime }/U$ has finitely many non-real poles, part (ii) of Theorem REF follows from Theorem REF .", "Assume henceforth that $S = Re^P = U/\\psi $ is as in the hypotheses of part (iii), in particular with $P$ a non-constant polynomial, and let $2 \\le m \\in \\mathbb {N}$ .", "Since $\\psi $ maps $H$ into itself, $\\psi $ has a series representation $\\psi (z) = Az + B + \\sum _{k \\in K} A_k \\left( \\frac{1}{x_k - z} - \\frac{1}{x_k} \\right) ,$ with $A, B, A_k \\in \\mathbb {R}$ , $A \\ge 0$ , $A_k > 0$ and $\\sum _{k \\in K} A_k x_k^{-2} < + \\infty $ [29].", "Write, using (REF ), $U^{(m)} = S \\sum _{j=0}^m a_j \\psi ^{(j)} = S a_0 \\phi ,\\quad a_j = { m \\atopwithdelims ()j } \\frac{S^{(m-j)}}{S} = a_0 b_j,\\quad \\phi = \\phi _m = \\sum _{j=0}^m b_j \\psi ^{(j)} .$ Here $a_m = b_0 = 1$ and, by standard estimates based on the formulas $\\frac{S^{\\prime }(z)}{S(z)} = \\frac{R^{\\prime }(z)}{R(z)} + P^{\\prime }(z), \\quad \\frac{S^{(p+1)}(z)}{S(z)} = \\frac{S^{(p)}(z)}{S(z)} \\cdot \\frac{S^{\\prime }(z)}{S(z)} + \\frac{d}{dz}\\left( \\frac{S^{(p)}(z)}{S(z)} \\right),$ the real rational functions $a_j, b_j$ satisfy $a_j(z) = { m \\atopwithdelims ()j } P^{\\prime }(z)^{m-j} \\left( 1 + O \\left( \\frac{1}{|z|} \\right) \\right) , \\quad b_j(z) = { m \\atopwithdelims ()j } P^{\\prime }(z)^{-j} \\left( 1 + O \\left( \\frac{1}{|z|} \\right) \\right)$ as $z \\rightarrow \\infty $ .", "The key to the proof of Theorem REF is the following.", "Proposition 8.1 Let $s_0$ be a large positive real number and let $I \\subseteq \\mathbb {R}\\setminus [-s_0, s_0]$ be an open interval which contains no poles of $\\psi $ .", "Then the number of zeros of $U^{(m)}$ in $I$ , counting multiplicities, is at most 1 if $m$ is even, and at most 2 if $m$ is odd.", "Next, let $k \\in K$ be such that $|k|$ is large, and let $n_{k,m}$ be the number of zeros of $U^{(m)}$ , counting multiplicities, in $(x_k, x_{k+1})$ .", "If $m$ is even then $n_{k,m} = 1$ , while if $m$ is odd then $n_{k,m} \\in \\lbrace 0, 2 \\rbrace $ .", "Proof.", "Suppose first that $m$ is even, set $b_{m+1} = 0$ and recall that $b_0 = 1$ .", "Since $P$ is a real polynomial, $S$ and $a_0$ do not change sign on $I$ and so (REF ) implies that $U^{(m)}$ has the same number of zeros in $I$ as $\\phi $ .", "Thus to prove the first assertion it suffices to show that if $s_0$ is sufficiently large then the derivative $\\phi ^{\\prime }$ is positive on $I$ , where $\\phi ^{\\prime }$ is given by $\\phi ^{\\prime } = \\sum _{j=0}^m \\left( b_j \\psi ^{(j+1)} + b_j^{\\prime } \\psi ^{(j)} \\right) =\\sum _{j=0}^m b_j \\psi ^{(j+1)} + \\sum _{j=1}^{m+1} b_{j}^{\\prime } \\psi ^{(j)} =\\sum _{j=0}^{m} c_j \\psi ^{(j+1)} ,$ in which the $c_j$ satisfy, as $z \\rightarrow \\infty $ , by (REF ), $c_j (z) = b_{j}(z) + b_{j+1}^{\\prime }(z)= { m \\atopwithdelims ()j } P^{\\prime }(z)^{-j} \\left( 1 + O \\left( \\frac{1}{|z|} \\right) \\right) .$ For $x \\in I$ let $X_k = P^{\\prime }(x) (x_k-x) \\in \\mathbb {R}\\setminus \\lbrace 0 \\rbrace $ and let $Q_m$ be as in Lemma REF .", "Then (REF ), (REF ), (REF ) and the fact that $c_0 (\\infty ) = 1$ deliver the following, in which the $o(1)$ terms are uniformly small for $x \\in I$ , provided $s_0$ is large enough: $\\phi ^{\\prime }(x)&=& \\sum _{j=0}^{m} c_j (x) \\frac{d^{j+1}}{dx^{j+1}}\\left( Ax + B + \\sum _{k \\in K} A_k \\left( \\frac{1}{x_k - x} - \\frac{1}{x_k} \\right) \\right) \\\\&\\ge & \\sum _{j=0}^{m} c_j (x) \\frac{d^{j+1}}{dx^{j+1}}\\left( \\sum _{k \\in K} A_k \\left( \\frac{1}{x_k - x} - \\frac{1}{x_k} \\right) \\right) \\\\&=&\\sum _{k \\in K} A_k \\sum _{j=0}^{m} c_j (x) \\frac{d^{j+1}}{dx^{j+1}}\\left( \\frac{1}{x_k - x} \\right)\\\\&=&\\sum _{k \\in K} A_k \\sum _{j=0}^{m} { m \\atopwithdelims ()j } P^{\\prime }(x)^{-j} (1+o(1))\\left( \\frac{(j+1)!", "}{(x_k - x)^{j+2}} \\right) \\\\&=&\\sum _{k \\in K} \\frac{A_k}{ (x_k-x)^{2} }\\sum _{j=0}^{m} (1+o(1)) { m \\atopwithdelims ()j } \\frac{(j+1)!", "}{X_k^{j} } \\\\&=&\\sum _{k \\in K} \\frac{A_k }{(x_k-x)^{2}} \\left( Q_m\\left(\\frac{1}{X_k}\\right) +\\sum _{j=0}^{m} \\frac{o(1) }{X_k^j } \\right)\\\\&\\ge &\\sum _{k \\in K} \\frac{A_k }{ (x_k-x)^{2}} (d_m - o(1)) \\max \\lbrace 1, X_k^{-m} \\rbrace > 0.$ This proves the first assertion of Proposition REF when $m$ is even, and the case of odd $m$ follows from Rolle's theorem and the above reasoning applied to $U^{(m+1)}$ .", "To prove the second assertion, observe that (REF ) and (REF ) deliver $U^{(m)}(z) \\sim S(z)\\psi ^{(m)}(z) \\sim \\frac{S(z) m!", "A_k}{(x_k-z)^{m+1}}$ as $z \\rightarrow x_k$ , in which $A_k > 0$ and $S(x)$ has no zeros in $[x_k, x_{k+1}]$ .", "Thus $n_{k,m}$ has the opposite parity to $m$ , and the result follows.", "$\\Box $ Assume henceforth that $U^{(m)}$ has finitely many non-real zeros.", "Since all but finitely many zeros and poles of $U$ are real, [28] and Proposition REF imply that, as $r \\rightarrow + \\infty $ , $\\mathfrak {T}(r, U^{\\prime }/U) = O( \\log r ) \\quad \\hbox{and} \\quad N(r, 1/U^{(m)}) \\le 2 N(r, U) + O( \\log r) \\le (2+o(1)) N(r, U) .$ Lemma 8.1 $U$ has finite order.", "Proof.", "Suppose first that $m$ is even.", "Proposition REF gives $k_0 \\in \\mathbb {N}$ such that if $k \\in K$ and $|k| \\ge k_0$ then $U^{(m)}$ has a simple zero $t_k$ with $x_k < t_k < x_{k+1}$ and $x_k t_k > 0$ .", "Moreover, by assumption and Proposition REF , all but finitely many zeros of $U^{(m)}$ belong to the set $\\lbrace t_k \\rbrace $ .", "Hence the product $\\Pi _1(z) =\\prod _{k \\in K, |k| \\ge k_0}\\frac{1-z/t_k}{1-z/x_k}$ converges and maps $H$ into $H$ (by the same proof as in Lemma REF ), and $\\frac{U^{(m)}}{U} = \\Pi _1 U_1,$ where $U_1$ has finitely many zeros and all but finitely many poles of $U$ are poles of $U_1$ .", "The first estimate of (REF ) and standard properties of the Tsuji characteristic together lead to $\\mathfrak {T}(r, U^{(m)}/U) = O( \\log r )$ as $r \\rightarrow + \\infty $ .", "Applying (REF ) to $\\Pi _1$ , combined with the same inequality of Levin and Ostrovskii [30] as used in Lemma REF then gives, as $R \\rightarrow + \\infty $ , $\\frac{T(R, 1/U_1 )}{2 R^2} &\\le & \\int _R^\\infty \\frac{ T(t, 1/U_1)}{t^3} \\, dt \\\\&\\le & \\int _R^\\infty \\frac{ m(t, 1/U_1)}{t^3} \\, dt + O \\left( \\frac{\\log R}{R^2} \\right)\\\\&\\le & \\int _R^\\infty \\frac{m(t, U/U^{(m)}) + m(t, \\Pi _1)}{t^3} \\, dt + O \\left( \\frac{\\log R}{R^2} \\right)\\\\&\\le & 2 \\int _R^\\infty \\frac{\\mathfrak {T}(t, U/U^{(m)})}{t^2} \\, dt + O \\left( \\frac{\\log R}{R^2} \\right)= O \\left( \\frac{\\log R}{R} \\right).$ Thus $U_1$ has order at most 1 in the plane, and so, by (REF ) applied to $\\psi $ , $T(r, U) = m(r, U) + N(r, U) \\le m(r, e^P) + O( \\log r ) + N(r, U_1) ,$ which implies that the order of $U$ is at most the degree of $P$ .", "When $m$ is odd the argument is slightly more complicated.", "In this case, there exists $k_0 \\in \\mathbb {N}$ such that if $k \\in K$ and $|k| \\ge k_0$ then $U^{(m)}$ has in $(x_k, x_{k+1})$ either (a) no zeros at all, or (b) two zeros $u_k, v_k$ , these possibly coinciding but having the same sign as $x_k$ .", "This time let $\\Pi _1(z) =\\left(\\prod \\frac{1-z/u_k }{1-z/x_k} \\right)\\left(\\prod \\frac{1-z/v_k }{1-z/x_k} \\right)$ with the products over those $k \\in K$ with $|k| \\ge k_0$ such that case (b) arises, and each mapping $H$ into $H$ .", "Applying (REF ) twice then gives $m(r, \\Pi _1) = O( \\log r) $ as $r \\rightarrow + \\infty $ .", "Now define $U_1$ by (REF ): again $U_1$ has finitely many zeros and, since $m \\ge 3$ , all but finitely many poles of $U$ are poles of $U_1$ .", "The remainder of the proof then proceeds as before.", "$\\Box $ Lemma 8.2 Let $K_\\varepsilon = \\lbrace z \\in \\mathbb {C}: |z| \\ge 1, \\varepsilon \\le | \\arg z | \\le \\pi - \\varepsilon \\rbrace $ , where $\\varepsilon $ is small and positive, and let $n \\in \\mathbb {N}$ .", "Then $U$ satisfies on $K_\\varepsilon $ , $T_n(z) = \\frac{ U^{(n)}(z)}{U(z)} = P^{\\prime }(z)^n (1 + o(1)) \\quad \\hbox{as $z \\rightarrow \\infty $.", "}$ Proof.", "This is standard and is proved by induction on $n$ .", "For $n=1$ , (REF ) is an immediate consequence of (REF ) and (REF ).", "Next, it may be assumed that $n \\ge 1$ and (REF ) holds on $K_{\\varepsilon /2}$ , so that (REF ) for $n+1$ follows from Cauchy's estimate for derivatives and the relation $T_{n+1} = T_n^{\\prime } + T_n T_1$ .", "$\\Box $ Lemma 8.3 Let $\\delta , \\sigma \\in (0, 1)$ .", "Then $U$ satisfies $(m+1 - \\delta ) N(r, U) \\le N(r, 1/U^{(m)})$ as $r \\rightarrow \\infty $ in a set of lower logarithmic density at least $1 - \\sigma $ .", "Proof.", "Since $T_m$ has finite order of growth $\\rho (T_m)$ , Lemma REF gives a positive constant $C_1$ , depending only on $\\sigma $ and $\\rho (T_m)$ , such that $T(2r, 1/T_m) \\le T(2r, T_m) + O(1) \\le C_1 T(r, T_m)$ for all $r$ in a set $F_1 \\subseteq [1, \\infty )$ having lower logarithmic density at least $1-\\sigma $ .", "Let $N_0(r,1/U^{(m)} )$ count common zeros of $U^{(m)}$ and $U$ , each such zero counted only once.", "The fact that all but finitely many poles and zeros of $U$ are real, simple and interlaced implies that $(m+1) N(r, U) &=& m N(r, U) + N(r, 1/U) + O( \\log r) \\\\&\\le & N(r, T_m) + N_0(r,1/U^{(m)}) + O( \\log r) \\\\&\\le & T(r, 1/T_m) + N_0(r,1/U^{(m)}) + O( \\log r) \\\\&=& m(r, U/U^{(m)}) + N(r, U/U^{(m)}) + N_0(r,1/U^{(m)}) + O( \\log r) \\\\&\\le & N(r, 1/U^{(m)}) + m(r, U/U^{(m)}) + O( \\log r)$ as $r \\rightarrow + \\infty $ .", "Now let $\\varepsilon $ be small and positive: then (REF ) implies that the contribution to $m(r, U/U^{(m)})$ from $K_\\varepsilon $ is bounded as $r \\rightarrow + \\infty $ .", "Apply Lemma REF to $1/T_m = U/U^{(m)}$ , with $R = 2r$ and $\\mu (r) = 4 \\varepsilon $ .", "In view of (REF ) this shows that, as $r \\rightarrow \\infty $ in $F_1$ , $(m+1) N(r, U) &\\le & N(r, 1/U^{(m)}) + O( \\log r) + 88 \\varepsilon \\left( 1 +\\log \\frac{1}{4 \\varepsilon } \\right) T(2r, 1/T_m ) \\\\&\\le & N(r, 1/U^{(m)}) + O( \\log r) + 88 \\varepsilon \\left( 1 +\\log \\frac{1}{4 \\varepsilon } \\right) C_1 T(r, T_m ) \\\\&\\le & N(r, 1/U^{(m)}) + O( \\log r) + 88 \\varepsilon \\left( 1 +\\log \\frac{1}{4 \\varepsilon } \\right) C_1 N(r, T_m ) \\\\&\\le & N(r, 1/U^{(m)}) + O( \\log r) + 88 \\varepsilon \\left( 1 +\\log \\frac{1}{4 \\varepsilon } \\right) C_1 (m+1) N(r, U ) .$ Because $\\varepsilon $ may be chosen arbitrarily small, while $C_1$ does not depend on $\\varepsilon $ , (REF ) follows.", "$\\Box $ Since $m \\ge 2$ , (REF ) and (REF ) together give a contradiction, and Theorem REF is proved.", "$\\Box $" ], [ "Proof of Theorem ", "Let $f$ be as in the hypotheses and assume that $f^{\\prime \\prime }$ has finitely many non-real zeros.", "Denote by $X$ the set of poles and zeros of $f$ .", "If $X$ is neither bounded above nor bounded below, using a translation makes it possible to assume that the poles $x_k$ and zeros $y_k$ satisfy $x_k < y_k < x_{k+1}$ and $x_k/y_k > 0$ for each $k$ .", "Hence $f = \\psi e^h$ where $\\psi $ is defined as in (REF ) and maps $H$ into itself, while $h$ is an entire function.", "If $h$ is constant then (REF ) follows from [28], as does (REF ) if $f^{\\prime \\prime }$ has only real zeros.", "Furthermore, if $h$ is non-constant then a contradiction arises via part (ii) or (iii) of Theorem REF .", "It remains only to consider the case where $X$ is bounded above or below, and here it may be assumed that all zeros and poles of $f$ are positive.", "If $\\min X$ is a pole of $f$ then the argument of the previous paragraph goes through unchanged, and delivers (REF ) or (REF ), neither of which is compatible with $X$ being bounded below.", "Finally, if $\\min X$ is a zero of $f$ then again $f = \\psi e^h$ with $h$ entire and $\\psi (H) \\subseteq H$ , but this time $-1/\\psi $ is formed as in (REF ) using the poles and zeros of $1/f$ , and again the previous argument may be deployed.", "$\\Box $ School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK [email protected]" ] ]
2105.11762
[ [ "BoundarySqueeze: Image Segmentation as Boundary Squeezing" ], [ "Abstract This paper proposes a novel method for high-quality image segmentation of both objects and scenes.", "Inspired by the dilation and erosion operations in morphological image processing techniques, the pixel-level image segmentation problems are treated as squeezing object boundaries.", "From this perspective, a novel and efficient \\textbf{Boundary Squeeze} module is proposed.", "This module is used to squeeze the object boundary from both inner and outer directions, which contributes to precise mask representation.", "A bi-directionally flow-based warping process is proposed to generate such squeezed feature representation, and two specific loss signals are designed to supervise the squeezing process.", "The Boundary Squeeze module can be easily applied to both instance and semantic segmentation tasks as a plug-and-play module by building on top of some existing methods.", "Moreover, the proposed module is light-weighted, and thus has potential for practical usage.", "Experiment results show that our simple yet effective design can produce high-quality results on several different datasets.", "Besides, several other metrics on the boundary are used to prove the effectiveness of our method over previous work.", "Our approach yields significant improvement on challenging COCO and Cityscapes datasets for both instance and semantic segmentation, and outperforms previous state-of-the-art PointRend in both accuracy and speed under the same setting.", "Codes and models will be published at \\url{https://github.com/lxtGH/BSSeg}." ], [ "Introduction", "Image segmentation tasks comprehensively understand image content by densely predicting binary masks for each object (instance segmentation) or each category (semantic segmentation) [47], [10], [27].", "As for humans, image segmentation is precisely annotated in another paradigm, where the efforts are focused on mask boundaries.", "In this paper, we aim to improve image segmentation by also emphasizing precise localization of mask boundaries as human labeling.", "Figure: Illustration of the boundary squeezing process.", "In (a), the red arrows mean squeeze boundary features from the object's outside (contraction process), while the blue arrows mean squeeze boundary features from inside of the object (expansion process).", "These two processes complement each other and help generate accurate boundary (green line in (b)).", "Best viewed on screen and zoom in.Recent state-of-the-art works for both instance segmentation and semantic segmentation are based on fully convolution network (FCN) [47], [78], [11], [27], [38].", "With FCN, features are densely generated on a spatial grid, where mask predictions are accomplished by classifying features in a Region of Interest (RoI) or whole image.", "These methods treat all pixels equally, and ignore the challenge of generating discrimative features for boundary pixels.", "Accordingly, non-uniform representations [15], [38] are proposed to process boundary pixels in a special way.", "Boundary-Preserving Mask R-CNN [15] adds a boundary branch to refine the mask, while PointRend [38] adaptively selects a non-uniform set of points to refine during the upsampling process.", "However, both works have several shortcomings.", "For the former, directly learning the object boundaries is hard.", "For the latter, point-based sampling and rendering along the boundary can not guarantee the structural failure case of objects.", "As shown in the first row of Fig.", "REF , the baseball bat has a similar appearance to the background wall.", "The rendering module in PointRend fails to classify the background and foreground things since there is no boundary bound and the points for rendering are sampled from coarse features without guidance.", "Figure: Example result pairs: (a) Left: from Mask R-CNN  vs. PointRend  and vs. Our Boundary Squeeze, using ResNet-50  backbone.", "(b) Right: from DeeplabV3+  PointRend  vs. Our Boundary Squeeze.", "Note that our model predicts masks with more structural object boundaries where the PointRend fails to predict the finer details.As mentioned previously, in this work, we want to unify the two different segmentation problems via boundary bound.", "We propose a conceptually new approach named Boundary Squeeze for modeling segmentation representation.", "Our insight is squeezing the object boundary from two different directions, leading to a much finer boundary feature representation.", "The illustration is shown in Fig.", "REF .", "Such precise boundary feature representation benefits the segmentation tasks and achieves significant improvement on mask quality.", "Compared with the previous rendering based [38], our method can preserve more structural information and avoid the shortcomings of randomly selected points.", "The segmentation results of our method are shown in the third and fifth columns of Fig.", "REF .", "Our approach is inspired by the previous traditional binary segmentation, such as differential edge detection and morphological processing [22], [51].", "Our approach has two opposite directions to squeeze the boundary: one from outer parts and the other from the inner parts.", "We design a novel flow-based warping module to let the network learn to achieve these purposes.", "In particular, there are two independent modules supervised by two different mask labels.", "One is the dilation (or contraction) results of original masks, while the other is erosion (or expansion) results of original masks.", "The dual supervisions from both maps work complementary and result in much more satisfactory boundary results.", "We term our new network module as the Boundary Squeeze module.", "Our Boundary Squeeze is a general module that allows many implementations on different frameworks.", "It can be a plug-in module for the current state-of-the-art network to refine object boundary in an end-to-end manner.", "For example, it can be appended into Mask R-CNN-like network [27], [4] by replacing the mask head with our proposed Boundary Squeeze mask head for instance segmentation task.", "It can also be appended at the end of modern semantic segmentation networks such as DeeplabV3+ [11] for refining the semantic segmentation.", "These will be detailed in the latter section.", "We evaluate our Boundary Squeeze on the instance and semantic segmentation tasks using COCO [44], Cityscapes [16], BDD [72] benchmarks.", "From qualitative results, our module outputs sharp and more structural boundaries, which are better than recent state-of-the-art PointRend [38] as illustrated in Fig.", "REF .", "We also observe quantitative improvements over the PointRend through various metrics, including standard intersection-over-union based metrics for these tasks (mask AP and mIoU).", "Since these metrics are biased towards object-interior pixels and are relatively insensitive to boundary improvements [14], we also compute the boundary-aware metrics (F-score and recent proposed boundary AP [14]).", "Our method also achieves improvement over PointRend under the same setting [67] while running more efficiently.", "These results prove the effectiveness of our Boundary Squeeze.", "To summarize, our contributions have the following aspects: We propose Boundary Squeeze, a novel and efficient module by treating the segmentation as boundary squeezing process.", "The module is supervised via different labels, which can be obtained from mask annotation for free.", "Boundary Squeeze can be performed as a plug-and-play module by easily being deployed into current state-of-the-art segmentation methods, including Mask R-CNN [27] and DeeplabV3+ [11].", "Extensive experiments and analysis have verified the effectiveness of the proposed method.", "Our approach can produce finer boundary results.", "It has more structural outputs than PointRend [38].", "We achieve better results than PointRend and Boundary-Preserving Mask R-CNN [15] over different metrics on COCO [44] and Cityscapes datasets while running more efficiently.", "The proposed approach is also verified for the semantic segmentation task on Cityscapes [16] and BDD [72] datasets.", "It also improves strong DeeplabV3+ models [11] by a significant margin." ], [ "Related Work", "In this section, we will review the related work in three different aspects: boundary processing, instance segmentation and semantic segmentation.", "Boundary Processing: Modeling boundary has a long history in computer vision community.", "Boundary detection has been a fundamental computer vision task as it provides an essential cue for recognition.", "In the era of deep learning, some CNN-based methods have significantly pushed the development of this field, such as [71], [48], [39], [28], [20], [57].", "Snake [34] and Deep Snake [53] refine the initial object contours recursively through specific operations such as circular convolution.", "CASENet [74] proposes a challenging task of category-aware boundary detection.", "InstanceCut [37] adopts boundaries to partition semantic segmentation into instance-level segmentation.", "Different from previous works, our methods treat object segmentation as a boundary squeezing process where the boundary of the segmentation mask is squeezed into a thinner representation via specific supervisions.", "Instance Segmentation: Instance Segmentation aims to detect and segment each instance [17], [26].", "The two-stage pipeline Mask R-CNN and its variants [27], [30], [7], [4], [66], [58] first generate object proposals using Region Proposal Network (RPN) [55] and then predict boxes and masks on each RoI feature.", "Further improvements have been made to boost its accuracy.", "PANet [45] introduces bottom-up path to enrich FPN features, and Mask Scoring R-CNN [30] addresses the misalignment between the confidence score and localization accuracy of predicted masks.", "HTC [7] extends the Cascade Mask R-CNN [4] by augmenting a semantic segmentation branch.", "SCNet [61] balances the IoU distribution of the samples for both training and inference.", "Several single-stage methods [69], [12], [60], [6], [77], [3], [63], [64] achieve significant progress and comparable results with two-stage pipelines.", "Meanwhile, there are several bottom-up approaches [50], [19], [46].", "Accurate boundary localization can explicitly contribute to the mask prediction for instance segmentation, and there are many recent work [38], [15], [13] focus on boundary modeling.", "In particular, PointRend [38] handles the image segmentation upsampling procedure as a rendering process via a shared multiple-layer network.", "Boundary preserving Mask R-CNN [15] proposes to predict instance-level boundaries to augment the mask head.", "Although those work achieves better segmentation results, the extra computation results in slow inference speed which will be shown in experiment section.", "Rather than direct fusing edge information into mask head, our method treats the instance segmentation as a boundary squeezing process and can fully utilize the limited mask annotation.", "Semantic Segmentation: Semantic Segmentation is required to assign a semantic label for each pixel.", "Fully convolutional networks (FCNs) [47] are the foundation of modern semantic segmentation approaches.", "Recent approaches try to overcome the limited receptive field of FCNs by multi-scale pooling [78], [68], dilated convolution [73], [9], [10], [11], non-local operators [62], [23], [41], [32], [76], [75].", "There are also several works on modeling semantic segmentation boundaries [1], [2], [35], [8], [59], [40].", "Previous works obtain better boundary localization by structure modeling, such as boundary neural fields [1], affinity field [35], random walk [2].", "The work [8], [24] adopts edge information to refine network output by predicting edge maps from intermediate CNN layers.", "Zhu et al.", "[82] uses boundary relation loss to utilize coarse predicted segmentation labels for data augmentation.", "Gated-SCNN [59] adds a boundary stream to learn detailed low-level information by gated convolution.", "Our method can be a plug-in method and can be easily extended into semantic segmentation methods.", "We show our method can improve the state-of-the-art semantic segmentation method on different datasets." ], [ "Method", "Overview: In this section, we will first describe the motivation of our approach.", "Then we will introduce some notations of our method.", "After that, the details of our Boundary Squeeze Module will be presented.", "In particular, we take instance segmentation setting for the illustration.", "Then the supervision signals of our method will be presented.", "Finally, we will describe how to deploy our module on two segmentation tasks: instance segmentation and semantic segmentation.", "Figure: Network Architecture of our proposed Boundary Squeeze.", "(a), The entire pipeline of our Boundary Squeeze network with Mask R-CNN as the example.", "(b), The details of our proposed Boundary Squeeze Module.", "SFG: Squeeze Feature Generator.", "(c), The details of Squeezed Feature Generator.", "Best view it in color and zoom in." ], [ "Boundary Squeeze Module (BSM)", "Motivation: As mentioned in the previous section, our method is inspired by traditional morphological processing [22], [51].", "The boundary squeezing process is the simulation of the dilation and erosion processes via a learned neural network.", "Rather than regular processing of the binary image, we process the feature map, which makes the entire process into an end-to-end manner, and such a network can be trained in a data-driven way.", "The second advantage of the processing feature is that this design is general and flexible and can be applied in different segmentation tasks, such as semantic segmentation and instance segmentation.", "Notation: We introduce several notations for the following sections in this part.", "We use the Mask R-CNN-based instance segmentation setting to illustrate our method.", "$F_{RoI}$ is produced using RoIAlign [27] from P2-P5 FPN [42] features.", "We also use RoIAlign and P2 FPN features to produce $F_{RoI}^{^{\\prime }}$ .", "Note that the size of $F_{RoI}$ is 14 $\\times $ 14, while the original size of $F_{RoI}^{^{\\prime }}$ is 28 $\\times $ 28 then one 1 $\\times $ 1 convolution layer is used to downsample the resolution of $F_{RoI}^{^{\\prime }}$ to 14 $\\times $ 14.", "We denote the features for dilation (or called contraction) branch as $F_{contraction}$ and the features for erosion (or called expansion) branch as $F_{expansion}$ .", "The loss function for each branch is $L_{contraction}$ and $L_{expansion}$ , respectively.", "The squeezed feature of the boundary branch is denoted as $F_{boundary}$ , and the boundary branch is supervised by $L_{boundary}$ .", "The ground truth of segmentation branch, boundary branch, contraction branch, and expansion branch are $G_s$ , $G_b$ , $G_c$ , and $G_e$ separately.", "Details of Boundary Squeeze Module: As shown in Fig.", "REF (b), the BSM takes two distinct RoI features ($F_{RoI}$ and $F_{RoI}^{^{\\prime }}$ ) as inputs and outputs three different features, including $F_{boundary}$ , $F_{contraction}$ , and $F_{expansion}$ .", "First, $F_{RoI}$ and $F_{RoI}^{^{\\prime }}$ are added together to get $F_{sum}$ .", "The choice of $F_{RoI}^{^{\\prime }}$ follows the design of the previous works [11], [38], [6], by introducing low-level and high-frequency details into the $F_{RoI}$ , which benefits the boundary squeezing process.", "Then $F_{sum}$ is fed into two parallel branches: contraction branch and expansion branch.", "In both branches, two 3 $\\times $ 3 convolution layers are first used to generate its unique features $F_{sum}^{^{\\prime }}$ , after that, a Squeezed Feature Generator (SFG) is utilized to produce squeezed feature ($F_{contraction}$ or $F_{expansion}$ ).", "$F_{contraction}$ and $F_{expansion}$ are used to squeeze boundary features from the outside and inside of an object, respectively.", "SFG will be detailed in the following parts.", "Finally, both squeezed features and $F_{RoI}$ are summed together in a residual manner [29] as the final squeezed boundary feature $F_{boundary}$ , formulated as $F_{boundary} = F_{contraction} + F_{expansion} + F_{RoI}$ .", "Squeezed Feature Generator: We propose a flow-based approach to generate the squeezed features.", "The inputs of SFG are $F_{sum}^{^{\\prime }}$ and $F_{RoI}$ , and the output is one squeezed feature i.e.", "$F_{contraction}$ or $F_{expansion}$ .", "Specifically, following [21], we first concatenate $F_{sum}^{^{\\prime }}$ and $F_{RoI}$ then adopt one $3 \\times 3$ convolution layer to generate the flow field $\\delta \\in \\mathbb {R}^{14 \\times 14 \\times 2}$ for each RoI.", "With $\\delta $ , features of each position $p_{l}$ on the standard spatial grid $F_{sum}^{^{\\prime }}$ are mapped to a new position $\\hat{p}$ via $p_{l}+\\delta _{l}(p_{l})$ .", "We use the differentiable bilinear sampling mechanism to achieve this process.", "This sampling mechanism, proposed in the spatial transformer networks [33], [81], linearly interpolates the values of the four nearest neighbor pixels of $p_{l}$ from $F_{sum}^{^{\\prime }}$ .", "This process can be formulated using Equ.", "REF .", "$F_{squeeze}(p_{x}) = \\sum _{p\\in \\mathcal {N}(p_{l})} w_pF_{sum}^{^{\\prime }}(p)$ where $w_p$ calculated from flow map $\\delta $ , represents bilinear kernel weights on the warped spatial gird.", "$\\mathcal {N}$ represents the number of involved neighboring pixels.", "The whole process of SFG is illustrated in Fig.", "REF (c).", "Essentially, the warping process is the deformation of the entire RoI feature, and one can also choose other methods to achieve the same purpose, like deformable convolution [18].", "However, though deformable convolution results in a similar performance, it needs more inference time, which can be found in the experiment section.", "Figure: Visualization of Contraction branch Ground Truth (CGT) and Expansion branch Ground Truth (EGT).", "White pixels are positive while black pixels are negative.", "Boundary pixels are labeled as red to distinguish positive pixels and negative pixels.Supervision Signals: The different supervision signals control the warping operation's direction of the contraction branch and expansion branch.", "Given a binary mask annotation $G_s$ , we use Equ.", "REF to generate $G_c$ and $G_e$ where $H_{dilation}$ is the dilation operation in morphological processing which can extend the $G_s$ 's boundary into the background, and $H_{erosion}$ is the erosion operation which is used to shrink $G_s$ from its original border.", "$K$ is the kernel size of $H_{dilation}$ and $H_{erosion}$ .", "$\\left\\lbrace \\begin{array}{lr}G_{c} = H_{dilation}(G_s, K) - G_{s}, & \\\\G_{e} = G_{s} - H_{erosion}(G_s, K)&\\end{array}\\right.$ The visualization of some $G_c$ and $G_e$ can be found in Fig.", "REF .", "Following [15], we use the Laplacian operator to generate $G_b$ .", "The supervision signal of the segmentation branch is the original $G_s$ .", "Figure: Visual Interpretation of Our proposed BSM.", "(a) The top sub-figure shows the comparison results on instance feature representation of boundary.", "(Boundary Only supervision and our BSM results).", "GT means the Ground Truth mask.", "Our BSM outputs thinner and more precise boundaries, and it also contains more structural information.", "(b) The bottom sub-figure visualizes the two types of learned flow fields.", "Best view it on the screen and zoom in.Visual Interpretation: To better understand our boundary squeezing process, we present two types of visual interpretations in Fig.", "REF .", "The first is the instance-aware feature visualization and the second is the visualization of two flow fields learned from two independent branches.", "As shown at the top of Fig.", "REF , compared with directly predict boundary, like [1] does, our BSM produces a clearer and thinner boundary feature representation $F_{boundary}$ .", "Note that we utilize Principal Component Analysis (PCA) [65] to reduce the number of $F_{boundary}$ 's dimension into three for visualization.", "This comparison result proves our motivation: squeezed boundary feature results in a more discriminative and structural-preserving representation.", "The bottom parts of Fig.", "REF show two different flow fields.", "For the contraction branch, the direction of learned flow points is to the inner parts of instance objects, while the direction of learned flow mainly expands to the outside of instance objects in the expansion branch.", "Both flows squeeze the object boundary in a complementary way and benefit each other, proven in the experiment section." ], [ "Boundary Squeeze on Segmentation Task", "In this section, we will describe how to deploy our proposed BSM on two segmentation tasks.", "Instance Segmentation: For instance segmentation task, we adopt the Mask R-CNN [27] as the baseline method.", "As shown in Fig.", "REF (a), the proposed BSM is inserted after the four consecutive 3 $\\times $ 3 convolution layers of Mask R-CNN head.", "And the final prediction head One deconvolution layer with 2 $\\times $ 2 kernel size, one ReLU layer, and one 1 $\\times $ 1 convolution layer.", "(denote as Pred in Fig.", "REF (a)) is the same with Mask R-CNN.", "The kernel size of dilation and erosion operations is set to 5 and the relevant experiments can be found in supplementary.", "The total loss $L_{instance}$ is the combination of multi-tasks learning, formulated as $L_{instance}=L_{detection}+L_{mask}$ .", "$L_{detection}$ is the loss of detection branch and $L_{mask}$ = $L_{segmentation}$ + $L_{boundary}$ + $L_{contraction}$ + $L_{expansion}$ .", "$L_{segmentation}$ is Binary Cross-Entropy loss (BCE), $L_{boundary}$ is the combination of BCE and Dice loss [49], $L_{contraction}$ and $L_{expansion}$ are Dice loss.", "During inference, we take the output of the segmentation branch as the model's output.", "Semantic Segmentation: For the semantic segmentation task, we choose DeeplabV3+ [11] as our baseline.", "In particular, the proposed BSM is appended at the end of DeeplabV3+ head (after the output of Atrous Spatial Pyramid Pooling).", "We treat the entire feature map as one RoI in instance segmentation task.", "The low-level feature is the same as the DeeplabV3+ design.", "For supervision signal generation, since most semantic segmentation datasets [16], , [72] have no separate background class, to simulate the expansion and contraction process, we adopt the binary setting for each class.", "We treat each semantic class as one binary mask, which means each class in the semantic segmentation label map is one RoI in the instance segmentation setting.", "Then we can fully utilize the same setting as instance segmentation.", "The kernel size of dilation and erosion operations is set to 15.", "The entire training process is supervised by both original Cross-Entropy loss and three extra losses (boundary loss and two independent Dice losses on two squeezing branches), which are the same as the instance segmentation task." ], [ "Experiment: Instance Segmentation", "Overview: We first conduct experiments on the instance segmentation task.", "We perform extensive ablation experiments on the challenging COCO dataset [44].", "We also present experimental results on the Cityscapes [16] and LVIS [25].", "Table: Ablation studies.", "We first verify the effectiveness of each branch of our model in (a).", "Then (b) provides some experiments with different loss functions of contraction and expansion branch.", "After that, we explore the influence of low-level RoI features in (c), and investigate the impact of information fusion of different branches in (d).", "All models are tested on COCO val2017.Dataset: (a) COCO contains 118k images for training, 5k images for validation, and 20k images for testing.", "Following the common practice [27], [15], [43], we train our models using the training set (train2017) and report results on the validation set (val2017) for ablation studies.", "We also report the results on the test set (test-dev2017) for comparing with other methods.", "(b) Cityscapes has 2975 images, 500 images, and 1525 images for training, validation, and test, respectively.", "For instance segmentation task, Cityscapes has 8 instance categories.", "(c) LVIS is a recently proposed dataset with the same images as COCO but has more than 1K categories.", "The annotations quality of Cityscapes and LVIS are much higher than COCO, especially on the boundary.", "Metrics: For these three datasets, we use the standard mask AP, the average precision over different IoU thresholds (from 0.5 to 0.95) as the primary metric.", "To better evaluate the boundary quality of the instance segmentation results, we use the newly proposed boundary AP [14] further in some experiments.", "Implementation details: We use the PyTorch library [52] and Detectron2 [67] to implement all the models.", "We use the ResNet-50 pre-trained on ImageNet [56] with FPN [42] as the backbone network, unless otherwise stated.", "All models are trained using 8 GPUs with stochastic gradient descent (SGD).", "Momentum and weight decay are set as 0.9 and 0.0001, respectively.", "For the COCO dataset, the training images are resized to a shorter side from 640 to 800 with a step of 32 pixels and their longer side less or equal to 1333.", "At inference time, images are resized to the short side of 800 pixels.", "In the ablation study, all models are trained for 90k iterations.", "The learning rate is initialized to 0.02 and reduced by a factor of 10 at iteration 60k and 80k.", "We use a mini-batch of 16 images (2 images per GPU).", "For the Cityscapes dataset, we use a mini-batch of 8 images (1 image per GPU).", "All models are trained on the training set for 24k iterations with the learning rate initialized to 0.01, then reduced to 0.001 at iteration 18k and tested on the validation set.", "At training time, images are resized randomly to a shorter edge from 800 to 1024 pixels with a step of 32 pixels, and their longer edge is less or equal to 2048 pixels.", "At inference time, images are resized to the shorted edge of 1024 pixels.", "For the LVIS dataset, we use LVIS$^*$v0.5 version.", "Following [38], the LVIS$^*$v0.5 dataset is constructed by keeping only the 80 COCO categories from LVISv0.5.", "All models are trained on COCO $train2017$ set with standard 1$\\times $ schedule [67] and tested on the validation set of LVIS$^*$v0.5 where images are resized to the short edge of 800 pixels.", "Each branch in BSM: We first verify the effectiveness of each branch of BSM using Mask R-CNN [27] as the baseline in Tab.", "REF (a).", "After adding boundary branch, contraction branch, and expansion branch separately, AP is improved by 0.8, 0.6 and 0.9, respectively.", "These results prove the effectiveness of each branch.", "Adding both contraction branch and expansion branch leads to 1.3 AP gain, which means these two branches can complement each other.", "Adding all three branches results in 1.7 AP gain, which indicates the effectiveness of using boundary supervision explicitly.", "Loss function design of contraction and expansion branch: Then we explore the choices of loss function on contraction and expansion branch in Tab.", "REF (b).", "Using Binary Cross-Entropy (BCE), weighted Binary Cross-Entropy (weighted BCE), Dice Loss [49] severally, the final AP is 36.2, 36.6, 36.9, respectively.", "The latter two achieve better performance since they can better handle the class imbalance problems.", "After combining Dice Loss with BCE or weighted BCE, there are slight performance drops, partially because cross-entropy loss focuses more on pixel-level difference [20] while contraction and expansion branch need to concentrate more on the similarity of two sets of image pixels.", "So we use Dice Loss as the loss function for contraction and expansion branches.", "Low-level RoI features: We present an ablation study on the importance of adding low-level RoI features $F_{RoI}^{^{\\prime }}$ .", "Low-level features can provide detailed positional information to improve the segmentation quality proven in previous work [11].", "We expect this conclusion can apply to our method as well.", "As shown in Tab.", "REF (c), using low-level RoI features can contribute to 0.4 AP gain.", "Information fusion among different features: We further investigate the effectiveness of information fusion among different branches' features.", "Tab.", "REF (d) offers some results of this aspect: adding mask features to the boundary branch in a residual connection manner results in 0.6 AP gain, which indicates this process can produce better feature representation for boundary prediction.", "Since there is some information gap between the final boundary feature and mask feature, we add one 1 $\\times $ 1 convolution layer when merging boundary features to mask features, and this process leads to 0.4 AP gain.", "After using the above two operations, our model gains 0.9 AP.", "Table: More analysis studies.", "(a) offers the results using different feature deformation methods.", "(b) is used to show the generality of our method where R means ResNet and X means ResNeXt .", "All results are reported on COCO val2017.", "Inference times are tested using one V100 GPU with single scale testing.Different feature deformation methods: In our implementation, we use feature warping to generate contraction feature and expansion feature.", "We also compared some other similar methods to produce these features and provide results in Tab.", "REF (a).", "Our baseline model does not use any feature deformation method where the AP is 36.0.", "After using DGMN [76], we find 0.6 AP gain with an extra 8ms inference time.", "DCNV2 [80] and our feature warping method achieve a similar performance (36.8 and 36.9), but DCNV2 uses more inference time.", "So we choose the flow-based warping in our implementation.", "Different backbone networks: In addition to ResNet50, we carry out experiments on other backbone networks, including ResNet101 and ResNeXt101 [70] in Tab.", "REF (b).", "With the ResNet101 backbone, our method results in a gain over the baseline of 1.5 AP.", "We find 1.6 AP gain of our approach on the ResNeXt101 backbone.", "These decent results prove the generality of our method.", "Table: Comparison results with baseline and related methods on COCO val2017.", "All the models are trained and tested using the same setting for fair comparison.Table: Comparisons with state-of-the-art methods on COCO test-dev2017.", "All models are trained on COCO train2017.", "Aug. means using multi-scale training data augmentation.", "Sched.", "means learning rate schedule during training.", "1×\\times is 90K90K iterations, 2×\\times is 180K180K iterations, 3×\\times is 270K270K iterations and so on, 18e means 18 epochs.", "R means ResNet and X means ResNext .", "* ^* means our implementation.Efficiency and Effectiveness: Both accuracy and efficiency are important factors to consider in practical use.", "We further compare the inference time, mask AP (AP$^{\\text{mask}}$ ), and recently proposed boundary AP (AP$^{\\text{boundary}}$ )  [14] of our method with baseline Mask R-CNN and other related works including, PointRend [38], BMask R-CNN [15], and B2Inst-BlendMask [36].", "Compared with AP$^{\\text{mask}}$ , AP$^{\\text{boundary}}$ can better evaluate the accuracy of the boundary.", "All models use ResNet50 as backbone and use the same inference setting except B2Inst-BlendMask$\\footnote {Since the authors of B2Inst do not release their codes and models, all results of B2Inst-BlendMask come from the B2Inst paper.", "}$ .", "Results are shown in Tab.", "REF .", "Our method has the fastest inference speed among related methods and outperforms other works on AP$^{\\text{mask}}$ and AP$^{\\text{boundary}}$ .", "Table: Experiments results of BSM and another three methods on Cityscapes and LVIS datasets.", "AP and AP 50 _{50} are reported on the validation set of Cityscapes, while AP * ^* and AP 50 * ^*_{50} are reported on the validation set of LVIS * ^*v0.5." ], [ "Main Results", "Comparison with state-of-the-art methods on COCO: We compare BSM to the state-of-the-art instance segmentation methods on COCO $\\textit {test-dev2017}$ in Tab.", "REF .", "BSM uses 3$\\times $ learning rate schedule with ResNet101 as the backbone network achieves 40.4 AP, which is much better than other state-of-the-art methods.", "When using cascade architecture, BSM also conducts state-of-the-art results.", "Visual results comparison: We also provide some visual results on COCO datasets in Fig.", "REF .", "Our model can generate better segmentation results than Mask R-CNN and PointRend.", "PointRend focuses on refining the segmentation results around boundary pixels.", "BSM can segment instances well in three challenging situations: (a) the appearance of instances is very similar to its surrounding background, (b) complicated scenes, (c) instances overlapping.", "We argue that the reason why our model can produce better segmentation results than the other two methods is that our method can generate accurate boundaries.", "And these precise boundaries can help locate instances and distinguish different instances well.", "Results on Cityscapes and LVIS datasets: Besides the COCO dataset, we also verify the effectiveness and generalization of BSM on Cityscape [16] and LVIS [25] datasets.", "The annotations of both datasets have significantly higher quality.", "We compare our method with the baseline model Mask R-CNN [27] and another two recent methods, including PointRend [38] and BMask R-CNN [15].", "Tab.", "REF shows the results on both datasets.", "Our BSM achieves the best results with ResNet-50 or ResNet-101 as the backbone network on both datasets.", "Figure: Visualization and comparison results of Mask R-CNN , PointRend  and our BSM on the COCO 𝑣𝑎𝑙2017\\textit {val2017} using ResNet-50  with FPN  as backbone.", "Compared with the other two methods, BSM can produce better segmentation results.", "Best viewed on screen and zoom in.Overview We also carry out experiments on semantic segmentation task to verify the generality of our BSM.", "We report results based on DeeplabV3+ [11].", "Dataset: (a) Cityscapes semantic segmentation set [16] , compared with Cityscapes instance segmentation set introduced in Sec.", "REF , Cityscapes semantic segmentation set has 19 categories.", "Both datasets have the same training, validation and test images.", "(b) BDD [72] is a recent proposed road scene dataset with 10000 images in total.", "For dataset split, 7000, 1000, and 2000 images are used for train, validation and test, respectively.", "For both datasets, we train our models on their training set and report results on validation set.", "Metrics: In semantic segmentation, mean IoU (mIoU) is the most commonly used metric.", "Besides, to evaluate the segmentation quality on boundary, we also report F-score on boundary [59], [54] with four thresholds.", "Implementation details: We use the DeeplabV3+ [11] as the baseline model and add our approach to it.", "Following the original paper, the output stride of backbone network is set to 8.", "We use the same training and testing setting for different methods (details are in the supplementary) for fair comparison.", "The architecture of BSM is the same as that for instance segmentation.", "We use the res2 features as low-level features to provide detailed location information.", "Main Results Quantitatively, we compare BSM with DeeplabV3+ and PointRend in Tab.", "REF .", "For Cityscapes dataset, with ResNet50 and ResNet101 as backbone network, our method obtains 1.0 and 1.1 mIoU gain over PointRend.", "With the reduction of F-score threshold, our method achieves a larger performance gain, which means our model can generate more precise boundaries.", "For BDD dataset, our BSM obtains consistent performance improvements as well.", "Visual Results In Fig.", "REF , we give the visualization on semantic segmentation task.", "As shown in Fig.", "REF (b) and (c), we observe the clear and thin boundary feature and boundary prediction.", "The two opposite flow fields, are shown in Fig.", "REF (d) and (e).", "Final results are shown in Fig.", "REF (f).", "Besides, as shown in the right part of Fig.", "REF , BSM has better visual results than PointRend on two different types of objects: things and scenes.", "Both visual examples prove our motivation: boundary squeezing process can generate more accurate boundary via learned flow field warping and improve segmentation result.", "Table: Comparison results of DeeplabV3+, PointRend and BSM.", "All results are reported on the validation set.", "X-px means X pixels along the boundaries.", "All the models are trained in the same setting." ], [ "Conclusion", "In this paper, we propose a conceptually new idea to model the segmentation process as boundary squeezing.", "We design the opposite flow-based modules named Boundary Squeeze Module (BSM).", "The supervisions of BSM can be obtained via dilation operator, erosion operator of the existing annotations.", "The boundary squeezing process can be achieved via flow-based warping.", "After BSM, the feature contains precise boundary information, and the mask quality is much better.", "We verify our proposed BSM on two different tasks, including instance segmentation and semantic segmentation.", "Extensive results prove that our BSM outperforms previous work PointRend in various settings.", "We also show significant gains over various baselines on both COCO and Cityscapes datasets.", "One shortcoming of BoundarySqueeze is the limited resolution problem which will be our future work." ] ]
2105.11668
[ [ "Guiding the Growth: Difficulty-Controllable Question Generation through\n Step-by-Step Rewriting" ], [ "Abstract This paper explores the task of Difficulty-Controllable Question Generation (DCQG), which aims at generating questions with required difficulty levels.", "Previous research on this task mainly defines the difficulty of a question as whether it can be correctly answered by a Question Answering (QA) system, lacking interpretability and controllability.", "In our work, we redefine question difficulty as the number of inference steps required to answer it and argue that Question Generation (QG) systems should have stronger control over the logic of generated questions.", "To this end, we propose a novel framework that progressively increases question difficulty through step-by-step rewriting under the guidance of an extracted reasoning chain.", "A dataset is automatically constructed to facilitate the research, on which extensive experiments are conducted to test the performance of our method." ], [ "Introduction", "Although the state-of-the-art Question Answering (QA) systems can achieve competitive performance in answering simple questions that can be trivially solved via word and context matching , , their performance substantially declines when answering harder questions that require deep reasoning.", "To solve this issue, Deep Question Generation (DQG) aims at generating questions that require reasoning over rich context.", "It is an emerging topic that attracts more and more attention due to its great potential in strengthening the reasoning ability of AI systems.", "Existing approaches on DQG suffer from the problem of weak control over generation process , or requiring more structured data other than free text as input , .", "In our work, we explore the task of DQG in a more controllable manner.", "Specifically, we focus on Difficulty Controllable Question Generation (DCQG) from free text under a answer-unaware setting (i.e., instead of given in advance, the answer is generated in pair with the question).", "DCQG can help to generate question and answer pairs of various difficulty levels without extra human labeling.", "In addition, controllability over the difficulty of generated data facilitate training with certain curriculum-learning-based methods, and designing reading comprehension exams of various difficulty levels for educational purpose.", "However, there exist three major challenges in DCQG: Figure: An example of generating a complex question via step-by-step rewriting based on the extracted reasoning chain.First, how to define the difficulty level?", "Previous works on generating deep questions roughly divide questions into two classes, e.g., whether the given context gives sufficient information to answer the question , or whether a QA model can correctly answer it .", "However, such classification may not consist with human perspective and lack interpretability.", "Our work explicitly defines the difficulty level of a question as the number of inference steps required to answer it.", "Intuitively, a complex question requires more hops of reasoning to answer than a simpler one.", "For example, in Fig.", "REF , $Q_2$ is more difficult than $Q_1$ as it requires to infer “Top Gun” is “the film directed by Tony Scott” before answering “Who starred Top Gun”.", "Second, how to generate questions of different difficulty levels in a controlled manner?", "Previous works implement controllable QG simply by incorporating a difficulty variable into the hidden states before decoding .", "In our work, to more explicitly model the question difficulty and better control the generation process, we propose a step-by-step rewriting mechanism to progressively increase difficulties of the generated questions.", "Specifically, we first transform the given free text into a context graph.", "Then, we sample the answer and the reasoning chain of the question from this graph.", "Finally, we design a question generator and a question rewriter, to generate an initial simple question with the selected inputs and step-by-step rewrite it into more complex ones.", "As shown in Fig.REF , “Tom Cruise” is the selected answer, and $Q_1$ is the generated initial simple question.", "We progressively adapt $Q_1$ into $Q_5$ by incorporating more information in the reasoning chain (e.g., “Top Gun is directed by Tony Scott.” and “Top Gun is a 1986 action film.”).", "By clearly modeling the reasoning chain of the question and applying it to our step-by-step rewriting mechanism, we well control the difficulty of the generated question, as well as the logical reasoning process to answer it.", "Third, how to build a suitable training dataset for DCQG?", "Instead of building a dataset from scratch with intensive human efforts, we design effective strategies to automatically construct the training data from existing QA datasets.", "Specifically, we utilize HotpotQA , a QA dataset where most questions require two inference steps to answer.", "By decomposing each question into two sub-questions and utilizing one of them as an intermediate result, each case of HotpotQA can be factorized into a chain of questions to serve as the training data for our task.", "Note that although the questions in HotpotQA are mostly of two-hop reasoning complexity, our framework is not restricted to only generating questions with complexities up to two hops.", "The question rewriter learns how to transform an $n$ -hop question to ($n+1$ )-hop, empowering us to generate out-of-distribution questions of more than two hops.", "Human evaluation results show that, compared with baseline models, our system can generate deep questions in a more well-formed and concise way.", "In addition, we successfully boost performance of multi-hop QA systems, by augmenting the HotpotQA dataset with a large scale of question-answer pairs generated from our method.", "Further experiments show that our method can also generate out-of-distribution questions of more hops.", "1. why is controllable QG difficult?", "2. summarize contributions" ], [ "Related Work", "Deep Question Generation Most of the previous QG researches , , mainly focus on generating single-hop questions like the ones in SQuAD .", "However, such questions are well-studied and can be easily answered through word and context matching.", "In the hope that AI systems could provoke more in-depth interaction with humans, Deep Question Generation (DQG) aims at generating more challenging questions that require deep reasoning.", "Many recent works attempted to conquer this task with graph-based neural architecture.", "and generated complex questions based on knowledge graphs, but their methods could not be directly applied to QG from free text, which lacks clear logical structure.", "In sequential question generation, tended to use a dual-graph interaction to better capture context dependency.", "However, they considered all the tokens as nodes, which led to a very complex graph.", "tried to generate multi-hop questions from free text with the help of entity graphs constructed by external tools.", "Our work shares a similar setting with , and we further explore the problem of how to generate deep questions in a more controllable way.", "Missing: , Difficulty Controllable Question Generation DCQG is a relatively new task.", "classified questions as easy or hard according to whether they could be correctly answered by a BERT-based QA model, and controlled the question difficulty by modifying the hidden states before decoding.", "Another research on QG from knowledge graphs estimated the question difficulty based on popularity of the named entity and manipulated the generation process by incorporating the difficulty level into the input embedding of the Transformer-based decoder.", "In our work, we control the question difficulty based on the number of its reasoning hops, which is more explainable.", "Figure: An overview of our proposed framework.", "The selected reasoning chain is marked as light blue nodes.Question Rewriting It is another emerging trend in the recent researches, demonstrating benefits to both QG and QA tasks.", "With rewriting, QG models produced more complex questions by incorporating more context information into simple questions , , QA pipelines could also decompose the original complex question into multiple shorter questions to improve model performance , ." ], [ "Method", "Given input context text $\\mathcal {C}$ and a specific difficulty level $d$ , our objective is to generate a (question, answer) pair $({\\mathcal {Q}},{\\mathcal {A}})$ , where ${\\mathcal {A}}$ is a sub-span of $\\mathcal {C}$ and $\\mathcal {Q}$ requires $d$ -hop reasoning to answer.", "Fig.", "REF and Algorithm  give an overview of our proposed framework.", "First, we construct a context graph $\\mathcal {G}_{CG}$ corresponding to the given context, from which a subgraph $\\mathcal {G}_T$ is selected to serve as the reasoning chain of the question.", "Next, with the reasoning chain and other contextual information as input, a GPT2-based QG model (QG$_{Initial}$ ) generates the initial simple question $\\mathcal {Q}_1$ .", "Then, $\\mathcal {Q}_1$ is fed to a question rewriting module (QG$_{Rewrite}$ ) to iteratively rewrite it into a more complex question $\\mathcal {Q}_i$ $(i=2,3,\\dots ,d)$ .", "In what follows, we will introduce the whole generation process in more details.", "Context Graph Construction We follow the method proposed by   to build the context graph $\\mathcal {G}_{CG}$ .", "Specifically, we first apply open information extraction to extract $\\langle subject, relation, object\\rangle $ triples from context sentences.", "Each triple is then transformed into two nodes connected with a directed edge, like A Perfect Murder $\\stackrel{is}{\\longrightarrow }$ a 1998 American crime film in Fig.", "REF .", "The two nodes respectively represent subject and object, and the edge describes their relation.", "Coreference resolution is applied to merge nodes referring to the same entity.", "For instance, A Perfect Murder is merged with It in Fig.", "REF .", "Reasoning Chain Selection With the context graph constructed, we sample a connected subgraph $\\mathcal {G}_T$ consisting of $d+1$ nodes from it to serve as the reasoning chain of the generated question.", "A node $\\mathcal {N}_{0}$ is first sampled as the answer of the question, if it is, or linked with, a named entity that has more than one node degree.", "Next, we extract from $\\mathcal {G}_{CG}$ a maximum spanning tree $\\mathcal {G}_L$ , with $\\mathcal {N}_{0}$ as its root node of $\\mathcal {G}_L$ , e.g., the tree structure shown in Fig.", "REF .", "$\\mathcal {G}_{CG}$ is temporarily considered as an undirected graph at this step.", "We then prune $\\mathcal {G}_L$ into $\\mathcal {G}_T$ to keep only $d+1$ nodes.", "During pruning, we consider the sentence position where each node is extracted in order to make the reasoning chain relevant to more context.", "In the following, we will denote the node in $\\mathcal {G}_T$ as $\\mathcal {N}_i$ $(i=0,1,\\dots ,d)$ , where each node is subscripted by preorder traversal of $\\mathcal {G}_T$ , and $\\mathcal {N}_{P(i)}$ as the parent of $\\mathcal {N}_i$ .", "[t] Procedure of Our DCQG Framework 1.2 [1] context $\\mathcal {C}$ , difficulty level $d$ $({\\mathcal {Q}}, {\\mathcal {A}})$ $\\mathcal {G}_{CG} \\leftarrow \\mathbf {BuildCG}(\\mathcal {C})$ $\\mathcal {N}_0 \\leftarrow \\mathbf {SampleAnswerNode}(\\mathcal {G}_{CG})$ $\\mathcal {G}_L \\leftarrow \\mathbf {MaxTree}(\\mathcal {G}_{CG}, \\mathcal {N}_0)$ $\\mathcal {G}_T \\leftarrow \\mathbf {Prune}(\\mathcal {G}_L, d)$ $\\mathcal {N}_i$ in $\\mathbf {PreorderTraversal}(\\mathcal {G}_T)$ if $i=0$ then continue $\\mathcal {N}_{P(i)} = \\mathbf {Parent}(\\mathcal {N}_i)$ $\\mathcal {S}_i = \\mathbf {ContextSentence}(\\mathcal {C},\\mathcal {N}_i,\\mathcal {N}_{P(i)})$ $\\mathcal {R}_i \\leftarrow \\left\\lbrace \\begin{array}{ll}\\text{\\emph {Bridge}} & \\text{if } {\\mathcal {N}_i \\text{=} \\mathbf {FirstChild}(\\mathcal {N}_{P(i)})}\\\\\\text{\\emph {Intersection}} & \\text{else}\\\\\\end{array} \\right.$ $\\mathcal {Q}_i \\leftarrow \\left\\lbrace \\begin{array}{l}\\mathbf {QG}_{Initial}(\\mathcal {N}_i,\\mathcal {N}_{P(i)}, \\mathcal {S}_i) \\text{\\ \\quad \\qquad if } {i=1}\\\\\\mathbf {QG}_{Rewrite}(\\mathcal {Q}_{i-1}, \\mathcal {N}_i,\\mathcal {N}_{P(i)}, \\mathcal {S}_i,\\mathcal {R}_i ) \\text{ else}\\\\\\end{array} \\right.$ return $(\\mathcal {Q}_d, \\mathcal {N}_0)$ Step-by-step Question Generation Our step-by-step QG process is described at lines 5-11 in Algorithm .", "The following notations are defined for clearer illustration: ${\\mathcal {Q}}_i$ $(i=1,2,\\dots ,d)$ represents the question generated at each step, where ${\\mathcal {Q}}_d$ is the final question ${\\mathcal {Q}}$ , and ${\\mathcal {Q}}_{i+1}$ is rewritten from ${\\mathcal {Q}}_{i}$ by adding one more hop of reasoning.", "$\\mathcal {S}_{i}$ represents the context sentence from which we extract triple $\\mathcal {N}_i \\rightarrow \\mathcal {N}_{P(i)}$ .", "$\\mathcal {R}_i$ is the rewriting type of $\\mathcal {Q}_i$ $(i=2,3,\\dots ,d)$ .", "Specifically, we consider two types of rewriting patterns in this work: Bridge and Intersection.", "As shown in Fig.", "REF , Bridge-style rewriting replaces an entity with a modified clause, while Intersection adds another restriction to an existing entity in the question.", "These two types can be distinguished by whether $\\mathcal {N}_i$ is the first child of its parent node, i.e., whether its parent node has already been rewritten once in Bridge style.", "To generate the final question with the required difficulty level $d$ , we first use a question generator QG$_{Initial}$ to generate an initial simple question based on $\\mathcal {N}_1$ , $\\mathcal {N}_0$ , and the corresponding context sentence $\\mathcal {S}_1$ .", "Then, we repeatedly (for $d-1$ times) use QG$_{Rewrite}$ to rewrite question $\\mathcal {Q}_{i-1}$ into a more complex one $\\mathcal {Q}_i$ , based on node $\\mathcal {N}_i$ and its parent node, context sentence $\\mathcal {S}_i$ , and the rewriting type $\\mathcal {R}_i$ $(i=2,3,\\dots ,d)$ .", "Formally, the generation process of QG$_{Initial}$ and the rewriting process of QG$_{Rewrite}$ can be defined as: ${\\mathcal {Q}}_1 = \\mathop {\\arg \\max }_{\\bar{\\mathcal {Q}}_1} P(\\bar{\\mathcal {Q}}_1 | \\mathcal {N}_1,\\mathcal {N}_0, \\mathcal {S}_1)$ ${\\mathcal {Q}}_i = \\mathop {\\arg \\max }_{\\bar{\\mathcal {Q}}_i} P(\\bar{\\mathcal {Q}}_i | \\mathcal {Q}_{i-1}, \\mathcal {N}_i,\\mathcal {N}_{P(i)}, \\mathcal {S}_i,\\mathcal {R}_i )$ where $i=2,3,\\dots ,d$ .", "In our implementation, both QG$_{Initial}$ and QG$_{Rewrite}$ are initialized with pre-trained GPT2-small model , and then fine-tuned on our constructed dataset (see Section ).", "The encoder of QG$_{Rewrite}$ , as illustrated in Fig.", "REF , is similar to  .", "If $\\mathcal {N}_i$ points to $\\mathcal {N}_{P(i)}$ , then the input sequence is organized in the form of “$\\left< \\text{\\emph {bos}}\\right>$ $\\mathcal {S}_i$ $\\left< \\text{\\emph {nodeC}}\\right>$ $\\mathcal {N}_{i}$ $\\left< \\text{\\emph {edge}}\\right>$ $\\mathcal {E}_i$ $\\left< \\text{\\emph {nodeP}} \\right>$ $\\mathcal {N}_{P}^{(i)}$ $\\left< \\text{\\emph {type}} \\right>$ $\\mathcal {R}_{i}$ $\\left< \\text{\\emph {subq}}\\right>$ ${\\mathcal {Q}}_{i-1}$ $\\left< \\text{\\emph {eos}}\\right>$”, where $\\mathcal {E}_i$ is the edge from $\\mathcal {N}_{i}$ to $\\mathcal {N}_{P}^{(i)}$ .", "The positions of “$\\left< \\text{\\emph {nodeC}}\\right>$ $\\mathcal {N}_{i}$” and “$\\left< \\text{\\emph {nodeP}} \\right>$ $\\mathcal {N}_{P}^{(i)}$” will be exchanged if $\\mathcal {N}_{P(i)}$ points to $\\mathcal {N}_i$ .", "As for QG$_{Initial}$ , its input is organized in the same way except without “$\\left< \\text{type} \\right>$ $\\mathcal {R}_{i}$ $\\left< \\text{subq}\\right>$ ${\\mathcal {Q}}_{i-1}$”.", "The segment embedding layer is utilized to identify different segments.", "For those parts in $\\mathcal {S}_i$ and $\\mathcal {Q}_{i-1}$ that are the same as, or refer to the same entity as $\\mathcal {N}_{P(i)}$ , we replace their segment embeddings with the one of $\\mathcal {N}_{P(i)}$ , considering that the parent node of $\\mathcal {N}_i$ plays an important role in denoting what to ask about, or which part to rewrite, as shown in Fig.", "REF ." ], [ "Automatic Dataset Construction", "Manually constructing a new dataset for our task is difficult and costly.", "Instead, we propose to automatically build a dataset from existing QA datasets without extra human annotation.", "In our work, the training data is constructed from HotpotQA , in which every context $\\mathcal {C}$ consists of two paragraphs $\\lbrace \\mathcal {P}_1, \\mathcal {P}_2\\rbrace $ , and most of the questions require two hops of reasoning, each concerning one paragraph.", "HotpotQA also annotates supporting facts $\\mathcal {F}$ , which are the part of the context most relevant to the question.", "In addition to the information already available in HotpotQA, we also need the following information to train QG$_{Initial}$ and QG$_{Rewrite}$ : $(\\mathcal {Q}_1,\\mathcal {A}_1)$ , the simple intermediate question and its answer, which are used to train QG$_{Initial}$ ; $\\mathcal {R}_2$ , the type of rewriting from ${\\mathcal {Q}}_1$ to ${\\mathcal {Q}}_2$ ; $\\lbrace \\mathcal {N}_0, \\mathcal {N}_1, \\mathcal {N}_2\\rbrace $ , the reasoning chain of ${\\mathcal {Q}}_2$ ; $\\mathcal {S}_i$ $(i=1,2)$ , the context sentences where we extract $\\mathcal {N}_0$ , $\\mathcal {N}_1$ and $\\mathcal {N}_2$ .", "[t] 1.2 Procdure of Data Construction [1] context $\\mathcal {C} = \\lbrace \\mathcal {P}_1, \\mathcal {P}_2\\rbrace $ , QA pair $({\\mathcal {Q}}_2, {\\mathcal {A}}_2)$ , supporting facts $\\mathcal {F}$ $\\mathcal {R}_1, ({\\mathcal {Q}}_1, {\\mathcal {A}}_1),\\mathcal {S}_1, \\mathcal {S}_2, \\lbrace \\mathcal {N}_0, \\mathcal {E}_1, \\mathcal {N}_1, \\mathcal {E}_2, \\mathcal {N}_2\\rbrace $ $\\mathcal {R}_1 \\leftarrow \\mathbf {TypeClassify}({\\mathcal {Q}}_2)$ if $\\mathcal {R}_1 \\notin \\lbrace $Bridge, Intersection$\\rbrace $ then return $subq_1,subq_2 \\leftarrow \\mathbf {DecompQ}({\\mathcal {Q}}_2)$ $suba_1,suba_2 \\leftarrow \\mathbf {QA}(subq_1), \\mathbf {QA}(subq_2)$ ${\\mathcal {Q}}_1, {\\mathcal {A}}_1 \\leftarrow \\left\\lbrace \\begin{array}{ll}subq_2, suba_2 & \\text{if } {\\mathcal {A}_2 = suba_2}\\\\subq_1, suba_1 & \\text{else}\\\\\\end{array} \\right.$ $\\mathcal {S}_1, \\mathcal {S}_2 \\leftarrow \\left\\lbrace \\begin{array}{ll}\\mathcal {F} \\cap \\mathcal {P}_1, \\mathcal {F} \\cap \\mathcal {P}_2 & \\text{if } {\\mathcal {Q}}_1 \\text{ concerns } \\mathcal {P}_1\\\\\\mathcal {F} \\cap \\mathcal {P}_2, \\mathcal {F} \\cap \\mathcal {P}_1 & \\text{else}\\\\\\end{array} \\right.$ $\\mathcal {N}_2 \\leftarrow \\mathbf {FindNode}(\\mathcal {A}_2)$ $\\mathcal {N}_0, \\mathcal {E}_1, \\mathcal {N}_1, \\mathcal {E}_2 \\leftarrow \\mathbf {Match}(subq_1, subq_2)$ Algorithm  describes our procedure to obtain the above information.", "The construction process is facilitated with the help of a reasoning type classifier ($\\mathbf {TypeClassify}$ ) and a question decomposer ($\\mathbf {DecompQ}$ ), referring to .", "For each question in HotpotQA (i.e.", "${\\mathcal {Q}}_2$ ), we first distinguish its reasoning type, and filter out those that are not Bridge and Intersection.", "The reasoning type here corresponds to the rewriting type $\\mathcal {R}_i$ .", "Then, $\\mathbf {DecompQ}$ decomposes ${\\mathcal {Q}}_2$ into two sub-questions, $subq_1$ and $subq_2$ , based on span prediction and linguistic rules.", "For example, the $\\mathcal {Q}_2$ in Fig.", "REF will be decomposed into $subq_1$ =“To which film A Perfect Murder was a modern remake?”, and $subq_2$ =“Who directed Dial M for Murder?”.", "After that, an off-the-shelf single-hop QA model is utilized to acquire the answer of the two sub-questions, which should be “Dial M for Murder” and “Alfred Hitchcock” in the example.", "As for ${\\mathcal {Q}}_1$ , it is one of the sub-questions.", "When ${\\mathcal {Q}}_2$ is of the Intersection type, ${\\mathcal {Q}}_1$ can be either $subq_1$ or $subq_2$ .", "For the Bridge type, it is the sub-question that shares the same answer as ${\\mathcal {A}}_2$ .", "For the example above, ${\\mathcal {Q}}_1$ is $subq_2$ because $suba_2 = {\\mathcal {A}}_2$ .", "The context sentence $\\mathcal {S}_i$ is supposed to provide supporting facts contained in the paragraph $\\mathcal {F}$ that concerns ${\\mathcal {Q}}_i$ $(i=1,2)$ .", "For the reasoning chain, it is selected from the local context graph by first locating $\\mathcal {N}_2$ and then finding $\\mathcal {N}_0, \\mathcal {N}_1$ through text matching with the two sub-questions." ], [ "Experimental Setup", "The constructed dataset described in Section  is made up of 55,553/6,944/6,944 samples for training/validation/test.", "We have made it publicly available to facilitate future research.$\\,$https://tinyurl.com/y44k7ekj During dataset construction, the coreference resolution toolkit we utilized is from AllenNLP 1.0.0 , and the open information extraction toolkit is provided by the Plasticity developer API.$\\,$https://www.plasticity.ai/ The question decomposer and reasoning type classifier follow the implementations of .", "QG$_{Initial}$ and QG$_{Rewrite}$ are both initialized with the GPT2-small model from the HuggingFace Transformer library , and fine-tuned on our constructed dataset.", "They are trained for 8 epochs and 10 epochs, respectively, with batch size 16.", "We apply top-p nucleus sampling with $p = 0.9$ during decoding.", "AdamW is used as optimizer, with the initial learning rate set as $6.25\\times 10^{-5}$ /.", "We will release the code after publication.", "To evaluate quality of generation results from our framework, we heuristically sample the answers and reasoning chains from contructed context graphs of the texts in HotpotQA to generate 150,305 two-hop questions with the trained QG$_{Initial}$ and QG$_{Rewrite}$ .", "Quality of the generated data is evaluated in Sections REF , REF and REF .", "Table: Comparison of QG GPT2 _{GPT2} and baseline methods on the test set of HotpotQA short _{short}.Table: Results of human evaluation." ], [ "Baseline Models", "The following multi-hop QG systems are compared with automatic evaluation metrics: NQG++ is a seq2seq model based on bi-directional Gate Recurrent Unit (GRU), with enriched features of answer position and lexical information.", "ASs2s is a seq2seq model based on Long Short-term Memory (LSTM), which separately encodes answer and context.", "SRL-Graph and DP-Graph are two state-of-the-art QG systems.", "They integrate graph-level and document-level information as inputs, encoded with an attention-based Graph Neural Network (GNN) and a bi-directional GRU respectively.", "Their difference is that SRL-Graph constructs semantic graph via semantic role labelling, and DP-Graph through dependency parsing.", "QG$_{GPT2}$ is a vanilla GPT2-based QG model, similar to the implementation of the two QG models in our framework, but without context graph construction or multi-stage generation.", "Its input is concatenation of the context and sampled answer, and in the segment embedding layer, the position where the answer appears in the context segment is denoted.", "We compare performance of these baselines under the same setting as , where the dataset is a shortened version of HotpotQA (HotpotQA$_{short}$ ), with each context abbreviated to only include the supporting facts and the part that overlaps with the question.", "The training, validation and test sets contain 84,368/6,072/6,072 examples, respectively.", "QG$_{GPT2}$ is initialized with GPT2-small model and finetuned on HotpotQA$_{short}$ for 7 epochs with batch size 16, and conduct top-p nucleus sampling with $p = 0.9$ when decoding.", "For DP-Graph, we use their released model and code for the experiment.", "For the other three baselines, we directly refer to the experiment results from .", "The generation results are evaluated with BLEU4 , METEOR , and ROUGE-L .", "As shown in Table REF , we can see that QG$_{GPT2}$ achieves the best performance in terms of BLEU4 and METEOR, while DP-Graph is the best in terms of ROUGE-L. We did not directly compare our DCQG method with the above baselines in this experiment, because the input of the baseline models and our framework are different.", "To generate multi-hop questions in a controlled manner, the reasoning chain needs to be sampled from context graphs as part of our framework input, which is not directly provided in HotpotQA$_{short}$ .", "In comparison, the baseline methods only need context text as input.", "Instead, we perform human evaluation to evaluate the data quality of our model and the two strongest baselines (i.e., DP-Graph and QG$_{GPT2}$ ), as well as test its ability in augmenting QA systems." ], [ "Human Evaluation", "We randomly sampled 200 questions in the original HotpotQA dataset, 200 questions from scalable generation results of our method, and 200 questions respectively from DP-Graph and QG$_{GPT2}$ generated in the experiment described in Section REF .", "Then, they are manually evaluated by eight human annotators, who are graduate students, majoring in English Literature, Computer Science, or Electronic Engineering.", "They voluntarily offer to help without being compensated in any form.", "Before annotation, they are noticed with very detailed annotation manuals.", "The sampled questions are scored in the following five dimensions: Well-formed: whether a question is semantically and grammatically correct.", "Annotators score the question as yes, accptable, or no.", "acceptable is selected if the question is not completely grammatically correct, but its meaning is still inferrable.", "Relevant: whether a question is answerable according to the given context (yes or no).", "A question cannot be relevant if it is not well-formed.", "Right Answer: whether the given answer of a well-formed question is its correct answer (yes or no).", "Multi-hop: whether the question requires reasoning over multiple text spans (yes or no).", "Concise: This metric is only evaluated for those questions already classified as multi-hop, to see whether the QG models are overfitted, generating multi-hop questions by inappropriately adding too many modifiers.", "Annotators will score the question as yes if no single word can be deleted, acceptable if it is a little lengthy but still in a natural way, and no if it is abnormally verbose.", "The human evaluation results are shown in Table.", "REF .", "We can see that except in the dimension of right answer, our method performs consistently better than the two baseline models and is comparable to the reference questions of HotpotQA.", "The matching rate of the questions and answers generated by our method is 63.5%, slightly lower than QG$_{GPT2}$ (65.5%).", "However, it should be emphasized that QG$_{GPT2}$ and DP-Graph directly take the answer and most relevant content given by HotpotQA$_{short}$ as inputs, while our method conducts content selection from the whole context on its own.", "Our method performs especially well in terms of multi-hop, well-formed and concise.", "For concise, it is even better than the reference questions in the HotpotQA.", "For reference, the average number of words of the questions in HotpotQA, and those generated by our method, QG$_{GPT2}$ , and DP-Graph are 17.44, 17.18, 19.26 and 19.32, respectively.", "A very possible explanation is that the enriched graph information and multi-stage generation largely control the question's structure and content.", "In comparison, we find that the questions generated by QG$_{GPT2}$ and DP-Graph tend to unreasonably pile too many modifiers and subordinate clauses.", "Figure: Performance of the DistilBERT-based QA system on HotpotQA short _{short}, augmented with different quantities of generated data" ], [ "Boosting Multi-hop QA Performance", "We further evaluate whether the generated data can boost QA performance.", "The dataset created by our framework contains 150,305 questions.", "As a comparison, we utilize QG$_{GPT2}$ to generate the same amount of data with the same sampled answers and contextual sentences used by our method.Some low-quality questions are filtered out if its number of words is not between 6$\\sim $ 30 (4.7% for our method and 9.2% for QG$_{GPT2}$ ), or the corresponding answer appears in it (2.7% for our method and 2.4% for QG$_{GPT2}$ ).", "Finally, we randomly sample 100,000 QA pairs with contextual sentences for each of the two methods, and augment the HotpotQA$_{short}$ dataset with them, respectively.", "A DistilBERT-based QA model is implemented.", "It takes concatenation of the context and question as input to predict the answer span.", "During training, the original cases from Hotpot$_{short}$ are oversampled to ensure that they are at least four times as the generated data.", "We use Adam as the optimizer, with mini-batch size as 32.", "The learning rate is initially set to be $3\\times 10^{-5}$ and adaptively decays during training.", "The configurations are the same for all the QA experiments, except that the training datasets are different combinations of the original HopotQA$_{short}$ and the generated data.", "The validation and test sets are the same with HotpotQA$_{short}$ , and the evaluation metrics include Exact Match (EM) and F1.", "We test the impact of the generated data under both high-resource (using the whole training set of HotpotQA$_{short}$ ) and low-resource settings (using only 25% of the data randomly sampled from HotpotQA$_{short}$ ).", "Fig.", "REF compares the QA performance , augmented with different quantities of the data generated by our method and by QG$_{GPT2}$ .", "We can see that under both settings, our method achieves better performance than QG$_{GPT2}$ .", "Under low-resource setting, the promotion effect of our generated data is more significant and obviously better than that of QG$_{GPT2}$ .", "The performance of the QA model steadily improves when the training dataset is augmented with more of our generated data.", "EM and F1 of the QA model are improved by 2.56% and 1.69%, respectively, when 100,000 cases of our generated data are utilized.", "DP-Graph is not compared here, as their graph construction method, concerning every context token, is too slow to perform generation on a large scale.", "Besides, it is highly unlikely to be better in boosting QA performance due to its much lower right answer score in human evaluation.", "Table: Performance of the BERT SQuAD _{SQuAD} QA model on different generated datasets." ], [ "Controllability Analysis", "Controllability of our DCQG framework is analyzed with the help of a QA model (BERT$_{SQuAD}$ ),$\\,$https://tinyurl.com/y5a5u6by which is initialized with BERT and fine-tuned on SQuAD , a one-hop QA dataset.", "Its performance is tested in answering the two-hop questions generated by our framework (Ours$_{twohop}$ ) and their corresponding intermediate questions (Ours$_{onehop}$ ), as well as the question sets generated by DP-Graph and QG$_{GPT2}$ on the test set of HotpotQA$_{short}$ for comparison.", "The results are shown in Table REF .", "Since the question and answer pairs in the generated data are not completely matched, we list their Right Answer (RA) scores for reference, and re-calculate their estimated EM, assuming that the QA model completely fails on the questions matched with incorrect answers (EM$_{RA}$ ).", "We can see that EM and F1 of BERT$_{SQuAD}$ decline by 28.96% and 25.82%, respectively, from Ours$_{twohop}$ to Ours$_{onehop}$ , demonstrating that our framework is able to generate questions of different difficulty levels.", "Furthermore, though according to human evaluation results, the questions in Ours$_{twohop}$ are more well-formed and concise, and the RA score is also very close to the one of QG$_{GPT2}$ , BERT$_{SQuAD}$ achieves better performance on QG$_{GPT2}$ than Ours$_{twohop}$ (4.80% higher in EM and 7.50% in F1).", "On DP-Graph, BERT$_{SQuAD}$ also performs better in terms of EM$_{RA}$ .", "It indicates that the questions in $QG_{GPT2}$ are easier overall, mixed with both single- and multi-hop questions, as it cannot controllably manipulate question difficulty during generation.", "Figure: Two examples of generating three-hop questions based on the extracted reasoning chains.Case study is conducted for further analysis.", "Fig.", "REF shows two examples of three-hop question generation process, with which we want to illustrate three points.", "First, our framework is able to generate out-of-distribution questions of more than two hops, unconstrained by the limitation of its training data.", "Second, the context texts and the answers of these two questions are the same, but two different questions with different underlying logic are generated, under control of the extracted reasoning chains.", "Third, we can also see that the intermediate questions, serving as springboards, are efficiently used by QG$_{Rewrite}$ to generate more complex questions by rewriting part of the content.", "We also experiment with generation of more complex questions.", "But we find that for generation of more than three hops, quality of the generated questions drastically declines.", "The semantic errors become more frequent, and they tend to repeat content in the last intermediate questions, probably because the input of QG$_{Rewrite}$ has become too long to be precisely encoded by the GPT2-small model due to the growing length of the question." ], [ "Conclusion", "We explored the task of multi-hop QG with control over the question's answer, reasoning chain, and inference hops.", "A step-by-step generation framework was proposed to accomplish this objective, with an input sampler to extract the reasoning chain, a question generator to produce a simple question, and a question rewriter to further adapt it into a complex one.", "A dataset was automatically constructed based on HotpotQA to facilitate the research on our task.", "Experiments showed that our method not only generated multi-hop questions in a more well-formed and concise manner than baseline models, but was also able to control its difficulty." ] ]
2105.11698
[ [ "Bias-Robust Bayesian Optimization via Dueling Bandits" ], [ "Abstract We consider Bayesian optimization in settings where observations can be adversarially biased, for example by an uncontrolled hidden confounder.", "Our first contribution is a reduction of the confounded setting to the dueling bandit model.", "Then we propose a novel approach for dueling bandits based on information-directed sampling (IDS).", "Thereby, we obtain the first efficient kernelized algorithm for dueling bandits that comes with cumulative regret guarantees.", "Our analysis further generalizes a previously proposed semi-parametric linear bandit model to non-linear reward functions, and uncovers interesting links to doubly-robust estimation." ], [ "Introduction", "Bayesian optimization [29] is a model-based approach for zero-order global optimization with noisy feedback.", "It has been successfully applied to many applications such as hyper-parameter tuning of machine learning models, robotics and chemical design.", "Some variants such as Expected-Improvement [10] or the GP-UCB algorithm [34] come with theoretical guarantees, ensuring convergence to the global optimum in finite time under suitable regularity assumptions.", "Closely related is the field of bandit algorithms [24], in particular the linear bandit model [3], [12], [2].", "Most linear bandit algorithms and Bayesian optimization approaches require that the true function is realized in a known reproducing kernel Hilbert space (RKHS) and rely on unbiased evaluations of the objective.", "The regularity assumptions raise questions of robustness to miss-specification and adversarial attacks, and addressing these limitations has been the content of several recent works [28], [9], [8].", "We study a setting where the learner's objective is to maximize an unknown function $f: \\rightarrow $ with additive confounded feedback, yt = f(xt) + bt + t , where $x_t$ is the evaluation point chosen by the learner at time $t$ , $\\epsilon _t$ is $\\sigma ^2$ -sub-Gaussian (zero-mean) observation noise, and $b_t$ is an additive confounding term.", "We assume that $b_t$ is chosen by an adversary, but does not depend on the input $x_t$ .", "The bias term allows to model the influence of an unobserved and uncontrolled covariate, or a perturbation of the feedback signal imposed by an adversary.", "One can also interpret () as a contextual model, where $b_t$ captures the effect of a changing (unobserved) context on the reward.", "We discuss further examples and applications in Section REF below.", "The proposed feedback model () generalizes the semi-parametric contextual bandit model studied by [23], where $f(x_t) = \\langle x_t, \\theta \\rangle $ is a linear function defined by a parameter $\\theta \\in ^d$ .", "They show that a doubly-robust least-squares estimator allows to recover reward differences $\\langle x - x^{\\prime }, \\theta \\rangle $ for inputs $x, x^{\\prime } \\in ^d$ despite the confounding.", "They further propose an elimination-style algorithm, bandit orthogonalized semiparametric estimation (BOSE), which is based on sampling actions from a distribution that minimizes the variance of the estimator.", "However, finding low variance distributions requires solving a convex-quadratic feasibility problem, which is computationally demanding and in general leads to sampling distributions with support that spans $^d$ ." ], [ "Contributions", "Our first contribution is two reductions of the confounded feedback model () to the dueling bandit setting.", "This allows us to leverage existing algorithms for dueling bandits in the confounded observation setting.", "We then propose the first efficient algorithm for kernelized dueling bandits that comes with theoretical guarantees on the cumulative regret.", "The approach is based on information-directed sampling (IDS), which was recently studied in the context of linear partial monitoring by [21].", "In particular, we propose an efficient approximation of IDS, that reduces the computation complexity from $(||^4)$ to $(||)$ on finite action sets $$ .", "For continuous action sets, the proposed algorithm requires to optimize a (non-convex) acquisition function over the input space, akin to standard Bayesian optimization." ], [ "Motivating Examples", "We start by motivating our problem through applications." ], [ "Range-Adjusting Measurement Devices", "In many real-world optimization tasks, the observed feedback arises from a physical sensing device.", "Such measurement devices can be subject to calibration errors or might automatically adjust the output range for better sensitivity.", "For example, in optimization of free electron lasers [19], [13], the target signal is measured with a gas-detector, which exploits a physical law to amplify the signal.", "An input voltage is used to control the amplification factor, which requires re-adjustmentTranslating the range-adjusted signal into an absolute value is possible, but not straightforward and comes with other limitations.", "with increasing target signal such that the physical relationship between the target and the measured output stays approximately linearMore precisely, the relationship depends on the photon energy, pulse intensity, and physical properties of the gas involved.", "[33], [16].", "Our feedback model allows optimization that is robust to absolute changes in the target signal occurring at any time." ], [ "Distributed Optimization of Additive Functions", "In high-dimensional settings, previous work on Bayesian optimization often uses structural assumptions to reduce the dependence of the sample complexity on the dimension.", "One popular choice is additivity [17], for example coordinate-wise $f(x) = \\sum _{i \\in [d]} f_i(x^i)$ .", "To optimize additive functions, we can apply $d$ individual learners to optimize each 1-dimensional component $f_i$ separately.", "Note that the learners cannot directly evaluate $f_i(x_i)$ , but only obtain the global noisy feedback $f(x) + \\epsilon $ , which depends on the choices $x = (x^1,\\dots , x^d)$ of all learners.", "When the learners act in parallel and there is no communication possible, the feedback of each learner is confounded as in () by the other learner's choices.", "Our robust approach guarantees that the learners are able to optimize each component successfully, despite the confounding." ], [ "Adversarial Attacks", "Robustness to adversarial attacks was studied recently in the context of bandit algorithms [28], [9] and Bayesian optimization [8].", "In all previous work that we are aware of, the corruption of the feedback is allowed to depend on the actions, with varying assumptions of whether or not the adversary observes the action choice of the learner.", "Note that our feedback model is more stringent, as it does not allow for action dependence.", "On the other hand, our model allows for sublinear regret even with a constant corruption in every round, whereas in previous work, the regret scales with the total amount of corruption." ], [ "Related Work", "There is a vast amount of literature on bandit algorithms [24] and Bayesian optimization [29], [34], [32], [14].", "Our feedback model is a generalization of the semi-parametric linear bandit setting proposed by [23] with applications for example in mobile health [38].", "A variant of Thompson sampling was analyzed in the same setting by [18], which they show outperforms the BOSE approach by [23], but the frequentist regret bound they derive has an extra factor in the dimension.", "The Thompson sampling variant is also computationally more efficient, but requires to explicitly compute the probabilities that each action is optimal under the posterior distribution.", "For Bayesian optimization, robust variants have been considered recently, for example with adversarial perturbations of the input [7], corruption of the output [8] and distributionally robust optimization [20].", "In the context of adversarial attacks, there is an increasing body of work [28], [25], [27], [15], [8], [9], however the feedback model differs from ours, see also the discussion in Section REF .", "Of particular relevance to our work is the (stochastic) dueling bandit setting [41], [37], [6] and kernelized variants [35], [36].", "Early work by [40] applied the dueling bandit model to the optimization setting with continuous action sets and concave reward functions, and established a connection to gradient-based optimization.", "However, to the best of our knowledge, none of the previous works provide bounds on the cumulative regret in the kernelized (non-concave) setting.", "A kernelized algorithm with theoretical guarantees that requires point evaluations and dueling feedback is by [39].", "Closely related is also the work by [31] on linear dueling bandits with possibly infinite input spaces.", "They establish a connection to the generalized linear bandit model (GLM) and propose an algorithm which, similar to ours, relies on finding an informative action pair with low regret.", "However, gap-dependent bounds and a kernelized variant was not provided, and for finite action sets $$ , their algorithm requires $(||^2)$ computation steps per round.", "Recent work by [4] considers the finite-armed dueling bandit setting with adversarial corruptions of the feedback.", "Lastly, we remark that most previous work on dueling bandits considers binary feedback, whereas here we are interested in quantitative feedback on the reward-difference between the chosen action pair.", "Formally, we consider sub-Gaussian dueling feedback which includes the Bernoulli likelihood, but does not exploit the heteroscedasticity of binary observations." ], [ "Setting", "Let $\\subset ^d$ be a compact input space and $f: \\rightarrow $ a fixed and unknown objective function.", "In each round $t=1, \\dots , n$ , the learner chooses an action $x_t \\in $ and observes the confounded outcome yt = f(xt) + bt + t , where $\\epsilon _t$ is $\\sigma ^2$ -sub-Gaussian, conditionally independent noise, and $b_t \\in ^d$ is an unobserved and possibly time-dependent confounding term.", "We assume that $b_t$ does not depend on the current action $x_t$ chosen by the learner, and satisfies one of the following assumptions: The bias $b_t$ is bounded, $|b_t| \\le C_{max}$ and fixed at the beginning of round $t$ , but can otherwise arbitrarily depend on $(x_s, y_s)_{s=1}^{t-1}$ .", "The difference between two consecutive bias terms is bounded, $|b_t - b_{t-1}| \\le D_{max}$ and $b_t$ is fixed at the beginning of round $t-1$ , but can otherwise arbitrarily depend on $(x_s, y_s)_{s=1}^{t-2}$ .", "Which assumption is used is specified in the relevant context.", "Let $x^* \\in _{x \\in } f(x)$ be the optimal action.", "The suboptimality gap is $\\Delta (x) = f(x^*) - f(x)$ .", "The learner's objective is to maximize the cumulative reward $\\sum _{t=1}^n f(x_t)$ , or equivalently minimize the regret, Rn = t=1n f(x*) - f(xt) = t=1n (xt) .", "For the analysis, we assume that the function $f$ is in a known reproducing kernel Hilbert space (RKHS) $$ with associated kernel $k : \\times \\rightarrow $ and bounded Hilbert norm $\\Vert f\\Vert _{} \\le B$ .", "This is a standard assumption in Bayesian optimization [34], [11] and justifies the use of kernelized least-squares regression, formally introduced in Section .", "We further require that the kernel function is bounded, $k(x,x) \\le 1$ ." ], [ "Reduction to Dueling Bandits", "It is clear from the observation model () that any additive shift of the objective, i.e.", "$\\tilde{f}(x) = f(x) + c$ for $c \\in $ , can be absorbed in the unobserved confounding terms $b_t$ , hence rendering the observation sequences for $f(x)$ and $\\tilde{f}(x)$ indistinguishable.", "In particular, the learner can only hope to recover the true function up to an additive constant.", "Fortunately, to determine the best action $x^* = _{x \\in } f(x)$ , it suffices to estimate reward differences $f(x^{1}) - f(x^{2})$ for actions $x^1, x^2 \\in $ , which is indeed possible.", "To do so, previous work in the linear setting relies on doubly-robust estimation [23], [18].", "Here we take a different approach, and propose a generic reduction of the feedback model () to the dueling bandit model [41], [37].", "The reduction has the advantage that we can leverage existing algorithms for dueling bandits, which also eliminates the need to find low-variance sampling distributions for doubly-robust estimation.", "In dueling bandits, the learner chooses two actions $x_t^{1},x_t^{2}\\in $ and obtains (noisy) feedback on which of the two actions has higher reward.", "Meanwhile, the learner suffers regret for both actions, but the reward of each action is not observed.", "While most previous work on dueling bandits uses a binary feedback model, here, we are concerned with a quantitative version of the same setting, which is a special case of linear partial monitoring [26].", "Specifically, for a reward function $f : \\rightarrow $ and actions $x_t^{1},x_t^{2}\\in $ , we define quantitative dueling feedback as follows: dt = f(xt1) - f(xt2) + t , where $\\xi _t$ is $\\rho ^2$ -sub-Gaussian observation noise and $\\rho $ is known to the learner.", "The distribution of $\\xi _t$ is allowed to depend on $(x_t^{1}, x_t^{2})$ as long as the sub-Gaussian tail assumption is satisfied uniformly over all actions.", "Note that this includes binary feedback typically used for dueling bandits, and, more generally, bounded noise distributions.", "Next, we present two reductions schemes to generate the dueling bandit feedback () from confounded observations ()." ], [ "Two-Point Reduction", "The first scheme uses two confounded observations to construct the dueling bandit feedback.", "Given two inputs $x_t^{1},x_t^{2}\\in $ in round $t$ , we evaluate both points according to (), where the order of evaluation is uniformly randomized.", "The two observations are yt1 = f(xt1) + b2t+it + 2t+it , yt2 = f(xt2) + b2t + 1- it + 2t+1 - it , where $i_t \\sim \\text{Bernoulli}(0.5)$ .", "We then define dt = yt1 - yt2 .", "Assuming that $b_{2t}$ and $b_{2t+1}$ are fixed before either of $x_t^{1},x_t^{2}$ is chosen by the learner and using that the observation noise $\\epsilon _t$ is zero-mean, one easily confirms that $[d_t] = f(x_t^{1})- f(x_t^{2})$ .", "We further use the following properties of sub-Gaussian random variables.", "Any random variable $X$ such that $X \\in [-B,B]$ is $B^2$ -sub-Gaussian.", "For two independent random variables $X_1, X_2$ that are $B_1^2$ - and $B_2^2$ -sub-Gaussian respectively, $X_1 + X_2$ is $(B_1^2 + B_2^2)$ -sub-Gaussian.", "Hence if $|b_{2t} - b_{2t+1}| \\le D_{\\max }$ , it follows that t = dt - [dt] = b2t+it + 2t+it - (b2t + 1- it + 2t+1 - it) is $(D_{\\max }^2 + 2\\sigma ^2)$ -sub-Gaussian." ], [ "One-Point Reduction", "Perhaps surprisingly, one can also construct the dueling bandit feedback from a single observation using randomization.", "For given inputs $x_t^{1},x_t^{2}\\in $ we choose one point uniformly at random and evaluate the confounded function () to obtain a single observation yt = f(xt(1+it)) + bt + t , where $i_t \\sim \\text{Bernoulli}(0.5)$ .", "The dueling bandit feedback is dt = (-1)it   2 yt  .", "Again, we get an unbiased observation of the reward difference, $[d_t] = f(x_t^{1}) - f(x_t^{2})$ .", "Further, if $|b_t| \\le C_{\\max }$ , then $\\xi _t = d_t - [d_t]$ is $4(C_{\\max }^2 + \\sigma ^2)$ -sub-Gaussian.", "Compared to the two-point reduction, here the sub-Gaussian variance $\\rho $ depends on the absolute value $|b_t|$ of the confounding term instead of the difference $|b_t - b_{t+1}|$ .", "On the other hand, the one-point sampling scheme only requires $b_t$ to be fixed before the choice of $x_t$ , but may depend on all previous actions and observations." ], [ "Information-Directed Sampling", "With the reduction to dueling feedback, we are set to readily apply any dueling bandit algorithm in the confounded setting.", "Furthermore, dueling bandits (as defined in ()) are a special case of partial monitoring, for which also several algorithms exist.", "In the following, we adapt the information-directed sampling (IDS) approach [30], more specifically, the version proposed by [21] for linear partial monitoring.", "The main reason for this choice is that IDS works with quantitative dueling feedback, whereas most other work on dueling bandits focuses on settings with Bernoulli likelihood.", "Also, IDS can be formulated as a kernelized algorithm, and comes with theoretical guarantees on the regret.", "However, a direct adaptation of IDS to the dueling setting as proposed by [21] requires $(||^4)$ computational steps per round for finite action sets.", "In the following, we introduce an approximation of IDS, which obtains the same theoretical guarantees and only requires to optimize a simple score function over the action set.", "The resulting efficient kernelized dueling bandit algorithm may be of independent interest.", "On a high level, IDS samples actions from a distribution that minimizes a trade-off between an estimate of the regret and an information gain, as we elaborate below." ], [ "Kernel Regression for Dueling Feedback", "The first step is to set up kernelized least-squares regression for the dueling bandit feedback.", "Recall that we assume that $$ is a RKHS with kernel function $k: \\times \\rightarrow $ and $f\\in $ with $\\Vert f\\Vert _\\le B$ .", "In round $t$ , the learner has already collected data $_t = \\lbrace (x_s^{1},x_s^{2}, d_s)\\rbrace _{s=1}^{t-1}$ , where $x_s^{1},x_s^{2}\\in $ is the input pair chosen at step $s$ , and $d_s$ is quantitative dueling bandit feedback defined in Eq. ().", "The kernel least-squares estimator with regularizer $\\lambda > 0$ is ft = f s=1t-1 (f(xs1) - f(xs2) - ds)2 + f2 The solution corresponds to the posterior mean of the Gaussian process model with kernel $k$ and prior variance $\\lambda ^{-1}$ and can be computed in closed form.", "Let $\\mathbf {d}_t = [d_1, \\dots , d_{t-1}]^ be the vector which collects the observations and define $ Kt t-1t-1$ and $ kt(x) t-1$ for $ x $ as follows:\\begin{align*}[K_t]_{ij} &\\triangleq k(x_i^1,x_j^1) - k(x_i^1,x_j^2) - k(x_i^2,x_j^1) + k(x_i^2, x_j^2)\\\\[k_t(x)]_i &\\triangleq k(x,x_i^1) - k(x,x_i^2)\\end{align*}The least-squares solution $ ft$ evaluated at any $ x $ is $ ft(x) = kt(x)Kt + )-1 dt$, where $$ is the identity matrix in the appropriate dimension.", "To compute uncertainty estimates, we further define\\begin{align}k_t(x,y) &\\triangleq k(x,y) - k_t(x)^K_t + \\lambda )^{-1}k_t(y)\\,, \\nonumber \\\\\\psi _t(x,z) &\\triangleq k_t(x,x)^2 + k_t(z,z)^2 - 2 k_t(x,z)\\,.", "\\end{align}For any $ t 1$, $ x1,x2 $, the estimate $ ft$ satisfies with probability at least $ 1-$,\\begin{align}&\\left|\\hat{f}_t(x^1) - \\hat{f}_t(x^2) - \\big (f(x^1) - f(x^2)\\big )\\right|^2\\nonumber \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\le \\beta _{t,\\delta } \\psi _t(x^1, x^2)\\,,\\end{align}where the \\emph {confidence coefficient} is chosen as follows,\\begin{align*}\\beta _{t,\\delta }^{1/2} &\\triangleq \\rho \\sqrt{\\log \\det ( + \\lambda ^{-1} K_t) + 2\\log \\tfrac{1}{\\delta }} + \\sqrt{\\lambda }B\\,.\\end{align*}The confidence bound is by \\cite {AbbasiYadkori2012}, which improves upon earlier results by \\cite {Srinivas2009}.$" ], [ "Gap Estimate", "We use $\\hat{f}_t$ to define an estimate $\\hat{\\Delta }_t(x)$ of the suboptimality gaps $\\Delta (x) = f(x^*) - f(x)$ for all $x \\in $ .", "Let $\\hat{x}^*_t = _{x \\in } \\hat{f}_t(x)$ be the empirical estimate of the maximizer.", "Note that $\\hat{x}^*_t$ is always defined, despite the fact that in general we can determine $\\hat{f}_t$ only up to a constant shift (which does not affect the maximizer).", "Define t z ft(z) - ft(x*t) + t,1/2 t(x*t, z)1/2 , where $\\psi _t(x,z)$ is defined in ().", "Intuitively, $\\delta _t$ is the largest plausible regret that the learner can occur from playing $\\hat{x}^*_t$ given the confidence estimates.", "The gap estimate is defined for any $x \\in $ as follows: t(x) t + ft(x*t) - ft(x)  .", "The gap estimate satisfies an upper bound on the true gaps, provided that () holds, as summarized in the next lemma.", "With probability $1-\\delta $ , for all $x \\in $ and $t \\ge 1$ , (x) 2 t(x) .", "With probability at least $1-\\delta $ , (x) = f(x*) - f(x) = z f(z) - f(x*t) + f(x*t) - f(x) (i) z ft(z) - ft(x*t) + t,1/2 t(z, x*t)1/2             + ft(x*t) - ft(x) + t,1/2 t(x*t, x)1/2 (ii)=t(x) + t,1/2t(x*t, x)1/2 .", "Here, $(i)$ uses () twice and $(ii)$ is by definition of $\\hat{\\Delta }_t(x)$ .", "On the other hand, t(x) = z ft(z) - ft(x) + t,1/2t(z, x*t)1/2 t,1/2t(x*t, x)1/2 .", "We used that $\\psi _t(\\hat{x}^*_t, x) = \\psi _t(x, \\hat{x}^*_t)$ in the last step.", "The claim follows with the last two displays combined.", "Note that this gap estimate is different from the choice proposed by [21], and importantly, allows us to reduce computational complexity." ], [ "Information Gain", "IDS further requires an information gain function $I_t(x^{1},x^{2})$ defined for each action, which in our case consists of an input pair, $x^{1},x^{2}\\in $ .", "Several choices were proposed by [21].", "Here we use the differential of the log-determinant potential, It(x1,x2) (1 + t(x1, x2)) , = (+ Kt+1)(+ Kt) This choice of information gain corresponds to the Bayesian mutual information $_t\\big (d_t;f|(x_t^{1},x_t^{2})=(x^{1},x^{2})\\big )$ in the Gaussian process model.", "In particular, the total information gain for (REF ) resembles the Gaussian entropy n=t=1n It(xt1, xt2) = (n + Kn+1) .", "The log-determinant depends on the kernel function and upper bounds are known for many popular choices.", "For example, the linear kernel, $k(x,x^{\\prime }) = \\langle x,x^{\\prime }\\rangle $ , satisfies $\\gamma _n \\le (d \\log (n))$ and the RBF kernel, $k(x,x^{\\prime }) = \\exp (-\\Vert x-x^{\\prime }\\Vert _2^2/2)$ , satisfies $\\gamma _n \\le \\log (n)^{d+1}$ [34].", "ruled [t] InputInput Approx.", "IDS for Dueling Feedback $t=1,2,3, \\dots $ $\\hat{x}^*_t \\leftarrow _{x \\in } \\hat{f}_t(x)$ $\\hat{\\Delta }_t(x) \\leftarrow \\delta _t + \\hat{f}_t(\\hat{x}^*_t) - \\hat{f}_t(x)$ *Eq.", "(REF ) $I_t(x) \\leftarrow \\log (1 + \\psi _t(\\hat{x}^*_t, x))$ *Eq.", "(REF ) $x_t, p \\leftarrow _{x \\in ,p \\in [0,1]} \\frac{\\left((1-p)\\delta _t + p \\hat{\\Delta }_t(x)\\right)^2}{pI_t(x)}$ $B_t\\sim \\text{Bernoulli}(p)$ $B_t == 1$ , $(x_t^{1},x_t^{2}) \\leftarrow (\\hat{x}^*_t, x_t)$ $d_t \\leftarrow $DuelingFeedback($x_t^{1},x_t^{2}$ ) $(x_t^{1},x_t^{2}) \\leftarrow (\\hat{x}^*_t, \\hat{x}^*_t)$ *No feedback" ], [ "(Approximate) Information-Directed Sampling", "Given the gap estimate $\\hat{\\Delta }_t(x)$ and information gain $I_t(x^1, x^2)$ , we optimize the following trade-off jointly over $\\times [0,1]$ , zt, pt z ,p [0,1] ((1-p)t + p t(z))2pIt(x*t, z) .", "Note that for fixed $z$ , the optimal trade-off probability is $p(z) = \\min \\left(\\frac{\\delta _t}{\\hat{\\Delta }_t(z) - \\delta _t}, 1\\right)$ .", "Consequently, IDS samples the pair $(\\hat{x}^*_t, z_t)$ with probability $p_t$ , and otherwise, with probability $1-p_t$ , the greedy pair $(\\hat{x}^*_t, \\hat{x}^*_t)$ .", "Note that choosing the same action for dueling feedback provides no information, and in particular the least-squares estimate remains unchanged.", "The approach is summarized in Algorithm REF ." ], [ "Computational Complexity", "As usual, computing the kernel estimates requires to invert the kernel matrix $K_t$ .", "With incremental updates, the exact kernel estimate and all related quantities can be computed in $(d^2n^3)$ steps in total over $n$ rounds.", "Note that $\\hat{x}^*_t$ and $\\delta _t$ can be computed in $(||)$ steps assuming that all kernel quantities have been pre-computed.", "The trade-off (REF ) can be computed in $(||)$ steps, since it only requires to evaluate the gap estimates $\\hat{\\Delta }(x)$ and information gain $I_t(\\hat{x}^*_t, x)$ for all $x \\in $ .", "Therefore the overall complexity is $(d^2n^3||)$ .", "Of course, Algorithm REF can also be applied in the linear setting without kernelization, in which case the overall complexity is $(d^2n||)$ .", "We remark that (REF ) is an approximation of the IDS trade-off proposed by [21], which requires to optimize a similar quantity over distributions $(\\times )$ .", "This is also possible, but the direct implementation suggested in [21] requires $(||^4)$ compute steps per round to calculate the gap estimates and to find the IDS distribution." ], [ "Regret Bounds", "In the language of linear partial monitoring, the dueling feedback () is a so-called locally observable game, which informally means that any reward difference $f(x) - f(x^{\\prime })$ can be estimated from playing actions which have no more regret than playing either $x$ or $x^{\\prime }$ alone [21].", "For dueling bandits, IDS (without the approximation and sampling scheme that we introduce here) has regret at most $R_n \\le (\\sqrt{n \\beta _n \\gamma _n})$ , see [21].", "Here we show that Algorithm REF satisfies a similar result.", "Note that the regret guarantee applies generally to settings with quantitative dueling feedback, where we define regret as follows: Rdueln = t=1n (xt1) + (xt2) .", "For $\\rho ^2$ -sub-Gaussian dueling bandit feedback (), Algorithm REF satisfies with probability at least $1-\\delta $ , Rdueln (n n, (n + 1)) .", "Note that the learner requires knowledge of the sub-Gaussian variance $\\rho ^2$ and the Hilbert norm bound $\\Vert f\\Vert _{} \\le B$ , which appear in the definition of the confidence coefficient $\\beta _{n,\\delta }$ .", "The regret guarantee has the same scaling as the best known bound for GP-UCB in standard Bayesian optimization [34], [11].", "Note that $\\beta _{n,\\delta } = \\left(\\gamma _n + \\log \\frac{1}{\\delta }\\right)$ , hence combined with bounds for the total information gain $\\gamma _n$ , we can derive bounds for specific choices of the kernel function.", "For example in the linear setting, we get $R^\\text{\\textit {duel}}_n \\le \\left(\\rho \\sqrt{n} (d\\log (n) + \\log \\frac{1}{\\delta })\\right)$ , which is the same as for LinUCB [2].", "For the RBF kernel, we get $R^\\text{\\textit {duel}}_n \\le \\left(\\rho \\sqrt{n} (\\log (n)^{2d+2} + \\log \\frac{1}{\\delta })\\right)$ .", "Applied to the confounded setting with either the one- or two-point reduction, we get the following result.", "Note that depending on whether the learner requires one or two evaluations per round, the timescale differs by a factor of two.", "In the confounded setting () with $\\sigma ^2$ -sub-Gaussian observation noise and dueling feedback obtained via the one-point reduction (REF ), the regret of Algorithm REF satisfies with probability at least $1-\\delta $ , Rn ((C + )n n, (n + 1)) , assuming that $\\max _{t\\in [n]} b_t \\le C_{\\max }$ and the adversary is allowed to choose $b_t$ depending on all previous actions and observations, $\\lbrace x_s,y_s\\rbrace _{s=1}^{t-1}$ .", "With the two-point reduction (REF ), Algorithm REF satisfies with probability at least $1-\\delta $ , Rn ((D + )n n, (n + 1)) , assuming that $\\max _{t\\in [n]} |b_{2t} - b_{2t +1}| \\le D_{\\max }$ and the adversary is allowed to choose $b_t$ depending on all but the last two actions and observations, $\\lbrace x_s,y_s\\rbrace _{s=1}^{t-2}$ .", "Note that the algorithm requires knowledge of the bound $C_{\\max }$ or $D_{\\max }$ respectively, which is needed to compute $\\beta _{t,\\delta }$ .", "This is in line with the previous work in the linear setting [23], [18].", "Removing or weakening this assumption is an interesting direction for future work.", "Note that in the linear case with the one-point reduction method, our regret bound is $R_n \\le \\tilde{}(C_{\\max } d \\sqrt{n})$ , which matches the result by [23].", "The dependence on $d$ and $n$ cannot be improved even in the un-confounded linear bandit setting for general $$ [24].", "The results for the two-point reduction requires a stronger assumption on the sequence $(b_t)_{t=1}^{2n}$ , but the regret only depends on the differences $|b_{2t} - b_{2t+1}|$ .", "Hence, the result assures sub-linear regret even for settings where the confounding terms are unbounded, for example when the objective function is subject to drift.", "[Proof of Theorem REF ] First, by Lemma REF with probability at last $1-\\delta $ , $\\Delta (x_t) \\le 2 \\hat{\\Delta }_t(x_t)$ .", "We extend the definition of the gap estimate to two points, $\\hat{\\Delta }_t(x^1, x^2) = \\hat{\\Delta }(x^1) + \\hat{\\Delta }(x^2)$ .", "For a sampling distribution $\\mu \\in (\\times )$ , denote the expected gap $\\hat{\\Delta }_t(\\mu ) \\triangleq _{x^1,x^2 \\sim \\mu }[\\hat{\\Delta }_t(x^1, x^2)]$ and the expected information gain $I_t(\\mu ) = _{x^1,x^2 \\sim \\mu }[I_t(x^1,x^2)]$ .", "The information ratio is defined as follows: t() t()2It()  .", "Let $(\\mu _t)_{t=1}^n$ be the sequence of sampling distributions $\\mu _t \\in (\\times )$ defined by Algorithm REF , $\\mu _t = (1-p_t) e_{(\\hat{x}^*_t, \\hat{x}^*_t)} + p_t e_{(\\hat{x}^*_t, z_t)}\\,,$ where $e_x$ denotes a Dirac on $x \\in $ .", "By [21], with probability $1-\\delta $ , Rdueln t=1n t(t) ( n + (1)) + (n) .", "The claim in the theorem follows if we show that the information ratio is bounded such that $\\Psi _t(\\mu _t) \\le (\\beta _{n, \\delta })$ .", "To this end, note that t(t) = ((1-pt) 2 t + pt (t(zt) + t))2pIt(x*t, zt) x , p [0,1] 4 ((1-p)t + p t(x))2p It(x*t, x) x 4 t(x)2It(x*t, x)  , where the first inequality follows from $\\delta _t \\le \\hat{\\Delta }_t(x)$ and the definition of $z_t$ and $p_t$ , and the second inequality sets $p=1$ .", "On the other hand, using that the kernel is bounded, $k(x,x) \\le 1$ , one easily checks that $\\psi _t(x^1, x^2) \\le 4$ .", "With $a \\le 3 \\log (1 + a)$ for all $a \\in [0,4]$ we find for $x^{1},x^{2}\\in $ t(x1,x2) 3 (1 + t(x1,x2)) = 3It(x1,x2) .", "Next, define $\\tilde{z}_t = _{x \\in } \\hat{f}_t(x) + \\beta _{t,\\delta }^{1/2} \\psi _t(\\hat{x}_t, x)^{1/2}$ and observe that $\\hat{\\Delta }_t(\\tilde{z}_t) = \\beta _{t,\\delta }^{1/2} \\psi _t(\\hat{x}_t, \\tilde{z}_t)^{1/2}$ .", "The claim follows with (REF ) from noting that t(t) 4 t(zt)2It(zt) 12 t, t(xt, zt) t(xt, zt) = 12 t, .", "The result follows from the fact that $\\beta _{t,\\delta }$ is monotonically increasing.", "Algorithm REF also satisfies a gap-dependent bound for finite action sets $$ .", "Let $\\Delta _{\\min } = \\min _{x \\ne x^*} \\Delta (x)$ be the smallest non-zero gap and assume that $x^*$ is unique.", "The theorem applies to the confounded setting via the reduction method similar to Corollary REF .", "Assuming that $x^*$ is unique, the regret of Algorithm REF satisfies with probability at least $1-\\delta $ , Rdueln (-1 n,( n + n)) .", "For linear bandits, the regret bound reads $R^\\text{\\textit {duel}}_n \\le (\\Delta _{\\min }^{-1}d^2 \\log (n)^2)$ and for the RKHS setting with RBF kernel, the bound is $R^\\text{\\textit {duel}}_n \\le (\\Delta _{\\min }^{-1} \\log (n)^{2d+2})$ .", "[Proof of Theorem REF ] Our proof uses the strategy introduced by [22] that relies on finding an instance-dependent bound on the information ratio.", "Let $\\Psi _t(\\mu _t)$ be the information ratio defined in (REF ).", "We apply [21] to the scaled information gain $\\tilde{I}_t = \\beta _{t,\\delta } I_t$ , to find with probability $1-\\delta $ , t=1n t(xt1, xt2)t=1n t(t)t, (n,( n + (n))) We condition now on the event that the previous equation and the confidence estimate in () hold simultaneously.", "As before, let $\\tilde{z}_t = _{x \\in } \\hat{f}_t(x) + \\beta _{t,\\delta }^{1/2} \\psi _t(\\hat{x}_t, x)^{1/2}$ .", "We may assume that $\\tilde{z}_t \\ne \\hat{x}_t$ , since otherwise $\\delta _t = 0$ and therefore $\\Psi _t(\\mu _t) = 0$ .", "On the other hand, this implies that $2 \\hat{\\Delta }_t(\\tilde{z}_t) \\ge \\Delta _{\\min }$ by (REF ) and using that $x^*$ is unique.", "Next, we reuse the inequality leading to (REF ) to find t(t) p [0,1] 4 ((1-p)t + p t(zt))2 p It(xt, zt) .", "First, consider the case where $2 \\delta \\le \\hat{\\Delta }_t(\\tilde{z}_t)$ .", "Computing the minimizer of the previous display and using (REF ), we find t(t) 16 t t(zt) It(xt, zt) 48 t t, t(zt) 96 t t,  .", "For the other case where $2\\delta _t > \\hat{\\Delta }_t(\\tilde{z}_t)$ , using (REF ) directly gives t(t) 12 t, 24 t t, t(zt) 48 t t,  .", "Finally, note that $\\delta _t \\le \\frac{1}{2}\\hat{\\Delta }_t(x_t^{1},x_t^{2})$ .", "Using the bound on the information ratio, and solving for the regret, we find t=1n t(xt1, xt2) (-1 n,( n + n)) .", "The claim follows with Lemma REF by noting that, Rdueln t=1n 2 t(xt1, xt2) ." ], [ "A Connection to Doubly-Robust Estimation", "In the finite-dimensional, linear case, the objective function is $f(x) = \\langle x, \\theta \\rangle $ for a fixed parameter $\\theta \\in ^d$ .", "To obtain an estimate $\\hat{\\theta }_t$ of the unknown parameter $\\theta $ directly from confounded data $\\lbrace (x_s, y_s=\\langle x_s, \\theta \\rangle + b_s + \\epsilon _s)\\rbrace _{s=1}^{t-1}$ , [23] use a randomized policy $x_t \\sim \\mu _t$ and a doubly-robust estimation approach.", "For centered feature vectors $\\bar{x}_t = _{x \\sim \\mu _t}[x]$ and regularizer $\\lambda > 0$ , they define t s=1t-1 (xt - xt)(xt - xt) d , t t-1s=1t-1 (xt - xt)yt.", "They further derive the following high-probability bound for doubly-robust estimator: t - t2 (d (n) + (n/) + ) .", "Interestingly, when $\\mu _t = \\text{Uniform}(\\lbrace x^1,x^2\\rbrace )$ is chosen to randomize between two actions $x^1,x^2 \\in $ , then this estimator coincides with the least-squares estimator that we obtain for the dueling bandit feedback.", "This follows immediately from noting that $2(x_t - \\bar{x}_t) = x^1 - x^2$ .", "Further, the concentration bounds are on the same quantity, but the reduction avoids the detour to prove the (more general) concentration bound (REF ) and leads tighter bounds.", "We remark that in general, the BOSE algorithm by [23] requires to compute a sampling distributions $\\mu _t$ supported on $d+1$ points.", "We also note that [18] propose another variant of the doubly-robust estimator (REF ).", "This estimator coincides with our estimation scheme in the same way.", "Figure: Benchmarks with randomly sampled action sets of size 20 and d=4d=4.", "The confidence region shows 2×\\times standard error over 50 repetition.", "Without confounding, UCB performs best, and is closely followed by IDS and Semi-TS.", "With additive bias the performance of UCB degrades significantly, whereas the robust methods maintain sublinear regret.", "Note that in the drift experiment (bottom left), the bias is unbounded, which leads to linear regret for all methods expect IDS-two.", "With bounded bias (right column), IDS-one and Semi-TS show similar performance, whereas the performance of IDS-two is varying.", "The BOSE algorithm shows sublinear behavior but is much more conservative.Figure: Performance of kernelized algorithms on the camelback benchmark.", "The confidence region shows 2×\\times standard error over 50 repetitions.", "With bounded confounding that simulates a re-calibration process (left), GP-UCB is surprisingly competitive with IDS-two, followed by IDS-one.", "When the feedback is confounded by an unbounded, periodic drift (right), we observe linear regret of GP-UCB and IDS-one (because the bias is unbounded), whereas IDS-two maintains good performance despite the confounding." ], [ "Experiments", "We evaluate the proposed method with the one-point reduction (IDS-one) and the two-point reduction (IDS-two) in two numerical experiments with confounded observations.", "To allow a fair comparison with the two-sample scheme, we account for the regret of both evaluations and scale the x-axis appropriately." ], [ "UCB", "For the linear setting, LinUCB [5] is implemented as in [2] using a regularizer $\\lambda = 1$ and confidence coefficient $\\beta _{t,\\delta }^{1/2} = \\sqrt{\\log \\det V_t + 2 \\log \\frac{1}{\\delta }} + 1$ .", "In the kernelized setting, we use GPUCB [34] with an empirically tuned confidence coefficient $\\beta _{n, \\delta } = 1$ .", "As shown by [8], increasing the confidence coefficient to a larger value as required in the stochastic setting can lead to better robustness, although we did not see an improvement of performance in our experiments." ], [ "BOSE", "The BOSE algorithm [23] uses the doubly-robust least-squares estimator (REF ).", "We set the required concentration coefficient $\\beta _{t,\\delta }^{\\text{DR}}$ to t,DR = d (1 + t/d) + 2(t) + 1 , where we drop (conservative) constants required for the theoretical results in favor of better empirical performance.", "BOSE requires to solve a convex-quadratic feasibility problem on the space of sampling distributions over the remaining plausible actions, and no specific computation method was recommended by the authors.", "We compute the sampling distribution by solving the saddle point problem stated in [23] using exponentiated gradient descent." ], [ "SemiTS", "The semi-parametric Thompson sampling is implemented as in [18], with a less conservative over-sampling parameter $v=\\sqrt{2 \\log (t/\\delta )}$ .", "Our choice improves performance over the theoretical value.", "We also remark that SemiTS requires to compute the probability of each action being optimal under a Gaussian perturbation of the mean parameter.", "We do so by computing the empirical sampling probabilities from 1000 random samples per round, the alternative being to compute Gaussian integrals over $d$ -dimensional polytopes $_x = \\lbrace \\nu \\in ^d : \\langle x, \\nu \\rangle \\ge \\max _{x^{\\prime } \\in } \\langle x^{\\prime }, \\nu \\rangle \\rbrace $ .", "While SemiTS is significantly faster than BOSE in our implementation, computing the posterior probabilities accurately for larger action sets remains challenging.", "In all experiments we set confidence level $\\delta =0.05$ ." ], [ "Linear Reward", "In the first experiment, we use a linear reward function $f(x) = \\langle x, \\theta \\rangle $ .", "For each repetition we sample $k=20$ actions uniformly on the $d=4$ dimensional unit sphere.", "We add Gaussian observation noise with variance $\\sigma ^2=1$ , that is $\\epsilon _t \\sim (0, 1)$ in ().", "In this setting we compare to BOSE, SemiTS and LinUCB [5], [2], where the latter does not directly deal with the confounding.", "We consider four different types of confounding: a) no bias; b) the adversary repeats the last observation with a minus sign, $b_t = -y_{t-1}$ , which makes it much harder to identify the best action (negative repeat); c) a continues drift, $b_t = -0.1t$ , i.e.", "unbounded confounding; and d) same as the previous, but with compensated drift, $b_t = -0.1t + y_{t-1}$ , thereby making the bias terms bounded but dependent on the previous observation.", "The result is shown in Figure REF .", "As expected, in the unconfounded setting UCB works best, followed by both IDS variants and SemiTS with reasonable performance.", "With confounding, the regret of UCB is increased by a lot, whereas BOSE shows sublinear behaviour but is relatively inefficient.", "In the example with unbounded bias (drift) only IDS-two performs well, as in fact the theoretical assumptions for all other methods are invalidated.", "With bounded bias (i.e.", "negative repeat and compensated drift), SemiTS and IDS-one are competitive, while IDS-two clearly outperforms the baselines in the negative repeat experiment." ], [ "Camelback", "Our second experiment is in the non-linear, kernelized setting with observation noise variance $\\sigma ^2=0.1$ .", "As benchmark we choose the camelback function on the domain $[-2,2] \\times [-1,1]$ , f(x1, x2) = - (x12 (4 - 2.1x12 + x143.)", "+ x1x2 + x22(4x22-4), 2.5) We discretize the input space using 30 points per dimension.", "The only direct competitor that we are aware of is the method of [8].", "This method is, however, equivalent to GP-UCB [34] with an up-scaled confidence coefficient.", "This suggests that the UCB approach is inherently robust up to a certain degree of corruption, which is also visible in our experiment.", "For both algorithms, we use an RBF kernel with lengthscale $0.2$ and regularizer $\\lambda = 1$ , and set $\\beta _{n,\\delta }=1$ in favor of better empirical performance.", "We use two types of confounding that we expect is relevant in applications: a) a calibration process, which monitors a moving average over the last 10 observations and adjusts the output range to $[-0.1, 0.1]$ whenever the average is no longer in this range; and b) periodic drift of the objective, $b_t = \\text{sin}(0.2t) - 0.1t$ .", "Results are shown in Figure REF .", "In the first variant GPUCB works surprisingly well despite the confounding and is on-par with IDS-two.", "With unbounded drift, both GPUCB and IDS-one obtain linear regret, whereas the performance of IDS-two in unaffected." ], [ "Conclusion", "We introduced randomized evaluation schemes based on pair-wise comparisons that make dueling bandit algorithms applicable to robust optimization with additive confounding.", "Moreover, we derived a kernelized dueling bandit algorithm based on recent ideas by [21].", "The resulting algorithm satisfies worst-case and gap-dependent regret bounds on the cumulative regret and could be of broader interest in the dueling bandit setting.", "Our numerical experiments validate the theoretical findings." ], [ "Acknowledgements", "This research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement No 815943." ] ]
2105.11802
[ [ "There is no data like more data -- current status of machine learning\n datasets in remote sensing" ], [ "Abstract Annotated datasets have become one of the most crucial preconditions for the development and evaluation of machine learning-based methods designed for the automated interpretation of remote sensing data.", "In this paper, we review the historic development of such datasets, discuss their features based on a few selected examples, and address open issues for future developments." ], [ "Introduction", "In the era of machine learning – more specifically: deep learning – the availability of annotated datasets has become one of the most crucial preconditions for the development and evaluation of new methods for the automated interpretation of remote sensing data.", "While it was possible to train shallow learning approaches on comparably small datasets, deep learning requires large-scale data to reach the desired generalization performance.", "The main goal of general computer vision is the analysis of every-day images containing every-day objects, such as furniture, animals, or road signs.", "Thus, extremely large image databases, such as ImageNetAs a prime example for an annotated computer vision dataset, ImageNet contains more than 14 million images depicting objects from more than 20,000 categories., have been created already more than 10 years ago and form the backbone of many modern machine learning developments.", "In contrast to that, the annotation of remote sensing data is much more complicated due to the dependence on several factors such as sensor technology and target application.", "To provide a hypothetical example: A dataset for the detection of water surfaces from synthetic aperture radar (SAR) imagery will contain observations and annotations that are very different from the observations and annotations contained in a dataset for the semantic segmentation of urban land cover types from multi-spectral optical data.", "This lack of generality has led to the generation of uncountable remote sensing datasets.", "With this paper, we intend to review those developments in order to provide readers with an overview of what is available so far, and what will be needed in the future." ], [ "History of Remote Sensing Datasets", "Of course, datasets have always existed in remote sensing.", "Even before the machine learning era, it was necessary to validate novel signal and image processing algorithms on dedicated test data.", "The generation – and publication – of data dedicated to the training of machine learning algorithms, however has probably only started about 15 years ago, when the IEEE-GRSS Data Fusion Contest was created to foster research in remote sensing data fusion.", "Already from the second contest (organized in 2007) on, typical machine learning tasks (here: pixel-wise classification of land cover) were in the center of interest.", "The beginnings of ML applied to remote sensing were centered on the analysis of individual study sites.", "The datasets were comparably small and samples for training, validation, and testing were often taken from the same image.", "Figure REF shows the development of datasets over time, illustrating also approximate dataset sizes and purposes.", "In the context of this paper, we look at the size of datasets from two perspectives: Size in terms of spatial pixels: For this size measure, we count the number of pixels in the highest available image resolution, while ignoring multi-band, multi-channel and multi-sensor data.", "That is, pixels are only counted once to get a feeling for some form of spatial coverage provided by the dataset.", "Volume in terms of data storage: The amount of disk space required for a dataset serves as an indicator for the provided multitude of modalities (e.g.", "in the form of multi- or hyperspectral bands or several sensor types), as well as the resolution of the imagery.", "Figure: Evolution of remote sensing datasets dedicated to machine learning tasks.", "Since dataset “size” being a hard-to-define measure, it is represented in two ways: The vertical axis relates to the actual data volume, while the circle size relates to the number of spatial pixels covered by the dataset.", "This way, size is connected to both the spatial dimension as well as the overall information content in terms of implicit features such as resolution, sensors modalities, numbers of bands/channels etc.In this context, it is important to mention that we have tried our best to collect information about as many datasets as possible.", "However, it is clear that there will always be datasets we have not yet become aware of.", "Besides, for some datasets, we were not able to gather all required information.", "In spite of that, we believe that several interesting insights can be drawn from the timeline: Besides the data provided in the frame of the IEEE GRSS Data Fusion Contests, there are a few other pioneering datasets, which have certainly fostered research of machine learning applied to remote sensing data in its early stages.", "Those are the UC Merced dataset as the first dataset dedicated to scene classification [1]; the ISPRS Vaihingen/Potsdam dataset, which was originally intended to benchmark semantic segmentation approaches tailored to aerial imagery [2], but has also been used to train methods for other tasks, e.g.", "single-image height reconstruction in the meantime; and the SZTAKI-INRIA dataset designed for object detection [3].", "While more and more datasets have been released starting from 2015, 2018 seems to be the year from which on datasets grew larger and larger both in terms of spatial extent and multi-modal information content.", "As also confirmed by the statistics displayed in Fig.", "REF , most pixels are available for the task of object detection, which correlates to its popularity in deep learning-oriented remote sensing research.", "When it comes to data volume, however, semantic segmentation and classification lead the way, which indicates that in contrast to object detection here more multi-modal data are used.", "As of the time of writing of this paper, we were aware of 181 datasets aiming at the combination of machine learning and remote sensing, of which we were able to consider 141 due to all necessary information being available.", "All of those datasets provide different features, address different sensors, different resolutions and different tasks.", "Thus, there is not yet the single go-to dataset that is used for pre-training most newly developed models, or for benchmarking certain tasks against the state-of-the-art." ], [ "Two Important Examples", "In this section, we describe both the oldest/first and the largest currently available remote-sensing oriented machine learning datasets to provide examples giving a more detailed view of the peculiarities of such datasets and the developments during the last years." ], [ "DFC2007", "As mentioned above, to our knowledge the data provided in the frame of the 2007 IEEE-GRSS Data Fusion Contest was the first dataset at the intersection of remote sensing and machine learning that was openly published for the benefit of the community [4].", "It is a typical old-school dataset sampled from just a single study scene (namely the city of Pavia, Italy).", "It contains two multi-spectral Landsat images (acquired in 1994 and 2000), as well as a time series of 9 ERS SAR images (acquired between August 1992 and July 1995), as well as a sparse map annotating several pixel patches into four generic urban land cover classes.", "Due to the limitation to one scene, the dataset enables the investigation of scene-specific models, which can either be from classical image processing or employ shallow learning.", "Figure: Distribution of dataset sizes over the typical remote sensing tasks: (a) Dataset size expressed in number of pixels; (b) dataset size expressed in data volume." ], [ "Functional Maps of the World (FMoW)", "According to our timeline in Fig.", "REF , the Functional Maps of the World dataset [5] currently is the largest available dataset both in terms of pixels and data volume.", "It consists of 1,047,691 images from 207 countries and is made for the development of machine learning models for the prediction of the functional purposes of buildings and land use from temporal sequences of satellite images and corresponding metadata features about location, time, sun angles, physical sizes etc.", "All the image data of FMoW stem from the Digital Globe constellation and were gathered in pairs, consisting of 4-band (Quickbird-2 or GeoEye-1) or 8-band (Worldview-2/3) multispectral imagery in the visible to near-infrared region, as well as a pan-sharpened RGB image that represents a fusion of the high-resolution panchromatic image and the RGB bands from the lower-resolution multispectral image." ], [ "Current Status and Open Issues", "As mentioned in Section , the amount of datasets for machine learning in remote sensing continues to grow, as new tasks and new sensors, combined with ever-improving possibilities to handle big geospatial data, require new materials for training and evaluating new solutions.", "On the downside, a one-for-all go-to solution – a remote sensing-oriented ImageNet – is still not in sight.", "While this is not really a problem with respect to the generation of sensor- and task-specific models, it introduces a significant overhead in terms of data preparation for every new undertaking in algorithm development; and it also hinders the comparability of newly developed methods, as transparent benchmarks do not really exist.", "We thus hope that the future will bring joint endeavors aiming at the establishment of such a standard database.", "This could either be realized from scratch, or build upon one of the larger existing datasets.", "In any case, it is important that this standard database meets the following criteria: Its data should be sampled across the globe and throughout the year to cover as many cultural and climatic regions as possible.", "The dataset should contain data from as many modalities as possible.", "Starting with freely available satellite imagery, this refers to at least the different Sentinel satellites and the Landsat mission.", "Of course, a possibility to add higher resolution satellite or even aerial data would be highly desirable.", "Ideally, the dataset would try to cover several remote sensing tasks.", "Instead of just focusing on object detection or scene classification, a multi-use annotation would enhance re-usability significantly.", "Since manual labeling of large amounts of remote sensing imagery is very expensive, time-consuming, difficult, and error-prone, robust globally transferable land cover schemes that address multiple semantic scales have to be defined.", "In addition, research has to be invested in reliable ways to source the required annotations.", "Options include automated labeling from existing geodata as well as crowd-sourced mapping – or a combination thereof." ], [ "Summary and Conclusion", "In this paper, we have summarized the developments in datasets for machine learning applied to remote sensing problems.", "We have shown the historic timeline, addressed the pecularities of remote sensing data illustrated by some example datasets, and discussed the current status and open issues.", "While an increase in the availability of annotated data is observable, due to the heterogeneity of remote sensing measurements and tasks, there is still not a single go-to dataset, which could serve the purpose of transparent benchmarking and standardized pre-training.", "However, the increasing importance of open data satellite missions such as the Sentinels of the European Copernicus program or “AI for social good”-based open data initiatives by companies such as Microsoft pave the way to fill this gap in the future.", "Ideally, existing large-scale datasets are built upon and extended for this purpose to benefit from existing expertise." ], [ "Michael Schmitt would like to acknowledge support by the Open Data Impact Award 2020 from the German Stifterverband." ] ]
2105.11726
[ [ "Optimal ANN-SNN Conversion for Fast and Accurate Inference in Deep\n Spiking Neural Networks" ], [ "Abstract Spiking Neural Networks (SNNs), as bio-inspired energy-efficient neural networks, have attracted great attentions from researchers and industry.", "The most efficient way to train deep SNNs is through ANN-SNN conversion.", "However, the conversion usually suffers from accuracy loss and long inference time, which impede the practical application of SNN.", "In this paper, we theoretically analyze ANN-SNN conversion and derive sufficient conditions of the optimal conversion.", "To better correlate ANN-SNN and get greater accuracy, we propose Rate Norm Layer to replace the ReLU activation function in source ANN training, enabling direct conversion from a trained ANN to an SNN.", "Moreover, we propose an optimal fit curve to quantify the fit between the activation value of source ANN and the actual firing rate of target SNN.", "We show that the inference time can be reduced by optimizing the upper bound of the fit curve in the revised ANN to achieve fast inference.", "Our theory can explain the existing work on fast reasoning and get better results.", "The experimental results show that the proposed method achieves near loss less conversion with VGG-16, PreActResNet-18, and deeper structures.", "Moreover, it can reach 8.6x faster reasoning performance under 0.265x energy consumption of the typical method.", "The code is available at https://github.com/DingJianhao/OptSNNConvertion-RNL-RIL." ], [ "Introduction", "As a representative of artificial intelligence methods, deep learning has begun to exceed or approach human performance in various tasks, including image classification, natural language processing, and electronic sports [11], [2], [1].", "But this success is at the cost of high energy consumption.", "Recently, neuromorphic hardware, including TrueNorth, SpiNNaker, Loihi, and so on [5], [19], [4], is attracting more and more researchers due to their high temporal resolution and low power budget.", "This kind of hardware runs Spiking Neural Networks (SNNs) instead of Artificial Neural Networks (ANNs).", "With unique memory and communication designs, an SNN implemented on SpikNNaker can achieveachieves the power consumption of 0.3W for MNIST classification [26].", "For object detection, the spiking version of a YOLO model is estimated to be at least 100 times more energy-efficient than that on GPU [16].", "Such dedicated hardware and algorithms are highly appealing for mobile applications like autonomous vehicles and other power-limited scenes.", "Figure: Illustration of the ANN-SNN converison.Nevertheless, training high-performance SNNs is a nontrivial problem.", "The neurons in SNNs emit discrete spikes, which disables the direct backpropagation training.", "Up to now, the training algorithms for SNNs can be summarised into four methodologies: supervised backpropagation through time [28], [29], unsupervised STDP learning [15], [6], ANN-SNN conversion [3], [7], [22], and other mixture methods [17], [27], [20].", "For deep SNNSNNs training, ANN-SNN conversion requires less GPU computing than supervised training with surrogate gradients.", "Meanwhile, it has yielded the best performance in large-scale networks and datasets among methodologies.", "Therefore, ANN-SNN conversion has become the first choice for deep SNN training, which is also the focus of this paper.", "As illustrated in Fig.", "REF , ANN-SNN conversion is to map the parameters of a pre-trained ANN to an SNN with low accuracy loss.", "Cao et al.", "cao2015spiking started the study of ANN-SNN conversion.", "They found the equivalence between the ReLU activation and the a spiking neurons'neuron's firing rate, which is the foundation of later rate-based methods.", "Diehl et al.", "diehl2015fast attributed the performance loss to inappropriate activation of neurons and proposed Weight Normalization methods (model-based and data-based) to scale the ANN weights.", "Rueckauer et al.", "rueckauer2017conversion gave a detailed theoretical explanation of ANN-SNN conversion and proposed a new reset-by-subtraction neuron to overcome accuracy degradation.", "They also extended the use of bias and Batch Normalization (BN) and proposed the Max Normalization algorithm (Max Norm for short), which uses maximum values of activation as scaling factors.", "Kim et al.", "kim2020spiking suggested to use enhanced normalization and channel-wise normalization for convolutional nets.", "Different from conversion using ANN activation, Sengupta et al.", "sengupta2019going proposed SpikeNorm, which makes use of spiking statistics to set the thresholds.", "In a sense, setting the threshold is equivalent to scaling parameters.", "To enable more accurate conversion, Rueckauer et al.", "rueckauer2017conversion further proposed Robust Normalization (Robust Norm for short), where the scaling factor changes from maximum activation value to 99.9% of activation.", "Yet this is not the first attempt to manually manipulate the threshold or factor.", "Cao et al.", "cao2015spiking set the firing thresholds based on spike density.", "These practices allow the firing rates of some neurons to be constant 1.", "Here we refer to this phenomenon as spike saturation.", "However, both the maximum and 99.9% are rigid, which inspires us to explore a trainable way to achieve low conversion loss.", "The conversion methods mentioned above all incur long simulation time when applied to deeper networks and more complicated datasets.", "That is, converted SNNs need a longer time to rival the original ANNs in precision.", "This restricts the practical promotion, such as real-time tracking and detection.", "Robust Normalization somewhat mitigates this problem by increasing the firing rates.", "Spike saturation actually causes the subsequent layers to infer faster.", "Based on this observation, Han et al.", "han2020rmp started to improve the inference latency by scaling the SpikeNorm thresholds.", "Han et al.", "then gave a theoretical analysis of the scale (setting the firing threshold as the expectation of weights $\\times $ spikes), but they used the manually set value eventually.", "Nevertheless, this inspires us that inference latency and parameters can establish associations on the model.", "A hybrid training scheme also helps.", "Rathi et al.", "rathi2020enabling realized fewer inference time-steps by conversion-based initialization and spike-timing-dependent backpropagation.", "Other methods concern coding schemes to achieve fast inference (i.e.", "shorter inference time), including Temporal-Switch Coding [9], FS-conversion coding [25].", "However, simply bypassing rate coding is not that rational.", "Though rate coding is not the perfect coding scheme, it is partially in line with observations inof the visual cortex [23].", "Therefore, fast inference for rate-encoding deep SNNs is an important research direction.", "But it still lacks instructive principles and theories.", "In this paper, we propose an ANN-SNN conversion method that enables high accuracy and low latency.", "The main contributions of this paper are summarized as follows: We theoretically analyze ANN-SNN conversion and derive the sufficient conditions of the optimal conversion.", "Based on this, we proposed the Rate Norm Layer to replace the ReLU activation function in source ANN, enabling direct conversion from a trained ANN to an SNN.", "without any operation.", "This will reduce the potential loss of information caused by normalization.", "We propose an optimal fit curve to quantify the fit between the activation value of source ANN and the actual firing rate of target SNN, and derive one upper bound of this convergent curve.", "We show that based on the Squeeze Theorem, the inference time can be reduced by optimizing the coefficient in the upper bound.", "These results can not only systematically explain previous findings that reasonable scaling of the threshold can speed up inference, but also give a proper theoretical basis for fast inference research.", "We demonstrate the utility of the proposed method with near loss-less conversion in deep network architectures on the MNIST, CIFAR-10, CIFAR-100 datasets.", "Moreover, it achieves $8.6\\times $ faster reasoning under $0.265\\times $ energy consumption of the typical method." ], [ "Methods", "In this section, athe theory for ANN-SNN conversion is first introduced.", "Based on this, the Rate Norm Layer with trainable threshold is thus proposed.", "Then, we analyse the reason for slow inference and suggest optimization for fit of firing rates.", "Finally, we present athe stage-wise learning strategy for accurate and fast SNN." ], [ "Theory for Conversion from ANN to SNN", "The fundamental principle of ANN-SNN conversion is to match analog neurons' activation an analog neuron's activation with spiking neurons' firing ratethe firing rate of a spiking neuron.", "One common way is to convert ReLU nonlinearity activation to the Integrate-and-Fire (I&F) neuron.", "To be specific, for analog neurons in layer $l$ ($l=1,2,...,L$ ), the ReLU activation can be described by: $\\mathbf {a}_l = \\max (W_{l-1}\\mathbf {a}_{l-1}+\\mathbf {b}_{l-1},0),$ where vector $\\mathbf {a}_l$ is the output of all ReLU-based artificial neurons in layer $l$ , $W_{l-1}$ and $\\mathbf {b}_{l-1}$ is the weight and the bias term for the neurons in layer $l-1$ .", "As for the I&F neuron, the membrane potential $v_{l}^i(t)$ for the $i$ -th neuron in layer $l$ is formulated by: $\\frac{dv^i_{l}(t)}{dt} = \\sum _j \\sum _{t_j\\in T_j} W^{ij}_{l-1} \\delta (t-t_j)+{b}_{l-1}^i,$ where $b_{l-1}^i$ denotes the biasinput current to the $i$ -th neuron, $W^{ij}_{l-1}$ denotes the synaptic weight between the $j$ -th presynaptic neuron in layer $l-1$ and the $i$ -th neuron in layer $l$ .", "$\\delta (\\cdot )$ is the delta function.", "$T_j$ denotes the set of the spike times of the $j$ -th presynaptic neuron, i.e., $T_j=\\lbrace t_j^{(1)},t_j^{(2)},\\dots ,t_j^{(K)}\\rbrace $ .", "When the membrane potential $v_{l}^i(t)$ exceeds the firing threshold $v_{th,l}$ in layer $l$ , a spike is generated and the membrane potential $v_{l}^i(t)$ is reset to the rest value $v_{rest}<v_{th,l}$ .", "To match analog neurons' activationan analog neuron’s activation with spiking neurons'a spiking neuron's firing rate, we discretize and vectorize Eq.", "REF into time-steps and obtain the spikingreset-by-subtraction neuron model for layer $l$ .", "$\\mathbf {m}_l(t) &= \\mathbf {v}_l(t-1) + W_{l-1} \\mathbf {s}_{l-1}(t)+\\mathbf {b}_{l-1}, \\nonumber \\\\\\mathbf {s}_l(t) &= U(\\mathbf {m}_l(t)-{v}_{th,l}),\\\\\\mathbf {v}_l(t) &= \\mathbf {m}_l(t)-{v}_{th,l}\\mathbf {s}_l(t), \\nonumber $ where $\\mathbf {m}_l(t)$ and $\\mathbf {v}_l(t)$ represent the membrane potential of all I&F neurons in layer $l$ after neuronal dynamics and after the trigger of a spike at time $t$ , $U(\\cdot )$ is the Heaviside Step Function, $\\mathbf {s}_l(t)$ denotes the vector of the binary spikes, the element inof which equals 1 if there is a spike and 0 otherwise.", "$\\mathbf {b}_{l-1}$ is the vector of ${b}_{l-1}^i$ , and $W_{l-1}$ is the weight matrix.", "Note that here we use the \"soft reset\" [10] instead of the \"hard reset\".", "At the moment of a spike, the membrane potential $\\mathbf {v}_l(t)$ is reducedreduces by an amount equal to the firing threshold $v_{th,l}$ , instead of going back to the reset value.", "Based on these definitionsdefinition, we can derive the relationship between the firing rate $\\mathbf {r}_l(t)$ of spiking neurons in layer $l$ and $\\mathbf {r}_{l-1}(t)$ of neurons in layer $l-1$ , which is depicted in Lemma REF .", "The proof can be found in the Appendix.", "Lemma 1 For a spiking neural network consistingconsists of the reset-by-subtraction neurons mentioned in Eq.", "REF , assume that $W_{l-1}$ and $\\mathbf {b}_{l-1}$ are the parameters for layer $l-1$ .", "Then when $t \\rightarrow \\infty $ , the relation of the firing rate $\\mathbf {r}_l(t)$ and $\\mathbf {r}_{l-1}(t)$ is given by: $\\mathbf {r}_l = {\\rm clip} \\left(\\frac{W_{l-1}\\mathbf {r}_{l-1}+\\mathbf {b}_{l-1}}{v_{th,l}},0,1 \\right),$ where ${\\rm clip}(x,0,1)=x$ when $x\\in [0,1]$ , ${\\rm clip}(x,0,1)=1$ when $x > 1$ , and ${\\rm clip}(x,0,1)=0$ when $x < 0$ .", "With Lemma 1 and Eq.", "REF , We can derive the theorem for conversion from ANN to SNN: Theorem 1 For an $L$ -layered ANN with the ReLU activation and an $L$ -layered SNN with the reset-by-subtraction neurons, assume that $W^{ \\text{ANN}}_{l-1},\\mathbf {b}^{\\text{ANN}}_{l-1}$ are the parameters for layer $l-1$ of thean ANN, and $W^{\\text{SNN}}_{l-1},\\mathbf {b}^{\\text{SNN}}_{l-1}$ are the parameters for layer $l-1$ of thean SNN.", "$\\max _{l}$ is the maximum activation of layer $l$ in ANN, and $v_{th,l}$ is the firing threshold of layer $l$ in SNN.", "The ANN can be converted to the SNN when $t \\rightarrow \\infty $ (Eq.", "REF equals Eq.", "REF ) if for $l=1,2,...,L$ , the following equations hold: $\\frac{W^{\\text{SNN}}_{l-1}}{v_{th,l}} = W^{\\text{ANN}}_{l-1} \\frac{\\max _{l-1}}{\\max _{l}},~~~\\frac{\\mathbf {b}^{\\text{SNN}}_{l-1}}{v_{th,l}} = \\frac{\\mathbf {b}^{\\text{ANN}}_{l-1} }{\\max _{l}}.$ The proof of Theorem REF is presented in the Appendix.", "Eq.", "REF implies that scaling operations are necessary to convert ANN to SNN, either scaling weights (i.e.", "weight normalization) or setting thresholds (i.e.", "threshold balancing).", "In this sense, weight normalization is equivalent to threshold balancing." ], [ "Rate Norm Layer", "The choices of scaling factors are often empirical, and post-training [21], [10].", "To overcome this, we propose Rate Norm Layer (RNL) to replace the ReLU activation in ANN.", "The idea is to use a clip function with a trainable upper bound to output the simulated firing rate, which is the limitation of actual firing rates in SNN when inference time $T \\rightarrow \\infty $ as time step increases.", "Here we denote the simulated firing rate as $\\hat{\\mathbf {r}}_l$ , Rate Norm Layer can be formally expressed as follows: ${\\theta }_l &= p_l \\cdot \\max (W_{l-1} {\\hat{\\mathbf {r}}_{l-1}}{\\mathbf {z}_{l-1}}+\\mathbf {b}_{l-1}), \\nonumber \\\\\\mathbf {z}_l &={\\rm clip}(W_{l-1} {\\hat{\\mathbf {r}}_{l-1}}{\\mathbf {z}_{l-1}}+\\mathbf {b}_{l-1},0,{\\theta }_l), \\\\\\hat{\\mathbf {r}}_l &= \\frac{\\mathbf {z}_l}{\\theta _l} {\\quad \\lbrace \\textbf {simulated~firing~rate~in~ANN} \\rbrace }, \\nonumber $ where $p_l$ is a trainable scalarparameter $(p_l \\in [0,1])$ , and $\\theta _l$ is the threshold of the $l$ -th layer$\\theta _l$ is a dynamic threshold of the $l$ -th layer.", "$p_l$ is a trainable parameter and is restricted in $[0,1]$ .", "With Theorem REF satisfied ($v_{th,l}=\\theta _l$ ) and $p_l=1$ , one can find that Eq.", "REF is equivalent to Eq.", "REF Comparing Eq.", "REF and Eq.", "REF , one can find that an ANN with RNL can be directly converted to an SNN without any scaling.", "In this caseWhen $p_l=1$ , RNL will degenerate to the Max Norm algorithm, which scales the weight $W_{l-1}$ by $\\frac{\\max _{l-1}}{\\max _{l}}$ and the bias $\\mathbf {b}_{l-1}$ by $\\frac{1}{\\max _{l}}$ .", "A diagram comparison of different scaling schemes can be seen in Fig.", "REF .", "Analyses in Section REF The following analysis will show that if the optimization for fast inferenceof inference time is deployed, the upper limit of $p_l$ may be unnecessary.", "A diagram comparison of different scaling schemes can be seen in the appendix.", "For mini-batch training, different batches have different maximum outputs.", "To reduce the perturbation caused by data sampling, $\\rm running\\_max(W_{l-1} \\hat{\\mathbf {r}}_{l-1}+\\mathbf {b}_{l-1})$${\\rm running\\_max}(\\mathbf {z}_l)$ is used instead of $\\max (W_{l-1} \\hat{\\mathbf {r}}_{l-1}+\\mathbf {b}_{l-1})$$\\max (\\mathbf {z}_l)$ .", "Figure: Response of simulated firing rate of Rate Normalization Layer with regard to ANN activation.", "The trainable threshold can adapt during training, and manual setting can be cancelled.The design of the Rate Norm Layer mainly considers the following three factors: Compared with directly cutting the simulated firing rate to 1, the backpropagation becomes more effective in RNL training.", "$\\max (W_{l-1} \\hat{\\mathbf {r}}_{l-1}+\\mathbf {b}_{l-1})$ and $\\rm running\\_max(W_{l-1} \\hat{\\mathbf {r}}_{l-1}+\\mathbf {b}_{l-1})$$\\max (\\mathbf {z}_l)$ and ${\\rm running\\_max}(\\mathbf {z}_l)$ can enable the gradient to flow out smoothly.", "Their participation in threshold calculating is similar to Batch Norm (BN) in mini-batch training.", "However, RNL cannot replace BN, because BN rescales the data to a normal distribution, which is related to the characteristics of ANN.", "RNL more guarantees the similarity with ReLU at the conversion level.", "The threshold $\\theta _l$ enables better generalization.", "Existing solutions mainly focus on using a subset of the training data for offline normalization.", "This will potentially influence the generalization of the scaled SNN for data out of subset.", "In contrast, $\\theta _l$ uses all data in training, which can be used directly in SNN inference.", "The threshold $\\theta _l$ is determined in training.", "For faster inference, Robust Norm requires empirical percentile, which RNL doesn't need, as $p_l$ is trainable.", "Certainly, just using ANN loss will not guide the model to reduce the inference delay.", "This also requires additional loss design, which will be shown later." ], [ "Optimization for Fast Inference", "RateThe rate-based SNN models takes rate coding as input, and the time average of the output spikes as output.", "In the conversion methods, the coding scheme mainly consists of two ways.", "One is Poisson coding, of which spikes obey the Poisson process.", "The other is constant coding, to cooperate with the reset-by-subtraction neuron.", "Constant coding is not a new thing, it can be regarded as an integrating ADC in the signal processing [8].", "The two primary rate coding forms take time for the firing rate to approach its expectations.", "For the constant coding, its accumulated current needs to be rounded down when converted to spike counts.", "So the firing rate will be jagged and approach the analog value (Fig.", "REF ).", "Using both codings will bring about unpredictable rate output in the first few time-steps.", "Following the suggestions of previous literature, constant coding is chosen as the primary scheme in this paper.", "The time when the output firing rate of an SNN matches the analog output of an ANNSNN is referred to as “inference time”, “inference latency” or “inference delay” [18].", "Fig.", "REF implies that for both rate coding schemes, there will be an output delay.", "In deep neural networks, the stacking of layer-by-layer delays will bring greater inference delay.", "For example, ResNet-44 requires 350 time-steps to achieve the best accuracy [13].", "This problem is not limited to ANN-SNN Conversion.", "The BPTTdirectly trained SNN model also has similar issues.", "Now that the reason for slow inference is attributed to the deviation of the encoder accumulation.", "To further analyze the characteristics, we propose to use $K(\\hat{\\mathbf {r}},\\mathbf {r}(t))$ (K curve) to quantify the relationship between the simulated firing rate $\\hat{\\mathbf {r}}$ of ANN and the real firing rate $\\mathbf {r}(t)$ of SNN after conversion.", "$K(\\hat{\\mathbf {r}},\\mathbf {r}(t))= \\frac{{\\Vert \\mathbf {r}(t) - \\hat{\\mathbf {r}} \\Vert }_2^2}{{\\Vert \\hat{\\mathbf {r}} \\Vert }_2^2}.$ Note that the design of $K$ resembles the chi-square tests in hypothesis testing.", "$\\hat{\\mathbf {r}}$ and $\\mathbf {r}(t)$ denote the firing rates of all neuronsthe neuron in a certain layer.", "$\\Vert \\cdot \\Vert _2$ indicates the $L^2$ norm.", "The denominator ${{\\Vert \\hat{\\mathbf {r}} \\Vert }_2^2}$ makes $K(\\hat{\\mathbf {r}},\\mathbf {r}(t))$ have scale invariance.", "Therefore, we can compare the fitting of the firing rate between different layers.", "Ideally, given enough “inference time”, $K(\\hat{\\mathbf {r}},\\mathbf {r}(t))$ will converge to 0.", "We believe K curve is an adequate metric as the neuron population encoding information is considered rather than any single neuron.", "Figure: Layer-wise K curves over time of a VGG16 model.", "(a) The result of SNN using the Max Norm algorithm; (b) The result of scaling the thresholds by 0.8.Specifically, Fig.", "REF gives an example to illustrate how the K curve fits between different layers of a VGG16 and a converted SNN.", "An image is used for reasoning and calculating K curves.", "As the layer deepens, the convergence speed of the K curve becomes slower.", "The time $t_s$ at which the deep curve begins to fall continues to be delayed.", "It indicates that not until $t_s$ do the spikes generate.", "By accelerating the convergence of the K curve, the inference can speed up.", "Here we derive one of the upper bound for $K(\\hat{\\mathbf {r}},\\mathbf {r}(t))$ .", "Theorem 2 For layer $l$ in an ANN and the converted SNN with constant coding, given the simulated firing rate $\\hat{\\mathbf {r}}_l$ and the real firing rate $\\mathbf {r}_l$ , we have: $K_l < \\frac{2\\Omega _l}{t},$ where $K_l$ denotes the abbreviation of $K(\\hat{\\mathbf {r}}_l,\\mathbf {r}_l(t))$ in layer $l$ , $\\Omega _l=\\frac{ \\Vert \\hat{\\mathbf {r}}_l \\Vert _1 }{\\Vert \\hat{\\mathbf {r}}_l \\Vert _2^2} $ .", "$\\Vert \\cdot \\Vert _p$ denotes $L^p$ norm.", "The detailed proof of Theorem REF is described in the Appendix.", "$\\Omega _l$ is named as Rate Inference Loss (RIL) of the $l$ -th layer.", "Eq.", "REF indicates that $K_l$ is less than an inverse proportional curve to $t$ .", "For the convergence of the last layer (the $L$ -th layer), if $\\Omega _L$ reduces, $K_L$ will converge faster to 0 due to the Squeeze Theorem.", "That is, the real firing rate of the SNN approaches the simulated value of the ANN more faster, leading to faster and more stable outputs of SNN.", "However, considering that the network actually has inference delays layer by layer, a better solution is to reduce the average value of $\\Omega _l (l=1,2,\\dots ,L)$ .", "Thus, the overall training loss is composed of the loss related to the task and RIL multiplied by hyperparameter $\\lambda $ .", "${L}^{\\prime }(f(\\mathbf {x}),\\mathbf {y})={L}(f(\\mathbf {x}),\\mathbf {y})+ \\lambda \\frac{\\sum {\\Omega _l}}{L},$ where $(\\mathbf {x},\\mathbf {y})$ is data tuple for training and $f(\\cdot )$ is the network with L Rate Norm Layers.", "Based on Eq.", "REF , we are able to calculate the partial derivative of $\\Omega _L$ w.r.t.", "$p_L$ , and obtain: $\\frac{\\partial \\Omega _L}{\\partial p_L} &= \\sum _i{\\frac{\\partial \\Omega _L}{\\partial \\hat{\\mathbf {r}}_{L,i}} \\frac{\\partial \\hat{\\mathbf {r}}_{L,i}}{\\partial p_L} } \\nonumber \\\\&= \\sum _i{\\bigg [ \\bigg ( \\frac{ \\Vert \\hat{\\mathbf {r}}_{L}\\Vert _2^2 -2\\hat{\\mathbf {r}}_{L,i} \\Vert \\hat{\\mathbf {r}}_{L,i} \\Vert _1 }{\\Vert \\hat{\\mathbf {r}}_{L} \\Vert _2^4} \\bigg ) \\bigg ( - \\frac{\\hat{\\mathbf {r}}_{L,i}}{p_L} \\bigg ) \\bigg ] } \\nonumber \\\\&= \\frac{\\Vert \\hat{\\mathbf {r}}_L \\Vert _1 }{p_L \\Vert \\hat{\\mathbf {r}}_L \\Vert _2^2 },$ where $\\hat{\\mathbf {r}}_{L,i}$ denotes the $i$ -th element of $\\hat{\\mathbf {r}}_{L}$ .", "Eq.", "REF implies that the partial derivative of $\\Omega _L$ w.r.t.", "$p_L$ is positive.", "Simply minimizing $\\Omega _L$ will reduce the neurons’ $p_l$ .", "The upper limit of $p_l$ is unnecessary in this sense.", "Nevertheless, this will lead more neurons to saturation state and loseloss model accuracy.", "Thus, we jointly optimize the two losses and tune the hyper-parameter $\\lambda $ to reach the optimal trade-off between model accuracy and conversion loss.Thus, joint learning of task loss and rate inference loss will greatly alleviate the problem.", "So far, the current theories and analyses can also systematically explain the findings of Han et al.", "That is, reasonable scaling of the threshold can speed up the inference.", "If the threshold is set smaller, the Rate Inference Loss will decrease.", "Since the curve value and time are inversely proportional, it is equivalent to accelerating the decrease of the curve.", "On the contrary, if the threshold is too large, many neurons in the SNN will have a low firing rates to produce accurate output.", "Of course, the threshold should not be too small to prevent more neurons from saturating and losing information.", "Fig.", "REF (b) shows the K curve after adjusting the threshold using the algorithm of Han et al." ], [ "Training for Accurate and Fast SNN", "In Section REF , the $p_l$ of the Rate Norm Layer is restricted to $[0,1]$ .", "This means if the Rate Norm Layer is directly trained, the simulated firing rate after clipping and scaling will inevitably bear information loss.", "The information loss here indicates that the output distribution due to cropping is different from the original distribution.", "This will make it difficult to take advantage of ANN's performance.", "So the training strategy needs to be carefully designed.", "When $p_l=1$ is fixed, $\\mathbf {z}_l$ is clipped with ${\\rm running\\_max}(\\mathbf {z}_l)$ , which has less information loss.", "AfterWhen the network synaptic parameters are fixed, the neuron threshold starts training.", "The training goal at this stage is to reduce information loss and to reduce Rate Inference Loss.", "As the output $f(\\mathbf {x})$ of an ANN is $\\hat{\\mathbf {r}}_L$ , the goal is to optimize (according to Eq.", "REF ): ${L}^{\\prime }(\\hat{\\mathbf {r}}_L,\\mathbf {y})={L}(\\hat{\\mathbf {r}}_L,\\mathbf {y})+\\lambda \\frac{\\sum {\\Omega _L}}{L},$ we decompose this goal into two stage goals (shown in Fig.", "REF ): Stage 1 is accuracy training, when network outputs $\\hat{\\mathbf {r}}^*_L$ .", "The target is: $\\min _{W,b} {L}(\\hat{\\mathbf {r}}^*_L,\\mathbf {y} ).$ Stage 2 is for fast inference.", "Assume the output of this stage is $\\hat{\\mathbf {r}}^{\\prime }_L$ .", "Then the target is: $&\\min _{\\theta } {T}(\\hat{\\mathbf {r}}_j^*,\\hat{\\mathbf {r}}_j^{\\prime }), \\nonumber \\\\{T}(\\hat{\\mathbf {r}}_j^*,\\hat{\\mathbf {r}}_j^{\\prime }) =& 1- Cos(\\hat{\\mathbf {r}}^*_L,\\hat{\\mathbf {r}}^{\\prime }_L) + \\lambda \\frac{\\sum {\\Omega _L}}{L}.$ The cos distance is used to maintain neuron information.", "The detailed training and converting algorithm is described in Algorithm REF in the AppendixThe detailed training algorithm is described in the appendix." ], [ "Experiment Implementation", "We validate our methods on the image recognition benchmarks, namely the MNISThttp://yann.lecun.com/exdb/mnist/, CIFAR-10, CIFAR-100https://www.cs.toronto.edu/ kriz/cifar.html datasets.", "For MNIST, we consider a 7-layered CNN and AlexNet.", "For CIFAR-10, we use VGG-16 and PreActResNet-18 network structures.", "It is worth noting that we did not use the common ResNet as we think PreActResNet will help the training of Rate Norm Layers [12].", "For CIFAR-100, VGG-16, PreActResNet-18 and PreActResNet-34 are used.", "The choice of all hyperparameters is shown in Table S1 in the Appendix.", "We present the simulation results and analysis in the following subsections.", "Table: Best Accuracy Performance comparing with related methods.", "Values in the table represent the best accuracy and the accuracy loss of conversion (Acc ANN -Acc SNN \\text{Acc}_{\\text{ANN}} - \\text{Acc}_{\\text{SNN}}).", "The structure of 7-Layered CNN is 32C3-P2-32C3-P2-32C3-P2-32FC10, which is different from the one with asterisk in the table." ], [ "Accuracy Performance", "We first evaluate the effectiveness of the proposed Rate Norm Layer.", "A networkNetwork with Rate Norm Layers is trainable with backpropagation.", "When testing SNN performance, all Rate Norm Layers are converted to I&F neuron models with $\\theta _l$ as the threshold.", "Table REF shows the best accuracy of the converted SNNs compared with typical works.", "The converted SNNs achieve the state-of-the-art performance on MNIST and CIFAR-100 datasets, and reach a similar performance level on CIFAR-10.", "For VGG-16 trained by CIFAR-100, the proposed method reaches top-1 accuracy 75.02%, whereas the state-of-the-art ANN-SNN algorithm reaches 70.93%.", "Conversion loss is considered here to evaluate the quality of the ANN-SNN conversion.", "As illustrated in Table REF , ANN-SNN conversion with the Rate Norm Layers has low conversion loss, or even negative conversion loss, indicating that the converted SNN may outperformoutperforms the original ANN.", "In contrast, the loss is usually positive for other methods, meaning that the performance of the converted SNN is not as good as ANN." ], [ "Fast Inference Performance", "We test whether the proposed Rate Inference Loss can speed up inference, that is, speed up the convergence of the K curve.", "Fig.", "REF (a) and (b) show how the K curve and accuracy change over latency, where the dotted line in Fig.", "REF (a) represents the Max Norm method.", "As can be seen from Fig.", "REF (a), the K curves of the proposed method converge to 0 quickly, which are much faster than those of the Max Norm method.", "Thus the proposed method can implement fast inference.", "The inference performance can be observed in Fig.", "REF (b).", "The proposed method reaches an accuracy of 85.40% using 32 time-steps, whereas the methods of Max Norm, Robust Norm, and RMP-SNN reach 10.00%, 43.03% and 63.30% at the end of 32 time-steps.", "Moreover, the proposed method achieves an accuracy above 90% using only 52 time-steps, which is 8.6 times faster than Max Norm that uses 446 time-steps.", "Detailed accuracy comparison on time T is shown in Table REF .", "The threshold and $\\Omega $ of VGG-16 are visualized in Fig.", "REF (c).", "For the Max Norm method, as all the I&F neurons use the same threshold 1, we regard the maximum value of ReLU outputs as the equivalent threshold.", "It can be found that the distribution gap of the threshold is relatively large.", "But when paying attention to the $\\Omega $ distribution, the $\\Omega $ after the threshold scaling (orange column) is usually smaller than that of Max Norm (blue column).", "In contrast, training with Rate Inference Loss will keep $\\Omega $ at a relatively low and average level, and thus benefit inference.", "Table: Fast Inference Performance comparing with related methods.", "Values in the table represent the instant accuracy of latency T. All the networks are trained on CIFAR-10." ], [ "Energy Estimation of Neuromorphic Hardware", "SNNs have a considerable potential on neuromorphic chips.", "One of the benefits is to reduce the energy budget.", "To study the energy efficiency of fast reasoning, we use the energy model proposed by Cao et al.", "[3] to model the energy consumption on neuromorphic chips.", "Assumed that a spike activity would bring about the energy consumption of $\\alpha $ Joules and 1 time-step takes $1ms$ .", "Then the power model is defined as: $P = \\frac{total \\, spikes}{1\\times 10^{-3}} \\times \\alpha \\, (Watts)$ The previous energy analysis mainly focused on total energy consumption during inference.", "However, in real application scenarios, the total energy consumed before the model reaches reliable accuracy is more important.", "In this regard, we evaluate the performance of the proposed method and the Max Norm method on energypower.", "Fig.", "REF (a) is the power histogram over time.", "Due to the lower threshold, the power of our model is relatively high.", "The integral of the deep color area represents the energy consumed to achieve 90% accuracy.", "The energy consumption of the proposed model is only 0.265 times of the Max Norm method when it reaches 90% accuracy.", "This means that the 8.6 $\\times $ reasoning speedup will not bring much more energy consumption.", "Fig.", "REF (b) shows the logarithmic ratio of energy and inference speedup.", "Our model exhibits the properties of “fast reasoning” and “energy efficiency”.", "Figure: Energy analysis of VGG-16 on neuromorphic chips.", "(a) is the power histogram.", "The integral of the deep color area is the energy for accuracy to reach 90%.", "(b) is the logarithmic ratio of energy consumption and inference speedup.", "`p' is for the proposed method, and `b' is for the baseline Max Norm method." ], [ "Conclusions", "This paper proposes a method to directly convert conventional ANNs to SNNs.", "The Rate Norm Layer is introduced to replace ReLU for optimal conversion.", "Besides, we quantify the fit between the ANN activation and the firing rate of the converted SNN by an optimal fit curve.", "The inference time can be reduced by optimizing the coefficient of the upper bound of the fit curve, namely Rate Inference Loss.", "Thus, a two-stagedjoint learning scheme is proposed to obtain fast and accurate deep SNNs.", "Experimental results demonstrate that our methods achieve low accuracy loss and fast reasoning with deep structures such as VGG and PreActResNet." ], [ "Acknowledgment", "This work was supported by the National Natural Science Foundation of China (62027804, 61825101, 62088102 and 61961130392).", "Here we provide the theoretic proofs of theorems and lemmas in this paper.", "Lemma 1 For a spiking neural network consistings of the reset-by-subtraction neurons mentioned in Eq.", "3, assume that $W_{l-1}$ and $\\mathbf {b}_{l-1}$ are the parameters for layer $l-1$ .", "Then when $t \\rightarrow \\infty $ , the relation of the firing rate $\\mathbf {r}_l(t)$ and $\\mathbf {r}_{l-1}(t)$ is given by: $\\mathbf {r}_l = {\\rm clip} \\left(\\frac{W_{l-1}\\mathbf {r}_{l-1}+\\mathbf {b}_{l-1}}{v_{th,l}},0,1 \\right),$ where ${\\rm clip}(x,0,1)=x$ when $x\\in [0,1]$ , ${\\rm clip}(x,0,1)=1$ when $x>1$ , and ${\\rm clip}(x,0,1)=0$ when $x<0$ .", "For the reset-by-subtraction spiking neurons formulated by Eq.", "3 in the main text, we can stack the equations and get the discrete function between spikes of layer $l$ and layer $l-1$ : $\\mathbf {v}_l(t)-\\mathbf {v}_l(t-1)=W_{l-1}\\mathbf {s}_{l-1}(t)+\\mathbf {b}_{l-1}-v_{th,l}\\mathbf {s}_l(t).$ By summing the left and right expressions over time and dividing $t v_{th,l}$ on the both sidesdividing by $t v_{th,l}$ , the equation can be reformulated as: $\\mathbf {r}_l(t) = \\frac{\\sum ( \\mathbf {s}_l(t))}{t} = \\frac{W_{l-1} \\mathbf {r}_{l-1}(t)+\\mathbf {b}_{l-1}}{v_{th,l}} - \\frac{\\mathbf {v}_l(t)}{t v_{th,l}},$ where $\\mathbf {r}_l (t)$ denotes the firing rates of all neurons in layer $l$ .", "From the membrane potential updating function (Eq.", "3), $\\mathbf {v}_l (t)$ is in the range of $[0, v_{th,l}]$ , thus we have: $\\lim _{t\\rightarrow \\infty } \\frac{\\mathbf {v}_l (t)}{tv_{th,l}}= 0.$ As the value of $\\mathbf {s}_l(t)$ can only be 0 or 1, the firing rate $\\mathbf {r}_l (t)$ is strictly restricted in $[0,1]$ .", "When $t \\rightarrow \\infty $ , it's straightforward to conclude that $\\mathbf {r}_l = {\\rm clip} \\left(\\frac{W_{l-1}\\mathbf {r}_{l-1}+\\mathbf {b}_{l-1}}{v_{th,l}},0,1 \\right).$ Theorem 1 For an $L$ -layered ANN with the ReLU activation and an $L$ -layered SNN with the reset-by-subtraction neurons, assume that $W^{ \\text{ANN}}_{l-1},\\mathbf {b}^{\\text{ANN}}_{l-1}$ are the parameters for layer $l-1$ of thean ANN, and $W^{\\text{SNN}}_{l-1},\\mathbf {b}^{\\text{SNN}}_{l-1}$ are the parameters for layer $l-1$ of thean SNN.", "$\\max _{l}$ is the maximum activation of layer $l$ in ANN, and $v_{th,l}$ is the firing threshold of layer $l$ in SNN when $t \\rightarrow \\infty $ .", "The ANN can be converted to the SNN (Eq.", "1 equals Eq.", "4) if for $l=1,2,...,L$ , the following equations hold: $\\frac{W^{\\text{SNN}}_{l-1}}{v_{th,l}} = W^{\\text{ANN}}_{l-1} \\frac{\\max _{l-1}}{\\max _{l}},~~~\\frac{\\mathbf {b}^{\\text{SNN}}_{l-1}}{v_{th,l}} = \\frac{\\mathbf {b}^{\\text{ANN}}_{l-1} }{\\max _{l}}.$ The conversion of ANN to SNN is built on the basis of the equivalent of SNN firing rate and ANN activation.", "Considering spiking neurons of layer $l$ and $l-1$ in an SNN, the relationship between the firing rate $\\mathbf {r}_l(t)$ and $\\mathbf {r}_{l-1}(t)$ is: $\\mathbf {r}_l = {\\rm clip} \\left(\\frac{W^{\\text{SNN}}_{l-1} \\mathbf {r}_{l-1}+\\mathbf {b}^{\\text{SNN}}_{l-1}}{v_{th,l}},0,1 \\right).$ Note that the firing rate $\\mathbf {r}_{i} (i=1,2,\\dots ,L)$ in Eq.", "REF is restricted in $[0,1]$ .", "But the ANN activation (ReLU) only satisfy $a_i \\ge 0$ (Eq.", "1 in the main text).", "In fact, for countable limited dataset, the activation generated by the network is also upper bounded.", "Assume that the upper bound for the output of all ReLU-based artificial neurons in layer $i$ is $\\max _i$ , we have: $0 \\le \\mathbf {a}_i \\le {\\rm max}_i.$ Let $\\mathbf {z}_i=\\frac{\\mathbf {a}_i}{\\max _i}$ , then $0 \\le \\mathbf {z}_i \\le 1 \\, (i=1,2,\\dots ,L)$ .", "According to Eq.", "1, the activation of layer $l$ and $l-1$ satisfy: $\\mathbf {a}_l = \\max \\left( W^{\\text{ANN}}_{l-1} \\mathbf {a}_{l-1} +\\mathbf {b}^{\\text{ANN}}_{l-1} ,0 \\right)$ As the $\\mathbf {a}_l$ clipped by $\\max _l$ equals the original $\\mathbf {a}_l$ .", "Then, $\\mathbf {a}_l = {\\rm clip} \\left(W^{\\text{ANN}}_{l-1} \\mathbf {a}_{l-1} +\\mathbf {b}^{\\text{ANN}}_{l-1} ,0,{\\rm max}_l \\, \\right)$ By dividing $\\max _l$ on both sides of Eq.", "REF and substituting $\\mathbf {a}_i$ by $\\mathbf {z}_i\\max _i$ , we have: $\\mathbf {z}_l = {\\rm clip} \\left( \\frac{ W^{\\text{ANN}}_{l-1} \\mathbf {z}_{l-1}\\max _{l-1} +\\mathbf {b}^{\\text{ANN}}_{l-1} }{\\max _l} ,0,1 \\right)$ Comparing Eq.", "REF and Eq.", "REF , We can conclude that Eq.", "REF equals Eq.", "REF if for $l=1,2,...,L$ , the following equations hold: $\\frac{W^{\\text{SNN}}_{l-1}}{v_{th,l}} = W^{\\text{ANN}}_{l-1} \\frac{\\max _{l-1}}{\\max _{l}},~~\\frac{\\mathbf {b}^{\\text{SNN}}_{l-1}}{v_{th,l}} = \\frac{\\mathbf {b}^{\\text{ANN}}_{l-1} }{\\max _{l}}$ Theorem 2 For layer $l$ in an ANN and the converted SNN with constant coding, given the simulated firing rate $\\hat{\\mathbf {r}}_l$ and the real firing rate $\\mathbf {r}_l$ , we have: $K_l < \\frac{2\\Omega _l}{t},$ where $K_l$ denotes the abbreviation of $K(\\hat{\\mathbf {r}}_l,\\mathbf {r}_l(t))$ in layer $l$ , $\\Omega _l=\\frac{ \\Vert \\hat{\\mathbf {r}}_l \\Vert _1 }{\\Vert \\hat{\\mathbf {r}}_l \\Vert _2^2} $ .", "$\\Vert \\cdot \\Vert _p$ denotes $L^p$ norm.", "Consider that firing is a cumulative and rounded firing process.", "When $\\hat{\\mathbf {r}}_l$ is given, ${\\mathbf {r}}_l(t)$ is approximate as $\\frac{\\lfloor \\hat{\\mathbf {r}}_l t \\rfloor }{t}$ , also $\\hat{\\mathbf {r}}_{l,i}-\\frac{1}{t} < \\frac{\\lfloor \\hat{\\mathbf {r}}_{l,i} t \\rfloor }{t} \\le \\hat{\\mathbf {r}}_{l,i}$ , where $\\hat{\\mathbf {r}}_{l,i}$ denotes the $i$ -th element of the vector $\\hat{\\mathbf {r}}_{l}$ .", "For the $l$ -th layer, we have: $\\begin{split}\\bigg \\Vert \\mathbf {r}_l (t)- \\hat{\\mathbf {r}}_l \\bigg \\Vert _2^2 &=\\bigg \\Vert \\frac{\\lfloor \\hat{\\mathbf {r}}_l t \\rfloor }{t} - \\hat{\\mathbf {r}}_l \\bigg \\Vert _2^2 =\\sum _i \\left[ \\frac{\\lfloor \\hat{\\mathbf {r}}_{l,i} t \\rfloor }{t} - \\hat{\\mathbf {r}}_{l,i} \\right]^2 \\\\&= \\sum _i \\left[ \\left( \\frac{\\lfloor \\hat{\\mathbf {r}}_{l,i} t \\rfloor }{t} \\right)^2 - 2 \\hat{\\mathbf {r}}_{l,i} \\frac{\\lfloor \\hat{\\mathbf {r}}_{l,i} t \\rfloor }{t} + \\hat{\\mathbf {r}}^2_{l,i} \\right]\\\\&< \\sum _i \\left[ \\hat{\\mathbf {r}}_{l,i}^2 - 2 \\hat{\\mathbf {r}}_{l,i} \\left( \\hat{\\mathbf {r}}_{l,i}-\\frac{1}{t} \\right) + \\hat{\\mathbf {r}}^2_{l,i}\\right]\\\\& = \\sum _i \\left[ \\hat{\\mathbf {r}}^2_{l,i} - 2\\hat{\\mathbf {r}}^2_{l,i} +2\\hat{\\mathbf {r}}_{l,i} \\frac{1}{t} + \\hat{\\mathbf {r}}^2_{l,i} \\right]\\\\& = \\sum _i \\frac{2 \\hat{\\mathbf {r}}_{l,i}}{t} = \\frac{2 \\Vert \\hat{\\mathbf {r}}_l \\Vert _1}{t}\\end{split}$ The last equality holds as simulated firing rate $\\hat{\\mathbf {r}}_l \\ge 0$ .", "Thus the sum of all items in $\\hat{\\mathbf {r}}_l$ equals its $L^1$ norm.", "Now we conclude that: $\\begin{split}K_l &= \\frac{\\Vert \\mathbf {r}_l(t) - \\hat{\\mathbf {r}}_l \\Vert _2^2 }{\\Vert \\hat{\\mathbf {r}}_l \\Vert _2^2} < \\frac{2 \\Vert \\hat{\\mathbf {r}}_{l} \\Vert _1 }{\\Vert \\hat{\\mathbf {r}}_l \\Vert _2^2 t} = \\frac{2\\Omega _l}{t}\\end{split}$ Mini-batch training and converting a spiking neural network from a source ANN with Rate Norm Layers.", "`r_max' is short for `running_max'.", "Require:A spiking neural network $f_{SNN}$ with thresholds {$v_{th,k}$ |$k$ =$1,2,\\cdots ,L$ }; A network with $L$ Rate Norm Layers $f_{ANN}$ ; Dataset $D$ for training; Number of epochs for Stage 1 $epoch1$ ; Number of epochs for Stage 2 $epoch2$ Ensure: $p_k$ =$1.0$ , ${\\rm r\\_max}_k$ =$1.0$ $(k=1,2,\\dots ,L)$ ; momentum parameter $m=0.1$ , $\\lambda =0.5$ [1] {Stage 1: Accuracy Training for $f_{ANN}$ } e = 1 to $epoch1$ length of Dataset $D$ Sample minibatch $(\\mathbf {x},\\mathbf {y})$ from $D$ $\\hat{\\mathbf {r}}^*_0=\\mathbf {x}$ k = 1 to L ${\\rm r\\_max}_k$ = (1-$m$ ) ${\\rm r\\_max}_k$ +$m$ $\\max (\\hat{\\mathbf {r}}^*_{k-1})$ $\\theta _k$ = $p_k$ ${\\rm r\\_max}_k$ $\\hat{\\mathbf {r}}^*_{k}$ = clip[$(W_{k-1} \\hat{\\mathbf {r}}^*_k + \\mathbf {b}_{k-1})/\\theta _k$ ,0,1] Loss = ${L}(\\hat{\\mathbf {r}}^*_{L};y)$ Backward propagation through network Update $W_k$ and $\\mathbf {b}_k (k=1,2,\\dots ,L-1)$ {Stage 2: Fast Inference Training for $f_{ANN}$ } e = 1 to $epoch2$ length of Dataset $D$ Sample minibatch $(\\mathbf {x},\\mathbf {y})$ from $D$ $\\hat{\\mathbf {r}}^*_0=\\hat{\\mathbf {r}}^{\\prime }_0=\\mathbf {x}$ k = 1 to L ${\\rm r\\_max}_k$ = (1-$m$ ) ${\\rm r\\_max}_k$ +$m$ $\\max (\\hat{\\mathbf {r}}^*_{k-1})$ $\\theta _k$ = $p_k$ ${\\rm r\\_max}_k$ $\\hat{\\mathbf {r}}^*_{k}$ = clip[$(W_{k-1} \\hat{\\mathbf {r}}^*_k + \\mathbf {b}_{k-1})/\\theta _k$ ,0,1] k = 1 to L ${\\rm r\\_max}_k$ = (1-$m$ ) ${\\rm r\\_max}_k$ +$m$ $\\max (\\hat{\\mathbf {r}}^{\\prime }_{k-1}$ ) $\\theta _k$ = $p_k$ ${\\rm r\\_max}_k$ $\\hat{\\mathbf {r}}^{\\prime }_{k}$ = clip[$(W_{k-1} \\hat{\\mathbf {r}}^{\\prime }_k + \\mathbf {b}_{k-1})/\\theta _k$ ,0,1] Loss = ${T}(\\hat{\\mathbf {r}}_L^*,\\hat{\\mathbf {r}}_L^{\\prime })$ (Eq.REF ) Backward propagation through network Update $p_k (k=1,2,\\dots ,L)$ {Converting SNN $f_{SNN}$ from pre-trained $f_{ANN}$ } k = 1 to L $f_{SNN}.W_{k}=f_{ANN}.W_{k}$ $f_{SNN}.\\mathbf {b}_{k}=f_{ANN}.\\mathbf {b}_{k}$ $f_{SNN}.v_{th,k}=f_{ANN}.\\theta _{k}$ $f_{SNN}$ The training and converting algorithm is described in Algorithm REF .", "In Stage 2, since each trainable $p_i$ will scale the output, to reduce the instability of threshold training on the deep model, all layers share the same $p_i$ when training.", "Besides, to limit $p_i \\in [0,1]$ of Rate Norm Layer, use $sigmoid(p^{\\prime }_i)$ in place of $p_i$ .", "The experiments are conducted on the PyTorch platform.", "The GPU used in training is NVIDIA GeForce RTX 2080 Ti." ] ]
2105.11654
[ [ "The Higgs-Graviton Couplings: from Amplitudes to the Action" ], [ "Abstract In this paper we study the coupling of scalar (Higgs) particles ($\\phi$) with gravitons ($h$) and their possible effects.", "The general form of the 3-point interaction $\\phi(p) h(1)h(2)$ can be derived using the scaling behavior of the spinor variables under the little group; the resulting vertices exhibit such simplicity, that some simplifications should be hidden in the expressions obtained from the extended scalar action.", "To investigate this, we study an extended Einstein-Hilbert action that besides the minimal coupling, it also includes terms of the form $\\phi R^2$, $\\phi R^{\\mu\\nu} R_{\\mu\\nu}$ and $\\phi R^{\\mu\\nu\\rho\\sigma} R_{\\mu\\nu\\rho\\sigma}$, as well as the term $\\epsilon_{\\mu\\nu \\alpha\\beta} \\phi_5 R^{\\mu\\nu}_{\\rho\\sigma} R^{\\alpha\\beta\\rho\\sigma}$ for the case of a pseudo-scalar ($\\phi_5$).", "The resulting vertices satisfy KLT-type relations, i.e., they can be written as the square of the coupling of the Higgs with gluons.", "We find that the amplitude for the Higgs decay into a pair of gravitons (on-shell) only receives a contribution coming from the square of the Riemann tensor.", "Similar results are obtained for the 3-body decay $\\phi \\to h h^* (\\to XX)$, with an off-shell graviton ($h^*$) that goes into the final state $XX$.", "One could expect that these quadratic terms can produce new loop effects, however we find that the new contribution from this non-minimal coupling to the graviton self-energy, also vanishes for on-shell gravitons." ], [ "Introduction", "Understanding the properties of gravity at the quantum level has been the subject of intensive research for almost a century.", "Some progress has been gained in the infra-red (IR) domain, where one considers quantum gravity (QG) on the same footing as the other fundamental interactions, with gravity being mediated by a massless spin-2 quantum field, the graviton [1], [2], [3], [4].", "This understanding of weakly-coupled gravity was obtained using standard QFT methods developed during the 60's and early 70's, such as quantization, renormalization, and regularization of Yang-Mills theories [5].", "It was learned then that pure QG is finite at one-loop, but divergences start appearing at two-loop level in pure gravity, while the inclusion of matter interacting with gravity presents divergences already at one-loop, leading to the conclusion that QG is non-renormalizable [6].", "On the other hand, the progress made on effective field theories, which brought a deeper understanding of its philosophy and calculation machinery, allowed to perform loop-calculations even for non-renormalizable theories, such as chiral perturbation theory [8].", "This in turn paved the way to perform reliable calculations for perturbative quantum gravity (pQG) at the loop-level [9], [10], [11], [12].", "More recently, pQG has benefited from the progress made on modern helicity amplitude methods [13], [14], [15].", "Within the common approach to QFT, as it is presented in most text-book, the kinematics involves the energy-momenta 4-vectors, while the wave function for spin-1/2 and vector particles involves the Dirac spinors and polarization vectors.", "However, within the Helicity formalism, one works with a unified framework, where both kinematical variables and the wave functions for external particles are dealt using spinor variables, i.e.", "Weyl spinors.", "A variety of results that one could call \"perturbative jewels\" have been discovered over the years, which include, for example, the derivation of simple expressions for the maximal-helicity violating amplitudes for gluons in Yang-Mills theories (Parke-Taylor), and also for gravitons from pQG [16].", "One of the most precious result is that some amplitudes involving gravitons can be expressed as the square of the corresponding amplitude for gluons, a result that is expressed as: $GR=YM\\times YM$ , which summarizes the relationship between gravity and Yang-Mills amplitudes.", "This result was derived first at tree-level using results in string theory [17], and is known as the KLT relations.", "More recently, it was found that some relations hold at loop-level, which is known as the double-copy formula or Bern-Carrasco-Johansson (BCJ) relation [18], [19].", "Most progress has been made for massless theories, but more recently the massive case has also been tackled [20], [21].", "On the other hand, scalar particles play an important role in particle physics and in cosmology, with one case being experimentally confirmed in the HEP front: the SM Higgs boson detected in 2012 at the LHC; measuring its properties is a main target of current and future colliders.", "But plenty of other hypothetical scalars have been proposed in the literature, ranging from extra Higgs doublets, axion particles, dilaton, inflatons, to name a few.", "In this regard, understanding the coupling of scalar particles with gravity seems worth studying.", "This has been discussed before in many contexts, such as Einstein gravity extended with minimaly-coupled scalars [22].", "This was discussed in connection with the equivalence theorem and for the identification of divergences in quantum gravity [23].", "The quantum corrections to the Newton potential resulting between gravity interacting scalars have been discussed too [24].", "More recently, the possible corrections to Newton law coming from the squares of the Ricci scalar and Ricci tensor have been discussed [25], where it is found that all amplitudes with two heavy scalars and an arbitrary number of gravitons are not affected by those quadratic interactions.", "Possible scalar resonances appearing in graviton scattering has been studied too [26].", "In this paper we are interested in studying a related problem that also involves the coupling of a scalar with gravitons, including those quadratic terms in curvature.", "Namely, we want to study the decay of a scalar into gravitons as a possible effect of extended gravity; here, the scalar we are referring to could correspond to any Higgs boson that arises within the SM or some extension, it could even be the inflaton or some other super-heavy scalar particle.", "One of our goals is to identify possible differences in the contributions coming from the quadratic terms involving the Ricci scalar, as well as the Ricci and Riemann tensors.", "The coupling of the Higgs boson ($\\phi $ ) with the graviton ($h$ ) will be considered in full generality, for both scalar and pseudo-scalar cases.", "The organization of this paper is the following.", "First, in Section 2 we consider the scaling behavior of the spinor variables under the little group to derive the general form of the 3-point vertex $\\phi (p) h(1)h(2)$ , identifying the allowed helicity combinations of the graviton, namely: $h_i (1), h_i (2) = +2, +2$ or $-2, -2$ .", "Then, in Section 3 we look at the Higgs-graviton action, and derive the Higgs-graviton vertices by expanding the gravitational field around Minkowski space-time, i.e., $g_{\\mu \\nu }= \\eta _{\\mu \\nu }+ \\kappa _G h_{\\mu \\nu }$ .", "For this, we consider an effective Lagrangian that contains the Ricci scalar, Ricci tensor, and Riemann tensor, and expand them up to $O(h^2)$ in the graviton field.", "We discuss then the mass-dimension of the resulting coupling, the coupling of the scalar particle with the those terms; and also consider the interactions of a pseudoscalar with gravitons.", "We show that the resulting vertices satisfy the KLT relations, and can indeed be written as the square of the YM terms.", "Then in Section 4 we present our calculation of the Higgs decay width, which could be used to discriminate among the different terms of the effective action; however, we find that only the term quadratic in the Riemann tensor contributes to the amplitude for the decay $\\phi \\rightarrow hh$ ; this remains valid also for the 3-body decay $\\phi \\rightarrow h h^* (\\rightarrow XX)$ , with an off-shell graviton that goes into the final state $XX$ .", "These results suggest that may be in order to find such differences, we may have to consider loop effects, which are discussed in Section 5.", "In particular, we calculate the new scalar contribution to the graviton self-energy, i.e.", "the Feynman diagrams with one scalar and one graviton in the loop, but we find that this also vanishes when the external graviton is on-shell.", "Conclusions are presented in section 6." ], [ "Constructing the Higgs-Graviton vertices from Amplitudes ", "Although the use of Lagrangians, Feynman diagrams and Feynman rules, seemed irreplaceable in QFT, this view has changed with the surge of the constructible program [27] (also called the “bootstrap\" [28]), where one can derive the fundamental interaction from the general properties of amplitudes and spinors that appear in the S-matrix.", "Within this approach, the fundamental kinematical variables are the spinors, which can be considered as the building blocks for the amplitudes of massless particles [16].", "The starting point is to express the 4-momenta, $p_{\\mu }$ , as a $2\\times 2$ matrix: $ p_{\\alpha \\dot{\\alpha }} = \\sigma ^{\\mu }_ {\\alpha \\dot{\\alpha }} p_{\\mu }$ ($\\alpha =1,2$ and $\\dot{\\alpha }=\\dot{1},\\dot{2}$ ), using the using the Pauli matrices $\\sigma ^{\\mu }$ .", "For massless particles $det(p)=0$ , and then the momentum matrix can be factorized in terms of Weyl spinors, as: $ p_{\\alpha \\dot{\\alpha }} = \\chi _{\\alpha } \\tilde{\\chi }_{\\dot{\\alpha }}$ .", "It can be convenient to use Dirac notation, with $\\chi _{\\alpha } (p_i) \\rightarrow | i \\rangle $ and $\\tilde{\\chi }_{\\dot{\\alpha }} (p_i) \\rightarrow [ i | $ , both for the momentum and for the wave functions associated with external fermions.", "The inner products of spinors are defined as: $\\langle i j \\rangle = \\chi ^\\alpha \\chi _\\alpha $ and $ [i j ] = \\tilde{\\chi }_{\\dot{\\alpha }} \\tilde{\\chi }^{\\dot{\\alpha }}$ , such that: $\\langle j i \\rangle [i j ] = 2 p_i \\cdot p_j$ .", "One also need to have expressions for the polarization vectors in terms of spinors.", "These are: $\\epsilon ^+_{\\alpha \\dot{\\alpha }} = -\\sqrt{2} \\eta _{\\alpha } \\tilde{\\chi }_{\\dot{\\alpha }} / \\langle \\eta \\chi \\rangle $ , and $\\epsilon ^-_{\\alpha \\dot{\\alpha }} = -\\sqrt{2} \\chi _{\\alpha } \\tilde{\\eta }_{\\dot{\\alpha }} / [\\tilde{\\chi } \\tilde{\\eta }]$ , where $\\eta , \\tilde{\\eta }$ are two reference spinors that are unphysical and thus they should disappear at the end of the calculations.", "Thus, one has a unified framework, where both the kinematics and the dynamics are described with the same type of variables.", "Within the constructive program for QFT [28], one attempts to work out all the $n$ -point amplitudes in terms of the basic 3-point amplitudes.", "The master formulae for the coupling of three particles with helicities $(h_1,h_2,h_3)$ , has been derived by working out the scaling properties of spinors, which amounts to a consideration of their transformation properties under the Little group.", "Namely, under the little group the momentum does not change, and this is respected provided that the spinors transform as: $\\chi _i \\rightarrow t_i \\chi _i$ , and $\\tilde{\\chi }_i \\rightarrow t^{-1}_i \\tilde{\\chi }_i$ , while $\\epsilon ^+_{\\alpha \\dot{\\alpha }} \\rightarrow t^{-2} \\epsilon ^+_{\\alpha \\dot{\\alpha }}$ and $\\epsilon ^-_{\\alpha \\dot{\\alpha }} \\rightarrow t^2 \\epsilon ^-_{\\alpha \\dot{\\alpha }}$ .", "An amplitude, partly composed of external lines which depend upon the momenta of the particles, itself transforms under little group transformations by a change of scale; the amplitude follows the same transformation law as the spinors themselves and scales homogeneously for each of the particles in question.", "This scaling behavior produces the master formula for the 3-point amplitude, which is given by: $A ( 1^{h_1}, 2^{h_2},3^{h_3} ) = \\left\\lbrace \\begin{array}{ll}c_{123} \\, \\langle 12\\rangle ^{h_3-h_2-h_1} \\langle 13 \\rangle ^{h_2-h_1-h_3} \\langle 23 \\rangle ^{h_1-h_2-h_3},\\hspace{5.69054pt} & h_1+h_2+h_3 < 0 \\nonumber \\\\\\tilde{c}_{123} \\, [ 12 ]^{h_1+h_2-h_3} [13 ]^{h_1+h_3-h_2} [ 23 ]^{h_2+h_3-h_1},\\hspace{5.69054pt} & h_1+h_2+h_3 > 0\\end{array}\\right.$ where $c_{123}$ , $\\tilde{c}_{123}$ are constants to be determined, and $\\langle ij\\rangle $ and $[ij]$ denote the spinor products for particles of momenta $i,j$ .", "We can apply this formula to derive the coupling of a scalar $\\phi $ with a pair of gravitons $h$ ; for the scalar particle we have $h_1=0$ , while for the gravitons one has $h_{2,3} = \\pm 2$ .", "Then, for the combination of helicities $(h_1,h_2, h_3)=(0,-2,-2)$ , one gets: $A(1^{0}, 2^{-2},3^{-2}) = c \\langle 23\\rangle ^{4}.$ On the other hand, for the other allowed helicity combination $(h_1,h_2,h_3)= (0,+2,+2)$ , the result for the amplitude is: $A(1^{0}, 2^{+2},3^{+2}) = \\tilde{c} [23]^{4}.$ In deriving these results we have kept only the terms associated with local terms in the corresponding Lagrangian.", "The simplicity of these results is quite remarkable, especially when one compares it with the derivation of graviton interactions based on the traditional approach, i.e., from the Einstein-Hilbert action, where the graviton couplings (and wave function) involve a massive proliferation of space-time indices.", "Furthermore, one can also look at the coupling of the Higgs boson with a pair of gluons, which are given by: $A(1^{0}, 2^{-1},3^{-1}) = c_0 \\langle 23\\rangle ^{2}$ , and $A (1^{0}, 2^{+1},3^{+1}) = c_0 [23]^{2}$ .", "We can see that the Higgs-graviton couplings (modulo the coefficients $c,\\tilde{c},c_0$ ) can be expressed as the square of the corresponding Higgs-gluon couplings, i.e., $A (\\phi hh) = A( \\phi gg) * A( \\phi gg)$ .", "Thus, the Higgs-graviton couplings obey the KLT-type relations." ], [ "The Higgs-graviton couplings from the Action", "We are interested in the coupling of a Higgs particle ($\\phi $ ) (Higgs, dilaton, inflaton, etc) with gravity.", "At the lowest order, the minimal coupling of gravitons and scalars is described by the following Lagrangian: ${\\cal {L}}_{mc} = \\sqrt{-g} [ \\frac{1}{2} g^{\\mu \\nu } \\partial _\\mu \\phi \\partial _\\nu \\phi - \\frac{m^2}{2} \\phi ^2]$ This Lagrangian induces several scalar-graviton vertices, including the 3-point vertex: $\\phi \\phi h$ and 4-point vertex:$\\phi \\phi h h$ , which can contribute to the graviton self-energy.", "Thus, in order to have the 3-point vertex with one scalar and two gravitons $\\phi hh$ we have to consider a Lagrangian with non-minimal couplings.", "These couplings could be induced at one-loop if one includes a quartic term for the scalar particle, which produces a cubic scalar self-interaction after SSB, and then the triangle or bubble loop-diagrams can induce the interaction $\\phi hh$ of our interest.", "In this paper we shall consider an effective Lagrangian that includes the non-minimal scalar-graviton couplings.", "We shall denote by $\\phi $ the neutral component of an scalar multiplet $\\Phi $ that transforms under some gauge symmetry.", "The effective Lagrangian shall contains terms with a quadratic factor, $c_i |\\Phi |^2$ , which after SSB takes the form: $c_i |\\Phi |^2= c_i (V^2 + 2 V \\phi + \\phi ^2) + ... $ , where $V$ denotes the v.e.v.", "of the Higgs field, and only the linear term $ 2 V c_i \\phi = \\kappa _i \\phi $ will be considered.", "We shall consider terms involving the scalar curvature ($R$ ), its square ($R^2$ ), the square of the Ricci tensor ($R_{\\mu \\nu }R^{\\mu \\nu }$ ) and the square of the Riemann tensor $R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta }$ .", "Thus, the effective Lagrangian is given as follows: ${\\cal {L} }_{nm} = \\frac{\\sqrt{-g}}{\\kappa ^2} \\left( \\kappa _1\\phi R + \\kappa _2 \\phi R^2 + \\kappa _3 \\phi R_{\\mu \\nu }R^{\\mu \\nu } + \\kappa _4 \\phi R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta } +\\dots \\right),$ where $\\kappa ^2= 8\\pi G_N$ , and $G_N$ is Newton's gravitational constant.", "Alternatively, one could consider a singlet field ($\\chi $ ), and then include the linear factor $ c_i \\chi $ , with $c_i$ denoting a constant.", "For pure gravity in 4D, the Gauss-Bonnet term satisfies: $R^2 - 4R_{\\mu \\nu }R^{\\mu \\nu }+ R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta }= \\partial _\\mu X^\\mu $ , i.e., it is a total derivative, and thus it suffices to take into account only two of those terms in the effective lagrangian.", "However, when one couples gravity with a scalar, it is not possible to eliminate its effects in Equation (REF ), and in principle we can include all of these terms in the action[22].", "Then, following the usual approach to perturbative quantum gravity, one expands the metric ($g_{\\mu \\nu }$ ) upon a Minkowski background ($\\eta _{\\mu \\nu }$ ), with the flcutuations identified as the graviton ($h_{\\mu \\nu }$ ), and substitute this expansion in the curvature invariant terms making up the EH action (and beyond).", "Here, in order to clarify some issues such as the link between the classical action, the Feynman rules for the gravitons and the amplitudes, the corresponding expansion is carried up to second order in $\\kappa $ .", "Namely: $g_{\\mu \\nu } = \\eta _{\\mu \\nu } + \\kappa h_{\\mu \\nu } + \\frac{\\kappa ^2}{2} h_{\\mu \\alpha }h^{\\alpha }_\\nu +\\dots ,$ One can check the dimension of the couplings $\\kappa _i$ that appear in hereafter.", "Notice that the metric is dimensionless, while mass-dimension of the graviton field is $[h_{\\mu \\nu }]=1$ , while $[\\kappa ]=-1$ , then $[g_{\\mu \\nu }]=[\\kappa h_{\\mu \\nu }]=0$ .", "Being the derivative of the metric, the Chirstoffel symbols have mass dimension $[\\Gamma ]=1$ , while the Riemman tensor, the Ricci tensor and the Ricci scalar (being the second derivative of the metric), have mass dimensions $[R]=2$ .", "Thus, we have: $[\\kappa _1]=-1$ , while $[\\kappa _{2,3,4}]=-3$ .", "Then, we can redefine $\\kappa _i$ in order to introduce a dimensionless coupling, and play with scales (which we take as $V$ and $\\kappa > V$ ).", "We can play with two possibilities for $\\kappa _1$ , namely: $\\kappa _1= \\lambda _1/V$ or $\\kappa _1= \\lambda _1 \\kappa $ .", "For $\\kappa _{j}$ ($j=2,3,4$ ) we have more options, like $\\kappa _{j} = \\lambda _j \\kappa ^3$ or $\\kappa _{j} = \\lambda _j \\kappa ^2/V$ , with the last case being the coupling of the Higgs boson with gravitons, when this coupling arises at one-loop level.", "What about the case of a pseudoscalar?", "Now, the coupling of a pseudoscalar $\\phi _5$ with gravitons will arise from the term in the action: ${\\mathcal {L}}_5 = \\frac{1}{4\\kappa ^2} \\kappa _5 \\phi _5 \\sqrt{-g} \\epsilon _{\\mu \\nu \\rho \\sigma }R^{\\mu \\nu }_{\\hspace{8.53581pt}\\alpha \\beta }R^{\\rho \\sigma \\alpha \\beta }.$ We present next the main results of the expansion for each of the terms in the action foe the scalar and pseudo-scalar, as well as the resulting interaction vertices written in momentum space." ], [ " $\\phi hh$ interaction from {{formula:be18584a-066a-4f25-8d99-39ff8bce7b08}} (2nd order) ", "We start by considering the expansion of the Ricci scalar $R$ .", "The first term in the expansion is given by: $R^{(1)} = \\partial _{\\mu }\\partial _{\\nu }h^{\\mu \\nu } - \\partial ^2 h$ where $h= h^{\\mu }_{\\, \\, \\mu }$ .", "The superscript in the scalar curvature denotes the order in $\\kappa $ that is being considered.", "One can see that the first term in the expansion contains only one graviton, and thus it does not induce the vertex $\\phi hh$ .", "But when one considers the next order, we do get two gravitons; namely: $\\begin{split}(\\sqrt{-g}R)^{(2)} &= R^{(2)} + \\frac{1}{2}h R^{(1)} \\\\&= \\frac{1}{4}h^{\\mu \\nu }\\partial ^2h_{\\mu \\nu } + \\frac{1}{2}\\partial _{\\mu }h^{\\mu \\nu }\\partial _{\\beta }h_{\\nu }^{\\beta } +\\frac{1}{2}h\\partial _{\\mu }\\partial _{\\nu }h^{\\mu \\nu } -\\frac{1}{4}h\\partial ^2h \\\\&+ \\partial _{\\mu }\\Big (-\\frac{1}{4}h\\partial ^{\\mu }h - \\frac{1}{4}h_{\\alpha \\beta }\\partial ^{\\mu }h^{\\alpha \\beta } + h^{\\mu \\beta }\\partial _{\\beta }h - h^{\\mu \\beta }\\partial _{\\nu }h^{\\nu }_{\\beta }\\Big ),\\end{split}$ Then, we can find the expression for the corresponding vertex $\\phi (p) h(k_1) h(k_2)$ in momentum space.", "We find: $V^{\\lbrace \\phi hh\\rbrace }_{\\mu \\nu \\rho \\sigma }(k,k^{\\prime }) = \\kappa _1 \\left(-\\frac{1}{4}{k^{\\prime }}^{2}\\eta _{\\rho \\mu }\\eta _{\\sigma \\nu } + \\frac{1}{4}{k^{\\prime }}^2\\eta _{\\mu \\nu }\\eta _{\\rho \\sigma } + k_{\\nu }k^{\\prime }_{\\sigma }\\eta _{\\rho \\mu } - \\frac{1}{2}k^{\\prime }_{\\rho }k^{\\prime }_{\\sigma }\\eta _{\\mu \\nu }\\right).$" ], [ "$\\phi hh$ interaction from {{formula:adad9c50-db78-45a4-bedc-cdbbc60155c5}}", "The interaction of the Higgs with a pair of gravitons can arise from the term $\\sqrt{-g} R^2$ .", "In this case, from the very definition, one can see that the resulting vertex can be expressed as the product of two similar expressions, namely: $(\\sqrt{-g}R^2)^{(2)} = (R^2)^{(2)} = (\\partial _{\\mu }\\partial _{\\nu }h^{\\mu \\nu } - \\partial ^{2}h)(\\partial _{\\alpha }\\partial _{\\beta }h^{\\alpha \\beta } - \\partial ^{2}h).$ The corresponding vertex $\\phi (p) h(k_1) h(k_2)$ in momentum space, is given by: $V^{\\lbrace \\phi hh\\rbrace }_{\\mu \\nu \\rho \\sigma }(k,k^{\\prime }) = \\kappa _2 \\Big (2k_{\\mu }k_{\\nu }k^{\\prime }_{\\rho }k^{\\prime }_{\\sigma } -4k^{2}k^{\\prime }_{\\rho }k^{\\prime }_{\\sigma }\\eta _{\\mu \\nu } + 2k^{2}{k^{\\prime }}^{2}\\eta _{\\mu \\nu }\\eta _{\\rho \\sigma }\\Big ).$" ], [ " $\\phi hh$ interaction from {{formula:e42d61b1-c74e-4a72-be10-814bbaef1401}}", "Similarly, the expansion of the scalar invariant formed as the contraction of the Ricci tensor and its inverse, is given as: $\\begin{split}(\\sqrt{-g} R_{\\mu \\nu }R^{\\mu \\nu })^{(2)} &= \\frac{1}{4}\\partial _{\\mu }\\partial _{\\nu }h\\partial ^{\\mu }\\partial ^{\\nu }h + \\frac{1}{4}\\partial ^2h_{\\mu \\nu }\\partial ^2h^{\\mu \\nu } + \\frac{1}{2}\\partial _{\\mu }\\partial _{\\nu }h\\partial ^2h^{\\mu \\nu } \\\\&- \\partial _{\\mu }\\partial _{\\nu }h\\partial _{\\gamma }\\partial ^{\\nu }h^{\\gamma \\mu } - \\partial ^2h_{\\mu \\nu }\\partial _{\\gamma }\\partial ^{\\nu }h^{\\gamma \\mu } \\\\&+ \\frac{1}{2}\\partial _{\\mu }\\partial _{\\nu }h^{\\mu }_{\\beta }(\\partial _{\\alpha }\\partial ^{\\nu }h^{\\alpha \\beta } + \\partial _{\\alpha }\\partial ^{\\beta }h^{\\alpha \\nu }).\\end{split}$ As in the previous case, we can show that this expression can be written as the product of two terms, namely: $( \\sqrt{-g} R_{\\mu \\nu }R^{\\mu \\nu } )^{(2)}& = & \\Big [ \\frac{1}{2} ( h^{\\lambda }_{\\nu ,\\mu ,\\lambda } - h^{\\lambda }_{\\lambda ,\\nu ,\\mu } + h^{\\lambda }_{\\mu ,\\nu ,\\lambda } -h^{ \\, \\, \\lambda }_{\\lambda \\mu \\nu } ) \\Big ] \\\\& \\times & \\Big [ \\frac{1}{2} ( h^{\\lambda \\nu , \\mu }_{\\hspace{14.22636pt},\\lambda } - h^{\\lambda ,\\nu ,\\mu }_{\\lambda } +h^{\\lambda \\mu ,\\nu }_{\\hspace{14.22636pt},\\lambda } - h^{\\mu \\nu ,\\lambda }_{\\hspace{14.22636pt},\\lambda } ) \\Big ]$ The corresponding vertex $\\phi (p) h(k_1) h(k_2)$ , in momentum space, is given by: $V^{ \\lbrace \\phi hh \\rbrace }_{\\mu \\nu \\rho \\sigma }(k,k^{\\prime }) &=& \\kappa _{3}\\Big [ \\frac{1}{2} (k \\cdot k^{\\prime })^2 \\eta _{\\mu \\nu } \\eta _{\\rho \\sigma } + \\frac{1}{2}k^2{k^{\\prime }}^{2}\\eta _{\\rho \\nu }\\eta _{\\sigma \\mu } -2(k \\cdot k^{\\prime })k_{\\sigma }k^{\\prime }_{\\rho }\\eta _{\\mu \\nu } + (k \\cdot k^{\\prime })k_{\\nu }{k^{\\prime }}_{\\sigma }\\eta _{\\rho \\mu } \\\\& &+ k_{\\rho } k_{\\sigma }{k^{\\prime }}^{2} \\eta _{\\mu \\nu } - 2k_{\\nu }k_{\\sigma }{k^{\\prime }}^{2}\\eta _{\\rho \\mu } + k_{\\mu }k_{\\rho }{k^{\\prime }}_{\\nu }{k^{\\prime }}_{\\sigma }\\Big ]$" ], [ "$\\phi hh$ interaction from {{formula:ef66722e-562f-4c99-9644-2f8c87055029}}", "The expansion of the squared Riemann tensor, gives the following terms: $\\begin{split}(\\sqrt{-g} R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta })^{(2)} &= \\partial _{\\alpha }\\partial _{\\beta }h_{\\mu \\nu }\\partial ^{\\alpha }\\partial ^{\\beta }h^{\\mu \\nu } - 2\\partial _{\\mu }\\partial _{\\nu }h_{\\alpha \\beta }\\partial ^{\\mu }\\partial ^{\\beta }h^{\\alpha \\nu } \\\\&+ \\partial _{\\mu }\\partial _{\\nu }h_{\\alpha \\beta }\\partial ^{\\alpha }\\partial ^{\\beta }h^{\\mu \\nu }.\\end{split}$ Again, we can work this expression and show that it can be written as the product of two terms, namely: $\\begin{split}(\\sqrt{-g}R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta })^{(2)}&= \\Big [\\frac{1}{2}(h_{\\alpha \\nu ,\\beta ,\\mu } - h_{\\beta \\alpha ,\\nu ,\\mu } + h_{\\beta \\mu ,\\nu ,\\alpha } - h_{\\mu \\nu ,\\beta ,\\alpha })\\Big ] \\\\&\\times \\Big [\\frac{1}{2}(h^{\\alpha \\nu ,\\beta ,\\mu } - h^{\\beta \\alpha ,\\nu ,\\mu } + h^{\\beta \\mu ,\\nu ,\\alpha } - h^{\\mu \\nu ,\\beta ,\\alpha })\\Big ].\\end{split}$ The corresponding vertex $\\phi (p) h(k_1) h(k_2)$ in momentum space, is given as: $V^{\\lbrace \\phi hh\\rbrace }_{\\mu \\nu \\rho \\sigma }(k,k^{\\prime }) = \\kappa _4 \\Big (2k_{\\rho }k_{\\sigma }k^{\\prime }_{\\mu }k^{\\prime }_{\\nu } -4(k \\cdot k^{\\prime })k_{\\sigma }k^{\\prime }_{\\nu }\\eta _{\\rho \\mu } + 2(k \\cdot k^{\\prime })(k \\cdot k^{\\prime })\\eta _{\\rho \\mu }\\eta _{\\sigma \\nu }\\Big ).$" ], [ "$\\phi _5 hh$ interaction for a pseudoscalar", "The expansion of Equation (6), gives the following result: $\\begin{split}\\Big (\\frac{\\sqrt{-g}}{4} \\epsilon _{\\mu \\nu \\rho \\sigma }R^{\\mu \\nu }_{\\hspace{8.53581pt}\\alpha \\beta }R^{\\rho \\sigma \\alpha \\beta }\\Big )^{(2)} &= -\\frac{1}{2}\\epsilon _{\\mu \\rho \\sigma \\alpha }\\partial ^{\\alpha }\\partial _{\\beta }h_{\\nu }^{\\sigma }\\partial ^{\\rho }\\partial ^{\\nu }h^{\\beta \\mu } + \\frac{1}{2}\\epsilon _{\\mu \\rho \\sigma \\alpha }\\partial ^{\\alpha }\\partial _{\\nu }h_{\\beta }^{\\sigma }\\partial ^{\\rho }\\partial ^{\\nu }h^{\\beta \\mu } \\\\&+ \\frac{1}{4}\\epsilon _{\\nu \\rho \\sigma \\alpha }\\partial ^{\\alpha }\\partial ^{\\sigma }h_{\\beta \\mu }\\partial ^{\\rho }\\partial ^{\\nu }h^{\\beta \\mu }.\\end{split}$ This term leads to the following expression for the vertex $\\phi _5(p) h(k_1) h(k_2)$ : $V^{\\lbrace \\phi _5 hh\\rbrace }_{\\mu \\nu \\rho \\sigma }(k,k^{\\prime }) = \\kappa _5 \\Big (\\epsilon _{\\beta \\delta \\nu \\sigma }k^{\\beta }k_{\\rho }{k^{\\prime }}^{\\delta }{k^{\\prime }}_{\\mu } - \\epsilon _{\\beta \\gamma \\nu \\sigma }(k \\cdot k^{\\prime })k^{\\beta }{k^{\\prime }}^{\\gamma }\\eta _{\\rho \\mu }\\Big ).$" ], [ "The Higgs decay into gravitons ", "In this section we use the vertices $\\phi hh $ from the previous section to calculate the decay width for the mode $\\phi (p) \\rightarrow h (k) h (k^{\\prime })$ .", "Besides obtaining the amplitude for the decay from the extended action, we also want to verify that the results agree with the ones derived from the constructible approach (Sect.", "2).", "We work with the spinor helicity method, which permits to convert the tensorial amplitude into a function of spinor products.", "In order to build the corresponding amplitudes, one has to multiply the interaction vertices $\\phi (p) h(k_1) h(k_2)$ with the graviton polarization tensors, which can be expressed as the product of polarization vectors, i.e., $\\epsilon _{\\mu \\nu } (k_i) = \\epsilon _{\\mu } (k_i) \\epsilon _{ \\nu } (k_i)$ .", "Within the spinor helicity formalism, for a vector particle of momentum $k$ , the polarization vectors are given by: $ \\epsilon ^{\\mu }_{+}(k;q)= -\\frac{\\langle q |\\gamma ^{\\mu }|k]}{\\sqrt{2} \\langle qk \\rangle }$ , and $ \\epsilon ^{\\mu }_{-}(k;q)= -\\frac{\\langle k |\\gamma ^{\\mu }| q ]}{\\sqrt{2} [ qk ]}$ , where $q$ is a light-like reference momentum ($q^2=0$ ), it is arbitrary provided that $q \\ne k$ , and it can be chosen in such a way that the calculations are simplified.", "We also need to consider the on-shell conditions for the three particles, i.e.", "$k^2 = {k^{\\prime }}^2= 0$ and $p^2=m^2_{\\phi }$ , as well as the transversality conditions for the polarization vectors.", "The first important lesson that we obtain is that the amplitudes for the decay $\\phi \\rightarrow hh$ , only get contributions from the term containing the square of the Riemann tensor, while the amplitude constructed from the terms $\\phi R$ (2nd order), $\\phi R^2$ and $\\phi R^{\\mu \\nu } R_{\\mu \\nu }$ vanish for on-shell gravitons.", "On the other hand, we find that for the pseudo-scalar case, the term considered in eq.", "6 does contribute to the corresponding amplitude." ], [ "Helicity amplitudes for $\\phi \\rightarrow hh $ (Scalar)", "Thus, we have found that for on-shell gravitons, the only contributions to the amplitude for the decay of the scalar into gravitons comes from the squared Riemann tensor, i.e., $ \\frac{1}{\\kappa ^2}\\sqrt{-g}\\phi R_{\\mu \\nu \\alpha \\beta }R^{\\mu \\nu \\alpha \\beta }$ .", "Then, the total amplitude for graviton with polarizations $h_1 = h_2 = +2$ , is given by: $\\mathcal {A}_{3}(\\phi h^{+}h^{+}) = \\frac{\\kappa _4 }{2}[kk^{\\prime }]^4,$ and the total amplitude for the opposite helicity combination is given by: $\\mathcal {A}_{3} (\\phi h^{-}h^{-}) = \\frac{\\kappa _4}{2}\\langle kk^{\\prime } \\rangle ^4.$ Any other helicity combination is zero for all interaction terms; for example, $\\mathcal {A}_{3}(\\phi h^{+}h^{-}) = 0$ .", "These results are summarize in table 1.", "Table: Helicity Amplitudes for the 2-body Higgs decay φ→hh\\phi \\rightarrow hh.We notice that this result based on the expansion of the extended gravity action, agrees with the 3-point amplitude found in section 2, which was obtained based on the constructive approach.", "This is what was expected, but one can not stop wondering about the huge simplification that was obtained by bootstrapping the amplitudes." ], [ "Helicity amplitudes for $\\phi _5 \\rightarrow hh $ (Pseudoscalar)", "On the other hand, for the pseudo-scalar case we have studied the graviton expansion of the term $\\sqrt{-g} \\phi _5 \\epsilon _{\\mu \\nu \\rho \\sigma }R^{\\mu \\nu }_{\\hspace{8.53581pt}\\alpha \\beta }R^{\\rho \\sigma \\alpha \\beta }$ .", "After multiplying the resulting vertex with the polarization tensors, we find a non-vanishing amplitude, and for for the helicities $h_1 = h_2 = +2$ it is given by: $\\mathcal {A}_{3}^{P.S.", "}(\\phi _{5}h^{+}h^{+}) = \\kappa _5 \\frac{[kk^{\\prime }]^2}{4\\langle kk^{\\prime } \\rangle ^2}\\epsilon _{\\beta \\delta \\nu \\sigma }k^{\\beta }{k^{\\prime }}^{\\delta }\\langle {k^{\\prime }} |\\gamma ^{\\nu }|k]\\langle k|\\gamma ^{\\sigma }|{k^{\\prime }}].$ The complex conjugate of Equation (REF ) is: $\\mathcal {A}_{3}^{P.S.", "}(\\phi _{5}h^{+}h^{+})^{*} = \\kappa _5 \\frac{\\langle k^{\\prime }k \\rangle ^2}{4[{k^{\\prime }}k]^2}\\epsilon _{\\alpha \\lambda \\mu \\rho }k^{\\alpha }{k^{\\prime }}^{\\lambda }\\langle k|\\gamma ^{\\mu }|{k^{\\prime }}]\\langle {k^{\\prime }}|\\gamma ^{\\rho }|k].$ As a result one gets the following expression for the squared amplitude: $\\begin{split}|\\mathcal {A}_{3}^{P.S.", "}(\\phi _{5}h^{+}h^{+})|^2 &= \\mathcal {A}_{3}^{P.S.", "}(\\phi _5 h^{+}h^{+})\\mathcal {A}_{3}^{P.S.", "}(\\phi _5 h^{+}h^{+})^{*} \\\\&= \\frac{\\kappa ^2_5}{16}\\epsilon _{\\beta \\delta \\nu \\sigma }\\epsilon _{\\alpha \\lambda \\mu \\rho }k^{\\beta }{k^{\\prime }}^{\\delta }k^{\\alpha }{k^{\\prime }}^{\\lambda }\\langle {k^{\\prime }}|\\gamma ^{\\nu }|k]\\langle k|\\gamma ^{\\sigma }|{k^{\\prime }}]\\langle k|\\gamma ^{\\mu }|{k^{\\prime }}]\\langle {k^{\\prime }}|\\gamma ^{\\rho }|k].\\end{split}$ We simplify the previous expression by using the following identity for the product of the Levi-Civita symbols (obtained with the help of Mathematica): $\\begin{split}\\epsilon _{\\beta \\delta \\nu \\sigma }\\epsilon _{\\alpha \\lambda \\mu \\rho } &= -\\eta _{\\alpha \\sigma }\\eta _{\\rho \\beta }\\eta _{\\mu \\delta }\\eta _{\\lambda \\nu } + \\eta _{\\alpha \\nu }\\eta _{\\rho \\beta }\\eta _{\\mu \\delta }\\eta _{\\lambda \\sigma } + \\eta _{\\alpha \\sigma }\\eta _{\\rho \\beta }\\eta _{\\lambda \\delta }\\eta _{\\mu \\nu } - \\eta _{\\alpha \\delta }\\eta _{\\rho \\beta }\\eta _{\\lambda \\sigma }\\eta _{\\mu \\nu } - \\eta _{\\alpha \\nu }\\eta _{\\rho \\beta }\\eta _{\\lambda \\delta }\\eta _{\\mu \\sigma } \\\\&+\\eta _{\\alpha \\delta }\\eta _{\\rho \\beta }\\eta _{\\lambda \\nu }\\eta _{\\mu \\sigma } + \\eta _{\\alpha \\sigma }\\eta _{\\mu \\beta }\\eta _{\\rho \\delta }\\eta _{\\lambda \\nu } - \\eta _{\\alpha \\nu }\\eta _{\\mu \\beta }\\eta _{\\rho \\delta }\\eta _{\\lambda \\sigma } - \\eta _{\\alpha \\sigma }\\eta _{\\lambda \\beta }\\eta _{\\rho \\delta }\\eta _{\\mu \\nu } + \\eta _{\\alpha \\beta }\\eta _{\\rho \\delta }\\eta _{\\lambda \\sigma }\\eta _{\\mu \\nu } \\\\&+ \\eta _{\\alpha \\nu }\\eta _{\\lambda \\beta }\\eta _{\\rho \\delta }\\eta _{\\mu \\sigma } - \\eta _{\\alpha \\beta }\\eta _{\\rho \\delta }\\eta _{\\lambda \\nu }\\eta _{\\mu \\sigma } - \\eta _{\\alpha \\sigma }\\eta _{\\mu \\beta }\\eta _{\\lambda \\delta }\\eta _{\\rho \\nu } + \\eta _{\\alpha \\delta }\\eta _{\\mu \\beta }\\eta _{\\lambda \\sigma }\\eta _{\\rho \\nu } + \\eta _{\\alpha \\sigma }\\eta _{\\lambda \\beta }\\eta _{\\mu \\delta }\\eta _{\\rho \\nu } \\\\&-\\eta _{\\alpha \\beta }\\eta _{\\mu \\delta }\\eta _{\\lambda \\sigma }\\eta _{\\rho \\nu } - \\eta _{\\alpha \\delta }\\eta _{\\lambda \\beta }\\eta _{\\mu \\sigma }\\eta _{\\rho \\nu } + \\eta _{\\alpha \\beta }\\eta _{\\lambda \\delta }\\eta _{\\mu \\sigma }\\eta _{\\rho \\nu } + \\eta _{\\alpha \\nu }\\eta _{\\mu \\beta }\\eta _{\\lambda \\delta }\\eta _{\\rho \\sigma } - \\eta _{\\alpha \\delta }\\eta _{\\mu \\beta }\\eta _{\\lambda \\nu }\\eta _{\\rho \\sigma } \\\\&-\\eta _{\\alpha \\nu }\\eta _{\\lambda \\beta }\\eta _{\\mu \\delta }\\eta _{\\rho \\sigma } + \\eta _{\\alpha \\beta }\\eta _{\\mu \\delta }\\eta _{\\lambda \\nu }\\eta _{\\rho \\sigma } + \\eta _{\\alpha \\delta }\\eta _{\\lambda \\beta }\\eta _{\\mu \\nu }\\eta _{\\rho \\sigma } - \\eta _{\\alpha \\beta }\\eta _{\\lambda \\delta }\\eta _{\\mu \\nu }\\eta _{\\rho \\sigma },\\end{split}$ Then, after applying some spinor product identities, we finally get the result: $|\\mathcal {A}_{3}^{P.S.", "}(\\phi _{5}h^{+}h^{+})|^2 = \\frac{\\kappa ^2_5}{16}\\langle kk^{\\prime } \\rangle ^4 [kk^{\\prime }]^4.$ Thus, we found that the expression for the squared amplitude is actually simpler than for the amplitude itself.", "Then, one would be tempted to write the corresponding amplitude as follows: $\\mathcal {A}_{3}^{P.S.", "}(\\phi _{5}h^{+}h^{+}) = \\frac{\\kappa _5}{4} [kk^{\\prime }]^4,$ which would also be of the form derived in section 2." ], [ "The decay width for $\\phi \\rightarrow hh $ ", "In order to obtain the decay width for the two-body mode $\\phi \\rightarrow h \\, h$ , one has to square the amplitudes (20) and (21) and add them.", "Thus, the squared amplitude, summed over polarizations ($h_1=\\pm 2$ and $h_2=\\pm 2$ ), takes the form: $\\langle |\\mathcal {M} (\\phi \\rightarrow hh)|^2 \\rangle = \\sum _{h_i} \\kappa ^2_4 |A_3(\\phi h^{h_i} h^{h_i}) |^2 = \\frac{\\kappa ^2_4}{2} m^8_{\\phi }.$ The expression for the decay width of the Higgs (in the rest frame) is given as follows: $\\Gamma (\\phi \\rightarrow hh)=\\frac{S | \\vec{p} |}{8\\pi m_{\\phi }^2}\\langle |\\mathcal {M}(\\phi \\rightarrow hh)|^2\\rangle ,$ where $|\\vec{ p }|$ is the magnitude of the momentum of either outgoing graviton, and $S$ is a factor to account for identical particles in the final state; here, $S = 1/2$ .", "The final expression for the decay width is given by: $\\Gamma (\\phi \\rightarrow hh)=\\frac{\\kappa _{4}^2m_{\\phi }^7}{64\\pi }.$ The decay width for the SM Higgs depends on the value of $\\kappa _4$ , which arises starting at one loop-level [29]; for instance, the contribution from the top quark is of the order: $\\kappa _4 \\simeq G_{N}m^2_t / (v m^2_\\phi ) \\simeq m^2_t / ( \\kappa ^2 v m^2_{\\phi }) \\simeq 1/ (v \\kappa ^2)$ , where $m_t$ and $v$ denote the top quark mass and the electroweak vev, respectively.", "This expression is in agreement with our estimate based on dimensional analysis presented in Sect.", "2.", "This formula produces an extremely small decay width, of order $10^{-70} GeV$ , which is clearly not observable.", "However, for other scalars with heavier masses the decay width may be more sizable.", "Similarly, we can study the corresponding decay width for the pseudo-scalar case, the difference will appear when one considers the one-loop contribution of the top quark to the value of $\\kappa _5$ ." ], [ "The 3-body decay width for $\\phi \\rightarrow hh^* $", "Part of our interest in studying the Higgs decay into gravitons was the hope that it could help to distinguish the different contributions associated with each term in the effective action, but we found that only the square of the Riemann tensor contributes to the decay amplitude for on-shell gravitons.", "Thus it is valid to ask ourselves whether we could get some contribution from those terms in the effective action when we consider some process with one gravitons off-shell, whether a N-body decay width or a cross-section.", "Here we look into this question by considering the decay of the Higgs into one real graviton and one off-shell graviton, which in turn decays into other particles, such as a pair of scalars, fermions or gauge bosons.", "As this coupling of the graviton with matter through the energy-momentum tensor, we can use this in order to discuss the process as general as possible.", "The amplitude for the decay $\\phi (p) \\rightarrow h (p_1) + X(p_3) X(p_3)$ is given by: $M^{h_i} = \\epsilon ^{\\mu \\nu } (p_1) V_{\\mu \\nu \\rho \\sigma } (p,p_1, q) \\frac{iP^{\\rho \\sigma \\alpha \\beta }}{q^2} T_{\\alpha \\beta }(XX).$ where $T_{\\alpha \\beta }(XX)$ denotes the energy-momentum tensor for X matter.", "The tensor $P^{\\rho \\sigma \\alpha \\beta }$ appears in the graviton propagator, which is taken in de Donder or harmonic gauge.", "Now, we shall discuss first the contribution from the Ricci scalar (2nd order), as well as the squares of the Ricci Scalar, Ricci tensor and Riemann tensor.", "First, we consider the contribution from the Ricci scalar expanded up to second order in $h$ , we get that: $\\epsilon ^{\\mu \\nu }(k) _{+}V_{\\mu \\nu \\rho \\sigma } = -\\frac{\\kappa _{1}}{8\\langle q p_1 \\rangle ^2} p^{2}_1 \\langle q|\\gamma _{\\rho }|p_1]\\langle q|\\gamma _{\\sigma }|p_1],$ which vanishes when we take the on-shell condition of the first graviton.", "For the squared Ricci scalar, we also find that $\\epsilon ^{\\mu \\nu }_{+}(p_1)V_{\\mu \\nu \\rho \\sigma } = 0$ , so this term would also not contribute in any 3-body decay of the Higgs.", "Next, we consider the contribution coming from the squared Ricci tensor; for a graviton helicity $ h = +2$ , we find that: $\\epsilon ^{\\mu \\nu }_{+}(k)V_{\\mu \\nu \\rho \\sigma } = \\frac{\\kappa _3}{4 \\langle qp_1 \\rangle ^2}p^2_1 q^{2} \\langle q|\\gamma _{\\rho }|p_1]\\langle q|\\gamma _{\\sigma }|p_1],$ which vanishes upon taking the on-shell condition for the first graviton, i.e, $p^2_1 = 0$ .", "Finally, we consider the amplitude obtained from squared Riemann tensor.", "One needs to contract the polarization tensor with the corresponding vertex, we find gain that this is the only non vanishing term.", "Namely, the contraction of the vertex with the tensor $P^{\\rho \\sigma \\alpha \\beta }$ , gives the result: $\\epsilon ^{\\mu \\nu } V_{\\mu \\nu \\rho \\sigma } P^{\\rho \\sigma \\alpha \\beta } =(q\\cdot \\epsilon ^{i})^2 p^{\\rho }_1 p^{\\sigma }_1 + \\frac{1}{4} q\\cdot \\epsilon ^{i} q\\cdot p_1(p^{\\rho }_1 \\epsilon ^{i\\sigma } + p^{\\sigma }_1 \\epsilon ^{i\\rho }) + (q\\cdot p_1)^2 \\epsilon ^{i\\rho } \\epsilon ^{i\\sigma }$ Thus, the amplitude for the 3-body decay $\\phi (p) \\rightarrow h(p_1) + X(p_2) X(p_3)$ , only gets a contribution from the squared Riemann tensor.", "For a graviton polarizations $h_i=+2$ or $-2$ , which are described by the tensor $\\epsilon ^{h_i}_{\\alpha \\beta }= \\epsilon ^{i}_\\alpha \\epsilon ^{i}_\\beta $ , with $i=+1$ or $-1$ , the resulting amplitude is given as: $M^{h_i} = 2 \\frac{\\kappa \\kappa _4}{q^2} [ (q\\cdot \\epsilon ^{i})^2 p^{\\rho }_1 p^{\\sigma }_1 + \\frac{1}{4} q\\cdot \\epsilon ^{i} q\\cdot p_1(p^{\\rho }_1 \\epsilon ^{i\\sigma } + p^{\\sigma }_1 \\epsilon ^{i\\rho }) + (q\\cdot p_1)^2 \\epsilon ^{i\\rho } \\epsilon ^{i\\sigma }] T_{\\rho \\sigma }$ Then, as an example we consider the decay into a pair of massless scalars ($s$ ), which amounts to consider the corresponding Energy-momentum tensor, i.e.", "$T_{\\rho \\sigma } = p_{2\\rho }p_{3\\sigma } + p_{3\\rho }p_{2\\sigma } - \\frac{1}{2} p_2\\cdot p_3 \\eta _{\\rho \\sigma }$ .", "The squared amplitud, summed over polarizations, is given by: $\\sum _i |M^{h_i}|^2 = 2 (\\kappa \\kappa _4)^2 (p_1 \\cdot p_2)^2[ p_1 \\cdot p_2 + \\frac{1}{4} p_1 \\cdot q]^2$ We can then evaluate the decay width using the formulae for the 3-body phase-space, which can be expressed in terms of the energy fractions $x_2=2 p^0_2/m$ , $x_3=2 p^0_3/m$ , such that the differential decay width is given as follows: $\\frac{d^2 \\Gamma }{dx_2dx_3} = \\frac{m}{256 \\pi ^3} \\sum _i |M^{h_i}|^2$ we can then integrate this expression, and get numerical results for the decay width, but we refrain from doing so, since we only wanted to find which of the curvature terms in the action contribute to the 3-body amplitude.", "Nevertheless, it is interesting to show that even the 3-body amplitude simplifies this much with the use of helicity amplitude methods." ], [ "Loop effects - graviton self energy", "We have seen that only the squared Riemann tensor contributes to the tree-level amplitude for the decays $\\phi \\rightarrow h h, \\, h h^*$ , and we want to discuss here if this pattern also holds at loop levels.", "For this we shall discuss the scalar contributions to the graviton self-energy.", "From the early work of t'Hooft and Veltman [6], it is known that pure quantum gravity is finite at one-loop.", "This is due to the fact that the possible counter-term is proportional to the Gauss-Bonnet term, which is a total derivative in 4D and it can be eliminated from the action.", "The contribution of scalars to the graviton vacuum polarization has also been studied in detail [30], considering the Feynman graphs generated with the 3-point ($\\phi \\phi h$ ) and 4-point vertex ($\\phi \\phi h h$ ), from the minimal coupling action.", "These contributions contain a divergence that requires a counter-term involving $ R, R^2$ and $R_{\\mu \\nu }R^{\\mu \\nu }$ .", "This term is not contained in the original Lagrangian, thus reflecting the fact that gravity is not renormalizable.", "The scalar Lagrangian with minimal coupling does not induce a 3-point vertex $\\phi hh$ , which can be induced only when non-minimal couplings are included, i.e.", "for Higgs multiplets with SSB or scalar singlets.", "In principle such vertices could arise from the squares of the Ricci Scalar, Ricci tensor and Riemann tensor, as well as the second order expansion of the Ricci scalar.", "Thus, we can try to evaluate the contribution from each of these terms to the graviton self-energy, but rather than considering all possible diagrams, here we shall only consider the contribution form new type of diagrams, where both an scalar and one graviton circulate in the loop (see fig.", "1).", "Figure: Contribution of φhh\\phi hh insertion to the graviton self energyBecause the loop involves two vertices of the type $h-h^*-\\phi $ , $V_{\\mu \\nu \\alpha \\beta }(k,q)$ and $V_{\\gamma \\eta \\rho \\sigma } (q, k)$ , with the virtual graviton $h^*$ and scalar ($\\phi $ ) circulating the the loop, we have that the contributions to the loop coming from the Ricci scalar (to second order) and the squares of the Ricci Scalar, Ricci tensor will vanish.", "Thus the only possible non-zero contribution of this type will come from the squared Riemann tensor.", "The amplitude for this diagram is given by: $\\Pi _{\\mu \\nu \\rho \\sigma } (k_1) = \\int \\frac{d^4q}{(2\\pi )^4} V_{\\mu \\nu \\alpha \\beta } (k,q)\\frac{iP^{\\alpha \\beta \\gamma \\eta }}{q^2}\\frac{i}{(q+k_1)^2-m^2 } V_{\\gamma \\eta \\rho \\sigma } (q, k)$ where $m$ is the scalar mass.", "However, after contracting with polarization tensor for on-shell gravitons ($k^2=0$ ), we find that the corresponding amplitude for this diagram also vanishes.", "Thus, it appears that the physical content of this type of theory is an issue that may be deeper than we thought." ], [ "Conclusions", "In this paper we studied the coupling of scalar particles ($\\phi $ ) with a pair of gravitons ($h$ ).", "First, using the scaling behavior of the spinor variables, we derived the general form of the 3-point amplitude $\\phi (p) h(1)h(2)$ for the specific gravitons helicities $h_{i} (1,2) $ ; the resulting amplitudes for the allowed combinations $h_i (1,2) = +2, +2$ or $-2, -2$ turned out to be only of two types, proportional to the spinor products $[kk^{\\prime }]^4$ or $\\langle kk^{\\prime } \\rangle ^4$ , which are of such simplicity that it motivated us to look for the simplifications that are suspected to be hidden in the Higgs-graviton action.", "We then consider an extended action that contains the Ricci scalar, Ricci tensor, and Riemann tensor, and expanded these terms up to $O(h^2)$ in the graviton field, i.e.", "$g_{\\mu \\nu }= \\eta _{\\mu \\nu }+ \\kappa h_{\\mu \\nu }$ .", "We also considered the interactions of a pseudoscalar with gravitons.", "We found that the resulting vertices satisfy the KLT relations, and can indeed be written as the square of the YM terms.", "We then constructed the amplitude for decay $\\phi \\rightarrow hh$ , and we found that only the term quadratic in the Riemann tensor contributes to the decay amplitude; and this remains valid also when we considered the 3-body decay $\\phi \\rightarrow h h^* (\\rightarrow XX)$ , with an off-shell graviton that goes into the final state $XX$ .", "Then, in order to find some effect from the quadratic terms in curvature, we considered loop effects.", "In particular, we calculated the contribution to the vertex $h-\\phi -h$ to the graviton self-energy, i.e.", "with Feynman diagrams having one scalar and one graviton in the loop, and we found that this term also vanishes when the external graviton is on-shell.", "Although the initial motivation of our work, was to use some phenomenology to search for physical effects of the interaction of scalars with gravitons, as we made progress we found that there are extra aspects that one has to consider [31].", "It appears that the physical content of this type of theory that mixes scalars with higher-order gravity, is an issue that may be deeper than we thought.", "Acknowledgments.", "Two of us (A.A.-L. and J.L.D.-C.) would like to thank the support of CONACYT and SNI (Mexico)." ] ]
2105.11684
[ [ "Einstein-Yang-Mills Theory: Gauge Invariant Charges and Linearization\n Instability" ], [ "Abstract We construct the gauge-invariant electric and magnetic charges in Yang-Mills theory coupled to cosmological General Relativity (or any other geometric gravity), extending the flat spacetime construction of Abbott and Deser.", "For non-vanishing background gauge fields, the charges receive non-trivial contribution from the gravity part.", "In addition, we study the constraints on the first order perturbation theory and establish the conditions for linearization instability: that is the validity of the first order perturbation theory." ], [ "INTRODUCTION", "In 3+1 dimensions neither General Relativity [2] nor pure Yang-Mills theory [3], [4] has solitonic solutions.", "However, the coupled theory, the Einstein-Yang-Mills theory, with or without a cosmological constant has various solitons.", "See [5] for the first noted example in asymptotically flat spacetimes, and [6] for asymptotically anti-de Sitter spacetimes.", "See [7] for monopole type solutions in $R^{2}$ gravity.", "In this work we will work out the conserved charges of this coupled system and also find the constraints in the linearization instability of the first order perturbation theory.", "Conserved quantities in asymptotically flat spacetimes for pure gravity was famously given in [8], which was generalized to asymptotically (anti) de Sitter spacetimes in [9] and generalized to higher derivative gravity theories in [10].", "On the other hand, conserved gauge invariant charges in pure Yang-Mills theory was constructed by Abbott and Deser [1].", "Here in the first part of this work we follow the Abbott-Deser construction for a dynamical curved background with generically an asymptotically (A)dS behavior.", "The second problem we study is the question of the validity of the perturbation theory in the Einstein-Yang-Mills system.", "It is well known [11], [12], [13], [14], [15], [16], [17], [18], [20], [21], [19] that not all perturbative solutions come from the linearization of a possible exact solution.", "If that happens, one speaks of linearization instability and the perturbation thus fails.", "To have a linearization stable theory the first order perturbative solution must satisfy an integral constraint.", "We shall find this for the Einstein-Yang-Mills theory.", "Before we study the Einstein-Yang-Mills system in full detail, let us give our conventions [22] and recap the flat space construction [1].", "We will work in $D=3+1$ dimensions exclusively, but the discussion can be extended to other dimensions with the caveat that both pure Yang-Mills theory and pure General Relativity might have solitonic solutions for $D>3+1$ .", "We use the mostly plus signature $(-+++)$ and assume a compact Lie group ${\\mathsf {G}}$ with the Lie algebra ${\\cal {G}}$ given as $\\left[T_{a},T_{b}\\right]=iC_{abc}T^{c},$ with $C_{abc}$ real.", "In the adjoint representation we write $(T_{a}^{Ad})^{b}\\,_{c}:=-iC^{b}\\,_{ca}$ ; and defining $(D_{\\mu }\\psi )_{n}:=\\partial _{\\mu }\\psi _{n}-iA_{\\mu }^{a}(T_{a})_{n}^{k}\\psi _{k}$ in flat spacetime we have ${L}=-\\frac{1}{4}F_{\\mu \\nu }^{a}F_{a}^{\\mu \\nu }+\\text{${L}$}_{matter}(\\psi ,D_{\\mu }\\psi ),$ with the field equations $\\partial _{\\mu }F_{a}^{\\mu \\nu }=-J_{a}^{\\nu }.$ The current $J_{a}^{\\mu }=-F_{c}^{\\mu \\nu }C_{cab}A_{b\\nu }-i\\frac{\\partial \\text{${L}$}_{matter}}{\\partial (D_{\\mu }\\psi )_{n}}(T_{a})_{n}^{k}\\psi _{k},$ is partially conserved $\\partial _{\\mu }J_{a}^{\\mu }=0,$ and hence yields the conserved charges $Q_{a}:=\\int J_{a}^{0}d^{3}x.$ But these charges are gauge-covariant, not gauge-invariant.", "To get gauge–invariant charges, one can employ the AD technique [9] which is based on the following observation.", "Assuming the Yang-Mills coupling $g_{YM}=1$ , without loss of generality, we can define the matrix valued gauge field and the field strength $\\hat{A}_{\\mu }:=T^{a}A_{\\mu }^{a},~~~~~~~~~~~~~~\\hat{F}_{\\mu \\nu }:=T^{a}F_{\\mu \\nu }^{a}.$ Let the unitary matrix $\\hat{U}$ be in the same representation as $T^{a}$ , then the gauge transformed gauge field reads $\\hat{A}_{\\mu }^{\\hat{U}}=\\hat{A}_{\\mu }+\\hat{U}^{-1}D_{\\mu }\\hat{U},$ with the gauge-covariant derivative defined as $D_{\\mu }\\hat{U}:=\\nabla _{\\mu }\\hat{U}+[\\hat{A}_{\\mu },\\hat{U}],$ and the field strength transforms as usual $\\hat{F}_{\\mu \\nu }^{\\hat{U}}=\\hat{U}^{-1}\\hat{F}_{\\mu \\nu }\\hat{U}.$ Under an infinitesimal transformation $\\hat{U}\\cong 1+\\hat{\\xi }$ , one gets $\\delta \\hat{A}_{\\mu }=D_{\\mu }\\hat{\\xi },~~~~~~~~~~~~~~\\delta \\hat{F}_{\\mu \\nu }=[\\hat{F}_{\\mu \\nu },\\hat{\\xi }].$ So clearly, $D_{\\mu }\\hat{\\xi }=0$ defines the symmetries of a “background” field $\\hat{A}_{\\mu }$ which we shall denote as $\\bar{\\hat{A}}_{\\mu }$ from now on; clearly $[\\hat{F}_{\\mu \\nu },\\hat{\\xi }]=0$ .", "In the space of gauge fields, $\\hat{\\xi }$ acts like a Killing vector akin to the spacetime Killing vectors $\\delta g_{\\mu \\nu }=\\nabla _{\\mu }X_{\\nu }+\\nabla _{\\nu }X_{\\mu }=0$ .", "As there can be more than one solution to $D_{\\mu }\\hat{\\xi }=0$ , we shall put an index to denote the elements of the symmetry set and write as $\\hat{\\xi }^{s}$ , which is exactly the correct matrix that will turn $J_{a}^{\\mu }$ to be a gauge invariant current since $\\text{Tr}(\\hat{\\xi }\\hat{J}_{\\mu })$ is gauge-invariant for $\\hat{J}_{\\mu }=J_{\\mu }^{a}T^{a}$ .", "But this procedure requires a choice of background gauge-field and hence it must be done carefully.", "Instead of repeating the full details of the flat space construction, we now study the curved space version which also includes the flat space as a special case." ], [ "Construction of the conserved charges in the Einstein-Yang-Mills\nsystem", "The following construction works for any gravity theory based on Riemannian geometry with a Lagrangian of the generic form, but for the sake of concreteness, we shall take the gravity sector to be given as the Einstein-Hilbert Lagrangian.", "The coupled action reads $S=_{{\\mathcal {M}}}d^{4}x\\sqrt{-g}\\,\\Bigg (\\frac{R-2\\Lambda }{2\\kappa }-\\frac{1}{4}F_{\\mu \\nu }^{a}F_{a}^{\\mu \\nu }+\\text{${L}$}_{matter}\\Bigg ).$ As long as the Yang-Mills and the matter fields decay sufficiently fast at spatial infinity, the conserved energy, momentum, angular momentum as constructed, say in [1], [10] are intact, so we will not repeat these well-established discussions here, but work out the Yang-Mills part in some detail.", "Variation with respect to $\\hat{A}_{\\mu }$ yields $D_{\\mu }\\hat{F}^{\\mu \\nu }=\\nabla _{\\mu }\\hat{F}^{\\mu \\nu }+[\\hat{A}_{\\mu },\\hat{F}^{\\mu \\nu }]=\\hat{J}^{\\nu },$ where $\\hat{F}^{\\mu \\nu }:=\\nabla ^{\\mu }\\hat{A}^{\\nu }-\\nabla ^{\\nu }\\hat{A}^{\\mu }+[\\hat{A}^{\\mu },\\hat{A}^{\\nu }].$ The field strength satisfies the Bianchi identity $D_{\\alpha }\\hat{F}_{\\mu \\nu }+D_{\\mu }\\hat{F}_{\\nu \\alpha }+D_{\\nu }\\hat{F}_{\\alpha \\mu }=0,$ and with the normalization $\\text{Tr}(T^{a}T^{b})=\\frac{1}{2}\\delta ^{ab}$ , one has in components $F_{\\mu \\nu }^{a}=\\partial _{\\mu }A_{\\nu }^{a}-\\partial _{\\nu }A_{\\mu }^{a}+C^{a}\\,_{bc}A_{\\mu }^{b}A_{\\nu }^{c}.$ Using (REF ) we will construct the gauge-invariant electric field, while the magnetic charge will follow from the Bianchi identity.", "Assume now that for $\\hat{J}^{\\nu }=0$ , the background matrix $\\bar{\\hat{A}}_{\\mu }$ solves the source-free equation $\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }=0$ ; and we expand the field equations about this solution as Please see Appendix-A for an extended discussion of the expansion of the field equations up to and including second order in perturbation theory.", "$\\hat{A}_{\\mu }=\\bar{\\hat{A}}_{\\mu }+\\lambda \\hat{a}_{\\mu }+\\frac{\\lambda ^{2}}{2}\\hat{b}_{\\mu }+{\\mathcal {O}}(\\lambda ^{3}),$ where $\\lambda $ is a small parameter.", "As we are in a dynamical background spacetime, the metric also receives perturbations which we shall write as $g_{\\mu \\nu }=\\bar{g}_{\\mu \\nu }+\\tau h_{\\mu \\nu }+\\frac{\\tau ^{2}}{2}k_{\\mu \\nu }+{\\mathcal {O}}(\\tau ^{3}),$ with $\\tau $ being a different small parameter.", "Under these expansions, the full equation split as $D_{\\mu }\\hat{F}^{\\mu \\nu }=\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }+(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}+(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}+...=J^{\\nu },$ and by assumption the zeroth order term vanishes in the absence of a source $\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }=\\bar{\\nabla }_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }+[\\bar{\\hat{A}}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }]=0.$ At the linear order one finds $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}=\\bar{D}_{\\mu }\\left(\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu }+\\frac{1}{2}\\bar{\\hat{F}}^{\\mu \\nu }h)\\right)+\\lambda [\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }].$ Similarly, the second order expansion, $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}$ , reads $& & (D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}=\\bar{D}_{\\mu }\\left((\\hat{F}^{\\mu \\nu })^{(2)}+\\frac{\\tau }{2}(\\hat{F}^{\\mu \\nu })^{(1)}h+\\frac{\\tau ^{2}}{4}\\bar{\\hat{F}}^{\\mu \\nu }(k-h_{\\rho \\sigma }h^{\\rho \\sigma })\\right)\\nonumber \\\\& & ~~~~~~~~~~~~~~~~~~-\\frac{\\tau }{2}h\\bar{D}_{\\mu }(\\hat{F}^{\\mu \\nu })^{(1)}+\\lambda [\\hat{a}_{\\mu },(\\hat{F}^{\\mu \\nu })^{(1)}]+\\frac{\\lambda ^{2}}{2}[\\hat{b}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }].$ Moving all the higher order terms to the right-hand side, we can recast (REF ) as $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}=\\hat{{\\mathcal {J}}}^{\\nu },$ where the current $\\hat{{\\mathcal {J}}}^{\\nu }:=\\hat{J}^{\\nu }-(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}-...$ is composed of the matter current as well as all the terms beyond the linear one coming from the expansion.", "The crucial point is that this current is covariantly conserved with respect to the background connection explicitly, $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=0$ .To see the direct computation for the conservation of the current see Appendix-B.", "Finally, substituting (REF ) in (REF ), one finds $\\bar{D}_{\\mu }\\left(\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu }+\\frac{1}{2}\\bar{\\hat{F}}^{\\mu \\nu }h)\\right)+\\lambda [\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }]=\\hat{{\\mathcal {J}}}^{\\nu }.$ Covariantly conserved current does not immediately yield a conserved charge; to get a partially conserved current, we appeal to the symmetries of the background gauge field as discussed in the previous section.", "So we assume the existence of some (but at least one) background gauge covariant matrices $\\bar{\\hat{\\xi }}^{s}$ such that $\\bar{D}_{\\mu }\\bar{\\hat{\\xi }}^{s}=\\bar{\\nabla }_{\\mu }\\bar{\\hat{\\xi }}^{s}+[\\bar{\\hat{A}}_{\\mu },\\bar{\\hat{\\xi }}^{s}]=0,$ which yields $\\left[\\bar{D}_{\\nu },\\bar{D}_{\\mu }\\right]\\bar{\\hat{\\xi }}^{s}=0=[\\bar{\\hat{F}}_{\\nu \\mu },\\bar{\\hat{\\xi }}^{s}].$ Since $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=0$ and $\\bar{D}_{\\nu }\\bar{\\hat{\\xi }}^{s}=0$ , we can write $\\sqrt{-\\bar{g}}\\bar{D}_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\sqrt{-\\bar{g}}\\bar{\\nabla }_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\partial _{\\nu }\\left(\\sqrt{-\\bar{g}}\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })\\right)=0,$ which can be used to express the conserved electric chargesFor the details of the calculation see Appendix-C. for each background gauge symmetry as: $Q_{\\text{E}}^{s}:=\\frac{1}{4\\pi }_{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{0}),$ where we assumed that the four dimensional spacetime ${\\mathcal {M}}$ is diffeomorphic to $\\Sigma \\times \\mathbb {R}$ and $\\bar{\\gamma }$ denotes the induced metric on the spatial hypersurface.", "Using the explicit form of the current and employing the Stokes' theorem, one arrives at $Q_{\\text{E}}^{s}=\\frac{1}{4\\pi }_{\\partial \\Sigma }d^{2}x~\\sqrt{\\bar{\\beta }}\\sigma _{i}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right),$ where $\\bar{\\beta }$ is the two dimensional induced metric on the boundary of the hypersurface and $\\sigma _{i}$ is its unit one form.", "Observe that if the background gauge field is chosen to be pure gauge or zero, then the order $\\tau $ term in the charge expression vanishes and the gauge-invariant electric charges have the same form as their flat spacetime versions [1], while generically gravity contributes in a nontrivial way.", "Magnetic charge discussion follows similarly but now one employs the Bianchi identity which can be written with the help of the dual of the field strength as $D_{\\mu }\\,^{\\star }\\hat{F}^{\\mu \\nu }=0,$ where $^{\\star }\\hat{F}^{\\mu \\nu }:=\\frac{1}{2\\sqrt{-g}}\\epsilon ^{\\mu \\nu \\rho \\sigma }\\hat{F}_{\\rho \\sigma }$ .", "More explicitly the identity can be written as $\\frac{1}{2\\sqrt{-g}}\\epsilon ^{\\mu \\nu \\rho \\sigma }\\left(\\partial _{\\mu }\\hat{F}_{\\rho \\sigma }+[\\hat{A}_{\\mu },\\hat{F}_{\\rho \\sigma }]\\right)=0,$ which is the same as the expression in the flat spacetime case.", "So, expanding the gauge field about a background $\\bar{\\hat{A}}_{\\mu }$ , and the metric tensor about $\\bar{g}_{\\mu \\nu }$ one arrives at $^{\\bar{\\star }}\\hat{\\mathcal {J}}^{\\nu }=\\bar{D}_{\\mu }\\,^{\\star }(\\hat{F}^{\\mu \\nu })^{\\left(1\\right)}+\\lambda [\\hat{a}_{\\mu },{}^{\\bar{\\star }}\\bar{\\hat{F}}^{\\mu \\nu }],$ with the linear part of the dual field strength given as $^{\\star }(\\hat{F}^{\\mu \\nu })^{(1)}=\\frac{1}{2\\sqrt{-\\bar{g}}}\\bar{\\epsilon }^{\\mu \\nu \\rho \\sigma }\\lambda (\\bar{D}_{\\rho }\\hat{a}_{\\sigma }-\\bar{D}_{\\sigma }\\hat{a}_{\\rho })$ and the background dual field as $^{\\bar{\\star }}\\bar{\\hat{F}}^{\\mu \\nu }=\\frac{1}{2\\sqrt{-\\bar{g}}}\\bar{\\epsilon }^{\\mu \\nu \\rho \\sigma }(\\bar{D}_{\\rho }\\bar{\\hat{A}}_{\\sigma }-\\bar{D}_{\\sigma }\\bar{\\hat{A}}_{\\rho })$ .", "Then the conserved magnetic charges can be written as $Q_{M}^{s}=\\frac{1}{4\\pi }_{\\partial \\Sigma }d^{2}x\\,\\sigma _{i}\\,\\text{Tr}\\left(\\bar{\\xi }^{s}\\,^{\\bar{\\star }}(F^{i0})^{\\left(1\\right)}\\right),$ which has the same form as its flat space version [1].", "The magnetic charges are topological: as can be seen from comparing equations (29) and (33) the metric $\\sqrt{\\bar{\\beta }}$ does not explicitly appear in (33).", "Instead the Hodge dual appears which just is used to define the magnetic field.", "Hence we can equivalently express (REF ) as follows $Q_{M}=\\frac{1}{4\\pi }_{\\partial }d^{2}x\\,\\sigma _{i}\\,B^{(s)i},$ where $B^{(s)i}=Tr(\\bar{\\xi }^{(s)}\\,{}^{\\bar{*}}(F^{i0})^{(1)}).$" ], [ "Linearization Instability", "In nonlinear theories, there are some cases for which the first order perturbation theory is constrained at the second order.", "When this happens, one speaks of the theory having a linearization instability about the zeroth order (or the background solution).", "This topic is rather extensive: see [11], [12], [13], [14], [15], [17], [18], [19], [20]; and for a relevant review of the literature we would like to refer the reader to the recent PhD thesis [21], where the issue is elaborated in sufficient detail.", "Here let us study the linearization instability issue in the Gravity-Yang-Mills system.", "[Einstein gravity can be taken as a concrete example, but generic gravity theories can exhibit nontrivial linearization instability behavior as discussed in [19], [20].]", "We first assume a spacetime with noncompact hypersurfaces and at the end concentrate on the case of compact hypersurfaces without a boundary.", "Let us go back to the Yang-Mills equation with $J^{\\nu }=0$ , expand again up to second order in the gauge-field and the metric perturbation to get $(\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu })\\cdot (\\bar{\\hat{A}},\\bar{g})+\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(1)}\\cdot (\\hat{a},h)+\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(1)}\\cdot (\\hat{b},k)+\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)+...=0,$ where the center dot notation means, for example, $\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(1)}$ operator is evaluated at the first order expansion of the gauge field and the metric tensor $(\\hat{a},h)$ .", "By assumption, we have $(\\bar{D}_{\\mu }\\bar{F}^{\\mu \\nu })\\cdot (\\bar{\\hat{A}},\\bar{g})=0$ , which together with the gravity sector, determine the background solutions $(\\bar{A}_{\\mu },\\bar{g}_{\\mu \\nu })$ up to gauge degrees of freedom, of course.", "Similarly, by assumption, we have $\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(1)}\\cdot (\\hat{a},h)=0$ , which together with the linearized part of the gravity sector, determine the linearized solutions $(\\hat{a}_{\\mu },h_{\\mu \\nu })$ , again up to gauge transformations.", "So the second order terms are determined from the equation $\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(1)}\\cdot (\\hat{b},k)+\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)=0,$ which basically says that once $(\\hat{a}_{\\mu },h_{\\mu \\nu })$ are found from the linearized equations, $-\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)$ acts like a source term for the second order perturbations $(\\hat{b}_{\\mu },k_{\\mu \\nu })$ .", "If this happens then the first order perturbation theory is intact and improvable and moreover, linearized solutions obtained from the linearized equations can come from the linearization of some exact solutions.", "Please see the diagram in [19] that depicts this commumativity.", "So the necessary and sufficient condition for linearization stability is that (REF ) should not constrain the first order solutions $(\\hat{a}_{\\mu },h_{\\mu \\nu })$ and it should determine the second order solutions $(\\hat{b}_{\\mu },k_{\\mu \\nu })$ up to gauge transformations.", "But clearly this is very hard to check for all linear solutions of the theory, so in what follows let us find a weaker (necessary) condition.", "This condition will be in the form of an integral whose purely gravitational analog is called the Taub charge [23] and see the following recent discussion [24].", "From (REF ), we have $_{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\,\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}(D_{\\mu }\\hat{F}^{\\mu 0})^{(1)}\\cdot (\\hat{b},k)+\\bar{\\hat{\\xi }}^{s}(D_{\\mu }\\hat{F}^{\\mu 0})^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)\\right)=0.$ The first term in the integrand is of the same form as the first order term $\\left(D_{\\mu }F^{\\mu 0}\\right)^{(1)}\\cdot (\\hat{a},h)$ , albeit now evaluated at the second order fields instead of the first order ones.", "So, obviously this piece can be written as a boundary term as (REF ) with the substitution $(\\hat{a},h)\\rightarrow (\\hat{b},k)$ .", "The second term in the integrand requires more work, it is not clear at all if it can be written as a boundary integral.", "Nevertheless, to write some parts of $(D_{\\mu }\\hat{F}^{\\mu 0})^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)$ as a boundary term, we use the explicit form of the second order expansion (REF ): $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h) & = & \\bar{D}_{\\mu }\\left((\\hat{F}^{\\mu \\nu })^{(2)}+\\frac{\\tau }{2}(\\hat{F}^{\\mu \\nu })^{(1)}h+\\frac{\\tau ^{2}}{4}\\bar{\\hat{F}}^{\\mu \\nu }(k-h_{\\rho \\sigma }h^{\\rho \\sigma })\\right)\\nonumber \\\\& & -\\frac{\\tau }{2}h\\bar{D}_{\\mu }(\\hat{F}^{\\mu \\nu })^{(1)}+\\lambda [\\hat{a}_{\\mu },(\\hat{F}^{\\mu \\nu })^{(1)}]+\\frac{\\lambda ^{2}}{2}[\\hat{b}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }],$ where $& & (\\hat{F}^{\\mu \\nu })^{(2)}=\\frac{\\lambda ^{2}}{2}\\left(\\bar{D}^{\\mu }\\hat{b}^{\\nu }-\\bar{D}^{\\nu }\\hat{b}^{\\mu }+2\\left[\\hat{a}^{\\mu },\\hat{a}^{\\nu }\\right]\\right)+\\tau \\lambda \\left(h^{\\nu \\sigma }(\\bar{D}_{\\sigma }\\hat{a}^{\\mu }-\\bar{D}^{\\mu }\\hat{a}_{\\sigma })+h^{\\mu \\sigma }(\\bar{D}^{\\nu }\\hat{a}_{\\sigma }-\\bar{D}_{\\sigma }\\hat{a}^{\\nu })\\right)\\nonumber \\\\& & ~~~~~~~~~~~+\\frac{\\tau ^{2}}{2}\\left(\\bar{\\hat{F}}^{\\mu \\sigma }(2h^{\\nu \\lambda }h_{\\lambda \\sigma }-k_{\\sigma }^{\\nu })-\\bar{\\hat{F}}^{\\nu \\sigma }(2h^{\\mu \\lambda }h_{\\lambda \\sigma }-k_{\\sigma }^{\\mu })+2\\bar{\\hat{F}}_{\\sigma \\rho }h^{\\mu \\sigma }h^{\\nu \\rho }\\right).$ Inserting this expression into (REF ) obtains $\\left(D_{\\mu }F^{\\mu \\nu }\\right)^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)=\\bar{D}_{\\mu }\\hat{\\mathcal {X}}^{\\mu \\nu }-\\frac{\\tau }{2}h\\bar{D}_{\\mu }(\\hat{F}^{\\mu \\nu })^{(1)}+\\lambda [\\hat{a}_{\\mu },(\\hat{F}^{\\mu \\nu })^{(1)}],$ where we have introduced an antisymmetric field, $\\hat{\\mathcal {X}}^{\\mu \\nu }$ , to express the result in a more compact form.", "Direct calculation yields $\\hat{\\mathcal {X}}^{\\mu \\nu } & =\\frac{\\tau }{2}(\\hat{F}^{\\mu \\nu })^{(1)}h+\\lambda ^{2}\\left[\\hat{a}^{\\mu },\\hat{a}^{\\nu }\\right]-\\frac{\\tau ^{2}}{4}\\bar{\\hat{F}}^{\\mu \\nu }h_{\\rho \\sigma }h^{\\rho \\sigma }+\\tau ^{2}\\left(\\bar{\\hat{F}}^{\\mu \\sigma }h^{\\nu \\lambda }h_{\\lambda \\sigma }-\\bar{\\hat{F}}^{\\nu \\sigma }h^{\\mu \\lambda }h_{\\lambda \\sigma }+\\bar{\\hat{F}}_{\\sigma \\rho }h^{\\mu \\sigma }h^{\\nu \\rho }\\right)\\nonumber \\\\& \\ ~~\\ ~~\\ ~\\ ~~\\ ~+\\tau \\lambda \\left(h^{\\nu \\sigma }(\\bar{D}_{\\sigma }\\hat{a}^{\\mu }-\\bar{D}^{\\mu }\\hat{a}_{\\sigma })+h^{\\mu \\sigma }(\\bar{D}^{\\nu }\\hat{a}_{\\sigma }-\\bar{D}_{\\sigma }\\hat{a}^{\\nu })\\right).$ Then from (REF ), one finds $& & \\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}(D_{\\mu }\\hat{F}^{\\mu 0})^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)\\right)=\\bar{\\nabla }_{i}\\text{Tr}\\left(\\bar{\\xi }^{s}\\hat{\\mathcal {X}}^{i0}\\right)-\\frac{\\tau }{2}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}h\\bar{D}_{i}(\\hat{F}^{i0})^{(1)}\\right)\\nonumber \\\\& & ~~\\ ~~\\ ~\\ ~~\\ ~~\\ ~~\\ ~\\ ~~\\ ~~\\ ~~\\ ~\\ ~~\\ ~~\\ ~~\\ ~\\ ~~\\ +\\lambda \\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}[\\hat{a}_{i},(\\hat{F}^{i0})^{(1)}]\\right).$ Since $\\text{Tr}\\bigl (\\bar{\\hat{\\xi }}^{s}[\\hat{a}_{i},(\\hat{F}^{i0})^{(1)}]\\bigr )=\\text{Tr}\\bigl ([(\\hat{F}^{i0})^{(1)},\\bar{\\hat{\\xi }}^{s}]\\hat{a}_{i}\\bigr )$ , we have $& _{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}(D_{\\mu }\\hat{F}^{\\mu 0})^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)\\right)=_{\\Sigma }d^{3}x~\\partial _{i}\\left(\\sqrt{\\bar{\\gamma }}\\text{Tr}\\bigl (\\bar{\\hat{\\xi }}^{s}\\hat{\\mathcal {X}}^{i0}\\bigr )\\right)\\nonumber \\\\& \\ ~~\\ ~~\\ ~~\\ ~\\ ~+\\lambda _{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left([(\\hat{F}^{i0})^{(1)},\\bar{\\hat{\\xi }}^{s}]\\hat{a}_{i}\\right)-\\frac{\\tau }{2}_{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}h\\bar{D}_{i}(\\hat{F}^{i0})^{(1)}\\right).$ If we use the first order equation $\\bar{D}_{i}(\\hat{F}^{i0})^{(1)}=\\frac{\\tau }{2}\\bar{D}_{i}\\bigl (h\\bar{\\hat{F}}^{i0}\\bigr )+\\lambda [\\hat{a}_{i},\\bar{\\hat{F}}^{i0}],$ (REF ) reduces to $& _{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}(D_{\\mu }\\hat{F}^{\\mu 0})^{(2)}\\cdot (\\hat{a}^{2},h^{2},\\hat{a}h)\\right)=_{\\Sigma }d^{3}x~\\partial _{i}\\left(\\sqrt{\\bar{\\gamma }}\\text{Tr}\\bigl (\\bar{\\hat{\\xi }}^{s}\\hat{\\mathcal {X}}^{i0}\\bigr )\\right)\\nonumber \\\\& \\ ~~\\ ~~\\ ~~\\ ~\\ ~+\\lambda _{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left([(\\hat{F}^{i0})^{(1)},\\bar{\\hat{\\xi }}^{s}]a_{i}\\right)-\\frac{\\tau ^{2}}{8}_{\\Sigma }d^{3}x~\\partial _{i}\\left(\\sqrt{\\hat{\\gamma }}\\text{Tr}\\bigl (\\bar{\\hat{\\xi }}^{s}h^{2}\\bar{\\hat{F}}^{i0}\\bigr )\\right).$ So from (REF ) we arrive at $\\lambda _{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left([(\\hat{F}^{i0})^{(1)},\\bar{\\hat{\\xi }}^{s}]a_{i}\\right)=\\int _{\\partial \\Sigma }d^{2}x~\\sqrt{\\bar{\\beta }}{\\mathcal {I}},$ where we know ${\\mathcal {I}}$ from (REF ) and (REF ) explicitly so we need not depict it again.", "Consider now the case for which all the fields decay sufficiently fast, such that the boundary term on the right-hand side vanishes, or the case when the hyperspace is compact without a boundary ($\\partial \\Sigma =0$ ), then we get an integral constraint in the bulk for the linearized solutions: $_{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left([(\\hat{F}^{i0})^{(1)},\\bar{\\hat{\\xi }}^{s}]\\hat{a}_{i}\\right)=0,$ which reads explicitly as $_{\\Sigma }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\left([\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i},\\bar{\\hat{\\xi }}^{s}]\\hat{a}_{i}\\right)=0.$ This is not satisfied for generic solutions.", "Hence in a spacetime for closed hypersurfaces, the theory is generically linearization unstable." ], [ "Conclusions", "We have constructed the gauge-invariant conserved electric and magnetic charges in Yang-Mills theory in a dynamical curved background generalizing the flat spacetime construction of Abbott-Deser [1].", "Electric charges arise from the field equations, while the magnetic charges arise from the Bianchi identity.", "The crucial ingredient is the symmetry of the background gauge field that solves the curved space Yang-Mills equation.", "For the gravity part one can take any geometric theory of gravity based on the Riemannian geometry, but to be concrete we chose the cosmological General Relativity.", "To be able to define the electric and magnetic charges, besides the mentioned symmetry of the background gauge field, as defined by $\\delta _{\\xi }A_{\\mu }=\\bar{D}_{\\mu }\\xi =0$ , one also needs a time-like Killing vector for the spacetime which we assumed.", "Our results in curved spacetime reduces to the flat spacetime expressions in the correct limit.", "We have also studied the linearization instability issue in the Gravity-Yang-Mills theory and established a second order integral constraint that must be satisfied by any solution of linearized Yang-Mills theory in a spacetime with closed (compact without boundary) spatial hypersurfaces.", "We have not discussed the linearization instability in the gravity sector as it was recently done in [19] and described in great detail in the thesis [21]." ], [ "Acknowledgments", "The works of E.A.", "and E.K.", "are partially supported by the TUBITAK Grant No.", "120F253.", "The work of E.K.", "is partially supported by the TUBITAK Grant No.", "119F241." ], [ "Appendix a: First and second order Expansions of the field equations", "Here, we consider the expansion of the Yang-Mills fields and equations about the background quantities, the background metric and background gauge field, up to the cubic terms.", "While the first order terms will be used to construct the conserved charges, the quadratic terms will give us the integral constraint on solutions of the linearized equations.", "Let us start with the gauge field and assume that it can be expanded about the background field $\\bar{\\hat{A}}_{\\mu }$ up to the third order terms as $\\hat{A}_{\\mu }=\\bar{\\hat{A}}_{\\mu }+\\lambda \\hat{a}_{\\mu }+\\frac{\\lambda ^{2}}{2}\\hat{b}_{\\mu }.$ Here $\\lambda $ denotes the expansion parameter, $\\hat{a}_{\\mu }$ and $\\hat{b}_{\\mu }$ are the first and the second order expansions respectively.", "The background gauge field $\\bar{\\hat{A}}_{\\mu }$ satisfies the background field equations without a source $\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }=\\bar{\\nabla }_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }+[\\bar{\\hat{A}}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }]=0.$ We express the expansion of a generic tensor field $T$ about its background value $\\bar{T}$ as $T=\\bar{T}+(T)^{(1)}+(T)^{(2)}+...,$ where $(T)^{(1)}$ denotes the linearized $T$ tensor and $(T)^{(2)}$ denotes the second order expansion of it.", "For example, explicitly the field strength is expanded as $\\hat{F}_{\\mu \\nu }=\\bar{\\hat{F}}_{\\mu \\nu }+(\\hat{F}_{\\mu \\nu })^{(1)}+(\\hat{F}_{\\mu \\nu })^{(2)}+...$ up to the third order.", "Assuming a Riemann connection, we have $\\hat{F}_{\\mu \\nu }=\\partial _{\\mu }\\hat{A}_{\\nu }-\\partial _{\\nu }\\hat{A}_{\\mu }+[\\hat{A}_{\\mu },\\hat{A}_{\\nu }].$ The decomposition of the field strength at first order reads $(\\hat{F}_{\\mu \\nu })^{(1)}=\\lambda (\\bar{D}_{\\mu }\\hat{a}_{\\nu }-\\bar{D}_{\\nu }\\hat{a}_{\\mu }),$ and at second order one arrives at $(\\hat{F}_{\\mu \\nu })^{(2)}=\\frac{\\lambda ^{2}}{2}\\left(\\bar{D}_{\\mu }\\hat{b}_{\\nu }-\\bar{D}_{\\nu }\\hat{b}_{\\mu }+2[\\hat{a}_{\\mu },\\hat{a}_{\\nu }]\\right).$ Now we can compute the expansion of $\\hat{F}^{\\mu \\nu }$ .", "For this purpose, we use perturbation of the spacetime metric about a background metric $\\bar{g}_{\\mu \\nu }$ $g_{\\mu \\nu }=\\bar{g}_{\\mu \\nu }+\\tau h_{\\mu \\nu }+\\frac{\\tau ^{2}}{2}k_{\\mu \\nu },$ and its inverse $g^{\\mu \\nu }=\\bar{g}^{\\mu \\nu }-\\tau h^{\\mu \\nu }+\\frac{\\tau ^{2}}{2}\\left(2h^{\\mu \\sigma }h_{\\sigma }^{\\nu }-k^{\\mu \\nu }\\right).$ The field strength with upper indices, $\\hat{F}^{\\mu \\nu }=g^{\\mu \\sigma }g^{\\nu \\rho }\\hat{F}_{\\sigma \\rho }$ , at the first order yields $(\\hat{F}^{\\mu \\nu })^{(1)}=\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu }),$ which at second order reads $& & (\\hat{F}^{\\mu \\nu })^{(2)}=\\frac{\\lambda ^{2}}{2}\\left(\\bar{D}^{\\mu }\\hat{b}^{\\nu }-\\bar{D}^{\\nu }\\hat{b}^{\\mu }+2\\left[\\hat{a}^{\\mu },\\hat{a}^{\\nu }\\right]\\right)+\\tau \\lambda \\left(h^{\\nu \\sigma }(\\bar{D}_{\\sigma }\\hat{a}^{\\mu }-\\bar{D}^{\\mu }\\hat{a}_{\\sigma })+h^{\\mu \\sigma }(\\bar{D}^{\\nu }\\hat{a}_{\\sigma }-\\bar{D}_{\\sigma }\\hat{a}^{\\nu })\\right)\\nonumber \\\\& & ~~~~~~~~~~~+\\frac{\\tau ^{2}}{2}\\left(\\bar{\\hat{F}}^{\\mu \\sigma }(2h^{\\nu \\lambda }h_{\\lambda \\sigma }-k_{\\sigma }^{\\nu })-\\bar{\\hat{F}}^{\\nu \\sigma }(2h^{\\mu \\lambda }h_{\\lambda \\sigma }-k_{\\sigma }^{\\mu })+2\\bar{\\hat{F}}_{\\sigma \\rho }h^{\\mu \\sigma }h^{\\nu \\rho }\\right).$ Now we can expand $D_{\\mu }\\hat{F}^{\\mu \\nu }$ .", "Using $D_{\\mu }\\hat{F}^{\\mu \\nu }=\\partial _{\\mu }\\hat{F}^{\\mu \\nu }+\\Gamma _{\\mu \\sigma }^{\\mu }\\hat{F}^{\\sigma \\nu }+[\\hat{A}_{\\mu },\\hat{F}^{\\mu \\nu }]$ together with the previous expressions one obtains $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}=\\bar{D}_{\\mu }\\left(\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu })+\\frac{\\tau }{2}\\bar{\\hat{F}}^{\\mu \\nu }h\\right)+\\lambda [\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }],$ where $h=\\bar{g}^{\\mu \\nu }h_{\\mu \\nu }$ .", "Similarly, the second order expansion gives us the following $& & (D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}=\\bar{D}_{\\mu }\\left((\\hat{F}^{\\mu \\nu })^{(2)}+\\frac{\\tau }{2}(\\hat{F}^{\\mu \\nu })^{(1)}h+\\frac{\\tau ^{2}}{4}\\bar{\\hat{F}}^{\\mu \\nu }(k-h_{\\rho \\sigma }h^{\\rho \\sigma })\\right)\\\\& & ~~~~~~~~~~~~~~~~~~-\\frac{\\tau }{2}h\\bar{D}_{\\mu }(\\hat{F}^{\\mu \\nu })^{(1)}+\\lambda [\\hat{a}_{\\mu },(\\hat{F}^{\\mu \\nu })^{(1)}]+\\frac{\\lambda ^{2}}{2}[\\hat{b}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }],\\nonumber $ with $k=\\bar{g}^{\\mu \\nu }k_{\\mu \\nu }$ .", "In order to construct the conserved charges of the theory, we will not use the explicit form of the second order expansion.", "But this result will become important in linearization instability discussion.", "The field equations are expanded as $D_{\\mu }\\hat{F}^{\\mu \\nu }=\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }+(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}+(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}+...=\\hat{J}^{\\nu }$ where $\\bar{D}_{\\mu }\\bar{\\hat{F}}^{\\mu \\nu }=0$ by assumption.", "We put all the higher order terms to the right hand side of the equation and define a new current $\\hat{{\\mathcal {J}}}^{\\nu }:=\\hat{J}^{\\nu }-(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(2)}-...\\,.$ Then we express the linearized field equations as $(D_{\\mu }\\hat{F}^{\\mu \\nu })^{(1)}=\\hat{{\\mathcal {J}}}^{\\nu }.$ Substituting (REF ) in the last equation, one finds $\\bar{D}_{\\mu }\\left(\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu })+\\frac{\\tau }{2}\\bar{\\hat{F}}^{\\mu \\nu }h\\right)+\\lambda [\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }]=\\hat{{\\mathcal {J}}}^{\\nu }.$ Using the last equation one can prove the conservation of the new current $\\hat{{\\mathcal {J}}}^{\\nu }$ ." ], [ "APPENDIX B: CONSERVATION OF THE NEW CURRENT", "For the consistency of the construction, the new current have to be conserved.", "To prove the conservation let us consider an antisymmetric rank two tensor, say $X^{\\mu \\nu }$ .", "We first calculate the commutator $\\left[\\bar{D}_{\\nu },\\bar{D}_{\\mu }\\right]X^{\\mu \\nu }$ to make the construction easier.", "Explicitly we write $[\\bar{D}_{\\nu },\\bar{D}_{\\mu }]X^{\\mu \\nu }=\\bar{D}_{\\nu }\\bar{D}_{\\mu }X^{\\mu \\nu }-\\bar{D}_{\\mu }\\bar{D}_{\\nu }X^{\\mu \\nu },$ which yields $[\\bar{D}_{\\nu },\\bar{D}_{\\mu }]X^{\\mu \\nu }=[\\bar{\\nabla }_{\\nu },\\bar{\\nabla }_{\\mu }]X^{\\mu \\nu }+[\\bar{\\nabla }_{\\nu }\\bar{\\hat{A}}_{\\mu }-\\bar{\\nabla }_{\\mu }\\bar{\\hat{A}}_{\\nu },X^{\\mu \\nu }]+[\\bar{\\hat{A}}_{\\nu },[\\bar{\\hat{A}}_{\\mu },X^{\\mu \\nu }]]-[\\bar{\\hat{A}}_{\\mu },[\\bar{\\hat{A}}_{\\nu },X^{\\mu \\nu }]],$ where $[\\bar{\\nabla }_{\\nu },\\bar{\\nabla }_{\\mu }]X^{\\mu \\nu }=0$ .", "Using the Jacobi identity, the last two terms in the last equation yields $[\\bar{\\hat{A}}_{\\nu },[\\bar{\\hat{A}}_{\\mu },X^{\\mu \\nu }]]-[\\bar{\\hat{A}}_{\\mu },[\\bar{\\hat{A}}_{\\nu },X^{\\mu \\nu }]]=-[X^{\\mu \\nu },[\\bar{\\hat{A}}_{\\nu },\\bar{\\hat{A}}_{\\mu }]].$ Then equation (REF ) reduces to $[\\bar{D}_{\\nu },\\bar{D}_{\\mu }]X^{\\mu \\nu }=[\\bar{\\nabla }_{\\nu }\\bar{\\hat{A}}_{\\mu }-\\bar{\\nabla }_{\\mu }\\bar{\\hat{A}}_{\\nu },X^{\\mu \\nu }]-[X^{\\mu \\nu },[\\bar{\\hat{A}}_{\\nu },\\bar{\\hat{A}}_{\\mu }]].$ Since $\\bar{\\nabla }_{\\nu }\\bar{\\hat{A}}_{\\mu }-\\bar{\\nabla }_{\\mu }\\bar{\\hat{A}}_{\\nu }=\\bar{\\hat{F}}_{\\nu \\mu }-[\\bar{\\hat{A}}_{\\nu },\\bar{\\hat{A}}_{\\mu }]$ , one can re-express the commutator as $[\\bar{D}_{\\nu },\\bar{D}_{\\mu }]X^{\\mu \\nu }=[\\bar{\\hat{F}}_{\\nu \\mu },X^{\\mu \\nu }].$ Due to antisymmetry of the tensor field $X^{\\mu \\nu }$ , the last expression also yields the following identity $\\bar{D}_{\\nu }\\bar{D}_{\\mu }X^{\\mu \\nu }=\\frac{1}{2}[\\bar{\\hat{F}}_{\\nu \\mu },X^{\\mu \\nu }].$ For the special case $X^{\\mu \\nu }=\\hat{F}^{\\mu \\nu }$ , one has $\\bar{D}_{\\nu }\\bar{D}_{\\mu }\\hat{F}^{\\mu \\nu }=\\frac{1}{2}[\\bar{\\hat{F}}_{\\nu \\mu },\\hat{F}^{\\mu \\nu }]=0.$ Note that $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }$ includes these type of terms and the above identities will be useful when we prove the conservation of the new current.", "From equation (REF ), we write $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=\\bar{D}_{\\nu }\\bar{D}_{\\mu }\\left(\\lambda (\\bar{D}^{\\mu }\\hat{a}^{\\nu }-\\bar{D}^{\\nu }\\hat{a}^{\\mu })+\\tau (\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu })+\\frac{\\tau }{2}\\bar{\\hat{F}}^{\\mu \\nu }h\\right)+\\lambda \\bar{D}_{\\nu }[\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }].$ Using the identity (REF ) it can be rewritten as $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=\\frac{\\lambda }{2}[\\bar{D}^{\\mu }\\hat{a^{\\nu }}-\\bar{D}^{\\nu }\\hat{a}^{\\mu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\frac{\\tau }{2}[\\bar{\\hat{F}}^{\\sigma \\mu }h_{\\sigma }^{\\nu }-\\bar{\\hat{F}}^{\\sigma \\nu }h_{\\sigma }^{\\mu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\frac{\\tau }{4}h[\\bar{\\hat{F}}^{\\mu \\nu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\lambda [\\bar{D}_{\\nu }\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }],$ and then it becomes $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=\\lambda [\\bar{D}^{\\mu }\\hat{a}^{\\nu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\tau h_{\\sigma }^{\\nu }[\\bar{\\hat{F}}^{\\sigma \\mu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\frac{\\tau }{4}h[\\bar{\\hat{F}}^{\\mu \\nu },\\bar{\\hat{F}}_{\\mu \\nu }]+\\lambda [\\bar{D}_{\\nu }\\hat{a}_{\\mu },\\bar{\\hat{F}}^{\\mu \\nu }].$ The first and the last term on the right vanish from the antisymmetry of the indices.", "Also we have proved the vanishing of the third term in equation (REF ).", "There remains the second term only $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=\\tau h_{\\sigma }^{\\nu }[\\bar{\\hat{F}}^{\\sigma \\mu },\\bar{\\hat{F}}_{\\mu \\nu }].$ Renaming the indices $\\nu $ and $\\sigma $ , vanishing of this term is obvious.", "So, one ends up with $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=0$ , which is the expected result." ], [ "APPENDIX C: DEFINITION OF THE CONSERVED CHARGES", "Using the expressions $\\bar{D}_{\\nu }\\hat{{\\mathcal {J}}}^{\\nu }=0$ and $\\bar{D}_{\\nu }\\bar{\\hat{\\xi }}^{s}=0$ , we can write $\\bar{D}_{\\nu }(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=0=\\bar{\\nabla }_{\\nu }(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })+[\\bar{\\hat{A}}_{\\nu },\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu }].$ But we need a quantity which is conserved in the ordinary sense instead of the covariant conservation.", "Following the flat spacetime case we write $\\bar{D}_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\bar{\\nabla }_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })+[\\bar{\\hat{A}}_{\\nu },\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })]=0,$ where $[\\bar{\\hat{A}}_{\\nu },\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })]=0$ and so we have $\\bar{D}_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\bar{\\nabla }_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=0.$ Multiplying with $\\sqrt{-\\bar{g}}$ and using $\\sqrt{-\\bar{g}}\\bar{\\nabla }_{\\nu }X^{\\nu }=\\partial _{\\nu }\\left(\\sqrt{-\\bar{g}}X^{\\nu }\\right)$ we express $\\sqrt{-\\bar{g}}\\bar{D}_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\sqrt{-\\bar{g}}\\bar{\\nabla }_{\\nu }\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })=\\partial _{\\nu }\\left(\\sqrt{-\\bar{g}}\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{\\nu })\\right),$ from which we can define the total charges as $Q^{s}:=\\frac{1}{4\\pi }d^{4}x~\\partial _{0}\\left(\\sqrt{-\\bar{g}}\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{0})\\right).$ Using the Stokes theorem this can be written as $Q^{s}:=\\frac{1}{4\\pi }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}(\\bar{\\hat{\\xi }}^{s}\\hat{{\\mathcal {J}}}^{0}).$ Note that $\\bar{\\gamma }$ denotes the induced metric on the hypersurface.", "Using the explicit form of the linearized field equations $\\hat{{\\mathcal {J}}}^{0}$ reads $\\hat{{\\mathcal {J}}}^{0}=\\bar{D}_{i}\\left(\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\right)+\\lambda [\\hat{a}_{i},\\bar{\\hat{F}}^{i0}].$ We have $\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}[\\hat{a}_{i},\\bar{\\hat{F}}^{i0}]\\right)=\\text{Tr}\\left([\\bar{\\hat{\\xi }}^{s},\\bar{\\hat{F}}^{i0}]\\hat{a}_{i}\\right)=0,$ where the first equality comes form the cyclic property of trace and the second one is obtained from $[\\bar{D}_{\\mu },\\bar{D}_{\\nu }]\\bar{\\hat{\\xi }}^{s}=0$ .", "Then inserting (REF ) in equation (REF ), the conserved charges can be written as $Q^{s}:=\\frac{1}{4\\pi }d^{3}x~\\sqrt{\\bar{\\gamma }}\\text{Tr}\\bar{D}_{i}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right).$ To be able to use the Stokes theorem again, we need to convert the background gauge covariant derivative to the tensorial covariant derivative.", "The gauge covariant derivative and trace commute with each other.", "So we can express $& & 4\\pi Q^{s}:=\\frac{}{}d^{3}x~\\sqrt{\\bar{\\gamma }}\\bar{\\nabla }_{i}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right)\\nonumber \\\\& & +d^{3}x~\\sqrt{\\bar{\\gamma }}\\left[\\bar{\\hat{A}}_{i},\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right)\\right],$ where the terms in the second line of the last equation vanish automatically.", "Then we arrive at $Q^{s}:=\\frac{1}{4\\pi }d^{3}x~\\partial _{i}\\left\\lbrace \\sqrt{\\bar{\\gamma }}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right)\\right\\rbrace .$ After applying the Stokes theorem one more time the last equation yields the following expression for the conserved charges $Q^{s}:=\\frac{1}{4\\pi }d^{2}x~\\sqrt{\\bar{\\beta }}\\bar{\\sigma }_{i}\\text{Tr}\\left(\\bar{\\hat{\\xi }}^{s}\\Bigl (\\lambda (\\bar{D}^{i}\\hat{a}^{0}-\\bar{D}^{0}\\hat{a}^{i})+\\tau (\\bar{\\hat{F}}^{0i}h_{0}^{0}+\\bar{\\hat{F}}^{ki}h_{k}^{0}+\\bar{\\hat{F}}^{0k}h_{k}^{i}+\\frac{h}{2}\\bar{\\hat{F}}^{i0})\\Bigr )\\right).$ Here $\\bar{\\beta }$ denotes the two dimensional induced metric on the boundary of the hypersurface and $\\bar{\\sigma }_{i}$ is its unit one form." ] ]
2105.11744
[ [ "Dynamic Dual Sampling Module for Fine-Grained Semantic Segmentation" ], [ "Abstract Representation of semantic context and local details is the essential issue for building modern semantic segmentation models.", "However, the interrelationship between semantic context and local details is not well explored in previous works.", "In this paper, we propose a Dynamic Dual Sampling Module (DDSM) to conduct dynamic affinity modeling and propagate semantic context to local details, which yields a more discriminative representation.", "Specifically, a dynamic sampling strategy is used to sparsely sample representative pixels and channels in the higher layer, forming adaptive compact support for each pixel and channel in the lower layer.", "The sampled features with high semantics are aggregated according to the affinities and then propagated to detailed lower-layer features, leading to a fine-grained segmentation result with well-preserved boundaries.", "Experiment results on both Cityscapes and Camvid datasets validate the effectiveness and efficiency of the proposed approach.", "Code and models will be available at \\url{x3https://github.com/Fantasticarl/DDSM}." ], [ "Introduction", "Semantic segmentation, which entails assigning a label to each pixel of an image, is useful in a growing number of applications, including augmented reality, surveillance, and autonomous driving.", "With the development of deep FCN networks [1], [2], [3], the related works mainly focus on two aspects: global context modeling [2], [4] and local details modeling [5].", "The former models the long-range dependencies among pixels on the higher level of the network by overcoming the limited receptive field of the convolution network.", "The latter imports extra components such as lower-level features [6] or includes edge supervision [7] for finer and detailed results.", "The feature pyramids encode different scaled features where the higher layers contain coarse semantics while the lower layers represent fine details [8], [9].", "However, the interrelationship between semantic context and local details is not well explored.", "In this paper, we focus on exploring the interrelationship between two different layers.", "Since the semantic gaps [10], our solution enhances lower-layer features based on its affinity with the higher layer instead of directly adding features in FPN [8], successfully propagating the semantic context to local details via dynamic sampling of representative pixels and channels in higher layers.", "Figure: Visualizations of the prediction and error map.", "Our model outputs finer boundaries and small objects.", "(a) Input, (b) Ground truth, (c) Output of UPerNet , (d) Output of ours, (e) Error map of UPerNet , (f) Error map of Ours.It is noted that the existing affinity modeling methods, including self-attention based [12] or graph-based [13] models, require expensive pixel-wised computation across the whole image.", "For instance, FPT [14] uses a transformer to model the adjacent features' affinity, leading to immense resource cost.", "There is a rather high redundancy since the natural image meets the piece-wise smoothness constraint that the pixels within the same segment share certain visual characteristics.", "Accordingly, inspired by dynamic graph modeling [15], [16] and deformable convolutions [17], [18], we propose a dynamic affinity modeling method to avoid redundancy and achieve efficient feature propagation.", "Rather than using full pixels, we sample representative pixels to form adaptive compact support.", "Furthermore, we propose a dynamic sampler based on DCNv2 [18] instead of sampling fixed neighborhood pixels to fit the orientation distribution of image structures.", "Moreover, the channel encodes corresponding class-specific information, and several works [3] have shown the advantages of considering spatial and channel simultaneously to enhance class-aware information in feature representation.", "Hence, we also apply the dynamic sampler in feature channels and conduct channel-wise dynamic affinity modeling.", "After obtaining the sampled pixels/channels, we calculate both spatial and channel-wise affinities for each pixel/channel in the lower-layer.", "Finally, relevant semantics is aggregated according to the affinities and propagated to detailed lower-layer features.", "In summary, we propose a Dynamic Dual Sampling Module(DDSM), which contains spatial-wise and channel-wise dynamic affinity modeling.", "The representation of the pixels /channels in the lower layer is enhanced by some dynamically sampled pixels/channels from the higher layer.", "We validate DDSM's effectiveness on two typical networks, UPerNet [11] and Deeplabv3+ [6].", "Notably, our method could provide fine-grained segmentation results with well-preserved boundaries.", "Fig.", "REF shows the error map refined by DDSM based on UPerNet [11].", "After inserting DDSM into UPerNet [11], it achieves competitive results on Cityscapes [19] and Camvid [20].", "In particular, DDSM outperforms DAnet [3] with only 30% computation during the inference." ], [ "Method", "In this section, we first describe our dynamic sampler, which is inspired by DCNv2 [18].", "Then, we give a detailed introduction of our proposed Dynamic dual sampling module, which dynamically samples pixels and channels simultaneously.", "Finally, we deploy our proposed module into two frameworks." ], [ "Dynamic Sampler", "Given the input features $\\mathbf {x}$ with dimension $C \\times H \\times W$ , the sampler dynamically samples $N = k \\times k$ pixels from $\\mathbf {x}$ for each position $p$ , as Fig.", "REF shows.", "Specifically, a regular grid $\\mathcal {R}_{k \\times k}=\\lbrace p_n|n=1,2,...,N\\rbrace $ is defined to get an initial sampling area of $p$ .", "Then, we use a $1 \\times 1$ convolution layer instead of $3 \\times 3$ in DCN [17] to learn an offset for each position in the grid $\\mathcal {R}_{k \\times k}$ , and then an offset map with dimension of $2N \\times H \\times W$ is obtained, in which $N$ 2D offsets $\\Delta p_{n} = (q_x, q_y), n=1,2, \\cdots , N$ are learned.", "With the offset map, we use bilinear interpolation to compute the sampled features $\\mathbf {x}(p+p_{n}+\\Delta p_{n})$ of each sampled position $p+p_{n}+\\Delta p_{n}$ .", "To learn the offset map more flexibly and further boost the performance, a learnable scalar $\\Delta m_n$ is added following the work of DCNv2 [18].", "Given the dynamic sampler $\\mathbf {F}$ , the sampled features can be formulated as Eq.REF : $\\mathbf {F}(\\mathbf {x}(p))=\\lbrace \\mathbf {x}(p+p_n+\\Delta p_{n})\\Delta m_n|n=1,2,..., N \\rbrace .$ The output ($C \\times H \\times W \\times N$ ) gives the features at $N$ positions sampled from $\\mathbf {x}$ for each position $p$ in the $H \\times W$ feature map." ], [ "Dynamic Dual Sampling Module(DDSM)", "The Dynamic Dual Sampling Module consists of two parts, namely spatial-wise dynamic affinity modeling and channel-wise dynamic affinity modeling.", "The final output is the summation of the output features from both parts.", "Spatial-wise Dynamic Affinity Modeling: This part dynamically assigns features of $N$ pixels sampled from the higher layer for each pixel in the lower layer.", "As shown in Fig.", "REF (a), for the low-level features $\\mathbf {x}_l$ and the high-level features $\\mathbf {x}_h$ , we first upsample $\\mathbf {x}_h$ to $H \\times W$ , same as $\\mathbf {x}_l$ .", "Note that position information is crucial in feature fusion.", "A common method of introducing position information is to summarize features and positional encodings as input [21].", "We add learnable positional embeddings [21] $\\mathbf {e}_{pl}$ , $\\mathbf {e}_{ph}$ to the features $\\mathbf {x}_l$ , $\\mathbf {\\hat{x}}_h$ to disambiguate different spatial positions.", "Then we concatenate both $\\mathbf {x}_h$ and $\\mathbf {x}_l$ into features $ \\mathbf {x}_{cat}=(\\mathbf {x}_l+\\mathbf {e}_{pl})||(\\mathbf {\\hat{x}}_h+\\mathbf {e}_{ph})$ .", "Then, we use three $1 \\times 1$ convolution layers to do dimension reduction on $\\mathbf {x}_{cat}$ , $\\mathbf {x}_l+\\mathbf {e}_{pl}$ , and $\\mathbf {\\hat{x}}_h+\\mathbf {e}_{ph}$ , forming a new feature set as $\\mathbf {x}_l^{\\theta } = W_{\\theta }( \\mathbf {x}_l+\\mathbf {e}_{pl})$ , $\\mathbf {x}_{cat}^{\\phi } = W_{\\phi }\\mathbf {x}_{cat}$ , $\\mathbf {x}_h^{g} = W_{g} (\\mathbf {\\hat{x}}_h+\\mathbf {e}_{ph})$ .", "Similar to the definition in the work [22], $\\mathbf {x}_l^{\\theta }$ , $\\mathbf {x}_{cat}^{\\phi }$ and $\\mathbf {x}_h^{g}$ correspond to Query, Key, and Value function.", "The work [22] uses the entire feature map to calculate the affinity map.", "Nevertheless, we use the dynamic sampler $\\mathbf {F}$ to sample $N$ pixels in the Key for each position in the Query to obtain the affinity map.", "We sample $N$ pixels from $ \\mathbf {x}_{cat}^{\\phi }$ to form sampled features for each position $p$ in $\\mathbf {x}_l^{\\theta }$ .", "Matrix multiplication ${red}{X}^{1 \\times C} \\times {blue}{X}^{C \\times N}$ is performed between the features of each position in $\\mathbf {x}_l^{\\theta }$ and the transposed sampled features to form the affinity map with softmax normalization.", "Then, matrix multiplication ${red}{X}^{1 \\times N} \\times {blue}{X}^{N \\times C}$ is performed between the affinity map and sampled features from $\\mathbf {x}_h^{g}$ .", "The above two processes are executed $N$ times to obtain the aggregation result of $N$ sampled features, which will be assigned to the low-level features through a summation operation.", "The spatial-wise dynamic affinity modeling is formulated as Eq.REF , $\\mathbf {x}_{\\mathbf {S}out}(p)=\\sum \\nolimits _{n=1}^{N} \\delta [\\mathbf {x}_l^{\\theta }(p) \\mathbf {F}(\\mathbf {x}_{cat}^{\\phi }(n))^{\\top } ] \\mathbf {F}(\\mathbf {x}_h^{g}(n)),$ where $\\mathbf {x}_{\\mathbf {S}out}(p)$ is the augmented feature, $p$ is a position in $\\mathbf {x}_l$ and $\\delta $ is Softmax.", "Fig.", "REF (a) gives the detailed pipline.", "Channel-wise Dynamic Affinity Modeling: The channel-wise dynamic affinity modeling is built to explore interdependencies along channels since the channel encodes class-specific information.", "Different from previous works [3], [23], our module dynamically samples the channels of high-level features and aggregates them according to affinity to enhance low-level features.", "The input $\\mathbf {x}_l$ and $\\mathbf {x}_h$ are both average pooled to $a \\times a$ to reduce the spatial resolution.", "$\\mathbf {x}_l$ and $\\mathbf {x}_h$ is concatenated and then reshaped to $c \\times a^2$ to perform channel-wise dynamic sampling.", "As Fig.", "REF (b) shows, in order to reuse the dynamic sampler $\\mathbf {F}$ , we reshape the features with $c \\times a^2$ dimension into $a^2 \\times \\sqrt{c} \\times \\sqrt{c}$ (we set channel $c$ as a square number 64 in this work), and then reshape back to $c \\times a^2 \\times N_c$ for affinity calculation after the sampling of $N_c$ channels.", "For each channel in the pooled low-level feature, the affinity map is calculated with the corresponding sampled $N_c$ channels.", "Finally, the high-level features are also dynamically sampled in the channel dimension and propagated to the corresponding channel of the low-level features according to the affinity map.", "Note that we downsample $\\mathbf {x}_h$ to $16 \\times 16$ to reduce computation here.", "The whole process can be formulated as Eq.", "REF : $\\mathbf {x}_{\\mathbf {C}out}(c)=\\sum \\nolimits _{n=1}^{N_c} \\delta [\\mathbf {x}_l^{\\alpha }(c) \\mathbf {F}(\\mathbf {x}_{cat}^{\\beta }(n))^{\\top } ] \\mathbf {F}(\\mathbf {x}_h^{\\gamma }(n)),$ where $c$ is the channel in low-level features, $\\mathbf {x}_l^{\\alpha } = W_{\\alpha } \\mathbf {x}_l^p$ , $\\mathbf {x}_{cat}^{\\beta } = W_{\\beta }(\\mathbf {x}_l^p||\\mathbf {x}_h^p$ ), $\\mathbf {x}_h^{\\gamma } = W_{\\gamma } \\mathbf {x}_h^d$ , $\\mathbf {x}_l^p$ and $\\mathbf {x}_h^p$ are the pooled low-level and high-level features, $\\mathbf {x}_h^d$ is the downsampled high-level features, $\\delta $ means Softmax, $W_{\\alpha }$ ,$W_{\\beta }$ ,$W_{\\gamma }$ are implemented with $1\\times 1$ convolution layers.", "Plugin into Two Architectures: Our proposed DDSM is end-to-end trainable, and it can dynamically propagate rich semantic information between adjacent features.", "We use DDSM in UPerNet [11], which contains FPN[24] with Pyramid Pooling Module(PPM)[2], and Deeplabv3+[6].", "In UPerNet [11], we replace the upsample module with the proposed DDSM.", "Let $\\lbrace \\mathbf {x}_{s} | s=2,3,4,5\\rbrace $ be the output of each stage $s$ in encoder, e.g.", "ResNet [25].", "The enhanced higher-level features $\\mathbf {\\widetilde{x}}_s$ and the corresponding $\\mathbf {x}_{s-1}$ are passed into DDSM to form their enhanced bottom level features $\\mathbf {\\widetilde{x}}_{s-1}$ .", "For Deeplabv3+ [6], we insert one DDSM module which dynamically assigns the output of ASPP $aspp(\\mathbf {x}_{5})$ to the low-level $\\mathbf {x}_{2}$ to form $\\mathbf {\\widetilde{x}}_{2}$ for final segmentation." ], [ "Experiment", "In order to verify the effectiveness of our proposed DDSM, we conduct thorough experiments on Cityscapes[19] and CamVid[20].", "The mean Intersection over Union (mIoU) is adopted as the evaluation metric in all experiments, and F-Score[26] is used to measure the boundary performance.", "Implementation details: Our method is implemented using the Pytorch framework.", "For all our experiments, an SGD is used as the optimizer, momentum and weight decay are set to 0.9 and 5e-4, respectively.", "The learning rate is set as 0.01 and is decayed by multiplying $(1-\\frac{\\text{epoch}}{\\text{max\\_epoch}})^{0.9}$ .", "For data augmentation in training, we employ a random horizontal flip, a random resize with scale range [0.75,2], and then a random crop of $1024\\times 1024$ for Cityscapes ($720\\times 720$ for Camvid).", "Ablation study: To verify the effectiveness of each component of our method, we conduct multiple sets of experiments on Cityscapes, including whether to use DCN [17], spatial-wise and channel-wise dynamic affinity modeling or not.", "We insert two DDSMs into the second and third stages of UPerNet [11].", "The number of sampled spatial positions and channels are both set to 9.", "As shown in Table REF , based on UPerNet [11], both spatial and channel-wise dynamic modules will bring more benefits than DCN [17].", "A mix of both modules improves the performance by 1.26%.", "Simultaneously, to verify the applicability of DDSM in different frameworks, we insert one DDSM in Deeplabv3+ [6].", "Table REF (right) also shows the performance improvement of DDSM on Deeplabv3+.", "Table: Ablation Study on Cityscapes val set.", "S: Spatial-wise dynamic affinity modeling, C: Channel-wise dynamic affinity modeling.", "All networks use ResNet50 as backbone.", "And Δ\\Delta (%) means the absolute numerical improvement.Analysis of Boundary F-Score and Visualizations: To show the advantages of our model at the boundaries, we adopt the boundary F-Score [26] to measure the segmentation accuracy at the boundaries.", "Table REF shows the boundary F-Score under different thresholds.", "Our model is entirely ahead of the baseline, proving its advantages at the boundaries.", "We also visualize the input and output features of DDSM, as shown in Fig.", "REF .", "All feature maps are averaged along channels for display.", "The visualizations show that DDSM can dynamically propagate high-level semantic information to detailed low-level features from spatial and channel domain.", "Table: Boundary F-Score on UPerNet .Comparison with previous work: The mIoU of our results on the Cityscapes test set reaches 81.7%, which performs favorably against state-of-the-art segmentation methods.", "Meantime, our method has shown advantages in terms of computational consumption, which is only about 30% of DAnet's [3], as Table REF shows.", "We sample 25 pixels and 9 channels here to obtain further performance improvement according to the ablation study.", "We insert three DDSMs in UPerNet [11] to form $\\mathbf {\\widetilde{x}}_4$ , $\\mathbf {\\widetilde{x}}_3$ , $\\mathbf {\\widetilde{x}}_2$ .", "Multi-scale testing is conducted following CCNet [27].", "Table REF also shows the advantages of our method over the Non-Local based methods.", "Table: Comparison on Cityscapes test set.", "Only the methods that merely use the fine dataset are listed.", "GFlops is measured by 1024 ×\\times 1024 inputs.Experiments on CamVid: To further verify the effectiveness of DDSM, we also conduct experiments on the CamVid dataset.", "Table REF shows our results on CamVid.", "Our model without Cityscapes pre-training outperforms the others.", "After pre-training, the mIoU performance is improved to 80.6%.", "Table: Comparison on CamVid test set.", "We do not adopt multi-scale testing or other tricks." ], [ "Conclusion", "We propose an end-to-end trainable Dynamic Dual Sampling Module for both spatial-wise dynamic affinity modeling and channel-wise dynamic affinity modeling between two different features.", "Thus lower layer features are dynamically enhanced by the features of representative pixels and channels from the higher layer simultaneously.", "Through lots of experiments, our proposed DDSM is verified to be effective on different networks.", "Our model achieves advanced performance on the Cityscapes and Camvid datasets while significantly reducing computational consumption." ] ]
2105.11657
[ [ "Learning an Overlap-based Observation Model for 3D LiDAR Localization" ], [ "Abstract Localization is a crucial capability for mobile robots and autonomous cars.", "In this paper, we address learning an observation model for Monte-Carlo localization using 3D LiDAR data.", "We propose a novel, neural network-based observation model that computes the expected overlap of two 3D LiDAR scans.", "The model predicts the overlap and yaw angle offset between the current sensor reading and virtual frames generated from a pre-built map.", "We integrate this observation model into a Monte-Carlo localization framework and tested it on urban datasets collected with a car in different seasons.", "The experiments presented in this paper illustrate that our method can reliably localize a vehicle in typical urban environments.", "We furthermore provide comparisons to a beam-end point and a histogram-based method indicating a superior global localization performance of our method with fewer particles." ], [ "Introduction", "All mobile systems that navigate autonomously in a goal-directed manner need to know their position and orientation in the environment, typically with respect to a map.", "This task is often referred to as localization and can be challenging especially at the city-scale and in the presence of a lot of dynamic objects, e.g., vehicles and humans.", "Over the last decades, a wide range of localization systems have been developed relying on different sensors.", "Frequently used sensors are GPS receivers, inertial measurement units, cameras, and laser range scanners.", "All sensing modalities have their advantages and disadvantages.", "For example, GPS does not work indoors, cameras do not work well at night or under strong appearance changes, and LiDARs are active sensors and are still rather expensive.", "Figure: The image in the lower part shows the trajectories of the dataset used in this paper, overlayed on OpenStreetMap.The blue trajectory represents the sequence used to generate a map for localization.The green and the orange trajectories represent two different test sequences.The 3D point clouds shown in the upper part show LiDAR scans of the same place, once during mapping and once during localization.", "Since the LiDAR data was collected in different seasons, the appearance of the environment changed quite significantly due to changes in the vegetation but also due to parked vehicles at different places.Most autonomous robots as well as cars have a 3D LiDAR sensor onboard to perceive the scene and directly provide 3D data.", "In this paper, we consider the problem of vehicle localization only based on a 3D LiDAR sensor.", "For localization, probabilistic state estimation techniques such as extended Kalman filters (EKF) [8], particle filters [12], or incremental factor graph optimization approaches [24] can be found in most localization systems today.", "Whereas EKFs and most optimization-based approaches track only a single mode, particle filters inherently support multi-modal distributions.", "Furthermore, PFs do not restrict the motion or observation model to follow a specific distribution such as a Gaussian.", "The motion model can often be defined quite easily based on the physics of vehicle motion.", "The observation model, however, is trickier to define.", "Often, these hand-designed models are used and they strongly impact the performance of the resulting localization system.", "Frequently used observation models for LiDARs are the beam-end point model, also called the likelihood field [27], the ray-casting model [9], or models based on handcrafted features [26], [35].", "Recently, researchers also focused on learning such models completely from data [15], [30].", "In this paper, we address the problem of learning observation models for 3D LiDARs and propose a deep neural network-based approach for that.", "We explore the possibilities of learning an observation model based on OverlapNet [3] that predicts the overlap between two LiDAR scans, see Fig.", "REF , where the overlap is defined as the ratio of points that can be seen from both LIDAR scans.", "In this work, we investigate this concept for 3D LiDAR-based observation models.", "The main contribution of this paper is a novel observation model for 3D LiDAR-based localization.", "Our model is learned from range data using a deep neural network.", "It estimates the overlap and yaw angle offset between a query frame and map data.", "We use this information as the observation model in Monte-Carlo localization (MCL) for updating the importance weights of the particles.", "Based on our novel observation model, our approach achieves online localization using 3D LiDAR scans over extended periods of time with a comparably small number of particles.", "The source code of our approach is available at: https://github.com/PRBonn/overlap_localization" ], [ "Related Work", "Localization is a classical topic in robotics [27].", "For localization given a map, one often distinguishes between pose tracking and global localization.", "In pose tracking, the vehicle starts from a known pose and the pose is updated over time.", "In global localization, no pose prior is available.", "In this work, we address global localization using 3D laser scanners without assuming any pose prior from GPS or other sensors.", "Thus, we focus mainly on LiDAR-based approaches in this section.", "Traditional approaches to robot localization rely on probabilistic state estimation techniques.", "These approaches can still be found today in several localization systems [19], [6].", "A popular framework is Monte-Carlo localization [9], [28], [11], which uses a particle filter to estimate the pose of the robot.", "In the context of autonomous cars, there are many approaches building and using high-definition (HD) maps for localization, i.e., tackling the simultaneous localization and mapping problem [25] and additionally adding relevant information for the driving domain.", "Levinson et al.", "[17] utilize GPS, IMU, and LiDAR data to build HD maps for localization.", "They generate a 2D surface image of ground reflectivity in the infrared spectrum and define an observation model that uses these intensities.", "The uncertainty in intensity values has been handled by building a prior map [18], [32].", "Barsan et al.", "[15] use a fully convolutional neural network (CNN) to perform online-to-map matching for improving the robustness to dynamic objects and eliminating the need for LiDAR intensity calibration.", "Their approach shows a strong performance but requires a good GPS prior for operation.", "Based on this approach, Wei et al.", "[30] proposed a learning-based compression method for HD maps.", "Merfels and Stachniss [20] present an efficient chain graph-like pose-graph for vehicle localization exploiting graph optimization techniques and different sensing modalities.", "Based on this work, Wilbers et al.", "[31] propose a LiDAR-based localization system performing a combination of local data association between laser scans and HD map features, temporal data association smoothing, and a map matching approach for robustification.", "Other approaches aim at performing LiDAR-based place recognition to initialize localization.", "For example, Kim et al.", "[16] transform point clouds into scan context images and train a CNN based on such images.", "They generate scan context images for both the current frame and all grid cells of the map and compare them to estimate the current location as the cell presenting the largest score.", "Yin et al.", "[34] propose a Siamese network to first generate fingerprints for LiDAR-based place recognition and then use iterative closest points to estimate the metric poses.", "Cop et al.", "[7] propose a descriptor for LiDAR scans based on intensity information.", "Using this descriptor, they first perform place recognition to find a coarse location of the robot, eliminate inconsistent matches using RANSAC, and then refine the estimated transformation using iterative closest points.", "In contrast to approaches that perform place recognition first, our approach integrates a neural network-based observation model into an MCL framework to estimate the robot pose.", "Recently, several approaches exploiting semantic information for 3D LiDAR localization have been proposed.", "In our earlier work [5], we used a camera and a LiDAR to detect victims and localize the robot in an urban search and rescue environment.", "Ma et al.", "[19] combine semantic information such as lanes and traffic signs in a Bayesian filtering framework to achieve accurate and robust localization within sparse HD maps.", "Yan et al.", "[33] exploit buildings and intersections information from a LiDAR-based semantic segmentation system [21] to localize on OpenStreetMap.", "Schaefer et al.", "[23] detect and extract pole landmarks from 3D LiDAR scans for long-term urban vehicle localization.", "Tinchev et al.", "[29] propose a learning-based method to match segments of trees and localize in both urban and natural environments.", "Dubé et al.", "[10] propose to perform localization by extracting segments from 3D point clouds and matching them to accumulated data.", "Whereas, Zhang et al.", "[35] utilize both ground reflectivity features and vertical features for localizing autonomous car in rainy conditions.", "In our previous work [4], we also exploit semantic information [21], [1] to improve the localization and mapping results by detecting and removing dynamic objects.", "Different to the above discussed methods [15], [19], [30], which use GPS as prior for localization, our method only exploits LiDAR information to achieve global localization without using any GPS information.", "Moreover, our approach uses range scans without explicitly exploiting semantics or extracting landmarks.", "Instead, we rely on CNNs to predict the overlap between range scans and their yaw angle offset and use this information as an observation model for Monte-Carlo localization.", "The localization approach proposed in this paper is based on our previous work called OverlapNet [3], which focuses on loop-closing for 3D LiDAR-based SLAM." ], [ "Our Approach", "The key idea of our approach is to exploit the neural network, OverlapNet [3] that can estimate the overlap and yaw angle offset of two scans for building an observation model for localization in a given map.", "To this end, we first generate virtual scans at 2D locations on a grid rendered from the aggregated point cloud of the map.", "We train the network completely self-supervised on the map used for localization (see Sec.", "REF ).", "We compute features using our pre-trained network that allows us to compute the overlap and yaw angles between a query and virtual scans (see Sec.", "REF ).", "Finally, we integrate an observation model using the overlap (see Sec.", "REF ) and a separate observation model for the yaw angle estimates (see Sec.", "REF ) in a particle filter to perform localization (see Sec.", "REF )." ], [ "OverlapNet", "We proposed the so-called OverlapNet to detect loop closures candidates for a 3D LiDAR-based SLAM.", "The idea of overlap has its origin in the photogrammetry and computer vision community [14].", "The intuition is that to successfully match two images and calculate their relative pose, the images must overlap.", "This can be quantified by defining the overlap percentage as the percentage of pixels in the first image, which can successfully be projected back into the second image.", "In OverlapNet, we use the idea of overlap for range images generated from 3D LiDAR scans exploiting the range information explicitly.", "First, we generate from the LiDAR scans a range-image like input tensor $\\mathbf {I} \\in \\mathbb {R}^{H \\times W \\times 4}$ , where each pixel $(i,j)$ in the range-image like representation corresponds to the depth and the corresponding normal, i.e., $\\mathbf {I}(i,j) = (\\mathbf {r}, \\mathbf {n}_x, \\mathbf {n}_y, \\mathbf {n}_z)$ , where $\\mathbf {r}= ||\\mathbf {p}||_2$ .", "The indices for inserting points and estimated normals are computed using a spherical projection that maps points $\\mathbf {p} \\in \\mathbb {R}^3$ to two-dimensional coordinates $(i,j)$ , also see [3], [4].", "OverlapNet uses a siamese network structure and takes the tensors $\\mathbf {I}_1$ and $\\mathbf {I}_2$ of two scans as input for the legs to generate feature volumes, $\\mathbf {F}_1$ and $\\mathbf {F}_2$ .", "These two feature volumes are then used as inputs for two heads.", "One head $\\mathcal {H}_{\\text{overlap}}(\\mathbf {F}_1, \\mathbf {F}_2)$ estimates the overlap percentage of the scans and the other head $\\mathcal {H}_{\\text{yaw}}(\\mathbf {F}_1, \\mathbf {F}_2)$ estimates the relative yaw angle.", "For training, we determine ground truth overlap and yaw angle values using known poses estimated by a SLAM approach [2].", "Given these poses, we can train OverlapNet completely self-supervised.", "For more details on the network structure and the training procedure, we are referring to our prior work [3]." ], [ "Map of Virtual Scans", "OverlapNet requires two LiDAR scans as input.", "One is the current scan and the second one has to be generated from the map.", "Thus, we build a map of virtual LiDAR scans given an aggregated point cloud by using a grid of locations with grid resolution $\\gamma $ , where we generate virtual LiDAR scans for each location.", "The grid resolution is a trade-off between the accuracy and storage size of the map.", "Instead of storing these virtual scans, we just need to use one leg of the OverlapNet to obtain a feature volume $\\mathbf {F}$ using the input tensor $\\mathbf {I}$ of this virtual scan.", "Storing the feature volume instead of the complete scan has two key advantages: (1) it uses more than an order less space than the original point cloud (ours: $0.1\\,$ Gb/km, raw scans: $1.7\\,$ Gb/km) and (2) we do not need to compute the $\\mathbf {F}$ during localization on the map.", "The features volumes of the virtual scans are then used to compute overlap and yaw angle estimates with a query scan that is the currently observed LiDAR point cloud in our localization framework." ], [ "Monte-Carlo Localization", "Monte-Carlo localization (MCL) is a localization algorithm based on the particle filter proposed by Dellaert et al.", "[9].", "Each particle represents a hypothesis for the robot's or autonomous vehicle's 2D pose $\\mathbf {x}_t = (x, y, \\theta )_t$ at time $t$ .", "When the robot moves, the pose of each particle is updated with a prediction based on a motion model with the control input $\\mathbf {u}_t$ .", "The expected observation from the predicted pose of each particle is then compared to the actual observation $\\mathbf {z}_t$ acquired by the robot to update the particle's weight based on an observation model.", "Particles are resampled according to their weight distribution and resampling is triggered whenever the effective number of particles, see for example [13], drops below 50% of the sample size.", "After several iterations of this procedure, the particles will converge around the true pose.", "MCL realizes a recursive Bayesian filtering scheme.", "The key idea of this approach is to maintain a probability density $p(\\mathbf {x}_t\\mid \\mathbf {z}_{1:t},\\mathbf {u}_{1:t})$ of the pose $\\mathbf {x_t}$ at time $t$ given all observations $\\mathbf {z}_{1:t}$ up to time $t$ and motion control inputs $\\mathbf {u}_{1:t}$ up to time $t$ .", "This posterior is updated as follows: $&p(\\mathbf {x}_t\\mid \\mathbf {z}_{1:t},\\mathbf {u}_{1:t}) = \\eta ~p(\\mathbf {z}_t\\mid \\mathbf {x}_{t}) \\cdot \\nonumber \\\\&\\;\\;\\int {p(\\mathbf {x}_t\\mid \\mathbf {u}_{t}, \\mathbf {x}_{t-1})~p(\\mathbf {x}_{t-1} \\mid \\mathbf {z}_{1:t-1},\\mathbf {u}_{1:t-1})\\ d\\mathbf {x}_{t-1}},$ where $\\eta $ is a normalization constant, $p(\\mathbf {x}_t\\mid \\mathbf {u}_{t}, \\mathbf {x}_{t-1})$  is the motion model, and $p(\\mathbf {z}_t\\mid \\mathbf {x}_{t})$  is the observation model.", "This paper focuses on the observation model.", "For the motion model, we follow a standard odometry model for vehicles [27].", "We split the observation model into two parts: $p(\\mathbf {z}_t\\mid \\mathbf {x}_{t}) &= p_{L} \\left(\\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right)~p_{O} \\left( \\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right) ,$ where $\\mathbf {z}_{t}$ is the LiDAR observation at time $t$ , $p_{L} \\left(\\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right)$ is the probability encoding the location $(x, y)$ agreement between the current query LiDAR scan and the virtual scan at the grid where the particle locate and $p_{O} \\left( \\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right)$ is the probability encoding the yaw angle $\\theta $ agreement between the same pairs of scans." ], [ "Overlap Observation Model", "Given a particle $i$ with the state vector $(x_i, y_i, \\theta _i)$ , the overlap estimates encode the location agreement between the query LiDAR scan and virtual scans of the grid cells where particles locate.", "It can be directly used as the probability: $p_{L} \\left(\\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right) &\\propto f\\left(\\mathbf {z}_{t}, \\mathbf {z}_{i}; \\mathbf {w} \\right),$ where $f$ corresponds to the neural network providing the overlap estimation between the input scans $\\mathbf {z}_{t}, \\mathbf {z}_{i}$ and $\\mathbf {w}$ is the pre-trained weights of the network.", "$\\mathbf {z}_{t}$ and $\\mathbf {z}_{i}$ are the current query scan and a virtual scan of one $(x, y)$ location respectively.", "Note that no additional hyperparameter is needed to formulate our observation model for localization.", "For illustration purposes, Fig.", "REF shows the probabilities of all grid cells in a local area calculated by the overlap observation model.", "The blue car in the figure shows the current location of the car.", "The probabilities calculated by the overlap observation model can represent well the hypotheses of the current location of the car.", "Typically, a large number of particles are used, especially when the environment is large.", "However, a large amount of particles will increase the computation time linearly.", "When applying the overlap observation model, particles could still obtain relatively large weights as long as they are close to the actual pose, even if not in the exact same position.", "This allows us to use fewer particles to achieve a high success rate of global localization.", "Furthermore, the overlap estimation only encodes the location hypotheses.", "Therefore, if multiple particles locate in the same grid area, only a single inference against the nearest virtual scan of the map needs to be done, which can further reduce the computation time." ], [ "Yaw Angle Observation Model", "Given a particle $i$ with the state vector $(x_i, y_i, \\theta _i)$ , the yaw angle estimates encode the orientation agreement between the query LiDAR scan and virtual scans of the corresponding grids where particles locate.", "We formulate the orientation probability as follows: $p_{O} \\left(\\mathbf {z}_{t} \\mid \\mathbf {x}_{t} \\right) &\\propto \\exp {\\left( -\\frac{1}{2} \\frac{\\Big (g\\left(\\mathbf {z}_{t}, \\mathbf {z}_{i}; \\mathbf {w} \\right) - \\theta _i \\Big )^2}{\\sigma ^2_\\theta }\\right)} ,$ where $g$ corresponds to the neural network providing the yaw angle estimation between the input scans $\\mathbf {z}_{t}, \\mathbf {z}_{i}$ and $\\mathbf {w}$ is the pre-trained weights of the network.", "$\\mathbf {z}_{t}$ and $\\mathbf {z}_{i}$ are the current query scan and a virtual scan of one particle respectively.", "When generating the virtual scans of the grid map, all virtual scans will be set facing the absolute 0 yaw angle direction.", "By doing this, the estimated relative yaw angle between the query scan and the virtual scan indicates the absolute yaw angle of the current query scan.", "Eq.", "(REF ) assumes a Gaussian measurement error in the heading.", "By combining overlap and yaw angle estimation, the proposed observation model will correct the weights of particles considering agreements between the query scan and the map with the full pose $(x, y, \\theta )$ ." ], [ "Experimental Evaluation", "In this paper, we use a grid representation and generate virtual $360{}$ scans for each grid point.", "The resolution of the grid is $\\gamma =20\\,$  cm.", "When generating the virtual scans, we set the yaw angle to $0{}$ .", "Therefore, when estimating the relative yaw angle between the query frame and the virtual frames, the result will indicate the absolute yaw of the query frame.", "For the yaw angle observation model in Eq.", "(REF ), we set $\\sigma _{\\theta } = 5{}$ .", "To achieve global localization, we train a new model only based on the map scans and the generated virtual scans.", "The main focus of this work is a new observation model for LiDAR-based localization.", "Therefore, when comparing different methods, we only change the observation model $p(\\mathbf {z}_t|\\mathbf {x}_t)$ of the MCL framework and keep the particle filter-based localization the same.", "The motion model is the typical odometry model [27]." ], [ "Car Dataset", "The dataset used in this paper was collected using a self-developed sensor platform illustrated in Fig.", "REF .", "To test LiDAR-based global localization, a large-scale dataset has been collected in different seasons with multiple sequences repeatedly exploring the same crowded urban area.", "For our car dataset, we performed a 3D LiDAR SLAM [2] combined with GPS information to create near ground truth poses.", "During localization, we only use LiDAR scans for global localization without using GPS.", "The dataset has three sequences that were collected at different times of the year, sequence 00 in September 2019, sequence  01 in November 2019, and sequence 02 in February 2020.", "The whole dataset covers a distance of over 10 km.", "We use LiDAR scans from sequence 02 to build the virtual scans and use sequence 00 and 01 for localization.", "As can be seen from Fig.", "REF , the appearance of the environment changes quite significantly since the dataset was collected in different seasons and in crowded urban environments, including changes in vegetation, but also cars at different locations and moving people.", "Figure: Sensor setup used for data recording: Ouster OS1-64 LiDAR sensor plus GNSS information from a Emilid Reach RS2." ], [ "Different Observation Models", "In the following experiments, we use the same MCL framework and only exchange the observation models.", "We compare our observation model with two baseline observation models: the typical beam-end model [27] and a histogram-based model derived from the work of Röhling et al.", "[22].", "The beam-end observation model is often used for 2D LiDAR data.", "For 3D LiDAR scans, it needs much more particles to make sure that it converges to the correct pose, which causes the computation time to increase substantially.", "In this paper, we implement the beam-end model with a down-sampled point cloud map using voxelization with a resolution of 10 cm.", "Our second baseline for comparison is inspired by Röhling et al.", "[22], which proposed a fast method to detect loop closures through the use of similarity measures on histograms extracted from 3D LiDAR data.", "The histogram contains the range information.", "We use a similar idea, but integrate it into the MCL framework as a baseline observation model.", "We employ the same grid map and virtual frames as used for our method with the histogram-based observation model.", "When updating the weights of particles, we will first generate the histograms of the current query scan and the virtual scans of grids where the particles locate.", "Then, we use the same Wasserstein distance to measure the similarity between them and update the weights of particles as follows: $p(\\mathbf {z}_t|\\mathbf {x}_{t}) \\propto d \\left( h(\\mathbf {z}_{t}), h(\\mathbf {z}_{i} \\right)),$ where $d$ represents the Wasserstein distance between histograms $h(\\mathbf {z}_{t}), h(\\mathbf {z}_{i})$ of LiDAR scan $\\mathbf {z}_{t}, \\mathbf {z}_{i}$ .", "Fig.", "REF shows the 2D heatmaps in $x$ and $y$ calculated for the different observation models.", "As can be seen, the proposed observation model tends to give higher weights to the positions along the road, which leads to a higher success rate when the vehicle aims to localize in an urban environment.", "We will show the numerical results which verify that our method can achieve a high success rate with much fewer particles in the next sections.", "Figure: Trajectory from sequence 01" ], [ "Localization Performance", "The experiment presented in this section is designed to show the performance of our approach and to support the claim that it is well suited for global localization.", "First of all, we show the general localization results tested with two sequences in Fig.", "REF .", "The qualitative results show that, after applying our sensor-model, the proposed method can well localize in the map with only LiDAR data collected in highly dynamic environments at different times.", "For quantitative results, we calculate the success rate for different methods with different particle numbers comparing our approach to different methods, as shown in Fig.", "REF .", "The x-axis represents the number of particles used during localization, while the y-axis is the success rate of different setups.", "The success rate for a specific setup of one method is calculated using the number of success cases divided by the total numbers of the tests.", "To decide whether one test is successful or not, we check the location error by every 100 frames after converging.", "If the location error is smaller than a certain threshold, we count this run as a success case.", "We test our method together with two baselines using five different numbers of particles $N=$  {$1,000$  ; $5,000$  ; $10,000$  ; $50,000$  ; $100,000$ }.", "For each setup, we sample 10 trajectories and perform global localization.", "Quantitative results of localization accuracy are shown in Tab.", "REF and Tab.", "REF .", "Tab.", "REF shows the location error of all methods tested with both sequences.", "The location error is defined as the root mean square error (RMSE) of each test in terms of $(x,\\,y)$ Euclidean error with respect to the ground truth poses.", "Tab.", "REF shows the mean and the standard deviation of the error for each observation model.", "Note that the location error is only calculated for success cases.", "Tab.", "REF shows the yaw angle error.", "It is the RMSE of each test in terms of yaw angle error with respect to the ground truth poses.", "The table shows the mean and the standard deviation of the error for each observation model.", "As before, the yaw angle error is also only calculated for cases in which the global localization converged.", "Table: Location resultsTable: Yaw angle resultsAs can be seen from the results, our method achieves higher success rates with a smaller number of particles compared to the baseline methods, which also makes the proposed method faster than baseline methods.", "Furthermore, our method converges already with $100,000$  particles in all cases, whereas the other observation models still need more particles to sufficiently cover the state space.", "Moreover, the proposed method gets similar performance in location error comparing to the baseline methods but it achieves better results in yaw angle estimation.", "This is because the proposed method decouples the location and yaw angle estimation and, therefore, can exploit more constraint in yaw angle corrections.", "To sum up, the proposed method outperforms the baselines method in terms of success rate, while getting similar results in terms of location error.", "Moreover, our method outperforms baseline methods in yaw angle estimation, because of the proposed de-coupled observation model.", "Furthermore, our method is faster than the baseline method.", "The runtime details will be shown in the next experiment.", "Figure: Number of observation model evaluations for updating the weights at each timestep with 100,000100,000 particles.", "The beam end model needs to be evaluated for each and every particle individually.", "The histogram-based method is more computationally efficient, while our proposed method still needs the fewest evaluations." ], [ "Runtime", "In this experiment, we show the number of observation model evaluations necessary for updating the weights at each time step in Fig.", "REF .", "This is a fairer way to compare the computational cost of different methods, since our neural network-based method uses GPU to concurrently updating the weights of particles, while the other methods only use CPU.", "As can be seen, our method needs a smaller number of observation model evaluations to update the weights for all particles.", "This is because we only need to perform the network inference for all particles which are localized in the same grid cell once.", "For one incoming frame and the virtual frame of that grid cell, the inputs of the network and thus the outputs are the same for all particles in that cell.", "We tested our method on a system equipped with an Intel i7-8700 with 3.2 GHz and an Nvidia GeForce GTX 1080 Ti with 11 GB of memory.", "For initializing in the large-scale map, the worst case will take around $43\\,$ s to process one frame.", "However, after converging, the proposed method takes only 1 s on average to process one frame with $10,000$ particles." ], [ "Conclusion", "In this paper, we presented a novel observation model and integrated it into an MCL framework to solve the global localization problem.", "Our method exploits OverlapNet to estimate the overlap and yaw angle between the current frame and the virtual frames generated at each particle using the pre-built map.", "This allows us to successfully update the weights of particles with agreements of both location and orientation.", "We implemented and evaluated our approach on an urban dataset collected with a car in different seasons and provided comparisons to other existing techniques.", "The experiments suggest that our approach can achieve a similar performance as other approaches in global localization while obtaining a higher success rate and lower computational time.", "In future work, we will test our method on more datasets with different types of LiDAR sensors.", "We also plan to test our method with high definition maps and want to exploit semantic information." ] ]
2105.11717
[ [ "On the Erd\\H{o}s-P\\'osa property for long holes in $C_4$-free graphs" ], [ "Abstract We prove that there exists a function $f(k)=\\mathcal{O}(k^2 \\log k)$ such that for every $C_4$-free graph $G$ and every $k \\in \\mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$.", "This answers a question of Kim and Kwon [Erd\\H{o}s-P\\'osa property of chordless cycles and its applications.", "JCTB 2020]." ], [ "Introduction", "A classic theorem of Erdős and Pósa [5] asserts that there exists a function $f: \\mathbb {N}\\rightarrow \\mathbb {R}$ such that for every graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint cycles, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no cycle.", "The Erdős-Pósa theorem has since been extensively generalized, where the objects to be packed are not necessarily cycles, the containment relation is not necessarily the subgraph relation, and the host class is not necessarily the class of all graphs.", "These results are too numerous to cite properly, but we refer the interested reader to a recent survey of Raymond and Thilikos [15].", "In this paper, we are interested in the induced subgraph relation.", "A graph is $H$ -free if it has no induced subgraph isomorphic to $H$ .", "A family $\\mathcal {F}$ of graphs has the (induced) Erdős-Pósa property if there exists a function $f:\\mathbb {N}\\rightarrow \\mathbb {R}$ such that for every graph $G$ and every $k\\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint (induced) subgraphs each isomorphic to a graph in $\\mathcal {F}$ , or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no (induced) subgraph isomorphic to a graph in $\\mathcal {F}$ .", "For each $\\ell \\ge 3$ , we let $C_\\ell $ denote the cycle of length $\\ell $ .", "A hole in a graph is an induced $C_\\ell $ with $\\ell \\ge 4$ .", "A graph is chordal if it has no hole.", "The class of chordal graphs is a widely studied class of graphs, in part because many NP-complete problems can be efficiently solved when restricted to chordal graphs.", "Given a graph $G$ and $k \\in \\mathbb {N}$ as input, the Chordal Deletion problem asks whether there exists a set $X$ of at most $k$ vertices in $G$ such that $G-X$ is chordal.", "Motivated by the Chordal Deletion problem, Jansen and Pilipczuk [8] asked whether cycles of length at least 4 have the induced Erdős-Pósa property.", "This was recently answered in the affirmative by Kim and Kwon [10].", "Theorem 1.1 (Kim and Kwon [10]) There exists a function $g(k)=\\mathcal {O}(k^2 \\log k)$ such that for every graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes, or a set $X$ of at most $g(k)$ vertices such that $G-X$ has no hole.", "Whenever the Erdős-Pósa property holds for a family of objects, it is natural to ask if an analogous “long” version also holds.", "That is, does the Erdős-Pósa property still hold if we insist that the objects are of length at least $\\ell $ , for some constant $\\ell $ ?", "For example, by Menger's theorem [12], the Erdős-Pósa property holds for ($S$ -$T$ )-paths (with $f(k)=k$ ) and Montejano and Neumann-Lara [13] showed that it also holds for ($S$ -$T$ )-paths of length at least $\\ell $ .", "The original Erdős-Pósa theorem was extended to long cycles by Birmelé, Bondy, and Reed [1] (see [6], [14] for improved bounds).", "Kakimura, Kawarabayashi, and Marx [9] showed that $S$ -cycles (cycles with at least one vertex in a prescribed vertex set $S$ ) have the Erdős-Pósa property, and this was extended to long $S$ -cycles by Bruhn, Joos, and Schaudt [2].", "Huynh, Joos, and Wollan [7] showed that $(S,T)$ -cycles (cycles that use at least one vertex from each of two prescribed vertex sets $S$ and $T$ ) and long $(S,T)$ -cycles both have the Erdős-Pósa property.", "These results might lead one to conjecture that whenever the Erdős-Pósa property holds, the “long” Erdős-Pósa property also holds.", "However, this turns out to be false for holes, in the following strong sense.", "Let $\\alpha \\in \\mathbb {N}$ and $\\mathcal {F}$ be a family of graphs.", "We say that $\\mathcal {F}$ has the $\\frac{1}{\\alpha }$ -integral (induced) Erdős-Pósa property if there exists a function $f: \\mathbb {N}\\rightarrow \\mathbb {R}$ so that for every graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint (induced) subgraphs, each isomorphic to a graph in $\\mathcal {F}$ and such that every vertex of $G$ is contained in at most $\\alpha $ of these (induced) subgraphs, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no (induced) subgraph in $\\mathcal {F}$ .", "Kim and Kwon [10] proved that for all $\\alpha \\in \\mathbb {N}$ and $\\ell \\ge 5$ , cycles of length at least $\\ell $ do not have the $\\frac{1}{\\alpha }$ -integral induced Erdős-Pósa property.", "Theorem 1.2 (Kim and Kwon [10]) Let $\\alpha \\in \\mathbb {N}$ and $\\ell \\ge 5$ .", "There is no function $f: \\mathbb {N}\\rightarrow \\mathbb {R}$ such that for every graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least $\\ell $ such that every vertex of $G$ is contained in at most $\\alpha $ of these holes, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $\\ell $ .", "Theorem REF is rather surprising, since there are many objects that do not have the Erdős-Pósa property but do have the $\\frac{1}{2}$ -integral Erdős-Pósa property.", "For example, odd cycles do not have the Erdős-Pósa property [4] but they do have the $\\frac{1}{2}$ -integral Erdős-Pósa property [16].", "An important observation is that the examples from Theorem REF have large complete bipartite graphs as induced subgraphs.", "Therefore, Kim and Kwon asked whether the induced Erdős-Pósa property holds for long cycles, when we restrict to $C_4$ -free graphs.", "Question 1 (Kim and Kwon [10]) For fixed $\\ell \\ge 6$ , does there exist a function $f: \\mathbb {N}\\rightarrow \\mathbb {R}$ such that for every $C_4$ -free graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least $\\ell $ , or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $\\ell $ ?", "The main result of this paper is that Question REF is true for $\\ell =6$ .", "Theorem 1.3 There exists a function $f(k)=\\mathcal {O}(k^2 \\log k)$ such that for every $C_4$ -free graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least 6, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least 6.", "Moreover, there is a polynomial-time algorithm that, given a $C_4$ -free graph $G$ and a positive integer $k$ , either finds $k$ vertex-disjoint holes of length at least 6 in $G$ , or a set $X$ of at most $\\mathcal {O}(k^2 \\log k)$ vertices such that $G-X$ has no hole of length at least 6.", "Note that Theorem REF implies Theorem REF as follows.", "We claim that we may take $g(k)=5(k-1)+f(k)$ , where $f(k)$ is the function from Theorem REF .", "Let $G$ be a graph and $k \\in \\mathbb {N}$ .", "Let $\\mathcal {U}$ be a collection of vertex-disjoint holes of $G$ , where each $H \\in \\mathcal {U}$ has length 4 or 5, and $| \\mathcal {U}|$ is maximum with this property.", "If $|\\mathcal {U}| \\ge k$ , then we are done.", "Otherwise, let $X$ be the set of vertices in $G$ covered by the holes in $\\mathcal {U}$ .", "Since $G - X$ is $C_4$ -free, by Theorem REF , $G-X$ either contains $k$ vertex-disjoint holes of length at least 6, or a set $X^{\\prime }$ of at most $f(k)$ vertices such that $(G-X)-X^{\\prime }$ has no hole of length at least 6.", "If the former holds, we are done.", "If the latter holds, then $G-(X \\cup X^{\\prime })$ has no holes and $|X \\cup X^{\\prime }| \\le 5(k-1)+f(k)$ .", "For brevity, we call a hole of length at least 6 a long hole.", "Our proof of Theorem REF follows the same general strategy as [10], but with several simplifications.", "Indeed, even though we need to deal with several additional technical difficulties in finding long holes, our proof is still shorter than the proof for holes.", "If one only cares about Theorem REF , it is easy to rewrite our proof to obtain an extremely succinct proof for holes.", "We begin by reducing Theorem REF to the case where there is a given shortest long hole $C$ such that $G-V(C)$ has no long hole and $C$ has $c^{\\prime }k\\log k$ vertices for some constant $c^{\\prime }$ .", "We show that every vertex in $G-V(C)$ either has at most three neighbors that are consecutive on $C$ , or it `almost' dominates $C$ .", "We let $D$ be the set of vertices that almost dominate $C$ .", "These vertices play a similar role as the $C$ -dominating vertices in [10].", "As in [10], we can show that $D$ is a clique.", "In [10], holes are divided into four different types, depending on whether they intersect $D$ or not, and whether they are contained in the closed neighborhood of $C$ or not.", "For each of these four types, Kim and Kwon [10] find either $k$ vertex-disjoint holes of that type, or a small vertex set hitting all holes of that type.", "In our work, we unify and simplify many of the steps in their proof.", "In particular, we do not separately consider long holes all of whose vertices are close to $C$ .", "We simply distinguish long holes depending on whether they intersect $D$ or not.", "For each of these two types, we find either $k$ vertex-disjoint long holes of that type, or a small vertex set hitting all holes of that type." ], [ "Paper Outline", "In Section , we introduce some basic definitions.", "We reduce Theorem REF to the restricted setting discussed above (Theorem REF ) in Section , and show that our main result follows from Theorem REF .", "The rest of the paper is dedicated to proving Theorem REF .", "In Section , we introduce almost $C$ -dominating vertices, and prove some structural lemmas relative to the cycle $C$ .", "In Section , we find a greedy packing of long holes contained in the union of $D$ and the third closed neighborhood of $C$ in $G-D$ .", "We then find a hitting set of size $\\mathcal {O}(k\\log k)$ for such holes if the greedy packing does not find $k$ vertex-disjoint long holes.", "In Section , we find a `sparse' ear decomposition $ of $ G$.", "We argue that the number of branching points (vertices of degree $ 3$) of $ is $\\mathcal {O}(k\\log k)$ , otherwise we can find $k$ vertex-disjoint long holes.", "We remove all branching points of $ together with some vertices of $ C$ close to the branching points.", "We complete the proof of Theorem~\\ref {thm:main2} in Section~\\ref {sec:avoiding}.", "Long holes that intersect $ D$ are handled in Subsection~\\ref {subsec:avoding}, and long holes that avoid $ D$ are handled in Subsection~\\ref {subsec:traversing}.", "We end the paper with some open problems in Section~\\ref {sec:openproblems}.$" ], [ "Basic Definitions", "All graphs in this paper are finite, undirected, and have no loops and parallel edges.", "Let $G$ and $H$ be graphs.", "We denote by $V(G)$ and $E(G)$ the vertex set and the edge set of $G$ , respectively.", "We let $G \\cup H$ and $G \\cap H$ be the graphs with vertex set $V(G) \\cup V(H)$ (respectively, $V(G) \\cap V(H)$ ) and edge set $E(G) \\cup E(H)$ (respectively, $E(G) \\cap E(H)$ ).", "If $H$ is a subgraph of $G$ and $S \\subseteq V(G)$ , we let $H-S$ be the graph obtained from $H$ by removing $S \\cap V(H)$ and all edges with at least one end in $S$ .", "We abbreviate $H-\\lbrace v\\rbrace $ as $H-v$ , and we let $H[S]:=H - (V(G) \\setminus S)$ .", "A subgraph of $G$ is induced if it is equal to $G[S]$ for some $S\\subseteq V(G)$ .", "We say that a set $S$ of vertices in $G$ is a clique if every pair of vertices in $S$ is adjacent.", "We say that $u$ is a neighbor of $v$ if $uv \\in E(G)$ .", "The open neighborhood of $A \\subseteq V(G)$ is the set of vertices in $V(G)\\setminus A$ having a neighbor in $A$ , and is denoted $N_G(A)$ .", "The set $N_G[A]:=N_G(A)\\cup A$ is the closed neighborhood of $A$ .", "For $v \\in V(G)$ , we let $N_G(v):=N_G(\\lbrace v\\rbrace )$ and $N_G[v]:=N_G[\\lbrace v\\rbrace ]$ .", "For a subgraph $H$ of $G$ , we let $N_G(H):=N_G(V(H))$ and $N_G[H]:=N_G[V(H)]$ .", "The degree of $v$ , denoted $\\deg _G(v)$ , is $|N_G(v)|$ .", "We let $\\mathbb {N}$ be the set of positive integers, and for all $n\\in \\mathbb {N}$ , let $[n]:=\\lbrace 1, \\dots , n\\rbrace $ .", "The length of a path in $G$ is its number of edges.", "For two vertices $x$ and $y$ in $G$ , an $(x,y)$ -path is a path in $G$ whose ends are $x$ and $y$ .", "The distance between $x$ and $y$ is the length of a shortest $(x,y)$ -path, and is denoted $\\operatorname{dist}_G(x,y)$ .", "If there is no $(x,y)$ -path in $G$ , then $\\operatorname{dist}_G(x,y):=\\infty $ .", "The distance between two vertex sets $X,Y\\subseteq V(G)$ , written as $\\operatorname{dist}_G(X,Y)$ , is the minimum $\\operatorname{dist}_G(x,y)$ over all $x\\in X$ and $y\\in Y$ .", "For $A \\subseteq V(G)$ and $r \\in \\mathbb {N}$ , we let $N_G^r[A]$ denote the set of all vertices $w$ such that $\\operatorname{dist}_G(\\lbrace w\\rbrace , A)\\le r$ .", "We abbreviate $\\operatorname{dist}_G(\\lbrace x\\rbrace ,Y)$ as $\\operatorname{dist}_G(x,Y)$ , and $N_G^r[\\lbrace x\\rbrace ]$ as $N_G^r[x]$ .", "For a subgraph $H$ of $G$ , let $N_G^r[H]:=N_G^r[V(H)]$ .", "For a path $P$ and two vertices $x$ and $y$ in $P$ , we denote by $xPy$ the subpath of $P$ from $x$ to $y$ .", "For two walks $P$ and $Q$ such that the last vertex of $P$ is the first vertex of $Q$ , we let $PQ$ be the concatenation of $P$ and $Q$ .", "In the special case that $P$ and $Q$ are paths with $x,y \\in V(P)$ and $y,z \\in V(Q)$ , we abbreviate $(xPy)(yQz)$ as $xPyQz$ .", "For $A \\subseteq V(G)$ , an $A$ -path is a path in $G$ such that its ends are in $A$ , and all other vertices are contained in $V(G)\\setminus A$ .", "For a subgraph $H$ of $G$ , a $V(H)$ -path is simply called an $H$ -path.", "We will need the following result of Simonovitz [17], which is useful for finding many vertex-disjoint cycles in a graph of maximum degree 3.", "We define $s_k$ for $k\\in \\mathbb {N}$ as $s_k={\\left\\lbrace \\begin{array}{ll}4k(\\log k + \\log \\log k +4) \\quad &\\text{if } k\\ge 2\\\\2 &\\text{if } k=1.\\end{array}\\right.", "}$ Theorem 2.1 (Simonovitz [17]) Let $G$ be a graph all of whose vertices have degree 3 and let $k \\in \\mathbb {N}$ .", "If $\\vert V(G)\\vert \\ge s_k$ , then $G$ contains $k$ vertex-disjoint cycles.", "Moreover, these $k$ cycles can be found in polynomial time." ], [ "An Equivalent Formulation", "Recall that a hole of $G$ is long if it has length at least 6.", "In this section, we show that to prove Theorem REF , it suffices to prove the following theorem.", "Theorem 3.1 Let $G$ be a $C_4$ -free graph, $k \\in \\mathbb {N}$ , and $C$ be a cycle in $G$ such that $C$ is a shortest long hole of $G$ , $|V(C)| > \\mu _k:=88575k+24003s_k$ , and $G-V(C)$ does not contain a long hole.", "Then there exists a polynomial-time algorithm that finds $k$ vertex-disjoint long holes of $G$ or a set $X_{\\ref {thm:main2}}$ of at most $\\mu _k$ vertices in $G$ such that $G-X_{\\ref {thm:main2}}$ has no long hole.", "We first need an algorithm to detect a shortest long hole, if one exists.", "Lemma 3.2 Let $G$ be a graph, $Q$ be a $(u, v)$ -induced path in $G$ of length at least 4 for some $u,v\\in V(G)$ , and $P$ be a shortest $(u, v)$ -path in $G-(N_G[V(Q) \\setminus \\lbrace u, v\\rbrace ] \\setminus \\lbrace u, v\\rbrace )$ .", "Then $P \\cup Q$ is a long hole.", "Moreover, among all long holes of $G$ containing $Q$ , $P \\cup Q$ is shortest.", "Since $P$ is a shortest path, it is induced.", "Moreover, since $u$ and $v$ are not adjacent, $P$ has length at least 2, and hence $P \\cup Q$ has length at least 6.", "Since there is no edge from $V(Q) \\setminus \\lbrace u, v\\rbrace $ to $V(P) \\setminus \\lbrace u,v\\rbrace $ in $G$ , $P \\cup Q$ is a long hole of $G$ containing $Q$ .", "To see that $P \\cup Q$ is a shortest such hole, observe that if $H$ is any hole of $G$ containing $Q$ , then $H - (V(Q) \\setminus \\lbrace u, v\\rbrace )$ is a $(u, v)$ -path in $G-(N_G[V(Q) \\setminus \\lbrace u, v\\rbrace ] \\setminus \\lbrace u, v\\rbrace )$ .", "Lemma 3.3 Given a graph $G$ , a shortest long hole of $G$ (if one exists) can be found in time $\\mathcal {O}(\\vert V(G)\\vert ^7)$ .", "We guess a 5-tuple of vertices $\\mathcal {X}:=(x_1, x_2, x_3, x_4, x_5)$ that form an induced path $x_1x_2x_3x_4x_5$ , and find a shortest $(x_5,x_1)$ -path $P_{\\mathcal {X}}$ in $G-(N_G[\\lbrace x_2, x_3, x_4\\rbrace ] \\setminus \\lbrace x_1, x_5\\rbrace )$ (if one exists).", "By Lemma REF , this process finds the shortest long hole of $G$ containing the path $x_1x_2x_3x_4x_5$ (if such a hole exists).", "Therefore, if $P_{\\mathcal {X}}$ does not exist for all $\\mathcal {X}$ , then $G$ does not contain a long hole.", "Otherwise, if we choose $\\mathcal {X}=(x_1, x_2, x_3, x_4, x_5)$ , such that $|P_\\mathcal {X}|$ is minimum, then $x_5 P_{\\mathcal {X}}x_1x_2x_3x_4x_5$ is a shortest long hole of $G$ .", "Since shortest paths can be computed in time $\\mathcal {O}(|V(G)|^2)$ , the above algorithm runs in time $\\mathcal {O}(\\vert V(G)\\vert ^7)$ .", "We are now ready to prove Theorem REF assuming that Theorem REF holds.", "This argument has been repeatedly used in various Erdős-Pósa type results.", "By repeatedly applying Lemma REF , we can construct a sequence of graphs $G_1,\\ldots , G_{\\ell +1}$ and a sequence of cycles $C_1,\\ldots , C_{\\ell }$ in polynomial time such that $G_1=G$ , for each $i\\in [\\ell ]$ , $C_i$ is a shortest long hole of $G_i$ , for each $i\\in [\\ell ]$ , $G_{i+1}=G_i-V(C_i)$ , and $G_{\\ell +1}$ has no long holes.", "If $\\ell \\ge k$ , then we found a packing of $k$ long holes.", "Thus, we may assume that $\\ell \\le k-1$ .", "The following claim will complete the proof.", "Claim For each $j \\in [\\ell +1]$ , we can find in polynomial time either $k$ vertex-disjoint long holes of $G$ , or a vertex set $X_{j}$ of $G_{j}$ of size at most $(\\ell +1-j)\\mu _k$ so that $G_j-X_j$ has no long holes.", "We proceed by reverse induction on $j$ .", "The base case of $j=\\ell +1$ holds with $X_{\\ell +1}=\\emptyset $ since $G_{\\ell +1}$ has no long holes.", "Fix $j\\le \\ell $ .", "By induction, there is a subset of vertices $X_{j+1}$ of $G_{j+1}$ of size at most $(\\ell -j)\\mu _k$ hitting all long holes of $G_{j+1}$ .", "Since $G_{j+1}=G_j-V(C_j)$ , $C_{j}$ is a shortest long hole of $G_{j}-X_{j+1}$ , and $\\left( G_{j}-X_{j+1} \\right)-V(C_{j})$ has no long holes.", "If $C_{j}$ has length at most $\\mu _k$ , then we set $X_{j}:=X_{j+1}\\cup V(C_{j})$ .", "Clearly, $\\vert X_j\\vert \\le (\\ell +1-j)\\mu _k$ .", "Otherwise, by applying Theorem REF to $(G_{j}-X_{j+1}, k, C_{j})$ , we can find in polynomial time either $k$ vertex-disjoint long holes of $G_{j}-X_{j+1}$ , or a vertex set $X$ of size at most $\\mu _k$ of $G_{j}-X_{j+1}$ hitting all long holes.", "In the former case, we output $k$ vertex-disjoint holes, and we are done.", "In the latter case, we set $X_{j}:=X_{j+1}\\cup X$ .", "Observe that $G_{j}- X_j$ has no long holes and $\\vert X_{j}\\vert \\le (\\ell +1-j)\\mu _k$ as claimed.", "In particular, the set $X_1$ hits every long hole of $G_1=G$ and $\\vert X_1\\vert \\le \\ell \\mu _k\\le (k-1)\\mu _k=\\mathcal {O}(k^2\\log k).", "$" ], [ "Structure Relative to the Cycle $C$", "For the remainder of the paper, we fix a $C_4$ -free graph $G$ , $k \\in \\mathbb {N}$ , and a cycle $C$ in $G$ such that $C$ is the shortest long hole of $G$ , $|V(C)| > \\mu _k$ , and $G-V(C)$ does not contain a long hole.", "Our goal is to prove Theorem REF .", "We begin by analyzing the structure of $G$ relative to $C$ ." ], [ "Almost $C$ -dominating vertices", "We say that a vertex $u \\notin V(C)$ is almost $C$ -dominating if the neighbors of $u$ on $C$ can be enumerated $u_1, \\dots , u_\\ell $ (with $u_{\\ell +1}=u_1$ ) such that $\\ell \\ge 2$ and $\\operatorname{dist}_C(u_i, u_{i+1}) \\in \\lbrace 1,3\\rbrace $ for all $i \\in [\\ell ]$ .", "This definition is motivated by the following lemma.", "Lemma 4.1 For every vertex $v \\in V(G)\\setminus V(C)$ , either $v$ is almost $C$ -dominating or $\\vert N_G(v)\\cap V(C)\\vert \\le 3$ and the vertices in $N_G(v)\\cap V(C)$ are consecutive along $C$ .", "Towards a contradiction, let $v \\in V(G)\\setminus V(C)$ be a vertex for which neither outcome occurs.", "Since $v$ is not almost $C$ -dominating and $G$ is $C_4$ -free, there is a $(v_1, v_2)$ -subpath $P$ of $C$ such that $v_1, v_2 \\in N_G(v)\\cap V(C)$ , no internal vertex of $P$ is in $N_G(v)\\cap V(C)$ , and $|V(P)| \\ge 5$ .", "If $|V(C)|-|V(P)|=0$ , then the second outcome occurs, which contradicts that $v$ is a counterexample.", "Suppose $V(C) \\setminus V(P):=\\lbrace x\\rbrace $ .", "If $x \\in N_G(v)$ , then the second outcome occurs again, which is a contradiction.", "If $x \\notin N_G(v)$ , then $vv_1xv_2v$ is an induced $C_4$ of $G$ , which is also a contradiction.", "Thus, $|V(C)|-|V(P)| \\ge 2$ .", "But now, $v_1Pv_2vv_1$ is a long hole of length strictly less than $|V(C)|$ , contradicting the assumption that $C$ is a shortest long hole in $G$ .", "We next show that the set of almost $C$ -dominating vertices is a clique in $G$ .", "Lemma 4.2 The set of almost $C$ -dominating vertices is a clique.", "Let $a$ and $b$ be almost $C$ -dominating vertices and towards a contradiction suppose that $a$ and $b$ are not adjacent.", "Let $P_1, P_2, P_3$ be subpaths of $C$ such that each $P_i$ has length 2 and $\\operatorname{dist}_C(P_i, P_j)\\ge 2$ for distinct $i,j\\in [3]$ .", "We can choose such paths because $\\mu _k\\ge 11$ and $\\vert V(C)\\vert >\\mu _k$ .", "By the definition of an almost $C$ -dominating vertex, $a$ has a neighbor in each of $P_1, P_2, P_3$ , and $b$ has a neighbor in each of $P_1, P_2, P_3$ .", "If $a$ and $b$ have common neighbors in two of the $P_i$ , then $G$ has a hole of length 4, a contradiction.", "Therefore, by symmetry, we may assume that $a$ and $b$ have no common neighbors in $V(P_1 \\cup P_2)$ .", "For $i\\in [2]$ , let $P_i^{ab}$ be a shortest $(a,b)$ -path in $G[\\lbrace a,b\\rbrace \\cup V(P_i)]$ .", "Then $C^{\\prime }:=P_1^{ab} \\cup P_2^{ab}$ is a hole of length at least 6 and at most $8 < \\mu _k < \\vert V(C)\\vert $ , which contradicts the assumption that $C$ is a shortest long hole.", "For the remainder of the paper, we let $D$ be the set of almost $C$ -dominating vertices of $G$ .", "Since every hole of $G$ contains at most two vertices from each clique of $G$ , we have the following corollary.", "Corollary 4.3 Every hole of $G$ contains at most two vertices of $D$ ." ], [ "Cycle-distance lemma", "We now give a lower bound on the length of a path in $G-D$ such that its ends are on $C$ and are far apart in $C$ .", "Lemmas REF and REF for $C$ -paths are essentially shown in [10], with a similar assumption.", "We extend the argument for $C$ -paths to all paths in $G-D$ in Lemma REF .", "This lemma will be used repeatedly throughout the paper.", "Lemma 4.4 If $x$ and $y$ are vertices of $C$ such that $\\operatorname{dist}_C(x,y)\\ge 4$ , then every $(x,y)$ -path in $G-D$ has length at least 4.", "Towards a contradiction, suppose there is an $(x,y)$ -path $R$ in $G-D$ of length at most 3.", "Let $P$ and $Q$ be the two $(x,y)$ -paths in $C$ such that $\\vert V(P)\\vert \\le \\vert V(Q)\\vert $ .", "Since $\\mu _k\\ge 16$ , $Q$ has length at least 8.", "Since $C$ is a hole and $\\operatorname{dist}_C(x,y)\\ge 4$ , $R$ has an internal vertex contained in $V(G)\\setminus V(C)$ .", "By Lemma REF , every vertex of $V(R)\\setminus V(C)$ has at most 3 neighbors on $C$ and these neighbors are consecutive along $C$ .", "So, $R$ has length 3.", "If one of the internal vertices of $R$ is in $C$ , then the other internal vertex has two neighbors in $C$ that have distance at least 3 in $C$ , contradicting Lemma REF .", "Therefore, every internal vertex of $R$ is not contained in $C$ .", "Now, $G[V(Q)\\cup V(R)]$ contains a hole $C^{\\prime }$ of length at least $8-4+3=7$ .", "Since $P$ contains at least three internal vertices while $R$ contains at most two internal vertices, $C^{\\prime }$ is shorter than $C$ , a contradiction.", "Lemma 4.5 Let $m \\in \\mathbb {N}$ , and $P$ be a $C$ -path in $G-D$ with ends $x$ and $y$ .", "If $\\operatorname{dist}_C(x,y)\\ge 4m$ , then $P$ has length at least $m+3$ .", "We proceed by induction on $m$ .", "The base case $m=1$ follows from Lemma REF .", "Fix $m\\ge 2$ and towards a contradiction let $P:=p_1p_2 \\cdots p_n$ be a $C$ -path of length at most $m+2$ in $G-D$ such that $\\operatorname{dist}_C(p_1,p_n)\\ge 4m$ .", "We may clearly assume that $P$ is a shortest such path.", "In particular, $P$ is an induced path.", "Note that $n-1\\le m+2$ and thus, $m\\ge n-3$ .", "Since $\\operatorname{dist}_C(x,y)\\ge 4m\\ge 4$ , Lemma REF implies that $P$ has length at least 4.", "We distinguish cases depending on whether $\\lbrace p_j : j\\in [n-2]\\setminus [2]\\rbrace $ contains a vertex in $N_G(C)$ or not.", "Case 1.", "$\\lbrace p_j : j\\in [n-2]\\setminus [2]\\rbrace $ contains a vertex in $N_G(C)$ .", "We choose an integer $i\\in [n-2]\\setminus [2]$ such that $p_i\\in N_G(C)$ , and choose a neighbor $z$ of $p_i$ in $C$ .", "Since $p_1p_2 \\cdots p_iz$ is a $C$ -path of length $i<m$ , by the induction hypothesis, $\\operatorname{dist}_C(p_1, z)< 4(i-2)$ .", "Since $zp_ip_{i+1} \\cdots p_{n-1}p_n$ is a $C$ -path of length $n-i+1\\le m+4-i\\le m+1$ , by the induction hypothesis, we have $\\operatorname{dist}_C(z, p_n)<4(n-i+1-2)=4(n-i-1)$ .", "Therefore, we have $\\operatorname{dist}_C(p_1, p_n)\\le \\operatorname{dist}_C(p_1, z)+\\operatorname{dist}_C(z, p_n)<4(n-3)\\le 4m,$ a contradiction.", "Case 2.", "$\\lbrace p_j : j\\in [n-2]\\setminus [2]\\rbrace $ contains no vertices in $N_G(C)$ .", "Let $Q$ be a shortest path from $N_G(p_2)\\cap V(C)$ to $N_G(p_{n-1})\\cap V(C)$ in $C$ , and let $q, q^{\\prime }$ be its ends.", "Let $C^{\\prime }:= p_2Pp_{n-1}q^{\\prime }Qqp_2$ .", "By assumption, there are no edges between $\\lbrace p_j : j\\in [n-2]\\setminus [2]\\rbrace $ and $V(Q)$ .", "By the minimality of $|V(Q)|$ there are no edges between $\\lbrace p_2, p_{n-1}\\rbrace $ and $V(Q)$ .", "Finally, since $P$ and $Q$ are induced paths, it follows that $C^{\\prime }$ is a hole.", "Moreover, because neither $p_2$ nor $p_{n-1}$ are in $D$ , Lemma REF implies $\\operatorname{dist}_C(q,q^{\\prime }) \\ge \\operatorname{dist}_C(p_1, p_n)-4 \\ge 4m -4 \\ge 4$ .", "Therefore, $|V(C^{\\prime })| \\ge 6$ .", "Observe that $\\operatorname{dist}_C(p_1,p_n)\\le \\frac{\\vert V(C)\\vert }{2}$ and $m\\le \\frac{\\operatorname{dist}_C(p_1,p_n)}{4}\\le \\frac{\\vert V(C)\\vert }{8}$ .", "Thus, $P$ has length at most $m+2\\le \\frac{\\vert V(C)\\vert }{8}+2$ .", "Therefore, $ \\vert V(C^{\\prime })\\vert =\\vert V(Q)\\vert +\\vert (V(P)\\setminus \\lbrace p_1, p_n\\rbrace )\\vert \\le \\frac{\\vert V(C)\\vert }{2}+ 1 + \\frac{\\vert V(C)\\vert }{8} +1 < \\vert V(C)\\vert .$ This contradicts that $C$ is a shortest long hole of $G$ .", "This concludes the proof.", "Lemma 4.6 Let $m \\in \\mathbb {N}$ , and $P$ be a path in $G-D$ with ends $x$ and $y$ in $C$ .", "If $\\operatorname{dist}_C(x,y)\\ge 4m$ , then $P$ has length at least $m+3$ .", "We proceed by induction on $m$ .", "The base case $m=1$ follows from Lemma REF .", "Fix $m \\ge 2$ and towards a contradiction let $P=p_1p_2 \\cdots p_n$ with $p_1=x, p_n=y$ , $n \\le m+3$ , and $\\operatorname{dist}_C(x,y)\\ge 4m$ .", "We may assume that $P$ contains a vertex of $C$ as an internal vertex, otherwise $P$ is a $C$ -path, and we are done by Lemma REF .", "Let $i\\in [n-1]\\setminus [1]$ be such that $p_i\\in V(C)$ .", "By the induction hypothesis, $\\operatorname{dist}_C(p_1, p_i)< 4(i-3)$ , and $\\operatorname{dist}_C(p_i, p_n)< 4(n-i-2)$ .", "Thus, $\\operatorname{dist}_C(p_1, p_n)<4(i-3)+4(n-i-2)=4(n-5)<4m$ ." ], [ "First Greedy Packing of Long Holes", "In this section we construct a first greedy packing of long holes in $G$ .", "For a vertex $v \\in V(C)$ and a positive integer $r$ , let $\\mathcal {S}^r_v:= N^r_{G-(D \\cup (V(C)\\setminus \\lbrace v\\rbrace ))} [v]$ .", "For $X \\subseteq V(C)$ and $r \\in \\mathbb {N}$ , let $\\mathcal {S}^r_X:= \\bigcup _{v \\in X} \\mathcal {S}^r_v$ and for a subgraph $H$ of $C$ , let $\\mathcal {S}^r_H:=\\mathcal {S}^r_{V(H)}$ .", "Algorithm.", "(First greedy packing) Choose a vertex $v_{\\mathsf {init}} \\in V(C)$ to hit $C$ .", "Initialize $C_1=C-v_{\\mathsf {init}}$ , $D_1=D$ , and $M_1=\\lbrace v_{\\mathsf {init}}\\rbrace $ .", "Choose a previously unchosen pair $(A,B)$ , where $A$ is a subset of size at most 2 of $D_i$ (possibly, $A=\\emptyset $ ) and $B$ is a set of at most 60 consecutive vertices of $C$ contained in $C_i$ .", "If there are no remaining pairs, then proceed to step (6).", "Test if $G[A \\cup \\mathcal {S}^3_B]$ contains a long hole.", "If no, then proceed to step (2).", "If yes, let $H_i$ be a long hole of $G[A \\cup \\mathcal {S}^3_B]$ and set $D_{i+1}:=D_i \\setminus A$ and $C_{i+1}:=C_i - N_C^{15}[B]$ and $M_{i+1}:=M_i\\cup A\\cup N_C^{15}[B]$ , then proceed to step (1).", "Let $\\ell $ be the largest index for which $D_{\\ell }$ exists.", "Define $X_{\\ref {lem:greedypacking}}:=(M_{\\ell }\\cap D)\\cup N_C^{60}[M_{\\ell }\\cap V(C)]$ .", "Lemma 5.1 Let $\\ell $ be the largest index for which $D_\\ell $ exists in the first greedy packing algorithm.", "Then $\\lbrace H_i:i\\in [\\ell -1]\\rbrace $ is a set of vertex-disjoint long holes of $G$ and $\\vert X_{\\ref {lem:greedypacking}}\\vert \\le 212\\ell $ .", "Moreover, for all subpaths $Q$ of $C$ on at most 60 vertices, no long hole of $G-X_{\\ref {lem:greedypacking}}$ is contained in $\\mathcal {S}^3_Q\\cup D$ .", "Let $i,j$ be distinct indices in $[\\ell -1]$ and let $(A_i, B_i)$ and $(A_j, B_j)$ be the pairs considered by the algorithm when the holes $H_i$ and $H_j$ were constructed, respectively.", "By construction, $A_i\\cap A_j=\\emptyset $ and $\\operatorname{dist}_C(B_i, B_j) \\ge 16$ .", "If $\\mathcal {S}^3_{B_i}$ and $\\mathcal {S}^3_{B_j}$ have a common vertex, then there is a $C$ -path in $G-D$ of length at most 6 that starts at $B_1$ and ends at $B_2$ .", "Therefore, by Lemma REF , $\\operatorname{dist}_C(B_1, B_2)<16$ , which is a contradiction.", "Thus, $\\mathcal {S}^3_{B_1}$ and $\\mathcal {S}^3_{B_2}$ are disjoint, and so $H_i$ and $H_j$ are vertex-disjoint.", "It is straightforward to check that $\\vert M_{\\ell }\\cap D\\vert \\le 2(\\ell -1)$ , $\\vert M_{\\ell }\\cap V(C)\\vert \\le 90(\\ell -1)+1$ , and $C[M_{\\ell }\\cap V(C)]$ consists of at most $\\ell $ connected components.", "Therefore, $\\vert X_{\\ref {lem:greedypacking}}\\vert \\le 212\\ell $ .", "For the second statement, suppose that $G-X_{\\ref {lem:greedypacking}}$ has a long hole $H$ contained in $\\mathcal {S}^3_Q\\cup D$ for some subpath $Q$ of $C$ on at most 60 vertices.", "Because $G-V(C)$ has no long holes, $H$ contains a vertex in $V(C)\\setminus X_{\\ref {lem:greedypacking}}$ .", "Since $N^{60}_C[M_{\\ell }\\cap V(C)] \\subseteq X_{\\ref {lem:greedypacking}}$ , $\\operatorname{dist}_C(V(H)\\cap V(C), M_{\\ell }\\cap V(C))\\ge 61, $ and $Q$ is disjoint from $M_{\\ell }\\cap V(C)$ .", "Therefore, $Q$ is fully contained in $C_{\\ell }$ .", "By considering the pair $(V(H)\\cap D, V(Q) )$ in the algorithm, we should have proceeded one more step, which is a contradiction.", "The next lemma extends the second part of Lemma REF , by showing that for all subpaths $Q$ of $C-X_{\\ref {lem:greedypacking}}$ , there is no long hole of $G-X_{\\ref {lem:greedypacking}}$ contained in $\\mathcal {S}^3_Q$ .", "We stress that there could still be a subpath $Q$ of $C$ such that $G-X_{\\ref {lem:greedypacking}}$ has a long hole contained $\\mathcal {S}^3_Q$ .", "However, these remaining holes will be dealt with in Section .", "Figure: The paths H 1 H_1 and H 2 H_2 in Lemma .Lemma 5.2 For all subpaths $Q$ of $C-X_{\\ref {lem:greedypacking}}$ , there is no long hole of $G-X_{\\ref {lem:greedypacking}}$ contained in $\\mathcal {S}^3_Q$ .", "For a contradiction, suppose that $G-X_{\\ref {lem:greedypacking}}$ contains a long hole $H$ in $\\mathcal {S}^3_Q$ for some subpath $Q$ of $C-X_{\\ref {lem:greedypacking}}$ .", "We assume that $Q$ is a shortest such path.", "By Lemma REF , $\\vert V(Q)\\vert \\ge 61$ .", "Let $Q:=q_1q_2 \\cdots q_m$ .", "By the minimality of $|V(Q)|$ , $(\\mathcal {S}^3_{q_1}\\setminus \\mathcal {S}^3_{Q-q_1})\\cap V(H)\\ne \\emptyset \\quad \\text{and} \\quad (\\mathcal {S}^3_{q_m}\\setminus \\mathcal {S}^3_{Q-q_m})\\cap V(H)\\ne \\emptyset .$ Let $a\\in (\\mathcal {S}^3_{q_1}\\setminus \\mathcal {S}^3_{Q-q_1})\\cap V(H)$ and $b\\in (\\mathcal {S}^3_{q_m}\\setminus \\mathcal {S}^3_{Q-q_m})\\cap V(H)$ .", "Clearly, we have $a \\ne b$ .", "Let $H_1$ and $H_2$ be the two $(a,b)$ -paths in $H$ .", "See Figure REF for an illustration.", "Let $R:=q_{21}q_{22} \\cdots q_{40}$ .", "Let $R_1$ and $R_2$ be the two components of $Q-V(R)$ containing $q_1$ and $q_m$ respectively.", "If $a\\in \\mathcal {S}^3_R$ , then $G-D$ contains a $(q_1, r)$ -path of length at most 6, for some $r \\in V(R)$ .", "However, since $\\operatorname{dist}_C(q_1, V(R))\\ge 16$ , this contradicts Lemma REF .", "For the same reason, $b\\notin \\mathcal {S}^3_R$ .", "We claim that for each $i\\in [2]$ , $V(H_i)\\cap \\mathcal {S}^3_R\\ne \\emptyset $ .", "Suppose for a contradiction that $V(H_i)\\cap \\mathcal {S}^3_R= \\emptyset $ .", "Because $\\mathcal {S}^3_{R_1}\\cap V(H_i)\\ne \\emptyset $ , $\\mathcal {S}^3_{R_2}\\cap V(H_i)\\ne \\emptyset $ , and $V(H_i)\\cap \\mathcal {S}^3_R= \\emptyset $ , there must be an edge $uv$ in $H_i$ such that $u\\in \\mathcal {S}^3_{R_1}\\cap V(H_i)$ and $v\\in \\mathcal {S}^3_{R_2}\\cap V(H_i)$ .", "Thus, $G-D$ contains an $(r_1, r_2)$ -path of length at most 7, for some $r_1 \\in V(R_1)$ and $r_2 \\in V(R_2)$ .", "However, this contradicts Lemma REF , since $\\operatorname{dist}_C(R_1, R_2)\\ge 20$ .", "For each $i\\in [2]$ , let $x_i$ be the vertex in $\\mathcal {S}^3_R\\cap V(H_i)$ such that $\\operatorname{dist}_{H_i}(a, x_i)$ is minimum.", "Let $H^{\\prime }:=x_1H_1aH_2x_2$ .", "If $\\operatorname{dist}_{H_i}(a, x_i)\\le 1$ , then there is a $(q_1, r)$ -path of length at most 7 in $G-D$ , for some $r \\in V(R)$ .", "Since $\\operatorname{dist}_C(q_1, R)\\ge 20$ , this contradicts Lemma REF .", "Thus, $H^{\\prime }$ has length at least 4.", "Let $P_i$ be a path of length at most 3 from $x_i$ to $R$ in $\\mathcal {S}^3_R$ for each $i\\in [2]$ .", "Observe that there is no edge between $(P_1-x_1)\\cup (P_2-x_2)\\cup R$ and $H^{\\prime }-\\lbrace x_1, x_2\\rbrace $ , because of the choice of $x_1$ and $x_2$ .", "Therefore, if we let $P$ be a shortest $(x_1, x_2)$ -path in $G[V(P_1)\\cup V(P_2)\\cup V(R)]$ , then $H^{\\prime } \\cup P$ is a long hole of $G-X_{\\ref {lem:greedypacking}}$ .", "However, $H^{\\prime } \\cup P$ is contained in $\\mathcal {S}^3_{q_1q_2 \\cdots q_{40}}$ , contradicting Lemma REF ." ], [ "Sparse Ear Decompositions", "A sparse ear decomposition in $G$ is an ear decomposition $C \\cup P_1 \\cup \\dots \\cup P_\\ell $ of a subgraph $ of $ G$ satisfying the following conditions.$ $C$ is the cycle of $G$ given by Theorem REF , $ has maximum degree $ 3$, and\\item for each $ i []$, $ Pi$ has an end $ p0$ in $ C$ such that $ Pi:=p0p1 pm$ with $ m 6$, $ Pi - {p0, pm}$ is an induced path in $ G$, andno vertex in $ {p2,p3,p4}$ has a neighbor in $ CP1Pi-1$.$ A vertex of degree 3 in a sparse ear decomposition $ is called a \\emph {branching point} of $ .", "Let $C \\cup P_1 \\cup \\dots \\cup P_\\ell $ be a sparse ear decomposition in $G$ .", "An $-path $ P$ \\emph {extends} $ if $C \\cup P_1 \\cup \\dots \\cup P_\\ell \\cup P$ is a sparse ear decomposition.", "We say that $ is \\emph {maximal} if there does not exist an $ -path extending $.We say that $ is good if for every $i \\in [\\ell -1]$ , $P_i$ extends $C\\cup P_1\\cup \\cdots \\cup P_{i-1}$ , subject to (1), $\\vert V(P_i)\\cap V(C)\\vert $ is maximum, and subject to (1) and (2), $P_i$ is shortest.", "Lemma 6.1 A maximal good sparse ear decomposition $ in $ G$ can be constructed in polynomial time.$ Suppose we are given a good sparse ear decomposition $C \\cup P_1 \\cup \\dots \\cup P_\\ell $ .", "We claim that in polynomial time, we can find a path $P$ such that $P$ extends $ and $ C P1 ...PP$ is good, or determine that no such path $ P$ exists.To do so, for all $ 6$-tuples of vertices $ (p0, p1, p2, p3, p4, pm)$ of $ G$ we define a potential candidate $ P(p0, p1, p2,p3,p4, pm)$ for $ P$ by first testing if:\\begin{itemize}\\item p_0\\in V(C), p_m\\in V(\\setminus \\lbrace p_0\\rbrace , and p_0 and p_m are not branching points of ,\\item p_0p_1\\in E(G), p_1p_2p_3p_4 is an induced path in G-V(, and \\lbrace p_2,p_3,p_4\\rbrace \\cap N_G[=\\emptyset .\\end{itemize}$ If the above conditions fail, $P(p_0, p_1, p_2,p_3,p_4, p_m)$ is undefined.", "If the above conditions hold, but there is no path from $p_4$ to $p_m$ in $G^{\\prime }:=G-((V(\\cup N_G[\\lbrace p_1, p_2, p_3\\rbrace ]) \\setminus \\lbrace p_m\\rbrace ),$ then $P(p_0, p_1, p_2,p_3,p_4, p_m)$ is undefined.", "Otherwise, we choose a shortest path $Q$ from $p_4$ to $p_m$ in $G^{\\prime }$ , and define $P(p_0, p_1, p_2,p_3,p_4, p_m):=p_0p_1p_2p_3p_4Q$ .", "If $P(p_0, p_1, p_2,p_3,p_4, p_m)$ is undefined for all ($p_0, p_1, p_2,p_3,p_4, p_m)$ , then no such $P$ exists.", "Otherwise, we choose $P$ to be the shortest path among all $P(p_0, p_1, p_2,p_3,p_4, p_m)$ which contain the maximum number of vertices of $C$ .", "To complete the proof of the lemma we begin with the good sparse ear decomposition $C$ and repeatedly apply the above claim to obtain a set of paths $P_1, \\dots , P_\\ell $ such that for all $i \\in [\\ell ]$ , $C \\cup P_1 \\cup \\dots \\cup P_i$ is a good sparse ear decomposition, $P_i$ extends $C \\cup P_1 \\cup \\dots \\cup P_{i-1}$ , and there is no path $P$ such that $P$ extends $C \\cup P_1 \\cup \\dots \\cup P_{\\ell }$ and $C \\cup P_1 \\cup \\dots \\cup P_{\\ell } \\cup P$ is good.", "Then $C \\cup P_1 \\cup \\dots \\cup P_{\\ell }$ is a maximal good sparse ear decomposition of $G$ , and can clearly be found in polynomial time.", "For the remainder of the paper, we fix a maximal good sparse ear decomposition $C\\cup P_1\\cup \\cdots \\cup P_\\ell .$ Lemma 6.2 If $F$ is a cycle of $, then $ G[V(F)]$ contains a long hole.Moreover, if $ has at least $s_k$ branching points, then we can output $k$ vertex-disjoint long holes of $G$ in polynomial time.", "For the first part, we may assume that $F$ is a cycle of $ other than $ C$, because $ C$ is a long hole.Since $ F$ is a cycle of $ other than $C$ , $F$ contains an internal vertex of $P_i$ for some $i\\in [\\ell ]$ .", "Let $i$ be the maximum integer in $[\\ell ]$ such that $F$ contains an internal vertex of $P_i$ .", "Observe that if $P_j$ has an end which is an internal vertex of $P_i$ , then $j>i$ .", "Therefore, by the choice of $i$ , $V(P_i) \\subseteq V(F)$ .", "Let $P_i=p_0p_1 \\cdots p_m$ be such that $p_0 \\in V(C)$ and no vertex in $\\lbrace p_2, p_3, p_4\\rbrace $ has a neighbor in $C\\cup P_1\\cup \\cdots \\cup P_{i-1}$ .", "Then $p_1p_2p_3p_4p_5$ is an induced path and no vertex in $\\lbrace p_2, p_3, p_4\\rbrace $ has a neighbor in $V(F) \\setminus \\lbrace p_j: j\\in [5]\\rbrace $ .", "So, by Lemma REF , $G[V(F)]$ contains a long hole.", "For the second part, assume that $ contains $ sk$ branching points.By Theorem~\\ref {thm:simonovitz},we can find a set $ {Fi: i[k]}$ of $ k$ vertex-disjoint cycles of $ in polynomial time.", "By the first part, $G[V(F_j)]$ contains a long hole $H_j$ for all $j \\in [k]$ .", "Finally, by Lemma REF , we can find long holes in $\\lbrace H_i: i\\in [k]\\rbrace $ in polynomial time.", "Let $B$ be the set of branching points of $, and define$$X_{\\ref {lem:sparsealgo}}:=N^{31}_C[B\\cap V(C)] \\cup (B\\setminus V(C)).$$$ By Lemma REF , we may assume that $\\vert B\\vert < s_{k}$ .", "We now prove a special property of $ to beused in Section~\\ref {sec:avoiding}.$ Lemma 6.3 For each $j \\in [\\ell ]$ , let $j=C \\cup P_1 \\cup \\cdots \\cup P_j$ .", "Let $v\\in V(C)\\setminus X_{\\ref {lem:sparsealgo}}$ , $u$ be an internal vertex of $P_j$ for some $j\\in [\\ell ]$ , and $R=v_0v_1 \\cdots v_s$ be an $j$ -path with $v_0=v$ and $v_s=u$ .", "Then $s\\ge 6$ .", "Let $R=v_0v_1 \\cdots v_s$ be a counterexample with $s$ minimum.", "By the minimality of $s$ , if $s\\ge 2$ , then there is no edge between $\\lbrace v_i:i\\in \\lbrace 0\\rbrace \\cup [s-2]\\rbrace $ and the set of internal vertices of $P_j$ .", "Let $P_j=u_0u_1 \\cdots u_p$ , where $u_0 \\in V(C)$ and no vertex in $\\lbrace u_2, u_3, u_4\\rbrace $ has a neighbor in $C\\cup P_1\\cup \\cdots \\cup P_{j-1}$ .", "Since we added the set $N^{31}_C[B\\cap V(C)]$ to $X_{\\ref {lem:sparsealgo}}$ , we have $\\operatorname{dist}_C(v_0, B\\cap V(C))\\ge 32$ .", "Let $t$ be the minimum integer such that $v_{s-1}$ is adjacent to $u_{t}$ .", "First assume that $v_{s-1}$ has a neighbor in $\\lbrace u_1, u_2, u_3, u_4\\rbrace $ .", "Then there is a $C$ -path from $v_0$ to $u_0$ of length at most $(s-1)+5\\le 9$ .", "By Lemma REF , $\\operatorname{dist}_C(v_0, u_0)<28$ , which contradicts $\\operatorname{dist}_C(v_0, B\\cap V(C))\\ge 32$ .", "So, $v_{s-1}$ has no neighbor in $\\lbrace u_1, u_2, u_3, u_4\\rbrace $ .", "Therefore $Q:=u_0P_ju_tv_{s-1}Rv_0$ is a path extending ${j-1}$ with $|V(Q) \\cap V(C)|=2$ .", "Since $ is good, this implies $ up V(C)$.$ If $p-t>s$ , then $Q$ is shorter than $P_j$ , a contradiction.", "So, $p-t\\le s$ , and thus, there is a $C$ -path of length at most $s+(p-t)\\le 2s\\le 10$ from $v_0$ to $u_p$ .", "By Lemma REF , $\\operatorname{dist}_C(v_0, u_p)<32$ , which contradicts $\\operatorname{dist}_C(v_0, B\\cap V(C))\\ge 32$ .", "We conclude that $s\\ge 6$ , as required.", "Before we proceed further, we deal with cycles with some additional constraint similar to extensions for $, that intersect $ C$ on exactly one vertex.A cycle $ F=v0v1v2 vnv0$ with $ n5$ is an \\emph {appendage} of $ if $v_0\\in V(C)\\setminus B$ and $V(F)\\cap V(=\\lbrace v_0\\rbrace $ , $v_1v_2 \\cdots v_n$ is an induced path of $G$ , and no vertex in $\\lbrace v_2,v_3,v_4\\rbrace $ has a neighbor in $.$ The vertex $v_0$ is called the tip of this appendage.", "Clearly, an appendage contains a long hole, because $v_1v_2v_3v_4v_5$ is an induced path and there are no edges between $\\lbrace v_2, v_3, v_4\\rbrace $ and $V(F)\\setminus \\lbrace v_i: i\\in [5]\\rbrace $ .", "We now show that if there are two appendages of $ whose tips have distance at least $ 20$ in $ C$,then they are vertex-disjoint.$ Lemma 6.4 Let $F_1$ and $F_2$ be appendages of $ with tips $ x1$ and $ x2$, respectively.If $ distC(x1, x2)20$, then $ V(F1)V(F2)=$.$ Suppose $\\operatorname{dist}_C(x_1, x_2)\\ge 20$ .", "Let $F_1=v_0v_1v_2 \\cdots v_{m_1}v_0$ and $F_2=w_0w_1w_2 \\cdots w_{m_2}w_0$ with $x_1=v_0$ and $x_2=w_0$ such that no vertex in $\\lbrace v_2, v_3, v_4\\rbrace $ has a neighbor in $V(F_1)\\setminus \\lbrace v_2, v_3, v_4\\rbrace $ , and no vertex in $\\lbrace w_2, w_3, w_4\\rbrace $ has a neighbor in $V(F_2)\\setminus \\lbrace w_2, w_3, w_4\\rbrace $ .", "Towards a contradiction, suppose that $V(F_1)\\cap V(F_2)\\ne \\emptyset $ .", "We choose a minimum integer $i$ such that $v_i$ has a neighbor in $F_2$ .", "Let $j$ be the minimum integer such that $v_i$ is adjacent to $w_j$ .", "Then $X=v_0v_1 \\cdots v_iw_jw_{j-1} \\cdots w_0$ is a $C$ -path of length $i+j+1$ , and $X-\\lbrace v_0, w_0\\rbrace $ is an induced path.", "Since $\\operatorname{dist}_C(x_1, x_2)\\ge 20$ , by Lemma REF , we have $i+j+1\\ge 8$ , and $i$ or $j$ must be at least 4.", "So, $X$ includes one of $\\lbrace v_2, v_3, v_4\\rbrace $ or $\\lbrace w_2, w_3, w_4\\rbrace $ , which implies that $X$ is an $-path extending $ .", "This contradicts the maximality of $.$ Using Lemma REF , we can find either $k$ vertex-disjoint appendages or a set hitting all appendages.", "Lemma 6.5 In polynomial time, we can find either $k$ vertex-disjoint appendages of $ or a vertex set $ XREF $ of size at most $ 20k$ hitting all appendages.$ Initialize $S=\\emptyset $ .", "For each $v\\in V(C)\\setminus B$ , we test whether there is an appendage with tip $v$ .", "For this, we guess a path $p_mp_0p_1 \\cdots p_4$ in $G-(V(\\setminus \\lbrace v\\rbrace )$ such that $p_m$ has no neighbor in $\\lbrace p_1, \\ldots , p_4\\rbrace $ , $p_1p_2p_3p_4$ is induced, and every vertex in $\\lbrace p_2, p_3, p_4\\rbrace $ has no neighbor in $.For such a path,we test whether there is a path from $ pm$ to $ p4$ in$$G- ((V(\\cup N_G[\\lbrace p_1, p_2, p_3\\rbrace ]) \\setminus \\lbrace p_4\\rbrace ).$$If there is a path, then by finding a shortest path,we can output an appendage with tip $ v$.Also, in that case, we add $ v$ to $ S$.$ Let $X_{\\ref {lem:narrowalgo}}$ be the final set $S$ .", "By construction, $X_{\\ref {lem:narrowalgo}}$ hits all appendages.", "Therefore, we may assume $\\vert X_{\\ref {lem:narrowalgo}}\\vert > 20k$ .", "Since $\\vert X_{\\ref {lem:narrowalgo}}\\vert > 20k$ , there exists a subset $N$ of $X_{\\ref {lem:narrowalgo}}$ such that $|N|=k$ and $\\operatorname{dist}_C(x_1,x_2) \\ge 20$ for all distinct $x_1,x_2 \\in N$ .", "Let $\\mathcal {F}$ be the set of appendages output by the algorithm with a tip in $N$ .", "By Lemma REF , the appendages in $\\mathcal {F}$ are vertex-disjoint, so we can output $k$ vertex-disjoint appendages, as required.", "Hitting long holes In the previous sections, we have defined $X_{\\ref {lem:greedypacking}}, X_{\\ref {lem:sparsealgo}}$ , and $X_{\\ref {lem:narrowalgo}}$ .", "Let $X_{\\ref {lem:tunnellemma4}}:=X_{\\ref {lem:greedypacking}} \\cup X_{\\ref {lem:sparsealgo}}\\cup X_{\\ref {lem:narrowalgo}}.$ In this section, we complete the proof of Theorem REF by either finding $k$ vertex-disjoint long holes of $G$ or sets $X_{\\ref {prop:Davoid}}$ and $X_{\\ref {prop:traversing}}$ such that $G- (X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long holes and $|X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}|=\\mathcal {O}(k \\log k)$ .", "We divide long holes into two types, depending on whether a long hole intersects $D$ or not.", "We say that a long hole is $D$ -avoiding if it contains no vertex of $D$ , and $D$ -traversing, otherwise.", "We first consider $D$ -avoiding holes in Subsection REF , and consider $D$ -traversing holes in Subsection REF .", "Before doing so, we require a few more definitions related to the cycle $C$ .", "For these definitions, we regard $C:=a_0a_1a_2 \\cdots a_ma_0$ , as a directed cycle where $a_0$ is directed towards $a_1$ .", "For a subpath $Q$ of $C$ , let $\\mathsf {bd}_L(Q)$ and $\\mathsf {bd}_R(Q)$ be the ends of $Q$ such that $Q$ is directed from $\\mathsf {bd}_L(Q)$ to $\\mathsf {bd}_R(Q)$ .", "For an integer $i\\ge 2$ , let $\\mathsf {bd}^i_L(Q):=N^{i-1}_Q[ \\mathsf {bd}_L(Q)]$ , $\\mathsf {bd}^i_R(Q):=N^{i-1}_Q[ \\mathsf {bd}_R(Q)]$ , $\\mathsf {bd}^i(Q):=\\mathsf {bd}^i_L(Q)\\cup \\mathsf {bd}^i_R(Q)$ and $\\mathsf {int}^i(Q):=V(Q)\\setminus \\mathsf {bd}^i(Q)$ .", "The following two lemmas will be useful to find a long hole.", "Lemma 7.1 Let $Q$ be a path on more than 160 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ , and let $Q^*$ be a subpath of $Q$ on 20 vertices.", "If $P$ is a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ contained in $G[\\mathcal {S}^{3}_{Q}]$ , then $P$ intersects $\\mathcal {S}^3_{Q^*}\\setminus \\mathcal {S}^3_{Q-V(Q^*)}$ .", "By assumption, $P$ intersects $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ and $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ .", "Thus, we may assume that $Q-V(Q^*)$ has two distinct connected components.", "Let $Q_1$ and $Q_2$ be the connected components of $Q-V(Q^*)$ , where $\\mathsf {bd}_L(Q)\\in V(Q_1)$ .", "Let $p_1$ and $p_2$ be the ends of $P$ such that $p_1\\in \\mathcal {S}^3_{\\mathsf {bd}^{20}_L(Q)}$ .", "Suppose for a contradiction that $P$ does not contain a vertex of $\\mathcal {S}^3_{Q^*}\\setminus \\mathcal {S}^3_{Q-V(Q^*)}$ .", "Observe that $p_1\\in \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ and $p_2\\in \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ , and by Lemma REF , $p_1\\notin \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ and $p_2\\notin \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ .", "Thus, there is an edge $uv$ of $P$ where $u\\in \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ and $v\\in \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ .", "But since $\\operatorname{dist}_C(V(Q_1), V(Q_2))\\ge 20$ , this is not possible by Lemma REF .", "Lemma 7.2 Let $Q$ be a path on 60 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ .", "If $P_1$ and $P_2$ are two vertex-disjoint paths from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ contained in $G[\\mathcal {S}^{3}_{Q}]-V(Q)$ such that there are no edges between $P_1$ and $P_2$ , then $\\mathcal {S}^3_{Q}$ contains a long hole.", "Let $A_1:=\\mathsf {bd}^{20}_L(Q)$ and $A_2:=\\mathsf {bd}^{20}_R(Q)$ .", "We may assume that for each $i, j\\in [2]$ , $\\vert V(P_i)\\cap \\mathcal {S}^3_{A_j}\\vert =1$ .", "For $i, j\\in [2]$ , let $p_{i,j}$ be the end of $P_i$ contained in $\\mathcal {S}^3_{A_j}$ .", "By Lemma REF , $p_{i,1}\\ne p_{i,2}$ and $P_i$ has length at least 2.", "For each $i, j\\in [2]$ , let $P_{i,j}$ be a path of length at most 3 in $\\mathcal {S}^3_{A_j}$ from $p_{i,j}$ to $A_j$ .", "For each $j\\in [2]$ , let $U_j$ be a shortest path from $p_{1,j}$ to $p_{2,j}$ in $G[V(P_{1,j}) \\cup V(P_{2,j})\\cup A_j]$ .", "By the choice of $P_1$ and $P_2$ , no vertex of $(P_1-p_{1,1})\\cup (P_2-p_{2,1})$ is contained in $\\mathcal {S}^3_{A_1}$ .", "This implies that no internal vertex of $U_1$ has a neighbor in $(P_1-p_{1,1})\\cup (P_2-p_{2,1})$ .", "Similarly, no internal vertex of $U_2$ has a neighbor in $(P_1-p_{1,2})\\cup (P_2-p_{2,2})$ .", "Also, by Lemma REF , there is no edge between $U_1$ and $U_2$ .", "Since $P_1$ and $P_2$ have length at least 2, $U_1\\cup U_2\\cup P_1\\cup P_2$ is a long hole contained in $\\mathcal {S}^3_{Q}$ .", "$D$ -avoiding long holes In this section, we show that by taking at most 380 vertices in each connected component of $C-X_{\\ref {lem:tunnellemma4}}$ , we can hit all $D$ -avoiding long holes.", "We begin with the following structural property.", "Lemma 7.3 The graph $G-(X_{\\ref {lem:tunnellemma4}} \\cup D)$ has no induced path $P:=p_1p_2 \\cdots p_\\ell $ of length at least 4 such that $p_1\\in \\mathcal {S}^2_q$ for some $q\\in V(C)\\setminus X_{\\ref {lem:tunnellemma4}}$ , $p_2,p_3,p_4\\notin \\mathcal {S}^2_C$ , $\\lbrace p_i:i\\in [\\ell -1]\\rbrace \\cap V(=\\emptyset $ , and $p_\\ell \\in V($ .", "Suppose for a contradiction that such an induced path $P$ exists.", "Since $p_1\\in \\mathcal {S}^2_q$ and $p_2\\notin \\mathcal {S}^2_C$ , $p_1$ has no neighbor in $C$ .", "Choose a vertex $q^{\\prime }$ in $\\mathcal {S}^2_q$ such that $qq^{\\prime }p_1$ is a path.", "We claim that no vertex in $\\lbrace p_2,p_3,p_4\\rbrace $ has a neighbor in $.Suppose not.", "Since $ p2,p3,p4$ are not in $ S1C$, one of $ {p2,p3,p4}$ has a neighbor in $ V(V(C)$.Thus, there is an $ -path from $q$ to a vertex in $V(\\setminus V(C)$ that has length at most 5, which contradicts Lemma REF .", "We observe that $R=qq^{\\prime }p_1Pp_\\ell $ is either an extension or an appendage of $.Assume that $ q'$ has a neighbor in $ P-{p1, p}$.Choose a neighbor $ z$ in $ P-{p1, p}$ such that $ distP(p1, z)$ is minimum.As $ p2,p3,p4S2C$, $ q'$ has no neighbor in $ {p2, p3, p4}$.Thus, $ distP(p1, z)4$.", "This implies that the cycle $ q'p1Pzq'$ is a long hole in $ G-V(C)$, a contradiction.Therefore, $ q'$ has no neighbor in $ P-{p1, p}$ and in particular, $ R-{q, p}$ is induced.$ As no vertex in $\\lbrace p_2, p_3, p_4\\rbrace $ has a neighbor in $,we conclude that $ R$ is an extension or an appendage of $ .", "This contradicts that $ is maximal and $ XREF $ hits all appendages.$ Figure: The setting in Lemma .", "We show that the ends of PP have to be contained in distinct sets of 𝒮 L 1 3 \\mathcal {S}^{3}_{L_1} and 𝒮 R 1 3 \\mathcal {S}^{3}_{R_1}.Lemma 7.4 Let $Q$ be a path on more than 160 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ .", "Let $H$ be a $D$ -avoiding long hole in $G-X_{\\ref {lem:tunnellemma4}}$ such that $H$ contains no vertices in $\\mathsf {bd}^{80}(Q)$ , and it contains a vertex $v$ in $\\mathsf {int}^{80}(Q)$ .", "Then the connected component of $H\\cap G[\\mathcal {S}^{3}_{\\mathsf {int}^{20}(Q)}]$ containing $v$ is a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{40}_L(Q)\\setminus \\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{40}_R(Q)\\setminus \\mathsf {bd}^{20}_R(Q)}$ .", "Because $X_{\\ref {lem:greedypacking}}\\subseteq X_{\\ref {lem:tunnellemma4}}$ , by Lemma REF , $H$ is not contained in $\\mathcal {S}^3_Q$ .", "Let $P$ be the connected component of $H\\cap G[\\mathcal {S}^{3}_{\\mathsf {int}^{20}(Q)}]$ containing $v$ .", "Then $P$ is a path.", "Let $\\overline{Q}$ be the other subpath of $C$ with the same ends as $Q$ .", "For convenience, let $L_1:=\\mathsf {bd}^{40}_L(Q)\\setminus \\mathsf {bd}^{20}_L(Q)$ , $L_2:=\\mathsf {bd}^{80}_L(Q)\\setminus \\mathsf {bd}^{60}_L(Q)$ , $R_1:=\\mathsf {bd}^{40}_R(Q)\\setminus \\mathsf {bd}^{20}_R(Q)$ , $R_2:=\\mathsf {bd}^{80}_R(Q)\\setminus \\mathsf {bd}^{60}_R(Q)$ , and $M:=\\mathsf {int}^{20}(Q)$ .", "See Figure REF .", "We claim that both ends of $P$ are contained in $\\mathcal {S}^3_{L_1\\cup R_1}$ .", "Suppose the contrary, and let $w$ be an end of $P$ such that $w\\in \\mathcal {S}^3_M\\setminus \\mathcal {S}^3_{L_1\\cup R_1}$ .", "Let $w^{\\prime }$ be a neighbor of $w$ in $H$ that is not in $P$ .", "First assume that $w^{\\prime }\\in \\mathcal {S}^3_{C}$ .", "Since $w^{\\prime }\\notin \\mathcal {S}^3_M$ , $w^{\\prime }$ is contained in $\\mathcal {S}^3_{\\overline{Q}}\\cup \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Therefore, there is a $C$ -path of length at most 7 in $G-D$ with one end in $V(\\overline{Q})\\cup \\mathsf {bd}^{20}(Q)$ and the other end in $\\mathsf {int}^{40}(Q)$ .", "However, since $\\operatorname{dist}_C( V(\\overline{Q})\\cup \\mathsf {bd}^{20}(Q), \\mathsf {int}^{40}(Q))\\ge 20$ , we contradict Lemma REF .", "Thus, we may assume that $w^{\\prime }\\notin \\mathcal {S}^3_{C}$ .", "Since $w^{\\prime } \\notin \\mathcal {S}^3_{C}$ and $w \\in \\mathcal {S}^3_{C}$ , we have $w\\in \\mathcal {S}^3_{C}\\setminus \\mathcal {S}^2_{C}$ .", "Following the direction from $w$ to $v$ in $P$ , let $a$ be the first vertex so that $a\\in \\mathcal {S}^2_{C}$ .", "Since $a$ is in $\\mathcal {S}^3_M$ , again by Lemma REF , $a \\notin \\mathcal {S}^3_{\\overline{Q}}$ .", "Thus, $a\\in \\mathcal {S}^2_q$ for some $q\\in V(Q)$ .", "Observe that $q\\notin X_{\\ref {lem:tunnellemma4}}$ .", "Let $w^{\\prime \\prime }$ be the neighbor of $w^{\\prime }$ in $C$ that is not $w$ .", "Now, in the path $aPww^{\\prime }w^{\\prime \\prime }$ , $a\\in \\mathcal {S}^2_q$ while the next three vertices are not contained in $\\mathcal {S}^2_{C}$ .", "By Lemma REF , no vertex in $aPww^{\\prime }w^{\\prime \\prime }$ is contained in $.", "Following the direction from $ w$ to $ w'$ in $ H$, let $ u$ be the first vertex contained in $ .", "Then the subpath of $H$ from $a$ to $u$ containing $w$ satisfies the conditions of Lemma REF , which yields a contradiction.", "So, we conclude that both ends of $P$ are contained in $\\mathcal {S}^3_{L_1\\cup R_1}$ .", "Figure: The construction of a long hole in Lemma  when the two ends of PP are contained in 𝒮 L 1 3 \\mathcal {S}^3_{L_1}.To conclude the lemma, we claim that one end of $P$ is in $\\mathcal {S}^3_{L_1}$ and the other end is in $\\mathcal {S}^3_{R_1}$ .", "Suppose not.", "By symmetry, we may assume that both ends of $P$ are in $\\mathcal {S}^3_{L_1}$ .", "See Figure REF for an illustration.", "Let $P_1$ and $P_2$ be the two subpaths of $P$ from $v$ to the ends of $P$ .", "Applying Lemma REF to $P_1$ and $P_2$ , we deduce that for each $i\\in [2]$ , $P_i$ contains a vertex of $\\mathcal {S}^3_{L_2}$ .", "For each $i\\in [2]$ , let $Q_i$ be a shortest subpath of $P_i$ from $\\mathcal {S}^3_{L_1}$ to $\\mathcal {S}^3_{L_2}$ respectively.", "By Lemma REF , all the internal vertices of $Q_i$ are contained in $\\mathcal {S}^3_{\\mathsf {bd}^{60}_L(Q) \\setminus \\mathsf {bd}^{40}_L(Q)}$ , and $Q_i$ is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{80}_L(Q) \\setminus \\mathsf {bd}^{20}_L(Q)}$ .", "By Lemma REF , $G\\left[\\mathcal {S}^3_{\\mathsf {bd}^{80}_L(Q) \\setminus \\mathsf {bd}^{20}_L(Q)}\\right]$ contains a long hole.", "This contradicts Lemma REF .", "Proposition 7.5 There is a polynomial-time algorithm to find a vertex set $X_{\\ref {prop:Davoid}}\\subseteq V(G)\\setminus X_{\\ref {lem:tunnellemma4}}$ of size at most $380 \\vert V(C)\\cap X_{\\ref {lem:tunnellemma4}}\\vert $ such that $X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}$ hits all $D$ -avoiding long holes.", "Recall that $C=a_0a_1a_2 \\cdots a_ma_0$ and consider it as a directed cycle where $a_0$ is directed to $a_1$ .", "For a subpath $Q$ of $C$ and two vertices $a_i$ and $a_j$ in $Q$ , we denote by $a_i\\preccurlyeq _Q a_j$ if there is a directed path from $a_i$ to $a_j$ .", "We construct a set $S$ as follows.", "We iterate the following over all components $Q$ of $C-X_{\\ref {lem:tunnellemma4}}$ .", "For each $i\\in [7]$ , let $L_i:=\\mathsf {bd}^{20(i+1)}_L(Q)\\setminus \\mathsf {bd}^{20i}_L(Q)$ , $R_i:=\\mathsf {bd}^{20(i+1)}_R(Q)\\setminus \\mathsf {bd}^{20i}_R(Q)$ , and $M:=\\mathsf {int}^{20}(Q)$ .", "If $Q$ has at most 320 vertices, then we add all vertices of $Q$ to $S$ .", "Assume that $Q$ has more than 320 vertices.", "We first add $\\mathsf {bd}^{160}(Q)$ to $S$ .", "Let $F_L$ be the connected component of $G[\\mathcal {S}^3_{M}]-M$ intersecting both $\\mathcal {S}^3_{L_1}$ and $\\mathcal {S}^3_{L_3}$ , if one exists.", "Similarly, let $F_R$ be the connected component of $G[\\mathcal {S}^3_M]-M$ intersecting both $\\mathcal {S}^3_{R_1}$ and $\\mathcal {S}^3_{R_3}$ , if one exists.", "If $F_L$ or $F_R$ does not exist, then we skip this component $Q$ .", "(If there are two components intersecting both $\\mathcal {S}^3_{L_1}$ and $\\mathcal {S}^3_{L_2}$ , then there is a long hole contained in $\\mathcal {S}^3_{L_1\\cup L_2\\cup L_3}$ by Lemma REF , which contradicts Lemma REF .", "Thus, if such a component exists, then the component uniquely exists, and the same argument holds for $\\mathcal {S}^3_{R_1}$ and $\\mathcal {S}^3_{R_3}$ .)", "Now, assume that both $F_L$ and $F_R$ exist.", "If $F_L=F_R$ , then we skip this component $Q$ .", "If $F_L\\ne F_R$ , then let $q$ be the vertex of $Q$ where $\\mathcal {S}^3_{q}\\cap V(F_L)\\ne \\emptyset $ and it is largest with respect to the $\\preccurlyeq _Q$ relation.", "We add the set $\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ to $S$ .", "Let $X_{\\ref {prop:Davoid}}$ be the final set $S$ .", "Clearly, $\\vert X_{\\ref {prop:Davoid}}\\vert \\le 380\\vert V(C)\\cap X_{\\ref {lem:tunnellemma4}}\\vert $ and it can be computed in polynomial time.", "It remains to show that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -avoiding long hole.", "Suppose that there is a $D$ -avoiding long hole $H$ in $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ .", "We choose such a hole with minimum $\\vert V(H)\\cap V(C)\\vert $ .", "Since $G-V(C)$ has no long hole, $H$ must contain a vertex of $C$ .", "Let $Q$ be a connected component of $C-X_{\\ref {lem:tunnellemma4}}$ containing a vertex of $H$ , say $v$ .", "Observe that when we constructed $X_{\\ref {prop:Davoid}}$ , we added $\\mathsf {bd}^{160}(Q)$ to $X_{\\ref {prop:Davoid}}$ , and therefore, $v$ is contained in $\\mathsf {int}^{160}(Q)$ .", "So, by Lemma REF , the connected component of $H\\cap G[\\mathcal {S}^{3}_M]$ containing $v$ is a path from $\\mathcal {S}^{3}_{L_1}$ to $\\mathcal {S}^{3}_{R_1}$ .", "Let $P$ be the path.", "Clearly its ends are contained in $F_L$ and $F_R$ , respectively.", "We separately consider the two cases depending on whether $F_L=F_R$ or not.", "Case 1.", "$F_L\\ne F_R$ .", "As $\\mathsf {int}^{160}(Q)$ is non-empty, $Q$ has more than 320 vertices.", "Let $q$ be the vertex of $Q$ where $\\mathcal {S}^3_{q}\\cap V(F_L)\\ne \\emptyset $ and it is largest with respect to the $\\preccurlyeq _Q$ relation.", "By the construction of $X_{\\ref {prop:Davoid}}$ , the set $\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ was added to $X_{\\ref {prop:Davoid}}$ .", "Let $U:=\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ .", "Observe that if $q$ is contained in $\\mathsf {bd}^{100}_R(Q)$ , then by Lemma REF , each of $F_L$ and $F_R$ contains a path from $\\mathsf {bd}^{100}_{R_5}(Q)$ to $\\mathsf {bd}^{100}_{R_7}(Q)$ , and by Lemma REF , there is a long hole contained in $\\mathsf {bd}^{100}_{R_5\\cup R_6\\cup R_7}(Q)$ , a contradiction.", "So, $q$ is not contained in $\\mathsf {bd}^{100}_R(Q)$ .", "Also, the path $P$ is a path from $\\mathcal {S}^3_{L_1}$ to $\\mathcal {S}^3_{R_1}$ , $q$ is not contained in $\\mathsf {bd}^{160}_L(Q)$ .", "Therefore, $Q-U$ has two connected components.", "Let $Q_1$ and $Q_2$ be the connected components of $Q-U$ such that $\\mathsf {bd}^{20}(Q)\\in V(Q_1)$ .", "As $P$ is a path from $\\mathcal {S}^{3}_{L_1}$ to $\\mathcal {S}^{3}_{R_1}$ , $P$ is a path between $\\mathcal {S}^3_{Q_1}$ and $\\mathcal {S}^3_{Q_2}$ .", "Let $P^{\\prime }$ be a shortest subpath of $P$ from $\\mathcal {S}^3_{Q_1}$ to $\\mathcal {S}^3_{Q_2}$ .", "By the minimality, $P^{\\prime }$ does not contain a vertex of $Q$ , because every neighbor of a vertex in $Q_1$ is in $\\mathcal {S}^3_{Q_1}$ and every neighbor of a vertex in $Q_2$ is in $\\mathcal {S}^3_{Q_2}$ .", "Thus, $P^{\\prime }$ is fully contained in $G[\\mathcal {S}^3_{M}]-M$ .", "On the other hand, since $\\mathcal {S}^3_{Q_2}\\cap V(F_L)=\\emptyset $ while $P^{\\prime }$ intersects $\\mathcal {S}^3_{Q_2}$ , $P^{\\prime }$ is not contained in $F_L$ .", "So, each of $P^{\\prime }$ and $F_L$ contains a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(G[U])}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(G[U])}$ .", "By Lemma REF , there is a long hole contained in $\\mathcal {S}^3_U$ , which contradicts the consequence of Lemma REF .", "Case 2.", "$F_L= F_R$ .", "By Lemma REF , $P$ intersects $\\mathcal {S}^3_{L_7}\\cap V(F_L)$ and $\\mathcal {S}^3_{R_7} \\cap V(F_L)$ .", "We choose a vertex $w_1$ of $P$ in $\\mathcal {S}^3_{L_7}\\cap V(F_L)$ and $w_2$ of $P$ in $\\mathcal {S}^3_{R_7} \\cap V(F_L)$ .", "Let $Z$ be a shortest path from $w_1$ to $w_2$ in $F_L$ .", "Let $v_1v_2v_3v_4v_5$ be a subpath of $H$ such that $v_3\\in \\mathcal {S}^3_{L_2}\\setminus \\mathcal {S}^3_{L_1}$ and $v_3\\in V(P)$ .", "Such a vertex $v_3$ exists by Lemma REF .", "Then $\\lbrace v_2, v_3, v_4\\rbrace $ is contained in $\\mathcal {S}^3_{L_1\\cup L_2\\cup L_3}$ .", "If $Z$ contains a vertex in $\\mathcal {S}^3_{\\mathsf {bd}^{100}_L(Q)}$ , then there are two subpaths of $P$ from $\\mathcal {S}^3_{L_5}$ to $\\mathcal {S}^3_{L_7}$ contained in $\\mathcal {S}^3_{L_5\\cup L_6\\cup L_7}$ .", "Then by Lemma REF , there is a long hole contained in $\\mathcal {S}^3_{L_5\\cup L_6\\cup L_7}$ , a contradiction.", "Therefore, $Z$ contains no vertex in $\\mathcal {S}^3_{\\mathsf {bd}^{100}_L(Q)}$ , and by Lemma REF , no vertex in $\\lbrace v_2, v_3, v_4\\rbrace $ has a neighbor in $Z$ .", "Let $H^{\\prime }$ be the path from $v_1$ to $v_5$ that does not contain $v_2$ .", "Let $H^{\\prime \\prime }:=(H^{\\prime }- (V(P)\\setminus \\lbrace w_1, w_2\\rbrace ))\\cup Z$ .", "Clearly, $H^{\\prime \\prime }$ is connected.", "Let $H^{\\prime \\prime \\prime }$ be a shortest path from $v_1$ to $v_5$ in $H^{\\prime \\prime }$ .", "Observe that there are no edges between $\\lbrace v_2, v_3, v_4\\rbrace $ and $V(H^{\\prime \\prime \\prime })\\setminus \\lbrace v_1, v_5\\rbrace $ in $G$ .", "So, by Lemma REF , $v_1v_2v_3v_4v_5(H^{\\prime \\prime \\prime })v_1$ is a $D$ -avoiding long hole.", "As $v\\in V(P)$ and $v\\notin Z$ , this long hole has vertices of $C$ strictly less than $H$ .", "This contradicts the minimality of $\\vert V(H)\\cap V(C)\\vert $ .", "We conclude that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -avoiding long hole.", "$D$ -traversing long holes It remains to deal with $D$ -traversing long holes.", "We begin by constructing a second greedy packing of long holes.", "This step is similar to the first greedy packing in Section , but now we also consider the vertex set $V(G)\\setminus (D \\cup \\mathcal {S}^3_C$ ).", "Algorithm.", "(Second greedy packing) Set $G_1:=G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ and initialize $C_1=C-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ , $D_1=D\\setminus (X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ , and $M_1=\\emptyset $ .", "Choose a previously unchosen pair $(A,v)$ , where $A$ is a non-empty subset of size at most 2 of $D_i$ and $v$ is a vertex of $C_i$ .", "If there are no remaining pairs, then proceed to step (6).", "Test if $G_1-M_i$ contains a long hole such that it contains $v$ and its intersection on $D_i$ is contained in $A$ .", "If no, then proceed to step (1).", "If yes, then set $D_{i+1}:=D_i \\setminus A$ and $C_{i+1}:=C_i - N^{90}_C[v]$ and $M_{i+1}:=M_i\\cup A\\cup N^{90}_C[v]$ , then proceed to step (2).", "Let $\\ell $ be the largest index for which $D_{\\ell }$ exists.", "Define $X_{\\ref {prop:traversing}}:=M_{\\ell }$ .", "Figure: An example of a long hole HH in Lemma , where Q=G[N C 35 [v]]Q=G[N^{35}_C[v]].We first show that when we consider the pair $(A, v)$ , if a long hole $H$ is detected, then the connected component of $H\\cap G[\\mathcal {S}^3_Q]$ containing $v$ is a path, where $Q=G[N^{35}_C[v]]$ .", "See Figure REF for an example.", "Figure: The paths QQ and RR in Lemma .", "The vertex dd may have a neighbor in R-xR-x.Lemma 7.6 Let $(A, v)$ be a pair considered in the $i$ -th step of the second greedy packing such that $G_1-M_i$ contains a $D$ -traversing long hole $H$ with $v\\in V(H)$ and $V(H)\\cap D_i\\subseteq A$ .", "Let $Q=G[N^{35}_C[v]]$ .", "Then the connected component of $H\\cap G[\\mathcal {S}^3_Q]$ containing $v$ is a path $P$ such that $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ , and If $a$ is an end of $P$ and $b$ is the neighbour of $a$ in $H$ such that $b \\notin V(P)$ , then $b \\in V(G)\\setminus \\mathcal {S}^3_C$ .", "Let $\\overline{Q}$ be the other subpath of $C$ with the same ends as $Q$ .", "Since $H$ is $D$ -traversing and $D\\cap \\mathcal {S}^3_Q=\\emptyset $ , $P$ is a path.", "We first show that $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ .", "Suppose for a contradiction that $P$ contains a vertex $w$ of $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Following the path $P$ from $v$ to $w$ , let $x$ be the first vertex contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Let $R_1=q_1q_2 \\cdots q_t$ be the path $vPx$ where $q_1=v$ and $q_t=x$ .", "Since $x\\in \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ , there is a path $R_2$ of length at most 3 from $x$ to $\\mathsf {bd}^{20}(Q)$ in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "See Figure REF for an illustration.", "Note that no vertex of $(V(R_2)\\setminus \\lbrace x\\rbrace ) \\cup \\mathsf {bd}^{20}(Q)$ has a neighbor in $R_1-x$ , because $x$ is the first vertex that is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Let $d\\in A\\cap V(H)$ .", "Since $d$ is a vertex of $H$ , it has no neighbor in the internal vertices of $R_1$ .", "As $d$ is almost $C$ -dominating, $d$ has a neighbor in $\\mathsf {bd}^{20}(Q)$ and has a neighbor in $N_C[v]$ .", "Let $T$ be a shortest path from $N_G(d)\\cap (\\mathsf {bd}^{20}(Q) \\cup V(R_2))$ to $N_G(d)\\cap N_C[v]$ in $ G[\\mathsf {bd}^{20}(Q)\\cup N_C[v]\\cup V(R_1) \\cup V(R_2)].", "$ We claim that $T$ contains at least 3 internal vertices of $R_1$ .", "Suppose not.", "Then the distance from $N_C[v]$ to $\\mathsf {bd}^{20}(Q) \\cup V(R_2)$ in $T$ is at most 3, and $T\\cup R_2$ contains a path of length at most 6 from $N_C[v]$ to $\\mathsf {bd}^{20}(Q)$ .", "By Lemma REF , $\\operatorname{dist}_C(N_C[v], \\mathsf {bd}^{20}(Q))<16$ .", "However, this contradicts $\\operatorname{dist}_C(v, \\mathsf {bd}^{20}(Q))\\ge 16$ .", "We conclude that $T$ contains at least 3 internal vertices of $Q$ .", "Thus, $G[V(T)\\cup \\lbrace d\\rbrace ]$ is a long hole contained in $\\mathcal {S}^3_Q\\cup D$ .", "But such a long hole had to be considered during the first greedy packing.", "This is a contradiction.", "Now, we verify the second statement.", "Let $a$ be an end of $P$ and $b$ be the neighbour of $a$ in $H$ such that $b \\notin V(P)$ .", "Assume for a contradiction that $b\\in \\mathcal {S}^3_C$ .", "Because $P$ is a connected component of $H\\cap G[\\mathcal {S}^3_Q]$ , $b$ is contained in $\\mathcal {S}^3_{\\overline{Q}}\\setminus \\mathcal {S}^3_Q$ .", "By Lemma REF , $a$ is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}\\setminus \\mathcal {S}^3_{\\mathsf {int}^{20}(Q)}$ .", "But this is not possible because $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ .", "Proposition 7.7 Let $\\ell $ be the largest index for which $D_\\ell $ exists in the second greedy packing algorithm.", "Then $G$ contains $\\ell -1$ vertex-disjoint long holes and $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183\\ell $ .", "Moreover, $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long hole.", "It is straightforward to verify that $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183\\ell $ .", "To prove that $G$ contains $\\ell -1$ vertex-disjoint long holes, it suffices to prove that the holes constructed by the second greedy packing are vertex-disjoint.", "Let $H_1$ and $H_2$ be two $D$ -traversing long holes constructed in the algorithm, and let $(A_1, v_1)$ and $(A_2, v_2)$ be the considered pairs when $H_1$ and $H_2$ are constructed, respectively.", "For each $i\\in [2]$ , let $P_i:=H_i-A_i$ , $Q_i:= G[N^{35}_C[v_i]]$ , and let $P_i^*$ be the connected component of $H_i\\cap G[\\mathcal {S}^3_{Q_i}]$ containing $v_i$ .", "By Step (5) of the second greedy packing, we have $\\operatorname{dist}_C(V(Q_1), V(Q_2))\\ge 20$ and by Lemma REF , $\\mathcal {S}^3_{Q_1}$ and $\\mathcal {S}^3_{Q_2}$ are vertex-disjoint.", "In particular, $P_1^*$ and $P_2^*$ are vertex-disjoint.", "Suppose that $V(H_1) \\cap V(H_2) \\ne \\emptyset $ .", "Since $A_1\\cap A_2=\\emptyset $ , we have $V(P_1) \\cap V(P_2) \\ne \\emptyset $ .", "Let $w_1$ be a vertex of $P_1$ that has a neighbor in $P_2$ such that $\\operatorname{dist}_{P_1}(v_1, w_1)$ is minimum, and let $w_2$ be a neighbor of $w_1$ in $P_2$ such that $\\operatorname{dist}_{P_2}(v_2, w_2)$ is minimum.", "By the choice of $w_1$ and $w_2$ , $v_1P_1w_1w_2P_2v_2$ is an induced path.", "Let $R:=v_1P_1w_1w_2P_2v_2$ .", "We claim that for some $i\\in [2]$ , $R$ contains a subpath of $P_i^*$ from $v_i$ to an end of $P_i^*$ and contains the next two vertices in $H_i$ .", "If not, then by Lemma REF there is a path of length at most 3 in $G-D$ between $\\mathcal {S}^3_{\\mathsf {int}^{20}(Q_1)}$ and $\\mathcal {S}^3_{\\mathsf {int}^{20}(Q_2)}$ , and there is a path of length at most 9 in $G-D$ , whose ends are contained in $\\mathsf {int}^{20}(Q_1)$ and $\\mathsf {int}^{20}(Q_2)$ , respectively.", "Since $\\operatorname{dist}_C(\\mathsf {int}^{20}(Q_1), \\mathsf {int}^{20}(Q_2))\\ge 28$ , this contradicts Lemma REF .", "Without loss of generality, $R$ contains a subpath of $P_1^*$ from $v_1$ to an end of $P_1^*$ , say $z$ , and contains the next two vertices $z_1$ and $z_2$ in $H_1$ .", "Following the direction from $z$ to $z_1$ in $R$ , we choose the first vertex $u$ contained in $.$ By Lemma REF , $z_1$ is contained in $V(G)\\setminus \\mathcal {S}^3_C$ and $z$ is contained in $\\mathcal {S}^3_C\\setminus \\mathcal {S}^2_C$ .", "Following the direction from $z$ to $v_1$ in $P_1^*$ , let $y$ be the first vertex contained in $\\mathcal {S}^2_C$ .", "Observe that $yRu$ is an induced path and the next three vertices of $yRu$ after $y$ are not in $\\mathcal {S}^2_C$ .", "However, such a path $yRu$ does not exist by Lemma REF .", "Thus, the holes constructed by the second greedy packing are vertex-disjoint.", "It remains to prove that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long hole.", "By Proposition REF , $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -traversing long hole.", "Thus, it suffices to prove that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no $D$ -avoiding long hole.", "Since $D$ is a clique by Lemma REF and $H$ is a hole, $H$ contains at most 2 vertices of $D$ , and if it contains two vertices of $D$ , then they are adjacent in $H$ .", "So, $H-D$ is a path.", "Since $G-V(C)$ has no long hole, $H$ contains a vertex of $C$ .", "Let $v$ be a vertex of $H$ in $C$ .", "By considering the pair $(V(H)\\cap D, v)$ in the second greedy packing, we should have proceeded one more step, which is a contradiction.", "We are now ready to prove Theorem REF .", "We begin by running the first greedy packing to obtain $X_{\\ref {lem:greedypacking}}$ .", "We may assume that the largest index $\\ell $ in the first greedy packing is at most $k$ , else we obtain $k$ vertex-disjoint long holes by Lemma REF .", "Next, we construct a maximal good sparse ear decomposition $ using Lemma~\\ref {lem:sparsealgo}.By Lemma~\\ref {lem:narrowalgo}, we can either find $ k$ vertex-disjoint appendages of $ , or a vertex set $X_{\\ref {lem:narrowalgo}}$ of size at most $20k$ hitting all appendages.", "Since each appendage contains a long hole, we may assume that $X_{\\ref {lem:narrowalgo}}$ exists.", "We then apply Proposition REF to find $X_{\\ref {prop:Davoid}}$ such that $X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}$ hits all $D$ -avoiding long holes.", "Lastly, we run the second greedy packing to obtain $X_{\\ref {prop:traversing}}$ .", "We may assume that the largest index $\\ell $ in the second greedy packing is at most $k$ , else we are done by Proposition REF .", "Since each of the above steps is performed in polynomial time, the entire algorithm is a polynomial-time algorithm.", "We claim that we may take $X_{\\ref {thm:main2}}= X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}$ .", "By Proposition REF , $X_{\\ref {thm:main2}}$ hits all long holes of $G$ , so it only remains to bound the size of $X_{\\ref {thm:main2}}$ .", "Recall that $X_{\\ref {lem:tunnellemma4}}=X_{\\ref {lem:greedypacking}} \\cup X_{\\ref {lem:sparsealgo}}\\cup X_{\\ref {lem:narrowalgo}}$ and $X_{\\ref {lem:sparsealgo}}=N^{31}_C[B\\cap V(C)] \\cup (B\\setminus V(C))$ , where $B$ is the set of branching points of $.", "By Lemma~\\ref {lem:narrowalgo}, $ |XREF | 20k$ .", "If $ has at least $s_k$ branching points, then by Lemma REF , we can output $k$ vertex-disjoint long holes.", "Thus, we may assume that $|B| < s_k$ .", "Therefore, $|X_{\\ref {lem:sparsealgo}}|=|N^{31}_C[B\\cap V(C)] \\cup (B\\setminus V(C))| \\le 63s_k.$ By Lemma REF , $\\vert X_{\\ref {lem:greedypacking}}\\vert \\le 212k$ , and by Proposition REF , $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183k$ .", "Finally, by Proposition REF , $\\vert X_{\\ref {prop:Davoid}}\\vert \\le 380\\vert X_{\\ref {lem:tunnellemma4}}\\vert $ .", "Putting this all together, $\\vert X_{\\ref {thm:main2}}\\vert =\\vert X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}\\vert \\le 381(212k+63s_k+20k)+183k = \\mu _k.", "$ Open Problems In this paper, we proved that holes of length at least 6 have the Erdős-Pósa property in $C_4$ -free graphs.", "Our proof is also the shortest known proof that holes have the Erdős-Pósa property.", "Our first open problem is to answer Question REF for larger values of $\\ell $ .", "By making our proof much longer, we believe our methods could answer Question REF affirmatively for $\\ell =7$ , but new ideas are needed for $\\ell \\ge 8$ .", "Next, it is unclear if the $\\mathcal {O}(k^2 \\log k)$ bound in Theorem REF is optimal.", "We conjecture that Theorem REF holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "Conjecture 8.1 There exists a function $f(k)=\\mathcal {O}(k \\log k)$ such that for every $C_4$ -free graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least 6, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least 6.", "By the argument in the introduction, if Conjecture REF is true, then the induced Erdős-Pósa property for cycles of length at least 4 also holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "As noted in [10], this would be tight via a reduction to the $\\Omega (k \\log k)$ lower bound that holds for cycles.", "In general, smaller bounding functions give improved approximation factors for the corresponding packing and covering problems.", "For example, the $\\mathcal {O}(k \\log k)$ bounding function for planar minors proved in [3], currently gives the best approximation algorithm for packing planar minors.", "Our final open problem concerns what other classes of graphs have the induced Erdős-Pósa property.", "For a fixed graph $H$ , the class of $H$ -subdivisions is the class of graphs that can be obtained from $H$ by repeatedly subdividing edges.", "Theorem REF says that the the class of $C_4$ -subdivisions has the induced Erdős-Pósa property, and Theorem REF says that for all $\\ell \\ge 5$ , the class of $C_\\ell $ -subdivisions does not have the induced Erdős-Pósa property.", "Kwon and Raymond [11] investigated for which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property.", "They designed three different types of constructions to show that for various graphs $H$ , the class of $H$ -subdivisions does not have the induced Erdős-Pósa property.", "In each of their constructions, a large complete bipartite induced subgraph always appears.", "This is another reason why investigating the induced Erdős-Pósa property in $C_4$ -free graphs is particularly interesting.", "Our main theorem asserts that $C_6$ -subdivisions have the induced Erdős-Pósa property in the class of $C_4$ -free graphs.", "Question 2 For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property?", "For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property in $C_4$ -free graphs?", "Note that it is not true that for all graphs $H$ , the class of $H$ -subdivisions has the induced Erdős-Pósa property in $C_4$ -free graphs.", "Kwon and Raymond [11] proved that if a forest $F$ has a connected component having at least two vertices of degree at least 3, then the class of $F$ -subdivisions does not have the induced Erdős-Pósa property in $C_4$ -free graphs." ], [ "Hitting long holes", "In the previous sections, we have defined $X_{\\ref {lem:greedypacking}}, X_{\\ref {lem:sparsealgo}}$ , and $X_{\\ref {lem:narrowalgo}}$ .", "Let $X_{\\ref {lem:tunnellemma4}}:=X_{\\ref {lem:greedypacking}} \\cup X_{\\ref {lem:sparsealgo}}\\cup X_{\\ref {lem:narrowalgo}}.$ In this section, we complete the proof of Theorem REF by either finding $k$ vertex-disjoint long holes of $G$ or sets $X_{\\ref {prop:Davoid}}$ and $X_{\\ref {prop:traversing}}$ such that $G- (X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long holes and $|X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}|=\\mathcal {O}(k \\log k)$ .", "We divide long holes into two types, depending on whether a long hole intersects $D$ or not.", "We say that a long hole is $D$ -avoiding if it contains no vertex of $D$ , and $D$ -traversing, otherwise.", "We first consider $D$ -avoiding holes in Subsection REF , and consider $D$ -traversing holes in Subsection REF .", "Before doing so, we require a few more definitions related to the cycle $C$ .", "For these definitions, we regard $C:=a_0a_1a_2 \\cdots a_ma_0$ , as a directed cycle where $a_0$ is directed towards $a_1$ .", "For a subpath $Q$ of $C$ , let $\\mathsf {bd}_L(Q)$ and $\\mathsf {bd}_R(Q)$ be the ends of $Q$ such that $Q$ is directed from $\\mathsf {bd}_L(Q)$ to $\\mathsf {bd}_R(Q)$ .", "For an integer $i\\ge 2$ , let $\\mathsf {bd}^i_L(Q):=N^{i-1}_Q[ \\mathsf {bd}_L(Q)]$ , $\\mathsf {bd}^i_R(Q):=N^{i-1}_Q[ \\mathsf {bd}_R(Q)]$ , $\\mathsf {bd}^i(Q):=\\mathsf {bd}^i_L(Q)\\cup \\mathsf {bd}^i_R(Q)$ and $\\mathsf {int}^i(Q):=V(Q)\\setminus \\mathsf {bd}^i(Q)$ .", "The following two lemmas will be useful to find a long hole.", "Lemma 7.1 Let $Q$ be a path on more than 160 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ , and let $Q^*$ be a subpath of $Q$ on 20 vertices.", "If $P$ is a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ contained in $G[\\mathcal {S}^{3}_{Q}]$ , then $P$ intersects $\\mathcal {S}^3_{Q^*}\\setminus \\mathcal {S}^3_{Q-V(Q^*)}$ .", "By assumption, $P$ intersects $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ and $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ .", "Thus, we may assume that $Q-V(Q^*)$ has two distinct connected components.", "Let $Q_1$ and $Q_2$ be the connected components of $Q-V(Q^*)$ , where $\\mathsf {bd}_L(Q)\\in V(Q_1)$ .", "Let $p_1$ and $p_2$ be the ends of $P$ such that $p_1\\in \\mathcal {S}^3_{\\mathsf {bd}^{20}_L(Q)}$ .", "Suppose for a contradiction that $P$ does not contain a vertex of $\\mathcal {S}^3_{Q^*}\\setminus \\mathcal {S}^3_{Q-V(Q^*)}$ .", "Observe that $p_1\\in \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ and $p_2\\in \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ , and by Lemma REF , $p_1\\notin \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ and $p_2\\notin \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ .", "Thus, there is an edge $uv$ of $P$ where $u\\in \\mathcal {S}^3_{Q_1}\\setminus \\mathcal {S}^3_{Q^*}$ and $v\\in \\mathcal {S}^3_{Q_2}\\setminus \\mathcal {S}^3_{Q^*}$ .", "But since $\\operatorname{dist}_C(V(Q_1), V(Q_2))\\ge 20$ , this is not possible by Lemma REF .", "Lemma 7.2 Let $Q$ be a path on 60 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ .", "If $P_1$ and $P_2$ are two vertex-disjoint paths from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(Q)}$ contained in $G[\\mathcal {S}^{3}_{Q}]-V(Q)$ such that there are no edges between $P_1$ and $P_2$ , then $\\mathcal {S}^3_{Q}$ contains a long hole.", "Let $A_1:=\\mathsf {bd}^{20}_L(Q)$ and $A_2:=\\mathsf {bd}^{20}_R(Q)$ .", "We may assume that for each $i, j\\in [2]$ , $\\vert V(P_i)\\cap \\mathcal {S}^3_{A_j}\\vert =1$ .", "For $i, j\\in [2]$ , let $p_{i,j}$ be the end of $P_i$ contained in $\\mathcal {S}^3_{A_j}$ .", "By Lemma REF , $p_{i,1}\\ne p_{i,2}$ and $P_i$ has length at least 2.", "For each $i, j\\in [2]$ , let $P_{i,j}$ be a path of length at most 3 in $\\mathcal {S}^3_{A_j}$ from $p_{i,j}$ to $A_j$ .", "For each $j\\in [2]$ , let $U_j$ be a shortest path from $p_{1,j}$ to $p_{2,j}$ in $G[V(P_{1,j}) \\cup V(P_{2,j})\\cup A_j]$ .", "By the choice of $P_1$ and $P_2$ , no vertex of $(P_1-p_{1,1})\\cup (P_2-p_{2,1})$ is contained in $\\mathcal {S}^3_{A_1}$ .", "This implies that no internal vertex of $U_1$ has a neighbor in $(P_1-p_{1,1})\\cup (P_2-p_{2,1})$ .", "Similarly, no internal vertex of $U_2$ has a neighbor in $(P_1-p_{1,2})\\cup (P_2-p_{2,2})$ .", "Also, by Lemma REF , there is no edge between $U_1$ and $U_2$ .", "Since $P_1$ and $P_2$ have length at least 2, $U_1\\cup U_2\\cup P_1\\cup P_2$ is a long hole contained in $\\mathcal {S}^3_{Q}$ ." ], [ "$D$ -avoiding long holes", "In this section, we show that by taking at most 380 vertices in each connected component of $C-X_{\\ref {lem:tunnellemma4}}$ , we can hit all $D$ -avoiding long holes.", "We begin with the following structural property.", "Lemma 7.3 The graph $G-(X_{\\ref {lem:tunnellemma4}} \\cup D)$ has no induced path $P:=p_1p_2 \\cdots p_\\ell $ of length at least 4 such that $p_1\\in \\mathcal {S}^2_q$ for some $q\\in V(C)\\setminus X_{\\ref {lem:tunnellemma4}}$ , $p_2,p_3,p_4\\notin \\mathcal {S}^2_C$ , $\\lbrace p_i:i\\in [\\ell -1]\\rbrace \\cap V(=\\emptyset $ , and $p_\\ell \\in V($ .", "Suppose for a contradiction that such an induced path $P$ exists.", "Since $p_1\\in \\mathcal {S}^2_q$ and $p_2\\notin \\mathcal {S}^2_C$ , $p_1$ has no neighbor in $C$ .", "Choose a vertex $q^{\\prime }$ in $\\mathcal {S}^2_q$ such that $qq^{\\prime }p_1$ is a path.", "We claim that no vertex in $\\lbrace p_2,p_3,p_4\\rbrace $ has a neighbor in $.Suppose not.", "Since $ p2,p3,p4$ are not in $ S1C$, one of $ {p2,p3,p4}$ has a neighbor in $ V(V(C)$.Thus, there is an $ -path from $q$ to a vertex in $V(\\setminus V(C)$ that has length at most 5, which contradicts Lemma REF .", "We observe that $R=qq^{\\prime }p_1Pp_\\ell $ is either an extension or an appendage of $.Assume that $ q'$ has a neighbor in $ P-{p1, p}$.Choose a neighbor $ z$ in $ P-{p1, p}$ such that $ distP(p1, z)$ is minimum.As $ p2,p3,p4S2C$, $ q'$ has no neighbor in $ {p2, p3, p4}$.Thus, $ distP(p1, z)4$.", "This implies that the cycle $ q'p1Pzq'$ is a long hole in $ G-V(C)$, a contradiction.Therefore, $ q'$ has no neighbor in $ P-{p1, p}$ and in particular, $ R-{q, p}$ is induced.$ As no vertex in $\\lbrace p_2, p_3, p_4\\rbrace $ has a neighbor in $,we conclude that $ R$ is an extension or an appendage of $ .", "This contradicts that $ is maximal and $ XREF $ hits all appendages.$ Figure: The setting in Lemma .", "We show that the ends of PP have to be contained in distinct sets of 𝒮 L 1 3 \\mathcal {S}^{3}_{L_1} and 𝒮 R 1 3 \\mathcal {S}^{3}_{R_1}.Lemma 7.4 Let $Q$ be a path on more than 160 vertices in $C-X_{\\ref {lem:tunnellemma4}}$ .", "Let $H$ be a $D$ -avoiding long hole in $G-X_{\\ref {lem:tunnellemma4}}$ such that $H$ contains no vertices in $\\mathsf {bd}^{80}(Q)$ , and it contains a vertex $v$ in $\\mathsf {int}^{80}(Q)$ .", "Then the connected component of $H\\cap G[\\mathcal {S}^{3}_{\\mathsf {int}^{20}(Q)}]$ containing $v$ is a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{40}_L(Q)\\setminus \\mathsf {bd}^{20}_L(Q)}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{40}_R(Q)\\setminus \\mathsf {bd}^{20}_R(Q)}$ .", "Because $X_{\\ref {lem:greedypacking}}\\subseteq X_{\\ref {lem:tunnellemma4}}$ , by Lemma REF , $H$ is not contained in $\\mathcal {S}^3_Q$ .", "Let $P$ be the connected component of $H\\cap G[\\mathcal {S}^{3}_{\\mathsf {int}^{20}(Q)}]$ containing $v$ .", "Then $P$ is a path.", "Let $\\overline{Q}$ be the other subpath of $C$ with the same ends as $Q$ .", "For convenience, let $L_1:=\\mathsf {bd}^{40}_L(Q)\\setminus \\mathsf {bd}^{20}_L(Q)$ , $L_2:=\\mathsf {bd}^{80}_L(Q)\\setminus \\mathsf {bd}^{60}_L(Q)$ , $R_1:=\\mathsf {bd}^{40}_R(Q)\\setminus \\mathsf {bd}^{20}_R(Q)$ , $R_2:=\\mathsf {bd}^{80}_R(Q)\\setminus \\mathsf {bd}^{60}_R(Q)$ , and $M:=\\mathsf {int}^{20}(Q)$ .", "See Figure REF .", "We claim that both ends of $P$ are contained in $\\mathcal {S}^3_{L_1\\cup R_1}$ .", "Suppose the contrary, and let $w$ be an end of $P$ such that $w\\in \\mathcal {S}^3_M\\setminus \\mathcal {S}^3_{L_1\\cup R_1}$ .", "Let $w^{\\prime }$ be a neighbor of $w$ in $H$ that is not in $P$ .", "First assume that $w^{\\prime }\\in \\mathcal {S}^3_{C}$ .", "Since $w^{\\prime }\\notin \\mathcal {S}^3_M$ , $w^{\\prime }$ is contained in $\\mathcal {S}^3_{\\overline{Q}}\\cup \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Therefore, there is a $C$ -path of length at most 7 in $G-D$ with one end in $V(\\overline{Q})\\cup \\mathsf {bd}^{20}(Q)$ and the other end in $\\mathsf {int}^{40}(Q)$ .", "However, since $\\operatorname{dist}_C( V(\\overline{Q})\\cup \\mathsf {bd}^{20}(Q), \\mathsf {int}^{40}(Q))\\ge 20$ , we contradict Lemma REF .", "Thus, we may assume that $w^{\\prime }\\notin \\mathcal {S}^3_{C}$ .", "Since $w^{\\prime } \\notin \\mathcal {S}^3_{C}$ and $w \\in \\mathcal {S}^3_{C}$ , we have $w\\in \\mathcal {S}^3_{C}\\setminus \\mathcal {S}^2_{C}$ .", "Following the direction from $w$ to $v$ in $P$ , let $a$ be the first vertex so that $a\\in \\mathcal {S}^2_{C}$ .", "Since $a$ is in $\\mathcal {S}^3_M$ , again by Lemma REF , $a \\notin \\mathcal {S}^3_{\\overline{Q}}$ .", "Thus, $a\\in \\mathcal {S}^2_q$ for some $q\\in V(Q)$ .", "Observe that $q\\notin X_{\\ref {lem:tunnellemma4}}$ .", "Let $w^{\\prime \\prime }$ be the neighbor of $w^{\\prime }$ in $C$ that is not $w$ .", "Now, in the path $aPww^{\\prime }w^{\\prime \\prime }$ , $a\\in \\mathcal {S}^2_q$ while the next three vertices are not contained in $\\mathcal {S}^2_{C}$ .", "By Lemma REF , no vertex in $aPww^{\\prime }w^{\\prime \\prime }$ is contained in $.", "Following the direction from $ w$ to $ w'$ in $ H$, let $ u$ be the first vertex contained in $ .", "Then the subpath of $H$ from $a$ to $u$ containing $w$ satisfies the conditions of Lemma REF , which yields a contradiction.", "So, we conclude that both ends of $P$ are contained in $\\mathcal {S}^3_{L_1\\cup R_1}$ .", "Figure: The construction of a long hole in Lemma  when the two ends of PP are contained in 𝒮 L 1 3 \\mathcal {S}^3_{L_1}.To conclude the lemma, we claim that one end of $P$ is in $\\mathcal {S}^3_{L_1}$ and the other end is in $\\mathcal {S}^3_{R_1}$ .", "Suppose not.", "By symmetry, we may assume that both ends of $P$ are in $\\mathcal {S}^3_{L_1}$ .", "See Figure REF for an illustration.", "Let $P_1$ and $P_2$ be the two subpaths of $P$ from $v$ to the ends of $P$ .", "Applying Lemma REF to $P_1$ and $P_2$ , we deduce that for each $i\\in [2]$ , $P_i$ contains a vertex of $\\mathcal {S}^3_{L_2}$ .", "For each $i\\in [2]$ , let $Q_i$ be a shortest subpath of $P_i$ from $\\mathcal {S}^3_{L_1}$ to $\\mathcal {S}^3_{L_2}$ respectively.", "By Lemma REF , all the internal vertices of $Q_i$ are contained in $\\mathcal {S}^3_{\\mathsf {bd}^{60}_L(Q) \\setminus \\mathsf {bd}^{40}_L(Q)}$ , and $Q_i$ is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{80}_L(Q) \\setminus \\mathsf {bd}^{20}_L(Q)}$ .", "By Lemma REF , $G\\left[\\mathcal {S}^3_{\\mathsf {bd}^{80}_L(Q) \\setminus \\mathsf {bd}^{20}_L(Q)}\\right]$ contains a long hole.", "This contradicts Lemma REF .", "Proposition 7.5 There is a polynomial-time algorithm to find a vertex set $X_{\\ref {prop:Davoid}}\\subseteq V(G)\\setminus X_{\\ref {lem:tunnellemma4}}$ of size at most $380 \\vert V(C)\\cap X_{\\ref {lem:tunnellemma4}}\\vert $ such that $X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}$ hits all $D$ -avoiding long holes.", "Recall that $C=a_0a_1a_2 \\cdots a_ma_0$ and consider it as a directed cycle where $a_0$ is directed to $a_1$ .", "For a subpath $Q$ of $C$ and two vertices $a_i$ and $a_j$ in $Q$ , we denote by $a_i\\preccurlyeq _Q a_j$ if there is a directed path from $a_i$ to $a_j$ .", "We construct a set $S$ as follows.", "We iterate the following over all components $Q$ of $C-X_{\\ref {lem:tunnellemma4}}$ .", "For each $i\\in [7]$ , let $L_i:=\\mathsf {bd}^{20(i+1)}_L(Q)\\setminus \\mathsf {bd}^{20i}_L(Q)$ , $R_i:=\\mathsf {bd}^{20(i+1)}_R(Q)\\setminus \\mathsf {bd}^{20i}_R(Q)$ , and $M:=\\mathsf {int}^{20}(Q)$ .", "If $Q$ has at most 320 vertices, then we add all vertices of $Q$ to $S$ .", "Assume that $Q$ has more than 320 vertices.", "We first add $\\mathsf {bd}^{160}(Q)$ to $S$ .", "Let $F_L$ be the connected component of $G[\\mathcal {S}^3_{M}]-M$ intersecting both $\\mathcal {S}^3_{L_1}$ and $\\mathcal {S}^3_{L_3}$ , if one exists.", "Similarly, let $F_R$ be the connected component of $G[\\mathcal {S}^3_M]-M$ intersecting both $\\mathcal {S}^3_{R_1}$ and $\\mathcal {S}^3_{R_3}$ , if one exists.", "If $F_L$ or $F_R$ does not exist, then we skip this component $Q$ .", "(If there are two components intersecting both $\\mathcal {S}^3_{L_1}$ and $\\mathcal {S}^3_{L_2}$ , then there is a long hole contained in $\\mathcal {S}^3_{L_1\\cup L_2\\cup L_3}$ by Lemma REF , which contradicts Lemma REF .", "Thus, if such a component exists, then the component uniquely exists, and the same argument holds for $\\mathcal {S}^3_{R_1}$ and $\\mathcal {S}^3_{R_3}$ .)", "Now, assume that both $F_L$ and $F_R$ exist.", "If $F_L=F_R$ , then we skip this component $Q$ .", "If $F_L\\ne F_R$ , then let $q$ be the vertex of $Q$ where $\\mathcal {S}^3_{q}\\cap V(F_L)\\ne \\emptyset $ and it is largest with respect to the $\\preccurlyeq _Q$ relation.", "We add the set $\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ to $S$ .", "Let $X_{\\ref {prop:Davoid}}$ be the final set $S$ .", "Clearly, $\\vert X_{\\ref {prop:Davoid}}\\vert \\le 380\\vert V(C)\\cap X_{\\ref {lem:tunnellemma4}}\\vert $ and it can be computed in polynomial time.", "It remains to show that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -avoiding long hole.", "Suppose that there is a $D$ -avoiding long hole $H$ in $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ .", "We choose such a hole with minimum $\\vert V(H)\\cap V(C)\\vert $ .", "Since $G-V(C)$ has no long hole, $H$ must contain a vertex of $C$ .", "Let $Q$ be a connected component of $C-X_{\\ref {lem:tunnellemma4}}$ containing a vertex of $H$ , say $v$ .", "Observe that when we constructed $X_{\\ref {prop:Davoid}}$ , we added $\\mathsf {bd}^{160}(Q)$ to $X_{\\ref {prop:Davoid}}$ , and therefore, $v$ is contained in $\\mathsf {int}^{160}(Q)$ .", "So, by Lemma REF , the connected component of $H\\cap G[\\mathcal {S}^{3}_M]$ containing $v$ is a path from $\\mathcal {S}^{3}_{L_1}$ to $\\mathcal {S}^{3}_{R_1}$ .", "Let $P$ be the path.", "Clearly its ends are contained in $F_L$ and $F_R$ , respectively.", "We separately consider the two cases depending on whether $F_L=F_R$ or not.", "Case 1.", "$F_L\\ne F_R$ .", "As $\\mathsf {int}^{160}(Q)$ is non-empty, $Q$ has more than 320 vertices.", "Let $q$ be the vertex of $Q$ where $\\mathcal {S}^3_{q}\\cap V(F_L)\\ne \\emptyset $ and it is largest with respect to the $\\preccurlyeq _Q$ relation.", "By the construction of $X_{\\ref {prop:Davoid}}$ , the set $\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ was added to $X_{\\ref {prop:Davoid}}$ .", "Let $U:=\\lbrace v\\in V(Q) : v\\preccurlyeq _Q q, \\operatorname{dist}_Q(v, q)\\le 59 \\rbrace $ .", "Observe that if $q$ is contained in $\\mathsf {bd}^{100}_R(Q)$ , then by Lemma REF , each of $F_L$ and $F_R$ contains a path from $\\mathsf {bd}^{100}_{R_5}(Q)$ to $\\mathsf {bd}^{100}_{R_7}(Q)$ , and by Lemma REF , there is a long hole contained in $\\mathsf {bd}^{100}_{R_5\\cup R_6\\cup R_7}(Q)$ , a contradiction.", "So, $q$ is not contained in $\\mathsf {bd}^{100}_R(Q)$ .", "Also, the path $P$ is a path from $\\mathcal {S}^3_{L_1}$ to $\\mathcal {S}^3_{R_1}$ , $q$ is not contained in $\\mathsf {bd}^{160}_L(Q)$ .", "Therefore, $Q-U$ has two connected components.", "Let $Q_1$ and $Q_2$ be the connected components of $Q-U$ such that $\\mathsf {bd}^{20}(Q)\\in V(Q_1)$ .", "As $P$ is a path from $\\mathcal {S}^{3}_{L_1}$ to $\\mathcal {S}^{3}_{R_1}$ , $P$ is a path between $\\mathcal {S}^3_{Q_1}$ and $\\mathcal {S}^3_{Q_2}$ .", "Let $P^{\\prime }$ be a shortest subpath of $P$ from $\\mathcal {S}^3_{Q_1}$ to $\\mathcal {S}^3_{Q_2}$ .", "By the minimality, $P^{\\prime }$ does not contain a vertex of $Q$ , because every neighbor of a vertex in $Q_1$ is in $\\mathcal {S}^3_{Q_1}$ and every neighbor of a vertex in $Q_2$ is in $\\mathcal {S}^3_{Q_2}$ .", "Thus, $P^{\\prime }$ is fully contained in $G[\\mathcal {S}^3_{M}]-M$ .", "On the other hand, since $\\mathcal {S}^3_{Q_2}\\cap V(F_L)=\\emptyset $ while $P^{\\prime }$ intersects $\\mathcal {S}^3_{Q_2}$ , $P^{\\prime }$ is not contained in $F_L$ .", "So, each of $P^{\\prime }$ and $F_L$ contains a path from $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_L(G[U])}$ to $\\mathcal {S}^{3}_{\\mathsf {bd}^{20}_R(G[U])}$ .", "By Lemma REF , there is a long hole contained in $\\mathcal {S}^3_U$ , which contradicts the consequence of Lemma REF .", "Case 2.", "$F_L= F_R$ .", "By Lemma REF , $P$ intersects $\\mathcal {S}^3_{L_7}\\cap V(F_L)$ and $\\mathcal {S}^3_{R_7} \\cap V(F_L)$ .", "We choose a vertex $w_1$ of $P$ in $\\mathcal {S}^3_{L_7}\\cap V(F_L)$ and $w_2$ of $P$ in $\\mathcal {S}^3_{R_7} \\cap V(F_L)$ .", "Let $Z$ be a shortest path from $w_1$ to $w_2$ in $F_L$ .", "Let $v_1v_2v_3v_4v_5$ be a subpath of $H$ such that $v_3\\in \\mathcal {S}^3_{L_2}\\setminus \\mathcal {S}^3_{L_1}$ and $v_3\\in V(P)$ .", "Such a vertex $v_3$ exists by Lemma REF .", "Then $\\lbrace v_2, v_3, v_4\\rbrace $ is contained in $\\mathcal {S}^3_{L_1\\cup L_2\\cup L_3}$ .", "If $Z$ contains a vertex in $\\mathcal {S}^3_{\\mathsf {bd}^{100}_L(Q)}$ , then there are two subpaths of $P$ from $\\mathcal {S}^3_{L_5}$ to $\\mathcal {S}^3_{L_7}$ contained in $\\mathcal {S}^3_{L_5\\cup L_6\\cup L_7}$ .", "Then by Lemma REF , there is a long hole contained in $\\mathcal {S}^3_{L_5\\cup L_6\\cup L_7}$ , a contradiction.", "Therefore, $Z$ contains no vertex in $\\mathcal {S}^3_{\\mathsf {bd}^{100}_L(Q)}$ , and by Lemma REF , no vertex in $\\lbrace v_2, v_3, v_4\\rbrace $ has a neighbor in $Z$ .", "Let $H^{\\prime }$ be the path from $v_1$ to $v_5$ that does not contain $v_2$ .", "Let $H^{\\prime \\prime }:=(H^{\\prime }- (V(P)\\setminus \\lbrace w_1, w_2\\rbrace ))\\cup Z$ .", "Clearly, $H^{\\prime \\prime }$ is connected.", "Let $H^{\\prime \\prime \\prime }$ be a shortest path from $v_1$ to $v_5$ in $H^{\\prime \\prime }$ .", "Observe that there are no edges between $\\lbrace v_2, v_3, v_4\\rbrace $ and $V(H^{\\prime \\prime \\prime })\\setminus \\lbrace v_1, v_5\\rbrace $ in $G$ .", "So, by Lemma REF , $v_1v_2v_3v_4v_5(H^{\\prime \\prime \\prime })v_1$ is a $D$ -avoiding long hole.", "As $v\\in V(P)$ and $v\\notin Z$ , this long hole has vertices of $C$ strictly less than $H$ .", "This contradicts the minimality of $\\vert V(H)\\cap V(C)\\vert $ .", "We conclude that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -avoiding long hole.", "$D$ -traversing long holes It remains to deal with $D$ -traversing long holes.", "We begin by constructing a second greedy packing of long holes.", "This step is similar to the first greedy packing in Section , but now we also consider the vertex set $V(G)\\setminus (D \\cup \\mathcal {S}^3_C$ ).", "Algorithm.", "(Second greedy packing) Set $G_1:=G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ and initialize $C_1=C-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ , $D_1=D\\setminus (X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ , and $M_1=\\emptyset $ .", "Choose a previously unchosen pair $(A,v)$ , where $A$ is a non-empty subset of size at most 2 of $D_i$ and $v$ is a vertex of $C_i$ .", "If there are no remaining pairs, then proceed to step (6).", "Test if $G_1-M_i$ contains a long hole such that it contains $v$ and its intersection on $D_i$ is contained in $A$ .", "If no, then proceed to step (1).", "If yes, then set $D_{i+1}:=D_i \\setminus A$ and $C_{i+1}:=C_i - N^{90}_C[v]$ and $M_{i+1}:=M_i\\cup A\\cup N^{90}_C[v]$ , then proceed to step (2).", "Let $\\ell $ be the largest index for which $D_{\\ell }$ exists.", "Define $X_{\\ref {prop:traversing}}:=M_{\\ell }$ .", "Figure: An example of a long hole HH in Lemma , where Q=G[N C 35 [v]]Q=G[N^{35}_C[v]].We first show that when we consider the pair $(A, v)$ , if a long hole $H$ is detected, then the connected component of $H\\cap G[\\mathcal {S}^3_Q]$ containing $v$ is a path, where $Q=G[N^{35}_C[v]]$ .", "See Figure REF for an example.", "Figure: The paths QQ and RR in Lemma .", "The vertex dd may have a neighbor in R-xR-x.Lemma 7.6 Let $(A, v)$ be a pair considered in the $i$ -th step of the second greedy packing such that $G_1-M_i$ contains a $D$ -traversing long hole $H$ with $v\\in V(H)$ and $V(H)\\cap D_i\\subseteq A$ .", "Let $Q=G[N^{35}_C[v]]$ .", "Then the connected component of $H\\cap G[\\mathcal {S}^3_Q]$ containing $v$ is a path $P$ such that $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ , and If $a$ is an end of $P$ and $b$ is the neighbour of $a$ in $H$ such that $b \\notin V(P)$ , then $b \\in V(G)\\setminus \\mathcal {S}^3_C$ .", "Let $\\overline{Q}$ be the other subpath of $C$ with the same ends as $Q$ .", "Since $H$ is $D$ -traversing and $D\\cap \\mathcal {S}^3_Q=\\emptyset $ , $P$ is a path.", "We first show that $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ .", "Suppose for a contradiction that $P$ contains a vertex $w$ of $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Following the path $P$ from $v$ to $w$ , let $x$ be the first vertex contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Let $R_1=q_1q_2 \\cdots q_t$ be the path $vPx$ where $q_1=v$ and $q_t=x$ .", "Since $x\\in \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ , there is a path $R_2$ of length at most 3 from $x$ to $\\mathsf {bd}^{20}(Q)$ in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "See Figure REF for an illustration.", "Note that no vertex of $(V(R_2)\\setminus \\lbrace x\\rbrace ) \\cup \\mathsf {bd}^{20}(Q)$ has a neighbor in $R_1-x$ , because $x$ is the first vertex that is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}$ .", "Let $d\\in A\\cap V(H)$ .", "Since $d$ is a vertex of $H$ , it has no neighbor in the internal vertices of $R_1$ .", "As $d$ is almost $C$ -dominating, $d$ has a neighbor in $\\mathsf {bd}^{20}(Q)$ and has a neighbor in $N_C[v]$ .", "Let $T$ be a shortest path from $N_G(d)\\cap (\\mathsf {bd}^{20}(Q) \\cup V(R_2))$ to $N_G(d)\\cap N_C[v]$ in $ G[\\mathsf {bd}^{20}(Q)\\cup N_C[v]\\cup V(R_1) \\cup V(R_2)].", "$ We claim that $T$ contains at least 3 internal vertices of $R_1$ .", "Suppose not.", "Then the distance from $N_C[v]$ to $\\mathsf {bd}^{20}(Q) \\cup V(R_2)$ in $T$ is at most 3, and $T\\cup R_2$ contains a path of length at most 6 from $N_C[v]$ to $\\mathsf {bd}^{20}(Q)$ .", "By Lemma REF , $\\operatorname{dist}_C(N_C[v], \\mathsf {bd}^{20}(Q))<16$ .", "However, this contradicts $\\operatorname{dist}_C(v, \\mathsf {bd}^{20}(Q))\\ge 16$ .", "We conclude that $T$ contains at least 3 internal vertices of $Q$ .", "Thus, $G[V(T)\\cup \\lbrace d\\rbrace ]$ is a long hole contained in $\\mathcal {S}^3_Q\\cup D$ .", "But such a long hole had to be considered during the first greedy packing.", "This is a contradiction.", "Now, we verify the second statement.", "Let $a$ be an end of $P$ and $b$ be the neighbour of $a$ in $H$ such that $b \\notin V(P)$ .", "Assume for a contradiction that $b\\in \\mathcal {S}^3_C$ .", "Because $P$ is a connected component of $H\\cap G[\\mathcal {S}^3_Q]$ , $b$ is contained in $\\mathcal {S}^3_{\\overline{Q}}\\setminus \\mathcal {S}^3_Q$ .", "By Lemma REF , $a$ is contained in $\\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}\\setminus \\mathcal {S}^3_{\\mathsf {int}^{20}(Q)}$ .", "But this is not possible because $V(P)\\cap \\mathcal {S}^3_{\\mathsf {bd}^{20}(Q)}=\\emptyset $ .", "Proposition 7.7 Let $\\ell $ be the largest index for which $D_\\ell $ exists in the second greedy packing algorithm.", "Then $G$ contains $\\ell -1$ vertex-disjoint long holes and $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183\\ell $ .", "Moreover, $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long hole.", "It is straightforward to verify that $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183\\ell $ .", "To prove that $G$ contains $\\ell -1$ vertex-disjoint long holes, it suffices to prove that the holes constructed by the second greedy packing are vertex-disjoint.", "Let $H_1$ and $H_2$ be two $D$ -traversing long holes constructed in the algorithm, and let $(A_1, v_1)$ and $(A_2, v_2)$ be the considered pairs when $H_1$ and $H_2$ are constructed, respectively.", "For each $i\\in [2]$ , let $P_i:=H_i-A_i$ , $Q_i:= G[N^{35}_C[v_i]]$ , and let $P_i^*$ be the connected component of $H_i\\cap G[\\mathcal {S}^3_{Q_i}]$ containing $v_i$ .", "By Step (5) of the second greedy packing, we have $\\operatorname{dist}_C(V(Q_1), V(Q_2))\\ge 20$ and by Lemma REF , $\\mathcal {S}^3_{Q_1}$ and $\\mathcal {S}^3_{Q_2}$ are vertex-disjoint.", "In particular, $P_1^*$ and $P_2^*$ are vertex-disjoint.", "Suppose that $V(H_1) \\cap V(H_2) \\ne \\emptyset $ .", "Since $A_1\\cap A_2=\\emptyset $ , we have $V(P_1) \\cap V(P_2) \\ne \\emptyset $ .", "Let $w_1$ be a vertex of $P_1$ that has a neighbor in $P_2$ such that $\\operatorname{dist}_{P_1}(v_1, w_1)$ is minimum, and let $w_2$ be a neighbor of $w_1$ in $P_2$ such that $\\operatorname{dist}_{P_2}(v_2, w_2)$ is minimum.", "By the choice of $w_1$ and $w_2$ , $v_1P_1w_1w_2P_2v_2$ is an induced path.", "Let $R:=v_1P_1w_1w_2P_2v_2$ .", "We claim that for some $i\\in [2]$ , $R$ contains a subpath of $P_i^*$ from $v_i$ to an end of $P_i^*$ and contains the next two vertices in $H_i$ .", "If not, then by Lemma REF there is a path of length at most 3 in $G-D$ between $\\mathcal {S}^3_{\\mathsf {int}^{20}(Q_1)}$ and $\\mathcal {S}^3_{\\mathsf {int}^{20}(Q_2)}$ , and there is a path of length at most 9 in $G-D$ , whose ends are contained in $\\mathsf {int}^{20}(Q_1)$ and $\\mathsf {int}^{20}(Q_2)$ , respectively.", "Since $\\operatorname{dist}_C(\\mathsf {int}^{20}(Q_1), \\mathsf {int}^{20}(Q_2))\\ge 28$ , this contradicts Lemma REF .", "Without loss of generality, $R$ contains a subpath of $P_1^*$ from $v_1$ to an end of $P_1^*$ , say $z$ , and contains the next two vertices $z_1$ and $z_2$ in $H_1$ .", "Following the direction from $z$ to $z_1$ in $R$ , we choose the first vertex $u$ contained in $.$ By Lemma REF , $z_1$ is contained in $V(G)\\setminus \\mathcal {S}^3_C$ and $z$ is contained in $\\mathcal {S}^3_C\\setminus \\mathcal {S}^2_C$ .", "Following the direction from $z$ to $v_1$ in $P_1^*$ , let $y$ be the first vertex contained in $\\mathcal {S}^2_C$ .", "Observe that $yRu$ is an induced path and the next three vertices of $yRu$ after $y$ are not in $\\mathcal {S}^2_C$ .", "However, such a path $yRu$ does not exist by Lemma REF .", "Thus, the holes constructed by the second greedy packing are vertex-disjoint.", "It remains to prove that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no long hole.", "By Proposition REF , $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}})$ has no $D$ -traversing long hole.", "Thus, it suffices to prove that $G-(X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}})$ has no $D$ -avoiding long hole.", "Since $D$ is a clique by Lemma REF and $H$ is a hole, $H$ contains at most 2 vertices of $D$ , and if it contains two vertices of $D$ , then they are adjacent in $H$ .", "So, $H-D$ is a path.", "Since $G-V(C)$ has no long hole, $H$ contains a vertex of $C$ .", "Let $v$ be a vertex of $H$ in $C$ .", "By considering the pair $(V(H)\\cap D, v)$ in the second greedy packing, we should have proceeded one more step, which is a contradiction.", "We are now ready to prove Theorem REF .", "We begin by running the first greedy packing to obtain $X_{\\ref {lem:greedypacking}}$ .", "We may assume that the largest index $\\ell $ in the first greedy packing is at most $k$ , else we obtain $k$ vertex-disjoint long holes by Lemma REF .", "Next, we construct a maximal good sparse ear decomposition $ using Lemma~\\ref {lem:sparsealgo}.By Lemma~\\ref {lem:narrowalgo}, we can either find $ k$ vertex-disjoint appendages of $ , or a vertex set $X_{\\ref {lem:narrowalgo}}$ of size at most $20k$ hitting all appendages.", "Since each appendage contains a long hole, we may assume that $X_{\\ref {lem:narrowalgo}}$ exists.", "We then apply Proposition REF to find $X_{\\ref {prop:Davoid}}$ such that $X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}$ hits all $D$ -avoiding long holes.", "Lastly, we run the second greedy packing to obtain $X_{\\ref {prop:traversing}}$ .", "We may assume that the largest index $\\ell $ in the second greedy packing is at most $k$ , else we are done by Proposition REF .", "Since each of the above steps is performed in polynomial time, the entire algorithm is a polynomial-time algorithm.", "We claim that we may take $X_{\\ref {thm:main2}}= X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}$ .", "By Proposition REF , $X_{\\ref {thm:main2}}$ hits all long holes of $G$ , so it only remains to bound the size of $X_{\\ref {thm:main2}}$ .", "Recall that $X_{\\ref {lem:tunnellemma4}}=X_{\\ref {lem:greedypacking}} \\cup X_{\\ref {lem:sparsealgo}}\\cup X_{\\ref {lem:narrowalgo}}$ and $X_{\\ref {lem:sparsealgo}}=N^{31}_C[B\\cap V(C)] \\cup (B\\setminus V(C))$ , where $B$ is the set of branching points of $.", "By Lemma~\\ref {lem:narrowalgo}, $ |XREF | 20k$ .", "If $ has at least $s_k$ branching points, then by Lemma REF , we can output $k$ vertex-disjoint long holes.", "Thus, we may assume that $|B| < s_k$ .", "Therefore, $|X_{\\ref {lem:sparsealgo}}|=|N^{31}_C[B\\cap V(C)] \\cup (B\\setminus V(C))| \\le 63s_k.$ By Lemma REF , $\\vert X_{\\ref {lem:greedypacking}}\\vert \\le 212k$ , and by Proposition REF , $\\vert X_{\\ref {prop:traversing}}\\vert \\le 183k$ .", "Finally, by Proposition REF , $\\vert X_{\\ref {prop:Davoid}}\\vert \\le 380\\vert X_{\\ref {lem:tunnellemma4}}\\vert $ .", "Putting this all together, $\\vert X_{\\ref {thm:main2}}\\vert =\\vert X_{\\ref {lem:tunnellemma4}}\\cup X_{\\ref {prop:Davoid}}\\cup X_{\\ref {prop:traversing}}\\vert \\le 381(212k+63s_k+20k)+183k = \\mu _k.", "$ Open Problems In this paper, we proved that holes of length at least 6 have the Erdős-Pósa property in $C_4$ -free graphs.", "Our proof is also the shortest known proof that holes have the Erdős-Pósa property.", "Our first open problem is to answer Question REF for larger values of $\\ell $ .", "By making our proof much longer, we believe our methods could answer Question REF affirmatively for $\\ell =7$ , but new ideas are needed for $\\ell \\ge 8$ .", "Next, it is unclear if the $\\mathcal {O}(k^2 \\log k)$ bound in Theorem REF is optimal.", "We conjecture that Theorem REF holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "Conjecture 8.1 There exists a function $f(k)=\\mathcal {O}(k \\log k)$ such that for every $C_4$ -free graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least 6, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least 6.", "By the argument in the introduction, if Conjecture REF is true, then the induced Erdős-Pósa property for cycles of length at least 4 also holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "As noted in [10], this would be tight via a reduction to the $\\Omega (k \\log k)$ lower bound that holds for cycles.", "In general, smaller bounding functions give improved approximation factors for the corresponding packing and covering problems.", "For example, the $\\mathcal {O}(k \\log k)$ bounding function for planar minors proved in [3], currently gives the best approximation algorithm for packing planar minors.", "Our final open problem concerns what other classes of graphs have the induced Erdős-Pósa property.", "For a fixed graph $H$ , the class of $H$ -subdivisions is the class of graphs that can be obtained from $H$ by repeatedly subdividing edges.", "Theorem REF says that the the class of $C_4$ -subdivisions has the induced Erdős-Pósa property, and Theorem REF says that for all $\\ell \\ge 5$ , the class of $C_\\ell $ -subdivisions does not have the induced Erdős-Pósa property.", "Kwon and Raymond [11] investigated for which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property.", "They designed three different types of constructions to show that for various graphs $H$ , the class of $H$ -subdivisions does not have the induced Erdős-Pósa property.", "In each of their constructions, a large complete bipartite induced subgraph always appears.", "This is another reason why investigating the induced Erdős-Pósa property in $C_4$ -free graphs is particularly interesting.", "Our main theorem asserts that $C_6$ -subdivisions have the induced Erdős-Pósa property in the class of $C_4$ -free graphs.", "Question 2 For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property?", "For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property in $C_4$ -free graphs?", "Note that it is not true that for all graphs $H$ , the class of $H$ -subdivisions has the induced Erdős-Pósa property in $C_4$ -free graphs.", "Kwon and Raymond [11] proved that if a forest $F$ has a connected component having at least two vertices of degree at least 3, then the class of $F$ -subdivisions does not have the induced Erdős-Pósa property in $C_4$ -free graphs." ], [ "Open Problems", "In this paper, we proved that holes of length at least 6 have the Erdős-Pósa property in $C_4$ -free graphs.", "Our proof is also the shortest known proof that holes have the Erdős-Pósa property.", "Our first open problem is to answer Question REF for larger values of $\\ell $ .", "By making our proof much longer, we believe our methods could answer Question REF affirmatively for $\\ell =7$ , but new ideas are needed for $\\ell \\ge 8$ .", "Next, it is unclear if the $\\mathcal {O}(k^2 \\log k)$ bound in Theorem REF is optimal.", "We conjecture that Theorem REF holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "Conjecture 8.1 There exists a function $f(k)=\\mathcal {O}(k \\log k)$ such that for every $C_4$ -free graph $G$ and every $k \\in \\mathbb {N}$ , $G$ either contains $k$ vertex-disjoint holes of length at least 6, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least 6.", "By the argument in the introduction, if Conjecture REF is true, then the induced Erdős-Pósa property for cycles of length at least 4 also holds with a $\\mathcal {O}(k \\log k)$ bounding function.", "As noted in [10], this would be tight via a reduction to the $\\Omega (k \\log k)$ lower bound that holds for cycles.", "In general, smaller bounding functions give improved approximation factors for the corresponding packing and covering problems.", "For example, the $\\mathcal {O}(k \\log k)$ bounding function for planar minors proved in [3], currently gives the best approximation algorithm for packing planar minors.", "Our final open problem concerns what other classes of graphs have the induced Erdős-Pósa property.", "For a fixed graph $H$ , the class of $H$ -subdivisions is the class of graphs that can be obtained from $H$ by repeatedly subdividing edges.", "Theorem REF says that the the class of $C_4$ -subdivisions has the induced Erdős-Pósa property, and Theorem REF says that for all $\\ell \\ge 5$ , the class of $C_\\ell $ -subdivisions does not have the induced Erdős-Pósa property.", "Kwon and Raymond [11] investigated for which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property.", "They designed three different types of constructions to show that for various graphs $H$ , the class of $H$ -subdivisions does not have the induced Erdős-Pósa property.", "In each of their constructions, a large complete bipartite induced subgraph always appears.", "This is another reason why investigating the induced Erdős-Pósa property in $C_4$ -free graphs is particularly interesting.", "Our main theorem asserts that $C_6$ -subdivisions have the induced Erdős-Pósa property in the class of $C_4$ -free graphs.", "Question 2 For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property?", "For which graphs $H$ does the class of $H$ -subdivisions have the induced Erdős-Pósa property in $C_4$ -free graphs?", "Note that it is not true that for all graphs $H$ , the class of $H$ -subdivisions has the induced Erdős-Pósa property in $C_4$ -free graphs.", "Kwon and Raymond [11] proved that if a forest $F$ has a connected component having at least two vertices of degree at least 3, then the class of $F$ -subdivisions does not have the induced Erdős-Pósa property in $C_4$ -free graphs." ] ]
2105.11799
[ [ "Automatic Dynamic Parallelotope Bundles for Reachability Analysis of\n Nonlinear Systems" ], [ "Abstract Reachable set computation is an important technique for the verification of safety properties of dynamical systems.", "In this paper, we investigate reachable set computation for discrete nonlinear systems based on parallelotope bundles.", "The algorithm relies on computing an upper bound on the supremum of a nonlinear function over a rectangular domain, which has been traditionally done using Bernstein polynomials.", "We strive to remove the manual step of parallelotope template selection to make the method fully automatic.", "Furthermore, we show that changing templates dynamically during computations cans improve accuracy.", "To this end, we investigate two techniques for generating the template directions.", "The first technique approximates the dynamics as a linear transformation and generates templates using this linear transformation.", "The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates.", "We have implemented our approach in a Python-based tool called Kaa and improve its performance by two main enhancements.", "The tool is modular and use two types of global optimization solvers, the first using Bernstein polynomials and the second using NASA's Kodiak nonlinear optimization library.", "Second, we leverage the natural parallelism of the reachability algorithm and parallelize the Kaa implementation.", "We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems." ], [ "Introduction", "One of the most widely-used techniques for performing safety analysis of nonlinear dynamical systems is reachable set computation.", "The reachable set is defined to be the set of states visited by at least one of the trajectories of the system starting from an initial set.", "Computing the reachable set for nonlinear systems is challenging primarily due to two reasons: First, the tools for performing nonlinear analysis are not very scalable.", "Second, computing the reachable set using set representations involves wrapping error.", "That is, the overapproximation acquired at a given step would increase the conservativeness of the overapproximation for all future steps.", "One of the techniques for computing the overapproximation of reachable sets for discrete time nonlinear systems is to use parallelotope bundles.", "Here, the reachable set is represented as a parallelotope bundle, an intersection of several parallelotopes.", "One of the advantages of this technique is its utilization of a special form of nonlinear optimization problem to overapproximate the reachable set.", "The usage of a specific form of nonlinear optimization mitigates the drawback involved with the scalability of nonlinear analysis.", "However, wrapping error still remains to be a problem for reachability using parallelotope bundles.", "The template directions for specifying these parallelotopes are provided as an input by the user.", "Often, these template directions are selected to be either the cardinal axis directions or some directions from octahedral domains.", "However, it is not clear that the axis directions and octagonal directions are optimal for computing reachable sets.", "Also, even an expert user of reachable set computation tools may not be able to provide a suitable set of template directions for computing reasonably accurate over-approximations of the reachable set.", "Picking unsuitable template directions would only cause the wrapping error to increase, thus increasing the conservativeness of the safety analysis.", "In this paper, we investigate techniques for generating template directions automatically and dynamically.", "That is, instead of providing the template directions to compute the parallelotope, the user just specifies the number of templates and the algorithm automatically generates the template directions.", "We study two techniques for generating the template directions.", "First, we compute a local linear approximation of the nonlinear dynamics and use the linear approximation to compute the templates.", "Second, we generate a specific set of sample trajectories from the set and use principal component analysis (PCA) over these trajectories.", "We observe that the accuracy of the reachable set can be drastically improved by using templates generated using these two techniques.", "For standard nonlinear benchmark systems, we show that generating templates in a dynamic fashion improves the accuracy of the reachable set by two orders of magnitude.", "We demonstrate that even when the size of the initial set increases, our template generation technique returns more accurate reachable sets than both manually-specified and random template directions." ], [ "Related Work", "Reachable set computation of nonlinear systems using template polyhedra and Bernstein polynomials has been first proposed in [9].", "In [9], Bernstein polynomial representation is used to compute an upper bound of a special type of nonlinear optimization problem [17].", "Several improvements to this algorithm were suggested in [10], [27] and [7] extends it for performing parameter synthesis.", "The representation of parallelotope bundles for reachability was proposed in [13] and the effectiveness of using bundles for reachability was demonstrated in [12], [14].", "However, all of these papers used static template directions for computing the reachable set.", "Using template directions for reachable set has been proposed in [26] and later improved in [8].", "Leveraging the principal component analysis of sample trajectories for computing reachable set has been proposed in [30], [5], [28].", "More recently, connections between optimal template directions for reachability of linear dynamical systems and bilinear programming have been highlighted in [19].", "For static template directions, octahedral domain directions [6] remain a popular choice." ], [ "Preliminaries", "The state of a system, denoted as $x$ , lies in a domain $D \\subseteq {\\mathbb {R}}^n$ .", "A discrete-time nonlinear system is denoted as $x^{+} = f(x)$ where $f: {\\mathbb {R}}^{n} \\rightarrow {\\mathbb {R}}^{n}$ is a nonlinear function.", "The trajectory of a system that evolves according to Equation REF , denoted as $\\xi (x_0)$ is a sequence $x_0, x_1, \\ldots $ where $x_{i+1} = f(x_i)$ .", "The $k^{th}$ element in this sequence $x_k$ is denoted as $\\xi (x_0,k)$ .", "Given an initial set $\\Theta \\subseteq {\\mathbb {R}}^n$ , the reachable set at step $k$ , denoted as $\\Theta _k$ is defined as $\\Theta _k = \\lbrace \\xi (x,k)\\: | \\: x \\in \\Theta \\rbrace $ A parallelotope $P$ , denoted as a tuple $\\langle a, G \\rangle $ where $a \\in {\\mathbb {R}}^n$ is called the anchor and $G$ is a set of vectors $\\lbrace g_1, g_2, \\ldots , g_n\\rbrace $ , $\\forall _{1 \\le i \\le n} ~ g_i \\in {\\mathbb {R}}^n$ called generators, represents the set $P = \\lbrace x\\:|\\: \\exists \\alpha _1, \\ldots , \\alpha _n, \\mbox{ such that } 0 \\le \\alpha _i \\le 1, x = a + \\sum _{i=1}^n \\alpha _i g_i \\rbrace .$ We call this representation as the generator representation of the parallelotope.", "We refer to a generator of a specific parallelotope $P$ using dot notation, for example $P.g_1$ .", "For readers familiar with zonotopes [18], [2], a parallelotope is a special form of zonotope where the number of generators $n$ equals the dimensionality of the set.", "One can also represent the parallelotope as a conjunction of half-space constraints.", "In half-space representation, a parallelotope is represented as a tuple $\\langle {\\cal T}, c_{l}, c_{u} \\rangle $ where ${\\cal T}\\in {\\mathbb {R}}^{n \\times n}$ are called template directions and $c_{l}, c_{u} \\in {\\mathbb {R}}^{n}$ such that $\\forall _{1 \\le i \\le n} ~ c_{l}[i] \\le c_{u}[i]$ are called bounds.", "The half-space representation defines the set of states $P = \\lbrace \\: x\\: | \\: c_{l} \\le {\\cal T}x \\le c_{u} \\rbrace .$ Intuitively, the $i^{th}$ constraint in the parallelotope corresponds to an upper and lower bound on the function ${\\cal T}_{i} x$ .", "That is, $c_{l}[i] \\le {\\cal T}_{i}x \\le c_{u}[i]$ .", "The half-plane representation of a parallelotope can be converted into the generator representation by computing $n+1$ vertices $v_1, v_2, \\ldots , v_{n+1}$ of the parallelotope in the following way.", "The vertex $v_1$ is obtained by solving the linear equation $\\Lambda x = c_{l}$ .", "The $j+1$ vertex is obtained by solving the linear equation $\\Lambda x = \\mu _{j}$ where $\\mu _{j}[i] = c_{l}[i]$ when $i \\ne j$ and $\\mu _{j}[j] = c_{u}[j]$ .", "The anchor $a$ of the parallelotope is the vertex $v_1$ and the generator $g_{i} = v_{i+1} - v_{1}$ .", "Consider the xy-plane and the parallelotope $P$ given in half-plane representation as $0 \\le x-y \\le 1$ , $0 \\le y \\le 1$ .", "This is a parallelotope with vertices at $(0,0)$ , $(1,0)$ , $(2,1)$ , and $(1,1)$ .", "In the half-space representation, the template directions of the parallelotope $P$ are given by the directions $[1, -1]$ and $[0, 1]$ .", "The half-space representation in matrix form is given as follows: $\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\le \\begin{bmatrix} 1 & -1 \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} \\le \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}.", "$ To compute the generator representation of $P$ , we need to compute the anchor and the generators.", "The anchor is obtained by solving the linear equations $x-y = 0, y = 0$ .", "Therefore, the anchor $a$ is the vertex at origin $(0,0)$ To compute the two generators of the parallelotope, we compute two vertices of the parallelotope.", "Vertex $v_1$ is obtained by solving the linear equations $x - y = 1, y = 0$ .", "Therefore, vertex $v_1$ is the vertex $(1,0)$ .", "Similarly, vertex $v_2$ is obtained by solving the linear equations $x-y = 0, y = 1$ .", "Therefore, $v_2$ is the vertex $(1,1)$ .", "The generator $g_1$ is the vector $v_1 - a$ , that is $(1,0)- (0,0) = (1,0)$ The generator $g_2$ is the vector $v_2 - a$ , that is $(1,1) - (0,0) = (1,1)$ .", "Therefore, all the points in the paralellotope can be written as $(x,y) = (0,0) + \\alpha _1 (1,0) + \\alpha _2(1,1)$ , $0 \\le \\alpha _1, \\alpha _2 \\le 1$ .", "A parallelotope bundle $Q$ is a set of parallelotopes $\\lbrace P_1, \\ldots , P_m\\rbrace $ .", "The set of states represented by a parallelotope bundle is given as the intersection $Q = \\bigcap _{i=1}^m P_i.$ Often, the various parallelotopes in a bundle share common template directions.", "In such cases, the conjunction of all the parallelotope constraints in a bundle $Q$ is written as $c_{l}^Q \\le T^Q x \\le c_{u}^Q$ .", "Notice that the number of upper and lower bound half-space constraints in this bundle are stricly more than $n$ in these cases, i.e., $T^Q \\in R^{m \\times n}$ where $m>n$ .", "Each parallelotope in such a bundle is represented as a subset of constraints in $c_{l}^Q \\le T^Q x \\le c_{u}^Q$ .", "These types of bundles are often considered in the literature.", "[9], [10], [14] Alternatively, we consider parallelotope bundles where the consisting parallelotopes do not share template directions.", "We consider such bundles because we generate the $n$ template directions automatically at each step.", "The basic building block in this work is a conservative overapproximation to a constrained nonlinear optimization problem with a box domain.", "Consider a nonlinear function $h : {\\mathbb {R}}^n \\rightarrow {\\mathbb {R}}$ and the optimization problem denoted as $\\mathsf {optBox(h)}$ as $\\max ~ h(x) \\\\s.t.", "~~ x \\in [0,1]^{n}.\\nonumber $ For computing the reachable set of a nonlinear system, we need an upper bound for the optimization problem.", "Several techniques using interval arithmetic and Bernstein polynomials have been developed in the recent past [1], [17], [22], [29]." ], [ "Reachability Algorithm", "In this work, we develop parallelotope reachability algorithms that are automatic with dynamic parallelotopes.", "The state of the art, in contrast, is manual, where the user specifies a set of parallelotope directions at the beginning.", "The parallelotopes are also static and do not change during the course of the computation.", "In this section, we detail the modifications to the algorithm and present their correctness arguments." ], [ "Manual Static Algorithm", "We first present the original algorithm[10] where the user manually specifies the number of parallelotopes and a set of static directions for each parallelotope.", "Recall the system is $n$ -dimensional with dynamics function $f: {\\mathbb {R}}^{n} \\rightarrow {\\mathbb {R}}^{n}$ .", "The parallelotope bundle $Q$ is specified as a collection of $m$ template directions ${\\cal T}^{Q} \\in {\\mathbb {R}}^{m \\times n} (m > n)$ and the set of constraints that define each of the member parallelotopes.", "Another input to the algorithm is the initial set, given as a parallelotope $P_0$ .", "When the initial set is a box, $P_0$ consists has axis-aligned template directions.", "The output of the algorithm is, for each step $k$ , the set $\\overline{\\Theta }_k$ , which is an overapproximation of the reachable set at step $k$ , $\\Theta _k \\subseteq \\overline{\\Theta }_k$ .", "The high-level pseudo-code is written in Algorithm REF .", "The algorithm simply calls TransformBundle for each step, producing a new parallelotope bundle computed from the previous step's bundle.", "To compute the image of $Q$ , the algorithm computes the upper and lower bounds of $f(x)$ with respect to each template direction.", "Since computing the maximum value of $f(x)$ along each template direction on $Q$ is computationally difficult, the algorithm instead computes the maximum value over each of the constituent parallelotopes and uses the minimum of all these maximum values.", "The TransformBundle operation works as follows.", "Consider a parallelotope $P$ in the bundle $Q$ .", "From the definition, it follows that $Q \\subseteq P$ .", "Given a template direction ${\\cal T}_i$ , the maximum value of ${\\cal T}_{i} f(x)$ for all $x \\in Q$ is less than or equal to the maximum value of ${\\cal T}_{i} f(x)$ for all $x \\in P$ .", "Similar argument holds for the minimum value of $T_{i} f(x)$ for all $x \\in Q$ .", "To compute the upper and lower bounds of each template direction ${\\cal T}_{i} f(x)$ , for all $x \\in P$ , we perform the following optimization.", "$\\max ~ {\\cal T}_i^{P} \\cdot f(x) \\\\s.t.", "~~ x \\in P.\\nonumber $ Given that $P$ is a parallelotope, all the states in $P$ can be expressed as a vector summation of anchor and scaled generators.", "Let $\\langle a, G \\rangle $ be the generator representation of $P$ .", "The optimization problem given in Equation REF would then transform as follows.", "$\\max ~ {\\cal T}_i \\cdot f(a + \\Sigma _{i=1}^{n} \\alpha _i g_i) \\\\s.t.", "~~ \\overline{\\alpha }\\in [0,1]^{n}.\\nonumber $ Equation REF is a form of $\\mathsf {optBox({\\cal T}_{i} \\cdot f)}$ over $[0,1]^n$ .", "One can compute an upper-bound to the constrained nonlinear optimization by invoking one of the Bernstein polynomial or interval-arithmetic-based methods.", "Similarly, we compute the lowerbound of ${\\cal T}_{i}f(x)$ for all $x \\in P$ by computing the upperbound of $-1 \\times {\\cal T}_{i}f(x)$ .", "We iterate this process (i.e., computing the upper and lower bound of $T_{i}f(x)$ ) for each parallelotope in the bundle $Q$ .", "Therefore, the tightest upper bound on $T_{i}f(x)$ over $Q$ is the least of the upper bounds computed from each of the parallelotopes.", "A similar argument holds for lower bounds of $T_{i}f(x)$ over $Q$ .", "Therefore, the image of the bundle $Q$ will be the bundle $Q^{\\prime }$ where the upper and lower bounds for templates directions are obtained by solving several constrained nonlinear optimization problems.", "The parallelotope bundle $Q^{\\prime }$ computed using TransformBundle (Algorithm REF ) is a sound overapproximation of the image of bundle $Q$ w.r.t the dynamics $x^{+} = f(x)$ .", "[t] KwInputInput KwOutputOutput AppendPCAAppendNewPCATemplates AppendLinAppendNewLinAppTemplates TransformBundTransformBundle UpdateTempUpdateTemplates GetSuppGetSupportPoints PropPointsPropagatePointsOneStep PCAPCA ApproxLinearTransApproxLinearTrans SetLifeSpanSetLifeSpan AddTemptoBundAddTemplateToBundle RemoveTempRemoveTempFromBund FunProc: Dynamics $f$ , Initial Parallelotope $P_0$ , Step Bound $S$ , Template Dirs ${\\cal T}$ , indexes for parallelotopes $I$ Reachable Set Overapproximation $\\overline{\\Theta }_k$ for each step $k$ $Q_0 = \\lbrace P_0 \\rbrace $ for$k \\in [1, 2, \\ldots , S]$ $Q_k$ = ($f$ , $Q_{k-1}$ , ${\\cal T}$ ) $\\overline{\\Theta }_k = Q_k$ $\\overline{\\Theta }_1 \\ldots \\overline{\\Theta }_S$ $f$ , $Q$ , ${\\cal T}$ $Q^{\\prime } \\leftarrow \\lbrace \\rbrace $ ; $c_{u} \\leftarrow +\\infty $ ; $c_{l} \\leftarrow -\\infty $ foreach parallelotope $P$ in $Q$ $\\langle a, G \\rangle \\leftarrow \\mathsf {generatorRepresentation}(P)$ foreach template direction ${\\cal T}_i$ in the template directions ${\\cal T}$ $c_{u}^{\\prime }[i] \\leftarrow \\mathsf {min}\\lbrace \\mathsf {optBox({\\cal T}_{i} \\cdot f)}, c_{u}^{\\prime }[i] \\rbrace $ (Equation REF ) $c_{l}^{\\prime }[i] \\leftarrow \\mathsf {max}\\lbrace -1 \\times \\mathsf {optBox(-1\\times {\\cal T}_{i} \\cdot f)}, c_{l}^{\\prime }[i] \\rbrace $ Construct parallelotopes $P_{1}^{\\prime }, \\ldots , P_{k}^{\\prime }$ from ${\\cal T}, c_{l}^{\\prime }, c_{u}^{\\prime }$ and indexes from $I$ $Q^{\\prime } \\leftarrow \\lbrace P_{1}^{\\prime }, \\ldots , P_{k}^{\\prime }\\rbrace $ $Q^{\\prime }$ Reachable set computation using manual and static templates." ], [ "Automatic Dynamic Algorithm", "The proposed automatic dynamic algorithm does not require the user to provide the set of template directions ${\\cal T}$ ; instead it generates these templates directions automatically at each step.", "We use two techniques to generate such template directions, first: computing local linear approximations of the dynamics and second, performing principal component analysis (PCA) over sample trajectories.", "To do this, we first sample a set of points in the parallelotope bundle called support points and propagate them to the next step using the dynamics $f$ .", "Support points are a subset of the vertices of the parallelotope that either maximize or minimize the template directions.", "Intuitively, linear approximations can provide good approximations when the dynamics function is a time-discretization of a continuous system.", "In this case, for small time steps a nonlinear function can be approximated fairly accurately by a linear transformation.", "We use the support points as a data-driven approach to find the best-fit linear function to use.", "If the dynamics of a system is linear, i.e., $x^{+} = Ax$ , the image of the parallelotope $c_{l} \\le {\\cal T}x \\le c_{u}$ , is the set $c_{l} \\le {\\cal T}\\cdot A^{-1} x \\le c_{u}$ .", "Therefore, given the template directions of the initial set as ${\\cal T}_0$ , we compute the local linear approximation of the nonlinear dynamics and change the template directions by multiplying them with the inverse of the approximate linear dynamics.", "The second technique for generating template directions performs principal component analysis over the images of the support points.", "Using PCA is a reasonable choice as it produces orthonormal directions that can construct a rotated box for bounding the points.", "Observe that in general, the dynamics is nonlinear and therefore, the reachable set could be non-convex.", "On the other hand, a parallelotope bundle is always a convex set.", "To mitigate this discrepancy, we can improve accuracy of this representation by considering more template directions.", "For this purpose, we use a notion of template lifespan, where we use the linear approximation and/or PCA template directions not only from the current step, but also from the previous $L$ steps.", "We will demonstrate the effectiveness and tune each of the options (PCA / linear approximation as well as lifespan option) in our evaluation.", "The new approach is given in Algorithm REF .", "In this algorithm, instead of fixing the set of templates, we compute one set of templates (that is, a collection of $n$ template directions), using linear approximation of the dynamics and PCA.", "The algorithm makes use of helper function hstack, which converts column vectors into a matrix (as shown in Equation REF provided in Example ).", "The notation $M_{*,i}$ is used to refer to the $i^{th}$ column of matrix $M$ .", "The Maximize function takes in a parallelotope bundle $Q$ and direction vector $v$ (one of the template directions), and returns the point $p \\in Q$ that maximizes the dot product $v \\cdot p$ (for computing support points).", "This can be computed efficiently using linear programming.", "The ApproxLinearTrans function computes the best approximation of a linear transformation given a list of points before and after the one-step transformation $f$ .", "More specifically, given a matrix $X$ of points before applying the transformation $f$ , a matrix of points after the transformation $X^{\\prime }$ , we perform a least-squares fit for the linear transition matrix $A$ such that $X^{\\prime } \\approx AX$ .", "This can be computed by $A = X^{\\prime } X^\\dagger $ , where $X^\\dagger $ is the Moore-Penrose pseudoinverse of $X$ .", "The PCA function returns a set of orthogonal directions using principal component analysis of a set of points.", "Finally, TransformBundle is the same as in Algorithm REF .", "[t] KwInputInput KwOutputOutput CreatePCACreatePCA CreateLinCreateLin TransformBundTransformBundle UpdateTempUpdateTemplates GetSuppGetSupportPoints PropPointsPropagatePointsOneStep PCAPCA ApproxLinearTransApproxLinearTrans SetLifeSpanSetLifeSpan ExtractDirectionsExtractDirections AddTemptoBundAddTemplateToBundle MaximizeMaximize hstackhstack RemoveTempRemoveTempFromBund FunProc: Dynamics $f$ , Initial Parallelotope $P_0$ , Step Bound $S$ Reachable Set Overapproximation $\\overline{\\Theta }_k$ at each step $k$ $Q_0 = \\lbrace P_0 \\rbrace $ ${\\cal T}= (P_0.", "{\\cal T}_1, \\ldots , P_0.", "{\\cal T}_n)$ Init Template Directions for$k \\in [1, 2, \\ldots , S]$ $P_{supp}$ = ($Q_{k-1}$ ) (support points of $Q_{k-1}$ ) $P_{prop}$ = ($P_{supp}$ , $f$ ) (image of support points) $A$ = ($P_{supp}$ , $P_{prop}$ ) ${\\cal T}= {\\cal T}\\cdot A^{-1}$ ${\\cal T}_k^\\text{lin} = \\lbrace \\lbrace {\\cal T}_{*,1} , \\ldots , {\\cal T}_{*,n} \\rbrace \\rbrace $ ${\\cal T}_k^\\text{pca} = \\lbrace (P_{prop}) \\rbrace $ ${\\cal T}_k = {\\cal T}_k^\\text{lin} \\cup {\\cal T}_k^\\text{pca}$ For lifespan $L$ , instead call with ${\\cal T}_k \\cup {\\cal T}_{k-1} \\cup \\ldots \\cup {\\cal T}_{k-L}$ $Q_k$ = ($f$ , $Q_{k-1}, {\\cal T}_k$ ) $\\overline{\\Theta }_k \\leftarrow Q_k$ $\\overline{\\Theta }_1 \\ldots \\overline{\\Theta }_S$ $Q$ $P_{supp} = \\emptyset $ for$P \\in Q$ for$i \\in [1, 2, \\ldots , n]$ $P_{supp} = P_{supp} \\cup ~ (Q, P.T_i) \\cup ~ (Q, -P.T_i)$ $P_{supp}$ Automatic, Dynamic Reachability Algorithm Algorithm REF computes the dynamic templates for each time step $k$ .", "Line REF computes the linear approximation of the nonlinear dynamics and this linear approximation is used to compute the new template directions according to this linear transformation in Line REF .", "The PCA directions of the images of support points is computed in line REF .", "For the time step $k$ , the linear and PCA templates are given as ${\\cal T}_{k}^{lin}$ and ${\\cal T}_{k}^{pca}$ , respectively.", "To improve the accuracy of the reachable set, we compute the overapproximation of the reachable set with respect to not just the template directions at the current step, but with respect to other template directions for time steps that are within the lifespan $L$ ." ], [ "Evaluation", "We evaluate the efficacy of our dynamic parallelotope bundle strategies with our tool, Kaa [21].", "Kaa is written in Python and relies on the numpy library for matrix computations, sympy library for all symbolic subsitution, and scipy, matplotlib for plotting the reachable sets and computing the volume for lower-dimensional systems.", "The optimization procedure for finding the direction offets is performed through the Kodiak library.", "Finally, parallelization of the offset calculation procedures is implemented through the multiprocessing module.", "To estimate volume of reachable sets, we employ two techniques for estimating volume of individual parallelotope bundles.", "For systems of dimension fewer than or equal to three, we utilize scipy's convex hull routine.", "For higher-dimensional systems, we employ the volume of the tightest enveloping box around the parallelotope bundle.", "The total volume estimate of the overapproximation will be the sum of all the bundles' volume estimates." ], [ "Model Dynamics", "For benchmarking, we select six non-linear models with polynomial dynamics.", "Benchmarks against more general dynamics can be found in the appendix of the expanded verison.", "Many of these models are also implemented in Sapo [12], a previous tool exploring reachability with static parallelotope bundles.", "In these cases, we directly compare the performance of our dynamic strategies with the Sapo's static parallelotopes.", "To provide meaningful comparisions, we set the number of dynamic parallelotopes to be equal to the number of static ones excluding the initial box.", "Here, diagonal directions are defined to be vectors created by adding and subtracting distinct pairs of unit axis-aligned vectors from each other.", "By diagonal parallelotopes, we refer to parallelotopes defined only by axis-aligned and diagonal directions.", "Similarly, diagonal parallelotope bundles are parallelotope bundles solely consisting of diagonal parallelotopes.", "Sapo primarily utilizes static diagonal parallelotope bundles to perform its reachability computation.", "Note that the initial box, which is defined only through the axis-aligned directions, is contained in every bundle.", "For our experiments, we are concerned with the effects of additional static or dynamic parallelotopes added alongside the initial box.", "We refer to these parallelotopes as non-axis-aligned parallelotopes.", "In two dimensions, $\\mathbb {R}^2$ , we have the two unit axis-aligned directions, $[1,0]^T, [0,1]^T$ .", "The diagonal directions will then be $ [1,1]^T, \\; [1,-1]^T$ Consequently, the diagonal parallelotopes will precisely be defined by unique pairs of these directions, giving us a total ${4 \\atopwithdelims ()2} = 6$ diagonal parallelotopes.", "Table REF summarizes five standard benchmarks used for experimentation.", "The last seven-dimensional COVID supermodel is explained in the subsequent subsection below.", "Table: Benchmark models and relevant informationCOVID Supermodel: We benchmark our dynamic strategies with the recently introduced COVID supermodel [3], [25].", "This model is a modified SIR model accounting for the possibility of asymptomatic patients.", "These patients can infect susceptible members with a fixed probability.", "The dynamics account for this new group and its interactions with the traditional SIR groups.", "$\\begin{split}S_A^{\\prime } & = S_A -(\\beta S_A(A+I))\\cdot \\Delta \\\\S_I^{\\prime } & = S_I -(\\beta S_I (A + I))\\cdot \\Delta \\\\A^{\\prime } & = A + (\\beta S_I(A+I) - \\gamma I)\\cdot \\Delta \\\\I^{\\prime } & = I + (\\beta S_I (A+I) - \\gamma I)\\cdot \\Delta \\\\R_A^{\\prime } & = R_A + (\\gamma A)\\cdot \\Delta \\\\R_I^{\\prime } & = R_I + (\\gamma I)\\cdot \\Delta \\\\D^{\\prime } & = D + (\\eta I)\\cdot \\Delta \\end{split}$ where the variables denote the fraction of a population of individuals designated as Susceptible to Asymptomatic $(S_A)$, Susceptible to Symptomatic $(S_I)$, Asymptomatic (A), Symptomatic (I), Removed from Asymptomatic $(R_A)$, Removed from Symptomatic $(R_I)$, and Deceased (D).", "We choose the parameters ($\\beta = 0.25, \\gamma =0.02, \\eta =0.02$ ) where $\\beta $ is the probablity of infection, $\\gamma $ is the removal rate, and $\\eta $ is the mortality rate.", "The parameters are set based on figures shown in [3].", "The discretization step is chosen to be $\\Delta = 0.1$ and the initial box is set to be following dimensions: $S_A \\in [0.69, 0.7], \\, S_I \\in [0.09, 0.1], \\, A \\in [0.14, 0.15], \\, I \\in [0.04, 0.05], \\, R_A = 0,\\, R_I = 0, \\, D = 0$ ." ], [ "Accuracy of Dynamic Strategies", "The results of testing our dynamic strategies against static ones are summarized in Table REF .", "For models previously defined in Sapo, we set the static parallelotopes to be exactly those found in Sapo.", "If a model is not implemented in Sapo, we simply use the static parallelotopes defined in a model of equal dimension.", "To address the unavailability of a four-dimensional model implemented in Sapo, we sampled random subsets of five static non-axis-aligned parallelotopes and chose the flowpipe with smallest volume.", "A cursory analysis shows that the number of possible templates with diagonal directions grows with $O(n^n)$ with the number of dimensions and hence an exhaustive search on optimal template directions is infeasible.", "From our experiments, we conclude there is no universal optimal ratio between the number of dynamic parallelotopes defined by PCA and Linear Approxiation directions which perform well on all benchmarks.", "In Figure REF , we demonstrate two cases where varying the ratio imparts differing effects.", "Observe that using parallelotopes defined by linear approximation directions is more effective than those defined by PCA directions in the Vanderpol model whereas the Neuron model shows the opposite trend.", "Figure: Effect of varying ratio between the number of PCA and Linear Approximation parallelotopes.", "The Vanderpol (left) and the FitzHugh-Nagumo Neuron (right) phase plots are shown to illustrate differing effects of varying the PCA/LinApp ratio.", "The initial set for the Vanderpol model is set to x∈[0,0.05],y∈[1.95,2]x \\in [0,0.05], \\, y \\in [1.95,2]" ], [ "Performance under Increasing Initial Sets", "A key advantage of our dynamic strategies is the improved ability to control the wrapping error naturally arising from larger initial sets.", "Figure REF presents charts showcasing the effect of increasing initial sets on the total flowpipe volume.", "We vary the initial box dimensions to gradually increase the box's volume.", "We then plot the total flowpipe volume after running the benchmark.", "The same initial boxes are also used in computations using Sapo's static parallelotopes.", "The number of parallelotopes defined by PCA and Linear Approximation directions were chosen based on best performance as seen in Table REF .", "We remark that our dynamic strategies perform better than static ones in controlling the total flowpipe volume as the initial set becomes larger.", "On the other hand, the performance of static parallelotopes tends to degrade rapidly as we increase the volume of the initial box." ], [ "Performance against Random Static Templates", "We additionally benchmark our dynamic strategies against static random parallelotope bundles.", "We sample such parallelotopes in $n$ dimensions by first sampling a set of $n$ directions uniformly on the surface of the unit $(n-1)$ -sphere, then defining our parallelotope using these sampled directions.", "We sample twenty of these parallelotopes for each trial and average the total flowpipe volumes.", "As shown in Figure REF , our best-performing dynamic strategies consistently outperform static random strategies for all tested benchmarks." ], [ "Conclusions", "In this paper, we investigated two techniques for generating templates dynamically: first using linear approximation of the dynamics, and second using PCA.", "We demonstrated that these techniques improve the accuracy of reachable set by an order of magnitude when compared to static or random template directions.", "We also observed that both these techniques improve the accuracy of the reachable sets for different benchamrks.", "In future, we intend to investigate Koopman linearization techniques for computing alternative linear approximation template directions [4].", "We also wish to investigate the use of a massively parallel implementation using HPC hardware such as GPUs for optimizing over an extremely large number of parallelotopes and their template directions.", "This is inspired by the approach behind the recent tool PIRK [11]." ], [ "Acknowledgements", "Parasara Sridhar Duggirala and Edward Kim acknowledge the support of the Air Force Office of Scientific Research under award number FA9550-19-1-0288 and FA9550-21-1-0121 and National Science Foundation (NSF) under grant numbers CNS 1935724 and CNS 2038960.", "Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force or the National Science Foundation.", "Table: Tables presenting upper bounds on the total reachable set volume by strategy.", "The static directions are retrieved and/or inspired from Sapo models of equal dimension for benchmarking.", "The best performing strategy is highlighted in bold." ] ]
2105.11796
[ [ "Fluctuation-dissipation relations for thermodynamic distillation\n processes" ], [ "Abstract The fluctuation-dissipation theorem is a fundamental result in statistical physics that establishes a connection between the response of a system subject to a perturbation and the fluctuations associated with observables in equilibrium.", "Here we derive its version within a resource-theoretic framework, where one investigates optimal quantum state transitions under thermodynamic constraints.", "More precisely, we first characterise optimal thermodynamic distillation processes, and then prove a relation between the amount of free energy dissipated in such processes and the free energy fluctuations of the initial state of the system.", "Our results apply to initial states given by either asymptotically many identical pure systems or arbitrary number of independent energy-incoherent systems, and allow not only for a state transformation, but also for the change of Hamiltonian.", "The fluctuation-dissipation relations we derive enable us to find the optimal performance of thermodynamic protocols such as work extraction, information erasure and thermodynamically-free communication, up to second-order asymptotics in the number $N$ of processed systems.", "We thus provide a first rigorous analysis of these thermodynamic protocols for quantum states with coherence between different energy eigenstates in the intermediate regime of large but finite $N$." ], [ "Introduction", "Thermodynamics has been profoundly triumphant by impacting the natural sciences and allowing the development of technologies that go from coolers to spaceships.", "As a theory of macroscopic systems in equilibrium, it presents us with a compelling picture of what state transformations are allowed in terms of a small number of macroscopic quantities, such as work and entropy [1], [2].", "The drawback of the macroscopic description is that thermodynamics inevitably deals with average quantities, and as systems get smaller, fluctuations of these quantities become increasingly relevant, requiring a new description [3], [4].", "Going beyond the original scenario of equilibrium thermodynamics led scientists to investigate fluctuations around these averages and their impact on the system dynamics.", "This line of research dates back to Einstein and Smoluchowski, who derived the connection between fluctuations and dissipation effects for Brownian particles [5], [6].", "Now, it is well known that near-equilibrium, linear response theory provides a general proof of the fluctuation-dissipation theorem, which states that the response of a given system when subject to an external perturbation is expressed in terms of the fluctuation properties of the system in thermal equilibrium [7], [8].", "The theoretical description underlying the fluctuation-dissipation relations is usually expressed in terms of the stochastic character of thermodynamic variables.", "This approach is strongly motivated since it is experimentally viable [9], [10].", "On the other hand, a complementary approach is based on resource theories [11], [12], [13], [14].", "It aims to go beyond the thermodynamic limit and the assumption of equilibrium, and is often presented as an extension of statistical mechanics to scenarios with large fluctuations, the so-called single-shot statistical mechanics [15], [16].", "A natural question is then whether fluctuation-dissipation relations are present in such a resource-theoretic description.", "Although important insights have been obtained in trying to connect the information-theoretic and fluctuation theorem approaches [17], [18], they have, so far, not been explicitly related to dissipation.", "Resource-theoretic analysis of dissipation was performed independently [19], [20], [21], [22], where the authors investigated irreversibility of thermodynamic processes due to finite-size effects.", "However, these results were obtained for quasi-classical case of energy-incoherent states, and so they are not able to account for quantum effects that come into play when dealing with even smaller systems, when fluctuations around thermodynamic averages are no longer just thermal in their origin.", "This work makes a step forward towards a genuinely quantum framework characterising optimal thermodynamic state transformations and links fluctuations with free energy dissipation.", "We investigate a special case of state interconversion processes known as thermodynamic distillations.", "These are thermodynamic processes in which a given initial quantum system is transformed, with a given transformation error, to a pure energy eigenstate of the final system.", "In particular, we focus on the initial system consisting of $N$ non-interacting subsystems that are either energy-incoherent and non-identical (in different states and with different Hamiltonians), or pure and identical.", "Within this setting, our main results are given by two fluctuation-dissipation theorems.", "The main message behind both theorems is a precise relation between the free energy fluctuations of the initial state and the minimal amount of free energy dissipated in a thermodynamic distillation process for a given transformation error (see Box ).", "The first theorem applies to arbitrary number of independent energy-incoherent states, while the second one holds for asymptotically many identical pure states.", "Furthermore, our findings also provide new tools to study approximate transformations and corresponding asymptotic rates.", "Here, we not only extend previous distillation results [20] to non-identical systems, but also to genuinely quantum states in superposition of different energy eigenstates.", "[label=myautocounter]Fluctuation-dissipation relation for thermodynamic distillation processes In the optimal thermodynamic process of $\\epsilon $ -approximate transformation from many independent non-equilibrium systems into systems without fluctuations of free energy, the dissipated free energy satisfies $F_{\\rm diss}^{\\rm tot}= a(\\epsilon )\\, \\sigma ^{\\rm tot}(F),$ where $\\sigma ^{\\rm tot}(F)$ is the free energy fluctuation in the initial state, and $a(\\epsilon )=-\\Phi ^{-1}(\\epsilon )$ with $\\Phi ^{-1}(x)$ being an inverse of a Gaussian cumulative distribution function.", "[node distance=0pt] [ width=0.9height=0.6xmin=0, xmax=1, xtick=0,0.5,1, ymin=-2.2, ymax=2.2, ytick=-1,0,1, xlabel=$\\epsilon $ ,ylabel=$a(\\epsilon )$ , ] +[ color=black, no marks, line width = 1pt ] table[x=x,y=y]boxplotdata.txt; ); [thick, line width=0.4pt, dashed,gray] coordinates (0, 0) (1, 0) node (VL); We have proved such a statement for many independent systems in arbitrary incoherent states, as well as for many independent and identical systems in the same pure state.", "We conjecture that this is true for independent systems in arbitrary mixed states.", "Our results allow us for a rigorous study of important thermodynamic protocols.", "First of all, we extend the analysis of work extraction to the regime of not necessarily identical incoherent states and to pure states.", "By directly applying our main results, we obtain a second-order asymptotic expression for the optimal transformation error while extracting a given amount of work per copy of the initial subsystem.", "Moreover, we also verify the accuracy of the obtained expression by comparing it with the numerically optimised work extraction process.", "As a second application, we analyse the optimal cost of erasing $N$ independent bits prepared in an arbitrary state.", "In this case, we obtained the optimal transformation error of the erasure process as a function of invested work.", "The last application we consider is the optimal thermodynamically-free communication rate, i.e., the optimal encoding of information into a quantum system without using any extra thermodynamic resources.", "Applying our theorems gives us the optimal number of messages that can be encoded into a quantum system in a thermodynamically free-way, which we show to be directly related to the non-equilibrium free energy of the system.", "This result can be interpreted as the inverse of the Szilard engine, as in this process we use the ability to perform work to encode information.", "Furthermore, our results connect the fluctuations of free energy and the optimal average decoding error.", "The paper is organised as follows.", "We start with recalling the resource-theoretic approach to thermodynamics in Section  and introducing the necessary concepts used in the applications.", "In Section , we state our two main results concerning the fluctuation-dissipation relation for incoherent and coherent states, discuss their thermodynamic interpretation and apply them to three thermodynamic protocols of work extraction, information erasure and thermodynamically-free communication.", "The derivation of the main results can be found in Section .", "Finally, we conclude with an outlook in Section .", "In order to formally define the thermodynamic distillation process, we first need to identify the set of thermodynamically-free states and transformations.", "By definition, a state of the system that is in equilibrium with a thermal environment $E$ at inverse temperature $\\beta $ is a free state.", "Therefore, for a system described by a Hamiltonian $H$ , the only free state is given by the thermal Gibbs state $\\gamma = \\frac{e^{-\\beta H}}{Z}, \\qquad Z = \\mathrm {Tr}\\left( e^{-\\beta H} \\right) \\, .$ The set of free transformations that we consider is given by thermal operations [11], [13], [23], which act on the system as $ {\\cal E} (\\rho )=\\mathrm {Tr}_{E^{\\prime }}\\left( U\\left(\\rho \\otimes \\gamma _E\\right)U^{\\dagger } \\right),$ where $U$ is a joint unitary acting on the system and the thermal environment $E$ that is described by a Hamiltonian $H_E$ and is prepared in a thermal Gibbs state $\\gamma _E$ at inverse temperature $\\beta $ .", "Moreover, $U$ is commuting with the total Hamiltonian of the system and bath, $[U, H\\otimes \\mathbb {1}_E+ \\mathbb {1}\\otimes H_E] = 0$, and we discard any subsystem $E^{\\prime }$ of the joint system of the considered system and environment.", "A thermodynamic distillation process is a thermodynamically free transformation from a general initial system described by a Hamiltonian $H$ and prepared in a state $\\rho $ , to a target system described by a Hamiltonian $\\tilde{H}$ and in a state $\\tilde{\\rho }$ that is an eigenstate of $\\tilde{H}$ .In fact, all of our results apply to a slightly more general setting with target states being proportional to the Gibbs state on their support, e.g.", "for $\\tilde{\\rho }=\\frac{\\tilde{\\gamma }_k}{\\tilde{\\gamma }_k+\\tilde{\\gamma }_l}|\\tilde{E}_k\\rangle \\!\\langle \\tilde{E}_k|+\\frac{\\tilde{\\gamma }_l}{\\tilde{\\gamma }_k+\\tilde{\\gamma }_l}|\\tilde{E}_l\\rangle \\!\\langle \\tilde{E}_l|$, where $| {\\tilde{E}_i} \\rangle $ denotes the eigenstate of $\\tilde{H}$ and $\\tilde{\\gamma }_i$ is its thermal occupation.", "An $\\epsilon $ -approximate thermodynamic distillation process from $(\\rho ,H)$ to $(\\tilde{\\rho },\\tilde{H})$ is a thermal operation that transforms the initial system $(\\rho ,H)$ to the final system with Hamiltonian $\\tilde{H}$ and in a state $\\epsilon $ away from $\\tilde{\\rho }$ in the infidelity distance $\\delta $ , $\\delta (\\rho _1,\\rho _2):=1-\\left(\\mathrm {Tr}{\\sqrt{\\sqrt{\\rho _1}{\\rho _2}\\sqrt{\\rho _1}}}\\right)^2.$ We say that $\\rho $ is energy incoherent if it is a convex combination of eigenstates of $H$ .", "In this paper, we will study the distillation process from $N$ independent initial systems to arbitrary target systems, e.g., to $\\tilde{N}$ independent target systems as illustrated in Fig.", "REF .", "In particular, we will be interested in the asymptotic behaviour for large $N$ .", "Thus, our distillation setting is specified by a family of initial and target systems indexed by a natural number $N$ .", "Each initial system $(\\rho ^N,H^N)$ consists of $N$ non-interacting subsystems with the total Hamiltonian $H^N$ and a state $\\rho ^N$ given by $H^N=\\sum _{n=1}^N H^N_{n},\\qquad \\rho ^N=\\bigotimes _{n=1}^N \\rho ^N_{n},$ while each target system is described by an arbitrary Hamiltonian $\\tilde{H}^N$ and a state $\\tilde{\\rho }^N=|\\tilde{E}_k^{N}\\rangle \\!\\langle \\tilde{E}_k^{N}|$ , with $| {\\tilde{E}_k^{N}} \\rangle $ being some eigenstate $k$ of $\\tilde{H}^N$ .", "A typical example of this setting is when initial and target systems are given by copies of independent and identical subsystems.", "More precisely, in this case, the family of initial systems is given by $H^N$ with $H^N_n=H$ and $\\rho ^N=\\rho ^{\\otimes N}$ , while the family of target systems is given by $\\tilde{N}$ subsystems, each with a Hamiltonian $\\tilde{H}$ and in a state $|\\tilde{E}_k\\rangle \\!\\langle \\tilde{E}_k|$ .", "One is then interested in the optimal distillation rate $\\tilde{N}/N$ as $N$ tends to infinity.", "However, we will investigate a more general setting, allowing the subsystems to differ in both state and Hamiltonian, as long as the initial state is uncorrelated.", "Figure: Thermodynamic distillation process.", "The arrow depicts the existence of a thermal operation transforming NN independent initial systems to N ˜\\tilde{N} independent target systems.", "The colours representing the initial and target systems indicate that each subsystem is described by a different Hamiltonian and prepared in a different state." ], [ "Work extraction", "One of the manifestations of the second law of thermodynamics is that for a system interacting with a bath in thermal equilibrium, the maximum amount of work that it can perform (that can be extracted from the system) is bounded by the difference $\\Delta F$ between its initial and final free energy.", "Traditionally, the free energy $F = U-S/\\beta $ has been defined only for states at thermal equilibrium, with $U$ denoting the internal energy and $S$ the entropy of the system.", "However, taking into account its operational meaning, one can extend its definition to investigate also the case of non-equilibrium states.", "More precisely, the relative entropy, $D(\\rho \\Vert \\gamma ):=\\mathrm {Tr}\\left( \\rho (\\log \\rho -\\log \\gamma ) \\right),$ can be interpreted as a non-equilibrium generalisation of the free energy difference between a state $\\rho $ and a thermal state $\\gamma $ .", "It quantifies the maximum amount of work that can be extracted on average from the system in an out-of-equilibrium state [24], [25].", "Generally, work extraction protocols are based on controlling and changing the external parameters that define the Hamiltonian of the system [26], [27].", "Within a resource-theoretic treatment [23], [28], however, we avoid using an external agent.", "Therefore, we explicitly model the ancillary battery system $B$ , intending to transform it from an initial pure energy state to another pure energy state with higher energy, see Fig.", "REF .", "A continuous Hamiltonian usually describes the battery, but we can as well choose a Hamiltonian with the discrete spectrum, as long as its energy differences coincide with the amount of work we want to extract.", "Without loss of generality, we focus on a two-level battery system described by a Hamiltonian $H^N_B$ with eigenstates $| {0} \\rangle _B$ and $| {1} \\rangle _B$ corresponding to energies 0 and $W^N_{\\rm ext}$ , respectively.", "The possibility of extracting the amount of work equal to $W^N_{\\rm ext}$ from $N$ subsystems described by a Hamiltonian $H^N$ and prepared in a state $\\rho ^N$ is then equivalent to the existence of a thermodynamic distillation process $ {\\cal E} (\\rho ^N \\otimes |0\\rangle \\!\\langle 0|_B) = |1\\rangle \\!\\langle 1|_B \\,,$ from $(N+1)$ initial subsystems described by a Hamiltonian $H^N+H^N_B$ to a target subsystems with a Hamiltonian $H^N_B$ .", "If only an $\\epsilon $ -approximate distillation with transformation error $\\epsilon _N$ is possible, then $\\epsilon _N$ directly measures the quality of extracted work, i.e., with probability $1-\\epsilon _N$ we end up with a battery system in an excited state of energy $W^N_{\\rm ext}$ .", "Figure: Work extraction process.", "Extraction of work W ext N W^N_{\\rm ext} from NN subsystems described by a Hamiltonian H N H^N and prepared in a state ρ N \\rho ^N can be seen as a particular case of thermodynamic distillation process ℰ {\\cal E} involving a battery system BB.", "The battery is modelled by a two-level system with energy levels |0〉 B | {0} \\rangle _B and |1〉 B | {1} \\rangle _B corresponding to energies 0 and W ext N W^N_{\\rm ext}, respectively.", "The initial system is given by the investigated NN subsystems with a battery in the ground state |0〉 B | {0} \\rangle _B, while the target system is given just by the battery in the excited state |1〉 B | {1} \\rangle _B." ], [ "Information erasure", "The connection between information and thermodynamics is as old as the thermodynamic theory itself, going back to the thought experiment known as Maxwell's Demon [29].", "It suggests that if one has information about the particles' positions and momenta, one can reduce the entropy of a gas of particles without investing work, and thus violate the second law of thermodynamics.", "However, the recognition of the thermodynamic significance of information is perhaps best captured by the Szilard's engine [30], a simple setup that converts information into work.", "As in the Maxwell's demon example, the Szilard engine can overcome the second law of thermodynamics whenever some information about the state of the system is available.", "During the resolution of this puzzle, it became clear that thermodynamics imposes physical constraints on information processing.", "In particular, the second law can be reformulated as a statement that no thermodynamic process can result solely in the erasure of information.", "Every time information is erased, the erasure process is accompanied by a fundamental heat cost, i.e., an entropy increase in the environment [31].", "Alternatively, the Landauer's Principle [32] tells us that the erasure process has an unavoidable energetic cost, with the minimum possible amount of energy required to erase a completely unknown bit of information given by $\\log 2 / \\beta $ (see Ref.", "[33] in this context, where a more nuanced view on Szilard engine and Landauer erasure is presented).", "Similarly to the case of work extraction, the erasure process can also be formulated as a particular type of thermodynamic distillation process.", "The $N$ bits of information that one wants to erase can be represented by $N$ two-level systems in a state $\\rho ^N$ with a trivial Hamiltonian.", "We also add the two-level battery system $B$ initially in an excited state $| {1} \\rangle _B$ of energy $W^N_{\\rm cost}$ to measure the energetic cost of erasure.", "Then, the erasure process resetting the state $\\rho ^N$ to a fixed state $| {0} \\rangle ^{\\otimes N}$ is possible while investing $W^N_{\\rm cost}$ work, if there exists the following distillation process: $ {\\cal E} (\\rho ^N \\otimes |1\\rangle \\!\\langle 1|_B) = |0\\rangle \\!\\langle 0|^{\\otimes N}\\otimes |0\\rangle \\!\\langle 0|_B \\,,$ with the initial and target Hamiltonians being identical.", "The transformation error quantifies the quality of erasure, and the process is illustrated in Fig.", "REF .", "Figure: Information erasure.", "The NN bits of information to be erased are represented by NN subsystems in a state ρ N \\rho ^N with a trivial Hamiltonian H N H^N.", "The process is performed by attaching a battery system BB in an excited state |1〉 B | {1} \\rangle _B with energy W cost N W^N_{\\rm cost}, which measures the energetic cost of erasure.", "The erasure process resets the state ρ N \\rho ^N to a fixed state |0〉 ⊗N | {0} \\rangle ^{\\otimes N}, and de-excites the battery system." ], [ "Thermodynamically-free communication", "Since thermodynamics is closely linked with information processing, one can also study thermodynamic aspects of communication.", "A traditional communication scenario in which Alice wants to encode and transmit classical information to Bob over a quantum channel consists of the following three steps [34].", "First, she encodes a message $m \\in \\lbrace 1, ..., M \\rbrace $ by preparing a quantum system in a state $\\rho _m$ .", "Then, she sends it to Bob via a noisy quantum channel $\\mathcal {N}$ .", "Finally, Bob decodes the original message by performing an optimal measurement on $\\mathcal {N}(\\rho _m)$ .", "Crucially, in this standard scenario, both Alice and Bob are completely unconstrained, meaning that they can employ all encodings and decodings for free, and the only thing beyond their control is the noise channel $ {\\cal N} $ .", "Recently, a modification of this scenario was introduced that allows one to quantify the thermodynamic cost of communication [35], [36].", "More precisely, it is assumed for simplicity that Alice and Bob are connected via a noiseless channel, and Bob's decoding is still unconstrained.", "However, Alice is constrained to thermodynamically-free encodings, meaning that encoded states $\\rho _m$ can only arise from thermal operations acting on a given initial state $\\rho $ , interpreted as an information carrier.", "Physically, this means that Alice obeys the second law of thermodynamics, in the sense that the encoding channel is constrained to use no thermodynamic resources other than what the information carrier $\\rho $ initially has.", "We illustrate this process in Fig.", "REF .", "Now, the central question is: what is the optimal number of messages $M(\\rho ,\\epsilon _{\\mathrm {avg}})$ that can be encoded into $\\rho $ in a thermodynamically-free way, so that the average decoding error is smaller than $\\epsilon _{\\mathrm {avg}}$ ?", "We will investigate the case when the information carrier is given by $N$ independent systems in a state $\\rho ^N$ and with a Hamiltonian $H^N$ , as specified in Eq.", "(REF ).", "Then, instead of asking for $M(\\rho ^N,\\epsilon _{\\mathrm {avg}})$ , we can equivalently ask for the optimal encoding rate: $R(\\rho ^N, \\epsilon _{\\mathrm {avg}}) := \\frac{\\log [M(\\rho ^N, \\epsilon _{\\mathrm {avg}})]}{N} \\,.$ As we will explain later in the paper, the optimal thermodynamically-free encodings (i.e., the ones that allow one to achieve the optimal rate $R$ ) can be chosen to be given by thermodynamic distillation processes.", "Through this connection and our results on optimal distillation processes, we will derive second-order asymptotic expansion of $R(\\rho ^N, \\epsilon _{\\mathrm {avg}})$ working for large $N$ .", "Figure: Thermodynamically-free encoding.", "The thermal encoding of information can be captured by a thermodynamic distillation process by considering NN independent subsystems in a state ρ N \\rho ^N and with a Hamiltonian H N H^N as an information carrier.", "The sender encodes a message m∈{1,...,M}m \\in \\lbrace 1, ..., M \\rbrace into it by applying a thermal operation ℰ m {\\cal E} _{m}, and the receiver decodes the original message by performing a measurement on ℰ m (ρ N ) {\\cal E} _m(\\rho ^N)." ], [ "Information-theoretic notions and their thermodynamic interpretation", "Finally, before we proceed to present our results, let us introduce the necessary information-theoretic quantities together with their thermodynamic interpretation.", "For any $d$ -dimensional quantum state $\\rho $ , we define the relative entropy $D$ between $\\rho $ and a thermal Gibbs state $\\gamma $ , together with the corresponding relative entropy variance $V$ and the function $Y$ related to relative entropy skewness [37], [20], [38]: $D(\\rho \\Vert \\gamma ):=&\\mathrm {Tr}\\left( \\rho \\left(\\log \\rho -\\log \\gamma \\right) \\right),\\\\V(\\rho \\Vert \\gamma ):=&\\mathrm {Tr}\\left( \\rho \\left(\\log \\rho -\\log \\gamma -D(\\rho \\Vert \\gamma )\\right)^2 \\right),\\\\Y(\\rho \\Vert \\gamma ):=&\\mathrm {Tr}\\left( \\rho \\left|\\log \\rho -\\log \\gamma - D(\\rho \\Vert \\gamma )\\right|^3 \\right).$ It is clear from the above definitions that we are dealing with the average, variance and the absolute third moment of the random variable $\\log \\rho -\\log \\gamma $.", "As already mentioned, the average of this random variable, $D(\\rho \\Vert \\gamma )$ , can be interpreted as the non-equilibrium free energy of the system since $\\frac{1}{\\beta }D(\\rho \\Vert \\gamma )= \\mathrm {Tr}\\left( \\rho H \\right)-\\frac{S(\\rho )}{\\beta }+\\frac{\\log Z}{\\beta },$ where $S(\\rho ):=&-\\mathrm {Tr}\\left( \\rho \\log \\rho \\right)$ is the von Neumann entropy.", "The higher moments can then be understood as fluctuations of the non-equilibrium free energy content of the system.", "This is most apparent for pure states $\\rho =|\\psi \\rangle \\!\\langle \\psi |$ , as $V$ then simply describes energy fluctuations of the system: $\\frac{1}{\\beta ^2}V(|\\psi \\rangle \\!\\langle \\psi |\\Vert \\gamma )&=\\langle {\\psi } | H^2| {\\psi } \\rangle -\\langle {\\psi } | H| {\\psi } \\rangle ^2.$ Moreover, as noted in Ref.", "[20], when $\\rho =\\gamma ^{\\prime }$ is a thermal distribution at some different temperature $T^{\\prime }\\ne T$, the expression for $V$ becomes $V(\\gamma ^{\\prime }\\Vert \\gamma )=\\left(1-\\frac{T^{\\prime }}{T}\\right)^2 \\cdot \\frac{c_{T^{\\prime }}}{k_B},$ where $c_{T^{\\prime }}=\\frac{\\partial }{\\partial T^{\\prime }} \\mathrm {Tr}\\left( \\gamma ^{\\prime }H \\right)$ is the specific heat capacity of the system in a thermal state at temperature $T^{\\prime }$ , and $k_B$ is the Boltzmann constant.", "Now, for the initial system $(\\rho ^N,H^N)$ , we introduce the following notation for averaged free energy and free energy fluctuations: $\\bar{F}^N := \\frac{1}{\\beta N}\\sum _{n=1}^N D(\\rho ^{N}_n\\Vert \\gamma ^{N}_n),\\\\\\sigma ^2(F^N) := \\frac{1}{\\beta ^2 N}\\sum _{n=1}^N V(\\rho ^{N}_n\\Vert \\gamma ^{N}_n),\\\\\\kappa ^3(F^N) := \\frac{1}{\\beta ^3 N}\\sum _{n=1}^N Y(\\rho ^{N}_n\\Vert \\gamma ^{N}_n).$ We also introduce $F^N_{\\rm diss} := \\frac{1}{\\beta N}\\left(\\sum _{n=1}^N D(\\rho ^N_n\\Vert \\gamma ^N_n)-D(\\tilde{\\rho }^N\\Vert \\tilde{\\gamma }^N)\\right),$ which describes the average amount of free energy that is dissipated in the distillation process per subsystem of the initial system (note that if $F^N_{\\rm diss}$ is negative then the free energy instead of being dissipated is added to the system).", "For the quantities introduced in Eqs.", "(REF )-() and (REF ) we will drop the superscript $N$ to denote their value in the aysmptotic limit $N\\rightarrow \\infty $ , e.g.", "$\\bar{F}:=\\lim _{N\\rightarrow \\infty }\\bar{F}^N$.", "Let us also make two final technical comments.", "First, we only consider families of initial systems for which the limits of $\\sigma (F^N)$ and $\\kappa (F^N)$ as $N\\rightarrow \\infty $ are well-defined and non-zero.", "Second, in what follows, we will use a shorthand notation with $\\simeq $ , $\\lesssim $ and $\\gtrsim $ denoting equalities and inequalities up to terms of order $o(1/\\sqrt{N})$ ." ], [ "Fluctuation-dissipation relations", "Our first main result connects the optimal transformation error $\\epsilon $ of a thermodynamic distillation process from incoherent systems with the amount of free energy dissipated during that process and the free energy fluctuations of the initial state of the system.", "Theorem 1 (Fluctuation-dissipation relation for incoherent states) For a distillation setting with energy-incoherent initial states, the transformation error $\\epsilon _N$ of the optimal $\\epsilon $ -approximate distillation process in the asymptotic limit is given by $\\lim _{N\\rightarrow \\infty }\\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(-\\frac{F^N_{\\rm diss}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right),$ where $\\Phi $ denotes the cumulative normal distribution function.", "Moreover, for any $N$ there exists an $\\epsilon $ -approximate distillation process with the transformation error $\\epsilon _N$ bounded by $\\epsilon _N&\\le \\Phi \\left(-\\frac{F^N_{\\rm diss}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right)+\\frac{C\\kappa ^3(F^N)}{\\sigma ^3(F^N)}\\cdot \\frac{1}{\\sqrt{N}},$ where $C$ is a constant from the Berry-Esseen theorem that is bounded by $0.4097\\le C \\le 0.4748.$ We prove the above theorem in Sec.", "REF , and here we will briefly discuss its scope and consequences.", "First, from Eq.", "(REF ) it is clear that if the amount of dissipated free energy per subsystem, $F^N_{\\rm diss}$ , vanishes faster than $1/\\sqrt{N}$ , then the optimal transformation error $\\epsilon _N\\rightarrow 1/2$ .", "On the other hand, if $F^N_{\\rm diss}$ vanishes slower than $1/\\sqrt{N}$ , then the error either vanishes (when the target system has lower free energy than the initial one) or approaches 1 (in the opposite case).", "Thus, the only non-trivial behaviour of the optimal transformation error happens when $F^N_{\\rm diss}=\\frac{\\alpha }{\\sqrt{N}}+o\\left(\\frac{1}{\\sqrt{N}}\\right)$ for some constant $\\alpha $ describing the level of free energy dissipation.", "For the sake of interpretation we may now write the error in terms of total dissipated free energy $F^{\\rm tot}_{\\rm diss}=N F^N_{\\rm diss}$ and total free energy fluctuation $\\sigma ^{\\rm tot}(F)=\\sqrt{N}\\sigma (F^N)$ , to arrive at $\\lim _{N\\rightarrow \\infty }\\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(-\\frac{F_{\\rm diss}^{\\rm tot}}{\\sigma ^{\\rm tot}(F)}\\right).$ Then, both quantities scale as $\\sqrt{N}$ (since $F^{\\rm tot}_{\\rm diss}\\simeq \\alpha \\sqrt{N}$).", "Thus, the error is specified by the ratio between dissipated free energy and free energy fluctuations.", "As a result, the same level of free energy dissipation in two optimal distillation processes will lead to a smaller transformation error for the process, with the initial state exhibiting smaller free energy fluctuations.", "Alternatively, for two processes with the same optimal success probability, a distillation process from a state with smaller free energy fluctuations will lead to smaller free energy dissipation.", "As a particular example consider a battery-assisted distillation process, i.e.", "a thermodynamic transformation from $(\\rho ^N\\otimes |1\\rangle \\!\\langle 1|_B,H^N+H_B)$ to $(\\tilde{\\rho }^N\\otimes |0\\rangle \\!\\langle 0|_B,H^N+H_B)$, where the energy gap of the battery system $B$ is $W^N_{\\mathrm {cost}}$ .", "Now, the quality of transformation from $\\rho ^N$ to $\\tilde{\\rho }^N$ (measured by transformation error $\\epsilon _N$ ) depends on the amount of work $W^N_{\\mathrm {cost}}$ that we invest into the process.", "As expected, to achieve $\\epsilon \\le 1/2$ , we need to invest at least the difference of free energies $[D(\\tilde{\\rho }^N\\Vert \\tilde{\\gamma }^N)-D(\\rho ^N\\Vert \\gamma ^N)]/\\beta $ .", "However, Theorem REF tells us how much more work is needed to decrease the transformation error to a desired level: the more free energy fluctuations there were in $\\rho ^N$ , the more work we need to invest.", "Let us also compare Theorem REF to the results presented in Ref. [20].", "There, the authors studied the incoherent thermodynamic interconversion problem between identical copies of the initial system, $\\rho ^{\\otimes N}$ , and identical copies of the target system, $\\tilde{\\rho }^{\\otimes \\tilde{N}}$ .", "Here, for the price of the reduced generality of the target state (it has to be an eigenstate of the target Hamiltonian), we obtained a three-fold improvement.", "First, our result applies to general independent systems, not only to identical copies.", "Second, the Hamiltonians of the initial and target systems can vary, which is particularly important for applications like work extraction or thermodynamically-free communication.", "Finally, we went beyond the second-order asymptotic result and found a single-shot upper bound on the optimal transformation error $\\epsilon _N$ , Eq.", "(REF ), that holds for any finite $N$ .", "Thus, even in the finite $N$ regime, one can get a guarantee on the transformation error that is approaching the asymptotically optimal value as $N\\rightarrow \\infty $ .", "Our second main result connects the optimal transformation error $\\epsilon _N$ of a thermodynamic distillation process from $N$ identical copies of a pure quantum system with the amount of free energy dissipated during that process and the energy fluctuations of the initial state of the system.", "To formally state it, we first need to introduce a technical notion of a Hamiltonian with incommensurable spectrum.", "Given any two energy levels, $E_i$ and $E_j$ , of such a Hamiltonian, there does not exist natural numbers $m$ and $n$ such that $m E_i=n E_j$ .", "We then have the following result.", "Theorem 2 (Fluctuation-dissipation relation for identical pure states) For a distillation setting with $N$ identical initial systems, each in a pure state $|\\psi \\rangle \\!\\langle \\psi |$ and described by the same Hamiltonian $H$ with incommensurable spectrum, the transformation error $\\epsilon _N$ of the optimal $\\epsilon $ -approximate distillation process in the asymptotic limit is given by $\\lim _{N\\rightarrow \\infty }\\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(-\\frac{F^N_{\\rm diss}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right),$ where $\\Phi $ denotes the cumulative normal distribution function.", "Moreover, the result still holds if both the initial and target systems get extended by an ancillary system with an arbitrary Hamiltonian $H_A$ , with the initial and target states being some eigenstates of $H_A$ .", "We prove the above theorem in Sec.", "REF , and here we will only add one comment to the previous discussion.", "Namely, since for a pure state the free energy fluctuations are just the energy fluctuations (recall Eq.", "(REF )), and because in the considered scenario all pure states are identical, we have $\\sigma ^2(F^N)=\\left\\langle \\psi \\vphantom{H^2\\psi } \\right| H^2 \\left| \\psi \\vphantom{\\psi H^2} \\right\\rangle -\\left\\langle \\psi \\vphantom{H\\psi } \\right| H \\left| \\psi \\vphantom{\\psi H} \\right\\rangle ^2=:\\sigma ^2(H).$ Analogously to the incoherent case, the only non-trivial behaviour of the optimal transformation error happens when $F^N_{\\rm diss}$ is of the form from Eq.", "(REF ).", "Thus, the optimal transformation error is specified by the ratio $\\alpha /\\sigma (H)$ between the level of dissipated free energy in the distillation process and energy fluctuations of the initial state." ], [ "Optimal work extraction", "As the first application of our fluctuation-dissipation relations, we focus on work extraction process from a collection of $N$ non-interacting subsystems with Hamiltonians $H^N_n$ and in incoherent states $\\rho ^N_n$ .", "As already described in Sec.", "REF , this is just a particular case of a thermodynamic distillation process.", "We only need to note that the pure battery state does not contribute to fluctuations $\\sigma $ and $\\kappa $ , and that the difference between non-equilibrium free energies of the ground and excited battery states is just the energy difference $W^N_{\\rm ext}$ .", "Then, Theorem REF tells us that, in the asymptotic limit, the optimal transformation error for extracting the amount of work $w^N_{\\rm ext}:=W^N_{\\rm ext}/N$ per copy of the initial subsystem is $\\lim _{N\\rightarrow \\infty } \\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(\\frac{w_{\\rm ext}^N-\\bar{F}^N}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right).$ We thus clearly see that again, we have three cases dependent on the amount of dissipated work per subsystem, $(w_{\\rm ext}^N-\\bar{F}^N)$ .", "To get the asymptotic error different from 0, 1 or 1/2, the extracted work $w^N_{\\rm ext}$ has to be of the form $w^N_{\\rm ext}=\\bar{F}^N - \\frac{\\alpha }{\\sqrt{N}}+o\\left(\\frac{1}{\\sqrt{N}}\\right),$ for some constant $\\alpha $ .", "Combining the above two equations yields the following second-order asymptotic expression for the extracted work per copy of the system: $w_{\\rm ext}\\simeq \\bar{F} + \\frac{\\sigma (F)}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon ).$ Thus, for a fixed quality of extracted work measured by $\\epsilon $ , more work can be extracted from states with smaller free energy fluctuations (assuming that the average free energy $\\bar{F}$ is fixed).", "This is a direct generalisation of the result obtained in Ref.", "[20] to a scenario with non-identical initial systems and with a cleaner interpretation of the error in the battery system.", "We present the comparison between our bounds and the numerically optimised work extraction processes in Fig.", "REF .", "Similarly, by employing Theorem REF , we can investigate optimal work extraction process from a collection of $N$ non-interacting subsystems with identical Hamiltonians $H$ and each in the same pure state $|\\psi \\rangle \\!\\langle \\psi |$ .", "We simply need to choose the ancillary system $A$ to be the battery $B$ with energy splitting $W^N_{\\mathrm {ext}}$ and the initial and target states to be given by $| {0} \\rangle _B$ and $| {1} \\rangle _B$ .", "Also, since all systems are in identical pure states and have the same Hamiltonian, we have $\\bar{F}&=\\bar{F}^N=\\langle H\\rangle _\\psi +\\frac{\\log Z}{\\beta },\\\\\\sigma (F)&=\\sigma (F^N)=\\langle H^2\\rangle _\\psi -\\langle H\\rangle ^2_\\psi ,$ where we used a shorthand notation $\\langle \\cdot \\rangle _\\psi =\\left\\langle \\psi \\vphantom{\\cdot \\psi } \\right| \\cdot \\left| \\psi \\vphantom{\\psi \\cdot } \\right\\rangle $.", "As a result, the optimal amount of work extracted per one copy of a pure quantum system up to second-order asymptotic expansion is given by: $w_{\\rm ext}\\simeq \\langle H\\rangle _\\psi +\\frac{\\log Z}{\\beta } + \\frac{\\langle H^2\\rangle _\\psi -\\langle H\\rangle _\\psi ^2}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon ).$" ], [ "Optimal cost of erasure", "In order to obtain the optimal work cost of erasing $N$ two-level systems prepared in incoherent states $\\rho ^N_n$ , we apply Theorem REF analogously as in the previous section, but this time to the scenario described in Sec.", "REF .", "We then get the optimal transformation error in the erasure process given by $\\lim _{N\\rightarrow \\infty } \\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(\\frac{\\frac{1}{\\beta }s^N-w^N_{\\mathrm {cost}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right),$ where $s^N:=\\frac{1}{N}S(\\rho ^N)$ is the average entropy of the initial state, and $w^N_{\\mathrm {cost}}=W^N_{\\mathrm {cost}}/N$ is the invested work cost per subsystem.", "Using analogous reasoning as in the case of work extraction, we can now obtain the second-order asymptotics for the cost of erasure: $w_{\\mathrm {cost}}\\simeq \\frac{s}{\\beta } - \\frac{\\sigma (F)}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon ),$ where $s:=\\lim _{N\\rightarrow \\infty } s^N$ .", "Let us make two brief comments on the above result.", "First, we only considered the application of the incoherent result, Theorem REF , as in the case of trivial Hamiltonians, the erasure of a pure state $|\\psi \\rangle \\!\\langle \\psi |^{\\otimes N}$ is free (because all unitary transformations are then thermodynamically-free).", "Of course, our results straightforwardly extend to non-trivial Hamiltonians, but we believe that the simple case we described above is most illustrative and recovers the spirit of the original Landauer's erasure scenario.", "Second, since the maximally mixed initial state has vanishing free energy fluctuations, $\\sigma (F^N)=0$ , we cannot directly apply our result (that relates fluctuations of the initial state to dissipation) to get the erasure cost of $N$ completely unknown bits of information.", "However, using the tools described in Sec.", ", it is straightforward to show that in this case, the exact expression (working for all $N$ ) for the erasure cost is given by $w^N_{\\mathrm {cost}}= \\frac{1}{\\beta }\\left[ \\log 2 - \\frac{\\log (1-\\epsilon )}{N}\\right].$ Thus, for the case of zero error one recovers the Landauer's cost of erasure [39]." ], [ "Optimal thermodynamically-free communication rate", "Finally, we now explain how our fluctuation-dissipation relations, Theorems REF  and REF , allow one to obtain the optimal thermodynamically-free encoding rate into a collection of $N$ identical subsystems in either incoherent or pure states.", "We simply choose the target system to be a single $M$ -dimensional quantum system with a trivial Hamiltonian $\\tilde{H}=0$ that is prepared in any of the degenerate eigenstates of $\\tilde{H}$ .", "Note that the non-equilibrium free energy of such a target system is given by $\\frac{1}{\\beta }D(\\tilde{\\rho }^N\\Vert \\tilde{\\gamma }^N)=\\frac{1}{\\beta }\\log M.$ Our theorems then tell us that in the asymptotic limit, the optimal transformation error $\\epsilon $ in the considered distillation process is given by $\\lim _{N\\rightarrow \\infty } \\epsilon _N = \\lim _{N\\rightarrow \\infty }\\Phi \\left(\\frac{\\frac{\\log M}{N}-\\beta \\bar{F}^N}{\\beta \\sigma (F^N)}\\cdot \\sqrt{N}\\right).$ Rewriting the above, we get the following second-order asymptotic behaviour: $\\frac{\\log M}{N}\\simeq \\beta \\bar{F}+\\frac{\\beta \\sigma (F)}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon ).$ Now, the distillation process above can be followed by unitaries that map between $M$ degenerate eigenstates of $\\tilde{H}$ that we will simply denote $| {1} \\rangle ,\\dots ,| {M} \\rangle $ .", "Crucially, note that such unitaries are thermodynamically-free because they act in a fixed energy subspace.", "Such a protocol then allows one to encode $M$ messages into $M$ states $\\sigma _i$ , each one being $\\epsilon $ -close in infidelity to $| {i} \\rangle $ for $i\\in \\lbrace 1,\\dots ,M\\rbrace $ .", "Decoding the message using a measurement in the eigenbasis of $\\tilde{H}$ then leads to the average decoding error $\\epsilon _{\\mathrm {avg}}$ satisfying: $1-\\epsilon _{\\mathrm {avg}}:=\\frac{1}{M}\\sum _{i=1}^M \\left\\langle i \\vphantom{\\sigma _ii} \\right| \\sigma _i \\left| i \\vphantom{i\\sigma _i} \\right\\rangle =1-\\epsilon ,$ so that $\\epsilon _{\\mathrm {avg}}=\\epsilon $ .", "Using the communication protocol described above, we then get the following asymptotic lower bound on the optimal thermodynamically-free encoding rate into a state $\\rho ^N$ (recall Eq.", "(REF )): $R(\\rho ^N,\\epsilon _{\\mathrm {avg}}) \\gtrsim \\beta \\bar{F}+\\frac{\\beta \\sigma (F)}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon _{\\mathrm {avg}}).$ The above lower bound is exactly matching the upper bound for $R(\\rho ^N,\\epsilon _{\\mathrm {avg}})$ recently derived in Ref.", "[36] for a slightly different scenario with $\\rho ^N_n=\\rho $ and $H^N_n=H$ for all $n$ , with $\\tilde{H}^N=H^N$ , and with Gibbs-preserving operation instead of thermal operations.", "However, the proof presented there can be easily adapted to work in the current case if we keep the first restriction, i.e., when the initial state is $\\rho ^N=\\rho ^{\\otimes N}$ and all initial subsystems have equal Hamiltonians.", "We explain in detail how to adapt that proof in Appendix , where we also explain what technical result concerning hypothesis testing relative entropy needs to be proven in order to make the proof also work when subsystems are not identical.", "Here we conclude that $R(\\rho ^{\\otimes N},\\epsilon _{\\mathrm {avg}}) \\simeq D(\\rho \\Vert \\gamma )+\\frac{\\sqrt{V(\\rho \\Vert \\gamma )}}{\\sqrt{N}}\\Phi ^{-1}(\\epsilon _{\\mathrm {avg}}),$ where $\\rho $ is either a pure or incoherent state.", "The above result can be thermodynamically interpreted as the inverse of the Szilard engine.", "While the Szilard engine converts bits of information into work, the protocol studied here employs the free energy of the system (i.e., the ability to perform work) to encode bits of information.", "While the asymptotic result was recently proven in Ref.", "[35], here we proved that this relation is deeper as it also connects fluctuations of free energy to the optimal average decoding error.", "In what follows, we first introduce the mathematical formalism used to study the incoherent distillation process.", "We then use it to prove Theorem REF .", "Finally, we also prove Theorem REF by first mapping it to an equivalent incoherent problem and then using the formalism of incoherent distillations." ], [ "Distillation conditions via approximate majorisation", "A state of a $d$ -dimensional quantum system $\\rho $ will be called energy-incoherent if it commutes with the Hamiltonian of the system, i.e., when it is block-diagonal in the energy eigenbasis.", "Such a state can be equivalently represented by a $d$ -dimensional probability vector $p$ given by the eigenvalues of $\\rho $ .", "Since the thermal Gibbs state $\\gamma $ is energy-incoherent, it can be represented by a vector of thermal occupations $\\gamma $ .", "Moreover, an energy eigenstate $|E_k\\rangle \\!\\langle E_k|$ can be represented by a sharp state $s_k$ , with $(s_k)_j=\\delta _{jk}$ .", "In order to formulate the solution to the thermodynamic interconversion problem for incoherent states we will need two concepts: approximate majorisation and embedding.", "First, given two $d$ -dimensional probability vectors $p$ and $q$ , we say that $p$ majorises $q$ , and write $p \\succ q$ , if and only if [40] $\\forall k : \\:\\: \\sum _{j=1}^k p_j^\\downarrow \\ge \\sum _{j=1}^k q_j^\\downarrow ,$ where $p^{\\downarrow }$ denotes the vector $p$ rearranged in a decreasing order.", "Moreover, we say that $p$ $\\epsilon $ -post-majorises $q$  [20], and write $p \\succ _\\epsilon q$ , if $p$ majorises $r$ which is $\\epsilon $ -close in the infidelity distance to $q$ , i.e., $1-F(q,r)\\le \\epsilon , \\qquad F(q,r):=\\left(\\sum _{j=1}^d \\sqrt{q_j r_j}\\right)^2.$ Second, we express the thermal distribution $\\gamma $ as a probability vector with rational entries, $\\gamma &=\\left[\\frac{D_1}{D},\\dots ,\\frac{D_{d}}{D}\\right],$ with $D$ and $D_k$ being integers.", "Now, the embedding map is defined as a transformation that sends a $d$ -dimensional probability distribution $p$ to a $D$ -dimensional probability distribution $\\hat{p}$ in the following way [13]: $\\hat{p}=\\left[\\phantom{\\frac{i}{i}}\\!\\!\\!\\!\\right.\\underbrace{\\frac{p_1}{D_1},\\dots ,\\frac{p_1}{D_1}}_{D_1\\mathrm {~times}},\\,\\dots \\,,\\underbrace{\\frac{p_{d}}{D_{d}},\\dots ,\\frac{p_{d}}{D_{d}}}_{D_{d}\\mathrm {~times}}\\left.\\phantom{\\frac{i}{i}}\\!\\!\\!\\!\\right].", "$ Observe that the embedded version of a thermal state $\\gamma $ is a maximally mixed state over $D$ states $\\eta :=\\frac{1}{D}[1,\\dots ,1];$ and the embedded version of a sharp state $s_k$ is a flat state $f_k$ that is maximally mixed over a subset of $D_k$ entries, with zeros otherwise: $\\!\\!\\hat{s}_{k}=f_{k}:=\\left[\\phantom{\\frac{i}{i}}\\!\\!\\!\\!\\right.\\underbrace{0,\\dots ,0}_{\\sum _{j=1}^{k-1}D_j},\\,\\underbrace{1\\dots 1}_{D_k}\\,,\\underbrace{0,\\dots ,0}_{\\sum _{j=k+1}^d D_j}\\left.\\phantom{\\frac{i}{i}}\\!\\!\\!\\!\\right]\\!.", "$ We can now state the crucial theorem based on Ref.", "[13] and concerning thermodynamic interconversion for incoherent states.", "Theorem 3 (Corollary 7 of Ref.", "[20]) For the initial and target system with the same thermal distribution $\\gamma $ , there exists a thermal operation mapping between energy-incoherent states $p$ and a state $\\epsilon $ -close to $q$ in infidelity distance, if and only if $\\hat{p}\\succ _\\epsilon \\hat{q}$ .", "Despite the fact that in our case, we want to study the general case of initial and final systems with different Hamiltonians, with a little bit of ingenuity we can still use the above theorem.", "Namely, we consider a family of total systems composed of the first $N$ subsystems with initial Hamiltonians $H^N_n$ , and the remaining part described by the target Hamiltonian $\\tilde{H}^N$ .", "We choose initial states of the total system on the first $N$ subsystems to be a general product of incoherent states $p^N_n$ , while the remaining part to be prepared in a thermal equilibrium state $\\tilde{\\gamma }^N$ corresponding to $\\tilde{H}^N$ .", "Since Gibbs states are free, this setting is thermodynamically equivalent to having just the first $N$ systems with Hamiltonians $H^N_n$ and in states $p_n^N$ .", "Moreover, for target states of the total system, we choose thermal equilibrium states $\\gamma ^N_n$ for the first $N$ subsystems, and sharp states $\\tilde{s}_{k}^N$ of the Hamiltonian $\\tilde{H}^N$ for the remaining part.", "Again, this is thermodynamically equivalent to having just the system with Hamiltonian $\\tilde{H}^N$ and in a state $\\tilde{s}_{k}^N$ .", "Thus, employing Theorem REF , an $\\epsilon $ -approximate distillation process for incoherent states exists if and only if: $\\left(\\bigotimes _{n=1}^N \\hat{p}^N_n \\otimes \\hat{\\tilde{\\gamma }}^N\\right) \\succ _\\epsilon \\left(\\bigotimes _{n=1}^{N}\\hat{\\gamma }^N_n\\otimes \\hat{\\tilde{s}}_{k}^N \\right) .$ This way, using a single fixed Hamiltonian, we can encode transformations between different Hamiltonians.", "Let us introduce the following shorthand notation: $\\hat{P}^N:=\\bigotimes _{n=1}^N \\hat{p}^{N}_n,\\qquad \\hat{G}^N:=\\bigotimes _{n=1}^N {\\hat{\\gamma }}^N_n=\\bigotimes _{n=1}^N {\\eta }^N_n.", "$ Then, we can use the previous facts on the embedding map to conclude with the following statement: there exists an $\\epsilon $ -approximate thermodynamic distillation process from $N$ systems with Hamiltonians $H^N_n$ and in energy-incoherent states $p^N_n$ to a system with a Hamiltonian $\\tilde{H}^N$ and in a sharp energy eigenstate $\\tilde{s}_{k}^N$ if and only if $\\hat{P}^N \\otimes \\tilde{\\eta }^N \\succ _{\\epsilon } \\hat{G}^N \\otimes \\tilde{f}^N_k \\, .$" ], [ "Information-theoretic intermission", "Before we proceed, we need to make a short intermission for a few important comments concerning information-theoretic quantities introduced in Eqs.", "(REF )-().", "For incoherent states $\\rho $ and $\\gamma $ represented by probability vectors $p$ and $\\gamma $ , these simplify and take the following classical form: $D(p\\Vert \\gamma ):=&\\sum _i p_i\\left(\\log \\frac{p_i}{\\gamma _i}\\right),\\\\V(p\\Vert \\gamma ):=&\\sum _i p_i \\left(\\log \\frac{p_i}{\\gamma _i} - D(p\\Vert \\gamma )\\right)^2,\\\\Y(p\\Vert \\gamma ):=&\\sum _i p_i \\left|\\log \\frac{p_i}{\\gamma _i} - D(p\\Vert \\gamma )\\right|^3.$ Moreover, by direct calculation, one can easily show that the above quantities are invariant under embedding, i.e., $D(p\\Vert \\gamma )=D(\\hat{p}\\Vert \\eta )$, and the same holds for $V$ and $Y$ .", "Therefore $D(p\\Vert \\gamma )=&D(\\hat{p}\\Vert \\eta )=\\log D-H(\\hat{p}),\\\\V(p\\Vert \\gamma )=&V(\\hat{p}\\Vert \\eta )=V(\\hat{p}),\\\\Y(p\\Vert \\gamma )=&Y(\\hat{p}\\Vert \\eta )=Y(\\hat{p}),$ where $H(p):=&\\sum _i p_i(-\\log p_i),\\\\V(p):=&\\sum _i p_i (\\log p_i + H(p))^2,\\\\Y(p):=&\\sum _i p_i \\left|\\log p_i + H(p)\\right|^3,$ and note that $V(p)=0$ if and only if $p$ is a flat state." ], [ "Optimal error for a distillation process", "In order to transform the approximate majorisation condition from Eq.", "(REF ) into an explicit expression for the optimal transformation error, we will start from the following result proven by the authors of Ref. [20].", "Lemma 4 (Lemma 21 of Ref.", "[20]) Let $\\mathbf {p}$ and $\\mathbf {q}$ be distributions with $V(\\mathbf {q})=0$ .", "Then $\\min \\left\\lbrace \\epsilon | \\mathbf {p}\\succ _\\epsilon \\mathbf {q} \\right\\rbrace = 1-\\sum _{i=1}^{\\exp H(\\mathbf {q})}p_i^{\\downarrow }.$ Applying the above lemma to Eq.", "(REF ) yields the following expression for the optimal error $\\epsilon _N$ : $\\epsilon _N = 1-\\sum _{i=1}^{\\exp [H(\\hat{G}^{N})+H(\\tilde{f}_k^N)]} \\left(\\hat{P}^N\\otimes \\tilde{\\eta }^N\\right)_i^\\downarrow .$ Now, for an arbitrary distribution $p$ and any flat state $f$ , we make two observations: the size of the support of $f$ is simply $\\exp (H(f))$ , and the entries of $p\\otimes f$ are just the copied and scaled entries of $p$ .", "As a result, the sum of the $l$ largest elements of $p$ can be expressed as $\\sum _{i=1}^l p_i^\\downarrow = \\sum _{i=1}^{l \\exp (H(f))} (p\\otimes f)_i^\\downarrow .$ Inverting the above expression we can write $\\sum _{i=1}^{l} (p\\otimes f)_i^\\downarrow = \\sum _{i=1}^{l\\exp (-H(f))} p_i^\\downarrow ,$ where the summation with non-integer upper limit $x$ should be interpreted as: $\\sum _{i=1}^x p_i :=\\sum _{i=1}^{\\lfloor x \\rfloor } p_i +(x-\\lfloor x \\rfloor ) p_{\\lceil x \\rceil }.$ Since $\\tilde{\\eta }$ is a flat state, we conclude that $\\epsilon _N = 1- \\sum _{i=1}^{\\exp [H(\\hat{G}^{N})+H(\\tilde{f}_k^N)-H(\\tilde{\\eta }^N)]} (\\hat{P}^N)_i^\\downarrow \\, .$ We see that the error depends crucially on partial ordered sums as above.", "To deal with these kind of sums, we introduce the function $\\chi _{p}$ defined implicitly by the following equation $\\sum _{i=1}^{\\chi _{p}(l)} p_i^\\downarrow =\\sum _i \\lbrace p_i | p_i\\ge 1/l\\rbrace .$ In words: $\\chi _{p}(l)$ counts the number of entries of $p$ that are larger than $1/l$ .", "Now, we have the following lemma that will be crucial in proving our theorems.", "Lemma 5 Every $d$ -dimensional probability distribution $p$ satisfies the following for all $l\\in \\lbrace 1,\\dots ,d\\rbrace $ and for all $\\alpha \\ge 1$ : $\\sum _{i=1}^{l} p_i^\\downarrow &\\ge \\sum _{i=1}^{\\chi _{p}(l)} p_i^\\downarrow ,\\\\\\sum _{i=1}^{l} p_i^\\downarrow &\\le \\sum _{i=1}^{\\chi _{p}(\\alpha l)/c} p_i^\\downarrow ,$ where $c=\\sqrt{\\alpha }\\sum _{i=\\chi _{p}(\\sqrt{\\alpha }l)}^{\\chi _{p}(\\alpha l)} p_i^\\downarrow .$ The first inequality is very easily proven by observing that the number of entries larger than $1/l$ , i.e., $\\chi _{p}(l)$ , is bounded from above by $l$ due to normalisation.", "Now, to prove the second inequality, we start from the following observation: $\\sum _{i=1}^{\\chi _{p}(\\sqrt{\\alpha }l)} \\left(p_i^\\downarrow -\\frac{1}{\\sqrt{\\alpha }l}\\right)\\ge \\sum _{i=1}^{\\chi _{p}(\\alpha l)}\\left(p_i^\\downarrow -\\frac{1}{\\sqrt{\\alpha }l}\\right),$ which comes from the fact that all extra terms on the right hand side of the above are negative by definition.", "By rearranging terms we arrive at $\\chi _{p}(\\alpha l)-\\chi _{p}(\\sqrt{\\alpha }l)\\ge c l,$ which obviously implies $l \\le \\frac{\\chi _{p}({\\alpha }l)}{c}.$" ], [ "Proof of Theorem ", "We start by introducing the following averaged entropic quantities for the total initial distribution $\\hat{P}^N$ : $\\!\\!", "h_N:=&\\frac{1}{N} H(\\hat{P}^N)=\\frac{1}{N}\\sum _{n=1}^N H(\\hat{p}^N_n)=:\\frac{1}{N}\\sum _{n=1}^N h^N_n,\\\\\\!\\!", "v_N:=&\\frac{1}{N} V(\\hat{P}^N)=\\frac{1}{N}\\sum _{n=1}^N V(\\hat{p}^N_n)=:\\frac{1}{N}\\sum _{n=1}^N v^N_n,\\\\\\!\\!", "y_N:=&\\frac{1}{N} Y(\\hat{P}^N)=\\frac{1}{N}\\sum _{n=1}^N Y(\\hat{p}^N_n)=:\\frac{1}{N}\\sum _{n=1}^N y^N_n.$ Note that the above $v_N$ and $y_N$ are, up to temperature rescaling, incoherent versions of $\\sigma ^2(F^N)$ and $\\kappa ^3(F^N)$ defined in Eqs. ()-().", "We also define the function $l$ : $l(z):=\\exp \\left(N h_N+z\\sqrt{N v_N}\\right).$ We now rewrite the upper summation limit appearing in Eq.", "(REF ) employing the above function: $\\exp [H(\\hat{G}^{N})+H(\\tilde{f}^N_k)-H(\\tilde{\\eta }^N)]=l(x)$ so that $x &= \\frac{D(\\hat{P}^N\\Vert \\hat{G}^{N})- D(\\tilde{f}^N_k\\Vert \\tilde{\\eta }^N)}{\\sqrt{V(\\hat{P}^N)}}.$ This can be further transformed by employing the invariance of relative entropic quantities under embedding, Eqs.", "(REF )-(), to arrive at $x&=\\frac{\\sum \\limits _{n=1}^N D(p_{n}^N\\Vert \\gamma _n^N)-D(\\tilde{s}_{k}^N\\Vert \\tilde{\\gamma }^N)}{\\left(\\sum \\limits _{n=1}^N V(p_{n}^N\\Vert \\gamma _{n}^N)\\right)^{\\frac{1}{2}}},$ which is precisely the argument of $\\Phi $ appearing in the statement of Theorem REF in Eq.", "(REF ): $x = \\frac{F_{\\mathrm {diss}}^N}{\\sigma (F^N)}\\cdot \\sqrt{N}.$ We conclude that with the above $x$ we can then rewrite the expression for the optimal transformation error, Eq.", "(REF ), as $\\epsilon _N = 1- \\sum _{i=1}^{l(x)} (\\hat{P}^N)_i^\\downarrow \\, .$ Next, we will find an upper bound for the error employing Eq.", "(REF ): $\\epsilon _N&\\le 1-\\sum _{i=1}^{\\chi _{\\hat{P}^N}(l(x))}(\\hat{P}^N)_i^\\downarrow \\nonumber \\\\&=1- \\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x)} \\right\\rbrace \\!.$ In order to evaluate the above sum, consider $N$ discrete random variables $X_n$ taking values $-\\log (\\hat{p}^N_n)_i$ with probability $(\\hat{p}^N_n)_i$ , so that $\\langle X_n \\rangle & = h^N_{n},\\\\\\langle (X_n-\\langle X_n \\rangle )^2 \\rangle & = v^N_{n},\\\\\\langle \\left| X_n-\\langle X_n \\rangle \\right|^3\\rangle & = y^N_{n}, $ where the average $\\langle \\cdot \\rangle $ is taken with respect to the distribution $\\hat{p}_{n}^N$ .", "We then have the following $&\\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x)} \\right\\rbrace \\nonumber \\\\&\\quad =\\sum _{i_1,\\dots ,i_N}\\left\\lbrace \\prod _{n=1}^{N}(\\hat{p}^N_{n})_{i_n} |\\prod _{n=1}^{N}(\\hat{p}^N_{n})_{i_n}\\ge \\frac{1}{l(x)} \\right\\rbrace \\nonumber \\\\&\\quad =\\sum _{i_1,\\dots ,i_N}\\left\\lbrace \\prod _{n=1}^{N}(\\hat{p}^N_{n})_{i_n} |-\\sum _{n=1}^{N}\\log (\\hat{p}^N_{n})_{i_n}\\le \\log l(x) \\right\\rbrace \\nonumber \\\\&\\quad =\\Pr \\left[\\sum _{n=1}^N X_n\\le Nh_N+x\\sqrt{Nv_N}\\right] \\nonumber \\\\&\\quad =\\Pr \\left[\\frac{\\sum _{n=1}^N (X_n-\\langle X_n\\rangle )}{\\sqrt{\\sum _{n=1}^N \\langle (X_n-\\langle X_n \\rangle )^2 \\rangle }}\\le x\\right].$ Now, the Berry-Esseen theorem [41], [42] tells us that $\\!\\!\\!\\!", "\\left| \\Pr \\!\\left[\\!\\frac{\\sum _{n=1}^N (X_n-\\langle X_n\\rangle )}{\\sqrt{\\sum _{n=1}^N \\langle (X_n\\!-\\!\\langle X_n \\rangle )^2 \\rangle }}\\le x\\!\\right]\\!", "-\\!", "\\Phi (x)\\right|\\!\\le \\!\\frac{C y_N}{\\sqrt{Nv_N^3}},\\!\\!$ where $C$ is a constant that was bounded in Refs.", "[43], [44] by $0.4097\\le C \\le 0.4748.$ We thus have $\\left|\\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x)} \\right\\rbrace - \\Phi (x)\\right| \\le \\frac{Cy_N}{\\sqrt{Nv_N^3}},$ and so we conclude that the error $\\epsilon _N$ is bounded from above by $\\epsilon _N\\le \\Phi \\left(-\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right)+\\frac{C\\kappa ^3(F^N)}{\\sigma ^3(F^N)}\\cdot \\frac{1}{\\sqrt{N}},$ which proves the single-shot upper bound on transformation error, Eq.", "(REF ), presented in Theorem REF .", "We now switch to proving the asymptotic behaviour of the optimal transformation error captured by Eq.", "(REF ).", "First, from Eq.", "(REF ), it is clear that if $\\lim _{N\\rightarrow \\infty } v_n$ and $\\lim _{N\\rightarrow \\infty } y_n$ are well-defined and non-zero (as we assume), then $\\lim _{N\\rightarrow \\infty } \\epsilon _N \\le 1- \\lim _{N\\rightarrow \\infty } \\Phi \\left(\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right).$ Next, in order to lower bound the expression for the optimal error in the asymptotic limit we will apply Eq.", "() with $\\alpha =\\exp (\\delta \\sqrt{N})$ and $\\delta >0$ to Eq.", "(REF ): $\\epsilon _N \\ge 1- \\!\\!", "\\sum _{i=1}^{\\frac{\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}}l(x))}{c}} \\!\\!", "(\\hat{P}^N)_i^\\downarrow = 1- \\!\\!\\sum _{i=1}^{\\frac{\\chi _{\\hat{P}^N}(l(x+\\delta ))}{c}} \\!\\!", "(\\hat{P}^N)_i^\\downarrow ,$ where $\\!\\!\\!\\!", "c&=e^{\\frac{\\delta \\sqrt{N}}{2}} \\sum _{i=\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}/2}l(x))}^{\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}}l(x))} (\\hat{P}^N)_i^\\downarrow \\nonumber \\\\\\!\\!\\!\\!", "&=e^{\\frac{\\delta \\sqrt{N}}{2}} \\sum _{i=\\chi _{\\hat{P}^N}(l(x+\\delta /2))}^{\\chi _{\\hat{P}^N}(l(x+\\delta ))} (\\hat{P}^N)_i^\\downarrow \\nonumber \\\\\\!\\!\\!\\!", "&=e^{\\frac{\\delta \\sqrt{N}}{2}} \\left( \\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x+\\delta )} \\right\\rbrace \\right.\\nonumber \\\\\\!\\!\\!\\!", "&\\qquad \\qquad \\left.", "-\\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x+\\delta /2)}\\right\\rbrace \\right).$ Using Eq.", "(REF ) we can bound the above expression from below as $c \\ge e^{\\frac{\\delta \\sqrt{N}}{2}}\\left(\\Phi (x+\\delta )-\\Phi (x+\\delta /2)-\\frac{2Cy_N}{\\sqrt{Nv_N^3}}\\right).$ Now, for any finite $\\delta >0$ it is clear that there exists $N_0$ such that for all $N\\ge N_0$ we have $c>1$ .", "From this and Eq.", "(REF ) we get that for $N\\ge N_0$ we have $\\epsilon _N &\\ge 1- \\sum _{i=1}^{\\chi _{\\hat{P}^N}(l(x+\\delta ))} (\\hat{P}^N)_i^\\downarrow \\, \\nonumber \\\\&=1- \\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}^N_i \\ge \\frac{1}{l(x+\\delta )}\\right\\rbrace \\nonumber \\\\&\\ge 1-\\Phi (x+\\delta )-\\frac{Cy_N}{\\sqrt{Nv_N^3}},$ where in the last line we used Eq.", "(REF ) again.", "It is thus clear that $\\lim _{N\\rightarrow \\infty } \\epsilon _N\\ge 1-\\lim _{N\\rightarrow \\infty } \\Phi (x+\\delta )=\\lim _{N\\rightarrow \\infty } \\Phi (-x-\\delta )$ and, since it works for any $\\delta >0$ , we conclude that $\\lim _{N\\rightarrow \\infty } \\epsilon _N\\ge \\lim _{N\\rightarrow \\infty } \\Phi \\left(-\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right).$ Combining the above with the bound obtained in Eq.", "(REF ), we arrive at $\\lim _{N\\rightarrow \\infty }\\epsilon _N=\\lim _{N\\rightarrow \\infty } \\Phi \\left(-\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right),$ which completes the proof." ], [ "Proof of Theorem ", "The proof of Theorem REF will be divided into three parts.", "First, we will show that a thermodynamic distillation process from a general state $\\rho $ can be reduced to a distillation process from an incoherent state that is a dephased version of $\\rho $ .", "Employing this observation, we will recast the problem under consideration in terms of approximate majorisation and thermomajorisation as described in Sec.", "REF .", "Then, in the second part of the proof, we will derive the upper bound for the optimal transformation error $\\epsilon _N$ .", "Finally, in the third part, we will provide a lower bound for $\\epsilon _N$ and show that it is approaching the derived upper bound in the asymptotic limit." ], [ "Reducing the problem to the incoherent case", "The thermodynamic distillation problem under investigation is specified as follows.", "The family of initial systems consists of a collection of $N$ identical subsystems, each with the same Hamiltonian $H=\\sum _{i=1}^d E_i |E_i\\rangle \\!\\langle E_i|,$ and an ancillary system with an arbitrary Hamiltonian $H_A$ (note that the ancillary system can always be ignored by simply choosing its dimension to be 1).", "The family of initial states is given by $\\rho ^N= \\psi ^{\\otimes N} \\otimes |E^A_0\\rangle \\!\\langle E^A_0|,$ where $\\psi =|\\psi \\rangle \\!\\langle \\psi |,\\quad | {\\psi } \\rangle =\\sum _{i=1}^d \\sqrt{p_i} e^{i\\phi _i} | {E_i} \\rangle ,$ is an arbitrary pure state and $| {E^A_0} \\rangle $ is an eigenstate of $H_A$ with energy $E^A_0$ .", "The family of target systems is composed of subsystems described by arbitrary Hamiltonians $\\tilde{H}^N$ and a subsystem described by the Hamiltonian $H_A$ .", "The family of target states is given by $\\tilde{\\rho }^N= |\\tilde{E}^{N}_k\\rangle \\!\\langle \\tilde{E}^{N}_k| \\otimes |E^A_1\\rangle \\!\\langle E^A_1|,$ where $| {\\tilde{E}^{N}_k} \\rangle $ is some eigenstate of $\\tilde{H}^N$ and $| {E^A_1} \\rangle $ is an eigenstate of $H_A$ with energy $E^A_1$ .", "We are thus interested in the existence of a thermal operation $ {\\cal E} $ satisfying $ {\\cal E} (\\psi ^{\\otimes N} \\otimes |E^A_0\\rangle \\!\\langle E^A_0|) = |\\tilde{E}^{N}_k\\rangle \\!\\langle \\tilde{E}^{N}_k| \\otimes |E^A_1\\rangle \\!\\langle E^A_1|.$ We now have the following simple, but very useful, lemma.", "Lemma 6 Every incoherent state $\\sigma $ achievable from a state $\\rho $ through a thermal operation is also achievable from $ {\\cal D} (\\rho )$ , where $ {\\cal D} $ is the dephasing operation destroying coherence between different energy subspaces: $\\exists {\\cal E} : {\\cal E} (\\rho ) = \\sigma \\quad \\Leftrightarrow \\quad {\\cal E} ( {\\cal D} (\\rho )) = \\sigma .$ First, for a given $\\rho $ and incoherent $\\sigma $ , assume that there exists a thermal operation $ {\\cal E} $ such that $ {\\cal E} (\\rho ) = \\sigma $ .", "Now, employing the fact that every thermal operation is covariant with respect to time-translations [45], and using the fact that incoherent $\\sigma $ by definition satisfies $\\mathcal {D}(\\sigma ) = \\sigma $ , we get $ {\\cal E} ( {\\cal D} (\\rho ))= {\\cal D} ( {\\cal E} (\\rho ))= {\\cal D} (\\sigma )=\\sigma .$ Likewise, the reverse implication holds by noting that the dephasing operation is a thermal operation.", "Because the target state in our case is incoherent, we can use the above result to restate our problem as the existence of a thermal operation $ {\\cal E} $ satisfying $ {\\cal E} ( {\\cal D} (\\psi ^{\\otimes N} \\otimes |E^A_0\\rangle \\!\\langle E^A_0|)) = |\\tilde{E}^{N}_k\\rangle \\!\\langle \\tilde{E}^{N}_k| \\otimes |E^A_1\\rangle \\!\\langle E^A_1|.$ Since $ {\\cal D} (\\psi ^{\\otimes N} \\otimes |E^A_0\\rangle \\!\\langle E^A_0|)= {\\cal D} (\\psi ^{\\otimes N}) \\otimes |E^A_0\\rangle \\!\\langle E^A_0|,$ our problem further reduces to understanding the structure of the incoherent state $ {\\cal D} (\\psi ^{\\otimes N})$ .", "It is block-diagonal in the energy eigenbasis and can be diagonalised using thermal operations (since unitaries in a fixed energy subspace are free operations).", "After such a procedure, we end up with an incoherent state that is described by the probability distribution $P^N$ over the multi-index set $k$ $P^N_{k} =\\binom{N}{k_1,...,k_d}\\prod _{i=1}^d p^{k_i}_i.$ Note that $P^N_{k}$ specifies the probability of $k_1$ systems being in energy state $E_1$ , $k_2$ systems being in energy state $E_2$ , and so on; and that we made a technical assumption that energy levels are incommensurable, so that each vector $k$ corresponds to a different value of total energy.", "We have thus reduced the problem of thermodynamic distillation from pure states to thermodynamic distillation from incoherent states.", "More precisely, let us denote the sharp distributions corresponding to $| {E^A_i} \\rangle $ by $s^A_i$ and the corresponding flat states after embedding by $f^A_i$ .", "As before, we also use $\\tilde{s}_k^N$ and $\\tilde{f}_k^N$ to denote distributions related to the sharp state $| {\\tilde{E}^{N}_k} \\rangle $ and its corresponding flat state.", "The embedded Gibbs state corresponding to $H^N$ will be again denoted by $\\hat{G}^N$ , however now it has an even simpler form than in Eq.", "(REF ), as the initial systems have identical Hamiltonians: $\\hat{G}^N=\\hat{\\gamma }^{\\otimes N} = {\\eta }^{\\otimes N}.$ Similarly, $\\hat{P}^N$ will be used to denote the embedded initial state (even though it now has a different form than in Eq.", "(REF )): $\\hat{P}^N_{k, g_{k}} = \\binom{N}{k_1,...,k_d}\\prod ^{d}_{i=1}\\frac{p^{k_i}_i}{D^{k_i}_i},$ with $g_{k} \\in \\bigg \\lbrace 1, ..., \\prod _{i=1}^d D^{k_i}_i \\bigg \\rbrace $ indexing the degeneracy coming from embedding.", "With the notation set, our distillation problem can now be written as ${\\hat{P}}^N \\otimes f^A_0 \\otimes \\tilde{\\eta } \\succ _{\\epsilon } \\hat{G}^N \\otimes f^A_1 \\otimes \\tilde{f_k} \\, .$" ], [ "Upper bound for the transformation error", "We begin by observing that our target distribution in Eq.", "(REF ) is flat, and so $V(\\hat{G}^N \\otimes f^A_1 \\otimes \\tilde{f_k})=0$.", "Thus, we can employ Lemma REF and Eq.", "(REF ) to get the following expression for the optimal transformation error: $\\epsilon _N= 1-\\sum _{j=1}^{L}(\\hat{P}^N)^{\\downarrow }_j$ where $L$ is given by $L & = \\exp [H(\\hat{G}^{N})+H(f^A_1)+H(\\tilde{f}_k)-H(f^A_0)-H(\\tilde{\\eta })]\\nonumber \\\\& = \\exp [H(\\hat{G}^{N})-D(\\tilde{f}_k\\Vert \\tilde{\\eta })-\\beta (E_1^A-E_0^A)].", "$ Notice that in the current case $F_{\\text{diss}}^N$ , defined in Eq.", "(REF ), is given by $\\!\\!\\!", "F_{\\text{diss}}^N&\\!=\\!\\frac{1}{\\beta N}\\Big (D(\\psi ^{\\otimes N}\\Vert \\gamma ^{\\otimes N})+ D(|E_0^A\\rangle \\!\\langle E_0^A|\\Vert \\gamma _A)\\nonumber \\\\\\!\\!&\\qquad \\quad -D(|\\tilde{E}_{k}\\rangle \\!\\langle \\tilde{E}_{k}|\\Vert \\tilde{{\\gamma }}) -D(|E_1^A\\rangle \\!\\langle E_1^A|\\Vert \\gamma _A)\\Big )\\nonumber \\\\\\!\\!\\!", "&\\!=\\!", "\\frac{1}{\\beta N}\\Big (ND(\\psi \\Vert \\gamma )\\!-\\!D(\\tilde{f}_k\\Vert \\tilde{\\eta })\\!-\\!\\beta (E_1^A\\!-\\!E_0^A)\\Big ).$ Using the above we can then rewrite $L$ as $\\log L =& \\beta NF_{\\text{diss}}^N+H(\\hat{G}^N)-ND(\\psi \\Vert \\gamma ).$ Now, employing Eq.", "(REF ) and the above, we provide the upper bound for $\\epsilon _N$ : $\\!\\!\\!\\epsilon _N&\\le 1- \\sum _{k,g_{k}}\\left\\lbrace \\hat{P}^N_{k,g_{k}} |\\hat{P}^N_{k,g_{k}}\\ge \\frac{1}{L} \\right\\rbrace \\nonumber \\\\\\!\\!\\!", "&= 1- \\sum _{k}\\left\\lbrace {P}^N_{k} |{P}^N_{k}\\ge \\frac{\\prod _{i=1}^d D_i^{k_i}}{L} \\right\\rbrace \\nonumber \\\\\\!\\!\\!&= 1- \\sum _{k}\\left\\lbrace {P}^N_{k} |\\log {P}^N_{k}\\ge \\sum _{i=1}^d k_i\\log D_i-\\log L \\right\\rbrace \\nonumber \\\\\\!\\!\\!&= 1- \\sum _{k}\\bigg \\lbrace {P}^N_{k} \\Bigg |\\frac{\\log {P}^N_{k}}{N}\\ge \\sum _{i=1}^d \\frac{k_i}{N}\\log \\gamma _i+D(\\psi \\Vert \\gamma )\\nonumber \\\\\\!\\!\\!&\\phantom{= 1- \\sum _{k}\\bigg \\lbrace {P}^N_{k} \\Bigg |\\log {P}^N_{k}\\ge }-\\beta F^N_{\\text{diss}} \\Bigg \\rbrace .$ To simplify the calculation of the upper bound of $\\epsilon _N$ , we rewrite each $k$ as a function of a multi-parameter vector $s$ such that $k=k(s)=Np+\\sqrt{N}s,$ with $\\sum _{i=1}^d s_i=0$ .", "We then note that $D(\\psi \\Vert \\gamma )=-\\sum _{i=1}^d p_i \\log \\gamma _i$ and so the condition in Eq.", "(REF ) can be rewritten as $\\frac{\\log {P}^N_{k(s)}}{N}&\\ge \\frac{1}{\\sqrt{N}}\\sum _{i=1}^d s_i\\log \\gamma _i-\\beta F^N_{\\text{diss}} \\nonumber \\\\& = -\\frac{\\beta }{\\sqrt{N}}\\sum _{i=1}^d s_iE_i-\\beta F^N_{\\text{diss}}.$ As we rigorously argue in Appendix , the left-hand side of the above vanishes much quicker than the right-hand side when $N\\rightarrow \\infty $ , leading to $&\\!\\!\\!\\!\\lim _{N\\rightarrow \\infty }\\epsilon _N\\nonumber \\\\&\\!\\!\\!\\!\\quad \\le 1\\!-\\!", "\\lim _{N\\rightarrow \\infty }\\sum _{s}\\bigg \\lbrace {P}^N_{k(s)} \\Bigg |\\frac{1}{\\sqrt{N}} \\sum _{i=1}^d s_iE_i\\ge - F^N_{\\text{diss}}\\Bigg \\rbrace .\\!$ Our goal is then to calculate the sum of $P^N_{k(s)}$ in the limit $N\\rightarrow \\infty $ subject to the following hyper-plane constraint $\\frac{1}{\\sqrt{N}} s\\cdot E\\ge - F_{\\text{diss}}^N,$ where $E$ is a vector of energies (eigenvalues of $H$ ).", "First, we approximate the multinomial distribution $P^N$ specified in Eq.", "(REF ) by a multivariate normal distribution $\\mathcal {N}^{(\\mu ,\\Sigma )}$ with mean vector $\\mu =Np$ and covariance matrix $\\Sigma = N(\\text{diag }(p)-pp^{T})$ : $\\mathcal {N}^{(\\mu ,\\Sigma )}_{k(s)}&= \\frac{1}{\\sqrt{(2\\pi )^d|\\Sigma |}}\\exp \\left(-\\frac{1}{2}({k}-{\\mu })^T{\\Sigma }^{-1}({k}-{\\mu })\\right)\\nonumber \\\\&=\\frac{1}{\\sqrt{(2\\pi )^d|\\Sigma |}}\\exp \\left(-\\frac{1}{2}s^T{N\\Sigma }^{-1}s\\right).$ As we explain in Appendix , such an approximation can always be made with an error approaching to 0 as $N\\rightarrow \\infty $ .", "Next, we standardise the multivariate normal distribution $\\mathcal {N}^{(\\mu ,\\Sigma )}$ using rotation and scaling transformations: $\\Sigma = \\Theta ^T\\sqrt{\\Lambda }\\sqrt{\\Lambda }\\Theta ,$ where $\\Lambda $ is a diagonal matrix with the eigenvalues of $\\Sigma $ and $\\Theta $ is an orthogonal matrix with columns given by the eigenvectors of $\\Sigma $ .", "We illustrate this process for a three-level system (so described by $s_1$ and $s_2$ since $\\sum _i s_i=0$ ) in Fig.", "REF .", "This rotation and scaling of co-ordinates allows us to write $\\mathcal {N}^{(\\mu ,\\Sigma )}$ as a product of univariate standard normal distribution $\\phi (y_i)$ : $\\!\\!\\!\\!\\mathcal {N}^{(\\mu ,\\Sigma )}_{k(s(y))}&\\!=\\!\\frac{1}{\\sqrt{(2\\pi )^d|\\Sigma |}}\\exp \\left(\\!-\\frac{1}{2}y^Ty\\!\\right)=\\prod _{i=1}^d \\phi (y_i),\\!$ where $y= \\sqrt{N}(\\Theta ^T\\sqrt{\\Lambda })^{-1} s.$ Figure: Standardising the bivariate normal distribution.", "The points with equal probability density for the bivariate normal distribution are represented by a red ellipsis centred at the origin, and the black dashed line corresponds to the constraining hyper-plane.", "The upper bound on ϵ N \\epsilon _N is given by the probability mass within the area depicted in grey.", "In order to calculate it, we first apply a rotation and scaling transformation, making the ellipsis symmetric with respect to the origin.", "Then, using the rotational symmetry of the standard bivariate normal distribution, one can rotate it such that the hyper-plane becomes parallel to x 1 x_1.We then can equivalently write the equation specifying the hyper-plane, Eq.", "(REF ), as $(\\Theta \\sqrt{\\Lambda }y)\\cdot E\\ge -N F^N_{\\text{diss}}.$ Observe that the standard normal distribution given in Eq.", "(REF ) is rotationally invariant about the origin.", "One can thus choose a coordinate system $x=\\lbrace x_1,\\ldots , x_d\\rbrace $ by applying a suitable rotation $R$ on $y=\\lbrace y_1,\\ldots , y_d\\rbrace $, so that the hyper-plane specified in Eq.", "(REF ) becomes parallel to a particular coordinate axis $x_1$ .", "Eq.", "(REF ) can then be rewritten in the following form $\\!\\!\\!\\!\\mathcal {N}^{(\\mu ,\\Sigma )}_{k(s(x))}&\\!=\\!\\frac{1}{\\sqrt{(2\\pi )^d|\\Sigma |}}\\exp \\left(\\!-\\frac{1}{2}x^Tx\\!\\right)=\\prod _{i=1}^d \\phi (x_i).\\!$ As we have $x=Ry,$ we can use Eq.", "(REF ) and Eq.", "(REF ) to rewrite Eq.", "(REF ) as $\\frac{1}{N} \\Theta ^T\\sqrt{\\Lambda }R^Tx\\cdot E\\ge - F_{\\text{diss}}^N.$ To calculate the right hand side of the inequality given in Eq.", "(REF ) in the limit $N\\rightarrow \\infty $ , we integrate Eq.", "$(\\ref {MultivariateNormalmean0onx})$ from $-\\infty $ to $d_{\\mathcal {O}}$ along $x_1$ , and from $-\\infty $ to $+\\infty $ along any other $x_i\\ne x_1$ , where $d_\\mathcal {O}$ is the signed distance of the hyper-plane given in Eq.", "(REF ) from the origin (see Fig.", "REF ).", "This distance can be explicitly calculated as $d_{\\mathcal {O}} &=& \\frac{F_{\\text{diss}}^N}{\\sqrt{\\frac{1}{N^2}E\\cdot (R\\sqrt{\\Lambda }\\Theta )^T(R\\sqrt{\\Lambda }\\Theta )E}} \\nonumber \\\\&=& \\frac{\\sqrt{N}F_{\\text{diss}}^N}{\\sqrt{\\frac{1}{N}E\\cdot (\\Sigma E)}}= \\frac{\\sqrt{N}F^N_{\\rm diss}}{\\sigma (F^N)},$ where we have used the definition of $\\sigma (F^N)$ from Eq.", "() in the last line.", "Thus, the upper bound on $\\epsilon _N$ in the limit $N\\rightarrow \\infty $ given in Eq.", "(REF ) can be calculated as $&1\\!-\\!\\lim _{N\\rightarrow \\infty }\\int _{-\\infty }^{+\\infty }\\!dx_d\\phi (x_d)\\ldots \\int _{-\\infty }^{+\\infty }\\!dx_2\\phi (x_2)\\int _{-\\infty }^{d_\\mathcal {O}}\\!dx_1\\phi (x_1)\\nonumber \\\\&= 1\\!-\\!\\lim _{N\\rightarrow \\infty }\\Phi (d_\\mathcal {O}) = \\lim _{N\\rightarrow \\infty }\\Phi \\left(-\\frac{F^N_{\\rm diss}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right).", "$" ], [ "Lower bound for the error of transformation", "We start by writing $L$ from Eq.", "(REF ) as $L= \\exp \\left(AN+x\\sqrt{N v_N}\\right)=:L(x),$ where $A =&H(\\eta )-D(\\psi \\Vert {\\gamma }),\\qquad x=\\frac{\\sqrt{N}F_{\\text{diss}}^N}{\\sigma (F^N)}.$ In the previous section we have exactly calculated the right hand side of Eq.", "(REF ) in the limit $N\\rightarrow \\infty $ (see Eq.", "(REF )).", "Using Eqs.", "(REF )-(REF ), we can equivalently rewrite this as $\\lim _{N\\rightarrow \\infty }\\sum _{k,g_{k}}\\left\\lbrace \\hat{P}^N_{k,g_{k}} |\\hat{P}^N_{k,g_{k}}\\ge \\frac{1}{L(x)} \\right\\rbrace = \\lim _{N\\rightarrow \\infty }\\Phi (x),$ where $x$ depends on $N$ as per Eq.", "(REF ).", "We shall use this result to derive the lower bound on $\\epsilon _N$ .", "To prove the lower bound on transformation error $\\epsilon _N$ we start with the exact expression, Eq.", "(REF ), and use the inequality from Eq. ().", "Taking $\\alpha =\\exp (\\delta \\sqrt{N})$ and $\\delta >0$ we then get $\\epsilon _N \\ge 1- \\!\\!\\!\\!", "\\sum _{i=1}^{\\frac{\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}}L(x))}{c}} \\!\\!", "(\\hat{P}^N)_i^\\downarrow = 1- \\!\\!\\sum _{i=1}^{\\frac{\\chi _{\\hat{P}^N}(L(x+\\delta ))}{c}} \\!\\!", "(\\hat{P}^N)_i^\\downarrow ,$ where $c$ can be evaluated similarly as before.", "So $c&=e^{\\delta \\sqrt{N}/2} \\sum _{i=\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}/2}L(x))}^{\\chi _{\\hat{P}^N}(e^{\\delta \\sqrt{N}}L(x))} (\\hat{P}^N)_i^\\downarrow \\nonumber \\\\&=e^{\\delta \\sqrt{N}/2} \\sum _{i=\\chi _{\\hat{P}^N}(L(x+\\delta /2))}^{\\chi _{\\hat{P}^N}(L(x+\\delta ))} (\\hat{P}^N)_i^\\downarrow \\nonumber \\\\&=e^{\\delta \\sqrt{N}/2} \\left( \\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}_i^N \\ge \\frac{1}{L(x+\\delta )} \\right\\rbrace \\right.\\nonumber \\\\&\\quad \\qquad \\qquad \\left.", "-\\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}_i^N \\ge \\frac{1}{L(x+\\delta /2)}\\right\\rbrace \\right).$ Using Eq.", "(REF ), we see that the limiting behaviour of $c$ from Eq.", "(REF ) is given by $\\lim _{N\\rightarrow \\infty } c = \\left(\\Phi (x+\\delta )-\\Phi (x+\\delta /2)\\right) \\lim _{N\\rightarrow \\infty } e^{\\delta \\sqrt{N}/2}.$ Thus, for any finite $\\delta >0$ , there always exists $N_0$ such that for all $N\\ge N_0$ we have $c>1$ .", "Combining this observation with Eq.", "(REF ) we obtain $\\!\\!\\!", "\\lim _{N\\rightarrow \\infty }\\epsilon _N &\\ge 1- \\lim _{N\\rightarrow \\infty }\\sum _{i=1}^{\\chi _{\\hat{P}^N}(L(x+\\delta ))} (\\hat{P}^N)_i^\\downarrow \\, \\nonumber \\\\\\!\\!\\!", "&=1- \\lim _{N\\rightarrow \\infty }\\sum _{i}\\left\\lbrace \\hat{P}^N_i |\\hat{P}_i^N \\ge \\frac{1}{L(x+\\delta )}\\right\\rbrace \\nonumber \\\\\\!\\!\\!&= 1-\\lim _{N\\rightarrow \\infty }\\Phi (x+\\delta )=\\lim _{N\\rightarrow \\infty }\\Phi (-x-\\delta ),$ where the first equality in the last line follows from Eq.", "(REF ).", "Since the above inequality holds for any $\\delta >0$ , taking the limit $\\delta \\rightarrow 0$ we conclude that $\\lim _{N\\rightarrow \\infty } \\epsilon _N\\ge \\lim _{N\\rightarrow \\infty } \\Phi \\left(-\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right).$ Finally, combining the above with the bound obtained in Eq.", "(REF ), we have $\\lim _{N\\rightarrow \\infty }\\epsilon _N=\\lim _{N\\rightarrow \\infty } \\Phi \\left(-\\frac{F^N_{\\mathrm {diss}}}{\\sigma (F^N)}\\cdot \\sqrt{N}\\right),$ which completes the proof." ], [ "Outlook", "In this paper, we have derived a version of the fluctuation-dissipation theorem for state interconversion under thermal operations, where for a fixed transformation error we have established the relation between fluctuations of free energy in the initial state of the system and average dissipation of free energy, i.e.", "the difference in free energy between the initial and final states.", "We addressed and solved the problem in two different scenarios: for initial states being either energy-incoherent or pure, with the target state in both cases being an energy eigenstate, and with the possibility to change the Hamiltonian in the process.", "For the case of finitely many independent but not necessarily identical energy-incoherent systems, we have provided the single-shot upper bound on the optimal transformation error as a function of average dissipated free energy and free energy fluctuations.", "Moreover, in the asymptotic regime we obtained the optimal transformation error up to second order asymptotic corrections, which extends previous results of Ref.", "[20] to the regime of non-identical initial systems and varying Hamiltonians.", "For the first time we have also performed the asymptotic analysis of the thermodynamic distillation process from quantum states that have coherence in the energy eigenbasis.", "As a result, we expressed the optimal transformation error from identical pure states up to second order asymptotic corrections as a function of average dissipated free energy and free energy fluctuations.", "Our work can be naturally extended in the following directions.", "Firstly, one could generalise our analysis to arbitrary initial states.", "We indeed believe that an analogous result to ours will hold for such general mixed states with coherence.", "That is because dephasing into fixed energy subspaces leads to average free energy change of the order $O(\\frac{\\log N}{N})$ , which is negligible compared to the second order asymptotic corrections of the order $O(1/\\sqrt{N})$ that we focus on.", "In other words, the contribution of coherence to free energy vanishes faster with growing $N$ than what we are interested while studying second order corrections.", "Secondly, it would be extremely interesting to generalise the thermodynamic state interconversion problem to arbitrary final states, and see how the interplay between the fluctuations of the initial and target states affects dissipation.", "For energy-incoherent initial and final states one can infer from Ref.", "[22] that appropriately tuned fluctuations can significantly reduce dissipation, however nothing is known for states with coherence.", "Unfortunately, since thermal operations are time-translation covariant, such that coherence and athermality form independent resources [46], [45], [47], it seems unlikely that the current approach can be easily generalised.", "Thirdly, one could try to extend our results on pure states to allow for non-identical systems and to derive a bound working for all $N$ , not only for $N\\rightarrow \\infty $ (i.e., replace the proving technique based on central limit theorem by the one based on a version of Berry-Esseen theorem).", "In our work we have also provided a number of physical applications of our fluctuation-dissipation theorems by considering several scenarios and explaining how our results can be useful to describe fundamental and well-known thermodynamic and information-theoretic processes.", "We derived the optimal value of extractable work in a thermodynamic distillation process as a function of the transformation error associated to the work quality.", "This could potentially be used to clarify the notion of imperfect work [27], [48], [49], and to construct a comparison platform allowing one to continuously distinguish between work-like and heat-like forms of energy.", "We have also showed how our results yield the optimal trade-off between the work invested in erasing $N$ independent bits prepared in arbitrary states, and the erasure quality measured by the infidelity distance between the final state and the fully erased state.", "This can of course be straightforwardly extended to higher-dimensional systems and arbitrary final erased state (not necessarily the ground state).", "Finally, we have investigated the optimal encoding rate into a collection of non-interacting subsystems consisting of energy-incoherent or pure states using thermal operations.", "We derived the optimal rate (up to second-order asymptotics) of encoding information with a given average decoding error and without spending thermodynamic resources.", "This provides an operational interpretation of the resourcefulness of athermal quantum states for communication scenarios under the restriction of using thermal operations.", "We would also like to point out to some possible technical extensions of our results.", "Firstly, we used infidelity as our quantifier of transformation error, but we expect that similar results could be derived using other quantifiers, e.g., the trace distance.", "Secondly, our investigations were performed in the spirit of small-deviation analysis (where we look for constant transformation error and total free energy dissipation of the order $O(\\sqrt{N})$ ), but possibly other interesting fluctuation-dissipation relations could be derived within the the moderate and large deviation regimes.", "Thirdly, our result for pure states is limited to Hamiltonians with incommensurable spectrum, but we believe this is just a technical nuisance that one should be able to get rid of.", "Lastly, within the framework of general resource theories, it might be possible to derive analogous fluctuation-dissipation relations, but with free energy replaced by a resource quantifier relevant for a given resource theory." ], [ "Acknowledgements", "KK would like to thank Chris Chubb and Marco Tomamichel for helpful discussions.", "KK and AOJ acknowledge financial support by the Foundation for Polish Science through TEAM-NET project (contract no.", "POIR.04.04.00-00-17C1/18-00).", "TB and MH acknowledge support from the Foundation for Polish Science through IRAP project co-financed by EU within the Smart Growth Operational Programme (contract no.2018/MAB/5)." ], [ "Optimality of the communication rate", "The following derivation will closely follow the proof of Lemma 1 of Ref. [36].", "Let us assume that for a system $(\\rho ^N,H^N)$ it is possible to encode $M$ messages in a thermodynamically-free way so that the average decoding error is $\\epsilon $ .", "It means that there exists a set of $M$ encoding thermal operations $\\lbrace {\\cal E} _i\\rbrace _{i=1}^M$ and a decoding POVM $\\lbrace \\Pi _i \\rbrace _{i=1}^M$ such that $1-\\epsilon =\\frac{1}{M}\\sum _{i=1}^M \\mathrm {Tr}\\left( {\\cal E} _i(\\rho ^N) \\Pi _i \\right).$ Note that every thermal operation $ {\\cal E} _i$ between the initial system $(\\rho ^N,H^N)$ and a target system $(\\tilde{\\rho }^N,\\tilde{H}^N)$ preserves the thermal equilibrium state, i.e., $ {\\cal E} _i(\\gamma ^N)=\\tilde{\\gamma }^N.$ Now, let us introduce the following three states $\\tau :=&\\frac{1}{M} \\sum _{i=1}^M |i\\rangle \\!\\langle i|\\otimes {\\cal E} _i(\\rho ^N),\\\\\\zeta :=&\\frac{1}{M} \\sum _{i=1}^M |i\\rangle \\!\\langle i|\\otimes \\gamma ^N,\\\\\\tilde{\\zeta }:=&\\frac{1}{M} \\sum _{i=1}^M |i\\rangle \\!\\langle i|\\otimes \\tilde{\\gamma }^N.$ The hypothesis testing relative entropy $D_H^{\\epsilon }$ between $\\tau $ and $\\tilde{\\zeta }$ , defined by [50], [51], [52] $\\!\\!\\!", "D_H^{\\epsilon }(\\tau \\Vert \\tilde{\\zeta }) \\!", ":= - \\log \\inf \\big \\lbrace \\mathrm {Tr}\\left( Q\\tilde{\\zeta } \\right)\\ \\!\\big |\\ &0 \\le Q \\le \\mathbb {1},\\nonumber \\\\\\!\\!\\!\\!\\!", "&\\mathrm {Tr}\\left( Q\\tau \\right) \\ge 1-\\epsilon \\big \\rbrace ,\\!$ satisfies the following $D_H^{\\epsilon }(\\tau \\Vert \\tilde{\\zeta })\\ge -\\log \\mathrm {Tr}\\left( Q\\tilde{\\zeta } \\right)$ for $Q = \\sum _{i=1}^M |i\\rangle \\!\\langle i| \\otimes \\Pi _i.$ This is because the above (potentially suboptimal) choice of $Q$ clearly satisfies $0\\le Q\\le \\mathbb {1}$ and also $\\mathrm {Tr}\\left( Q\\tau \\right)= \\frac{1}{M} \\sum _{i=1}^M \\mathrm {Tr}\\left( {\\cal E} _i(\\rho )\\Pi _i \\right)\\ge 1-\\epsilon ,$ due to our assumption in Eq.", "(REF ).", "At the same time we have $\\mathrm {Tr}\\left( Q\\tilde{\\zeta } \\right)=\\frac{1}{M} \\sum _{i=1}^M \\mathrm {Tr}\\left( \\tilde{\\rho }^N \\Pi _i \\right)=\\frac{1}{M},$ so that $\\log M\\le D_H^{\\epsilon }(\\tau \\Vert \\tilde{\\zeta }).$ Next, we introduce the following encoding channel $ {\\cal F} :=\\sum _{i=1}^{M} |i\\rangle \\!\\langle i| \\otimes {\\cal E} _i,$ which satisfies $ {\\cal F} (\\zeta )=\\tilde{\\zeta }.$ Employing the data-processing inequality twice, we get the following sequence of inequalities: $D_H^{\\epsilon }(\\tau \\Vert \\tilde{\\zeta })&=D_H^{\\epsilon }\\left( {\\cal F} \\left(\\frac{1}{M}\\sum _{i=1}^M |i\\rangle \\!\\langle i| \\otimes \\rho ^N\\right)\\bigg \\Vert {\\cal F} ({\\zeta })\\right)\\nonumber \\\\&\\le D_H^{\\epsilon }\\left(\\frac{1}{M}\\sum _{i=1}^M |i\\rangle \\!\\langle i| \\otimes \\rho ^N\\bigg \\Vert \\zeta \\right)\\nonumber \\\\&\\le D_H^{\\epsilon }\\big ( \\rho ^N \\big \\Vert \\gamma ^N \\big ).$ Combining this with Eq.", "(REF ), we arrive at $\\log M\\le D_H^{\\epsilon }\\big ( \\rho ^N \\big \\Vert \\gamma ^N \\big ).$ Finally, for the case of identical initial subsystems, $\\rho ^N=\\rho ^{\\otimes N}$ and $\\gamma ^N=\\gamma ^{\\otimes N}$ , we can use the known second order asymptotic expansion of the hypothesis testing relative entropy [53], $\\frac{1}{N}D^\\epsilon _H(\\rho ^{\\otimes N}\\Vert \\gamma ^{\\otimes N})&\\simeq D(\\rho \\Vert \\gamma )+\\sqrt{\\frac{V(\\rho \\Vert \\gamma )}{N}}\\Phi ^{-1}(\\epsilon ),$ leading to $\\frac{\\log M}{N}\\lesssim D(\\rho \\Vert \\gamma )+\\sqrt{\\frac{V(\\rho \\Vert \\gamma )}{N}}\\Phi ^{-1}(\\epsilon ).$ For the above proof to work also in the case of non-identical subsystems, one would need to prove the following asymptotic behaviour of the hypothesis testing relative entropy: $\\!&\\!\\frac{1}{N}D^\\epsilon _H\\left(\\bigotimes _{n=1}^N \\rho _n\\bigg \\Vert \\bigotimes _{n=1}^N \\gamma _n\\right) \\nonumber \\\\ \\!&\\!~\\simeq \\frac{1}{N}\\sum \\limits _{n=1}^N D(\\rho _n\\Vert \\gamma _n)+\\sqrt{\\frac{\\frac{1}{N}\\!\\sum \\limits _{n=1}^N \\!V(\\rho _n\\Vert \\gamma _n)}{N}}\\Phi ^{-1}(\\epsilon ).\\!$" ], [ "Eliminating the logarithmic term", "We start with the following lemma that will be needed to prove our claim.", "Lemma 7 For a fixed value of $b>0$ and any $s$ , such that $\\Vert s\\Vert =\\sqrt{\\sum _{i=1}^ds_i^2}\\le b$ , we have $\\log P^N_{k(s)}= O(\\log N),$ where $P^N_{k}$ and $k(s)$ are defined by Eqs.", "(REF ) and (REF ), respectively.", "We start by using the definition, $P^N_{k(s)} = \\binom{N}{k_1(s_1),...,k_d(s_d)}p^{k_1(s_1)}_1 ... p^{k_d(s_d)}_d,$ to write $\\log P^N_{k(s)}$ as $\\!\\!\\!\\log P^N_{k(s)}\\!=\\!\\log N!", "-\\sum _{i=1}^d \\log k_i(s_i)!+\\sum _{i=1}^d k_i(s_i)\\log p_i.\\!$ Employing Stirling inequality, $&\\log \\sqrt{N}+N\\log N-N\\nonumber \\\\&\\qquad \\le \\log N!\\le \\nonumber \\\\&\\qquad \\qquad 1+\\log \\sqrt{N}+N\\log N-N,$ we first provide a lower bound for $\\log P^N_{k(s)}$ , $\\log P^N_{k(s)}\\ge & (\\log \\sqrt{N}+N\\log N-N)+\\sum _{i=1}^d k_i(s_i)\\log p_i\\nonumber \\\\&-\\sum _{i=1}^d(1\\!+\\!\\log \\sqrt{k_i(s_i)}\\!+\\!k_i(s_i)\\log k_i(s_i)\\!-\\!k_i(s_i)) \\nonumber \\\\=&-\\sum _{i=1}^d k_i(s_i)\\log \\Big (\\frac{k_i(s_i)}{Np_i}\\Big )-d\\nonumber \\\\&+\\frac{1}{2}\\Big (\\log N-\\sum _{i=1}^d\\log k_i(s_i)\\Big ).$ Recall that $k_i(s_i)=Np_i+\\sqrt{N}s_i$ , which implies $\\sum _{i=1}^d s_i=0$ .", "To simplify the above further, we lower bound the first term by employing the inequality $\\log (1+g)<g$ for $g>-1$ in the following way: $&\\sum _{i=1}^d k_i(s_i)\\log \\Big (\\frac{k_i(s_i)}{Np_i}\\Big )=\\sum _{i=1}^d k_i(s_i)\\log \\left(1+\\frac{s_i}{\\sqrt{N}p_i}\\right)\\nonumber \\\\&\\qquad \\le \\sum _{i=1}^d k_i(s_i)\\frac{s_i}{\\sqrt{N}p_i}=\\sqrt{N}\\sum _{i=1}^d(1+\\frac{s_i}{\\sqrt{N}p_i})s_i\\nonumber \\\\&\\qquad \\qquad =\\sum _{i=1}^d \\frac{s_i^2}{p_i}\\le \\sum _{i=1}^d \\frac{s_i^2}{p_{\\min }}\\le \\frac{b^2}{p_{\\min }},$ where $p_{\\min }=\\text{min }\\lbrace p_1,\\ldots , p_d\\rbrace $ .", "Moreover, observing that $\\log N \\ge \\left(\\log N-\\sum _{i=1}^d\\log k_i(s_i)\\right) \\ge -(d-1)\\log N ,$ we can conclude that $\\log N-\\sum _{i=1}^d\\log k_i(s_i)=O(\\log N).$ Putting it together, we further simplify the bound given in Eq.", "(REF ) as $\\!\\!\\!\\log P^N_{k(s)}\\ge -d-\\frac{b^2}{p_{\\min }}-\\frac{(d-1)}{2}\\log {N}=O(\\log N).$ Similarly, by employing the Stirling inequality from Eq.", "(REF ), we also prove an upper bound for $\\log P^N_{k(s)}$ as follows $\\log P^N_{k(s)}\\le & (1+\\log {\\sqrt{N}}+N\\log N-N)+\\sum _{i=1}^dk_i(s_i)\\log p_i\\nonumber \\\\&-\\sum _{i=1}^d\\Big (\\log \\sqrt{k_i(s_i)}+k_i(s_i)\\log k_i(s_i)-k_i(s_i)\\Big )\\nonumber \\\\=&-\\sum _{i=1}^dk_i(s_i)\\log \\Big (\\frac{k_i(s_i)}{Np_i}\\Big )+1\\nonumber \\\\&+\\frac{1}{2}(\\log N-\\sum _{i=1}^d\\log k_i(s_i)).$ Using the inequality $\\log (1+g)\\ge \\frac{g}{1+g}$ for $g>-1$ , we have $&\\sum _{i=1}^dk_i(s_i)\\log \\Big (\\frac{k_i(s_i)}{Np_i}\\Big )= \\sum _{i=1}^dk_i(s_i)\\log \\Big (1+\\frac{s_i}{\\sqrt{N}p_i}\\Big )\\nonumber \\\\&\\qquad \\ge \\sum _{i=1}^dk_i(s_i)\\frac{s_i}{s_i+\\sqrt{N}p_i}=\\sqrt{N}\\sum _{i=1}^d s_i = 0.$ The above inequality together with Eq.", "(REF ) imply that $\\log P^N_{k(s)}\\le O(\\log N)$ which completes the proof.", "Using Lemma REF , we will now be able to prove our claim that is captured by the following result.", "Lemma 8 The following limits are equal: $\\!\\!\\!\\!\\!\\!&\\lim _{N\\rightarrow \\infty }\\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{1}{N}\\log (P^N_{k(s)})+\\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i\\nonumber \\\\\\!\\!\\!\\!\\!\\!&\\phantom{\\lim _{N\\rightarrow \\infty }\\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big |}+F_{\\rm diss}^N\\ge 0\\Big \\rbrace \\nonumber \\\\\\!\\!\\!\\!\\!\\!", "&~~=\\lim _{N\\rightarrow \\infty }\\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i+F_{\\rm diss}^N\\ge 0\\Big \\rbrace .$ We start by introducing the following notation $&A(N) := \\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{1}{N}\\log (P^N_{k(s)})+\\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i\\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\quad +F_{\\text{diss}}^N\\ge 0\\Big \\rbrace ,\\nonumber \\\\&A(b,N) := \\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{1}{N}\\log (P^N_{k(s)})+\\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i\\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\quad +F_{\\text{diss}}^N\\ge 0\\text{ such that } \\Vert s\\Vert \\le b\\Big \\rbrace ,\\nonumber \\\\&B(N) := \\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i+F_{\\text{diss}}^N\\ge 0\\Big \\rbrace ,\\nonumber \\\\&B(b,N) := \\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\frac{\\beta }{\\sqrt{N}}\\sum _{i}s_iE_i+F_{\\text{diss}}^N\\ge 0\\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\quad \\text{ such that } \\Vert s\\Vert \\le b\\Big \\rbrace ,\\nonumber \\\\&\\Omega (b,N):= \\sum _{s}\\Big \\lbrace P^N_{k(s)}\\Big | \\text{ such that } \\Vert s\\Vert \\ge b\\Big \\rbrace .$ Our goal is to show that $\\lim _{N\\rightarrow \\infty } A(N)=\\lim _{N\\rightarrow \\infty } B(N).$ From the definition it follows that $A(N)-\\Omega (b,N)&\\le & A(b,N)\\le A(N),\\\\B(N)-\\Omega (b,N)&\\le & B(b,N)\\le B(N).$ Taking the limit $N\\rightarrow \\infty $ of Eqs.", "(REF ) and (), we have $\\lim _{N\\rightarrow \\infty }\\Big (A(N)-\\Omega (b,N)\\Big )&\\le \\lim _{N\\rightarrow \\infty } A(b,N)\\le \\lim _{N\\rightarrow \\infty } A(N),\\\\\\lim _{N\\rightarrow \\infty }\\Big (B(N)-\\Omega (b,N)\\Big )&\\le \\lim _{N\\rightarrow \\infty }B(b,N)\\le \\lim _{N\\rightarrow \\infty }B(N).$ Now, let us define $\\lim _{N\\rightarrow \\infty }\\Omega (b,N)=:\\epsilon (b).$ As the multinomial distribution concentrates around mean for $N\\rightarrow \\infty $ , it follows that $\\lim _{b\\rightarrow \\infty }\\epsilon (b)=0$ .", "Therefore, taking the limit $b\\rightarrow \\infty $ in Eq.", "(REF ) we have $& \\lim _{N\\rightarrow \\infty }A(N)\\le \\lim _{b\\rightarrow \\infty } \\lim _{N\\rightarrow \\infty }A(b,N)\\le \\lim _{N\\rightarrow \\infty }A(N)\\nonumber \\\\&\\qquad \\Rightarrow \\quad \\lim _{b\\rightarrow \\infty } \\lim _{N\\rightarrow \\infty }A(b,N)= \\lim _{N\\rightarrow \\infty }A(N).$ Analogously, taking the limit $b\\rightarrow \\infty $ in Eq.", "() we can show that $\\lim _{b\\rightarrow \\infty } \\lim _{N\\rightarrow \\infty }B(b,N)=\\lim _{N\\rightarrow \\infty }B(N).$ Moreover, for any fixed $b$ , by employing Lemma REF , we see that $\\frac{1}{N}\\log (P^N_{k(s)})=O(\\frac{\\log N}{N})$ , which vanishes as $N\\rightarrow \\infty $.", "Therefore, we have $\\lim _{N\\rightarrow \\infty } A(b,N) =\\lim _{N\\rightarrow \\infty } B(b,N),$ and so by taking the limit $b\\rightarrow \\infty $ , we arrive at $\\lim _{b\\rightarrow \\infty }\\lim _{N\\rightarrow \\infty } A(b,N) =\\lim _{b\\rightarrow \\infty }\\lim _{N\\rightarrow \\infty } B(b,N).$ Combining the above with Eqs.", "(REF )-(REF ) we have $\\lim _{N\\rightarrow \\infty } A(N) =\\lim _{N\\rightarrow \\infty } B(N),$ which completes the proof." ], [ "Central limit theorem for multinomial distribution", "Lemma 9 The multinomial distribution with mean $\\mu =Np$ and a covariance matrix $\\Sigma $ can be approximated in the asymptotic limit by a multivariate normal distribution $\\mathcal {N}^{(\\mu ,\\Sigma )}$ with mean $\\mu $ and a covariance matrix $\\Sigma $ .", "Assume $X_1\\ldots X_N$ are independent and identically distributed random vectors each of them with the following distribution $\\!\\!\\!\\text{Prob} (X=x) ={\\left\\lbrace \\begin{array}{ll}\\prod _{i=1}^d p_i^{x_i} &\\quad \\!\\text{if $x$ is unit vector},\\\\0 &\\quad \\!\\text{otherwise}.\\end{array}\\right.", "}$ Then, the mean vector of $X$ is $p$ and the covariance matrix $\\frac{1}{N}\\Sigma =\\text{ diag }(p)-pp^T$ .", "Define $S_N:=X_1+\\ldots +X_N$ .", "Then $\\text{Prob}(S_N=k) &=&\\binom{N}{k_1\\ldots k_d}p_1^{k_1}\\ldots p_d^{k_d}.$ We thus see that a multinomial distribution arises from a sum of independent and identically distributed random variables.", "Therefore, using the central limit theorem, we obtain that the distribution of $k$ approaches the distribution $\\mathcal {N}^{(\\mu ,\\Sigma )}$ arbitrarily well for $N\\rightarrow \\infty $, which completes the proof." ] ]
2105.11759
[ [ "Timing measurements with a 3D silicon sensor on Timepix3 in a 180 GeV/c\n hadron beam" ], [ "Abstract Test beam measurements have been carried out with a 3D sensor on a Timepix3 ASIC and the time measurements are presented.", "The measurements are compared to those of a thin planar sensor on Timepix3.", "It is shown that for a perpendicularly incident beam the time resolution of both detectors is dominated by the Timepix3 front-end.", "The 3D detector is dominated by the time-to-digital conversion whereas the analog front-end jitter also gives a significant contribution for the thin planar detector.", "The 3D detector reaches an overall time resolution of 567(6)ps compared to 683(8)ps for the thin planar detector.", "For a grazing angle beam, however, the thin planar detector achieves a better time resolution because it has a lower pixel capacitance, and therefore suffers less from jitter in the analog front-end for the low charge signals that mainly occur in this type of measurement.", "Finally, it is also shown that the 3D and thin planar detector can achieve time resolutions for large clusters of about 100ps and 250ps, respectively, by combining many single hit measurements." ], [ "Introduction", "Future experiments at the High Luminosity LHC [1] will see a further increase in the number of concurrent events per bunch crossing leading to pile-up.", "A possible solution that will enable them to cope with the increased pile-up is 4D tracking [2], [3] in which precise temporal information of the tracks helps the reconstruction algorithm to distinguish spatially overlapping vertices.", "Therefore, it is foreseen that precise time measurements will become crucial for vertex and tracking detectors used in particle physics experiments.", "New sensor technologies are being explored to achieve the time resolution required for 4D tracking as conventional “thick” planar silicon pixel sensors provide inadequate resolution.", "One strategy to improve the time resolution of a sensor is to decrease the drift distance of charge carriers.", "This can be done, for example, by making thinner sensors [4].", "In doing so, however, the amount of signal charge is also reduced, which leads to an increase in jitter due to a decrease in signal-to-noise ratio.", "On the contrary, 3D silicon sensor technology [5] also reduces the charge carrier drift distance, but does not suffer from a reduction of the signal charge.", "However, the readout electrodes of these sensors typically have a larger capacitance, which also decreases the signal-to-noise ratio, but this time due to an increase in the noise instead.", "In this paper timing measurements obtained with a 3D-silicon sensor bump bonded to a Timepix3 ASIC [6] are presented, and compared to measurements obtained with a thin planar sensor also bonded to Timepix3.", "After a description of the sensors and the measurement setup, the time measurement mechanism of Timepix3 is discussed in detail.", "Then the results for particles crossing the sensors perpendicularly are discussed.", "After this, the results for particles at a grazing angle of incidence are presented, and finally the possibility of improving the time resolution by combining multiple hits on a track is explored." ], [ "Description of sensors", "The 3D sensor technology differs from the planar technology by the geometry of the electrodes.", "This can be seen in the schematic diagrams of the sensors used in this study, which are shown in figure REF .", "In a planar sensor, the pixel electrodes are made by implanting dopants at the bulk surface whereas in a 3D sensor the electrodes penetrate into the bulk.", "For both sensors, the backside is a single electrode where the bias potential is applied, and the frontside electrodes are connected to individual readout channels on the ASIC.", "When depleted, charge carriers in a planar sensor drift perpendicularly to the sensor surface under influence of the electric field.", "In a 3D sensor the charge carriers drift mostly parallel to the sensor surface towards (or away from) the electrodes that are connected to the readout channels.", "The 3D and thin planar sensors used in this study collect holes and electrons, respectively, at the readout electrodes, which are connected to the ASIC.", "The $\\textnormal {n}^{+}$ electrode of the 3D sensor is referred to as the field electrode.", "The double-sided 3D sensor that is used in this study has been fabricated at IMB-CNM [7].", "The electrode regions were etched into the bulk material using an inductively coupled plasma.", "The high aspect ratio of the electrodes was achieved by a process of alternating etch and passivation cycles.", "The electrodes were then formed by filling the etched holes with doped polysilicon.", "This process was repeated for both the front- and backside of the sensor to make the $\\textnormal {p}^{+}$ and $\\textnormal {n}^{+}$ doped electrodes, respectively.", "Double sided processing of the wafer has multiple advantages over single sided processing: (i) producing electrodes with different types of doping is more difficult on a single surface, (ii) it makes it simpler to apply the bias potential, and (iii) the electrodes don't penetrate through the whole sensor which means that there is still some active sensor material above the electrodes, which improves the efficiency [8].", "The double sided processing does, however, require an alignment step which increases the cost.", "The thin planar sensor used in this study was fabricated at Advacam [9] and is an active-edge sensor [10] originally produced for the CLIC vertex detector [11].", "Both sensors are bonded to a Timepix3 ASIC that is read out by a SPIDR readout system [12], [13].", "Figure: Schematic diagrams of a double sided 3D (left) and a thin planar sensor (right).", "Dimensions and layout from measurements presented in this paper and , , .", "The local reference frame of the devices under test is also shown.", "The beam points along the negative zz-direction for a perpendicular incidence." ], [ "Measurement setup", "The measurements for this study were performed at the H8 beam line of the CERN Super Proton Synchrotron (SPS) using the LHCb VELO Timepix3 Telescope [15], [16], which provides the track reconstruction by measuring the position of each particle on eight detector planes (figure REF ).", "Subsequently the track position is interpolated with a resolution of about 1.6 to a device under test (DUT), which is located at the centre of the telescope.", "The DUT is mounted on two translation stages to align it with respect to the telescope planes, and a rotation stage to allow for angle studies.", "The particle beam consists of mixed hadrons ($p$ , $\\pi $ , $K$ ) of about 180.", "The hadrons are delivered in spills that are repeated every [range-phrase=–, range-units=single]2030 and contain a few million particles that are distributed over a duration of typically 4.5.", "Two independent reference time measurements are provided for each particle by two fast scintillators with an active area of ${1.5\\times {1.5}{\\square }}$ .", "They are located up- and downstream of the telescope, and are equipped with constant fraction discriminators (CFD).", "Their signals are registered by the onboard time-to-digital converter (TDC) of the SPIDR readout system [12], [13].", "The up- and downstream scintillators have time resolutions of respectively 381(8) and 182(4) [16].", "Figure: Diagram of the LHCb VELO Timepix3 Telescope .", "The distance between the scintillators is about 1, the distance between the outer telescope planes is about 48, and the distance between adjacent planes is about 2.5." ], [ "Timepix3 calibration", "For both DUTs the Timepix3 ASICs were operated in the ToA & ToT acquisition mode in which both the time of arrival (ToA) and time over threshold (ToT) are measured for each hit.", "Figure REF illustrates how these measurements are performed in Timepix3.", "When the preamplifier output crosses a threshold value, a local voltage controlled oscillator (VCO) is started which has a frequency of 640.", "The pixel logic determines the so-called fine time of arrival (fToA) by counting the number of clock cycles from the VCO until a rising edge of the 40 system clock arrives.", "Meanwhile, the pixel logic also registers the number of 40 clock cycles, which is called the coarse time of arrival (cToA).", "From the fToA and the cToA, the overall time of arrival of each hit can be determined with a granularity of about 1.56.", "For the same hit, the time over threshold is determined with a granularity of 25 by counting the 40 clock while the preamplifier output is above the threshold value.", "Figure: Diagram of the ToA and ToT measurements in Timepix3 for two hits with a different signal amplitude .When a minimum ionising particle (MIP) crosses the sensor, the generated electron-hole pairs induce a transient current signal on the pixel implants, which is subsequently integrated by the charge sensitive preamplifier in the analog front-end of the corresponding pixel [17].", "The preamplifier output signal is proportional to the integrated current, and thus to the number of electron-hole pairs that were generated in the sensor.", "The integrated current is then discharged at a constant rate by the Krummenacher feedback of the preamplifier [18], and the output signal will therefore decay linearly.", "As a result, the ToT is roughly proportional to the amount of charge in the signal.", "To convert the ToT measurement to charge, a test-pulse calibration is performed.", "A controlled amount of charge can be injected into the analog front-end of each pixel using the built-in test-pulse circuitry.", "This is done for charge values up to about ${18}{}$ in steps of approximately ${250}{}$ .", "The relationship between charge and the mean ToT response is then determined for each pixel by fitting the surrogate function [19] $\\textnormal {ToT} = p_0 + p_1 Q - \\frac{p_2}{Q - p_3}\\,,$ and the inverse relationship gives the conversion from ToT value to signal charge.", "The VCO divides the 25 period of the 40 clock into 16 TDC time bins of approximately 1.56 each.", "However, there are significant deviations in the widths of these bins as a consequence of (i) variations in the VCO frequency due to process variation in the fabrication of Timepix3, and (ii) variations in the signal propagation delay between a pixel and its corresponding VCO (which is shared by eight pixels) due to differences in the capacitive loading of the traces that connect them.", "The latter affects the width of the first time bin, which has an fToA value of zero.", "The time bins for fToA values 114 have a size that is mainly determined by the VCO frequency.", "The size of the last time bin, with $\\textnormal {fToA}=15$ , is determined by how much time in the 25 period remains after subtracting the total width of the other time bins.", "To correct timing errors introduced by the TDC time bin variations, their sizes are measured using externally timed digital test pulses, which bypass the analog front-end of the pixel, and directly go to the digital part instead.", "The external test pulses are generated by a pulse generator that is triggered on an edge of the 40 clock for synchronisation.", "The trigger delay is then varied in steps of 10 to scan the whole 25 period.", "For each value of the trigger delay, 1000 test pulses are sent to the pixels, and the resulting fToA values are recorded.", "Figure REF shows a part of such a delay scan for a single pixel of the 3D detector.", "For this pixel, a test pulse that is generated with a trigger delay of zero arrives in the $\\textnormal {fToA}=2$ time bin=a trigger delay of zero is not necessarily aligned with a 40 clock edge due to delays in the electronics and cabling.", "As the trigger delay is increased, the fToA decreases because the time to the first subsequent 40 edge decreases.", "After an fToA value of zero, the test pulse arrives in the $\\textnormal {fToA}=15$ bin of the next 25 period.", "Figure: Test pulse delay scan of a single pixel of the 3D detector.", "The plot shows the number of hits with even and odd fToA values as a function of the delay configuration of the pulse generator.For each fToA value $n$ , the time bin size is determined by fitting the number of hits in its corresponding time bin with $N_n\\!\\left(d\\right) =\\frac{N_{\\textnormal {total}}}{2}\\left[\\operatorname{erf}\\!\\left(\\frac{d - l}{\\sqrt{2}\\,\\sigma _{\\textnormal {j}}}\\right)- \\operatorname{erf}\\!\\left(\\frac{d - l - w}{\\sqrt{2}\\,\\sigma _{\\textnormal {j}}}\\right)\\right]\\, ,$ where $N_{\\textnormal {total}}$ is the total number of test pulses per delay value, $d$  is the trigger delay, $\\sigma _{\\textnormal {j}}$ is the jitter in the measured arrival time of the test pulses with respect to the 40 clock, and finally, $l$ and $w$ are the lower edge and size of the time bin, respectively.", "Figure REF shows the distribution of the time bin size for both DUTs.", "It can be seen that the first and last time bins deviate the most from their design value of 1.56.", "The results also show that some pixels only have 15 non-zero time bins: 0.49 and 24.8 for the 3D and thin planar detector, respectively.", "The difference between the two detectors is explained by the fact that they use different versions of Timepix3.", "The 3D detector uses the first iteration of Timepix3 whereas the thin planar detector uses the second iteration in which a power distribution issue was addressed.", "The results for the 3D detector are in agreement with test pulse delay scans performed on other first-iteration Timepix3 chips [20].", "For each hit, the time difference between the rising edge of the 40 clock and the centre location of the time bin for the corresponding fToA value is subtracted from the cToA to correct for the unequal sizes of the TDC time bins.", "Figure: Distribution of the TDC time bin size for the 3D and thin planar detector.", "The left plot shows the first and last bins which vary the most in size, and the right plot shows the time bins that lie in between.", "There is a spike at zero in the left plots due to bins with a size of zero.As was mentioned above, each VCO is shared by a group of eight pixels called a superpixel.", "This can lead to the scenario in which the oscillator is already running when a hit arrives in a pixel.", "The arrival time of the earlier hit that started the oscillator, lies somewhere in a 1.56 range (assuming an ideal VCO).", "The exact arrival time in this range is (of course) unknown, and the location of the time bin of the second hit is therefore also unknown because it depends on when the oscillator was started.", "As a consequence, the time bins will have a difference in time resolution: For the first hit, the time binning contribution $\\sigma _{\\textsc {tdc}}$ to the overall time resolution is ${1.56}{}/\\sqrt{12}$ because the first time measurement is described by a rectangular distribution.", "The time measurement of the second hit, however, has a (symmetric) triangular distribution with a base of $2\\times {1.56}{}$ due to the unknown arrival time of the first hit, and therefore it has a time binning resolution of $2\\times {1.56}{}/\\sqrt{24}$ , which is a factor $\\sqrt{2}$ worse than that of the first hit.", "Since the second hit is typically also associated with more timewalk, and therefore also more timing jitter (due to a lower signal to noise ratio in the analog front-end), only the time measurements of the first hits in each superpixel are used in this study.", "For each pixel, the time of arrival within the 25 period is determined as the centre location of the TDC time bin with respect to the 40 clock.", "However, this time calibration only corrects for timing errors within a single clock phase.", "A pixel can still have an overall time offset due to (i) phase differences among pixels in the 40 clock due to the clock distribution, and (ii) variations in the speed of the analog front-end due to the power distribution over the pixel matrix.", "These offsets cannot be measured with test pulses because they themselves suffer from (unknown) differences in arrival time over the pixels due to their routing delays in the chip.", "Test beam data is used to determine these overall time offsets as the mean time-residual with respect to the downstream scintillator for each pixel.", "These values are then used as corrections and subtracted from all time measurements in those pixels.", "Figure REF shows the pixel time offsets for both DUTs.", "The pixel time offsets of the 3D and thin planar detectors approximately follow Gaussian distributions with standard deviations of 0.64 and 0.56, respectively.", "The difference in their global behaviour over the pixel matrix is attributed to the fact that they use different versions of Timepix3.", "Figure: Pixel time offsets of the 3D (left) and thin planar detector (right) after correcting the timing errors from VCO variations." ], [ "Perpendicular incidence", "In this section the timing performance of the DUTs is assessed with a perpendicularly incident beam with particles crossing the sensor in the negative $z$ -direction (see figure REF for reference).", "To achieve a perpendicular incidence, the mean cluster size was measured for various angular positions of the rotation stage ranging from -8 to 8, and the angular position corresponding to a minimum in the mean cluster size, and thus true perpendicular incidence, was determined by fitting a second degree polynomial.", "The time resolution of a sensor typically improves with an increase in charge carrier velocity; therefore, the focus in this section is mainly on measurements performed at the highest reverse bias potential that allowed operation of the detector without breakdown—a state in which the leakage current increases exponentially.", "The 3D and thin planar detectors were operated at 60 and 90, respectively.", "The threshold values for detecting a hit were set at 800 and 700, respectively.", "First, the timing behaviour within a pixel cell for both DUTs is discussed.", "This is then shown to strongly depend on the typical signal size, which is substantially different for both sensors.", "Then the efficacy of two different types of timewalk corrections that can be applied to the time measurements is discussed.", "Finally the hit time resolution of both DUTs will be presented.", "The track information provided by the telescope is used to the determine the track intercept with the DUT for each track.", "Events are then collected based on the intrapixel coordinates of the track intercept into (overlapping) circular bins of 1 that are placed on a 0.2 square grid.", "This spacing is significantly smaller than the 1.6 resolution of the track intercept, and is chosen so to study the transition between the electrode regions and the region between them.", "For each bin the relative delay is determined as the mean difference between the hit time measurements on the Timepix3 ASIC and their corresponding reference time, which are defined as the weighted means of the up- and downstream scintillator measurements.", "It should be noted here that the overall time offset between the hits and the scintillators has been subtracted, and that a relative delay of zero therefore corresponds to this overall offset.", "The result is shown in figure REF for both DUTs.", "The electrodes in the 3D sensor are clearly visible as regions that have a large positive delay.", "Furthermore, the readout electrode appears to be slightly off-centre and the time delay seems to increase more gradually on its bottom left side.", "For the thin planar sensor it can be seen that it is mainly slower near the pixel corners.", "This will be explained in terms of signal charge below.", "Figure: Relative delay within a pixel cell of the 3D detector (left) and the thin planar detector (right).", "The shaded regions indicate where the relative delay is longer than 4.Figure REF shows the cluster charge distributions obtained in these measurements for both DUTs together with fits of a Landau distribution convolved with a Gaussian distribution.", "The left plot shows the charge distribution for particles going through different region of the 3D sensor.", "Events from particles going through an electrode have less charge because only energy that is deposited in the bulk silicon is converted into electron-hole pairs that drift so that they induce a signal.", "The cluster charge resulting from a particle going between the electrodes has a most probable value (MPV) of about 22, which is in agreement with a MIP crossing 300 of silicon.", "The cluster charge in the thin planar sensor has an MPV of about 3.3 for particles going through the central area of a pixel (defined as the region where the track intercept is at least 2 away from the nearest pixel edge).", "This value is as can be expected for a MIP crossing 50 of silicon.", "It can also be seen that the MPV of the cluster charge from particles crossing the sensor close to the boundary between two pixels is lower.", "This is expected because charge is being shared by two (or more) pixels, and sometimes not all pixels collect enough charge to reach the threshold level for registering a hit, causing this charge to escape detection and resulting in a lower cluster charge measurement.", "Furthermore, it seems that the thin planar detector has hits that are below threshold, but this is probably a problem with the charge calibration (section REF ) for small charges due to the ToT distribution being partially cut off, which leads to a mean ToT value that is not representative for measurements of charges close to the threshold value, which in turn affects the fit of the surrogate function.", "Figure: Distributions of the collected charge per cluster for the 3D detector (left) and the thin planar detector (right) normalised to the peak values of their corresponding fits.Figure REF shows the time residuals for the 3D detector after applying various corrections.", "In the left plot the time measurements are only corrected for the systematic timing offsets in the pixel matrix as described above in section REF .", "It can be seen that there is a tail in the time residual distribution that is dominated by hits from particles going through one of the electrodes.", "The middle plot shows the same time residuals after they are also corrected for timewalk by first collecting hits into bins based on their charge measurement, and subsequently subtracting the mean of the time residuals in each bin.", "This correction significantly narrows the readout and field electrode distributions because these measurements suffer more from timewalk due to the lower signal charge as was shown in figure REF .", "It can also be seen that the two residual distributions of the electrode events are not aligned after the timewalk correction.", "Somewhat surprisingly, the readout electrode region appears to be slower.", "As will be shown below, there is a slow region above the readout electrodes that can explain why these events are late.", "The right plot shows the result of a method that also corrects time variations that are not due to signal size variations.", "This method works by also binning hits on the track intercept within the pixel (in addition to binning on charge) leading to a lookup table of corrections in terms of the $x$ -intercept, $y$ -intercept, and charge.", "This effectively corrects for spatial regions that are slower (or faster) than others.", "In the remainder of this paper these two types of corrections will be referred to as partial- and full timewalk corrections, respectively.", "The term “timewalk” is usually restricted to only describe those variations in time-to-threshold (the time it takes a signal to reach threshold value) that are due to variations in signal size, but for conciseness this definition is expanded to include also other systematic effects that affect the time-to-threshold, such as variations in signal induction affecting the signal shape (which also includes drift time effects).", "Although applying a full timewalk correction can be important in some cases [16], its impact on the time resolution is relatively small for these DUTs as will be shown shortly.", "Figure: Time residuals in three different regions of the 3D detector without timewalk correction (left), with a timewalk correction based on charge only (centre), and with a correction based on both charge as well as track intercept (right).", "This shows that the tail in the time residual distribution can be mostly corrected with only a charge-based timewalk correction.Figure REF shows the time residuals for the thin planar detector.", "The left plot, which contains the time residuals without timewalk correction, shows that the distribution is wider than that of the 3D detector.", "This is due to more severe timewalk effects because the signals in the thin planar sensor are typically much smaller (see figure REF ).", "Also, the effect is more pronounced for particles that cross the sensor close to the edge of a pixel due to charge sharing.", "The middle plot shows that the partial timewalk correction, which is based on charge only, narrows the distribution considerably.", "The distribution from events near the pixel edge is still slightly off-centre, which is most likely due to a slower signal induction as a consequence of the electric field shape as well as a nonuniform weighting field.", "This is caused by the relatively small size of the pixel implant, which has a diameter of 30 compared to the pixel pitch of 55.", "Applying a full timewalk correction corrects for the remaining time offsets.", "Figure: Time residuals in two different regions of the thin planar detector without timewalk correction (left), with a timewalk correction based on charge only (centre), and with a correction based on both charge as well as track intercept (right).", "The timewalk corrections improve the residuals significantly because the sensor has low charge signals.The resolution of the hit time measurement is determined using the two residuals with respect to the up- and downstream scintillators.", "Since the scintillators perform independent measurements, the covariance between the two residuals corresponds to the time resolution of the hit time measurement: $\\sigma _{t}^2=\\mathrm {cov}\\!\\left({t - t_{\\textnormal {d}},\\,t - t_{\\textnormal {u}}}\\right)$ where $t$ is the hit time, and $t_{\\textnormal {u}}$ and $t_{\\textnormal {d}}$ are the up- and downstream scintillator measurements [16].", "The covariance is determined by performing a maximum likelihood fit of a bivariate Gaussian distribution to the residuals.", "Figure REF shows the hit time resolution of both DUTs as a function of signal charge.", "It also shows the combined contribution to the overall time resolution of the analog and digital parts of the front end in Timepix3.", "This contribution represents the limit of what time resolution can be achieved with Timepix3 and depends on the pixel capacitance of the sensor.", "A measurement of the front-end contribution is briefly explained in the following paragraph before discussing the test beam results shown in the figure.", "The Timepix3 analog front-end time resolution was measured using the same method as the test pulse delay scan that was used to determine the sizes of the time bins in section REF , but instead of sending the test pulses directly to the digital front-end, they were sent to the analog front-end.", "The delay scan was repeated for test pulse amplitudes corresponding to injected charges $Q$ ranging from 2 up to 17 in steps of 1.", "The maximum amount of injected charge is limited by the internal DACs that provide the test pulse voltage.", "The parameter $\\sigma _{\\textnormal {j}}$ in equation (REF ) is now identified as the jitter contribution of the analog front-end: $\\sigma _{\\textnormal {j}} = \\frac{\\sigma _{\\textnormal {v}}}{dV\\!/dt}\\, ,$ where $\\sigma _{\\textnormal {v}}$ and $dV\\!/dt$ are the noise and slew rate of the preamplifier output signal at the threshold value.", "The combined time resolution of the front-end is obtained by adding the contribution of the digital front-end to the fit result as $\\sigma _{\\textnormal {fe}}^2 = \\left(\\frac{\\sigma _{\\textnormal {v}}}{dV\\!/dt}\\right)^2 + \\frac{1}{n}\\sum _{i=1}^n \\frac{w_i^2}{12}\\, ,$ where the sum is over the time bin sizes $w_i$ that were measured before.", "The second term thus describes the mean variance of rectangular distributions having widths $w_i$ , and taking its square root gives 461 and 473 for the 3D and thin planar detectors, respectively.", "The thin planar detector has a higher value due to its bigger time bins (see also figure REF ).", "The charge dependence of the front-end time resolution is modelled as $\\sigma _{\\textnormal {fe}}\\!\\left(Q\\right)^2 = \\left(\\frac{a}{Q-b}\\right)^2 + c^2\\, ,$ where $a$ , $b$ , and $c$ are fit parameters.", "The fits are shown for the front-ends of both DUTs.", "It can be seen that the thin planar detector has a lower analog front-end contribution to the time resolution because the sensor has a lower pixel capacitance than the 3D sensor.", "Figure REF shows that the hit time resolutions of both DUTs have a strong charge dependence.", "The time resolution of the 3D detector after partial and full timewalk correction is dominated by the analog front-end for signals that are larger than 10 and 11.5, respectively.", "For the thin planar detector the time resolution is dominated by the analog front-end for signals larger than 2.", "After partial and full timewalk corrections, the 3D detector achieves an overall resolution of 620(7) and 609(7), respectively.", "This is only marginally better than the standard 300 planar sensors of the telescope which achieve a resolution of 650(9) [16].", "When a minimum charge cut of 15 is applied, effectively rejecting events in the electrode regions, these figures improve to 573(6) and 567(6), respectively.", "However, this cut also reduces the efficiency to 75.2(15).", "For the thin planar detector a minimum charge cut of 1 is used.", "It achieves an overall time resolution of 683(8) after full timewalk correction.", "This time resolution is different than the time resolution of 0.86 found in [21] for the same sensor type and ASIC.", "The difference can be attributed to several factors: (i) in addition to pixel corrections, fToA corrections were applied as described in section REF ; (ii) the measurements in this study were performed at a higher bias potential; and (iii) in this study the timewalk correction has been determined using test beam data instead of test pulse measurements.", "Figure: The time resolution as a function of hit charge for both DUTs.", "The contribution of the Timepix3 front-end obtained from test pulses is also shown." ], [ "Grazing incidence", "For the measurements discussed in this section, the DUTs are rotated around their local $y$ -axes such that the beam points mostly in the $x$ -direction with a small positive $z$ -component (see figure REF ).", "This results in long clusters covering 192.0(11) and 145.7(24) for the 3D and thin planar sensor, respectively.", "This allows for the investigation of the timing behaviour at various depths in the sensor as particles traverse a $z$ -range of less then 2 in each pixel.", "A more detailed explanation of this method can be found in [22].", "First the charge collection as well as the relative delay within a pixel cell of both DUTs is discussed, after which the single hit time resolution is assessed.", "The final part of this section investigates the possibility of combining the hit time measurements of each cluster to obtain a more precise time measurement.", "Figure: Diagrams showing the beam direction in a pixel cell of the 3D detector (left) and the thin planar detector (right) for measurements performed with a grazing-incidence beam.Hits are collected into spatial bins based on the $y$ -, and $z$ -intercepts of the reconstructed track with the $yz$ -plane at the $x$ -centre of the corresponding pixel.", "They are collected into rectangular bins of $0.5\\times {2.5}{\\square }$ for the 3D detector, and square bins of $0.5\\times {0.5}{\\square }$ for the thin planar detector.", "For each spatial bin, a convolution of a Landau and a Gaussian distribution is fitted to the charge distribution by performing a $\\chi ^2$ minimisation.", "Figure REF shows the MPV of the charge distribution as a function of $y$ - and $z$ -intercept.", "Compared to the measurements that were performed at perpendicular incidence, the collected charge in a single hit of the 3D detector is now much smaller because each pixel now only collects the charge from a particle traversing the width of the pixel instead of the full sensor thickness.", "As before, particles crossing the electrode regions of the 3D sensor have smaller signals.", "It can also be seen that the readout electrodes are under a slight angle, which explains the behaviour of the relative delay near the readout electrode in figure REF .", "The signal size of the thin planar sensor is mostly uniform over the pixel except near the edges where charge is lost due to charge sharing.", "Figure: Lateral intrapixel MPV of the collected charge for single hits of the 3D detector (left) and the thin planar detector (right) measured with a grazing incidence beam at bias potentials of 60 and 90, respectively.", "The length of the pillar structures can be seen clearly from the abrupt changes in the charge distribution at about z=230z={230}{} and 60 for the readout- and field pillars, respectively.The relative delay in each spatial bin is determined as before in section REF .", "Figure REF shows the relative delay for the 3D detector after a partial timewalk correction was applied.", "It also shows a bias potential scan of the relative delay as a function of sensor depth $z$ for events that fall into a 0.2 window centred at $y={15}{}$ .", "Most notable is the region above above $z\\sim {250}{}$ where the relative delay keeps increasing with $z$ until hits start falling outside of the ${250}{}$ time window of the clustering algorithm in the telescope reconstruction software.", "This is most likely an indication that the sensor is not depleted in this region, and that the charge is therefore being collected only by diffusion, resulting in long charge collection times.", "It might be expected that these long collection times allow for charge carrier recombination, but figure REF shows that there is no significant decrease in the collected charge in the nondepleted region.", "The carrier lifetime in lowly doped silicon (less than $\\sim [retain-unity-mantissa = false]{1e16}{}$ ) is dominated by the Shockley-Read-Hall mechanism [23], [24] in which electron-hole pairs recombine through deep-level impurities [25].", "This mechanism depends on the number of impurities and crystal defects in the bulk silicon.", "The actual carrier lifetime is therefore difficult to predict, but can be in the order of milliseconds for good quality silicon [26].", "Figure: Relative delay in the 3D detector measured at a bias potential of 60 (left) and the relative delay at y=15y={15}{} as a function of sensor depth zz for various bias potentials (right).", "In both plots a partial timewalk correction is applied.", "The shaded region indicates where the relative delay is longer than 2.Figure: Hit charge MPV of the 3D detector as a function of depth for various bias potentials at y=15y={15}{}.The relative delay for the thin planar detector after a partial timewalk correction is shown in figure REF together with a bias potential scan of the relative delay as a function of sensor depth $z$ at $y={27.5}{}$ .", "There is an increase in the relative delay near the $y$ -edges of the pixel due to a slower signal induction.", "It can also be seen that the relative delay has a minimum near the $z$ -centre of the sensor.", "This minimum is located at $z\\sim {20}{}$ for a bias potential of 90.", "The fact that the sensor is faster in this region cannot be caused by variations in signal size since the MPV of the charge distribution is uniform along $z$ at $y={27.5}{}$ as can be seen in figure REF .", "Instead, it is probably due to a difference in charge carrier velocity.", "For particles crossing the pixel close to the implant, the induced signal is mostly due to holes moving away from the pixel implant.", "Electrons, which have a higher drift velocity, will contribute to the signal when the particle crosses the pixel further away from the implant (towards the $z$ -centre), resulting in a faster signal.", "As the particle crossing point moves even closer to the backside of the sensor, the relative delay increases again because the weighting field is lower near the back electrode (since the pixel does not have an ideal parallel plate geometry), and hence the charge has to drift for some time before significant induction appears as can be seen from the weighting potential shown in figure REF .", "Figure: Relative delay in the thin planar detector measured at a bias potential of 90 (left) and the relative delay at y=27.5y={27.5}{} as a function of sensor depth zz for various bias potentials (right).", "In both plots a partial timewalk correction is applied.Figure: Equipotentials of the analytic weighting potential (as derived in ) at y=27.5y={27.5}{} for a 30 square pixel implant.The hit time resolution is shown in figure REF for both DUTs.", "It can be seen that the 3D detector has the best time resolution in the region ${60}{}<z<{230}{}$ .", "A full timewalk correction improves the time resolution outside of this region because it corrects for the differences in time delay that are not only due to signal size variation.", "For the region ${60}{}<z<{230}{}$ , only the time resolution after a full timewalk correction is shown as the improvement over a partial timewalk correction is not clearly visible in this plot.", "At the most probable signal charge, the thin planar detector has a better time resolution than the 3D detector because the latter suffers more from jitter in the analog front-end due to a higher pixel capacitance.", "Figure: The time resolution as a function of hit charge for both DUTs.", "The contribution of the Timepix3 front-end obtained from test pulses is also shown." ], [ "Multi-hit time resolution", "A more precise time measurement might be obtained by combining the single hit time measurements.", "One way to do this is by calculating a weighted mean of the single hit measurements using the charge dependent time resolution (figure REF ) in determining the weights as $\\sigma _{\\textnormal {t}}^{-2}$ .", "The result is shown for both DUTs in figure REF .", "For the 3D sensor only hits in the region ${60}{}<z<{230}{}$ are used in the weighted mean.", "The dependence of the cluster time resolution on the number of hits $n$ is modelled as $\\sigma _{\\textnormal {cl}}^2\\left(n\\right) = \\left(\\frac{1-\\rho }{n} + \\rho \\right) \\sigma _{\\textnormal {hit}}^2$ where $\\rho $ can be thought of as the mean correlation between the hits.", "This expression is exact for performing $n$ measurements of equal time resolution $\\sigma _{\\textnormal {hit}}$ that are all correlated to each other by the same amount $\\rho $ .", "For a weighted mean as performed here, it becomes an approximation.", "Figure: The cluster time resolution for both DUTs as a function of the number of hits used to calculate the cluster time.The thin planar detector achieves a time resolution of about 100 for about 50 to 60 hits, and the 3D detector achieves a resolution of about 250 for about 40 hits.", "It can be seen that the measurements on both DUTs are not completely uncorrelated since they do not scale as $1/\\sqrt{n}$ .", "In principle, correlation between hits is expected since all measurements are performed with a single clock.", "However, variation in the TDC bin size and time offsets between pixels effectively misalign the TDC bins and act as a type of dither resulting in much less correlation than what would be expected for perfectly aligned TDC bins.", "Correlations due to clock jitter, on the other hand, are not reduced by this misalignment.", "The 3D detector suffers more from correlations between the hits than the thin planar detector.", "This could be because the thin planar detector has more variation in the size of its time bins (figure REF ).", "Detailed investigation of these remaining correlations is outside the scope of this paper, but could be interesting for applications that use ASICs from the Timepix family as a readout for microchannel plates [28]." ], [ "Conclusion", "It has been shown that the 3D and thin planar sensors exhibit significant variations in the relative time delay as a function of intrapixel position for particles crossing the detector at both perpendicular and grazing angles of incidence.", "These variations were corrected by a conventional timewalk correction based on the amount of charge collected in each hit.", "Using the track information to correct for time variations that are due to spatial dependence of the signal induction offers only about a 1 improvement in the overall time resolution in measurements performed at perpendicular incidence.", "However, the overall time resolution is dominated by the Timepix3 front-end.", "For the most probable signal charge in a single hit, the time resolution of the 3D detector is dominated by the TDC in the digital front-end.", "For the thin planar detector, jitter in the analog front-end also has a significant contribution in addition to the TDC.", "Using the track information to correct for time variations might be useful in a future 4D tracker that uses a faster front-end with a more precise TDC.", "At perpendicular incidence the 3D detector suffers least from timewalk because the 3D sensor generates signals that are about a factor six bigger than that of the thin planar sensor due to the difference in thickness.", "Small signals in the analog front-end of Timepix3 do not only result in an increased time delay, but the timing jitter also increases significantly because the preamplifier output signal crosses the threshold value of the discriminator with a lower slew rate.", "In spite of this, the thin planar detector has a better time resolution for signals of less than about ${16}{}$ because the 3D detector has a worse time resolution for particles crossing the electrode regions and it has more analog front-end jitter due to a higher pixel capacitance.", "Still, at their typical signal size, the 3D detector achieves a better overall time resolution of 567(6) compared to 683(8) for the thin planar detector.", "Using a grazing angle beam showed that there is a slow region near the backside of the 3D sensor which is probably due to this region not being depleted.", "The best time resolution for the grazing angle measurements was achieved by the thin planar detector because its lower pixel capacitance gives it an advantage over the 3D detector for small signals.", "Furthermore, for the thin planar sensor it has been shown that, in terms of relative delay, there is an optimum region in the middle of a pixel cell due to the difference in charge carrier velocity.", "It would be interesting to compare the thin planar n-in-p sensor that was used in this study to a p-in-n sensor because it can be expected that a p-in-n sensor will have faster signals for charge generated in the region close to the pixel implant since electrons would dominate in the signal induction instead of holes.", "The possibility of achieving a better time resolution by combining several hits of the same cluster was briefly explored.", "Although this can indeed give a better time resolution, there is a difference between the DUTs in the amount of correlation in the hit time measurements.", "To achieve a good time resolution it is vital to have minimum correlations, and a more careful analysis is therefore required to understand the cause of this difference.", "We express our gratitude to the CERN Linear Collider Detector group for providing us with a Timepix3 assembly with an active-edge sensor, and also to Wiktor Byczynski and Raphael Dumps at CERN for their vital support during the test beam period.", "We also thank our colleagues in the CERN accelerator departments for the excellent performance of the SPS.", "This research was funded by the Dutch Research Council (NWO)." ] ]
2105.11800
[ [ "High Resolution LAsMA $^{12}$CO and $^{13}$CO Observation of the G305\n Giant Molecular Cloud Complex : I. Feedback on the Molecular Gas" ], [ "Abstract We observed the G305 star forming complex in the $J=3\\text{-}2$ lines of $^{12}$CO and $^{13}$CO to investigate how molecular gas surrounding the central stellar clusters is being impacted by feedback.", "The APEX telescope's LAsMA multi-beam receiver was used to observe the region.", "Excitation temperatures and column density maps were produced.", "Combining our data with data from the SEDIGISM survey resulted in a $^{13}$CO $J=3\\text{-}2/2\\text{-}1$ excitation map.", "To verify whether feedback from stellar clusters is responsible for exciting the gas, the distribution of CO excitation was compared with that of 8$\\,\\mu\\rm{m}$ emission imaged with Spitzer, which is dominated by UV-excited emission from PAHs.", "Line centroid velocities, as well as stacked line profiles were examined to investigate the effect of feedback on the gas dynamics.", "Line profiles along radially outward directions demonstrate that the excitation temperature and $^{13}$CO $J=3\\text{-}2/2\\text{-}1$ ratio increase steeply by factors of $\\sim\\,2-3$ at the edge of the denser gas traced by $^{13}$CO that faces the hot stars at the center of the complex and steadily decreases away from it.", "Column density also increases at the leading edge, but does not always decrease steadily outward.", "Regions with higher 8$\\,\\mu\\rm{m}$ flux have higher median excitation temperatures, column densities and $^{13}$CO $J=3\\text{-}2/2\\text{-}1$ ratio.", "The centroid velocity probability distribution function of the region shows exponential wings, indicative of turbulence driven by strong stellar winds.", "Stacked spectra in regions with stronger feedback have higher skewness and narrower peaks with pronounced wings compared to regions with weaker feedback.", "Feedback from the stellar cluster in G305 has demonstrable effects on the excitation as well as on the dynamics of the giant molecular cloud." ], [ "Introduction", "Massive stars ($M>8{M_{\\odot }}$ ) are rare and usually form inside giant molecular clouds (GMCs) as the dominant members of young stellar clusters [33].", "They are short lived ($\\le 30\\,{Myr}$ ), but are known to inject large amounts of feedback into the interstellar medium in the form of stellar winds, ionizing radiation and supernovae [26].", "These feedback mechanisms can in turn trigger or disrupt the formation of the next generation of stars when they interact with the natal molecular cloud.", "They can sweep up the surrounding gas and create parsec-scale cavities around them [11], forming dense shells of gas as a result and thereby triggering star formation.", "Conversely, they can also completely disperse their surrounding molecular gas suppressing star formation [26].", "The ability to both constructively and destructively affect the formation of subsequent generation of stars means that high mass stars play a significant role in driving the evolution of GMCs [53].", "Here, we study the feedback of massive stars in the G305 HII region and molecular cloud complex.", "Figure: Three-color image (green = Spitzer-IRAC4 8 μm\\mu \\rm {m}, red = Midcourse Space Experiment (MSX) 21.3 μm\\mu \\rm {m}, blue = Herschel-PACS 70 μm\\mu \\rm {m}) of the G305 star-forming complex.", "The 21.3 μm\\mu \\rm {m} emission is dominated by hot dust in the HII region.", "The colder gas is traced by the 70 μm\\mu \\rm {m} emission.", "The interface between the ionized and molecular gas appears as a blend of green (strong 8 μm\\mu \\rm {m} emission from PAHs), blue (colder molecular gas) and occasionally red (interfaces very close to HII regions).", "The positions of Danks 1 and 2 clusters have been marked with the smaller and the larger yellow circles respectively and the wolf-rayet star WR48a has been marked as a filled white star." ], [ "G305: a brief history", "The Giant Molecular Cloud associated with G305 is one of the most massive and luminous in the Milky Way (Fig.", "REF ).", "It is located in the Galactic plane at $l\\sim {305}\\,,\\, b\\sim {0}$ and at a kinematic distance of $4\\,{kpc}$ (derived from a combination of radio and H$\\alpha $ observations by [8]; [10] measured its spectrophotometric distance to be $3.8\\pm 0.6\\,{kpc}$ and most recently [5] measured the Gaia DR2 average distance to be $3.7\\pm 1.2\\,{kpc}$ ), which places it in the Scutum-Crux spiral arm.", "Given this distance, the complex has a diameter of $\\sim 30\\,{pc}$ [8] and a molecular mass of $\\sim 6\\times 10^5\\,\\rm {M_{\\odot }}$ [20].", "The G305 complex consists of a large central cavity that has been cleared by the winds from massive stars belonging to two visible central clusters (Danks 1 and 2) and the Wolf-Rayet star (WR48a) [8], [10] and is surrounded by a thick layer of molecular gas [20], [22].", "Radio continuum observations by [21] have revealed that the cavity is filled with ionised gas and identified 6 ultra-compact HII (UC HII) regions and also one bright rimmed cloud (BRC) at the periphery of the cavity, indicating molecular gas irradiated by UV radiation [45], [46], that may cause implosion [4] or evaporation.", "A number of studies have reported star formation tracers (water and methanol masers, HII regions and massive young stellar objects, MYSOs) [8], [28], [50], [16], [17].", "Furthermore, [20] found the concentration of star formation tracers to be enhanced inside a clump of NH$_3$ bearing molecular gas that faces the ionising sources, which is consistent with the hypothesis that the star formation has been triggered.", "Analysis of the stellar clusters in the complex reveals them to have ages of $1.5\\,{yr}$ for Danks 1 and $3\\,{yr}$ for Danks 2, with the former possibly being triggered by the latter [10].", "Additionally, a diffuse population of evolved massive stars was also found to exist within the confines of the G305 complex that had formed around the same time as the two clusters [27], [43], [29], [10], [12], [5].", "How feedback from the massive stars affects molecular clouds is still poorly understood.", "However, the extensive amount of work done in identifying and characterising the ongoing star formation taking place in this complex (UC HIIs, deeply embedded protostars and protoclusters) and mapping the distribution of molecular and ionized gas, makes G305 the ideal laboratory to study the role of feedback in affecting molecular cloud structures and creating future generation of stars.", "We have divided our work into two separate parts.", "In this paper, we want to further our understanding of how feedback from the central population of stars and the UC H II regions in G305 have affected the molecular gas.", "In the second paper of this series we will decompose the GMC into clumps and investigate whether any discernible differences exist in properties of clumps experiencing feedback compared to those further away from the feedback region.", "So far, the evidence of feedback on the molecular gas in G305 has mostly been phenomenological and qualitative.", "In this paper, we study the excitation and kinematics of the gas in the complex since that is the most direct evidence of the feedback affecting the molecular cloud.", "[22] observed the $J=1\\text{-}0$ line of the CO main isotopologue to trace the distribution of molecular gas with a resolution of $> 30 {}$ .", "Since, the compressed layer of gas due to the feedback is expected to be thin, resolving it requires a higher angular resolution.", "The feedback region will also exhibit higher excitation of gas that needs to be verified with excitation studies.", "In addition, the 12CO becomes optically thick for moderately dense clouds ($\\sim 10^3\\,{}$ at $25\\,{}$ for the ($J=1\\text{--}0$ ) transition) and consequently, provides unreliable measurements.", "This necessitates the need for observations with rarer isotopologues of CO.", "Finally, observations of the dispersed gas requires sensitivity to very low column densities.", "Therefore, we decided to use the Large APEX sub-Millimeter Array (LAsMA) seven pixel array on the Atacama Pathfinder EXperiment (APEX) 12 meter sub-millimeter telescope to observe the G305 complex in the $J=3\\text{-}2$ rotational transition of both 12CO and its 13CO isotopologue.", "The paper is organized as follows.", "We describe the observation methods and data reduction techniques in Sect.", ".", "We then present the results and the following analyses of the data along with their discussions in Sects.", "- .", "In Sect.", "we summarize the findings of this paper." ], [ "Observations", "We mapped a $1\\times 1$ area centred on $(l,b)=(305.0,0.0)$ .", "The observations were done between 2017 and 2019 using the APEX telescope [19] under the project number M-099.F-9527A-2017.", "The APEX telescope is located at an altitude of $5100 \\, {}$ at Llano de Chajnantor, in Chile.This publication is based on data acquired with the Atacama Pathfinder EXperiment (APEX).", "APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory and the Onsala Space Observatory.", "The receiver employed for these observations was the 7 pixel LAsMA MPIfR PI instrument operating in the frequency range of $270-{370}{}$ [18].", "It is a hexagonal array of six pixels surrounding a central pixel.", "The outer array is separated from the central pixel by $\\sim 2$ FWHM.", "The pixels employ superconductor-insulator-superconductor (SIS) sideband separating mixers (2SB).", "The backend consisted of Fast Fourier Transform Spectrometers (FFTS4G) that cover an intermediate frequency (IF) bandwidth of 4–8 ${}$ instantaneously with 65536 spectral channels with a width of ${61.03}{}$ .", "In order to observe both ${^{13}CO}$ ($\\nu _{\\text{rest}}\\sim 330.588\\,{}$ ) and ${^{12}CO\\,(3-2)}$ ($\\nu _{\\text{rest}}\\sim 345.796\\,{}$ ) simultaneously the local oscillator frequency was set at ${338.190}{}$ .", "This frequency was chosen in order to avoid contamination of the ${^{13}CO}\\,(3\\text{-}2)$ lines due to bright ${^{12}CO}\\,(3\\text{-}2)$ emission from the image band.", "For this setup, the FWHM is $\\sim {19}{}$ , and the velocity resolution is ${0.053}{}$ .", "The whole 1 square degree region was divided into four sub-maps – one for each quadrant centered around $l={305.5;;} \\, , \\,b={0;;}$ .", "Each sub-map was further divided into 9 smaller sub-maps of size ${10}{} \\times {10}{}$ each.", "Observations were performed in a position switching on-the-fly (OTF) mode.", "The reference-positions were carefully chosen and tested to ensure no emission was present in the velocity range of $-150\\, \\text{to}\\, {50}{}$ (see Sect.", "for details about the reference positions).", "For each scan, the 7 pixel-array was rotated to an angle of ${19.1}$ in equatorial coordinates to ensure minimal overlap between the pixels and maximize sky coverage.", "For each OTF scan, data were dumped every ${7}{}$ scanned.", "On reaching the edge of every mapped area, the array was shifted perpendicular to the scan direction by half the beam-width (${9}{}$ ) and the scan was then carried out in the reverse direction.", "This process was repeated until the whole sub-map was observed.", "The same sub-map was then observed in a 90 rotated frame to ensure Nyquist sampling was achieved along both directions and also to reduce systematic scanning effects.", "During each scan, a calibration measurement was made every 10-15 minutes.", "This setup of a sample spacing of $\\sim {7}{}$ along with a sampling time of ${0.1}{\\sec }$ resulted in datacubes of size $10{} \\, \\times \\, 10{}$ in approximately 1 hour.", "The calibration of the spectral lines was carried out using the apexOfflineCalibrator pipeline [34], [39].", "It uses a three load chopper wheel method, which is an extension of the “standard” method used for millimetre observations [49] to calibrate the antenna temperature scale.", "To remove the spectral variations of the atmosphere across the bandpass, the ATM model [37] was used.", "The pipeline provides the flexibility to determine the opacity for the whole band pass as one number or to calculate a more accurate opacity channel by channel.", "We chose the latter for calibrating our data since the 13CO line lies at the edge of an atmospheric water line at 325 GHz and hence an accurate accounting for this line in the bandpass is essential.", "After the calibration, the intensities are obtained on the $T^*_A$ (corrected antenna temperature) scale.", "Apart from the atmospheric attenuation, this also corrects for rear spillover, blockage, scattering and ohmic losses.", "All intensities stated in this paper are in the $T^*_A$ scale, unless specifically stated otherwise.", "Regular observations of Jupiter were done during the commissioning of the instrument and the observations of G305, resulting in a beam efficiency of $\\eta _{mb}=0.74$ with an uncertainty of about $10\\%$ .", "This efficiency value was used to convert intensities from $T^*_A$ scale to the main beam brightness temperature ($T_{mb}$ )." ], [ "Data Reduction", "The data was reduced using the GILDAS packagehttp://www.iram.fr/IRAMFR/GILDAS.", "A velocity range of ${-200}{}$ to ${200}{}$ was extracted from every spectrum and resampled to an adequate velocity resolution of ${0.5}{}$ to reduce the noise.", "The velocity range ${-150}{}$ to ${50}{}$ was masked before fitting a 3rd order baseline to each spectrum.", "The reduced, calibrated data obtained from the different scans were then combined and gridded using a $6{}$ cell size.", "The gridding process includes a convolution with a Gaussian kernel with a FWHM size of one-third the telescope FWHM beam width.", "The data cubes obtained have a final angular resolution of $\\sim 20{}$ .", "Spectra from all the pixels were then averaged to find the portions of the spectra containing line emission, which were then masked.", "The process of baseline removal was again repeated for the spectra of each pixel to obtain a more stable baseline and a cleaner map.", "Figure: Moment-0 maps of 12CO (3-2) [left] and 13 ^{13}CO (3-2) [right] lines towards the G305 GMC complex.", "In both maps, the emission has been integrated over a velocity range from -70-70 to +10+10\\, {}.", "Pixels with no emission >5σ>5\\sigma for at least three consecutive velocity channels have been excluded (seen in white).", "Overlaid on top as green stars are the position of stars as reported in .", "The red circles show the positions of the Danks 1 and 2 clusters and the red star is WR48a.", "The numbered regions are discussed in the text." ], [ "Moment-0 Maps", "In order to create the integrated intensity (“moment-0”) maps of the G305 region, a noise map of each grid-element (referred to as pixel from now onwards) was first created for both the 12CO and 13CO $J=$ 3-2 transitions.", "This was done by calculating the standard deviation for a part of the spectrum consisting of 100 emission free channels.", "Fig.", "REF shows the pixel-wise noise of the whole region in both 12CO and 13CO $J=$ 3-2 data sets.", "The moment-0 map was produced by integrating the emission above 5$\\sigma $ between ${-70}{}$ and ${+10}{}$ .", "Fig.", "REF shows our moment-0 maps of the G305 star forming complex.", "The 12CO map traces the large scale, diffuse as well as hot gas very well.", "The 13CO map on the other hand is sensitive to higher column density gas and hence, traces the more compact, dense clumps in the complexDue to the high opacity of 12CO and a resulting much lower effective critical density, the 13CO transition probes significantly higher H2 volume densities in the clouds.", "We have divided the map into nine regions for ease of explanation.", "This demarcation of the regions was done visually to highlight relevant molecular gas features observed in the channel-maps.", "From the two integrated maps it can be seen that the central region of the giant molecular cloud has been cleared out of most of the high density gas.", "Although it is reasonable to believe that the central stars clusters are responsible for this, it still needs to be verified whether the amount of feedback from the stars in the complex can indeed carve out such a large hole.", "This will be explored in Sect..", "Looking at the gas content of the complex, we observe three bright dense molecular cloud complexes.", "One in the (Galactic) north (region V and VI in Fig.REF ), one in the south (region VIII) and the third in the east (region III) of the central cavity.", "The northern and the southern complex are connected by a thin strip of high(er) column density tenuous gas (region VII) towards the west.", "Both of these complexes also show a carved-out hole in their center where one O and two B stars have been observed [27], [5].", "The cloud to the east (III) is disconnected from the others but has a long very straight ($\\sim 20\\,\\times \\,2\\,$ pc) filamentary structure (III and I) trailing away from the cavity itself.", "It has a central velocity of $\\sim 40\\,{}$ and a gradient of $\\sim 0.25\\,{}\\text{pc}^{-1}$ .", "Such filaments are found to be associated with many star-forming regions and are believed to play a crucial role in star-formation [2].", "The north-east of the cavity appears as a wind-blown structure delineated by regions II, III, IV and V. Towards the south of the cavity (between regions III and VIII), the gas also appears to have been dispersed by the feedback." ], [ "Velocity structure", "Fig.", "REF shows the average spectrum over the whole region.", "There is no confusion in the foreground except for some local emission at $\\sim -4\\,{}$ in 12CO.", "The line profile is not Gaussian.", "All emission from the complex is found within the velocity range between $-50$ and $-26\\, {}$ for 13CO and $-55$ and $-7\\, {}$ for 12CO.", "Figure: Averaged spectrum of 12CO and 13CO over the whole G305 giant molecular cloud complex.", "The 13CO line has been scaled up by a factor of 3.Figure: Channel maps of 12CO(3-2) emission towards G305.", "The emission has been integrated over 55\\,{} around the velocities indicated in each panel.Fig.", "REF shows the channel maps of G305 for the range of velocities $-50 \\rm {\\ to} -25\\,{}$ with a spacing of $5\\,{}$ .", "Regions I and VIII are prominent in the range $-25$ to $-35\\,{}$ , while the other regions (II, III, IV, V, VI and VII) emit in the range $-35$ to $-45\\,{}$ .", "The observed velocity structure is most likely a consequence of the interplay of the morphology of the cloud and the feedback from the central star clusters on the natal molecular cloud.", "For a comprehensive analysis of the morphology of the cloud we refer to [22] who studied the region reported here in more detail by comparing the molecular emission with the morphology of the surfaces of molecular clouds illuminated by the far-ultraviolet (FUV) photons of the stellar sources, commonly refered to as photon dominated regions (PDR) [48].", "Our results are consistent with the picture presented by these authors." ], [ "LTE Analysis ", "The CO data were combined to calculate the excitation temperature, optical depth, and column density.", "We assume that the molecular gas can be described as a system in Local Thermodymanic Equilibrium (LTE) and the brightness temperature ($T_B$ ) is equal to the measured $T_{mb}$ .", "Then, solving the radiative transfer equation for an isothermal slab of CO radiating at a frequency $\\nu $ we obtain: $\\frac{T_B}{{}} = [J_{\\nu }(T_{ex})-J_{\\nu }(T_{bg})] \\cdot \\left(1-e^{-\\tau _{\\nu }}\\right)$ where $T_{ex}$ is the excitation temperature of the line; $T_{bg}$ is the temperature of the cosmic microwave background; $\\tau _{\\nu }$ is the optical depth; and, $J_{\\nu }(T)$ is the equivalent temperature of a black body at temperature $T$ which can be written as: $\\frac{J_{\\nu }(T)}{{}} = \\frac{h\\nu }{k_B} \\left( \\frac{1}{e^{h\\nu /k_BT}-1}\\right)$ $h$ is the Planck's constant and $k_B$ is the Boltzmann constant.", "All the calculations were done on each $l,b,v$ three-dimensional pixel (from here on referred to as voxel).", "The advantage of this methodology over that of deriving properties over the velocity integrated values is that all the subsequent properties derived are independent of any segmentation method used for source extraction.", "To calculate the $^{12}$ CO excitation temperature, we adopt the textbook formalism [52], which assumes that the $^{12}$ CO (3-2) line is optically thick.", "This was then used to determine the $^{13}$ CO optical depth and subsequently the $^{13}$ CO column density.", "Eq.", "REF is used to calculate the excitation temperature.", "Using a value of $2.7\\,{}$ for the (cosmic microwave radiation) background temperature, we obtain: $\\frac{T_{ex}}{{}} = 16.6 \\, \\left[\\ln {\\left(1+\\frac{16.6}{T_{12}+0.04}\\right)}\\right]^{-1},$ where we use $T_{12}$ and $T_{13}$ for the main-beam brightness temperature of $^{12}$ CO and $^{13}$ CO, respectively.", "All the voxels satisfied the condition $T_{12} > T_{13}$ ; so, we did not have any significant case of self-absorption in our data that would have rendered the excitation temperature derived from this method unreliable.", "The optical depth is derived for each voxel from the excitation temperature obtained from $^{12}$ CO and the $^{13}$ CO intensity using the following equation derived from Eq.", "REF : $ \\tau _{13} = -\\ln \\left[ 1-\\frac{T_{13}}{15.9}\\left( \\frac{1}{e^{15.9/T_{ex}}-1}-0.0028\\right)^{-1}\\right],$ where $\\tau _{13}$ denotes the optical depth of $^{13}$ CO $(3\\text{-}2)$ transition.", "Only those voxels that have an emission greater than 5$\\sigma $ are considered real.", "The column density of $^{13}$ CO $J=2$ level is calculated as: $ \\frac{N_{13} (J=2)}{{\\square }} = \\frac{8\\pi }{c^3} \\frac{g_2}{g_3}\\frac{\\nu ^3}{A_{32}}\\frac{1}{1-e^{(-h\\nu /k_BT_{ex})}} \\, \\tau _{\\nu }\\,dv ,$ where $g_2$ and $g_3$ are statistical weights of $J=2$ and $J=3$ rotational energy levels respectively; $A_{32} = 2.181\\times 10^{-6}\\,{}$ is the Einstein's A coefficient for the $^{13}$ CO $(3\\text{-}2)$ transition [41].", "The rotational partition function, $Z$ , can be approximated by $ Z \\approx \\frac{k_B}{hB} \\left( T_{ex} + \\frac{hB}{3k_B}\\right),$ where $B=h/(8\\pi ^2\\mathcal {I})$ is the rotation constant and is calculated using the moment of inertia $\\mathcal {I}=\\mu R_{CO}^2$ , and $\\mu $ is the reduced mass, and $R_{CO}=0.112\\,{}$ is the mean atomic separation of the CO molecule.", "Finally, using $N_{13}(J)$ and $Z$ one can calculate the total column density of $^{13}$ CO using: $ \\frac{N_{13}}{{\\square }} = N_{13}(J) \\frac{Z}{2J+1} \\exp {\\left[\\frac{hBJ(J+1)}{k_BT_{ex}}\\right]}$" ], [ "Sub-Thermal Excitation", "The analysis presented in this section is based on the assumption that LTE is applicable in all the voxels with detected emission.", "But for gas with densities below the critical density of CO (3-2) ($\\approx 10^4\\,{}$ ), the energy level populations will not follow the Boltzmann distribution and consequently the measured excitation temperature will be lower than the actual temperature of the gas.", "An underestimation of the excitation temperature will lead to an overestimation of the column density values according to Eq.REF .", "But, for optically thick emission the critical denisity can be effectively reduced (roughly by $1/\\tau $ ) thereby making it possible for the gas to reach LTE at lower densities.", "We tried to test whether the limit of large optical depth for 12CO holds in regions where 13CO is also detected.", "For this, we took the lowest detected 13CO column density at a $5\\sigma $ significance and estimated the 12CO column density adopting a 12CO/13CO abundance ratio of 60.", "Figure: Left: Mean T ex T_{ex} map of the G305 complex.", "For each pixel this was calculated by integrating over all the channels.", "Only those channels that had emission above 5σ\\sigma noise level were considered.", "Right Integrated 13 ^{13}CO Column Density map of G305.", "This was calculated by summing the 13 ^{13}CO total column density per channel over all the channels.", "Overlaid on top, are the positions of the HII regions (triangles), UC HII regions (diamonds) and the BRC (star) as determined by , .This value was calculated using the equation $12C/13C = 6.21D_{GC}+18.71$ from [32] where $D_{GC}\\approx 6.6\\,{kpc}$ is the galactocentric distance of G305.", "The estimated 12CO column density corresponding to the lowest detected 13CO column density is then $\\sim \\,2.46\\times 10^{17}\\,{\\square }$ .", "Eq.REF was then used to calculate the velocity integrated optical depth for 12CO(3-2) transition.", "We obtained $\\int \\tau _{\\nu }dv \\approx 149.193\\,{}$ for an excitation temperature of $15\\,{}$ .", "Assuming a Gaussian line profile with a standard deviation of $\\sim 8.5\\,{}$ (see Sect.REF for the justification of the chosen value for standard deviation), the optical depth at the centroid velocity is given by $\\tau _{12} = \\left( \\int \\tau _{\\nu }dv\\right) / \\left(\\sigma _v \\cdot \\sqrt{2\\pi }\\right) \\approx 7$ .", "So, for pixels with detected 13CO emission, the 12CO optical depth is greater than 1.", "This along with the findings of [20] that the volume density of clumps in G305 ranges between $10^3\\text{--}10^4\\,{}$ means that the emission from the pixels where both 12CO and 13CO are detected are most likely in LTE.", "In the other regions where only 12CO emission is detected, the condition of LTE may not always apply and should be kept in mind while interpreting the results derived from them." ], [ "Excitation Temperature and Column Density maps ", "The integrated spectrum over the whole G305 region (see Fig.", "REF ) does not show any obvious sign of confusion along the line of sight (the emission appears to come from one local standard of rest (LSR) velocity without foreground confusion at any other velocity channel).", "Hence, properties like excitation temperature and column densities integrated over the whole linewidth are very likely to be a good representation of the corresponding physical properties of the GMC.", "Fig.", "REF (left) shows the excitation temperature map of the complex where each pixel represents the mean $^{12}$ CO excitation temperature at the corresponding position.", "Fig.", "REF (right) shows the $^{13}$ CO column density of the region summed over all velocity channels on a logarithmic scale.", "The hot regions towards the northern, southern and western molecular clouds are coincident with the HII and UC HII regions existing in the region [21], [22].", "The UC HII regions are also located in regions of high column density surrounded by a region with relatively lower column density.", "One also notes some unconnected small regions where the column density seems to be much higher than in some of the densest regions inside the larger molecular clouds, notably at approximately $(l,b) \\sim (30542,+019), (30530,+013)$ and $(30510,-08)$ .", "Of these, only that at $\\sim (30530,+013)$ is associated with a diffuse H II region G305.4399+00.2103.", "For other sources we have not found any counterparts in the literature.", "In Fig.REF we notice that the edge of the denser clouds show a marked jump in their excitation temperature at the side facing the central stellar clusters, which then tapers off as we move away from the center.", "This represents clear evidence that feedback from the stars acting on the edge of the dense gas is heating it.", "The exposed gas then acts as a shield for the rest of the cloud, resulting in a temperature profile that decreases as we move away from the center.", "Fig.", "REF shows the profiles of 13CO column density (top panel row) and excitation temperature (second row of panels from the top) along two different arbitrarily selected directions leading away from the center as marked on Fig.", "REF .", "These profiles have been obtained after smoothing the excitation temperature and column density maps to a resolution of 30${}$ in order to compare them with other ancillary data as will be evident in the following sections.", "The left panel shows the profile along the line marked N and the right panel shows that along S. As is evident from the excitation temperature profiles, the gas is heated on the leading edge of the molecular cloud facing the central cavity.", "The feedback from the stars results in a steep increase in the temperature by a factor of $\\sim 3$ at the edge facing the center.", "The temperature then steadily drops as we move away from the center.", "As pointed out in Sect.", "the excitation temperature before the leading edge of the profile may have been underestimated.", "So the increase in its value at the leading edge can be considered to be an upper limit.", "For the column density profiles, we notice that the northern profile also shows a pronounced evidence for compression as the column density at the leading edge is enhanced significantly compared to the trailing edge.", "However, for the southern profile even if the leading edge of the cloud has a higher column density than the rest of the cloud, the decrease in column density as we move away from the center is not as pronounced.", "Due to the absence of significant 13CO emission between the stars and the dense gas boundary, we were only able to determine a lower limit on the extent of the region that has an increased column density at the edge of the cloud (based on the minimum column density ($\\approx {4.1e15}\\,{\\square }$ corresponding to $5\\sigma $ times the rms noise value probed by our observations).", "This lower limit factor of $\\gtrsim 5$ for S and $\\gtrsim 10$ for N still represents a large increase in column density at the edge of the cloud.", "Figure: 13 ^{13}CO Column Density (grey), Excitation Temperature (orange), 13CO J=3-2J=3\\text{-}2/J=2-1J=2\\text{-}1 ratio (green), and 8μm\\mu m Flux (purple) profiles for two separate directions in G305.", "The profiles were plotted outwards from the center of the complex." ], [ "Rotational excitation ", "The energy from the feedback being deposited onto the molecular cloud from the stars may be used up to excite the CO molecules to their higher $J$ rotational levels.", "In this section, we investigate whether feedback in G305 affects the rotational excitation in the molecular cloud.", "We complemented our data with $^{13}$ CO $J=2\\text{-}1$ data from the SEDIGISM [42] survey to create rotational excitation maps for the region.", "For this, the LAsMA $^{13}$ CO $J=3\\text{-}2$ map was first smoothed to the 30 angular resolution of the SEDIGISM map.", "In order to trace the excitation, the $^{13}$ CO  $J=3\\text{-}2$ /$J=2\\text{-}1$ intensity ratio was calculated over the whole map.", "Pixels with a signal-to-noise ratio (S/N from now on) of less than 5 were blanked.", "Warm and dense gas is excited to higher $J$ levels resulting in a higher $^{13}$ CO ($J=3\\text{-}2$ )/($J=2\\text{-}1$ ) ratio.", "Figure: 13CO J=3-2J=3\\text{-}2/J=2-1J=2\\text{-}1 ratio map of the G305 complex.", "The dashed lines show the direction along which the profiles of the excitation ratio was plotted in Fig.", ".", "The black circles show the locations of the Danks 1 and 2 clusters.", "The black star shows the position of WR48a, and the red stars show the positions of the stars from .Fig.", "REF shows the map of the ratio of the two line intensities.", "It is evident that this ratio is higher on the side of the cloud facing the central cavity.", "We also show two directional profiles cut through the excitation map in Fig.REF (third row of panels from the top).", "Both these profiles demonstrate the effect of feedback very well.", "In a very narrow region facing the center we see the excitation ratio almost equal to 1.", "For linear molecules, the rotational excitation can either increase by higher temperatures or larger volume densities or a combination of both, indicating a higher pressure at the cloud surfaces.", "As we then move away from the center, the excitation ratio decreases fast, indicating that the front end of the cloud is acting as a shield for the rest of the gas trailing it." ], [ "Energetics of Feedback", "In this subsection, we try to explore whether the energy input from the central stars can account for the observed effects on the molecular gas morphology and excitation.", "Firstly, the OB stars inside the cavity of G305 can efficiently ionize the surrounding gas with the large amounts of ionizing photons they emit leading to photoevaporation of the cloud.", "Here, we estimate if this can lead to the observed size of the cavity given the age and type of stars in the complex.", "We follow the model used by [51] for a simple case of a spherical cloud of uniform density $n_H$ being illuminated by a central stellar population emitting photons isotropically at rate $\\mathcal {N}_i$ .", "The radius of the central hole carved out by these photons in time $t$ given an electron density $n_e$ is given by the following equation.", "$R_{ion} = \\left(\\frac{3\\mathcal {N}_i}{4 \\pi \\alpha _B n_e^2}[1-\\text{exp}(-n_e^2 \\alpha _B t/n_H)]\\right)^{\\frac{1}{3}},$ where $\\alpha _B$ is the recombination coefficient.", "Considering the stars from the Danks1 and Danks2 as well as those outside the two clusters as reported in [5] as the ionizing sources, we obtain $\\mathcal {N}_i\\approx 9.4\\times 10^{50}\\,{}$ using the stellar classification from [10], [5] and calculating the ionizing flux for different stellar types using [36] and [9].", "We adopt $n_H\\approx 10^4\\,{}$ and $\\alpha _B=2.7\\times 10^{-13}\\,{}$ [20], [22], $n_e=[100-5000]\\,{}$ .", "Figure REF shows the diameter of the ionized bubble as a function of time for a range of $n_e$ values.", "Given the diameter of the central cavity is $\\sim \\,10\\,\\text{pc}$ wide along the north-south direction (between regions V and VIII) and $\\sim \\,30\\,\\text{pc}$ across the east-west direction (between regions III and VII), this would imply that the population of visible stars inside the complex can drive such a cavity via photoionization given an $n_e\\sim \\, 10^2\\,{}$ .", "But, this is a very simplistic view, as the starts in the cavity are not located at its center but are spread out over $\\sim 20\\,{pc}$ .", "Also, electron density inside HII regions is not constant over time and varies from $10^2 \\text{--} 10^5\\,{}$ depending on the diameter of the cavity as $n_e\\propto D^{-1}$ [23], [15].", "A more realistic scenario would be one where the expansion of the cavity is initially driven by the Danks 2 system for the first $1.5\\,{Myr}$ which swepng out some of the ionized gas via strong winds (aided by the WR stars outside the Danks cluster) and thereby lowering the $n_e$ for the next generation of stars in Danks 1 and the others.", "The subsequent generations of stars could then clear out the gas more effectively.", "Figure: Time evolution of the ionization diameter of a spherical cloud of density 10 4 10^4\\,{} being illuminated by Danks1 and Danks2 star clusters at its center for different electron density values.", "The vertical dotted line is the minimum age of the Danks1 cluster take from .Next, we estimate whether the amount of energy input from the stars is large enough to cause the observed trend in the excitation line profiles, even if the nature of the profiles strongly suggest that feedback from the stars is heating the gas.", "For this we first calculate the external pressure exterted by the stars on the cloud surface.", "Assuming that most of the energy input from stellar radiation comes from the ionizing photons, the pressure from stars can be estimated as $P_{\\rm {star}} = \\mathcal {N}_i \\, \\langle h\\nu \\rangle / 4\\pi {R_s}^2 \\, c \\, k$ , where $\\langle h\\nu \\rangle $ is the mean photon energy of an O-star (assumed to be $\\sim 15\\,\\rm {eV}$ [38]), $R_s$ is the distance of the cloud from the emitting star, $c$ is the speed of light and $k$ is the Boltzmann's constant.", "We assume that the O5-O6V, B0-B1V and B2-B3V stars found by [27] in the G305.3+0.2 cluster are the stars responsible for the observed excitation profile N and Danks1 is responsible for that in S. But the spectral classification of stars in Danks1 by [10] spans a large range of ionising flux values.", "In order to obtain a lower limit on the ionising flux, we used the spectral type with the lowest ionising flux, i.e.", "O6V for O4-6, O8V for O6-8/8If, B3V for O8-B3, B3I for O8-B3I and WN9 for WNLh.", "We estimate a distance of 2.5 pc between the edge of the profile N and the stars and $\\sim 4$  pc between Danks1 and the edge of S. Using these values we obtain the external pressure from the stars for N and S to be $P_{\\rm {star, N}} \\sim 2.5\\times 10^5\\,{}$ , and $P_{\\rm {star, S}} \\sim 2.6\\times 10^5\\,{}$ respectively.", "We then calculate the thermal pressure at the edge of the cloud facing the cavity and that at the far end away from the cavity.", "The pressure exerted by thermal motion inside a cloud can be estimated as $P_{th}=n(H_2)\\,T\\,{}$ .", "For the far end of the cloud in both cases N and S, we assume the density to be equal to the average value of clump densities from Table 5 in [22], i.e.", "$n(H_2)\\sim 1.5\\times 10^3\\,{}$ .", "For the temperature we adopt a value of $T\\sim 10\\,{}$ for both N and S (see Fig.REF ).", "So, for both N and S the thermal pressure in the far end of the cloud is $P^{far}_{th} = 1.5\\times 10^4\\,{}$ .", "For the side of the profiles N and S facing the cavity, we need to deduce the value of $n_{H_2}$ .", "For this we assume that the pressure from the stars only compresses the cloud in a direction along the plane of the sky.", "We can then equate the proportional increase in column density to that of the observed volume density.", "From Fig.REF we calculate the proportional increase in column density at the edge facing the center compared to that away from the cavity to be $\\sim 4$ for N and $\\sim 1.6$ for S. Now calculating the thermal pressure at the edge of the profiles facing the central cavity we obtain $P^{near}_{th,N}=28\\cdot 4\\cdot 1.5\\times 10^3 = 1.68\\times 10^5\\,{}$ and $P^{near}_{th,S}=22\\cdot 1.6\\cdot 1.5\\times 10^3 = 5.28\\times 10^4\\,{}$ .", "The difference in the thermal pressure between the leading and trailing edge for N and S is therefore $\\Delta P_{th, N} \\sim 1.53\\times 10^5\\,{}$ and $\\Delta P_{th, S} \\sim 3.78\\times 10^4\\,{}$ respectively.", "This difference in thermal pressure accounts for $\\sim 61\\%$ of $P_{\\rm {star, N}}$ and less than $\\sim 14.5\\%$ of $P_{\\rm {star, S}}$ .", "Hence, for the profile N about $61\\%$ of the radiation pressure is going towards heating up the gas whereas for S this is less than $15\\%$ ." ], [ "Characterizing Feedback in G305", "So far, the evidence of feedback on the molecular gas in G305 has been mostly based on morphology, with specific examples of profiles along selected directions.", "In this section we attempt to quantitatively study the effect of feedback on the excitation of the gas at a global level over the whole cloud complex.", "In order to see this global nature of the gas excitation in feedback regions, we first need to identify as well as quantify where these feedback regions are." ], [ "Identifying feedback regions : GLIMPSE 8$\\mu \\rm {m}$ map", "The 8$\\,{}$ map obtained from the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire [3], [7] is a very useful tool to identify the regions of stellar feedback.", "Figure: GLIMPSE 88\\,{} map of the G305 regions.", "The black contours correspond to the 12 ^{12}CO J=3–2 integrated intensities of 30, 70 and 120\\,{} (50, 120 and 200σ\\sigma , respectively).", "The dashed lines show the direction along which the profiles of the excitation ratio was plotted in Fig.", ".PDRs mark the boundary between the ionized and neutral gas in a molecular cloud [40].", "The FUV photons from stars excite polycyclic aromatic hydrocarbons (PAHs) on the surface of dense molecular clouds at the interface between the ionization front and the molecular gas [47].", "The PAHs absorb FUV radiation from hot stars and re-emit fluorescently in several broad bands at near and mid infrared (IR) wavelengths [1].", "The most luminous of these is within the $6.5 \\text{ to }9\\,{}$ range covered by the 8$\\,\\mu {}$ filter of the Infrared Array Camera [14] aboard the Spitzer Space Observatory that was employed for GLIMPSE.", "Hence, bright regions in 8$\\,{}$ Spitzer/IRAC images correspond to regions subjected to large amounts of radiative feedback to FUV radiation from the stars.", "Fig.", "REF shows the Spitzer 8$\\,\\mu {}$ image of the G305 region.", "The molecular gas in the region seems to be coextensive with the bright 8$\\,\\mu {}$ emission.", "All of the CO emission in the region is associated with IR emission.", "In order to demonstrate the validity of the 8$\\,\\mu {}$ flux as a tracer for feedback we plotted its profiles along the same direction as the excitation temperature, column density and the $^{13}$ CO $J=3\\text{-}2$ /$J=2\\text{-}1$ intensity ratio in Fig.", "REF .", "Before plotting the profile, the 8$\\,{}$ map was first smoothed to the same resolution as the other excitation maps of 30${}$ .", "At the edges of the higher column density clouds, we observe a sharp increase in the 8$\\,\\mu {}$ flux (around a depth of $7\\,{pc}$ for N and $1.5\\,{pc}$ for S), marking the brightest parts of the PDRs of the clouds.", "This increase coincides with the increase in the gas excitation properties.", "Afterwards, all three properties decrease along with the 8$\\,\\mu {}$ flux.", "The decreasing profile as we move away from the center also demonstrates that the PDR is effectively shielding the molecular cloud, absorbing the bulk of the FUV photons.", "Inside the denser cloud along N, the 8$\\,\\mu {}$ profile also shows a second local peak (around a depth of $13\\,{pc}$ along the profile and at a depth of $\\sim \\,6\\,{pc}$ from the edge of the high density cloud) before sharply falling of.", "This also coincides with the local peak in the column density as well as the rotational excitation (third panel in Fig.REF ) profiles.", "But, the excitation temperature (first panel) does not show a distinct peak.", "An additional peak also exists at a depth of about $4\\,{pc}$ along N. This along with the peak coincident with the edge of the high density cloud appears to be part of a bubble shaped structure in Fig.REF .", "There is very little dense molecular material left around the former peak and the diffuse gas does not appear to be impacted from the feedback as can be seen from the excitation temperature profile corresponding to this peak.", "Some pockets of higher column density gas appear to be coincident with this peak.", "The feedback from the stars reported in [27] which are located inside the bubble are most likely responsible for this structure.", "This 8 $\\mu {}$ map was used as a template to identify regions of feedback based on their integrated flux.", "In order to quantify the feedback itself we assumed the 8$\\,{}$ intensity to be a proxy for the feedback by radiation, i.e.", "higher intensity corresponds to a stronger feedback.", "The properties of the molecular CO emission were then investigated in these feedback regions." ], [ "Molecular Gas Properties vs 8 $\\mu $ m Flux", "In this section we investigate how the excitation temperature derived from 12CO emission, the 12CO 3–2/2–1 intensity ratio, and the 13CO column density vary with the degree of feedback traced by the integrated 8$\\mu $ m flux.", "For this, the regions within a given interval of $8~{}$ integrated flux were masked.", "For every 8$\\,\\mu {}$ flux interval, we investigated the aforementioned properties of the molecular emission of pixels within that mask.", "Since the GLIMPSE image does not cover the whole range of latitude of the LAsMA map, only the overlapping range of latitude was considered for the analysis.", "Moreover, the blanked pixels in the excitation map were ignored.", "A pixel scatter plot was then made for the 8$\\,\\mu $ m flux versus each of the three properties (see Fig.REF ).", "Only those pixels with S/N>5 for all the properties being investigated were considered.", "This also ensured that the pixels with possible sub-thermal emission were avoided (these pixels are shown as translucent gray scatter plots in the figure).", "The color of the scatter points show the density of points in its vicinity.", "The mean values of the properties are plotted as a function of 8$\\,\\mu $ m flux on top of the pixel scatter plot.", "A power-law function was then fitted to the mean values for all the three properties and the results of the fits are also shown in the figure.", "As is evident from Fig.REF excitation and rotational temperatures increase with increasing 8$\\,\\mu $ m flux.", "However, we only see a very modest increase in the median column density values for higher 8$\\,\\mu {}$ flux.", "Additionally, we also see a set of scatter points which have a much steeper slope between 50 and 150$\\,$ MJy.sr$^{-1}$ for column density.", "This appears to be mostly from the regions which are far away from the central stars and receive very little feedback from them.", "Only one of these regions i.e.", "G305.4399+00.2103 is known to be a diffuse H II region in literature [50].", "The other regions could also be pre-stellar sources collapsing under gravitation and therefore show high column densities.", "It is beyond the scope of this paper to explore the reason for the observed trend at lower 8$\\,\\mu $ m flux.", "Figure: A pixel by pixel scatter plot of different gas properties versus 8μ8\\,\\mu m flux.", "The colors represents the probability density of the scatter points obtained by a kernel density estimate using Gaussian kernels.", "Overlaid on top are the mean values of the quantity probed along with their standard deviations plotted as a function of 8μ\\,\\mu m flux.", "The blue line shows the power law fit to the mean values vs 8μ\\,\\mu m flux.", "The results of the fit are shown at the top right corner of each panel.", "The gray translucent scatter points in the excitation temperature plot (left) correspond to the pixels which do not have a corresponding 13CO detection and are not included in the power law fit." ], [ "Dynamics of gas under feedback ", "In the preceding sections we have investigated the effects of feedback on the morphology of the gas and its excitation.", "In this section we will investigate the dynamical signature of this feedback.", "The shape of the spectrum in Fig.", "REF is the sum of the spectra over all the pixels in the region.", "As has been seen already, the G305 region consists of various clouds moving at different velocities with respect to us.", "Additionally, the shape of the profile varies from pixel to pixel.", "It is impossible to disentangle the contribution of the line centroid velocities and the shape of the line profile from each pixel on the overall shape of the profile in Fig.", "REF .", "In the following sections we will try to study these line characteristics in a statistical way." ], [ "Velocity Centroid Probability Distribution Function", "The probability distribution function (PDF) of velocities in observational or simulated datasets can be used to characterize a cloud's velocity structure.", "PDFs can show the degree of intermittency in the turbulent molecular cloud [13] through the shapes of their wings.", "Increasing intermittency causes a transition from Gaussian to exponential wings in velocity PDFs.", "In this section, we estimate the velocity PDF of G305 in order to reveal further effects of feedback on the molecular gas.", "It is not possible to know the true velocity PDFs of the clouds from their observations, owing to the fact that the complete velocity information is not available for all three dimensions.", "Historically, two methods have been employed to deduce the true velocity PDF from observational data of spectral lines: [24], [30], [31], [35] used the distribution of the line centroid velocities to deduce the velocity PDFs.", "Alternatively, [13] estimated the velocity PDFs from high S/N observations of single line profiles.", "[35] tested the realm of validity of both of these approaches and concluded that if the size of the observed map is larger or comparable to the depth of the cloud, the centroid velocity PDF reproduces the correct velocity distribution.", "In contrast, for small maps, the average line profiles comprise of a more comprehensive sampling of velocities, given their more comprehensive line-of-sight sampling and hence, provide a better approximation to the true velocity structure of the cloud.", "The G305 complex is believed to have a flattened geometry [22] in the plane of the sky.", "Hence, instead of the average line profile shown in Fig.", "REF , the centroid velocity PDF of the complex should be a good tracer of the actual velocity distribution.", "An added benefit of using the velocity centroid PDFs over using average line profiles is that local phenomena such as outflows that could bias the determination of the global velocity structure do not affect velocity centroid PDFs, since the broad wings of their line profiles leave the centroid velocity unaffected.", "The centroid velocity is effectively the moment-1 value of each pixel.", "Its values were calculated from the 12CO line using the formula, $ \\frac{v_{c}}{{}} = \\frac{\\sum _{i=1}^{N_{chan}} T_i \\cdot v_i}{\\sum _{i=1}^{N_{chan}}T_i}$ where, $N_{chan}$ is the total number of channels and $T_i$ is the observed antenna temperature of the corresponding channel.", "Similarly to the moment-0 maps, only channels with signal greater than $5\\sigma $ were included in the calculation.", "Once the centroid velocity for each pixel was obtained, the PDF was estimated using a normal histogram (i.e.", "the sum of the PDF $P$ is normalized to unity).", "Pixels were partially weighted based on their S/N.", "Pixels with a S/N greater than 100 were assigned an S/N of 100 in order to prevent pixels with significant emission, with S/N $>$ , say, 10 (but $<< 100$ ) to be unreasonably downweighted.", "Figure: Probability density distribution of centroid velocities of 12CO for the G305 molecular cloud complex.", "The error bars are statistical errors calculated as the square root of the histogram amplitudes.Fig.", "REF shows the centroid velocity PDF for the G305 complex.", "The shape of the wings in the velocity PDF suggest that the velocity distribution in the complex is not consistent with a Gaussian which would appear as a parabola on a log-lin plot.", "To quantify the shape of the distribution we calculate its statistical moments.", "The most frequently used moments are the mean ($\\langle v_c \\rangle $ ), variance ($\\sigma ^2$ ), and the Kurtosis ($K$ ) of the distribution defined as below: $& \\frac{\\langle v_c \\rangle }{{}} = \\int _{-\\infty }^{+\\infty } dv_c P(v_c) v_c \\\\& \\frac{\\sigma ^2}{{\\square \\square }} = \\int _{-\\infty }^{+\\infty } dv_c P(v_c) [v_c - \\langle v_c \\rangle ]^2 \\\\& K = \\frac{1}{\\sigma ^4} \\int _{-\\infty }^{+\\infty } dv_c P(v_c) [v_c - \\langle v_c \\rangle ]^4 .$ The variance is a measure of the total turbulent mixing energy.", "The Kurtosis is a measure of the deviation from a Gaussian distribution.", "It assumes a value of three for a Gaussian distribution and six for a distribution with exponential wings.", "For our PDF we obtain the following values for the aforementioned moments : $& \\langle v_c \\rangle = -37 \\pm 7 \\, {} \\\\& \\sigma ^2 = 4.0 \\pm 0.9 \\, {\\square \\square } \\\\& K = 5.1 \\pm 1.0$ The Kurtosis of $5.1 \\pm 1.0$ indicates that the velocity PDF of the complex has exponential wings.", "Using two-dimensional Burgers turbulence simulations, neglecting pressure forces, [6] showed that the velocity PDFs are Gaussian for models of decaying turbulence and have exponential wings for models driven by strong stellar winds." ], [ "Stacked Spectra", "The velocity centroid PDFs as seen in Sect.", "REF contain information only about the central velocity of the gas.", "In this section we investigate the effect of feedback on the line profiles of the G305 molecular cloud complex, especially whether the shape of the line profile in the regions where we expect feedback to be present is significantly different from those where we see little evidence of feedback.", "We also try to characterize the deviations, if any, as a function of the strength of the feedback.", "We used the same method as in section REF to quantify the strength of the feedback.", "Different 8$\\,\\mu {}$ intensities were logarithmically distributed.", "These values were used as a threshold to create masks on the G305 region.", "Regions with 8$\\,\\mu {}$ flux greater than the threshold were labeled as inside feedback zone and those outside were labeled outside feedback zone.", "All the maps were reprojected onto the same two dimensional grid over the galactic longitude and latitude.", "All the spectra were then translated to a common central velocity based on their velocity centroids (see section REF for how to calculate the velocity centroids).", "After aligning, a number of pixels equal to the square root of the total pixels in the respective zones were randomly selected.", "For these randomly chosen pixels the spectra were averaged to obtain an average spectrum representing the inside- and outside feedback zones.", "The spectra obtained by this method will be referred to as average stacked spectrum from now on.", "This process was then repeated 500 times to avoid any biases and obtain a more representative set of spectra for each region.", "Fig.", "REF shows the stacked spectra corresponding to the inside- and outside feedback zones for the G305 molecular cloud complex.", "We also derived a median stacked spectrum for each zone corresponding to all thresholds.", "This was obtained by calculating the median value for each channel over the 500 iterations performed.", "A few things become evident from Fig.", "REF .", "The spectra from the inside feedback zones are brighter than outside feedback zones.", "With increasing feedback strength the peaks of the former also get brighter indicating that with stronger feedback more gas gets excited to the higher $J$ transitions.", "There is not much overlap between the individual stacked spectra from the outside- and inside feedback zones.", "Hence, the median profiles for the two regions for each threshold which look very different from each other are indeed representative of the actual differences between the two regions and are not in fact biased by any individual stacked spectra.", "This was also confirmed by running a Kolmogorov-Smirnov (KS) test [25], [44] between the spectra inside and outside the thresholds.", "We placed $p<0.05$ constraint to reject the null hypothesis that the spectra are drawn from the same population.", "The p-value measures the probability of random chance being responsible for the observed difference between the two spectra.", "The test rejected the null hypothesis with overwhelming certainty ($p<10^{-5}$ ).", "The median stacked spectrum from the inside feedback zones appear to be blue shifted compared to that from the outside feedback zones.", "This is an consequence of the translation of the spectra to a common velocity using their moment-1 values.", "The spectra inside the feedback zones have a peak component that is at the same velocity as that outside the feedback zone.", "But in addition the feedback from the stars seems to have pushed some gas away from us which shows up as the broad positively skewed part of the spectrum inside the feedback zone.", "This makes the moment-1 value of the overall spectrum to be redshifted from the true peak of the spectrum.", "Consequently, in the process of aligning all spectra along their moment-1 value, those inside the feedback zone appear to be blue shifted.", "Thus, the apparent blue shift of the stacked spectra from inside the feedback zone is evidence of the stellar feedback pushing gas away from us.", "In order to characterize the stacked spectra in Fig.", "REF we calculated their statistical moments.", "The moments used for these profiles are the variance ($\\sigma ^2)$ , skewness (S), and Kurtosis (K) : $& \\frac{\\sigma ^2}{{\\square \\square }} = \\left(\\int _{-\\infty }^{+\\infty } dv P(v) [v - \\langle v \\rangle ]^2\\right)/\\left(\\int _{-\\infty }^{+\\infty } dv P(v)\\right) \\\\& S = \\frac{1}{\\sigma ^3} \\cdot \\left(\\int _{-\\infty }^{+\\infty } dv P(v) [v - \\langle v \\rangle ]^3\\right)/\\left(\\int _{-\\infty }^{+\\infty } dv P(v)\\right) \\\\& K = \\frac{1}{\\sigma ^4} \\cdot \\left(\\int _{-\\infty }^{+\\infty } dv P(v) [v - \\langle v \\rangle ]^4\\right)/\\left(\\int _{-\\infty }^{+\\infty } dv P(v)\\right) $ Here, $P(v)$ is the antenna temperature of the spectrum at the velocity $v$ , and $\\langle v \\rangle $ is the expectation value for the velocity of the spectrum given by $\\langle v \\rangle = \\left(\\int _{-\\infty }^{+\\infty } dv P(v) v \\right)/\\left(\\int _{-\\infty }^{+\\infty } dv P(v)\\right)\\,{}$ .", "Fig.", "REF demonstrates how these moments depend on the choice of the 8$\\,\\mu {}$ flux threshold.", "A few things stand out from this result.", "We expect turbulence to result in broader line profiles.", "But contrary to expectations, the variance of the average stacked spectra from the inside feedback zones are mostly smaller than that from the outside feedback zone.", "For the outside feedback zones the variance decreases until $\\sim 80 \\, {}$ and then stays constant with increasing threshold.", "Looking at the variances of average stacked spectra for the inside feedback zones, we observe that the variance stays constant irrespective of the choice of threshold until $\\sim 180 \\, {}$ when it starts decreasing.", "We observe that the stacked spectra are consistently more skewed inside the feedback regions than outside irrespective of the choice of threshold.", "A positive skewness implies that the profile's red wing is enhanced compared to the blue wing.", "As explained before, this hints at the gas being pushed away from us.", "For small threshold values, the stacked spectra in outside feedback zones have skewness close to zero.", "But, it increases with increasing threshold levels.", "Kurtosis of the stacked spectra in the inside feedback zones are mostly larger than those in the outside feedback zone indicating that the profiles have more pronounced wings for the gas impacted with feedback.", "For 8$\\,\\mu {}$ flux thresholds above $\\sim 180 \\, {}$ , the Kurtosis of the spectra inside the feedback zone increases.", "For very high threshold values the Kurtosis shows a large spread.", "These regions correspond to those very close to the UC HII regions and hence show that the feedback in these regions is very effective at pushing gas away.", "This is also evident from the stacked spectra in Fig.REF where secondary peaks are clearly visible showing gas clumps being pushed out.", "The Kurtosis of the spectra outside the feedback zone almost stays in the same range.", "We also observed that the trend followed by the Kurtosis is exactly opposite to the one followed by the variances for both inside and outside feedback zones.", "When the variances of stacked spectra decrease, the Kurtosis increases and vice versa." ], [ "Stacked Spectra for 8$\\,\\mu {}$ Flux Intervals", "In the preceding section, the stacked spectra inside the feedback zone for a given threshold include the cumulative sum of the contributions from all pixels with 8$\\,\\mu {}$ flux greater than the threshold value.", "It becomes difficult to identify the spectra representative of a given interval of 8$\\,\\mu {}$ flux.", "For this, we decided to investigate the stacked spectra corresponding to intervals of 8$\\,\\mu {}$ flux.", "Pixels corresponding to the flux intervals were masked and their stacked spectra were derived by randomly picking pixels corresponding to each mask, similar to that done in Sec.REF .", "The process was repeated 500 times.", "A median spectrum was derived from the set of 500 stacked spectra corresponding to each mask by taking the median intensity of each channel.", "Fig.", "REF shows the stacked spectra corresponding to the pixels in 8$\\,\\mu {}$ flux intervals.", "The peaks of the profiles do increase with increasing flux as expected.", "In order to look at their properties we calculated the statistical moments of each of the stacked spectra using Eq.REF – .", "Their median along with the 5 to 95 percentile range of values spanned by the spread were then plotted as a function of the median value of the 8$\\,\\mu {}$ flux interval.", "Fig.", "REF shows the statistical moments as a function of 8$\\,\\mu {}$ flux.", "The standard deviation initially decreases with increasing 8$\\,\\mu {}$ flux but starts increasing with flux from $\\sim 100 \\, {}$ up until $\\sim 215 \\, {}$ .", "After this the standard deviation of the stacked spectra decreases.", "The Kurtosis of the stacked spectra show the exact opposite trends to the standard deviation: where if the standard deviation decreases, the Kurtosis increases and vice versa.", "At the lowest range of probed 8$\\,\\mu {}$ flux the stacked spectra seem to have a skewness approaching 0 and a Kurtosis close to 3, which hints at a Gaussian shape of its profile.", "The skewness of the stacked spectra then increases to 0.6 for an 8$\\,\\mu {}$ flux of $\\sim 100 \\, {}$ , while it increases only a little (to 0.8) for higher values, around $\\sim 600 \\, {}$ .", "To investigate possible causes for the trends we find in the statistical moments of the stacked spectra, we plotted the contours of 8$\\,\\mu {}$ flux on top of 13CO integrated intensity map.", "The relevant 8$\\,\\mu {}$ flux values at which the statistical moments change directions are 84, 123, 178, 260 and 378$\\,{}$ .", "Fig.", "REF shows these contours overlaid on top of the 13CO moment 0 map.", "Almost all of the dense gas traced by the 13CO is contained inside the $123\\,{}$ contour.", "We expect the feedback from the central clusters to be deposited on the surface of the dense gas.", "As we examine the higher flux contours, they appear to surround the HII and UC HII regions in the complex [21].", "It is possible that the high column density gas surface interacting with the feedback from the central clusters is responsible for the observed increase in skewness and Kurtosis and the decrease in the standard deviation at $\\sim 100 \\, {}$ , as this is where we expect the feedback from the central stars to be deposited.", "As the 8$\\,\\mu {}$ flux increases, the gas is impacted mostly by the stellar winds and ionizing radiation from embedded HII and UC HII regions which appears to be responsible for the increase in skewness as well as kurtosis and quite interestingly, the decrease in standard deviation of the stacked line profiles.", "The 13CO emission from our observation suffers from a lack of completeness as we do not sample a lot of low column density gas for a given 8$\\,\\mu {}$ flux.", "Most of the spectra outside the masks are dominated by noise.", "In addition, the wings of the line profiles are also dominated by noise in many cases even when they are significantly detected in 12CO.", "Since, these wings correspond to the gas being expelled, it is difficult to trace them using 13CO.", "Due to these reasons, it was not possible to repeat the stacked spectra analyses with 13CO lines to gain any further insights into the causes of the observed trends.", "Examining the literature, we would like to emphasize that apparently this kind of analysis has not yet been performed, neither on any actual nor on simulated observations of molecular cloud complexes undergoing feedback.", "So far, this is a stand alone result and it needs to be seen if these observed trends in stacked spectra (if real) are unique to G305 or occur in other GMCs as well before making any meaningful speculations on the causes of these trends.", "Figure: 13CO J=3-2J=3\\text{-}2 moment 0 map with GLIMPSE 8μ\\,\\mu {} contours corresponding to 84(purple), 123(brown), 178(blue), 260(green) and 550.", "(orange)\\, {} overlaid on top.", "The HII and UC HII regions are marked by cyan triangles and red diamonds respectively." ], [ "Summary ", "We observed the G305 star forming giant molecular cloud with the APEX telescope in the 12CO and 13CO $J=3\\text{-}2$ transitions in order to study the effects of feedback from the hot, luminous stars at the center of the complex on the molecular gas.", "We summarize our finding below: The central region of the complex has been cleared out of most of the high column density gas as traced by the 13CO emission.", "The calculations of energy input from the visible stars in the complex showed that they have enough energy input to drive the size of the cavity observed if the electron density ($n_e$ ) in the region is less than $500\\,{}$ .", "A sequential formation of Danks 2 followed by Danks 1 and the other stars in the complex can explain the size of the observed cavity.", "12CO excitation temperature and 13CO column density maps of the region were produced under the assumption of LTE.", "Ratio maps of the rotational excitation were also made using 13CO $J=2\\text{-}1$ data from SEDIGISM and our 13CO $J=3\\text{-}2$ data.", "The validity of LTE assumption was also tested and it was concluded that regions with simultaneous emission from both 12CO and 13CO are most likely in LTE.", "Excitation temperature maps as well as ratio maps of the rotational excitation show that the feedback is being deposited in a narrow region at the edge of the dense gas facing the central stellar complex heating it up.", "The gas then shows a decline in temperature as one moves away from the center.", "The column density also shows a marked increase at the edge of the denser gas, but unlike the excitation temperature, does not always decrease drastically as one moves away from the center.", "Line profiles along two directions were chosen at random to test if the energy input from the stars is responsible for this increased excitation of the gas.", "For the profile towards the north of the complex $\\sim 61\\%$ of the radiation pressure from the nearby stars is being used up to heat and compress the gas at the surface of the cloud; whereas for that towards the south of the complex, $<14.5\\%$ of the input radiation pressure from the star is going towards heating the gas.", "The GLIMPSE 8$\\,\\mu {}$ flux, which is dominated by FUV-excited PAH emission, was used as a proxy to the feedback strength.", "The regions with higher 8$\\,\\mu {}$ flux have higher median excitation temperatures, 13CO column density as well as a higher median 13CO $J=3\\text{-}2/2\\text{-}1$ ratio.", "Investigating the impact of feedback on the dynamics of the gas showed that the centroid velocity probability distribution function of the pixels in the region showed exponential wings indicative of turbulence driven by strong stellar feedback.", "This was followed by stacking the spectra of the pixels and plotting the average stacked profiles.", "On assuming a certain 8$\\,\\mu {}$ flux threshold, the stacked spectra with 8$\\,\\mu {}$ flux above this threshold (assumed to indicate stronger feedback) showed systematically more skewed line profiles than the stacked spectra representing regions with 8$\\,\\mu {}$ flux less than the threshold.", "The stacked spectra of regions with stronger feedback on an average had narrower but much more winged profiles when compared to those from regions with weaker feedback.", "This positive skew is most likely indicative of parts of the cloud complex being pushed away from us, which results in a positive wing of the overall stacked spectra.", "We also note that the standard deviation and the kurtosis of the stacked profiles show opposing trends when plotted as a function of 8$\\,\\mu {}$ flux.", "We thank the staff of the APEX telescope for their assistance in observations.", "We also thank the anonymous referee whose invaluable suggestions have significantly improved the quality of the paper.", "This work acknowledges support by The Collaborative Research Council 956, sub-project A6, funded by the Deutsche Forschungsgemeinschaft (DFG).", "DC acknowledges support by the German Deutsche Forschungsgemeinschaft, DFG project number SFB956A.", "Parts of this work are based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory (JPL), California Institute of Technology under a contract with NASA.", "This publication also made use of data products from the Midcourse Space Experiment.", "Processing of the data was funded by the Ballistic Missile Defense Organization with additional support from NASA Office of Space Science.", "This research has also made use of the NASA/ IPAC Infrared Science Archive, which is operated by the JPL, under contract with NASA." ] ]
2105.11703
[ [ "Permutohedra for knots and quivers" ], [ "Abstract The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations.", "However, this correspondence is not a bijection: more than one quiver may be assigned to a given knot and encode the same information.", "In this work we study this phenomenon systematically and show that it is generic rather than exceptional.", "First, we find conditions that characterize equivalent quivers.", "Then we show that equivalent quivers arise in families that have the structure of permutohedra, and the set of all equivalent quivers for a given knot is parameterized by vertices of a graph made of several permutohedra glued together.", "These graphs can be also interpreted as webs of dual 3d $\\mathcal{N}=2$ theories.", "All these results are intimately related to properties of homological diagrams for knots, as well as to multi-cover skein relations that arise in counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds." ], [ "Introduction", "Knots and quivers play an important role in high energy theoretical physics.", "Knots often arise in the context of topological invariance and can be related to physical objects – such as Wilson loops, defects, and Lagrangian branes – in gauge theories and topological string theory.", "Quivers may encode interactions of BPS states assigned to their nodes, or the structure of gauge theories.", "These two seemingly different entities have been recently related by the so-called knots-quivers correspondence [1], [2], which identifies various characteristics of knots with those of quivers and moduli spaces of their representations.", "The knots-quivers correspondence follows from properties of appropriately engineered brane systems in the resolved conifold that represent knots, thus it is intimately related to topological string theory and Gromov-Witten theory [3], [4], and has been further generalized to branes in other Calabi-Yau manifolds [5], [6], see also [7].", "Other aspects and proofs (for two-bridge and arborescent knots and links) of the knots-quivers correspondence are discussed in [8], [9], [10], [11].", "If there is a correspondence between two types of objects, such as knots and quivers, an important immediate question is how unique both sides of this correspondence are.", "Examples of two different quivers of the same size that correspond to the same knot were already identified in [2], which means that the knots-quivers correspondence is not a bijection.", "In this paper we study this phenomenon systematically and find conditions that characterize equivalent quivers (i.e.", "different quivers that correspond to the same knot).", "It turns out that these conditions lead to interesting local and global structure of the set of equivalent quivers.", "We stress that equivalent quivers that we consider in this paper are of the same size $m$ , such that their nodes are in one-to-one correspondence with generators of HOMFLY-PT homology of a given knot.", "One can always use certain $q$ -identities to construct quivers of larger size that encode the same generating functions of knot polynomials, however this phenomenon has already been studied (see [2], [4]) and it is not of our primary interest.", "Let us thus consider a matrix $C$ of size $m$ (equal to the number of HOMFLY-PT homology generators of a given knot), such that entries $C_{ij}$ are numbers of arrows between nodes $i$ and $j$ of a symmetric quiver corresponding to this knot.", "We characterize the local equivalence of quivers by showing that some of the quivers equivalent to $C$ are encoded in matrices $C^{\\prime }$ , such that $C$ and $C^{\\prime }$ differ only by a transposition of two elements $C_{ab}$ and $C_{cd}$ , whose values differ by one and which satisfy a few additional conditions.", "From each such equivalent matrix $C^{\\prime }$ one can determine another set of equivalent matrices $C^{\\prime \\prime }$ , etc.", "This procedure produces a closed and connected network of equivalent quivers in a finite number of steps.", "It follows that any two equivalent quivers from this network differ simply by a sequence of transpositions of elements of their matrices.", "Figure: Permutohedron Π 4 \\Pi _4.", "Its vertices are labeled by permutations of elements {1,2,3,4}\\lbrace 1,2,3,4\\rbrace , and different colors of edges correspond to different types of transpositions (ij)(i\\ j) (for 1≤i<j≤41\\le i < j\\le 4).", "Vertices connected by an edge differ by one transposition of neighboring elements.Furthermore, we find that the network of such equivalent quivers has an interesting global structure.", "We show that equivalent quivers arise in families that form permutohedra.", "Recall that a permutohedron $\\Pi _n$ is the $(n-1)$ -dimensional polytope, whose vertices are labeled by permutations $\\sigma \\in S_n$ and edges correspond to transpositions of adjacent elements.", "Permutohedron $\\Pi _2$ consists of two vertices connected by an edge, $\\Pi _3$ is a hexagon, and $\\Pi _4$ is a truncated octahedron shown in figure REF .", "In our context, each vertex of a permutohedron represents a quiver matrix and each edge connects equivalent quivers (which are related by a transposition of two appropriate elements).", "Every permutohedron arises from a particular pattern of transpositions of elements of quiver matrices, or equivalently from some particular way of writing a generating function of colored superpolynomials for a given knot.", "For a given knot, there are typically several ways of writing a generating function of colored superpolynomials, which lead to different permutohedra connected by quivers they share.", "Examples of such graphs for torus knots $9_1$ and $11_1$ are shown in figures REF and REF , and we call them permutohedra graphs.", "We find that the above mentioned conditions that characterize equivalent quivers have interesting interpretation in both knot theory and topological string theory.", "In the knot theory these conditions are related to the structure of the (uncolored and $S^2$ -colored) HOMFLY-PT homology of a knot in question, and they have a nice graphical manifestation at the level of homological diagrams: they are the center of mass conditions for homology generators.", "On the other hand, these conditions can be also expressed in terms of multi-cover skein relations that arise in counting of holomorphic curves with boundaries on a Lagrangian brane in Calabi-Yau three-folds.", "These connections provide a new link between homological invariants of knots, Gromov-Witten theory, and moduli spaces of quiver representations.", "Moreover, equivalent quivers corresponding to a given knot represent dual 3-dimensional theories with $\\mathcal {N}=2$ supersymmetry, analogously as discussed in [12], [13], [14], [3].", "One can therefore interpret permutohedra graphs as webs of dual 3d $\\mathcal {N}=2$ theories.", "As mentioned above, the appearance of permutohedra can be interpreted at the level of generating functions of colored superpolynomials.", "More precisely, we show that each of them can be decomposed into a piece that encodes a given permutohedron, coupled to another piece that itself has a structure of a motivic generating function for a smaller quiver that we refer to as a prequiver.", "All equivalent quivers corresponding to a given permutohedron are obtained from the same prequiver in the procedure of splitting that involves specifying some particular permutation – so this is the reason why permutohedra arise.", "Figure: Permutohedra graph for 9 1 9_1 torus knot.", "It consists of two series of permutohedra Π 2 \\Pi _2, Π 3 \\Pi _3 and Π 4 \\Pi _4 connected in the middle, and several other permutohedra Π 2 \\Pi _2.Figure: Permutohedra graph for 11 1 11_1 torus knot.", "It consists of two series of permutohedra Π 2 \\Pi _2, Π 3 \\Pi _3, Π 4 \\Pi _4 and Π 5 \\Pi _5 connected in the middle, and several other permutohedra Π 2 \\Pi _2 and Π 3 \\Pi _3.From the above introductory remarks, or simply from figures REF and REF , it follows that the appearance of equivalent quivers is not an exception, but rather a common and abundant phenomenon.", "This also means that one should regard as a knot invariant the whole set of equivalent quivers, rather than one particular quiver from this class; moduli spaces of all such equivalent quivers encode the same information about the corresponding knot.", "The number of equivalent quivers that satisfy the above mentioned conditions grows fast with the size of the homological diagram: it appears that the unknot and trefoil are the only knots such that corresponding quivers are unique, while some knots with 6 or 7 crossings already have over $100\\, 000$ such equivalent quivers (see the last column of table REF ).", "For a given knot, the number of equivalent quivers that we consider is of the order of the size of the largest permutohedron in the permutohedra graph.", "For example, we find that the largest permutohedra for $(2,2p+1)$ torus knots are two $\\Pi _p$ , which means that the number of equivalent quivers for this family grows factorially as $2 p!$ .", "Apart from the number of equivalent quivers, in table REF we also present the number of pairings and symmetries for various knots that we analyze in the paper.", "By pairings we mean quadruples of generators in the homological diagram that satisfy the center of mass condition mentioned above; this is a necessary, but not sufficient, condition of local equivalence (i.e.", "equivalence of quiver matrices that differ by one transposition of their elements).", "On the other hand, by symmetries we mean quadruples of homology generators that satisfy sufficient conditions of local equivalence – the presence of symmetry means that an appropriate transposition of matrix elements indeed produces an equivalent quiver.", "In particular, we conjecture (and verify to high $p$ ) that the numbers of pairings and symmetries for $(2,2p+1)$ torus knots are respectively $p^2 (p - 1)/2$ and $p (p^2 - 1)/3$ .", "Finally, we also extend our analysis to quivers for knot complements [11], which encode $\\hat{Z}$ invariants for knot complements (also referred to as $F_K$ invariants) [15], [16], [17].", "We show that for $(2,2p+1)$ torus knots equivalence conditions that we find in this paper yield an interesting relation between quivers discussed above (that arise in the original knots-quivers correspondence) and quivers for knot complements.", "Table: The number of pairings, symmetries, and equivalent quivers that we have found for (2,2p+1)(2, 2 p + 1) torus knots, twist knots, and 6 2 ,6 3 ,7 3 6_2, 6_3, 7_3 knots.Note that in principle there might exist other equivalent quivers, which are not related by a series of transpositions that we mentioned above (e.g.", "they might be related by a cyclic permutation of length larger than 2, such that some transpositions of elements of quiver matrix, which arise from a decomposition of such a permutation, do not preserve the partition function).", "However, based on the evidence discussed in what follows, we conjecture that such equivalent quivers do not arise.", "This paper is structured as follows.", "Section  provides necessary background on knot homologies, knots-quivers correspondence, and multi-cover skein relations.", "In section  we focus on local equivalences and formulate the local equivalence theorem, which states that appropriate transpositions of elements of a given quiver matrix lead to equivalent quivers.", "In section  we discuss how these local equivalences lead to the global structure: we show that equivalent quivers arise in families that form permutohedra which are glued into larger graphs that parametrize all equivalent quivers for a given knot.", "In section  we present examples of such a global structure and illustrate how permutohedra of equivalent quivers arise and are glued together for various knots.", "In turn, in section  we consider examples of local equivalences and determine them for some particular quivers for infinite families of $(2,2p+1)$ torus knots and twist knots, as well as $6_2, 6_3$ and $7_3$ knots.", "Section  reveals relations of our results to knot complement quivers and $F_K$ invariants.", "In the appendix we present the lists of all equivalent quiver matrices for knots $5_2$ and $7_1$ , as well as particular choices of quiver matrices for infinite classes of twist knots." ], [ "Prerequisites", "In this section we summarize the background material on knot homologies, knots-quivers correspondence, and multi-cover skein relations, as well as introduce the notation that will be used throughout the paper." ], [ "Knot homologies", "The knots-quivers correspondence, which is of our main interest in this work, is inherently related to knot homologies.", "Let us therefore present first a few basic facts about them.", "We are especially interested in colored HOMFLY-PT homologies, denoted $\\mathcal {H}^R_{ijk}(K)$ for a knot $K$ , where $R$ is a representation (labeled by a Young diagram) referred to as the color [18], [19].", "In this paper we only consider symmetric representations $R=S^r$ , and in various formulae we simply use the label $r$ instead of $S^r$ .", "In particular, by ${G}_r(K)$ we denote the set of generators of the $S^r$ -colored homology.", "While explicit construction of colored HOMFLY-PT homologies has not been provided to date, strong constraints on their structure follow from conjectural properties of associated differentials that relate various generators.", "In particular, these constraints enable computation of colored superpolynomials and HOMFLY-PT polynomials for various knots.", "Colored superpolynomials are defined as follows: $P_r(a,q,t) = \\sum _{i,j,k} a^i q^j t^k \\textrm {dim} \\mathcal {H}^{S^r}_{ijk}(K) \\equiv \\sum _{i\\in {G}_r(K)} a^{a^{(r)}_i} q^{q^{(r)}_i} t^{t^{(r)}_i},$ where variables $a$ and $q$ are those that appear in HOMFLY-PT polynomials, $t$ is the refinement (Poincaré) parameter, and we refer to triples $(a^{(r)}_i,q^{(r)}_i,t^{(r)}_i)$ as homological degrees of the generator $i\\in {G}_r(K)$ .", "In the uncolored case $r=1$ we simply write $(a_i,q_i,t_i)\\equiv (a^{(1)}_i,q^{(1)}_i,t^{(1)}_i)$ .", "For a large class of knots the linear combination $t_i-a_i-q_i/2$ is constant for each $i\\in {G}_1(K)$ ; such knots are called thin [18].", "For a given color $r$ , it is useful to plot colored HOMFLY-PT generators on a planar diagram, such that the generator $i\\in {G}_r(K)$ is represented by a dot in position $(q^{(r)}_i,a^{(r)}_i)$ (and possibly decorated by the value $t^{(r)}_i$ ).", "The structure of differentials mentioned above also imposes constraints on the form of such diagrams.", "In particular, in the uncolored case all generators are assembled into two types of structures, referred to as a zig-zag and a diamond [19].", "The zig-zag consists of an odd number of generators, while each diamond consists of four generators.", "The homological diagram for each knot consists of one zig-zag and some number of diamonds.", "For example, homological diagrams for $(2,2p+1)$ torus knots consist of only one zig-zag made of $2p+1$ generators, while a diagram for $4_1$ knot consists of one diamond and a zig-zag made of only one dot.", "We will present examples of homological diagrams for these and other knots in what follows.", "For $t=-1$ colored superpolynomials reduce to colored HOMFLY-PT polynomials that take form of the Euler characteristic $P_r(a,q) = P_r(a,q,-1) = \\sum _{i,j,k} a^i q^j (-1)^k \\textrm {dim} \\mathcal {H}^{S^r}_{ijk}(K).$ We stress that by $P_r(a,q,t)$ and $P_r(a,q)$ we denote reduced polynomials (equal to 1 for the unknot).", "We use this normalization throughout the paper except section , where using the unreduced normalization is more appropriate.", "We also consider generating functions of colored superpolynomials and colored HOMFLY-PT polynomials for defined by $P_K(x,a,q,t) = \\sum _{r=0}^{\\infty } \\frac{x^r}{(q^2;q^2)_r} P_r(a,q,t), \\qquad \\quad P_K(x,a,q) = \\sum _{r=0}^{\\infty } \\frac{x^r}{(q^2;q^2)_r} P_r(a,q).", "$ Including $q$ -Pochhammer symbols $(q^2;q^2)_r=\\prod _{i=1}^{r}(1-q^{2i})$ in denominators provides a proper normalization for the knots-quivers correspondence as defined in [1], [2]." ], [ "Knots-quivers correspondence", "The knots-quivers correspondence is the statement that to a given knot one can assign a quiver in such a way that various characteristics of the knot are expressed in terms of invariants of this quiver (or invariants of moduli spaces of its representations).", "As already noticed in [2], this correspondence is not a bijection, and several quivers may correspond to the same knot.", "In this work we explain how to identify all such equivalent quivers and reveal the intricate structure they form.", "However, let us first present relevant background on quiver representation theory, and explain how it relates to knots.", "A quiver $Q=(Q_0,Q_1)$ consists of a set of nodes $Q_0$ and a set of arrows $Q_1$ .", "Each arrow connects either two different nodes, or a node to itself – in the latter case it is called a loop.", "We denote by $C_{ij}$ the number of arrows from the node $i$ to the node $j$ , and treat it as an element of a matrix $C$ .", "Quivers that arise in knots-quivers correspondence are symmetric, which means that for each arrow $i\\rightarrow j$ for $i,j\\in Q_0$ there exists an arrow in the opposite direction $j\\rightarrow i$ ; in this case the matrix $C$ is symmetric.", "In quiver representation theory one is interested in the structure of moduli spaces of quiver representations.", "Let us consider a symmetric quiver $Q$ with $m$ nodes and arrows determined by a matrix $C$ .", "We assign to each node $i$ a complex vector space of dimension $d_i$ ; the $m$ -tuple $d=(d_1,\\ldots ,d_m)$ is referred to as the dimension vector.", "Furthermore, for such a quiver we construct the following generating series $P_Q(x,q)= \\sum _{d}(-q)^{d\\cdot C \\cdot d}\\frac{x^{d}}{(q^2;q^2)_{d}} \\equiv \\sum _{d_1,\\ldots ,d_m\\ge 0}(-q)^{\\sum _{i,j=1}^m C_{ij} d_i d_j} \\frac{x_1^{d_1}\\cdots x_m^{d_m}}{(q^2;q^2)_{d_1}\\cdots (q^2;q^2)_{d_m}} , $ where $x=(x_1,\\ldots ,x_m)$ are referred to as quiver generating parameters.", "It turns out that this generating function encodes motivic Donaldson-Thomas invariants $\\Omega _{d_1,\\ldots ,d_m;j}$ of quiver $Q$ , i.e.", "appropriately defined intersection Betti numbers of moduli spaces of representations of $Q$ , for all dimension vectors $d$ .", "These invariants are encoded in the following product decomposition of (REF ): $P_Q(x,q) = \\prod _{(d_1,\\ldots ,d_m)\\ne 0} \\prod _{j\\in \\mathbb {Z}} \\prod _{k\\ge 0} \\Big (1 - (x_1^{d_1}\\cdots x_m^{d_m}) q^{2k+j+1} \\Big )^{(-1)^{j+1} \\Omega _{d_1,\\ldots ,d_m;j}}.$ It was postulated in [20] and proven in [21] that motivic Donaldson-Thomas invariants $\\Omega _{d_1,\\ldots ,d_m;j}$ are non-negative integers.", "The knots-quivers correspondence was motivated by the observation that generating series of colored knot polynomials (REF ) can be written in the form (REF ) for appropriate specialization of generating parameters $x_i$ .", "This statement was proven in various examples in [2], for two-bridge knots in [9], and for arborescent knots in [10].", "The relation between (REF ) and (REF ) has various interesting consequences.", "For example, it follows that Ooguri-Vafa invariants of a knot [22] are expressed in terms of motivic Donaldson-Thomas invariants; as the latter invariants are proven to be integer, it follows that that Ooguri-Vafa invariants are also integer, as has been suspected for a long time.", "On the other hand, if all colored superpolynomials can be expressed in the form (REF ), it follows that all of them are encoded in a finite number of parameters, i.e.", "the elements of the matrix $C$ and additional parameters that arise in the specialization of $x_i$ .", "Let us now formulate the knots-quivers correspondence in all details, in a way appropriate for the perspective of this work.", "Definition 1 We say that the quiver $Q$ corresponds to the knot $K$ if $Q$ is symmetric and there exists a bijection $Q_{0} \\ni i\\longleftrightarrow i \\in {G}_1(K)$ such that $\\left.P_{Q}(x,q)\\right|_{(-q)^{C_{ii}}x_{i}=xa^{a_{i}}q^{q_{i}}t^{t_{i}}}=P_{K}(x,a,q,t) \\quad and \\quad C_{ii}=t_{i}.$ The substitution $(-q)^{C_{ii}}x_{i}=xa^{a_{i}}q^{q_{i}}t^{t_{i}}$ following the bijection (REF ) is called the knots-quivers change of variables.", "Denoting $a^{a_{i}}q^{q_{i}-C_{ii}}(-t)^{C_{ii}}$ as $\\lambda _{i}$ , we can write it shortly as $ x_{i}=x\\lambda _{i} \\qquad \\textrm {or}\\qquad x=x\\lambda .$ The above correspondence can be also reduced to the level of HOMFLY-PT polynomials, simply by putting $t=-1$ in the knots-quivers change of variables.", "Note that the above formulation differs from the original one [1], [2] that does not require bijectivity, only the existence of $\\lbrace a_i,q_i\\rbrace _{i\\in Q_0}$ allowing (REF ).", "In consequence, transformations enlarging the quiver and preserving the generating function – forbidden by definition REF – are allowed in [1], [2].", "Therefore, $Q$ corresponding to $K$ in the sense of the definition REF is the minimal quiver in the original sense of [1], [2].", "One can also define a generalized knots-quivers correspondence [3], which allows for $x_{i}=x^{n_{i}}\\lambda _{i}$ (possibly with $n_i>1$ ), but we do not consider it here." ], [ "Multi-cover skein relations and quivers", "Let us now change perspective to that of curve counting for topological strings.", "It is natural to view holomorphic curves in a Calabi-Yau three-fold with boundary on a Lagrangian $L$ as deforming Chern-Simons theory on $L$ (see [23]).", "In [24] this perspective was used to give a new mathematical approach to open curve counts.", "Then, [4] showed that the invariance of generalized holomorphic curve counts under bifurcations of basic disks – called multi-cover skein relation – generates quiver degeneracies, i.e.", "implies the existence of different quivers corresponding to the same knot.", "One can visualize the multi-cover skein relation as resolving the intersection between disk boundaries, see figure REF .", "Using the language of [3], it can be adapted to quivers as the equality of motivic generating series of two quivers shown at the bottom of figure REF , where each basic disc corresponds to the quiver node, and the linking number corresponds to the number of arrows.", "Physically, it corresponds to the duality between two 3d $\\mathcal {N}=2$ theories and has an interesting relations with the wall-crossing from [25], [20].", "More details can be found in [4].", "Figure: Multi-cover skein relationon linking disks (top) and dual quiver description (bottom) .The phenomenon presented in figure REF is the simplest example of unlinking.", "From the perspective of BPS states, it corresponds to reinterpreting the bound state made of two basic states as an independent basic state.", "In terms of quivers, it means removing one pair of arrows which encode the interaction leading to a bound state and adding a new node.", "Adapting [4] to our notation, we define the general case of unlinking in the following way: Definition 2 Consider a symmetric quiver $Q$ and fix $a,b\\in Q_{0}$ .", "The unlinking of nodes $a,b$ is defined as a transformation of $Q$ leading to a new quiver $\\widetilde{Q}$ such that: There is a new node $n$ : $\\widetilde{Q}_{0}=Q_{0}\\cup n$ .", "The number of arrows of the new quiver is given by $\\widetilde{C}_{ab} & =C_{ab}-1, & \\widetilde{C}_{nn} & =C_{aa}+2C_{ab}+C_{bb}-1,\\\\\\widetilde{C}_{in} & =C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi}, & \\widetilde{C}_{ij} & =C_{ij}\\quad \\textrm {for all other cases,} \\nonumber $ where $\\delta _{ij}$ is a Kronecker delta.", "One can check that quivers on the left- and right-hand side of figure REF correspond respectively to $C=\\left[\\begin{array}{cc}C_{aa} & C_{ab}\\\\C_{ba} & C_{bb}\\end{array}\\right]=\\left[\\begin{array}{cc}0 & 1\\\\1 & 0\\end{array}\\right]\\qquad \\longrightarrow \\qquad \\widetilde{C}=\\left[\\begin{array}{ccc}\\widetilde{C}_{aa} & \\widetilde{C}_{ab} & \\widetilde{C}_{an}\\\\\\widetilde{C}_{ba} & \\widetilde{C}_{bb} & \\widetilde{C}_{bn}\\\\\\widetilde{C}_{na} & \\widetilde{C}_{nb} & \\widetilde{C}_{nn}\\end{array}\\right]=\\left[\\begin{array}{ccc}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 1\\end{array}\\right].$ For us, the most important result of [4] is the following statement: Theorem 3 (Ekholm, Kucharski, Longhi) The unlinking accompanied by the substitution $x_{n}=q^{-1}x_{a}x_{b}$ preserves the motivic generating function of the quiver: $P_{Q}(x,q)=\\left.P_{\\widetilde{Q}}(x,q)\\right|_{x_{n}=q^{-1}x_{a}x_{b}}.$ In section REF we use it to prove the local equivalence theorem." ], [ "Local equivalence of quivers", "In this section we show that for a given quiver of size $m$ (equal to the number of HOMFLY-PT generators of the corresponding knot), encoded in a symmetric matrix $C$ , there exist equivalent quivers such that their matrices differ from $C$ only by a transposition of two non-diagonal elements $C_{ab}$ and $C_{cd}$ , as long as the values of these two elements differ by 1 and certain extra conditions are met.", "This is the phenomenon that we refer to as local equivalence of quivers.", "In the next sections we show that these local equivalences give rise to an intricate global structure whose building blocks are permutohedra, and provide various examples of this phenomenon.", "We start from introducing an equivalence relation that describes quiver degeneracies in a natural way.", "Definition 4 Assume that quiver $Q$ corresponds to the knot $K$ and quiver $Q^{\\prime }$ corresponds to the knot $K^{\\prime }$ in the sense of the definition REF .", "Then we define $Q\\sim Q^{\\prime } \\Longleftrightarrow K \\textrm { and } K^{\\prime } \\textrm { have the~same colored HOMFLY-PT homology.", "}$ In the rest of the paper we refer to the simplest and most common version of (REF ), namely $K=K^{\\prime }$ .", "However, each time we write that two (or more) quivers correspond to the same knot, we keep in mind that another knot with the same colored HOMFLY-PT homology would lead to the same equivalence class of quivers." ], [ "Analysis of possible equivalences", "Let us study when two quivers $Q$ and $Q^{\\prime }$ can correspond to the same knot $K$ .", "Using definition REF , we start from $P_{K}(x,a,q,t)=\\left.P_{Q}(x,q)\\right|_{x=x\\lambda }=\\left.P_{Q^{\\prime }}(x,q)\\right|_{x=x\\lambda ^{\\prime }}$ with $\\lambda _{i}=\\lambda ^{\\prime }_{i}=a^{a_{i}}q^{q_{i}-C_{ii}}(-t)^{C_{ii}},\\quad C_{ii}=t_i \\quad \\forall i\\in Q_{0}=Q^{\\prime }_{0}.$ We will analyze equation  (REF ) order by order in $x$ .", "The linear one holds automatically, so let us focus on terms proportional to $x^2$ : $\\begin{split}\\frac{P_{2}(a,q,t)x^{2}}{(1-q^{2})(1-q^{4})} & =\\sum _{i\\in Q_{0}}\\frac{(-q)^{4C_{ii}}x^{2}\\lambda _{i}^{2}}{(1-q^{2})(1-q^{4})} +\\sum _{i,j\\in Q_{0},i\\ne j}\\frac{(-q)^{C_{ii}+2C_{ij}+C_{jj}}x^{2}\\lambda _{i}\\lambda _{j}}{(1-q^{2})(1-q^{2})}\\\\& =\\sum _{i\\in Q^{\\prime }_{0}}\\frac{(-q)^{4C_{ii}}x^{2}\\lambda _{i}^{2}}{(1-q^{2})(1-q^{4})} +\\sum _{i,j\\in Q^{\\prime }_{0},i\\ne j}\\frac{(-q)^{C_{ii}+2C^{\\prime }_{ij}+C_{jj}}x^{2}\\lambda _{i}\\lambda _{j}}{(1-q^{2})(1-q^{2})},\\end{split}$ where we used (REF ) to write $\\lambda _{i}=\\lambda ^{\\prime }_{i}$ and $C_{ii}=C^{\\prime }_{ii}$ .", "In consequence, the only difference between $Q$ and $Q^{\\prime }$ can appear in non-diagonal terms $C_{ij}$ and $C^{\\prime }_{ij}$ .", "Since equation (REF ) needs to hold for all $a$ and $t$ (which are independent from $C_{ij}$ and $C^{\\prime }_{ij}$ ), we require the equality between coefficients of each monomial in these variables.", "The only possibility of having $Q\\ne Q^{\\prime }$ satisfying (REF ) comes from $C_{ij}\\ne C^{\\prime }_{ij}$ which however lead to the same coefficient of each monomial in $a$ and $t$ on both sides.", "The way $q$ -monomials on both sides are matched can be described by permutations of terms in the coefficient of each monomial in $a$ and $t$ .", "Let us focus on the simplest non-trivial case.", "We assume that each coefficient of monomials in $a$ and $t$ has only one term except from the expression corresponding to $\\lambda _{a}\\lambda _{b}$ and $\\lambda _{c}\\lambda _{d}$ .", "This means that we require $\\lambda _{a}\\lambda _{b}=q^{2s}\\lambda _{c}\\lambda _{d}$ for some $s\\in \\mathbb {Z}$ and $\\lambda _{a}$ , $\\lambda _{b}$ , $\\lambda _{c}$ , $\\lambda _{d}$ being pairwise different.", "(Note that for thin knots we immediately know that $s=0$ .)", "Therefore, we get $C_{ij}=C^{\\prime }_{ij}\\; \\forall i,j\\in Q_0\\backslash \\lbrace a,b,c,d\\rbrace $ and (REF ) can be reduced to $\\lambda _{a}\\lambda _{b}(-q)^{C_{aa}+C_{bb}}\\bigg (q^{2C_{ab}}+q^{-2s+2C_{cd}}\\bigg )=\\lambda _{a}\\lambda _{b}(-q)^{C_{aa}+C_{bb}}\\left(q^{2C^{\\prime }_{ab}}+q^{-2s+2C^{\\prime }_{cd}}\\right),$ where we used $C_{aa}+C_{bb}=C_{cc}+C_{dd}$ that comes from the comparison of $t$  powers in $\\lambda _{a}\\lambda _{b}=q^{2s}\\lambda _{c}\\lambda _{d}$ .", "In consequence, there is only one non-trivial way to satisfy (REF ), namely $C^{\\prime }_{ab}= C_{cd}-s,\\quad C^{\\prime }_{cd}= C_{ab}+s.$ Using the language of permutations of terms in the generating function, this corresponds to the transposition $\\lambda _{a}\\lambda _{b}(-q)^{C_{aa}+2C_{ab}+C_{bb}}$ $\\leftrightarrow $ $\\lambda _{c}\\lambda _{d}(-q)^{C_{cc}+2C_{cd}+C_{dd}}$ .", "For $s=0$ it translates to the transposition of matrix entries $C_{ab}\\leftrightarrow C_{cd}$ .", "Let us continue the analysis of the simplest non-trivial case and check what conditions come from the cubic order of (REF ).", "In order to save space, we start from examining where differences between $\\left.P_{Q}(x,q)\\right|_{x_{i}=x\\lambda _{i}}$ and $\\left.P_{Q^{\\prime }}(x,q)\\right|_{x_{i}=x\\lambda _{i}}$ can arise.", "The general formula reads $\\begin{split}\\frac{P_{3}(a,q,t)x^{3}}{(1-q^{2})(1-q^{4})(1-q^{6})} & =\\sum _{i\\in Q_{0}}\\frac{(-q)^{9C_{ii}}x^{3}\\lambda _{i}^{3}}{(1-q^{2})(1-q^{4})(1-q^{6})}\\\\& +\\sum _{i,j\\in Q{}_{0},i\\ne j}\\frac{(-q)^{4C_{ii}+4C_{ij}+C_{jj}}x^{3}\\lambda _{i}^{2}\\lambda _{j}}{(1-q^{2})(1-q^{4})(1-q^{2})}\\\\& +\\sum _{i,j,k\\in Q{}_{0},i\\ne j\\ne k}\\frac{(-q)^{C_{ii}+2C_{ij}+C_{jj}+2C_{jk}+C_{kk}+2C_{ik}}x^{3}\\lambda _{i}\\lambda _{j}\\lambda _{k}}{(1-q^{2})(1-q^{2})(1-q^{2})},\\end{split}$ so we have to look for terms containing $\\lambda _{a}\\lambda _{b}$ or $\\lambda _{c}\\lambda _{d}$ .", "They are given by $\\begin{split}\\frac{x^{3}\\lambda _{a}\\lambda _{b}}{(1-q^{2})(1-q^{4})(1-q^{6})} & \\Big [(-q)^{4C_{aa}+4C^{\\prime }_{ab}+C_{bb}}\\lambda _{a}+(-q)^{4C_{bb}+4C^{\\prime }_{ab}+C_{aa}}\\lambda _{b}\\\\& +(1+q^{2})(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bc}+C_{cc}+2C_{ac}}\\lambda _{c}\\\\& +(1+q^{2})(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bd}+C_{dd}+2C_{ad}}\\lambda _{d}\\\\& +(1+q^{2})\\sum _{i\\in Q_{0}\\backslash \\lbrace a,b,c,d\\rbrace }(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bi}+C_{ii}+2C_{ai}}\\lambda _{i}\\Big ]\\end{split}$ and $\\begin{split}\\frac{x^{3}\\lambda _{c}\\lambda _{d}}{(1-q^{2})(1-q^{4})(1-q^{6})} & \\Big [(-q)^{4C_{cc}+4C^{\\prime }_{cd}+C_{dd}}\\lambda _{c}+(-q)^{4C_{dd}+4C^{\\prime }_{cd}+C_{dd}}\\lambda _{d}\\\\& +(1+q^{2})(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}}\\lambda _{a}\\\\& +(1+q^{2})(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{bd}+C_{bb}+2C_{bc}}\\lambda _{b}\\\\& +(1+q^{2})\\sum _{i\\in Q_{0}\\backslash \\lbrace a,b,c,d\\rbrace }(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{di}+C_{ii}+2C_{ci}}\\lambda _{i}\\Big ]\\end{split}$ for $\\left.P_{Q^{\\prime }}(x,q)\\right|_{x=x\\lambda }$ and analogous terms without prime symbols for $\\left.P_{Q}(x,q)\\right|_{x=x\\lambda }$ .", "Since $\\lambda _{a}\\lambda _{b}=q^{2s}\\lambda _{c}\\lambda _{d}$ , imposing the equality between $\\left.P_{Q^{\\prime }}(x,q)\\right|_{x=x\\lambda }$ and $\\left.P_{Q}(x,q)\\right|_{x=x\\lambda }$ implies conditions for sums of terms from both (REF ) and (REF ) for $\\lambda _{a}$ , $\\lambda _{b}$ , $\\lambda _{c}$ , $\\lambda _{d}$ and each $\\lambda _{i}$ , $i\\in Q_{0}\\backslash \\lbrace a,b,c,d\\rbrace $ : $\\begin{split}\\lambda _{a} & \\Big [(-q)^{4C_{aa}+4C^{\\prime }_{ab}+C_{bb}+2s}+(1+q^{2})(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}}\\Big ]\\\\& =\\lambda _{a}\\Big [(-q)^{4C_{aa}+4C_{ab}+C_{bb}+2s}+(1+q^{2})(-q)^{C_{cc}+2C_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}}\\Big ],\\end{split}$ $\\begin{split}\\lambda _{b} & \\Big [(-q)^{4C_{bb}+4C^{\\prime }_{ab}+C_{aa}+2s}+(1+q^{2})(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{bd}+C_{bb}+2C_{bc}}\\Big ]\\\\& =\\lambda _{b}\\Big [(-q)^{4C_{bb}+4C_{ab}+C_{aa}+2s}+(1+q^{2})(-q)^{C_{cc}+2C_{cd}+C_{dd}+2C_{bd}+C_{bb}+2C_{bc}}\\Big ],\\end{split}$ $\\begin{split}\\lambda _{c} & \\Big [(-q)^{4C_{cc}+4C^{\\prime }_{cd}+C_{dd}}+(1+q^{2})(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bc}+C_{cc}+2C_{ac}+2s}\\Big ]\\\\& =\\lambda _{c}\\Big [(-q)^{4C_{cc}+4C_{cd}+C_{dd}}+(1+q^{2})(-q)^{C_{aa}+2C_{ab}+C_{bb}+2C_{bc}+C_{cc}+2C_{ac}+2s}\\Big ],\\end{split}$ $\\begin{split}\\lambda _{d} & \\Big [(-q)^{4C_{dd}+4C^{\\prime }_{cd}+C_{cc}}+(1+q^{2})(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bd}+C_{dd}+2C_{ad}+2s}\\Big ]\\\\& =\\lambda _{d}\\Big [(-q)^{4C_{dd}+4C_{cd}+C_{cc}}+(1+q^{2})(-q)^{C_{aa}+2C_{ab}+C_{bb}+2C_{bd}+C_{dd}+2C_{ad}+2s}\\Big ],\\end{split}$ $\\begin{split}\\lambda _{i} & \\Big [(-q)^{C_{aa}+2C^{\\prime }_{ab}+C_{bb}+2C_{bi}+C_{ii}+2C_{ai}+2s}+(-q)^{C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{di}+C_{ii}+2C_{ci}}\\Big ]\\\\& =\\lambda _{i}\\Big [(-q)^{C_{aa}+2C_{ab}+C_{bb}+2C_{bi}+C_{ii}+2C_{ai}+2s}+(-q)^{C_{cc}+2C_{cd}+C_{dd}+2C_{di}+C_{ii}+2C_{ci}}\\Big ].\\end{split}$ In each equation we have to match three $q$ -monomials on both sides in a non-trivial way.", "For example, in (REF ) we must take $4C_{aa}+4C^{\\prime }_{ab}+C_{bb}+2s &= C_{cc}+2C_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}+2, \\nonumber \\\\C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac} &= 4C_{aa}+4C_{ab}+C_{bb}+2s,\\\\C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}+2 &=C_{cc}+2C_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}, \\nonumber \\\\&\\textrm {or} \\nonumber \\\\4C_{aa}+4C^{\\prime }_{ab}+C_{bb}+2s &= C_{cc}+2C_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}, \\nonumber \\\\C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac} &=C_{cc}+2C_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}+2 ,\\nonumber \\\\C_{cc}+2C^{\\prime }_{cd}+C_{dd}+2C_{ad}+C_{aa}+2C_{ac}+2 &= 4C_{aa}+4C_{ab}+C_{bb}+2s.$ Analogous matching for equations for (REF -REF ), combined with $C_{aa}+C_{bb}=C_{cc}+C_{dd}$ and (REF ), leads to two possible ways for non-trivial pairwise cancellation: $\\begin{aligned}C_{ab}+s=&\\ C_{cd}-1, \\\\C_{aa}+C_{cd}=&\\ C_{ad}+C_{ac}+s+1, \\\\C_{bb}+C_{cd}=&\\ C_{bd}+C_{bc}+s+1, \\\\C_{ab}+C_{cc}+s=&\\ C_{bc}+C_{ac}, \\\\C_{ab}+C_{dd}+s=&\\ C_{bd}+C_{ad}\\end{aligned}\\qquad \\text{or} \\qquad \\begin{aligned}C_{ab}+s=&\\ C_{cd}+1, \\\\C_{aa}+C_{cd}=&\\ C_{ad}+C_{ac}+s, \\\\C_{bb}+C_{cd}=&\\ C_{bd}+C_{bc}+s, \\\\C_{ab}+C_{cc}+s=&\\ C_{bc}+C_{ac}+1, \\\\C_{ab}+C_{dd}+s=&\\ C_{bd}+C_{ad}+1.\\end{aligned}$ Combining (REF ) with $C_{aa}+C_{bb}=C_{cc}+C_{dd}$ , we deduce that $s=0$ .", "Putting it in equations (REF )-(REF ) and performing the analogous matching of terms, we learn that: $C_{cd} & =C_{ab}-1, &C_{ci}+C_{di} & =C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi}\\quad \\forall i\\in Q_{0}\\\\\\textrm { or }\\qquad \\quad C_{ab} &=C_{cd}-1,&C_{ai}+C_{bi} & =C_{ci}+C_{di}-\\delta _{ci}-\\delta _{di}\\quad \\forall i\\in Q_{0}.$ These conditions are required for the transposition $C_{ab}\\leftrightarrow C_{cd}$ to lead to an equivalent quiver.", "Now, let us slightly modify our assumptions to $\\lambda _a=q^{2s_1}\\lambda _c$ , $\\lambda _b=q^{2s_2} \\lambda _d$ , and requirement that $q^{2C_{ab}}\\lambda _a\\lambda _b+q^{2C_{cd}}\\lambda _c\\lambda _d+q^{2C_{ad}}\\lambda _a\\lambda _d+q^{2C_{bc}}\\lambda _b\\lambda _c$ corresponds to the only monomial in $a$ and $t$ which coefficient has more than one $q$ -monomial at the level of $x^2$ .", "Let us consider all types of permutations of these terms by focusing on which is equal to $q^{2C_{ab}}\\lambda _a\\lambda _b$ in $P_{Q^{\\prime }}$ .", "If it is $q^{2C^{\\prime }_{ab}}\\lambda _a\\lambda _b$ , then $C_{ab}=C^{\\prime }_{ab}$ , if it is $q^{2C^{\\prime }_{cd}}\\lambda _c\\lambda _d$ , then we have a situation that was described earlier in this section.", "The only truly different case comes from equating $q^{2C_{ab}}\\lambda _a\\lambda _b$ with $q^{2C^{\\prime }_{ad}}\\lambda _a\\lambda _d$ or $q^{2C^{\\prime }_{bc}}\\lambda _b\\lambda _c$ .", "In the first case the analogs of equations (REF ) and (REF ) imply $s=0$ and $C_{bi}=C_{di}$ for every $i \\in Q_0\\backslash \\lbrace a, b, d \\rbrace $ .", "This means that nodes $b$ and $d$ are indistinguishable and the transposition $C_{ab}\\leftrightarrow C_{ad}$ can be understood as a relabeling $b \\leftrightarrow d$ .", "The second case is completely analogous and can be understood as a relabeling $a \\leftrightarrow c$ .", "Now we would like to analyze the possibility of composing transpositions satisfying conditions (REF ) or () into a bigger cycle.", "Let us therefore assume that $\\lambda _a\\lambda _b=\\lambda _c\\lambda _d=\\lambda _e\\lambda _f$ , all lambdas – as well as $C_{ab}$ , $C_{cd}$ , $C_{ef}$ – are pairwise different, and equations (REF ) or () – as well as their counterparts for $c,d,e,f$ – are satisfied.", "Among them there is an equation $C_{ac} + C_{bc} = C_{cc} + C_{cd}$ (if $C_{ab}<C_{cd}$ ) or $C_{ac} + C_{bc} = C_{cc} + C_{cd}-1$ (if $C_{ab}>C_{cd}$ ) which becomes violated after the transposition $C_{cd}\\leftrightarrow C_{ef}$ .", "Similarly, performing the transposition $C_{ab} \\leftrightarrow C_{cd}$ causes the violation of an analogous equation required for $C_{cd}\\leftrightarrow C_{ef}$ .", "In consequence, we see that after composing transpositions which preserve the generating function into a bigger cycle, we always get an inequivalent quiver.", "Moreover, an analogous argument implies that the composition of transpositions $C_{ab} \\leftrightarrow C_{cd}$ and $C_{de}\\leftrightarrow C_{fg}$ (both of which involve the same node $d$ ) leads to an inequivalent quiver.", "We have not yet excluded all non-trivial ways of matching terms in (REF ) – for example one may think about a permutation that leads to an equivalent quiver, but is composed of transpositions that change the partition function.", "However, based on the evidence discussed below, it appears that such permutations are little likely to arise, and thus we make the following conjecture: Conjecture 5 Consider a quiver $Q$ corresponding to the knot $K$ .", "If there exists another symmetric quiver $Q^{\\prime }$ such that $Q^{\\prime }\\sim Q$ in the sense of the definition REF , then either $Q^{\\prime }=Q$ or they are related by a sequence of disjoint transpositions, each exchanging non-diagonal elements $C_{ab}\\leftrightarrow C_{cd}, \\qquad C_{ba}\\leftrightarrow C_{dc},$ for some pairwise different $a,b,c,d,\\in Q_{0}$ , such that $\\lambda _{a}\\lambda _{b} = \\lambda _{c}\\lambda _{d}$ and $C_{ab} = C_{cd}-1,\\qquad \\quad C_{ai}+C_{bi}=C_{ci}+C_{di}-\\delta _{ci}-\\delta _{di},\\quad \\forall i\\in Q_{0},$ or $C_{cd} = C_{ab}-1,\\qquad \\quad C_{ci}+C_{di}=C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi},\\quad \\forall i\\in Q_{0}.$ For the simplest thin knots we verify this conjecture in the following way.", "Since $a_i$ and $t_i$ fix $q_i$ and $C_{ii}$ , permutations of terms in coefficients of monomials in $a$ and $t$ are in one-to-one correspondence with permutations of $C_{ij}$ .", "Therefore, we just need to find all incident products $\\lambda _a \\lambda _b=\\lambda _c \\lambda _d=\\lambda _e \\lambda _f=\\ldots $ and for each of them check all permutations of the set $\\lbrace C_{ab},C_{cd},C_{ef},\\ldots \\rbrace $ .", "Using this procedure, we verified conjecture REF for quivers corresponding to $3_1$ , $4_1$ , and $5_1$ knot.", "For thin knots we can also give another general argument supporting conjecture REF – we can exclude those 3-cycles that are not necessarily composed of transpositions preserving the generating function.", "To this end, let us assume that $\\lambda _a \\lambda _b=\\lambda _c \\lambda _d=\\lambda _e \\lambda _f$ , these terms are the only instance of multiple $q$ -monomials in the coefficient of $a$ and $t$ monomials in (REF ), and $Q^{\\prime }$ arises from $Q$ by performing the 3-cycle $(C_{ab}\\; C_{cd}\\; C_{ef})$ or $(C_{ab}\\; C_{ef}\\; C_{cd})$ with $C_{ab},\\,C_{cd},\\,C_{ef}$ being all distinct.", "Then, in the qubic term (REF ), we have multiple ways to cancel the terms in front of $\\lambda _a,\\lambda _b,\\ldots ,\\lambda _f$ .", "In total, it results in $44^3$ non-trivial systems of 30 linear equations, which we treated with the help of computer and confirmed that together with the center of mass conditions they cannot be satisfied in a non-trivial way.", "In the next section we formulate and prove the theorem which is an analog of conjecture REF with a reversed direction of implication.", "Together, they provide a complete description of quiver equivalences." ], [ "Local equivalence theorem", "Theorem 6 Consider a quiver $Q$ corresponding to the knot $K$ and another symmetric quiver $Q^{\\prime }$ such that $Q^{\\prime }_0=Q_0$ and $\\lambda ^{\\prime }_i=\\lambda _i\\;\\forall i\\in Q_0$ ($\\lambda _i$ comes from the knots-quivers change of variables).", "If $Q$ and $Q^{\\prime }$ are related by a sequence of disjoint transpositions, each exchanging non-diagonal elements $C_{ab}\\leftrightarrow C_{cd}, \\qquad C_{ba}\\leftrightarrow C_{dc},$ for some pairwise different $a,b,c,d,\\in Q_{0}$ , such that $\\lambda _{a}\\lambda _{b} = \\lambda _{c}\\lambda _{d}$ and $C_{ab} = C_{cd}-1,\\qquad \\quad C_{ai}+C_{bi}=C_{ci}+C_{di}-\\delta _{ci}-\\delta _{di},\\quad \\forall i\\in Q_{0},$ or $C_{cd} = C_{ab}-1,\\qquad \\quad C_{ci}+C_{di}=C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi},\\quad \\forall i\\in Q_{0},$ then $Q$ and $Q^{\\prime }$ are equivalent in the sense of the definition REF .", "In order to apply this theorem to various knots and quivers, we usually start from looking for $\\lambda _a,\\lambda _b,\\lambda _c,\\lambda _d$ that satisfy the condition $\\lambda _a\\lambda _b=\\lambda _c\\lambda _d$ .", "We call a quadruple of pairwise different $a,b,c,d\\in Q_{0}$ such that this equation holds a pairing.", "Note that only some pairings generate transpositions (REF ) leading to equivalent quiver – if this is the case, we call them symmetries.", "If a symmetry is consistent with constraints (REF ) or (REF ) we call it non-trivial; if it follows from $C^{\\prime }_{ij}=C_{ij}$ we call it trivial.", "Furthermore, symmetries of quivers are tightly related to homological diagrams for knots, providing a neat illustration of the aforementioned conditions.", "After the change of variables (REF ), each pairing $\\lambda _a\\lambda _b=\\lambda _c\\lambda _d$ gives the vector identity $\\vec{v}_a+\\vec{v}_b=\\vec{v}_c+\\vec{v}_d$ , where $\\vec{v}_i=(q_i,a_i)$ is a vector of homological degrees of the generator $i$ .", "This identity can be interpreted as a requirement that the centers of mass for pairs of nodes $\\lbrace a,b\\rbrace $ and $\\lbrace c,d\\rbrace $ coincide (assuming that masses of all nodes are equal).", "We visualize it as a parallelogram with the diagonals $ab$ and $cd$ , see figure REF .", "Figure: The set of generators of the uncolored HOMFLY-PT homology for 4 1 4_1 knot and the parallelogram corresponding to the pairing λ 2 λ 5 =λ 3 λ 4 \\lambda _2\\lambda _5=\\lambda _3\\lambda _4The remaining constraints (REF ) or (REF ) also have a nice pictorial representation in terms of generators of the $S^{r}$ -colored HOMFLY-PT homology.", "The case $r=1$ corresponds to the uncolored homology, encoded in the linear term of the quiver generating series and thus depending only on the numbers of loops in $Q$ .", "It suits well for visualizing the pairing, but not the rest of constraints.", "However, the case $r=2$ involves the quadratic term of the quiver series and therefore depends on all entries of the quiver matrix.", "Moreover, there exists a well-defined surjective map $Q_0\\times Q_0 \\rightarrow {G}_{2}$ coming from the knots-quivers change of variables.", "Figure: The set of generators of the S 2 S^2-colored HOMFLY-PT homology for 4 1 4_1 knot(the labels x i x j x_ix_j are consistent with the labels in figure ).For example, the $S^2$ -colored homology for $4_1$ knot is shown in figure REF .", "There are 3 kinds of generators: 5 black nodes are in one-to-one correspondence with $x_i^2,\\ i=1\\dots 5$ .", "Blue and purple nodes correspond to $x_ix_j$ with $i\\ne j$ , and for each pair $(i,j)$ there are exactly 2 generators, which we connect by an arc.", "The distinction between blue and purple nodes is justified by taking the common denominator in the quadratic term of the quiver series.", "Each term $x_ix_j$ is multiplied by $(1+q^2)$ , therefore contributing twice to the colored superpolynomial.", "The blue node has the $q$ -degree $q_i+q_j+C_{ii}+2C_{ij}+C_{jj}$ , while the purple one is shifted by two: $q_i+q_j+C_{ii}+2C_{ij}+C_{jj}+2$ .", "Having in mind the pairing condition inducing cancellations of all terms except those corresponding to arrows between different nodes ($2C_{ij}$ ), we can visualize any constraint of the form $C_{is}+C_{js}=C_{ks}+C_{ls}$ as a parallelogram connecting nodes with the same color.", "For example, the constraint $C_{12}+C_{15}=C_{13}+C_{14}$ is visualized in figure REF .", "Figure: The constraint C 12 +C 15 =C 13 +C 14 C_{12}+C_{15}=C_{13}+C_{14} as a parallelogram rule.", "There are cancellationswhen equating the sums of the qq- and aa-degrees of x 1 x 2 ,x 1 x 5 x_1x_2,x_1x_5 and x 1 x 3 ,x 1 x 4 x_1x_3,x_1x_4, since λ 2 λ 5 =λ 3 λ 4 \\lambda _2\\lambda _5=\\lambda _3\\lambda _4 implies q 2 +q 5 =q 3 +q 4 q_2+q_5=q_3+q_4 and a 2 +a 5 =a 3 +a 4 a_2+a_5=a_3+a_4.The constraint holds only if the corresponding sums of vectors agree(simultaneously for the blue and purple quadruples of nodes)." ], [ "Proof of the local equivalence theorem", "Let us prove the theorem REF .", "Since disjoint transpositions described there are independent, we can consider a general form of one such transposition and show that it preserves the generating function.", "This automatically implies that if $Q$ and $Q^{\\prime }$ are connected by a sequence of such transformations, then they correspond to the same knot.", "Therefore, without loss of generality, we assume that $Q$ corresponds to $K$ , $Q^{\\prime }_0=Q_0$ , $\\lambda ^{\\prime }_i=\\lambda _i$ $\\forall i\\in Q_0$ , and we have $C^{\\prime }_{ij}=C_{ij}$ except one transposition $C_{ab}\\leftrightarrow C_{cd}$ for some pairwise different $a,b,c,d\\in Q_{0}$ .", "We also require $\\lambda _{a}\\lambda _{b}=\\lambda _{c}\\lambda _{d},\\quad C_{cd}=C_{ab}-1, \\quad C_{ci}+C_{di}=C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi},\\quad i\\in Q_{0}$ and analogous constraints for $C^{\\prime }$ (the case $C_{ab}=C_{cd}-1$ can be covered by changing labels $ab\\leftrightarrow cd$ in the whole argument).", "We want to show that $Q^{\\prime }$ also corresponds to $K$ .", "We will do it by connecting $Q^{\\prime }$ with $Q$ by transformations preserving the motivic generating functions, namely unlinking nodes $a,b$ in $Q$ and nodes $c,d$ in $Q^{\\prime }$ (the invariance of generating function under these transformations is assured by theorem REF ).", "From definition REF we have $\\widetilde{C}_{ij} & =C_{ij}\\quad \\forall i,j\\in Q_{0}\\backslash \\lbrace a,b\\rbrace & \\widetilde{C}\\,^{\\prime }_{ij} & =C^{\\prime }_{ij}\\quad \\forall i,j\\in Q_{0}\\backslash \\lbrace c,d\\rbrace \\nonumber \\\\\\widetilde{C}_{ab} & =C_{ab}-1 & \\widetilde{C}\\,^{\\prime }_{cd} & =C^{\\prime }_{cd}-1 \\\\\\widetilde{C}_{in} & =C_{ai}+C_{bi}-\\delta _{ai}-\\delta _{bi}, & \\widetilde{C}\\,^{\\prime }_{in} & =C^{\\prime }_{ci}+C^{\\prime }_{di}-\\delta _{ci}-\\delta _{di},\\nonumber \\\\\\widetilde{C}_{nn} & =C_{aa}+2C_{ab}+C_{bb}-1, & \\widetilde{C}\\,^{\\prime }_{nn} & =C^{\\prime }_{cc}+2C^{\\prime }_{cd}+C^{\\prime }_{dd}-1.", "\\nonumber $ In consequence $\\widetilde{C}\\,^{\\prime }_{ab} & =C^{\\prime }_{ab}=C_{cd}=C_{ab}-1=\\widetilde{C}_{ab},\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{cd} & =C^{\\prime }_{cd}-1=C_{ab}-1=C_{cd}=\\widetilde{C}{}_{cd},\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{an} & =C^{\\prime }_{ac}+C^{\\prime }_{ad}=C_{ac}+C_{ad}=C_{aa}+C_{ab}-1=\\widetilde{C}_{an},\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{bn} & =C^{\\prime }_{bc}+C^{\\prime }_{bd}=C_{bc}+C_{bd}=C_{ab}+C_{bb}-1=\\widetilde{C}_{bn}, \\nonumber \\\\\\widetilde{C}^{\\prime }_{cn} & =C^{\\prime }_{cc}+C^{\\prime }_{cd}-1=C^{\\prime }_{ac}+C^{\\prime }_{bc}=C_{ac}+C_{bc}=\\widetilde{C}_{cn}, \\\\\\widetilde{C}\\,^{\\prime }_{dn} & =C^{\\prime }_{cd}+C^{\\prime }_{dd}-1=C^{\\prime }_{ad}+C^{\\prime }_{bd}=C_{ad}+C_{bd}=\\widetilde{C}_{dn},\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{in} & =C^{\\prime }_{ci}+C^{\\prime }_{di}=C_{ci}+C_{di}=C_{ai}+C_{bi}=\\widetilde{C}_{in},\\quad \\forall i\\in Q_{0}\\backslash \\lbrace a,b,c,d\\rbrace ,\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{nn} & =C^{\\prime }_{cc}+2C^{\\prime }_{cd}+C^{\\prime }_{dd}-1=C_{cc}+2C_{ab}+C_{dd}-1=C_{cc}+2C_{ab}+C_{dd}-1=\\widetilde{C}_{nn},\\nonumber \\\\\\widetilde{C}\\,^{\\prime }_{ij} & =C^{\\prime }_{ij}=C_{ij}=\\widetilde{C}_{ij}\\quad \\textrm {for all other cases,}\\nonumber $ which can be summarized simply as $\\widetilde{Q}\\,^{\\prime }=\\widetilde{Q}$ .", "In our unlinking of $Q^{\\prime }$ and $Q$ we have a freedom to choose the knots-quivers change of variables for the new nodes (for the old ones we have $\\lambda ^{\\prime }_{i}=\\lambda _{i}$ ).", "We take $\\widetilde{\\lambda }^{\\prime }_{n}=q^{-1}\\lambda _{c}\\lambda _{d}=q^{-1}\\lambda _{a}\\lambda _{b}=\\widetilde{\\lambda }_{n},$ and use theorem REF to get $\\left.P_{Q^{\\prime }}(x,q)\\right|_{x_{i}=x\\lambda ^{\\prime }_{i}}=\\left.P_{\\widetilde{Q}^{\\prime }}(x,q)\\right|_{x_{i}=x\\lambda ^{\\prime }_{i},\\;x_{n}=x\\widetilde{\\lambda }^{\\prime }_{n}}=\\left.P_{\\widetilde{Q}}(x,q)\\right|_{x_{i}=x\\lambda _{i},\\;x_{n}=x\\widetilde{\\lambda }_{n}}=\\left.P_{Q}(x,q)\\right|_{x_{i}=x\\lambda _{i}}.$ Therefore $\\left.P_{Q^{\\prime }}(x,q)\\right|_{x=x\\lambda ^{\\prime }}=\\left.P_{Q}(x,q)\\right|_{x=x\\lambda }=P_{K}(x,a,q,t),$ so $Q^{\\prime }$ also corresponds to $K$ , as we wanted to show." ], [ "Global structure and permutohedra graphs", "In the previous section we found transformations that produce equivalent quivers and conditions they satisfy.", "This fact enables us to systematically determine equivalent quivers for a given knot: starting from some particular quiver we can consider all possible transpositions of its matrix elements, and identify those that satisfy conditions of theorem REF and thus yield equivalent quivers.", "Repeating this procedure for each newly found equivalent quiver, after a finite number of steps we obtain a closed and connected network with an intricate structure.", "(Recall that in principle there might exist other equivalent quivers, which are not related by a series of transpositions from theorem REF – e.g.", "they might be related by a cyclic permutation of length larger than 2, such that some transpositions of elements of quiver matrix, which arise from a decomposition of such a permutation, do not preserve the partition function.", "However, we conjectured that such equivalent quivers do not arise, and we do not focus on them in the rest of this work.)", "In order to reveal the structure of the network of equivalent quivers mentioned above, it is of advantage to assemble these quivers in one graph, such that each vertex of this graph corresponds to one quiver, and two vertices are connected by an edge if two corresponding quivers differ by one transposition of non-diagonal elements.", "Examples of such graphs are shown in figures REF and REF (for knots $9_1$ and $11_1$ ), and in section  for several other knots.", "One immediately observes that these graphs are built from smaller building blocks, which are combinatorial structures known as permutohedra.", "Various permutohedra are glued to each other and form a connected graph representing all equivalent quivers, which we refer to as a permutohedra graph in what follows.", "In this section we explain why equivalent quivers arise in families that form permutohedra, and how their structure follows from local properties revealed in theorem REF .", "In the next section we illustrate these structures in detail in several explicit examples." ], [ "Permutohedra – what they are and why they arise", "To start with, recall that a permutohedron of order $n$ , denoted $\\Pi _n$ , is an $(n-1)$ -dimensional polytope whose vertices represent permutations of $n$ objects $\\lbrace 1,\\ldots ,n\\rbrace $ and edges correspond to flips (transpositions) of adjacent neighbors [26], [27].", "The permutohedron $\\Pi _n$ has thus $n!$ vertices and each vertex has $n-1$ immediate neighbors.", "$\\Pi _n$ has also $(n-1)n!/2$ edges; each edge corresponds to one of $n(n-1)/2$ types of flips $(i\\ j)$ (for $1\\le i<j\\le n$ ).", "We call these operations flips in order to distinguish them from transpositions of elements of quiver matrices; as we will see, transpositions in quiver matrices are simply manifestations of certain underlying flips.", "The permutohedron $\\Pi _3$ is a hexagon, see figure REF .", "$\\Pi _4$ is a (3-dimensional) truncated octahedron that consists of $4!=24$ vertices.", "It has 36 edges of 6 different types, such that 3 edges meet at each vertex, and its faces form 6 quadrangles and 8 hexagons, see figure REF .", "Planar realizations of $\\Pi _n$ for $n=1,2,3,4$ are shown in figure REF .", "Figure: Planar realizations of permutohedra Π n \\Pi _n of orders 1,2,3,4.", "One quadrangular face of Π 4 \\Pi _4 is represented by an external region.", "Three-dimensional representation of permutohedron Π 4 \\Pi _4 is shown in figure .Let us explain now why certain families of equivalent quivers form permutohedra.", "To get some intuition, it is of advantage to understand it first as a consequence of a particular structure of generating functions of colored superpolynomials; in section REF we show how this structure arises from the local properties revealed in theorem REF .", "We find that instead of writing a generating function of colored superpolynomials in a form of the generating series (REF ) for a quiver of size $m$ , it can be written in an intermediate form $ P_K(x,a,q,t) = \\sum _{\\check{d}_1,\\ldots ,\\check{d}_{m-n}\\ge 0}(-q)^{\\sum _{i,j}\\check{C}_{ij}\\check{d}_i\\check{d}_j}\\frac{\\check{x}_1^{\\check{d}_1}\\cdots \\check{x}_{m-n}^{\\check{d}_{m-n}}}{(q^{2};q^{2})_{\\check{d}_1}\\cdots (q^{2};q^{2})_{\\check{d}_{m-n}}}\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n} \\Big |_{\\check{x}_i=x\\check{\\lambda }_i},$ for $2n\\le m$ and with the following properties.", "The first terms under the sum take the same form as the summand in the usual quiver generating series (REF ), however they are associated to a novel quiver of size $m-n$ that we call a prequiver and denote its matrix by $\\check{C}$ .", "Then, it is the factor $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ which is responsible for the appearance of all equivalent quivers associated to a particular permutohedron; note that it has only $n$ labels $\\check{d}_1,\\ldots ,\\check{d}_n$ , and we require that (combined with the first $n$ $q$ -Pochhammers from the denominator) it has the structure $ \\frac{\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}}{(q^{2};q^{2})_{\\check{d}_1}\\cdots (q^{2};q^{2})_{\\check{d}_{n}}} = \\sum _{\\check{d}_1=\\alpha _1+\\beta _1}\\cdots \\sum _{\\check{d}_n=\\alpha _n+\\beta _n} \\frac{ (-q)^{2\\sum _{i<j}\\beta _{i}\\alpha _{j}+ \\pi _2(\\alpha _1,\\ldots ,\\alpha _n;\\beta _1,\\ldots ,\\beta _n)}\\kappa ^{\\beta _1+\\ldots +\\beta _n}}{(q^{2};q^{2})_{\\alpha _{1}}(q^{2};q^{2})_{\\beta _{1}}\\cdots (q^{2};q^{2})_{\\alpha _{n}}(q^{2};q^{2})_{\\beta _{n}}},$ where $\\pi _2(\\alpha _1,\\ldots ,\\alpha _n;\\beta _1,\\ldots ,\\beta _n)$ is a purely quadratic polynomial in $\\alpha _i$ 's, $\\beta _j$ 's, and other $\\check{d}_{k}$ 's (for $k>n)$ , that is symmetric in $(\\alpha _1,\\ldots ,\\alpha _n)$ and (independently) in $(\\beta _1,\\ldots ,\\beta _n)$ ; $\\kappa $ is an extra parameter.", "Furthermore, we impose the invariance of the above expression under any permutation $\\sigma \\in S_n$ of indices $\\lbrace 1,\\ldots ,n\\rbrace $ , so that the whole $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ is symmetric in all $\\check{d}_1,\\ldots ,\\check{d}_n$ .", "Note that most of the above expression on the right-hand side, i.e.", "the terms symmetric in $\\alpha _i$ 's and $\\beta _j$ 's, as well as the defining relations $\\check{d}_i=\\alpha _i+\\beta _i$ , are already invariant under permutation of the indices.", "The only non-invariant term is $\\sum _{i<j} \\beta _i \\alpha _j$ , so in other words we impose that the above expression is invariant if we replace this term by $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ , for any permutation $\\sigma $ .", "Below we provide specific forms of $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ , including symmetric polynomials $\\pi _2$ , that have the above properties.", "At this stage let us stress that it is the form of the term $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ that uniquely determines a permutation $\\sigma $ and is responsible for the appearance of a permutohedron.", "First, a permutation $\\sigma $ is determined by a set of its inversions, i.e.", "a set of all pairs $(\\sigma (i),\\sigma (j))$ , such that $i<j$ and $\\sigma (i)>\\sigma (j)$ .", "We can therefore treat symbols $\\beta $ and $\\alpha $ as determining respectively the first and the second element of a given pair $(\\sigma (i),\\sigma (j))$ .", "For example, the term $\\sum _{i<j} \\beta _{i} \\alpha _{j}$ encodes the trivial permutation.", "Any other permutation can be uniquely encoded by inverting labels in appropriate summands in $\\sum _{i<j} \\beta _{i} \\alpha _{j}$ .", "Therefore, if we insist that (REF ) is invariant under all permutations of indices $\\lbrace 1,\\ldots ,n\\rbrace $ , this means that in fact we can consider $n!$ expressions that are in one-to-one correspondence with permutations encoded in the terms $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ , and can be associated to vertices of a permutohedron $\\Pi _n$ .", "Such a permutohedron has $n(n-1)/2$ types of edges (denoted by different colors in various figures in this paper), which correspond to all transpositions $(k\\ l)$ , for $1\\le k < l\\le n$ .", "However, at a given vertex, corresponding to the permutation $\\sigma $ and the term $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ , only $n-1$ edges meet.", "They correspond to transpositions of adjacent elements that change only one summand in the expression $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ .", "Let us see it on the example of a vertex corresponding to the trivial permutation, represented by $\\sum _{i<j} \\beta _{i} \\alpha _{j}$ , and $n-1$ edges corresponding to transpositions of neighboring elements $\\tau =(k\\ (k+1)),\\;k=1,\\ldots ,n-1$ .", "In that case the only difference between $\\sum _{i<j} \\beta _{i} \\alpha _{j}$ and $\\sum _{i<j} \\beta _{\\tau (i)} \\alpha _{\\tau (j)}$ amounts to replacing precisely one summand $\\beta _k\\alpha _{k+1}$ by $\\beta _{k+1}\\alpha _k$ .", "This is why a transformation of one term $\\beta _k\\alpha _{k+1}$ into $\\beta _{k+1}\\alpha _k$ (for $k=1,\\ldots ,n-1$ ) in (REF ) is represented by one edge of a permutohedron.", "Similarly, $n-1$ edges meeting at any other vertex that represents a permutation $\\sigma $ , correspond to those transpositions $(k\\ l)$ that affect precisely one term in $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ .", "All this is also a manifestation of the well known fact that a permutohedron is the Hasse diagram of a set of appropriately ordered inversions.", "Furthermore, let us explain how the prequiver $\\check{C}$ introduced in (REF ), combined with $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ , gives rise to the original quiver $C$ of size $m$ and a number of its equivalent companions.", "First, in the expression (REF ) there are $(m-n)$ $q$ -Pochhammers $(q^2;q^2)_{\\check{d}_i}$ .", "In (REF ), $n$ of them are combined with $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ and get split into pairs $(q^2;q^2)_{\\alpha _i}(q^2;q^2)_{\\beta _i}$ .", "This produces $n$ new $q$ -Pochhammers, and altogether we get $m$ independent $q$ -Pochhammers that correspond to $m$ nodes of a quiver $C$ that we are after.", "The prequiver term $(-q)^{\\sum _{i,j}\\check{C}_{ij}\\check{d}_i\\check{d}_j}$ in (REF ) together with $(-q)^{2\\sum _{i<j} \\beta _i\\alpha _j + \\pi _2(\\alpha _1,\\ldots ,\\alpha _n;\\beta _1,\\ldots ,\\beta _n)}$ give rise to an overall quadratic expression that defines the full quiver matrix $C$ .", "The terms $\\kappa ^{\\beta _1+\\ldots +\\beta _n}$ get absorbed into the first $n$ generating parameters: $\\check{x}_1^{\\check{d}_1}\\cdots \\check{x}_{n}^{\\check{d}_{n}} \\kappa ^{\\beta _1+\\ldots \\beta _n}= \\check{x}_1^{\\alpha _1}(\\check{x}_1\\kappa )^{\\beta _1}\\cdots \\check{x}_{n}^{\\alpha _{n}}(\\check{x}_{n}\\kappa )^{\\beta _{n}}$ .", "In this way we obtain a quiver generating function for the quiver of size $m$ encoded in a matrix $C$ that we are interested in.", "To see it more clearly and to make contact with the notation in (REF ), we can rename summation variables: for example identify all $\\check{d}_k$ ($k=n+1,\\ldots ,m-n$ ) with $d_{n+k}$ , and let $d_{2i-1}\\equiv \\alpha _i$ and $d_{2i}\\equiv \\beta _i$ ; in addition, identify $\\check{x}_k$ with $x_{n+k}$ for $k=n+1,\\ldots ,m-n$ , and let $x_{2i-1}\\equiv \\check{x}_i$ and $x_{2i} \\equiv \\check{x}_i \\kappa $ .", "This gives rise to generating parameters as in (REF ).", "We refer to the process of replacing first $n$ nodes by $2n$ nodes, which is a manifestation of (REF ), as splitting, while the remaining $(m-2n)$ nodes of the quiver $C$ we call spectators.", "Under this relabeling, for a vertex representing the permutation $\\sigma $ , a flip of the term $\\beta _k\\alpha _{l}$ (in the sum $\\sum _{i<j} \\beta _{\\sigma (i)}\\alpha _{\\sigma (j)})$ into $\\beta _{l}\\alpha _k$ translates into a flip of $d_{2k}d_{2l-1}$ into $d_{2l}d_{2k-1}$ , which encodes a transposition of elements $C_{2k,2l-1}$ and $C_{2l,2k-1}$ (that we considered in theorem REF ) at the level of the matrix $C$ .", "For each vertex there are $n-1$ of such transpositions, which on one hand correspond to $n-1$ equivalent matrices related by one transposition to a given matrix $C$ , and on the other hand correspond to $n-1$ edges meeting at each vertex of a permutohedron $\\Pi _n$ .", "Note that we can make any other identification of indices that would amount to a permutation of all variables $d_i$ , and thus would yield a permutation of rows and columns of the matrix $C$ ; in particular, in section  we identify a prequiver part as corresponding to the last $n$ rather than first $n$ indices as above.", "Let us also note the following interesting feature.", "Not only the generating function of colored HOMFLY-PT polynomials, but also the generating function of colored superpolynomials is expected to take form (REF ).", "This means that the full dependence on the parameter $a$ , as well as $t$ , is captured by the parameter $\\kappa $ that appears in the factor $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ in (REF ), and in $\\check{\\lambda }_i$ that enter the identification of generating parameters $\\check{x}_i=x\\check{\\lambda }_i$ .", "Note that $\\check{\\lambda }_i$ are just a subset of all $\\lambda _j$ , so that $\\lambda _j=\\check{\\lambda }_i$ for appropriate values of $i$ , and the remaining $\\lambda _j$ arise from a simple rescaling $\\lambda _j=\\kappa \\check{\\lambda }_k$ (for appropriate $k$ and $j$ ).", "As we will see in what follows, $\\kappa $ is a monomial of the form $\\kappa = a^{\\kappa _a}q^{\\kappa _q}(-t)^{\\kappa _t}$ .", "Also note that $\\check{\\lambda }_i$ are different for various realizations (REF ) (corresponding to various permutohedra) for a given knot, because they correspond to various subsets of all $\\lambda _i$ that are associated to the nodes that arise in a given prequiver.", "In consequence, the values of $\\kappa $ are also different for various representations (REF ) of the same knot.", "It would be interesting to understand better why a dependence on $a$ and $t$ is simply captured by $\\kappa = a^{\\kappa _a}q^{\\kappa _q}(-t)^{\\kappa _t}$ and $\\check{\\lambda }_i$ , and possibly how it arises from properties of HOMFLY-PT homology.", "To sum up, after above identifications we obtain a family of quiver generating functions for various quivers $C$ of size $m$ in the standard form (REF ), and with parameters $x_i$ appropriate for the knots-quivers correspondence.", "The family of quivers that we obtain is parametrized by all permutations $\\sigma \\in S_n$ : the combinations $\\sum _{i<j} \\beta _{\\sigma (i)} \\alpha _{\\sigma (j)}$ for various $\\sigma $ that appear in the exponent of $(-q)$ affect the form of the matrix $C$ that we read off from quadratic terms, and thus give rise to $n!$ different but equivalent quivers, labeled by permutations of $n$ elements.", "This is why we can assign these quivers to vertices of permutohedron $\\Pi _n$ .", "An edge of such a permutohedron that represents a flip (transposition) of two elements from the set $\\lbrace 1,\\ldots ,n\\rbrace $ , at the same time corresponds to a transposition of certain two elements $C_{2k,2l-1}$ and $C_{2l,2k-1}$ of the matrix $C$ that we analyzed in theorem REF .", "The above analysis focuses on one permutohedron.", "However, typically we can write a generating function of colored superpolynomials for a given knot in the form (REF ) in several different ways, with different prequivers and terms $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}$ for various choices of nodes.", "This gives rise to several permutohedra that encode all equivalent quivers for a given knot.", "Some of these quivers are common between two (or more) permutohedra, therefore we obtain a large connected graph made of several permutohedra glued together." ], [ "Permutohedra from colored superpolynomials", "Let us now provide an explicit form of (REF ).", "We stress that expressions given below naturally occur in formulae for colored superpolynomials, so it is useful to understand their role from the perspective of equivalent quivers.", "First, we consider a special case that arises from the identification $\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n} = (\\xi ;q^2)_{\\check{d}_1+\\ldots +\\check{d}_n}$ , which is indeed familiar from various expressions for colored superpolynomials.", "We then have $\\frac{(\\xi ;q^2)_{\\check{d}_1+\\ldots +\\check{d}_n}}{(q^2;q^2)_{\\check{d}_{1}}\\cdots (q^2;q^2)_{\\check{d}_{n}}}&= \\sum \\limits _{\\alpha _{1}+\\beta _{1}=\\check{d}_{1}} \\cdots \\sum \\limits _{\\alpha _{n}+\\beta _{n}=\\check{d}_{n}} (-q)^{\\beta _{1}^2+\\ldots +\\beta _{n}^2+2\\sum _{i=1}^{n-1} \\beta _{i+1}(\\check{d}_{1}+\\ldots + \\check{d}_{i})} \\times \\nonumber \\\\& \\quad \\times \\frac{\\big (\\xi q^{-1}\\big )^{\\beta _{1}+\\cdots +\\beta _n}}{(q^2;q^2)_{\\alpha _{1}}(q^2;q^2)_{\\beta _{1}}\\cdots (q^2;q^2)_{\\alpha _{n}}(q^2;q^2)_{\\beta _{n}}},$ which is proven in [2].", "The left-hand side is explicitly symmetric in $\\check{d}_1,\\ldots ,\\check{d}_n$ , so the above equality proves that the right-hand side is also invariant under permutations of $\\lbrace 1,\\ldots ,n\\rbrace $ .", "In the exponent of $(-q)$ we have $\\sum _{i=1}^{n-1} \\beta _{i+1}(\\check{d}_{1}+\\ldots + \\check{d}_{i})=\\sum _{i>j} \\beta _i \\alpha _j+\\sum _{i>j} \\beta _i \\beta _j$ , so the first term $\\sum _{i>j} \\beta _i \\alpha _j$ is responsible for the permutohedron structure, while $\\sum _{i>j} \\beta _i \\beta _j$ is the second elementary symmetric polynomial, which is symmetric in all $\\beta _i$ in agreement with (REF ).", "If $\\xi $ is just a constant (independent of $\\check{d}_k$ 's), we identify $\\kappa =\\xi q^{-1}$ .", "An interesting version of (REF ), that also appears in expressions for colored superpolynomials, arises for $\\xi =\\kappa q^{2(h_{n+1} \\check{d}_{n+1} + \\ldots + h_{m-n} \\check{d}_{m-n}) + 2k( \\check{d}_1+\\ldots +\\check{d}_n)+1},$ where $h_s$ are fixed coefficients.", "Substituting such $\\xi $ to (REF ) also produces an exponent of $q$ that is a quadratic function, symmetric in $\\alpha _i$ 's and $\\beta _j$ 's.", "For brevity, let us type the corresponding version of (REF ) that involves just two summation variables $\\check{d}_{i}$ and $\\check{d}_{j}$ (which would correspond to a single transposition) and one spectator node corresponding to the variable $\\check{d}_{s}$ and the coefficient $h_s$ $ \\frac{(\\kappa q^{2h_{s}\\check{d}_{s}+2k(\\check{d}_{i}+\\check{d}_{j})+1};q^{2})_{\\check{d}_{i}+\\check{d}_{j}}}{(q^{2};q^{2})_{\\check{d}_{i}}(q^{2};q^{2})_{\\check{d}_{j}}}=\\sum \\limits _{\\alpha _{i}+\\beta _{i}=\\check{d}_{i}} & \\sum \\limits _{\\alpha _{j}+\\beta _{j}=\\check{d}_{j}}(-q)^{\\beta _{i}^{2}+\\beta _{j}^{2}+2\\beta _{i}(\\alpha _{j}+\\beta _{j})} \\kappa ^{\\beta _i+\\beta _j}\\nonumber \\\\& \\times \\frac{q^{(2h_{s}\\check{d}_{s}+2k(\\check{d}_{i}+\\check{d}_{j}))(\\beta _{i}+\\beta _j)}}{(q^{2};q^{2})_{\\alpha _{i}}(q^{2};q^{2})_{\\beta _{i}}(q^{2};q^{2})_{\\alpha _{j}}(q^{2};q^{2})_{\\beta _{j}}}\\\\= \\sum \\limits _{\\alpha _{i}+\\beta _{i}=\\check{d}_{i}} & \\sum \\limits _{\\alpha _{j}+\\beta _{j}=\\check{d}_{j}}(-q)^{(2k+1)\\beta _{i}^{2}+(2k+1)\\beta _{j}^{2}+2(k+1)\\beta _{i}\\alpha _{j}+2(2k+1)\\beta _{i}\\beta _{j}}\\nonumber \\\\& \\times \\frac{(-q)^{2k(\\beta _{i}\\alpha _{i}+\\beta _{j}\\alpha _{i}+\\beta _{j}\\alpha _{j})+2h_{s}(\\beta _{i}\\check{d}_{s}+\\beta _{j}\\check{d}_{s})}\\kappa ^{\\beta _{i}+\\beta _{j}}}{(q^{2};q^{2})_{\\alpha _{i}}(q^{2};q^{2})_{\\beta _{i}}(q^{2};q^{2})_{\\alpha _{j}}(q^{2};q^{2})_{\\beta _{j}}}.\\nonumber $ From the powers of $(-q)$ in the last two lines above one can read off appropriate elements of the resulting matrix $C$ .", "Note that using indices $i$ and $j$ is helpful in understanding the invariance of the right-hand side of the above expression under a flip: if we identify $i=1$ and $j=2$ or $i=2$ and $j=1$ , then the left-hand side is clearly invariant, while the only change on the right amounts respectively to replacing $\\beta _1\\alpha _2$ by $\\beta _2 \\alpha _1$ .", "Finally, the most general form of (REF ) arises from introducing an arbitrary number of spectators and a parameter $l$ in addition to $k$ in (REF ) as follows: $ \\begin{split}\\frac{\\Pi _{\\check{d}_1,\\ldots ,\\check{d}_n}}{(q^{2};q^{2})_{\\check{d}_1}\\cdots (q^{2};q^{2})_{\\check{d}_{n}}}& =\\sum \\limits _{\\alpha _{1}+\\beta _{1}=\\check{d}_{1}}\\cdots \\sum \\limits _{\\alpha _{n}+\\beta _{n}=\\check{d}_{n}}\\frac{\\kappa ^{\\beta _1+\\ldots +\\beta _n}}{(q^{2};q^{2})_{\\alpha _{1}}(q^{2};q^{2})_{\\beta _{1}}\\cdots (q^{2};q^{2})_{\\alpha _{n}}(q^{2};q^{2})_{\\beta _{n}}}\\\\&\\phantom{=}\\times (-q)^{2\\sum _{i<j}\\beta _{i}\\alpha _{j}+(2\\sum _{s=n+1}^{m-n} h_{s}\\check{d}_{s}+2k(\\alpha _{1}+\\ldots +\\alpha _{n})+l(\\beta _{1}+\\ldots +\\beta _{n}))(\\beta _{1}+\\ldots \\beta _{n})},\\end{split}$ which is also invariant under permutations of indices $1,\\ldots ,n$ , affecting the form of the term $\\sum _{i<j}\\beta _i \\alpha _j$ .", "If $l=2k+1$ , the above expression reduces to (REF ) (generalized to $n$ summations), and then it can be written concisely using the $q$ -Pochhammer symbol.", "For $l\\ne 2k+1$ we do not know if there is such a concise manifestly symmetric representation, however we do not necessarily need it – the crucial property is invariance of the above expression under permutations of indices $\\lbrace 1,\\ldots ,n\\rbrace $ .", "In what follows we prove that (REF ) is indeed invariant under such permutations." ], [ "Permutohedra from local equivalence", "In turn, we now show how permutohedra arise from the local equivalence of quivers revealed in theorem REF , and in particular explain how (REF ) arises from this theorem (and thus has the required symmetry properties).", "Suppose that conditions (REF ) of the theorem REF are satisfied, so that two quivers related by a transposition of elements $C_{ab}$ and $C_{cd}$ are equivalent.", "We now write the quiver matrix $C$ in a form that automatically implements these conditions.", "To this end, we focus first on the $4\\times 4$ submatrix of $C$ with elements $C_{ij}$ for $i,j=a,b,c,d$ , and rewrite is as follows: $\\left(\\begin{array}{cc:cc}C_{aa} & C_{ad} & C_{ac} & {amber}{C_{ab}} \\\\C_{ad} & C_{dd} & {amber}{C_{cd}} & C_{bd} \\\\C_{ac} & {amber}{C_{cd}} & C_{cc} & C_{bc} \\\\{amber}{C_{ab}} & C_{bd} & C_{bc} & C_{bb}\\end{array}\\right) = \\left(\\begin{array}{cc:cc}C_{aa} & C_{aa} + k & C_{ac} & {amber}{C_{ac} + k} \\\\C_{aa} + k & C_{aa} + l & {amber}{C_{ac} + k + 1} & C_{ac}+l \\\\C_{ac} & {amber}{C_{ac} + k + 1} & C_{cc} & C_{cc} + k \\\\{amber}{C_{ac} + k} & C_{ac} + l & C_{cc} + k & C_{cc} + l\\end{array}\\right).$ In order to get the right-hand side we introduced two parameters $k,l\\in \\mathbb {Z}$ , defined such that $C_{ad} = C_{aa} + k$ and $C_{dd} = C_{aa} + l$ .", "From the second equation in (REF ) with $i=a$ we then get $C_{ab} = C_{ac}+ C_{ad}-C_{aa} = C_{ac}+k$ .", "Similarly, the second equation in (REF ) with $i=b$ takes form $C_{ad} + C_{bd} = C_{dd} + C_{cd} - 1$ , and combined with the first equation in (REF ) and the above relations it yields $C_{bd} = C_{ac} + l$ .", "Analogously, (REF ) with $i=c$ and $i=d$ implies respectively $C_{cb} = C_{cc} + k$ and $C_{bb} = C_{cc} + l$ .", "The right-hand side of (REF ) follows from these relations and we rewrite it further as $\\left(\\begin{array}{cc}C_{aa} & C_{ac} \\\\C_{ac} & C_{cc}\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}1 & 1 \\\\1 & 1\\end{array}\\right) + \\left(\\begin{array}{cc}1 & 1 \\\\1 & 1\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & k \\\\k & l\\end{array}\\right) +\\left[ \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0\\end{array}\\right) +\\left(\\begin{array}{cc}0 & 0 \\\\1 & 0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0\\end{array}\\right) \\right]$ The terms in this expression turn out to have familiar interpretation.", "The fist matrix is (an appropriate part of) the prequiver $\\check{C}$ .", "In particular, if we rename summation variables as $(d_a,d_d,d_c,d_b) = (\\alpha _a,\\beta _a,\\alpha _c,\\beta _c)$ and $\\check{d}_a=\\alpha _a+\\beta _a$ and $\\check{d}_c=\\alpha _c+\\beta _c$ , consistently with earlier conventions, the composition of these vectors with the first term in (REF ) can be written as $\\left(\\begin{array}{c}d_a \\\\d_d \\\\d_c \\\\d_b \\\\\\end{array}\\right)^T\\left(\\begin{array}{cc:cc}C_{aa} & C_{aa} & C_{ac} & C_{ac} \\\\C_{aa} & C_{aa} & C_{ac} & C_{ac} \\\\C_{ac} & C_{ac} & C_{cc} & C_{cc} \\\\C_{ac} & C_{ac} & C_{cc} & C_{cc}\\end{array}\\right)\\left(\\begin{array}{c}d_a \\\\d_d \\\\d_c \\\\d_b \\\\\\end{array}\\right)=\\left(\\begin{array}{c}\\check{d}_a \\\\\\check{d}_c \\\\\\end{array}\\right)^T\\left(\\begin{array}{cc}C_{aa} & C_{ac} \\\\C_{ac} & C_{cc} \\\\\\end{array}\\right)\\left(\\begin{array}{c}\\check{d}_a \\\\\\check{d}_c \\\\\\end{array}\\right)= C_{aa} \\check{d}_a^2 + 2 C_{ac} \\check{d}_a \\check{d}_c + C_{cc} \\check{d}_c^2,$ so that $(-q)$ raised to the above power indeed provides the contribution from the prequiver (i.e.", "the first factor in the summand) in (REF ).", "Analogous contribution from the second term in (REF ) takes form $\\left(\\begin{array}{c}d_a \\\\d_d \\\\d_c \\\\d_b \\\\\\end{array}\\right)^T\\left(\\begin{array}{cc:cc}0 & k & 0 & k \\\\k & l & k & l \\\\0 & k & 0 & k \\\\k & l & k & l\\end{array}\\right)\\left(\\begin{array}{c}d_a \\\\d_d \\\\d_c \\\\d_b \\\\\\end{array}\\right)= \\big (2k (\\alpha _a + \\alpha _c) + l (\\beta _a + \\beta _c)\\big ) (\\beta _a + \\beta _c),$ which we recognize as $k$ - and $l$ -dependent contribution in (REF ).", "Finally, analogous contribution from the last term (in round brackets) in (REF ) takes form $2\\beta _a\\alpha _c$ , which is nothing but the term in (REF ) that is responsible for the permutohedron structure.", "In this case it is $\\Pi _2$ and the flip $\\tau =(a\\ c)$ , realized by $2\\beta _{\\tau (a)}\\alpha _{\\tau (c)}=2\\beta _c\\alpha _a$ , corresponds to the transposition of non-diagonal terms $C_{ab}\\leftrightarrow C_{cd}$ , which gives the quiver matrix equivalent to (REF ): $\\left(\\begin{array}{cc:cc}C_{aa} & C_{ad} & C_{ac} & {amber}{C_{cd}} \\\\C_{ad} & C_{dd} & {amber}{C_{ab}} & C_{bd} \\\\C_{ac} & {amber}{C_{ab}} & C_{cc} & C_{bc} \\\\{amber}{C_{cd}} & C_{bd} & C_{bc} & C_{bb}\\end{array}\\right) = \\left(\\begin{array}{cc:cc}C_{aa} & C_{aa} + k & C_{ac} & {amber}{C_{ac} + k +1} \\\\C_{aa} + k & C_{aa} + l & {amber}{C_{ac} + k} & C_{ac}+l \\\\C_{ac} & {amber}{C_{ac} + k} & C_{cc} & C_{cc} + k \\\\{amber}{C_{ac} + k + 1} & C_{ac} + l & C_{cc} + k & C_{cc} + l\\end{array}\\right).$ We already can see how the local constraints of theorem REF give rise to the expression (REF ).", "There is just one more term in (REF ) that we should reconstruct: the one that involves spectator nodes.", "To this end we enlarge (REF ) by two rows and columns, still assuming that $C_{ab}$ and $C_{cd}$ can be exchanged, and write such a matrix in the form: $\\begin{small}\\left(\\begin{array}{cc:cc:cc}C_{aa} & C_{ad} & C_{ac} & {amber}{C_{ab}} & C_{ae} & C_{af} \\\\C_{ad} & C_{dd} & {amber}{C_{cd}} & C_{bd} & C_{de} & C_{df} \\\\C_{ac} & {amber}{C_{cd}} & C_{cc} & C_{bc} & C_{ce} & C_{cf} \\\\{amber}{C_{ab}} & C_{bd} & C_{bc} & C_{bb} & C_{be} & C_{bf} \\\\C_{ae} & C_{de} & C_{ce} & C_{be} & C_{ee} & C_{ef} \\\\C_{af} & C_{df} & C_{cf} & C_{bf} & C_{ef} & C_{ff} \\\\\\end{array}\\right)= \\left(\\begin{array}{cc:cc:cc}C_{aa} & C_{aa} + k & C_{ac} & {amber}{C_{ac} + k} & C_{ae} & C_{af} \\\\C_{aa} + k & C_{aa} + l & {amber}{C_{ac} + k + 1} & C_{ac} + l & C_{ae} + h_e & C_{af} + h_f \\\\C_{ac} & {amber}{C_{ac} + k + 1} & C_{cc} & C_{cc} + k & C_{ce} & C_{cf} \\\\{amber}{C_{ac} + k} & C_{ac} + l & C_{cc} + k & C_{cc} + l & C_{ce} + h_e & C_{cf} + h_f \\\\C_{ae} & C_{ae} + h_e & C_{ce} & C_{ce} + h_e & C_{ee} & C_{ef} \\\\C_{af} & C_{af} + h_f & C_{cf} & C_{cf} + h_f & C_{ef} & C_{ff} \\\\\\end{array}\\right)\\end{small}$ The top-left $4\\times 4$ submatrix is expressed in terms of $k$ and $l$ in the same way as in (REF ).", "In addition, if we denote $C_{de} - C_{ae} = h_e$ and substitute to the second constraint in (REF ) with $i=e$ , we get $C_{be} = C_{ce} + h_e$ .", "Analogously, for $C_{df} - C_{af} = h_f$ we get $C_{bf} = C_{cf} + h_f$ , and altogether we obtain the matrix on the right.", "It follows that the contribution of these extra rows and columns to the quiver generating function reads $(-q)^{\\sum _s h_s \\check{d}_s}$ , which yields an appropriate term in (REF ) that we were after.", "To sum up, we have shown how the formula (REF ) arises from local constraints of theorem REF in the presence of one symmetry, which thus yields a permutohedron $\\Pi _2$ .", "Let us now illustrate how permutohedron $\\Pi _3$ arises if we assume that in addition to the symmetry involving $C_{ab}$ and $C_{cd}$ , there is also another symmetry that involves $C_{be}$ and $C_{cf}$ .", "Such two symmetries may exist in a matrix of size $6\\times 6$ , which we write in the form $\\begin{small}\\left(\\begin{array}{cc:cc:cc}C_{aa} & C_{ad} & C_{ac} & {amber}{C_{ab}} & C_{ae} & {red}{C_{af}} \\\\C_{ad} & C_{dd} & {amber}{C_{cd}} & C_{bd} & {red}{C_{de}} & C_{df} \\\\C_{ac} & {amber}{C_{cd}} & C_{cc} & C_{bc} & C_{ce} & {blue}{C_{cf}} \\\\{amber}{C_{ab}} & C_{bd} & C_{bc} & C_{bb} & {blue}{C_{be}} & C_{bf} \\\\C_{ae} & {red}{C_{de}} & C_{ce} & {blue}{C_{be}} & C_{ee} & C_{ef} \\\\{red}{C_{af}} & C_{df} & {blue}{C_{cf}} & C_{bf} & C_{ef} & C_{ff} \\\\\\end{array}\\right)= \\left(\\begin{array}{cc:cc:cc}C_{aa} & C_{aa} + k & C_{ac} & {amber}{C_{ac} + k} & C_{ae} & {red}{C_{ae} + k} \\\\C_{aa} + k & C_{aa} + l & {amber}{C_{ac} + k + 1} & C_{ac} + l & {red}{C_{ae} + k + 1} & C_{ae} + l \\\\C_{ac} & {amber}{C_{ac} + k + 1} & C_{cc} & C_{cc} + k & C_{ce} & {blue}{C_{ce} + k} \\\\{amber}{C_{ac} + k} & C_{ac} + l & C_{cc} + k & C_{cc} + l & {blue}{C_{ce} + k + 1} & C_{ce} + l \\\\C_{ae} & {red}{C_{ae} + k + 1} & C_{ce} & {blue}{C_{ce} + k + 1} & C_{ee} & C_{ee} + k \\\\{red}{C_{ae} + k} & C_{ae} + l & {blue}{C_{ce} + k} & C_{ce} + l & C_{ee} + k & C_{ee} + l \\\\\\end{array}\\right)\\end{small}$ where the right-hand side is expressed in terms of parameters $k$ and $l$ and arises from solving the constraints of theorem REF analogously as above.", "Note that two symmetries of original quiver: $C_{ab}\\leftrightarrow C_{cd}$ and $C_{be}\\leftrightarrow C_{cf}$ correspond to transpositions $(1\\ 2)$ and $(2\\ 3)$ acting on the element $(1,2,3)$ ; highlights in (REF ) match colors in figure REF .", "After performing one of these transformations we obtain a new quiver (with $+1$ in the other highlighted entry, like in (REF )), which also has two symmetries.", "One is an inverse of the transformation we just performed, the other is a transposition $C_{de}\\leftrightarrow C_{af}$ , denoted in red in (REF ).", "This behavior is perfectly consistent with the structure of $\\Pi _3$ – the new symmetry corresponds to transposition $(1\\ 3)$ , denoted in red in figure REF .", "Using theorem REF , one can check that the whole structure of $\\Pi _3$ is preserved: there are six equivalent versions of the matrix (REF ) connected by three symmetries, but only two of them can be applied to each representant of the class.", "Furthermore, the right-hand side of (REF ) can be written in the form $\\left(\\begin{array}{ccc}C_{aa} & C_{ac} & C_{ae} \\\\C_{ac} & C_{cc} & C_{ce} \\\\C_{ae} & C_{ce} & C_{ee}\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}1 & 1 \\\\1 & 1\\end{array}\\right) +\\left(\\begin{array}{ccc}1 & 1 & 1 \\\\1 & 1 & 1 \\\\1 & 1 & 1\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & k \\\\k & l\\end{array}\\right)+\\left(\\begin{array}{ccc}0 & 1 & 1 \\\\0 & 0 & 1 \\\\0 & 0 & 0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0\\end{array}\\right) +\\left(\\begin{array}{ccc}0 & 0 & 0 \\\\1 & 0 & 0 \\\\1 & 1 & 0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0\\end{array}\\right)$ where the $3\\times 3$ matrix in the first term is a prequiver.", "Straightforward generalization of the above procedure to more symmetries leads to prequivers of arbitrary size and corresponding permutohedra, or – equivalently – the general form of (REF ): Definition 7 A $(k,l)$ -splitting of $n$ nodes with permutation $\\sigma \\in S_n$ in the presence of $m-2n$ spectators (with corresponding integer shifts $h_s$ ) and with a multiplicative factor $\\kappa $ is defined as the following transformation of a quiver $\\check{C}$ and a change of variables $\\check{\\lambda }$ .", "For any two split nodes $i$ and $j$ , $i<j$ , and any spectator $s$ , we transform the matrix $\\check{C}$ in the following way (depending on the presence of inversion in permutation $\\sigma $ ): every picture/.style=line width=0.75pt [x=0.75pt,y=0.75pt,yscale=-1,xscale=1] [-stealth](394,184.4+20) – (472.06,164.89) ; [-stealth](394,344.4-20) – (472.06,363.91) ; (394,264.4) node $\\left(\\begin{array}{c:c:c:c:c}\\check{C}_{ss} & \\cdots & \\check{C}_{si} & \\cdots & \\check{C}_{sj}\\\\ \\vdots & \\ddots & \\vdots & & \\vdots \\\\ \\check{C}_{is} & \\cdots & \\check{C}_{ii} & \\cdots & \\check{C}_{ij}\\\\ \\vdots & & \\vdots & \\ddots & \\vdots \\\\ \\check{C}_{js} & \\cdots & \\check{C}_{ji} & \\cdots & \\check{C}_{jj}\\end{array}\\right)$ ; (684,164.4) node $\\left(\\begin{array}{ c : c : c : c : c : c : c }\\check{C}_{ss} & \\cdots & \\check{C}_{si} & \\check{C}_{si} +h_{s} & \\cdots & \\check{C}_{sj} & \\check{C}_{sj} +h_{s}\\\\ \\vdots & \\ddots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ \\check{C}_{is} & \\cdots & \\check{C}_{ii} & \\check{C}_{ii} +k & \\cdots & \\check{C}_{ij} & [rgb]{0.96,0.65,0.14}{\\check{C}_{ij} +k}\\\\ \\check{C}_{is} +h_{s} & \\cdots & \\check{C}_{ii} +k & \\check{C}_{ii} +l & \\cdots & [rgb]{0.96,0.65,0.14}{\\check{C}_{ij} +k+1} & \\check{C}_{ij} +l\\\\ \\vdots & & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ \\check{C}_{js} & \\cdots & \\check{C}_{ji} & [rgb]{0.96,0.65,0.14}{\\check{C}_{ji} +k+1} & \\cdots & \\check{C}_{jj} & \\check{C}_{jj} +k\\\\ \\check{C}_{js} +h_{s} & \\cdots & [rgb]{0.96,0.65,0.14}{\\check{C}_{ji} +k} & \\check{C}_{ji} +l & \\cdots & \\check{C}_{jj} +k & \\check{C}_{jj} +l\\end{array}\\right)$ ; (684,364.4) node $\\left(\\begin{array}{c:c:c:c:c:c:c}\\check{C}_{ss} & \\cdots & \\check{C}_{si} & \\check{C}_{si} +h_{s} & \\cdots & \\check{C}_{sj} & \\check{C}_{sj} +h_{s}\\\\ \\vdots & \\ddots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ \\check{C}_{is} & \\cdots & \\check{C}_{ii} & \\check{C}_{ii} +k & \\cdots & \\check{C}_{ij} & [rgb]{0.96,0.65,0.14}{\\check{C}_{ij} +k+1}\\\\ \\check{C}_{is} +h_{s} & \\cdots & \\check{C}_{ii} +k & \\check{C}_{ii} +l & \\cdots & [rgb]{0.96,0.65,0.14}{\\check{C}_{ij} +k} & \\check{C}_{ij} +l\\\\ \\vdots & & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ \\check{C}_{js} & \\cdots & \\check{C}_{ji} & [rgb]{0.96,0.65,0.14}{\\check{C}_{ji} +k} & \\cdots & \\check{C}_{jj} & \\check{C}_{jj} +k\\\\ \\check{C}_{js} +h_{s} & \\cdots & [rgb]{0.96,0.65,0.14}{\\check{[rgb]{0.96,0.65,0.14}{C}}_{ji} +k+1} & \\check{C}_{ji} +l & \\cdots & \\check{C}_{jj} +k & \\check{C}_{jj} +l\\end{array}\\right)$ ; (394+50,144.4) node $\\sigma ( i) < \\sigma ( j)$ ; (394+50,384.4) node $\\sigma ( i) >\\sigma ( j)$ ; whereas for any permutation the change of variables is transformed as follows: $\\left(\\begin{array}{c}\\check{\\lambda }_s\\\\\\vdots \\\\\\check{\\lambda }_i\\\\\\vdots \\\\\\check{\\lambda }_j\\end{array}\\right)\\longrightarrow \\left(\\begin{array}{c}\\check{\\lambda }_s\\\\\\vdots \\\\\\check{\\lambda }_i\\\\\\check{\\lambda }_i\\kappa \\\\\\vdots \\\\\\check{\\lambda }_j\\\\\\check{\\lambda }_j\\kappa \\end{array}\\right).$ Clearly, the top right matrix above (corresponding to $\\sigma (i)<\\sigma (j)$ $\\rightarrow $ no inversion) is encoded in the quadratic terms in the powers of $(-q)$ in (REF ).", "The bottom right matrix (corresponding to $\\sigma (i)>\\sigma (j)$ $\\rightarrow $ inversion) arises after exchanging labels $i$ and $j$ in (REF ).", "Moreover, in the language of the definition REF , the $(k,2k+1)$ -splitting is a manifestation of the formula (REF ).", "For $k=0$ it specializes to $(0,1)$ -splitting that is a manifestation of the basic formula (REF ) with $\\xi =\\kappa q$ .", "Definition 8 If the inverse of splitting – for any parameters from definition REF – can be applied to a given quiver $C$ and associated change of variables $\\lambda $ , we call the target of this operation a prequiver $\\check{C}$ , and the associated change of variables is denoted $\\check{ \\lambda }$ .", "Conversely, splitting the nodes of a prequiver produces the quiver: $\\check{C}\\;\\longrightarrow \\; C, \\qquad \\qquad \\check{\\lambda }\\;\\longrightarrow \\; \\lambda .$ For clarity, let us see how $(k,l)$ -splitting looks for a full matrix in which we split the first $n$ nodes in the presence of $m-2n$ spectators with shifts $h_{1},\\ldots ,h_{m-2n}$ and trivial permutation: $\\begin{tiny}\\left[\\begin{array}{c:c:c:c:ccc}\\check{C}_{11} & \\check{C}_{12} & \\ldots & \\check{C}_{1n} & \\check{C}_{1,n+1} & \\ldots & \\check{C}_{1,m-n}\\\\ \\check{C}_{21} & \\check{C}_{22} & \\ldots & \\check{C}_{2n} & \\check{C}_{2,n+1} & \\ldots & \\check{C}_{2,m-n}\\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & & \\vdots \\\\ \\check{C}_{n1} & \\check{C}_{n2} & \\ldots & \\check{C}_{nn} & \\check{C}_{n,n+1} & \\ldots & \\check{C}_{n,m-n}\\\\ \\check{C}_{n+1,1} & \\check{C}_{n+1,2} & \\ldots & \\check{C}_{n+,1n} & \\check{C}_{n+1,n+1} & \\ldots & \\check{C}_{n+1,m-n}\\\\\\vdots & \\vdots & & \\vdots & \\vdots & \\ddots & \\vdots \\\\\\check{C}_{m-n,1} & \\check{C}_{m-n,2} & \\ldots & \\check{C}_{m-n,n} & \\check{C}_{m-n,n+1} & \\ldots & \\check{C}_{m-n,m-n}\\end{array}\\right]\\nonumber \\end{tiny}$ $\\downarrow $ $\\begin{tiny}\\left[\\begin{array}{cc:cc:c:cc:ccc}\\check{C}_{11} & \\check{C}_{11}+k & \\check{C}_{12} & \\check{C}_{12}+k & \\ldots & \\check{C}_{1n} & \\check{C}_{1n}+k & \\check{C}_{1,n+1} & \\ldots & \\check{C}_{1,m-n}\\\\\\check{C}_{11}+k & \\check{C}_{11}+l & \\check{C}_{12}+k+1 & \\check{C}_{12}+l & \\ldots & \\check{C}_{1n}+k+1 & \\check{C}_{1n}+l & \\check{C}_{1,n+1}+h_{1} & \\ldots & \\check{C}_{1,m-n}+h_{m-2n}\\\\ \\check{C}_{21} & \\check{C}_{21}+k+1 & \\check{C}_{22} & \\check{C}_{22}+k & \\ldots & \\check{C}_{2n} & \\check{C}_{2n}+k & \\check{C}_{2,n+1} & \\ldots & \\check{C}_{2,m-n}\\\\\\check{C}_{21}+k & \\check{C}_{21}+l & \\check{C}_{22}+k & \\check{C}_{22}+l & \\ldots & \\check{C}_{2n}+k+1 & \\check{C}_{2n}+l & \\check{C}_{2,n+1}+h_{1} & \\ldots & \\check{C}_{2,m-n}+h_{m-2n}\\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & & \\vdots & & \\vdots \\\\ \\check{C}_{n1} & \\check{C}_{n1}+k+1 & \\check{C}_{n2} & \\check{C}_{n2}+k+1 & \\ldots & \\check{C}_{nn} & \\check{C}_{nn}+k & \\check{C}_{n,n+1} & \\ldots & \\check{C}_{n,m-n}\\\\\\check{C}_{n1}+k & \\check{C}_{n1}+l & \\check{C}_{n2}+k & \\check{C}_{n2}+l & \\ldots & \\check{C}_{nn}+k & \\check{C}_{nn}+l & \\check{C}_{n,n+1}+h_{1} & \\ldots & \\check{C}_{n,m-n}+h_{m-2n}\\\\ \\check{C}_{n+1,1} & \\check{C}_{n+1,1}+h_{1} & \\check{C}_{n+1,2} & \\check{C}_{n+1,2}+h_{1} & \\ldots & \\check{C}_{n+1,n} & \\check{C}_{n+1,n}+h_{1} & \\check{C}_{n+1,n+1} & \\ldots & \\check{C}_{n+1,m-n}\\\\\\vdots & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\\check{C}_{m-n,1} & \\check{C}_{m-n,1}+h_{m-2n} & \\check{C}_{m-n,2} & \\check{C}_{m-n,2}+h_{m-2n} & \\ldots & \\check{C}_{m-n,n} & \\check{C}_{m-n,n}+h_{m-2n} & \\check{C}_{m-n,n+1} & \\ldots & \\check{C}_{m-n,m-n}\\end{array}\\right].\\nonumber \\end{tiny}$ It is straightforward to check that the constraints from theorem REF are satisfied for the above matrix, and that it is consistent with (REF ) and (REF )." ], [ "Examples – global structure", "In this section we analyze in detail equivalent quivers and the structure of their permutohedra graphs for knots $3_1$ , $4_1$ , $5_1$ , $5_2$ , $6_1$ , $7_1$ , and the whole series of $(2,2p+1)$ torus knots." ], [ "Trefoil knot, $3_1$ ", "The generating function of superpolynomials of the knot $3_1$ is given by [28] $P_{3_1}(x,a,q,t)=\\sum _{r=0}^{\\infty } \\frac{x^ra^{2r}q^{-2r}}{(q^2;q^2)_r}\\sum _{k=0}^{r} \\left[\\begin{array}{c}r\\\\k\\end{array}\\right] q^{2k(r+1)}t^{2k} (-a^2q^{-2}t;q^2)_k,$ where we use the $q$ -binomial $\\left[\\begin{array}{c}r\\\\k\\end{array}\\right]=\\frac{(q^2;q^2)_r}{(q^2;q^2)_{r-k}(q^2;q^2)_k}.$ Linear order ($r=1$ ) of (REF ) encodes the uncolored superpolynomial $P_1(a,q,t)=a^2 q^{-2} + a^2 q^2 t^2 + a^4 t^3$ .", "Its homological diagram consists of one zig-zag made of 3 nodes, see figure REF .", "Figure: Homology diagram and a quiver matrix for 3 1 3_1 knot.The labels 0, 2 and 3 are tt-degrees of generators, while blueλ i {blue}{\\lambda _i} arise in specialization of quiver generating parameters.", "For 3 1 3_1 knot the quiver is unique, so the permutohedra graph consists of one vertex (shown in red).Let us rederive the trefoil quiver following section .", "We start from noticing that if we keep the $q$ -Pochhammer $(-a^2q^{-2}t;q^2)_k$ on the side, the remaining part of $P_{3_1}(x,a,q,t)$ can be easily rewritten in the quiver form.", "First, we express the $q$ -binomial as in (REF ) and cancel $(q^2;q^2)_r$ : $\\sum _{r=0}^{\\infty } \\frac{x^ra^{2r}q^{-2r}}{(q^2;q^2)_r}\\sum _{k=0}^{r} \\left[\\begin{array}{c}r\\\\k\\end{array}\\right] q^{2k(r+1)}t^{2k}=\\sum _{r=0}^{\\infty } x^ra^{2r}q^{-2r}\\sum _{k=0}^{r} \\frac{1}{(q^2;q^2)_{r-k}(q^2;q^2)_k} q^{2k(r+1)}t^{2k}.$ Then, we define new summation variables: $\\check{d}_1=r-k$ and $\\check{d}_2=k$ , which allows us rewrite (REF ) as a motivic generating function of a prequiver: $\\begin{split} \\sum _{\\check{d}_1,\\check{d}_2 \\ge 0}(-q)^{2\\check{d}_1\\check{d}_2+2\\check{d}_2^2} \\frac{\\left(xa^2q^{-2}\\right)^{\\check{d}_1}\\left(xa^2(-t)^{2}\\right)^{\\check{d}_2}}{(q^2;q^2)_{\\check{d}_1}(q^2;q^2)_{\\check{d}_2}}&=\\left.", "\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C} \\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^2;q^2)_{\\check{d}}}\\right|_{\\check{x}=x\\check{\\lambda }}, \\\\\\check{C}=\\left[\\begin{array}{c:c}0 & 1\\\\1 & 2\\end{array}\\right], \\phantom{\\qquad } & \\phantom{\\qquad } \\check{\\lambda } = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2 (-t)^2\\end{array}\\right].\\end{split}$ Now we put $(-a^2q^{-2}t;q^2)_k$ back with $k=\\check{d}_2$ and apply a variant of formula (REF ) for splitting one node (because only one $\\check{d}_i$ enters $k$ ): $\\frac{(\\xi ;q^2)_{\\check{d}_{i}}}{(q^2;q^2)_{\\check{d}_{i}}}= \\sum \\limits _{\\alpha _{i}+\\beta _{i}=\\check{d}_{i}} (-q)^{\\beta _{i}^2}\\frac{\\left(\\xi q^{-1}\\right)^{\\beta _{i}}}{(q^2;q^2)_{\\alpha _{i}}(q^2;q^2)_{\\beta _{i}}},$ with $\\xi =-a^2q^{-2}t$ and $i=2$ .", "This leads to $\\begin{split}P_{3_1}(x,a,q,t)=\\sum _{d_1,\\alpha _2,\\beta _2 \\ge 0} & \\frac{\\left(xa^2q^{-2}\\right)^{d_1}\\left(xa^2(-t)^{2}\\right)^{\\alpha _2}\\left(xa^4q^{-3}(-t)^{2}\\right)^{\\beta _2}}{(q^2;q^2)_{d_1}(q^2;q^2)_{\\alpha _2}(q^2;q^2)_{\\beta _2}}\\\\ &\\times (-q)^{2d_1\\alpha _2+2d_1\\beta _2+2\\alpha _2^2+ 2\\alpha _2\\beta _2+3\\beta _{2}^2},\\end{split}$ which is equal to $\\left.", "P_Q(x,q)\\right|_{x=x\\lambda }$ for $C=\\left[\\begin{array}{c:cc}0 & 1 & 1\\\\1 & 2 & 2\\\\1 & 2 & 3\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2 (-t)^2\\\\a^4 q^{-3} (-t)^3\\end{array}\\right].$ This is the quiver found in [1], [2]; in the language of definition REF it arises from (REF ) by $(0,1)$ -splitting of the second node, with trivial permutation $\\sigma (2)=2$ , $h_1=0$ , and $\\kappa =-a^2q^{-3}t$ : $\\begin{split}\\check{C}=\\left[\\begin{array}{c:c}0 & 1\\\\1 & 2\\end{array}\\right]&\\;\\longrightarrow \\;C=\\left[\\begin{array}{c:cc}0 & 1 & 1+0\\\\1 & 2 & 2+0\\\\1+0 & 2+0 & 2+1\\end{array}\\right], \\\\\\check{\\lambda } = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2 (-t)^2\\end{array}\\right]& \\; \\longrightarrow \\;\\lambda = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2 (-t)^2\\\\a^2 (-t)^2 \\times a^2q^{-3}(-t)\\end{array}\\right].\\end{split}$ In the above process we did not have to make any choices, therefore we expect that the above quiver is unique.", "This is indeed the case: since the trefoil knot is thin, all quiver equivalences come from permutations of non-diagonal matrix entries, but there are no possible pairings that could lead to non-trivial permutations of non-diagonal entries.", "In consequence, the conjecture REF holds for the trefoil knot." ], [ "Figure-eight knot, $4_1$", "For the figure-eight knot two corresponding quivers have been already found in [2], [4].", "Let us rederive this result and check that there are no other equivalent quivers.", "The generating function of superpolynomials of the figure-eight knot reads [28]: $P_{4_1}(x,a,q,t)= \\sum _{r=0}^{\\infty }\\sum _{k=0}^{r} \\frac{x^r (-1)^k a^{-2k} t^{-2k} q^{-k^2+3k}(q^{-2r};q^2)_k}{(q^2;q^2)_r (q^{2};q^2)_k} (-a^{2}q^{-2}t;q^{2})_k(-a^{2}q^{2r}t^{3};q^{2})_k.$ For $r=1$ we obtain the superpolynomial $P_1(a,q,t) = 1 + a^{-2} t^{-2} + q^{-2} t^{-1} + q^2 t + a^2 t^2$ .", "The corresponding homological diagram consists of a degenerate zig-zag made of one node and a diamond, see figure REF .", "Figure: Homological diagram for 4 1 4_1 knot, with labels λ i \\lambda _i assigned to various nodes (top).", "In the bottom the two equivalent quivers are shown, which differ by a transposition of elements C 2,5 C_{2,5} and C 3,4 C_{3,4} of the quiver matrix (shown in yellow, together with their symmetric companions).", "The positions of these elements are encoded in combinations λ 2 λ 5 \\lambda _2\\lambda _5 and λ 3 λ 4 \\lambda _3\\lambda _4, which are equal to each other (satisfy the center of mass condition).", "The permutohedra graph is given by Π 2 \\Pi _2 that consists of two vertices connected by one edge.In order to find equivalent quivers we follow section  again.", "We use the relation $(q^{-2r};q^2)_k = (-1)^k q^{-2rk+k(k-1)}\\frac{(q^2;q^2)_r}{(q^2;q^2)_{r-k}}$ , as well as (REF ) for $(-a^{2}q^{2r}t^{3};q^{2})_k / (q^2;q^2)_k$ , to rewrite $ \\sum _{0\\le k \\le r} \\frac{x^r (-1)^k a^{-2k} t^{-2k} q^{-k^2+3k}(q^{-2r};q^2)_k}{(q^2;q^2)_r (q^{2};q^2)_k} (-a^{2}q^{2r}t^{3};q^{2})_k&=\\left.", "\\sum _{\\check{d}}\\frac{(-q)^{\\check{d}\\cdot \\check{C} \\cdot \\check{d}}\\check{x}^{\\check{d}}}{(q^2;q^2)_{\\check{d}}}\\right|_{\\check{x}=x\\check{\\lambda }},\\nonumber \\\\\\check{C}=\\left[\\begin{array}{c:c:c}0 & -1 & 0 \\\\-1 & -2 & -1 \\\\0 & -1 & 1\\end{array}\\right], \\phantom{\\qquad } \\phantom{\\qquad } \\check{\\lambda } & = \\left[\\begin{array}{c}1 \\\\a^{-2} q^2 (-t)^{-2}\\\\q (-t)\\end{array}\\right],$ where we substitute $r-k=\\check{d}_1$ and $k=\\check{d}_2+\\check{d}_3$ .", "In addition, we rewrite the remaining term $(-a^2q^{-2}t;q^2)_k\\equiv (-a^2q^{-2}t;q^2)_{\\check{d}_2+\\check{d}_3}$ , using (REF ) for $n=2$ : $\\begin{split}\\frac{(\\xi ;q^2)_{\\check{d}_{i}+\\check{d}_j}}{(q^2;q^2)_{\\check{d}_{i}}(q^2;q^2)_{\\check{d}_{j}}}= \\sum \\limits _{\\alpha _{i}+\\beta _{i}=\\check{d}_{i}}\\sum \\limits _{\\alpha _{j}+\\beta _{j}=\\check{d}_{j}} & (-q)^{\\beta _{i}^2+\\beta _{j}^2+2\\beta _{i}(\\alpha _{j}+\\beta _{j})} \\\\& \\times \\frac{\\left(\\xi q^{-1}\\right)^{\\beta _{i}}}{(q^2;q^2)_{\\alpha _{i}}(q^2;q^2)_{\\beta _{i}}} \\frac{\\left(\\xi q^{-1}\\right)^{\\beta _{j}}}{(q^2;q^2)_{\\alpha _{j}}(q^2;q^2)_{\\beta _{j}}}.\\end{split}$ Now the two equivalent quivers arise from two possible specializations of $(i,j)$ in the term $\\beta _i\\alpha _j$ in the above expression.", "For $(i,j)=(2,3)$ , from the quadratic terms in the exponent of $(-q)$ we read off the following quiver matrix: $C=\\left[\\begin{array}{c:cc:cc}0 &- 1 & -1 & 0 & 0\\\\-1 & -2 & -2 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {-1};\\phantom{-1}\\\\-1 & -2 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {0};\\phantom{0} & 0\\\\0 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {0};\\phantom{0} & 1 & 1\\\\0 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {-1};\\phantom{-1} & 0 & 1 & 2\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}1 \\\\a^{-2} q^2 (-t)^{-2}\\\\q^{-1} (-t)^{-1}\\\\q (-t)\\\\a^2 q^{-2} (-t)^2\\end{array}\\right]$ which is consistent with the result in [2] (up to a permutation of rows and columns) and corresponds to the red dot in figure REF .", "On the other hand, setting $(i,j)=(3,2)$ yields $C=\\left[\\begin{array}{c:cc:cc}0 &- 1 & -1 & 0 & 0\\\\-1 & -2 & -2 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {0};\\phantom{0}\\\\-1 & -2 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {-1};\\phantom{-1} & 0\\\\0 & -1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {-1};\\phantom{-1} & 1 & 1\\\\0 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {0};\\phantom{0} & 0 & 1 & 2\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}1 \\\\a^{-2} q^2 (-t)^{-2}\\\\q^{-1} (-t)^{-1}\\\\q (-t)\\\\a^2 q^{-2} (-t)^2\\end{array}\\right]$ which is consistent with the second, equivalent quiver found in [4].", "The two above quivers are also presented in figure REF and they differ by a transposition of elements shown in yellow.", "This transposition corresponds to a single possible inversion encoded in the term $\\beta _i\\alpha _j$ in (REF ).", "In the language of definition REF , quivers (REF ) and (REF ) arise from the prequiver (REF ) by $(0,1)$ -splitting of nodes number 2 and 3.", "Since we split two nodes, there are 2 possible permutations.", "For the identity permutation ($\\sigma (2)=2$ , $\\sigma (3)=3$ ) we obtain (REF ) $\\check{C}=\\left[\\begin{array}{c:c:c}0 & -1 & 0 \\\\-1 & -2 & -1 \\\\0 & -1 & 1\\end{array}\\right]\\overset{\\sigma (2)<\\sigma (3)}{\\longrightarrow }C=\\left[\\begin{array}{c:cc:cc}0 &- 1 & -1+0 & 0 & 0+0\\\\-1 & -2 & -2+0 & -1 & {amber}{-1+0}\\\\-1+0 & -2+0 & -2+1 & {amber}{-1+0+1} & -1+1\\\\0 & -1 & {amber}{-1+0+1} & 1 & 1+0\\\\0+0 & {amber}{-1+0} & -1+1 & 1+0 & 1+1\\end{array}\\right].$ On the other hand, for a transposition $\\sigma =(2\\ 3)$ (i.e.", "$\\sigma (2)=3$ , $\\sigma (3)=2$ ) we get $\\check{C}=\\left[\\begin{array}{c:c:c}0 & -1 & 0 \\\\-1 & -2 & -1 \\\\0 & -1 & 1\\end{array}\\right]\\overset{\\sigma (2)>\\sigma (3)}{\\longrightarrow }C=\\left[\\begin{array}{c:cc:cc}0 &- 1 & -1+0 & 0 & 0+0\\\\-1 & -2 & -2+0 & -1 & {amber}{-1+0+1}\\\\-1+0 & -2+0 & -2+1 & {amber}{-1+0} & -1+1\\\\0 & -1 & {amber}{-1+0} & 1 & 1+0\\\\0+0 & {amber}{-1+0+1} & -1+1 & 1+0 & 1+1\\end{array}\\right].$ In both cases we have $h_1=0$ and $\\kappa =-a^2q^{-3}t$ .", "The quiver matrices (REF ) and (REF ) are related by a transposition of non-diagonal entries.", "The condition $\\lambda _2 \\lambda _5=\\lambda _3 \\lambda _4$ from theorem REF is satisfied, so it is a symmetry.", "The permutohedra graph is given by $\\Pi _2$ that consists of two vertices connected by an edge, as shown in figure REF .", "Since the $4_1$ knot is thin, all equivalent quivers come from permutations of non-diagonal elements of $C$ .", "However, we checked that there are no more pairings apart from $\\lambda _2 \\lambda _5=\\lambda _3 \\lambda _4$ , so we found the whole equivalence class and the conjecture REF holds for the figure-eight knot." ], [ "Cinquefoil knot, $5_1$", "In turn, we anlayze $5_1$ knot.", "The generating function of its colored superpolynomials is given by [28] $P_{5_1}(x,a,q,t)=\\sum _{r=0}^{\\infty } \\frac{x^r a^{4r}q^{-4r}}{(q^2;q^2)_r}\\sum _{0\\le k_2 \\le k_1 \\le r} \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] &(-a^2q^{-2}t;q^2)_{k_1}\\\\\\times &\\, q^{2[(2r+1)(k_1+k_2)-r k_1-k_1 k_2]}t^{2(k_1+k_2)}, \\nonumber $ which for $r=1$ encodes the superpolynoimal $P_1(a,q,t)=a^4 q^{-4} + a^4 t^2 + a^6 q^{-2} t^3 + a^4 q^{4} t^4 + a^6 q^{2} t^5 $ .", "The homological diagram is a a zig-zag made of 5 nodes, see figure REF .", "Figure: Two copies of the homological diagram for 5 1 5_1 knot are shown on top.", "On each copy we denoted a parallellogram that encodes a symmetry, i.e.", "a transposition of two matrix elements that yields an equivalent quiver.", "In total there are 3 equivalent quivers, shown in bottom, which correspond to 3 vertices of the permutohedra graph.", "The permutohedra graph is made of two Π 2 \\Pi _2 that share a common vertex (in red).In analogy to the case of $4_1$ , we rewrite the summand in (REF ) as a product of the motivic generating series for the prequiver and $(-a^2q^{-2}t;q^2)_{k_1}$ with $k_1=(k_1-k_2)+k_2=\\check{d}_2+\\check{d}_3$ $\\begin{split}P_{5_{1}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{x^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{-2}t;q^{2})_{\\check{d}_{2}+\\check{d}_{3}}\\Big |_{\\check{x}=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c}0 & 1 & 3\\\\1 & 2 & 3\\\\3 & 3 & 4\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^4q^{-4}\\\\a^4q^{-2}(-t)^2\\\\a^4 (-t)^4\\end{array}\\right].\\end{split}$ Then, the application of (REF ) leads to $(0,1)$ -splitting of nodes number 2 and 3 (the node number 1 is a spectator with $h_1=0$ ; $\\kappa =-a^2q^{-3}t$ ), which can be done in two ways.", "The identity permutation $(\\sigma (2)=2,\\sigma (3)=3)$ yields $C=\\left[\\begin{array}{c:cc:cc}0 & 1 & 1 & 3 & 3\\\\1 & 2 & 2 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3}\\\\1 & 2 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4} & 4\\\\3 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4} & 4 & 4\\\\3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3} & 4 & 4 & 5\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^4q^{-4}\\\\a^4q^{-2}(-t)^2\\\\a^6q^{-5}(-t)^3\\\\a^4 (-t)^4\\\\a^6q^{-3}(-t)^5\\end{array}\\right]$ whereas the transposition $\\sigma =(2\\ 3)$ gives $C=\\left[\\begin{array}{c:cc:cc}0 & 1 & 1 & 3 & 3\\\\1 & 2 & 2 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4}\\\\1 & 2 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3} & 4\\\\3 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3} & 4 & 4\\\\3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4} & 4 & 4 & 5\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^4q^{-4}\\\\a^4q^{-2}(-t)^2\\\\a^6q^{-5}(-t)^3\\\\a^4 (-t)^4\\\\a^6q^{-3}(-t)^5\\end{array}\\right].$ Comparing with theorem REF , it is clear that this symmetry comes from the pairing $\\lambda _3\\lambda _4 = \\lambda _2 \\lambda _5$ (shown in orange in figure REF ).", "However, for the cinquefoil knot we find another pairing $\\lambda _1 \\lambda _5 = \\lambda _2 \\lambda _3$ (shown in green in figure REF ), which also leads to a non-trivial symmetry.", "Using definitions REF and REF we can see that the quiver from (REF ) admits not only the inverse of $(0,1)$ -splitting analyzed above, but also the inverse of $(1,3)$ -splitting.In fact it admits also the inverse of $(1,2)$ -splitting with $h_1=0$ and $h_1=2$ , but they capture the same symmetries.", "This phenomenon is characteristic for all instances of splitting two nodes, when it is possible to interpret $\\lambda _a \\lambda _b=\\lambda _c \\lambda _d$ as $\\lambda _a$ , $\\lambda _c$ coming from splitting node $a$ and $\\lambda _d$ , $\\lambda _b$ coming from splitting node $d$ or $\\lambda _a$ , $\\lambda _d$ coming from splitting node $a$ and $\\lambda _c$ , $\\lambda _b$ coming from splitting node $c$ .", "More precisely, $P_{5_1}$  can be rewritten as $\\begin{split}P_{5_{1}}(x,a,q,t)&=\\left.\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{2r}t^{3};q^{2})_{\\check{d}_{2}+\\check{d}_{3}}\\right|_{\\check{x}=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c}4 & 3 & 3\\\\3 & 0 & 1\\\\3 & 1 & 2\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^4 (-t)^4\\\\a^4q^{-4}\\\\a^4q^{-2}(-t)^2\\end{array}\\right],\\end{split}$ which leads to (REF ) by $(1,3)$ -splitting of nodes number 2 and 3 (the node number 1 is a spectator with $h_1=1$ ) with permutation $\\sigma =(2\\ 3)$ and $\\kappa =-a^2q^{-1}t^3$ .", "This automatically implies that there exists another equivalent quiver, arising from $(1,3)$ -splitting of (REF ) with the trivial permutation $C=\\left[\\begin{array}{c:cc:cc}4 & 3 & 4 & 3 & 4\\\\3 & 0 & 1 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2}\\\\4 & 1 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3} & 4\\\\3 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {3};\\phantom{3} & 2 & 3\\\\4 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2} & 4 & 3 & 5\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^4 (-t)^4\\\\a^4q^{-4}\\\\a^6q^{-5}(-t)^3\\\\a^4q^{-2}(-t)^2\\\\a^6q^{-3}(-t)^5\\end{array}\\right],$ which is the quiver on the left-hand side in figure REF (up to a permutation of nodes).", "To sum up, we have found 3 equivalent quivers for $5_1$ , and from quiver (REF ) we can obtain either of the other two, by appropriate transpositions of elements of the quiver matrix.", "However, since these transpositions are not disjoint, we cannot compose them.", "In consequence the permutohedra graph, shown in figure REF , consists of two permutohedra $\\Pi _2$ that share a common vertex (in red) that represents quiver (REF ).", "Using an argument analogous to the one for the figure-eight knot, we can check that since there are no pairings other than those depicted in figure REF , we have found all equivalent quivers.", "In consequence, the conjecture REF holds for the cinquefoil knot." ], [ "$5_2$ knot", "The knot $5_2$ is a more involved example.", "Having identified one quiver for this knot (e.g.", "the one found in [2]) and considering all possible local equivalences following theorem REF , we found 12 equivalent quivers for this knot (they are listed explicitly in appendix ).", "It turns out these quivers form an interesting structure of three permutohedra $\\Pi _3$ glued along their edges.", "Let us explain how this structure arises.", "We start from the following generating function of superpolynomials [2] $\\nonumber P_{5_2}(x,a,q,t)=\\sum _{r=0}^\\infty \\frac{x^r }{(q^2;q^2)_r}\\sum _{0\\le k_2\\le k_1\\le r} & \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right](-1)^{r+k_1} (-a^{2}q^{-2}t;q^{2})_{k_{1}}(-a^2q^{2r}t^{3};q^2)_{k_1}\\\\&\\times a^{2k_2}q^{k_1^2+k_1+2(k_2^2-k_2-rk_1)}t^{2k_2-r}~.$ At linear order we find the superpolynomial $P_1(a,q,t) = a^2q^2t^2 + a^2q^{-2} + a^4 t^3 + a^2t + a^4q^2t^4 + a^4q^{-2} t^2 + a^6 t^5$ .", "The homological diagram consists of a diamond and a zig-zag of length three, see figure REF .", "Figure: Homology diagram for 5 2 5_2 knot; labels λ i \\lambda _i are consistent with ().The generating function (REF ) can be rewritten in the form $\\begin{split}P_{5_{2}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{x^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{-2}t;q^{2})_{\\check{d}_{2}+\\check{d}_{3}+\\check{d}_{4}}\\Big |_{x=x\\check{\\lambda }},\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c}0 & 0 & 1 & 1\\\\0 & 1 & 1 & 2\\\\1 & 1 & 2 & 2\\\\1 & 2 & 2 & 4\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2q^{-1}(-t)\\\\a^2(-t)^2\\\\a^4q^{-2}(-t)^4\\end{array}\\right].\\end{split}$ Then, (0,1)-splitting of nodes number 2, 3, 4 with trivial permutation, $h_1=0$ , and $\\kappa =-a^2q^{-3}t$ leads to $C=\\left[\\begin{array}{c:cc:cc:cc}0 & 0 & 0 & 1 & 1 & 1 & 1\\\\0 & 1 & 1 & 1 & 1 & 2 & 2\\\\0 & 1 & 2 & 2 & 2 & 3 & 3\\\\1 & 1 & 2 & 2 & 2 & 2 & 2\\\\1 & 1 & 2 & 2 & 3 & 3 & 3\\\\1 & 2 & 3 & 2 & 3 & 4 & 4\\\\1 & 2 & 3 & 2 & 3 & 4 & 5\\\\\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^2q^{-2}\\\\a^2q^{-1}(-t)\\\\a^4q^{-4}(-t)^2\\\\a^2(-t)^2\\\\a^4q^{-3}(-t)^3\\\\a^4q^{-2}(-t)^4\\\\a^6q^{-5}(-t)^5\\end{array}\\right].$ Because the splitting involves three nodes, it gives rise to a permutohedron $\\Pi _3$ , which is a hexagon.", "Furthermore, (REF ) can be rewritten in another form $P_{5_{2}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{x^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}\\Pi _{\\check{d}_2,\\check{d}_3,\\check{d}_4}\\Big |_{x=x\\check{\\lambda }}\\nonumber \\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c}1 & 0 & 1 & 1\\\\0 & 0 & 1 & 1\\\\1 & 1 & 2 & 2\\\\1 & 1 & 2 & 3\\\\\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^2q^{-1}(-t)\\\\a^2q^{-2}\\\\a^2(-t)^2\\\\a^4q^{-3}(-t)^3\\\\\\end{array}\\right],$ $\\Pi _{\\check{d}_2,\\check{d}_3,\\check{d}_4}=\\sum \\limits _{\\alpha _{2}+\\beta _{2}=\\check{d}_{2}}&\\sum \\limits _{\\alpha _{3}+\\beta _{3}=\\check{d}_{3}}\\sum \\limits _{\\alpha _{4}+\\beta _{4}=\\check{d}_{4}}(-q)^{2\\check{d}_{1}(\\beta _{2}+\\beta _{3}+\\beta _{4})+2(\\beta _{2}+\\beta _{3}+\\beta _{4})^{2}+2(\\beta _{2}\\alpha _{3}+\\beta _{2}\\alpha _{4}+\\beta _{3}\\alpha _{4})}\\nonumber \\\\& \\times \\frac{(a^{2}q^{-2}t^{2})^{(\\beta _{2}+\\beta _{3}+\\beta _{4})}(q^{2};q^{2})_{\\check{d}_{2}}(q^{2};q^{2})_{\\check{d}_{3}}(q^{2};q^{2})_{\\check{d}_{4}}}{(q^{2};q^{2})_{\\alpha _{2}}(q^{2};q^{2})_{\\beta _{2}}(q^{2};q^{2})_{\\alpha _{3}}(q^{2};q^{2})_{\\beta _{3}}(q^{2};q^{2})_{\\alpha _{4}}(q^{2};q^{2})_{\\beta _{4}}}.\\nonumber $ In this case the factor $\\Pi _{d_2,d_3,d_4}$ encodes $(0,2)$ -splitting of the last three nodes with trivial permutation, $h_1=1$ , and $\\kappa =a^2q^{-2}t^2$ , which leads to a rearrangment of quiver (REF ): $C=\\left[\\begin{array}{c:cc:cc:cc}1 & 0 & 1 & 1 & 2 & 1 & 2\\\\0 & 0 & 0 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {1};\\phantom{1} & 1 & 1\\\\1 & 0 & 2 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2} & 3 & 2 & 3\\\\1 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2} & 2 & 2 & 2 & 2\\\\2 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {1};\\phantom{1} & 3 & 2 & 4 & 3 & 4\\\\1 & 1 & 2 & 2 & 3 & 3 & 3\\\\2 & 1 & 3 & 2 & 4 & 3 & 5\\\\\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^2q^{-1}(-t)\\\\a^2q^{-2}\\\\a^2(-t)^2\\\\a^4q^{-4}(-t)^2\\\\a^4q^{-2}(-t)^4\\\\a^4q^{-3}(-t)^3\\\\a^6q^{-5}(-t)^5\\end{array}\\right].$ This means that the corresponding permutohedron is also $\\Pi _3$ , and one of its vertices corresponding to the above matrix is shared with the previous permutohedron (there is also another quiver common to these two permutohedra).", "Note that $(0,2)$ -splitting of prequiver (REF ) with permutation $\\sigma =(2\\ 3)$ yields the quiver for $5_2$ knot found in [2]: $C=\\left[\\begin{array}{c:cc:cc:cc}1 & 0 & 1 & 1 & 2 & 1 & 2\\\\0 & 0 & 0 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2} & 1 & 1\\\\1 & 0 & 2 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {1};\\phantom{1} & 3 & 2 & 3\\\\1 & 1 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {1};\\phantom{1} & 2 & 2 & 2 & 2\\\\2 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {2};\\phantom{2} & 3 & 2 & 4 & 3 & 4\\\\1 & 1 & 2 & 2 & 3 & 3 & 3\\\\2 & 1 & 3 & 2 & 4 & 3 & 5\\\\\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^2q^{-1}(-t)\\\\a^2q^{-2}\\\\a^2(-t)^2\\\\a^4q^{-4}(-t)^2\\\\a^4q^{-2}(-t)^4\\\\a^4q^{-3}(-t)^3\\\\a^6q^{-5}(-t)^5\\end{array}\\right].$ Elements that are transposed between (REF ) and (REF ) are highlighted in yellow.", "Furthermore, the quiver (REF ) also admits the inverse of another splitting, which corresponds to the following rewriting of (REF ): $\\begin{split}P_{5_{2}}(x,a,q,t)&=\\left.\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{x^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{2r}t^{3};q^{2})_{\\check{d}_{2}+\\check{d}_{3}+\\check{d}_{4}}\\right|_{x=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c}2 & 1 & 1 & 1\\\\1 & 0 & 0 & 0\\\\1 & 0 & 1 & 1\\\\1 & 0 & 1 & 2\\end{array}\\right] \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^2(-t)^2\\\\a^2q^{-2}\\\\a^2q^{-1}(-t)\\\\a^4q^{-4}(-t)^2\\end{array}\\right].\\end{split}$ In this case $(1,3)$ -splitting of the last three nodes with permutation $\\sigma =(2\\ 3)$ , $h_1=1$ , and $\\kappa =-a^2q^{-1}t^3$ leads to $C=\\left[\\begin{array}{c:cc:cc:cc}2 & 1 & 2 & 1 & 2 & 1 & 2\\\\1 & 0 & 1 & 0 & 2 & 0 & 1\\\\2 & 1 & 3 & 1 & 3 & 2 & 3\\\\1 & 0 & 1 & 1 & 2 & 1 & 2\\\\2 & 2 & 3 & 2 & 4 & 3 & 4\\\\1 & 0 & 2 & 1 & 3 & 2 & 3\\\\2 & 1 & 3 & 2 & 4 & 3 & 5\\\\\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^2(-t)^2\\\\a^2q^{-2}\\\\a^4q^{-3}(-t)^3\\\\a^2q^{-1}(-t)\\\\a^4q^{-2}(-t)^4\\\\a^4q^{-4}(-t)^2\\\\a^6q^{-5}(-t)^5\\end{array}\\right],$ which is also a reordering of  (REF ).", "This means that (REF ) captures the third permutohedron $\\Pi _3$ , and the quiver (REF ) (or its reordered version (REF )) corresponds to the vertex that is shared with the previous $\\Pi _3$ .", "Figure: The permutohedra graph for 5 2 5_2 knot consists of three Π 3 \\Pi _3 (shown schematically in bottom together with the formulas they correspond to) glued along common edges.", "The edges in this graph correspond to 6 types of transpositions arising from various quadruples of homology generators, which are also shown in various colors on the homological diagrams.Following the above analysis we find that the permutohedra graph for $5_2$ has the structure shown in figure REF .", "The permutohedron arising from 6 permutations associated to $(0,1)$ -splitting of the prequiver (REF ) lies on the top of the graph.", "The bottom-right $\\Pi _3$ comes from all possible $(0,2)$ -splittings of the prequiver (REF ).", "Finally, $(1,3)$ -splittings of (REF ) lead to the bottom-left hexagon.", "The quiver (REF ) (or its reordered form (REF )) is denoted by the green dot.", "The red dot represents the quiver (REF ) (or its reordered form (REF )) found in [2].", "The symmetry connecting these two quivers is denoted by the blue edge.", "Moreover, we find that each pair of permutohedra $\\Pi _3$ identified above has 2 common quivers, which are connected by a transposition that is also common to such two permutohedra.", "Altogether, the permutohedra graph takes form of 3 permutohedra $\\Pi _3$ glued along their edges, as shown in figure REF .", "The triangle in the middle of the graph represents two transpositions whose composition is also a transposition (not a 3-cycle), so it does not contradict the argument in section REF .", "In the figure we also show how various symmetries (transpositions of matrix elements that relate various equivalent quivers, which correspond to edges of the permutohedra graph) arise from quadruples of homology generators, and denote them in various colors.", "According to conjecture REF , we expect that figure REF presents the whole equivalence class of quivers." ], [ "$7_1$ knot", "Another interesting example is $7_1$ knot.", "Applying theorem REF systematically, we find 13 equivalent quivers, which we list explicitly in the appendix .", "More detailed analysis reveals that they form two permuthohedra $\\Pi _3$ that share one common vertex (corresponding to a common quiver), and each of these $\\Pi _3$ in addition shares a common vertex with one of the two permutohedra $\\Pi _2$ .", "The generating function of colored superpolynomials takes the form [13], [12] $P_{7_1}(x,a,q,t)=\\sum _{r=0}^{\\infty } \\frac{x^r a^{6r}q^{-6r}}{(q^2;q^2)_r}\\sum _{0 \\le k_3 \\le k_2 \\le k_1 \\le r} \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right]\\left[\\begin{array}{c}k_2\\\\k_3\\end{array}\\right](-a^2q^{-2}t;q^2)_{k_1}\\\\\\times \\, q^{2[(2r+1)(k_1+k_2+k_3)-r k_1-k_1 k_2-k_2 k_3]}t^{2(k_1+k_2+k_3)}.", "\\nonumber $ For $r = 1$ we get the uncolored superpolynomial $P_1(a,q,t) = a^6 q^{-6} + a^6 q^{-2} t^2 + a^8 q^{-4} t^3 + a^6 q^{2} t^4 + a^8 t^5 + a^6 q^6 t^6 + a^8 q^4 t^7$ .", "The corresponding homological diagram consists of one zig-zag made of 7 nodes, see figure REF .", "Figure: Homology diagram for 7 1 7_1 knot; labels λ i \\lambda _i are consistent with ().First, we rewrite (REF ) as follows: $\\begin{split}P_{7_{1}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{-2}t;q^{2})_{\\check{d}_{2}+\\check{d}_{3}+\\check{d}_{4}}\\Big |_{\\check{x}=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c}0 & 1 & 3 & 5\\\\1 & 2 & 3 & 5\\\\3 & 3 & 4 & 5\\\\5 & 5 & 5 & 6\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^6q^{-2} (-t)^4\\\\a^6(-t)^6\\end{array}\\right].\\end{split}$ The (0,1)-splitting of the last three nodes with trivial permutation, $h_1=0$ , and $\\kappa =-a^2q^{-3}t$ leads to $C=\\left[\\begin{array}{c:cc:cc:cc}0 & 1 & 1 & 3 & 3 & 5 & 5\\\\1 & 2 & 2 & 3 & 3 & 5 & 5\\\\1 & 2 & 3 & 4 & 4 & 6 & 6\\\\3 & 3 & 4 & 4 & 4 & 5 & 5\\\\3 & 3 & 4 & 4 & 5 & 6 & 6\\\\5 & 5 & 6 & 5 & 6 & 6 & 6\\\\5 & 5 & 6 & 5 & 6 & 6 & 7\\\\\\end{array}\\right], \\qquad \\lambda = \\left[\\begin{array}{c}a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^8q^{-7}(-t)^3\\\\a^6q^{-2} (-t)^4\\\\a^8q^{-5}(-t)^5\\\\a^6(-t)^6\\\\a^8q^{-3}(-t)^7\\end{array}\\right],$ which reproduces the quiver from [2].", "More generally, splitting these three nodes with all possible permutations yields one permutohedron $\\Pi _3$ .", "Furthermore, we can also rewrite (REF ) as $\\begin{split}P_{7_{1}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{2r}t^{3};q^{2})_{\\check{d}_{2}+\\check{d}_{3}+\\check{d}_{4}}\\Big |_{\\check{x}=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c}6 & 5 & 5 & 5\\\\5 & 0 & 1 & 3\\\\5 & 1 & 2 & 3\\\\5 & 3 & 3 & 4\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^6(-t)^6\\\\a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^6q^{-2} (-t)^4\\end{array}\\right].\\end{split}$ In this case $(1,3)$ -splitting of the last three nodes with permutation $\\sigma =(2\\ 4)$ , $h_1=1$ , and $\\kappa =-a^2q^{-1}t^3$ gives a rearrangment of the quiver (REF ): $C=\\left[\\begin{array}{c:cc:cc:cc}6 & 5 & 6 & 5 & 6 & 5 & 6\\\\5 & 0 & 1 & 1 & 3 & 3 & 5\\\\6 & 1 & 3 & 2 & 4 & 4 & 6\\\\5 & 1 & 2 & 2 & 3 & 3 & 5\\\\6 & 3 & 4 & 3 & 5 & 4 & 6\\\\5 & 3 & 4 & 3 & 4 & 4 & 5\\\\6 & 5 & 6 & 5 & 6 & 5 & 7\\\\\\end{array}\\right], \\qquad \\lambda = \\left[\\begin{array}{c}a^6(-t)^6\\\\a^6q^{-6}\\\\a^8q^{-7}(-t)^3\\\\a^6q^{-4}(-t)^2\\\\a^8q^{-5}(-t)^5\\\\a^6q^{-2} (-t)^4\\\\a^8q^{-3}(-t)^7\\end{array}\\right],$ and analogous splittings with all other permutations give rise to another permutohedron $\\Pi _3$ .", "Therefore we have identified two permutohedra that share a common vertex, which represents the quiver matrix  (REF ) (or its reordered form (REF )).", "Let us now focus on $\\Pi _3$ arising from the prequiver (REF ).", "One can check that almost all quivers represented by its other vertices cannot be obtained from other prequivers.", "The only exception is $C=\\left[\\begin{array}{c:cc:cc:cc}0 & 1 & 1 & 3 & 3 & 5 & 5\\\\1 & 2 & 2 & 3 & 4 & 5 & 6\\\\1 & 2 & 3 & 3 & 4 & 5 & 6\\\\3 & 3 & 3 & 4 & 4 & 5 & 6\\\\3 & 4 & 4 & 4 & 5 & 5 & 6\\\\5 & 5 & 5 & 5 & 5 & 6 & 6\\\\5 & 6 & 6 & 6 & 6 & 6 & 7\\\\\\end{array}\\right], \\qquad \\lambda = \\left[\\begin{array}{c}a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^8q^{-7}(-t)^3\\\\a^6q^{-2} (-t)^4\\\\a^8q^{-5}(-t)^5\\\\a^6(-t)^6\\\\a^8q^{-3}(-t)^7\\end{array}\\right],$ that arises from (0,1)-splitting of (REF ) with permutation $\\sigma =(2\\ 4)$ .", "Indeed, $(0,1)$ -splitting of the last two nodes of the prequiver $\\check{C}=\\left[\\begin{array}{ccc:c:c}0 & 1 & 5 & 1 & 3\\\\1 & 2 & 6 & 2 & 4\\\\5 & 6 & 7 & 6 & 6\\\\1 & 2 & 6 & 3 & 4\\\\3 & 4 & 6 & 4 & 5\\\\\\end{array}\\right], \\qquad \\qquad \\check{\\lambda } = \\left[\\begin{array}{c}a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^8q^{-3}(-t)^7\\\\a^8q^{-7}(-t)^3\\\\a^8q^{-5}(-t)^5\\end{array}\\right],$ with permutation $\\sigma =(4\\ 5)$ , $h_1=2,\\,h_2=1,\\,h_3=0$ , and $\\kappa =-a^{-2}q^5t$ leads to $C=\\left[\\begin{array}{ccc:cc:cc}0 & 1 & 5 & 1 & 3 & 3 & 5\\\\1 & 2 & 6 & 2 & 3 & 4 & 5\\\\5 & 6 & 7 & 6 & 6 & 6 & 6\\\\1 & 2 & 6 & 3 & 3 & 4 & 5\\\\3 & 3 & 6 & 3 & 4 & 4 & 5\\\\3 & 4 & 6 & 4 & 4 & 5 & 5\\\\5 & 5 & 6 & 5 & 5 & 5 & 6\\\\\\end{array}\\right], \\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^6q^{-6}\\\\a^6q^{-4}(-t)^2\\\\a^8q^{-3}(-t)^7\\\\a^8q^{-7}(-t)^3\\\\a^6q^{-2} (-t)^4\\\\a^8q^{-5}(-t)^5\\\\a^6q^{-6}\\end{array}\\right],$ which is a rearrangement of (REF ).", "This means that the quiver (REF ) (or its reordered form (REF )) is a gluing point of permutohedra $\\Pi _3$ and $\\Pi _2$ .", "Figure: The permutohedra graph for 7 1 7_1 knot consists of two Π 3 \\Pi _3 and two Π 2 \\Pi _2 appropriately glued.", "Altogether it has 13 vertices representing equivalent quivers, and 8 symmetries corresponding to various quadruples of homology generators (and represented by different colors of the edges in the graph).An analogous phenomenon occurs for the second $\\Pi _3$ , which is also connected to another permutohedron $\\Pi _2$ .", "Altogether, the permutohedra graph consists of two $\\Pi _3$ and two $\\Pi _2$ , as shown in figure REF .", "The quiver (REF ) (or equivalently (REF )), also found in [2], is common to the two $\\Pi _3$ and it is represented by the red dot.", "The $\\Pi _3$ on the left arises from the prequiver (REF ), whereas the one on the right corresponds to the prequiver (REF ).", "The quiver (REF ) (or its reordered form (REF )) is represented by the green node, and it glues the left $\\Pi _3$ with $\\Pi _2$ coming from the prequiver (REF ).", "Analogous gluing point is present on the right-hand side of the graph.", "In total we found 8 non-trivial symmetries shown in figure REF in various colors, and 13 equivalent quivers that we list explicitly in the appendix .", "Using the procedure described in section REF we checked that there are no other equivalent quivers.", "According to conjecture REF , we expect that figure REF presents the whole equivalence class of quivers." ], [ "$6_1$ knot", "Another example that we consider is $6_1$ knot.", "We have found 141 equivalent quivers, which form quite complicated permutohedra graph, shown in figure REF .", "These quivers are related to each other by 16 symmetries (transpositions of various pairs of quiver matrices).", "Figure: The permutohedra graph for 6 1 6_1 knot has 141 vertices that represent equivalent quivers (left).", "Excluding symmetries that involve λ 1 \\lambda _1 reduces the whole graph to a cube-like shape (right).", "Each face of this cube is one Π 4 \\Pi _4 (a bit squashed), and neighboring Π 4 \\Pi _4's are glued along a square, which is a common face to both Π 4 \\Pi _4's.", "The red vertex represents the quiver () (or its reordered form ()).The generating function of colored superpolynomials for $6_1$ knot reads [13]: $P_{6_1}(x,a,q,t)=\\sum _{r=0}^{\\infty } \\frac{x^r}{(q^2;q^2)_r} \\sum _{0\\le k_2\\le k_1 \\le r} & \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right](-a^{-2}q^{2}t^{-1};q^{-2})_{k_1}(-a^{-2}q^{-2r}t^{-3};q^{-2})_{k_1} \\nonumber \\\\&\\times a^{2(k_1+k_2)} t^{2(k_1+k_2)} q^{2(k_1^2+k_2^2-k_1-k_2)}.$ Linear order of this equation gives the uncolored superpolynomial $P_1(a,q,t)= 1 + a^{-2}t^{-2} + q^{2} t + q^{-2}t^{-1} + a^{2} t^{2} + 1 + a^{2} q^{2} t^{3} + a^{2} q^{-2}t + a^{4} t^{4}$ .", "The corresponding homological diagram, shown in figure REF , consists of 2 diamonds and a degenerate zig-zag made of one node that coincides with one vertex of the upper diamond, so that $\\lambda _1= \\lambda _6$ .", "Figure: Homology diagram for 6 1 6_1 knot; labels λ i \\lambda _i are consistent with ().First, we rewrite (REF ) as $\\begin{split}P_{6_{1}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}(-a^{2}q^{-2}t;q^{2})_{\\check{d}_{2}+\\check{d}_{3}+\\check{d}_{4}+\\check{d}_{5}}\\Big |_{\\check{x}=x\\check{\\lambda }}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c:c}0 & -1 & -1 & -1 & -1\\\\-1 & -2 & -2 & -2 & -1\\\\-1 & -2 & -1 & -2 & -1\\\\-1 & -2 & -2 & 0 & 0\\\\-1 & -1 & -1 & 0 & 1\\end{array}\\right], \\phantom{\\qquad } \\phantom{\\qquad } \\check{\\lambda } = \\left[\\begin{array}{c}1\\\\a^{-2} q^2 (-t)^{-2}\\\\q^{-1} (-t)^{-1}\\\\1 \\\\a^2 q^{-3} (-t)\\\\\\end{array}\\right].\\end{split}$ Then $(1,3)$ -splitting of the last four nodes with permuation $\\sigma =(2\\ 4\\ 5\\ 3)$ , $h_1=1$ , and $\\kappa =-a^{2} q^{-1} t^{3}$ leads to the quiver found in [2]: $C=\\left[\\begin{array}{c:cc:cc:cc:cc}0& -1& 0& -1& 0& -1& 0& -1& 0\\\\-1& -2& -1& -2& -1& -2& 0& -1& 0\\\\0& -1& 1& 0& 1& -1& 1& 1& 2\\\\-1& -2& 0& -1& 0& -2& 0& -1& 1\\\\0& -1& 1& 0& 2& -1& 1& 0& 2\\\\-1& -2& -1& -2& -1& 0& 1& 0& 1\\\\0& 0& 1& 0& 1& 1& 3& 2& 3\\\\-1& -1& 1& -1& 0& 0& 2& 1& 2\\\\0& 0& 2& 1& 2& 1& 3& 2& 4\\end{array}\\right], \\phantom{\\qquad } \\phantom{\\qquad }\\lambda = \\left[\\begin{array}{c}1 \\\\a^{-2} q^2 (-t)^{-2}\\\\q (-t)\\\\q^{-1} (-t)^{-1}\\\\a^{2} q^{-2} (-t)^{2}\\\\1 \\\\a^{2} q^{-1} (-t)^{3}\\\\a^{2} q^{-3} (-t)\\\\a^{4} q^{-4} (-t)^{4}\\end{array}\\right].$ On the other hand, we can rewrite (REF ) in the form $\\begin{split}P_{6_{1}}(x,a,q,t)&=\\sum _{\\check{d}}(-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^{2};q^{2})_{\\check{d}}}\\Pi _{\\check{d}_2,\\check{d}_3,\\check{d}_4,\\check{d}_5}\\Big |_{\\check{x}=x\\check{\\lambda }}\\\\\\Pi _{\\check{d}_2,\\check{d}_3,\\check{d}_4,\\check{d}_5}&=\\sum \\limits _{\\alpha _{2}+\\beta _{2}=\\check{d}_{2}}\\sum \\limits _{\\alpha _{3}+\\beta _{3}=\\check{d}_{3}}\\sum \\limits _{\\alpha _{4}+\\beta _{4}=\\check{d}_{4}}\\sum \\limits _{\\alpha _{5}+\\beta _{5}=\\check{d}_{5}} \\prod _{i=2}^5 \\frac{(a^{2}q^{-2}t^{2})^{\\beta _{i}}(q^2;q^2)_{\\check{d}_i}}{(q^2;q^2)_{\\alpha _i}(q^2;q^2)_{\\beta _i}} \\\\& \\times (-q)^{2(\\beta _{2}+\\beta _{3}+\\beta _{4}+\\beta _{5})^{2}+2(\\alpha _2\\beta _3+\\alpha _2\\beta _4+\\alpha _2\\beta _5+\\alpha _3 \\beta _4+\\alpha _3\\beta _5+\\alpha _4\\beta _5)}\\\\\\check{C}&=\\left[\\begin{array}{c:c:c:c:c}0 & -1 & -1 & 0 & 0\\\\-1 & -2 & -2 & -1 & -1\\\\-1 & -2 & -1 & 0 & 0\\\\0 & -1 & 0 & 1 & 1\\\\0 & -1 & 0 & 1 & 2\\end{array}\\right], \\phantom{\\qquad } \\phantom{\\qquad } \\check{\\lambda } = \\left[\\begin{array}{c}1\\\\a^{-2} q^2 (-t)^{-2}\\\\q^{-1} (-t)^{-1}\\\\q (-t)\\\\a^{2} q^{-2} (-t)^{2}\\\\\\end{array}\\right],\\end{split}$ Then, $(0,2)$ -splitting of the last four nodes with permutation $\\sigma =(2\\ 5)(3\\ 4)$ , $h_1=0$ , and $\\kappa =a^{2} q^{-2} t^{2}$ leads to $C=\\left[\\begin{array}{c:cc:cc:cc:cc}0& -1& -1& -1& -1& 0& 0& 0& 0\\\\-1& -2& -2& -2& -1& -1& 0& -1& 0\\\\-1& -2& 0& -2& 0& -1& 1& -1& 1\\\\-1& -2& -2& -1& -1& 0& 1& 0& 1\\\\-1& -1& 0& -1& 1& 0& 2& 0& 2\\\\0& -1& -1& 0& 0& 1& 1& 1& 2\\\\0& 0& 1& 1& 2& 1& 3& 1& 3\\\\0& -1& -1& 0& 0& 1& 1& 2& 2\\\\0& 0& 1& 1& 2& 2& 3& 2& 4\\end{array}\\right], \\phantom{\\qquad } \\phantom{\\qquad }\\lambda = \\left[\\begin{array}{c}1 \\\\a^{-2} q^2 (-t)^{-2}\\\\1 \\\\q^{-1} (-t)^{-1}\\\\a^{2} q^{-3} (-t)\\\\q (-t)\\\\a^{2} q^{-1} (-t)^{3}\\\\a^{2} q^{-2} (-t)^{2}\\\\a^{4} q^{-4} (-t)^{4}\\end{array}\\right].$ which is a rearrangement of (REF ).", "This means that the above quiver is common to two permutohedra $\\Pi _4$ , and it is represented by the red dot in figure REF and REF .", "In figure REF , which shows a planar projection of a part of the permutohedra graph, $\\Pi _4$ coming from the prequiver (REF ) is oriented along axis $\\nearrow $ , whereas $\\Pi _4$ oriented along $\\nwarrow $ corresponds to the prequiver (REF ).", "All other quiver matrices that we found are listed in the Mathematica file attached to the arXiv submission.", "According to conjecture REF , we expect that there are no more equivalent quivers and figure REF presents the whole equivalence class.", "Figure: Planar projection of a part of the permutohedra graph for 6 1 6_1 knot.", "In homological diagrams (on left and right) it is indicated how some of its symmetries, corresponding to edges of the graph, arise from quadruples of homology generators.", "The positions of two two permutohedra Π 4 \\Pi _4 mentioned in the text are indicated schematically in the bottom." ], [ "$(2,2p+1)$ torus knots", "The last example we consider is a series of $(2,2p+1)$ torus knots.", "For this class the number of equivalent quivers grows rapidly; for $p=1,...,7$ we have found respectively 1, 3, 13, 68, 405, 2684 and 19557 equivalent quivers, which permutohedra graphs have an interesting structure.", "For $p=1$ there is just one corresponding quiver, see section REF ; for $p>1$ the permutohedra graph consists of two series of larger and larger permutohedra $\\Pi _2,\\ldots ,\\Pi _p$ (and several additional permutohedra of small size that do not belong to these series).", "In each of these two series each permutohedron $\\Pi _i$ is connected to $\\Pi _{i-1}$ and $\\Pi _{i+1}$ (for $i=3,\\ldots ,p-1$ ), and the two largest permutohedra $\\Pi _p$ from both series are also connected.", "Such a structure is present for $5_1$ , $7_1$ , $9_1$ and $11_1$ knots in figures REF , REF , REF , and REF , respectively.", "In this section we explain how two largest permutohedra $\\Pi _p$ for $(2,2p+1)$ torus knot arise.", "To start with, note that the generating function of superpolynomials for $(2,2p+1)$ -torus knot can be written, among others, in the following two equivalent ways, which correspond to different grading conventions for the $S^r$ -colored HOMFLY-PT homologies [12], [13]: $P_{T_{2,2p+1}}(x,a,q,t) &=\\sum _{r\\ge 0} x^r a^{2pr}q^{-2pr}\\sum _{0\\le k_{p}\\le \\ldots \\le k_{2}\\le k_{1}\\le r}\\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] \\cdots \\left[\\begin{array}{c}k_{p-1}\\\\k_{p}\\end{array}\\right] \\\\&\\quad \\times q^{2\\sum _{i=1}^{p}((2r+1)k_{i}-k_{i-1}k_{i})}t^{2(k_{1}+k_{2}+\\ldots +k_{p})}(-a^2q^{-2}t;q^2)_{k_1} =\\nonumber \\\\&= \\sum _{r\\ge 0} x^r a^{2pr}q^{-2pr}\\sum _{0\\le k_{p}\\le \\ldots \\le k_{2}\\le k_{1}\\le r}\\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] \\cdots \\left[\\begin{array}{c}k_{p-1}\\\\k_{p}\\end{array}\\right]\\\\&\\quad \\times q^{2\\sum _{i=1}^{p}((2r+1)k_{i}-k_{i-1}k_{i})}t^{2(k_{1}+k_{2}+\\ldots +k_{p})}(-a^2q^{2r}t^3;q^2)_{r-k_p}.\\nonumber $ For $p=1$ , i.e.", "$3_1$ knot, the above expressions reduce to $\\begin{split}\\sum _{0\\le k_{1}\\le r} \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] q^{2k_1(r+1)}t^{2k_1} & (-a^2q^{-2}t;q^2)_{k_1}\\\\&= \\sum _{0\\le k_{1}\\le r} \\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] q^{2k_1(r+1)}t^{2k_1}(-a^2q^{2r}t^3;q^2)_{r-k_1},\\end{split}$ and the two permutohedra consist of one vertex.", "They are in fact identified, so that the full permutohedra graph consists just of one $\\Pi _1$ .", "In general, both (REF ) and () can be rewritten in form (REF ) using the formula $\\left[\\begin{array}{c}r\\\\k_1\\end{array}\\right]\\left[\\begin{array}{c}k_1\\\\k_2\\end{array}\\right] \\cdots \\left[\\begin{array}{c}k_{p-1}\\\\k_{p}\\end{array}\\right]=\\frac{(q^2;q^2)_r}{(q^2;q^2)_{r-k_1}(q^2;q^2)_{k_1-k_2}\\cdots (q^2;q^2)_{k_{p-1}-k_p}(q^2;q^2)_{k_p}}.$ In case of (REF ) we set $\\check{d}_1 &= r-k_1, &\\check{d}_2 &= k_1 - k_2, &\\check{d}_3 &= k_2 - k_3, &\\check{d}_4 &= k_3 - k_4,\\\\&\\ldots &\\check{d}_{i+1} &= k_{i} - k_{i+1}, && \\ldots &\\check{d}_{p+1} &= k_p,$ which leads to $& P_{T_{2,2p+1}}(x,a,q,t) = \\sum _{\\check{d}} (-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^2;q^2)_{\\check{d}}}(-a^2q^{-2}t;q^2)_{\\check{d}_2+\\dots + \\check{d}_{p+1}}\\Big |_{\\check{x}=x\\check{\\lambda }} \\\\& \\check{C}=\\left[\\begin{array}{c:c:c:c:c:c:c}0 & 1 & 3 & 5 & \\dots & 2p-3 & 2p-1 \\\\1 & 2 & 3 & 5 & \\dots & 2p-3 & 2p-1 \\\\3 & 3 & 4 & 5 & \\dots & 2p-3 & 2p-1 \\\\5 & 5 & 5 & 6 & \\dots & 2p-3 & 2p-1 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\2p-3 & 2p-3 & 2p-3 & 2p-3 & \\dots & 2p-2 & 2p-1 \\\\2p-1 & 2p-1 & 2p-1 & 2p-1 & \\dots & 2p-1 & 2p \\\\\\end{array}\\right]\\left.\\begin{array}{c}\\check{d}_1 \\\\\\check{d}_2 \\\\\\check{d}_3 \\\\\\check{d}_4 \\\\\\vdots \\\\\\check{d}_{p} \\\\\\check{d}_{p+1}\\end{array}\\right., \\quad \\check{\\lambda } = \\left[\\begin{array}{c}a^{2p}q^{-2p}\\\\a^{2p}q^{-2(p-1)}(-t)^2\\\\a^{2p}q^{-2(p-2)} (-t)^4\\\\a^{2p}q^{-2(p-3)} (-t)^6\\\\\\vdots \\\\a^{2p}q^{-2}(-t)^{2p+2} \\\\a^{2p}(-t)^{2p}\\end{array}\\right].", "\\nonumber $ The $(0,1)$ -splitting of the nodes corresponding to $\\check{d}_2,\\dots ,\\check{d}_{p+1}$ with trivial permutation, $h_1=0$ , and $\\kappa =\\xi q^{-1}=-a^2q^{-3}t$ produces the quiver found in [2]: $C=\\left[\\begin{array}{c:cc:cc:c:cc}0 & 1 & 1 & 3 & 3 & \\dots & 2p-1 & 2p-1 \\\\1 & 2 & 2 & 3 & 3 & \\dots & 2p-1 & 2p-1 \\\\1 & 2 & 3 & 4 & 4 & \\dots & 2p & 2p \\\\3 & 3 & 4 & 4 & 4 & \\dots & 2p-1 & 2p-1 \\\\3 & 3 & 4 & 4 & 5 & \\dots & 2p & 2p \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\2p-1 & 2p-1 & 2p & 2p-1 & 2p & \\dots & 2p & 2p \\\\2p-1 & 2p-1 & 2p & 2p-1 & 2p & \\dots & 2p & 2p+1 \\\\\\end{array}\\right],\\quad \\lambda = \\left[\\begin{array}{c}a^{2p}q^{-2p}\\\\a^{2p}q^{-2(p-1)}(-t)^2\\\\a^{2(p+1)}q^{-2(p-1)-3}(-t)^3 \\\\a^{2p}q^{-2(p-2)}(-t)^4\\\\a^{2(p+1)}q^{-2(p-2)-3}(-t)^5\\\\\\vdots \\\\a^{2p}(-t)^{2p} \\\\a^{2(p+1)}q^{-3}(-t)^{2p+1}\\end{array}\\right].$ On the other hand, for the expression () we introduce $& & \\check{d}_1 &= r-(r-k_p) = k_p, & & &\\\\\\check{d}_{2} &= r-k_1, &\\check{d}_{3} &= k_{1} - k_{2}, &&\\ldots &\\check{d}_{p+1} &= k_{p-1}-k_p,$ and then find $& P_{T_{2,2p+1}}(x,a,q,t) =\\sum _{\\check{d}} (-q)^{\\check{d}\\cdot \\check{C}\\cdot \\check{d}}\\frac{\\check{x}^{\\check{d}}}{(q^2;q^2)_{\\check{d}}}(-a^2q^{2r}t^3;q^2)_{\\check{d}_2+\\dots + \\check{d}_{p+1}}\\Big |_{\\check{x}=x\\check{\\lambda }} \\\\&\\check{C}=\\left[\\begin{array}{c:c:c:c:c:c:c}2p & 2p-1 & 2p-1 & 2p-1 & \\dots & 2p-1 & 2p-1 \\\\2p-1 & 0 & 1 & 3 & \\dots & 2p-5 & 2p-3 \\\\2p-1 & 1 & 2 & 3 & \\dots & 2p-5 & 2p-3 \\\\2p-1 & 3 & 3 & 4 & \\dots & 2p-5 & 2p-3 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\2p-1 & 2p-5 & 2p-5 & 2p-5 & \\dots & 2p-4 & 2p-3 \\\\2p-1 & 2p-3 & 2p-3 & 2p-3 & \\dots & 2p-3 & 2p-2 \\\\\\end{array}\\right]\\left.\\begin{array}{c}\\check{d}_1 \\\\\\check{d}_2 \\\\\\check{d}_3 \\\\\\check{d}_4 \\\\\\vdots \\\\\\check{d}_{p} \\\\\\check{d}_{p+1}\\end{array}\\right., \\ \\check{\\lambda } = \\left[\\begin{array}{c}a^{2p}(-t)^{2p} \\\\a^{2p}q^{-2p}\\\\a^{2p}q^{-2(p-1)}(-t)^2\\\\a^{2p}q^{-2(p-2)} (-t)^4 \\\\\\vdots \\\\a^{2p}q^{-4}(-t)^{2p-4} \\\\a^{2p}q^{-2}(-t)^{2p-2}\\end{array}\\right].", "\\nonumber $ One can check that the $(1,3)$ -splitting of the nodes corresponding to $\\check{d}_2,\\dots ,\\check{d}_{p+1}$ with permutation $\\sigma =(2\\ \\ (p+1))$ , $h_1=1$ , and $\\kappa =-a^2q^{-1}t^3$ yields the same quiver as in (REF ).", "Note that both prequivers given above are the same up to reordering of nodes, however the two splittings are different.", "This is why we obtain two different permutohedra $\\Pi _p$ , respectively left (for (REF )) and right (()) in figures REF , REF , REF , and  REF .", "These two permutohedra share the quiver matrix (REF ), which can be obtained from appropriate splittings of corresponding prequivers, as explained above.", "An interested reader may conduct careful analysis of other permutohedra in these graphs." ], [ "Examples – local structure", "In the previous section we presented permutohedra graphs for simple knots and discussed in detail the structure of glued permutohedra embedded in these graphs.", "In this section we take the opposite perspective and study the local structure: we choose some particular quiver and identify all equivalent quivers related to it by a single transposition of matrix elements (a single symmetry, to which we refer as local).", "We also provide interpretation of such equivalences in terms of homological diagrams.", "We conduct such an analysis for infinite families of $(2,2p+1)$ torus knots (also denoted $T_{2,2p+1}$ ), $TK_{2|p|+2}$ and $TK_{2p+1}$ twist knots, and in addition $6_2,6_3$ and $7_3$ knots.", "The quivers that we analyze are those found in [2] (apart from the quiver for $7_3$ knot that was found in [9]), and they are indicated by red vertices in permutohedra graphs in figures  REF , REF , REF , and REF .", "The symmetries that we analyze in this section are represented by edges adjacent to these red vertices.", "Recall that: Quiver matrices for $(2,2p+1)$ torus knots that we consider are given in (REF ).", "A homological diagram for $(2,2p+1)$ torus knot consists of one zig-zag made of $2p+1$ generators.", "Quiver matrices for twist knots $TK_{2|p|+2}$ (i.e.", "$4_1,6_1,8_1,\\dots $ knots) are given in the appendix .", "A homological diagram for $TK_{2|p|+2}$ knot consists of $p$ diamonds and a zig-zag made of one generator, so altogether it has $4p+1$ generators.", "Quiver matrices for twist knots $TK_{2p+1}$ (i.e.", "$3_1,5_2,7_2,\\dots $ knots) are also given in the appendix .", "A homological diagram for $TK_{2p+1}$ knot consists of $p-1$ diamonds and a zig-zag of length 3, so altogether it has $4p-1$ generators.", "Figure: Enumeration of wedges and diamonds in the homology diagrams, from left to right: T 2,2p+1 T_{2,2p+1}, TK 2|p|+2 TK_{2|p|+2}, TK 2p+1 TK_{2p+1}.In this section we fix the ordering of homological generators (and correspondingly quiver nodes) as shown in figure REF .", "In what follows we call a part of a zig-zag consisting of three consecutive nodes that form a shape $\\wedge $ a wedge.", "We enumerate diamonds and wedges by $r,r^{\\prime },r^{\\prime \\prime },r^{\\prime \\prime \\prime },\\dots $ , such that $r\\le r^{\\prime }\\le r^{\\prime \\prime }\\le r^{\\prime \\prime \\prime }\\dots $ ; for a wedge or a zig-zag labeled by $r$ , we enumerate the generators it consists of as in the bottom of figure REF .", "We write pairings $\\lambda _a\\lambda _b=\\lambda _c\\lambda _d$ as column vectors with entries $a,b,c,d$ .", "Recall that we call such a paring a symmetry if quiver matrices with elements $C_{ab}$ and $C_{cd}$ exchanged are equivalent.", "We also call the requirements $C_{ai}+C_{bi}=C_{ci}+C_{di}$ (for $i\\ne a,b,c,d$ ) spectator constraints.", "Theorem 9 For infinite families of knots $T_{2,2p+1},TK_{2|p|+2},TK_{2p+1},p=1,2,3,\\dots $ , quiver matrices given respectively in (REF ) and in appendix , have the following local symmetries: $\\begin{aligned}T_{2,2p+1}: &\\ \\begin{bmatrix}2r \\\\2r^{\\prime }+3 \\\\2r+3 \\\\2r^{\\prime }\\end{bmatrix},\\ \\begin{bmatrix}2r+3 \\\\2r^{\\prime }+2 \\\\2r+2 \\\\2r^{\\prime }+3\\end{bmatrix}\\\\TK_{2|p|+2}: &\\ \\begin{bmatrix}4r-1 \\\\4r^{\\prime } \\\\4r \\\\4r^{\\prime }-1\\end{bmatrix},\\ \\begin{bmatrix}4r-1 \\\\4r^{\\prime }-2 \\\\4r-2 \\\\4r^{\\prime }-1\\end{bmatrix},\\ \\begin{bmatrix}4r+1 \\\\4r^{\\prime } \\\\4r \\\\4r^{\\prime }+1\\end{bmatrix},\\ \\begin{bmatrix}4r+1 \\\\4r^{\\prime \\prime }-2 \\\\4r^{\\prime }+1 \\\\4r^{\\prime }-2\\end{bmatrix},\\ \\begin{bmatrix}4 \\\\4p-1 \\\\5 \\\\4p-2\\end{bmatrix}\\\\TK_{2p+1}: &\\ \\begin{bmatrix}2 \\\\4r^{\\prime }+3 \\\\3 \\\\4r^{\\prime }+2\\end{bmatrix},\\ \\begin{bmatrix}2 \\\\4r^{\\prime }+1 \\\\1 \\\\4r^{\\prime }+2\\end{bmatrix},\\ \\begin{bmatrix}2 \\\\4p+1 \\\\3 \\\\4p\\end{bmatrix}\\bigcup \\mathcal {T}\\left(TK_{2|p|+2} \\setminus \\begin{bmatrix}4 \\\\4p-1 \\\\5 \\\\4p-2\\end{bmatrix}\\right)\\end{aligned}$ where $r^{\\prime }=r+1$ , $r^{\\prime \\prime }=r+2$ , and $\\mathcal {T}\\left(TK_{2|p|+2} \\setminus \\begin{bmatrix}4 \\\\4p-1 \\\\5 \\\\4p-2\\end{bmatrix}\\right):=\\begin{bmatrix}4r+2 \\\\4r^{\\prime }+1 \\\\4r+1 \\\\4r^{\\prime }+2\\end{bmatrix},\\ \\begin{bmatrix}4r \\\\4r^{\\prime }+1 \\\\4r+1 \\\\4r^{\\prime }\\end{bmatrix},\\ \\begin{bmatrix}4r+2 \\\\4r^{\\prime }+3 \\\\4r+3 \\\\4r^{\\prime }+2\\end{bmatrix},\\ \\begin{bmatrix}4r \\\\4r^{\\prime \\prime }+3 \\\\4r^{\\prime } \\\\4r^{\\prime }+3\\end{bmatrix}.$ Figure: The local symmetries for T 2,2p+1 T_{2,2p+1} torus knots, r=0,⋯,p-1r=0,\\dots ,p-1(the symmetry exists only for r ' =r+1r^{\\prime }=r+1)Figure: The local symmetries for twist knots.", "The symmetries which are shared between TK 2|p|+2 TK_{2|p|+2} and TK 2p+1 TK_{2p+1} twist knots do not have blue labels (any choice of λ\\lambda 's ordering from figure  is valid for them).", "The top-right symmetry is signature for the TK 2|p|+2 TK_{2|p|+2} twist knots, whereas the three bottom ones – for TK 2p+1 TK_{2p+1} twist knots.Recall again that entries of the vectors given above are labels of appropriate quadruples of quiver nodes or homology generators.", "For $(2,2p+1)$ torus knots, the condition $r^{\\prime }=r+1$ means that these generators belong to two consecutive wedges, see figure REF .", "For twist knots, generators that encode a symmetry belong to various diamonds or the wedge, see figure REF .", "Below we give a proof of theorem REF divided into three parts, each corresponding to one of the infinite families of knots.", "It is followed by the analysis of $6_2,6_3,7_3$ knots.", "Figure: The four local symmetries of quiver () corresponding to 6 1 6_1 knot, shown as the colorful thick edges" ], [ "$(2,2p+1)$ torus knots", "For this family of knots, the homology diagram is a chain of $p$ wedges joined together.", "The wedges are labeled by $r=0,1,2,\\dots ,p-1$ as in figure REF , and the labeling of all generators is shown explicitly in figure REF .", "Note that what we label as the zeroth node corresponds to the quiver series parameter $x_1$ , while the $i$ -th node for $i>1$ corresponds to $x_i$ .", "This notation is convenient, since in the formulas we can let $r=0$ referring to the first wedge, so we do not have to treat it separately.", "If $r$ and $r^{\\prime }$ label two wedges and $r^{\\prime }=r+1$ , they share the common node labeled by $2r+2=2r^{\\prime }$ .", "Note that the quiver matrix (REF ) (its special cases are given in (REF , REF , REF )) has elements $C_{ij}$ such that $\\begin{aligned} \\text{$i,j$ both odd or even:} \\qquad &\\ C_{ij} = j-1, & i = j: \\qquad &\\ C_{jj} = j, \\\\\\text{$i$ odd, $j$ even:} \\qquad &\\ C_{ij} = j, & \\text{$j$ even:} \\qquad &\\ C_{1j} = j-1, \\\\\\text{$i$ even, $j$ odd:} \\qquad &\\ C_{ij} = j - 2 + \\delta _{i+1,j},\\qquad & \\text{$j$ odd:} \\qquad &\\ C_{1j} = j-2.", "\\\\\\end{aligned}$ Figure: Pairings between the two wedges: type 2A.1 (left) and 2A.2 (right)We now use theorem REF to determine symmetries of this quiver.", "First, suppose that a pairing is made of generators from only two wedges, which are located in a generic position and not necessarily joined together, see figure REF .", "A direct check of conditions from theorem REF shows that the two pairings in figure REF are the symmetries if $r^{\\prime }=r+1$ , see figure REF .", "In order to confirm that there are no other symmetries, we label the four wedges by $r,r^{\\prime },r^{\\prime \\prime },r^{\\prime \\prime \\prime }$ such that $r<r^{\\prime }<r^{\\prime \\prime }<r^{\\prime \\prime \\prime }$ (see figure REF ).", "Figure: Pairings between 3 and 4 wedges, which are not symmetries for the quiver matrix ().", "From top to bottom: 3A.1, 3A.2, 4A.1, 4A.2.In consequence equation (REF ) leads to the following pairings: $\\begin{aligned}\\text{3A.1:} &\\quad C_{ab} = 2r^{\\prime \\prime }+1,\\; C_{cd} = 2r^{\\prime }+1 &\\text{3A.2:} &\\quad C_{ab} = 2r^{\\prime \\prime }+2,\\; C_{cd} = 2r^{\\prime }+1 \\\\\\text{4A.1:} &\\quad C_{ab} = 2r^{\\prime \\prime \\prime }+1,\\; C_{cd} = {\\left\\lbrace \\begin{array}{ll} 2r^{\\prime }+1,\\; r^{\\prime \\prime }=r^{\\prime }+1 \\\\ 2r^{\\prime \\prime },\\; r^{\\prime \\prime }>r^{\\prime }+1 \\end{array}\\right.}", "&\\text{4A.2:} &\\quad C_{ab} = 2r^{\\prime \\prime \\prime }+2,\\; C_{cd} = 2r^{\\prime \\prime }+1\\end{aligned}$ It follows that the condition $|C_{ab}-C_{cd}|=1$ from theorem REF cannot be met in all these cases, so the only symmetries are indeed those in figure REF .", "Figure: The local symmetries of quivers () for T 2,2p+1 T_{2,2p+1} torus knots." ], [ "Twist knots $TK_{2|p|+2}$ : {{formula:b91638c8-1ca4-4a1f-b472-abbf823fe4db}}", "We now conduct analogous analysis for a family of twist knots $TK_{2|p|+2}$ .", "Recall that a homological diagram for such a knot – for a given $p$ – consists of $p$ diamonds and an extra dot.", "Consider a quadruple of diamonds with labels $(r,r^{\\prime },r^{\\prime \\prime },r^{\\prime \\prime \\prime })$ , such that $1\\le r\\le r^{\\prime }\\le r^{\\prime \\prime }\\le r^{\\prime \\prime \\prime } \\le p$ .", "We classify all pairings by the number of diamonds and their relative position.", "Tables in figure REF provide such classification, while all possible pairings between two diamonds are depicted in figure REF .", "Figure: The complete classification of pairings between diamonds in a homological diagram.", "Only Greengreen pairings, which arise for some specific values of r,r ' ,r '' r,r^{\\prime },r^{\\prime \\prime } for a given pp, produce local symmetries shown in figure .We now show that the green pairings in figure REF are indeed local symmetries.", "The detailed analysis of four of them is given in figure REF and REF .", "Notice that the rightmost pairing in figure REF is a particular case of $\\begin{bmatrix}4r+1 \\\\4r^{\\prime \\prime }-2 \\\\4r^{\\prime }+1 \\\\4r^{\\prime }-2\\end{bmatrix}$ Indeed, from the sub-matrix $\\begin{aligned}a = &\\ 4r+1 \\\\b = &\\ 4r^{\\prime }-2 \\\\c = &\\ 4r+5 \\\\d = &\\ 4r^{\\prime }-6\\end{aligned}\\qquad \\begin{pmatrix}2r-4 & 2r-2 & 2r-4 & 2r-2-\\delta _{r+1,r^{\\prime }} \\\\2r-2 & 2r^{\\prime } & 2r & 2r^{\\prime }-2 \\\\2r-4 & 2r & 2r-2 & 2r-\\delta _{r+2,r^{\\prime }} \\\\2r-2-\\delta _{r+1,r^{\\prime }} & 2r^{\\prime }-2 & 2r-\\delta _{r+2,r^{\\prime }} & 2r^{\\prime }-2\\end{pmatrix}$ we see that $r^{\\prime }=r+2$ is the only candidate for a symmetry (otherwise the condition $|C_{ab}-C_{cd}|=1$ fails).", "To stress again, in the examples above (figure REF and REF ) the crucial condition for the symmetry is $r^{\\prime }=r+1$ , i.e.", "pairing of the two neighboring diamonds.", "Among the green candidates in table REF there is only one case left: $\\begin{aligned}a = &\\ 4r+1 \\\\b = &\\ 4r^{\\prime }-2 \\\\c = &\\ 4r \\\\d = &\\ 4r^{\\prime }-1\\end{aligned}\\qquad \\begin{pmatrix}2r-4 & 2r-2 & 2r-3 & 2r-3 \\\\2r-2 & 2r^{\\prime } & 2r & 2r^{\\prime }-2 \\\\2r-3 & 2r & 2r-1 & \\boxed{2r-1} \\\\2r-3 & 2r^{\\prime }-2 & \\boxed{2r-1} & 2r^{\\prime }-3\\end{pmatrix}$ $\\begin{aligned}& s=4\\ &\\ (-1) + (2)\\ {red}{\\ne }\\ & (1) + (1) \\\\& s=5\\ &\\ (-2) + (0)\\ {red}{\\ne }\\ & (0) + (-1) \\\\& \\underline{r<r^{\\prime \\prime }<r^{\\prime }:} \\\\& s=4r-1\\ &\\ (2r-4) + (2r-1) = &\\ (2r-2) + (2r-3) \\\\& s=4r-2\\ &\\ (2r-3) + (2r) = &\\ (2r-1) + (2r-2) \\\\& s=4r^{\\prime }\\ &\\ (2r-2) + (2r^{\\prime }-1) = &\\ (2r-1) + (2r^{\\prime }-2) \\\\& s=4r^{\\prime }+1\\ &\\ (2r-4) + (2r^{\\prime }-3) = &\\ (2r-3) + (2r^{\\prime }-4) \\\\& s=4r^{\\prime \\prime }\\ &\\ (2r-2) + (2r^{\\prime \\prime }) = &\\ (2r-1) + (2r^{\\prime \\prime }-1) \\\\& \\underline{r<r^{\\prime }<r^{\\prime \\prime }:} \\\\& s=4r^{\\prime \\prime }-1\\ &\\ (2r-3) + (2r^{\\prime }-2)\\ {red}{\\ne }\\ &\\ (2r-1) + (2r^{\\prime }-3) \\\\& s=4r^{\\prime \\prime }-2\\ &\\ (2r-2) + (2r^{\\prime })\\ {red}{\\ne }\\ &\\ (2r) + (2r^{\\prime }-1) \\\\\\end{aligned}$ Due to the failure of the four spectators (${red}{\\ne }$ ), the case (REF ) gives a symmetry if and only if $r=1$ and $r^{\\prime }=p$ , which means that the bottom diamond interacts with the top diamond.", "For example, if $r=p=1$ , the pairing (REF ) turns into the only symmetry for $4_1$ knot, figure REF .", "We have thus shown that all five cases in the first row of figure REF are indeed non-trivial symmetries.", "It turns out that all other pairings listed in table REF fail to be a (non-trivial) symmetry.", "This happens because of the two reasons: when $C_{ab}\\ne C_{cd}$ , either the condition $|C_{ab}-C_{cd}|=1$ fails in general, or it is satisfied only when some diamonds collide, which brings us back to the case of two diamonds.", "On the other hand, any pairing between two diamonds which is not in our “top five” fails due to spectator constraints (which we verified in Mathematica).", "To sum up, only five cases give a symmetry: four of them involve a pair of diamonds, and one involves a triple (the “vertical” pairing)." ], [ "Twist knots $TK_{2p+1}$ : {{formula:10b33117-45cf-4940-b2e3-286f86b98c5b}}", "For this family of twist knots, a large portion of symmetries determined by the pairings originating from diamonds is the same as for the previous family of twist knots $TK_{2|p|+2}$ .", "The reason is a structural similarity between their HOMFLY-PT homologies.", "To be more specific, the main building blocks (diamonds) are the same for both families.", "The difference is in the form of a zig-zag, which for $TK_{2|p|+2}$ knots is degenerated to a dot, while for $TK_{2p+1}$ knot it takes form of a single wedge (of length 3).", "Therefore, at this stage we only need to study how this wedge interacts with diamonds.", "In total, there are five potential pairings: $\\begin{bmatrix}2 \\\\4r+1 \\\\3 \\\\4r\\end{bmatrix},\\ \\begin{bmatrix}1 \\\\4r+2 \\\\3 \\\\4r\\end{bmatrix},\\ \\begin{bmatrix}2 \\\\4r+3 \\\\3 \\\\4r+2\\end{bmatrix},\\ \\begin{bmatrix}1 \\\\4r+3 \\\\3 \\\\4r+1\\end{bmatrix},\\ \\begin{bmatrix}1 \\\\4r+2 \\\\2 \\\\4r+1\\end{bmatrix},$ where $r=1,\\ldots p-1$ enumerates diamonds.", "One of these cases turns out to be trivial: $\\begin{aligned}a = &\\ 1 \\\\b = &\\ 4r+2 \\\\c = &\\ 3\\\\d = &\\ 4r\\end{aligned}\\qquad \\begin{pmatrix}2 & 1 & 2 & 1 \\\\1 & 2r-2 & 2 & 2r-3 \\\\2 & 2 & 3 & 1 \\\\1 & 2r-3 & 1 & 2r-3\\end{pmatrix}$ The other four cases are investigated below in detail, see tables in figure REF .", "For the top-left case the only possibility for a symmetry is $r=p-1$ .", "This proves the bottom-right symmetry in figure REF .", "Another non-trivial symmetry arises from the top-right case in figure REF .", "The spectator constraints are satisfied for $1<r<r^{\\prime }$ , so we get the symmetry between the wedge and the first diamond, which is depicted in figure REF (bottom-left).", "Likewise, the rightmost pairing in (REF ) is a symmetry as well, see figure REF (bottom-middle).", "However, $\\lambda _1\\lambda _{4r+3}=\\lambda _{3}\\lambda _{4r+1}$ does not lead to a symmetry because of the spectator constraint for $s=2$ .", "That is why we end up with only three local symmetries between the wedge and a diamond.", "Figure: The non-trivial pairings between the wedge and a diamond" ], [ "$6_2,6_3,7_3$ knots", "Finally, with the support of the attached Mathematica code, we determine local symmetries for three other knots $6_2, 6_3$ and $7_3$ , for some particular quivers found in [2] and [9]." ], [ "$6_2$ knot", "Let us start from the knot $6_2$ .", "The quiver from [2] reads $C=\\left[\\begin{array}{ccccccccccc}-2 & -2 & -1 & -1 & -1 & -1 & 0 & -1 & 1 & 1 & 1 \\\\-2 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\\\-1 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\\\-1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 1 \\\\-1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\\\-1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\\\0 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 2 \\\\-1 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 3 & 3 \\\\1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 \\\\1 & 2 & 2 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & 3 \\\\1 & 2 & 2 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & 4 \\\\\\end{array}\\right],\\qquad \\qquad \\lambda = \\left[\\begin{array}{c}q^{-2} (-t)^{-2}\\\\a^2 q^{-4} (-t)^{-1}\\\\a^2 q^{-2} \\\\q^2\\\\a^2 (-t)\\\\a^2 (-t)\\\\a^2 q^2 (-t)^2\\\\a^4 q^{-2} (-t)^2\\\\a^4 (-t)^3\\\\a^2 q^4 (-t)^3\\\\a^4 q^{2} (-t)^4\\\\\\end{array}\\right].$ There are eight local symmetries associated to (REF ) for the following pairings: $\\lambda _1\\lambda _7&=\\lambda _3\\lambda _4, &\\lambda _1\\lambda _{11}&=\\lambda _4\\lambda _8,&\\lambda _5\\lambda _{11}&=\\lambda _8\\lambda _{10}, &\\lambda _6\\lambda _{11}&=\\lambda _8\\lambda _{10},\\\\\\lambda _1\\lambda _9&=\\lambda _3\\lambda _5, &\\lambda _1\\lambda _9&=\\lambda _3\\lambda _6,&\\lambda _2\\lambda _7&=\\lambda _3\\lambda _5, &\\lambda _2\\lambda _7&=\\lambda _3\\lambda _6.$ Their graphical representation, together with the homology diagram, is given in figure REF .", "Figure: Homology diagram and local symmetries for 6 2 6_2 knot, each picture marked with * corresponds to two symmetries, due to double-valued nodes λ 5 \\lambda _5 and λ 6 \\lambda _6." ], [ "$6_3$ knot", "For $6_3$ the quiver matrix from [2] is given by $C=\\left[\\begin{array}{ccccccccccccc}0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & -1 \\\\0 & 1 & 0 & -1 & -2 & 1 & 0 & -1 & -2 & 1 & 1 & 0 & -1 \\\\0 & 0 & 0 & -1 & -2 & 1 & 0 & 0 & -2 & 1 & 1 & 0 & 0 \\\\-1 & -1 & -1 & -2 & -3 & 0 & -1 & -2 & -3 & -1 & 0 & -2 & -2 \\\\-1 & -2 & -2 & -3 & -3 & -1 & -1 & -2 & -3 & -1 & -1 & -2 & -2 \\\\0 & 1 & 1 & 0 & -1 & 2 & 1 & 0 & -1 & 2 & 1 & 1 & -1 \\\\0 & 0 & 0 & -1 & -1 & 1 & 1 & 0 & -1 & 2 & 1 & 1 & 0 \\\\-1 & -1 & 0 & -2 & -2 & 0 & 0 & -1 & -2 & 0 & 0 & -1 & -2 \\\\-1 & -2 & -2 & -3 & -3 & -1 & -1 & -2 & -2 & 0 & -1 & -1 & -2 \\\\0 & 1 & 1 & -1 & -1 & 2 & 2 & 0 & 0 & 3 & 2 & 1 & 0 \\\\0 & 1 & 1 & 0 & -1 & 1 & 1 & 0 & -1 & 2 & 2 & 1 & 0 \\\\-1 & 0 & 0 & -2 & -2 & 1 & 1 & -1 & -1 & 1 & 1 & 0 & -1 \\\\-1 & -1 & 0 & -2 & -2 & -1 & 0 & -2 & -2 & 0 & 0 & -1 & -1 \\\\\\end{array}\\right],\\quad \\lambda = \\left[\\begin{array}{c}1\\\\a^2 q^{-2} (-t)\\\\1\\\\q^{-4} (-t)^{-2}\\\\a^{-2} q^{-2} (-t)^{-3}\\\\a^2 (-t)^2\\\\q^2 (-t)\\\\q^{-2} (-t)^{-1}\\\\a^{-2} (-t)^{-2}\\\\a^2 q^2 (-t)^3\\\\q^{4} (-t)^2\\\\1\\\\a^{-2} q^{2} (-t)^{-1}\\end{array}\\right].$ For (REF ) there are six local symmetries for the following pairings: $\\lambda _2\\lambda _8&=\\lambda _4\\lambda _6, &\\lambda _2\\lambda _{12}&=\\lambda _4\\lambda _{10}, &\\lambda _3\\lambda _8&=\\lambda _4\\lambda _7, \\\\\\lambda _3\\lambda _9&=\\lambda _5\\lambda _7, &\\lambda _3\\lambda _{13}&=\\lambda _5\\lambda _{11}, &\\lambda _6\\lambda _{13}&=\\lambda _8\\lambda _{11},$ which graphical representation, together with the homology diagram, is given in figure REF .", "Figure: Homology diagram and local symmetries for 6 3 6_3 knot." ], [ "$7_3$ knot", "As the last isolated example we consider the $7_3$ knot.", "The quiver from [9] reads $C = \\left[\\begin{array}{ccccccccccccc}2 & 0 & 3 & 2 & 1 & 5 & 4 & 3 & 3 & 2 & 5 & 4 & 3 \\\\0 & 0 & 1 & 1 & 0 & 3 & 3 & 2 & 1 & 1 & 3 & 3 & 2 \\\\3 & 1 & 4 & 2 & 2 & 5 & 4 & 4 & 4 & 2 & 5 & 4 & 4 \\\\2 & 1 & 2 & 2 & 1 & 3 & 3 & 3 & 3 & 2 & 3 & 3 & 3 \\\\1 & 0 & 2 & 1 & 1 & 3 & 2 & 2 & 2 & 1 & 3 & 2 & 2 \\\\5 & 3 & 5 & 3 & 3 & 6 & 4 & 4 & 6 & 4 & 6 & 4 & 4 \\\\4 & 3 & 4 & 3 & 2 & 4 & 4 & 3 & 5 & 4 & 5 & 4 & 3 \\\\3 & 2 & 4 & 3 & 2 & 4 & 3 & 3 & 4 & 3 & 5 & 4 & 3 \\\\3 & 1 & 4 & 3 & 2 & 6 & 5 & 4 & 5 & 3 & 6 & 5 & 4 \\\\2 & 1 & 2 & 2 & 1 & 4 & 4 & 3 & 3 & 3 & 4 & 4 & 3 \\\\5 & 3 & 5 & 3 & 3 & 6 & 5 & 5 & 6 & 4 & 7 & 5 & 5 \\\\4 & 3 & 4 & 3 & 2 & 4 & 4 & 4 & 5 & 4 & 5 & 5 & 4 \\\\3 & 2 & 4 & 3 & 2 & 4 & 3 & 3 & 4 & 3 & 5 & 4 & 4 \\\\\\end{array}\\right],\\qquad \\qquad \\lambda = \\left[\\begin{array}{c}a^6 q^{-4} (-t)^2\\\\a^4 q^{-4} \\\\a^6 (-t)^4\\\\a^4 (-t)^2\\\\a^{4} q^{-2} (-t)\\\\a^6 q^4 (-t)^6\\\\a^4 q^4 (-t)^4\\\\a^4 q^2 (-t)^3\\\\a^{8} q^{-2} (-t)^5\\\\a^{6} q^{-2} (-t)^3\\\\a^{8} q^{2} (-t)^7\\\\a^{6} q^{2} (-t)^5\\\\a^{6} (-t)^4\\\\\\end{array}\\right].$ For (REF ) there are seven local symmetries for the following pairings: $\\lambda _1\\lambda _{10}&=\\lambda _2\\lambda _9, &\\lambda _2\\lambda _{11}&=\\lambda _3\\lambda _{10}, &\\lambda _3\\lambda _{10}&=\\lambda _4\\lambda _9, &\\lambda _3\\lambda _{12}&=\\lambda _4\\lambda _{11}, \\\\\\lambda _4\\lambda _{13}&=\\lambda _5\\lambda _{12}, &\\lambda _6\\lambda _{12}&=\\lambda _7\\lambda _{11}, &\\lambda _7\\lambda _{13}&=\\lambda _8\\lambda _{12}.", "&$ Their graphical representation, together with the homology diagram, is given in figure REF .", "Figure: Homology diagram and local symmetries for 7 3 7_3 knot." ], [ "$F_K(x,a,q)$ invariants and knot complement quivers", "In the last section we broaden our perspective and show that the equivalence criteria from theorem REF can be used to relate quivers that we considered so far to another type of quivers, which in [11] have been associated to $F_K(x,a,q)$ invariants of knot complements, constructed in [15], [16], [17].", "In this section we focus on $T_{2,2p+1}$ torus knots and show that for each $p$ , a quiver associated to $F_K(x,a,q)$ invariant is equivalent to a subquiver of a quiver for unreduced colored HOMFLY-PT polynomials construced in [2].", "Before presenting this relation, let us recall how the knots-quivers correspondence works in the unreduced normalization (which we denote by adding a bar to all quantities) defined for HOMFLY-PT generating functions by $\\bar{P}_{K}(x,a,q)=\\sum _{r=0}^{\\infty }x^r a^{-r}q^r \\frac{(a^2;q^2)_r}{(q^2;q^2)_r}P_{r}(a,q).$ The presence of $(a^2;q^2)_r$ in the numerator in the summand (relative to the reduced normalization) implies that the unreduced quiver matrix $\\bar{C}_{IJ}$ can be obtained from the reduced one (given by $C_{ij}$ ) by the following relation [2]In our convention $\\alpha _i \\leftrightarrow \\beta _i$ with respect to [2].", ": $\\sum _{I, J = 1}^{2 m} \\bar{C}_{IJ} \\bar{d}_I \\bar{d}_J = \\sum _{i, j = 1}^m \\left[C_{ij} \\alpha _i \\alpha _j + (C_{ij} + 1) \\beta _i \\beta _j \\right] + 2 \\sum _{i \\le j} C_{ij} \\alpha _i \\beta _j + 2\\sum _{i < j} (C_{ij} + 1) \\alpha _i \\beta _j,$ where $\\alpha _i$ and $\\beta _i$ are the new summation indices for the quiver motivic generating series.", "They are related to the summation indices of the reduced normalization by $d_i = \\alpha _i + \\beta _i$ and $\\bar{d}_I$ can be thought of as the entries of a vector $\\bar{d} = (\\alpha _1, \\alpha _2, \\dots , \\alpha _m, \\beta _1, \\beta _2, \\dots , \\beta _m).$ Then the unreduced quiver matrix takes the form of a $2 m \\times 2 m$ block matrix $\\bar{C} = \\left[\\begin{array}{l|r}C \\; & \\; C \\\\\\hline C & C\\end{array}\\right]+\\left[\\begin{array}{l|r}0 \\; & \\; 0 \\\\\\hline 0 & 1\\end{array}\\right]+\\left[\\begin{array}{c | c}\\; 0 \\; & \\; \\theta \\\\\\hline \\theta ^T & \\; 0\\end{array}\\right]\\begin{array}{l}\\rbrace \\alpha \\\\\\rbrace \\beta \\\\\\end{array}$ where 1 and 0 are the matrices with only ones or zeros respectively, and the matrix $\\theta $ is defined as $\\theta _{ij} = {\\left\\lbrace \\begin{array}{ll}0, & j \\ge i \\\\1, & j < i\\end{array}\\right.", "}\\quad {\\rm with} \\quad i, \\, j = 1, 2, \\dots , m.$ Note that going from $d_i$ to $\\alpha _i$ and $\\beta _i$ can be understood as an example of splitting.", "It follows from the fact that switching between the reduced and unreduced normalization corresponds to multiplication by $a^{-r}q^r (a^2;q^2)_r$ .", "Since $r=\\sum _i d_i$ , we split all nodes, and $a^{-r}q^r$ enters the change of variables.", "The only difference with splitting presented in section  lies in the ordering.", "There we put $\\alpha _i$ next to $\\beta _i$ , here we start from all alphas and then write all betas to match the convention in [2]." ], [ "Trefoil knot complement", "Let us focus on the simplest example of the trefoil.", "The “standard\" and knot complement quivers are given by: $\\bar{C}_{3_1} =\\left[\\begin{array}{ccc|ccc}0 & 1 & 1 & 0 & 2 & 2 \\\\1 & 2 & 2 & 1 & 2 & 3 \\\\1 & 2 & 3 & 1 & 2 & 3 \\\\\\hline 0 & 1 & 1 & 1 & 2 & 2 \\\\2 & 2 & 2 & 2 & 3 & 3 \\\\2 & 3 & 3 & 2 & 3 & 4\\end{array}\\right],\\qquad \\qquad C_{F_{3_1}} =\\left[\\begin{array}{cccc}3 & 2 & 3 & 2 \\\\2 & 2 & 3 & 2 \\\\3 & 3 & 4 & 3 \\\\2 & 2 & 3 & 3\\end{array}\\right].$ Let us exchange $x_2 \\leftrightarrow x_4$ in $\\bar{C}_{3_1}$ and then remove the first pair of nodes (interestingly, they look like the redundant pair of nodes [4], but they have a different change of variables).", "After relabeling its vertices to $(x^{\\prime }_1,x^{\\prime }_2,x^{\\prime }_3,x^{\\prime }_4)$ , we permute it into $(x^{\\prime }_3,x^{\\prime }_2,x^{\\prime }_4,x^{\\prime }_1)$ .", "This gives: $\\begin{array}{c}x_1 \\\\x_2 \\\\x_3 \\\\x_4 \\\\x_5 \\\\x_6\\end{array}\\left[\\begin{array}{ccc|ccc}0 & 1 & 1 & 0 & 2 & 2 \\\\1 & 2 & 2 & 1 & 2 & 3 \\\\1 & 2 & 3 & 1 & 2 & 3 \\\\\\hline 0 & 1 & 1 & 1 & 2 & 2 \\\\2 & 2 & 2 & 2 & 3 & 3 \\\\2 & 3 & 3 & 2 & 3 & 4\\end{array}\\right]\\rightsquigarrow \\begin{array}{c}x_1 \\\\x_4 \\\\x_3 \\\\x_2 \\\\x_5 \\\\x_6\\end{array}\\left[\\begin{array}{cccccc}0 & 0 & 1 & 1 & 2 & 2 \\\\0 & 1 & 1 & 1 & 2 & 2 \\\\1 & 1 & 3 & 2 & 2 & 3 \\\\1 & 1 & 2 & 2 & 2 & 3 \\\\2 & 2 & 2 & 2 & 3 & 3 \\\\2 & 2 & 3 & 3 & 3 & 4\\end{array}\\right]\\rightsquigarrow \\begin{array}{c}x^{\\prime }_1 \\\\x^{\\prime }_2 \\\\x^{\\prime }_3 \\\\x^{\\prime }_4\\end{array}\\left[\\begin{array}{cccccc}3 & 2 & 2 & 3 \\\\2 & 2 & 2 & 3 \\\\2 & 2 & 3 & 3 \\\\3 & 3 & 3 & 4\\end{array}\\right]\\rightsquigarrow \\begin{array}{c}x^{\\prime }_3 \\\\x^{\\prime }_2 \\\\x^{\\prime }_4 \\\\x^{\\prime }_1\\end{array}\\left[\\begin{array}{cccc}3 & 2 & 3 & 2 \\\\2 & 2 & 3 & 2 \\\\3 & 3 & 4 & 3 \\\\2 & 2 & 3 & 3\\end{array}\\right].$ After framing by $-3$ , the rightmost quiver in (REF ) agrees with the quiver associated to the trefoil complement in [11].", "We can also illustrate this relation at the level of formulas.", "The $F_K$  invariant reads [17] $\\begin{split}F_{3_{1}}(x,a,q)&=\\sum _{k=0}^{\\infty }x^{k} q^{2k}\\frac{(x;q^{-2})_{k}(a^2q^{-2};q^2)_{k}}{(q^2;q^2)_{k}}\\\\&=\\sum _{k=0}^{\\infty }x^{2k} q^{3k}\\frac{(x^{-1};q^{2})_{k}(a^2q^{-2};q^2)_{k}}{(q^2;q^2)_{k}}(-q)^{-k^2}.\\end{split}$ On the other hand, the unreduced HOMFLY-PT generating function is given by [28] $\\begin{split}\\bar{P}_{3_1}(x,a,q)&=\\sum _{r=0}^{\\infty }x^ra^rq^{-r} \\frac{(a^2;q^2)_r}{(q^2;q^2)_r}\\sum _{k=0}^{r} \\left[\\begin{array}{c}r\\\\k\\end{array}\\right] q^{2k(r+1)} (a^2q^{-2};q^2)_k\\\\&=\\sum _{k=0}^{\\infty }x^{k}a^{k}q^{k} \\frac{(a^2;q^2)_k (a^2q^{-2};q^2)_k}{(q^2;q^2)_k} q^{2k^2} \\sum _{l=0}^{\\infty }x^{l}a^{l}q^{-l}\\frac{(a^2 q^{2k};q^2)_l }{(q^2;q^2)_l } q^{2kl}.\\end{split}$ Comparing (REF ) with (REF ), we can see that the structure of $q$ -Pochhammers indexed by $k$ is exactly the same.", "The net difference $-3k^2$ in $q$ powers corresponds to the framing change, whereas all powers linear in $k$ enter the change of variables and do not interfere with the general structure.", "Finally, the whole sum over $l=r-k$ contributes to the removed pair of nodes." ], [ "Cinquefoil knot complement", "For the knot $5_1$ , the two quivers are given by $\\bar{C}_{5_1} =\\left[\\begin{array}{ccccc|ccccc}0 & 1 & 1 & 3 & 3 & 0 & 2 & 2 & 4 & 4 \\\\1 & 2 & 2 & 3 & 3 & 1 & 2 & 3 & 4 & 4 \\\\1 & 2 & 3 & 4 & 4 & 1 & 2 & 3 & 5 & 5 \\\\3 & 3 & 4 & 4 & 4 & 3 & 3 & 4 & 4 & 5 \\\\3 & 3 & 4 & 4 & 5 & 3 & 3 & 4 & 4 & 5 \\\\\\hline 0 & 1 & 1 & 3 & 3 & 1 & 2 & 2 & 4 & 4 \\\\2 & 2 & 2 & 3 & 3 & 2 & 3 & 3 & 4 & 4 \\\\2 & 3 & 3 & 4 & 4 & 2 & 3 & 4 & 5 & 5 \\\\4 & 4 & 5 & 4 & 4 & 4 & 4 & 5 & 5 & 5 \\\\4 & 4 & 5 & 5 & 5 & 4 & 4 & 5 & 5 & 6\\end{array}\\right],\\qquad \\qquad C_{F_{5_1}} = \\left[\\begin{array}{cccccccc}5 & 4 & 5 & 4 & 4 & 4 & 5 & 4 \\\\4 & 4 & 5 & 4 & 3 & 3 & 5 & 4 \\\\5 & 5 & 6 & 5 & 4 & 4 & 5 & 5 \\\\4 & 4 & 5 & 5 & 3 & 3 & 4 & 4 \\\\4 & 3 & 4 & 3 & 3 & 2 & 3 & 2 \\\\4 & 3 & 4 & 3 & 2 & 2 & 3 & 2 \\\\5 & 5 & 5 & 4 & 3 & 3 & 4 & 3 \\\\4 & 4 & 5 & 4 & 2 & 2 & 3 & 3\\end{array}\\right]$ We repeat similar steps as in the trefoil case, exchanging $x_2 \\leftrightarrow x_6$ in $\\bar{C}_{5_1}$ and permuting $(x^{\\prime }_1,x^{\\prime }_2,x^{\\prime }_3,x^{\\prime }_4,x^{\\prime }_5,x^{\\prime }_6,x^{\\prime }_7,x^{\\prime }_8) \\mapsto (x^{\\prime }_7,x^{\\prime }_2,x^{\\prime }_8,x^{\\prime }_3,x^{\\prime }_5,x^{\\prime }_4,x^{\\prime }_6,x^{\\prime }_1)$ to obtain: $\\begin{array}{c}x_1 \\\\x_2 \\\\x_3 \\\\x_4 \\\\x_5 \\\\x_6 \\\\x_7 \\\\x_8 \\\\x_9 \\\\x_{10}\\end{array}\\left[\\begin{array}{ccccc|ccccc}0 & 1 & 1 & 3 & 3 & 0 & 2 & 2 & 4 & 4 \\\\1 & 2 & 2 & 3 & 3 & 1 & 2 & 3 & 4 & 4 \\\\1 & 2 & 3 & 4 & 4 & 1 & 2 & 3 & 5 & 5 \\\\3 & 3 & 4 & 4 & 4 & 3 & 3 & 4 & 4 & 5 \\\\3 & 3 & 4 & 4 & 5 & 3 & 3 & 4 & 4 & 5 \\\\\\hline 0 & 1 & 1 & 3 & 3 & 1 & 2 & 2 & 4 & 4 \\\\2 & 2 & 2 & 3 & 3 & 2 & 3 & 3 & 4 & 4 \\\\2 & 3 & 3 & 4 & 4 & 2 & 3 & 4 & 5 & 5 \\\\4 & 4 & 5 & 4 & 4 & 4 & 4 & 5 & 5 & 5 \\\\4 & 4 & 5 & 5 & 5 & 4 & 4 & 5 & 5 & 6\\end{array}\\right]\\rightsquigarrow \\begin{array}{c}x^{\\prime }_1 \\\\x^{\\prime }_2 \\\\x^{\\prime }_3 \\\\x^{\\prime }_4 \\\\x^{\\prime }_5 \\\\x^{\\prime }_6 \\\\x^{\\prime }_7 \\\\x^{\\prime }_8\\end{array}\\left[\\begin{array}{cccccccc}3 & 4 & 4 & 2 & 2 & 3 & 5 & 5 \\\\4 & 4 & 4 & 3 & 3 & 4 & 4 & 5 \\\\4 & 4 & 5 & 3 & 3 & 4 & 4 & 5 \\\\2 & 3 & 3 & 2 & 2 & 3 & 4 & 4 \\\\2 & 3 & 3 & 2 & 3 & 3 & 4 & 4 \\\\3 & 4 & 4 & 3 & 3 & 4 & 5 & 5 \\\\5 & 4 & 4 & 4 & 4 & 5 & 5 & 5 \\\\5 & 5 & 5 & 4 & 4 & 5 & 5 & 6\\end{array}\\right]\\rightsquigarrow \\begin{array}{c}x^{\\prime }_7 \\\\x^{\\prime }_2 \\\\x^{\\prime }_8 \\\\x^{\\prime }_3 \\\\x^{\\prime }_5 \\\\x^{\\prime }_4 \\\\x^{\\prime }_6 \\\\x^{\\prime }_1\\end{array}\\left[\\begin{array}{cccccccc}{5} & 4 & 5 & 4 & 4 & 4 & 5 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {5};\\phantom{5} \\\\4 & {4} & 5 & 4 & 3 & 3 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4} & 4\\\\5 & 5 & 6 & 5 & 4 & 4 & 5 & 5 \\\\4 & 4 & 5 & 5 & 3 & 3 & 4 & 4 \\\\4 & 3 & 4 & 3 & 3 & 2 & 3 & 2 \\\\4 & 3 & 4 & 3 & 2 & 2 & 3 & 2 \\\\5 & [overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {4};\\phantom{4} & 5 & 4 & 3 & 3 & {4} & 3 \\\\[overlay]\\node [fill=yellow!50,draw=black,inner sep=2pt, anchor=text, rectangle, rounded corners=1mm] {5};\\phantom{5} & 4 & 5 & 4 & 2 & 2 & 3 & {3}\\end{array}\\right]$ If we now subtract the result from $C_{F_{5_1}}$ , we get $\\left[\\begin{array}{cccc|cccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$ The two quivers would agree if we swap $C_{2,7}\\leftrightarrow C_{1,8}$ in the rightmost matrix of (REF ).", "Fortunately, it turns out to be an example of the quiver equivalence from theorem REF , so the relation between two kinds of quivers holds." ], [ "General $T_{2,2p+1}$ knot complement", "We now compare the two recursive formulas for $T_{2,2p+1}$ torus knots.", "Starting with the “standard\" quiver defined using unreduced colored HOMFLY-PT polynomials $\\bar{C}_{T_{2,2p+1}}$ , we propose an algorithm of transforming it into a quiver $C_{F_{T_{2,2p+1}}}$ associated to the respective knot complement: Label the vertices of $\\bar{C}_{T_{2,2p+1}}$ upside down as $x_1,\\dots ,x_{4p+2}$ .", "Exchange $x_2 \\leftrightarrow x_{2p+2}$ .", "Remove the first two nodes $(x_1,x_{2p+2})$ with the smallest number of self-loops.", "The resulting diagonal is of the form: $(3,\\dots ,2p+1,\\ 2,3,\\dots ,2p+1,2p+2)$ .", "Relabel these entries as $(x^{\\prime }_1,\\dots ,x^{\\prime }_{4p})$ .", "Permute the $x^{\\prime }_i$ 's: $(x^{\\prime }_1,\\dots ,x^{\\prime }_{4p}) \\mapsto (x^{\\prime }_{4p-1},\\dots , x^{\\prime }_1)$ in order to get the diagonal: $(2p+1, 2p, 2p+2, 2p+1,\\ \\dots \\ ,\\ 3, 2, 4, 3)$ .", "Such permutation is fixed uniquely for each $p$ .", "For example, $p = 1,2,3$ leads respectively to: $\\begin{aligned}(x^{\\prime }_1,\\dots , x^{\\prime }_4) \\mapsto & \\ (x^{\\prime }_4, x^{\\prime }_2, x^{\\prime }_3, x^{\\prime }_1),\\\\(x^{\\prime }_1,\\dots , x^{\\prime }_8) \\mapsto & \\ (x^{\\prime }_7,x^{\\prime }_2,x^{\\prime }_8,x^{\\prime }_3,x^{\\prime }_5,x^{\\prime }_4,x^{\\prime }_6,x^{\\prime }_1), \\\\(x^{\\prime }_1, \\dots , x^{\\prime }_{12}) \\mapsto & \\ (x^{\\prime }_{11}, x^{\\prime }_4, x^{\\prime }_{12}, x^{\\prime }_5, x^{\\prime }_9, x^{\\prime }_2, x^{\\prime }_{10}, x^{\\prime }_3, x^{\\prime }_7, x^{\\prime }_6, x^{\\prime }_8, x^{\\prime }_1).", "\\\\\\end{aligned}$ Figure: The block structure and transpositions that relate the “standard\" sub-quiver based on unreduced HOMFLY-PT polynomials for T 2,2p+1 T_{2,2p+1} torus knots, to the knot complement quiver (only the upper part is shown, since it is symmetric).After these steps, we compare the resulting quiver matrix to $C_{F_{T_{2,2p+1}}}$ .", "It turns out that the results almost agree, up to transpositions of certain non-diagonal entries, indicated in figure REF .", "Each block in this figure has size $4\\times 4$ : the diagonal blocks represent framed knot complement quivers for the trefoil, while the off-diagonal part differs from them by a transposition of elements, each time appearing in the top-right corner of each upper-diagonal block, and extending to lower diagonal blocks by symmetry.", "This suggests that the two formulas agree, up to the quiver equivalence relation.", "Another argument comes from the fact that transforming the quiver from reduced to unreduced normalization corresponds to splitting all nodes, which (as discussed in section ) can be done in many ways, all of which lead to equivalent quivers.", "We checked that transpositions depicted in figure REF are indeed symmetries for $T_{2,2p+1}$ torus knots up to $p=3$ .", "We conjecture that it is always the case, which means that in the equivalence class of quivers corresponding to the $T_{2,2p+1}$ torus knot in the unreduced normalization there exists a representative such that the knot complement quiver is its subquiver." ], [ "Acknowledgements", "We thank Tobias Ekholm, Angus Gruen, Sergei Gukov, Pietro Longhi, Sunghyuk Park, and Marko Stošić for insightful discussions and comments on the manuscript.", "The work of J.J. was supported by the Polish National Science Centre (NCN) grant 2016/23/D/ST2/03125.", "P.K.", "is supported by the Polish Ministry of Education and Science through its programme Mobility Plus (decision no. 1667/MOB/V/2017/0).", "The work of H.L., D.N.", "and P.S.", "is supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00)." ], [ "Equivalent quivers for knots $5_2$ and {{formula:9536aaac-cf72-45e7-8363-62ee2768cbc2}}", "In this appendix we present equivalent quivers that we found for knots $5_2$ and $7_1$ .", "Quiver matrices given below correspond to appropriate vertices in the permutohedra graphs, as indicated by their labels; the same labeling is used in the attached Mathematica file." ], [ "Quiver matrices for twist knots", "In this appendix we provide quiver matrices for twist knots, which were found in [2].", "Interestingly, for each of the two families of twist knots $TK_{2|p|+2}$ and $TK_{2p+1}$ , such a matrix can be presented in a universal way.", "The quiver matrix for $TK_{2|p|+2}$ twist knot found in [2] takes form $C^{TK_{2|p|+2}}=\\left[\\begin{array}{ccccccc}F_{0} & F & F & F & \\cdots & F & F\\\\F^{T} & D_{1} & R_{1} & R_{1} & \\cdots & R_{1} & R_{1}\\\\F^{T} & R_{1}^{T} & D_{2} & R_{2} & \\cdots & R_{2} & R_{2}\\\\F^{T} & R_{1}^{T} & R_{2}^{T} & D_{3} & \\cdots & R_{3} & R_{3}\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\F^{T} & R_{1}^{T} & R_{2}^{T} & R_{3}^{T} & \\cdots & D_{|p|-1} & R_{|p|-1}\\\\F^{T} & R_{1}^{T} & R_{2}^{T} & R_{3}^{T} & \\cdots & R_{|p|-1}^{T} & D_{|p|}\\end{array}\\right],$ where $F_{0}=\\left[0\\right],\\qquad \\qquad F=\\left[\\begin{array}{cccc}0 & -1 & 0 & -1\\end{array}\\right],$ and $D_{k}=\\left[\\begin{array}{cccc}2k & 2k-2 & 2k-1 & 2k-3\\\\2k-2 & 2k-3 & 2k-2 & 2k-4\\\\2k-1 & 2k-2 & 2k-1 & 2k-3\\\\2k-3 & 2k-4 & 2k-3 & 2k-4\\end{array}\\right], \\qquad R_{k}=\\left[\\begin{array}{cccc}2k & 2k-2 & 2k-1 & 2k-3\\\\2k-1 & 2k-3 & 2k-2 & 2k-4\\\\2k & 2k-1 & 2k-1 & 2k-3\\\\2k-2 & 2k-3 & 2k-2 & 2k-4\\end{array}\\right].$ The element $F_0$ represents a zig-zag of length 1, i.e.", "a single homology generator, while the diagonal blocks $D_k$ represent diamonds (up to a permutation of homology generators and an overall shift).", "The identification with $\\lambda _i$ 's in figure REF is as follows: $\\begin{array}{c|cccc}& [rgb]{0.82,0.01,0.11}{\\lambda _{4r-2}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r-1}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r+1}}\\\\\\hline [rgb]{0.82,0.01,0.11}{\\lambda _{4r-2}} & 2r & 2r-2 & 2r-1 & 2r-3\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r-1}} & 2r-2 & 2r-3 & 2r-2 & 2r-4\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r}} & 2r-1 & 2r-2 & 2r-1 & 2r-3\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r+1}} & 2r-3 & 2r-4 & 2r-3 & 2r-4\\end{array} \\qquad \\begin{array}{c|cccc}& [rgb]{0.82,0.01,0.11}{\\lambda _{4r-2}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r-1}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r}} & [rgb]{0.82,0.01,0.11}{\\lambda _{4r+1}}\\\\\\hline [rgb]{0.82,0.01,0.11}{\\lambda _{4r^{\\prime }-2}} & 2r & 2r-1 & 2r & 2r-2\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r^{\\prime }-1}} & 2r-2 & 2r-3 & 2r-1 & 2r-3\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r^{\\prime }}} & 2r-1 & 2r-2 & 2r-1 & 2r-2\\\\[rgb]{0.82,0.01,0.11}{\\lambda _{4r^{\\prime }+1}} & 2r-3 & 2r-4 & 2r-3 & 2r-4\\end{array}$ This means that $D_k$ encodes interactions of nodes nodes within one diamond, while $R_k$ encodes interactions of nodes from two diamonds labelled by $r,r^{\\prime }$ .", "Quiver matrices for $TK_{2p+1}$ twist knots found in [2] read $C^{TK_{2p+1}}=\\left[\\begin{array}{ccccccc}D_{1} & R_{1} & R_{1} & R_{1} & \\cdots & R_{1} & R_{1}\\\\R_{1}^{T} & D_{2} & R_{2} & R_{2} & \\cdots & R_{2} & R_{2}\\\\R_{1}^{T} & R_{2}^{T} & D_{3} & R_{3} & \\cdots & R_{3} & R_{3}\\\\R_{1}^{T} & R_{2}^{T} & R_{3}^{T} & D_{4} & \\cdots & R_{4} & R_{4}\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\R_{1}^{T} & R_{2}^{T} & R_{3}^{T} & R_{4}^{T} & \\cdots & D_{p-1} & R_{p-1}\\\\R_{1}^{T} & R_{2}^{T} & R_{3}^{T} & R_{4}^{T} & \\cdots & R_{p-1}^{T} & D_{p}\\end{array}\\right],$ where the block elements in the first row and column are $D_{1}=\\left[\\begin{array}{ccc}2 & 1 & 2\\\\1 & 0 & 1\\\\2 & 1 & 3\\end{array}\\right],\\qquad \\qquad R_{1}=\\left[\\begin{array}{cccc}1 & 2 & 1 & 2\\\\0 & 2 & 0 & 1\\\\1 & 3 & 2 & 3\\end{array}\\right], $ and all other elements, for $k>1$ , take the form $D_{k}=\\left[\\begin{array}{cccc}2k-3 & 2k-2 & 2k-3 & 2k-2\\\\2k-2 & 2k & 2k-1 & 2k\\\\2k-3 & 2k-1 & 2k-2 & 2k-1\\\\2k-2 & 2k & 2k-1 & 2k+1\\end{array}\\right], \\qquad \\quad R_{k}=\\left[\\begin{array}{cccc}2k-3 & 2k-2 & 2k-3 & 2k-2\\\\2k-1 & 2k & 2k-1 & 2k \\\\2k-2 & 2k & 2k-2 & 2k-1\\\\2k-1 & 2k+1 & 2k & 2k+1\\end{array}\\right].$ In this case $D_1$ represents a zig-zag of the same form as for the trefoil knot, and $D_k$ (for $k>1$ ) represent diamonds (up to a permutation of homology generators and an overall constant shift)." ] ]
2105.11806
[ [ "Efimov effect evaporation after confinement" ], [ "Abstract The continuous confinement of quantum systems can be described by means of the $d$-method, where the dimension $d$ is taken as a continuous parameter.", "In this work we describe in detail how this method can be used to obtain the root mean square radii for a squeezed three-body system.", "These observables are used to investigate the disappearance of the Efimov states around the two-body threshold during a progressive confinement of the system from three to two dimensions.", "We illustrate how the disappearance takes place through the loss of one of the particles, whereas the other two remain bound." ], [ "Efimov effect evaporation after confinementThis work has been partially supported by the Spanish Ministry of Science, Innovation and University MCIU/AEI/FEDER,UE (Spain) under Contract No.", "PGC2018-093636-B-I00.", "E. Garrido A.S. Jensen E. Garrido Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain [email protected] A.S: JensenDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark The continuous confinement of quantum systems can be described by means of the $d$ -method, where the dimension $d$ is taken as a continuous parameter.", "In this work we describe in detail how this method can be used to obtain the root mean square radii for a squeezed three-body system.", "These observables are used to investigate the disappearance of the Efimov states around the two-body threshold during a progressive confinement of the system from three to two dimensions.", "We illustrate how the disappearance takes place through the loss of one of the particles, whereas the other two remain bound.", "The properties of few-body systems are to a large extent determined by the dimension of the space where they are allowed to move.", "The simplest example is perhaps the case of a two-body system, always bound in two dimensions (2D) by any infinitesimally small attractive potential, but not necessarily bound in three dimensions (3D) due to the different behavior of the centrifugal barrier [1], [2].", "This fact has two immediate consequences.", "The first one is that the Efimov effect, present in three-body systems where at least two of the three pair-interactions have nearly zero energy [3], does not exist in 2D [4], [5], [6], [7], [8].", "The second one is that, when confining an unbound two-body system from 3D to 2D, there must necessarily be a point in the confinement process where the two-body system is precisely at the zero-energy threshold.", "From this last result one can also conclude that, given an unbound three-body system containing at least two identical particles, when confining from 3D to 2D there will always be a point in the confinement process where the Efimov conditions are fulfilled.", "This fact has recently been investigated in Ref.", "[9], where the confinement of the unbound three-body state was implemented by using the dimension $d$ as a parameter that moves continuously from $d=3$ to $d=2$ , in such a way that for some particular value, $d=d_E$ , the Efimov effect shows up.", "One of the main results in [9] is that the Efimov states appear and disappear around the two-body threshold extremely fast within a very small dimension interval around $d=d_E$ .", "The equivalence between the $d$ -method, using $d$ as a continuous parameter, and the more natural approach of confining the system by means of an external squeezing potential included explicitly in a 3D calculation, has been investigated in Refs.", "[10], [11], [12].", "The main conclusion is that it is possible to establish a connection between a non-integer value of $d$ and a specific strength of the squeezing potential.", "In this way, the $d$ -method, much simpler to implement numerically, appears as an efficient tool to describe confined systems.", "The price to pay when using the $d$ -formalism is that the calculation of a given observable, unavoidably measured in an integer-dimensional space, requires a translation of the $d$ -dimensional wave function into the ordinary 3D space.", "This is done by considering the $d$ -wave function as a wave function in the ordinary 3D space, but deformed along the squeezing direction [10], [11], [12].", "The purpose of this letter is twofold.", "First, to clarify how the three-body wave functions obtained within the $d$ -method can in practice be used to compute observables, in particular root mean square radii, which are very relevant in order to understand the spatial distribution of the three constituents.", "The squeezing of the system along the $z$ -axis (3D to 2D squeezing) will be considered.", "And second, to employ these observables to visualize how, during the squeezing process from 3D to 2D, the disappearance of the Efimov states close to the two-body threshold does actually take place.", "The results will be illustrated with one of the systems detailed in Ref.", "[9], namely the case of two heavy and one light particles with mass ratio $m_H/m_L=133/6$ , and with heavy-light potential such that $d_E=2.75$ .", "A three-body system in a $d$ -dimensional space has, after removal of the center of mass motion, $2d$ degrees of freedom, which can be described by means of the well-known Jacobi coordinates, $\\mathbf {x}$ and $\\mathbf {y}$ .", "As usual, from these coordinates one can construct the hyperradius, $\\rho =(x^2+y^2)^{1/2}$ , and the hyperangle, $\\alpha =\\arctan (x/y)$ , which, together with the $2(d-1)$ angles defining the directions of $\\mathbf {x}$ and $\\mathbf {y}$ , constitute the $2d$ hyperspherical coordinates necessary to describe the system.", "Using these coordinates, the $d$ -dimensional wave function can be obtained by means of the hyperspherical adiabatic expansion method, described in detail in Ref.", "[13], in such a way that the wave function is written as: $\\Psi _d(\\rho ,\\Omega _d)=\\frac{1}{\\rho ^{\\frac{2d-1}{2}}} \\sum _n f_n^{(d)}(\\rho ) \\Phi _n^{(d)}(\\rho , \\Omega _d),$ where $\\Omega _d$ collects the $2d-1$ hyperangles.", "The angular functions $\\Phi _n^{(d)}(\\rho , \\Omega _d)$ are the eigenfunctions of the angular part of the Faddeev (or Schrödinger) equations, whereas the functions $f_n^{(d)}(\\rho )$ are obtained after solving a coupled set of radial, Schrödinger-like, differential equations where the eigenvalues of the angular part enter as effective potentials.", "As already mentioned, the use Eq.", "(REF ) in order to compute whatever observable, requires a translation of the $\\Psi _d$ wave function into the 3D space in which the observable is measured.", "This connection is done by considering the $d$ -wave function (REF ) as a wave function in the ordinary 3D space, but deformed along the squeezing direction (that we choose along the $z$ -axis).", "In other words, the Jacobi coordinates, $\\mathbf {x}$ and $\\mathbf {y}$ in the $d$ -space, will be taken as ordinary 3D vectors, written as $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ .", "Each of them has therefore the usual Cartesian components, where $\\tilde{x}_z$ and $\\tilde{y}_z$ shall denote the corresponding components along the $z$ -axis, and $\\tilde{x}_\\perp $ and $\\tilde{y}_\\perp $ the projection of $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ on the $xy$ -plane.", "The key point is that in order to incorporate the deformation produced by the external field, the $\\tilde{x}_z$ and $\\tilde{y}_z$ components along the squeezing direction do not represent the actual value of the coordinate, but instead, the actual value deformed by means of a scale parameter $s$ , i.e., $\\tilde{x}_z=x_z/s$ and $\\tilde{y}_z=y_z/s$ .", "In the same way, the perpendicular components along the non-squeezed directions do provide the true value along these directions, i.e., $\\tilde{x}_\\perp =x_\\perp $ and $\\tilde{y}_\\perp =y_\\perp $ .", "Therefore, for squeezing along the $z$ -axis we can write: $\\tilde{x}^2=\\tilde{x}_\\perp ^2+\\tilde{x}_z^2=x_\\perp ^2+\\frac{x_z^2}{s^2},$ $\\tilde{y}^2=\\tilde{y}_\\perp ^2+\\tilde{y}_z^2=y_\\perp ^2+\\frac{y_z^2}{s^2},$ where the scale parameter $s$ can take values within the range $0 \\le s \\le 1$ .", "For $s=0$ only the values $x_z=0$ and $y_z=0$ are allowed (otherwise the value of the coordinates diverges), and the system is fully squeezed into the 2D plane.", "For $s=1$ we have $\\tilde{x}^2=x_\\perp +x_z^2$ and $\\tilde{y}^2=y_\\perp +y_z^2$ , and $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ become the usual coordinates in a non-squeezed space.", "Equations (REF ) and (REF ) can be easily generalized to other squeezing scenarios, like from 3 to 1 dimensions, or from 2 to 1 dimensions [11].", "Using the 3D $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ Jacobi coordinates we can construct the usual 3D hyperspherical coordinates, which together with the angles $\\Omega _{\\tilde{x}} \\equiv \\lbrace \\theta _{\\tilde{x}}, \\varphi _{\\tilde{x}} \\rbrace $ and $\\Omega _{\\tilde{y}} \\equiv \\lbrace \\theta _{\\tilde{y}}, \\varphi _{\\tilde{y}} \\rbrace $ giving the directions of $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ , contain the hyperradius $\\tilde{\\rho }$ and the hyperangle $\\tilde{\\alpha }$ : $\\tilde{\\rho }^2=\\tilde{x}^2+\\tilde{y}^2, \\hspace*{28.45274pt} \\tilde{\\alpha }=\\arctan \\left( \\frac{\\tilde{x}}{\\tilde{y}} \\right),$ or, equivalently, $\\tilde{x}=\\tilde{\\rho }\\sin \\tilde{\\alpha }, \\hspace*{28.45274pt} \\tilde{y}=\\tilde{\\rho }\\cos \\tilde{\\alpha }.$ Also, from the definitions above it is simple to see that the ordinary 3D volume element can be written as: $dV=x_\\perp dx_\\perp dx_z d\\varphi _x y_\\perp dy_\\perp dy_z d\\varphi _y=s^2 \\tilde{x}_\\perp d\\tilde{x}_\\perp d\\tilde{x}_z d\\varphi _{\\tilde{x}} \\tilde{y}_\\perp d\\tilde{y}_\\perp d\\tilde{y}_z d\\varphi _ {\\tilde{y}}=s^2 d\\tilde{V}.$ Once the $d$ -wave function $\\Psi _d$ in Eq.", "(REF ) is interpreted as a deformed wave function in the ordinary 3D space, it is necessary to normalize correctly the wave function in three dimensions.", "Thanks to the relation above we can obtain the normalization constant as: $C(s)^2=\\int dV \\left| \\Psi _d(x_\\perp , x_z, y_\\perp , y_z,s) \\right|^2=s^2 \\int d\\tilde{V} \\left| \\Psi _d(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}})\\right|^2,$ where the last integral does not depend on $s$ .", "The wave function $\\tilde{\\Psi }_d=\\Psi _d/C(s)$ is correctly normalized to 1 in the ordinary 3D space, and it can therefore be employed to compute the expectation value of whatever observable of interest.", "In particular, we shall focus here on $\\langle x^2 \\rangle $ , which is given by: $\\langle x^2\\rangle = \\int dV x^2 \\left| \\tilde{\\Psi }_d(x_\\perp , x_z, y_\\perp , y_z, s)\\right|^2=\\int dV (x_\\perp ^2+x_z^2) \\left| \\tilde{\\Psi }_d(x_\\perp , x_z, y_\\perp , y_z, s)\\right|^2,$ which by use of Eq.", "(REF ), and keeping in mind that $\\tilde{x}_\\perp =x_\\perp $ and $\\tilde{x}_z=x_z/s$ , can be written as: $\\langle x^2\\rangle =s^2 \\int d\\tilde{V} \\tilde{x}_\\perp ^2 \\left| \\tilde{\\Psi }_d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2+s^4 \\int d\\tilde{V} \\tilde{x}_z^2 \\left| \\tilde{\\Psi }_d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2.$ Since $\\tilde{\\Psi _d}=\\Psi _d/C(s)$ and $C(s)^2$ is given by Eq.", "(REF ), the equation above becomes: $\\langle x^2\\rangle =\\frac{\\int d\\tilde{V} \\tilde{x}_\\perp ^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2+s^2 \\int d\\tilde{V} \\tilde{x}_z^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2}{\\int d\\tilde{V} \\left| \\Psi _d(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}})\\right|^2}.$ The integrals in Eq.", "(REF ) do not depend on $s$ .", "This actually implies that, in terms of the $\\tilde{\\mathbf {x}}$ and $\\tilde{\\mathbf {y}}$ coordinates, there is no preferred direction in the 3D space, and we therefore have that: $\\int d\\tilde{V} \\tilde{x}_i^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2=\\frac{1}{3} \\int d\\tilde{V} \\tilde{x}^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2,$ where $i$ is any of the Cartesian components of the $\\tilde{\\mathbf {x}}$ vector in the 3D space.", "Using Eq(REF ) we can then finally write: $\\langle x^2\\rangle = \\frac{2+s^2}{3} \\frac{\\int d\\tilde{V} \\tilde{x}^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2}{\\int d\\tilde{V} \\left| \\Psi _d(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}})\\right|^2},$ and similarly: $\\langle y^2\\rangle = \\frac{2+s^2}{3} \\frac{\\int d\\tilde{V} \\tilde{y}^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2}{\\int d\\tilde{V} \\left| \\Psi _d(\\tilde{\\mathbf {x}}, \\tilde{\\mathbf {y}})\\right|^2}.$ When only $s$ -waves are included in the calculation, the wave function $\\Psi _d$ does not depend on $\\Omega _{\\tilde{x}}$ and $\\Omega _{\\tilde{y}}$ , and making use of Eqs.", "(REF ) and (REF ) we can write: $\\langle x^2\\rangle = \\frac{2+s^2}{3} \\frac{\\int \\tilde{\\rho }^7 d\\tilde{\\rho } \\sin ^4\\tilde{\\alpha }\\cos ^2\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho },\\tilde{\\alpha })\\right|^2}{\\int \\tilde{\\rho }^5 d\\tilde{\\rho } \\sin ^2\\tilde{\\alpha }\\cos ^2\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho }, \\tilde{\\alpha })\\right|^2}$ and $\\langle y^2\\rangle = \\frac{2+s^2}{3} \\frac{\\int \\tilde{\\rho }^7 d\\tilde{\\rho } \\sin ^2\\tilde{\\alpha }\\cos ^4\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho },\\tilde{\\alpha })\\right|^2}{\\int \\tilde{\\rho }^5 d\\tilde{\\rho } \\sin ^2\\tilde{\\alpha }\\cos ^2\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho }, \\tilde{\\alpha })\\right|^2}.$ Note that the ratio $\\frac{ \\langle y^2 \\rangle }{\\langle x^2 \\rangle }=\\frac{\\int d\\tilde{V} \\tilde{y}^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2}{\\int d\\tilde{V} \\tilde{x}^2 \\left| \\Psi _d(\\tilde{\\mathbf {x}},\\tilde{\\mathbf {y}})\\right|^2}=\\frac{\\int \\tilde{\\rho }^7 d\\tilde{\\rho } \\sin ^2\\tilde{\\alpha }\\cos ^4\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho },\\tilde{\\alpha })\\right|^2}{\\int \\tilde{\\rho }^7 d\\tilde{\\rho } \\sin ^4\\tilde{\\alpha }\\cos ^2\\tilde{\\alpha } d\\tilde{\\alpha }\\left| \\Psi _d(\\tilde{\\rho },\\tilde{\\alpha })\\right|^2}$ does not depend on the scale parameter.", "The Jacobi coordinates do not provide the actual relative coordinates between the particles, $\\mathbf {r}_x$ and $\\mathbf {r}_y$ , but they are related through proportionality factors depending on the mass of the particles [13].", "Taking this into account, one can easily get: $\\frac{ \\langle r_y^2 \\rangle ^{1/2} }{\\langle r_x^2 \\rangle ^{1/2}}=\\sqrt{ \\frac{m_k(m_i+m_j) }{m_i+m_j+m_k } \\frac{m_i+m_j }{m_i m_j } }\\frac{ \\langle y^2 \\rangle ^{1/2}}{\\langle x^2 \\rangle ^{1/2}},$ where $r_x$ is the distance between the particles $i$ and $j$ , connected by the $x$ -coordinate, and $r_y$ is the distance between particle $k$ and the center of mass of the $ij$ -system, which are connected by the $y$ -coordinate.", "We shall consider the case of two identical spinless heavy particles with mass $m_H$ and one spinless light particle with mass $m_L$ .", "Only relative $s$ -waves will be included in the calculation.", "The interaction between the two heavy particles is put equal to zero.", "The interaction between the light and the heavy particles is such that the heavy-light two-body system and the three-body system are both unbound in three dimensions.", "As mentioned, it is known that in two dimensions any infinitesimal attraction is able to support at least one bound two-body state [1], [2].", "Therefore, along the squeezing process, from three to two dimensions, the heavy-light system unavoidably becomes bound for some particular strength of the external squeezing potential, or, equivalently, within the $d$ -formalism, for some specific value of the non-integer dimension, $d=d_E$ .", "For this dimension, $d_E$ , the two-body binding energy is equal to zero, and, since only relative $s$ -waves are considered, the Efimov conditions are fulfilled [9].", "The energy separation between the Efimov states depends crucially on the $m_H/m_L$ mass ratio, in such a way that the larger the ratio, the closer the Efimov states.", "In here we shall use the heavy-light Gaussian potential specified in Ref.", "[9] (the range of the potential is taken as length unit), which determines the appearance of the Efimov states for $d=d_E=2.75$ .", "We shall consider the mass ratio $m_H/m_L=133/6$ , which, as shown in [9], corresponds to an energy scale factor between the Efimov states of $46.5$ .", "In Ref.", "[9] it is also shown (Fig.", "5 in [9]) that, when squeezing to dimensions smaller than $d_E$ , the three lowest Efimov states survive as bound three-body states down to $d=2$ [14].", "The other infinitely many Efimov states disappear very fast around the two-body threshold.", "This fact was interpreted as a process where, when decreasing $d$ , one of the heavy particles is becoming less and less bound with respect to the bound heavy-light two-body system, being eventually lost [9], [12].", "Figure: Definition of the three possible Jacobi sets.Making use of Eqs.", "(REF ) and (REF ) one can easily visualize how the disappearance process takes place.", "To do so, we define the three possible Jacobi sets as shown in Fig.", "REF .", "If the interpretation made in [9], [12] is correct, when close to the state disappearance, one of the heavy particles, particle 2 in Fig.", "REF , should be pretty far from the bound heavy-light system (particles 1 and 3).", "This means that, when close to the disappearance of the state, if the first Jacobi set is used (Fig.", "REF a) we should get $r_y/r_x \\approx 0.5$ (since the two heavy particles are taken identical, the center of mass of the heavy-heavy system is at half the distance between them).", "In the same way, if the second Jacobi set is employed, Fig.", "REF b, we should get $r_y/r_x \\rightarrow \\infty $ .", "And finally, in the third Jacobi set, Fig.", "REF c, we should obtain $r_y/r_x \\approx 1$ , since the center of mass of the heavy-light system is pretty close to the center of mass of the heavy particle.", "Note however that, since the two heavy particles are identical, the second and third Jacobi sets are indistinguishable.", "Therefore, the use of any of them will simultaneously contain the features corresponding to the two Jacobi sets.", "The consequence is that, when using the second, or third, Jacobi set to describe the three-body system, only the divergence of the $r_y/r_x$ ratio will be visible.", "Figure: The outer panel shows, as a function of dd, the ratio 〈r y 2 〉 1/2 /〈r x 2 〉 1/2 \\langle r_y^2\\rangle ^{1/2} /\\langle r_ x^2 \\rangle ^{1/2} when computed in the second Jacobi set, as depicted in Fig. b.", "The insetshows the same ratio when computed in the first Jacobi set, Fig. a.", "The arrows indicate value of the dimension d E d_E for which the Efimov conditions are fulfilled.To check the results discussed in the previous paragraphs, we consider the third excited Efimov state for the three-body system under consideration (the $m_H/m_L=133/6$ case in Ref.[9]).", "This is the last of the Efimov states disappearing during the squeezing process for $d<d_E$ .", "For this state, we show in Fig.", "REF the ratio $\\langle r_y^2\\rangle ^{1/2}/\\langle r_x^2\\rangle ^{1/2}$ , Eq.", "(REF ), as a function of the dimension $d$ .", "The curves in the inner and outer panels, show, respectively, the result when the first and second Jacobi sets are used to describe the system.", "As we can see, for $d=d_E=2.75$ , where the Efimov effect takes place, the radius ratios take values of $\\sim 1.64$ and $\\sim 0.57$ in the first and second Jacobi sets, respectively.", "These values are consistent with an isosceles triangular spatial geometry, where the light particle is located in the vertex connecting the two equal sides of the triangle.", "The distance between each of the heavy particles and the light one is about 1.75 times the distance between the two heavy particles.", "When reducing the dimension, the geometry changes, approaching more and more the one depicted in Fig.", "REF , with the light particle close to one of the heavy ones, and the second heavy particle moving far apart.", "In fact, as we can see in Fig, REF , when close to the dimension at which the Efimov states disappears ($d\\approx 2.51$ ), the curve corresponding to the first (inner panel) and second (outer panel) Jacobi sets approaches the value of 0.5 and diverges to $+\\infty $ , respectively.", "As mentioned when discussing Fig.", "REF , this is the fingerprint of the disappearance process anticipated in Refs.", "[9], [12], namely, one of the heavy particles is moving far apart, being progressively less and less bound with respect to the bound heavy-light system, and lying eventually into the continuum.", "Figure: Contour plots of the density function in Eq.", "() for dimension values d=2.75d=2.75, 2.67, 2.60, 2.55, and 2.52, panels (a), (b), (c), (d), and (e), respectively.", "On each panel the upper andlower plots correspond to the calculations in the first and second Jacobi sets, respectively.", "The lengths are given in units of the range of the Gaussian interaction between the two heavy particles.This behaviour can also be seen in Fig.", "REF , where we show, for different values of the dimension $d$ , the contour plots of the density function $F(r_x,r_y) = \\int r_x^2 r_y^2 d\\Omega _{x} d\\Omega _{y} \\left| \\tilde{\\Psi }_d(\\mathbf {x},\\mathbf {y},s) \\right| ^2,$ where, for simplicity, since it does not affect our illustration purpose, we have taken $s=1$ for all the values of $d$ .", "In the figure, the panels (a), (b), (c), (d), and (e) correspond to the dimension $d=d_E=2.75$ , $d=2.67$ , $d=2.60$ , $d=2.55$ , and $d=2.52$ , respectively.", "In all the panels the upper figure shows the result obtained in the first Jacobi set, where $r_x=r_{HH}$ and $r_y=r_{L,HH}$ , respectively, whereas the lower figure is the result in the second Jacobi set, where $r_x=r_{HL}$ and $r_y=r_{H,HL}$ .", "The distances are given in units of the range of the Gaussian interaction between the two heavy particles.", "In the upper part of Fig.", "REF a, which corresponds to $d=d_E$ , we can see that, although the density function is peaked at value matching the ratio $r_{L,HH}/r_{HH}\\approx 0.5$ , the wave function shows however a long tail, mainly in the $r_{L,HH}$ direction, which increases pretty much the expectation value of $\\langle r_{L,HH}^2 \\rangle $ .", "This leads to the ratio $\\langle r_{L,HH}^2 \\rangle ^{1/2}/\\langle r_{HH}^2 \\rangle ^{1/2}\\approx 1.64$ shown in the inner part of Fig.", "REF for $d=d_E$ .", "When squeezing the system (decreasing $d$ ), we can observe, by following the upper figures in the five panels from left to right, that the tail of the wave function disappears progressively, in such a way that for $d=2.52$ (upper part of Fig.", "REF e), close to the disappearance of the state, the wave function reduces to a very narrow stripe that follows very closely the axis $r_{L,HH}=r_{HH}/2$ , very consistent with the geometry shown in Fig.,REF a, where one of the heavy particles is very far from the bound heavy-light system.", "A similar conclusion is reached when analyzing the lower plots in Fig.", "REF , which correspond to the results in the second Jacobi set.", "For $d=d_E=2.75$ (lower plot in Fig.", "REF a) the wave function is peaked at a value of $r_{H,HL}$ a bit smaller than $r_{HL}$ , consistent with the isosceles structure where the two heavy particles are closer to each other than the light and the heavy particles (keep in mind that $r_{H,HL}\\approx r_{HH}$ ).", "Furthermore, the wave function is clearly more extended along the $r_{HL}$ -axis than along the vertical axis, which leads to the ratio $\\langle r_{H,HL}^2 \\rangle ^{1/2}/\\langle r_{HL˘}^2 \\rangle ^{1/2}\\approx 0.57$ shown in the outer part of Fig.", "REF for $d=d_E$ .", "When squeezing the system, as we can see when moving to the right in the lower plots in Fig.", "REF , the peak is moving towards small values of $r_{HL}$ , and, at the same time, the central part of the density distribution is progressively disappearing.", "The result is that, when close to the disappearance of the state, $d=2.55$ and $2.52$ , it can be clearly seen that the density function presents two very distinct components, one of them with $r_{HL}\\ll r_{H,HL}$ , consistent with the structure shown in Fig.", "REF b, and a second component with $r_{HL}\\approx r_{H,HL}$ , consistent with the structure in Fig.", "REF c. The simultaneous presence of these two components is a consequence of the fact that the second and third Jacobi sets are indistinguishable.", "In previous works the $d$ -method was shown to be an efficient tool in order to describe squeezed two- and three-body systems, essentially due to the fact that the external squeezing potential does not enter explicitly [9], [10], [11], [12].", "This fact strongly simplifies the calculation of the wave functions, but at the expense of complicating the calculation of expectation values, since the wave function in $d$ dimensions has to be translated into the ordinary three-dimensional space.", "In this work we have focused on three-body systems squeezed from three to two dimensions, showing how the corresponding root mean square radii can be extracted after interpretation of the $d$ -wave function as a 3D wave function deformed along the squeezing direction.", "This procedure can be similarly applied to other observables, as well as other squeezing scenarios, like from three to one, or from two to one, dimensions.", "The computed radii have been then used to investigate how the Efimov states disappear around the two-body threshold during the squeezing process.", "We have considered a three-body system made of two identical heavy particles and a light particle.", "Neither the three-body system, nor any of the two-body subsystems, is bound in 3D.", "As shown in [9], during the squeezing process the system must, for some particular squeezing ($d=d_E$ ), fulfill the Efimov conditions, but, except a few of them that survive down to $d=2$ , the rest of the Efimov states disappear very fast around the two-body threshold.", "In this work we have visualized how this disappearance process takes place.", "We have shown that this happens through the loss of one of the heavy particles, which becomes progressively less and less bound with respect to the bound heavy-light system, being then eventually released into the continuum." ] ]
2105.11745
[ [ "Effects of interactivity and presentation on review-based explanations\n for recommendations" ], [ "Abstract User reviews have become an important source for recommending and explaining products or services.", "Particularly, providing explanations based on user reviews may improve users' perception of a recommender system (RS).", "However, little is known about how review-based explanations can be effectively and efficiently presented to users of RS.", "We investigate the potential of interactive explanations in review-based RS in the domain of hotels, and propose an explanation scheme inspired by dialog models and formal argument structures.", "Additionally, we also address the combined effect of interactivity and different presentation styles (i.e.", "using only text, a bar chart or a table), as well as the influence that different user characteristics might have on users' perception of the system and its explanations.", "To such effect, we implemented a review-based RS using a matrix factorization explanatory method, and conducted a user study.", "Our results show that providing more interactive explanations in review-based RS has a significant positive influence on the perception of explanation quality, effectiveness and trust in the system by users, and that user characteristics such as rational decision-making style and social awareness also have a significant influence on this perception." ], [ "Introduction", "Explaining the recommendations generated algorithmically by a recommender system (RS) has been shown to offer significant benefits for users with respect to factors such as transparency, decision support, or trust in the system [53], [54].", "Many approaches to explaining the products or services suggested by an RS have been based on ratings provided by other users or properties of the recommended items, approaches related to collaborative and content-based filtering methods [24], [56].", "More recently, fueled by the advances in natural language processing, user-written reviews have received considerable attention as rich sources of information about an item’s benefits and disadvantages, which can be utilized for explanatory purposes.", "Reviews are, however, subjective, and may be inconsistent with the overall rating given by the user.", "Even when overcoming the challenge of processing noisy review texts, the question of which review-based information to show and how to present it is still largely open, partly due to the lack of empirical conclusions on how to best present review-based explanations, just as there is a general lack of user-centric evaluations of explanatory RS [43].", "While as yet no overall theoretical model of explainable recommendations has been established, we propose to analyze explanations through the lens of argumentation theory which has produced a wide range of models of argumentation [5].", "One class of argumentation models defines - with many variations - logical structures of argumentative elements such as claims, evidence or facts supporting a claim, rebuttals and other components.", "A recommendation issued by a RS can be considered a specific form of a claim, namely that the user will find the recommended item useful or pleasing [18].", "The role of an explanation is thus to provide supportive evidence (or rebuttals) for this claim.", "Claims are, however, also present in the individual user’s rating and opinions, which may require explaining their grounds as well, thus creating a complex multi-level argumentative structure in an explainable RS, a concern also raised in [21].", "A different branch of argumentation theories [61] have abandoned the idea of static argumentation models and propose a dialectical approach to argumentation, focusing more on the process of exchanging arguments as part of a dialogue between two parties.", "This approach has been applied in the formulation of models [26], [60], [37] that take into account the social aspect of the explanatory process (an explainer transfers knowledge to an explainee) [39]: a set of interactions, an exchange of questions and answers.", "However, the practical application of these conceptual frameworks and their actual benefit from the users' perspective is yet to be determined, mostly due to the lack of user-centered evaluations of implementations based on such frameworks.", "Thus, we propose an interactive approach to argumentative explanations based on reviews, which allows users to actively explore explanatory information, while providing answers to some of their potential questions at different levels of detail (i.e.", "why is this [item] recommended?, how customers rated [additional features]?, what was reported on [feature]?).", "Moreover, we provide empirical evidence of the effect that an implementation of this approach may have on users' perception, particularly in the hotels domain.", "More specifically, we aimed to answer: RQ1: How do users perceive review-based explanations with different degrees of interactivity, in terms of explanation quality, and of the transparency, efficiency and trust in the system?", "Additionally, we also aimed to test the combined effect of explanation interactivity and different presentation styles, particularly: using only text, using a bar chart or using a table, to show, among others, the distribution of positive and negative comments on the quality of an item.", "Thus: RQ2: How do different presentation styles influence users’ perception of review-based explanations with different degrees of interactivity?", "Furthermore, we also addressed the influence that different user characteristics might have on the perception of the proposed approach.", "Regardless of its type, an explanation may not satisfy all possible explainees [50] .", "Moreover, individual user characteristics can lead to different perceptions of a RS [29], [65], for which we assumed that this would also be the case for explanations, as discussed by [6], [30], [25].", "Since a main objective of providing explanations is to support users in their decision-making, investigating the effect of different personal styles to perform such a process is of particular interest to us.", "Particularly, we focus on the moderating effect of the rational and intuitive decision making styles [23], the former characterized as a propensity to search for information and evaluate alternatives exhaustively, and the latter by a quick processing based mostly on hunches and feelings.", "Furthermore, since review-based explanations rely on the expressed opinions of other users, we investigated the effects of the extent to which users are inclined to adopt the perspective of others when making decisions, a trait defined as social awareness by [10].", "We also considered visualization familiarity, i.e.", "the extent to which a user is familiar with graphical or tabular representations of information.", "Consequently, RQ3: How do individual differences in decision-making styles, social awareness or visualization familiarity moderate the perception of review-based explanations with different degrees of interactivity and presentation styles?", "Finally, the contributions of this paper can be summarized as follows: We formulate a scheme for explanations as interactive argumentation in review-based RS, inspired by dialogue models and formal argument structures.", "To test our research questions, we implemented an interface based on the proposed scheme, and a RS based on a matrix factorization model (i.e.", "EFM, [68]), and sentiment-based aspect detection, using the state of art natural language processing model BERT ([15]).", "We provide empirical evidence of the effect of review-based interactive explanations on users’ perception, as well as the influence of user characteristics on such perception." ], [ "Related work", "Review-based explanatory methods leverage user generated content, rich in detailed evaluations on item features, which cannot be deduced from the general ratings, thus enabling the generation of more detailed explanations, compared to collaborative filtering (e.g.", "“Your neighbors’ ratings for this movie” [24]) and content-based approaches (e.g.", "[56]).", "Review-based methods allow to provide: 1) verbal summaries of reviews, using abstractive summarization from natural language generation (NLG) techniques [8], [14], 2) a selection of helpful reviews (or excerpts) that might be relevant to the user, detected using deep learning techniques and attention mechanisms [11], [17], 3) a statistical view of the pros and cons of item features, usually using topic modelling or aspect-based sentiment analysis [64], [68], [16], information that is integrated to RS algorithms like matrix or tensor factorization [68], [4], [62]) to generate both recommendations and aspect-based explanations.", "Our evaluation is based on the third approach, and is particularly related to the model proposed by [68], since it facilitates getting statistical information on users’ opinions, which has been proven to be useful for users [40], [25], and can be provided in explanations with different presentation styles (strictly verbal or visual).", "Yet, the optimal way of presenting such information, either in a textual (short summaries) or a graphical form (e.g., different types of bar charts) remains unclear.", "In addition to information display factors, a second factor could also influence users' perception of the explanations: the possibility of interacting with the system, to better understand the rationale for its predictions.", "Interactive explanations have been already addressed in the field of explainable artificial intelligence (XAI) (although to a much lesser extent compared to static explanations [1]).", "Here, the dominant trend has been to provide mechanisms to check the influence that specific features, points or data segments may have on the final predictions of a machine learning (ML) algorithm, as in the work of [31], [13], [49].", "However, despite the progress of XAI interactive approaches, their impact and possibilities in explainable RS remain largely unexplored, as well as the empirical validation of their effects on users.", "More specifically, the dominant ML interactive approach differs from ours in at least two ways: 1) we use non-discrete and non-categorical sources of information, subjective in nature and unstructured, which, however, can be used to generate both textual and visual structured arguments 2) such approach is designed to meet the needs of domain experts, i.e.", "users with prior knowledge of artificial intelligence, while we aim to target the general public.", "Therefore, we explore in this paper an interactive explanation approach that facilitates the exploration of arguments that support claims made by the system (why an item is recommended).", "To this end, we adopted the definition of interactivity stated by Steuer [52]: “extent to which users can participate in modifying the form and content of mediated environment in real time”, and characterized the degree of interactivity of proposed explanations through the Liu and Shrum dimensions of interactivity [34]: active control and two-way communication.", "The first is characterized by voluntary actions that can influence the user experience, while the second refers to the ability of two parties to communicate to one another.", "Active control is reflected in our proposal by the possibility to use hyperlinks and buttons, that allow users to access explanatory information at different levels of detail at will, while two-way communication is represented by the ability to indicate the system (through pre-defined questions) which are their most relevant features, so the presentation of the explanatory content (answers) is adjusted accordingly.", "In order to formulate and test an interactive flow for review-based explanations, we set our focus on argumentative models that may enable the two-way communication desideratum.", "In contrast to static approaches to explanation, dialog models have been formulated conceptually [58], [2], [37], [45], allowing arguments over initial claims in explanations, within the scope of an interactive exchange of statements.", "Despite the potential benefit of using these models to increase users' understanding of intelligent systems [39], [63], their practical implementation in RS (and in XAI in general) still lacks sufficient empirical validation [50], [39], [37].", "This dialogical approach contrasts with other argumentative - though static - explanation approaches [9], [3], [32], [67], [25] based on static schemes of argumentation (e.g.", "[55], [22]), where little can be done to indicate to the system that the explanation has not been fully understood or accepted, and that additional information is still required.", "Consequently, we formulated a scheme of interactive explanations in review-based RS, combining elements from dialogue models and static argument schemes (section 3), and conducted a user study to test the effect of the proposed interactive approach.", "Effects of interactivity have been studied widely in fields like online shopping and advertising [34], [51], and more specifically in the evaluation of critique-based RS, where users are able to specify preferences for the system to recalculate recommendations, which has been found to be beneficial for user experience [12], [36], [35].", "Despite the intuitive advantages that interactivity can bring, interactivity does not always translate into a more positive attitude towards the system, since it also depends on the context and the task performed [34].", "Nevertheless, it has also been shown that higher active control is beneficial in environments involving information needs, and a clear goal in mind [34], which is actually our case (i.e.", "deciding which hotel to book).", "Furthermore, we hypothesized (in line with [34]) that a number of user characteristics may moderate the effect of interactive functionalities, on the perception of explanations.", "Particularly, we aimed to test the moderating effect of decision-making styles and social awareness.", "In regard to the former, research has shown that it is determined significantly by preferences and abilities to process available information [19].", "Particularly, we believe that users with a predominant rational decision making style would better perceive explanations with a higher degree of interactivity, than explanations with less possibility of interaction, given their tendency to thoroughly explore information when making decisions [23].", "On the other hand, more intuitive users may not find the interactive explanations very satisfactory, given their tendency to make decisions through a quicker process [23], so that a first explanatory view would be sufficient, and it would not be necessary to navigate in depth the arguments that the system can offer.", "As for social awareness, and in line with results reported by [25], we hypothesize that users with a higher social awareness may perceive explanations with higher interactivity more positively, given their tendency to take into account the opinions of others, and to adjust their own using those of others, while choosing between various alternatives [48], which has been proved to be beneficial during decision making [66], and is facilitated by our approach.", "Finally, in regard to presentation styles, visual arguments (a combination of visual and verbal information) may have a greater \"rhetorical power potential\" than verbal arguments, due (among others) to their greater immediacy (possibility of quick processing) [7].", "This could especially benefit users with a predominantly intuitive decision-making style, due to their usually quick manner of making decisions, based mostly on first impressions [23].", "However, users with lower visual abilities might benefit less from a presentation based on images or graphics [47], [27].", "Consequently, we believe that when exposed to graphic-based explanation formats, higher interactive explanations may be beneficial to users with lower visual familiarity, as they could access additional information to better understand the explanations provided." ], [ "Scheme for explanations as interactive argumentation in review-based RS", "In order to evaluate our research questions, we designed an interaction scheme for the exploration of explanatory arguments in review-based RS.", "This scheme is inspired by dialog-based explanation models [59], [60], [37], in which instead of a single issue of explanatory utterances, an explanation process is regarded as an interaction, where a user could indicate when additional arguments are required, to increase their understanding of system claims.", "Walton [59], [60] modeled the possible actions as explanation requests or attempts, the former representing user questions, and the latter characterized as a set of assertions as system response.", "On the other hand, Madumal et al.", "[37] noted that argumentation may occur within explanation, and modeled the shift between explanatory and argumentative dialogue, as well as the explanatory loops that can be triggered, when follow-up questions arise.", "While this type of models may help to define the moves allowed within an explanatory interaction, they offer little indication of how the arguments within the interaction moves should be structured, to increase their acceptance by users.", "To this end, we rely on the scheme by Habernal et al.", "[22], an adaptation of the Toulmin model of argumentation [55], and formulated to better represent the kind of arguments usually found in user-generated content.", "This scheme involves: claim (conclusion of the argument), premise (a general reason to accept a claim), backing (specific information or additional evidence to support the claim), rebuttal (statement that attacks the claim) and refutation (statement that attacks the rebuttal).", "Our proposed scheme is shown in Figure 1.", "Unlike Walton, who modeled explanatory movements as explanation requests and attempts, we considered an explanation process as a sequence of argumentation attempts (the system intends to provide arguments to explain something) followed by argument requests (the user ask the system to provide - follow-up - arguments that support the claim that user will find the recommended item useful).", "The realization of such a scheme as a user interface developed for validation with users is depicted in Figure 2.", "Figure: Scheme for explanations as interactive argumentation in review-based RS.", "Blue boxes represent argumentation attempts by the system, green boxes the argument requests by users.Figure: Screenshots of implemented system for user study.", "Orange arrows depict the sequence of allowed moves, pointing towards the next interface provided.", "Steps c,d,e are enabled only in study condition interactivity “high”.", "a) List of recommended hotels, and first level argumentation attempt; when clicking on “More on why recommended”, system displays: b-c) a second level argumentation attempt; when clicking on “What was reported?”, system shows d) a third level argumentation attempt on the chosen feature; when clicking on an feature button, system shows e) only statements on the fine-grained chosen feature.As noted by [39] and [37], an explanatory dialogue can take place both through verbal interactions and through a visual interface (non-verbal communication, or a combination of verbal and visual elements), which applies to both questions and answers.", "As for argument presentation styles, while arguments are usually associated with oral or written speech, arguments can also be communicated using visual representations (e.g.", "graphics or images) [7].", "Thus, we considered the following styles for the argumentation attempt “% of positive and negative opinions”: 1) Table (Figures 3a, 3b), bar chart (Figures 3c, 3d), and text (Figures 3e, 3f), the latter using the template proposed by [25], which facilitates the display of rebuttal statements, which can hardly be represented graphically.", "Figure: Manipulation of presentation style in combination with interactivity, in user study.", "Top: table, middle bar chart, bottom text.", "Left: interactivity high, right: interactivity low." ], [ "User study", "To answer our research questions, we implemented a RS that reflects the scheme described in section 3, and conducted a user study to compare users’ perception of the overall system (in terms of transparency, effectiveness and trust), and of the explanations (in terms of explanation confidence, transparency, persuasiveness, satisfaction and sufficiency).", "We considered the factor interactivity, and the values “high” (users could make all possible argument requests (see Figures 1 and 2), and “low” (users could only make the initial argument request “more on why recommended?”).", "To validate the effect of factor presentation style, we considered the values: table (Figures 3 a,b), bar chart (Figures 3 c,d) and text (Figures 3 e,f).", "The study follows a 3x2 between-subjects design, and each participant was assigned randomly to one of six conditions (combination of interactivity and presentation style).", "We hypothesized: H1: Users' perception of the system and its explanations is more positive when they are given explanations with higher interactivity.", "H2: Users with a predominantly rational decision style perceive explanations with higher interactivity more positively than less rational decision makers.", "H3: Less intuitive users perceive explanations with higher interactivity more positively, compared to more intuitive users.", "H4: Users with greater social awareness perceive higher interactive explanations more positively than users with less social awareness.", "H5a: Users with a predominantly intuitive decision-making style or H5b a greater visualization familiarity will prefer bar chart explanations over text explanations, regardless of interactivity.", "H6: Users who are less familiar with data visualization will perceive explanations with higher interactivity more positively, particularly in the case of more challenging visualizations such as bar charts." ], [ "Participants", "We recruited 170 participants (66 female, mean age 37.61 and range between 18 and 72) through Amazon Mechanical Turk.", "We restricted the execution of the task to workers located in the U.S, with a HIT (Human Intelligence Task) approval rate greater than 98%, and a number of HITs approved greater than 500.", "We applied a quality check to select participants with quality survey responses (we asked validation questions to check attentiveness within questionnaires, and questions related to the content of the system).", "We discard participants with less than 10 (out of 12) correct answers, or no effective interaction with the system (checked in logs).", "The responses of 27 of the 197 initial participants were then discarded for a final sample of 170 subjects (statistical power of 90%, $\\alpha $ =0.05).", "Participants were rewarded with $1.4 plus a bonus up to $0.40 depending on the quality of their response to the question “Why did you choose this hotel?” set at the end of the survey.", "Time devoted to the task by participants (in minutes): M=10.88, SD= 1.62." ], [ "Dataset and Implemented system", "Dataset and aspect annotation: ArguAna [57], includes hotel reviews and ratings from TripAdvisor; sentiment and explicit features are annotated sentence wise.", "We categorized the explicit features in 10 general features (room, price, staff, location, facilities, bathroom, ambience, food and beverages, comfort and checking), with the help of 2 annotators (Krippendorff's alpha of 0.72), aiming to train a classifier to detect the main aspect addressed in a sentence (e.g.", "“I loved the bedding” would be classified as room).", "Aspect-based sentiment detection: We trained a BERT classifier [15] to detect the general feature addressed within a sentence: we used a 12-layer model (BertForSequenceClassification), 6274 training sentences, 1569 test sentences, F-score 0.84 (macro avg.).", "We also trained a BERT classifier to detect the sentiment polarity, using a 12-layer model (BertForSequenceClassification), 22674 training sentences, 5632 test sentences, F-score 0.94 (macro avg.).", "Classifier was used to 1) consolidate the quality of hotels and relevance of aspects to users (see Figures 2b, 2d), and 2) to present participants with negative and positive excerpts from reviews regarding a chosen feature (Figures 2d, 2e).", "Explainable RS method: We implemented the Explicit Factor Model (EFM) [68], a review-based matrix factorization (MF) method to generate both recommendations and explanations.", "The rating matrix consisted of 1284 items and 884 users extracted from the ArguAna dataset (only users with at least 5 written reviews were included), for a total of 5210 ratings.", "Item quality and user preferences matrices were consolidated using the sentiment detection described previously.", "The number of explicit features was set to 10.", "Model-specific hyperparameters were selected via grid-search-like optimization.", "After 100 iterations, we reached an RMSE of 1.27.", "Finally, values of predicted rating matrix were used to sort the list of recommendations and also shown within explanations (average hotel rating represented with 1-5 green circles).", "Values of the quality matrix were also used to calculate the percentages of positive and negative comments regarding different features (Figure 3).", "Personalization mechanism: To reduce implications of the cold start problem [46] (system does not have enough information about the user to generate an adequate profile and thus, personalized recommendations), participants were asked for the five hotel features that mattered most to them, in order of importance.", "The system calculated a similarity measure, to detect users within the EFM preference matrix with a similar order of preferences.", "Then the most similar user was used as a proxy to generate recommendations, i.e.", "we selected the predicted ratings of this proxy user, and used them to sort recommendations and features within explanations." ], [ "Questionnaires", "Evaluation: We utilized items from [44] to evaluate the perception of system transparency (construct transparency, user understands why items were recommended), of system effectiveness [29] (internal reliability Cronbach’s $\\alpha $ = 0.85, construct perceived system effectiveness, system is useful and helps the user to make better choices), and of trust in the system [38] ($\\alpha $ = 0.90, constructs trusting beliefs, user considers the system to be honest and trusts its recommendations; and trusting intentions, user willing to share information).", "We used the user experience items (UXP) of [30] to address explanations reception, which we will refer to as explanation quality ($\\alpha $ = 0.82), comprising: explanation confidence (user is confident that she/he would like the recommended item), explanation transparency (explanation makes the recommendation process clear), explanation satisfaction (user would enjoy a system if recommendations are presented this way), and explanation persuasiveness (explanations are convincing).", "We added an item adapted from [17] (explanations provided are sufficient to make a decision) to evaluate explanation sufficiency.", "All items were measured with a 1-5 Likert-scale (1: Strongly disagree, 5: Strongly agree).", "User characteristics: We used all the items of the Rational and Intuitive Decision Styles Scale [23] (internal reliability Cronbach’s $\\alpha $ = 0.84 and $\\alpha $ = 0.92, respectively), the scale of the social awareness competency proposed by [10] ($\\alpha $ = 0.70), and the visualization familiarity items proposed by [30] ($\\alpha $ = 0.86).", "All items were measured with a 1-5 Likert-scale (1: Strongly disagree, 5: Strongly agree)." ], [ "Procedure", "Instructions indicated that a list of hotels reflecting the results of a hypothetical hotels’ search and within the same price range would be presented (i.e no filters to search hotels were offered to participants), and that they could click on the name of a desired hotel to see general information about it.", "However, we asked, as we were more interested in their views on the explanations given for each recommendation, to click on the \"More on why recommended\" links of hotels they might be interested in, and to explore the information provided.", "No further instructions were given regarding how to interact with the different interaction options, since we were interested to address to what extent the users used them or not.", "Users were instructed to indicate which hotel they would finally choose, and to write a few sentences reporting their reasons for it, for which a bonus up to $0.4 would be paid, depending on the quality of this response, with the aim of achieving a more motivated choice by the participants, as well as to encourage a more effective interaction with the system.", "We then presented a cover story, which sought to establish a common starting point in terms of travel motivation (a holiday trip).", "Next, we presented to the participants the system showing a list of 30 recommended hotels (sorted by predicted rating), and their corresponding personalized explanations (system implementation details in section 4.2).", "Finally, evaluation and validation questions were presented, plus an open-ended one, asking for general opinions and suggestions about the explanations." ], [ "Data analysis", "We evaluated the effect that interactivity and presentation style (independent variables IVs) may have on 2 different levels: 1) overall system perception, and 2) perception of specific aspects of explanations, and to what extent user characteristics (regarded as moderators or covariates) could influence such perception (rational and intuitive decision-making style, social awareness and visualization familiarity).", "In case 1) the dependent variables (DVs) are evaluation scores on: system transparency (user understands why items were recommended), effectiveness (system helps user to make better decisions), trust (user considers the system to be honest and trusts its recommendations) and explanation quality, a variable calculated as the average of scores reported on specific aspects of explanations: satisfaction, transparency, persuasiveness, confidence and sufficiency.", "In case 2) the DVs are addressed explanation-wise: confidence (explanation makes user confident that she/he will like the recommendation), explanation transparency (explanation makes the recommendation process clear), satisfaction (user would enjoy a system if recommendations are presented this way), persuasiveness (explanations are convincing), and sufficiency (explanations provided are sufficient to make a decision).", "Scores of the rational and the intuitive decision making styles, social awareness and visualization familiarity for each individual as the average of the reported values for the items of every scale.", "Internal consistency (Cronbach’s alpha) was checked for system evaluation and user characteristics constructs (see section 4.3).", "Overall system perception: Given that DVs are continuous and correlated (see Table 1), a MANCOVA analysis was performed.", "Subsequent ANCOVA were performed to test main effects of IVs and covariates, as well as the effect of interactions between them.", "Q-Q plots of residuals were checked to validate the adequacy of the analysis.", "Perception of explanations: DVs are ordinal (scores are the reported answers to single questionnaire items), thus we performed ordinal logistic regressions to test influence on DVs by predictor variables (IVs and covariates), no multicollinearity was tested, as well as Q-Q plots of residuals.", "DVs are also correlated (see Table 2), so significant tests were conducted using Bonferroni adjusted alpha levels of .01 (.05/5).", "Use of interactive options: Calculated based on system activity logs.", "A Mann-Whitney U test was used to compare distributions of users characteristics who used or not use such options." ], [ "Evaluation and User Characteristics Scores", "The average evaluation scores by presentation style and interactivity are shown in Tables 1 and 2.", "Distributions of the scores of rational (M = 4.35, SD= 0.50) and intuitive (M = 2.59, SD= 0.98) decision making styles, social awareness (M = 4.04, SD= 0.53) and visualization familiarity (M = 3.23, SD= 0.95) are depicted in Figure 4a.", "Table: Mean values and standard deviations of perception on the overall system, per presentation style and interactivity (n=170), p<<0.05*, p<<0.01**; values reported with a 5-Likert scale; higher mean values correspond to a positive perception of the overall RS.", "Pearson correlation matrix, p<<0.001 for all coefficients.", "Table: Mean values and standard deviations of perception on explanation specific aspects, per presentation style and interactivity (n=170), p<<0.05*, p<<0.01**; values reported with a 5-Likert scale; higher mean values correspond to a positive perception on the explanations.", "Pearson correlation matrix, p<<0.001 for all coefficients." ], [ "Overall System Perception", "Interactivity: We found a significant multivariate effect of interactivity on overall system perception F(4,157) = 2.68, p = .034.", "Univariate tests revealed that interactivity significantly influences the perception of explanation quality F(1,168) = 9.76, p = .002, effectiveness F(1,168) = 4.02, p = .047, and trust F(1,168) = 4.63, p = 0.033.", "In all these cases, the average of every variable was significantly higher for the high condition than for low condition (see Table 1).", "Presentation style: We found no significant main effect of presentation style.", "Rational decision-making style: We found a significant multivariate effect of rational style, F(4,157) = 7.55, p $<$ .001.", "Univariate tests revealed a main effect of rational decision-making style on explanation quality, F(1,168) = 20.27, p $<$ .001, system transparency F(1,168) = 8.25, p = .005, effectiveness, F(1,168) = 26.76, p $<$ .001 and trust, F(1,168) = 24.94, p $<$ .001.", "In all these cases, a positive trend was observed between these variables and the rational decision-making style, i.e.", "the higher the rational decision-making score, the higher the perceived explanation quality, the transparency, the effectiveness and the trust, independent of the style or interactivity (see Figure 4b).", "Social awareness: We found a significant multivariate effect of social awareness, F(4,157) = 6.41, p $<$ .001.", "Univariate tests revealed a main effect of social awareness on explanation quality F(1,168) = 17.25, p $<$ .001, system transparency F(1,168) = 12.57, p $<$ .001, effectiveness F(1,168) = 22.85, p $<$ .001 and trust F(1,168) = 18.02, p $<$ .001.", "In all these cases, a positive trend was observed between these variables and social awareness, i.e.", "the higher the social awareness score, the higher the perceived explanation quality, the transparency, the effectiveness and the trust, independent of the style or interactivity (see Figure 4c).", "Figure: a) Kernel density estimate of user characteristics scores: rational and intuitive decision making styles, social awareness and visualization familiarity.", "b) Effect of rational decision-making style on the perception of the overall system (fitted means of individual scores).", "c) Effect of social awareness on the perception of the overall system (fitted means of individual scores)." ], [ "Perception of Explanations", "Interactivity: We found a main significant effect of interactivity; here, the odds of participants reporting higher values of explanation sufficiency when interactivity high was 2.30 (95% CI, 1.26 to 4.29) times that of interactivity low, a statistically significant effect, Wald $\\chi $ 2(1) = 7.32, p = .007.", "We observed a similar pattern in relation to explanation confidence (p = .017), explanation transparency (p = .043) and explanation satisfaction (p = .041).", "However, this association (despite p $<$ .05) is non-significant after Bonferroni correction (corrected p $<$ 0.01).", "Presentation style: We found no significant main effect of presentation style.", "Additionally, we observed a possible interaction (p$<$ = 0.05, although non-significant after Bonferroni correction, corrected p$<$ 0.01) between: Rational decision-making style and interactivity: An increase in rational decision-making score was associated with an increase in the odds of participants under interactive high condition reporting higher values of explanation sufficiency, with an odds ratio of 3.20 (95% CI, 0.99 to 10.65), Wald $\\chi $ 2(1) = 3.81, p = .051 (Figure 5a).", "Intuitive decision-making style and presentation style: An increase in intuitive decision-making score was associated with an increase in the odds of participants under bar chart condition reporting higher values of explanation satisfaction, with an odds ratio of 2.40 (95% CI, 1.14 to 5.18), Wald $\\chi $ 2(2) = 5.67, p = .023, compared to participants under text condition (see Figure 5b).", "Social awareness and interactivity: An increase in social awareness score was associated with an increase in the odds of participants under interactive high condition reporting higher values of explanation persuasiveness, with an odds ratio of 3.83 (95% CI, 1.20 to 12.34), Wald $\\chi $ 2(1) = 5.17, p = .023 (Figure 5c).", "Visualization familiarity and interactivity: An increase in visualization familiarity score was associated with an increase in the odds of participants under interactive high condition reporting higher values of explanation satisfaction, odds ratio of 1.91 (95% CI, 1.03 to 3.58), Wald $\\chi $ 2(1) = 4.24, p = .039 (Fig.", "5d).", "Figure: Interaction plots (fitted means of individual scores) for perception of explanation: a) sufficiency, interaction between interactivity and rational decision-making style.", "b) satisfaction, interaction between presentation and intuitive decision-making.", "c) persuasiveness, interaction between interactivity and social awareness.", "d) satisfaction, interaction between interactivity and visualization familiarity." ], [ "Use of interaction options", "48% of the users assigned to the interactivity high conditions used at least one of the interaction options provided.", "48.15% of participants used the ‘more features’ option when explanations were displayed using table, 26.92% using bar chart and 33.3% using text.", "55.56% of participants used the ‘what was reported’ option when explanations were displayed as table, 50% as bar chart and 3.7% as text.", "And 22.22% of participants used the ‘comments on specific features’ option when explanations were displayed as table, 19.23% as bar chart and 3.7% as text.", "Additionally, a Mann-Whitney U test revealed that the average of visualization familiarity scores of users who used the interaction options (M = 2.98, SD = 1.05) is significantly lower than the score of those that did not use them (M = 3.41, SD = 0.85) , U(Nused=41, Nnot used=44) = 678.50, p = .024)." ], [ "Discussion", "Our results show that greater interactivity has a significantly positive effect on users’ perception, in terms of system effectiveness and trust, as well as of explanation quality, compared to explanations with lower interactivity, thus confirming our H1.", "We believe that the interactivity aspects addressed in our proposal could play a determining role in the observed effect, namely: active control and two-way communication.", "The former, by enabling users to be in control of which argumentative content to display; the latter by enabling them to indicate the system which argumentative statements require further elaboration, and which features are of real relevance at the time of making the decision, an approach that might contribute to a better acceptance and understanding of explanations, as predicted by dialog models of explanation [60], [26].", "However, the benefit and actual use of interactive options in review-based explanations might be influenced by individual differences, as discussed by [34] for the scope of online advertising and shopping.", "In particular, we found that the way people process information when making decisions would play an important role in the perception of interactive review-based explanations.", "More precisely, and in line with H2, we found that greater interactivity might have a more positive effect on the perception of explanation sufficiency by more rational users, which is explained by the propensity of people with a predominant rational decision-making style, to search for information and evaluate alternatives exhaustively [23].", "However, and contrary to our expectations, we observed that the degree of intuitive decision style did not moderate the effect of interactivity on users' perception, so we cannot confirm our H3.", "In this regard, despite the predominant quick process based mostly on hunches that characterize more intuitive decision-makers [23], we believe that looking at some verbatim excerpts from other users' reviews may also be of benefit to them, as they could corroborate whether their hunches are aligned with the system's assertions, although they may not do so as extensively as less intuitive users would do.", "Additionally, in line with our H4 and results reported by [25], we observed that social awareness might moderate the effect of interactivity on explanation persuasiveness.", "Here, results suggest that participants with a higher disposition to listen and take into account others’ opinions, tend to perceive higher interactive explanations as more persuasive, which seems a consequence of the possibility to read reports of personal experiences by customers, who have already made use of the recommended items.", "This represents a potential advantage in the evaluation of experience goods like hotels, which is characterized by a greater reliance on word-of-mouth [41], [28].", "In line with H5a, our observations suggest that intuitive decision style might mediate the effect of presentation on explanation satisfaction, independent of interactivity.", "Particularly, explanatory arguments presented as a bar chart seemed to be perceived as more satisfactory to more intuitive users, than the presentation using a table or only text, presumably due to their greater immediacy [7], thus facilitating the rapid decision-making process that characterizes more intuitive users.", "However, contrary to our expectations, we cannot conclude that users with more visualization familiarity will perceive the bar chart explanations better than the text-based ones (H5b).", "One possible reason could be that a text-based format makes it easier to visualize argumentative components as rebuttal and refutation, which could lead to a higher acceptance of an argument, as advocated by argumentation theory ([22]), but could hardly be expressed through graph-based formats.", "Additionally, although users with lower visualization familiarity tended to use the interaction options more, we cannot confirm our hypothesis that those users would perceive graphic-based explanations (i.e.", "bar chart) better when more interactive options are offered, (H6).", "Actually, we found that users with more experience with data visualization reported a more positive perception for explanations with higher interactivity, independent of presentation style.", "We believe this is not due to difficulties understanding the explanations (as we thought would be the case for users with less visualization familiarity), but because higher interactivity facilitated a structured navigation and more appealing display of the data, which would not be as easy to process or useful if presented on a single static explanation.", "Overall, we observed a main effect of rational decision-making style and social awareness in the perception of the system and all the proposed explanations.", "This suggests that review-based explanations seem to benefit more the users who tend to evaluate information thoroughly and take into account the opinions of others when making decisions, compared to users who use a more shallow information-seeking process.", "Interactivity and transparency perception.", "Despite the main effect of interactivity on the overall perception of the system and its explanations, the mean perception of system transparency (user understands why items were recommended) is only slightly higher for the interactivity high condition than for the low condition.", "We believe that the reason might be two-fold: 1) Walton's [60] suggests to include an explicit mechanism to confirm effective understanding by the user, so that if this has not yet been achieved, the iterative cycle of user questions and system responses may continue.", "In consequence, we believe that a more flexible approach in which the user could, for example, write their own questions, rather than the bounded link-based options, might contribute in this regard.", "And 2) users may be also interested in understanding the reasons why the hotel x is better than hotel y.", "This would not only be in line with the view of authors who claim that the why-questions ask for a contrastive explanation (“why P rather than Q?”) [26], [33], [39], but also concurs with some participants’ suggestions, that options for comparison would be very useful, e.g.", "“It'd be easier if information wasn't on each separate page, too.", "I'd like an option to compare and contrast options”.", "Use of interaction options.", "We observed that almost half of participants under the condition interactivity “high” actually used the interaction options, although participants were not explicitly instructed to use them, so it can reasonably be inferred that their use was mainly voluntary.", "It is critical, however, that these options are named appropriately, indicating clearly their destinations (as stated by [20] guidelines), to increase the probability of their use, as evidenced by the lack of use of the option to read reviews excerpts in the text condition (Fig.", "3e).", "Additionally, some of the users assigned to the low interactivity condition pointed to 1) the lack of access to additional information in connection to the explanations (particularly customer reviews) as a disadvantage, with about a quarter of those participants writing suggestions on the subject, e.g.", "“I would prefer to read the actual reviews and understand why ratings were what they were”, or 2) insufficiency of aggregated percentages of positive and negative opinions to adequately explain recommendations, e.g.", "“I feel they maybe could have a lot more information more on SPECIFICALLY what they say about the room instead of just an overall aggregation”.", "In this regard, it is important to note though, that participants of all conditions had access to the full hotel reviews (they were included in the general view of each hotel).", "Practical implications.", "Our approach was specifically tested in hotels domain, however, since it allows users to navigate from aggregated accounts of other users’ opinions to detailed extracts of individual reviews, we believe it might generalize adequately to domains that involve the evaluation of experience goods, i.e.", "those whose attributes can only be evaluated when the product has already been experienced [42], and where the search for information is characterized by a greater reliance on word-of-mouth [41], [28] for example restaurants, movies or books.", "Additionally, our findings lead to the following practical implications, to be considered in the design of review-based explanations in RS involving experience goods: Providing interactive explanations resembling an argumentative communication between system and user could contribute to a better perception of the system.", "This could be implemented using web navigation options, e.g.", "links or buttons that indicate explicitly their destination, indicating if possible, a why or what question that will be answered by the system afterwards.", "Presenting both aggregated opinion statistics and excerpts of comments filtered by feature, as part of an interactive explanation, is a beneficial way to provide explanations sufficient in content, while avoiding overwhelming users with irrelevant data in a single step or screen.", "Given the practical difficulty of detecting user characteristics (e.g., decision-making style or visualization familiarity) by the system, we suggest interactive options to be considered, not only to provide in-depth arguments or to detect the relevance of features to the user, but also to modify the presentation style of argument components." ], [ "Conclusions and future work", "In this paper, we have presented a scheme for explanations as interactive argumentation in review-based RS, inspired by dialogue explanation models and formal argument schemes, that allows users to navigate from aggregated accounts of other users’ opinions to detailed extracts of individual reviews, in order to facilitate a better understanding of the claims made by the RS.", "We tested an implementation of the proposed scheme in the hotels domain, and found that more interactive explanations contributed to a more positive perception of effectiveness and trust in the system.", "We also found that individual differences in terms of user characteristics (e.g.", "decision-making style, social awareness and visualization familiarity) may lead to differences in the perception of the proposed implementation.", "While our proposal suggests a first step towards an effective implementation of interactive explanations for review-based RS, some important improvements can still be considered, to increase users' perception of transparency, as pointed out in the previous section.", "Here, the provision of links with predefined why, how or what questions, while practical, could be improved, for example, with the possibility for the user to ask more flexible questions, even in natural language.", "Thus, as future work, we plan to leverage advances of conversational agents (i.e.", "chatbots), natural language processing and natural language generation techniques, such as question answering and automatic summarization, to enhance the implementation proposed in this paper.", "It is important to note that our scheme entails an explanatory dialogue on a single-item level.", "However, we plan in the future to investigate the effect of contrastive dialogue-based explanations of the type “Why P rather than not-P?”.", "In this respect, we believe that this type of explanation can be leveraged to enable users further possibilities to influence the recommendation process itself, e.g.", "requesting for a more refined set of recommendations that better suit their preferences, based on an explanatory contrast between the different options.", "The above might result in greater satisfaction with the overall system, as has been proven with interactive RS in the past, but this time from the explanations as such." ], [ "Acknowledgements", "This work was funded by the German Research Foundation (DFG) under grant No.", "GRK 2167, Research Training Group “User-Centred Social Media”." ] ]
2105.11794
[ [ "The Evolution of Rapid Optical/X-ray Timing Correlations in the Initial\n Hard State of MAXI J1820+070" ], [ "Abstract We report on a multi-epoch campaign of rapid optical/X-ray timing observations of the superbright 2018 outburst of MAXI J1820+070, a black hole low-mass X-ray binary system.", "The observations spanned 80 days in the initial hard-state, and were taken with NTT/ULTRACAM and GTC/HiPERCAM in the optical (ugriz filters at time resolutions of 8--300 Hz) and with ISS/NICER in X-rays.", "We find (i) a growing anti-correlation between the optical and X-ray lightcurves, (ii) a steady, positive correlation at an optical lag of 0.2 s (with a longer lag at longer wavelengths) present in all epochs, and (iii) a curious positive correlation at \\textit{negative} optical lags in the last, X-ray softest epoch, with longer wavelengths showing a greater correlation and a more negative lag.", "To explain these we postulate the possible existence of two synchrotron-emitting components; a compact jet and a hot flow.", "In our model, the significance of the jet decreases over the outburst, while the hot flow remains static (thus, relatively, increasing in significance).", "We also discuss a previously discovered quasi-periodic oscillation and note how it creates coherent optical time lags, stronger at longer wavelengths, during at least two epochs." ], [ "Introduction", "Low-Mass X-ray Binaries (LMXBs) are highly variable systems involving accretion on to either a neutron star or black hole.", "Over the past few decades, there have been many efforts to study this variability and detail its behaviour, and there is an expanding body of literature detailing this (see [5], , and many others).", "But why is the study of this variability important?", "In short, because these systems are complex, and unresolvable with current telescopes.", "LMXBs host a compact object accreting via a disc of material transferred from a Roche-lobe-filling companion star.", "The environment is complex, with an outer disc, hot inner flow/corona, and compact, relativistic jets (to name just a few), which all emit across the electromagnetic spectrum.", "And, during transient, violent outbursts that occur every few years or decades and can last for weeks to months, the scale, presence, and behaviour of these regions can change significantly.", "Their compact nature means that physically important timescales can span $\\sim $ microseconds in the inner zones, to decades at the other extreme.", "The goal of multiwavelength timing studies of these sources is to understand the physical processes in these components, and thus the system as a whole.", "Over the relatively short history of multiwavelength astronomy, better technology and new telescopes have improved the temporal resolution of such studies, and with it, our understanding has advanced; , and were some of the earliest reports on rapid stochastic multiwavelength variability down to millisecond scales, while was one of the works that showed intriguing relations between the rapid optical and X-ray variability for the first time.", "This interband relationship was then found to vary between systems, each time showing complex behaviour, interpreted as a varying dominance of the inflowing or outflowing plasma through the disc, the inner flow and the jet (e.g.", ", , , , , , ).", "The true importance of these studies is in providing novel quantitative constraints of the physical scales and interactions between the accreting plasma components.", "For instance, a rapid optical/infrared lag of $\\sim $  100 ms relative to X-rays has now been observed in several systems and appears to be an important constraint for models of jet launching and acceleration .", "Yet, these studies rely on the source being both sufficiently bright and well-observed at multiple wavelengths simultaneously, the former being rare and the latter being marred by the inherent unpredictability of these outbursts.", "As such, these studies have so far been few and far between, and rarely carried out multiple times over the same outburst - though there are hints at an evolution of processes at different stages of the outburst , .", "Solutions are not yet unique, with processes such as a jet and a hot flow invoked to explain certain signatures on intermediate timescales , .", "We still remain severely data-limited in terms of high-quality strictly simultaneous multiwavelength time series in order to make progress.", "In 2018, one particular X-ray binary was discovered.", "It became bright enough and observed well enough that a good picture of its initial, several-week-long hard state – including evolving inter-band correlations and Fourier components, observed at over 100 Hz – has been made possible.", "Discovered first as optical transient ASASSN-18ey on 6 March 2018 , and then as an X-ray source on 11 March , MAXI J1820+070 (hereafter J1820) was detected during the rapid outburst rise.", "It quickly rose to a brightness of $\\sim $ 4 Crab , becoming the brightest extra-solar object in the X-ray sky by the time it peaked on 23 March .", "By this point, its brightness had led to observations at many sites (e.g.", "[3], , [4], ); not only did these observations quickly identify it as a likely Black Hole LMXB [3], , , but they also revealed rapid optical flaring (, ) and even a significant optical/X-ray correlation .", "Later, a QPO would be first identified in this source at around this peak , , and would be seen to evolve over the next few months .", "This stage of the outburst was the `hard state', where it is believed that the inner edge of the accretion disc is recessed, and a relativistic jet is present .", "After the peak, J1820 entered a gradual decline in X-ray flux.", "In early July 2018, it transitioned rapidly to the soft state , where the accretion disc extends to the Innermost Stable Circular Orbit (ISCO) and the jet is quenched.", "During this time, a unique blackbody X-ray emission signature was detected, which has been suggested as originating from within the ISCO, the so-called 'plunge region' .", "J1820 remained in this state until late September 2018, when it transitioned back to the hard state .", "It has since undergone a series of small `rebrightenings' , , , , , [1], but as of yet, it has not undergone a second outburst.", "Fig.", "REF shows a timeline of the hard state outburst at X-ray and radio wavelengths, using data from the Neil Gehrels Swift Observatory, MAXI, and AMI-LA.", "Radio parallax measurements have since constrained J1820 to a distance of $2.96\\pm 0.33$  kpc [2], and the optical parallax found using Gaia EDR3 gives a distance of $2.94^{+0.87}_{-0.55}$  kpc (calculated using the recommended zero-point correction – ), which improves on the previous estimate reported in Gaia DR2 .", "J1820's brightness led to several multiwavelength campaigns using high-time-resolution instrumentation over the course of its outburst.", "In , we discussed the optical/X-ray correlations taken from a single night, using HiPERCAM and NICER during the rising hard accretion state.", "Therein, we noted the presence of a sub-second optical lag of order $\\sim $  100 ms between the bands dependent upon wavelength, which we attributed to structure within the compact jet, and presence of an anti-correlation, which we put in the context of the hot accretion flow.", "Together with GX 339–4 and V404 Cyg , these results make J1820 the third well-studied black hole XRB to show the aforementioned sub-second lag.", "The above results all highlight the importance of J1820 as a benchmark for understanding accretion.", "Here, we expand on these results to trace the timing properties through the primary hard state, including four new observations between NICER and another optical instrument, ULTRACAM, as well as a second correlated HiPERCAM/NICER observation later in the outburst.", "All observations were taken during the initial hard state, cover time resolutions from 8 Hz to 300 Hz, and cover a span of roughly 80 days in total.", "We construct a picture of the evolving optical/X-ray correlations over this period, and discuss to what processes they may relate.", "ULTRACAM is a fast-timing optical camera on the 3.58 m New Technology Telescope (NTT) in La Silla, Chile.", "It was built for the purpose of fast optical timing in multiple wavebands.", "To this end, it includes three channels for simultaneous multiwavelength monitoring (with replaceable filters).", "It can also observe at frame-rates well above 100 Hz; this is achieved by the lack of a physical shutter, and frame-transfer CCDs that can rapidly shift charge into a storage area for reading out, freeing up the original pixels for observation and thereby achieving low dead times .", "We used ULTRACAM to observe J1820's initial outburst peak in the early mornings of 2018 March 16, March 22, April 12, and April 17.", "All observations were carried out with the $u_s$ , $g_s$ , and $i_s$ SDSS filters, except for the first, which used the $r_s$ filter instead of $i_s$ .", "Unlike most observations of this type, the times were not explicitly chosen to coincide with X-ray observations – instead, the overlaps were purely coincidental and the serendipitous result of near-constant monitoring of J1820 by multiple telescopes.", "ULTRACAM was used in two-window mode (one each for the target and comparison star), with both window sizes of 50 x 50 pixels with a 2 x 2 binning for sensitivity and speed.", "See Table REF for observational details.", "J1820 was very faint in $u_s$ , and so ULTRACAM's on-chip co-adding feature was used; this provides a longer exposure time in $u_s$ so as to increase signal-to-noise ratio.", "The data were reduced using the ULTRACAM pipeline v9.14 .", "The bias was subtracted from each frame, and flat field corrections were also applied.", "Aperture sizes scaled to the instantaneous seeing were used, with radii between 0$$ 7 and 3$$ 5, with an annulus of between 12$$ and 6$$ 3 to calculate the background.", "These apertures had variable centre positions that tracked the centroids of the sources on each frame, with a two-pass iteration (where an initial pass is made to track the sources on the CCD before a second photometry pass) used for accuracy.", "Our times were then adjusted to Barycentric Dynamical Time (BJD_TDB) using methods given in .", "Our comparison star is located at RA = 18 20 26.43, Dec = 07 10 11.7 (J2000), and is listed in the PanSTARRS survey catalog with $g_s$ /$i_s$ magnitudes of 13.3083/12.233 respectively.", "The star was taken to be constant, and was used for photometric calibration.", "For the $u_s$ observations, the comparison star was too faint to perform photometry within a single frame.", "Hence, we used the measured zero-point magnitude for the $u_s$ band in photometric conditions for ULTRACAM (Vik Dhillon, priv.", "communication) in order to calibrate our observations.", "We extracted the J1820 and comparison star magnitudes using aperture photometry with a variable aperture size that was dictated by the seeing conditions.", "The aperture also tracked the centroid of the source of interest by using a bright star in the field as a reference.", "For the $u_s$ observations, we used J1820 as the reference object itself so as to not lose tracking within the field." ], [ "GTC/HiPERCAM – Optical", "High-speed multi-colour photometry of J1820 was carried out using HiPERCAM on the 10.4 m Gran Telescopio Canarias on La Palma.", "HiPERCAM uses 4 dichroic beamsplitters to image simultaneously 5 optical channels covering the $u_sg_sr_si_sz_s$ -bands (respectively, central wavelengths 3526, 4732, 6199, 7711 and 9156 Å).", "The CCDs were binned by a factor of 8 and used in the highest-speed drift mode.", "We orientated the instrument (PA = 58$^{\\circ }$ ) and used two windows (96x72 pixels each), one centered on J1820, and another on a comparison star, APASS–34569459 .", "The observations discussed here were taken on 2018 April 17 from 03:26–06:11 UT, and 2018 June 07 from 04:41–05:39, coordinated with NICER.", "The exposure time was 2 ms, the cadence 2.9 ms, the median seeing 2.2.", "The sky was affected by mild cirrus on both dates.", "We used the HiPERCAM pipeline softwarehttps://github.com/HiPERCAM/hipercam to de-bias, flat-field and extract the target count rates using aperture photometry with a seeing-dependent circular aperture tracking the centroid of the source.", "Sky background was removed using the clipped mean of an annular region around the target.", "The target was brighter than all stars in the field.", "We thus used the raw target counts for the analyses presented herein; note that our primary results are not affected when using photometry relative to the comparison star." ], [ "ISS/NICER – X-ray", "NICER (Neutron star Interior Composition ExploreR) is an X-ray instrument aboard the International Space Station (ISS).", "It comprises 52 functioning X-ray concentrator optics and silicon drift detector pairs, arranged in seven groups of eight.", "Individual photons between 0.2-12 keV, and their energies, can be detected to a time resolution of 40 ns .", "J1820 was observed with an intensive monitoring program during the initial hard state of its outburst.", "Data reduction of ObsIDs 1200120105, 1200120107, 1200120127, 1200120131, and 1200120172, were completed using nicerdas, a collection of NICER-specific tools, and part of HEASARChttps://heasarc.gsfc.nasa.gov.", "Full Level2 calibration and screening was conducted with nicerl2, which calibrated, checked the time intervals, merged, and cleaned the data.", "Barycentric correction was carried out using barycorr, then the photon events (all between 0.2-12 keV) were binned to the times of the optical light-curve.", "Our analysis of the optical and X-ray data involves creating simultaneous lightcurves, Cross-Correlation Functions (CCFs) and Fourier analysis.", "In the following we detail the methodology used." ], [ "Simultaneous Lightcurves", "Simultaneous lightcurves are plotted in Fig.", "REF .", "The optical and X-ray data are not, by default, binned simultaneously.", "However, while the optical data were taken in discrete time bins by both instruments, NICER is a photon-counting instrument and thus records the arrival time of each photon.", "Therefore, we create simultaneous lightcurves by binning the photons directly to the optical time bins, after barycentering both datasets.", "Since the optical lightcurves have a constant deadtime (time between the bins in which no data were recorded) the X-ray photons observed during this time are disregarded.", "For X-rays, the square root of the counts per bin was used to determine the error for each bin.", "Since the $u_s$ band data were sampled at a different rate to the other optical bands, a separate X-ray lightcurve was created.", "This lightcurve is not plotted in Fig.", "REF , but was used in creating the Cross-Correlation functions and in the Fourier analysis for the $u_s$ band data in epochs 1–3 and 5." ], [ "Cross-Correlation Functions", "Cross-correlations are plotted in Fig.", "REF & REF .", "Cross-correlations are measurements of how much one lightcurve (or any time-series) varies dependent on another as a function of lag.", "In these cases, we create optical vs. X-ray cross-correlations; the figures therefore show the response of the optical lightcurves to variations in the X-ray lightcurve, as a function of time lag.", "Positive values indicate a net correlation at that lag, and negative values a net anti-correlation, each normalised so that 1 and -1 indicate perfect correlations and anti-correlations.", "The cross-correlations were produced by splitting the simultaneous lightcurves into segments of equal length.", "Each segment was then `pre-whitened' by removing a linear trend.", "A Cross-Correlation Function (CCF) was then run on each segment, using the methodology of .", "The mean CCF was then determined and the standard error on each bin was calculated.", "To probe variations on different timescales we compute CCFs using segment sizes of 10 s (Fig.", "REF ) and 2 s (Fig REF )." ], [ "Fourier Analysis", "Fourier analysis is presented in Fig.", "REF –REF .", "These involved computing the Fourier transform of the lightcurves and then analysing them at each frequency.", "The power spectra represent the amplitude of the variability at each Fourier frequency.", "The coherence represents the relative magnitude of the complex-valued cross-spectrum, i.e.", "a measure of how the bands are correlated at that frequency.", "The phase lags represent the relative phase angle of the complex-valued cross-spectrum, i.e.", "a measure of the lag between the bands at each frequency as a function of phase (measured in radians).", "The time lags show the same data as the phase lags, but converted into the time domain.", "This analysis made use of the Stingrayhttps://github.com/StingraySoftware/stingray python package .", "Values for the intrinsic coherence, and errors on those values, were determined using methods described in , where our data fit into the category of `High powers, high measured coherence'.", "Good Time Intervals (GTIs) were used based on the individual epochs of X-ray observation, and then cross-spectra were computed over independent lightcurve segments and averaged.", "The segment lengths were $2^{12}$ bins for epochs 1–3 and 5, and $2^{14}$ bins for epochs 4 and 6.", "For observations with co-adding in $u_s$ , the nearest multiple of 2 was used as the bin length, so that the lightcurve segments were of similar size compared to the other filters of the same observation.", "These segment sizes were selected to balance frequency range against statistics, making sure that all bands were averaged over at least 5 segments (aside from the $u_s$ bands in epoch 3 and 5, which had only 3 and 4 segments respectively).", "Root-mean-squared (rms$^2$ ) normalisation was applied to the power spectra .", "The white noise was fitted and removed from the power spectra before calculating the coherence (see Section REF for details).", "In Figures 5–8 the frequency-dependent products were binned logarithmically in frequency; for the power spectra the factor was 1.1, while for the coherence, time lags, and phase lags the factor was 1.3 (these were chosen to balance the clarity of features with the size of the uncertainties).", "Time lags were calculated by dividing the phase lags by $2 \\pi f$ , where $f$ is the frequency of the bin.", "Since the conversion is ambiguous and could be $\\pm 2\\pi $ , we assumed that the phase lags of the frequency bins around 1 Hz were correct, based on their relationship to the sub-second time lag seen in Figure REF .", "Each time lag was then arbitrarily shifted based on what would cause the fewest discontinuities." ], [ "Results", "In the Figs.", "REF –REF , the violet plot on the left shows the timeline of the outburst in MJD, seen by Swift/BAT (see Figure REF ) – the stronger the colour, the brighter J1820 was in hard X-rays.", "The epochs are marked.", "Each plot shows the variation in all bands.", "The colour key is as follows: $u_s$ (blue), $g_s$ (green/teal), $r_s$ (red), $i_s$ (dark red/brown), $z_s$ (black), and X-rays (violet)." ], [ "Lightcurves", "Fig.", "REF shows portions of the lightcurves from each of the epochs in all optical bands as well as in X-rays.", "The lightcurves show a lot of similarities – in the optical there are numerous sub-second flares with an increase of a factor $\\sim $ 1.5–2 in flux.", "A common property of all lightcurves is that the variations tend to be far stronger in the red than in the blue, and is particularly true of the sub-second flares – this is also seen in other hard-state LMXBs , .", "These flares become less frequent as the epochs continue, but are still present in epoch 6.", "Interestingly, the lightcurves are sometimes anti-correlated during these flares, with optical activity rising while X-ray activity decreases - see, for example, the inset to Epoch 4." ], [ "Cross-Correlation Functions", "CCFs from 10 s segments can be seen in Figure REF .", "Each epoch shows a sub-second correlation peak.", "Epochs 1–5 also show some form of a `precognition dip'; i.e.", "an anti-correlation at negative lags, which means that either the optical lightcurve dips a few seconds before an X-ray flare, or that optical flares occur before an X-ray dip.", "We can also see how the CCFs evolve over time.", "Between epochs 1–5, the correlation steadily decreases at positive lags, perhaps caused by an anti-correlation component becoming more significant in the data.", "Additionally, from epoch 3 onwards a new positively-correlated component appears, peaking at negative lags, which is stronger at longer wavelengths.", "This feature evolves from $\\sim $ -3 s in epoch 3 to -1 s in epoch 6.", "To probe the more rapid variations, we also created CCFs from 2 s segments which can be seen in Figure REF .", "These reveal several details.", "Firstly, the sub-second correlation is strongest just before the peak of the outburst.", "Secondly, the lag of the CCF peak is fairly constant from epoch 2 onwards, even into epoch 6.", "Finally, the CCF peak has a `tail' extending from the initial peak out to 0.5–0.75 s and appears to shrink over time, or at least become less significant compared to some anti-correlated component; see, in particular, the difference between epochs 4 and 6.", "The sub-second correlation was previously reported in .", "In that work, we found that the lag was greater at longer wavelengths – this can be seen in several of these epochs, and will be shown more explicitly in Section REF , and finally discussed in Section REF .", "Our highest-resolution epochs, 4 and 6, also show a small spike in the CCF at 0 s lag.", "This is confirmed by CCFs created from 1 s lightcurve segments and below.", "However, our data do not have sufficient time resolution to study these particular features.", "A closer inspection shows that the correlated negative-lag component is present in epoch 6, but with curious results.", "Firstly, the peak is now at 0 s, not -1 s. Secondly, at negative lags the longer wavelengths have a stronger correlation, while the shortest wavelength ($u_s$ ) shows a trend towards anti-correlation.", "At positive lags, the reverse is true; it is the longer wavelengths that now show a trend towards anti-correlation.", "This shows either that this negative component affects the signals down to these rapid timescales, or that there are two components that affect these lags – this correlated negative component, and a new component that has appeared between epochs 5 and 6.", "These possibilities will be discussed in Sections REF and REF ." ], [ "Fourier Analysis", "To better understand the nature of different components contributing to the CCF, we perform the decomposition of the observed variability into different time scales using Fourier technique.", "Figs.", "REF – REF show various Fourier components in each optical band plotted against X-rays.", "Table: Fitting parameters for the power spectra.", "Each cell contains the following: Number of Lorentzians fitted; white noise level in fractional rms 2 ^2 units ×10 -5 \\times 10^{-5} (Reduced Chi 2 ^2 χ ν 2 \\chi ^2_\\nu )." ], [ "Power Spectra", "The power spectra can be seen in Fig.", "REF .", "In addition to the five optical bands (detailed at the beginning of Section ) the X-ray power spectra are also shown.Note that these are only for the X-ray data that are strictly simultaneous with our fast optical photometry (aside from the $u_s$ band in cases of co-adding, i.e.", "epochs 1–3 and 5), and not the full spectra from the NICER observations.", "Thus, these are not directly comparable to the power spectra in .", "The power is in fractional rms$^2$ units and is multiplied by the frequency.", "As noted in Section REF , the separate X-ray lightcurves for the co-added $u_s$ bands in epochs 1–3 and 5 are not shown.", "Additionally, the $u_s$ power spectrum in epoch 6 is not shown due to the poorer data quality.", "A mix of zero-centered and non-zero-centered Lorentzians along with a constant white noise component were fitted to each band.", "For these plots, that white noise component was removed and the fitted parameters can be seen in Table REF .", "The increased numbers of Lorentzians (and increased $\\chi ^2_\\nu $ values) for epochs 4 and 6 are due to the higher cadences, larger segment sizes, and lower noise levels in the HiPERCAM data; these lead to far lower uncertainties, and thus require more Lorentzians to fit numerous features in these bands.", "Regarding the evolution of the power: In all epochs, the power in the optical bands is consistently higher at longer wavelengths, although highest overall in X-rays.", "The manifestation of this can be seen in Figure REF , where one can see activity at longer wavelengths being much stronger than that at shorter ones.", "The power in each band evolves over the course of the outburst.", "At optical wavelengths, the power above $\\sim $ 3 Hz drops between epochs 1 and 6 by almost an order of magnitude.", "This is most evident when looking at 10 Hz in the $i_s$ -band power spectrum.", "However, at the lowest frequencies it appears more stable.", "This could be interpreted as a Lorentzian component peaking at $\\sim $ 1–2 Hz and becoming less significant as the outburst continues.", "However, this does not mean that the component disappears.", "Furthermore, a small feature is seen to peak at $\\sim $ 30–40 Hz in all optical bands in epochs 4 and 6 (the only bands that extend to this frequency with good statistics.", "Epochs 2 and 3 may show this too, but the uncertainties are too large to confirm this).", "Meanwhile, the X-ray power spectrum behaves in the opposite manner.", "It remains roughly constant between epochs at all frequencies except the lowest, where it drops by an order of magnitude between the earliest and latest epochs.", "All the power spectra show a break at around 1 Hz, and epochs 4 and 6 possibly show higher-frequency breaks at around 40 Hz.", "However, Lorentzian fitting could not sufficiently quantify these breaks, and therefore their validity and cause will instead be left as a topic for future work.", "Regarding the existence of a Quasi-Periodic Oscillation: In epochs 4 and 5, a Quasi-Periodic Oscillation (QPO)-like feature can be seen at $\\sim $ 0.1 Hz.", "While Lorentzian fitting did not significantly improve with an additional component at these frequencies for all bands, an X-ray QPO at these frequencies has been previously detected; the existence and effects of such a feature are discussed in Section REF ." ], [ "Coherence", "Fig.", "REF shows the coherence.", "Overall, this is generally low (<0.1) at all frequencies, which is typical for these sources .", "During all epochs and bands, the optical is more coherent with the X-rays at lower frequencies, and decreases with increasing frequency in every epoch.", "However, the coherence at lower frequencies decreases as the outburst continues, eventually dropping by over an order of magnitude by epoch 6.", "There are numerous smaller features here, but for this work, we will just note the peaks which occur in the later epochs – at 0.1 Hz in epochs 4 and 5, and at 0.3 Hz in epoch 6.", "These will be referred to later in Section REF in the context of a QPO.", "While there is no one relation for the dependence of coherence with optical band, there are discrete sections that do show clear trends.", "Saliently, in the 1–5 Hz range, shorter wavelengths tend to be more coherent than longer ones (particularly in the epochs with the best statistics, such as 4 & 6) – this will be discussed in Section REF .", "There are also sections where the opposite is true – spikes in coherence at the QPO frequency in epochs 4 and 6 are stronger at longer wavelengths.", "These, again, will be noted in Section REF ." ], [ "Phase Lags", "The phase lags can be seen in Figure REF .", "Those in the range 1–10 Hz are roughly the same across all observations, with a shift of +$\\pi $ appearing at around 3 Hz; these reflect the presence and stability of the positively-correlated peak.", "Above 10 Hz, there are few clear trends and it is difficult to make definitive claims; if this regime is dominated by components with <0.1 s delay, then we have many jumps from +$\\pi $ to -$\\pi $ over this period, and log binning would average out this behaviour.", "However, one difference is the behaviour of the phase lags below 0.5 Hz.", "In epoch 1, the phase lags are mostly constant at +$\\pi $ /4, and in epoch 2, they appear to increase towards lower frequencies.", "However, in epochs 3-5 (a month after outburst peak), phase lags change to roughly $\\pm \\pi $ – i.e.", "the two components are roughly in `anti-phase', where the peak of one component coincides with the trough of another (this is the Fourier representation of the anti-correlation component that appears in the CCF – see Figure REF ).", "The transition to this anti-correlation in the phase lags occurs at around 0.2 Hz, where there is a sudden discontinuity; analyses of epochs 4 and 5 are inconclusive in showing whether phase lags increase from $-\\pi $ , or decrease from $+\\pi $ at this discontinuity.", "It is perhaps worth noting that negative phase lags, sometimes approaching $\\pm \\pi $ , are seen at lower frequencies in multiple other sources , , , .", "This lower-frequency behaviour then changes again much later in the outburst during epoch 6, at which time the anti-correlation component is now bounded to a small section at roughly 0.3 Hz, with lower frequencies being generally above 0." ], [ "Time Lags", "The time lags can be seen in Figure REF .", "At frequencies below $\\sim $ 0.2 Hz in epochs 3–5, there is confusion as to whether the time lags are positive or negative – this depends on whether the phase lags are assumed to be positive or negative, which is unclear from Fig.", "REF , as this is the point at which the phase lags are close to $\\pm \\pi $ .", "Figure REF also presents insets over the 1–10 Hz range, showing the similarities over the epochs.", "Shorter frequencies almost uniformly have a smaller lag than longer frequencies over this range; this is only not the case in epochs with poorer statistics (i.e.", "epoch 3) or below 2.5 Hz in epoch 6.", "This wavelength dependence will be discussed in Section REF , with epoch 6 in particular discussed in Section REF .", "In epochs 4–6, a feature can be seen that is similar to a QPO, with significant effects in the coherence and the lags.", "In epochs 4 and 5, this feature is at roughly 0.1 Hz, which increases to 0.3 Hz in epoch 6.", "Each bin with this feature shows (i) an increase in the power spectra, (ii) higher overall coherence (sometimes by an order of magnitude, particularly in epochs 4 and 6), (iii) greater coherence at longer wavelengths, (iv) small error bars in the lags, and (v) negative time lags (changing from -4 s in epoch 4 to -1 s in epoch 6).", "These features are best seen in epochs 4 and 6, where the statistics are better than other epochs.", "This possible QPO can also be seen in the CCFs (Figure REF ).", "A positively correlated component can be seen between -4 and -3 s in epochs 3–5, and at -1 s in epoch 6, as indicated by the time lags seen in Fig.", "REF , often stronger at longer wavelengths.", "We briefly analysed the CCFs to test for the significance of this feature – see Section REF in the Appendix.", "As it turns out, a feature at this frequency is not a new discovery; showed the evolution of a QPO in X-rays over time that corresponds exactly with our feature described here.", "Therefore, there appears to be a connection with this QPO and the features, including a negative lag in the CCF, in our data.", "Indeed, QPOs have been associated with changes in the lags in other LMXB sources previously , , .", "Does this mean that the QPO shows optical variability preceding X-ray?", "Not necessarily; due to the periodic nature of phases (as discussed in Section REF ), phase lags between $\\pi $ –$2\\pi $ radians would be represented as negative lags between -$\\pi $ –0, and this might be occurring here.", "Additionally, the negative lags seen in the CCF could just be a result of the periodic nature of this component; epochs 4 and 6, for example, show a second feature at positive lags (5 s and 2 s respectively).", "These give a time period of 8 s and 3 s respectively between the two features; this matches the period of the QPO in both epochs (roughly 0.125 Hz and 0.3 Hz respectively)We also see this behaviour in epoch 3, where we do not see clear similar QPO features.", "However, the QPO is still detected by during this time at a similar frequency.", "Additionally, with only 222s of correlated time, epoch 3 has the poorest statistics of any of our epochs; this may explain why we do not see such QPO features.. See also Section REF , where we simulate the Fourier components of epoch 6, and show how both the positive and negative correlations disappear when the QPO's Fourier components are removed.", "There is no clear mechanism by which a QPO would directly cause X-ray emission to lag optical emission in this way, while there are a number of models that would show the opposite (e.g.", "see Section REF ); we thus consider the latter case to be the more likely one here." ], [ "The Oddity of Epoch 6", "The QPO described in the previous section cannot, by itself, explain all the lags in epoch 6; the phase lags that are significantly different from other epochs extend over the frequency range 0.08–2.5 Hz, not just around the QPO frequency.", "At these frequencies, shorter wavelengths have a consistently greater lag than longer wavelengths; this is the inverse trait of the sub-second lag seen between 1–5 Hz in other epochs (while this sub-second lag and wavelength dependence is still seen in epoch 6, note also how this new component supersedes it up to 2.5 Hz – Fig.", "REF ).", "The epoch 6 lightcurves show low coherence compared to other epochs ($\\sim $ 0.01 – the only exception here is the 0.3 Hz frequency bin coincident with the QPO).", "As for the lags, over this range, $g_{s}$ , $r_{s}$ , $i_{s}$ and $z_{s}$ bands even have negative lags with respect to X-rays, whereas $u_{s}$ almost always has positive lags at the same frequency.", "This behaviour is also evident in the 2 s CCFs (Figure REF ), where the longer-wavelength $r_{s}$ , $i_{s}$ and $z_{s}$ bands show a rising correlation at negative lags and peak at 0 s, while the shorter-wavelength $g_{s}$ band does not, and the $u_{s}$ band shows an anti-correlation.", "The QPO, along with this different behaviour component, are both strong features in epoch 6.", "To what magnitude, and in what ways, do they affect epoch 6's cross-correlation (Figs.", "REF & REF )?", "To find out, we simulated an approximation of the Fourier components of the X-rays and $i_s$ band of epoch 6, creating a lightcurve for each from these components, and then cross-correlated them.", "We then modified the Fourier components to remove both the Lorentzian responsible for the QPO and the negative lags; for the latter, we instead assumed an interpolated flat distribution of 2$\\pi $ /5 in the phase lags below 2 Hz.", "A CCF was made from these lightcurves as well, and the two results (as well as the inputs) are shown in Figure REF .", "Significantly, it can be seen how the cross-correlation is entirely different between -2 s and 3 s lags, no longer showing the negative correlations unique to epoch 6, nor the positive anti-correlation that is present in epochs 4 and 6.", "From this, we conclude that the QPO and the negative lags are the primary cause for the oddities we see in the epoch 6 CCF.", "For more information, including how each component affects the CCF individually and further evidence of the QPO influencing positive as well as negative lags, see Appendix REF .", "Figure: Two simulations of the i s i_{s} band with X-rays from epoch 6.", "Top: Input Fourier components.", "The red lines are a representation of the data as it was seen in Figs.", "–, and the blue lines are a modification that removes the QPO and the negative lags from the i s i_{s} band's Fourier components between 0.02–2 Hz.", "Bottom: CCFs made by converting the Fourier components into lightcurves and then cross-correlating the results.", "CCFs were averaged over multiple 10s segments.", "Note how the behaviour completely changes between -2 and +3 s, showing the significance of epoch 6's negative lags over this range.Analysis of our results has shown both features that are constant, and ones that are varying in specific ways over the course of the outburst.", "To summarise our main findings: In all epochs, J1820 shows rapid, sub-second red flares, and longer-scale variations that are stronger at longer wavelengths (Fig.", "REF ).", "Over the epochs, an anti-correlation component (stronger at longer wavelengths) around zero lag becomes increasingly significant – until late into the hard state decline, when it is superseded by a positive correlation at negative lags, again stronger at longer wavelengths (Fig.", "REF ).", "In all epochs, the CCF reveals a sub-second peak in the optical/X-ray correlation function at roughly 0.2 s. The peak maintains a roughly similar shape over the epochs, but appears to shrink in comparison to other features(Fig.", "REF ).", "Over time, optical bands become less variable (i.e.", "decrease in power) at higher frequencies, but the variability/power remains roughly constant at lower frequencies.", "For X-rays, this relation is inverted, showing an overall decrease of rms$^2$ power at low frequencies.", "The optical power spectra also have consistently higher rms$^2$ power at longer wavelengths (Fig.", "REF ).", "Coherence at lower frequencies drops as the outburst continues (Fig.", "REF ).", "The phase/time lags are mostly consistent between 1-10 Hz across epochs.", "At lower frequencies, they change from being near +$\\pi $ /4 to being near $\\pm \\pi $ as the outburst progresses.", "Epoch 6, however, fits neither of these trends (Fig.", "REF ).", "All epochs have an interval between Fourier frequencies 1–5 Hz where shorter wavelengths have shorter time lags.", "This behaviour is roughly consistent (aside from in epoch 6), and neither the lag nor separation by wavelength appear to change (Fig.", "REF ).", "In these observations lies evidence for evolving processes within the system.", "We will now address several key points and theories based on these observations." ], [ "Compact Jet", "Jet activity has already been found in this source (e.g.", ", ), and the presence of rapid red variations and a sub-second optical lag that we show in this paper can both result from jet activity .", "Radio data show the source to be relatively bright in the radio, and the long-term lightcurve approximates that in hard X-rays (Fig.", "REF ).", "Meanwhile, presented evidence that the optical emission was likely on the optically-thin tail of synchrotron power-law emission from a jet during April 2018.", "However, another interesting phenomenon ties in with this: the wavelength dependence of the sub-second optical lag.", "In we investigated the data shown in epoch 4, and it was first found that a component of the optical emission lagged the X-rays by roughly 170ms.", "It was also found that this lag was dependent on wavelength; shorter wavelengths lagged less, and longer wavelengths lagged more.", "In the previous paper, it was suggested that this feature is emission from a compact jet.", "In this interpretation, we consider material emitting in hard X-rays close to the compact object from a jet-emitting disk (; though it is also theorised that X-rays may come from the jet itself: , ).", "A portion of this material is then ejected as a jet; with a fluctuating ejection rate, this does not necessarily lead to a uniform stream along the jet, but instead an outflow that varies in density and/or Lorentz factor over time.", "We can interpret this as a series of discrete shells of matter; since these shells vary in speed, faster shells can thus collide with earlier, slower shells.", "When they do, they emit through synchrotron radiation.", "This is the internal shock model , , , the development of which has been motivated by research into Gamma-Ray Bursts (GRBs) and Active Galactic Nuclei (AGN) jets , , .", "found that the jet in this source is highly relativistic ($\\Gamma $ = 6.81); this would mean that a time delay of 170 ms corresponds to roughly 5$\\times 10^{4}$  km between the X-ray and this synchrotron emitting region.", "The energy of this synchrotron emission is dependent upon the variation in the Lorentz factor of the colliding material; a larger gradient produces higher energy dissipation.", "Collisions between larger gradients also occur closer to the compact object, and thus at shorter time lags.", "Since the regions close to the compact object are more compact, synchrotron emission from these regions is more self-absorbed and peaks at shorter wavelength.", "Thus we see shorter time lags for shorter wavelength.", "A difference of 20 ms between $z_s$ and $u_s$ peak lags would, for a highly relativistic jet, correspond to a spatial extent of 6$\\times 10^{3}$  km.", "With the new observations presented in this paper, we have found that this behaviour is also not only present across all our epochs between 1–5 Hz in Fourier frequency, but it also appears to be fairly consistent in that range (with the exception of epoch 6, where a different component has the opposite effect on wavelengths up to 2 Hz) and is independent of the shape of the X-ray power spectrum.", "However, while the behaviour stays more or less consistent, the relative contribution of this process to the overall variability appears to decrease over time; note the decreasing significance of the sub-second peak in Figures REF & REF .", "We also note the significantly changing phase lags; suggested that $\\pm \\pi $ phase lags at low (<$\\sim $ 1 Hz) frequencies could be a sign of Doppler-boosting of a jet in high-inclination systems, which was put forward by .", "However, our analysis (see Figure REF ) now shows that $\\pm \\pi $ phase lags are not a constant feature of this source, and only appear in the short timescales covered by epochs 3-5.", "Additionally, the Coherence also decreases over time as the lags change, similar to what has been seen in GX 339-4 .", "Over this same range, the X-ray power spectra at these frequencies also decrease in strength over time, with a sharp decrease between epochs 1 and 2, where there is also a sharp decrease seen in the CCFs.", "notes that the Lorentzians that can describe the X-ray power spectra move to higher frequency over an outburst, which leads to such decreases in power at low frequencies.", "This is interpreted as resulting from changes in the source geometry.", "What do we know of the evolution of the geometry of the source?", "found that the corona appears to shrink over the course of the hard state, based on a model that assumed a disc that extends to the innermost stable circular orbit.", "In our data, we see a broad anti-correlation, which is more often attributed to a hot flow inside a truncated disc (See Section REF ).", ", meanwhile, describes a radially decreasing corona and also features a truncated disc, inside which is a hot flow.", "In either scenario, an increasingly compact corona could mean that the X-ray emission from it would contribute less to variability at lower frequencies, and would also correspond with a decrease in the significance of the jet component (because both the corona and the jet are linked through fluctuations in accretion power, which heat the X-ray emitting corona and power the jet; thus, changes in one indicate changes in the other; ).", "Overall, the corona becoming more compact would, by itself and its effect on the jet, explain several of features that we see.", "The optical QPO could also be explained by a precessing jet.", "This geometrically-based interpretation has the corona, which is connected to the jet, precessing in such a way that it creates variability in the lightcurves.", "This has been demonstrated in, e.g., , though is still a matter of debate (see, e.g., , ).", "The QPO may also contribute to the anti-correlation around zero lag in epochs 3–5; the high coherence at the QPO frequency would mean that smooth oscillations would be seen in the CCF, and the anti-correlation occurs between the QPO correlation peaks of -3 s and 5 s (also worth noting is that the QPO is stronger at longer wavelengths, a fact which is also true of the anti-correlation).", "It is thus feasible that the QPO contributes to the strength of the anti-correlation, though it need not necessarily be the sole cause of it (for instance, an anti-correlation at negative lags is present in epochs 1 and 2, when no QPO could be seen in the optical power spectra)." ], [ "Truncated Disc and Inner Accretion Flow", "Is the disc truncated, and if so, does its inner radius evolve?", "noted that, using relativistic reflection models, the inner edge of the accretion disc appears to remain steady and close to ISCO during most of the hard state; however, noted the inner radius of the disc being much more truncated, and evolving over time, moving inwards overall (though perhaps in a stochastic fashion).", "Likewise, reported a truncated disc moving closer to the black hole as the hard state evolved.", "If a disc's innermost radius is recessed from the black hole, then there is potential for a hot accretion flow to form.", "Both the observed fast UV/X-ray timing and low optical polarization properties can be explained in terms of this (optically thin, geometrically thick) hot inner flow , , .", "Our observations show several features that could indicate this as a significant process.", "For instance, the CCFs in Figure REF show the presence of an anti-correlation in several of the epochs.", "The anti-correlation can be expected if the hot flow broadband spectrum has a pivoting point, e.g.", "if an increase of mass accretion rate leads to an increase in X-ray luminosity, at the same time causing higher synchrotron self-absorption within the flow (as a result of higher electron number density), thus leading to a drop in optical emission .", "In this scenario, the variability amplitude is higher at energies further away from the pivoting point, hence we expect to have stronger variability at longer wavelengths, as observed (Fig.", "REF ).", "In order to explain the complex anti-correlations at both positive and negative lags in epochs 4–6 in terms of the hot flow scenario, one needs to have two sources of both X-ray and optical emission ; X-rays would be produced by disc and synchrotron Comptonization, and optical by synchrotron emission in the hot flow and irradiated disc emission.", "These features may appear in the spectrum close to the state transition.", "The natural expectation of such scenario is the different shape of the correlation with soft and hard X-rays, which we indeed see (more details in Section REF ).", "The presence of a simultaneous QPO at X-ray and optical wavelengths is another expectation of the hot flow scenario , which seems to be confirmed by our data from epochs 4 and 6.", "A correlated QPO can significantly alter the shape of the CCF and can potentially explain some features of the epoch 6 CCF (see Section REF and Fig.", "REF for more discussion).", "On the other hand, the amplitude of phase lag at the QPO frequency, $\\sim -\\pi /2$ , is not consistent with the expectation of the linear theory, which suggests either 0 or $\\pi $ depending on the system orientation (; though it is worth noting that the QPO phase lag was closer to $\\pm \\pi $ at earlier epochs).", "Furthermore, the lag at the QPO frequency can be altered by the aperiodic component – however, quantitative conclusions on this possibility can only be drawn from dedicated simulations, which are beyond our present scope.", "Alternatively, if we assume that the true phase lag is positive (i.e., shifted by $2\\pi $ ), the reprocessing signal can contribute to the QPO : 0.3 Hz if within the range of frequencies at which the reprocessed QPO is not smeared out by the light travel delays.", "The hot accretion flow scenario can explain most of the changing components in the CCF from different epochs, but not the steady narrow peaks at sub-second lags.", "The fast optical correlation, most probably coming from a separate emission component, has to be added to the hot flow contribution to get the overall CCF shape consistent with the data." ], [ "Epoch 6 and the Emergence of Superhumps", "Towards the end of the hard state, a superhump modulation at a period of $\\sim $ 0.7 days was first reported in the optical light-curve of J1820 by , and then later expanded upon in .", "This signal appeared around day 87 (MJD 58275), with post-hoc analysis revealing that it may have appeared as early as MJD 58272.", "Epoch 6 took place on MJD 58276.2, very soon after the superhump appeared.", "Considering the times of maximum light noted in , and assuming a period of 0.7 days, a maximum occurred at MJD 58276.23, essentially concurrent with epoch 6.", "To date, there have been very few studies into the effect of superhumps on optical/X-ray correlations.", "Given that superhumps are considered to be a property of the outer disc (see ), the timescales involved will correspond to the light-travel time to the disc's tidal radius, which for J1820 will be $\\sim $ 10s, and hence any correlated variations are likely to be heavily smeared, compared to the timescales being studied here.", "Actually, optical/X-ray CCFs were constructed for the black hole LMXB Swift J1753.5–0127, and were found to be independent of the superhump period present in that system (see Section 3 of , and note that “orbital-like modulation” refers to superhumps).", "However, there have been no studies that examined phase lags in this scenario.", "Thus, this avenue of research would be valuable in investigating whether or not they contribute to the features we see in epoch 6, and, by extension, might be affecting the optical/X-ray correlations and variability of LMXB systems as a whole.", "Further studies of J1820's superhump properties can be found in Thomas et al.", "(2021; Subm.", ")." ], [ "A Combined Jet and Hot Flow Model", "Let us now link our findings to the various models presented.", "The source shows repeated rapid red flares, and a sub-second optical/X-ray correlation that has a larger lag at longer wavelengths.", "The components dominating the correlation at low frequencies change as the hard state evolves; the X-ray power spectra and the optical/X-ray coherence both decrease at these frequencies, and the phase lags move towards $\\pm \\pi $ .", "The source also becomes softer over time, and the sub-second lag in the cross-correlations becomes less significant.", "Meanwhile, the X-ray power and the coherence at higher frequencies remains static.", "We do not find that the donor star is an explanation for our features; while the star could theoretically produce a correlated component at positive lags in our CCFs through X-ray heating and reprocessing, combining mass and orbital period estimates from [2] and with Kepler's third law gives the distance between the compact object and donor to be $\\sim $ 16 light-seconds, and the effect in the lags would likely vary between epochs as we observe different phases, in disagreement with either the smooth evolution or constant nature of the correlated components we see.", "However, given the high system inclination ($\\sim 75$ , ), the shortest delays between X-ray and (reprocessed) optical photons from the near-side of the disc are expected to be about $\\sim 0.5$  s, with some additional smearing to longer lags due to light travel times across the face of the disc.", "Hence, it is possible that X-ray reprocessing off the accretion disc could be significant to the variability; this can be tested in future by comparing these results to similar soft-state observations, where the illuminating component should be more dominant.", "In all, we suggest a two-component model; one correlated, and the other anti-correlated.", "The correlated component we ascribe to a compact jet, which becomes less significant over time.", "The anti-correlated component, meanwhile, we ascribe to a hot flow, which remains static.", "A jet as the correlated component would explain the red flares, the optical/X-ray sub-second correlation , and the larger lag at longer wavelengths , .", "X-rays coming from the inflow would contribute more to the X-ray variability at the lowest (<0.1 Hz) and the highest (>1 Hz) frequencies.", "If the corona is contracting (Evidenced either by a change in the vertical extent, as in , or a change in the radial extent and a decreasing disc truncation radius, as in ), the variability of hard X-rays from that corona would decrease at lower frequencies, as would the optical/X-ray coherence over the same range – while the jet, closely linked to the corona, would also decrease in significance, leading to the decline of the sub-second correlation.", "The latter, anti-correlated, component we ascribe to the hot flow.", "This component stays mostly static, and thus, relatively, contributes more to the overall variability as the jet declines in significance.", "A hot flow scenario could feasibly also explain the QPO that we see in the data.", "The hot flow does not appear to increase in significance – note that the coherence does not increase." ], [ "Beyond the Jet and Hot Flow", " and reported the detection of optical and near-infrared winds respectively in J1820.", "The effect of winds on optical/X-ray timing correlations has not yet been explored in depth, however they would occur on similar timescales to those studied here.", "V404 Cyg is a similar system to J1820 (albeit with a much longer orbital period of 6.47 days and thus a larger physical scale; ); in that source, the wind launching zone was found to be on the order of a few $\\times 10^5$  km , or about 0.5 lightseconds.", "For a source inclination of 75$$ , and using eq.", "(4) in , we get minimum lags on the order of 0.01 s, so contribution of the wind to the CCF timescales that we probe is feasible from a timing standpoint.", "However, the shallowness of the P Cygni absorption feature (1–2 percent below the continuum level, ) implies that the wind is optically thin, which would mean that there would be minimal reprocessed emission due to the wind.", "Further investigation into this possibility would require better data on the optical depth and the ionization of the wind, combined with simulations." ], [ "Conclusions", "We have presented analysis of optical and X-ray lightcurves from the black hole Low-Mass X-ray Binary (LMXB) MAXI J1820+070 over the course of roughly 80 days.", "In doing so, we show an evolving Cross Correlation Function (CCF) at longer ($\\sim $ 10 second) timescales, a consistent sub-second correlation, and various changes in the Fourier components, including differences between different optical wavelengths.", "This paper thus shows both the dynamic and static nature of LMXBs, even over a single outburst.", "The shifting of phase lags at lower frequencies, the slowly climbing photon index, and the increasingly significant anti-correlation shows how the coherent components can change on a timescale of weeks.", "Meanwhile, the constant nature of the correlation at sub-second lags, mid-frequency time lags, and rapid red flares in the lightcurves show that other components are more stable, and can be present with broadly static properties more than two months apart.", "Additionally, it shows how a Quasi-Periodic Oscillation (QPO), travelling upwards through the Fourier frequencies, can change the resultant lags and correlation features.", "We discuss our findings in terms of two synchrotron-emitting components – a correlated jet and an anti-correlated hot flow – as major contributors to the overall variability.", "If we allow for the jet to dominate at the lowest (<0.1 Hz) and the highest (>1 Hz) frequencies, and the hot flow to dominate in between, the interaction of these components can create the features we observe in several epochs.", "A correlated component at negative lags can be seen in several epochs.", "Fourier analysis showed this component to be related to the frequency of a QPO in both the optical and X-ray lightcurves, previously reported in X-rays by .", "The lightcurves are consistently coherent at these frequencies, with greater coherence (and thus correlation) at longer wavelengths.", "As the QPO increases in frequency over the outburst, the lag also evolves, becoming less negative.", "We note that, due to the periodic nature of the QPO, this negative lag could easily be a Fourier artefact, and the true lag is positive, with X-ray variability leading optical by several seconds.", "Epoch 6 shows us features that are more difficult to understand.", "Between 0.08 to 2.5 Hz, there is some component that causes a drop in optical/X-ray phase lags.", "This component is more significant at longer wavelengths, and the lags become negative in most bands.", "The QPO mentioned earlier is in the middle of these frequencies, but there is no indication as to whether it is related or not.", "Further observations of LMXBs close to the intermediate state would be highly desirable to investigate this.", "The evolution of the optical/X-ray correlations over the course of an LMXB's outburst remains an area rich with possibility for new discoveries.", "This paper highlights the fact that further, more frequent investigations of an LMXB over its hard state (and, ideally, over the transition to the soft state) would be invaluable in further decoding the shifting phenomena inside these sources." ], [ "Data Availability", "The NICER data underlying this article are available in the HEASARC Data Archive (https://heasarc.gsfc.nasa.gov/docs/archive.html).", "The ULTRACAM and HiPERCAM data will be shared on reasonable request to the corresponding author.", "The Swift data from Fig.", "REF are available from the Swift Archive (https://www.swift.ac.uk/archive/).", "The AMI-LA data from Fig.", "REF are available from (https://www.nature.com/articles/s41550-020-1023-5).", "We acknowledge support from STFC and a UGC-UKIERI Thematic Partnership.", "We would like to thank the anonymous referee for their helpful comments.", "We would also like to thank Joe Bright, Piergiorgio Casella, Rob Fender, Adam Ingram, Sera Markoff, Sara Motta, Tom Russell, Gregory Sivakoff, and Alex Tetarenko for their helpful conversations.", "We thank our ULTRACAM observers Paul Chote, Martin Dyer, and Anna Pala.", "We also thank Keith Gendreau, Zaven Arzoumanian, and the rest of the NICER team for their assistance in coordinating observations.", "JAP is part supported by a University of Southampton Central VC Scholarship, and thanks D Ashton for spectral timing help, as well as A Stevens and D Huppenkothen for help with the Stingray software.", "TS thanks the Spanish Ministry of Economy and Competitiveness (MINECO; grant AYA2017-83216).", "KR acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.", "694745).", "AV acknowledges the Academy of Finland grant 309308 and the International Space Science Institute (ISSI) in Bern, Switzerland for support.", "This work was supported by the Programme National des Hautes Energies of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES.", "HiPERCAM and VSD were funded by the European Research Council (FP/2007–2013) under ERC-2013-ADG grant agreement no.", "340040.", "ULTRACAM and VSD are funded by the STFC.", "FMV acknowledges support from STFC under grant ST/R000638/1.", "HiPERCAM observations were made with the GTC telescope (Spanish Observatorio del Roque de los Muchachos, Instituto de Astrofísica de Canarias), under Director's Discretionary Time.", "SMARTNet helped to coordinate observations.", "We have made use of software and web tools from the High Energy Astrophysics Science Archive Research Center (HEASARC), and made use of data and the 'Build XRT Products' tool supplied by the UK Swift Science Data Centre at the University of Leicester." ] ]
2105.11769
[ [ "Wormhole on the Brane with Ordinary Matter: The Broader View" ], [ "Abstract In this paper we attempt to examine the possibility of construction of a traversable wormhole on the Randall-Sundrum braneworld with ordinary matter employing the Kuchowicz potential as one of the metric potentials.", "In this scenario, the wormhole shape function is obtained and studied, along with validity of Null Energy Condition (NEC) and the junction conditions at the surface of the wormhole are used to obtain a few of the model parameters.", "The investigation, besides giving an estimate for the bulk equation of state parameter, draws important constraints on the brane tension which is a novel attempt in this aspect and very interestingly the constraints imposed by a physically plausible traversable wormhole is in high confirmity with those drawn from more general space-times or space-time independent situations involved in fundamental physics.", "Also, we go on to claim that the possible existence of a wormhole may very well indicate that we live on a three-brane universe." ], [ "INTRODUCTION", "The idea of a wormhole can be achieved from a set of solutions of the Einstein Field Equation (EFE), materializing into a tunnel-like structure connecting two different regions of space-time of same universe or two different universe, can be traced back to the Einstein-Rosen bridge  [1].", "However, the term `wormhole' was coined by Fuller and Wheeler  [2], they showed that such solutions of the EFE were unstable as the throat of the wormhole would pinch-off, thus trapping any traversing signal in an infinite curvature region making the wormhole non-traversable.", "The modern idea of traversable wormholes was introduced first by Morris and Thorne [3], who were studying wormhole solutions as a tool for a simpler understanding of General Relativity (GR).", "They prescribed that in order to have stable traversable wormholes, the corresponding energy-momentum tensor for the wormhole must violate the null energy condition (NEC) of GR, such that the pinching-off of the throat can be prevented.", "Further, if the wormhole can be sustained, it can allow the possibility of time-travel, violating causality [4].", "Such `exotic' matter which violates the NEC has not been observed till date, but also finds existence in theories attempting to explain the present cosmic acceleration [5], [6], [7], [8].", "Traversable wormholes admitting exotic matter have been studied in [9], [10], [11], [12], [13], [14], [15].", "There have been a number of attempts to modify GR at higher energy scales [16], [17] in order to avoid the singularity problems [18] arising in cosmology and black hole physics, and also at energy scales corresponding to the present epoch of cosmic evolution to accommodate naturally the current cosmic acceleration without incorporating exotic matter [19], [20].", "The idea of braneworld [21], [21] involving a higher dimensional approach to modify GR was studied in [22] from a purely geometrical approach.", "Braneworlds have been used very effectively as a platform to investigate cosmological [23], [24], [25], [26], [27], [28], astrophysical [29], [30], [31], [32], [33] and collapse [34], [35], [36], [37] problems.", "Due to the non-locality and non-closure properties of the modified EFE on the brane, the Minimum Geometric Deformation (MGD) approach introduced in [38], [39] is a key tool for studying collapse problems on the brane.", "Wormhole solutions in the context of modified gravity without accounting for extra dimensions have been investigated in [40], [41], [42], [43], [44], [45], [46].", "In higher dimensional modified gravity approach,wormhole solutions can be found in [47], [48], [49], [50], [51], [52], [53].", "The possibility of existence of wormholes in the central and outer region of galactic halos has been investigated in [54] and  [55], [56], respectively.", "We attempt to investigate the possibility of construction of a traversable wormhole on the braneworld, such that the metric potential in the temporal direction is of the Kuchowicz type [57].", "Also,due to lack of experimental evidence in favour of exotic matter so far, we choose to consider a linear Equation of State (EOS) corresponding to perfect fluid matter on the brane, describing ordinary matter.", "In the present article we have studied the mathematical model of the wormhole on the brane including the modified EFE on the braneworld, solution for the wormhole shape function, validity of the NEC on the brane for ordinary perfect fluid matter and obtaining the possible numerical values of the constants from the junction conditions for plotting the physical parameters of interest pertaining to the wormhole.", "We will then discuss the physical relevance of the obtained solutions and plots for the physical parameters and draw conclusion based on it in the following section." ], [ "Mathematical Model of the Brane Wormhole", "In this section we will try to construct a detailed mathematical model of the brane wormhole for linear EOS, where one of the metric potentials is of the Kuchowicz type.", "We will start with the modified EFE on the Randall-Sundrum (RS) II braneworld." ], [ "Modified EFE on the Brane", "We begin with a the general form of the modified EFE on the 3-brane [22] and move on to obtain the set of EFEs for the case of a spherically symmetric, static metric.", "The RS II braneworld model considers a single 3-brane embedded in higher dimensional bulk having one extra dimension which is the compact space $S_1/\\bf {Z}_2$ , where $\\bf {Z}_2$ refers to the orbifold symmetry or the reflection symmetry.", "The general form of the modified EFE on the 3-brane is given as $G_{\\mu \\nu } = - \\Lambda g_{\\mu \\nu } + \\kappa ^{2} T_{\\mu \\nu }^{mod},$ which may be expanded as $G_{\\mu \\nu } = - \\Lambda g_{\\mu \\nu } + \\kappa ^{2} T_{\\mu \\nu }+ \\kappa _{5}^{4} S_{\\mu \\nu } - E_{\\mu \\nu },$ such that the modified stress-energy tensor $T_{\\mu \\nu }^{mod}$ contains the corrections to the standard EFE, besides the matter stress-energy tensor $T_{\\mu \\nu }$ on the brane.", "Here, the 4-dimensional constant $\\kappa ^{2}=\\frac{1}{6} \\sigma \\kappa _{5}^{4}$ ,where $\\sigma $ denotes the brane tension.", "In the high energy regime the brane tension is small and tends to zero for extremely high energies.", "We will take $\\kappa ^{2}=8\\pi G=1$ on the brane.", "The effective cosmological constant on the brane is given by $\\Lambda =\\frac{1}{2}\\kappa _5^2\\left(\\Lambda _5+\\frac{1}{6}\\kappa _5^2 \\sigma ^2\\right).$ In the RS II set-up the bulk is Anti-de Sitter ($AdS_5$ ),making the bulk cosmological constant $\\Lambda _5$ negative in magnitude,and it is fine-tuned with the brane tension in such a way as to give the value of $\\Lambda $ to be zero.", "The corrections to the EFE are of two types-(i) Local correction ($S_{\\mu \\nu }$ ) which comprises of terms quadratic in the stress-energy tensor on the brane,and (ii) Non-local correction $( E_{\\mu \\nu })$ which comprises of the projection of the bulk Weyl tensor on the brane, transferring the gravitational effect from the bulk.", "The local correction is given by $S_{\\mu \\nu } = \\frac{1}{12} T T_{\\mu \\nu } - \\frac{1}{4} T_{\\mu \\alpha } T^{\\alpha }_{\\nu } + \\frac{1}{24} g_{\\mu \\nu } \\left[ 3 T_{\\alpha \\beta } T_{\\alpha \\beta } - (T^{\\alpha }_{\\alpha })^{2} \\right], $ where $T = T^{\\alpha }_{\\alpha }$ and we consider matter on the brane with perfect fluid stress-energy tensor $T_{\\mu \\nu }=\\rho u_{\\mu } u_{\\nu }+ ph_{\\mu \\nu }.", "$ Here $u_{\\mu }$ is the 4-velocity,$h_{\\mu \\nu }=g_{\\mu \\nu }+u_{\\mu } u_{\\nu }$ is the induced metric on the brane, $\\rho $ and $p$ are the energy density and pressure.", "The non-local correction has the form ${E}_{\\mu \\nu } = C^{(5)}_{ACBD} n^{C} n^{D} g_{\\mu }^{A} g_{\\nu }^{B}=-\\frac{6}{\\sigma }\\left[Uu_{\\mu } u_{\\nu }+Pr_{\\mu }r_{\\nu }+h_{\\mu \\nu }\\left(\\frac{U-P}{3}\\right)\\right].$ where $C^{(5)}_{ACBD}$ is the bulk Weyl tensor, $n^C$ is unit normal spacelike vector, $r_{\\mu }$ denotes projected radial vector, $U$ and $P$ denote the bulk energy density and bulk pressure respectively.", "${E}_{\\mu \\nu }$ is symmetric and traceless.", "It has no components orthogonal to the brane.", "For a static, spherically symmetric matter distribution on the 3-brane, the line element is given by $ds^2=-e^{\\nu (r)}dt^2+e^{\\lambda (r)}dr^2+r^2(d\\theta ^2+sin^2\\theta d\\phi ^2).$ The modified EFE on the brane, given by Eq.", "(REF ) can be computed to be $e^{-\\lambda }\\left(\\frac{\\lambda ^\\prime }{r}-\\frac{1}{r^2}\\right)+\\frac{1}{r^2}&=&\\left(\\rho \\left( 1+\\frac{\\rho }{2 \\sigma } \\right) +{\\frac{6 U}{\\sigma }}\\right),\\\\e^{-\\lambda }\\left(\\frac{\\nu ^\\prime }{r}+\\frac{1}{r^2}\\right) -\\frac{1}{r^2}&=&\\left(p +{\\frac{\\rho \\left( p +\\frac{\\rho }{2} \\right)}{\\sigma }}+{\\frac{2U}{\\sigma }}+{\\frac{4 P}{\\sigma }}\\right),\\\\e^{-\\lambda }\\left(\\frac{\\nu ^{\\prime \\prime }}{2}-\\frac{\\lambda ^\\prime \\nu ^\\prime }{4}+\\frac{{\\nu ^\\prime }^2}{4}+\\frac{\\nu ^\\prime -\\lambda ^\\prime }{2r}\\right) &=& \\Bigg (p+{\\frac{\\rho \\left(p+\\frac{\\rho }{2}\\right)}{\\sigma }}+{\\frac{2U}{\\sigma }}-{\\frac{{2 P}}{\\sigma }}\\Bigg ).$ The brane energy-momentum tensor and the overall effective energy momentum tensor are both conserved separately.", "So,the conservation equation on the brane reads the same as in GR $ \\frac{dp(r)}{dr}=-\\frac{1}{2}\\frac{d\\nu (r)}{dr}(p(r)+\\rho (r)).$" ], [ "Solution for the Wormhole Shape Function", "The line element for a static,spherically symmetric wormhole can be written as $ds^2=-e^{\\nu (r)}dt^2+\\frac{dr^2}{1-\\frac{b(r)}{r}}+r^2(d\\theta ^2+sin^2\\theta d\\phi ^2), $ where $b(r)$ denotes the shape function of the wormhole.", "Rewriting the above field equations (REF )-() for a wormhole with static and spherically symmetric matter distribution,in terms shape function $b(r)$ we get eventually $\\frac{b^\\prime }{r^2}&=&\\rho \\left(1+\\frac{\\rho }{2\\sigma }\\right) +{\\frac{6(X\\rho +Y)}{\\sigma }}, \\\\\\left(1-\\frac{b}{r}\\right)\\left(\\frac{\\nu ^\\prime }{r}+\\frac{1}{r^2}\\right) -\\frac{1}{r^2}&=&p +{\\frac{\\rho \\left(2 p +\\rho \\right)}{2\\sigma }}+{\\frac{2(X\\rho +Y)}{\\sigma }}+{\\frac{4 \\omega (X\\rho +Y)}{\\sigma }}\\\\\\left(1-\\frac{b}{r}\\right)\\left(\\nu ^{\\prime \\prime } + {{\\nu ^\\prime }^2} +\\frac{\\nu ^\\prime }{r} \\right) -\\frac{b^\\prime -b}{2r}\\left({\\nu ^\\prime } +\\frac{1}{r} \\right)&=&p+{\\frac{\\rho \\left(2 p +\\rho \\right)}{2 \\sigma }}+{\\frac{2(X\\rho +Y)}{\\sigma }}-{\\frac{2 \\omega (X\\rho +Y)}{\\sigma }}.", "$ Here we have considered the bulk EOS $P=\\omega U$  [58] and the energy density of the brane and bulk are related by $U=X\\rho +Y$  [59].", "Now,we consider the metric potential $e^{\\nu (r)}$ to be of the Kuchowicz type [57] which is given as $e^{\\nu (r)}=e^{Br^2+2\\ln C},$ where $B$ and $C$ are arbitrary constants.", "The dimension of $B$ is $[L^{-2}]$ and $C$ is a dimensionless constant.", "This metric potential is singularity free and well behaved.", "As argued earlier,due to lack of observational confirmation of exotic matter which is usually preferred for constructing traversable wormholes, we chose to propose an ansatz for the EOS describing ordinary matter on the brane as follows $ p(r)=\\mu \\rho (r),$ where $\\mu >0$ .", "Using Eqs.", "(REF ), (REF ) and (REF ) we get $ \\rho (r)=C_1e^{-\\frac{(\\mu +1) Br^2}{2\\mu }},$ where $C_1$ is the integration constant which can be determined from the matching condition.", "The variation of the energy density $\\rho (r)$ on the brane along the radial distance $r$ is plotted in Fig.", "1.", "The figure clearly indicates that the matter density at the throat is much higher and gradually decreasing with respect to the radial parameter attains a minimum value at the boundary.", "Figure: Variation of the density with respect to rr.The obtained $\\rho $ can be plugged in along with Eq.", "(REF ) in the field Eq.", "() to obtain the shape function for the wormhole given by $&&b(r) =\\frac{2r e^{-2Br^2}}{Bk^4\\sigma }\\left(-\\frac{ \\mu {C_1}(\\sigma \\mu k^4-2 X (\\omega -1)){e^{\\frac{B r^2(3\\mu -1)}{2\\mu }}}}{(3\\mu -1)(2Br^2+1)}+\\frac{k^4}{(\\mu -1)(2Br^2+1)}\\right.\\nonumber \\\\&& \\left(-\\frac{{C_1}^2(2\\mu +1) \\mu { e^{{\\frac{Br^2(\\mu -1)}{\\mu }}}}}{4}\\left.+\\frac{\\sigma (\\mu -1)}{2} \\left( B (2 B r^2+1)+ Y (\\omega -1) \\right)e^{2Br^2}+C_2 B \\right) \\right) $ where $C_2$ is the integration constant.", "Figure: Variation of the shape function with respect to rr.The shape function is found to be dependent on all the brane and bulk parameters including the brane tension.", "The variation of the shape function $b(r)$ along the radial distance $r$ has been plotted in Fig.", "2.", "The properties of the shape function that can be inferred from the plot and its feasibility in construction of wormhole has been discussed later in the concluding section." ], [ "Validity of NEC", "One of the most important energy conditions appearing in GR is the null energy condition given by $T_{\\mu \\nu }k^{\\mu }k^{\\nu } \\ge 0 $ , where $k^{\\mu }$ denotes a null vector.", "This reduces to $\\rho +p \\ge 0 $ for a perfect fluid.", "So, for the ordinary matter we have considered on the brane having EOS of form given by Eq.", "(REF ), the NEC holds good.", "However, as we are concerned with the effective stress-energy tensor on the brane due to the local and non-local correction terms, so it is the effective energy density and pressure that we must consider for constructing the null energy condition on the brane.", "Hence,the NEC becomes $\\rho ^{eff}(r)+p_{r}^{eff}(r) \\ge 0 $ , where it is to be noted from Eqs.", "() and () that the effective pressures in the radial and transverse directions differ due to difference in contribution from the bulk pressure term.", "Here, $\\rho ^{eff}(r)=\\left[\\rho (r) \\left( 1+\\frac{\\rho (r) }{2 \\sigma } \\right) +{\\frac{6 U}{\\sigma }}\\right], $ and $p_{r}^{eff}(r)=\\left(p \\left( r \\right) +{\\frac{\\rho \\left( r \\right) \\left( p \\left( r \\right) +\\frac{\\rho (r)}{2} \\right)}{\\sigma }}+{\\frac{2U}{\\sigma }}+{\\frac{4 P}{\\sigma }}\\right).", "$ The sum of the effective energy density and effective radial pressure can be computed to be $&&\\rho _{eff}+p_{(r)eff}(r))\\nonumber \\\\&=&\\frac{1}{k^4\\sigma } \\left(k^4{C_1}^2 (\\mu +1)e^{-\\frac{(\\mu +1) Br^2}{\\mu }}+4C_1 \\left(\\frac{1}{4} \\sigma (\\mu +1) k^4+X(\\omega +2)\\right) e^{-\\frac{(\\mu +1) B r^2}{2\\mu }}+4Y (\\omega +2) \\right).", "$ Figure: Variation of the NEC with respect to rr.The variation of $\\rho _{eff}(r)+p_{r}^{eff}(r)$ along $r$ has been plotted in Fig.", "3.", "We find that the NEC is violated for effective matter distribution on the brane." ], [ "The Junction Conditions", "The exterior of the wormhole is completely vacuum.", "So the solution for the exterior spacetime line metric of wormhole must be Schwarzschild type can be written as $ds^2=\\left(1-\\frac{2M}{r}\\right)dt^2-\\left(1-\\frac{2M}{r}\\right)^{-1}dr^2-r^2(d\\theta ^2+\\sin ^2 \\theta d\\phi ^2).$ where M is the total mass of the wormhole.", "Presence of matter on the surface of wormhole must produces an extrinsic discontinuity which generates an intrinsic surface energy density and surface pressure.", "The surface (i.e., the shape function) behaves as the junction between the space-time of wormhole and exterior which makes the wormhole geodesically complete manifold with the ordinary matter and characterizes the configuration of wormhole.", "Thus according to the fundamental junction condition there should be a smooth matching between the two spacetimes at the junction.", "Again the metric coefficients are continuous at the junction surface do not confirm the continuity of derivatives of these coefficients there.", "So to determine the surface stress-energy $S_{ij}$ one can use formalism suggested by Darmois-Israel [60], [61].", "Now intrinsic surface stress energy tensor $S_{i}^{j}$ can be obtain from Lanczos equation [62], [63], [64], [65] in the following form $S_{j}^{i}=-\\frac{1}{8\\pi } (\\kappa _{j}^{i}-\\delta _{j}^{i} \\kappa _{k}^{k}),$ the discontinuity in the second fundamental form can be written as $\\kappa _{ij}=\\kappa _{ij}^{+}-\\kappa _{ij}^{-},$ whereas the second fundamental form is given by $\\kappa _{ij}^{\\pm }=-n_{\\nu }^{\\pm }\\left[\\frac{\\partial ^{2}X_{\\nu }}{\\partial \\xi ^{i}\\partial \\xi ^{j}}+\\Gamma _{\\alpha \\beta }^{\\nu }\\frac{\\partial X^{\\alpha }}{\\partial \\xi ^{i}}\\frac{\\partial X^{\\beta }}{\\partial \\xi ^{j}} \\right]|_S,$ where the unit normal vector $n_{\\nu }^{\\pm }$ are defined as $n_{\\nu }^{\\pm }=\\pm \\left|g^{\\alpha \\beta }\\frac{\\partial f}{\\partial X^{\\alpha }}\\frac{\\partial f}{\\partial X^{\\beta }}\\right|^{-\\frac{1}{2}}\\frac{\\partial f}{\\partial X^{\\nu }},$ with $n^{\\nu }n_{\\nu }=1$ .", "Here $\\xi ^{i}$ is the intrinsic coordinate of the surface of the wormhole and $f(x^{\\alpha }(\\xi ^{i}))=0$ is the parametric equation of it.", "Here $+$ and $-$ corresponds to exterior and interior spacetime respectively of the wormhole.", "Now surface stress energy tensor, for the spherically symmetric space time, can be written as $S_{i}^{j}= diag(-\\Sigma ,\\mathcal {P})$ .", "Where $\\Sigma $ and $\\mathcal {P}$ is the surface energy density and surface pressure respectively, which can be obtained at the surface of the wormhole (i.e at $r=R$ ) by the following equations $\\Sigma &=&-\\frac{1}{4\\pi R}\\bigg [\\sqrt{e^\\lambda }\\bigg ]_-^+\\nonumber \\\\&=&\\frac{1}{4\\pi R}\\left[\\sqrt{\\left(1-\\frac{2M}{R}\\right)} - \\sqrt{1-\\frac{2 e^{-2 BR^2}}{Bk^4\\sigma }\\left(-\\frac{\\mu C_1 (\\sigma \\mu k^4-2 X ( \\omega -1)){e^{\\frac{BR^2 (3\\mu -1) }{2\\mu }}}}{{(3\\mu -1)(2B R^2+1)}}+H_1\\right)} \\right]$ $\\mathcal {P} & =&\\frac{1}{16\\pi R } \\bigg [\\bigg (\\frac{2f-f^\\prime R}{\\sqrt{f}}\\bigg ) \\bigg ]_-^+ =\\frac{1}{16\\pi R}\\left[\\frac{2(1-\\frac{M}{R})}{ \\sqrt{1-\\frac{2M}{R}}}-\\frac{F1}{F2}\\right],$ where $H_1= -\\frac{{C_1}^2 (2\\mu +1) k^4\\mu {e^{{\\frac{BR^2 (\\mu -1)}{\\mu }}}}}{4(\\mu -1)(2B R^2+1)}+ \\frac{(4B^2k^4R^2\\sigma +Y(\\omega -1) ){ e^{2 BR^2}}+C_2 B k^4\\sigma }{4(2B R^2+1)}$ , and $&&F1=\\frac{1}{k^4\\sigma ( 3\\mu -1 )(\\mu -1)(2Br^2+1 )^{2}B}\\left(-4(\\sigma \\mu k^4-2X (\\omega -1))(\\mu -1)\\left((B^2r^4+\\frac{Br^2}{2}-1) \\mu + \\right.\\right.", "\\nonumber \\\\&&\\left.", "\\left.B^2 r^4+\\frac{Br^2}{2}\\right){C_1} e^{-{\\frac{Br^2( \\mu +1 )}{2\\mu }}}+\\left(-3 k^4{C_1}^2(2\\mu +1)\\left(\\left(2 B^2 r^4+Br^2-1\\right) \\mu +2B^2 r^4+Br^2\\right){e^{-{\\frac{Br^2(\\mu +1)}{\\mu }}}}+\\right.\\right.", "\\nonumber \\\\&& \\left.", "\\left.2(\\mu -1)\\left(C_2 B k^4\\sigma (2 B^2 r^4+2Br^2-1){e^{-2Br^2}}-Y(\\omega -1)\\right)\\right)(3\\mu -1)\\right),\\nonumber \\\\$ and $F2=\\sqrt{1-\\frac{2 e^{-2 BR^2}}{Bk^4\\sigma (2B R^2+1)}\\left(-\\frac{\\mu C_1 (\\sigma \\mu k^4-2 X ( \\omega -1)){e^{\\frac{BR^2 (3\\mu -1) }{2\\mu }}}}{{(3\\mu -1)}}+H_1\\right)}.\\nonumber \\\\$ Now for the static wormhole the surface pressure as well as the surface energy must vanishes at the boundary surface.", "So one must i.e., we have $\\sigma =\\mathcal {P}=0$ at boundary which eventually provides the condition $b(R)=2M.$ In order to calculate the different unknown constants we have used the following boundary conditions, (i) From the junction condition at the boundary (from Eq.", "(REF )) $b(R)=2M$ .", "(ii) Continuity of the metric potential $g_{tt}$ and its derivative $\\frac{\\delta g_{tt}}{\\delta r}$ at the surface boundary i.e., at $r=R$ .", "Now for best fit results we choose $\\mu = .501,\\ \\sigma = 10^3, M=1.7,\\ r_0=0.75,\\ R=4 $ , where $r_0$ is the throat radius.", "Making use of the above conditions, we obtain the values different parameters associated with the wormhole as: $X = -1943.448951, Y = 1684.121721, C_1 = 64.85766573, C_2 = -61.40896649,\\ \\omega = 0.9157485630$ .", "These values of the parameters have been used to plot Figs.", "(REF )-(REF ).", "All the plots have been started from the throat (i.e.", "$r=r_0)$ upto the surface boundary (i.e at $r=R$ )." ], [ "DISCUSSIONS AND CONCLUSION", "In this work, we have tried to construct a traversable wormhole on the brane without using exotic matter, as such matter has not yet been observed.", "To obtain physically valid solution for a traversable wormhole one must have the metric potential $g_{tt}$ finite through out the wormhole.", "On considering the Kuchowicz type metric potential, as discussed earlier,it automatically satisfies the desired condition.", "To obtain the shape function for a static, spherically symmetric wormhole on the brane we have solved Einstein field equations along with the Kuchowicz type metric potential and studied its properties in details to check the viability of our model.", "We have also estimated values of the model parameters from junction conditions.", "As we shall see, we can find a satisfactory estimate of the bulk EOS parameter besides drawing constraints on the brane tension.", "In this section we are going to conclude some of the important results that we have obtained from our present study.", "1.", "Matter density: By solving the conservation equation along with the EOS we have obtained the matter density of the wormhole and its variation has been shown in Fig.", "REF .", "From the plot of density with respect to radial distance confirms that the matter near the throat is much denser than the boundary.", "The variation shows that the density is gradually decreasing from throat to the boundary of the WH.", "2.", "Properties of Shape function: For the construction of a traversable wormhole, the shape function plays an important role.", "In order to obtain a physically acceptable traversable wormhole the shape function must satisfy a number of conditions [3].", "Here we discuss those conditions and corresponding behaviour for our model (i) First of all at the throat radius $r=r_0$ , the shape function $b(r)$ must be equal to the throat radius $r_0$ itself.", "From Fig.", "REF one can observe that our model satisfies this condition.", "(ii) The metric coefficient $(g_{rr})$ must be regular and well behaved i.e., $b(r)<r$ for $r>r_0$ .", "Again from Fig.", "REF we have observed this condition is also satisfied for our present study of wormhole.", "(iii) In addition to the above, the nature of the plot for the shape function must be such that if the curve is rotated with respect to the $b(r)$ axis, the $b(r)>0$ half for both positive and negative $r$ , must approximately resemble the upper half of a wormhole.", "Fig.", "REF tells us that it does so.", "3.", "Flaring-out condition: To have a stable wormhole, it must obey the flaring-out condition at the throat in order to ensure that there is no pinch-off making the wormhole non-traversable.", "The flaring-out condition states that the first order derivative of the shape function with respect to $r$ at the throat must be less than 1.", "Now, this implies that the null energy condition (NEC) must be violated.", "We see from Eq.", "(REF ) that it is not, but on the brane we are interested only about the effective matter distribution which as seen from Eq.", "(REF ) and Fig.", "REF violates NEC.", "This is a huge advantage of considering wormhole on the brane as contrasted to GR, because we find that there is no need to consider exotic matter but the effective matter description due to local and non-local higher dimensional corrections enforce violation of NEC even with ordinary matter on the brane.", "4.", "Constraints on brane tension and estimate for $\\omega $: Further the value of the bulk EOS parameter $\\omega $ which we obtain from the matching condition is 0.9157485630 which is close to +1.", "This justifies the bulk space to be $AdS_5$ (negative cosmological constant).", "We find two other new and interesting features from our wormhole model concerned with the brane tension: (i) As the value of the brane tension $\\sigma $ is lowered from the value $10^3 TeV^4$ as considered for our analysis,we see that on lowering $\\sigma $ upto $1 TeV^4$ ,our wormhole solution is valid but on lowering $\\sigma $ further in $0.01-0.99 TeV^4$ range, the essential conditions in 2) for wormhole formation are violated, also the condition for violation of the NEC for effective matter description on the brane no longer holding true.", "Thus, we can claim that from our wormhole model we draw a constraint on the brane tension such that always $\\sigma >1$ .", "This is exactly in confirmation with the constraint drawn on the brane tension from the perspective of fundamental investigations [16], [49].", "A possible explanation of this may arise from the fact that, as we approach extremely high energy scales $\\sigma \\rightarrow 0$ , as discussed before.", "So, for energy scales corresponding to $\\sigma < 1$ , quantum gravity effects must dominate and the effective description on the brane is no longer valid.", "(ii) If we vary $\\sigma $ further below 0, it turns out that the values of the parameters to be obtained from the matching conditions become imaginary,which strongly rules out the possibility of a negative tension brane.", "Such a brane is well known to exhibit instabilities from two different fundamental perspectives of black hole [66] and gravitational [67] physics.", "There have recently been attempts to test the idea of braneworld against observations [68], [69], [70], [71], [72], [73].", "There have also been a number of recent investigations on possible ways for detection of wormholes [74], [75], [76], [77], [78], [79].", "Our theoretical model of brane wormhole satisfies all the properties required to describe a traversable wormhole without requiring exotic matter.", "It also imposes two very important constraints on the brane tension,which is one of the most crucial parameters involved in the study of braneworld and more importantly, both the constraints are in confirmation with ones obtained from studies on different aspects of braneworld including fundamental aspects and also aspects related to black holes.", "Thus we conclude with the notion that the Kuchowicz metric potential provides physically acceptable and theoretical sound solutions and showing consistency with both wormhole and braneworld in presence of ordinary matter.", "More boldly we can claim from our analysis that if wormholes do exist in nature, then it must be true that our universe has a very high likelihood of being a 3-brane embedded in higher dimensions." ] ]
2105.11785
[ [ "Beyond-mean-field effects in Rabi-coupled two-component Bose-Einstein\n condensate" ], [ "Abstract We theoretically calculate and experimentally measure the beyond-mean-field (BMF) equation of state in a coherently-coupled two-component Bose-Einstein condensate (BEC) in the regime where averaging of the interspecies and intraspecies coupling constants over the hyperfine composition of the single-particle dressed state predicts the exact cancellation of the two-body interaction.", "We show that with increasing the Rabi-coupling frequency $\\Omega$, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang (LHY) scaling $\\propto n^{5/2}$ to an expansion in integer powers of density, where, in addition to a two-body BMF term $\\propto n^2 \\sqrt{\\Omega}$, there emerges a repulsive three-body contribution $\\propto n^3/\\sqrt{\\Omega}$.", "We experimentally evidence this two contributions, thanks to their different scaling with $\\Omega$, in the expansion of a Rabi-coupled two-component $^{39}$K condensate in a waveguide.", "By studying the expansion with and without Rabi coupling, we reveal an important feature relevant for observing BMF effects and associated phenomena in mixtures with spin-asymmetric losses: Rabi coupling helps preserve the spin composition and thus prevents the system from drifting away from the point of vanishing mean field." ], [ "Bogoliubov spectrum and BMF energy in coupled binary mixtures", "The expansion of the Hamiltonian (1) (hereafter, equation numbers without prefix S refer to the main text) up to quadratic terms in the fields $\\hat{\\phi }_{\\sigma ,{\\bf p}}$ (we switch to momentum space) can be written as $E_{\\rm MF}+E_0+\\hat{H}_2$ , where $E_{\\rm MF}$ is given by Eq.", "(2), $E_0=\\frac{1}{2}\\sum _{\\bf p}\\left(-p^2-\\frac{n_\\uparrow +n_\\downarrow }{\\sqrt{n_\\uparrow n_\\downarrow }}\\frac{\\Omega }{2}-g_{\\uparrow \\uparrow }n_\\uparrow - g_{\\downarrow \\downarrow }n_\\downarrow +\\frac{1}{p^2}\\sum _{\\sigma \\sigma ^{\\prime }}g^2_{\\sigma \\sigma ^{\\prime }}n_\\sigma n_{\\sigma ^{\\prime }}\\right),$ and $\\hat{H}_2=\\!\\frac{1}{2}(\\hat{\\phi }_\\uparrow ^\\dagger \\hat{\\phi }_\\downarrow ^\\dagger \\hat{\\phi }_\\uparrow \\hat{\\phi }_\\downarrow )\\!\\begin{pmatrix}\\end{pmatrix}\\hspace{0.0pt}\\frac{p^2}{2}+g_{\\uparrow \\uparrow }n_\\uparrow +\\frac{\\Omega }{2}\\sqrt{\\frac{n_\\downarrow }{n_\\uparrow }} & g_{\\uparrow \\downarrow }\\sqrt{n_\\uparrow n_\\downarrow }-\\frac{\\Omega }{2} & g_{\\uparrow \\uparrow }n_\\uparrow & g_{\\uparrow \\downarrow }\\sqrt{n_\\uparrow n_\\downarrow } \\\\$ gn n-2 p22+gn+2nn gn n gn gn gn n p22+gn+2nn gn n-2 gn n gn gn n-2 p22+gn+2nn .", "The first four terms in the brackets on the right-hand side of Eq.", "(REF ) arise as a compensation for the “incorrectly” ordered terms added to the quadratic form (REF ) in order to make it symmetric with respect the ordering of creation and annihilation operators.", "This symmetrized form is convenient since it stays symmetrized under the usual Bogoliubov transformation and diagonalizes into $\\hat{H}_2=\\frac{1}{2}\\sum _{{\\bf p},\\pm } E_{p,\\pm }(\\hat{b}_{{\\bf p},\\pm }^\\dagger \\hat{b}_{{\\bf p},\\pm }+ \\hat{b}_{{\\bf p},\\pm } \\hat{b}_{{\\bf p},\\pm }^\\dagger )=\\sum _{{\\bf p},\\pm } E_{p,\\pm }\\hat{b}_{{\\bf p},\\pm }^\\dagger \\hat{b}_{{\\bf p},\\pm } + \\frac{1}{2}\\sum _{{\\bf p},\\pm } E_{p,\\pm }.$ The last term in Eq.", "(REF ) is due to the standard renormalisation of the coupling constant $g_{\\sigma \\sigma ^{\\prime }}\\rightarrow g_{\\sigma \\sigma ^{\\prime }}(1+g_{\\sigma \\sigma ^{\\prime }}\\sum _p 1/p^2)$ in the MF term.", "The dispersion relations $E_{p,\\pm }$ of the Bogoliubov modes (created by operators $\\hat{b}^\\dagger _{{\\bf p},\\pm }$ ) can be written as $E_{p,\\pm }=\\sqrt{D_p\\pm \\sqrt{D_p^2-\\frac{p^2}{2}\\left(\\frac{p^2}{2}+\\frac{\\Omega }{2}\\frac{n_\\uparrow +n_\\downarrow }{\\sqrt{n_\\uparrow n_\\downarrow }}\\right)\\left[\\prod _\\sigma \\left(\\frac{p^2}{2}+2g_{\\sigma \\sigma }n_\\sigma +\\frac{\\Omega }{2}\\sqrt{\\frac{n_{\\bar{\\sigma }}}{n_\\sigma }}\\right)-\\left(2g_{\\uparrow \\downarrow }\\sqrt{n_\\uparrow n_\\downarrow }-\\frac{\\Omega }{2}\\right)^2\\right]}},$ where $D_p=\\frac{1}{2}\\sum _\\sigma \\left(\\frac{p^2}{2}+\\frac{\\Omega }{2}\\sqrt{\\frac{n_{\\bar{\\sigma }}}{n_\\sigma }}\\right)\\left(\\frac{p^2}{2}+2g_{\\sigma \\sigma }n_\\sigma +\\frac{\\Omega }{2}\\sqrt{\\frac{n_{\\bar{\\sigma }}}{n_\\sigma }}\\right)-\\frac{\\Omega }{2}\\left(2g_{\\uparrow \\downarrow }\\sqrt{n_\\uparrow n_\\downarrow }-\\frac{\\Omega }{2}\\right).$ From Eq.", "(REF ) one sees that the mode $E_{p,-}$ is gapless, while $E_{p,+}$ has a gap $\\sqrt{2D_0}\\ne 0$ .", "The former is just the Goldstone mode due to the breaking of the $U(1)$ symmetry in the condensed state, while the latter is related to the gap introduced by $\\Omega $ in having a different phase for the two spinor components.", "The gap vanishes for $\\Omega =0$ , since in this case there exist two Goldstone modes of the broken $U(1)\\times U(1)$ symmetry.", "The BMF energy is obtained by adding the vacuum part of Eq.", "(REF ) to the constant energy $E_0$ .", "The BMF energy thus explicitly reads $E_{\\rm BMF}=\\frac{1}{2}\\int {d^3p\\over (2\\pi )^3} \\left(E_{p,+}+E_{p,-}-p^2-\\frac{n_\\uparrow +n_\\downarrow }{\\sqrt{n_\\uparrow n_\\downarrow }}\\frac{\\Omega }{2}-g_{\\uparrow \\uparrow }n_\\uparrow - g_{\\downarrow \\downarrow }n_\\downarrow +\\frac{1}{p^2}\\sum _{\\sigma \\sigma ^{\\prime }}g^2_{\\sigma \\sigma ^{\\prime }}n_\\sigma n_{\\sigma ^{\\prime }}\\right),$ where we have replaced the sum over momenta by the integral.", "Equation (4) is obtained from Eq.", "(REF ) under the conditions (3) as follows.", "We switch to the integration variable $z=p^2$ and express the integral in Eq.", "(REF ) as a contour integral around the branch cut of $\\sqrt{z}$ along the positive real semiaxis.", "We then deform this contour towards the negative semiaxis such that it now goes around the branch cut of $E_{p,+}$ .", "This is a finite interval which we map onto $x\\in [0,1]$ ." ] ]
2105.11723
[ [ "Entangled quantum Unruh Otto engine is more efficient" ], [ "Abstract We propose a relativistic quantum Otto cycle between an entangled state of two qubits and their composite excited (or ground) state whose efficiency can be greater than the usual single qubit quantum Otto engine.", "The hot and cold reservoirs are constructed by providing uniform accelerations to these qubits along with the interaction between the background field and individual qubits.", "The efficiency, as measured from one of the qubits' frame, not only depends on the energy gap of the states but also the relative acceleration between them.", "For lower acceleration of our observer's qubit compared to the other one, the cycle is more efficient than the single qubit quantum Otto engine.", "Furthermore, a complete protocol to construct such a cycle is being provided." ], [ "Relation between proper times of the two uniformly accelerated frames", "This analysis is followed from [14] (see the discussion at the beginning of section 2 of this reference).", "Consider two uniformly accelerated detectors both are moving in the right Rindler wedge with acceleration parameters are given by $a_1$ and $a_2$ .", "For simplicity we take their motion along Minkowski $X$ -axis.", "The trajectories on the Minkowski spacetime are given by the following relations among the coordinates: $&&T = \\frac{e^{a_1\\xi _1}}{a_1}\\sinh (a_1\\eta _1)~;\\nonumber \\\\&&X = \\frac{e^{a_1\\xi _1}}{a_1}\\cosh (a_1\\eta _1)~,$ for first detector and $&&T=\\dfrac{e^{a_{2}\\xi _2}}{a_{2}}\\sinh (a_2\\eta _2)~;\\nonumber \\\\&& X=\\dfrac{e^{a_{2}\\xi _2}}{a_{2}}\\cosh (a_{2}\\eta _2)~,$ for the second detector.", "These relations in terms of the detector's proper time are given by $&&T = \\frac{1}{A_1}\\sinh (A_1\\tau _1)~;\\nonumber \\\\&&X = \\frac{1}{A_1}\\cosh (A_1\\tau _1)~,$ and $&&T=\\dfrac{1}{A_2}\\sinh (A_2\\tau _2)~;\\nonumber \\\\&& X=\\dfrac{1}{A_2}\\cosh (A_2\\tau _2)~,$ respectively.", "In the above $A_1$ and $A_2$ are the proper accelerations while $\\tau _1$ and $\\tau _2$ are their proper times.", "These are related to $a_1, a_2$ and $\\eta _1, \\eta _2$ by the following relations: $&&A_1 = a_1 e^{-a_1\\xi _1}; \\,\\,\\, A_2 = a_2 e^{-a_2\\xi _2}~;\\nonumber \\\\&&\\tau _1 = \\eta _1 e^{a_1\\xi _1}; \\,\\,\\, \\tau _2 = \\eta _2 e^{a_2\\xi _2}~.$ On a constant $T/X$ line both the detectors satisfy the following relation $A_1\\tau _1 = A_2\\tau _2~.$ For the choice $\\xi _1=0=\\xi _2$ in their respective frames, $a_1, a_2$ are identified as proper accelerations while $\\eta _1, \\eta _2$ are then their respective proper times.", "Thus for this simple choice we have the relation between the respective proper times as $\\tau _2 =\\alpha _a \\tau _1~,$ where $\\alpha _a = a_1/a_2$ .", "For constant velocity phase we can have $\\tau _2=\\alpha _v\\tau _1$ , where $\\alpha _{v}=\\sqrt{1-v_{rel}^{2}}$ where, $v_{rel}$ is the relative velocity between the detectors.", "If the detectors are moving with same constant velocity, then $v_{rel}=0$ and hence $\\alpha _v=1$ ." ], [ "Time evolution of density operator", "The evolution of density operatorwill find out in interaction picture.", "In the interaction picture we express the interaction Hamiltonian (REF ) as $H_{int}^{I}=\\mu \\Big (m_{1}^{I}\\Phi (X_{1}(\\tau _{1}))\\dfrac{d\\tau _{1}}{d\\tau }+m_{2}^{I}\\Phi (X_{2}(\\tau _{2}))\\dfrac{d\\tau _{2}}{d\\tau }\\Big )~,$ where, $m_{i}^{I}=\\mathcal {U}^{i\\dagger }m_{i}\\mathcal {U}^i$ with $\\mathcal {U}^i = \\exp (-i \\int H_id\\tau _i)$ .", "Here $H_i$ denotes part of (REF ) which corresponds to $i^{th}$ qubit.", "Note that when $\\rho $ changes, the level spacing $\\omega $ remains fixed.", "This is happening in stage 2 and stage 4.", "In that case $H_i$ is taken to be time independent and hence $\\mathcal {U}^i = \\exp (-i H_i\\Delta \\tau ^s_i)$ corresponding to $s^{th}$ stage.", "Since $m_i$ is given by (REF ), we reexpress (REF ) as $H_{int}^{I}=M_{1}\\dfrac{d\\tau _{1}}{d\\tau }\\Phi (X_{1}(\\tau _{1}))+M_{2}\\dfrac{d\\tau _{2}}{d\\tau }\\Phi (X_{2}(\\tau _{2}))~,$ where $M_{1}\\equiv \\mu m_{1}^{I}=\\mu \\mathcal {U}^{1\\dagger }m_{1}\\mathcal {U}^1=\\mu \\begin{pmatrix}0&0&e^{i\\omega \\Delta \\tau ^s_{1}}&0\\\\0&0&0&e^{i\\omega \\Delta \\tau ^s_{1}}\\\\e^{-i\\omega \\Delta \\tau ^s_{1}}&0&0&0\\\\0&e^{-i\\omega \\Delta \\tau ^s_{1}}&0&0\\\\\\end{pmatrix}~;$ and $M_{2}\\equiv \\mu m_2^I= \\mu \\mathcal {U}^{2\\dagger }m_{2}\\mathcal {U}^2=\\mu \\begin{pmatrix}0&e^{i\\omega \\Delta \\tau ^s_{2}}&0&0\\\\e^{-i\\omega \\Delta \\tau ^s_{2}}&0&0&0\\\\0&0&0&e^{i\\omega \\Delta \\tau ^s_{2}}\\\\0&0&e^{-i\\omega \\Delta \\tau ^s_{2}}&0\\\\\\end{pmatrix}~.", "$ The matrix forms are represented in the basis $\\lbrace {e_1,e_2},{e_1,g_2},{g_1,e_2},{g_1,g_2}\\rbrace $ .", "The initial density matrix for the collective system, composed of two qubits and the scalar field, in interaction picture is given by $\\rho _0^I =\\rho _0 = \\rho _{A_0}\\otimes \\rho _{f_0}~,$ with the initial state of the field is represented by density operator $\\rho _{f_0}={0_{M}}{0_{M}}$ by considering initially the field is in Minkowski vacuum state ${0_M}$ .", "The evolution of density operator is determined by the equation $i\\dfrac{d\\rho ^{I}(\\tau )}{d\\tau }=[H^{I}_{int},\\rho ^{I}(\\tau )]~.$ Here everything has to be measured from the first detector.", "So the proper time $\\tau $ in above has to be chosen as $\\tau _1$ .", "The solution of the Eq.", "(REF ) is normally achieved by well known perturbative approach.", "Till the second order in perturbation series it is given by $\\rho ^{I}(t)=\\underbrace{\\rho ^{I}(t_0)}_{\\mathcal {O}(\\mu ^0)} -\\underbrace{i\\int _{t_0}^{t}[H^{I}_{int}(t^{\\prime }),\\rho ^{I}(t_{0})]dt^{\\prime }}_{\\mathcal {O}(\\mu ^1)} - \\underbrace{T\\int _{t_0}^{t} dt^{\\prime }\\int _{t_0}^{t^{\\prime }} dt^{\\prime \\prime } [[H^{I}_{int}(t^{\\prime }),[[H^{I}_{int}(t^{\\prime \\prime }),\\rho ^{I}(t_{0})]]}_{\\mathcal {O}_(\\mu ^2)} +\\mathcal {O}(\\mu ^3)~,$ where, $T$ means time-ordered.", "product.", "By removing the time order product, the can also be expressed as $\\rho ^{I}(t)=\\rho ^{I}(t_{0})-i\\int _{t_{0}}^{t}[H^{I}_{int}(t^{\\prime }),\\rho ^{I}(t_{0})]dt^{\\prime }-\\dfrac{1}{2}\\int _{t_{0}}^{t}\\int _{t_{0}}^{t}[H^{I}_{int}(t^{\\prime }),[H^{I}_{int}(t^{\\prime \\prime }),\\rho ^{I}(t_{0})]]dt^{\\prime \\prime }dt^{\\prime }~.$ Here $t$ is the clock time of our observer.", "For the main purpose, we need to choose $t=\\tau _1$ , which we will consider later.", "As we are interested in the evolution of the two detector system, the field degrees of freedom must be traced out.", "At time $t=t_{0}$ , the composite state is represented by (REF ).", "The integrant in first order term of Eq.", "(REF ), after taking trace over all field states, yields $&&\\text{Tr}_{f}([H^{I}_{int}(t^{\\prime }),\\rho ^{I}(t_{0})])=\\sum _{\\lbrace {\\Theta }\\rbrace =\\text{set of all field states}}\\langle \\Theta \\vert [H^{I}_{int}(t^{\\prime }),\\rho ^{I}(t_{0})]\\vert \\Theta \\rangle \\nonumber \\\\&&=\\sum _{\\lbrace {\\Theta }\\rbrace }\\langle \\Theta \\vert \\Big (H_{int}^{I}(t^{\\prime })\\rho _{A_0}\\otimes {0_{M}}{0_{M}}-\\rho _{A_0}\\otimes {0_{M}}{0_{M}}H_{int}^{I}(t^{\\prime })\\Big )\\vert \\Theta \\rangle \\nonumber \\\\&&=\\langle 0_{M}\\vert \\Big (\\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}M_{1}(\\tau _{1}^{\\prime })\\Phi (X_{1}(\\tau _{1}^{\\prime }))+\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}M_{2}(\\tau _{2}^{\\prime })\\Phi (X_{2}(\\tau _{2}^{\\prime }))\\Big )\\rho _{A_0}{0_M}\\nonumber \\\\&&-{0_M}\\rho _{A_0}\\Big (\\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}M_{1}(\\tau _{1}^{\\prime })\\Phi (X_{1}(\\tau _{1}^{\\prime }))+\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}M_{2}(\\tau _{2}^{\\prime })\\Phi (X_{2}(\\tau _{2}^{\\prime }))\\Big )\\vert 0_{M}\\rangle ~.$ In the last step we substituted the explicit expression for interaction Hamiltonian (REF ).", "Now as $\\langle 0_{M}\\vert \\Phi (\\tau _{i}^{\\prime })\\vert 0_{M}\\rangle =0$ , one finds $\\text{Tr}_{f}([H^{I}_{int}(t^{\\prime }),\\rho ^{I}(t_{0})])=0$ ; i.e.", "first order term does not contribute to the perturbative solution.", "Let us now concentrate on the next order term in (REF ).", "The second-order term consists of several parts.", "For the present discussion, we drop the notation “$int$ ” and will use it only when necessary.", "The expansion of brackets in the integrant of the $\\mathcal {O}(\\mu ^2)$ terms gives $[H(\\tau ^{\\prime }),[H(\\tau ^{\\prime \\prime }),\\rho (\\tau _{0})]]&=&H(\\tau ^{\\prime })H(\\tau ^{\\prime \\prime })\\rho (\\tau _{0})-H(\\tau ^{\\prime })\\rho (\\tau _{0})H(\\tau ^{\\prime \\prime })\\nonumber \\\\&-&H(\\tau ^{\\prime \\prime })\\rho (\\tau _{0})H(\\tau ^{\\prime })+\\rho (\\tau _{0})H(\\tau ^{\\prime \\prime })H(\\tau ^{\\prime })~.$ Next using explicit expressions (REF ) and (REF ) we have to take trace over all field states.", "The first term of the above yields $\\begin{split}\\text{Tr}_{f}(H(\\tau ^{\\prime })H(\\tau ^{\\prime \\prime })\\rho ^{I}(\\tau _{0}))=\\sum _{\\lbrace {\\Theta }\\rbrace }\\langle \\Theta \\vert \\Big [\\Big (\\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}M_{1}(\\tau ^{\\prime })\\Phi (X_{1}(\\tau ^{\\prime }))+\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}M_{2}(\\tau ^{\\prime })\\Phi (X_{2}(\\tau ^{\\prime }))\\Big )\\\\\\times \\Big (\\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}M_{1}(\\tau ^{\\prime \\prime })\\Phi (X_{1}(\\tau ^{\\prime \\prime }))+\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}M_{2}(\\tau ^{\\prime \\prime })\\Phi (X_{2}(\\tau ^{\\prime \\prime }))\\Big )\\rho _{A_0}{0_M}{0_M}\\Big ]\\vert \\Theta \\rangle \\\\=M_{1}(\\tau _{1}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}+ M_{1}(\\tau _{1}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\\\ +M_{2}(\\tau _{2}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}+ M_{2}(\\tau _{2}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle \\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}~.\\end{split}$ In the above we used the notation ${0_M}{\\Phi (X_{i}(\\tau _{i}^{\\prime }))}{\\Phi (X_{j}(\\tau _{j}^{\\prime \\prime }))}{0_M}=\\langle \\Phi _{j}(\\tau ^{\\prime \\prime }_{j})\\Phi _{i}(\\tau ^{\\prime }_{i})\\rangle $ .", "Similarly, other terms lead to $\\begin{split}\\text{Tr}_{f}(H(\\tau ^{\\prime })\\rho ^{I}(\\tau _{0}) H(\\tau ^{\\prime \\prime }))=M_{1}(\\tau _{1}^{\\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })\\langle \\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\Phi _{1}(\\tau ^{\\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\\\+M_{1}(\\tau _{1}^{\\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })\\langle \\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\Phi _{1}(\\tau ^{\\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\\\+M_{2}(\\tau _{2}^{\\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })\\langle \\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\Phi _{2}(\\tau ^{\\prime }_{2})\\rangle \\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}+M_{2}(\\tau _{2}^{\\prime })\\rho _{A0}M_{2}(\\tau _{2}^{\\prime \\prime })\\langle \\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\Phi _{2}(\\tau ^{\\prime }_{2})\\rangle \\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}~;\\end{split}$ $\\begin{split}\\text{Tr}_{f}(H(\\tau ^{\\prime \\prime })\\rho ^{I}(\\tau _{0}) H(\\tau ^{\\prime }))=M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime })\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\\\+M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime })\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\\\+M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime })\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}+M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime })\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle \\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}~;\\end{split}$ and $\\begin{split}\\text{Tr}_{f}(\\rho ^{I}(\\tau _{0})H(\\tau ^{\\prime \\prime }) H(\\tau ^{\\prime }))=\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })\\langle \\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\Phi _{1}(\\tau ^{\\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\\\+\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{2}(\\tau _{2}^{\\prime })\\langle \\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\Phi _{2}(\\tau ^{\\prime }_{2})\\rangle \\dfrac{d\\tau _{1}^{\\prime \\prime }}{dt^{\\prime \\prime }}\\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\\\ +\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })\\langle \\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\Phi _{1}(\\tau ^{\\prime }_{1})\\rangle \\dfrac{d\\tau _{1}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}+\\rho _{A_0}M_{2}(\\tau _{1}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\langle \\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\Phi _{2}(\\tau ^{\\prime }_{2})\\rangle \\dfrac{d\\tau _{2}^{\\prime }}{dt^{\\prime }}\\dfrac{d\\tau _{2}^{\\prime \\prime }}{dt^{\\prime \\prime }}~.\\end{split}$ Substitution of these in (REF ) yields (REF ) with $\\delta \\rho $ is given by $\\rho (\\tau ^f_1) \\equiv \\rho ^I(\\tau ^f_{1},\\tau ^f_{2}(\\tau ^f_1))=\\rho _{A_0}+\\delta \\rho (\\tau ^f_{1},\\tau ^f_{2}(\\tau ^f_1))~,$ where $&&\\delta \\rho (\\tau ^f_{1},\\tau ^f_{2}(\\tau ^f_1))=-\\dfrac{1}{2}\\Big [\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\Gamma _{11}\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }+\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\Gamma _{12}^{1}\\langle \\Phi _{1}(\\tau ^{\\prime }_{1})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle d\\tau _{2}^{\\prime \\prime }d\\tau _{1}^{\\prime }\\nonumber \\\\&&+\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\Gamma _{12}^{2}\\langle \\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\Phi _{2}(\\tau ^{\\prime }_{2})\\rangle d\\tau _{2}^{\\prime }d\\tau _{1}^{\\prime \\prime }+\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\Gamma _{21}^{1}\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{1}(\\tau ^{\\prime \\prime }_{1})\\rangle d\\tau _{2}^{\\prime }d\\tau _{1}^{\\prime \\prime }\\nonumber \\\\&&+\\int _{\\tau ^i_{1}}^{\\tau ^f_{1}}\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\Gamma _{21}^{2}\\langle \\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\Phi _{1}(\\tau ^{\\prime }_{1})\\rangle d\\tau _{2}^{\\prime \\prime }d\\tau _{1}^{\\prime }+\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\int _{\\tau ^i_{2}}^{\\tau ^f_{2}}\\Gamma _{22}\\langle \\Phi _{2}(\\tau ^{\\prime }_{2})\\Phi _{2}(\\tau ^{\\prime \\prime }_{2})\\rangle d\\tau _{2}^{\\prime \\prime }d\\tau _{2}^{\\prime }\\Big ]~.", "$ In the above the final expression will be achieved after replacing $\\tau _2$ in terms of $\\tau _1$ by $\\tau _2=\\alpha \\tau _1$ .", "The explicit expressions for $\\Gamma $ 's are given by $&&\\Gamma _{11} = M_{1}(\\tau _{1}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0} - M_{1}(\\tau _{1}^{\\prime })\\rho _{A0}M_{1}(\\tau _{1}^{\\prime \\prime }) - M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime })\\nonumber \\\\&&+\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })~;\\\\&&\\Gamma _{12}^{1} = M_{1}(\\tau _{1}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0} - M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime })~;\\\\&&\\Gamma _{12}^{2} = \\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{2}(\\tau _{2}^{\\prime }) - M_{2}(\\tau _{2}^{\\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime }) ~;\\\\&&\\Gamma _{21}^{1} = M_{2}(\\tau _{2}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}- M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime })~;\\\\&&\\Gamma _{21}^{2} = \\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })-M_{1}(\\tau _{1}^{\\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime }) ~;\\\\&&\\Gamma _{22} = M_{2}(\\tau _{2}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0} - M_{2}(\\tau _{2}^{\\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime }) - M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime })\\nonumber \\\\&&+\\rho _{A_0}M_{2}(\\tau _{1}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })~.$" ], [ "Matrix representations of the operators appearing in $\\Gamma $ ,s", "The matrix forms of the operators, appearing in Eq.", "(REF ) to Eq.", "(), in the basis $\\lbrace {e_1,e_2},{e_1,g_2},{g_1,e_2},{g_1,g_2}\\rbrace $ can be obtained by using (REF ), (REF ) and (REF ).", "They are listed below.", "Below we denote $\\Delta \\tau _{1}^{\\prime }=\\tau _{1}^{\\prime }+\\tau _{a}$ and $\\Delta \\tau _{2}^{\\prime }=\\tau _{1}^{\\prime }+\\tau _{b}$ , where $-\\tau _{a}$ and $-\\tau _{b}$ , the initial proper times of first and second detectors respectively, signify the time durations of interaction with the field.", "Same goes for $\\Delta \\tau _{1}^{\\prime \\prime },\\Delta \\tau _{2}^{\\prime \\prime }$ respectively as well.", "$M_{1}(\\tau _{1}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}$ : $\\begin{pmatrix}pe^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&0\\\\0&q\\vert b_{1}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&q\\vert b_{2}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $M_{1}(\\tau _{1}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}$ : $\\begin{pmatrix}0&0&0&0\\\\0&q b_{1}^{*}b_{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0\\\\0&q \\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&qb_{1} b_{2}^{*}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0\\\\pe^{-i\\omega (\\Delta \\tau _{1}^{\\prime }+\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&0\\\\\\end{pmatrix}$ $M_{2}(\\tau _{2}^{\\prime })M_{1}(\\tau _{1}^{\\prime \\prime })\\rho _{A_0}$ : $\\begin{pmatrix}0&0&0&0\\\\0&q b_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&q\\vert b_{2}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&q \\vert b_{1}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&qb_{1} b_{2}^{*}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\pe^{-i\\omega (\\Delta \\tau _{2}^{\\prime }+\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&0\\\\\\end{pmatrix}$ $M_{2}(\\tau _{2}^{\\prime })M_{2}(\\tau _{2}^{\\prime \\prime })\\rho _{A_0}$ : $\\begin{pmatrix}pe^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&0\\\\0&q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&qb_{1}b_{2}^{*}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0\\\\0&qb_{1}^{*}b_{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })$ : $\\begin{pmatrix}pe^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&0\\\\0&q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })M_{2}(\\tau _{2}^{\\prime })$ : $\\begin{pmatrix}0&0&0&pe^{i\\omega (\\Delta \\tau _{1}^{\\prime \\prime }+\\Delta \\tau _{2}^{\\prime })}\\\\0&qb_{1}b_{2}^{*}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&q\\vert b_{1}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&0\\\\0&q\\vert b_{2}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&qb_{1}^{*}b_{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })M_{1}(\\tau _{1}^{\\prime })$ : $\\begin{pmatrix}0&0&0&pe^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }+\\Delta \\tau _{1}^{\\prime })}\\\\0&qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{1}^{\\prime })}&q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{1}^{\\prime })}&0\\\\0&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{1}^{\\prime })}&qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{1}^{\\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })M_{2}(\\tau _{2}^{\\prime })$ : $\\begin{pmatrix}pe^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&0&0&0\\\\0&q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&0\\\\0&qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime \\prime }-\\Delta \\tau _{2}^{\\prime })}&0\\\\0&0&0&0\\\\\\end{pmatrix}$ $M_{1}(\\tau _{1}^{\\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })$ : $\\begin{pmatrix}q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&qb_{1}^{*}b_{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }+\\Delta \\tau _{1}^{\\prime \\prime })}\\\\0&0&0&0\\\\0&0&pe^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\qb_{1}b_{2}^{*}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }+\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}\\\\\\end{pmatrix}$ $M_{1}(\\tau _{1}^{\\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })$ : $\\begin{pmatrix}qb_{1}^{*}b_{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&q\\vert b_{2}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{1}^{\\prime }+\\Delta \\tau _{2}^{\\prime \\prime })}\\\\0&0&0&0\\\\0&pe^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0&0\\\\q\\vert b_{1}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }+\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&qb_{1}b_{2}^{*}e^{-i\\omega (\\Delta \\tau _{1}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}\\\\\\end{pmatrix}$ $M_{2}(\\tau _{2}^{\\prime })\\rho _{A_0}M_{1}(\\tau _{1}^{\\prime \\prime })$ : $\\begin{pmatrix}qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&q\\vert b_{1}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }+\\Delta \\tau _{1}^{\\prime \\prime })}\\\\0&0&pe^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}&0\\\\0&0&0&0\\\\q\\vert b_{2}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }+\\Delta \\tau _{1}^{\\prime \\prime })}&0&0&qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{1}^{\\prime \\prime })}\\\\\\end{pmatrix}$ $M_{2}(\\tau _{2}^{\\prime })\\rho _{A_0}M_{2}(\\tau _{2}^{\\prime \\prime })$ : $\\begin{pmatrix}q\\vert b_{1}\\vert ^{2}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&qb_{1}b_{2}^{*}e^{i\\omega (\\Delta \\tau _{2}^{\\prime }+\\Delta \\tau _{2}^{\\prime \\prime })}\\\\0&pe^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}&0&0\\\\0&0&0&0\\\\qb_{1}^{*}b_{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }+\\Delta \\tau _{2}^{\\prime \\prime })}&0&0&q\\vert b_{2}\\vert ^{2}e^{-i\\omega (\\Delta \\tau _{2}^{\\prime }-\\Delta \\tau _{2}^{\\prime \\prime })}\\\\\\end{pmatrix}$ We use them in Appendix to evaluate Eq.", "(REF )." ], [ "Structure of Wightman function", "Here we give the explicit forms of the Wightman functions, defined as $\\langle \\Phi _{i}(\\tau ^{\\prime }_{i})\\Phi _{j}(\\tau ^{\\prime \\prime }_{j})\\rangle =G_{ij}(\\tau _{i}^{\\prime },\\tau _{j}^{\\prime \\prime })~, $ for different situations as appeared in Eq.", "(REF ).", "Also we will find few properties among them which will be used in next Appendix.", "Here the expression in $(1+1)$ dimensions will be given.", "This is sufficient as similar properties also hold in $(1+3)$ dimensions.", "We provide the forms which are considered in [15] as we will see later that such is efficient to tackle the situation elegantly.", "For real massless scalar field, Wightman function, with respect to Rindler frame with ${0_M}$ as vacuum, is obtained by taking $\\beta \\rightarrow \\infty $ of those in [15].", "This is given by $G_{jl}(\\Delta \\xi ,\\Delta \\eta )&=&\\int _{-\\infty }^{\\infty }\\dfrac{dk}{8\\pi \\omega _{k}\\sqrt{\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{j}}\\Big )\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{l}}\\Big )}}\\Big [e^{ik\\Delta \\xi -i\\omega _{k}\\Delta \\eta }e^{\\frac{\\pi \\omega _{k}}{2}(\\frac{1}{a_{j}}+\\frac{1}{a_{l}})}\\nonumber \\\\&+& e^{ik\\Delta \\xi +i\\omega _{k}\\Delta \\eta }e^{-\\frac{\\pi \\omega _{k}}{2}(\\frac{1}{a_{j}}+\\frac{1}{a_{l}})}\\Big ]~.$ This expression is already in Rindler coordinates.", "Using the relation between the proper times of the detectors $\\tau _2 = \\alpha \\tau _1$ , we express the above one in terms of proper time of first detector.", "The Wightman functions, appearing in (REF ), come out to be in the following forms.", "$G_{12}(\\tau _{1}^{\\prime },\\tau _{2}^{\\prime \\prime })$ : $\\Delta \\xi =0,\\Delta \\eta =\\tau _{1}^{\\prime }-\\alpha _a\\tau _{1}^{\\prime \\prime }$ $G_{12}(\\tau _{1}^{\\prime },\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime })) &=& \\int _{-\\infty }^{\\infty }\\dfrac{dk}{8\\pi \\omega _{k}\\sqrt{\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{2}}\\Big )}}\\Big [e^{-i\\omega _{k}(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\nonumber \\\\ &+& e^{+i\\omega _{k}(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\Big ]~;$ $G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))$ : $\\Delta \\xi =0,\\Delta \\eta =\\tau _{1}^{\\prime \\prime }-\\alpha _{a}\\tau _{1}^{\\prime }$ $G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime })) &=& \\int _{-\\infty }^{\\infty }\\dfrac{dk}{8\\pi \\omega _{k}\\sqrt{\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{2}}\\Big )}}\\Big [e^{-i\\omega _{k}(\\tau _{1}^{\\prime \\prime }-\\alpha _{a}\\tau _{1}^{\\prime })}e^{\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\nonumber \\\\&+& e^{+i\\omega _{k}(\\tau _{1}^{\\prime \\prime }-\\alpha _{a}\\tau _{1}^{\\prime })}e^{-\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\Big ]~;$ $G_{21}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime })$ : $\\Delta \\xi =0,\\Delta \\eta =\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }$ $G_{21}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime }) &=& \\int _{-\\infty }^{\\infty }\\dfrac{dk}{8\\pi \\omega _{k}\\sqrt{\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{2}}\\Big )}}\\Big [e^{-i\\omega _{k}(\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\nonumber \\\\&+& e^{+i\\omega _{k}(\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\Big ]~;$ $G_{21}(\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),\\tau _{1}^{\\prime })$ : $\\Delta \\xi =0,\\Delta \\eta =\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime }$ $G_{21}(\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),\\tau _{1}^{\\prime }) &=& \\int _{-\\infty }^{\\infty }\\dfrac{dk}{8\\pi \\omega _{k}\\sqrt{\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi \\omega _{k}}{a_{2}}\\Big )}}\\Big [e^{-i\\omega _{k}(\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime })}e^{\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\nonumber \\\\&+& e^{+i\\omega _{k}(\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime })}e^{-\\frac{\\pi \\omega _{k}}{2a_{1}}(1+\\alpha _{a})}\\Big ]~.$ In the above $\\omega _{k}=\\vert k\\vert $ and hence we can rewrite them in the following forms as well: $&&G_{12}(\\tau _{1}^{\\prime },\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))=\\int _{-\\infty }^{0}\\dfrac{dk}{8\\pi (-k)\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{-\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{-ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]\\nonumber \\\\&+& \\int ^{\\infty }_{0}\\dfrac{dk}{8\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]\\nonumber \\\\&=& \\int _{\\infty }^{0}\\dfrac{d(-k)}{8\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]\\nonumber \\\\&+&\\int ^{\\infty }_{0}\\dfrac{dk}{8\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]\\nonumber \\\\&=& \\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]~.$ Similarly, others can be expressed as $G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))=\\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\tau _{1}^{\\prime \\prime }-\\alpha _{a}\\tau _{1}^{\\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\tau _{1}^{\\prime \\prime }-\\alpha _{a}\\tau _{1}^{\\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]~;$ $G_{21}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime })=\\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]~;$ and $G_{21}(\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),\\tau _{1}^{\\prime })=\\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sqrt{\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}}\\Big [e^{-ik(\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime })}e^{\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}+e^{ik(\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime })}e^{-\\frac{\\pi k}{2a_{1}}(1+\\alpha _{a})}\\Big ]~.$ These explicit expressions shows the following property: $G_{12}(\\tau _{1}^{\\prime },\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))=G_{21}(-\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),-\\tau _{1}^{\\prime })~,$ which will be used in the next Appendix.", "We now look at the structure of $G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })$ and $G(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))$ .", "Like the earlier way we find $G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })=\\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sinh \\Big (\\frac{\\pi k}{a_{1}}\\Big )}\\Big [e^{-ik(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{a_{1}}}+e^{ik(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{a_{1}}}\\Big ]~;$ and $G_{22}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))=\\int ^{\\infty }_{0}\\dfrac{dk}{4\\pi k\\sinh \\Big (\\frac{\\pi k}{a_{2}}\\Big )}\\Big [e^{-ik\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k}{a_{2}}}+e^{ik\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k}{a_{2}}}\\Big ]~.$ Using $\\dfrac{1}{a_{2}}=\\dfrac{\\alpha _{a}}{a_{1}}$ and changing the variable $\\alpha _{a} k=y$ we find $G_{22}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))=\\int ^{\\infty }_{0}\\dfrac{dy}{4\\pi y\\sinh \\Big (\\frac{\\pi y}{a_{1}}\\Big )}\\Big [e^{-iy(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi y}{a_{1}}}+e^{iy(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi y}{a_{1}}}\\Big ]~.$ Hence, we get a relation $G_{22}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))=G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })~.$ This property is also essential for the next Appendix.", "Same properties are also applicable to the (1+3) dimension structure of the Wightman function.", "The expression for (1+3) dimension structure as given in [15] is, $\\begin{split}G_{jl}(\\Delta \\eta _{jl})=\\int _{0}^{\\infty }d\\omega _{k}\\int \\dfrac{d^{2}k_{p}}{(2\\pi )^{4}}\\dfrac{2}{\\sqrt{a_{j}a_{l}}}\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{j}},\\dfrac{\\vert k_{p}\\vert }{a_{j}}\\Big ]\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{l}},\\dfrac{\\vert k_{p}\\vert }{a_{l}}\\Big ]\\\\\\times [e^{-i\\omega _{k}\\eta _{jl}}e^{\\frac{\\pi \\omega _{k}}{2}(\\frac{1}{a_{j}}+\\frac{1}{a_{l}})}+e^{i\\omega _{k}\\eta _{jl}}e^{-\\frac{\\pi \\omega _{k}}{2}(\\frac{1}{a_{j}}+\\frac{1}{a_{l}})}]\\end{split}$ where, $\\mathcal {K}[n, z]$ denotes the modified Bessel function of the second kind of order $n$ .", "This expression will also be used in following Appendices." ], [ "Derivation of Eq. (", "To calculate the efficiency, we have to take the trace of product between $\\delta \\rho _{A}^{\\alpha }$ and $h_{\\alpha ^{\\prime }}$ .", "As $h_{\\alpha ^{\\prime }}$ is time-independent (refer to Eq.", "(REF )), while taking the product, we can take it inside the integral of Eq.", "(REF ).", "There is a total of six terms corresponding to each $\\Gamma $ , and every term contains factors like $\\textrm {Tr}(\\Gamma h_{\\alpha })$ in the integrant.", "We will calculate these traces one by one.", "The matrix forms of $\\Gamma $ ,s can be evaluated by using results of Appendix in equations (REF ) to $(\\ref {A14})$ .", "We list below the values of the traces of each term in (REF ).", "In the following equations, $-\\tau _{a}$ is the initial time of first detector when the interaction is turned on.", "Trace with $\\Gamma _{12}^{1}$ $2q\\mu ^{2}\\alpha _{a}[b_{1}^{*}b_{2}e^{i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}b_{2}^{*}e^{-i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]~;$ Trace with $\\Gamma _{12}^{2}$ $2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}[b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]~;$ Trace with $\\Gamma _{21}^{1}$ $2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}[b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]~;$ Trace with $\\Gamma _{21}^{2}$ $2q\\mu ^{2}\\alpha _{a}[b_{1}b_{2}^{*}e^{-i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b^{*}_{1}b_{2}e^{i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]~;$ Trace with $\\Gamma _{11}$ $4\\mu ^{2}\\cos (\\omega (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))[p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2})]~.$ Trace with $\\Gamma _{22}$ $4\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}^{2}\\cos (\\omega \\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))[p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2})]~.$ Once we have the list of all the traces, we multiply the traces with their corresponding Wightman functions as appeared in Eq.", "(REF ).", "We consider the interaction to be on within the time range $[-\\tau _{a},\\tau _{a}]$ , measured by first detector's clock.", "The reason for this has been explained above Eq.", "(REF ) and the value is given by Eq.", "(REF ).", "We will use the property of the Wightman function, represented by Eq.", "(REF ) to simplify equations.", "We will simplify the expression pairwise.", "First we take terms corresponding to (REF ) and (REF ).", "These two lead to $\\begin{split}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}[b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau ^{\\prime \\prime }_{1}+\\tau _{a}))}]G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))\\\\+2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}[b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]G_{21}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime })\\Big ]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.\\end{split} $ We change the variables from $\\tau _{1}^{\\prime }\\rightarrow -\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime }\\rightarrow -\\tau _{1}^{\\prime \\prime }$ in the second term corresponding to $G_{21}(\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),\\tau _{1}^{\\prime \\prime })$ .", "This gives us, $\\begin{split}2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [[b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))\\\\+[b_{1}b_{2}^{*}e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{a})-(\\tau _{1}^{\\prime \\prime }-\\tau _{a}))}-b_{1}^{*}b_{2}e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{a})-(\\tau _{1}^{\\prime \\prime }-\\tau _{a}))}]G_{21}(-\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }),-\\tau _{1}^{\\prime \\prime })\\Big ]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }\\\\= 2q\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [b_{1}b_{2}^{*}[e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{a})-(\\tau _{1}^{\\prime \\prime }-\\tau _{a}))}-e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]\\\\+b_{1}^{*}b_{2}[e^{-i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }+\\tau _{a})-(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-e^{i\\omega (\\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{a})-(\\tau _{1}^{\\prime \\prime }-\\tau _{a}))}]\\Big ]G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.\\end{split} $ Rearranging the terms finally we get, $\\begin{split}-4iq\\mu ^{2}\\alpha ^{\\prime }\\alpha _{a}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}-1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}-1)\\tau _{a}}\\Big ]\\sin (\\omega (\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.\\end{split}$ Next we concentrate on terms corresponding to (REF ) and(REF ).", "They collectively yield $\\begin{split}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [2q\\mu ^{2}\\alpha _{a}[b_{1}^{*}b_{2}e^{i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b_{1}b_{2}^{*}e^{-i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]G_{12}(\\tau _{1}^{\\prime },\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }))\\\\+2q\\mu ^{2}\\alpha _{a}[b_{1}b_{2}^{*}e^{-i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}-b^{*}_{1}b_{2}e^{i\\omega ((\\tau _{1}^{\\prime }+\\tau _{a})-\\alpha _{a}(\\tau _{1}^{\\prime \\prime }+\\tau _{a}))}]G_{21}(\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),\\tau _{1}^{\\prime })\\Big ]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.\\end{split}$ In similar way the above simplifies to $-4iq\\mu ^{2}\\alpha _{a} \\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}-1)\\tau _{a}}+b^{*}_{1}b_{2}e^{-i\\omega (\\alpha _{a}-1)\\tau _{a}}\\Big ]\\sin (\\omega (\\tau _{1}^{\\prime }-\\alpha _{a}\\tau _{1}^{\\prime \\prime }))G_{21}(\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),\\tau _{1}^{\\prime })d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.$ Before going to other two terms, we now add terms (REF ) and (REF ).", "This leads to, upon using $\\tau ^{\\prime }_1\\rightarrow -\\tau ^{\\prime }_1$ and $\\tau ^{\\prime \\prime }_1\\rightarrow -\\tau ^{\\prime \\prime }_1$ , $\\begin{split}-4i\\alpha ^{\\prime }q\\mu ^{2}\\alpha _{a}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}-1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}-1)\\tau _{a}}\\Big ]\\sin (\\omega (\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }\\\\-4iq\\mu ^{2}\\alpha \\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\Big [b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}-1)\\tau _{a}}+b^{*}_{1}b_{2}e^{-i\\omega (\\alpha _{a}-1)\\tau _{a}}\\Big ]\\sin (\\omega (\\alpha _{a}\\tau _{1}^{\\prime \\prime }-\\tau _{1}^{\\prime }))G_{21}(-\\tau _{2}^{\\prime \\prime }(\\tau _{1}^{\\prime \\prime }),-\\tau _{1}^{\\prime })d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.", "\\end{split}$ Next using (REF ) and as $\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime }$ are integration variables, we can interchange them in the second expression.", "Therefore whole expression simplifies to $-4iq\\mu ^{2}\\alpha _{a}(1+\\alpha ^{\\prime })\\Big [b_{1}b_{2}^{*}e^{i\\omega (\\alpha _{a}-1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha _{a}-1)\\tau _{a}}\\Big ]\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\sin (\\omega (\\alpha _{a}\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.$ Now we will give attention to other two terms, containing (REF ) and (REF ).", "Adding them and then using Eq.", "(REF ), we obtain, $\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}4\\mu ^{2}[\\cos (\\omega (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2}))\\nonumber \\\\+\\alpha ^{\\prime }\\alpha _{a}^{2}\\cos (\\omega \\alpha _{a}(\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2}))]G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.$ Finally, accumulating Eq.", "(REF ) and Eq.", "(REF ) we obtain our desire expression (REF ).", "$\\text{Tr}(\\delta \\rho ^{\\alpha } h_{\\alpha ^{\\prime }}) &=& \\int _{-\\tau _a}^{\\tau _a}\\int _{-\\tau _a}^{\\tau _a}\\Big [2iq\\mu ^{2}\\alpha (1+\\alpha ^{\\prime })[b_{1}b_{2}^{*}e^{i\\omega (\\alpha -1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha -1)\\tau _{a}}]\\nonumber \\\\&&\\times \\sin (\\omega (\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))\\nonumber \\\\&&-2\\mu ^{2}[\\cos (\\omega (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2}))+\\nonumber \\\\&&\\alpha ^{\\prime }\\alpha ^{2}\\cos (\\omega \\alpha (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2}))]G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })\\Big ]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.$ In the above we denote $G_{ij} = \\langle \\Phi _i(\\tau _i^{\\prime })\\Phi _j(\\tau _{j}^{\\prime \\prime })\\rangle $ Use of this in (REF ) in principle gives us the explicit expression for efficiency of our Otto engine.", "Without evaluation of the integration one can not tell anything more about its properties.", "At this junction, let us look at a special case where $p=0$ and hence $q=1$ .", "It means, according to (REF ), our system is initially in the entangled state ${\\chi }$ .", "In this situation (REF ) simplifies to $\\text{Tr}(\\delta \\rho ^{\\alpha } h_{\\alpha ^{\\prime }}) &=&\\int _{-\\tau _a}^{\\tau _a}\\int _{-\\tau _a}^{\\tau _a}\\Big [2i\\mu ^{2}\\alpha (1+\\alpha ^{\\prime })[b_{1}b_{2}^{*}e^{i\\omega (\\alpha -1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha -1)\\tau _{a}}]\\nonumber \\\\&&\\times \\sin (\\omega (\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))\\nonumber \\\\&&-2\\mu ^{2}[\\cos (\\omega (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2})\\nonumber \\\\&&+\\alpha ^{\\prime }\\alpha ^{2}\\cos (\\omega \\alpha (\\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2})]G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })\\Big ]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }~.$ Now if ${\\chi }={s}$ , i.e.", "the initial state is chosen to be the symmetric entangled state, then we have $b_1=b_{2}=\\dfrac{1}{\\sqrt{2}}$ .", "This gives us a much more simpler result (REF )." ], [ "Explicit evaluation of $\\textrm {Tr}(\\delta \\rho ^\\alpha h_{\\alpha ^{\\prime }})$", "$\\text{Tr}(\\delta \\rho ^\\alpha h_{\\alpha ^{\\prime }})$ is given by (REF ).", "We approach the simplification stepwise.", "First we integrate the term corresponding to $G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))$ .", "Using (REF ) one finds $&&2i\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}\\sin (\\omega (\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime }))G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime }\\nonumber \\\\&&=\\int _{0}^{\\infty }\\dfrac{dk}{4\\pi kf(k)}\\int _{-\\tau _{a}}^{\\tau _{a}}\\int _{-\\tau _{a}}^{\\tau _{a}}[e^{i\\omega (\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}-e^{-i\\omega (\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}]\\nonumber \\\\&&\\times [e^{ik(\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{\\frac{\\pi k(1+\\alpha )}{2a_{1}}}+e^{-ik(\\alpha \\tau _{1}^{\\prime }-\\tau _{1}^{\\prime \\prime })}e^{-\\frac{\\pi k(1+\\alpha )}{2a_{1}}}]d\\tau _{1}^{\\prime \\prime }d\\tau _{1}^{\\prime } ~,$ where, $f(k)=\\sqrt{\\sinh \\Big (\\dfrac{k\\pi }{a_{1}}\\Big )\\sinh \\Big (\\dfrac{k\\pi }{a_{2}}\\Big )}$ .", "After integrating over $\\tau ^{\\prime \\prime }_1$ variable and then on $\\tau ^{\\prime }_1$ and also incorporating the pre-factor in (REF ) we find term corresponding to $G_{12}(\\tau _{1}^{\\prime \\prime },\\tau _{2}^{\\prime }(\\tau _{1}^{\\prime }))$ as $\\begin{split}\\dfrac{2\\mu ^{2}q(1+\\alpha ^{\\prime })}{\\pi }[b_{1}b_{2}^{*}e^{i\\omega (\\alpha -1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha -1)\\tau _{a}}]\\int _{0}^{\\infty }\\dfrac{dk\\sinh \\Big (\\dfrac{\\pi k(1+\\alpha )}{2a_{1}}\\Big )}{k\\sqrt{\\sinh \\Big (\\dfrac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\dfrac{\\pi k\\alpha }{a_{1}}\\Big )}}\\\\\\times \\Big [\\dfrac{\\sin ((k+\\omega )\\tau _{a})\\sin (\\alpha (k+\\omega )\\tau _{a})}{(k+\\omega )^{2}}-\\dfrac{\\sin ((k-\\omega )\\tau _{a})\\sin (\\alpha (k-\\omega )\\tau _{a})}{(k-\\omega )^{2}}\\Big ]~.\\end{split}$ Now, we simplify the terms corresponding to $G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })$ .", "Proceeding in the similar way one finds the total term corresponding to $G_{11}(\\tau _{1}^{\\prime },\\tau _{1}^{\\prime \\prime })$ as $\\begin{split}\\dfrac{2\\mu ^{2}}{\\pi }\\int _{0}^{\\infty }\\dfrac{dk}{k}\\coth \\Big (\\dfrac{\\pi k}{a_{1}}\\Big )\\Big \\lbrace (p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((k+\\omega )\\tau _{a})}{(k+\\omega )^{2}}+\\dfrac{\\sin ^{2}((\\omega -k)\\tau _{a})}{(\\omega -k)^{2}}\\Big ]\\\\+\\alpha ^{2}\\alpha ^{\\prime }(p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((k+\\alpha \\omega )\\tau _{a})}{(k+\\alpha \\omega )^{2}}+\\dfrac{\\sin ^{2}((\\alpha \\omega -k)\\tau _{a})}{(\\alpha \\omega -k)^{2}}\\Big ]\\Big \\rbrace ~.\\end{split}$ Finally collecting all these terms we find $\\begin{split}\\text{Tr}(\\delta \\rho ^{\\alpha }h_{\\alpha ^{\\prime }})=\\dfrac{2\\mu ^{2}q(1+\\alpha ^{\\prime })}{\\pi }[b_{1}b_{2}^{*}e^{i\\omega (\\alpha -1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha -1)\\tau _{a}}]\\int _{0}^{\\infty }\\dfrac{dk\\sinh \\Big (\\dfrac{\\pi k(1+\\alpha )}{2a_{1}}\\Big )}{k\\sqrt{\\sinh \\Big (\\dfrac{\\pi k}{a_{1}}\\Big )\\sinh \\Big (\\dfrac{\\pi k\\alpha }{a_{1}}\\Big )}}\\\\\\times \\Big [\\dfrac{\\sin ((k+\\omega )\\tau _{a})\\sin (\\alpha (k+\\omega )\\tau _{a})}{(k+\\omega )^{2}}-\\dfrac{\\sin ((k-\\omega )\\tau _{a})\\sin (\\alpha (k-\\omega )\\tau _{a})}{(k-\\omega )^{2}}\\Big ]\\\\-\\dfrac{2\\mu ^{2}}{\\pi }\\int _{0}^{\\infty }\\dfrac{dk}{k}\\coth \\Big (\\dfrac{\\pi k}{a_{1}}\\Big )\\Big \\lbrace (p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((k+\\omega )\\tau _{a})}{(k+\\omega )^{2}}+\\dfrac{\\sin ^{2}((\\omega -k)\\tau _{a})}{(\\omega -k)^{2}}\\Big ]\\\\+\\alpha ^{2}\\alpha ^{\\prime }(p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((k+\\alpha \\omega )\\tau _{a})}{(k+\\alpha \\omega )^{2}}+\\dfrac{\\sin ^{2}((\\alpha \\omega -k)\\tau _{a})}{(\\alpha \\omega -k)^{2}}\\Big ]\\Big \\rbrace ~.\\end{split}$ A similar calculation can be done for the (1+3) dimension structure of the Wightman's function.", "It can be noticed that the dependence of Wightman's function in both (1+1) and (1+3) dimensions on $\\tau ^{\\prime },\\tau ^{\\prime \\prime }$ is identical.", "Hence using the expression given in Eq.", "(REF ) we get that for (1+3) dimension structure, $\\begin{split}\\text{Tr}(\\delta \\rho ^{\\alpha }h_{\\alpha ^{\\prime }})= \\dfrac{q\\mu ^{2}(1+\\alpha ^{\\prime })}{\\pi ^{4}\\sqrt{a_{1}a_{2}}}[b_{1}b_{2}^{*}e^{i\\omega (\\alpha -1)\\tau _{a}}+b_{1}^{*}b_{2}e^{-i\\omega (\\alpha -1)\\tau _{a}}]\\int _{0}^{\\infty }d\\omega _{k}\\int d^{2}k_{p}\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{1}},\\dfrac{\\vert k_{p}\\vert }{a_{1}}\\Big ]\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{2}},\\dfrac{\\vert k_{p}\\vert }{a_{2}}\\Big ]\\\\\\times \\sinh \\Big (\\dfrac{\\pi \\omega _{k}(1+\\alpha )}{2a_{1}}\\Big )\\Big [\\dfrac{\\sin ((\\omega _{k}+\\omega )\\tau _{a})\\sin (\\alpha (\\omega _{k}+\\omega )\\tau _{a})}{(\\omega _{k}+\\omega )^{2}}-\\dfrac{\\sin ((\\omega _{k}-\\omega )\\tau _{a})\\sin (\\alpha (\\omega _{k}-\\omega )\\tau _{a})}{(\\omega _{k}-\\omega )^{2}}\\Big ]\\\\- \\dfrac{\\mu ^{2}}{a_{1}\\pi ^{4}}\\int _{0}^{\\infty }d\\omega _{k}\\int d^{2}k_{p}\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{1}},\\dfrac{\\vert k_{p}\\vert }{a_{1}}\\Big ]\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{1}},\\dfrac{\\vert k_{p}\\vert }{a_{1}}\\Big ]\\\\\\times \\cosh \\Big (\\dfrac{\\pi \\omega _{k}}{a_{1}}\\Big )\\Big \\lbrace (p+q(\\vert b_{1}\\vert ^{2}-\\vert b_{2}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((\\omega _{k}+\\omega )\\tau _{a})}{(\\omega _{k}+\\omega )^{2}}+\\dfrac{\\sin ^{2}((\\omega -\\omega _{k})\\tau _{a})}{(\\omega -\\omega _{k})^{2}}\\Big ]\\\\+\\alpha ^{2}\\alpha ^{\\prime }(p+q(\\vert b_{2}\\vert ^{2}-\\vert b_{1}\\vert ^{2}))\\Big [\\dfrac{\\sin ^{2}((\\omega _{k}+\\alpha \\omega )\\tau _{a})}{(\\omega _{k}+\\alpha \\omega )^{2}}+\\dfrac{\\sin ^{2}((\\alpha \\omega -\\omega _{k})\\tau _{a})}{(\\alpha \\omega -\\omega _{k})^{2}}\\Big ]\\Big \\rbrace \\end{split}$ Now as in our case $|b_1|=1/\\sqrt{2}, |b_2|=1/\\sqrt{2}$ and $p=0, q=1$ , then taking $\\alpha =\\alpha _{a_H}$ , $\\omega =\\omega _2$ , $a_1=a_{H_1}$ and $a_2 = a_{H_2} = a_{H_1}/\\alpha _{a_H}$ in (REF ) for (1+1) dimensions and (REF ) for (1+3) dimensions respectively, one finds $\\text{Tr}(\\delta \\rho ^{H}h_{\\alpha ^{\\prime }})=\\pm 2(1+\\alpha ^{\\prime })I_{1}~,$ where for (1+1) dimensions, $\\begin{split}I_{1}=\\dfrac{\\mu ^{2}}{\\pi } \\cos (\\omega _{2}(\\alpha _{a_H}-1)\\tau _{a})\\int _{0}^{\\infty }\\dfrac{dk\\sinh \\Big (\\dfrac{\\pi k(1+\\alpha _{a_H})}{2a_{H_1}}\\Big )}{k\\sqrt{\\sinh \\Big (\\dfrac{\\pi k}{a_{H_1}}\\Big )\\sinh \\Big (\\dfrac{\\pi k\\alpha _{a_H}}{a_{H_1}}\\Big )}}\\\\\\times \\Big [\\dfrac{\\sin ((k+\\omega _2)\\tau _{a})\\sin (\\alpha _{a_H}(k+\\omega _2)\\tau _{a})}{(k+\\omega _2)^{2}}-\\dfrac{\\sin ((k-\\omega _2)\\tau _{a})\\sin (\\alpha _{a_H}(k-\\omega _2)\\tau _{a})}{(k-\\omega _2)^{2}}\\Big ] \\equiv \\dfrac{\\mu ^{2}}{\\pi } I~.\\end{split}$ and for (1+3) dimensions, $\\begin{split}I_{1}=\\dfrac{\\mu ^{2}\\sqrt{\\alpha _{a_{H}}}}{2\\pi ^{4}a_{H_{1}}}\\cos (\\omega _{2}(\\alpha _{a_{H}}-1)\\tau _{a})\\int _{0}^{\\infty }d\\omega _{k}\\int d^{2}k_{p}\\mathcal {K}\\Big [\\dfrac{i\\omega _{k}}{a_{H_{1}}},\\dfrac{\\vert k_{p}\\vert }{a_{H_{1}}}\\Big ]\\mathcal {K}\\Big [\\dfrac{i\\alpha _{a_{H}}\\omega _{k}}{a_{H_{1}}},\\dfrac{\\alpha _{a_{H}}\\vert k_{p}\\vert }{a_{H_{1}}}\\Big ]\\\\\\times \\sinh \\Big (\\dfrac{\\pi k(1+\\alpha _{a_{H}})}{2a_{H_{1}}}\\Big )\\Big [\\dfrac{\\sin ((\\omega _{k}+\\omega _{2})\\tau _{a})\\sin (\\alpha _{a_{H}}(\\omega _{k}+\\omega _{2})\\tau _{a})}{(\\omega _{k}+\\omega _{2})^{2}}\\\\-\\dfrac{\\sin ((\\omega _{k}-\\omega _{2})\\tau _{a})\\sin (\\alpha _{a_{H}}(\\omega _{k}-\\omega _{2})\\tau _{a})}{(\\omega _{k}-\\omega _{2})^{2}}\\Big ]=\\dfrac{\\mu ^{2}\\sqrt{\\alpha _{a_{H}}}}{2\\pi ^{4}a_{H_{1}}}I~.\\end{split}$ In the above equation, positive sign is for ${s}$ while negative sign is for ${a}$ ." ] ]
2105.11709
[ [ "Predicting Links on Wikipedia with Anchor Text Information" ], [ "Abstract Wikipedia, the largest open-collaborative online encyclopedia, is a corpus of documents bound together by internal hyperlinks.", "These links form the building blocks of a large network whose structure contains important information on the concepts covered in this encyclopedia.", "The presence of a link between two articles, materialised by an anchor text in the source page pointing to the target page, can increase readers' understanding of a topic.", "However, the process of linking follows specific editorial rules to avoid both under-linking and over-linking.", "In this paper, we study the transductive and the inductive tasks of link prediction on several subsets of the English Wikipedia and identify some key challenges behind automatic linking based on anchor text information.", "We propose an appropriate evaluation sampling methodology and compare several algorithms.", "Moreover, we propose baseline models that provide a good estimation of the overall difficulty of the tasks." ], [ "Introduction", "Hyperlinks are the backbone of the Internet.", "Placed within the hypertext of web pages, they allow users to explore the World Wide Web by clicking on anchor texts redirecting them to new contents of interest.", "Moreover, the global link structure of the Internet can be seen as a network carrying rich features for information retrieval systems such as web browsers [22], [16].", "On Wikipedia, collaborative editors are asked to follow specific guidelines https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Linking to ensure the quality of internal links (or wikilinks).", "Indeed, these links should highlight the concepts presented in an article while not overwhelming the readers with unnecessary connections.", "Figure: Example of an abstract taken from Wikipedia.", "Wikilinks are represented with the schema [anchor text|article title].", "We added nonexistent links, between curly brackets {|}, whose anchor texts were found in other articles using a string-matching heuristic.Previous works have proposed algorithms to automatically identify new links in document networks: a link exists between two documents if their contents are semantically related.", "In this paper, we examine this task in the specific case of Wikipedia internal hyperlinks.", "In our case, a document $A$ is linked to document $B$ if there is a sequence of words (an anchor text) in $A$ which refers directly to document $B$ .", "Therefore, given a mapping from anchor texts to documents, the task of searching for new candidates can be tackled with a simple string-matching technique.", "The difficulty of the problem becomes the relevance of potential links given the topic addressed by the article i.e.", "if a potential anchor text is representative enough of the context of an article to be made an hyperlink.", "Many sequences of words can be hyperlink candidates, e.g.", "when they match with titles of Wikipedia pages, however only a few of them will be actual links in the original document.", "This work is a first step towards automatically predicting relevant hyperlinks for new documents with respect to a network of Wikipedia articles.", "Our contributions are the following: we create and make publicly available https://github.com/brochier/wikipedia_hyperlink_networks several exports of Wikipedia centered around different topics in which the text content, the hyperlink network and the associated anchor texts of the articles are provided; we propose two evaluation protocols for a transductive and an inductive prediction tasks.", "In the former case, only a few links are missing from every articles while in the latter case, we aim at predicting all the links of previously unseen articles.", "Furthermore, we provide a strong evaluation sampling method that relies on false positive predictions of a simple string-matching algorithm; we present experimental results performed with text-based, graph-based and hybrid methods for this task." ], [ "Related Works", "Link prediction [18], [19] is a largely studied problem across various disciplines.", "In this paper, we focus on predicting links in a document network, i.e.", "when the nodes are associated with textual features.", "In this section, we first cover recent algorithms applied to the link prediction problem and then relate important works on Wikipedia's hyperlinks." ], [ "Link Prediction in Document Networks", "Early works on link prediction includes similarity-based methods [1], maximum likelihood models [30], [6] and probabilistic models [8], [14].", "More recently, network embedding (NE) methods have achieved better performance and scalability in several application domains.", "DeepWalk (DW) [24] and node2vec [11] are the most well-known NE algorithms.", "They train dense embedding vectors by predicting nodes co-occurrences through random walks by adapting the Skip-Gram model [21] initially designed for word embedding.", "However, in document networks, these previous models do not benefit from the information contained in the text content associated with the nodes.", "To address this, several methods [3], [10] propose to combine text and graph features to produce efficient embeddings.", "As such, Text-Associated DeepWalk (TADW) [32] extends DeepWalk to deal with textual attributes.", "They prove, following the work in [17], that Skip-Gram with hierarchical softmax can be equivalently formulated as a matrix factorization problem.", "TADW then consists in constraining the factorization problem with a pre-computed representation of the documents by using Latent Semantic Analysis (LSA) [7].", "Graph2Gauss (G2G) [2] is an approach that embeds each node as a Gaussian distribution instead of a vector.", "The algorithm is trained by passing node attributes (document-term frequencies) through a non-linear transformation via a deep neural network (encoder).", "Inductive Document Network Embedding (IDNE) [5] is a method that produces, via a topic attention mechanism, representations of documents that reflect their connections in a network.", "In this direction, some approaches [27], [4] specifically model the textual features underlying each link in the network.", "Finally, Graph Neural Networks (GNN) [26], [28], [13] have recently shown great performances in link prediction problems.", "However, it is often non-trivial to infer new representations from the features only, since their aggregators rely on the neighborhood of a node.", "In this paper, we focus on methods that can easily induce predictions for new documents with no known link to the original network.", "Document network embedding methods have been applied to a wide variety of networks such as citation networks [20], [9], question & answer websites [5] and newspaper networks [10].", "However, Wikipedia has particular characteristics, the main one being the fact that links are materialised in the documents by anchor texts.", "Moreover, the existence of a link is the result of an editorial choice meant to improve the reader's ability to explore a particular topic.", "In the following section, we cover some of the works that studied these characteristics." ], [ "Wikipedia Hyperlink Network", "Improving link coverage https://meta.wikimedia.org/wiki/Research:Improving_link_coverage is an important challenge for any website, both to improve the navigation within that website and to enhance its referencing by external browsers.", "A direction for automatically finding useful hyperlinks relies on the user activity logs [29], [23], [12].", "Given clickthrough rate of the hyperlinks on Wikipedia, it is possible to predict usefulness probabilities for candidate links.", "These candidates can be selected based on navigation paths, i.e.", "when users often navigate from a source page to a target page without using a direct hyperlink.", "Our approach differs since we seek to train a model to identify textual patterns that should produce a link where user logs are helpful to (1) identify candidates and (2) quantify the relevance of a link.", "Learning from text feature has the potential to generalize better particularly for real time prediction when a user publishes a new article.", "In this direction, [31] study the efficiency of a simple proper noun based matching system, similar to those we use as baseline models in our experiments.", "The idea is to extract proper nouns from the article content and to match these with existing article titles.", "As we will show in Section , this tends to achieve low precision scores since it produces too many false positive links.", "Finally, it is well-established that Wikipedia gathers very different communities with different linking behaviors [25].", "To take this into consideration, we evaluate the algorithms on several subgraphs centered around different topics of the full Wikipedia network." ], [ "Predicting anchor text hyperlinks", "The task of predicting if a sequence of words in a document is an anchor text linking to another document can be seen as a link prediction task in a network of documents.", "To perform this task, as presented in Section , classifiers are trained to predict a binary decision link/no link with textual and graph information.", "One common issue during this training phase is the collect of negative examples.", "Sampling on all nonexistent links can be computationally infeasible as the densities of the networks are small (below 1%, see Table REF ) and the number of potential documents pairs is the square of the number of documents.", "To reduce the amount of testing pairs, a usual methodology (used in [11]) consists in randomly sampling as many negative pairs as positive ones.", "This tends to produce extremely high and thus not meaningful scores as the link probability between two nodes in social networks is often highly correlated with the average path length between them.", "In this paper we are interested in a specific kind of document linking task as all links must start from some anchor text pointing to another document in the network.", "This is a strong constraint that can be used to select relevant negative samples based on a simple string-matching procedure, that is, we collect every anchor text responsible for a link in the network, and for each of them, we identify all the articles that contain the same sequence of words.", "By doing so, we generate negative samples that are hard to distinguish with positive ones in terms of text features.", "One of the goals of this paper is to compare generic document linking methods to specific ones taking into account the specificities of anchor text hyperlink prediction.", "First, we present the strong baselines we used to perform document linking and then we introduce some novel simple models based on heuristics specific to the task targeted in this paper." ], [ "Document linking with textual and graph information", "We compared several algorithms to perform document linking prediction.", "The following methods rely either or both on the textual and graph information contained in the document network.", "Note that for all methods, we construct representations of dimension $d=512$ : LSA: we use an SVD decomposition of TF-IDF vectors (term frequency inverse document frequency [15]); DeepWalk (DW): we use skip-gram with 10 negative samples (following [11]) and we apply a window of size 10 after performing 80 random walks per node of lengths 40.", "Since DW makes no use of the text features, it cannot be used in the inductive case; IDNE: we run all experiments with $n_t=32$ topic vectors performing 1000 iterations with mini-batches of 64 balanced positive and negative samples; Graph2gauss (G2G): we use term frequency representations of the documents as input; TADW: we follow the guidelines of the original paper by using 20 iterations and a penalty term $\\lambda =0.2$ .", "For induction, we follow the strategy presented in [5]." ], [ "Document linking in the context of anchor text hyperlink prediction", "In addition to these methods, we developed two heuristic techniques based on anchor texts (AT), named AT (title) and AT (anchor), that rely on a string-matching procedure.", "Both methods are given a mapping from strings to Wikipedia articles.", "They only differ in the way this mapping is constructed: AT (title): maps any article's title with its article.", "Moreover, any title redirecting to this article is also considered (e.g.", "both United Kingdom and UK will map the article United Kingdom).", "AT (anchor): maps any anchor text encountered in Wikipedia to the targeted article.", "This allows us to match all the hyperlinks of the datasets, ensuring the highest recall possible.", "However, this model achieves low precision as it tends to over-link.", "Given their respective mappings, these two algorithms predict the existence of a link between two articles if and only if the source page contains at least one of the strings mapping to the target page.", "As such, these methods output exact predictions (true of false).", "Finally, given that AT (anchor) achieves perfect recall, we propose a simple model, ATILP (Anchor Text Informed Link Prediction) that focuses on reducing the number of false positives retrieved by the former.", "This model selects all candidate links identified by AT (anchor), and extracts a representation of their anchor texts using LSA.", "Then, three scores are computed given the LSA vectors of the anchor texts $x_{at}$ , the source documents $x_{ds}$ and the target documents $x_{dt}$ : $s_1$ : is the cosine similarity between $x_{at}$ and $x_{ds}$ , representing how the anchor text is similar to the source document; $s_2$ : is the cosine similarity between $x_{at}$ and $x_{dt}$ , representing how the anchor text is similar to the target document; $s_3$ : is the cosine similarity between $x_{dt}$ and $x_{ds}$ , representing how the two documents are similar.", "Note that this score is directly used a prediction by the LSA model.", "We then train a least squares linear regression without neither normalization nor regularization on the previous scores, $(s_1, s_2, s_3)$ , to predict the probability of a link between a pair of documents.", "The training set is built by randomly sampling 1000 existing and 1000 nonexistent links selected with AT (anchor).", "The motivation behind this simple model is that (1) $s_1$ should capture if an anchor text represents the concepts of the source document, (2) $s_2$ should represent how much the anchor text specifically describes the target document and (3) $s_3$ should indicate if the two documents are semantically related." ], [ "Evaluation setup", "The evaluation of link prediction methods [33] needs careful design.", "Our objective is to quantify the ability of an algorithm to identify relevant links in Wikipedia articles.", "Given a network of documents, an algorithm is trained based on text/graph features and is then asked to produce the probabilities of links given pairs of documents.", "We consider two cases for which we provide experiment results: the transductive case: the algorithms are trained on a network and try to predict missing links within that network (increasing the number of links).", "To simulate this, we remove 10% of the links from the original network during training and evaluate the algorithms on the hidden links.", "Note that in this case, an algorithm can leverage the existing links connecting a document for its predictions.", "the inductive case: the algorithms are trained on a network and try to predict links for new documents outside of the network (increasing the number of documents).", "To simulate this, we remove 10% of the documents from the original network during training and evaluate the algorithms on the hidden documents.", "Note that in this case, an algorithm can only rely on the text features of a document to predict its links.", "In both cases, once the algorithms have produced link probabilities ($p \\in [0,1]$ ) or absolute predictions ($p \\in \\lbrace 0,1\\rbrace $ ), we compute the area under the precision-recall curve (AUC) given the true labels (1 if a link exists, 0 otherwise).", "Moreover, we report the precision (P) and the recall (R) of the predictions.", "When an algorithm outputs probabilities, we threshold them given the true number of positive samples in the test set, enforcing equal values of precision and recall.", "Evaluating both the transductive and inductive cases allows us to identify how well an algorithm can generalize from text features.", "However, we should only compare the rankings of the algorithms and not directly their scores since the ratio of negative to positive samples is higher in the transductive case than in the inductive one.", "This is due to the fact that we test only 10% of the existing links in the first case and test all existing links of the hidden documents in the second case.", "This has the effect of producing lower scores for the transductive task, even if it is an easier task to solve.", "For all experiments, we run 5 times the evaluations with the same splits for all algorithms and report the AUC, the precision and the recall in Section .", "For each dataset, we additionally report the scores obtained by a random algorithms for comparison." ], [ "Datasets", "To build the datasets, we download an XML dump of Wikipedia https://dumps.wikimedia.org/ from January 2021 and we extract the full unweighted directed network of wikilinks with the abstract of each article.", "Moreover, we extract the anchor text of each link in these abstracts.", "Then, we construct subgraphs centered around 4 articles dealing with a variety of topics, namely politics (Joe Biden), science (Science), literature (The Little Prince) and sport (Cristiano Ronaldo).", "The extraction is performed by computing the personalized PageRank scores of these articles and by then selecting the 1000 highest ranked articles.", "The main properties of the resulting networks are reported in Table REF .", "Table: Datasets properties: numbers of documents n V n_V, of links n E n_E, and of words in the vocabulary n W n_W, average (and standard deviation) of document lengths ℓ D \\ell _D and of the numbers of positive and negative samples per document n + n_+ and n - n_- used for the evaluations." ], [ "Experiment Results", "In Table REF , we report the results of the experiments.", "As expected, AT (anchor) has a perfect recall but has a low precision.", "AT (title) shows that (1) not all anchor texts are as trivial as article's titles since the model achieves only 50% recall and (2) many sequences of words corresponding to titles are not worth being made a link in the documents (low precision).", "Graph-based (DW) and text-based (LSA) models perform similarly across the datasets, with a small advantage for LSA.", "Hybrid methods (G2G, TADW and IDNE) do not bring any improvement.", "Our interpretation is that these methods are good for capturing general concepts, but they do not capture fine grained discriminative features capable of distinguishing two conceptually related pages from two actual linked pages.", "For Wikipedia, this distinction is hard to learn because it is mainly due to the editorial choice made by the collaborative users.", "Finally, ATILP achieves slightly better scores than the best models.", "The learned coefficients $c$ of the linear regression associated with the scores $s_1, s_2$ and $s_3$ have average values across all runs: $c_{s_1} = 0.10 (0.09) $ , $c_{s_2} = 0.36 (0.04)$ and $c_{s_3} = 1.06 (0.10)$ .", "Since no normalization is applied beforehand, these coefficients show how much the scores weight in the estimation of the probability of a link.", "Without surprise, the semantic similarity $s_3$ between the two documents accounts for the majority of the prediction.", "However, the similarity $s_2$ between the anchor text and the target document seems to help the model to improve the predictions.", "We hypothesize that the model identifies anchor texts representing too general concepts such as American in Figure REF , thus decreasing the predicted probability of a link when the similarity $s_2$ is low given a high $s_3$ .", "Table: Experiment results.", "The AUC, precision, recall and their standard deviations (in parenthesis) for each dataset, task and method are reported." ], [ "Conclusion and Future Work", "In this paper, we present the task of link prediction in the case of Wikipedia anchor text hyperlinks.", "We propose an evaluation procedure where the sampling method relies on a mapping from anchor texts to target documents.", "We evaluate several algorithms and we highlight that solving this problem requires modeling the interplay between the anchor texts, the source documents and the target documents.", "For future work, we want to use recent neural models in NLP to improve the modeling of these relations.", "This work has been partially funded by the Agence Nationale pour la Recherche (ANR) through the following programs: ANR-19-CE38-0011 (ARCHIVAL), ANR-16-CONV-0002 (ILCB) and ANR-11-IDEX-0001-02 (A*MIDEX)." ] ]
2105.11734
[ [ "An analysis of the time-frequency structure of several bursts from\n FRB121102 detected with MeerKAT" ], [ "Abstract We present a detailed study of the complex time-frequency structure of a sample of previously reported bursts of FRB 121102 detected with the MeerKAT telescope in September 2019.", "The wide contiguous bandwidth of these observations have revealed a complex bifurcating structure in some bursts at $1250$ MHz.", "When de-dispersed to their structure-optimised dispersion measures, two of the bursts show a clear deviation from the cold plasma dispersion relationship below $1250$ MHz.", "We find a differential dispersion measure of ${\\sim}1{-}2$ pc cm$^{-3}$ between the lower and higher frequency regions of each burst.", "We investigate the possibility of plasma lensing by Gaussian lenses of ${\\sim}10$ AU in the host galaxy, and demonstrate that they can qualitatively produce some of the observed burst morphologies.", "Other possible causes for the observed frequency dependence, such as Faraday delay, are also discussed.", "Unresolved sub-components in the bursts, however, may have led to an incorrect DM determination.", "We hence advise exercising caution when considering bursts in isolation.", "We analyse the presence of two apparent burst pairs.", "One of these pairs is a potential example of upward frequency drift.", "The possibility that burst pairs are echoes is also discussed.", "The average structure-optimised dispersion measure is found to be $563.5\\pm 0.2 (\\text{sys}) \\pm 0.8 (\\text{stat})$ pc cm$^{-3}$ $-$ consistent with the values reported in 2018.", "We use two independent methods to determine the structure-optimised dispersion measure of the bursts: the DM_phase algorithm and autocorrelation functions.", "The latter $-$ originally developed for pulsar analysis $-$ is applied to FRBs for the first time in this paper." ], [ "Introduction", "Discovered just over a decade ago , fast radio bursts (FRBs) are one of the newest astrophysical enigmas.", "Despite a limited number of detections (${\\sim }140$ published sources in the Transient Name Server (TNS)),Available at the https://www.wis-tns.org/.", "great strides have recently been made in narrowing down likely progenitors.", "Earlier this year, for example, an FRB-like event was associated with a Galactic magnetar [3], .", "However, due to the extensive range in energetics and activity levels of FRBs, not all can be attributed to a Milky Way-like population of magnetars [2], , , , .", "One such example is FRB 121102 , , whose prolific repetitions have made it one of the most well-studied FRBs to date.", "Targeted multi-wavelength campaigns have revealed coincident persistent radio and optical emission , .", "Using spectroscopic data from the optical source, calculated the redshift to be $z=0.19273(8)$ .", "FRB 121102 thus became the first FRB to be localised to a host galaxy: a low-metallicity dwarf.", "This lead many to consider a possible connection between FRBs and young magnetars born in rare superluminous supernovae events .", "High resolution optical imaging was then used to pin-point the FRB to a star-forming region in the galaxy [1], .", "As well as being well-localised, FRB 121102 goes through active phases, with a possible period of ${\\sim }157\\,$ days , .", "This has further facilitated targeted observing campaigns.", "Polarisation measurements have revealed the extreme and dynamic magneto-ionic environment of FRB 121102: emission was found to be nearly 100% linearly polarised with a rotation measure (RM) of $1.46\\times 10^5$  rad m$^{-2}$ that decreased to $1.33\\times 10^5$  rad m$^{-2}$ over a 7 month period .", "This rapid change in RM without a comparable change in the dispersion measure (DM) implies extreme variation in the line-of-sight projected magnetic field.", "As noted by , such large variation has only been seen near the Galactic center magnetar J1745$-$ 2900 .", "The source of the persistent radio emission is currently unknown.", "It may be from a weak active galactic nucleus or from a magnetised electron-ion nebula .", "Despite numerous follow-up searches, no prompt optical, X-ray or gamma-ray counterparts have been detected , , , .", "FRB 121102 has been observed over a broad range of radio frequencies: from 600 MHz , to 8 GHz , , .", "This has revealed a wide variety of time-frequency structures .", "A common feature of repeating FRBs is a downward drift in frequency, where sub-bursts that arrive at later times have lower central frequencies , [4], , .", "Higher frequency sub-bursts also appear to have shorter temporal durations , , , .", "Further, it has recently been shown that three repeating FRBs (FRB 121102, FRB 180916.J0158+65 and FRB 180814.J0422+73) have an inverse relationship between the frequency drift rate and temporal durations of sub-bursts .", "A number of models have been proposed to explain these phenomena, invoking intrinsic mechanisms, propagation effects, or a combination thereof .", "Intrinsic mechanisms include pulsar-like sparking and cosmic-comb models , , radius-to-frequency mapping in pulsars , the decreasing Lorentz factor of electrons near the surface of a neutron star , decelerating blastwaves from the flare ejecta of young magnetars , or the (potentially relativistic) motion of highly collimated FRB emission with respect to an observer .", "Propagation effects include scintillation , and plasma lensing .", ", however, find scintillation to be inconsistent with the measured drift rate of FRB 121102.", "Further, one would expect to observe upward and downward drift in roughly equal parts.", "Plasma lensing bares a similar shortfall: the lack of upward drift reported in repeating FRBs requires a (rather unlikely) single dominant lens.", "A definite example of upward drifting has yet to be reported, however it might be present in some pairs of closely-separated FRB bursts.", "Here, the second burst arrives at a higher frequency than the first, for example in FRB 180916.J0158+65 , , FRB 190611 , FRB 200428 [3], , and burst 03 in .", "In these cases it is unclear whether the sub-bursts are indeed emitted within the same burst envelope or are independent.", "show that the first two bursts of FRB 200428 were likely emitted within the same burst envelope and that the observed drift may be a result of scintillation.", "In the case of burst 03, however, there is no discernible scintillation.", "The upward drift may evidence lensing, but it is unclear whether the bursts are indeed from the same event.", "Since the burst morphology of FRB 121102, and some other repeating FRBs [4], , , evolves with frequency, there is ambiguity between burst structure and the DM .", "The emission of sub-bursts with different intrinsic central frequencies close in time, as well as propagation effects, can complicate accurate DM determination.", "For example, a burst that appears to have a different DM to other bursts may be made up of unresolved sub-bursts that drift down in frequency .", "To understand the mechanisms driving FRBs, it is essential that features intrinsic to FRBs are resolved.", "In maximising the frequency-averaged burst structure, one can determine sub-burst timescales and calculate frequency drift rates.", "In Paper I , we presented 11 detections of FRB 121102 made using the MeerTRAP system and single burst detection pipeline at the MeerKAT radio telescope .", "Observations were taken over a ${\\sim }3$ hour period on the 10th of September 2019 during the active phase of the FRB.", "Some of these bursts were observed to have complex frequency structure similar to those seen by , with a few showing downward drifting substructure.", "MeerKAT's wide band receiver ($900{-}1670$  MHz usable L-band range) allowed a detailed analysis of this complex frequency structure and frequency-dependant sub-burst drifting at a relatively low frequency.", "A number of intriguing features were noted, one of which is an apparent change in behaviour of some bursts at frequencies around 1250 MHz.", "Here, emission either became significantly fainter, exhibited a complex bifurcated substructure or appeared to deviate from the expected frequency-dependant arrival time ($t\\sim \\nu ^{-2}$ ).", "Two of the bursts (bursts 03 and 05) were each observed with a small `precursor' separated from the main burst by ${\\sim }28$  ms and ${\\sim }34$  ms, respectively, with the signal level between bursts equal to the noise floor.", "Three bursts had observable sub-bursts and the remaining six bursts had no discernible underlying structure.", "In this paper we provide the structure-optimised DMs of the bursts presented in and give an analysis of the observed time-frequency structures.", "The paper is organised as follows: in Section we briefly detail the data reduction and in Section we present the two algorithms used to calculate the structure-optimised DM – Auto-Correlation Functions , and DM_phase .", "Section provides the results and a discussion of the observed burst features, and Section concludes the paper." ], [ "Data reduction", "A detailed description of the data capture is given in Paper I .", "The data contain only total intensity information (Stokes I only), with 4096 channels over a 856 MHz bandwidth, and a time-resolution of $306\\,$ s, centered on 1284 MHz.", "We cleaned the data manually for each burst using pazi in PSRCHIVEAvailable http://psrchive.sourceforge.net.", "to remove corrupt frequency channels.", "This masked a total of ${\\sim }30\\%$ of the band." ], [ "Maximizing burst structure", " argued that a DM metric in which the frequency-averaged burst structure is maximised is more appropriate than maximizing the peak signal-to-noise (S/N).", "This structure-optimised DM corresponds to the DM value at which each sub-burst is correctly de-dispersed.", "calculate the optimal DMs by maximizing the steepness of peaks in the frequency-averaged profile; specifically, they find the DM that maximises the mean square of each profile's forward difference time derivative .", "The relatively low time-resolution of the MeerKAT data ($306.24\\,\\rm s$ ), necessitated a different approach; although we note the methods used here are applicable to high resolution data, too.", "We implement two different techniques.", "In the first, Auto-Correlation Functions (ACFs) are used to determine the widths of structures in each burst , .", "Here, the structure-optimised DM is that which minimises the widths.", "The second invokes DM_phase,Available https://github.com/danielemichilli/DM_phase.", "where the structure-optimised DM is found by maximising the coherent power across the bandwidth .", "For the analysis, the data were de-dispersed over a trial range of $540.0 \\le \\rm DM\\le 590.0 \\, {\\rm pc \\, cm^{-3}}$ with steps of $0.1 \\, {\\rm pc \\, cm^{-3}}$ .", "This step size was found to be suitable for bursts whose morphology evolved significantly with DM.", "This was not the case for all bursts – for example, see the top panel of Figure REF .", "Here, there is an unchanging time lag for numerous consecutive DM values.", "We chose not to increase the step size in such instances as it did not affect the results significantly." ], [ "Autocorrelation functions", "ACFs give the correlation of a signal with a delayed copy of itself over different delay times $\\tau =t_2-t_1$ .", "They are a useful tool in determining microstructure time-scales of pulsar signals , , , and prove to be appropriate for our analysis of FRB sub-structure.", "The frequency-averaged ACF of a single burst, $f(t)$ , is given by $\\text{ACF}(\\tau ) = \\int _{-\\infty }^\\infty f(t)\\overline{f(t-\\tau )} \\text{d}t \\;\\;,$ where $\\tau $ is the time lag and $\\overline{f(t)}$ denotes the complex conjugate of $f(t)$ .", "The narrow structures of the burst contribute to the ACF up to a scale that corresponds to their burst width ($t_s$ ).", "As such, the presence of narrow structure is evidenced by a flattening in the ACF, i.e.", "where the ACF flattens, the narrow features no longer contribute to it.", "The lower the time lag value at which the ACF flattens, the shorter the burst width of these narrow structures and the more enhanced the sub-structure.", "defines the point at which an ACF flattens as the point of intersection of tangents fitted to the first (narrow) ACF region and the following (broad) ACF region (Figure REF ).", "In a bid to automate this process, developed the Turn-Off Point (TOP) algorithm.", "Here, instead of fitting tangents by hand, a point of `significant flattening' is located by comparing the gradient of the ACF in different regions.", "We use the TOP algorithm and verify the results by fitting tangents to the ACFs by eye.", "In Figure REF we compare the ACF of burst 11 for $\\rm DM=\\pm 1\\,{\\rm pc \\, cm^{-3}}$ offset from the maximised $\\rm DM_{\\rm struct}=563.7\\,{\\rm pc \\, cm^{-3}}$ , where $\\rm DM_{\\rm struct}$ is shown to have the smallest time lag.", "Figure REF shows the corresponding dynamic spectra and frequency-averaged burst profiles (lower panel), and the DM vs time lag ($t_s$ ) obtained via the TOP algorithm (upper panel).", "Final results were obtained by interpolating the DM vs time lag (denoted $f_\\text{ACF}$ ), where $\\rm DM_{\\rm struct}$ corresponded to the minimum time lag of the curve (Figure REF ).", "Akin to the uncertainty estimation technique used in DM_phase (see next subsection), the standard deviation was calculated via the Taylor series: $\\sigma _\\text{DM} = \\sqrt{\\left|\\frac{2\\sigma _{f\\text{\\tiny {ACF}}}^2}{f_\\text{ACF}^{\\prime \\prime }(\\rm DM_{\\rm struct})} \\right|} \\;\\;,$ where $\\sigma _f$ is given by the residuals of the interpolation.", "Figure: a.", "A schematic of the frequency-averaged ACF for burst 11 with DM struct =563.7 pc cm -3 \\rm DM_{\\rm struct}=563.7 \\, {\\rm pc \\, cm^{-3}}.", "The sub-burst is depicted by the first bump, whose structure contributes up to a time-scale of t s t_s ms – the point at which the ACF first flattens.", "The tangents are fitted by eye to illustrate the concept.", "b.", "An example of ACFs for burst 11 de-dispersed to different DMs.", "The circles correspond to the points of flattening given by the TOP algorithmFigure: The top panel shows the DM vs time lag (t s t_s) for burst 11.", "The shaded region corresponds to the uncertainty of DM struct \\rm DM_{\\rm struct} (±0.6 pc cm -3 \\pm 0.6\\,{\\rm pc \\, cm^{-3}}).", "There is a sudden jump to higher time lag values at DM ≈564.3 pc cm -3 \\rm DM\\approx 564.3\\,{\\rm pc \\, cm^{-3}}, which is reflected in the behaviour of the ACFs in Figure .", "The bottom panel shows the frequency-averaged burst profiles and waterfall plots de-dispersed to the relevant DMs.", "The resolution of the spectra is decimated to 256 channels.", "Note that the first panel corresponds to the results from DM_phase.", "More sub-components appear to be resolved, as evidenced by the extra peak in the profile." ], [ "Coherent power spectra", "The DM_phase algorithm finds the structure-optimised DM of a burst by maximising the coherent power across the bandwidth : $P(\\omega ,DM) = \\omega ^2\\left|\\int \\frac{\\mathcal {F}\\left[ D(t^{\\prime },f) \\right]}{ \\left|\\mathcal {F}\\left[ D(t^{\\prime },f) \\right]\\right|} \\mathrm {d}f\\right|^2 \\;\\; ,$ where $\\mathcal {F}$ denotes the Fourier transform, $D(t^{\\prime },f)$ is the dynamic spectrum as a function of emission frequency and time, and $\\omega $ is the Fourier frequency.", "Uncertainties are found by converting the standard deviation of the coherent power spectrum into a standard deviation in DM via the Taylor series.", "For further detail, refer to the DM_phase Github documentationAvailable at https://github.com/danielemichilli/DM_phase/tree/master/docs.", "and Seymour et al.", "(in prep)." ], [ "Results and Discussion", "The structure-optimised DMs are presented in Table REF .", "The frequency spectra (`waterfall' plots) are shown in Figure REF .", "Where the structure-optimised DMs given by the ACF and DM_phase methods agree, bursts are de-dispersed to the mean of the two results.", "Where they differ, the most likely candidate DM is used (as discussed in Section REF ).", "Section REF provides a comparison of the two techniques, after which the average DM for the epoch is calculated.", "A number of caveats in determining $\\rm DM_{\\rm struct}$ are highlighted here.", "The burst properties are then presented and possible implications are discussed.", "Table: Structure-optimised DMs for the 11 FRB 121102 bursts.", "Due to their low fluxes, including/excluding the precursors in the analysis for bursts 03 and 05 did not affect the value of DM struct \\rm DM_{\\rm struct}.", "For bursts 07, 08 and 10, DM_phase gave multiple possible values, as discussed in Section .", "The last column gives the best estimate for each burst.", "Where the ACF method and DM_phase results agree, the mean of the results is used, rounded up to the nearest decimal value.", "Where results disagree, the most likely value is chosen, as discussed in Section .", "Asterisks denote the selected sample of best estimates used in the second calculation of the average DM.Figure: Dynamic spectra of the bursts detected with MeerKAT on the 10th of September 2019.", "The top panels show the frequency-averaged burst profile.", "The bottom panels show the frequency spectra with the resolution of each burst decimated to 256 channels to enhance visibility.", "The time resolution of the bursts is 306.24306.24\\,s.", "The RFI was removed from the data manually.", "The flux density scale is uncalibrated and shown in arbitrary units.", "Please refer to for full details.", "The bursts are de-dispersed to the structure-optimised DMs given in Table (the mean of ACF and DM_phase).", "Burst 01 is not shown, as a structure-optimised DM could not be established.", "Burst 02 is de-dispersed to 564.6 pc cm -3 564.6\\,{\\rm pc \\, cm^{-3}}, burst 03 to 565.9 pc cm -3 565.9\\,{\\rm pc \\, cm^{-3}}, burst 04 to 572.4 pc cm -3 572.4\\,{\\rm pc \\, cm^{-3}}, burst 05 to 564.5 pc cm -3 564.5\\,{\\rm pc \\, cm^{-3}}, burst 06 to 563.1 pc cm -3 563.1\\,{\\rm pc \\, cm^{-3}}, burst 07 to 563.0 pc cm -3 563.0\\,{\\rm pc \\, cm^{-3}}, burst 08 to 563.6 pc cm -3 563.6\\,{\\rm pc \\, cm^{-3}}, burst 09 to 565.1 pc cm -3 565.1\\,{\\rm pc \\, cm^{-3}}, burst 10 to 563.4 pc cm -3 563.4\\,{\\rm pc \\, cm^{-3}}, and burst 11 to 563.3 pc cm -3 563.3\\,{\\rm pc \\, cm^{-3}}." ], [ "Comparison of techniques", "The results from the ACF method and DM_phase largely agree to within a 1$\\sigma $ confidence level, and both methods did not find structure when bursts were particularly faint (i.e.", "burst 01 and the precursor of burst 03, the latter of which is discussed in Section REF ).", "DM_phase gives multiple values for bursts 07, 08 and 10, which necessitates further investigation.", "The multiple values are evidenced by multiple peaks in the coherent power spectra.", "For burst 07, three values were determined for $\\rm DM_{\\rm struct}$ using DM_phase (Figure REF ).", "The first, with $\\rm DM=562.9\\pm 0.2\\,{\\rm pc \\, cm^{-3}}$ , agrees with the ACF method.", "At this DM value, there are at least three distinct sub-bursts.", "We take this to be the structure-optimised DM.", "For the next two DM_phase values for $\\rm DM_{\\rm struct}$ , the sub-bursts begin to align with the main burst in time, and the structure in the frequency-averaged profile diminishes.", "While the results for burst 11 agree within the uncertainty margins, the burst profiles look significantly different at the central values (Figure REF ).", "We argue that the most representative structure-optimised DM is given by DM_phase ($\\rm DM_{\\rm struct}=562.8\\pm 0.3\\,{\\rm pc \\, cm^{-3}}$ ), where one can see an additional peak in the profile.", "Further, at this DM a bright sub-component of the second burst aligns with the main burst.", "The results from ACF and DM_phase also differ for burst 02.", "Here it is unclear which is most likely, as the burst profile changes so little (Figure REF ).", "We take the structure-optimised DM to be the mean of the two methods.", "Figure REF shows the two DM_phase values for $\\rm DM_{\\rm struct}$ for burst 08.", "The ACF method agrees with the second result, where $\\rm DM_{\\rm struct}=564.9\\pm 0.6\\,{\\rm pc \\, cm^{-3}}$ .", "Here, however, the profile structure has been washed out.", "At $\\rm DM_{\\rm struct}=563.6\\,{\\rm pc \\, cm^{-3}}$ , one can see the two peaks from the sub-bursts.", "Burst 10 is shown in Figure REF .", "Here, it is unclear whether or not the burst consists of two sub-bursts – the missing frequency bands at ${\\sim }1250$  MHz could indicate the appearance of two distinct bursts.", "Looking at the second panel, the top half of the burst does not align with the bottom half, which may suggest that they are sub-bursts.", "The behaviour and appearance of burst 10 is also similar to that of burst 07.", "This may imply $\\rm DM_{\\rm struct}=563.6\\pm 0.4\\,{\\rm pc \\, cm^{-3}}$ .", "One may also argue for the lower DM value by noting that it is better in line with previous DM measurements of FRB 121102 , , .", "Ultimately, however, the result is ambiguous.", "In summary, as a result of the comparison, we note that (i) DM_phase occasionally gives multiple possible values for $\\rm DM_{\\rm struct}$ , and it is important to manually check these, as the highest peak in the power-DM function does not necessarily correspond to the structure-optimised DM; (ii) the ACF method failed to identify structure where the burst separation is less than a millisecond (e.g.", "burst 08) and the analysis requires more manual intervention.", "While the two definitions of `maximum structure' give results that are largely consistent with each other, ambiguity still exists within the metric.", "For example, at 2$\\sigma $ , the two solutions for burst 8 are compatible, but visual inspection shows they are clearly alternative to each other.", "Care should be taken when performing these types of analyses and results should be accepted with a measure of caution.", "Going forward, it will be interesting to compare these methodologies with those of and for bursts with higher time resolution data.", "Figure: Burst 07 at the structure-optimised DMs identified by DM_phase.", "The resolution of the spectra is decimated to 256 channels.", "The first panel ( DM struct =562.9 pc cm -3 \\rm DM_{\\rm struct}=562.9\\,{\\rm pc \\, cm^{-3}}) agrees with the ACF method ( DM struct =563.1±0.4 pc cm -3 \\rm DM_{\\rm struct}=563.1\\pm 0.4\\,{\\rm pc \\, cm^{-3}}).", "Three (possibly four) sub-bursts are evidenced, which march down in frequency.", "The profile structure then decreases as the lower frequency bursts begin to sweep under the main burst.Figure: Burst 02 at the structure-optimised DMs identified by the ACF method (first pannel) and DM_phase (second pannel).", "There is little observable change in the structure.", "The resolution of the spectra is decimated to 256 channels.Figure: Burst 08 at the structure-optimised DMs identified by DM_phase.", "The resolution of the spectra is decimated to 256 channels.", "The second panel ( DM struct =564.9 pc cm -3 \\rm DM_{\\rm struct}=564.9\\,{\\rm pc \\, cm^{-3}}) agrees with the ACF result ( DM struct =564.6±0.4 pc cm -3 \\rm DM_{\\rm struct}=564.6\\pm 0.4\\,{\\rm pc \\, cm^{-3}}), however here the two sub-bursts are not reflected in the frequency-averaged profile and the bursts overlap each other.", "At DM struct =563.6 pc cm -3 \\rm DM_{\\rm struct}=563.6\\,{\\rm pc \\, cm^{-3}} the sub-bursts are distinct in the profile and show a downward frequency drift in the waterfall plot.Figure: Burst 10 at the structure-optimised DMs identified by DM_phase.", "The resolution of the spectra is decimated to 128 channels to enhance visibility.", "The first panel ( DM struct =563.6 pc cm -3 \\rm DM_{\\rm struct}=563.6\\,{\\rm pc \\, cm^{-3}}) agrees with the ACF result ( DM struct =563.3±0.4 pc cm -3 \\rm DM_{\\rm struct}=563.3\\pm 0.4\\,{\\rm pc \\, cm^{-3}}).", "In this case, there are two sub-bursts.", "In the second panel, the lower burst sweeps under the upper burst.", "The behaviour and appearance of burst 10 is similar to that of burst 07.", "Arguably, the missing frequency bands at ∼1250{\\sim }1250 MHz may create the illusion of two sub-bursts." ], [ "Average DM variation", "The average structure-optimised DM of the epoch is calculated in two ways.", "In the first, the average is taken over the 10 bursts, weighted by the errors, to give $\\rm DM_{\\rm struct}=564.8\\pm 0.6 (\\text{sys}) \\pm 2.5 (\\text{stat}) \\,{\\rm pc \\, cm^{-3}}$ and $\\rm DM_{\\rm struct}=564.4\\pm 0.6 (\\text{sys}) \\pm 2.9 (\\text{stat}) \\,{\\rm pc \\, cm^{-3}}$ using the ACF method and DM_phase, respectively.", "The first uncertainty is the systematic uncertainty given by the respective methods and the second is the statistical uncertainty given by the standard deviation of the data.", "The values of $\\rm DM_{\\rm struct}$ for some of the bursts fall outside of this region; most notable of which is burst 04, which is ${\\sim }8\\,{\\rm pc \\, cm^{-3}}$ greater than the average.", "We attribute this difference to insufficient S/N.", "Unresolved components in the bursts may also significantly influence the resultant $\\rm DM_{\\rm struct}$ .", "We thus urge caution when interpreting the DM change between bursts in this, and for that matter any, sample.", "We also note that the errors given by both methods are under-representations of the true uncertainty on the measurement, as they do not take into account potentially unresolved components.", "As such, even individual results with small uncertainties should be closely examined.", "Good examples of this are bursts 03 and 05, whose ambiguity is discussed in Section REF .", "Establishing a reliable mean DM for the epoch may best be achieved by only considering bursts whose sub-components appear to be reasonably resolved.", "As such, we recalculate the average DM with a selected sample of bursts and their best estimates.", "The final data set consists of burst 02 (mean of ACF and DM_phase), burst 08 (DM_phase), burst 07 (DM_phase) and burst 11 (DM_phase).", "This gives a structure-optimised DM of $563.5\\pm 0.2 (\\text{sys}) \\pm 0.8 (\\text{stat})\\,{\\rm pc \\, cm^{-3}}$ .", "Figure REF shows the bursts de-dispersed to $563.5\\,{\\rm pc \\, cm^{-3}}$ .", "An important question then is whether this single DM creates a cohesive picture of the burst sample.", "This is discussed in Section REF .", "The average structure-optimised DM is consistent with 2018 observations ($563.6\\pm 0.5 \\,{\\rm pc \\, cm^{-3}}$ ; and $563.5\\pm 1.3 \\,{\\rm pc \\, cm^{-3}}$ ; ) taken 1 year prior.", "The uncertainties make it unclear whether the average DM has indeed remained constant over this period, or whether it has increased or even decreased.", "A linear interpolation with 2016 observations reveals an average increase of ${\\sim }\\,1\\,{\\rm pc \\, cm^{-3}}$ (Figure REF ).", "This is roughly consistent the ${\\sim }1{-}3 \\,{\\rm pc \\, cm^{-3}}$ increase from 2012 to 2016 reported by , however more data is needed in our case to confirm whether the increase is indeed secular.", "There are a number of scenarios that may account for the apparent trend.", "A persistent increase in DM may, for example, be attributed to a young neutron star whose supernova ejecta expands into a high density interstellar medium , .", "The FRB may also be associated with a young star whose ionisation drives outward expansion into a surrounding H II region .", "Alternatively, if the FRB source is moving rapidly through an H II region due to – for instance – a supernova kick, the DM may increase or decrease depending on the direction of the kick .", "In the magnetar flare model by , the increase in DM may be attributed to the photoionisation of neutral gas by the UV and X-ray radiation from the shock.", "Here, an increase of $0.01{-}1\\,{\\rm pc \\, cm^{-3}}$ is expected on a time scale of days to months.", "For an in-depth discussion on the DM and RM evolution of FRB 121102 in the context of the supernova remnant models by and , see .", "Should the DM be shown to decrease in the future, plasma lensing may also be accountable (although the scenarios mentioned above would not be ruled out by this).", "In this case, plasma lensing would be local to the source (e.g.", "a nebula) or the host galaxy .", "It has been shown that non-local propagation effects, such as from Hubble expansion, gas density fluctuations in large-scale structure and gravitational potential fluctuations, cannot account for the observed DM variations of FRB 121102 .", "Figure: Bursts de-dispersed to an average structure optimised DM of ∼563.5 pc cm -3 {\\sim }563.5 \\,{\\rm pc \\, cm^{-3}}.", "Note the difference in the behaviour of the main bursts of 03 and 05 from that shown in Figure .", "Instead of showing an apparent deviation from the t∼ν -2 t\\sim \\nu ^{-2} relationship, the middle section of the main bursts are misaligned, and are thus possibly made up of unresolved downward drifting sub-bursts.Figure: Structure-optimised DMs measured for FRB 121102 between 2016 and 2019.", "The dashed grey line shows the linear interpolation, which gives an average increase of ∼1 pc cm -3 {\\sim }\\,1\\, {\\rm pc \\, cm^{-3}} per year.", "and use the maximum steepness method to determine DM struct \\rm DM_{\\rm struct}, and use DM_phase." ], [ "Sub-bursts", "Bursts 07 and 11 (and possibly 10) have a bifurcating structure around 1250 MHz.", "Similar behaviour (at a different central frequency) has been observed in FRB 121102 before and in FRB 180916.J0158+65 , where the right-most component of each burst appears to follow a different DM to the previous components.", "Particularly notable is latter: burst 11 in , where a bright component that aligns with the previous sub-bursts is embedded in the right-most sub-burst.", "This presents the possibility that the sub-bursts of bursts 07 and 11 are not single sub-bursts with a different DM, but rather comprise multiple unresolved components that drift down in frequency.", "We investigate the apparent change in DM between the sub-bursts by splitting the spectra at 1250 MHz and 1100 MHz, respectively.", "We note that the RFI affected (and thus removed) frequency bands at ${\\sim }1250$  MHz and the overlapping frequencies of the different components may affect the DM results.", "For burst 07, the DM for the higher frequency sub-burst is ${\\sim }1\\,{\\rm pc \\, cm^{-3}}$ lower than the lower frequency sub-burst (a 1$\\sigma $ difference; $563.3\\pm 0.7$  ${\\rm pc \\, cm^{-3}}$ vs $564.4\\pm 0.4$  ${\\rm pc \\, cm^{-3}}$ , using DM_phase).", "For burst 11, the higher frequency sub-burst is ${\\sim }2\\,{\\rm pc \\, cm^{-3}}$ lower than the lower frequency sub-burst ($562.7\\pm 0.4$  ${\\rm pc \\, cm^{-3}}$ vs $564.9\\pm 0.5$  ${\\rm pc \\, cm^{-3}}$ , using DM_phase).", "Interestingly, we note that if the bursts are multiple images caused by plasma lensing, the predicted observed difference in DM values could be as large as ${\\sim }1\\,{\\rm pc \\, cm^{-3}}$ .", "Other mechanisms can in principle be invoked to explain a difference in DM: e.g.", "the sub-bursts could be emitted from different parts of the magnetosphere, or they could be observed along different sight-lines through dense plasma , – for example the line-of-sight through a nebula will vary depending on the neutron star rotation phase at the time of emission .", "However, the differential DMs expected in these cases are too small to account for those observed in bursts 07 and 11.", "It will be interesting to see in higher resolution data going forward whether similar sub-bursts truly do misalign with previous sub-bursts or if the effect is a result of unresolved downward drifting sub-structure.", "This clearly has implications on the DM of FRB 121102 for this epoch and other epochs.", "If the right-most sub-bursts are made up of unresolved sub-bursts that align with the higher frequency sub-burst, then the DM of the burst is best described by the DM of the higher frequency sub-burst.", "Figure: Lensing simulations using an overdense (left) or underdense (right) Gaussian lens, as described in Section .", "Top panels show the magnification μ(t,f)\\mu (t, f), with a logarithmic colour bar extending from 10 -1 -10 2 10^{-1} - 10^{2}.", "The bottom panels show a mock FRB (modelled as an achromatic Gaussian (μ(f)=1\\mu (f)=1 with 0.5 ms width) with magnification, geometric time delay, and dispersive delays of the lensing field at a given time applied.", "The colourbar is saturated to magnifications between 0 and 5." ], [ "Burst pairs", "In bursts 03 and 05, a bright (main) burst is preceded by a faint (precursor) burst, with a separation time of ${\\sim }28$  ms and ${\\sim }34$  ms, respectively.", "FRB burst pairs are not connected by an emission bridge.", "This distinguishes burst pairs from sub-bursts .", "Numerous burst pairs have been observed in FRB 121102 before, as well as in other FRBs (both repeating and apparently non-repeating).", "A summary is provided in Table REF .", "Table: Burst pairs observed in FRBs.", "Where bursts are from repeaters, the burst name for the individual burst (given in italics) follows the naming convention of the relevant paper.", "If there is no convention, the name corresponds to the observation number XX in the relevant paper as BXX.", "Apparently one-off FRBs are named as per usual.We note that the waiting time between the burst pairs of apparently one-off bursts are significantly shorter than those of repeaters.", "Due to the small sample size, this may just be coincidence." ], [ "Burst Envelope", "Whether or not burst pairs are independent events (or even echoes; Section REF ) is currently an open question.", ", for example, propose that burst pairs may be broad bursts with only two resolvable components.", "In the case of a neutron star, if the source is active as a radio emitter for a duration similar to its rotation period, then we may expect to see pre- and postcursor bursts, as well as, occasionally, both of them.", "To date, no such triplets have been observed.", "The longest duration of a single burst reported for FRB 121102 is $39\\pm 2$  ms , which is comparable to the total time scales of bursts 03 and 05 (${\\sim }37$  ms and ${\\sim }39$  ms, respectively).", "As such, it is feasible that the bursts occurred within the same burst envelope, and hence that burst 03 shows upward drift, i.e.", "the main burst of 03 arrives at a higher frequency than the precursor (Figure REF ).", "This comparison, however, provides only tenuous evidence." ], [ "The DM of the Main Bursts", "When de-dispersed to their structure-maximised DMs, the main bursts of bursts 03 and 05 appear to change behaviour at ${\\sim }1250$  MHz, where the tails abruptly tilt to earlier times (see Figures REF and REF ).", "In our sample, this feature is exclusive to the bursts with precursors.", "In particular (using DM_phase), the lower and upper frequency bands of burst 03 have $\\rm DM_{\\rm struct}=564.7\\pm 0.7\\,{\\rm pc \\, cm^{-3}}$ and $\\rm DM_{\\rm struct}=567.1\\pm 0.5\\,{\\rm pc \\, cm^{-3}}$ , respectively; and the lower and upper frequency bands of burst 05 have $\\rm DM_{\\rm struct}=563.3\\pm 0.2\\,{\\rm pc \\, cm^{-3}}$ and $\\rm DM_{\\rm struct}=565.3\\pm 0.2\\,{\\rm pc \\, cm^{-3}}$ , respectively.", "This may indicate a deviation from the $\\nu ^{-2}$ law.", "On the other hand, it is possible that correctly de-dispersing the lower frequency parts of the bursts may give the most representative DM, even though that component is not dominant over the observed bandwidth.", "In this case, the upper part of the bursts would comprise unresolved downward drifting sub-bursts (the first panels of Figures REF and REF ).", "In support of this scenario, the lower DM values are more in line with other bursts in the sample and with the previously reported DM values for FRB 121102 , , .", "Interestingly, there is a differential DM of ${\\sim }1$  ${\\rm pc \\, cm^{-3}}$ between the main bursts of burst 03 and 05, as illustrated by their shapes at the DMs depicted in Figures REF and REF : burst 03 looks the same as burst 05 when it is de-dispersed to values ${\\sim }1$  ${\\rm pc \\, cm^{-3}}$ higher than burst 05.", "This further highlights the challenges in determining an average or representative DM for an epoch – there may be no single DM that best describes all bursts in a sample, and it is difficult to isolate genuine changes in DMs between bursts." ], [ "Plasma Lensing", "Here we consider the potential change in behaviour observed in the main bursts of 03 and 05 when de-dispersed to their individual structure-optimised DMs.", "The deviation from a $\\nu ^{-2}$ law could be caused by multi-path propagation, either through geometric delays (caused by the differing path lengths of light across frequency), or through differential DM (caused by the different electron column through the different paths across frequency).", "explored the possibility of plasma lensingAlso see for a discussion of possible plasma lensing in FRB 121102, evidenced by a large spectral peak at 7.1 GHz.", "of FRBs from lenses within the host galaxy, considering 1D overdense (divergent) Gaussian lens of width $a$ , extra column density $\\rm DM_l$ , and a distance between the source and lens of $d_{sl}$ .", "The focal length of a lens must be less than the distance to the observer from the lens for caustics to form, expressed as the constraint $0.65 \\left(\\frac{d_{sl}}{\\rm {pc}}\\right) \\left(\\frac{\\rm DM_{l}}{\\rm {pc}\\,\\rm {cm}^{-3}}\\right) \\left(\\frac{a}{\\rm {AU}}\\right)^{-2} \\left(\\frac{\\nu }{\\rm {GHz}} \\right)^{-2} \\ge 1 \\;\\;.$ The formation of caustics depends very strongly on small-scale variations of DM, since even a small $\\rm DM_{l}$ can form caustics with sufficiently small $a$ , due to the $a^{-2}$ dependence.", "As an example of this, strong lensing in the Black Widow pulsar B1957+20 is seen to occur in regions where $\\Delta \\rm DM\\sim 10^{-4}\\,{\\rm pc \\, cm^{-3}}$ over ${\\sim }1000\\,$ km .", "To make informed estimates of lensing occurring in FRB 121102, one would like to have measurements of the smaller scale DM variations.", "Lensing can occur in proximity of the source of the FRB or farther out in the host galaxy.", "In the first case, we can rely on the measured rotation measure (RM) variations of 2200 rad m$^{-2}$ over 3 days , coupled with an estimate of the magnetic field $B$ , in order to estimate the several day variations of DM.", "fit the measured RM variations with the model of , which assumes winds and flares from a young magnetar driving a constant-velocity expansion of a highly magnetized nebula.", "They try three different model conditions of the free magnetic energy of the magnetar, onset of the magnetar's active period, and radial velocity; their fit conditions and best fit values are given in Table 4 of .", "From the range of outflow velocities, magnetic energies, and best-fit nebular ages, one can derive a range of expanding shell radii of $R=0.04{-}0.12$  pc, an inferred magnetic field strength within the nebula of $B= 0.76{-}1.36$  G, and extra DM in this region of $\\rm DM= 0.09{-}0.16\\,{\\rm pc \\, cm^{-3}}$ .", "Using the range of magnetic fields, the inferred variation of DM over 3 days (from $\\Delta \\textrm {RM}\\sim 2200$  rad m$^{-2}$ over 3 days) is $\\Delta \\rm DM\\sim 0.002{-}0.0036\\,{\\rm pc \\, cm^{-3}}$ .", "With these estimates, it is not impossible to get lensing, but to satisfy the focal constraint of Eq.", "REF at our observing frequencies one would need larger values of DM, or fluctuations on smaller scales.", "While lensing could occur, there are not sufficient electrons to create differential DMs of ${\\sim }1\\,{\\rm pc \\, cm^{-3}}$ , and geometric time delays of lensing would be of order $$ s rather than several ms to explain the precursors as bursts preceding an echo.", "If lensing is occurring in the host galaxy (i.e.", "if the DM in the host is de-coupled from the region causing the large RM), then there is much more material able to cause lensing.", "As mentioned in , lensing in the host galaxy can create caustics, which could cause geometric time delays up to ${\\sim }10$  ms, with differential DM of ${\\sim } 1\\,{\\rm pc \\, cm^{-3}}$ .", "We confirm that lenses with $a \\sim 10$  AU satisfy the focal constraint of Eq.", "REF and are consistent with the measured DM variations ${\\sim }1 \\,{\\rm pc \\, cm^{-3}}$ .", "We performed simple geometric optics simulations following the ideas presented , for an overdense or underdense Gaussian lens.", "We place the lenses at a distance of $d_{sl} = 500\\,$ pc, with lens sizes of as $a \\sim 10$  AU with $\\Delta $ DM$\\,=1\\,{\\rm pc \\, cm^{-3}}$ for $DM(x_{\\rm lens})$ , and use the unknown relative velocity between the source and lens of $v=100\\,$ km/s.", "The phase across the lens is $\\phi (x_{\\rm lens}) = \\phi _{\\rm g}(x_{\\rm lens}) + \\phi _{\\rm DM}(x_{\\rm lens})$ , where $\\phi _{\\rm g}$ is the geometric phase; images are defined as positions of stationary phase (ie.", "where $\\frac{d\\phi (x_{\\rm lens})}{d x_{\\rm lens}} = 0$ ), and the magnification of each image is given by $\\frac{d^{2}\\phi (x_{\\rm lens})}{d x_{\\rm lens}^{2}}$ .", "The results of the simulation are shown in the top panels of Figure REF .", "Along with a magnification $\\mu (f)$ , each image has geometric delay $\\tau _{g}(f)$ and differential DM $DM(f)$ .", "The total magnification $\\mu (f, t)$ is computed as the incoherent sum of the images.", "To qualitatively assess how the morphology of bursts could be affected by lensing, we create mock bursts (an achromatic Gaussian with 0.5 ms width), and for a given time, apply $\\mu (f)$ , $\\tau _{\\rm g}(f)$ , and the time delays associated with DM($f$ ) – we show three examples of mock bursts for both a divergent and convergent lens in the bottom panels of Figure REF .", "Several features comparable to the observed burst structures can be produced, including dimming at lower frequencies, and chromatic DMs (or apparent differential DMs from $\\tau _{\\rm g}(f)$ ).", "Near a cusp caustic (or “catastrophe”), the flux could sharply decrease below the focal frequency, associated with a sharp increase in DM (eg.", "Figure REF , panel b, left), and in regions of multiple images, one can see multiple echoed copies of the burst with different apparent DM (eg.", "panel c, right).", "In this example, such features would last on ${\\sim }$ day–month timescales for lensing in the host galaxy, but this depends on the size and distance of the lens, the transverse velocity of the FRB, and the magnification of the caustics.", "However, we caution that these are highly simplified and idealistic simulations; a proper treatment would consider interference between images, and more realistic lenses.", "In geometric optics, the magnification formally diverges at caustic boundaries; in such regions, wave optics will become important , and we may instead see a smooth transition of intensity across frequency (see the lower part of Figures REF and REF ), and possibly interference effects.", "Interference effects could induce changes on much smaller timescales; for the above example, a very rough timescale for interference effects is $t_{\\rm diff} \\sim \\frac{\\lambda }{v_{\\rm sl}} \\frac{d_{\\rm sl}}{2c \\tau _{g}} \\sim 40$  s for the simulated lens values, and $\\lambda = 25$  cm, $\\tau _{g}=100\\mu $ s." ], [ "Polarisation", "FRB 121102 resides in an extreme magneto-ionic environment, so lensing scenarios may be distinguished using polarisation properties.", "The refractive index in a magnetised plasma for left (L) and right (R) circular polarisation states is $n_{L,R} = \\sqrt{1 - \\frac{f_{p}^{2}}{f(f \\mp f_{B,||}) }} \\approx 1 - \\frac{1}{2} \\frac{f_{p}^{2}}{f^{2}}\\left(1 \\pm \\frac{f_{B,||}}{f}\\right),$ where $f_{p} \\approx 9\\,\\mathrm {kHz} \\sqrt{n_{e} / cm^{-3}}$ is the plasma frequency, and $f_{B,||} \\approx 2.8\\,\\mathrm {MHz} \\sqrt{B_{||} / G}$ is the cyclotron frequency of the parallel magnetic field.", "Different refractive indices would imply a different group velocity between the two polarisation states, which results in a Faraday delay of $\\tau _{FR} \\approx 0.0572\\,\\mathrm {ns} \\left( \\frac{\\rm {RM}}{\\mathrm {rad}\\,\\rm {m}^{-2}} \\right) \\left(\\frac{f}{\\mathrm {GHz}} \\right)^{-3}.$ At 1 GHz, $\\rm {RM}\\approx 10^{5}$  rad m$^{-2}$ results in a delay of $\\approx 5.7$ s. Additionally, if lensing effects are important, the focal frequencies between the two polarisations will differ by the cyclotron frequency , $\\Delta f \\approx 2.8\\, \\mathrm {MHz}/G$ .", "These effects are unlikely to be seen in incoherent filterbank data (as those presented in this paper), but could potentially be revealed by coherently comparing the timestreams between polarisations.", "Searching for a coherent correlation may also reveal whether the precursors, which are qualitatively quite similar to the bursts they precede, are copies or echoes.", "An example of such techniques is shown in , who coherently correlate nearby giant bursts in PSR B1957+20.", "In addition, Faraday delays will be much more evident at lower frequencies, scaling as $\\nu ^{-3}$ .", "In such an environment, it may be possible to observe higher order effects such as Faraday conversion , .", "However, at lower frequencies the source is likely depolarized within individual channels, and it may only be possible to detect these effects coherently, using voltage data of bursts.", "Figure: Burst 03 de-dispersed to the structure-optimised DMs given by frequency bands below (panel 1) and above (panel 2) 1210 MHz.", "The resolution of the spectra is decimated to 256 channels.Figure: Burst 05 de-dispersed to the structure-optimised DMs given by frequency bands below (panel 1) and above (panel 2) 1210 MHz.", "The resolution of the spectra is decimated to 256 channels." ], [ "Dimming below 1250 MHz", "We do not expect that the observed dimming below 1250 MHz is caused by absorption, unless absorption is highly variable, since a burst was detected by CHIME at much lower frequencies.", "However, absorption could still play a role if conditions are changing along the line-of-sight.", "In fact, if FRB 121102 is in an orbit , absorption and lensing could easily be phase dependent.", "This may be assessed by looking for a phase dependence of burst properties." ], [ "Frequency drifts", "A common feature of repeating FRBs is a downward drift in frequency [4], , .", "In Paper I, the structure-optimised DMs reported here were used to characterise the drift-rates of bursts 07 and 11 using a 2D ACF method.", "The structure-optimised DM for burst 07 has since been updated, and thus we present the revised value for the frequency drift.", "Previously, burst 07 was sub-banded and the drift rate was measured at a center frequency of 1400 MHz over a bandwidth of 214 MHz.", "Here, we additionally measure the drift rate at a center frequency of 1284 MHz over 856 MHz.", "The drift rate of burst 08 is also measured at a center frequency of 1284 MHz over 856 MHz.", "Results are presented in Table REF .", "Table: Measured drift rates for various sub-bands of bursts 07, 08 and 11.These are consistent with those published between $600{-}6500$  MHz, with a slope of $\\alpha =-0.147\\pm 0.014\\, \\text{ms}^{-1}$ .", "No upward drift is reported.", "Where bursts consist of two sub-bursts (e.g.", "bursts 03 and 05), however, it is unclear whether the bursts are independent or occur within the same burst envelope.", "In the latter case, burst 03 may be an example of upward drifting: the second panel of Figure REF shows a faint precursor burst between ${\\sim }$ 1000–1200 MHz, followed by a main burst between ${\\sim }$ 1000–1700 MHz.", "The precursor, however, may just be intrinsically fainter overall than the second component or fainter at higher frequencies.", "Similar potential upward drifting behaviour has been reported before in repeating (periodic) FRB 180916.J0158+65 with a burst separation of ${\\sim }60$  ms , , in the apparently one-off FRB 190611 with a burst separation of ${\\sim }0.7$  ms and in the Galactic FRB 200428 with a burst separation of ${\\sim }29$  ms [3], ." ], [ "Conclusion", "In this paper, we calculated the structure-optimised DMs for 10 out of the 11 FRB 121102 bursts detected by the MeerKAT radio telescope originally presented in .", "Two independent methods were used to do so – ACFs and DM_phase .", "We find that while results largely agree, care should be taken when selecting an “optimal” DM: where results are ambiguous, it is not always clear which burst profile best represents the burst at origin.", "Potentially unresolved sub-components further complicate accurate DM determination.", "The main bursts of 03 and 05 illustrate this point well: while they appear to deviate from the standard $t\\sim \\nu ^{-2}$ relationship when de-dispersed to their structure-optimised DMs, it is possible that the bursts are actually composed of unresolved downward drifting sub-bursts.", "If we consider the main bursts of 03 and 05 (without considering any unresolved components) we find that at lower frequencies the DMs are ${\\sim }1{-}2$  ${\\rm pc \\, cm^{-3}}$ lower than at higher frequencies.", "This may imply lensing, which we show can plausibly account for such differences if the lensing occurs in the host galaxy.", "In such a scenario, no single DM can describe the intrinsic burst morphology.", "Two of the reported bursts have precursors (bursts 03 and 05).", "The time difference between bursts is comparable to the longest duration burst reported for FRB 121102 , and thus the two bursts may plausibly result from one event.", "Polarisation information and RMs are unfortunately unavailable.", "If it were, a coherent correlation between polarisations could reveal whether burst pairs are actually echoes .", "Determining the average DM for the epoch is not elementary.", "Potentially unresolved sub-components of the bursts would greatly influence the measured DM.", "As such, we found $\\langle \\rm DM\\rangle = 563.5\\pm 0.2 (\\text{sys}) \\pm 0.8 (\\text{stat})\\,{\\rm pc \\, cm^{-3}}$ by considering only bursts whose DM could be established with reasonable reliability.", "This is consistent with measurements of FRB 121102 by and taken one year prior to our sample.", "Interpolating using these results, as well as 2016 measurements by , we obtain a mean increase of 1 ${\\rm pc \\, cm^{-3}}$ per year.", "Future observations will help establish whether the increase is indeed persistent.", "To establish whether the potential deviation from the $t\\sim \\nu ^{-2}$ relationship observed in the main bursts of 03 and 05 is plausible, one might compare the individual structure-optimised DMs to the average DM of the epoch.", "At a 1$\\sigma $ confidence level, the DM of burst 05 ($564.5\\pm 0.3\\,{\\rm pc \\, cm^{-3}}$ and $564.4\\pm 0.3\\,{\\rm pc \\, cm^{-3}}$ for ACF and DM_phase, respectively) is consistent with the average.", "The structure-optimised DM of burst 03, however, is not.", "One would require a deviation of ${\\sim }1{-}2\\,{\\rm pc \\, cm^{-3}}$ from the average DM of the epoch.", "Such is possible (note that bursts 03 and 05 have different $\\rm DM_{\\rm struct}$ values for near-identical structures), but the more mundane option may be more likely: that the bursts do not exhibit a deviation from the expected cold plasma dispersion relationship but instead have downward drifting sub-components that are not resolved.", "Many of the issues highlighted in this paper are seen in our wide band, and thus some narrow band data may not have been sensitive to these effects.", "It will be interesting going forward to see if similar behaviour is observed in higher resolution, wide bandwidth data." ], [ "Acknowledgements", "The authors would like to thank the anonymous referee for their invaluable feedback and perspective.", "The authors would like to thank Daniele Michilli and Andrew Seymour for help with DM_phase, and thank Fang Xi Lin for help and advice with lensing simulations.", "The authors would also like to thank SARAO for the approval of the DDT MeerKAT request and the CAM/CBF and operator teams for their time and effort invested in the observations.", "The MeerKAT telescope is operated by the South African Radio Astronomy Observatory (SARAO), which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation.", "EP is supported by a L'Oréal-UNESCO For Women in Science Young Talents Fellowship and by a PhD fellowship from the South African National Institute for Theoretical Physics (NITheP).", "AW acknowledges funding from the South African Research Chairs Initiative funded by the National Research Foundation and Department of Science and Technology.", "MC, BWS, FJ, KR, VM and MCB acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 694745)." ], [ "Data Availability", "The data underlying this article will be shared on reasonable request to the corresponding author." ] ]
2105.11822
[ [ "Remarks on Liouville type theorems for the steady MHD and Hall-MHD\n equations" ], [ "Abstract In this note we investigate Liouville type theorems for the steady three dimensional MHD and Hall-MHD equations, and show that the velocity field $u$ and the magnetic field $B$ are vanishing provided that $B\\in L^{6,\\infty}(\\mathbb{R}^3)$ and $u\\in BMO^{-1}(\\mathbb{R}^3)$, which state that the velocity field plays an important role.", "Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three dimensional MHD equations." ], [ "Introduction", "Consider the following general system including the so called magnetohydrodynamics (MHD) and Hall-MHD equations: $\\left\\lbrace \\begin{array}{llll}\\displaystyle - \\Delta u + u \\cdot \\nabla u + \\nabla p = B \\cdot \\nabla B, ~~ \\rm {in} ~~ \\mathbb {R}^3, \\\\\\displaystyle - \\Delta B + u \\cdot \\nabla B - B \\cdot \\nabla u = \\alpha \\nabla \\times ((\\nabla \\times B) \\times B), ~~ \\rm {in} ~~ \\mathbb {R}^3,\\\\\\displaystyle {\\rm {div}} ~ u = 0, ~~~ {\\rm {div}} ~ B = 0, ~~ \\rm {in} ~~ \\mathbb {R}^3,\\end{array}\\right.$ where $u = u(x)=(u_1,u_2,u_3)$ is the velocity field of the fluid flows, $B = B(x) = (B_1,B_2,B_3)$ is the magnetic field, $p = p(x)$ is the pressure of the flows, and $\\alpha \\in \\mathbb {R}$ is an absolute constant.", "When $\\alpha = 0$ , the system (REF ) is the MHD system, which describes the steady state of the magnetic properties of electrically conducting fluids, including plasmas, liquid metals, etc; for the physical background and mathematical theory we refer to Schnack [26] and the references therein.", "When $\\alpha = 1$ , this system governs the dynamics plasma flows of strong shear magnetic fields as in the solar flares, and has many important applications in the astrophysics (for example, see Chae-Degond-Liu [5]).", "In this note, we focus on the Liouville type properties of the above system, which is motivated by the development of Navier-Stokes equations.", "When $B = 0$ , the system (REF ) reduces to the standard Navier-Stokes equations, and a very challenging open question is if there exists a nontrivial solution when the Dirichlet integral $\\int _{\\mathbb {R}^3}|\\nabla u|^2dx$ is finite, which dates back to Leray's celebrated paper [20] and is explicitly written in Galdi's book ([12], Remark X.", "9.4, pp.729; see also Tsai's book [31], pp.23).", "This uniqueness problem, or equivalently the Liouville type problem is widely open.", "Galdi proved the above Liouville type theorem by assuming $u\\in L^{\\frac{9}{2}}(\\mathbb {R}^3)$ in [12].", "Chae in [3] showed the condition $\\triangle u\\in L^{\\frac{6}{5}}(\\mathbb {R}^3)$ is sufficient for the vanishing property of $u$ by exploring the maximum principle of the head pressure.", "Also, Chae-Wolf gave an improvement of logarithmic form for Galdi's result in [7] by assuming that $\\int _{\\mathbb {R}^3} |u|^{\\frac{9}{2}}\\lbrace \\ln (2+\\frac{1}{|u|})\\rbrace ^{-1}dx<\\infty $ .", "Seregin obtained the conditional criterion $u\\in BMO^{-1}(\\mathbb {R}^3)$ in [28].", "More references, we refer to [18], [19], [29], [30], [2], [8], [4] and the references therein.", "Relatively speaking, the two-dimensional case is more easier due to the vorticity of the 2D Navier-Stokes equations satisfies a nice elliptic equation, to which the maximum principle applies.", "For example, Gilbarg-Weinberger [14] obtained Liouville type theorem provided the Dirichlet energy is finite.", "As a different type of Liouville property for the 2D Navier-Stokes equations, Koch-Nadirashvili-Seregin-Sverak [17] showed that any bounded solution is trivial solution, say $u \\equiv C$ , as a byproduct of their results on the non-stationary case (see also [1] for the unbounded velocity).", "However, for the MHD tpye system like (REF ), the situation is quite different.", "Due to the lack of maximum principle, there is not much progress in the study of MHD equation.", "For the two-dimensional MHD equations, Liouville type theorems were proved by assuming the smallness of the norm of the magnetic field in [33] and [32], and we refer to the recent paper in [10] by removing the smallness assumption.", "For the three-dimensional MHD equations, Chae-Weng [6] proved that if a smooth solution to (REF ) in $\\mathbb {R}^3$ with finite Dirichlet integral $\\int _{\\mathbb {R}^3} |\\nabla u|^2 + |\\nabla B|^2 dx < +\\infty ,$ and $u \\in L^3(\\mathbb {R}^3)$ , then the solutions $(u,B)$ is identically zero.", "Schulz [27] proved that if a smooth solution of the stationary MHD equations is in $L^6(\\mathbb {R}^3)$ and $u,B \\in BMO^{-1}(\\mathbb {R}^3)$ , then it is identically zero.", "Recently, Chae-Wolf [9] show that $L^6$ mean oscillations of the potential function of the velocity and magnetic field have certain linear growth by using the technique of [8].", "More references, we refer to [37], [21], [22], [36], [11] and the references therein.", "Motivated by some numerical experiments in [25], which seem to indicate that the velocity field should play a more important role than the magnetic field in the regularity theory of solutions to the MHD equations.", "Our goal is to get rid of the extra magnetic conditions in [6] or [27].", "Recall the definition of the space $BMO^{-1}(\\mathbb {R}^3)$ as follows.", "For a measurable set $E \\subset \\mathbb {R}^3$ , we denote by $|E|$ the 3-dimensional Lebesgue measure of $E$ , and for $f \\in L^1(E)$ we use the notation $f_E = \\frac{1}{|E|} \\int _E f dx,$ Combining (REF ) with Hölder's inequality, we know for all $f\\in L^p(E)$ $||f - f_E||_{L^p(E)} \\le C ||f||_{L^p(E)}.$ Then we say that $u \\in BMO^{-1}(\\mathbb {R}^3)$ , if there exists $\\mathbf {\\Phi } \\in C^\\infty (\\mathbb {R}^3,\\mathbb {R}^{3 \\times 3})$ such that $\\nabla \\cdot \\mathbf {\\Phi } = u$ , and for all $r>0$ , there holds $\\sup _{r>0}\\frac{1}{|B(r)|} \\int _{B(r)} |\\mathbf {\\Phi } - \\mathbf {\\Phi }_{B(r)}| dx <\\infty ,$ i.e.", "$\\mathbf {\\Phi }\\in BMO(\\mathbb {R}^3)$ .", "And a remarkable property of BMO space is $\\sup _{r>0}\\frac{1}{|B(r)|} \\int _{B(r)} |\\mathbf {\\Phi } - \\mathbf {\\Phi }_{B(r)}|^p dx <\\infty , \\quad \\forall ~~ 1 \\le p < +\\infty .$ Next, the definition of the space $L^{p,q}(\\mathbb {R}^n)$ is as follow: Let $\\Omega \\subset \\mathbb {R}^n$ and $1\\le p,l\\le \\infty $ , we say that a measurable function $f$ is in $L^{p,l}(\\Omega )$ if $\\Vert f\\Vert _{L^{p,l}(\\Omega )}<+\\infty $ , where $\\nonumber \\Vert f\\Vert _{L^{p,l}}(\\Omega )=\\left\\lbrace \\begin{array}{llll}\\left(\\int ^\\infty _0 \\lambda ^{l-1}|\\lbrace x\\in \\Omega ;|f|>\\lambda \\rbrace |^\\frac{l}{p} d\\lambda \\right)^\\frac{1}{l},\\quad ~{\\rm for}~r<\\infty ;\\\\sup_{\\lambda >0} \\lambda |\\lbrace x\\in \\Omega ;|f|>\\lambda \\rbrace |^\\frac{1}{p}, \\quad ~{\\rm for}~r=\\infty .\\end{array}\\right.$ We use Hölder inequality in Lorentz spaceas follows ([24]).", "Assume $1 \\le p_1,p_2 \\le \\infty $ , $1 \\le q_1,q_2 \\le \\infty $ and $u \\in L^{p_1,q_1}(\\Omega )$ , $v \\in L^{p_2,q_2}(\\Omega )$ .", "Then $uv \\in L^{p,q}(\\Omega )$ and $\\frac{1}{q} \\le \\frac{1}{q_1} + \\frac{1}{q_2}$ and the inequality $||uv||_{L^{p,q}(\\Omega )} \\le C ||u||_{L^{p_1,q_1}(\\Omega )}||v||_{L^{p_2,q_2}(\\Omega )}$ is valid,where $\\frac{1}{p} = \\frac{1}{p_1} + \\frac{1}{p_2}$ .", "Our main result is as follows: Theorem 1.1 Let $(u,B,p) \\in C^\\infty (\\mathbb {R}^3) \\times C^\\infty (\\mathbb {R}^3) \\times C^\\infty (\\mathbb {R}^3)$ be a solution of (REF ) with $B \\in L^{6,\\infty }(\\mathbb {R}^3)$ .", "Assume that $u \\in BMO^{-1}(\\mathbb {R}^3)$ .", "Then $u = B = 0$ .", "Remark 1.2 The above theorem shows that the velocity field should play a more important role than the magnetic field in the uniqueness theory, which is similar as in the regularity theory.", "For example, [16] and [34] have presented some regularity criterions to the MHD equations in terms of the velocity field only.", "Remark 1.3 The above result generalized Seregin's result in [28] or [29] to the MHD case or Hall-MHD case when $B\\equiv 0$ , which also improved Chae-Weng's result [6], where they assumed $\\nabla u,\\nabla B\\in L^2(\\mathbb {R}^3)$ and $u\\in L^3(\\mathbb {R}^3)$ .", "It also improved Schulz's theorem [27] by removing the additional conditions $B\\in BMO^{-1}(\\mathbb {R}^3)$ and $u \\in L^6(\\mathbb {R}^3)$ .", "Moreover, we relaxed the condition of $B$ to the Lorentz space of $L^{6,\\infty }(\\mathbb {R}^3)$ .", "A new observation is the use of delicate $L^q$ estimates of stationary Stokes system, which is independent of interest.", "For the MHD equations with partial viscosity or diffusivity, we have the following similar conclusions.", "Consider the MHD system (REF ) as follows: $\\left\\lbrace \\begin{array}{llll}\\displaystyle - \\lambda _1 \\partial _{11} u - \\lambda _2 \\partial _{22} u - \\lambda _3 \\partial _{33} u + u \\cdot \\nabla u + \\nabla p = B \\cdot \\nabla B, ~~ \\rm {in} ~~ \\mathbb {R}^3, \\\\\\displaystyle - \\mu _1 \\partial _{11} B - \\mu _2 \\partial _{22} B - \\mu _3 \\partial _{33} B + u \\cdot \\nabla B = B \\cdot \\nabla u, \\\\\\displaystyle {\\rm {div}} ~ u = 0, ~~~ {\\rm {div}} ~ B = 0, \\\\\\displaystyle \\lambda _i \\ge 0, ~~~ \\mu _i \\ge 0, i = 1,2,3.", "\\\\\\end{array}\\right.$ For the system (REF ), we have Theorem 1.4 Let $u,b$ be a smooth solutions of the system (REF ).", "Let's further assume that $B \\in L^6(\\mathbb {R}^3)$ and $u \\in L^3(\\mathbb {R}^3)$ , then $u, B \\equiv 0$ if the following condition holds $\\lambda _1 + \\lambda _2 + \\lambda _3 > 0, ~~~ \\mu _1 + \\mu _2 + \\mu _3 > 0.$ Remark 1.5 When $\\lambda _1 + \\lambda _2 + \\lambda _3=0$ , but $\\mu _1 + \\mu _2 + \\mu _3 > 0$ , one can derive that $B\\equiv 0$ and $u$ satisfies the three dimensional Euler equations.", "It is impossible to deduce that $u$ is vanishing from the bounded-ness of $L^q$ norm.", "For example, we can refer to the counterexample belonging to $C_0^\\infty (\\mathbb {R}^3)$ in [13].", "The same is true for another situation of $\\mu _1 + \\mu _2 + \\mu _3 = 0$ .", "In the proof, we need the following lemma (see, for example, Lemma A.5 [35]).", "Lemma 1.6 Let $f, g \\in \\dot{W}^{1,2}(\\mathbb {R}^3) \\cap BMO^{-1}(\\mathbb {R}^3)$ .", "Then there holds $||fg||_{L^2} \\le C \\left(||f||_{\\dot{W}^{1,2}}||g||_{BMO^{-1}} + ||f||_{BMO^{-1}}||g||_{\\dot{W}^{1,2}}\\right).$ We also need an interpolation inequality in Lorentz space (see, for example, Theorem 2.1 [23]).", "Lemma 1.7 Let $\\Omega $ be a domain in $\\mathbb {R}^n$ .", "Let $0 < q < p < r \\le +\\infty $ and $\\alpha > 0$ .", "If $f \\in L^{q,\\infty }(\\Omega ) \\cap L^{r,\\infty }(B(r))$ , then $f \\in L^{p,\\alpha }(\\Omega )$ and $||f||_{L^{p,\\alpha }(\\Omega )} \\le C ||f||_{L^{q,\\infty }(\\Omega )}^\\theta ||f||_{L^{r,\\infty }(\\Omega )}^{1-\\theta },$ where $C = C(q,r,p,\\alpha ) > 0$ and $\\frac{1}{p} = \\frac{\\theta }{q} + \\frac{1-\\theta }{r}.$ Remark 1.8 The lemma REF follows from Theorem 2.1 in [23], where the domain is the whole space $\\mathbb {R}^n$ .", "Note that $||f\\chi _\\Omega ||_{L^{q,\\infty }(\\mathbb {R}^n)} = ||f||_{L^{q,\\infty }(\\Omega )}$ , which show that the lemma holds for the general domain $\\Omega $ in $\\mathbb {R}^n$ .", "Throughout this article, $C(\\alpha _1,\\cdots ,\\alpha _n)$ denotes a constant depending on $\\alpha _1,\\cdots ,\\alpha _n$ , which may be different from line to line." ], [ "Proof of Theorem ", "Before starting the proof of Theorem REF , we state the following interior $L^q$ estimates for the steady Stokes system of an alternative version and sketch its proof.", "Lemma 2.9 Let $B(r) \\subset B(R)\\subset \\mathbb {R}^n$ with $n\\ge 2$ be concentric balls with $0 < r < R$ .", "Assume that $v$ is a very weak solution of the following Stokes system $- \\Delta v_i + \\partial _i p = g_i, \\quad {\\rm div}\\,v = 0 \\quad {\\rm in}~ B(R),$ where $g_i \\in L^q(B(R))$ , $1 < q < \\infty $ .", "Then $v \\in W_{loc}^{2,q}(B(R))$ and there exists a function $p\\in W_{loc}^{1,q}(B(R))$ .", "Moreover, there holds $||\\nabla ^2 v||_{L^q(B(r))} + ||\\nabla p||_{L^q(B(r))}\\le C(n,q) \\left(||g||_{L^q(B(R))} + (R-r)^{-2}||v||_{L^q(B(R))}\\right).$ Remark 2.10 A very weak solution $v$ of the system of (REF ) in $\\Omega \\subset \\mathbb {R}^n$ is defined as follows.", "The divergence-free vector field $v\\in L_{loc}^{1}(\\Omega )$ satisfies $\\int v\\cdot \\triangle \\zeta =-\\int g\\cdot \\zeta ,\\quad \\forall ~\\zeta \\in C_{c,\\sigma }^\\infty (\\Omega ),$ where $ C_{c,\\sigma }^\\infty (\\Omega )$ denotes any times differentiable divergence-free vector fields with compact support in $\\Omega .$ At first, recall Lemma 2.12 in [31] and we have $||\\nabla ^2 v||_{L^q(B(r))} \\le C(n,q,r,R) \\left(||g||_{L^q(B(R))} + ||v||_{L^1(B(R))}\\right).$ Specially, $||\\nabla ^2 v||_{L^q(B(1))} \\le C(n,q) \\left(||g||_{L^q(B(2))} + ||v||_{L^1(B(2))}\\right).$ Hence, for any $0<r<R$ and $2r\\le R$ , by scaling we get $||\\nabla ^2 v||_{L^q(B(r))} \\le C(n,q) \\left(||g||_{L^q(B(2r))} + r^{-n-2+\\frac{n}{q}} ||v||_{L^1(B(2r))}\\right).$ Secondly, for any fixed $r>0$ with $r<R$ , and $x_0 \\in B(r)$ , let $10d = R - r>0$ , then $\\bigcup _{x_0\\in B(r)}B(x_0,d)\\supset B(r),$ where $B(x_0,d)$ denotes the ball of radius $d$ centered at $x_0.$ Due to the lemma of Vitali, there exists finite points, $x_1,\\cdots ,x_k\\in B(r)$ such that $B(x_1,d),\\cdots , B(x_k,d)$ are disjoint balls satisfying $B(r)\\subset \\bigcup _{j=1}^k B(x_j,5d),$ and at most $K_0$ finite balls with $K_0\\le 6^n$ have a joint point in $ B(R)$ .", "Thence, by (REF ) and Hölder inequality we have $||\\nabla ^2 v||_{L^q(B(r))}^q&\\le & \\sum _{j=1}^k ||\\nabla ^2 v||_{L^q(B(x_j,5d))}^q \\\\&\\le & C(n,q) \\sum _{j=1}^k \\left(||g||_{L^q(B(x_j,10d))}^q + d^{-(n+2)q+n} ||v||_{L^1(B(x_j,10d))}^q\\right)\\\\&\\le & C(n,q) \\sum _{j=1}^k \\left(||g||_{L^q(B(x_j,10d))}^q + d^{-2q} ||v||_{L^q(B(x_j,10d))}^q\\right)$ Note that at most $K_1$ finite balls of $B(x_j,10d)$ with $K_1\\le 11^n$ have a joint point in $ B(R)$ .", "Then $||\\nabla ^2 v||_{L^q(B(r))}^q &\\le & C(n,q)11^n\\left(||g||_{L^q(B(R))}^q + d^{-2q} ||v||_{L^q(B(R))}^q\\right)\\\\&\\le & C(n,q)\\left(||g||_{L^q(B(R))}^q + (R-r)^{-2q} ||v||_{L^q(B(R))}^q\\right)$ by using $10d = R - r$ and $B(R)\\supset \\bigcup _{j=1}^k B(x_j,10d).$ At last, using the equations of (REF ), we get $||\\nabla p ||_{L^q(B(r))}&\\le & C(n,q)\\left(||g||_{L^q(B(R))}+ (R-r)^{-2} ||v||_{L^q(B(R))}\\right)$ The proof is complete.", "Next we prove Theorem REF .", "Step I. Cacciopolli type inequality.", "First, we introduce some cut-off functions of $\\phi _i$ with $i=1,\\cdots ,3$ for $r>1$ and $\\rho , \\tau >0$ , which will be used later.", "Assume that $\\phi _i(x) \\in C_0^{\\infty }(\\mathbb {R}^3)$ with $ 0 \\le \\phi _i \\le 1$ for $i=1,\\cdots ,3$ satisfying the following conditions: i) $r \\le \\rho < \\tau \\le 2r$ ; ii) $\\phi _1(x)=\\phi _1(|x|)=\\left\\lbrace \\begin{array}{llll}1,\\quad ~{\\rm in}~B_{\\rho };\\\\0, \\quad ~{\\rm in} ~ \\mathbb {R}^3 \\backslash B(\\frac{\\tau + 2\\rho }{3});\\\\\\end{array}\\right.$ iii) $\\phi _2(x)=\\phi _2(|x|)=\\left\\lbrace \\begin{array}{llll}1,\\quad ~{\\rm in}~B(\\frac{\\tau + 2\\rho }{3});\\\\0, \\quad ~{\\rm in} ~\\mathbb {R}^3 \\backslash B(\\frac{2\\tau + \\rho }{3});\\\\\\end{array}\\right.$ iv) $\\phi _3(x)=\\phi _3(|x|)=\\left\\lbrace \\begin{array}{llll}1,\\quad ~{\\rm in}~B(\\frac{2\\tau + \\rho }{3});\\\\0, \\quad ~{\\rm in} ~\\mathbb {R}^3 \\backslash B(\\tau );\\\\\\end{array}\\right.$ v) $|\\nabla \\phi _i| \\le \\frac{C}{\\tau - \\rho },\\quad |\\nabla ^2 \\phi _i| \\le \\frac{C}{(\\tau - \\rho )^2},\\quad i=1,\\cdots ,3.$ Secondly, multiply $(\\ref {eq:MHD})_1$ by $u \\phi _1$ , $(\\ref {eq:MHD})_2$ by $(B - b) \\phi _1$ , and integrate them over $B(2r)$ .", "Then by integration by parts we obtain $&&\\int _{B(2r)} |\\nabla u|^2 \\phi _1 + |\\nabla B|^2 \\phi _1 dx \\\\&=& \\frac{1}{2} \\int _{B(2r)} |u|^2 \\Delta \\phi _1 dx + \\frac{1}{2}\\int _{B(2r)} |u|^2 u \\cdot \\nabla \\phi _1 dx \\\\&&+ \\int _{B(2r)} (p - a) u \\cdot \\nabla \\phi _1 dx + \\frac{1}{2} \\int _{B(2r)} |B - b|^2 \\Delta \\phi _1 dx \\\\&&+ \\frac{1}{2}\\int _{B(2r)} |B - b|^2 u \\cdot \\nabla \\phi _1 dx - \\int _{B(2r)} u \\cdot (B - b) B \\cdot \\nabla \\phi _1 dx \\\\&&- \\alpha \\int _{B(2r)} ((\\nabla \\times (B - b)) \\times B) \\cdot ((B - b) \\times \\nabla \\phi _1) dx\\\\&\\doteq & I_1 + \\cdots + I_7,$ where $a$ and $b$ are constant vectors in $B(2r)$ , to be decided, and we used the following formula for the last term of the right hand $\\int \\nabla \\times F \\cdot G\\phi dx&=&\\int \\nabla \\times G \\cdot F\\phi dx+\\int G \\times F \\cdot \\nabla \\phi dx\\\\&=&\\int \\nabla \\times G \\cdot F \\phi dx - \\int F \\cdot (G \\times \\nabla \\phi ) dx.$ For the terms of $I_1$ and $I_2$ , it's easy to deduce that $|I_1| \\le C (\\tau - \\rho )^{-2} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho )} |u|^2 dx.$ and $|I_2| \\le C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho )} |u|^3 dx.$ For the term of $I_4$ , by (REF ) we have $|I_4| &\\le & C (\\tau - \\rho )^{-2} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho )} |B - b|^2 dx\\nonumber \\\\&\\le & C (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{2\\tau + \\rho }{3})\\backslash B(\\rho ))}^2|| 1 ||_{L^{\\frac{3}{2},1}(B(2r))}\\nonumber \\\\&\\le & C r^2 (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{2\\tau + \\rho }{3})\\backslash B(\\rho ))}^2.$ For the term $I_5$ , using $\\nabla \\cdot \\mathbf {\\Phi } = u$ , integration by parts and Hölder inequality, we have $|I_5| &=& \\left|\\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} |B - b|^2 \\partial _i(\\Phi _{ij} - \\bar{\\Phi }_{ij}) \\partial _j \\phi _1 dx\\right| \\\\&\\le & C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho )} |B - b| |\\nabla B| |\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}| dx\\\\&&+ C (\\tau - \\rho )^{-2} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho )} |B - b|^2 |\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}| dx \\\\&\\le & C (\\tau - \\rho )^{-1} ||B-b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho ))} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho ))} ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^{3,2}(B(\\frac{\\tau + 2\\rho }{3}) )} \\\\&&+ C (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho ))}^2 ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^{\\frac{3}{2},1}(B(\\frac{\\tau + 2\\rho }{3}))}.$ Choosing $\\bar{\\mathbf {\\Phi }} = \\mathbf {\\Phi }_{B(2r)}$ and $b=B_{B(\\frac{4r}{3})\\backslash B(r)}$ , by Lemma REF and (REF ), we have $ \\nonumber ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^{3,2}(B(2r))}&\\le & C ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^2(B(2r))}^\\frac{1}{3} ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^4(B(2r))}^\\frac{2}{3} \\le C r,$ $ \\nonumber ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^{\\frac{3}{2},1}(B(2r))}&\\le & C ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^\\frac{5}{4}(B(2r))}^\\frac{5}{12} ||\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}||_{L^\\frac{7}{4}(B(2r))}^\\frac{7}{12} \\le C r^2,$ and by $B \\in L^{6,\\infty }(\\mathbb {R}^3)$ $||b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3}) \\backslash B(\\rho )}&\\le &||b||_{L^{6,\\infty }(B(2r))}\\le Cr^{\\frac{1}{2}}|b|\\nonumber \\\\&\\le &Cr^{\\frac{1}{2}-3}||B||_{L^{6,\\infty }(B(2r))}||1||_{L^{\\frac{6}{5},1}(B(2r))}\\le C.$ Then we get $|I_5| &\\le & C r (\\tau - \\rho )^{-1} ||\\nabla B||_{L^2(B(\\frac{2\\tau + \\rho }{3})\\backslash B(\\rho ))} \\nonumber \\\\&&+ C r^2 (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{2\\tau + \\rho }{3})\\backslash B(\\rho ))}^2.$ The term of $I_6$ can be controlled by $|I_6| &=& \\left|\\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} \\partial _i(\\Phi _{ij} - \\bar{\\Phi }_{ij}) (B - b)_j B_k \\partial _k \\phi _1 dx\\right| \\\\&\\le & C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} |\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}| |B - b| |\\nabla B| dx \\\\&&+ C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} |\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}| |B| |\\nabla B| dx \\\\&&+ C (\\tau - \\rho )^{-2} \\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} |\\mathbf {\\Phi } - \\bar{\\mathbf {\\Phi }}| |B - b| |B| dx.$ Similarly as $I_5$ , we have $|I_6| &\\le &C r (\\tau - \\rho )^{-1} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} \\nonumber \\\\&& + C r^2 (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))}.$ Using (REF ) and (REF ), the term of $I_7$ can be controlled by $|I_7|&\\le & C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} |\\nabla B| |B| |B - b| dx \\nonumber \\\\&\\le & C (\\tau - \\rho )^{-1} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} ||B||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3}))} ||B - b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3}))} ||1||_{L^{6,2}(B(\\frac{\\tau + 2\\rho }{3}))}\\nonumber \\\\&\\le & C r^\\frac{1}{2} (\\tau - \\rho )^{-1} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))}.$ At last, we estimate the term of $I_3$ , and by $\\nabla \\cdot \\mathbf {\\Phi } = u$ we get $|I_3| &=& \\left|\\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} (p - a) \\partial _i(\\Phi _{ij} - \\bar{\\Phi }_{ij}) \\partial _j \\phi _1 dx\\right| \\\\&=& \\left|\\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} (p - a) (\\Phi _{ij} - \\bar{\\Phi }_{ij}) \\partial _{ij} \\phi _1 dx + \\int _{B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho )} \\partial _ip (\\Phi _{ij} - \\bar{\\Phi }_{ij}) \\partial _j \\phi _1 dx \\right| \\\\&\\le & C (\\tau - \\rho )^{-2} ||p - a||_{L^{\\frac{3s}{3-s}}(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} ||\\mathbf {\\Phi } - \\mathbf {\\bar{\\Phi }}||_{L^{\\frac{3s}{4s-3}}(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} \\\\&&+ C (\\tau - \\rho )^{-1} ||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} ||\\mathbf {\\Phi } - \\mathbf {\\bar{\\Phi }}||_{L^{\\frac{s}{s-1}}(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} \\\\&\\le & C r^{\\frac{4s-3}{s}}(\\tau - \\rho )^{-2} ||p - a||_{L^{\\frac{3s}{3-s}}(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} +C r^{\\frac{3s-3}{s}}(\\tau - \\rho )^{-1} ||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))},$ where $s \\in (1, 3)$ .", "At last, collecting (REF )-(REF ), (REF )-(REF ) and the estimates of $I_3$ , noting $r>1$ and $||B - b||_{L^{6,\\infty }(B(2r))}\\le C$ from (REF ) we arrive at the following Cacciopolli type inequality $ \\nonumber &&\\int _{B(\\rho )} |\\nabla u|^2 + |\\nabla B|^2 dx \\\\ \\nonumber &&\\le C (\\tau - \\rho )^{-2} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho )} |u|^2 dx + C (\\tau - \\rho )^{-1} \\int _{B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho )} |u|^3 dx\\\\\\nonumber &&+C r (\\tau - \\rho )^{-1} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} + C r^2 (\\tau - \\rho )^{-2} ||B - b||_{L^{6,\\infty }(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))}\\nonumber \\\\&&+C r^{\\frac{4s-3}{s}}(\\tau - \\rho )^{-2} ||p - a||_{L^{\\frac{3s}{3-s}}(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))} +C r^{\\frac{3s-3}{s}}(\\tau - \\rho )^{-1} ||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3})\\backslash B(\\rho ))}\\nonumber \\\\$ Step II.", "The bounded-ness of energy integral.", "Firstly, we estimate the first two terms of (REF ) by following the estimate of [8].", "Using $\\nabla \\cdot \\mathbf {\\Phi } = u$ , integration by parts and Hölder inequality, we have $&& \\int _{B(\\frac{2\\tau +\\rho }{3})} |u|^3 \\phi _2^3 dx \\\\&=& \\int _{B(\\frac{2\\tau +\\rho }{3})} \\partial _i (\\Phi _{ij} - (\\Phi _{ij})_{B(2r)}) u_j |u| \\phi _2^3 dx \\\\&=& - \\int _{B(\\frac{2\\tau +\\rho }{3})} (\\Phi _{ij} - (\\Phi _{ij})_{B(2r)}) \\partial _i (u_j |u|) \\phi _2^3 dx - \\int _{B(\\frac{2\\tau +\\rho }{3})} (\\Phi _{ij} - (\\Phi _{ij})_{B(2r)}) u_j |u| \\partial _i (\\phi _2^3) dx \\\\&\\le & C\\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{1}{2}} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\mathbf {\\Phi } - (\\mathbf {\\Phi })_{B(2r)}|^6 dx\\right)^{\\frac{1}{6}} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |u \\phi _2|^3 dx\\right)^{\\frac{1}{3}} \\\\&+& C(\\tau - \\rho )^{-1} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\mathbf {\\Phi } - (\\mathbf {\\Phi })_{B(2r)}|^3 dx\\right)^{\\frac{1}{3}} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |u \\phi _2|^3 dx\\right)^{\\frac{2}{3}},$ which implies from Young inequality that $\\int _{B(\\frac{2\\tau +\\rho }{3})} |u|^3 \\phi _2^3 dx \\le C r^{\\frac{3}{4}} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{3}{4}} + C r^3 (\\tau - \\rho )^{-3}.$ Then we have $\\nonumber (\\tau - \\rho )^{-1} ||u||_{L^3(B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho ))}^3 &\\le & C r^{\\frac{3}{4}} (\\tau - \\rho )^{-1} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{3}{4}} \\\\&& + C r^3 (\\tau - \\rho )^{-4}.$ By Hölder inequality, we get $\\nonumber &&(\\tau - \\rho )^{-2} ||u||_{L^2(B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho ))}^2 \\le C (\\tau - \\rho )^{-2} r ||u||_{L^3(B(\\frac{\\tau + 2\\rho }{3}) \\setminus B(\\rho ))}^2\\\\&\\le & C r^{\\frac{3}{2}} (\\tau - \\rho )^{-2} \\left(\\int _{B(\\frac{2\\tau +\\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{1}{2}} + C r^3 (\\tau - \\rho )^{-4}.$ Let's substitute (REF ) and (REF ) into (REF ).", "Note that $||B - b||_{L^{6,\\infty }(B(2r))}\\le C$ from (REF ) and by Young inequality we arrive at $\\int _{B(\\rho )} |\\nabla u|^2 + |\\nabla B|^2 dx &\\le & \\frac{1}{4} ||\\nabla u||_{L^2(B(\\frac{\\tau + 2\\rho }{3}))}^2+\\frac{1}{4} ||\\nabla B||_{L^2(B(\\frac{\\tau + 2\\rho }{3}))}^2 \\\\ \\nonumber && + C r^{\\frac{3s-3}{s}}(\\tau - \\rho )^{-1} ||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3}))} + C r^4 (\\tau - \\rho )^{-4} \\\\ \\nonumber && + C r^{\\frac{4s-3}{s}}(\\tau - \\rho )^{-2} ||p - a||_{L^{\\frac{3s}{3-s}}(B(\\frac{\\tau + 2\\rho }{3}))}, \\quad 1<s<3.$ Next we deal with the terms involving the pressure.", "By Lemma REF , the estimate of (REF ) ($n=3, q=s$ and $g=-u \\cdot \\nabla u + B \\cdot \\nabla B$ ) implies $||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3}))} &\\le & C(n,s) \\left(||u \\cdot \\nabla u - B \\cdot \\nabla B||_{L^s(B(\\frac{2\\tau +\\rho }{3}))} + (\\tau -\\rho )^{-2}||u||_{L^s(B(\\frac{2\\tau +\\rho }{3}))}\\right) \\nonumber \\\\&\\doteq & C(n,s) \\left(J_1 + J_2\\right).$ On one hand, for $J_1$ , using Hölder inequality we have $(J_1)^s &=& \\int _{B(\\frac{2\\tau + \\rho }{3})} |u \\cdot \\nabla u - B \\cdot \\nabla B|^s dx \\\\&\\le & C \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{s}{2}} \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |u|^\\frac{2s}{2-s} dx\\right)^{\\frac{2-s}{2}} \\\\&&+ C \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |\\nabla B|^2 dx\\right)^{\\frac{s}{2}} ||B||_{L^{6,\\infty }(B(\\frac{2\\tau + \\rho }{3}))}^s ||1||_{L^{\\frac{3}{3-2s},\\frac{2}{2-s}}(B(\\frac{2\\tau + \\rho }{3}))} \\\\&\\le & C \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |\\nabla u|^2 dx\\right)^{\\frac{s}{2}} \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |u|^\\frac{2s}{2-s} dx\\right)^{\\frac{2-s}{2}} \\\\&&+ C r^{3-2s} \\left(\\int _{B(\\frac{2\\tau + \\rho }{3})} |\\nabla B|^2 dx\\right)^{\\frac{s}{2}}, \\quad 1<s<\\frac{3}{2}.$ Due to the cut-off function (REF ), we have $\\int _{B(\\frac{2\\tau + \\rho }{3})} |u|^\\frac{2s}{2-s} dx \\le \\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx,$ hence, $&&\\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx \\\\&=& \\int _{B(\\tau )} \\partial _i(\\Phi _{ij} - (\\Phi _{ij})_{B(\\tau )}) u_j |u|^\\frac{4s - 4}{2-s} \\phi _3^\\frac{2s}{2-s} dx \\\\&=& - \\int _{B(\\tau )}(\\Phi _{ij} - (\\Phi _{ij})_{B(\\tau )}) \\partial _i (u_j |u|^\\frac{4s - 4}{2-s}) \\phi _3^\\frac{2s}{2-s} dx - \\int _{B(\\tau )}(\\Phi _{ij} - (\\Phi _{ij})_{B(\\tau )}) u_j |u|^\\frac{4s - 4}{2-s} \\partial _i(\\phi _3^\\frac{2s}{2-s}) dx \\\\&\\le & C \\left(\\int _{B(\\tau )} |\\nabla u|^2 dx\\right)^{\\frac{1}{2}} \\left(\\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx\\right)^{\\frac{2s-2}{s}} \\left(\\int _{B(\\tau )} |\\mathbf {\\Phi } - \\mathbf {\\Phi }_{B(\\tau )}|^{\\frac{2s}{4-3s}} dx\\right)^{\\frac{4-3s}{2s}} \\\\&&+ (\\tau - \\rho )^{-1} \\left(\\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx\\right)^{\\frac{3s-2}{2s}} \\left(\\int _{B(\\tau )} |\\mathbf {\\Phi } - \\mathbf {\\Phi }_{B(\\tau )}|^{\\frac{2s}{2-s}} dx\\right)^{\\frac{2-s}{2s}}$ holds for all $s\\in (1, \\frac{4}{3})$ .", "Using (REF ) and Young inequality, we have $\\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx \\le C r^{\\frac{12-9s}{2(2-s)}} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{s}{2-s}} + C r^3 (\\tau - \\rho )^{-\\frac{2s}{2-s}}.$ Then we have $(J_1)^s &\\le & C r^{\\frac{12-9s}{4}} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{3s}{2}} + C r^{\\frac{6-3s}{2}} (\\tau - \\rho )^{-s} ||\\nabla u||_{L^2(B(\\tau ))}^s \\\\&&+ C r^{3-2s} ||\\nabla B||_{L^2(B(\\tau ))}^s.$ Using Hölder inequality and (REF ), the term $J_2$ can be controlled by $(J_2)^s &=& (\\tau - \\rho )^{-2s} \\int _{B(\\frac{2\\tau +\\rho }{3})} |u|^s dx \\\\&\\le & C r^{\\frac{3s}{2}} (\\tau - \\rho )^{-2s} \\left( \\int _{B(\\tau )} |u \\phi _3|^\\frac{2s}{2-s} dx \\right)^{\\frac{2-s}{2}} \\\\&\\le & C r^{\\frac{12-3s}{4}} (\\tau - \\rho )^{-2s} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{s}{2}} + C r^3 (\\tau - \\rho )^{-3s},$ Combing the estimates of $J_1$ and $J_2$ , by (REF ) we have $ \\nonumber &&||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3}))} \\\\ \\nonumber &\\le & C r^{\\frac{12-9s}{4s}} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{3}{2}} + C r^{\\frac{6-3s}{2s}} (\\tau - \\rho )^{-1} ||\\nabla u||_{L^2(B(\\tau ))} \\\\ \\nonumber &&+ C r^{\\frac{3-2s}{s}} ||\\nabla B||_{L^2(B(\\tau ))} + C r^{\\frac{12-3s}{4s}} (\\tau - \\rho )^{-2} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{1}{2}} \\\\&&+ C r^\\frac{3}{s} (\\tau - \\rho )^{-3}, \\quad s\\in (1, \\frac{4}{3}).$ Furthermore, Poincaré inequality implies $||p - p_{B(\\frac{\\tau + 2\\rho }{3})}||_{L^{\\frac{3s}{3-s}}(B(\\frac{\\tau + 2\\rho }{3}))} \\le C||\\nabla p||_{L^s(B(\\frac{\\tau + 2\\rho }{3}))}.$ Then choosing $a = p_{B(\\frac{\\tau + 2\\rho }{3})}$ in (REF ), (REF ) and (REF ) imply $&&\\int _{B(\\rho )} |\\nabla u|^2 + |\\nabla B|^2 dx \\\\&\\le & \\frac{1}{2} \\int _{B(\\tau )} |\\nabla u|^2 + |\\nabla B|^2 dx + C r^4 (\\tau - \\rho )^{-4} \\\\&& +C r^{\\frac{7}{4}}(\\tau - \\rho )^{-2} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{3}{2}} + Cr^{\\frac{5}{2}}(\\tau - \\rho )^{-3} ||\\nabla u||_{L^2(B(\\tau ))} \\\\ \\nonumber &&+ Cr^{2}(\\tau - \\rho )^{-2} ||\\nabla B||_{L^2(B(\\tau ))} + Cr^{\\frac{13}{4}}(\\tau - \\rho )^{-4} ||\\nabla u||_{L^2(B(\\tau ))}^{\\frac{1}{2}}+ Cr^{4}(\\tau - \\rho )^{-5}$ Apply the iteration lemma (see, for example, Lemma 3.1 in [15]), we have $\\int _{B(\\rho )} |\\nabla u|^2 + |\\nabla B|^2 dx &\\le & C r^{8} (\\tau - \\rho )^{-8}.$ Choosing $\\rho = r$ and $\\tau = 2r$ , letting $r \\rightarrow +\\infty $ we have $\\int _{\\mathbb {R}^3} |\\nabla u|^2 + |\\nabla B|^2 dx \\le C.$ Step III.", "Vanishing property of Dirichlet integral.", "Next we prove the vanishing property of Dirichlet integral.", "It follows from (REF ) that $u,B \\in \\dot{W}^{1,2}(\\mathbb {R}^3)$ .", "Using Lemma REF and the condition $u \\in BMO^{-1}(\\mathbb {R}^3)$ , we have $|||u|^2||_{L^2(\\mathbb {R}^3)} \\le C ||u||_{\\dot{W}^{1,2}(\\mathbb {R}^3)} ||u||_{BMO^{-1}(\\mathbb {R}^3)},$ which means $u \\in L^4(\\mathbb {R}^3)$ .", "We claim that $u \\in L^p(\\mathbb {R}^3) \\quad {\\rm for ~~ all} \\quad p \\in [4,6].$ Indeed, for a new cut-off function $\\zeta $ satisfying $\\zeta = 1$ in $B(\\frac{4}{3} r)$ and $\\zeta = 0$ on $B(2r)^c$ , we get $\\left(\\int _{\\mathbb {R}^3} |u \\zeta |^6 dx\\right)^\\frac{1}{3} \\le \\int _{\\mathbb {R}^3} |\\nabla (u \\zeta )|^2 dx \\le C\\int _{B(2r)} |\\nabla u|^2 dx + C r^{-\\frac{1}{2}} \\left(\\int _{B(2r)} |u|^4 dx\\right)^\\frac{1}{2},$ which implies $u \\in L^6(\\mathbb {R}^3)$ .", "Similarly, $B \\in L^6(\\mathbb {R}^3),$ since $\\left(\\int _{\\mathbb {R}^3} |B \\zeta |^6 dx\\right)^\\frac{1}{3} \\le C\\int _{B(2r)} |\\nabla B|^2 dx + C r^{-2} ||B||_{L^{6,\\infty }(B(2r))}^2 ||1||_{L^{\\frac{3}{2},1}(B(2r))} <\\infty .$ Recall the Cacciopolli inequality (REF ), and choose $\\rho = r$ , $\\tau = 2r$ , $s=\\frac{3}{2}$ and $b=B_{B(\\frac{4}{3}r) \\setminus B(r)}$ .", "Then $\\int _{B(r)} |\\nabla u|^2 + |\\nabla B|^2 dx &\\le & C r^{-2} ||u||_{L^2(B(\\frac{4}{3} r) \\setminus B(r))}^2 + C r^{-1} ||u||_{L^3(B(\\frac{4}{3} r) \\setminus B(r))}^3 \\nonumber \\\\&&+ C ||B - b||_{L^{6,\\infty }(B(\\frac{4}{3}r)\\backslash B(r))} + C ||\\nabla B||_{L^2(B(\\frac{4}{3}r)\\backslash B(r))} \\nonumber \\\\&& + C ||\\nabla p||_{L^\\frac{3}{2} (B(\\frac{4}{3}r)\\backslash B(r))} + C ||p - a||_{L^{3}(B(\\frac{4}{3}r)\\backslash B(r))}.$ Using Hölder inequality, it follows that $r^{-2} ||u||_{L^2(B(\\frac{4}{3} r) \\setminus B(r))}^2 \\le C r^{-\\frac{1}{2}} ||u||_{L^4(B(\\frac{4}{3} r) \\setminus B(r))}^2\\rightarrow 0,$ and $r^{-1} ||u||_{L^3(B(\\frac{4}{3} r) \\setminus B(r))}^3 \\le C r^{-\\frac{1}{4}} ||u||_{L^4(B(\\frac{4}{3} r) \\setminus B(r))}^3\\rightarrow 0,$ as $r \\rightarrow \\infty $ since (REF ).", "Moreover, note that $||B - b||_{L^6(B(\\frac{4}{3} r) \\setminus B(r))} \\le C ||\\nabla B||_{L^2(B(\\frac{4}{3} r) \\setminus B(r))}\\rightarrow 0 \\quad {\\rm as} \\quad r \\rightarrow \\infty ,$ which implies the third and the fourth terms are vanishing as $r \\rightarrow \\infty $ .", "For the pressure, by (REF ) and (REF ) in Lemma REF , we have $||\\nabla p||_{L^\\frac{3}{2}(B(\\frac{4}{3}))}&\\le &C ||u||_{L^6(B(2r))} ||\\nabla u||_{L^2(B(2r))} + C||B||_{L^6(B(2r))} ||\\nabla B||_{L^2(B(2r))} \\nonumber \\\\&& +Cr^{-1/2} ||u||_{L^6(B(2r))}.$ Letting $r \\rightarrow \\infty $ , we obtain $\\nabla p \\in L^\\frac{3}{2}(\\mathbb {R}^3)$ , which yields $||\\nabla p||_{L^\\frac{3}{2} (B(\\frac{4}{3}r)\\backslash B(r))} \\rightarrow 0$ .", "Choosing $a = p_{B(\\frac{4}{3}r)\\backslash B(r)}$ , it immediately implies $||p - a||_{L^{3}(B(\\frac{4}{3}r)\\backslash B(r))} \\le ||\\nabla p||_{L^\\frac{3}{2} (B(\\frac{4}{3}r)\\backslash B(r))} \\rightarrow 0.$ Hence, (REF ) implies $\\int _{B(r)} |\\nabla u|^2 + |\\nabla B|^2 dx \\rightarrow 0 \\quad {\\rm as} \\quad r \\rightarrow \\infty ,$ which means $u \\equiv 0$ and $B \\equiv 0$ , since $u \\in L^{6}(\\mathbb {R}^3)$ and $B \\in L^{6,\\infty }(\\mathbb {R}^3)$ .", "The proof is complete." ], [ "Proof of Theorem ", "Assume that $\\eta (x) \\in C_0^{\\infty }(\\mathbb {R}^3)$ with $ 0 \\le \\eta \\le 1$ , satisfying that i) for $r > 1$ , $r \\le \\rho < \\tau \\le 2r$ , $ \\eta (x)=\\eta (|x|)=\\left\\lbrace \\begin{array}{llll}1,\\quad ~ {\\rm in} ~ B(\\rho );\\\\0, \\quad ~ {\\rm in} ~ \\mathbb {R}^3 \\backslash B(\\tau )\\end{array}\\right.$ ii) $&&|\\nabla \\eta | \\le \\frac{C}{\\tau - \\rho },\\quad |\\nabla ^2 \\eta | \\le \\frac{C}{(\\tau - \\rho )^2}.$ Since $B(\\tau )$ is a star-like domain(see P38, [12]), due to Theorem III 3.1 in [12], there exists a constant $C(s)$ and a vector-valued function $w : B(\\tau ) \\rightarrow \\mathbb {R}^3$ such that $w \\in W^{1,s}_0(B(\\tau ))$ , and $\\nabla \\cdot w = u \\cdot \\nabla \\eta $ satisfying $\\int _{B(\\tau )} |\\nabla w(x)|^s \\,dx dy\\le C(s) \\int _{B(\\tau )} |u \\cdot \\nabla \\eta |^s \\,dx, \\quad 1 < s < \\infty .$ Then multiplying $(\\ref {eq:VMHD})_1$ by $u \\eta -w $ , integrating it over $B(2r)$ and integration by parts yield that $ \\nonumber &&\\lambda _1 \\int _{B(2r)} |\\partial _1 u|^2 \\eta dx + \\lambda _2 \\int _{B(2r)} |\\partial _2 u|^2 \\eta dx + \\lambda _3 \\int _{B(2r)} |\\partial _3 u|^2 \\eta dx \\\\ \\nonumber &=& \\lambda _1 \\int _{B(2r)} \\partial _1 u \\cdot \\partial _1 w dx + \\lambda _2 \\int _{B(2r)} \\partial _2 u \\cdot \\partial _2 w dx + \\lambda _3 \\int _{B(2r)} \\partial _3 u \\cdot \\partial _3 w dx \\\\ \\nonumber &+& \\frac{1}{2} \\lambda _1 \\int _{B(2r)} |u|^2 \\partial _{11} \\eta dx + \\frac{1}{2} \\lambda _2 \\int _{B(2r)} |u|^2 \\partial _{22} \\eta dx + \\frac{1}{2} \\lambda _3 \\int _{B(2r)} |u|^2 \\partial _{33} \\eta dx \\\\ \\nonumber &-& \\int _{B(2r)} u \\cdot \\nabla w \\cdot u dx + \\frac{1}{2} \\int _{B(2r)} |u|^2 u \\cdot \\nabla \\eta dx + \\int _{B(2r)} B \\cdot \\nabla w \\cdot B dx \\\\&+& \\int _{B(2r)} B \\cdot \\nabla B \\cdot u \\eta dx.$ Similarly, by multiplying $(\\ref {eq:VMHD})_2$ by $B \\eta $ we derive $&&\\mu _1 \\int _{B(2r)} |\\partial _1 B|^2 \\eta dx + \\mu _2 \\int _{B(2r)} |\\partial _2 B|^2 \\eta dx + \\mu _3 \\int _{B(2r)} |\\partial _3 B|^2 \\eta dx \\\\ \\nonumber &=& \\frac{1}{2} \\mu _1 \\int _{B(2r)} |B|^2 \\partial _{11} \\eta dx + \\frac{1}{2} \\mu _2 \\int _{B(2r)} |B|^2 \\partial _{22} \\eta dx + \\frac{1}{2} \\mu _3 \\int _{B(2r)} |B|^2 \\partial _{33} \\eta dx \\\\ \\nonumber &+& \\frac{1}{2} \\int _{B(2r)} |B|^2 u \\cdot \\nabla \\eta dx - \\int _{B(2r)} B \\cdot \\nabla B \\cdot u \\eta dx - \\int _{B(2r)} B \\cdot u (B \\cdot \\nabla ) \\eta dx.$ Letting $G(r) = \\sum _{i=1}^3 \\lambda _i \\int _{B(r)} |\\partial _i u|^2 dx + \\sum _{i=1}^3 \\mu _i \\int _{B(r)} |\\partial _i B|^2 dx,$ and taking the sum of (REF ) and (REF ), we have $G(\\rho ) &\\le & \\lambda _1 \\int _{B(2r)} \\partial _1 u \\cdot \\partial _1 w dx + \\lambda _2 \\int _{B(2r)} \\partial _2 u \\cdot \\partial _2 w dx + \\lambda _3 \\int _{B(2r)} \\partial _3 u \\cdot \\partial _3 w dx \\\\ \\nonumber &+& \\frac{1}{2} \\lambda _1 \\int _{B(2r)} |u|^2 \\partial _{11} \\eta dx + \\frac{1}{2} \\lambda _2 \\int _{B(2r)} |u|^2 \\partial _{22} \\eta dx + \\frac{1}{2} \\lambda _3 \\int _{B(2r)} |u|^2 \\partial _{33} \\eta dx \\\\ \\nonumber &-& \\int _{B(2r)} u \\cdot \\nabla w \\cdot u dx + \\frac{1}{2} \\int _{B(2r)} |u|^2 u \\cdot \\nabla \\eta dx + \\int _{B(2r)} B \\cdot \\nabla w \\cdot B dx \\\\&+& \\frac{1}{2} \\mu _1 \\int _{B(2r)} |B|^2 \\partial _{11} \\eta dx + \\frac{1}{2} \\mu _2 \\int _{B(2r)} |B|^2 \\partial _{22} \\eta dx + \\frac{1}{2} \\mu _3 \\int _{B(2r)} |B|^2 \\partial _{33} \\eta dx \\\\ \\nonumber &+& \\frac{1}{2} \\int _{B(2r)} |B|^2 u \\cdot \\nabla \\eta dx - \\int _{B(2r)} B \\cdot u (B \\cdot \\nabla ) \\eta dx \\doteq \\sum _{i=1}^{14} I_i.$ Since the terms $I_1$ , $I_2$ and $I_3$ are similar, we compute the first term.", "Letting $T(\\rho ,\\tau ) = B(\\tau ) \\setminus B(\\rho )$ , using Hölder and Young inequality, (REF ) implies $|I_1| &\\le & \\lambda _1 \\int _{B(\\tau )} |\\partial _1 u| |\\partial _1 w| dx \\\\&\\le & \\varepsilon \\lambda _1 \\int _{B(\\tau )} |\\partial _1 u|^2 dx + C \\lambda _1 \\int _{B(\\tau )} |\\partial _1 w|^2 dx \\\\&\\le & \\varepsilon G(\\tau ) + C \\lambda _1 (\\tau - \\rho )^{-2} \\int _{T(\\rho ,\\tau )} |u|^2 dx \\\\&\\le & \\varepsilon G(\\tau ) + C \\lambda _1 r (\\tau - \\rho )^{-2} ||u||_{L^3(T(\\rho ,\\tau ))}^2,$ where $\\varepsilon >0$ , to be decided.", "Similarly, $|I_2|+|I_3| \\le \\varepsilon G(\\tau ) + C(\\lambda _2+ \\lambda _3) r (\\tau - \\rho )^{-2} ||u||_{L^3(T(\\rho ,\\tau ))}^2.$ The estimates of $I_4$ , $I_5$ and $I_6$ can be obtained directly from Hölder's inequality: $|I_4|+ |I_5|+|I_6|\\le C (\\lambda _1+ \\lambda _2+\\lambda _3) r (\\tau - \\rho )^{-2} ||u||_{L^3(T(\\rho ,\\tau ))}^2.$ Similarly, the terms of $I_{10}$ , $I_{11}$ and $I_{12}$ are also obtained: $|I_{10}|+ |I_{11}|+|I_{12}|\\le C ( \\mu _1+\\mu _2+\\mu _3) r^2 (\\tau - \\rho )^{-2} ||B||_{L^6(T(\\rho ,\\tau ))}^2.$ For the term $I_7$ , using (REF ) again, we have $|I_7| &\\le & \\int _{B(2r)} |u|^2 |\\nabla w| dx \\le \\left(\\int _{B(\\tau )} |u|^3 dx\\right)^{\\frac{2}{3}} \\left(\\int _{B(\\tau )} |\\nabla w|^3 dx\\right)^{\\frac{1}{3}} \\\\&\\le & C (\\tau - \\rho )^{-1} ||u||_{L^3(T(\\rho ,\\tau ))}||u||_{L^3(B(\\tau ))}^2,$ and $|I_8| \\le C (\\tau - \\rho )^{-1} ||u||_{L^3(T(\\rho ,\\tau ))}||u||_{L^3(B(\\tau ))}^2.$ Similarly, $|I_9| &\\le & \\int _{B(2r)} |B|^2 |\\nabla w| dx \\le C r ||B||_{L^6(B(\\tau ))}^2 \\left(\\int _{B(\\tau )} |\\nabla w|^3 dx\\right)^{\\frac{1}{3}} \\\\&\\le & C r (\\tau - \\rho )^{-1} ||B||_{L^6(B(\\tau ))}^2 ||u||_{L^3(T(\\rho ,\\tau ))}.$ The terms $I_{13}$ and $I_{14}$ can be treated as the term $I_9$ : $|I_{13}| + |I_{14}| \\le C r (\\tau - \\rho )^{-1} ||B||_{L^6(T(\\rho ,\\tau ))}^2 ||u||_{L^3(T(\\rho ,\\tau ))}.$ Collecting all the terms of $I_i$ , we arrive at $G(\\rho ) &\\le & 2\\varepsilon G(\\tau ) + C (\\lambda _1+\\lambda _2 +\\lambda _3) r (\\tau - \\rho )^{-2} ||u||_{L^3(T(\\rho ,\\tau ))}^2\\\\&&+ C (\\mu _1+\\mu _2+\\mu _3) r^2 (\\tau - \\rho )^{-2} ||B||_{L^6(T(\\rho ,\\tau ))}^2 \\\\&&+ C (\\tau - \\rho )^{-1} ||u||_{L^3(T(\\rho ,\\tau ))}||u||_{L^3(B(\\tau ))}^2 + C r (\\tau - \\rho )^{-1} ||B||_{L^6(B(\\tau ))}^2 ||u||_{L^3(T(\\rho ,\\tau ))}\\\\&\\le & 2\\varepsilon G(\\tau ) + C (\\lambda _1+\\lambda _2 +\\lambda _3) r (\\tau - \\rho )^{-2} ||u||_{L^3(T(r,2r))}^2\\\\&&+ C (\\mu _1+\\mu _2+\\mu _3) r^2 (\\tau - \\rho )^{-2} ||B||_{L^6(T(r,2r))}^2 \\\\&&+ C (\\tau - \\rho )^{-1} ||u||_{L^3(T(r,2r))}||u||_{L^3(B(\\tau ))}^2 + C r (\\tau - \\rho )^{-1} ||B||_{L^6(B(\\tau ))}^2 ||u||_{L^3(T(r,2r))}.$ Choosing $\\varepsilon = \\frac{1}{4}$ and using the iteration in [15], we have $G(\\rho ) &\\le & C (\\lambda _1+\\lambda _2 +\\lambda _3) r (\\tau - \\rho )^{-2} ||u||_{L^3(T(r,2r))}^2\\\\&&+ C (\\mu _1+\\mu _2+\\mu _3) r^2 (\\tau - \\rho )^{-2} ||B||_{L^6(T(r,2r))}^2 \\\\&&+ C (\\tau - \\rho )^{-1} ||u||_{L^3(T(r,2r))}||u||_{L^3(B(\\tau ))}^2 + C r (\\tau - \\rho )^{-1} ||B||_{L^6(B(\\tau ))}^2 ||u||_{L^3(T(r,2r))}.$ Taking $\\rho = r$ and $\\tau = 2r$ , letting $r \\rightarrow \\infty $ , we have $\\sum _{i=1}^3 \\lambda _i \\int _{\\mathbb {R}^3} |\\partial _i u|^2 dx + \\sum _{i=1}^3 \\mu _i \\int _{\\mathbb {R}^3} |\\partial _i B|^2 dx = 0,$ since $u \\in L^3(\\mathbb {R}^3)$ and $B \\in L^6(\\mathbb {R}^3)$ .", "Then there exist $i \\in \\lbrace 1,2,3\\rbrace $ and $j \\in \\lbrace 1,2,3\\rbrace $ , such that $\\partial _i u, \\partial _j B \\equiv 0$ due to the known condition (REF ).", "Without loss of generality, we assume that $\\partial _1 u = \\partial _1 B = 0$ , which means $u(x_1, x_2, x_3) = u(x_2, x_3)$ and $B(x_1, x_2, x_3) = B(x_2, x_3)$ .", "Then $\\int _{\\mathbb {R}^2} \\int _{-\\infty }^{+\\infty } |u(x_2, x_3)|^3dx_1 dx_2 dx_3 =\\infty ,$ if $u$ is a nontrivial solution, which is a contradiction with $u \\in L^3(\\mathbb {R}^3)$ .", "Thus we have $u\\equiv 0$ in $\\mathbb {R}^3$ .", "Using the same idea, we have $B \\equiv 0$ in $\\mathbb {R}^3$ .", "The proof is complete.", "Acknowledgments.", "W. Wang was supported by NSFC under grant 12071054 and 11671067." ] ]
2105.11667
[ [ "Rise and fall of laser-intensity effects in spectrally resolved Compton\n process" ], [ "Abstract The spectrally resolved differential cross section of Compton scattering, $d \\sigma / d \\omega' \\vert_{\\omega' = const}$, rises from small towards larger laser intensity parameter $\\xi$, reaches a maximum, and falls towards the asymptotic strong-field region.", "Expressed by invariant quantities: $d \\sigma /du \\vert_{u = const}$ rises from small towards larger values of $\\xi$, reaches a maximum at $\\xi_{max} = \\frac49 {\\cal K} u m^2 / k \\cdot p$, ${\\cal K} = {\\cal O} (1)$, and falls at $\\xi > \\xi_{max}$ like $\\propto \\xi^{-3/2} \\exp \\left (- \\frac{2 u m^2}{3 \\xi \\, k \\cdot p} \\right )$ at $u \\ge 1$.", "[The quantity $u$ is the Ritus variable related to the light-front momentum-fraction $s = (1 + u)/u = k \\cdot k' / k \\cdot p$ of the emitted photon (four-momentum $k'$, frequency $\\omega'$), and $k \\cdot p/m^2$ quantifies the invariant energy in the entrance channel of electron (four-momentum $p$, mass $m$) and laser (four-wave vector $k$).]", "Such a behavior of a differential observable is to be contrasted with the laser intensity dependence of the total probability, $\\lim_{\\chi = \\xi k \\cdot p/m^2, \\xi \\to \\infty} \\mathbb{P} \\propto \\alpha \\chi^{2/3} m^2 / k \\cdot p$, which is governed by the soft spectral part.", "We combine the hard-photon yield from Compton with the seeded Breit-Wheeler pair production in a folding model and obtain a rapidly increasing $e^+ e^-$ pair number at $\\xi \\lesssim 4$.", "Laser bandwidth effects are quantified in the weak-field limit of the related trident pair production." ], [ "Introduction", "Quantum Electro-Dynamics (QED) as pillar of the standard model (SM) of particle physics possesses a positive $\\beta $ function [1] which makes the running coupling strength $\\alpha (\\mathfrak {s} )$ increasingly with increasing energy/momentum scale $\\mathfrak {s}$ [2].", "In contrast, Quantum Chromo-Dynamics (QCD) as another SM pillar possesses a negative $\\beta $ function due to the non-Abelian gauge group [1], giving rise to the asymptotic freedom, $\\lim _{\\mathfrak {s} \\rightarrow \\infty } \\alpha _{QCD} (\\mathfrak {s}) \\rightarrow 0$ , i.e.", "QCD has a truly perturbative limit.", "In contrast, $\\lim _{\\mathfrak {s} \\rightarrow 0} \\alpha (\\mathfrak {s}) \\rightarrow 1/137.0359895(61)$ is not such a strict limit, nevertheless, QED predictions/calculations of some observables agree with measurements within 13 digits, see [5], [3], [4] for some examples.", "The situation in QED becomes special when considering processes in external (or background) fields: One can resort to the Furry (or bound-state) picture, where the (tree-level) interactions of an elementary charge (e.g.", "an electron) with the background are accounted for in all orders, and the interactions with the quantized photon field remains perturbatively in powers of $\\alpha $ .", "However, the Ritus-Narozhny (RN) conjecture [6], [7], [8], [9], [10] argues that the effective coupling becomes $\\alpha \\chi ^{2/3}$ , meaning that the Furry picture expansion beaks down at $\\alpha \\chi ^{2/3} > 1$ [11], [12], [13], [14] (for the definition of $\\chi $ see below) and one enters a genuinely non-perturbative regime.", "The latter requires adequate calculation procedures, as the lattice regularized approaches, which are standard since many years in QCD, e.g.", "in evaluations of observables in the soft sector where $\\alpha _{QCD} > 1$ , cf.", "[15].", "(In QED itself, an analog situation is meet in the Coulomb field of nuclear systems with proton numbers $Z > Z_{crit} \\approx 173$ : if $\\alpha Z_{crit} > 1$ the QED vacuum beak-down sets in; cf.", "[16] for the actual status of that field).", "With respect to increasing laser intensities the quest for the possible break-down of the Furry picture expansion in line with the RN conjecture becomes, besides its principal challenge, also of “practical\" interest, whether one can explore experimentally this yet uncharted regime of QED.", "(For other configurations, e.g.", "beam-beam interactions, cf.", "[17]).", "A prerequisite would be to find observables which display the typical dependence $\\propto \\alpha \\chi ^{2/3}$ , where we denote by $\\alpha $ the above quoted fine-structure constant at $\\mathfrak {s} \\rightarrow 0$ .", "In doing so we resort here to the lowest-order QED processes, that is nonlinear Compton and nonlinear Breit-Wheeler.", "Both processes seem to be investigated theoretically in depth in the past, however, enjoy currently repeated re-considerations w.r.t.", "refinements [18], [19], [20], [22], [21], [23], establishing approximation schemes [24], [25], [28], [27], [26] to be implemented in simulation codes [29], [30], [31] or to use them as building blocks in complex processes [32], with the starting point at trident [33], [34], [35].", "Beginning with Compton scattering ($p$ and $p^{\\prime }$ are the $in$ - and $out$ -electron four-momenta, $k$ the laser four-momentum, and $k^{\\prime }$ the $out$ -photon's four-momentum, respectively) the relevant (Lorentz and gauge invariant) variables of the entrance channel are [36] - (i) the classical intensity parameter of the laser: $\\xi = \\vert e \\vert {\\cal E} /(m \\omega )$ , here expressed by quantities in the lab.", ": ${\\cal E}$ - electric laser field strength, $\\omega $ - the central laser frequency; $- \\vert e \\vert $ and $m$ stand for the electron charge and mass, respectively, and $e^2 / 4 \\pi = \\alpha $ ,We employ natural units with $\\hbar = c = 1$ .", "- (ii) the available energy squared: $k \\cdot p / m^2 = (\\hat{s} /m^2 - 1)/2$ with $\\hat{s}$ as Mandelstam variable, - (iii) the quantum nonlinearity parameter: $\\chi = \\xi k \\cdot p/m^2$ .", "The latter quantity is often considered as the crucial parameter since in some limits it determines solely the probability of certain processes.", "$\\chi $ plays also a prominent role in the above mentioned discussion of the RN conjecture, where the Furry picture expansion of QED is argued to break down for $\\alpha \\chi ^{2/3} > 1$ .", "References [37], [38] point out that the large-$\\chi $ limits, facilitated by either large $\\xi $ (the high-intensity limit) or large $k \\cdot p/m^2$ (the high-energy limit), are distinctively different, with implications for approximation schemes in simulation codes.", "Figure REF exhibits a few selected curves $\\chi = const$ over the $\\xi $ vs. $k \\cdot p/m^2$ landscape to illustrate the current situation w.r.t.", "facilities where laser beams and electron beams are (or can be) combined.", "One has to add the options at E-320 at FACET-II/SLAC [39], [40] and electron beams which are laser-accelerated to GeV scales, e.g.", "[41], [42], [43].", "Usually, $\\xi = 1$ is said to mark the the onset of strong-field effects, and the corresponding processes at $\\xi > 1$ are termed by the attribute “nonlinear\".", "In this respect, the parameters provided by LUXE [44], [45] and FACET-II [39], [40] are interesting: $\\chi = {\\cal O}(1)$ and above and $\\xi > 1$ as well.Despite high intensities at XFELs, e.g.", "$I \\rightarrow 10^{22}$ W/cm${}^2$ , the intensity parameter $\\xi = \\frac{7.5 {\\rm eV}}{\\omega } \\sqrt{\\frac{I}{10^{20} {\\rm W/cm}^2}}$ [36] is small due to the high frequency, e.g.", "$\\omega =$ 1 - 25 keV.", "In the following we consider the LUXE kinematics (see Fig.", "REF ), $k \\cdot p/m^2 \\approx \\omega E_e (1 - \\cos \\Theta ) = 0.2078$ in head-on collisions ($\\cos \\Theta = -1$ ).", "Our aim is to quantify a simple observable as a function of $\\xi $ .", "To be specific, we select the invariant differential cross section $d \\sigma / du$ , where $u$ is the light-cone momentum-transfer of the $in$ -electron to the $out$ -photon, related to light-front momentum-fraction of the $out$ -photon $s = \\frac{u}{1 + u} = \\frac{k \\cdot k^{\\prime }}{ k \\cdot p}$ .", "(The mapping $u \\mapsto \\omega ^{\\prime }$ and $d \\sigma / du \\mapsto d \\sigma / d \\omega ^{\\prime }$ is discussed in [46], [47].)", "To make the meaning of the Ritus variable $u$ more explicit let us mention the relation $u = \\frac{e^{-\\zeta } \\nu ^{\\prime } (1 - \\cos \\Theta ^{\\prime })}{1 -e^{-\\zeta } \\nu ^{\\prime } (1 - \\cos \\Theta ^{\\prime })}$ , where $\\nu ^{\\prime } \\equiv \\omega ^{\\prime }/m$ , and the electron energy $E_e$ in lab.", "determines the rapidity $\\zeta $ via $E_e = m \\cosh \\zeta $ and $\\Theta ^{\\prime }$ denotes the polar lab.", "angle of the $out$ -photon.", "We call $d \\sigma / du$ a spectrally resolved observable.", "Figure: Curves of χ=const\\chi = const over the ξ\\xi vs. k·p/m 2 k \\cdot p/m^2 planefor χ=0.001,\\chi = 0.001, 0.01, 0.1, 1 and 10.The double-line depicts the curve αχ 2/3 =1\\alpha \\chi ^{2/3} = 1.Vertical thin lines are for maximum values of k·p/m 2 k \\cdot p/m^2 in reachat various electron accelerators(HZDR : E e =33E_e = 33 MeV,ELI : E e =600E_e = 600 MeV,LUXE , : E e =17.5E_e = 17.5 GeV)in combination with a high-intensity optical laser(we use ω=1.55\\omega = 1.55 eV as representative frequency).The vertical dotted line indicates a possible combination of theEuropean XFEL (ω=10\\omega = 10 keV) with a laser-accelerated electronbeam (E e =10E_e = 10 MeV) available in the high-energy density caveof the HIBEF collaboration .The horizontal delineation line ξ=1\\xi = 1 is thought to highlight theonset of the strong-field region above.Our note is organized as follows.", "In section we briefly recall a few approximations of the laser beam.", "Section is devoted to an analysis of the invariant differential cross section $d \\sigma /du$ and its dependence on the laser intensity parameter $\\xi $ in nonlinear Compton scattering.", "That is, we are going up and down on the vertical dashed line with label LUXE in Fig.", "REF around the point $\\chi = 1$ or $\\xi = 1$ .", "The discussion section (i) relates the cross section to the probability and (ii) considers a folding model which uses the hard-photon spectrum emerging from Compton scattering as seed for subsequent Breit-Wheeler pair production; a brief discussion of (iii) bandwidth effects relevant for sub-threshold trident pair production complements this section.", "We conclude in section .", "The appendix recalls a few basic elements of the one-photon Compton process." ], [ "Laser models", "In plane-wave approximationDue to the high symmetry, the exact solutions of the Dirac equation in a plane-wave background are in a comfortably compact form enabling an easy processing in evaluations of matrix elements.", "Without such a high symmetry, much more attempts are required [51], [52], [53].", "For some useful parameterizations of laser beams, see [54], [55], [56] and further citations therein.", "The relevance of the Fourier-zero mode of (non-)unipolar planar fields is mentioned in [57].", "the laser (circular polarization) can be described by the four-potential in axial gauge, $A = (0, \\vec{A})$ , with $\\vec{A} = f(\\phi ) \\left(\\vec{a}_x \\cos \\phi + \\vec{a}_y \\sin \\phi \\right)$ where $\\vec{a}_x^2 = \\vec{a}_y^2 = m^2\\xi ^2/e^2$ ; the polarization vectors $\\vec{a}_x$ and $\\vec{a}_y$ are mutually orthogonal.", "We ignore a possible non-zero value of the carrier envelope phase and focus on symmetric envelope functions $f(\\phi )$ w.r.t.", "the invariant phase $\\phi = k \\cdot x$ .", "One may classify the such a model class as follows.", "- 1) Laser pulses: $\\lim _{\\phi \\rightarrow \\pm \\infty } f(\\phi ) = 0$ , - 2) Monochromatic beam: $f(\\phi ) = 1$ , - 3) Constant cross field (ccf): $\\vec{A} = \\phi \\, \\vec{a}_x $ .", "The probabilities for the constant cross field option 3) coincide with certain limits of the plane-wave model 2) [58]; in [47] they are related to the large-$\\xi $ limit.", "Item 1) could be divided into several further sub-classes, such as 1.1): finite support region of the pulse, i.e.", "$f(\\vert \\phi \\vert > \\phi _{pulse \\, length}) = 0$ , $\\phi _{pulse \\, length} < \\infty $ , and 1.2): far-extended support region, i.e.", "$\\lim _{\\phi \\rightarrow \\pm \\infty } f(\\phi ) \\rightarrow 0$ , together with non-zero carrier envelope phase, asymmetric pulse shape, frequency chirping, polarization gating etc.", "A specific class is 1.1) with flat-top section, e.g.", "a box envelope $\\sqcap $ (cf.", "[59] for a recent explication) belonging to $C^0$ , or $\\cos ^2 \\otimes \\, \\sqcap $ belonging to $C^2$ or the construction in [60] belonging to $C^\\infty $ .", "Examples for item 1.2) are Gauss, super-Gauss (employed in [61], for instance), symmetrized Fermi function [62], [63], $1/\\cosh $ etc.", "The monochromatic beam, item 2), corresponds formally to an infinitely long flat-top “pulse\", abbreviated hereafter by IPA as acronym of infinite pulse approximation.", "It may be considered as special case of 1.1) with $\\phi _{pulse \\, length \\rightarrow \\infty }$ .", "FPA stands henceforth for the finite pulse-length plane-wave approximation.", "To be specific, we employ here 1.2) with $f(\\phi ) = 1 / \\cosh (\\phi /\\pi N)$ , where $N$ characterizes the number of oscillations in that pulse interval, where $f(\\phi )$ is significantly larger than zero (see [47], [62], [64] for the formalism), and IPA from 2).", "The laser model of class 1.1) is employed in subsection REF ." ], [ "Compton: differential invariant cross section ", "Let us consider the above pulse envelope function $f(\\phi ) = 1 / \\cosh (\\phi /(\\pi N))$ to elucidate the impact of a finite pulse duration and contrast it later on with the monochromatic laser beam model and some approximations thereof.", "Differential spectra $d \\sigma / du$ as a function of $u$ are exhibited in Fig.", "REF in the region $u \\le 3$ for several values of $\\xi \\le 1$ for the FPA (dashed curves) and IPA (solid curves) models recalled below.", "This complements figures 1 – 3 in [47].", "One observes that the harmonic structures (which would become more severe for linear polarization, see [47]) fade away at larger values of $u$ and $\\xi $ .", "Therefore, we are going to analyze that region in parameter space.", "There, IPA results represent reasonably well the trends of the more involved FPA calculations." ], [ "Numerical results", "In particular, we consider now $d \\sigma (u, \\xi , k \\cdot p/m^2) / du$ as a function of $\\xi $ for several constant values of $u$ , $u = 0.5,$ 1, 2, 4 and 8, for $k \\cdot p /m^2 = 0.2078$ – a value which is motivated by the LUXE opportunities [44], [45].", "The solid, dotted and dashed curves in Fig.", "REF are based on easily accessible models [58]: $\\bullet $ monochromatic model (IPA) (cf.", "equations (15, 16) in [47]): $ \\frac{d \\sigma _{IPA}}{du} &=& \\frac{ 2 \\pi \\alpha ^2}{k \\cdot p}\\frac{1}{(1+u)^2}\\sum _{n = 1}^\\infty \\, \\Theta (u_n - u) \\, F_n(z_n), \\\\F_n &=&- \\frac{2}{\\xi ^2} J_n^2(z_n) \\\\& + & A \\left(J_{n+1}^2(z_n) + J_{n-1}^2(z_n) - 2 J_n^2(z_n) \\right) \\nonumber $ with $A = 1 + \\frac{u^2}{2(1+u)}$ , $z_n(u, u_n) =\\frac{2 n \\xi }{\\sqrt{1 + \\xi ^2}} \\sqrt{\\frac{u}{u_n} (1 - \\frac{u}{u_n})}$ , and $u_n = \\frac{2 n k \\cdot p}{m^2 (1 + \\xi ^2)}$ ($J_n$ 's denote Bessel functions of first kind); $\\bullet $ IPA–large-$\\xi $ approximation (cf.", "equation (21) in [47]): $ \\frac{d\\sigma _{large-\\xi }}{du}&=& - \\frac{4 \\pi \\alpha ^2}{m^2 \\xi } \\frac{1}{\\chi } \\frac{1}{(1 + u)^2}{\\cal F}_C \\\\{\\cal F}_C & = &\\int _{z}^{\\infty } dy \\, \\Phi (y) + \\frac{2}{z}\\left[1 + \\frac{u^2 }{2 (1 + u)} \\right]\\Phi ^{\\prime }(z) $ where $\\Phi (z)$ and $\\Phi (z)^{\\prime }$ stand for the Airy function and its derivative with arguments $z = (u / \\chi )^{2/3}$ where $\\chi = \\xi k \\cdot p /m^2$ ; $\\bullet $ large-$\\xi $ -large-$u$ approximation (cf.", "equation (22) in [47]): $ \\frac{d\\sigma _{large-\\xi , \\, large-u}}{du} =\\frac{2 \\sqrt{\\pi } \\alpha ^2}{m^2 \\xi }\\chi ^{- 1/2} u^{- 3/2}\\exp \\left(- \\frac{2 u}{3 \\chi } \\right).$ Figure: Invariant differential cross sectiondσ(u=const,ξ,k·p/m 2 )/dud \\sigma (u = const, \\xi , k \\cdot p/m^2) / duas a function of ξ\\xi for several values of uu,u=0.5,u = 0.5, 1, 2, 4 and 8; for k·p/m 2 =0.2078k \\cdot p /m^2 = 0.2078.The asterisks depict results of short laser pulses withenvelope function f(φ)=1/cosh(φ/(πN))f(\\phi ) = 1 / \\cosh (\\phi /(\\pi N)) forN=10N = 10, dσ FPA /dud \\sigma _{FPA} / du, Eq.", "().The solid curves are for the monochromatic laser beam modeldσ IPA /dud \\sigma _{IPA} / du, Eq.", "(),while dashed (dotted) curves are basedon dσ large-ξ /dud \\sigma _{large-\\xi } / du, Eq.", "()(dσ large-ξ,large-u /dud \\sigma _{large-\\xi , \\, large-u}/du, Eq.", "()).The asterisks in Fig.", "REF depict results of the pulse model (FPA) described in [47], [62] (cf.", "equations (5 - 14) in [47]): $ \\frac{d \\sigma _{FPA}}{d u} &=& \\frac{\\alpha ^2}{k \\cdot p}\\frac{1}{(1+u)^2} \\int \\limits _0^\\infty d \\ell \\,\\Theta \\left(\\frac{u m^2}{2 k \\cdot p} - \\ell \\right) w (\\ell ) , \\\\w(\\ell , z) &=& \\int _0^{2 \\pi } d \\phi _{e^{\\prime }} \\left[- \\frac{2}{\\xi ^2} \\vert \\tilde{Y}_\\ell (z_\\ell ) \\vert ^2 \\right.$ $ + A \\left.\\left( \\vert Y_{\\ell - 1}(z_\\ell )\\vert ^2 + \\vert Y_{\\ell +1} (z_\\ell )\\vert ^2- 2 \\mbox{Re} \\tilde{Y}_\\ell (z_\\ell ) X_\\ell ^* (z_\\ell )) \\right) \\right] $ with $A$ as above, $z_\\ell (u, \\ell ) ={2\\ell \\xi }\\sqrt{\\frac{u}{u_\\ell }(1-\\frac{u}{u_\\ell })}$ and $u_\\ell = \\frac{2 \\ell k \\cdot p}{m^2}$ .", "The basic functions $Y_\\ell , \\tilde{Y}_\\ell , X_\\ell $ and their $\\phi _{e^{\\prime }}$ dependence are spelled out in [47], [62]; they depend crucially on the pulse envelope function $f(\\phi )$ .", "(Due to numerical accuracy reasons in integrating highly oscillating functions, these evaluations are constrained presently to not too large values of $\\xi $ .)", "The common basis of these models is sketched in Appendix .", "We consider in Fig.", "REF only $u > u_{KN} = 2 k \\cdot p /m^2$ , since at the Klein-Nishina (KN) edge the harmonic structures become severe, as seen in Fig.", "REF .", "The striking feature seen in Fig.", "REF is the pronounced $\\cap $ shape, which we coin “rise and fall\" of laser intensity effects.", "At small $\\xi $ , the realistic FPA ($N = 10$ ) results (asterisks) and the IPA model (solid curves) follow the same trends, consistent with Fig.", "REF .", "The large-$\\xi $ and large-$\\xi $ -large-$u$ approximations (dashed and dotted curves) are not supposed to apply in that region.", "However, they become useful representatives at large $\\xi $ ." ], [ "The rise", "Some guidance of the rising parts of the FPA results in Fig.", "REF can be gained by the monochromatic model.", "Casting Eq.", "() in the form $ F_n &=&- \\frac{2 (1 - \\Xi ^2)}{\\Xi ^2} J_n^2(\\Xi x_n) \\\\& + & A \\left(J_{n+1}^2(\\Xi x_n) + J_{n-1}^2(\\Xi x_n) - 2 J_n^2(\\Xi x_n) \\right) \\nonumber $ with $\\Xi ^2 = \\xi ^2 / (1 + \\xi ^2)$ and $x_n (u) = 2 n \\sqrt{\\frac{u}{u_n} (1 - \\frac{u}{u_n})}$ and expanding in powers of $\\Xi $ yields for the first terms $ F_1 &=& \\frac{1}{2} (2 A - x_1^2) - \\frac{x_1^2}{8} (8A -x_1^2 - 4) \\Xi ^2+ \\frac{x_1^4}{384} (90 A - 5 x_1^2 - 48) \\Xi ^4 + {\\cal O} (\\Xi ^6) , \\\\F_2 &=& \\hspace*{79.6678pt} \\frac{x_2^2}{32} (8 A - x_2^2) \\Xi ^2 - \\quad \\frac{x_2^4}{192} (18 A - x_2^2 - 6) \\Xi ^4 + {\\cal O} (\\Xi ^6) , \\\\F_3 &=& \\hspace*{179.25244pt} \\frac{x_3^4}{1152} (18 A - x_3^2) \\Xi ^4+ {\\cal O} (\\Xi ^6) .", "$ Due to the Heavyside $\\Theta $ function in Eq.", "(REF ), the leading-order power in $\\Xi $ depends on the value of $u$ , e.g.", "for $u < u_1$ , the series starts with ${\\cal O} (\\Xi ^0)$ and the coefficients of higher orders accordingly sum up.", "Higher values of $u$ facilitate higher orders of the leading terms, i.e.", "the rise becomes steeper since the respective leading-order term is $\\propto \\Xi ^{2 \\lfloor u/u_1 \\rfloor }$ .", "This statement is based on the structures in Eqs.", "(REF - ), suggesting $F_n = \\sum _{i = n - 1} F_{n}^{(2i)} \\, \\Xi ^{2 i}$ for the first terms $F_{n}^{(2i)}$ , and $\\sum _{n=1}^\\infty \\Theta (u_n - u) \\, F_n = \\sum _{n_{min}}^\\infty F_n$ with $n_{min} = 1 + \\lfloor u/u_1 \\rfloor $ .", "($\\lfloor \\cdot \\rfloor $ is the floor operation.)", "The series expansion of (REF ) in powers of $\\Xi $ ignores the sub-leading $\\Xi $ dependence in $x_n$ via $u_n = 2 n \\frac{k \\cdot p}{m^2} (1 - \\Xi ^2)$ but is suitable for $k \\cdot p = const$ .", "Analogously considerations apply to the pulse model, cf.", "section III.C in [62].", "An essential role is played by the Fourier transform of the pulse envelope in the limit $\\xi \\ll 1$ .", "It bridges to the IPA for long pulses." ], [ "The fall", "The maximum of the curves exhibited in Fig.", "REF at $u = {\\cal O}(1)$ is attained at $\\xi = {\\cal O}(1)$ , but moves towards larger values of $\\xi $ with increasing values of $u$ .", "Remarkably, the often discredited large-$\\xi $ and large-$\\xi $ –large-$u$ approximations (dotted and dashed curves) deliver results in fair agreement with the IPA results (solid curves) for $u > 1$ .", "That is, when being interested in the high-energy photon tails, the simple large-$\\xi $ –large-$u$ formula Eq.", "(REF ) [58] represents a fairly accurate description supposed $\\xi $ is sufficiently large.", "Obviously, at $\\xi < 1$ and $u < 2$ , such an approximation fails quantitatively.", "(In particular, at $u < 1$ the harmonic structures in IPA become severe.)", "Nevertheless, some estimate of the maximum position is provided by $\\xi _{max} \\approx \\frac{4}{9} u m^2 / k \\cdot p$ yielding $d \\sigma / du (\\xi _{max}) \\approx \\frac{2 \\sqrt{\\pi } \\alpha ^2}{m^2}\\frac{k \\cdot p}{m^2} \\left( \\frac{3}{2} \\right)^3 e^{- 3/2} \\, u^{-3}$ .", "The asymptotic fall is governed by $\\frac{d \\sigma }{du} \\propto \\frac{1}{m^2} \\frac{m^2}{k \\cdot p}\\xi ^{-3/2} u^{- 3/2} \\exp \\left( - \\frac{2 u m^2}{3 \\xi k \\cdot p}\\right)$ from Eq.", "(REF ).", "We emphasize the same pattern of rise and fall of $d \\sigma /d \\omega ^{\\prime } \\vert _{\\omega ^{\\prime } = const}$ , see Fig.", "REF , again for sufficiently large values of $\\omega ^{\\prime }$ .", "The spectrally resolved differential cross section $d \\sigma /d \\omega ^{\\prime } \\vert _{\\omega ^{\\prime } = const}$ is directly accessible in experiments.", "In the strong-field asymptotic region it displays a funneling behavior, i.e.", "the curves are squeezed into a narrow corridor already in the non-asymptotic $\\xi $ region, in contrast to $d \\sigma /d u\\vert _{u= const}$ in Fig.", "REF .", "Figure: The same as in Fig.", "but for dσ/dω ' d \\sigma /d \\omega ^{\\prime } as a functionof laser intensity parameter ξ\\xi for several constant values of ω ' \\omega ^{\\prime }." ], [ "Cross section vs. probability", "The cross section $\\sigma $ and Ritus probability rate $W$ are related as $W = \\frac{m^4}{4 \\pi \\alpha } \\frac{\\xi \\, \\chi }{q_0} \\sigma $ with $q_0$ denoting the energy component of the quasi-momentum of the $in$ -electron.", "This relation holds true for circular polarization and applies to respective differential quantities too.", "The different normalization modifies in particular the $\\xi $ dependence: The above emphasized “rise and fall\" of $d \\sigma /du$ corresponds to a monotonously rising probability $d W / du$ as a function of $\\xi $ .", "Having in mind Ritus' remark “the cross-sectional concept becomes meaningless\" since, at $\\xi \\rightarrow \\infty $ we have $\\sigma \\rightarrow 0$ while $W$ remains finite [58], we turn in this sub-section to the probability.", "In doing so we remind the reader of the subtle Ritus notation $W(\\chi ) = \\frac{1}{\\pi } \\int _0^{\\pi } d \\psi P(\\chi \\sin \\psi )$ in distinguishing the probabilities $W$ and $P$ .", "We also stress the varying behavior of total (cf.", "appendix A in [65]) and differential probabilities.", "Equation (REF ) emerges as a certain limit of the constant cross field probability [58] $\\frac{d P (\\chi , u)}{du} = - V (1 + u)^{-2} \\, {\\cal F}_C (z(\\chi , u), u)$ where the prefactor reads $V = \\alpha m^2 / \\pi q_0$ and ${\\cal F}_C$ is defined in Eq. ().", "With respect to the argument $z = (u/\\chi )^{2/3}$ of the Airy functions in Eq.", "(), the $\\chi $ - $u$ plane can be divided into the regions I (where $u > \\chi $ ) and II (where $u < \\chi $ ).", "Furthermore, the combination $1 + u$ suggests a splitting into $u < 1$ and $u > 1$ sub-domains.", "Picking up the leading-order terms in the related series expansions one arrives at $\\frac{d P}{du}\\vert _{u \\gg \\chi }&=& \\frac{V}{2 \\sqrt{\\pi }} \\chi ^{1/2}\\exp \\left( - \\frac{2u}{3 \\chi } \\right) \\nonumber \\\\& \\times &\\left\\lbrace \\begin{array}{l}u^{- 1/2} \\quad \\mbox{for} \\quad u \\ll 1, \\quad (I^{<})\\\\u^{- 3/2} \\quad \\mbox{for} \\quad u \\gg 1, \\quad (I^{>})\\\\\\end{array}\\right.", "\\\\\\frac{d P}{du}\\vert _{u \\ll \\chi }&=& - V \\Phi ^{\\prime }(0) \\chi ^{2/3} \\nonumber \\\\& \\times &\\left\\lbrace \\begin{array}{l}2 u^{- 2/3} \\quad \\mbox{for} \\quad u \\ll 1, \\quad (II^{<})\\\\u^{- 5/3} \\,\\,\\,\\quad \\mbox{for} \\quad u \\gg 1.", "\\quad (II^{>})\\\\\\end{array}\\right.", "$ It is the soft $u$ -part $II^{<}$ of the differential probability in Eq.", "() which essentially determines the total probability upon $u$ -integration,The value $\\chi = 1$ is special since the small-$u$ region of II and the large-$u$ region of I must be joint directly, $II^{<} \\otimes I^{>}$ , while for $\\chi < 1$ the small-$u$ region II and the small-$u$ region of I must be joint followed by the large-$u$ region of I, $II^{<} \\otimes I^{<} \\otimes I^{>}$ .", "In the opposite case of $\\chi > 1$ , the small-$u$ region of II must be joint with the large-$u$ region of II followed by the large-$u$ region of I, $II^{<} \\otimes II^{>} \\otimes I^{>}$ .", "This distinction becomes also evident when inspecting the differently shaped sections of the continuous curves $d P /du$ as a function of $u$ for several values of $\\chi $ .", "i.e.", "the celebrated result $P \\propto \\alpha \\chi ^{2/3}$ in Ritus notation [58] (stated as $\\lim _{\\chi , \\xi \\rightarrow \\infty } \\mathbb {P}\\propto \\alpha \\chi ^{2/3} m^2 / k \\cdot p$ in [37]), while the hard $u$ -part $I^{>}$ in Eq.", "(REF ) is at the origin of Eq.", "(REF ) when converting to cross section.", "In other words, one has to distinguish the $\\xi $ (or $\\chi $ ) dependence either of integrated or differential observables.", "In relation to the LUXE plans [44] we mention the photon detector developments [66] which should enable in fact the access to the differential spectra." ], [ "Secondary processes: Breit-Wheeler", "Instead transferring these considerations to the Breit-Wheeler (BW) process per se (cf.", "[67], [68], [69], [70], [71], [72], [74], [73] and further citations below), we estimate now the BW pair production seeded by the hard photons from the above Compton (C) process.", "The following folding model C$\\otimes $ BW is a pure two-step ansatz on the probabilistic level which mimics in a simple manner some part of the trident processFor recent work on the formalism, see [33], [34], [35], [77], [78], [79] which advance earlier investigations [80], [81].", "$e_L^- \\rightarrow {e_L^-}^{\\prime } + {e_L^-}^{\\prime \\prime } + e_L^+$ by ignoring (i) the possible off-shell effects and non-transverse components of the intermediate photon (that would be the one-step contribution) and (ii) the anti-symmetrization of the two electrons in the final state (that would be the exchange contribution).", "Such an ansatz is similar to the one in [75], where however bremsstrahlung$\\otimes $ BW has been analyzed.", "An analog approach has been elaborated in [76] for di-muon production.", "Specifically, we consider the two-step cascade process where a GeV-Compton photon with energy $\\omega ^{\\prime }$ is produced in the first step, and, in the next step, that GeV-photon interacts with the same laser field producing a BW-$e^+e^-$ pair.", "We estimate the number of pairs produced in one pulse by $ N^{e^+e^-} = \\int \\limits _0^\\infty d \\omega ^{\\prime } \\,\\int \\limits _0^{T_C} dt \\, \\frac{d \\Gamma _C (\\omega ^{\\prime }, t)}{d \\omega ^{\\prime }}\\int \\limits _t^{T_{BW}} dt^{\\prime } \\, \\Gamma _{BW} (\\omega ^{\\prime }, t^{\\prime }) ,$ where $d \\Gamma _C (\\omega ^{\\prime }, t) / d \\omega ^{\\prime }$ is the rate of photons per frequency interval $d \\omega ^{\\prime }$ (which corresponds to $d P /du$ in REF ) emerging from Compton at time $t \\in [0, T_C]$ , and $\\Gamma _{BW} (\\omega ^{\\prime }, t^{\\prime })$ is the rate of Breit-Wheeler pairs generated by a probe photon of frequency $\\omega ^{\\prime }$ at lab.", "frame time-distance $t^{\\prime } \\in [t, T_{BW}]$ .", "The underlying picture is that of an electron traversing a laser pulse in head-on geometry near to light cone.", "The passage time of an undisturbed electron would be $N T_0/2$ , $T_0 = 2 \\pi / \\omega $ .", "Neglecting spatio-temporal variations within the pulse, the final formula becomes $N^{e^+e^-} = F_t \\int _0^{E_e} d \\omega ^{\\prime } \\, N^e_0\\frac{d \\Gamma _C (\\omega ^{\\prime })}{d \\omega ^{\\prime }} \\, \\Gamma _{BW} (\\omega ^{\\prime })$ upon the restriction $\\omega ^{\\prime } < E_e$ and $F_t = T_C (T_{BW} - T_C / 2)$ .", "A crucial issue is the choice of the formation time(s) [80], [81].", "When gluing C$\\otimes $ BW on the amplitude level, such a time appears linearly in the pair rate [82]In laser pulses such an additional parameter is not needed [83].", "or quadratically in the net probability via overlap light-cone times (cf.", "[77], [78] for instance).", "The time-ordered double integral over the C and BW probabilities, yielding the cascade approximation (cf.", "equation (43) in [78]), is analog to our above formula by folding two rates, which facilitates $F_t \\propto T_0^2$ , if $T_{C, \\, BW} \\propto T_0$ .", "We attach to the Compton rate the number $N^e_0 = 6 \\times 10^9$ of primary electrons per bunch.", "In the numerical evaluation we employ the following convenient approximations: $ \\frac{d \\Gamma _C}{d \\omega ^{\\prime }} &=&- \\frac{\\alpha m^2}{\\pi \\,E_e^2}{\\cal F}_C(z, u),\\\\\\Gamma _{BW} (\\omega ^{\\prime }) &=& \\frac{\\alpha m^2}{\\omega ^{\\prime }} {\\cal F}_{BW} , \\\\{\\cal F}_{BW} &=& \\sqrt{\\frac{3^3}{2^9}}\\kappa \\exp \\left[-\\frac{8}{3\\kappa } \\left(1-\\frac{1}{15\\xi ^2}\\right) \\right] ,$ where $u=\\kappa /(\\chi - \\kappa )$ , $\\chi =(k\\cdot p)\\xi /m^2$ , $\\kappa =(k\\cdot k^{\\prime })\\xi /m^2$ , and ${\\cal F}_C (z,u)$ with $z = (u / \\chi )^{2/3}$ is defined in Eq. ().", "Note the rise of $\\Gamma _{BW}$ and the fall of $d \\Gamma _C / d \\omega ^{\\prime }$ as a function of $\\omega ^{\\prime }$ at fixed values of $\\xi $ and $k \\cdot p$ and laser frequency $\\omega $ , see Fig.", "REF .", "Increasing values of $\\xi $ lift $d \\Gamma _C / d \\omega ^{\\prime }$ somewhat and make it flatter in the intermediate-$\\omega ^{\\prime }$ region, thus sharpening the drop at $\\omega ^{\\prime } \\rightarrow E_e$ .", "Remarkable is the strong impact of increasing $\\xi $ on the BW rate.", "Figure: The BW rate Γ BW \\Gamma _{BW} from Eqs.", "(, ) (solid curves)and the dimensionless differential C rate dΓ C /dω ' d \\Gamma _C / d \\omega ^{\\prime }(dashed curves) fromEqs.", "(, )as a function of ω ' \\omega ^{\\prime } for ξ=1\\xi = 1 (red) and 3 (blue).One may replace Eq.", "() in (REF ) by the sum over harmonics expressed by Bessel functions [58] to get an improvement of accuracy in the small-$\\omega ^{\\prime }$ region:Strictly speaking, imposing a finite duration in the monochromatic laser beam model 2) turns it into a flat-top laser pulse model of class 1.1), exemplified in [59] and applied to the nonlinear Compton process.", "$\\frac{d \\Gamma _C^{IPA}}{d \\omega ^{\\prime }} &=& \\frac{d \\sigma _C^{IPA}}{d \\omega ^{\\prime }}\\frac{\\xi ^2 m^2}{4 \\pi \\alpha } \\frac{k \\cdot p}{q_0} ,\\\\\\frac{d \\sigma _C^{IPA}}{d \\omega ^{\\prime }} &=& \\frac{\\pi r_e^2 m^2}{\\xi ^2 k \\cdot p}\\sum _{n=1}^\\infty \\frac{\\Theta (u_n - u)}{P_n}\\left[ - 4 J_n^2 \\right.", "\\\\&+& \\left.", "\\xi ^2 (2 + \\frac{u^2}{1+u})\\left(J_{n-1}^2 + J_{n+1}^2 - 2 J_n^2 \\right) \\right] \\nonumber $ with $q_0 = \\sqrt{E_e^2 - m^2} + \\beta _p \\omega $ , $\\beta _p = \\xi ^2 \\frac{m^2}{2 q \\cdot k}$ , $q \\cdot k = k \\cdot p$ , and arguments $z = \\frac{2 n \\xi }{\\sqrt{1+\\xi ^2}} \\frac{1}{u_n} \\sqrt{u (u_n-u)}$ of the Bessel functions $J_n$ as well as $u_n = 2 n \\frac{k \\cdot p}{m^2} \\frac{1}{1 + \\xi ^2}$ , $u = \\frac{n \\omega - \\omega ^{\\prime }}{\\kappa _n - n \\omega + \\omega ^{\\prime }}$ , $\\kappa _n = n \\omega - \\frac{1}{2} m e^\\zeta + \\frac{1}{2} m (1+\\xi ^2) e^{- \\zeta }$ , $P_n = m \\vert n \\frac{\\omega }{m} - \\sinh \\zeta + \\frac{\\xi ^2}{2} e^{-\\zeta } \\vert $ .", "These equations make the dependence $u(\\omega ^{\\prime })$ explicit and relate again the differential cross section $d \\sigma / d \\omega ^{\\prime }$ with the differential rate $d \\Gamma / d \\omega ^{\\prime }$ .", "Since the BW rate is exceedingly small at small $\\omega ^{\\prime }$ (see Fig.", "REF ), improvements of the Compton rate by catching the details of harmonic structures there are less severe for the pair number Eq.", "(REF ).", "Equation () of the BW rate, however, is inappropriate at smaller values of $\\xi $ [75] and needs improvement.", "Instead of using the series expansion in Bessel functions [58], a convenient formula is ${\\cal F}_{BW} &=& \\frac{1}{4 \\pi } \\sum _{n = n_{min}}^\\infty \\int _1^{u_n} \\frac{du}{u^{3/2} \\sqrt{1 - u}} \\\\& \\times &\\exp \\left\\lbrace - 2 n (a - \\tanh a \\right\\rbrace \\frac{1 + 2 \\xi ^2 (2u-1) \\sinh ^2 a}{a \\tanh a} \\nonumber $ with $n_{min} = 2 \\xi (1 + \\xi ^2) / \\chi $ and $u_n = n / n_{min}$ .", "This representation emerges from the large-$n$ approximation of Bessel functions, $J_n(z) \\approx \\exp \\left\\lbrace - n (a- \\tanh a) \\right\\rbrace / \\sqrt{2 n \\pi \\tanh a}$ and $\\tanh a = \\sqrt{1 - z^2 / n^2}$ .", "In the large-$\\xi $ limit, one may replace the summation over $n$ by an integration to arrive, via a double saddle point approximation, at the famous Ritus expression (), which in turn is a complement of Eq.", "(REF ), see [58].", "Figure: The yield of e + e - e^+e^- pairs as a function of ξ\\xi forelectron energies 45 GeV (magenta dash-dotted curve),17.5 GeV (blue solid curve) and 8 GeV (red dashed curve)according to the probabilistic folding model C⊗\\otimes BWEq.", "().For N 0 e =6×10 9 N_0^e = 6 \\times 10^9 electrons per bunch and per laser shot of duration NT 0 N T_0.The special normalization F t =T 0 2 /2F_t = T_0^2/2 is chosen,as realized by T BW =(T 0 2 +T C 2 )/(2T C )T_{BW} = (T_0^2 + T_C^2)/(2 T_C).", "The choiceT C ≈T 0 T_C \\approx T_0 facilitates a Compton spectrumdN C /dω ' =T C dΓ C /dω ' d N_C/d \\omega ^{\\prime } = T_C \\, d \\Gamma _C / d \\omega ^{\\prime } which agrees,for ξ=𝒪(1)\\xi = {\\cal O} (1) andin the region ω ' <10\\omega ^{\\prime } < 10 GeV, with a bremsstrahlung spectrumgenerated by electrons of the same energy impinging on a foil withX/X 0 =0.01X/X_0 = 0.01 .It is the ξ\\xi dependence of the Compton spectrum(see Fig. )", "which makes the pair yield more rapidly risingwith ξ\\xi than thepair yield of the bremsstrahlung⊗\\otimes BW model in .Figure: The power pp as a function of ξ\\xi for E e =45E_e = 45 GeV(magenta dash-dotted curve), 17.5 GeV (blue solid curve) and 8 GeV(red dashed curve).", "The results exhibited in Fig.", "are described byN e + e - (ξ,E e )=N 0 e + e - (E e )ξ p(ξ,E e ) N^{e^+e^-} (\\xi , E_e) = N^{e^+e^-}_0 (E_e) \\, \\xi ^{p(\\xi , E_e)}.Numerical results are exhibited in Fig.", "REF for $E_e = 45$  GeV, 17.5 GeV and 8 GeV.", "One observes a stark rise of $N^{e^+ e^-}$ up to $\\xi \\sim 4$ , which turns for larger values of $\\xi $ into a modest rise.", "To quantify that rise one can employ the ansatz $N^{e^+e^-} (\\xi , E_e) = N^{e^+e^-}_0 (E_e) \\, \\xi ^{p(\\xi , E_e)}$ .", "Note that, by such a quantification of the $\\xi $ dependence, one gets rid of the normalization $F_t$ .", "For $E_e = 17.5$  GeV we find $p(\\xi \\approx 1) \\approx 20$ dropping to $p(\\xi \\approx 20) \\approx 2$ , see Fig.", "REF .", "Larger values of $E_e$ reduce $p$ , e.g.", "$p(\\xi \\approx 1)\\vert _{E_e = 45 \\, \\rm {GeV}} \\approx 10$ in agreement with [82], while $p(\\xi \\approx 1)\\vert _{E_e = 8 \\, \\rm {GeV}} > 40$ .", "At $\\xi \\rightarrow 20$ , a universal value of $p \\approx 2$ seems to emerge.", "The extreme nonlinear sensitivity of the pair number on the laser intensity parameter $\\xi $ at $\\xi < 10$ , and in particular at $\\xi \\approx 1$ , points to the request of a refined and adequately realistic modeling beyond schematic approaches." ], [ "Bandwidth effects in linear trident", "The threshold for linear trident, $e^- + \\gamma (1.55 \\, {\\mbox{e}V}) \\rightarrow {e^-}^{\\prime } + {e^-}^{\\prime \\prime } + e^+$ , is at $E_e = 337$  GeV, i.e.", "the LUXE kinematics is in the deep sub-threshold regime, where severe multi-photon effects build up the nonlinearity.", "However, also bandwidth effects can promote pair production in the sub-threshold region [84], [85], even at $\\xi \\rightarrow 0$ .", "The key is the cross section of linear trident $\\sigma _{ppT} (\\sqrt{\\hat{s}}, \\Delta \\phi )$ , which depends on the invariant energy $\\sqrt{\\hat{s}} = m \\sqrt{1 + 2 k \\cdot p/m^2}$ and the pulse duration $\\Delta \\phi $ for a given laser pulse.", "The quantity $\\sigma _{ppT} (\\sqrt{\\hat{s}}, \\Delta \\phi )$ is exhibited in Fig.", "REF as a function of $\\sqrt{\\hat{s}}$ for several values of $\\Delta \\phi $ .", "For definiteness, we employ the laser pulse model of class 1.1) with parameterization $\\vec{A} = f_{ppT} (\\phi ) \\, \\vec{a}_x \\, \\cos \\phi $ and envelope function $f_{ppT} = \\cos ^2 \\left( \\frac{\\pi \\phi }{2 \\Delta \\phi } \\right)\\sqcap (\\phi , 2 \\Delta \\phi )$ , i.e.", "the number of laser-field oscillations within the pulse is $N = \\Delta \\phi / \\pi $ .", "In contrast to the presentation above, we deploy results in this sub-section for linear polarization and the $\\cos ^2$ envelope.", "We employ the formalism in [79] and its numerical implementation, that is “pulsed perturbative QED\" in the spirit of Furry picture QED in a series expansion in powers of $\\xi $ .", "Applied to trident, the pulsed perturbative trident (ppT) arises from the diagrams $e^+$$e^-$ $-$ $e^-$$e^+$ (double lines: Volkov wave functions, vertical lines: photon propagator) as leading-order term surviving $\\xi \\rightarrow 0$ .", "The scaled number of pairs is $N_{tot} / N_0^e= 2 \\pi \\Gamma _{tot} / \\pi $ , where the probability rate is given by $\\Gamma _{tot} = \\sigma _{ppT} \\frac{\\omega }{2} \\frac{m^2 \\xi ^2}{4 \\pi \\alpha }\\int _{-\\infty }^{\\infty } d \\phi \\, f^2_{ppT} (\\phi ).$ The chosen pulse implies $\\int _{-\\infty }^{\\infty } d \\phi \\, f^2_{ppT} (\\phi ) =\\frac{3}{2} \\Delta \\phi $ and $N_{tot} / N_0^e = \\sigma _{ppT} \\frac{3 m^2 \\xi ^2}{16 \\alpha } \\Delta \\phi $ .", "Note the $\\xi ^2$ dependence from the “target density\" already entering in Eq.", "(REF ) (cf.", "[87], [84] for analog relations).", "This is in contrast to Fig.", "REF , where genuine nonlinear effects are at work and mix with a stronger $\\xi $ dependence for C$\\otimes $ BW.", "The $\\xi ^2$ dependence is characteristic for pair production by probe photons provided by an “external target\", such as in the bremsstrahlung-laser configuration of LUXE, cf.", "[75].", "Figure: Pulsed perturbative trident cross section σ ppT \\sigma _{ppT} as a functionof s ^/m\\sqrt{\\hat{s}} / m for several pulse lengths Δφ\\Delta \\phi .", "In the IPA limit,i.e.", "a monochromatic laser beam or Δφ→∞\\Delta \\phi \\rightarrow \\infty , the thresholdis at s ^=3m\\sqrt{\\hat{s}} = 3 m. Above the threshold, the dots depict a few points(cf.", "table 1 in )from perturbative trident without bandwidth effects.Bandwidth effects enable the pair production in the sub-threshold regions ^<3m\\sqrt{\\hat{s}} < 3 m.For the long laser pulses used in E-144 [88], [89], [90], such bandwidth effects are less severe." ], [ "summary", "In summary, inspired by the renewed interest in the Ritus-Narozhny conjecture and the new perspectives offered by the experimental capabilities of LUXE and E-320, we recollect a few features of elementary QED processes within the essentially known formalism.", "In particular, we focus on the $\\xi $ dependence.", "For nonlinear Compton scattering, we point out that, in the non-asymptotic region $\\chi = {\\cal O}(1)$ , $k \\cdot p \\lesssim m^2$ , the spectrally resolved cross section $d \\sigma /du \\vert _{u = const}$ as a function of the laser intensity parameter $\\xi $ displays a pronounced $\\cap $ shape for $u > u_{KN}$ (the “rise and fall\").", "This behavior is in stark contrast with the monotonously rising integrated probability $\\lim _{\\chi , \\xi \\rightarrow \\infty } \\mathbb {P} \\propto \\alpha \\chi ^{2/3} m^2 /k \\cdot p$ .", "That is, in different regions of the phase space, also different sensitivities of cross sections/rates/probabilities on the laser intensity impact can be observed.", "The soft (small-$u$ ) part, which determines the integrated cross section/probability, may behave completely different than the hard (large-$u$ ) contribution.An analog situation is known in QCD [91]: Tree-level diagrams are calculated primarily with constant $\\alpha _{QCD}$ .", "Renormalization improvement means then replacing $\\alpha _{QCD}$ by $\\alpha _{QCD} (\\mathfrak {s})$ which accounts at the same time for all vertices in the considered diagram by one global scale $\\mathfrak {s}$ .", "Inserting explicitly loop corrections leads finally to scale dependent couplings $\\alpha _{QCD} (\\mathfrak {s}_v)$ specific for each vertex $v$ separately.", "Transferred to certain approximations used in simulation codes such a behavior implies that one should test differentially where the conditions for applicability are ensured.", "The hard photons, once produced by Compton process in a laser pulse, act as seeds for secondary processes, most notably the Breit-Wheeler process.", "A folding model of type Compton$\\otimes $ Breit-Wheeler on the probabilistic level points to a rapidly increasing rate of $e^+ e^-$ production in the region $\\xi \\lesssim 4$ , when using parameters in reach of the planned LUXE set-up.", "The actual plans (see figure 2.10 in LUXE CDR [45]) uncover $\\xi = 2$ (40 TW, 8 $\\mu $ m laser), 6 (40 TW, 3 $\\mu $ m) and 16 (300 TW, 3 $\\mu $ m), and E-320 envisages $\\xi = 10$ .", "The folding model may be utilized as reference to identify the occurrence of the wanted one-step trident process in this energy-intensity regime.", "Furthermore, bandwidth effects in the trident process are isolated by considering the weak-field regime $\\xi \\rightarrow 0$ ." ], [ "Basics of nonlinear Compton process", "Following [92] we recall the basics of the underlying formalism of the nonlinear Compton process.", "Within the Furry picture the lowest-order, tree-level $S$ matrix element for the one-photon (four-momentum $k^{\\prime }$ , four-polarization $\\epsilon ^{\\prime }$ ) decay of a laser-dressed electron $e_L$ in the background field (REF ) $e_L (p) \\rightarrow e_L (p^{\\prime })+ \\gamma (k^{\\prime }, \\epsilon ^{\\prime })$ reads with suitable normalizations of the wave functions $ S_{fi} = - i e \\int d^4 x \\, J \\cdot {\\epsilon ^*}^{\\prime } \\, \\exp \\lbrace i k^{\\prime } \\cdot x \\rbrace $ where the current $J_\\mu (x) = \\bar{\\Psi }_{p^{\\prime }} \\gamma _\\mu \\Psi _p $ is built by the Volkov wave function $\\Psi _p = E_p u_p \\exp \\lbrace - i p \\cdot x \\rbrace \\exp \\lbrace - i f_p\\rbrace $ (spin indices are suppressed) and its adjoint $\\bar{\\Psi }$ with Ritus matrix $E_p = 1 + \\frac{e}{2 k \\cdot p} k A$ and phase function $f_p (\\phi ) = \\frac{e}{k \\cdot p}\\int _0^\\phi d \\phi ^{\\prime }\\left[p \\cdot A - \\frac{e}{2} A \\cdot A \\right]$ .", "We employ Feynman's slash notation and denote scalar products by the dot between four-vectors; $u_p$ is the free Dirac bi-spinor.", "Exploiting the symmetry of the background field, $A(\\phi = k \\cdot x)$ , Eq.", "(REF ) can be manipulated (cf.", "[92] for details) to arrive at $ S_{fi} = - i e (2 \\pi )^3 \\frac{2}{k_-} \\, \\delta ^{(3)}(\\underline{p} - \\underline{p^{\\prime }} - \\underline{k^{\\prime }}) \\,{\\cal M} (\\ell ) ,$ where $\\ell \\equiv (k^{\\prime }_- + p^{\\prime }_- - p_-)/ k_- = k^{\\prime } \\cdot p / k \\cdot p^{\\prime } $ accomplishes the balance equation $p + \\ell k - p^{\\prime } - k^{\\prime } = 0$ .", "(See [62] for a formulation with $S_{fi} = - i e (2 \\pi )^4 \\int \\frac{d \\ell }{2 \\pi } \\,\\delta ^{(4)} (p + \\ell k - p^{\\prime } -k^{\\prime }) \\, {\\cal M} (\\ell )$ .)", "Light-cone coordinates are useful here, e.g.", "$k_- = k^0 - k^3$ , $k_+ = k^0 + k^3$ , $k_\\perp = (k^1, k^2)$ , and $\\underline{k} = (k_+, k_\\perp )$ .", "Imposing gauge invariance yields the matrix element $ {\\cal M} = \\textstyle \\sum _{i=1}^3 J^{(i)} S^{(i)}$ with the pieces of the electron current $ \\hat{J}^{(0)} &=& \\bar{u}_{p^{\\prime }} {\\epsilon ^{\\prime }}^* u_p, \\\\\\hat{J}^{(1, 2)} &=& \\bar{u}_{p^{\\prime }} \\left(d_{p^{\\prime }} {\\epsilon _{\\pm }} k {\\epsilon ^{\\prime }}^* +d_{p} {{\\epsilon ^{\\prime }}}^* k {\\epsilon _{\\pm }} \\right) u_p, \\\\\\hat{J}^{(3)} &=& 4 k \\cdot {\\epsilon ^{\\prime }}^* d_{p^{\\prime }} d_p\\, \\bar{u}_{p^{\\prime }} k u_p, $ which combine to $J^{(1,2)} = \\hat{J}^{(1,2)} + \\frac{\\alpha _{\\pm }}{2 \\ell } \\hat{J}^{(0)}$ , $J^{(3)} = \\hat{J}^{(3)} + \\frac{\\beta }{\\ell } \\hat{J}^{(0)} $ .", "The following abbreviations are used: $d_p = \\frac{m \\xi }{4 k \\cdot p}$ and $d_{p^{\\prime }} = \\frac{m \\xi }{4 k \\cdot p^{\\prime }}$ , $\\alpha _\\pm = m \\xi \\left( \\frac{\\epsilon _\\pm \\cdot p}{k \\cdot p} -\\frac{\\epsilon _\\pm \\cdot p^{\\prime }}{k \\cdot p^{\\prime }}\\right)$ with $\\vec{\\epsilon }_\\pm = (\\vec{a}_x \\pm i \\vec{a}_y) e^2 / m^2 \\xi ^2$ and $\\beta = \\frac{m^2 \\xi ^2}{4} \\left( \\frac{1}{k \\cdot p} -\\frac{1}{k \\cdot p^{\\prime }} \\right)$ as well.", "The phase integrals $S^{(i)}$ are the remainders of the integration $d^4 x = d \\phi \\, dx_- \\, d^2 x_\\perp / k_-$ in Eq.", "(REF ): $S^{(1,2)} &=& \\int _{- \\infty }^\\infty d \\phi \\, f(\\phi ) \\,\\exp \\lbrace i (\\ell \\pm 1) \\phi - i (f_p (\\phi ) - f_{p^{\\prime }} (\\phi ) ) \\rbrace , \\nonumber \\\\S^{(3)} &=& \\int _{- \\infty }^\\infty d \\phi \\, f^2(\\phi )\\exp \\lbrace i \\ell \\phi - i (f_p (\\phi ) - f_{p^{\\prime }} (\\phi ) )\\rbrace \\\\&& \\times \\left[ 1 + \\cos 2 \\phi \\right].", "\\nonumber $ For a few special non-unipolar (plane-wave) fields and their envelopes $f(\\phi )$ , the phase integrals can be processed exactly by analytic means [93], [94], but in general a numerical evaluation is needed.", "The very special IPA case of $f(\\phi ) = 1$ allows for a simple representation of $f_p(\\phi )$ with subsequent decomposition of $S^{(i)}$ into Bessel functions, yielding final expressions as in Eqs.", "(REF , ).", "For identifying the weak-field limit, $\\xi \\rightarrow 0$ , it is necessary to recognize in Eq.", "(REF ) $J^{(0)} \\propto \\xi ^0$ , and $\\alpha _\\pm , \\, J^{(1, 2)} \\propto \\xi ^1$ , and $\\beta , \\, J^{(3)} \\propto \\xi ^2$ .", "Thus, $\\lim _{\\xi \\rightarrow 0} {\\cal M} = {\\cal M}_1 \\xi + {\\cal M}_2 \\xi ^2 + \\cdots $ .", "We emphasize the Fourier transform of the pulse envelope, $\\lim _{\\xi \\rightarrow 0} S^{(1,2)} = \\int _{- \\infty }^\\infty d \\phi \\, f(\\phi )\\exp \\lbrace i (\\ell \\pm 1) \\phi \\rbrace $ , entering ${\\cal M}_1$ , where only $S^{(1)}$ with $\\ell \\ge 0$ contributes to the wanted one-photon emission:The same reasoning applies in REF for one-pair emission, i.e.", "the finite-width Fourier transform of $f(\\phi )$ at $\\Delta \\phi < \\infty $ enables the sub-threshold pair production $e_L(p) \\rightarrow e_L(p^{\\prime }) + e_L(p^{\\prime \\prime }) + \\bar{e}_L(p^{\\prime \\prime \\prime })$ .", "${\\cal M}_1 (\\ell ) =J^{(1)} \\int _{- \\infty }^\\infty d \\phi \\, f(\\phi ) \\exp \\lbrace i (\\ell - 1) \\phi \\rbrace $ , $J^{(1)} (\\ell ) = \\frac{m}{2} \\epsilon _\\mu ^{\\prime }{}^* \\epsilon _{+ \\nu } \\bar{u}_{p^{\\prime }}\\left[\\frac{\\gamma ^\\mu k_\\ell \\gamma ^\\nu + 2 \\gamma ^\\mu p^\\nu }{2 k_\\ell \\cdot p}+\\frac{\\gamma ^\\nu k^{\\prime } \\gamma ^\\mu - 2 \\gamma ^\\nu p^\\mu }{2 k^{\\prime } \\cdot p}\\right] u_p$ with $k_\\ell \\equiv \\ell k$ .", "For pulses with broad support, $f (\\phi ) \\rightarrow 1$ , within the interval $\\Delta \\phi \\gg 1$ , one arrives at the standard Compton (Klein-Nishina) expression in leading order by using $\\xi \\rightarrow 2 e / m$ .", "The appearance of $\\delta (\\ell - 1)$ combined with the definition of $\\ell $ below Eq.", "(REF ) leads to the famous Compton formula via $\\delta \\left( \\omega ^{\\prime } - \\frac{\\omega m}{m + \\omega (1 - \\cos \\Theta ^{\\prime })} \\right)$ in the electron's rest frame by the subsequent phase space integration(s).", "The discussion of the large-$\\xi $ limit needs some care in general, cf.", "[26], [28].", "Useful limits are obtained for the IPA case [58] under the side condition $\\xi ^2 (1 - z^2/n^2) = const$ (cf.", "Eqs.", "(REF , )), e.g.", "$\\vert {\\cal M} \\vert ^2 \\vert _{u \\ll 1}\\propto (\\xi /u)^{2/3} $ and $\\vert {\\cal M}^2 \\vert _{u \\gg 1}\\propto u^{-1} (\\xi / u)^{1/2}\\exp \\lbrace - \\frac{2 u}{3 \\xi } \\frac{m^2}{k \\cdot p}\\rbrace \\rightarrow \\sqrt{\\xi }$ (cf.", "Eqs.", "() and (REF )), At $\\xi \\gg 1$ , the $\\xi $ and $u$ dependencies of resulting probabilities for circular polarization and constant cross field backgrounds coincide.", "These limits are at the heart of the “rise and fall\".", "The differential emission probability per $in$ -electron and per laser pulse follows from (REF ) by partial integration over the $out$ -phase space, $\\frac{d \\mathbb {P} }{d \\omega ^{\\prime } d \\Omega ^{\\prime }} =\\frac{e^2 \\omega ^{\\prime }}{64 \\pi ^3 \\, k \\cdot p \\, k \\cdot p^{\\prime }}\\vert {\\cal M} \\vert ^2 ,$ and may be transformed to other coordinates, e.g.", "$u$ or $\\ell $ etc.", "Having in mind the IPA limit, one should turn to the dimensionless differential rate [62] $\\frac{d \\Gamma _C}{d \\omega ^{\\prime } d \\Omega ^{\\prime }} =\\frac{e^2 \\omega ^{\\prime }}{32 \\pi ^2 \\, q_0 \\, k \\cdot p^{\\prime }}\\vert {\\cal M} \\vert ^2 .$ Spin averaging of the $in$ -electron, and spin summation of the $out$ -electron and summation over the $out$ -photon polarizations leads to $\\overline{\\vert {\\cal M} \\vert ^2}$ , unless one is interested in polarization effects as in [19], [20], [95].", "The above expressions (REF - REF ) can be further processed for special field envelopes, or $\\vert {\\cal M} \\vert ^2$ is numerically accessible, via Eq.", "(REF ), as mod-squared sum of complex number products provided by Eqs.", "(REF - REF ) which need afterwards explicit (numerical) spin and polarization summation/averaging to arrive at $\\overline{\\vert {\\cal M} \\vert ^2}$ .", "The cross section is obtained by normalization on the integrated laser photon flux: $d \\sigma = d \\Gamma _C \\frac{q_0}{k \\cdot p} \\frac{\\omega }{n_L}$ with $n_L = \\frac{m^2}{e^2} \\xi ^2 \\omega N_L$ , where $N_L = 1$ (IPA, quasi-momentum $q_0$ ) or $N_L = \\frac{1}{2 \\pi } \\int _{- \\infty }^\\infty d \\phi \\, f(\\phi )$ (FPA, $q_0 \\equiv p_0$ ).", "The circularly polarized laser background (REF ) is supposed in these relations.", "The authors gratefully acknowledge the collaboration with D. Seipt, T. Nousch, T. Heinzl, and useful discussions with A. Ilderton, K. Krajewska, M. Marklund, C. Müller, S. Rykovanov, and G. Torgrimsson.", "A. Ringwald and B.", "King are thanked for explanations w.r.t.", "LUXE.", "The work is supported by R. Sauerbrey and T. E. Cowan w.r.t.", "the study of fundamental QED processes for HIBEF." ] ]
2105.11758
[ [ "Use of Signed Permutations in Cryptography" ], [ "Abstract In this paper we consider cryptographic applications of the arithmetic on the hyperoctahedral group.", "On an appropriate subgroup of the latter, we particularly propose to construct public key cryptosystems based on the discrete logarithm.", "The fact that the group of signed permutations has rich properties provides fast and easy implementation and makes these systems resistant to attacks like the Pohlig-Hellman algorithm.", "The only negative point is that storing and transmitting permutations need large memory.", "Using together the hyperoctahedral enumeration system and what is called subexceedant functions, we define a one-to-one correspondance between natural numbers and signed permutations with which we label the message units." ], [ "Introduction", "The term cryptography refers to the study of techniques providing information security.", "Several mathematical objects have been used in this field to label the so-called message units.", "These objects are often integers.", "They may also be points or vectors on some curves.", "For this work, we shall use signed permutations.", "Let us denote by: $ [n] $ the set $ \\lbrace 1,\\cdots ,n\\rbrace $ , $ [\\pm n] $ the set $ \\lbrace -n,\\cdots ,-1,1,\\cdots ,n\\rbrace $ , $ \\mathcal {S}_n$ the symmetric group of degree $ n $ .", "Definition 1.1 A bijection $ \\pi :[\\pm n]\\longrightarrow [\\pm n] $ satisfying $ \\pi (-i)=-\\pi (i) $ for all $ i \\in [\\pm n] $ is called \"signed permutation\".", "We can also write a signed permutation $ \\pi $ in the form $\\pi = \\left( \\begin{array}{cccc} 1 & 2 & \\dots & n\\\\\\varepsilon _1 \\sigma _1 & \\varepsilon _2 \\sigma _2 & \\dots & \\varepsilon _n\\sigma _n \\end{array} \\right) \\;\\text{ with } \\sigma \\in \\mathcal {S}_n \\text{ and } \\varepsilon _i \\in \\lbrace \\pm 1\\rbrace \\; .$ Under the ordinary composition of mappings, all signed permutations of the elements of $ [n] $ form a group $ {\\mathcal {B}}_n $ called hyperoctahedral group of rank $ n $ .", "We write $ \\pi ^k=\\underbrace{\\pi \\circ \\cdots \\circ \\pi }_{k\\text{-times}} $ for an integer $ k $ and $ \\pi \\in {\\mathcal {B}}_n $ and when we multiply permutations, the leftmost permutation acts first.", "For example, $\\left( \\begin{array}{crcc} 1 & 2 & 3 & 4\\\\1 & -3 & 4 & 2 \\end{array} \\right)\\circ \\left( \\begin{array}{crcc} 1 & 2 & 3 & 4\\\\3 & -2 & 4 & 1 \\end{array} \\right)=\\left( \\begin{array}{crcr} 1 & 2 & 3 & 4\\\\3 & -4 & 1 & -2 \\end{array} \\right)\\; .$ Before the 1970's, to cipher or decipher a message, two users of cryptographic system must safely exchange information (the private key) which is not known by anyone else.", "New keys could be periodically distributed so as to keep the enemy guessing.", "In 1976, W. Diffie and M. Hellman [1] discovered an entirely different type of cryptosystem for which all of the necessary information to send an enciphered message is publicly available without enabling anyone to read the secret message.", "With this kind of system called public key, it is possible for two parties to initiate secret communications without exchanging any preliminary information or ever having had any prior contact.", "The security of public key cryptosystems is based on the hardness of some mathematical problems.", "As a public key cryptosystem consists of a private key (the deciphering key) that is kept secret and a public key (the enciphering key) which is accessible to the public, then the straightforward way to break the system is to draw the private key from the public key.", "Therefore, the required computation cost is equivalent to solving these difficult mathematical problems.", "For instance for the two particularly important examples of public key cryptosystems : RSA and Diffie-Hellman, both are connected with fundamental questions in number theory which are : difficulty of factoring a large composite integer whose prime factors are not known in advance and intractability of discrete logarithm in the multiplicative group of a large finite field, respectively.", "Definition 1.2 The discrete logarithm problem in the finite group $ G $ to the base $ g\\in G $ is the problem : given $ y\\in G $ , of finding an integer $ x $ such that $ g^x = y $ , provided that such an integer exists (in other words, provided that $ y $ is in the subgroup generated by $ g $ ).", "If we really want our random element $ y $ of $ G $ to have a discrete logarithm, $ g $ must be a generator of $ G $ .", "Example 1 Let $ G=(\\raisebox {0.5ex}{ \\mathbb {Z} }\\slash \\raisebox {-0.5ex}{ 19\\mathbb {Z} })^* $ be the multiplicative group of integers modulo 19.", "The successive powers of 2 reduced$ \\mod {19} $ are : $ 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1 $ .", "Let $ g $ be the generator 2, then the discrete logarithm of 9 to the base 2 is 8.", "In many ways, the group of signed permutations is analogous to the multiplicative group of a finite field.", "For this purpose, cryptosystems based on the latter can be translated to systems using the hyperoctahedral group.", "This work addresses public key cryptosystems based on the group of signed permutations and related to the discrete logarithm problem.", "We shall illustrate this in section by an hyperoctahedral group analogue for the Diffie-Hellman key exchange system then by describing two hyperoctahedral group public key cryptosystems for transmitting information.", "We shall also explain why there is no subexponential time algorithms to break the proposed systems.", "Before introducing the cryptosystems themselves, we give in section a method to define a bijection between integers and signed permutations.", "This time, we first convert the natural number in hyperoctahedral system and then we use the result to compute the corresponding signed permutation by means of subexceedant function." ], [ "Converting from one base to another", "Definition 2.1 Hyperoctahedral number system is a system that expresses all natural number $ n$ of $ \\mathbb {N}$ in the form : $n=\\sum _{i=0}^{k(n)} d_i.", "B_i \\; , \\text{ where }k(n)\\in \\mathbb {N},\\; d_i \\in \\lbrace 0,1,2,\\cdots ,2i+1 \\rbrace \\; \\text{ and }\\; B_i =2^i i!\\; .$ This definition is motivated by the fact that $ B_i $ is the cardinal of the hyperoctahedral group $ {\\mathcal {B}}_i $ .", "To denote the non-negative integer $ n $ of the equation (REF ) in the hyperoctahedral system, we use by convention the $ (k+1) $ -digits representation $ d_k d_{k-1} \\cdots d_2 d_1 d_0\\; .", "$ Theorem 2.2 Every positive integer has an unique representation in the hyperoctahedral system.", "Before giving the proof of this theorem, these are some properties of the hyperoctahedral system.", "Lemma 2.3 If $ n= d_k \\cdots d_1 d_0 $ is a number in hyperoctahedral system, then $ 0\\leqslant n\\leqslant B_{k+1}-1 $ Since $ 0\\leqslant d_i \\leqslant 2i+1 $ , we have $ 0\\leqslant \\sum _{i=0}^{k} d_i B_i \\leqslant \\sum _{i=0}^{k} (2i+1) B_i\\, .$ Recall that $ B_i =2^i i!", "$ , $(2i+1) B_i &=&(2(i+1-1)+1)B_i\\\\&=&(2(i+1)-1)B_i\\\\&=& 2(i+1)B_i-B_i\\\\&=& B_{i+1}-B_i\\, .$ Therefore, $ \\sum _{i=0}^{k} (2i+1) B_i =\\sum _{i=0}^{k} (B_{i+1}-B_i) =B_{k+1}-B_0=B_{k+1}-1 \\; .$ Lemma 2.4 Let $ n= d_k \\cdots d_1 d_0 $ be a number in hyperoctahedral system, then $ d_k B_k\\leqslant n < (d_k +1)B_k \\; .$ Let $ m= d_{k-1} \\cdots d_1 d_0 $ .", "From lemma REF , $ 0\\leqslant m < B_k $ .", "We also have $ n=d_k B_k +m $ , hence $ d_k B_k\\leqslant n < B_k +d_k B_k= (1+d_k )B_k \\; .", "$ Now, we can proceed to the demonstration of theorem REF which states the one-to-oneness between non negative integers and hyperoctahedral base numbers.", "Let $ a_n \\cdots a_ 1 a_0 $ with $ a_n \\ne 0 $ and $ b_m \\cdots b_1 b_0 , b_m \\ne 0 $ be two representations of a positive integer $ N $ in hyperoctahedral system.", "First, $ b_m \\ge 1 $ and $ a_n \\ge 1 $ imply $ B_m\\le b_mB_m\\le b_m \\cdots b_1 b_0\\;\\text{ and }B_n\\le a_n \\cdots a_ 1 a_0 .", "$ The relation $ n=m $ is immediate because if $ n<m $ then $B_m\\ge B_{n+1}>a_n \\cdots a_ 1 a_0 $ by lemma REF so $b_m \\cdots b_1 b_0>a_n \\cdots a_ 1 a_0 \\; .$ Similarly, if $ m<n $ then $a_n \\cdots a_ 1 a_0 > b_m \\cdots b_1 b_0\\; .$ We get a contradiction.", "Consequently $ n=m $ and $ a_i=b_i $ for all $ i\\in \\lbrace 0,1,\\ldots ,n\\rbrace $ by induction and by the unicity of the expression $ N=a_n B_n+r_n $ with $ r_n=a_{n-1} \\cdots a_ 1 a_0<B_n $ .", "To express a positive integer $n$ in the hyperoctahedral system, one proceeds with the following manner.", "Start by dividing $n$ by 2 and let $d_0$ be the rest $r_0 $ of the expression $n=r_0+(2)q_0 \\; .$ Divide $q_0$ by 4, and let $d_1$ be the rest $r_1 $ of the expression $q_0=r_1+(4)q_1 \\; .$ Continue the procedure by dividing $q_{i-1}$ by $2(i+1) $ and taking $d_i:=r_i$ of the expression $q_{i-1}=r_i+2(i +1)q_i \\; $ until $ q_l =0 $ for some $ l\\in \\mathbb {N}$ .", "In this way, we obtain $ n= d_l:d_{l-1}: \\cdots :d_1 :d_0 $ and we also have $n= d_0 + 2 \\left(d_1 + 4 \\cdot (d_2 + 2(3) \\cdot (d_3 + \\cdots ))\\right) .$ Now let $ n= d_{k-1}:d_{k-2}: \\cdots :d_1 :d_0 $ be a number in hyperoctahedral system.", "By definition REF , one way to convert $ n $ to the usual decimal system is to calculate $ d_{k-1} 2^{k-1}(k-1)!", "+\\cdots +d_1 .2+d_0 \\; .", "$ In practice, one can use this algorithm : Table: NO_CAPTION" ], [ "A bijection between the subexceedant functions and permutations", "Definition 2.5 A subexceedant function $ f $ on $ [n] $ is a map $ f : [n]\\longrightarrow [n] $ such that $ 1 \\leqslant f (i) \\leqslant i \\text{ for all } i\\in [n]\\; .$ We will denote by $ {\\mathcal {F}}_n $ the set of all subexceedant functions on $ [n] $ , and we will represent a subexceedant function $ f $ over $ [n] $ by the word $ f (1) f (2) \\cdots f (n) $ .", "Example 2 These are the sets $ {\\mathcal {F}}_n $ for $ n=1, 2, 3 $ : $&&{\\mathcal {F}}_1=\\lbrace 1\\rbrace \\\\&&{\\mathcal {F}}_2=\\lbrace 11,12\\rbrace \\\\&&{\\mathcal {F}}_3=\\lbrace 111,112,113,121,122,123\\rbrace .$ It is easy to verify that $ card\\; {\\mathcal {F}}_n = n!", "$ , since from each subexceedant $ f $ over $ [n-1] $ , one can obtain $ n $ distinct subexceedant functions over $ [n] $ by adding any integer $ i \\in [n] $ at the end of the word representing $ f $ .", "We will give a bijection between $ {\\mathcal {S}}_n $ and $ {\\mathcal {F}}_n $ .", "Let be the map $\\begin{array}[pos]{rcl}\\phi : {\\mathcal {F}}_n&\\longrightarrow &{\\mathcal {S}}_n\\\\f& \\longmapsto & \\sigma _f = (1\\, f(1))(2\\, f(2))\\cdots (n\\, f(n))\\end{array}.$ Notice that there is an abuse of notation in the definition of $ \\sigma _f $ .", "Indeed, if $ f (i) = i $ , then the cycle $ (i\\, f(i)) = (i) $ does not really denote a transposition but simply the identity permutation.", "Lemma 2.6 The map $ \\phi $ is a bijection from $ {\\mathcal {F}}_n $ onto $ {\\mathcal {S}}_n $ .", "Since $ {\\mathcal {S}}_n $ and $ {\\mathcal {F}}_n $ both have cardinality $ n!", "$ , it suffices to prove that $ f $ is injective.", "Let $ f $ and $ g $ be two subexceedant functions on $ [n] $ .", "Assume that $ \\phi ( f ) = \\phi (g) $ i.e.", "$ \\sigma _f=\\sigma _g $ .", "So we have : $(1\\, f(1))(2\\, f(2)) \\cdots (n\\, f(n)) = (1\\; g(1))(2\\; g(2)) \\cdots (n\\; g(n)) .$ Since $ \\sigma _f=\\sigma _g $ , then in particular $ \\sigma _f(n)=\\sigma _g(n) $ .", "By definition $ \\sigma _f (n)= f (n) $ and $ \\sigma _g(n)=g(n) $ , so $ f (n) = g(n) $ .", "Let us multiply both members of equation (REF ) on the right by the permutation $ (n\\; f(n))=(n\\; g(n)) $ , we obtain : $(1\\, f(1))(2\\, f(2)) \\cdots (n-1\\hspace{14.22636pt} f(n-1)) = (1\\; g(1))(2\\; g(2)) \\cdots (n-1\\hspace{14.22636pt} g(n-1)) .$ Now, if we apply the same process to this equation, we obtain $ f (n-1) = g(n-1) $ .", "By iterating, we conclude that $ f (i) = g(i) $ for all integers $ i\\in [n] $ and then $ f = g $ .", "Let $ \\sigma $ be a permutation of the symmetric group $ {\\mathcal {S}}_n $ and $ f $ be the inverse image of $ \\sigma $ by $ \\phi $ .", "Then $ f $ can be constructed as below : Set $ f (n) = \\sigma (n) $ .", "Multiply $ \\sigma $ on the right by $ (n\\; \\sigma (n)) $ (this operation consists in exchanging the image of $ n $ and the image of $ \\sigma ^{-1}(n) $ ), we obtain a new permutation $ \\sigma _1 $ having $ n $ as a fixed point.", "Thus $ \\sigma _1 $ can be considered as a permutation of $ {\\mathcal {S}}_{n-1} $ .", "Then set $ f (n-1) = \\sigma _1(n-1) $ .", "In order to obtain $ f (n-2) $ , apply now the same process to the permutation $ \\sigma _1 $ by multiplying $ \\sigma _1 $ by $ (n-1\\hspace{14.22636pt} \\sigma _1(n-1)) $ .", "Iteration determines $ f (i) $ for all integers $ i $ of $ [n] $ ." ], [ "Mapping hyperoctahedral base numbers to signed permutations", "Let $ d_{n-1}\\; d_{n-2}\\; \\cdots \\; d_1\\; d_0 $ be a $ n $ -digits number in hyperoctahedral system.", "That is $ d_i \\in \\lbrace 0,1,2,\\cdots ,2i+1 \\rbrace $ for $ i=0, \\cdots ,n-1 $ .", "Writing $ d_i =2q_i+r_i $ where $ r_i\\in \\lbrace 0,1\\rbrace $ and $ q_i\\in \\lbrace 0,\\ldots ,i\\rbrace $ , gives the subexceedant function $ f=f(1)\\cdots f(n) $ with $ f(i)=1+q_{i-1},\\; i=1,\\ldots ,n $ and the sequence $ (\\varepsilon _1,\\ldots ,\\varepsilon _n) $ with $ \\varepsilon _i=(-1)^{r_{i-1}} $ for $ i=1,\\ldots ,n $ .", "So to each integer $ d_{n-1}\\; d_{n-2}\\; \\cdots \\; d_1\\; d_0 $ , we associate the signed permutation $\\left( \\begin{array}{cccc} 1 & 2 & \\dots & n\\\\\\varepsilon _1 \\sigma _1 & \\varepsilon _2 \\sigma _2 & \\dots & \\varepsilon _n\\sigma _n \\end{array} \\right)$ where $ \\sigma = \\sigma _1 \\cdots \\sigma _n $ is the permutation associated to the subexceedant function $ f $ by the map $\\begin{array}[pos]{rcl}\\phi : {\\mathcal {F}}_n&\\longrightarrow &{\\mathcal {S}}_n\\\\f& \\longmapsto & \\sigma _f = (1 f (1))(2 f (2))\\cdots (n f (n))\\end{array}.$ Now, let $ \\pi =\\left( \\begin{array}{cccc} 1 & 2 & \\dots & n\\\\\\varepsilon _1 \\sigma _1 & \\varepsilon _2 \\sigma _2 & \\dots & \\varepsilon _n\\sigma _n \\end{array} \\right)$ be a signed permutation.", "As $ \\phi $ is a bijection, from $ \\pi $ we have $f=f(1)\\cdots f(n)=\\phi ^{-1}(\\sigma )\\text{ and }r_{i-1}={\\left\\lbrace \\begin{array}{ll}0& \\text{ if }\\varepsilon _i=1\\\\1& \\text{ if }\\varepsilon _i=-1\\end{array}\\right.", "}\\text{ for }i=1, \\ldots ,n.$ The digits $ d_i=2(f(i+1)-1)+r_i,\\; i=0,\\ldots ,n-1 $ form the hyperoctahedral number $ d_{n-1}\\; d_{n-2}\\; \\cdots \\; d_1\\; d_0 $ .", "It is easy to verify that $ d_i\\in \\lbrace 0,\\ldots ,2i+1\\rbrace $ .", "We have $& 1\\leqslant f(i)\\leqslant i &\\text{ for }i=1,\\ldots ,n\\\\& 0\\leqslant f(i+1)\\leqslant i+1 &\\text{ for }i=0,\\ldots ,n-1\\\\& 0\\leqslant 2(f(i+1)-1)\\leqslant 2i &\\text{ for }i=0,\\ldots ,n-1\\\\& 0\\leqslant d_i\\leqslant 2i+1&\\text{ because }0\\leqslant r_i\\leqslant 1.$" ], [ "Some cryptosystems based on hyperoctahedral group", "We suppose that we are using message units with signed permutations as equivalents (according to section ) in some publicly known large hyperoctahedral group $ {\\mathcal {B}}_n $ ." ], [ "Analog of the Diffie-Helman key exchange", "The Diffie-Hellman key exchange [1] which was the first public key cryptosystem originally used the multiplicative group of a finite field.", "It can be adapted for signed permutations group as follow.", "Suppose that two users Alice and Bob want to agree upon a secret key, a random element of $ {\\mathcal {B}}_n $ , which they will use to encrypt their subsequent messages to one another.", "They first choose a large integer $ n $ for the hyperoctahedral group $ {\\mathcal {B}}_n $ and select a signed permutation $ \\beta \\in {\\mathcal {B}}_n $ to serve as their \"base\" and make them public.", "Alice selects a random integer $ 0<a<B_n $ , which she keeps secret, and computes $ \\beta ^a\\in {\\mathcal {B}}_n $ which she makes public.", "Bob does the same.", "He selects a random integer $ 0<b<B_n $ , and transmits $ \\beta ^b $ to Alice over a public channel.", "Alice and Bob agree on the secret key $ \\beta ^{ab} $ by computing $ (\\beta ^b)^a $ and $ (\\beta ^a)^b $ respectively." ], [ "Analog of the ElGamal system", "Suppose user \"A\" requires sending a message $ \\mu $ to user \"B\".", "As in the key exchange system above, we start with a fixed publicly known hyperoctahedral group $ {\\mathcal {B}}_n $ for a large integer $ n $ and a signed permutation $ \\beta \\in {\\mathcal {B}}_n $ (preferably, but not necessarily, a generator) as base.", "Each user selects a random integer $ 0< u<|\\beta | $ (instead of $ u $ we take $ a $ for user \"A\" and $ b $ for user \"B\") which is kept secret, and computes $ \\beta ^u $ that he publishes as a public key.", "With the public key $ \\beta ^b $ , user \"A\" encrypt the message $ \\mu $ and sends the pair of signed permutations $ (m_1=\\beta ^a,m_2=\\mu .", "(\\beta ^{b})^a) $ to user \"B\".", "To recover $ \\mu $ , B computes $ (m_1^b)^{-1}=((\\beta ^a)^b)^{-1} $ and multiplies $ m_2 $ on the right by the result." ], [ "Analog of the Massey-Omura system for message transmission.", "We assume the same setup as in the previous subsection.", "Suppose that Alice wants to send Bob a message $ \\mu $ .", "She chooses a random integer $ c $ satisfying $ 0 < c < B_n $ and $ g.c.d.", "(c,B_n) = 1 $ , and transmits $ \\mu ^c $ to Bob.", "Next, Bob chooses a random integer $ d $ with the same properties, and transmits $ (\\mu ^c)^d $ to Alice.", "Then Alice transmits $ (\\mu ^{cd})^{c^{\\prime }}=\\mu ^d $ back to Bob, where $ c^{\\prime }c\\equiv 1\\mod {B_n} $ .", "Finally, Bob computes $ (\\mu ^d)^{d^{\\prime }} $ where $ d^{\\prime }d\\equiv 1\\mod {B_n} $ ." ], [ "Security", "For the Diffie-Hellman's key exchange above, a third party knows only the elements $ \\beta \\ ,\\beta ^a $ and $ \\beta ^b $ of $ {\\mathcal {B}}_n $ which are public knowledge.", "Obtaining $ \\beta ^{ab} $ knowing only $ \\beta ^a $ and $ \\beta ^b $ is as hard as taking the discrete logarithm $ a $ from $ \\beta $ and $ \\beta ^a $ (or $ b $ knowing $ \\beta $ and $ \\beta ^b $ ).", "Then an unauthorized third party must solve the discrete logarithm problem to the base $ \\beta \\in {\\mathcal {B}}_n $ to determine the key.", "Both ElGamal's [5] and Massey-Omura's [4] cryptosystems are essentially variants of Diffie-Hellman's key exchange system.", "Therefore, breaking either of the systems above requires the solution of the discrete logarithm problem for hyperoctahedral group.", "Definition 3.1 The hyperoctahedral discrete logarithm problem to the base $ \\beta \\in {\\mathcal {B}}_n $ is the problem, given $ \\pi \\in {\\mathcal {B}}_n $ , of finding an integer $ x $ such that $ \\pi = \\beta ^x $ if such $ x $ exists.", "Many improvements are made for solving the discrete logarithm problem in finite group.", "In hyperoctahedral group based cryptosystems of the sort discussed above, one does not work with the entire group $ {\\mathcal {B}}_n $ , but rather with cyclic subgroups : the group $ <\\beta > $ in the Diffie-Hellman system and ElGamal system and the group $ < \\mu > $ in the Massey-Omura system where we designate by $ <\\pi >=\\lbrace \\pi ^i\\, |\\, i=1,\\ldots ,|\\pi |\\rbrace $ the cyclic subgroup of $ {\\mathcal {B}}_n $ generated by $ \\pi \\in {\\mathcal {B}}_n $ , $ |\\pi | $ the order of the signed permutation $ \\pi $ .", "In other words, one works in the subgroup generated by the base of the discrete logarithm in each proposed system." ], [ "Choice of a suitable base", "The order of the cyclic subgroup of $ {\\mathcal {B}}_n $ has an important role to avoid an easy solution of the discrete logarithm.", "Recall that we can also represent a signed permutation $ \\pi $ in the form $\\pi = \\left( \\begin{array}{cccc} 1 & 2 & \\dots & n\\\\\\varepsilon _1 \\sigma (1) & \\varepsilon _2 \\sigma (2) & \\dots & \\varepsilon _n\\sigma (n) \\end{array} \\right) \\;\\text{ with } \\sigma \\in \\mathcal {S}_n \\text{ and } \\varepsilon _i \\in \\lbrace \\pm 1\\rbrace \\; .$ As stated in the work of Victor Reiner [2], a signed permutation decomposes uniquely into a product of commuting cycles just as permutations do.", "Example 3 The disjoint cycle form of $ \\left( \\begin{array}{ccrcrrc} 1 & 2 & 3& 4& 5& 6 & 7\\\\3 & 6 & -2& 7& -5& -1 & 4 \\end{array} \\right) $ is $ \\left( \\begin{array}{crcr} 1 & 3& 2& 6\\\\3 & -2& 6& -1 \\end{array} \\right)\\left( \\begin{array}{cc} 4 & 7\\\\7 & 4 \\end{array} \\right)\\left( \\begin{array}{c} 5\\\\-5 \\end{array} \\right)\\, .", "$ Definition 3.2 The order of a signed permutation $ \\pi $ is the smallest positive integer $ m $ such that $ \\pi ^m=\\iota $ where $ \\iota $ denotes the identity permutation.", "An $ \\ell $ -cycle $ C= \\left( \\begin{array}{ccccc} i_1 & i_2 & \\dots & i_{\\ell -1} & i_\\ell \\\\\\varepsilon _1 i_2 & \\varepsilon _2 i_3 & \\dots & \\varepsilon _{\\ell -1} i_\\ell & \\varepsilon _\\ell i_1 \\end{array} \\right) $ where $ \\varepsilon _i \\in \\lbrace \\pm 1\\rbrace $ has order $ \\ell $ .", "The proof of the following theorem can be found in [6].", "Theorem 3.3 The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of the cycles.", "Let us now assume that we have $ \\beta \\in {\\mathcal {B}}_n $ as base of the discrete logarithm.", "The order of $ <\\beta > $ can be very large for a large value of $ n $ , since the disjoint cycles of $ \\beta $ can be selected in such a way that their least common multiple be very large.", "Therefore brute force search is inefficient to solve the discrete logarithm problem to the base $ \\beta \\in {\\mathcal {B}}_n $ .", "$ <\\beta > $ is a cyclic group of finite order so it is commutative.", "Thus, in order to avoid an easy solution to the discrete logarithm problem using the techniques that apply to any finite abelian group (which take approximately $ \\sqrt{p} $ operations, where $ p $ is the largest prime dividing the order of the group), it is important for the order of the commutative group $ <\\beta > $ to be non smooth, that is, divisible by a large prime.", "Definition 3.4 Let $ n $ be a positive real number.", "we say that $ n $ is smooth if all of the prime factors of $ n $ are small.", "The method of Pohlig-Hellman [3] can efficiently computes the discrete logarithm in a group $ G $ if its order is $ B $ -smooth for a reasonably small $ B $ .", "Definition 3.5 Let $ B $ be a positive real number.", "An integer is said to be $ B $ -smooth if it is not divisible by any prime greater than $ B $ .", "The following theorem allows to generate the base $ \\beta $ so that the order of the subgroup $ <\\beta > $ of $ {\\mathcal {B}}_n $ have arbitrary smoothness.", "Theorem 3.6 Let $ p $ be a prime.", "For a large integer $ n $ , a cyclic subgroup $ G $ of $ {\\mathcal {B}}_n $ which its order is $ p $ -smooth and is not $ (p-1) $ -smooth can be constructed.", "Let $ n $ be a large integer and $ p\\le n $ a prime.", "There exists a $ p $ -cycle $ \\gamma _p \\in {\\mathcal {B}}_n $ .", "Let $ \\gamma _p , C_1 , C_2 , \\ldots , C_k\\in {\\mathcal {B}}_n $ be disjoint cycles of length respectively $ p , l_1 , l_2 , \\ldots , l_k $ with $ 1\\le l_i\\le p \\text{ for } i=1,\\ldots ,k \\text{ and } \\sum _{i=1}^{k}l_i\\le n-p\\; .", "$ Let us now consider the signed permutation $ \\pi =\\gamma _p C_1 C_2 \\ldots C_k\\in {\\mathcal {B}}_n $ of order $ |\\pi |=lcm(p , l_1 , l_2 , \\ldots , l_k)\\; .", "$ Let $ q $ be a prime such that $ q|lmc(p , l_1 , l_2 , \\ldots , l_k) $ .", "We show that $ q\\le p $ .", "Let us suppose that $ q>p $ .", "We obtain $ l_i<p<q $ which implies $ q\\nmid |\\pi | $ so $ q\\le p $ .", "We have just shown that every prime $ q $ dividing the order of $ \\pi $ is smaller than $ p $ , that is, $ |\\pi | $ is $ p $ -smooth.", "The order $ |<\\pi >|=|\\pi | $ of the cyclic subgroup $ <\\pi > $ of $ {\\mathcal {B}}_n $ generated by $ \\pi $ is $ p $ -smooth but it is not $ (p-1) $ -smooth because $ p\\mid |<\\pi >| $ and $ p>p-1 $ .", "This theorem provides high flexibility in selecting a subgroup of $ {\\mathcal {B}}_n $ on which cryptosystem resists to attacks by Silver-Pohlig-Hellman's algorithm.", "The cryptosystems that we have proposed are easy to implement by applying optimized method for exponentiation.", "Moreover, the multiplication on $ {\\mathcal {B}}_n $ which is the composition of mappings can be performed in time $ \\mathcal {O}(n) $ .", "However, as we must work in a very large hyperoctahedral group, the need of large memory from the point of view implementation requires improvements." ] ]
1612.05605
[ [ "Exact solutions for the denoising problem of piecewise constant images\n in dimension one" ], [ "Abstract In this paper we propose a method to determine explicitly the solution of the total variation denoising problem with an $L^p$ fidelity term, where $p>1$, for piecewise constant initial data in dimension one." ], [ "Introduction", "When an image is acquired it comes, unavoidably, with some distortion.", "Indeed, external conditions, other then defects or limitations of the instruments that are used to obtain them, affect the quality of the acquired data.", "Thus, in order to be able to perform any task on the image, it is important to be able to recover the clean version in the best possible way, i.e., with optimal fidelity.", "If we denote it by $u$ and the acquired, corrupted image by $f$ , it is usually assumed that the two are related via: $f=Au+n\\,,$ where $A$ is a bounded linear operator representing the blurring effect and $n$ is the implementation of the random noise.", "One of the aims of image reconstruction is deblurring and denoising $f$ in order to recover $u$ (see [8], [22]).", "Here we are interested in the denoising problem, i.e., when the operator $A$ is the identity and we have to remove the noise.", "Problem (REF ) is, in general, ill-posed (in the sense of Hadamard) and thus we need to regularize it (see [1], [44]).", "A widely used variational technique for this purpose was introduced by Rudin, Osher and Fatemi in [42], where they proposed to recover $u$ in an open set $\\Omega \\subset \\mathbb {R}^N$ via the minimization problem $\\min _{u\\in BV(\\Omega ),\\,\\,\\Vert u-f\\Vert ^2_{L^2}=\\sigma ^2}\\, |Du|(\\Omega )\\,,$ for some fixed $\\sigma >0$ , where $f$ is assumede to be in $L^2(\\Omega )$ and $|Du|(\\Omega )$ denotes the total variation of the function $u$ in $\\Omega $ .", "The choice of $BV(\\Omega )$ as the functional space where to perform the minimization is motivated by the fact that it allows for the presence of discontinuities in the solutions representing the sharp edges of the objects in the image and the so called staircase effect due to the Cantor part of the derivative that takes case of the fine texture.", "There are some interesting cases though, where the real image is represented by a function of bounded variation (see [30])).", "The minimization problem (REF ) has been shown to be equivalent to the following penalized minimization problem (known as the total variation denoising model with $L^2$ fidelity term) $\\min _{u\\in BV(\\Omega )} |Du|(\\Omega ) + \\lambda \\Vert u-f\\Vert ^2_{L^2(\\Omega )}\\,,$ for some Lagrange multiplier $\\lambda >0$ (see [17]).", "Today's literature on the study of problem (REF ) is extensive, and here we limit ourselves to recall that properties of the solutions have been studied, for instance, in [2], [3], [4], [9], [10], [14], [15], [20], [23], [26], [28], [31], [35], [41], [45], [46], the analysis of variants of (REF ) that use the generalized total variation have been performed in [11], [12], [38], [40], [39], anisotropic models are undertaken in [24], [27], [29], [34], while the effects of considering high-order models have been investigated in [19], [21], [25], [32], [39].", "Finally, other variants of (REF ) have been addressed in [6], [7], [37], and algorithmic considerations may be found in [13], [16], [18], [36].", "In this paper we study the one dimensional case where $f$ is a piecewise constant function, and we generalize the $L^2$ fidelity term to an $L^p$ fidelity term, with $p\\in [1,\\infty )$ .", "To be precise, we consider the minimization problem $\\min _{u\\in BV(\\Omega )}\\mathcal {G}(u)\\,,$ where $\\Omega :=(a,b)\\subset \\mathbb {R}$ and $\\mathcal {G}(u):=|Du|(\\Omega )+\\lambda \\Vert u-f\\Vert ^p_{L^p(\\Omega )}\\,,$ for a given initial piecewise constant data $f$ .", "Our aim is to provide a method for solving the minimization problem (REF ) in the case $p>1$ .", "We next explain the main idea behind the strategy we propose.", "The rigid structure of the initial data forces the solution to be piecewise constant itself, with jump set contained in the one of $f$ (see Corollary REF ).", "Moreover, a simple truncation argument shows that the solution takes values within the minimum and the maximum of $f$ .", "Hence, the minimization problem (REF ) with $f$ of the form $f(x)=\\sum _{i=1}^k f_i\\,\\chi _{(x_{i-1},x_i)}(x)\\,,\\quad \\quad f_i\\in \\mathbb {R}\\,,$ is equivalent to the following minimization problem $\\min _{v\\in Q}G(v)\\,,$ where $Q:=[\\min f, \\max f]^k$ and $G:\\mathbb {R}^k\\rightarrow \\mathbb {R}$ is the function defined as $G(v):=\\sum _{i=2}^k |v_i-v_{i-1}| + \\lambda \\sum _{i=1}^k L_i|f_i-v_i|^p\\,,$ with $v=(v_1,\\dots ,v_k)$ and $L_i:=x_i-x_{i-1}$ .", "The function $G$ is convex but it lacks differentiability on the hyperplanes where $\\lbrace v_{i-1}=v_i\\rbrace $ .", "Thus, in principle, one should minimize the function $G$ over several compact regions and then compare all the minimum values in order to find the global minimizer.", "Our method aims at overcoming this difficulty.", "We will be able, for each $\\lambda $ , to predict a priori - that is without knowing explicitly $u^\\lambda $ (the minimizer of $G$ corresponding to the parameter $\\lambda $ ) - what the relative position of each $u^\\lambda _i$ with respect to $u^\\lambda _{i-1}$ and $f_i$ will be.", "Knowing that, it is possible to look for the minimizer $u^\\lambda $ only in a specific region of $\\mathbb {R}^k$ , where the absolute values present in the expression of $G$ can be written explicitly.", "Hence, $u^\\lambda $ can be found by solving the appropriate Euler-Lagrange equation.", "We give a more detailed description of our method: the function $\\lambda \\mapsto u^\\lambda $ is continuous and $u^\\lambda \\rightarrow f$ as $\\lambda \\rightarrow \\infty $ (see Lemma REF ).", "Hence, for $\\lambda \\gg 1$ , we have that $u^\\lambda _i$ is very close to $f_i$ , and this allows us to tell the relative position of $u^\\lambda _i$ with respect to $u^\\lambda _{i-1}$ .", "Moreover, thanks to the qualitative properties of the solutions we will prove in Lemma REF and Proposition REF , we will also be able to tell the relative position of each $u_i$ with respect to $f_i$ .", "These information allow us to write explicitly the absolute values present in the expression of $G$ , as well as to write explicitly the Euler-Lagrange equation, whose solution will give us the minimizer $u^\\lambda $ .", "With this reasoning, we find the mininimizers for $\\lambda $ large (how large it has to be will be determined a posteriori).", "The idea now is to let $\\lambda $ decrease.", "Since $u^\\lambda $ is constant for small values of $\\lambda $ (see Lemma REF ), by continuity of $\\lambda \\mapsto u^\\lambda $ eventually two neighboring values $u^\\lambda _i$ and $u^\\lambda _{i-1}$ will happen to be the same.", "The main technical result (Theorem REF ) tells us that the same will be true for all smaller values of $\\lambda $ .", "As a result we now have to consider the function $G$ restricted to the subspace $\\lbrace v_{i-1}=v_i\\rbrace $ , thus reducing the number of variables.", "By continuity of $\\lambda \\mapsto u^\\lambda $ , it is then possible to predict the relative position of every $u_i$ with respect to $u_{i-1}$ , while the qualitative properties of the solutions will give us the relative position of $u_i$ with respect to $f_i$ .", "As a consequence, also in this case, we are able to write explicitly the Euler-Lagrange equation.", "We observe that price to pay for applying this method is that, in order to determine the solution of the minimization problem (REF ) for a certain value $\\bar{\\lambda }$ , we first need to know it for all $\\lambda >\\bar{\\lambda }$ .", "This, in the end, boils down to solve some equations, whose number can be roughly bounded above by $k(k+1)/2$ .", "Our result is related to the work of Strong and Chan (see [43]), where the authors consider the minimization problem (REF ) in the special case $p=2$ , but allowing the initial data $f$ to be a piecewise constant function with noise.", "Under certain conditions on the amplitude of the noise, they are able to determine the solution of the minimization problem (REF ) in the case $\\lambda \\gg 1$ .", "Just a couple of words about the case $p=1$ .", "The reason why the strategy described above fails for $p=1$ is because we cannot use the continuity of the map $\\lambda \\mapsto u^\\lambda $ .", "Indeed, even if for $p=1$ there is no uniqueness for the solution of the minimization problem (REF ) (see an example in Proposition REF ), there is always a solution taking only the values that $f$ takes (see Corollary REF ).", "But this jumping behavior of the solution prevents us to use continuity arguments, which are at the core of the strategy sketched above.", "Nonetheless, the possibility of obtaining an analytic method for computing the solution in the case $p=1$ is currently under investigation.", "The paper is organized as follows.", "After a brief recalling of the main properties of one dimensional functions of bounded variation in Section 2, we devote Section 3 to stating and proving basic results we will need in the sequel concerning the solutions of our minimization problem.", "In Section 4 we illustrate with a simple case the different behaviors of the solution in the cases $p=1$ and $p>1$ .", "Section 5 contains the main technical results needed to justify the strategy to determine the solution of the minimization problem (REF ) we describe in Section 6 we conclude with an explicit example." ], [ "Preliminaries", "In this section we review basic definitions of one dimensional functions of bounded variation.", "For more details, see [5], [33].", "Here $a,b\\in \\mathbb {R}$ with $a<b$ .", "Definition 2.1 Let $u:(a,b)\\rightarrow \\mathbb {R}$ .", "The pointwise variation of $u$ in $(a,b)$ is defined as $pV(u;a,b):=\\sup \\left\\lbrace \\, \\sum _{i=1}^{n-1}|u(x_{i+1})-u(x_i)| \\,:\\, a<x_1<\\dots <x_n<b \\,\\right\\rbrace \\,.$ Definition 2.2 For $u\\in L^1\\bigl ((a,b)\\bigr )$ its total variation in $(a,b)$ is given by $|Du|\\bigl ((a,b)\\bigr ):=\\sup \\left\\lbrace \\, \\int _a^b \\varphi ^{\\prime }u \\,\\mathrm {d}x \\,:\\, \\varphi \\in C^\\infty _0\\bigl ((a,b)\\bigr )\\,, |\\varphi |\\le 1 \\,\\right\\rbrace \\,.$ If $|Du|\\bigl ((a,b)\\bigr )<\\infty $ , we say that $u$ belongs to the space $BV\\bigl ((a,b)\\bigr )$ of functions of bounded variation in $(a,b)$ .", "In this case, $Du$ is a finite Radon measure on $(a,b)$ .", "Definition 2.3 Let $u\\in BV\\bigl ((a,b)\\bigr )$ .", "We define the jump set of $u$ as $J_u:=\\bigl \\lbrace \\, x\\in (a,b) \\,:\\, |Du|(\\lbrace x\\rbrace )\\ne 0 \\,\\bigr \\rbrace \\,.$ The relation among the total and the pointwise variation is given by the following result.", "Theorem 2.4 Let $u\\in L^1\\bigl ((a,b)\\bigr )$ and define the essential variation of $u$ as $eV(u;a,b):=\\inf \\lbrace \\, pV(v;a,b) \\,:\\, v=u\\quad L^1-a.e.", "\\text{ in } (a,b) \\,\\rbrace \\,.$ The infimum defining $eV(u;a,b)$ in (REF ) is achieved and it coincides with $|Du|\\bigl ((a,b)\\bigr )$ .", "Theorem (REF ) allows us to single out some well behaving representative of a BV function.", "Definition 2.5 Let $u\\in BV\\bigl ((a,b)\\bigr )$ .", "Any $v$ with $v=u$ $L^1$ -a.e.", "in $(a,b)$ such that $pV(v;a,b)=eV(u;a,b)=|Du|\\bigl ((a,b)\\bigr )$ is called a good representative of $u$ ." ], [ "The general structure of the solutions", "This section is devoted to stating and proving some basic results we need concerning the solution of the minimization problem (REF ).", "Albeit some of these properties may be known, we present here the proofs for the reader's convenience.", "We start by proving that a solution to the minimization problem (REF ) with a piecewise constant initial data $f$ needs to have the same structure as $f$ , i.e., it has to be a piecewise constant function with its jump set contained in the jump set of $f$ .", "In higher dimension, the inclusion $J_u\\subset J_f$ is well known (see [14] and [45]) in the case $p>1$ , while it is not always true if $p=1$ (see [20] and [28]).", "The following result has been proved, with a different argument, in [12].", "Theorem 3.1 Let $f\\in L^1\\bigl ((a,b)\\bigr )$ and let $u\\in BV\\bigl ((a,b)\\bigr )$ be a solution of (REF ).", "If $f$ is constant in $(c,d)\\subset (a,b)$ , then $u$ is constant in $(c,d)$ .", "Let $u\\in BV\\bigl ((a,b)\\bigr )$ and suppose it is a good representative such that $u(c)=\\lim _{y\\rightarrow c^-}u(y)\\,,\\quad \\quad u(d)=\\lim _{y\\rightarrow d^+}u(y)\\,.$ Define the function $\\widetilde{u}:=\\left\\lbrace \\begin{array}{ll}u & \\text{in } (a,b)\\!\\setminus \\!", "(c,d)\\,,\\\\t & \\text{in } (c,d)\\,,\\end{array}\\right.$ where $t:=_c^d u$ .", "We claim that $\\widetilde{u})\\le u)\\,,$ where equality holds if and only if $u\\equiv t$ in $(c,d)$ .", "We show that the above inequality holds separately for each term of the energy.", "The fact that the fidelity term decreases is due to Jensen's inequality.", "Indeed, recalling that $f$ is constant on $(c,d)$ , say $f\\equiv \\bar{f}$ in $(c,d)$ , we have that $\\Bigl |\\, _c^d u(y)\\;\\mathrm {d}y- \\bar{f} \\,\\Bigr |^p= \\Bigl |\\, _c^d \\bigl (u(y) - f\\bigr ) \\;\\mathrm {d}y\\,\\Bigr |^p\\le _c^d |u(y)-\\bar{f}|^p\\;\\mathrm {d}y\\,,$ and, integrating both sides on $(c,d)$ , we obtain $\\int _c^d|t-\\bar{f}|^p\\;\\mathrm {d}x\\le \\int _c^d|u(x)-\\bar{f}|^p\\;\\mathrm {d}x\\,,$ where the equality case holds if and only if $u\\equiv t$ in $(c,d)$ .", "We now consider the total variation term.", "We have that $|D\\widetilde{u}|([c,d])= |u(c)-t|+|u(d)-t|\\,,$ Suppose, without loss of generality, that $u(c)\\le u(d)$ .", "We will consider three cases: $t\\in [u(c),u(d)]$ , $t\\le u(c)$ and $t\\ge u(d)$ .", "In the first one, we simply notice that $|D\\widetilde{u}|([c,d]) = u(d)-u(c)\\le |Du|([c,d])\\,.$ If $t\\le u(c)$ , then there exists $x\\in [c,d)$ such that $u(x)\\le t$ .", "Thus, $|Du|([c,d])&\\ge \\bigl (u(c)-u(x)\\bigr )+\\bigl (u(d)-u(x)\\bigr )\\ge \\bigl (u(c)-t\\bigr )+\\bigl (u(d)-t\\bigr )\\\\&=|D\\widetilde{u}|([c,d])\\,.$ The case $t\\ge \\max \\lbrace u(c), u(d)\\rbrace $ can be treated similarly.", "This concludes the proof.", "The above result allows us to get the structure of minimizers of problem (REF ) in the case in which $f$ is a piecewise constant function.", "Corollary 3.2 Let $f$ be a piecewise constant function in $(a,b)$ , i.e., $f(x)=\\sum _{i=1}^k f_i\\,\\chi _{(x_{i-1},x_i)}(x)\\,,\\quad \\quad f_i\\in \\mathbb {R}\\,.$ Then any solution $u$ of the minimization problem (REF ) is of the form $u(x)=\\sum _{i=1}^k u_i\\,\\chi _{(x_{i-1},x_i)}(x)\\,,$ for some $(u_i)_{i=1}^k\\subset \\mathbb {R}\\,$ , not necessarily distinct from each other.", "In particular, a function $u$ of the form (REF ) is a solution of (REF ) if and only $\\bar{u}:=(u_1,\\dots ,u_k)\\in \\mathbb {R}^k$ is a solution of the minimization problem $\\min _{v\\in \\mathbb {R}^k}G(v)\\,,$ where $G:\\mathbb {R}^k\\rightarrow \\mathbb {R}$ is the function defined as $G(v):=\\sum _{i=2}^k |v_i-v_{i-1}| + \\lambda \\sum _{i=1}^k L_i|f_i-v_i|^p\\,,$ where $v=(v_1,\\dots ,v_k)$ and $L_i:=x_i-x_{i-1}$ .", "Thus, we now concentrate on the study of the minimization problem (REF ).", "The cases $p=1$ and $p>1$ turn out to be quite different.", "Heuristically, the difference lies in the fact that, in the first case, the two terms of the energy are of the same order while, for $p>1$ , the fidelity term is of higher order than the total variation one.", "This leads to very different behavior of the solutions in the two cases.", "One of the peculiar features of the case $p=1$ is the lack of uniqueness (see Proposition REF ).", "However, it is possible to identify a solution with a particular structure.", "Corollary 3.3 For $p=1$ , there exists a solution $u$ of the problem (REF ) such that $u_i\\in \\lbrace f_1,\\dots ,f_k\\rbrace $ for every $i=1,\\dots ,k$ .", "For any given quadruple of functions $s_1:\\lbrace 2,\\dots ,k\\rbrace \\rightarrow \\lbrace 0,1\\rbrace \\,,\\quad \\quad s_2:\\lbrace 1,\\dots ,k\\rbrace \\rightarrow \\lbrace 0,1\\rbrace \\,,$ $t_1:\\lbrace 2,\\dots ,k\\rbrace \\rightarrow \\lbrace 0,1\\rbrace \\,,\\quad \\quad t_2:\\lbrace 1,\\dots ,k\\rbrace \\rightarrow \\lbrace 0,1\\rbrace \\,,$ let us consider the set $\\mathcal {A}^{t_1,t_2}_{s_1,s_2}\\subset \\mathbb {R}^k$ such that $G(u)&=\\sum _{i=2}^k (-1)^{s_1(i)}t_1(i)(u_i-u_{i-1})+\\lambda \\sum _{i=1}^k (-1)^{s_2(i)}t_2(i)L_i(f_i-u_i) \\nonumber \\\\&=v^{s_1,s_2,t_1,t_2}_\\lambda \\cdot u + c^{s_1,s_2,t_1,t_2}_\\lambda \\,,$ for all $u\\in \\mathcal {A}^{t_1,t_2}_{s_1,s_2}$ , where $c^{s_1,s_2,t_1,t_2}_\\lambda \\in \\mathbb {R}$ and $v^{s_1,s_2,t_1,t_2}_\\lambda \\in \\mathbb {R}^k$ .", "The result then follows by noticing that $G$ restricted to any $\\mathcal {A}^{t_1,t_2}_{s_1,s_2}\\subset \\mathbb {R}^k$ is always minimized by a vector $u\\in \\mathbb {R}^k$ with $u_i=f_{\\sigma (i)}\\,,$ for some function $\\sigma :\\lbrace 1,\\dots ,k\\rbrace \\rightarrow \\lbrace 1,\\dots ,k\\rbrace $ and that $\\min _{\\mathbb {R}^k}G=\\min _{s_1,s_2,t_1,t_2}\\,\\min _{\\mathcal {A}^{t_1,t_2}_{s_1,s_2}}G_{|_{\\mathcal {A}^{t_1,t_2}_{s_1,s_2}}}\\,.$ Definition 3.4 We will denote by $u^\\lambda $ a solution of the minimization problem (REF ) corresponding to the value $\\lambda $ .", "This will be the solution, if $p>1$ , while, for $p=1$ , it will be understood as a solution whose structure is those given by the previous result.", "Remark 3.5 It is easy to see that $u_i\\in [\\min f, \\max f]$ for every solution $u$ .", "In the rest of this section we seek to understand the behavior of the solution $u^\\lambda $ in the limiting cases for $\\lambda $ , i.e., when $\\lambda \\ll 1$ and when $\\lambda \\gg 1$ .", "In the first case the predominant term of the energy is given by the total variation, thus we expect $u^\\lambda $ to minimizes it.", "Lemma 3.6 Fix $p\\ge 1$ , positive numbers $(L_i)_{i=1}^k$ and two constants $m<M$ .", "Then, there exists a constant $\\bar{\\lambda }>0$ , depending only on $p$ , $(L_i)_{i=1}^k$ , $m$ and $M$ , with the following property.", "For any piecewise constant function $f$ such that $f\\in [m,M]$ and any $\\lambda \\in (0,\\bar{\\lambda }]$ , we have that $u^\\lambda $ is constant.", "In particular, if $p>1$ then there exists $c\\in \\mathbb {R}$ such that $u^\\lambda _i\\equiv c$ for all $\\lambda \\in (0,\\bar{\\lambda }]$ and all $i=1,\\dots ,k$ .", "We first treat the case $p>1$ .", "Assume that $u^\\lambda $ is not constant and let $i\\in \\lbrace 1,\\dots ,k\\rbrace $ be such that $u^\\lambda _i=\\min \\lbrace u_j^\\lambda \\,:\\,j=1,\\dots ,k\\rbrace $ .", "Let $r:=\\inf \\lbrace j\\le i \\,:\\, u_s=u_i \\text{ for all } j\\le s\\le i\\rbrace \\,,$ $t:=\\sup \\lbrace j\\ge i \\,:\\, u_s=u_i \\text{ for all } i\\le s\\le j\\rbrace \\,.$ By hypothesis, either $r>1$ or $t<k$ .", "Consider, for $\\varepsilon >0$ , the vector $u^\\varepsilon \\in \\mathbb {R}^k$ defined as $u^\\varepsilon _j:=u_j+\\varepsilon $ for $j=r,\\dots ,t$ and $u^\\varepsilon _j:=u^\\lambda _j$ for all the other $j$ 's.", "Then, recalling that $u_j\\in [m,M]$ for all $j=1,\\dots ,k$ , we have that $\\lim _{\\varepsilon \\rightarrow 0^+}\\frac{G(u^\\varepsilon )-G(u^\\lambda )}{\\varepsilon }&=a+p\\lambda (-1)^{s_i}L_i|u_i-f_i|^{p-1} \\nonumber \\\\&\\le a+p\\lambda (M-m)^{p-1}\\max _{i=1,\\dots ,k} L_i\\,,$ where $a\\in \\lbrace -1,-2\\rbrace $ (in particular, $a=-1$ if $r=1$ or $t=k$ and $a=-2$ otherwise), and $s_i\\in \\lbrace 0,1\\rbrace $ .", "Let $\\bar{\\lambda }:=\\frac{1}{p(M-m)^{p-1}\\max _i L_i}\\,.$ If $\\lambda <\\bar{\\lambda }$ , from (REF ) we get that $G(u^\\varepsilon )<G(u^\\lambda )$ .", "This means that $u^\\lambda $ has to be constant for $\\lambda <\\bar{\\lambda }$ .", "Moreover, it is easy to see that the function $G$ restricted to the set $\\lbrace (u_1,\\dots ,u_k)\\in \\mathbb {R}^k \\,:\\, u_1=\\dots =u_k \\rbrace $ admits a unique minimizer, that is independent of $\\lambda $ .", "We now have to prove that $u^{\\bar{\\lambda }}$ is constant.", "Assume that $u^\\lambda _i\\equiv c$ for for all $\\lambda \\in (0,\\bar{\\lambda })$ and all $i=1,\\dots ,k$ .", "Let $\\bar{c}\\in \\mathbb {R}^k$ be the vector given by $\\bar{c}_i:=c$ .", "Then $G_\\lambda (c)<G_\\lambda (v)$ for all $v\\in \\mathbb {R}^k$ with $v\\ne \\bar{c}$ and all $\\lambda \\in (0,\\bar{\\lambda })$ , where the subscript $\\lambda $ is to underline the dependence of $G$ on $\\lambda $ .", "By letting $\\lambda \\nearrow \\bar{\\lambda }$ , we get $G_{\\bar{\\lambda }}(c)<G_{\\bar{\\lambda }}(v)$ for all $v\\in \\mathbb {R}^k$ and thus $u^{\\bar{\\lambda }}=\\bar{c}$ .", "Let us now treat the case $p=1$ .", "Suppose that $u^\\lambda $ is not constant.", "Recalling that $u^\\lambda _i\\in \\lbrace f_1,\\dots ,f_k\\rbrace $ , we have that $|D u^\\lambda |(\\Omega )\\ge \\min _i|f_i-f_{i-1}|\\,.$ On the other hand, for any function $v$ such that $v\\equiv c\\in [\\min f,\\max f]$ in $(a,b)$ , it holds that $G(v)\\le \\lambda k (\\max _i L_i)(M-m)\\,.$ Set $\\bar{\\lambda }:=\\frac{\\min _i|f_i-f_{i-1}|}{k (\\max _i L_i)(M-m)}\\,.$ For $\\lambda <\\bar{\\lambda }$ the above estimates show that $u^\\lambda $ must be constant.", "Finally, in order to prove that also $u^{\\bar{\\lambda }}$ is constant, we reason as follows: we know that $u^\\lambda =\\bar{c}^\\lambda $ for $\\lambda \\in (0,\\bar{\\lambda })$ , for some $\\bar{c}_\\lambda =(c_\\lambda ,\\dots ,c_\\lambda )\\in \\mathbb {R}^k$ .", "Take $\\lambda _n\\nearrow \\bar{\\lambda }$ .", "Since $c_{\\lambda _n}\\in [\\min f,\\max f]$ , up to a not relabelled subsequence we have that $c_{\\lambda _n}\\rightarrow c$ .", "We conclude that $G_{\\bar{\\lambda }}\\bigl ((c,\\dots ,c)\\bigr )\\le G_{\\bar{\\lambda }}(v)$ for all $v\\in \\mathbb {R}^k$ .", "We now consider the case $\\lambda \\gg 1$ .", "Since $\\lambda L_i |u^\\lambda _i-f_i|^p\\le G(u^\\lambda )\\le G(f)<\\infty \\,,$ we know that $u^\\lambda \\rightarrow f\\quad \\quad \\text{ as } \\lambda \\rightarrow \\infty \\,.$ The following results underline another important difference between the cases $p=1$ and $p>1$ .", "Indeed, if $p=1$ the limit (REF ) is reached for $\\lambda <\\infty $ , while if $p>1$ only asymptotically.", "Lemma 3.7 Let $p>1$ and assume that $f$ is not constant.", "Then $u^\\lambda \\in (\\min f,\\max f)$ for all $\\lambda >0$ .", "In particular, $f$ can never be a solution of the minimization problem (REF ).", "We first prove that $u^\\lambda $ cannot achieve the value $\\min f$ .", "Assume that $u^\\lambda _i=\\min f$ for some $i\\in \\lbrace 1,\\dots ,k\\rbrace $ .", "Let $r\\le i\\le s$ be such that $u_j=u_i$ for all $j=r,\\dots ,s$ .", "Consider, for $\\varepsilon >0$ , the vector $u^\\varepsilon \\in \\mathbb {R}^k$ given by $u^\\varepsilon _j:=u^\\lambda _j+\\varepsilon $ for $j=r,\\dots ,s$ and $u^\\varepsilon _j:=u^\\lambda _j$ for all other $j$ 's.", "Then $\\lim _{\\varepsilon \\rightarrow 0^+}\\frac{G(u^\\varepsilon )-G(u)}{\\varepsilon }=a-p\\lambda \\sum _{j=r}^s L_j(f_j-u^\\lambda _i)^{p-1}<0\\,,$ where $a\\in \\lbrace -1,-2\\rbrace $ .", "This is in contradiction with the minimality of $u^\\lambda $ .", "With a similar argument it is possible to show that $u$ cannot achieve $\\max f$ .", "Lemma 3.8 Let $p=1$ .", "Then there exists $\\bar{\\lambda }>0$ such that for all $\\lambda \\ge \\bar{\\lambda }$ the solution of the minimization problem (REF ) is unique and is given by $f$ itself.", "Suppose that there exists a sequence $\\lambda _j\\rightarrow \\infty $ for which $u^{\\lambda _j}_i\\ne f_i$ for all $j$ 's (this is possible, since $k$ is finite).", "By recalling that $u^{\\lambda _j}_i\\in \\lbrace f_1,\\dots ,f_k\\rbrace $ , setting $\\bar{\\lambda }:=\\frac{G(f)}{\\min _i L_i\\,\\min _i|f_i-f_{i-1}|}\\,,$ we have, for $\\lambda _j>\\bar{\\lambda }$ , that $G(u^{\\lambda _j})\\ge \\lambda _j L_i|u_i^{\\lambda _j}-f_i|>G(f)\\,,$ contradicting the minimality of $u^{\\lambda _j}$ ." ], [ "Explicit solutions in a simple case", "Here we study the case where $k=2$ .", "This analysis, albeit its simplicity, is important to underline some features that distinguish the behavior of the solution of the minimization problem (REF ) in the cases $p=1$ and $p>1$ .", "Proposition 4.1 Let $f_1<f_2$ .", "Then the solutions $u^\\lambda $ of the minimization problem (REF ) in the case $p=1$ are the following: if $L_1>L_2$ , set $\\lambda ^1_T:=\\frac{1}{L_2}$ .", "Then $\\left\\lbrace \\begin{array}{lll}u^\\lambda _1=u^\\lambda _2=f_1 & & \\text{ for } \\lambda <\\lambda ^1_T\\,,\\\\&\\\\u^\\lambda _1=f_1, u^\\lambda _2\\in [f_1,f_2] & & \\text{ for } \\lambda =\\lambda ^1_T\\,,\\\\&\\\\u^\\lambda _1=f_1, u^\\lambda _2=f_2 & & \\text{ for } \\lambda >\\lambda ^1_T\\,,\\\\\\end{array}\\right.$ if $L_1=L_2$ , set $\\lambda ^1_T:=\\frac{1}{L_1}$ .", "Then $\\left\\lbrace \\begin{array}{lll}u^\\lambda _1\\in [f_1,f_2], u^\\lambda _2\\ge u_1 & & \\text{ for } \\lambda \\le \\lambda ^1_T\\,,\\\\&\\\\u^\\lambda _1=f_1, u^\\lambda _2=f_2 & & \\text{ for } \\lambda >\\lambda ^1_T\\,,\\\\\\end{array}\\right.$ if $L_1<L_2$ , set $\\lambda ^1_T:=\\frac{1}{L_1}$ .", "Then $\\left\\lbrace \\begin{array}{lll}u^\\lambda _1=u^\\lambda _2=f_2 & & \\text{ for } \\lambda <\\lambda ^1_T\\,,\\\\&\\\\u^\\lambda _1\\in [f_1,f_2], u^\\lambda _2=f_2 & & \\text{ for } \\lambda =\\lambda ^1_T\\,,\\\\&\\\\u^\\lambda _1=f_1, u^\\lambda _2=f_2 & & \\text{ for } \\lambda >\\lambda ^1_T\\,,\\\\\\end{array}\\right.$ It is easy to see that we must have $f_1\\le u_1\\le u_2\\le f_2$ .", "Thus, we consider the region $\\mathcal {T}:=\\lbrace \\, (u_1, u_2)\\in \\mathbb {R}^2 \\;:\\; f_1\\le u_1\\le u_2\\le f_2 \\,\\rbrace \\,,$ and we rewrite the function $G$ in $\\mathcal {T}$ as $G(\\bar{u})=[\\lambda L_1-1]u_1 + [1-\\lambda L_2]u_2+\\lambda [f_2 L_2-f_1 L_1] =v_\\lambda \\cdot u+c_\\lambda \\,.$ When minimizing $G$ in $\\mathcal {T}$ , we can drop the term $c_\\lambda $ .", "Then, the minimizers, according to the position of the vector $\\frac{v_\\lambda }{|v_\\lambda |}$ (well defined for all $\\lambda $ 's, except in the case $L_1=L_2$ and $\\lambda =\\frac{1}{L_1}$ ), are the following: Figure: On the left it is displayed where the (renormalized) vector v λ v_\\lambda can vary: from v 1 v_1 for λ=0\\lambda =0 up tp (asymptotically) v ∞ :=arctanL 2 L 1 v_\\infty :=\\arctan \\frac{L_2}{L_1}.", "On the left the triangle where the vector uu can vary.Thus, by simply studying the sign of the components of $v_\\lambda $ , we obtain the desired result.", "Notice that the non uniqueness happens only when the vector $v_\\lambda $ is orthogonal to $\\lbrace x=y\\rbrace \\subset \\mathbb {R}^2$ .", "In the case $p>1$ the landscape of the solutions is quite different.", "Proposition 4.2 Let $f_1<f_2$ and let $p>1$ .", "Define $\\lambda ^p_T:=\\frac{1}{p}\\frac{(L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}})^{p-1}}{L_1 L_2(f_2-f_1)^{p-1}}\\,.$ The solution $u^\\lambda $ of the minimization problem (REF ) is the following: for $\\lambda \\le \\lambda _T^p$ $u^\\lambda _1=u^\\lambda _2=\\frac{L_1^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}\\,f_1 + \\frac{L_2^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}\\,f_2\\,,$ for $\\lambda >\\lambda _T^p$ $u^\\lambda _1= f_1+\\frac{1}{(p\\lambda L_1)^{\\frac{1}{p-1}}}\\,,\\quad \\quad u^\\lambda _2= f_2-\\frac{1}{(p\\lambda L_2)^{\\frac{1}{p-1}}}\\,.$ Recalling that $f_1\\le u_1\\le u_2\\le f_2$ , we just have to consider the region $\\mathcal {T}$ defined in (REF ) and to rewrite the function $G$ in that region as $G(u_1,u_2):=u_2-u_1+\\lambda L_1(u_1-f_1)^p+\\lambda L_2 (f_2-u_2)^p\\,.$ The critical point of $G$ is given by $u_1= f_1+\\frac{1}{(p\\lambda L_1)^{\\frac{1}{p-1}}}\\,,\\quad \\quad u_2= f_2-\\frac{1}{(p\\lambda L_2)^{\\frac{1}{p-1}}}\\,,$ and it belongs to the interior of $\\mathcal {T}$ , i.e., $u_1^\\lambda < u_2^\\lambda $ , only for $\\lambda >\\lambda ^p_T$ .", "Since $G$ is strictly convex, this critical value turns out to be the global minimizer of $G$ for $\\lambda >\\lambda _T^p$ .", "In the case $\\lambda \\le \\lambda _T^p$ , the point of minimum has to be on $\\partial \\mathcal {T}$ .", "Instead of performing all the computations for finding the minimum point in all of the three edges of $\\partial \\mathcal {T}$ and to compare them, we will use the following argument based on the continuity of the minimizer $u^\\lambda $ with respect to $\\lambda $ (see Lemma REF ), i.e., we invoke the fact that the function $\\lambda \\mapsto u^\\lambda $ is continuous.", "Notice that for $\\lambda \\searrow \\lambda _T^p$ we have $u_\\lambda \\rightarrow (\\bar{u},\\bar{u})\\,,$ where $\\bar{u}:=\\frac{L_1^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}\\,f_1 + \\frac{L_2^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}\\,f_2\\,,$ is independent of $\\lambda $ .", "By using the continuity of the solution, we can conclude that, for $\\lambda \\le \\lambda _T^p$ , the solution of the minimization problem is given by $(\\bar{u},\\bar{u})$ .", "Remark 4.3 We remark a couple of facts: we have that $\\lambda _T^p\\rightarrow \\lambda _T^1$ as $p\\rightarrow 1^+$ (in each of the cases for the definition of the second one).", "Indeed, suppose that $L_1<L_2$ .", "Then, $\\lim _{p\\rightarrow 1^+}\\lambda _T^p &=\\lim _{p\\rightarrow 1^+}\\frac{(L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}})^{p-1}}{L_1 L_2}\\\\&= \\frac{1}{L_1}\\lim _{p\\rightarrow 1^+}\\biggl (\\, 1+\\biggl (\\frac{L_1}{L_2}\\biggr )^{\\frac{1}{p-1}} \\,\\biggr )^{p-1}\\\\&= \\frac{1}{L_1}\\lim _{t\\rightarrow 0^+}exp\\left[\\, t\\log \\left[\\left(\\frac{L_1}{L_2}\\right)^{\\frac{1}{t}} +1\\right] \\,\\right]=\\frac{1}{L_2}=\\lambda _T^1\\,.$ Similar reasonings lead to the claimed result in the other two cases.", "In particular, notice that $\\lambda _T^p>\\lambda _T^1$ .", "The solutions that converge to a solution for $p=1$ , as $p\\searrow 1$ .", "Indeed, suppose $\\lambda >\\lambda _T^1$ , Then for $p$ sufficiently close to 1, from the above bullet point, we have that $\\lambda >\\lambda _T^p$ .", "Thus, the solution of the minimization problem for $p$ is given by (REF ).", "In this case, it is easy to see that the solution converges to $(f_1,f_2)$ , as $p\\searrow 1$ .", "In the case $\\lambda <\\lambda _T^1$ , we can assume as above that $p$ is so close to 1 that the solution of the minimization problem for $p$ is given by (REF ).", "If $L_1>L_2$ , then $\\frac{L_1^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}=\\frac{1}{\\Bigl ( \\frac{L_2}{L_1} \\Bigr )^{\\frac{1}{p-1}}+1}\\rightarrow 1\\,,\\quad \\quad \\text{ as } p\\rightarrow 1\\,,$ $\\frac{L_2^{\\frac{1}{p-1}}}{L_1^{\\frac{1}{p-1}}+L_2^{\\frac{1}{p-1}}}=\\frac{1}{\\Bigl ( \\frac{L_1}{L_2} \\Bigr )^{\\frac{1}{p-1}}+1}\\rightarrow 0\\,,\\quad \\quad \\text{ as } p\\rightarrow 1\\,.$ In the case $L_1=L_2$ , both coefficients are equal to $\\frac{1}{2}$ .", "Finally, in the case $\\lambda =\\lambda _T^1$ , since $\\lambda _T^p>\\lambda _T^1$ we have that the solution of the minimization problem is given by (REF ).", "The result follows by arguing as before.", "Remark 4.4 We expect a similar behavior for the minimization problem (REF ) in the case $p=1$ to hold also for general piecewise constant initial data $f$ .", "In particular, we believe that the non uniqueness of the solution happens only for a finite number of critical values of $\\lambda $ , where a continuum of solutions is present.", "This set of critical values will be the set whose elements are $\\lim _{p\\rightarrow 1^+}\\lambda ^p_i$ , where the $\\lambda ^p_i$ is the biggest value of $\\lambda $ for which the solution $u$ corresponding to the parameters $\\lambda $ and $p$ happens to have $u_i=u_{i+1}$ (see Theorem REF )." ], [ "The behavior of the solution for $p>1$", "This section contains the main result of this paper, namely Theorem REF , that is derived from the qualitative properties of the solutions proved in the following two lemmas and in Proposition REF .", "We start by proving the continuity of the solution $u^\\lambda $ with respect to $\\lambda $ .", "Lemma 5.1 Let $p>1$ .", "Then $\\lambda \\mapsto u^\\lambda $ is continuous and $\\lim _{\\lambda \\rightarrow \\infty }u^\\lambda =f$ .", "Fix $\\bar{\\lambda }>0$ and let $\\lambda _n\\rightarrow \\lambda $ .", "Then $G(u^{\\lambda _n})\\le G(v)$ for all $v\\in \\mathbb {R}^k$ , where equality holds if and only if $v=u^{\\lambda _n}$ .", "Since $|u^{\\lambda _n}|\\le \\sqrt{k}|\\max _i f_i|$ , up to a (not relabeled) subsequence, we have that $u^{\\lambda _n}\\rightarrow \\bar{v}$ .", "Using the continuity of $G$ in both $v$ and $\\lambda $ , we have that $G(\\bar{v})\\le G(v)$ for all $v\\in \\mathbb {R}^k$ .", "By the uniqueness of the solution, we deduce that $\\bar{v}=u^{\\lambda }$ , and that $u^{\\lambda _n}\\rightarrow u^\\lambda $ for all sequences $\\lambda _n\\rightarrow \\lambda $ .", "To prove the second part of the lemma, we reason as follows.", "Assume that $u^\\lambda $ does not converge to $f$ as $\\lambda \\rightarrow \\infty $ .", "Since $u_i\\in [\\min f,\\max f]$ , by compactness (up to a not relabelled subsequence) $u^\\lambda \\rightarrow v$ , for some $v\\ne f$ .", "In particular, there exists an index $i$ such that $|u^\\lambda _i-f_i|>\\varepsilon $ for $\\lambda \\gg 1$ , for some $\\varepsilon >0$ .", "So that $+\\infty >G(f)\\ge G(u^\\lambda )\\ge \\lambda |u^\\lambda _i-f_i|\\rightarrow \\infty \\,,$ as $\\lambda \\rightarrow \\infty $ .", "This is the desired contradiction.", "We now prove several qualitative properties regarding the behavior of the solution $u^\\lambda $ as $\\lambda $ varies.", "Some of the following results could be stated in a more inclusive way, but since they can be used to deduce qualitative properties of the solutions when no direct analysis can be performed, for clarity of exposition we opt to present each of them separately.", "Lemma 5.2 Let $p>1$ .", "Then, the following properties hold true: (i) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _i=u^\\lambda _{i+1}=\\dots =u^\\lambda _{i+r}=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{i-1}<\\bar{u}<u^\\lambda _{i+r+1} \\quad \\text{ or } \\quad u^\\lambda _{i-1}>\\bar{u}>u^\\lambda _{i+r+1}\\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\bar{u}^\\lambda $ is the solution of $\\min _{c\\in (u^\\lambda _{i-1}, u^\\lambda _{i+r+1})} \\sum _{j=i}^{i+r} L_j|c-f_j|^p\\,.$ In particular, $\\bar{u}^\\lambda $ is constant in $(\\lambda _1,\\lambda _2)$ .", "(ii) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _i=u^\\lambda _{i+1}=\\dots =u^\\lambda _{i+r}=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{i-1}\\,,\\, u^\\lambda _{i+r+1}<\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is increasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _i=f_i-\\Bigl ( \\frac{2}{p\\lambda L_i} \\Bigr )^{\\frac{1}{p-1}}\\,.$ (iii) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _i=u^\\lambda _{i+1}=\\dots =u^\\lambda _{i+r}=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{i-1}\\,,\\, u^\\lambda _{i+r+1}>\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is decreasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _i=f_i+\\Bigl ( \\frac{2}{p\\lambda L_i} \\Bigr )^{\\frac{1}{p-1}}\\,.$ (iv) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _1=u^\\lambda _2=\\dots =u^\\lambda _{r}=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{r+1}<\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is increasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _i=f_1-\\Bigl ( \\frac{1}{p\\lambda L_1} \\Bigr )^{\\frac{1}{p-1}}\\,.$ (v) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _1=u^\\lambda _{i+1}=\\dots =u^\\lambda _{r}=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{r+1}>\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is decreasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _i=f_1+\\Bigl ( \\frac{1}{p\\lambda L_1} \\Bigr )^{\\frac{1}{p-1}}\\,.$ (vi) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _{k-r}=\\dots =u^\\lambda _k=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{k-r-1}>\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is decreasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _k=f_k+\\Bigl ( \\frac{1}{p\\lambda L_k} \\Bigr )^{\\frac{1}{p-1}}\\,.$ (vii) Assume that, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ , there exists a function $\\lambda \\mapsto \\bar{u}^\\lambda $ such that, for some $r\\ge 0$ , $\\left\\lbrace \\begin{array}{c}u^\\lambda _{k-r}=\\dots =u^\\lambda _k=\\bar{u}^\\lambda \\,,\\\\\\\\u^\\lambda _{k-r-1}<\\bar{u}^\\lambda \\,.\\end{array}\\right.$ Figure: NO_CAPTIONThen $\\lambda \\mapsto \\bar{u}^\\lambda $ is increasing.", "In particular, in the case $r=0$ , we have $u^\\lambda _k=f_k-\\Bigl ( \\frac{1}{p\\lambda L_k} \\Bigr )^{\\frac{1}{p-1}}\\,.$ We start by proving property (i).", "Suppose that $u^\\lambda _{i-1}<\\bar{u}^\\lambda <u^\\lambda _{i+r+1}$ .", "In the other case we argue in a similar way.", "By hypothesis, the vector $u^\\lambda $ minimizes the function $G$ in the set $\\lbrace \\, (u_1,\\dots ,u_k)\\in \\mathbb {R}^k : u_{i-1}<u_i=\\dots =u_{i+r}<u_{i+r+1} \\,\\rbrace \\,,$ and in this set, the function $G$ can be written as $G(u)=\\widetilde{G}(u_1\\dots ,u_{i-1},u_{i+r+1},\\dots ,u_k)\\\\+\\lambda \\sum _{j=i}^{i+r} L_j|\\bar{u}-f_j|^p\\,.$ By keeping $u_1,\\dots ,u_{i-1}$ and $u_{i+r+1},\\dots ,u_k$ fixed, the claim follows by minimizing the above quantity with respect to $\\bar{u}$ .", "Since all the other properties can be proved with an argument whose general lines are similar, we just prove property (ii), leaving the details of the others proofs to the reader.", "In the hypothesis of (ii), it holds that $u^\\lambda $ is a minimizer of $G$ in the set $\\lbrace \\, (u_1,\\dots ,u_k)\\in \\mathbb {R}^k : u_{i-1},\\, u_{i+r+1}<u_i=\\dots =u_{i+r} \\,\\rbrace \\,.$ Restricted to this set, the function $G$ can be written as $G(u)=\\widetilde{G}(u_1\\dots ,u_{i-1},u_{i+r+1},\\dots ,u_k)\\\\+2\\bar{u}+\\lambda \\sum _{j=i}^{i+r} L_j|\\bar{u}-f_j|^p\\,.$ So, for $\\lambda \\in (\\lambda _1,\\lambda _2)$ and $u_1\\dots ,u_{i-1},u_{i+r+1},\\dots ,u_k$ fixed, $\\bar{u}^\\lambda $ is the minimizer of the strictly convex function $H(c):=2c+\\lambda \\sum _{j=i}^{i+r} L_j|c-f_j|^p$ in the set $(\\max \\lbrace {u^\\lambda _i}, u^\\lambda _{i+r}\\rbrace , \\max f)$ .", "To study the minimizer of $H$ , we can assume without loss of generality that $f_i<f_{i+1}<\\dots <f_{i+r}$ .", "Indeed, we notice that the order of the $f_j$ 's doesn't matter.", "Moreover, in the case in which $f_p=f_q$ for some $p\\ne q$ , we can simply collect the two terms in a single one and use $L_p+L_q$ as a corresponding factor in the above summation.", "We now want to prove that $\\lambda \\mapsto \\bar{u}$ is decreasing.", "Note that the function $H$ can be written as $H(c)=2c+\\lambda \\sum _{j=i}^{m}L_j(c-f_j)^p + \\lambda \\sum _{j=m+1}^{i+r}L_j(f_j-c)^p=:H_m(c)\\,,$ if $c\\in (f_m,f_{m+1}]$ , for some $m\\in \\lbrace i,\\dots ,i+r-1\\rbrace $ , and $H(c)=2c+\\lambda \\sum _{j=i}^{i+r}L_j(c-f_j)^p\\,,$ if $c\\in [f_{i+r},\\max f)$ .", "Consider the function $H_m$ in the interval $(f_m,f_{m+1})$ .", "We have that $H_m^{\\prime }(c)=2+p\\lambda \\left[\\, \\sum _{j=i}^{m}L_j(c-f_j)^{p-1} - \\sum _{j=m+1}^{i+r}L_j(f_j-c)^{p-1} \\,\\right]\\,.$ Here $H^{\\prime }_m(c)=0$ has a solution only if the term in the parenthesis is negative and if so, the let $\\lambda \\mapsto c^\\lambda $ be such a solution.", "It is easy to see that this function is regular in $(f_m,f_{m+1})$ .", "By differentiating the expression $H^{\\prime }_m(c^\\lambda )$ with respect to $\\lambda $ , we obtain $& p\\left[\\, \\sum _{j=i}^{m}L_j(c-f_j)^{p-1} - \\sum _{j=m+1}^{i+r}L_j(f_j-c)^{p-1} \\,\\right] \\\\& + \\lambda \\frac{\\mathrm {d}c^\\lambda }{\\mathrm {d}\\lambda }p(p-1)\\left[\\, \\sum _{j=i}^{m}L_j(c-f_j)^{p-2} + \\sum _{j=m+1}^{i+r}L_j(f_j-c)^{p-2} \\,\\right]=0\\,.$ Thus, by recalling that the term in the first parenthesis is negative, we get $\\frac{\\mathrm {d}c^\\lambda }{\\mathrm {d}\\lambda }<0$ , as desired.", "In the case in which the minimizer of the function $H$ is reached at a point $c=f_{m+1}$ , we simply consider the function $H_m$ and we apply the argument above.", "Finally, the same reasoning applies when $c\\in [f_{i+r},\\max f)$ .", "We are now in position to prove the fundamental result we will use to develop our strategy for finding the solution.", "Theorem 5.3 For each $i=1,\\dots ,k-1$ there exists $\\lambda _i\\in (0,\\infty )$ such that $u_i^\\lambda =u_{i+1}^\\lambda $ for $\\lambda \\le \\lambda _i$ , while $u_i^\\lambda \\ne u^\\lambda _{i+1}$ for $\\lambda >\\lambda _i$ .", "Step 1.", "We claim that if $u^{\\widetilde{\\lambda }}_i=u^{\\widetilde{\\lambda }}_{i+1}$ for some $\\widetilde{\\lambda }>0$ , then $u^\\lambda _i=u_{i+1}^\\lambda $ for all $\\lambda \\in (0,\\widetilde{\\lambda }]$ .", "Indeed, let $\\bar{\\lambda }:=\\min \\lbrace \\,\\lambda \\,:\\, u^\\mu _i=u^\\mu _{i+1} \\text{ fo all } \\mu \\in [\\lambda ,\\widetilde{\\lambda }]\\,\\rbrace \\,,$ and assume that $\\bar{\\lambda }>0$ .", "By continuity of $\\lambda \\mapsto u^\\lambda $ there exists $\\varepsilon >0$ such that $u_i^\\lambda \\ne u_{i+1}^\\lambda $ for $\\lambda \\in (\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ .", "Consider the case in which $u_i^\\lambda <u_{i+1}^\\lambda $ in $(\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ (the other case can be treated similarly).", "If $i=1$ , then property (v) of Lemma REF tells us that $\\lambda \\mapsto u_i^\\lambda $ is decreasing in $(\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ and thus it is not possible to have $u_i^{\\bar{\\lambda }}=u_{i+1}^{\\bar{\\lambda }}$ .", "If $i>1$ , we can focus, without loss of generality, only on the following two cases: $u_{i-1}^\\lambda >u_i^\\lambda $ and $u_{i-1}^\\lambda <u_i^\\lambda $ in $(\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ .", "In the first case, we get a contradiction since by property (iii) of Lemma REF , the map $\\lambda \\mapsto u^\\lambda _i$ is decreasing in $(\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ and thus, as above, we cannot have $u_i^{\\bar{\\lambda }}=u_{i+1}^{\\bar{\\lambda }}$ .", "In the other case, we have $u_{i-1}^\\lambda <u_i^\\lambda <u_{i+1}^\\lambda $ in $(\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ .", "By using property (i) of Lemma REF , we see that this is possible only if $u_i^\\lambda =f_i$ for all $\\lambda \\in (\\bar{\\lambda }-\\varepsilon ,\\bar{\\lambda })$ .", "This yields the desired contradiction.", "Step 2.", "Let us define $\\lambda _i:=\\max \\lbrace \\, \\lambda \\,:\\, u_i^\\mu =u_{i+1}^\\mu \\,,\\, \\text{ for all } \\mu \\le \\lambda \\,\\rbrace \\,.$ Step 1 and the continuity of $\\lambda \\mapsto u^\\lambda $ ensure that $\\lambda _i$ is well defined.", "Moreover, by Lemma REF , we also get that $\\lambda _i>0$ for all $i=i,\\dots ,k-1$ .", "Finally, the fact that $u^\\lambda \\rightarrow f$ as $\\lambda \\rightarrow \\infty $ , tells us that $\\lambda _i<\\infty $ for all $i=1,\\dots ,k-1$ .", "This concludes the proof.", "Remark 5.4 By a direct inspection of the proof of property (ii) of Proposition REF , we see that the function $\\lambda \\mapsto u^\\lambda $ is continuous.", "Moreover, we can also say that it is smooth for all $\\lambda \\in (0,\\infty )\\!\\setminus \\!S$ , where $S:=\\lbrace \\lambda _1,\\dots ,\\lambda _{k-1}\\rbrace \\cup T$ , where the $\\lambda _i$ 's are given by Theorem REF , and $T:=\\lbrace \\mu _1,\\dots ,\\mu _k\\rbrace $ where $\\mu _i:=\\inf \\lbrace \\lambda \\,:\\, u^\\lambda _i=f_i \\rbrace $ .", "Finally, we derive another consequence of Lemma REF that will ensure that the solution is monotone where $f$ is and with the same monotonicity.", "Proposition 5.5 Suppose that $f_i<f_{i+1}<\\dots <f_{i+r}$ .", "Then the solution $u$ of the minimization problem (REF ) is such that $u_i\\le u_{i+1}\\le \\dots \\le u_{i+r}$ .", "In particular, $u$ has the following structure: if $u_i\\ge f_{i+r}$ , then $u_j=u_i$ for all $j=i,\\dots ,i+r$ , if $u_{i+r}\\le f_i$ , then $u_j=u_{i+r}$ for all $j=i,\\dots ,i+r$ , otherwise, $u$ is of the form $u_j=\\left\\lbrace \\begin{array}{ll}u_i & \\text{ for } j=i,\\dots ,j_1\\,, \\\\f_j & \\text{ for } j=j_1+1,\\dots ,j_2-1 \\,, \\\\u_{i+r} & \\text{ for } j=j_2,\\dots ,k\\,, \\\\\\end{array}\\right.$ for some $f_{j_1}\\le u_i<f_{j_1+1}$ and $f_{j_2}\\le u_{i+r}<f_{j_2+1}$ , where $j_1<j_2$ .", "A similar statement holds in the case $f_i>f_{i+1}>\\dots >f_{i+r}$ .", "Step 1.", "We claim that $u_i\\le u_{i+1}\\le \\dots \\le u_{i+r}$ .", "Suppose that $u_{j-1}>u_j$ for some $j\\in \\lbrace i+1,\\dots ,i+r\\rbrace $ .", "We have to treat three cases: $u_j<f_j$ , $u_j=f_j$ and $u_j>f_j$ .", "In the first case, we get a contradiction with the minimality of $u^\\lambda $ since it is easy to see that $G(u^\\lambda _1,\\dots ,u^\\lambda _{j-1},u^\\lambda _j+\\varepsilon ,u^\\lambda _{j+1},\\dots ,u_k)<G(u^\\lambda )\\,,$ for $\\varepsilon >0$ small.", "Now, suppose $u_j>f_j$ and that $u_j>u_{j+1}$ .", "Then, for $\\varepsilon >0$ small, $G(u^\\lambda _1,\\dots ,u^\\lambda _{j-1},u^\\lambda _j-\\varepsilon ,u^\\lambda _{j+1},\\dots ,u_k)<G(u^\\lambda )\\,,$ yielding the desired contradiction.", "Finally, we can treat all the remaining cases (namely when $u_j=f_j$ or the case where $u_j>f_j$ and $u_{j+1}>u_j$ ) simultaneously as follows: let us denote by $j_m\\in \\lbrace i,\\dots ,j\\rbrace $ be the minimum index $r$ such that $u_r>u_{r+1}$ .", "In both cases we have $u_{j_m}>f_{j_m}$ , and thus, $G(u^\\lambda _1,\\dots ,u^\\lambda _{j_m-1},u^\\lambda _{j_m}-\\varepsilon ,u^\\lambda _{j_m+1},\\dots ,u_k)<G(u^\\lambda )\\,,$ for $\\varepsilon >0$ small.", "Step 2.", "Using Step 1, we have that $\\sum _{j=i+1}^{i+r} |u^\\lambda _j-u^\\lambda _{j-1}|=u^\\lambda _{i+r}-u^\\lambda _i\\,.$ Since this value is invariant under modification of $u^\\lambda _j$ for $j=i+1,\\dots ,i+r-1$ , if we keep $u_i$ and $u_{i+r}$ fixed, the minimality of $u^\\lambda $ implies that $\\sum _{j=i}^{i+r}|u_j-f_j|^p=\\min _\\mathcal {A}\\sum _{j=i}^{i+r}|v_i-f_i|^p\\,,$ where $\\mathcal {A}:=\\lbrace (v_{i+1},\\dots ,v_{i+r-1})\\in \\mathbb {R}^{i+r-2} \\,:\\, u_i\\le v_{i+1}\\le \\dots \\le v_{i+r-1}\\le u_{i+r}\\rbrace \\,.$ This proves the second part of the statement of the proposition." ], [ "A method for finding the solution.", "In this section we describe the method we propose in order to identify the solution of the minimization problem (REF ) in the case $p>1$ .", "The general idea is, for every $\\lambda >0$ , to be able to tell a priori the relative position of each $u^\\lambda _i$ with respect to $u^\\lambda _{i-1}$ and $f_i$ .", "Knowing that allows us to: (i) know if the minimization of $G$ has to take place in some subspace $\\lbrace v_{i_1-1}=v_{i_1}\\rbrace \\cap \\dots \\cap \\lbrace v_{i_r-1}=v_{i_r}\\rbrace $ , and hence if we have to reduce the number of variables $G$ depends on, (ii) write explicitly the absolute values present in the expression of $G$ .", "If we are able to do that, we can reduce the problem of minimizing the functional $G$ to the problem of minimizing a strictly convex functional of class $C^1$ , and thus the minimizer can be found by solving the appropriate Euler-Lagrange equation.", "Let us now explain how we are able to make our prediction.", "For the sake of clarity, let us assume that our initial data $f$ is like in the figure below.", "Figure: The initial data ff.Step 1: Solutions for $\\lambda $ large.", "By Proposition REF we know that for $\\lambda \\gg 1$ the solution $u^\\lambda $ is such that $u_i$ is very close to $f_i$ .", "In particular $u_{i-1}\\ne u_i$ for every $i=2,\\dots ,k$ .", "Moreover, we also know what the relative position of $u^\\lambda _i$ with respect to $u^\\lambda _{i-1}$ is.", "Using the properties given by Lemma REF and Proposition REF , will also allow us to know the relative position of each $u^\\lambda _i$ with respect to $f_i$ .", "Figure: The behavior of the solution for λ≫1\\lambda \\gg 1 as λ\\lambda decreases.Thus, we are able to determine $\\bar{r}_i,\\bar{s}_i,\\bar{t}_i\\in \\lbrace 0,1\\rbrace $ for which $G(u^\\lambda )=\\sum _{i=2}^k (-1)^{\\bar{s}_i}(u^\\lambda _i-u^\\lambda _{i-1})+\\lambda \\sum _{i=1}^k \\bar{r}_i L_i \\bigl ((-1)^{\\bar{t}_i}(f_i-u^\\lambda _i)\\bigr )^p\\,,$ for all $\\lambda \\gg 1$ .", "This tells us that we can find $u^\\lambda $ by solving the Euler-Lagrange equation of the function $G(v)=\\sum _{i=2}^k (-1)^{\\bar{s}_i}(v_i-v_{i-1})+\\lambda \\sum _{i=1}^k L_i \\bigl ((-1)^{\\bar{t}_i}(f_i-v_i)\\bigr )^p\\,.$ In particular, for all $i$ such that $\\bar{r}_i\\ne 0$ - for which we already know that $u^\\lambda _i=f_i$ - we get $u^\\lambda _i=f_i-(-1)^{\\bar{t}_i}\\left[\\, \\frac{(-1)^{\\bar{s}_i}-(-1)^{\\bar{s}_{i-1}}}{p\\lambda L_i} \\,\\right]^{\\frac{1}{p-1}}\\,.$ Notice that how big $\\lambda $ has to be in order to apply what we said above will be determined explicitly in the next step.", "Step 2.", "Solutions for all smaller $\\lambda $ 's.", "We now let $\\lambda $ decrease.", "Since for small $\\lambda $ 's we know that the solution $u^\\lambda $ is constant, the continuity of the function $\\lambda \\mapsto u^\\lambda $ implies that, eventually, a critical event will happen.", "That is, two neighboring values of $u^\\lambda $ will coincide.", "Notice that multiple critical events can happen simultaneously.", "Figure: The behavior of the solution for λ\\lambda after the first critical event as λ\\lambda decreases.Assume, for instance, that for some $\\lambda _j$ , $u^{\\lambda _j}_j$ happens to be equal to $u^{\\lambda _j}_{j-1}$ and that this is the only critical event taking place.", "By Theorem REF we know that the same will be true for all smaller $\\lambda $ 's, that is $u^\\lambda _j=u^\\lambda _{j-1}$ for all $\\lambda \\le \\lambda _j$ .", "So that we now have to consider the functional $G$ restricted to the subspace $\\lbrace v_j=v_{j-1}\\rbrace $ , that is $\\widetilde{G}(w_1,\\dots ,w_{k-1}):=G(w_1,\\dots ,w_{j-1},w_j,w_j,w_{j+1},\\dots ,w_{k-1})\\,.$ In particular, we get a reduction of the number of variables $G$ depends on.", "Notice that, for $\\lambda =\\lambda _j$ , the function $\\widetilde{G}$ has no issues of differentiability.", "So that the minimizer $u^\\lambda $ can be found by solving the Euler-Lagrange equation for $\\widetilde{G}$ .", "Step 3.", "Find all the solutions.", "We just repeat Step 2 until $u^\\lambda _i=u^\\lambda _i$ for all index $i=1,\\dots ,k-1$ .", "That would be the value of $\\lambda $ for which we stop, Indeed, by Lemma REF , we know that the solution will remain constant for all smaller values of $\\lambda $ .", "Notice that, after the first critical event described in Step 2, another kind of critical event can take place.", "Namely, it can happen that some $u^\\lambda _i$ will change its relative position with respect to $f_i$ .", "If that happens, we just have to change the corresponding $\\bar{t}_i$ and/or $\\bar{r}_i$ .", "Example.", "We illustrate the above strategy with an example.", "For simplicity, we will treat the case $p=2$ .", "Suppose that $k=6$ , take $L_1=L_3=L_5=1\\,,\\quad \\quad L_2=L_4=L_6=2\\,.$ Consider the initial data $f$ given by $f_1=2,\\quad f_2=1,\\quad f_3=3,\\quad f_4=5,\\quad f_5=6,\\quad f_6=4\\,.$ Figure: The initial data ff.For $\\lambda \\gg 1$ , we know that we have to consider the following functional $G(u_1,u_2,u_3,u_4,u_5,u_6)&:= u_1-2u_2+2u_5-u_6+\\lambda [\\,(2-u_1)^2+2(1-u_2)^2\\\\&\\hspace{14.22636pt}+|u_3-3|^2+2|u_4-5|^2+(6-u_5)^2+2(u_6-4)^2\\,]\\,.$ In particular, we obtain that the solution $u^\\lambda $ is given by $\\begin{array}{lllll}u_1^\\lambda :=2-\\frac{1}{2\\lambda }\\,,&&u_2^\\lambda :=1+\\frac{1}{2\\lambda }\\,,&&u_3^\\lambda :=3\\,,\\\\&&&&\\\\u_4^\\lambda :=5\\,,&&u_5^\\lambda :=6-\\frac{1}{\\lambda }\\,,&&u_6^\\lambda :=4+\\frac{1}{4\\lambda }\\,.\\end{array}$ for $\\lambda >1$ .", "Figure: The behavior of the solution for λ>1\\lambda >1 as λ\\lambda decreases.The first critical event happens for $\\lambda =1$ , when $u_1^\\lambda =u_2^\\lambda $ and $u_4^\\lambda =u^\\lambda _5$ .", "For smaller values of $\\lambda $ , we have to consider the functional $G(v_1,v_2,v_3,v_4)&:= 2v_3-v_1-v_4 + \\lambda [\\, (2-v_1)^2+2(v_1-1)^2+|v_3-3|^2\\\\&\\hspace{14.22636pt}+2(5-v_3)^2+(6-v_3)^2+2(v_4-4)^2 \\,]\\,.$ Here, the solution is given by $\\begin{array}{lllll}u_1^\\lambda :=\\frac{4}{3}+\\frac{1}{6\\lambda }\\,,&&u_2^\\lambda :=\\frac{4}{3}+\\frac{1}{6\\lambda }\\,,&&u_3^\\lambda :=3\\,,\\\\&&&&\\\\u_4^\\lambda :=\\frac{16}{3}-\\frac{1}{3\\lambda }\\,,&&u_5^\\lambda :=\\frac{16}{3}-\\frac{1}{3\\lambda }\\,,&&u_6^\\lambda :=4+\\frac{1}{4\\lambda }\\,.\\end{array}$ Figure: The behavior of the solution for λ∈(9 14,1]\\lambda \\in (\\frac{9}{14},1] as λ\\lambda decreases.Then, for $\\lambda =\\frac{9}{14}$ we have that $u_6^\\lambda =u_5^\\lambda $ .", "Hence, the new functional we have to consider is $G(v_1,v_2,v_3)&:= v_3-v_1 + \\lambda [\\, (2-v_1)^2+2(v_1-1)^2+|v_2-3|^2\\\\&\\hspace{14.22636pt}+2(5-v_3)^2+(6-v_3)^2+2(v_3-4)^2 \\,]\\,.$ Figure: The behavior of the solution for λ∈(,1 10,9 14]\\lambda \\in (,\\frac{1}{10},\\frac{9}{14}] as λ\\lambda decreases.The solution is now $\\begin{array}{lllll}u_1^\\lambda :=\\frac{4}{3}+\\frac{1}{6\\lambda }\\,,&&u_2^\\lambda :=\\frac{4}{3}+\\frac{1}{6\\lambda }\\,,&&u_3^\\lambda :=3\\,,\\\\&&&&\\\\u_4^\\lambda :=\\frac{16}{3}-\\frac{1}{6\\lambda }\\,,&&u_5^\\lambda :=\\frac{16}{3}-\\frac{1}{6\\lambda }\\,,&&u_6^\\lambda :=\\frac{16}{3}-\\frac{1}{6\\lambda }\\,.\\end{array}$ Notice that for $\\lambda =\\frac{1}{4}$ we have $u_1^\\lambda =f_1$ .", "Thus, for $\\lambda <\\frac{1}{4}$ , we have to consider the functional $G(v_1,v_2,v_3)&:= v_3-v_1 + \\lambda [\\, (v_1-2)^2+2(v_1-1)^2+|v_2-3|^2\\\\&\\hspace{14.22636pt}+2(5-v_3)^2+(6-v_3)^2+2(v_3-4)^2 \\,]\\,.$ Hence, the solution remains equal to the previous ones.", "For $\\lambda =\\frac{1}{10}$ we get $u_2^\\lambda =u_3^\\lambda $ .", "Then we consider the functional $G(v_1,v_2)&:= v_2-v_1 + \\lambda [\\, (2-v_1)^2+2(v_1-1)^2+|v_2-3|^2\\\\&\\hspace{14.22636pt}+2(5-v_2)^2+(6-v_2)^2+2(v_2-4)^2 \\,]\\,.$ Figure: The behavior of the solution for λ∈(9 122,1 10]\\lambda \\in (\\frac{9}{122},\\frac{1}{10}] as λ\\lambda decreases.Such a functional is minimized by $\\begin{array}{lllll}u_1^\\lambda :=\\frac{7}{4}+\\frac{1}{8\\lambda }\\,,&&u_2^\\lambda :=\\frac{7}{4}+\\frac{1}{8\\lambda }\\,,&&u_3^\\lambda :=\\frac{7}{4}+\\frac{1}{8\\lambda }\\,,\\\\&&&&\\\\u_4^\\lambda :=\\frac{24}{5}-\\frac{1}{10\\lambda }\\,,&&u_5^\\lambda :=\\frac{24}{5}-\\frac{1}{10\\lambda }\\,,&&u_6^\\lambda :=\\frac{24}{5}-\\frac{1}{10\\lambda }\\,.\\end{array}$ Finally, for $\\lambda \\le \\frac{9}{122}$ we have that the solution is given by $u_1^\\lambda =u_2^\\lambda =u_3^\\lambda =u_4^\\lambda =u_5^\\lambda =u_6^\\lambda :=\\frac{31}{9}\\,.$ Figure: The behavior of the solution for λ<9 122\\lambda <\\frac{9}{122}.Remark 6.1 The previous example allows us to identify some properties of the solution $u^\\lambda $ : it is not true that if $u_i^{\\bar{\\lambda }}=f_i$ , then $u_i^\\lambda =f_i$ for all $\\lambda \\ge \\bar{\\lambda }$ , the function $\\lambda \\mapsto u_i^\\lambda $ is not monotone in general.", "Nevertheless, a change in the monotonicity can happen only if $\\lambda =\\lambda _i$ or $\\lambda =\\lambda _{i-1}$ , Remark 6.2 Let us denote by $u^{\\lambda ,p}$ the solution of problem (REF ) corresponding to $p$ and $\\lambda $ .", "Although we know that, for every $\\lambda $ fixed, $u^{\\lambda ,p}\\rightarrow v$ as $p\\searrow 1$ , where $v$ is a solution of the problem (REF ) corresponding to $\\lambda $ and $p=1$ , we cannot apply directly our method to find $v$ , since analytic computations are difficult to perform in the case $p\\in (1,2)$ .", "Nevertheless, a finer analysis of the behavior of the solution $u^{\\lambda ,p}$ for $p\\in (1,2)$ is currently under investigation.", "Acknowledgments.", "The author wishes to thank Irene Fonseca for having introduced him to the study of this problem and for helpful discussions during the preparation of the paper.", "The author warmly thanks the Center for Nonlinear Analysis at Carnegie Mellon University for its support during the preparation of the manuscript.", "The research was funded by National Science Foundation under Grant No.", "DMS-1411646." ] ]
1612.05508
[ [ "Higher-order quantum bright solitons in Bose-Einstein condensates show\n truly quantum emergent behavior" ], [ "Abstract When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-Pitaevskii equation.", "Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with analytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscillations are absent.", "Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phenomenon." ], [ "unicode=true, a4paper=true, plainpages=false, pdftitle=Title of PDF, pdfauthor=Author of PDF, pdfsubject=Subject of PDF, colorlinks=true, citecolor=blue, rm Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent behavior Christoph Weiss [email protected] Joint Quantum Centre (JQC) Durham–Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom Lincoln D. Carr [email protected] Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-Pitaevskii equation.", "Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with analytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscillations are absent.", "Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phenomenon.", "05.60.Gg, 03.75.Lm, 03.75.Gg Bright soliton, Bose-Einstein condensation, Quantum many-body physics, Far-from-equilibrium quantum dynamics, Semiclassical breakdown The quantum-classical correspondence is well-established for single-particle quantum mechanics but is known to be problematic for some many-body quantum problems such as strongly correlated systems and even materials as simple as the antiferromagnet.", "A key macroscopic prediction of Bose-Einstein condensates (BECs) is the bright soliton, appearing as a localized robust ground state “lump” for attractive BECs.", "Based on the ubiquity of semiclassical limits for non-interacting and weakly interacting bosons, such as lasers and BECs, one might expect a well-defined emergent macroscopic classical behavior generically from such systems.", "To date, most aspects of matter-wave bright soliton experiments [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] seem to be explained on the semiclassical mean-field level via the Gross-Pitaevskii equation (GPE): thus they display quantum behavior on a single-particle level matching classical wave experiments such as nonlinear photonic crystals [12] and spin-waves in ferromagnetic films [13], [14].", "This statement is supported by the fact that quantum-quantum bright solitons [15], [16], [17], [18], [19], [20], [21], [22] — matter-wave bright solitons that display quantum behavior beyond the single-particle mean-field level — for many practical purposes show mean-field behavior predicted by the GPE emerging already for particle numbers as low as $N\\gtrapprox 3$  [23].", "So far, beyond-mean field effects only seem to play a role if two or more distinct bright solitons are involved: two matter-wave quantum bright solitons can behave quite differently from matter-wave mean-field bright solitons.", "Only the latter necessarily have a well-defined relative phase [24].", "Both the limit of well-defined phase [7] and the limit involving a superposition of many phases [25], [26] are experimentally relevant for matter-wave bright solitons [7], [26].", "In this Letter we show that truly quantum many-body effects are responsible for the dynamics of a single quantum-quantum bright soliton, a smoking-gun signal for quantum emergence in BEC experiments.", "For far-from equilibrium dynamics of beyond-ground state quantum bright solitons, we are only at the beginning of a journey similar to the case of quantum dark solitons.", "That scientific voyage required multiple lines of investigations [27], [28], [29], [30], [31] to arrive at the state-of-the art explanation that atom losses are necessary to obtain mean-field properties from many-body quantum solutions [32].", "Dark solitons were also realized experimentally in BECs [33] and have been further explored in detail over the years in comparison to such predictions, e.g. [34].", "In contrast, bright solitons to-date lack for instance a phase coherence measurement, let alone the kind of far-from-equilibrium dynamics we are predicting here.", "Thus we focus on a quantum bright soliton experiment easily accessible in current platforms.", "Specifically, one first prepares a single ground-state bright soliton and then rapidly changes the interaction, an “interaction quench” via a Feshbach resonance, a well-established experimental technique.", "For one-dimensional Bose gases recent work related to quenches includes positive-to-negative quenches [21], [35], and zero-to-positive quenches [36].", "Quenches involving dark-bright solitons [37], [38], quenched dynamics of two-dimensional solitary waves [39], and breathers in discrete nonlinear Schrödinger equations [40], [41], [42] were also investigated, as well as the breathing motion after a quench of the strength of a harmonic trap [43].", "For attractive BECs, there are very specific mean-field predictions [44]: in particular, for an interaction quench by a factor of four there are exact analytical mean-field results available that predict robust perfectly oscillatory behavior for all times [45].", "However, how quantum bright solitons would behave in such a situation is an open question which we address in the current Letter.", "One GPE interpretation of a higher order soliton is $N_s$ bound bright solitons, here $N_s=2$ , a kind of diatomic solitonic molecule in a nonlinear vibrational mode.", "One might therefore expect quantum fluctuations to cause the two solitons to unbind via e.g quantum tunneling out of a many-body potential, resulting in two equal-sized solitons moving away from each other [46].", "This is not at all what we find, and is inconsistent with exact results for the center-of-mass wave function [47].", "Moreover, our beyond-mean-field results are distinct from the GPE failing for strongly correlated systems like Mott insulators [48], [49], [50]; as well as from many-body systems on short timescales with differences disappearing for typical experimental parameters and large BECs [51].", "We will show that an interaction quench leaves a large soliton core with small emissions of single particles.", "Experimentally these dynamics will appear as a “fizzled” higher order bright soliton, a stationary soliton core with a small thermal cloud.", "Thus we establish a new kind of quantum macroscopicity in weakly interacting bosonic systems.", "The mean-field approach via the GPE is a powerful approximation which provides physical insight into weakly interacting ultracold atoms.", "In a quasi-one-dimensional wave guide [52], [1], [2], [3], [4], [6], [7], [8], [5], [9], [10], [11] the GPE reads $i\\hbar \\partial _t\\varphi (x,t) = -(\\hbar ^2/2m)\\partial _{xx}\\varphi (x,t)+(N-1)g_{1 \\rm D}|\\varphi (x,t)|^2 \\varphi (x,t),$ where $\\varphi (x,t)$ is a complex wave function normalized to unity and $N$ is the number of atoms of mass $m$ .", "The attractive interaction $g_{\\rm 1D} &=2\\hbar \\omega _{\\perp }a <0$ is proportional to the s-wave scattering length $a$ and the perpendicular angular trapping-frequency, $\\omega _{\\perp }$  [53].", "Some GPE predictions for repulsive BECs even become exact [54], [55] in the mean-field limit $g_{\\rm 1D}\\rightarrow 0,\\quad N\\rightarrow \\infty ,\\quad (N-1)g_{\\rm 1D} = \\rm const.$ While quantum bright solitons in their internal ground state in addition have a center-of-mass wavefunction (see Refs.", "[56], [57] and references therein), for measurements both many-body quantum physics [58], [17] and the GPE [59] predict bright solitons localized at $X_0$ with a single-particle density profile of form $\\varrho (x)\\equiv |\\varphi (x)|^2 = (2\\xi _N\\left\\lbrace \\cosh [(x-X_0)/(2\\xi _N)]\\right\\rbrace ^2)^{-1},$ where the soliton length $\\xi _N$ and the related soliton time $\\tau _N$ $\\xi _N \\equiv \\hbar ^2[m(N-1)\\left|g_{\\rm 1D}\\right|]^{-1}; \\quad \\tau _N\\equiv m\\xi _N^2/\\hbar .$ remain constant when approaching the mean-field limit (REF ).", "In this Letter we use an interaction quench $g_{\\rm 1D} (t)= \\left\\lbrace \\begin{array}{lcr}g_0 &:&t\\le 0\\\\\\eta g_0 &:&t>0\\end{array}\\right.", ";\\quad \\eta \\ge 1,\\quad g_0<0.$ After an interaction quench by a factor of $\\eta =4$ , the GPE yields the analytical result [45] $\\varrho (x,t) = \\left|{\\frac{\\cosh \\left[ 3x/(2\\xi _N) \\right] +3\\,{{\\rm e}^{-{\\rm i}t/\\tau _N}}\\cosh \\left[ x/(2\\xi _N) \\right] }{3\\,\\cos \\left( t/\\tau _N \\right) +4\\,\\cosh \\left( x/\\xi _N \\right) +\\cosh \\left( 2\\,x/\\xi _N \\right) }}\\right|^2,$ which is depicted in Fig.", "REF .", "For $3/2< \\eta ^{1/2} < 5/2$ mean-field predictions also are very specific: after losing a few atoms the system self-cools to the robust higher-order bright soliton of Eq.", "(REF ) [44].", "Figure: Semiclassical emergent dynamics.", "After an interaction quench by a factor of four, the GPE mean-field theorypredicts perfect oscillatory behavior,Eq. () .", "Is itrealistic to expect BEC experiments to reproduce these oscillations?", "(a) GPE density, normalized to its maximum for thefirst two oscillation periods as a function of space and time in soliton units.", "(b) 2D projection of (a).Within the text book derivation [59] of the GPE, the many-body wave function corresponding to the GPE is a Hartree-product state $\\psi _{\\rm GPE}(x_1,x_2,\\ldots ,x_N) = \\textstyle \\prod _{j=1}^N\\varphi (x_j).$ The mean-square difference between the position of two particles $\\langle (x_1-x_2)^2\\rangle _{\\rm GPE}(t)\\equiv \\textstyle \\int \\!", "dx_1\\!", "\\int dx_2\\,\\varrho (x_1,t) \\varrho (x_2,t)(x_1-x_2)^2,$ a measure that is distinct from and independent of the expansion of the center-of-mass wave function, can thus be calculated form the above analytical result to obtain $\\Delta _{1,2} \\equiv \\langle (x_1-x_2)^2\\rangle (t)/\\langle (x_1-x_2)^2\\rangle (0).$ While the mean-field prediction thus is a perfectly periodic function of period $2\\pi \\tau _N$ , the question is what we expect to find on the many-body quantum level .", "For fundamental considerations on the many-body level corresponding to the GPE, a very useful tool is the Lieb-Liniger Model (LLM) with the Hamiltonian [60], [61] $\\hat{H} = -(\\hbar ^2/2m)\\textstyle \\sum _{j=1}^N(\\partial ^2/\\partial x_j^2)+\\sum _{j=1}^{N-1}\\sum _{\\nu =j+1}^{N}g_{\\rm 1D}\\delta (x_j-x_{\\nu }),$ where $x_j$ denotes the position of particle $j$ of mass $m$ .", "The ground-state energy is given by $E_0(N)= -mg_{1\\rm D}^2 N(N^2-1)/(24\\hbar ^2).$ The energy eigenstates of excited states can be written as $E=\\textstyle \\sum _{r=1}^{N_{\\rm S}}\\left(E_0(N_r) +N_r \\hbar ^2 k_r^2/2m\\right), \\:\\:N=\\sum _{r=1}^{N_{\\rm S}} N_r,\\:\\: N_r>0$ corresponding to the intuitive interpretation of $N_{\\rm S}$ solitonlets — solitons that contain a fraction of the total number of particles — of size $N_r$ ($r=1,2,\\ldots ,N_{\\rm S}$ ) and their individual center-of-mass kinetic energy.", "Equation (REF ) is valid if the system size $L$ is large compared to even a two-particle soliton — this can be included by adding a diverging system size to the mean-field limit (REF ) to get [62], [63] $g_{\\rm 1D}\\rightarrow 0,\\; N\\rightarrow \\infty ,\\; L\\rightarrow \\infty ,\\; \\xi _N = {\\rm const.", "},\\; N/L= {\\rm const.", "}$ Reaching such a limit is a difficult numerical problem [64].", "However, by replacing the Hamiltonian (REF ) by the Bose-Hubbard model (BHM) used to model quantum bright solitons by e.g.", "[56], [65], [66], [57], we introduce thermalization mechanisms present in real experiments such as a weak imperfectly harmonic trap, or a one-dimensional waveguide embedded in a 3D geometry; for the BHM thermalization is due specifically to a lattice, in our case in the limit of very weak discretization.", "The BHM takes the form $\\hat{H}_{\\rm BHM} =~&-J\\textstyle \\sum _{j}\\left(\\hat{b}_j^{\\dag }\\hat{b}_{j+1}^{\\phantom{\\dag }}+\\hat{b}_{j+1}^{{\\dag }}\\hat{b}_{j}^{\\phantom{\\dag }}\\right)+ \\frac{1}{2}U\\sum _{j}\\hat{n}_j\\left(\\hat{n}_j-1\\right)$ where $\\hat{b}_j^{\\dag }$ ($\\hat{b}_{j}^{\\phantom{\\dag }}$ ) creates (annihilates) a particle on lattice site $j$ , $U<0$ quantifies the interaction energy of a pair of atoms and $\\hat{n}_j$ counts the number of atoms on lattice site $j$ .", "In order to use this in a way we can directly use the physical insight gained from the LLM (REF ), we choose for the hopping matrix element (cf.", "[56]) $J = \\hbar ^2/(2m {\\delta _{\\rm L}}^2)$ such that both models have the same single-particle dispersion in the long wavelength limit $k\\delta _{\\rm L}\\ll \\pi $ , with $\\delta _{\\rm L}$ the lattice constant.", "The interaction $U = g_\\mathrm {1D}(32\\,J{{\\hbar }}^{2}m+{m}^{2}{g}^{2}_\\mathrm {1D})^{1/2}/(4{{\\hbar }}^{2})$ is chosen such that the two-particle ground state has the same ground-state energy as Eq.", "(REF ) compared to the free gas [56].", "While both the weak lattice introduced by Eq.", "(REF )] and a weak harmonic trap [67], [68] break the integrability of the LLM, we can still approximately describe eigenstates by Eq.", "(REF ).", "Furthermore, from a modeling point of view, by choosing the lattice we avoid the divergence of the energy fluctuations of the initial state immediately after the interaction quench, caused by delta functions squared, in $\\langle \\Delta \\hat{H}^2_{\\rm new}\\rangle _0 = (\\eta -1)^2\\left[\\langle \\hat{H}_{\\rm int~old}^2\\rangle _0-\\langle \\hat{H}_{\\rm int~old}\\rangle _0^2\\right]$ .", "While in physics distributions with well-defined mean and diverging variance are well-known [69], a more severe reason for avoiding the LLM limit is that this limit seems to be mathematically ill-defined – an initial wave function with the wrong boundary conditions at $x_j=x_{\\ell }$ ($j\\ne \\ell $ ) has to be expressed in terms of eigenfunctions with the correct boundary conditions [17].", "Summarizing, we note that these energy fluctuations are consistent with the LLM predicting the presence of quantum superpositions involving many solitonlets in the initial state, but inconsistent with simple pictures predicting two large solitonlets that either oscillate [45] around each other or separate from each other [46] as the latter cannot happen rapidly [47].", "We use time-evolving block decimation (TEBD) [70] — a numerical method based on matrix product states [71], [72] — to solve the BHM (REF ).", "In order to exclude both boundary effects and effects introduced by additional traps, we start with a very weak harmonic trap, such that opening it hardly introduces atom losses [73] and thus the analytical result (REF ) remains valid.", "In our simulations, we switch this trap off at the same time as we introduce the interaction quench.", "If for $N\\gtrapprox 3$ quantum bright solitons indeed already show mean-field behavior [23], we should be able to see the mean-field oscillations depicted in Fig.", "REF already for $N=3$ .", "Figures REF and REF show that the mean-field oscillations are absent from the many-body TEBD data for $N=3$ and $N$ up to 16, respectively.", "For the BHM (REF ), we note that the relative particle measurement of Eq.", "(REF ) can be rewritten in second quantized form appropriate to TEBD by replacing $\\langle (x_1-x_2)^2\\rangle $ with $\\sum _{j,\\ell }(j-\\ell )^2\\langle \\hat{b}_j^\\dag \\hat{b}_j^{\\phantom{\\dag }}\\hat{b}_\\ell ^\\dag \\hat{b}_\\ell ^{\\phantom{\\dag }}\\rangle $ .", "Figure: Many-body quantum emergent dynamics.", "Following an interactionquench by a factor of η\\eta , the soliton shows no indication of special valuesfor the strength of the quench η\\eta (initial parameters used for BHM:N=3N=3, J=0.5J=0.5, U≃-0.5U\\simeq -0.5).", "(a) Singleparticle density as a function of both position and time;the mean-field oscillations displayed in Fig.", "for η=4\\eta =4 are absent in the TEBD data.", "(b) Square root of relative width Δ 1,2 \\sqrt{\\Delta _{1,2}} as a function of both time andquench strength η\\eta shows that the absence of mean-fieldoscillations are generic.", "(c) The condensate fraction provides afurther indicator of beyond-mean field behavior.", "(d) The sum overthe largest 11 eigenvalues λ j \\lambda _j of the single-particle density matrix(normalized to 1) shows that the system becomes even less mean-fieldfor longer times.Figure: Relative width measures for up to 16 particles.", "While the GPE predicts perfect oscillations, the predominant behavior in the matrix-product state numerics is that the relative distance of particles grows.", "(a) Relative width Δ 1,2 {\\Delta _{1,2}}as afunction of time: mean-field oscillations (lowest black dotted curve); TEBD results (N=3,4,8,16N=3,\\,4,\\,8,\\,16 blue, fuscia, brown, red curves from top to bottom); overall behavior is quadratic (fits shown as light blue curves).", "Here J=0.5J=0.5 and U≃-0.5,-0.33,-0.14,-0.07U\\simeq \\,-0.5,\\,-0.33,\\,-0.14,\\,-0.07.", "(b) Residual oscillations in panel (a) after subtracting the quadratic fit.", "(c) Relative errorcomparing TEBD data with distinct convergence parameters.", "From top tobottom: N=16N=16: χ max =60\\chi _{max}= 60 versus χ max =80\\chi _{max}= 80, N=8N=8:χ max =60\\chi _{max}= 60 vs. χ max =80\\chi _{max}= 80, N=4N=4: χ max =30\\chi _{max}= 30 vs. χ max =40\\chi _{max}= 40, N=3N=3 (too small to be visible): χ max =30\\chi _{max}= 30 vs. χ max =40\\chi _{max}= 40.Within the LLM, for relative distances large compared to the soliton length $\\xi _N$ , the leading-order contributions to excited states consists of $N_{\\rm S}$ solitonlets of terms corresponding to $N_{\\rm S}$ individual solitonlets moving apart [17] In order to obtain a physical understanding on the time scales on which these solitonlets can move apart if they initially sit on top of each other, we recall the text book result for the variance of an initially Gaussian single-particle wave function [74], $\\Delta X^2 = \\Delta X_0^2\\lbrace 1+[\\hbar t/(2M\\Delta X_0^2)]^2\\rbrace .$ For the relative motion the mass $M=Nm$ has to be replaced by the relative mass $m_{\\rm rel}\\equiv m\\,N_1N_2/(N_1+N_2)$ .", "If initially localized to $\\Delta X_0\\propto \\xi _N$ (much stronger localization leads to too high kinetic energies while much weaker localization leads to a too wide initial wave function) and for a relative mass independent of $N$ , the relative wave function will expand to a size larger than the initial wave function on time scales [cf. Eq.", "(REF )] $t\\propto m_r\\xi _N^2/\\hbar = [(N_1N_2)/(N_1+N_2)]\\tau _{N}.$ The Hartree product states (REF ) are also ideal to calculate mean energies which in the mean-field limit (REF ) become identical to the exact many-body quantum results [17].", "The kinetic energy prior to the interaction quench is $\\langle E_{\\rm kin}\\rangle _{\\rm old}=N^3mg_{1\\rm D}^2/(24\\hbar ^2) = -E_0^{\\rm old}$ , and the interaction energy $\\langle E_{\\rm int}\\rangle _{\\rm old}= 2E_0^{\\rm old}$ .", "Immediately after the interaction quench, the kinetic energy remains unchanged and the interaction energy is increased to $\\langle E_{\\rm int}\\rangle _{\\rm new}=\\eta \\langle E_{\\rm int}\\rangle _{\\rm old}=2\\eta E_0^{\\rm old}$ .", "In units of the new ground state energy $E_0^{\\rm new} = \\eta ^2 E_0^{\\rm old}$ we have a total average energy after the interaction quench of ${\\langle E\\rangle }= [(2\\eta -1)/\\eta ^2] {E_0^{\\rm new} }$ where $0< (2\\eta -1)/\\eta ^2<1$ for $\\eta >0.5$ and $\\eta \\ne 1$ .", "If ultracold attractive atoms are initially prepared in their ground state, by using the eigenstate thermalisation hypothesis [75], [76] we conjecture that an interaction quench by a factor of $\\eta $ will on short time scales lead to a final state consisting of a single bright soliton containing $N_1$ atoms, as given by thermodynamic predictions, and $N-N_1$ free atoms.", "In the mean-field limit (REF ), for bright solitons thermodynamic predictions read [77] $N_1 =\\left(\\frac{2\\eta -1}{\\eta ^2}\\right)^{1/3}N,\\;\\; \\xi _{N_1} =\\frac{1}{[\\eta (2\\eta -1)]^{1/3}}\\xi _N, \\;\\; \\eta >\\frac{1}{2},$ i.e., one large soliton with reduced particle number $N_1$ and reduced size $\\xi _{N_1}$ ; and $N-N_1$ single atoms which are not bound in molecules.", "To suggest that this might indeed be what happens seems counterintuitive at best, since following “thermalization” according to the eigenstate thermalization hypothesis, all energetically accessible eigenfunctions will be involved [75], [76]; violating both the Landau hypothesis [78], which at first glance seems to prevent co-existence of a large soliton and a free gas; argued against also by mean-field predictions [45], [44] as well as thermodynamic predictions for ultracold atoms in contact with a heat bath [63], [62].", "However, the Landau hypothesis is based on assumptions that are not fulfilled for bright solitons [77] and thermally isolated ultracold atoms, arguably realised in state-of-the-art experiments with bright solitons [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], behave quite differently from those in contact with a heat bath [77].", "Furthermore, contrary to rumours stating otherwise, one-dimensional Bose gases do thermalise, for example in the presence of a weak harmonic trap [67], [68].", "As depicted in Fig.", "REF , we conjecture that after an interaction quench to more negative interactions, the system will relax towards the situation predicted in thermal equilibrium: the co-existence between one large soliton and a free gas [77].", "The application of the eigenstate thermalisation hypothesis [75], [76] is supported by the fact that single atoms will move the initial cloud much faster than larger solitonlets [Eq.", "(REF )] that can continue to thermalize.", "Two macroscopic solitons sitting on top of each other would have to remain at the same position [47]; a freely expending gas passes the convergence test of Ref.", "[47] but energy conservation and Eq.", "(REF ) would require at least one soliton(let) to be present.", "Figure: Extrapolation to large particle number.", "By applying the eigenstate thermalisationhypothesis , we conjecturethat after an interaction quench to more negative interactions by afactor of η\\eta , the attractive BEC approaches the equilibriumpredictions for a thermally isolated gas of Ref. .", "(a) New soliton length, in units of the original soliton length ξ N, old \\xi _{N,\\rm old},for a soliton containing all atoms (lower curve) versus N 1 N_1 emitted free atoms(upper curve) as a function of the quench η\\eta .", "(b) Fraction of atoms in the soliton (uppercurve) and in the free gas (lower curve) [see also Eq.", "()].To conclude, we have combined evidence from three distinct models (GPE, LLM and BHM) to show that truly quantum emergent behaviour for attractive Bosons happens after an interaction quench to more attractive interactions.", "Combining the numerical evidence with general considerations based on the eigenstate thermalization hypothesis for larger particle numbers, we conjecture that the final many-body quantum state consists of one smaller bright soliton and lots of single atoms, thus yielding an ultimate example of a mean-field breakdown on time scales that remain experimentally relevant even in the mean-field limit (REF ).", "Our predictions are accessible to state-of-the-art experiments with thousands of atoms [1], [2], [3], [4], [6], [7], [8], [5], [9], [10], [11].", "Furthermore, the above conjecture offers an explanation as to why experiments that quasi-instantaneously switch from repulsive to attractive interactions (see for example Ref.", "[5]) while avoiding the “Bose-nova” collapse or modulational instability can nevertheless lead to one large matter-wave bright soliton (and a thermal cloud) being formed.", "We thank T. P. Billam, J.", "Brand, J. Cosme, S. A. Gardiner, B. Gertjerenken, and M. L. Wall for discussions.", "We thank the Engineering and Physical Sciences Research Council UK for funding (Grant No. EP/L010844/).", "This material is based in part upon work supported by the US National Science Foundation under grant numbers PHY-1306638, PHY-1207881, and PHY-1520915, and the US Air Force Office of Scientific Research grant number FA9550-14-1-0287.", "L.D.C.", "thanks Durham University and C.W.", "the Colorado School of Mines for hosting visits to support this research.", "Data will be available online soon [79]." ] ]
1612.05545
[ [ "Mean Field Games with Singular Controls" ], [ "Abstract This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls.", "We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls.", "Our existence and approximation results strongly hinge on the use of the Skorokhod $M_1$ topology on the space of c\\`adl\\`ag functions." ], [ "Introduction and overview", "Starting with the seminal papers [20], [27], the analysis of mean field games (MFGs) has received considerable attention in the stochastic control and financial mathematics literature.", "In a standard MFG, each player $i \\in \\lbrace 1, ..., N\\rbrace $ chooses an action from a given set of admissible controls that minimizes a cost functional of the form $J^i(u)=E\\left[\\int _0^Tf(t,X^i_t,\\bar{\\mu }^N_t,u^i_t)dt+g(X^i_T,\\bar{\\mu }^N_T)\\right]$ subject to the state dynamics $\\left\\lbrace \\begin{array}{ll}dX^i_t=b(t,X^i_t,\\bar{\\mu }^N_t,u^i_t)\\,dt+\\sigma (t,X^i_t,\\bar{\\mu }^N_t,u^i_t)\\,dW^i_t,\\\\X^i_0=x_0\\end{array} .\\right.$ Here $W^1, ..., W^N$ are independent Brownian motions defined on some underlying filtered probability space, $u=(u^1,\\cdots ,u^N)$ , $u^i=(u^i_t)_{t \\in [0,T]}$ is an adapted stochastic process, the action of player $i$ , and $\\bar{\\mu }^N_t:=\\frac{1}{N}\\sum _{j=1}^N\\delta _{X^j_t}$ denotes the empirical distribution of the individual players' states at time $t \\in [0,T]$ .", "In particular, all players are identical ex ante and each player interacts with the other players only through the empirical distribution of the state processes.", "The existence of approximate Nash equilibria in the above game for large populations has been established in [4], [20] using a representative agent approach.", "In view of the independence of the Brownian motions the idea is to first approximate the dynamics of the empirical distribution by a deterministic measure-valued process, and to consider instead the optimization problem of a representative player that takes the distribution of the states as given, and then to solve the fixed-point problem of finding a measure-valued process such that the distribution of the representative player's state process $X$ under her optimal strategy coincides with that process.The idea of decoupling local from global dynamic in large population has been applied to equilibrium models of social interaction in e.g.", "[17], [18].", "Following the representative agent approach, a MFG can then be formally described by a coupled optimization and fixed point problem of the form: $\\left\\lbrace \\begin{array}{ll}1.&\\textrm {fix a deterministic function }t\\in [0,T]\\mapsto \\mu _t\\in \\mathcal {P}(\\mathbb {R}^d);\\\\2.&\\textrm {solve the corresponding stochastic control problem}:\\\\&\\inf _{u} E\\left[\\int _0^Tf(t,X_t,\\mu _t,u_t)\\,dt+g(X_T,\\mu _T)\\right],\\\\&\\textrm {subject to}\\\\&dX_t =b(t,X_t,{\\mu }_t,u_t)\\,dt+\\sigma (t,X_t,{\\mu }_t,u_t)\\,dW_t\\\\&X_{0} =x_0,\\\\3.&\\textrm {solve the fixed point problem:} ~Law(X) =\\mu ,\\end{array}\\right.$ where $\\mathcal {P}(\\mathbb {R}^d)$ is the space of probability measures on $\\mathbb {R}^d$ and $Law(X)$ denotes the law of the process $X$ .", "There are essentially three approaches to solve mean field games.", "In their original paper [27], Lasry and Lions followed an analytic approach.", "They analyzed a coupled forward-backward PDE system, where the backward component is the Hamiltion-Jacobi-Bellman equation arising from the representative agent's optimization problem, and the forward component is a Kolmogorov-Fokker-Planck equation that characterizes the dynamics of the state process.", "A second, more probabilistic, approach was introduced by Carmona and Delarue in [4].", "Using a maximum principle of Pontryagin type, they showed that the fixed point problem reduces to solving a McKean-Vlasov forward-backward SDEs (FBSDEs for short).", "Other results based on probabilistic approaches include [1], [3], [5].", "Among them, [3], [5] consider linear-quadratic MFGs, while [1], [9] consider MFGs with common noise and with major and minor players, respectively.", "A class of MFGs in which the interaction takes place both through the state dynamics and the controls has recently been introduced in [8].", "In that paper the weak formulation, or martingale optimality principle, is used to prove the existence of a solution.", "A relaxed solution concept to MFGs was introduced by Lacker in [26].", "Considering MFGs from a more game-theoretic perspective, the idea is to search for equilibria in relaxed controls (“mixed strategies”) by first establishing the upper hemi-continuity of the representative agent's best response correspondence to a given $\\mu $ using Berge's maximum theorem, and then to apply the Kakutani-Fan-Glicksberg fixed point theorem in order to establish the existence of some measure-valued process $\\mu ^*$ such that the law of the agent's state process under a best response to $\\mu ^*$ coincides with that process.", "Relaxed controls date back to Young [31].", "They were later applied to stochastic control in, e.g.", "[14], [15], [25], to MFGs in [26], and to MFGs with common noise in [6].", "Applications of MFGs range from models of optimal exploitation of exhaustible resources [10], [12] to systemic risk [7], and from principal-agent problems [11] to problems of optimal trading under market impact [8], [22].", "Motivated by possible applications to optimal portfolio liquidation under strategic interaction that allow for both block trades and absolutely continuous trades as in [16], this paper provides a probabilistic framework for analyzing MFGs with singular controls.", "Extending [26], we consider MFGs with singular controls of the form $\\left\\lbrace \\begin{array}{ll}1.&\\textrm {fix a deterministic function }t\\in [0,T]\\mapsto \\mu _t\\in \\mathcal {P}(\\mathbb {R}^d);\\\\2.&\\textrm {solve the corresponding stochastic singular control problem}:\\\\&\\inf _{u,Z} E\\left[\\int _0^Tf(t,X_t,\\mu _t,u_t)\\,dt+g(X_T,\\mu _T)+\\int _0^Th(t)\\,dZ_t\\right],\\\\&\\textrm {subject to}\\\\&dX_t =b(t,X_t,{\\mu }_t,u_t)\\,dt+\\sigma (t,X_t,{\\mu }_t,u_t)\\,dW_t+c(t)\\,dZ_t,\\\\3.&\\textrm {solve the fixed point problem:~}Law(X) =\\mu ,\\end{array}\\right.$ where $u=(u_t)_{t \\in [0,T]}$ is the regular control, and $Z=(Z_t)_{t \\in [0,t]}$ is the singular control.", "When singular controls are admissible, the state process no longer takes values in the space of continuous functions, but rather in the Skorokhod space $\\mathcal {D}(0,T)$ of all càdlàg functions.", "The key is then to identify a suitable topology on the Skorokhod space with respect to which the compactness and continuity assumptions of the maximum and the fixed-point theorems are satisfied.", "There are essentially three possible topologies on the space of càdlàg functions: the (standard) Skorokhod $J_1$ topology ($J_1$ topology for short), the Meyer-Zheng topology (or pseudo-path topology), and the Skorokhod $M_1$ topology ($M_1$ topology for short).", "The $M_1$ topology seems to be the most appropriate one for our purposes.", "First, the set of bounded singular controls is compact in the $M_1$ topology but not in the $J_1$ topology.", "Second, there is no explicit expression for the metric corresponding to Meyer-Zheng topology.", "In particular, one cannot bound the value of a function at given points in time by the Meyer-Zheng topology.", "Third, the $M_1$ topology has better continuity properties than the $J_1$ topology.", "For instance, it allows for an approximation of discontinuous functions by continuous ones.", "This enables us to approximate solutions to certain classes of MFGs with singular controls by solutions to MFGs with only regular controls.", "Appendix B summarizes useful properties of the $M_1$ topology; for more details, we refer to the textbook of Whitt [30].", "To the best of our knowledge, ours is the first paper to establish the existence of solutions results to MFGs with singular controlsThe recent paper [13] only considers absolutely continuous singular controls.", "Our notion of singular controls is more general.. As a byproduct, we obtain a new proof for the existence of optimal (relaxed) controls for the corresponding class of stochastic singular control problems.", "A similar control problem, albeit with a trivial terminal cost function has been analyzed in [15].", "While the methods and techniques applied therein can be extended to non-trivial terminal cost functions after a modification of the control problem, they cannot be used to prove existence of equilibria in MFGs.", "In fact, in [15], it is assumed that the state space $\\mathcal {D}(0,T)$ is endowed with Meyer-Zheng topology, and that the spaces of admissible singular and regular controls are endowed with the topology of weak convergence and the stable topology, respectively.", "With this choice of topologies the continuity of cost functional and the upper-hemicontinuity of distribution of the representative agent's state process under the optimal control w.r.t.", "to a given process $\\mu $ cannot be established.", "As a second byproduct we obtain a novel existence of solutions result for a class of McKean-Vlasov stochastic singular control problems.", "Our second main contributions are two approximation results that allow us to approximate solutions to a certain class of MFGs with singular controls by the solutions to MFGs with only regular controls.", "The approximation result, too, strongly hinges on the choice of the $M_1$ topology.", "The rest of this paper is organized as follows: in Section 2, we recall the notion of relaxed controls for singular stochastic control problems, introduce MFGs with singular controls and state our main existence of solutions result.", "The proof is given in Section 3.", "In Section 4, we state and prove two approximation results for MFGs with singular controls by MFGs with regular controls.", "Appendix A recalls known results and definitions that are used throughout this paper.", "Append B reviews key properties of the $M_1$ topology." ], [ "Assumptions and the main results", "In this section we introduce MFGs with singular controls and state our main existence of solutions result.", "For a metric space $(E,\\varrho )$ we denote by $\\mathcal {P}_p(E)$ the class of all probability measures on $E$ with finite moment of $p$ -th order.", "For $p=0$ we write $\\mathcal {P}(E)$ instead of $\\mathcal {P}_0(E)$ .", "The set $\\mathcal {P}_p(E)$ is endowed with the Wasserstein distance $\\mathcal {W}_{p,(E,\\varrho )}$ ; see Definition REF .", "For a given interval $\\mathbb {I}$ we denote by $\\mathcal {D}(\\mathbb {I})$ the Skorokhod space of all $\\mathbb {R}^d$ -valued càdlàg functions on $\\mathbb {I}$ , by $\\mathcal {A}(\\mathbb {I}) \\subset \\mathcal {D}(\\mathbb {I})$ the subset of nondecreasing functions, by $\\mathcal {C}(\\mathbb {I})\\subset \\mathcal {D}(\\mathbb {I})$ the subset of continuous functions, and by $\\mathcal {U}(\\mathbb {I})$ the set of all measures on $\\mathbb {I} \\times U$ for some metric space $U$ , whose first marginal is the Lebesgue measure on $\\mathbb {I}$ , and whose second marginal belongs to $\\mathcal {P}(U)$ .", "For reasons that will become clear later we identify processes on $[0,T]$ with processes on the whole real line.", "For instance, we identify the space $\\mathcal {D}(0,T)$ with the space $\\widetilde{\\mathcal {D}}_{0,T}(\\mathbb {R})=\\lbrace x\\in \\mathcal {D}(\\mathbb {R}):x_t=0\\textrm { if }t<0\\textrm { and }x_t=x_T \\textrm { if }t>T\\rbrace .$ Likewise, we denote by $\\widetilde{\\mathcal {A}}_{0,T}(\\mathbb {R})$ and $\\widetilde{\\mathcal {C}}_{0,T}(\\mathbb {R})$ the subspace of $\\widetilde{\\mathcal {D}}_{0,T}(\\mathbb {R})$ with nondecreasing and continuous paths, respectively.", "Moreover, we denote by $\\widetilde{\\mathcal {U}}_{0,T}(\\mathbb {R})$ all measures $q(dt,du)$ on $\\mathbb {R}\\times U$ whose restriction to $[0,T]$ belongs to $\\mathcal {U}(0,T)$ , and whose restrictions to $(-\\infty ,0)$ and $(T,\\infty )$ are of the form $q(dt,du)=\\delta _{\\widetilde{u_0}}(du)dt$ and $q(dt,du)=\\delta _{\\widetilde{u_T}}(du)dt$ for some $\\widetilde{u_0}\\in U$ and $\\widetilde{u_T}\\in U$ , respectively.", "We occasionally drop the subscripts 0 and $T$ if there is no risk of confusion.", "Throughout this paper, $C > 0$ denotes a generic constant that may vary from line to line." ], [ "Singular stochastic control problems", "Before introducing MFGs with singular controls, we informally review stochastic singular control problems of the form: $\\left\\lbrace \\begin{array}{ll}\\inf _{u,Z} E\\left[\\int _0^Tf(t,X_t,u_t)\\,dt+g(X_T)+\\int _0^Th(t)\\,dZ_t\\right],\\\\\\textrm {subject to}\\\\dX_t=b(t,X_t,u_t)\\,dt+\\sigma (t,X_t,u_t)\\,dW_t+c(t)\\,dZ_t,\\\\X_{0-}=0.\\end{array}\\right.$ where all parameters are measurable in their respective arguments and are such that the control problem makes sense; see, e.g.", "[15] for details.Our specific assumptions on the model parameters are introduced in Section REF below.", "The regular control $u=(u_t)_{t \\in [0,T]}$ takes values in a compact metric space $U$ , and the singular control $Z=(Z_t)_{t \\in [0,T]}$ takes values in $\\mathbb {R}^d$ .", "For convenience we sometimes write $Z\\in \\widetilde{\\mathcal {A}}(\\mathbb {R})$ by which we mean that the sample paths of the stochastic process $Z$ belong to $\\widetilde{\\mathcal {A}}(\\mathbb {R})$ .", "Similarly, we occasionally write $X\\in \\widetilde{\\mathcal {D}}(\\mathbb {R})$ and $Y\\in \\widetilde{\\mathcal {C}}(\\mathbb {R})$ ." ], [ "Relaxed controls", "The existence of optimal relaxed controls to stochastic singular control problems has been addressed in [15] using the so-called compactification method.", "We use a similar approach to solve MFGs with singular controls, albeit in different topological setting.", "The following notion of relaxed controls follows [15] where we adopt our convention that all processes are extended to the whole real line.", "Definition 2.1 The tuple $r=(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z)$ is called a relaxed control if 1.", "$(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P})$ is a filtered probability space; 2.", "$\\mathbb {P}(X_{t}=0,Z_{t}=0, \\underline{Q}_t(du)=\\delta _{\\widetilde{u_0}}(du)\\textrm { if }t<0; X_{t}=X_T,Z_{t}=Z_T, \\underline{Q}_t(du)=\\delta _{\\widetilde{u_T}}(du)\\textrm { if }t>T)=1$ , for some $\\widetilde{u_0},\\widetilde{u_T}\\in U$ ; 3.", "$\\underline{Q}:\\mathbb {R}\\times \\Omega \\rightarrow \\mathcal {P}(U)$ is $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ progressively measurable, $Z$ is $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ progressively measurable and $Z\\in \\widetilde{\\mathcal {A}}(\\mathbb {R})$ ; 4.", "$X$ is a $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ adapted stochastic process, $X\\in \\widetilde{\\mathcal {D}}(\\mathbb {R})$ and for each $\\phi \\in \\mathcal {C}^2_b(\\mathbb {R}^d)$ , the space of all continuous and bounded functions with continuous and bounded first- and second-order derivatives, $\\mathcal {M}^{\\phi }$ is a well defined $\\mathbb {P}$ continuous martingale, where $\\begin{split}\\mathcal {M}^{\\phi }_t:=&~\\phi (X_t)-\\int _0^t\\int _U\\mathcal {L}\\phi (s,X_s,u)\\,\\underline{Q}_s(du)ds-\\int _0^t(\\partial _x\\phi (X_{s-}))^{\\top }c(s)dZ_s\\\\&-\\sum _{0\\le s\\le t}\\big (\\phi (X_s)-\\phi (X_{s-})-(\\partial _x\\phi (X_{s-}))^{\\top }\\triangle X_s\\big ),\\qquad t\\in [0,T]\\end{split}$ with $\\mathcal {L}\\phi (t,x,u):=\\frac{1}{2}\\sum _{ij}a_{ij}(t,x,u)\\frac{\\partial ^2\\phi (x)}{\\partial _{x_i}\\partial _{x_j}}+\\sum _ib_i(t,x,u)\\partial _{x_i}\\phi (x)$ and $a(t,x,u)=\\sigma \\sigma ^{\\top }(t,x,u)$ .", "The cost functional corresponding to a relaxed control $r$ is defined by $\\widetilde{J}(r)=E^{\\mathbb {P}}\\left[\\int _0^T\\int _Uf(t,X_t,u)\\,\\underline{Q}_t(du)dt+\\int _0^Th(t)\\,dZ_t+g(X_T)\\right].$ Let $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z)$ be a relaxed control.", "If the process $\\underline{Q}$ is of the form $\\underline{Q}_t(du)=\\delta _{u_t}(du)$ , for some progressively measurable $U$ -valued process $u$ , then we call $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,u,Z)$ a strict control.If there is no risk of confusion, then we call the processes $\\underline{Q}$ , respectively $u$ the relaxed, respectively strict control.", "In particular, any strict control corresponds to a relaxed control.", "Relaxed control can thus be viewed as a form of mixed strategies over strict controls.", "In particular, both the cost function and the state dynamics (more precisely, the martingale problem) are linear in relaxed controls.", "Furthermore, compactness w.r.t.", "relaxed controls is much easier to verify than compactness w.r.t.", "strict controls.", "Under suitable convexity conditions on the model data, the optimization problem over the set of relaxed controls is equivalent to the one over strict controls as shown by the following remark.", "Remark 2.2 For $(t,x)\\in [0,T]\\times \\mathbb {R}^d$ , let $K(t,x)=\\lbrace (a(t,x,u),b(t,x,u),e):~e\\ge f(t,x,u),~u\\in U\\rbrace .$ If $K(t,x)$ is convex for each $(t,x)\\in [0,T]\\times \\mathbb {R}^d$ , then it can be shown that for each relaxed control, there exists a strict control and a singular control with smaller or equal cost.", "Indeed, by the proof of [14], for any relaxed control $r=(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {P},X,\\underline{Q},Z)$ , there exists a progressively measurable $U$ -valued process $\\bar{u}$ and a $\\mathbb {R}^+$ -valued process $\\bar{v}$ such that for almost all $(t, \\omega ) \\in [0,T] \\times \\Omega $ , $\\begin{split}&\\left(\\int _U a(t,X_t(\\omega ),u)\\,\\underline{Q}_t(\\omega , du),\\int _U b(t,X_t(\\omega ),u)\\,\\underline{Q}_t(\\omega ,du),\\int _U f(t,X_t(\\omega ),u)\\,\\underline{Q}_t(\\omega ,du)\\right)\\\\=&~\\left(a(t,X_t(\\omega ),\\bar{u}_t(\\omega )),b(t,X_t(\\omega ),\\bar{u}_t(\\omega )),f(t,X_t(\\omega ), \\bar{u}_t(\\omega ))+ \\bar{v}_t(\\omega )\\right).\\end{split}$ Then $\\alpha =(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {P},X,\\bar{u},Z)$ is a strict control with smaller or equal cost.", "If $a(t,x,u)=K_{t,x}u^2$ , $b(t,x,u)=K_{t,x}^{\\prime }u^2$ , $f(t,x,u)=K_{t,x}^{\\prime \\prime }u^2$ , where $|K_{t,x}|,~|K_{t,x}^{\\prime }|,~|K_{t,x}^{\\prime \\prime }|\\le K$ for some positive constant $K$ , then the set $K(t,x)$ is convex for each $(t,x)\\in [0,T]\\times \\mathbb {R}^d$ ." ], [ "Canonical state space and disintegration", "In what follows, we always assume that $\\Omega $ is the canonical path space, i.e.", "$\\Omega =\\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\widetilde{\\mathcal {U}}(\\mathbb {R})\\times \\widetilde{\\mathcal {A}}(\\mathbb {R})$ and that the filtration $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ is generated by the coordinate projections $X,Q,Z$ .", "More precisely, for each $\\omega :=(x,q,z)\\in \\Omega $ , $X(\\omega )=x,\\qquad Q(\\omega )=q,\\qquad Z(\\omega )=z.$ and for $t\\in [0,T]$ , $\\mathcal {F}_t:=\\mathcal {F}^X_t\\times \\mathcal {F}^Q_t\\times \\mathcal {F}^Z_t$ , where $\\mathcal {F}^X_t=\\sigma (X_s,s\\le t),\\qquad \\mathcal {F}^Q_t=\\sigma (Q(S),S\\in \\mathcal {B}([0,t]\\times U)),\\qquad \\mathcal {F}^Z_t=\\sigma (Z_s,s\\le t);$ if $t<0$ , then $\\mathcal {F}_t:=\\lbrace \\Omega ,{Ø}\\rbrace $ and if $t>T$ , then $\\mathcal {F}_t:=\\mathcal {F}_T$ .", "The following argument shows that relaxed controls can be defined in terms of projection mappings.", "In fact, since $[0,T]$ and $U$ are compact, by the definition of $\\widetilde{\\mathcal {U}}(\\mathbb {R})$ , each $q\\in \\widetilde{\\mathcal {U}}(\\mathbb {R})$ allows for the disintegration $q(dt,du)=q_t(du)dt$ for some measurable $\\mathcal {P}(U)$ -valued function $q_t$ .", "By [26] and by definition of the space $\\widetilde{\\mathcal {U}}(\\mathbb {R})$ there exists a $\\mathcal {F}^Q_t$ -predictable $\\mathcal {P}(U)$ -valued process $\\Pi $ such that for each $q\\in \\widetilde{\\mathcal {U}}(\\mathbb {R})$ , $\\Pi _t(q)=q_t, ~a.e.", "~t\\in [0,T];\\qquad \\Pi _t(q)\\equiv \\delta _{\\widetilde{u_0}},~t<0;\\qquad \\Pi _t(q)\\equiv \\delta _{\\widetilde{u_T}},~t>T.$ Hence, the process $ Q^o_t:=\\Pi _t\\circ Q$ is $\\mathcal {F}_t$ -predictable.", "As a result, for each $\\omega =(x,q,z)$ , $Q(\\omega )(dt,du)=q(dt,du)=q_t(du)dt=\\Pi _t(q)(du)dt=\\Pi _t\\circ Q(\\omega )(du)dt=Q^o_t(\\omega )(du)dt.$ This yields an adapted disintegration of $Q$ in terms of the $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ progressively measurable process $Q^ o:\\mathbb {R}\\times \\Omega \\rightarrow \\mathcal {P}(U).$ and hence allows us to define control rules.", "We notice that it is not appropriate to replace $\\widetilde{\\mathcal {U}}(\\mathbb {R})$ in the definition of the canonical path space by the space of càdlàg $\\mathcal {P}(U)$ -valued functions as the definition of relaxed controls does not assume any path properties of $t \\mapsto \\underline{Q}_t.$ Definition 2.3 For the canonical path space $\\Omega $ , the canonical filtration $\\lbrace \\mathcal {F}_t, t \\in \\mathbb {R} \\rbrace $ and the coordinate projections $(X,Q,Z)$ introduced above, if $r=(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,{Q}^o,Z)$ is a relaxed control in the sense of Definition REF , then the probability measure $\\mathbb {P}$ is called a control rule.", "The associated cost functional is defined as $\\widehat{J}(\\mathbb {P}):=\\widetilde{J}(r).$ Let us denote by $\\mathcal {R}$ the class of all the control rules for the stochastic control problem (REF ).", "Clearly, $\\inf \\limits _{\\mathbb {P}\\in \\mathcal {R}}\\widehat{J}(\\mathbb {P})\\ge \\inf \\limits _{\\textrm {relaxed control }r}\\widetilde{J}(r).$ Conversely, for any relaxed control $r$ one can construct a control rule $\\mathbb {P}\\in \\mathcal {R}$ such that $\\widehat{J}(\\mathbb {P})=\\widetilde{J}(r).$ The proof is standard; it can be found in, e.g.", "[15].", "In other words, the optimization problems over relaxed controls and control rules are equivalent.", "It is hence enough to consider control rules.", "From now on, we let $(Q_t)_{t \\in \\mathbb {R}}:=(Q^o_t)_{t \\in \\mathbb {R}}$ for simplicity.", "Remark 2.4 In [15] - with the choice of different topologies and under suitable assumptions on the cost coefficients - it is shown that an optimal control rule exists if $g \\equiv 0$ .", "Their method allows for terminal costs only after a modification of the cost function; see [15] for details.", "As a byproduct (see Corollary REF ) of our analysis of MFGs, under the same assumptions on the coefficients as in [15] we establish the existence of an optimal control rule for terminal cost functions that satisfy a linear growth condition.", "In Section REF we furthermore outline a generalization of the stochastic singular control problem to problems of McKean-Vlasov-type." ], [ "Mean field games with singular controls", "We are now going to consider MFGs with singular controls of the form (REF ).", "We again restrict ourselves to relaxed controls.", "Throughout the paper, for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , put $\\mu _t=\\mu \\circ \\pi ^{-1}_t$ , where $\\pi _t:x\\in \\widetilde{\\mathcal {D}}(\\mathbb {R})\\rightarrow x_t$ .", "The first step of solving mean field games is to solve the representative agent's optimal control problem $\\left\\lbrace \\begin{array}{ll}\\inf _{u,Z} E\\left[\\int _0^Tf(t,X_t,\\mu _t, u_t)\\,dt+g(X_T,\\mu _T)+\\int _0^Th(t)\\,dZ_t\\right]\\\\\\mbox{subject to}\\\\dX_t=b(t,X_t,\\mu _t,u_t)\\,dt+\\sigma (t,X_t,\\mu _t,u_t)\\,dW_t+c(t)\\,dZ_t,\\\\X_{0-}=0\\end{array}\\right.$ for any fixed mean field measure $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ .", "The canonical path space for MFGs with singular controls is $\\Omega :=\\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\widetilde{\\mathcal {U}}(\\mathbb {R})\\times \\widetilde{\\mathcal {A}}(\\mathbb {R}).$ We assume that the spaces $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ and $\\widetilde{\\mathcal {A}}(\\mathbb {R})$ are endowed with the $M_1$ topology.", "We define a metric on the space ${\\mathcal {U}}(\\mathbb {R})$ induced by the Wasserstein distance on compact time intervals by $\\begin{split}d_{{\\mathcal {U}}(\\mathbb {R})}(q^1,q^2)&:=\\mathcal {W}_{p,[0,T]\\times U}\\left(\\frac{q^1}{T},\\frac{q^2}{T}\\right) \\\\& \\quad +\\sum _{n=0}^{\\infty } \\frac{1}{2^{n+1}}\\left\\lbrace \\mathcal {W}_{p,[-(n+1),-n]\\times U}(q^1,q^2)+\\mathcal {W}_{p,[T+n,T+n+1]\\times U}(q^1,q^2) \\right\\rbrace .\\end{split}$ The space $\\widetilde{\\mathcal {U}}(\\mathbb {R})$ endowed with the metric $d_{\\widetilde{\\mathcal {U}}(\\mathbb {R})}:=d_{\\mathcal {U}(\\mathbb {R})}$ is compact.", "Furthermore, it is well known [30] that the spaces $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ and $\\widetilde{\\mathcal {A}}(\\mathbb {R})$ are Polish spaces when endowed with the $M_1$ topology, and that the $\\sigma $ -algebras on $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ and $\\widetilde{\\mathcal {A}}(\\mathbb {R})$ coincide with the Kolmogorov $\\sigma $ -algebras generated by the coordinate projections.", "Definition 2.5 A probability measure $\\mathbb {P}$ is called a control rule with respect to $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ if 1.", "$(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P})$ is the canonical probability space and $(X,Q,Z)$ are the coordinate projections; 2. for each $\\phi \\in \\mathcal {C}^2_b(\\mathbb {R}^d)$ , $\\mathcal {M}^{\\mu ,\\phi }$ is a well defined $\\mathbb {P}$ continuous martingale, where $\\begin{split}\\mathcal {M}^{\\mu ,\\phi }_t:=&~\\phi (X_t)-\\int _0^t\\int _U\\mathcal {L}\\phi (s,X_s,\\mu _s,u)\\,Q_s(du)ds-\\int _0^t(\\partial _x\\phi (X_{s-}))^{\\top }c(s)dZ_s\\\\&-\\sum _{0\\le s\\le t}\\big (\\phi (X_s)-\\phi (X_{s-})-(\\partial _x\\phi (X_{s-}))^{\\top }\\triangle X_s\\big ),\\qquad t\\in [0,T]\\end{split}$ with $\\mathcal {L}\\phi (t,x,\\nu ,u):=\\frac{1}{2}\\sum _{ij}a_{ij}(t,x,\\nu ,u)\\frac{\\partial ^2\\phi (x)}{\\partial _{x_i}\\partial _{x_j}}+\\sum _ib_i(t,x,\\nu ,u)\\partial _{x_i}\\phi (x)$ and $a(t,x,\\nu ,u)=\\sigma \\sigma ^{\\top }(t,x,\\nu ,u)$ , for each $(t,x,\\nu ,u)\\in [0,T]\\times \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ .", "For a fixed measure $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , the corresponding set of control rules is denoted by $\\mathcal {R}(\\mu )$ , the cost functional corresponding to a control rule $\\mathbb {P}\\in \\mathcal {R}(\\mu )$ is $J(\\mu ,\\mathbb {P})=E^{\\mathbb {P}}\\left[\\int _0^T\\int _Uf(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _0^Th(t)\\,dZ_t+g(X_T,\\mu _T)\\right],$ and the (possibly empty) set of optimal control rules is denoted by $\\mathcal {R}^*(\\mu ):=\\textrm {argmin}_{\\mathbb {P\\in \\mathcal {R}(\\mu )}}J(\\mu ,\\mathbb {P}).$ If a probability measure $\\mathbb {P}$ satisfies the fixed point property $ \\mathbb {P}\\in \\mathcal {R}^*(\\mathbb {P}\\circ X^{-1}),$ then we call $\\mathbb {P}\\circ X^{-1}$ or $\\mathbb {P}$ or the associated tuple $(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {P},X,Q,Z)$ a relaxed solution to the MFG with singular controls (REF ).", "Moreover, if $\\mathbb {P}\\in \\mathcal {R}^*(\\mathbb {P}\\circ X^{-1})$ and $\\mathbb {P}(Q(dt,du)=\\delta _{\\bar{u}_t}(du)dt)=1$ for some progressively measurable process $\\bar{u}$ , then we call $\\mathbb {P}\\circ X^{-1}$ or $\\mathbb {P}$ or the associated tuple $(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {P},X,\\bar{u},Z)$ a strict solution.", "The following theorem gives sufficient conditions for the existence of a relaxed solution to our MFG.", "The proof is given in Section .", "Theorem 2.6 For some $\\bar{p}>p\\ge 1$ , we assume that the following conditions are satisfied: $\\mathcal {A}_1$ .", "There exists a positive constant $C_1$ such that $|b|\\le C_1$ and $|a|\\le C_1$ ; $b$ and $\\sigma $ are measurable in $t\\in [0,T]$ and continuous in $(x,\\nu ,u)\\in \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ ; moreover, $b$ and $\\sigma $ are Lipschitz continuous in $x\\in \\mathbb {R}^d$ , uniformly in $(t,\\nu ,u)\\in [0,T]\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ .", "$\\mathcal {A}_2$ .", "The functions $f$ and $g$ are measurable in $t\\in [0,T]$ and are continuous with respect to $(x,\\nu ,u)\\in \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ .", "$\\mathcal {A}_3$ .", "For each $(t,x,\\nu ,u)\\in [0,T]\\times \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ , there exist strictly positive constants $C_2$ , $C_3$ and a positive constant $C_4$ such that $-C_2\\left(1-|x|^{\\bar{p}}+\\int _{\\mathbb {R}^d}|x|^p\\,\\nu (dx)\\right)\\le g(x,\\nu )\\le C_3\\left(1+|x|^{\\bar{p}}+\\int _{\\mathbb {R}^d}|x|^p\\,\\nu (dx)\\right),$ and $|f(t,x,\\nu ,u)|\\le C_4\\left(1+|x|^p+|u|^p+\\int _{\\mathbb {R}^d}|x|^p\\,\\nu (dx)\\right).$ $\\mathcal {A}_4$ .", "The functions $c$ and $h$ are continuous and $c$ is strictly positive.", "$\\mathcal {A}_5$ .", "The functions $b$ , $\\sigma $ and $f$ are locally Lipschitz continuous with $\\mu $ uniformly in $(t,x,u)$ , i.e., for $\\varphi =b,~\\sigma ~\\textrm {and}~f$ , there exists $C_5>0$ such that for each $(t,x,u)\\in [0,T]\\times \\mathbb {R}^d\\times U$ and $\\nu ^1,\\nu ^2\\in \\mathcal {P}_p(\\mathbb {R}^d)$ there holds that $|\\varphi (t,x,\\nu ^1,u)-\\varphi (t,x,\\nu ^2,u)|\\le C_5\\Big (1+L(\\mathcal {W}_p(\\nu ^1,\\delta _0),\\mathcal {W}_p(\\nu ^2,\\delta _0))\\Big ) \\mathcal {W}_p(\\nu ^1,\\nu ^2),$ where $L(\\mathcal {W}_p(\\nu ^1,\\delta _0),\\mathcal {W}_p(\\nu ^2,\\delta _0))$ is locally bounded with $\\mathcal {W}_p(\\nu ^1,\\delta _0)$ and $\\mathcal {W}_p(\\nu ^2,\\delta _0)$ .", "$\\mathcal {A}_6$ .", "$U$ is a compact metrizable space.", "Under assumptions $\\mathcal {A}_1$ -$\\mathcal {A}_6$ , there exists a relaxed solution to the MFGs with singular controls (REF ).", "Remark 2.7 A typical example where assumption $\\mathcal {A}_3$ holds is $g(x,\\nu )=|x|^{\\bar{p}}+\\bar{g}(\\nu ),$ where $|\\bar{g}(\\nu )|\\le \\int _{\\mathbb {R}^d}|y|^p\\,\\nu (dy)$ .", "This assumption is not needed under a finite fuel constraint on the singular controls.", "It is needed in order to approximate MFGs with singular controls by MFGs with a finite fuel constraint.", "The assumption that $c>0$ is also only needed when passing from finite fuel constrained to unconstrained problems, see Lemma REF .", "Assumption $\\mathcal {A}_5$ is needed in order to prove the continuity of the cost function and the correspondence ${\\cal R}$ in $\\mu $ .", "A typical example for $\\mathcal {A}_5$ is $\\int |x|^p\\nu (dx)$ or $\\int |x|^p\\nu (dx)\\wedge K$ for some fixed constant $K$ if boundedness is required.", "Remark 2.8 If we assume for each $(t,x,\\nu )\\in [0,T]\\times \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)$ , $K(t,x,\\nu )$ is convex, where $K(t,x,\\nu )=\\lbrace (a(t,x,\\nu ,u),b(t,x,\\nu ,u),e):~e\\ge f(t,x,\\nu ,u),~u\\in U\\rbrace ,$ a strict solution to our MFG can be constructed from a relaxed solution.", "Let $r^*=(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {P}^*,X,Q,Z)$ is a relaxed solution to MFG.", "Let $a^*(t,x,u)=a(t,x,\\mu ^*_t,u)$ , $b^*(t,x,u)=b(t,x,\\mu ^*_t,u)$ and $f^*(t,x,u)=f(t,x,\\mu ^*_t,u)$ , where $\\mu ^*=\\mathbb {P}^*\\circ X^{-1}$ .", "Similar to Remark REF , there exist $U$ -valued process $\\bar{u}$ and $\\mathbb {R}^+$ -valued process $\\bar{v}$ such that (REF ) holds with $a,b,f$ replaced by $a^*,b^*,f^*$ , respectively.", "Define $\\alpha ^*=(\\Omega ,\\mathcal {F},\\mathcal {F}_t,\\mathbb {Q}^*,X,\\bar{u},Z),$ where $\\mathbb {Q}^*=\\mathbb {P}^*\\circ (X,\\delta _{\\bar{u}_t}(du)dt,Z)^{-1}$ .", "Then, $\\alpha ^*$ is a strict solution.", "The point is that the marginal distribution $\\mu ^*$ does not change when passing from $r^*$ to $\\alpha ^*$ ." ], [ "Proof of the main result", "The proof of Theorem REF is split into two parts.", "In Section REF we prove the existence of a solution to our MFG under a finite fuel constraint on the singular controls.", "The general case is established in Section REF using an approximation argument." ], [ "Existence under a finite fuel constraint", "In this section, we prove the existence of a relaxed solution to our MFG under a finite fuel constraint.", "That is, unless stated otherwise, we restrict the set of admissible singular controls to the set $\\widetilde{\\mathcal {A}}^m(\\mathbb {R}):=\\lbrace z\\in \\widetilde{\\mathcal {A}}(\\mathbb {R}):z_T\\le m\\rbrace , $ for some $m>0$ .", "By Corollary REF , the set $\\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ is $(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})$ compact.", "We start with the following auxiliary result on the tightness of the distributions of the solutions to a certain class of SDEs.", "The proof uses the definition of the distance $|x-[y,z]|$ of a point $x$ to a line segment $[y,z]$ and the modified strong $M_1$ oscillation function $\\widetilde{w}_s$ introduced in (REF ) and (REF ), respectively.", "Proposition 3.1 For each $n\\in \\mathbb {N}$ , on a probability space $(\\Omega ^n,\\mathcal {F}^n,\\mathbb {P}^n)$ , let $X^n$ satisfy the following SDE on $[0,T]$ : $dX_t^n={b}_n(t)\\,dt+\\,dM^n_t+d{c}_n(t),$ where the random coefficients ${b}_n$ is measurable and bounded uniformly in $n$ , $M^n$ is a continuous martingale with uniformly bounded and absolutely continuous quadratic variation, and ${c}_n$ is monotone and càdlàg in time a.s. and $\\sup _n E^{\\mathbb {P}^n}(|c_n(0)|\\vee |c_n(T)|)^{\\bar{p}}<\\infty $ .", "Moreover, assume that $X^n_t=0$ if $t<0$ and $X^n_t=X^n_T$ if $t>T$ .", "Then, the sequence $\\lbrace \\mathbb {P}^n\\circ (X^n)^{-1}\\rbrace _{n\\ge 1}$ is relatively compact as a sequence in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ .", "By the uniform boundedness of ${b}_n$ , $E^{\\mathbb {P}^n}(|c_n(0)|\\vee |c_n(T)|)^{\\bar{p}}$ and the quadratic variation of $M^n$ , there exists a constant $C$ that is independent of $n$ , such that $E^{\\mathbb {P}^n}\\sup _{0\\le t\\le T}|X^n_t|^{\\bar{p}}\\le C<\\infty .$ By [29] it is thus sufficient to check the tightness of $\\lbrace \\mathbb {P}^n\\circ (X^n)^{-1}\\rbrace _{n\\ge 1}$ .", "This can be achieved by applying Proposition REF .", "Indeed, the condition (REF ) holds, due to (REF ).", "Hence, one only needs to check that for each $\\epsilon >0$ and $\\eta >0$ , there exists $\\delta >0$ such that $\\sup _n\\mathbb {P}^n(\\widetilde{w}_s({X}^n,\\delta )\\ge \\eta )<\\epsilon .$ To this end, we first notice that for each $t$ and $t_1,~t_2,~t_3$ satisfying $0\\vee (t-\\delta )\\le t_1<t_2<t_3\\le (t+\\delta )\\wedge T$ , the monotonicity of $c_n$ implies $\\begin{split}&|{X}^n_{t_2}-[{X}^n_{t_1},{X}^n_{t_3}]|\\nonumber \\\\\\le &~ \\left|\\int _{t_1}^{t_2}{b}_n(s)\\,ds+{M}^n_{t_2}-M^n_{t_1}\\right|+\\left|\\int _{t_2}^{t_3}{b}_n(s)\\,ds+{M}^n_{t_3}-M^n_{t_2}\\right|\\\\&~+\\inf _{{0}\\le \\lambda \\le 1}\\left|{c}_n({t_2})-\\lambda c_n(t_1)-(1-\\lambda )c_n(t_3)\\right|\\nonumber \\\\=& \\left|\\int _{t_1}^{t_2}{b}_n(s)\\,ds+{M}^n_{t_2}-M^n_{t_1}\\right|+\\left|\\int _{t_2}^{t_3}{b}_n(s)\\,ds+{M}^n_{t_3}-M^n_{t_2}\\right|.\\end{split}$ Similarly, for $t_1$ and $t_2$ satisfying $0\\le t_1<t_2\\le \\delta $ , $|X^n_{t_1}-[0,X^n_{t_2}]|\\le \\left|\\int _{t_1}^{t_2}{b}_n(s)\\,ds+{M}^n_{t_2}-M^n_{t_1}\\right|.$ Therefore, $\\widetilde{w}_s({X},\\delta )\\le ~ 3\\sup _{ t}\\sup _{ t_1, t_2}\\left|\\int _{t_1}^{t_2}{b}_n(s)\\,ds+{M}^n_{t_2}-M^n_{t_1}\\right|,$ where the first supremum extends over $0\\le t\\le T$ and the second one extends over $0\\vee (t-\\delta )\\le t_1\\le t_2\\le T\\wedge (t+\\delta )$ .", "By the Markov inequality and the boundedness of ${b}_n$ and the quadratic variation, this yields $\\mathbb {P}^n(\\widetilde{w}_s(X^n,\\delta )\\ge \\eta )\\le \\frac{k(\\delta )}{\\eta },$ for some positive function $k(\\delta )$ that is independent of $n$ and $m$ with $\\lim _{\\delta \\rightarrow 0}k(\\delta )=0$ .", "The next result shows that the class of all possible control rules is relatively compact.", "In a subsequent step this will allow us to apply Berge's maximum theorem.", "Lemma 3.2 Under assumptions $\\mathcal {A}_1$ , $\\mathcal {A}_4$ and $\\mathcal {A}_6$ , the set $\\bigcup _{\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))}\\mathcal {R}(\\mu )$ is relatively compact in $\\mathcal {W}_{p}$ .", "Let $\\lbrace \\mu ^n\\rbrace _{n\\ge 1}$ be any sequence in $\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and $\\mathbb {P}^n\\in \\mathcal {R}(\\mu ^n), n\\ge 1$ .", "It is sufficient to show that $\\lbrace \\mathbb {P}^n\\circ X^{-1}\\rbrace _{ n\\ge 1}$ , $\\lbrace \\mathbb {P}^n\\circ Q^{-1}\\rbrace _{ n\\ge 1}$ and $\\lbrace \\mathbb {P}^n\\circ Z^{-1}\\rbrace _{n\\ge 1}$ are relatively compact.", "Since $U$ and $\\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ are compact by assumption and Corollary REF , respectively, $\\lbrace \\mathbb {P}^n\\circ Q^{-1}\\rbrace _{ n\\ge 1}$ and $\\lbrace \\mathbb {P}^n\\circ Z^{-1}\\rbrace _{ n\\ge 1}$ are tight.", "Since $\\widetilde{\\mathcal {U}}(\\mathbb {R})$ and $\\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ are compact, these sequences are relatively compact in the topology induced by Wasserstein metric; see [29].", "It remains to prove the relative compactness of $\\lbrace \\mathbb {P}^n\\circ X^{-1}\\rbrace _{n\\ge 1}$ .", "Since $\\mathbb {P}^n$ is a control rule associated with the measure $\\mu ^n$ , for any $n$ , it follows from Proposition REF that there exist extensions $(\\bar{\\Omega },\\bar{\\mathcal {F}},\\lbrace \\bar{\\mathcal {F}}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {Q}^n)$ of the canonical path spaces and processes $({X}^n,{Q}^n,{Z}^n,M^n)$ defined on it, such that $d{X}^n_t=\\int _U b(t,{X}^n_t,\\mu ^n_t,u)\\,{Q}^n_t(du)dt+\\int _U\\sigma (t,{X}^n_t,\\mu ^n_t,u)\\,M^n(du,dt)+c(t)\\,d{Z}^n_t$ and $\\mathbb {P}^n=\\mathbb {P}^n\\circ (X,Q,Z)^{-1}=\\mathbb {Q}^n\\circ ({X}^n,{Q}^n,{Z}^n)^{-1},$ where $M^n$ is a martingale measure on $(\\bar{\\Omega },\\bar{\\mathcal {F}},\\lbrace \\bar{\\mathcal {F}}_t\\in \\mathbb {R}\\rbrace ,\\mathbb {Q}^n)$ with intensity ${Q}^n$ .", "Relative compactness of $\\lbrace \\mathbb {P}^n\\circ X^{-1}\\rbrace _{n\\ge 1}$ now reduces to relative compactness of $\\lbrace \\mathbb {Q}^n\\circ (X^n)^{-1}\\rbrace _{n\\ge 1}$ , which is a direct consequence of the preceding Proposition REF .", "Remark 3.3 For the above result, the assumption $c>0$ is not necessary.", "To see this, we decompose ${X}^n$ as ${X}^n_{\\cdot }=\\int _0^{\\cdot }\\int _U b(t,{X}^n_t,\\mu ^n_t,u)\\,{Q}^n_t(du)dt+\\int _0^{\\cdot }\\int _U\\sigma (t,{X}^n_t,\\mu ^n_t,u)\\,M^n(du,dt)+\\int _0^{\\cdot }c^+(t)\\,d{Z}^n_t-\\int _0^{\\cdot }c^-(t)\\,d{Z}^n_t,$ where $c^+$ and $c^-$ are the positive and negative parts of $c$ , respectively.", "By the boundedness of $b$ and $\\sigma $ , we see that $E^{\\mathbb {Q}^n}|K^n_t-K^n_s|^4\\le C|t-s|^2,$ where $K^n_{\\cdot }:=\\int _0^{\\cdot }\\int _U b(t,{X}^n_t,\\mu ^n_t,u)\\,{Q}^n_t(du)dt+\\int _0^{\\cdot }\\int _U\\sigma (t,{X}^n_t,\\mu ^n_t,u)\\,M^n(du,dt).$ Kolmogorov weak tightness criterion implies that, for each $\\epsilon >0$ , there exists a compact set $\\mathcal {K}_1\\subseteq \\widetilde{\\mathcal {C}}(\\mathbb {R})$ such that $\\inf _n\\mathbb {Q}^n\\left(K^n\\in \\mathcal {K}_1\\right)\\ge 1-\\epsilon .$ Define $\\mathcal {K}_2:=\\left\\lbrace \\int _0^{\\cdot }c^+(s)\\,dz_s\\in \\widetilde{\\mathcal {A}}(\\mathbb {R}):~z_T\\in \\widetilde{\\mathcal {A}}^m(\\mathbb {R})\\right\\rbrace .$ and $\\mathcal {K}_3:=\\left\\lbrace -\\int _0^{\\cdot }c^-(s)\\,dz_s\\in \\widetilde{\\mathcal {D}}(\\mathbb {R}):z\\in \\widetilde{\\mathcal {A}}^m(\\mathbb {R})\\right\\rbrace .$ Thus, $\\begin{split}&~\\inf _n\\mathbb {P}^n\\lbrace \\omega \\in \\Omega :X_{\\cdot }(\\omega )\\in \\mathcal {K}_1+\\mathcal {K}_2+\\mathcal {K}_3\\rbrace \\\\\\ge &~\\inf _n\\mathbb {Q}^n\\left\\lbrace \\bar{\\omega }\\in \\bar{\\Omega }:K^n_{\\cdot }\\in \\mathcal {K}_1,\\int _0^{\\cdot }c^+(s)\\,d{Z}^n_s\\in \\mathcal {K}_2,-\\int _0^{\\cdot }c^-(s)\\,d{Z}^n_s\\in \\mathcal {K}_3\\right\\rbrace \\\\\\ge & ~ 1-\\epsilon .\\end{split}$ By Corollary REF , $\\mathcal {K}_2$ and $\\mathcal {K}_3$ are $M_1$ -compact subsets of $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ .", "By Remark REF (1), $\\mathcal {K}_1$ is also a $M_1$ compact subset.", "Note that the elements of $\\mathcal {K}_1$ do not jump and that $\\int _0^{\\cdot }c^+(s)\\,dz_s$ and $\\int _0^{\\cdot }c^-(s)\\,dz_s$ never jump at the same time.", "Thus, Proposition REF implies that $\\mathcal {K}_1+\\mathcal {K}_2+\\mathcal {K}_3$ is a $M_1$ -compact subset of $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ .", "The next result states that the cost functional is continuous on the graph $\\textrm {Gr}\\mathcal {R} :=\\lbrace (\\mu ,\\mathbb {P})\\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))\\times \\mathcal {P}_p(\\Omega ):~\\mathbb {P}\\in \\mathcal {R}(\\mu )\\rbrace .$ of the multi-function $\\mathcal {R}$ .", "This, too, will be needed to apply Berge's maximum theorem below.", "Lemma 3.4 Suppose that $\\mathcal {A}_1$ -$\\mathcal {A}_6$ hold.", "Then $J:\\textrm {Gr}\\mathcal {R}\\rightarrow \\mathbb {R}$ is continuous.", "For each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and $\\omega =(x,q,z)\\in \\Omega $ , set $ \\mathcal {J}(\\mu ,\\omega )=\\int _0^T\\int _Uf(t,x_t,\\mu _t,u)\\,q_t(du)dt+g(x_T,\\mu _T)+\\int _0^T h(t)\\,dz_t.$ Thus $J(\\mu ,\\mathbb {P})=\\int _{\\Omega }\\mathcal {J}(\\mu ,\\omega )\\mathbb {P}(d\\omega ).$ In a first step we prove that $\\mathcal {J}(\\cdot ,\\cdot )$ is continuous in the first variable; in a second step we prove continuity and a polynomial growth condition in the second variable.", "The joint continuity of $J$ will be proved in the final step.", "Step 1: continuity in $\\mu $ .", "Let $\\mu ^n\\rightarrow \\mu $ in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ and recall that $\\mu ^n_t=\\mu ^n\\circ \\pi _t^{-1}$ and $\\mu _t=\\mu \\circ \\pi _t^{-1}$ , where $\\pi $ is the projection on $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ .", "We consider the first two terms on the r.h.s.", "in (REF ) separately, starting with the first one.", "By assumption $\\mathcal {A}_5$ , $\\begin{split}&\\left|\\int _0^T\\int _Uf(t,x_t,\\mu ^n_t,u)\\,q_t(du)dt-\\int _0^T\\int _Uf(t,x_t,\\mu _t,u)\\,q_t(du)dt\\right|\\\\\\le & ~ C\\int _0^T\\left(1+L\\left(\\mathcal {W}_p(\\mu ^n_t,\\delta _0),\\mathcal {W}_p(\\mu _t,\\delta _0)\\right)\\right)\\mathcal {W}_p(\\mu ^n_t,\\mu _t)\\,dt\\\\\\le &~C\\left(\\int _0^T\\left(1+L\\left(\\mathcal {W}_p(\\mu ^n_t,\\delta _0),\\mathcal {W}_p(\\mu _t,\\delta _0)\\right)\\right)^{\\frac{p}{p-1}}\\,dt\\right)^{1-\\frac{1}{p}}\\left(\\int _0^T\\mathcal {W}_p(\\mu ^n_t,\\mu _t)^p\\,dt\\right)^{\\frac{1}{p}}.\\end{split}$ The convergence $\\mu ^n\\rightarrow \\mu $ in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ implies $\\mu ^n\\rightarrow \\mu $ weakly.", "By Skorokhod's representation theorem, there exists $\\bar{X}^n$ and $\\bar{X}$ defined on some probability space $(\\mathbb {Q},\\bar{\\Omega },\\bar{\\mathcal {F}})$ , such that $\\mu ^n=\\mathbb {Q}\\circ (\\bar{X}^n)^{-1}, \\quad \\mu =\\mathbb {Q}\\circ \\bar{X}^{-1}$ and $d_{M_1}(\\bar{X}^n,\\bar{X})\\rightarrow 0 \\quad \\mathbb {Q}\\textrm {-}a.s.$ Hence, (REF ) implies that $\\begin{split}&\\left|\\int _0^T\\int _Uf(t,x_t,\\mu ^n_t,u)\\,q_t(du)dt-\\int _0^T\\int _Uf(t,x_t,\\mu _t,u)\\,q_t(du)dt\\right|\\\\\\le & ~ C\\left(\\int _0^T\\left(1+L\\left(\\mathcal {W}_p(\\mathbb {Q}\\circ (\\bar{X}^n_t)^{-1},\\delta _0),\\mathcal {W}_p(\\mathbb {Q}\\circ \\bar{X}^{-1}_t,\\delta _0)\\right)\\right)^{\\frac{p}{p-1}}\\,dt\\right)^{1-\\frac{1}{p}}\\left(E^{\\mathbb {Q}}\\int _0^T|\\bar{X}^n_t-\\bar{X}_t|^p\\,dt\\right)^{\\frac{1}{p}}\\end{split}$ By Remark REF , we have $\\int _0^T|\\bar{X}^n_t-\\bar{X}_t|^p\\,dt\\rightarrow 0 \\qquad \\textrm { a.s. }\\mathbb {Q}.$ Moreover, we have $\\int _0^T|\\bar{X}^n_t-\\bar{X}_t|^p\\,dt\\le 2^pT\\left(d_{M_1}(\\bar{X}^n,0)^p+d_{M_1}(\\bar{X},0)^p\\right).$ On the other hand, $\\begin{split}E^{\\mathbb {Q}}\\left(d_{M_1}(\\bar{X}^n,0)^p+d_{M_1}(\\bar{X},0)^p\\right)=&~\\int _{\\mathcal {D}[0,T]}d_{M_1}(x,0)^p\\,\\mu ^n(dx)+\\int _{\\mathcal {D}[0,T]}d_{M_1}(x,0)^p\\,\\mu (dx)\\\\\\rightarrow &~2\\int _{\\mathcal {D}[0,T]}d_{M_1}(x,0)^p\\,\\mu (dx)<\\infty .\\end{split}$ Therefore, dominated convergence yields $E^{\\mathbb {Q}}\\int _0^T|\\bar{X}^n_t-\\bar{X}_t|^p\\,dt\\rightarrow 0.$ Since $\\sup _n\\mathcal {W}_p(\\mathbb {Q}\\circ (\\bar{X}^n_t)^{-1},\\delta _0)<\\infty $ it thus follows from the local boundedness of the function $L$ that $\\left|\\int _0^T\\int _Uf(t,x_t,\\mu ^n_t,u)\\,q_t(du)dt-\\int _0^T\\int _Uf(t,x_t,\\mu _t,u)\\,q_t(du)dt\\right|\\rightarrow 0,\\qquad \\textrm {uniformly in }\\omega .$ As for the second term on the r.h.s.", "in (REF ) recall first that $x^n\\rightarrow x$ in $M_1$ implies $x^n_t\\rightarrow x_t$ for each $t \\notin Disc(x)$ and $x^n_T\\rightarrow x_T$ .", "In particular, the mapping $x \\mapsto \\varphi (x_T)$ is continuous for any continuous real-valued function $\\varphi $ on $\\mathbb {R}^d$ .", "Since any continuous positive function $\\varphi $ on $\\mathbb {R}^d$ that satisfies $\\varphi (x)\\le C(1+|x|^p)$ , also satisfies $\\varphi (x_T)\\le C(1+|x_T|^p)\\le C(1+d_{M_1}(x,0)^p)$ we see that $\\begin{split}\\left|\\int _{\\mathbb {R}^d}\\varphi (x)\\,\\mu ^n_T(dx)-\\int _{\\mathbb {R}^d}\\varphi (x)\\,\\mu _T(dx)\\right| =\\left|\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}\\varphi (x_T)\\,\\mu ^n(dx)-\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}\\varphi (x_T)\\,\\mu (dx)\\right|\\stackrel{n \\rightarrow \\infty }{\\longrightarrow } 0.\\end{split}$ More generally, we obtain $\\mu ^n_T\\rightarrow \\mu _T$ from $\\mu ^n\\rightarrow \\mu $ , which also implies that $g(x_T,\\mu ^n_T)\\rightarrow g(x_T,\\mu _T)$ .", "Step 2: continuity in $\\omega $.", "If $\\omega ^n=(x^n,q^n,z^n)\\rightarrow \\omega =(x,q,z)$ , then $x^n_T\\rightarrow x_T$ .", "In particular, $g(x^n_T,\\mu _T)\\rightarrow g(x_T,\\mu _T).$ Moreover, $z^n\\rightarrow z$ in $M_1$ implies $z^n_t\\rightarrow z_t$ for for all continuity points of $z$ and $z^n_T\\rightarrow z_T$ .", "By the Portmanteau theorem this implies that $\\int _0^T h(t)\\,dz^n_t\\rightarrow \\int _0^Th(t)\\,dz_t.$ Next we show the convergence of $\\int _0^T\\int _U f(t,x^n_t,\\mu _t,u)\\,q^n_t(du)dt$ to $\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q_t(du)dt$ .", "By Assumption ${\\cal A}_2$ the convergence of $x^n$ to $x$ yields $f(t,x^n_t,\\mu _t,u)\\rightarrow f(t,x_t,\\mu _t,u)$ for each $t \\notin Disc(x)$ .", "From the compactness of $U$ it follows that $\\sup _{u\\in U}\\left|f(t,x^n_t,\\mu _t,u)-f(t,x_t,\\mu _t,u)\\right|\\rightarrow 0$ for each $t \\notin Disc(x)$ .", "Since $Disc(x)$ is at most countable this implies $\\begin{split}& \\left|\\int _0^T\\int _U f(t,x^n_t,\\mu _t,u)\\,q^n_t(du)dt-\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q^n_t(du)dt\\right| \\\\\\le & \\int _0^T \\sup _{u\\in U} \\left| f(t,x^n_t,\\mu _t,u)-f(t,x_t,\\mu _t,u) \\right| dt \\rightarrow 0.\\end{split}$ By [29], $q^n\\rightarrow q$ in $d_{\\widetilde{\\mathcal {U}}(\\mathbb {R})}$ implies $q^n\\rightarrow q$ weakly.", "Moreover, the first marginal of $q^n$ is Lebesgue measure.", "Thus, by [21], $q^n$ converges to $q$ in the stable topology (cf.", "[21] for the definition of the stable topology).", "For fixed $(x,\\mu )\\in \\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , the compactness of $U$ and the growth condition on $f$ implies the boundedness of $f$ .", "Hence the definition of stable topology yields that $\\lim _{n\\rightarrow \\infty }\\left|\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q^n_t(du)dt-\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q_t(du)dt\\right|=0.$ So we get the convergence $\\lim _{n\\rightarrow \\infty }\\left|\\int _0^T\\int _U f(t,x^n_t,\\mu _t,u)\\,q^n_t(du)dt-\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q_t(du)dt\\right|=0.$ Step 3: joint continuity of $J$.", "Thus far, we have established the separate continuity of the mapping $(\\mu ,\\omega )\\rightarrow \\mathcal {J}(\\mu ,\\omega )$ .", "We are now going to apply [29] to prove the joint continuity of $J$ .", "To this end, notice first that for each fixed $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , due to Assumption $\\mathcal {A}_3$ , $\\begin{split}&\\left|\\int _0^T\\int _U f(t,x_t,\\mu _t,u)\\,q_t(du)dt+\\int _0^T h(t)\\,dz_t\\right|\\\\\\le &~ C\\left(1+\\int _0^T\\int _U\\left(1+|x_t|^p+|u|^p+\\int _{\\mathbb {R}^d}|y|^p\\mu _t(dy)\\right)\\,q_t(du)dt+z_T\\right)\\\\\\le &~ C\\left(1+d_{M_1}(x,0)^p+\\mathcal {W}_{p,[0,T]\\times U}\\left(\\frac{q}{T},\\delta _0\\right)^p+d_{M_1}(z,0)+\\int _0^T\\int _{\\mathbb {R}^d}|y|^p\\,\\mu _t(dy)dt\\right)\\\\\\le &~ C\\left(1+d_{M_1}(x,0)^p+\\mathcal {W}_{p,[0,T]\\times U}\\left(\\frac{q}{T},\\delta _0\\right)^p+d_{M_1}(z,0)^p+\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}d_{M_1}(y,0)^p\\,\\mu (dy)\\right).\\end{split}$ Hence, using the uniform convergence (REF ), it follows from [29] that $(\\mu ^n,\\mathbb {P}^n)\\rightarrow (\\mu ,\\mathbb {P})$ implies that $\\begin{split}& \\left|E^{\\mathbb {P}^n}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t^n,u)\\,Q_t(du)dt+\\int _0^T h(t)\\,dZ_t\\right)-E^{\\mathbb {P}}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _0^Th(t)\\,dZ_t\\right)\\right| \\\\\\le & \\left|E^{\\mathbb {P}^n}\\left(\\int _0^T\\int _U f(t,X_t,\\mu ^n_t,u)\\,Q_t(du)dt+\\int _0^T h(t)\\,dZ_t\\right)-E^{\\mathbb {P}^n}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _0^T h(t)\\,dZ_t\\right)\\right|\\\\&+\\left|E^{\\mathbb {P}^n}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _0^T h(t)\\,dZ_t\\right)-E^{\\mathbb {P}}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _0^T h(t)\\,dZ_t\\right)\\right|\\\\\\rightarrow &0.\\end{split}$ Since the terminal cost functions is not necessarily Lipschitz continuous we need to argue differently in order to prove the continuous dependence of the expected terminal cost on $(\\mu ,\\mathbb {P})$ .", "First, we notice that for each $\\widetilde{p}>\\bar{p}$ , by the boundedness of $b$ , $\\sigma $ and $Z$ , we have that $\\sup _nE^{\\mathbb {P}^n}d_{M_1}(X,0)^{\\widetilde{p}} \\le C<\\infty ,$ which implies $\\lim _{K\\rightarrow \\infty }\\sup _n\\int _{\\lbrace x:d_{M_1}(x,0)>K\\rbrace } d_{M_1}(x,0)^{\\bar{p}}\\,\\mathbb {P}^n(dx)=0.$ By Assumption ${\\cal A}_3$ , $|g(x_T,\\mu _T)|\\le C\\left(1+|x_T|^{\\bar{p}}+\\int |y|^p\\mu _T(dy)\\right)\\le C\\left(1+|x_T|^{\\bar{p}}\\right).$ Together with (REF ) this implies, $E^{\\mathbb {P}^n}g(X_T,\\mu _T)\\rightarrow E^{\\mathbb {P}}g(X_T,\\mu _T).$ By the tightness of $\\lbrace \\mathbb {P}^n\\rbrace _{n\\ge 1}$ , for each $\\epsilon >0$ , there exists a compact set $K_{\\epsilon }\\subseteq \\widetilde{\\mathcal {D}}(\\mathbb {R})$ such that $\\begin{split} &~\\left|\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}g(x_T,\\mu ^n_T)\\mathbb {P}^n(dx)-\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}g(x_T,\\mu _T)\\mathbb {P}^n(dx)\\right|\\\\\\le &~\\int _{K_{\\epsilon }}|g(x_T,\\mu ^n_T)-g(x_T,\\mu _T)|\\mathbb {P}^n(dx)+\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})/K_{\\epsilon }}|g(x_T,\\mu ^n_T)-g(x_T,\\mu _T)|\\mathbb {P}^n(dx)\\\\\\le &~\\sup _{x\\in K_{\\epsilon }}|g(x_T,\\mu ^n_T)-g(x_T,\\mu _T)|+\\left(\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})/K_{\\epsilon }}|g(x_T,\\mu ^n_T)-g(x_T,\\mu _T)|^2\\mathbb {P}^n(dx)\\right)^{\\frac{1}{2}}\\left(\\sup _n\\mathbb {P}^n(\\widetilde{\\mathcal {D}}(\\mathbb {R})/K_{\\epsilon })\\right)^{\\frac{1}{2}}\\\\\\le &~\\sup _{x\\in K_{\\epsilon }}|g(x_T,\\mu ^n_T)-g(x_T,\\mu _T)|+C\\epsilon ^{\\frac{1}{2}}~~~(\\textrm {by }(\\ref {uniform-bounded-any-order})).\\end{split}$ Thus, $\\left|\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}g(x_T,\\mu ^n_T)\\mathbb {P}^n(dx)-\\int _{\\widetilde{\\mathcal {D}}(\\mathbb {R})}g(x_T,\\mu _T)\\mathbb {P}^n(dx)\\right|\\rightarrow 0.$ The convergence (REF ), (REF ) and (REF ) yield the joint continuity of $J(\\cdot ,\\cdot )$ .", "Remark 3.5 The preceding lemma shows that under a finite fuel constraint the cost functional $J$ is jointly continuous.", "In general, $J$ is only lower semi-continuous.", "In fact, for each positive constant $K$ , let $g_K(\\cdot ) := g(\\cdot ) \\wedge K$ and $\\begin{split}\\mathcal {J}_K(\\mu ,\\omega ):=&~\\int _0^T\\int _Uf(t,x_t,\\mu _t,u)\\,q_t(du)dt+g_K(x_T,\\mu _T)+\\int _0^T h(t)\\,dz_t\\end{split}$ By assumption $\\mathcal {A}_3$ , we have $|g_K(x,\\mu )|\\le 2K+C_2\\left(1+\\int _{\\mathbb {R}^d}|y|^p\\mu (dy)\\right)\\le C\\left(1+\\int _{\\mathbb {R}^d}|y|^p\\mu (dy)\\right).$ So (REF ) and (REF ) still hold with $g$ replaced by $g_K$ while (REF ) still holds for $f$ and $h$ .", "So $(\\mu ^n,\\mathbb {P}^n)\\rightarrow (\\mu ,P)$ implies $\\int _\\Omega \\mathcal {J}_K(\\mu ^n,\\omega ) \\mathbb {P}^n(d\\omega )\\rightarrow \\int _\\Omega \\mathcal {J}_K(\\mu ,\\omega ) \\mathbb {P}(d\\omega ).$ Thus, by monotone convergence theorem, we have $\\liminf _{n\\rightarrow \\infty }\\int _\\Omega \\mathcal {J}(\\mu ^n,\\omega ) \\mathbb {P}^n(d\\omega )\\ge \\int _\\Omega \\mathcal {J}(\\mu ,\\omega ) \\mathbb {P}(d\\omega ).$ We now recall from [15] an equivalent characterization for the set of control rules $\\mathcal {R}(\\mathcal {\\mu })$ .", "This equivalent characterization allows us to verify the martingale property of the state process by verifying the martingale property of its continuous part.", "Since it is difficult to locate the proof, we give a sketch one in Appendix .", "Proposition 3.6 A probability measure $\\mathbb {P}$ is a control rule with respect to the given $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ if and only if there exists an $\\mathcal {F}_t$ adapted process $Y\\in {\\mathcal {C}}(0,T)$ on the filtered canonical space $(\\Omega ,\\mathcal {F},\\mathcal {F}_t)$ such that (1) $\\mathbb {P}(\\omega \\in \\Omega :~X_{t}(\\omega )=Y_{t}(\\omega )+\\int _0^{t}c(s)\\,dZ_{s}(\\omega ),t\\in [0,T])=1$ ; (2) for each $\\phi \\in \\mathcal {C}^2_b(\\mathbb {R}^d)$ , $\\overline{\\mathcal {M}}^{\\mu ,\\phi }$ is a continuous $\\left(\\mathbb {P},\\mathcal {F}_t\\right)$ martingale, where $\\overline{\\mathcal {M}}^{\\mu ,\\phi }_t=\\phi (Y_t)-\\int _0^t\\int _U \\bar{\\mathcal {L}}\\phi (s,X_s,Y_s,\\mu _s,u)\\,Q_s(du)ds,\\qquad t\\in [0,T]$ with $\\bar{\\mathcal {L}}\\phi (s,x,y,\\nu ,u)=\\sum _i b_i(s,x,\\nu ,u)\\partial _{y_i}\\phi (y)+\\frac{1}{2}\\sum _{ij}a_{ij}(s,x,\\nu ,u)\\frac{\\partial ^2\\phi (y)}{\\partial _{y_i}\\partial _{y_j}}$ for each $(t,x,y,\\nu ,u)\\in [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d\\times \\mathcal {P}_p(\\mathbb {R}^d)\\times U$ .", "The previous characterization of control rules allows us to show that the correspondence $\\mathcal {R}$ has a closed graph.", "Proposition 3.7 Suppose that $\\mathcal {A}_1$ and $\\mathcal {A}_4$ -$\\mathcal {A}_6$ hold.", "For any sequence $\\lbrace \\mu ^n\\rbrace _{n\\ge 1}\\subseteq \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ with $\\mu ^n\\rightarrow \\mu $ in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ , if $\\mathbb {P}^n\\in \\mathcal {R}(\\mu ^n)$ and $\\mathbb {P}^n\\rightarrow \\mathbb {P}$ in $\\mathcal {W}_{p}$ , then $\\mathbb {P}\\in \\mathcal {R}(\\mu )$ .", "In order to verify conditions (1) and (2), notice first that, for each $n$ , there exists a stochastic process $Y^n\\in {\\mathcal {C}}(0,T)$ such that $\\mathbb {P}^n\\left(X_t=Y^n_t+\\int _0^t c(s)\\,dZ_s, t\\in [0,T] \\right)=1$ and such that the corresponding martingale problem is satisfied.", "In order to show that a similar decomposition and the martingale problem hold under the measure $\\mathbb {P}$ we apply Proposition REF .", "For each $n$ , there exits a probability space $(\\Omega ^n,\\mathcal {F}^n,\\mathbb {Q}^n)$ that supports random variables $(\\bar{X}^n,\\bar{Q}^n,\\bar{Z}^n)$ and a martingale measure $M^n$ with intensity $\\bar{Q}^n$ such that $\\mathbb {P}^n=\\mathbb {Q}^n\\circ (\\bar{X}^n,\\bar{Q}^n,\\bar{Z}^n)^{-1}$ and $d\\bar{X}^n_t=\\int _U b(t,\\bar{X}^n_t,\\mu ^n_t,u)\\,\\bar{Q}^n_s(du)ds+\\int _U \\sigma (t,\\bar{X}^n_t,\\mu ^n_t,u)\\,M^n(du,dt)+c(t)d\\bar{Z}^n_t.$ Thus, for each $0\\le s<t\\le T$ , $\\begin{split}E^{\\mathbb {P}^n}|Y^n_t-Y^n_s|^4=&~E^{\\mathbb {P}^n}\\left|\\left(X_t-\\int _0^tc(r)\\,dZ_r\\right)-\\left(X_s-\\int _0^sc(r)\\,dZ_r\\right)\\right|^4\\\\=&~E^{\\mathbb {Q}^n}\\left|\\left(\\bar{X}^n_t-\\int _0^tc(r)\\,d\\bar{Z}^n_r\\right)-\\left(\\bar{X}^n_s-\\int _0^sc(r)\\,d\\bar{Z}^n_r\\right)\\right|^4\\\\=&~E^{\\mathbb {Q}^n}\\left|\\int _s^t\\int _U b(r,\\bar{X}^n_r,\\mu ^n_r,u)\\,\\bar{Q}^n_r(du)dr+\\int _s^t\\int _U \\sigma (r,\\bar{X}^n_r,\\mu ^n_r,u)\\,M^n(du,dr)\\right|^4\\\\\\le &~ C|t-s|^2.\\end{split}$ Hence, Kolmogorov's weak compactness criterion implies the tightness of $Y^n$ .", "Therefore, taking a subsequence if necessary, the sequence $(X,Q,Z,Y^n)$ of random variables taking values in ${\\Omega } \\times {\\cal C}(0,T)$ has weak limit $(\\widehat{X},\\widehat{Q},\\widehat{Z},\\widehat{Y})$ defined on some probability space.", "By Skorokhod's representation theorem, there exists a probability space $(\\widetilde{\\Omega },\\widetilde{\\mathcal {F}},\\mathbb {Q})$ that supports random variables $(\\widetilde{X}^n,\\widetilde{Q}^n,\\widetilde{Z}^n,\\widetilde{Y}^n)$ and $(\\widetilde{X},\\widetilde{Q},\\widetilde{Z},\\widetilde{Y})$ such that ${Law}(\\widetilde{X}^n,\\widetilde{Q}^n,\\widetilde{Z}^n,\\widetilde{Y}^n)={Law}(X,Q,Z,Y^n),\\quad {Law}(\\widetilde{X},\\widetilde{Q},\\widetilde{Z},\\widetilde{Y})={Law}(\\widehat{X},\\widehat{Q},\\widehat{Z},\\widehat{Y})$ and $(\\widetilde{X}^n,\\widetilde{Q}^n,\\widetilde{Z}^n,\\widetilde{Y}^n)\\rightarrow (\\widetilde{X},\\widetilde{Q},\\widetilde{Z},\\widetilde{Y}) \\quad \\mathbb {Q}\\mbox{-a.s.}$ In particular, $\\tilde{Y}\\in \\mathcal {C}(0,T)$ as the uniform limit of a sequence of continuous processes, and $\\mathbb {Q} \\left(\\widetilde{X}_{t}=\\widetilde{Y}_{t} +\\int _0^{t}c(s)\\,d\\widetilde{Z}_s, t\\in [0,T]\\right)=1.$ Since $\\mathbb {P}^n\\rightarrow \\mathbb {P}$ , we have $\\mathbb {P}\\circ (X,Q,Z)^{-1}=\\mathbb {Q}\\circ (\\widetilde{X},\\widetilde{Q},\\widetilde{Z})^{-1}$ .", "Hence, there exists a stochastic process $Y\\in \\mathcal {C}(0,T)$ such that $\\mathbb {P} \\left( X_{t}=Y_{t}+\\int _0^{t}c(s)\\,dZ_s, t\\in [0,T] \\right)=1$ and $\\mathbb {P}\\circ (X,Q,Z,Y)^{-1}=\\mathbb {Q}\\circ (\\widetilde{X},\\widetilde{Q},\\widetilde{Z},\\widetilde{Y})^{-1}$ .", "Finally, for each $t\\in [0,T]$ , define $\\overline{\\mathcal {M}}^{n,\\mu ^n,\\phi }_t=\\phi (Y^n_t)-\\int _0^t\\int _U\\bar{\\mathcal {L}}(s,X_s,Y^n_s,\\mu ^n_s,u)\\,Q_s(du)ds,$ $\\widetilde{\\mathcal {M}}^{n,\\mu ^n,\\phi }_t=\\phi (\\widetilde{Y}^n_t)-\\int _0^t\\int _U\\bar{\\mathcal {L}}(s,\\widetilde{X}^n_s,\\widetilde{Y}^n_s,\\mu ^n_s,u)\\,\\widetilde{Q}^n_s(du)ds,$ and $\\widetilde{\\mathcal {M}}^{\\mu ,\\phi }_t=\\phi (\\widetilde{Y}_t)-\\int _0^t\\int _U\\bar{\\mathcal {L}}(s,\\widetilde{X}_s,\\widetilde{Y}_s,\\mu _s,u)\\,\\widetilde{Q}_s(du)ds.$ For each $0\\le s<t\\le T$ and each $F$ that is continuous, bounded and $\\mathcal {F}_s$ -measurable, we have $\\begin{split}0=&~E^{\\mathbb {P}^n}\\left(\\overline{\\mathcal {M}}^{n,\\mu ^n,\\phi }_t-\\overline{\\mathcal {M}}^{n,\\mu ^n,\\phi }_s\\right)F(X,Q,Z)=E^{\\mathbb {Q}}\\left(\\widetilde{\\mathcal {M}}^{n,\\mu ^n,\\phi }_t-\\widetilde{\\mathcal {M}}^{n,\\mu ^n,\\phi }_s\\right)F(\\widetilde{X}^n,\\widetilde{Q}^n,\\widetilde{Z}^n)\\\\&\\rightarrow E^{\\mathbb {Q}}\\left(\\widetilde{\\mathcal {M}}^{\\mu ^*,\\phi }_t-\\widetilde{\\mathcal {M}}^{\\mu ^*,\\phi }_s\\right)F(\\widetilde{X},\\widetilde{Q},\\widetilde{Z})=E^{\\mathbb {P}}\\left(\\overline{\\mathcal {M}}^{\\mu ,\\phi }_t-\\overline{\\mathcal {M}}^{\\mu ,\\phi }_s\\right)F(X,Q,Z).\\end{split}$ Remark 3.8 Note that the proof of Proposition REF , does not require the finite fuel constraint.", "The next corollary shows that the correspondence $\\mathcal {R}$ is continuous in the sense of [2].", "Corollary 3.9 Suppose that $\\mathcal {A}_1$ , $\\mathcal {A}_4$ -$\\mathcal {A}_6$ hold.", "Then, $\\mathcal {R}:\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))\\rightarrow 2^{\\mathcal {P}_p(\\Omega )}$ is continuous and compact-valued.", "The lower hemi-continuity of $\\mathcal {R}$ can be dealt with as [26] since $b$ and $\\sigma $ are Lipschitz continuous in $x$ .", "Lemma REF , Proposition REF and [2] imply that $\\mathcal {R}$ is upper hemi-continuous and compact-valued.", "Corollary 3.10 Under assumptions $\\mathcal {A}_1$ -$\\mathcal {A}_6$ , $\\mathcal {R}^*(\\mu )\\ne {Ø}$ for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and $\\mathcal {R}^*$ is upper hemi-continuous.", "By [23], for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ the set $\\mathcal {R}(\\mu )$ is nonempty.", "Corollary REF implies that $\\mathcal {R}$ is compact-valued and continuous.", "By Lemma REF , $J:Gr\\mathcal {R}\\rightarrow \\mathbb {R}$ is jointly continuous.", "Thus, [2] yields that $\\mathcal {R}^*$ is nonempty valued and upper hemi-continuous.", "Remark 3.11 Corollary REF in fact shows that the stochastic singular control problem (REF ) admits an optimal control rule in the sense of Definition REF .", "Using our method, we could have obtained Corollary REF under the same assumptions of the coefficients as in [15].", "We will generalize it to McKean-Vlasov case at the end of this section.", "Theorem 3.12 Under assumptions $\\mathcal {A}_1$ -$\\mathcal {A}_6$ and the finite-fuel constraint $Z \\in \\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ , there exists a relaxed solution to (REF ).", "From inequality (REF ) in the proof of Proposition REF , we see that for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and $\\mathbb {P}\\in \\mathcal {R}(\\mu )$ , there exists a nonnegative function $k(\\cdot )$ that is independent of $\\mu $ , such that $\\mathbb {P}(\\widetilde{w}_s({X},\\delta )>\\eta )\\le \\frac{k(\\delta )}{\\eta }$ and $\\lim _{\\delta \\rightarrow 0}k(\\delta )=0$ , where $\\widetilde{w}_s$ is the modified oscillation function defined in (REF ).", "Let us now define a set-valued map $\\psi $ by $\\psi :~~&\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R})) \\rightarrow 2^{\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R})},\\nonumber \\\\& \\mu \\mapsto \\lbrace \\mathbb {P}\\circ X^{-1}: \\mathbb {P}\\in \\mathcal {R}^*(\\mu )\\rbrace ,$ and let $S=\\left\\lbrace \\mathbb {P}\\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R})):\\textrm {for~ each}~ \\eta >0,~ \\mathbb {P}(\\widetilde{w}_s({X},\\delta )>\\eta )\\le \\frac{k(\\delta )}{\\eta }~\\textrm {and}~E^{\\mathbb {P}}\\sup _{0\\le t\\le T}|X_t|^{\\bar{p}}\\le C\\right\\rbrace $ where $C < \\infty $ denotes the upper bound in (REF ).", "It can be checked that $S$ is non-empty, relatively compact, convex, and that $\\psi (\\mu )\\subseteq S\\subseteq \\bar{S}$ , for each $\\mu \\in \\widetilde{\\mathcal {D}}(\\mathbb {R})$ .", "Hence, $\\psi : \\bar{S}\\rightarrow 2^{\\bar{S}}$ .", "Moreover, by Corollary REF , $\\psi $ is nonempty-valued and upper hemi-continuous.", "Therefore, [2] is applicable by embedding $\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ into $\\mathcal {M}(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , the space of all bounded signed measures on $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ endowed with weak convergence topology." ], [ "Existence in the general case", "In this section we establish the existence of a solution to MFGs with singular controls for general singular controls $Z \\in \\widetilde{\\mathcal {A}}(\\mathbb {R})$ .", "For each $m$ and $\\mu $ , define $\\Omega ^m=\\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\widetilde{\\mathcal {U}}(\\mathbb {R})\\times \\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ and denote by $\\mathcal {R}^m(\\mu )$ the control rules corresponding to $\\Omega ^m$ and $\\mu $ , that is, $\\mathcal {R}^m(\\mu )$ is the subset of probability measures in $\\mathcal {R}(\\mu )$ that are supported on $\\Omega ^m$ .", "Denote by MFG$^m$ the mean field games corresponding to $\\Omega ^m$ .", "The preceding analysis showed that there exists a solution $\\mathbb {P}^{m*}$ to MFG$^m$ , for each $m$ .", "In what follows, $\\mu ^{m*} :=\\mathbb {P}^{m*}\\circ X^{-1}.$ The next lemma shows that the sequence $\\lbrace \\mathbb {P}^{m*}\\rbrace _{m\\ge 1}$ is relatively compact; the subsequent one shows that any accumulation point is a control rule.", "Lemma 3.13 Suppose $\\mathcal {A}_1$ , $\\mathcal {A}_3$ , $\\mathcal {A}_4$ and $\\mathcal {A}_6$ hold.", "Then there exists a constant $K < \\infty $ such that $\\sup _m E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\le K<\\infty .$ As a consequence, the sequence $\\lbrace \\mathbb {P}^{m*}\\rbrace _{m\\ge 1}$ is relatively compact in $\\mathcal {W}_{p,\\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\widetilde{\\mathcal {U}}(\\mathbb {R})\\times \\widetilde{\\mathcal {A}}(\\mathbb {R})}$ .", "We recall that $c(\\cdot )$ is bounded away from 0.", "Hence, there exists a constant $C < \\infty $ such that, for all $m\\in \\mathbb {N}$ , $E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\le C\\left(1+E^{\\mathbb {P}^{m*}}|X_T|^{\\bar{p}}\\right)$ and $E^{\\mathbb {P}^{m*}}|X_t|^{{p}}\\le C\\left(1+E^{\\mathbb {P}^{m*}}|Z_T|^{{p}}\\right),~t\\in [0,T].$ Moreover, $\\begin{split}J(\\mu ^{m*},\\mathbb {P}^{m*})=&~E^{\\mathbb {P}^{m*}}\\left[\\int _0^T\\int _U f(t,X_t,\\mu ^{m*}_t,u)\\,Q_t(du)dt+g(X_T,\\mu ^{m*}_T)+\\int _0^T h(t)\\,dZ_t\\right]\\\\\\ge &~ -C\\left(1+\\int _0^T\\int _{\\mathbb {R}^d}|x|^p\\,\\mu ^{m*}_t(dx)dt+E^{\\mathbb {P}^{m*}}\\int _0^T|X_t|^p\\,dt+E^{\\mathbb {P}^{m*}}\\int _0^T\\int _U|u|^p\\,Q_t(du)dt\\right.\\\\&\\left.-E^{\\mathbb {P}^{m*}}|X_T|^{\\bar{p}}+\\int _{\\mathbb {R}^d}|x|^p\\,\\mu ^{m*}_T(dx)+E^{\\mathbb {P}^{m*}}\\left|\\int _0^Th(t)\\,dZ_t\\right|\\right)~~~(\\textrm {by assumption}~\\mathcal {A}_3)\\\\\\ge &~-C\\left(1+\\int _0^T\\int _{\\mathbb {R}^d}|x|^p\\,\\mu ^{m*}_t(dx)dt+E^{\\mathbb {P}^{m*}}\\int _0^T|X_t|^p\\,dt+E^{\\mathbb {P}^{m*}}\\int _0^T\\int _U|u|^p\\,Q_t(du)dt\\right.\\\\&~\\left.+\\int _{\\mathbb {R}^d}|x|^p\\,\\mu ^{m*}_T(dx)+E^{\\mathbb {P}^{m*}}\\left|\\int _0^Th(t)\\,dZ_t\\right|-E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\right)~~~(\\textrm {by } (\\ref {estimate-pm-zt-pbar})).\\end{split}$ Now choose any $\\mathbb {P}_0\\in \\mathcal {R}^m(\\mu ^{m*})$ such that $\\sup _mJ(\\mu ^{m*},\\mathbb {P}_0) < \\infty $ (e.g.", "$\\mathbb {P}_0\\in \\mathcal {R}(\\mu ^{m,*})$ such that $\\mathbb {P}_0(Q|_{[0,T]}\\equiv \\delta _{\\widetilde{u}}(du)dt|_{[0,T]},Z\\equiv 0)=1$ for some $\\widetilde{u}\\in U$ ).", "Then, $\\begin{split}E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\le &~ J(\\mu ^{m*},\\mathbb {P}^{m*})+C\\left(1+E^{\\mathbb {P}^{m*}}\\left|\\int _0^Th(t)\\,dZ_t\\right|+E^{\\mathbb {P}^{m*}}\\int _0^T|X_t|^p\\,dt+E^{\\mathbb {P}^{m*}}|X_T|^p\\right)\\\\\\le &~J(\\mu ^{m*},\\mathbb {P}_0)+C\\left(1+E^{\\mathbb {P}^{m*}}|Z_T|+E^{\\mathbb {P}^{m*}}|Z_T|^p\\right)~~~~(\\textrm {by }(\\ref {estimate-pm-xt-p})\\textrm { and the optimality of }\\mathbb {P}^{m*})\\\\\\le &~C\\left(1+E^{\\mathbb {P}^{m*}}|Z_T|+E^{\\mathbb {P}^{m*}}|Z_T|^p\\right).\\end{split}$ Since the measure $\\mathbb {P}^{m*}$ is supported on $\\Omega ^m$ , we see that $E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}$ is finite, for each $m$ .", "In order to see that there exists a uniform upper bound on $E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}$ , notice that, independently of $m$ we can choose $M > 0$ large enough such that $ E^{\\mathbb {P}^{m*}}|Z_T|^{{p_0}} \\le M + \\frac{1}{4C} E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\quad (p_0=1,p)$ Together with (REF ) this yields, $E^{\\mathbb {P}^{m*}}|Z_T|^{\\bar{p}}\\le 2C(1+M):=K.$ By [29] and Proposition REF , the relative compactness of $\\lbrace \\mathbb {P}^{m*}\\rbrace _{m\\ge 1}$ follows.", "The previous lemma shows that the sequence $\\lbrace \\mathbb {P}^{m*}\\rbrace _{m\\ge 1}$ has an accumulation point $\\mathbb {P}^*$ .", "Let $\\mu ^*=\\mathbb {P}^*\\circ X^{-1}$ .", "Clearly, $\\mu ^{m*}\\rightarrow \\mu ^*$ in $\\mathcal {W}_p$ along a subsequence.", "The following result is an immediate corollary to Proposition REF (see Remark REF ).", "Lemma 3.14 Suppose that $\\mathcal {A}_1$ and $\\mathcal {A}_3$ -$\\mathcal {A}_6$ hold, let $\\mathbb {P}^*$ be an accumulation point of the sequence $\\lbrace \\mathbb {P}^{m*}\\rbrace _{m\\ge 1}$ .", "Then, $\\mathbb {P}^*\\in \\mathcal {R}(\\mu ^*)$ .", "The next theorem establish the existence of relaxed MFGs solution to (REF ) in the general case, i.e.", "it proves Theorem REF .", "Theorem 3.15 Suppose $\\mathcal {A}_1$ -$\\mathcal {A}_6$ hold.", "Then $\\mathbb {P}^*\\in \\mathcal {R}^*(\\mu ^*)$ , i.e., for each $\\mathbb {P}\\in \\mathcal {R}(\\mu ^*)$ it holds that $J(\\mu ^*,\\mathbb {P}^*)\\le J(\\mu ^*,\\mathbb {P}).$ It is sufficient to prove that $J(\\mu ^*,\\mathbb {P}^*)\\le J(\\mu ^*,\\mathbb {P})$ for each $\\mathbb {P}\\in \\mathcal {R}(\\mu ^*)$ with $J(\\mu ^*,\\mathbb {P})<\\infty $ .", "By Proposition REF , there exists a filtered probability space $(\\bar{\\Omega },\\bar{\\mathcal {F}},\\bar{\\mathcal {F}}_t,\\bar{\\mathbb {P}})$ on which random variables $(\\bar{X},\\bar{Q},\\bar{Z},M)$ are defined such that $\\mathbb {P}=\\bar{\\mathbb {P}}\\circ (\\bar{X},\\bar{Q},\\bar{Z})^{-1}$ and $d\\bar{X}_t=\\int _U b(t,\\bar{X}_t,{\\mu }^*_t,u)\\,\\bar{Q}_t(du)dt+\\int _U \\sigma (t,\\bar{X}_t,{\\mu }^*_t,u)M(du,dt)+c(t)\\,d\\bar{Z}_t,$ where $M$ is a martingale measure with intensity $\\bar{Q}$ .", "Using the same argument as in the proof of Lemma REF we see that, $E^{{\\mathbb {P}}}{Z}_T^{\\bar{p}}=E^{\\bar{\\mathbb {P}}}\\bar{Z}_T^{\\bar{p}}<\\infty .$ Define $\\mathbb {P}^m=\\bar{\\mathbb {P}}\\circ (\\bar{X}^m,\\bar{Q},\\bar{Z}^m)\\in \\mathcal {R}^m(\\mu ^{m*})$ , such that $\\bar{X}^m$ is the unique strong solution to $d\\bar{X}^m_t=\\int _U b(t,\\bar{X}^m_t,{\\mu }^{m*}_t,u)\\,\\bar{Q}_t(du)dt+\\int _U \\sigma (t,\\bar{X}^m_t,{\\mu }^{m*}_t,u)M(du,dt)+c(t)\\,d\\bar{Z}^m_t,$ where for each $\\bar{\\omega }\\in \\bar{\\Omega }$ , $\\bar{Z}^m_t(\\bar{\\omega })=\\left\\lbrace \\begin{array}{ll}\\bar{Z}_t(\\bar{\\omega }),&\\textrm { if }t<\\tau ^m(\\bar{\\omega })\\\\m,&\\textrm { if }t\\ge \\tau ^m(\\bar{\\omega }),\\\\\\end{array}\\right.$ with $\\tau ^m(\\bar{\\omega })=\\inf \\lbrace t:\\bar{Z}_t{(\\bar{\\omega })}>m\\rbrace $ .", "Similarly, we can define $Z^m$ .", "Furthermore, if ${Z}$ is $\\widetilde{\\mathcal {A}}^m(\\mathbb {R})$ valued, we have ${Z}={Z}^m$ .", "Hence, $\\begin{split}&~E^{\\bar{\\mathbb {P}}}\\sup _{0\\le t\\le T}\\left|\\int _0^t c(s)\\,d\\bar{Z}_s-\\int _0^tc(s)\\,d\\bar{Z}^m_s\\right|\\\\=&~E^{{\\mathbb {P}}}\\sup _{0\\le t\\le T}\\left|\\int _0^t c(s)\\,d{Z}_s-\\int _0^tc(s)\\,d{Z}^m_s\\right|\\\\=&~\\int _{\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R})}\\sup _{0\\le t\\le T}\\left|\\int _0^tc(s)\\,d{Z}_s({\\omega })-\\int _0^tc(s)\\,d{Z}^m_s({\\omega })\\right|{\\mathbb {P}}(d{\\omega }).\\end{split}$ By Hölder's inequality, $\\begin{split}&~\\int _{\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R})}\\sup _{0\\le t\\le T}\\left|\\int _0^tc(s)\\,d{Z}_s(\\omega )-\\int _0^tc(s)\\,d{Z}^m_s({\\omega })\\right|{\\mathbb {P}}(d{\\omega })\\\\\\le &~\\left|\\int _{\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R})}\\int _0^Tc(t)\\,d{Z}_t({\\omega }){\\mathbb {P}}(d{\\omega })+\\int _{\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R})}\\int _0^Tc(t)\\,d{Z}^m_t({\\omega }){\\mathbb {P}}(d{\\omega })\\right|\\\\\\le &~ C \\left(E^{{\\mathbb {P}}}{Z}_T^{{p}}\\right)^{\\frac{1}{{p}}}{\\mathbb {P}}(\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R}))^{1-\\frac{1}{{p}}}+C \\left(E^{{\\mathbb {P}}}({Z}^m_T)^{{p}}\\right)^{\\frac{1}{{p}}}{\\mathbb {P}}(\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R}))^{1-\\frac{1}{{p}}}\\\\\\le &~ C \\left(E^{{\\mathbb {P}}}{Z}_T^{{p}}\\right)^{\\frac{1}{{p}}}{\\mathbb {P}}(\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R}))^{1-\\frac{1}{{p}}}.\\end{split}$ Since $\\widetilde{\\mathcal {A}}^m(\\mathbb {R})\\uparrow \\widetilde{\\mathcal {A}}(\\mathbb {R})$ implies ${\\mathbb {P}}(\\widetilde{\\mathcal {A}}(\\mathbb {R})\\backslash \\widetilde{\\mathcal {A}}^m(\\mathbb {R}))\\rightarrow 0$ we get, $E^{\\bar{\\mathbb {P}}}\\sup _{0\\le t\\le T}\\left|\\int _0^t c(s)\\,d\\bar{Z}_s-\\int _0^tc(s)\\,d\\bar{Z}^m_s\\right|\\rightarrow 0.$ Similarly, $E^{\\bar{\\mathbb {P}}}\\left|\\int _0^T h(t)\\,d\\bar{Z}_t-\\int _0^Th(t)\\,d\\bar{Z}^m_t\\right|\\rightarrow 0.$ By (REF ), (REF ) and (REF ), the Lipschitz continuity of $b$ and $\\sigma $ in $x$ and $\\mu $ and the Burkholder-Davis-Gundy inequality, standard estimate of SDE yields that $\\lim _{m\\rightarrow \\infty }E^{\\bar{\\mathbb {P}}}\\sup _{0\\le t\\le T}\\left|\\bar{X}^m_t-\\bar{X}_t\\right|= 0.$ By (REF ), (REF ), $\\mu ^{m*}\\rightarrow \\mu ^*$ in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ and the same arguments as in the proof of Lemma REF , we get $\\begin{split}&~ E^{\\bar{\\mathbb {P}}} \\left(\\int _0^T f(t,\\bar{X}^m_t,\\mu ^{m*}_t,u)\\,\\bar{Q}_t(du)dt+g(\\bar{X}^m_T,\\mu ^{m*}_T)+\\int _0^Th(t)\\,d\\bar{Z}^m_t\\right)\\\\\\rightarrow &~E^{\\bar{\\mathbb {P}}} \\left(\\int _0^T f(t,\\bar{X}_t,\\mu ^*_t,u)\\,\\bar{Q}_t(du)dt+g(\\bar{X}_T,\\mu ^*_T)+\\int _0^Th(t)\\,d\\bar{Z}_t\\right).\\end{split}$ This shows that $J(\\mu ^{m*},\\mathbb {P}^m)\\rightarrow J(\\mu ^*,\\mathbb {P}).$ Moreover, by Remark REF , $\\liminf _{m\\rightarrow \\infty }J(\\mu ^{m*},\\mathbb {P}^{m*})\\ge J(\\mu ^*,\\mathbb {P}^*)$ .", "Hence, $J(\\mu ^*,\\mathbb {P})= \\lim _{m\\rightarrow \\infty }J(\\mu ^{m*},\\mathbb {P}^m)\\ge \\liminf _{m\\rightarrow \\infty }J(\\mu ^{m*},\\mathbb {P}^{m*})\\ge J(\\mu ^*,\\mathbb {P}^*).$" ], [ "Related McKean-Vlasov stochastic singular control problem", "MFGs and control problems of McKean-Vlasov type are compared in [5].", "The literatures on McKean-Vlasov singular control focus on necessary conditions for optimality; the existence of optimal control is typically assumed; see e.g.", "[19].", "With the above method for MFGs, we can also establish the existence of an optimal control to the following McKean-Vlasov stochastic singular control problem: $\\min _{u,Z} J(u,Z)=\\min _{u,Z} E\\left[\\int _0^T f(t,X_t,Law(X_t),u_t)\\,dt+g(X_T,Law(X_T))+\\int _0^T h(t)\\,dZ_t\\right]$ subject to $\\begin{split}dX_t=b(t,X_t,Law(X_t),u_t)\\,dt+\\sigma (t,X_t,Law(X_t),u_t)\\,dW_t+c(t)\\,dZ_t, ~t\\in [0,T].\\end{split}$ To this end, we need to introduce relaxed controls and control rules similar to Section .", "Definition 3.16 We call $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z)$ a relaxed control to McKean-Vlasov stochastic singular control problem (REF )-(REF ) if it satisfies items 1, 2 and 3 in Definition REF and 4' $\\left(\\mathcal {M}^{\\mathbb {P},\\phi },\\lbrace \\mathcal {F}_t,t\\ge 0\\rbrace ,\\mathbb {P}\\right)$ is a well defined continuous martingale, where $\\begin{split}\\mathcal {M}^{\\mathbb {P},\\phi }_t=&~\\phi (X_t)-\\int _0^t\\int _U\\phi ^{\\prime }(X_s)b(s,X_s,\\mathbb {P}\\circ X_s^{-1},u)\\,\\underline{Q}_s(du)ds \\\\&-\\frac{1}{2}\\int _0^t\\int _U\\phi ^{\\prime }(X_s)a(s,X_s,\\mathbb {P}\\circ X_s^{-1},u)\\,\\underline{Q}_s(du)ds\\\\&-\\int _0^t\\phi ^{\\prime }(X_{s-})c(s)\\,dZ_s-\\sum _{0\\le s\\le t}\\left(\\phi (X_{s})-\\phi (X_{s-})-\\phi ^{\\prime }(X_{s-})\\Delta X_s\\right),~t\\in [0,T].\\end{split}$ For each relaxed control $r=(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z)$ , we define the corresponding cost functional by $J(r)=E^{\\mathbb {P}}\\left[\\int _0^T\\int _U f\\left(t,X_t,\\mathbb {P}\\circ X^{-1}_t,u\\right)\\,\\underline{Q}_t(du)dt+g\\left(X_T,\\mathbb {P}\\circ X^{-1}_T\\right)+\\int _0^T h(t)\\,dZ_t\\right].$ We still denote by $\\Omega :=\\widetilde{\\mathcal {D}}(\\mathbb {R})\\times \\widetilde{\\mathcal {U}}(\\mathbb {R})\\times \\widetilde{\\mathcal {A}}(\\mathbb {R})$ the canonical space, $\\mathcal {F}_t$ the canonical filtration and $(X,Q,Z)$ the coordinate projections with the associated predictable disintegration $Q^o$ , as introduced in Section .", "The notion of control rules can be defined similarly as that in Definition REF .", "Denote by $\\mathcal {R}$ all the control rules.", "For $\\mathbb {P}\\in \\mathcal {R}$ , the corresponding cost functional is defined as in (REF ).", "Using straightforward modifications of arguments given in the proof of [15] we see that our optimization problems over relaxed controls and over control rules are equivalent.", "Once the optimal control rule is established, under the same additional assumption as in Remark REF , we can establish a strict optimal control from the optimal control rule.", "The next two theorems prove the existence of an optimal control under a finite-fuel constraint.", "The existence results can then be extended to the general unconstraint case.", "We do not give a formal proof as the arguments are exactly the same as in the preceding subsection.", "Theorem 3.17 Suppose $\\mathcal {A}_4$ , $\\mathcal {A}_5$ hold and $\\mathcal {A}_1$ holds without Lipschitz continuity of $b$ and $\\sigma $ on $x$ .", "Under a finite-fuel constraint on the singular controls, $\\mathcal {R}\\ne \\emptyset $ .", "For each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ , there exists a solution to the martingale problem $\\mathcal {M}^{\\mu ,\\phi }$ , where $\\mathcal {M}^{\\mu ,\\phi }$ is defined in (REF ).", "Thus, we define a set-valued map $\\Phi $ on $\\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ with non-empty convex images by $\\Phi : \\mu \\rightarrow \\lbrace \\mathbb {P}\\circ X^{-1}: \\mathbb {P}\\in \\mathcal {R}(\\mu )\\rbrace ,$ where $\\mathcal {R}(\\mu )$ is the control rule with $\\mu $ as in the previous section.", "The compactness of $\\Phi (\\mu )$ for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and the upper hemi-continuity of $\\Phi $ are results of the compactness of $\\mathcal {R}(\\mu )$ for each $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ and upper hemi-continuity of $\\mathcal {R}(\\cdot )$ , respectively, which are direct results of Corollary REF .Note that we only need upper hemi-continuity of $\\mathcal {R}(\\cdot )$ , so Lipschtiz assumptions on $b$ and $\\sigma $ are not necessary.", "By analogy to the proof of Theorem REF we can define a non-empty, compact, convex set $\\bar{S} \\subset \\mathcal {P}_p(\\widetilde{\\mathcal {D}}(\\mathbb {R}))$ such that $\\Phi : \\bar{S} \\rightarrow 2^{\\bar{S}}$ .", "Hence, $\\Phi $ has a fixed point, due to [2].", "Theorem 3.18 Suppose $\\mathcal {A}_3$ -$\\mathcal {A}_6$ hold and that $\\mathcal {A}_1$ holds without Lipschitz assumptions on $b$ and $\\sigma $ in $x$ , and that $\\mathcal {A}_2$ holds with the continuity of $f$ and $g$ being replaced by lower semi-continuity.", "Under a finite-fuel constraint, there exist an optimal control rule, that is, there exists $\\mathbb {P}^*\\in \\mathcal {R}$ such that $J(\\mathbb {P}^*)\\le J(\\mathbb {P}) \\quad \\mbox{for all} \\quad \\mathbb {P}\\in \\mathcal {R}.$ It is sufficient to prove $\\mathcal {R}$ is compact and $J$ is lower semi-continuous.", "The former one can be achieved by the same way to Corollary REF .", "As for the lower semi-continuity, note that $f$ and $g$ can be approximated by continuous functions $f_N$ and $g_N$ increasingly.", "For $f_N$ and $g_N$ , by the same way as that in the proof of Lemma REF , one has $\\begin{split}&~\\liminf _{n\\rightarrow \\infty }E^{\\mathbb {P}^n}\\left[\\int _0^T\\int _U f_N(t,X_t,\\mathbb {P}^n\\circ X^{-1}_t,u)\\,Q_t(du)dt+g_N(X_T,\\mathbb {P}^n\\circ X^{-1}_T)+\\int _0^T h(t)\\,dZ_t\\right]\\\\\\rightarrow &~E^{\\mathbb {P}}\\left[\\int _0^T\\int _U f_N(t,X_t,\\mathbb {P}\\circ X^{-1}_t,u)\\,Q_t(du)dt+g_N(X_T,\\mathbb {P}\\circ X^{-1}_T)+\\int _0^T h(t)\\,dZ_t\\right].\\end{split}$ Thus, monotone convergence implies the lower semi-continuity of $J$ ." ], [ "MFGs with regular controls and MFGs with singular controls", "In this section we establish two approximation results for a class of MFGs with singular controls under finite-fuel constraints.", "For the reasons outlined in Remark REF below we restrict ourselves to MFGs without terminal cost or singular control cost.", "More precisely, we consider MFGs with singular controls of the form: $ \\left\\lbrace \\begin{array}{ll}1.&\\textrm { fix a deterministic measure }\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}_{0,T+\\epsilon }(\\mathbb {R}));\\\\2.&\\textrm { solve the corresponding stochastic singular control problem}:\\\\&\\inf _{u,Z } E\\left[\\int _0^Tf(t,X_t,{\\mu }_t,u_t)dt\\right] \\\\&\\textrm {subject to} \\\\&dX_t=b(t,X_t,{\\mu }_t,u_t)\\,dt+\\sigma (t,X_t,{\\mu }_t,u_t)dW_t+c(t)\\,dZ_t,~t\\in [0,T+\\epsilon ];\\\\3.&\\textrm {solve the fixed point problem: }{\\mu ={Law}(X),}\\end{array} \\right.$ for some fixed $\\epsilon >0$ under the finite-fuel constraint $Z\\in \\widetilde{{\\mathcal {A}}}^m_{0,T}(\\mathbb {R})$ .", "The reason we define the state process on the time interval $[0,T+\\epsilon ]$ is that we approximate the singular controls by absolutely continuous ones that are most naturally regarded as elements of $\\widetilde{\\cal D}_{0,T+\\epsilon }({\\mathbb {R}})$ rather than $\\widetilde{\\cal D}_{0,T}({\\mathbb {R}})$ ." ], [ "Solving MFGs with singular controls using MFGs with regular controls", "In this section we establish an approximation of (relaxed) solutions results for the MFGs (REF ) under a finite-fuel constraint by (relaxed) solutions to MFGs with only regular controls.", "To this end, we associate with each singular control $Z\\in \\widetilde{{\\mathcal {A}}}^m_{0,T}(\\mathbb {R})$ the sequence of absolutely continuous controls $Z^{[n]}_t=n\\int _{(t-\\frac{1}{n})}^t Z_s\\,ds \\quad \\left(t\\in \\mathbb {R}, n \\in \\mathbb {N} \\right).$ Then, $Z^{[n]}\\in \\widetilde{\\cal A}^m_{0,T+\\epsilon }({\\mathbb {R}})$ for all sufficiently large $n \\in \\mathbb {N}$ .", "Since each $Z^{[n]}$ is absolutely continuous and $Z$ is càdlàg we cannot expect convergence of $Z^n$ to $Z$ in the Skorokhod $J_1$ topology in general.", "However, by Proposition REF (3.)", "and the discussion before Proposition REF we do know that $Z^{[n]} \\rightarrow Z \\quad \\mbox{a.s. in} \\quad \\left(\\widetilde{\\cal D}_{0,T+\\epsilon }({\\mathbb {R}}), d_{M_1} \\right).$ For each $n$ , we consider the following finite-fuel constrained MFGs denoted by MFG$^{[n]}$: $\\left\\lbrace \\begin{array}{ll}1.&\\textrm { fix a deterministic measure }\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}_{0,T+\\epsilon }(\\mathbb {R}));\\\\2.&\\textrm { solve the corresponding stochastic control problem}:\\\\&\\inf _{u,Z}E\\left[\\int _0^Tf(t,X^{[n]}_t,{\\mu }_t,u_t)dt\\right] \\\\&\\textrm {subject to}\\\\&dX^{[n]}_t=b(t,X^{[n]}_t,{\\mu }_t,u_t)dt+\\sigma (t,X^{[n]}_t,{\\mu }_t,u_t)dW_t+c(t)\\,dZ^{[n]}_t,~ t\\in [0,T+\\epsilon ] \\\\&X^{[n]}_{0} = 0 \\\\&Z^{[n]}_t=n\\int _{(t-\\frac{1}{n})}^t Z_s\\,ds;\\\\3.", "&\\textrm {solve the fixed point problem: }{\\mu ={Law}(X^{[n]})}.\\end{array} \\right.$ Definition 4.1 We call the vector $r^n=(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z^{[n]})$ a relaxed control with respect to $\\mu $ for some $\\mu \\in \\mathcal {P}_p(\\widetilde{\\mathcal {D}}_{0,T+\\epsilon }(\\mathbb {R}))$ if $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,\\underline{Q},Z)$ satisfies 1.-3. in Definition REF with item 4 being replaced by $4^{\\prime }$ .", "$X$ is a $\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace $ adapted stochastic process and $X\\in \\widetilde{\\mathcal {D}}_{0,T+\\epsilon }(\\mathbb {R})$ such that for each $\\phi \\in \\mathcal {C}^2_b(\\mathbb {R}^d)$ , $\\mathcal {M}^{[n],\\mu ,\\phi }$ is a well defined $\\mathbb {P}$ continuous martingale, where $\\begin{split}\\mathcal {M}^{[n],\\mu ,\\phi }_t:=&~\\phi (X_t)-\\int _0^t\\int _U\\mathcal {L}\\phi (s,X_s,\\mu _s,u)\\,\\underline{Q}_s(du)ds-\\int _0^t(\\partial _x\\phi (X_s))^{\\top }c(s)\\,dZ^{[n]}_s,\\end{split}$ with $\\mathcal {L}$ defined as in Definition REF .", "The probability measure $\\mathbb {P}$ is called a control rule if $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P},X,{Q}^o,Z^{[n]})$ is a relaxed control with $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace )$ being the filtered canonical space with $\\Omega :={\\widetilde{\\mathcal {D}}}_{0,T+\\epsilon }(\\mathbb {{R}})\\times \\widetilde{{\\cal U}}_{0,T+\\epsilon }(\\mathbb {R}) \\times \\widetilde{\\mathcal {A}}^m_{0,T}(\\mathbb {R})$ and $(X,Q,Z)$ being the coordinate projections on $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace )$ and ${Q}^o$ being the disintegration of $Q$ as in Section REF .", "Remark 4.2 If $Z$ is discontinuous at $T$ , then $Z^{[n]}$ may not converge to $Z$ in $\\widetilde{{\\cal D}}_{0,T}(\\mathbb {R})$ but only in $\\widetilde{{\\cal D}}_{0,T+\\epsilon }(\\mathbb {R})$ .", "Likewise, the associated sequence of the state processes may only converge in $\\widetilde{{\\cal D}}_{0,T+\\epsilon }(\\mathbb {R})$ .", "The possible discontinuity at the terminal time $T$ is also the reason why there is no terminal cost and no cost from singular control in this section.", "If we assume that $T$ is always a continuous point, then terminal costs and costs from singular controls are permitted.", "In this case, one may as well allow unbounded singular controls.", "For each fixed $n$ and $\\mu $ , denote by $\\mathcal {R}^{[n]}(\\mu )$ the set of all the control rules for MFG$^{[n]}$, and define the cost functional corresponding to the control rule $\\mathbb {P}\\in \\mathcal {R}^{[n]}(\\mu )$ by $J^{[n]}(\\mu ,\\mathbb {P})=E^{\\mathbb {P}}\\left(\\int _0^T\\int _U f(t,X_t,\\mu _t,u)\\,Q_t(du)dt\\right).$ For each fixed $n$ and $\\mu $ , denote by $\\mathcal {R}^{[n]*}(\\mu )$ the set of all the optimal control rules.", "We can still check that $\\inf \\limits _{\\textrm {relaxed control }r^n}J^{[n]}(\\mu ,r^n)=\\inf \\limits _{\\mathbb {P}\\in \\mathcal {R}^{[n]}(\\mu )}J^{[n]}(\\mu ,\\mathbb {P}),$ which implies we can still restrict ourselves to control rules in analyzing MFG$^{[n]}$.", "The proof of the following theorem is very similar to that of Theorem REF and is hence omitted.", "Theorem 4.3 Suppose $\\mathcal {A}_1$ -$\\mathcal {A}_6$ hold.", "For each $n$ , there exists a relaxed solution $\\mathbb {P}^{[n]}$ to MFG$^{[n]}$.", "By Proposition REF , the sequence $\\left\\lbrace \\mathbb {P}^{[n]}\\right\\rbrace _{n\\ge 1}$ is relatively compact.", "Denote its limit (up to a subsequence) by $\\mathbb {P}^*$ and set $\\mu ^*=\\mathbb {P}^*\\circ X^{-1}$ .", "Then, $\\mu ^*$ is the limit of $\\mu ^{[n]}:=\\mathbb {P}^{[n]}\\circ X^{-1}$ .", "The following lemma shows that $\\mathbb {P}^*$ is admissible.", "Lemma 4.4 Suppose $\\mathcal {A}_1$ -$\\mathcal {A}_2$ , $\\mathcal {A}_4$ -$\\mathcal {A}_6$ hold.", "Then $\\mathbb {P}^*\\in \\mathcal {R}(\\mu ^*)$ .", "By Proposition REF there exists, for each $n$ , a $\\lbrace \\mathcal {F}_t,0\\le t\\le T+\\epsilon \\rbrace $ adapted continuous process $Y^n$ , such that $\\mathbb {P}^{[n]}\\left({X}_{t}=Y^n_{t}+\\int _0^{t}c(s)\\,d{Z}^{[n]}_{s},~t\\in [0,T+\\epsilon ]\\right)=1.$ Arguing as in the proof of Proposition REF , there exists a probability space $(\\widetilde{\\Omega },\\widetilde{\\mathcal {F}},\\mathbb {Q})$ supporting random varibales $(\\widetilde{X}^n,\\widetilde{Y}^n,\\widetilde{Q}^n,\\widetilde{Z}^n)$ and $(\\widetilde{X},\\widetilde{Y},\\widetilde{Q},\\widetilde{Z})$ such that $(\\widetilde{X}^n,\\widetilde{Y}^n,\\widetilde{Q}^n,\\widetilde{Z}^n) \\rightarrow (\\widetilde{X},\\widetilde{Y},\\widetilde{Q},\\widetilde{Z}) ~ \\mathbb {Q}\\mbox{-a.s.}$ and $\\mathbb {P}^{[n]}\\circ (X,Y^n,Q,Z)^{-1}=\\mathbb {Q}\\circ (\\widetilde{X}^n,\\widetilde{Y}^n,\\widetilde{Q}^n,\\widetilde{Z}^n)^{-1},$ which implies $\\mathbb {Q}\\left(\\widetilde{X}^n_{t}=\\widetilde{Y}^n_{t}+\\int _0^{t}c(s)\\,d\\widetilde{Z}^{[n],n}_s,~t\\in [0,T+\\epsilon ]\\right)=1,$ where $\\widetilde{Z}^{[n],n}_t=n\\int _{(t-1/n)}^t \\widetilde{Z}^n_s\\,ds$ .", "For each fixed $\\widetilde{\\omega }\\in \\widetilde{\\Omega }$ and for each $t$ which is a continuous point of $\\widetilde{Z}(\\widetilde{\\omega })$ , by (REF ) in Proposition REF , we have $\\begin{split}\\left|n\\int _{t-\\frac{1}{n}}^t\\widetilde{Z}^n_s(\\widetilde{\\omega })\\,ds-\\widetilde{Z}_t(\\widetilde{\\omega })\\right|\\le &~ n\\int _{t-\\frac{1}{n}}^t|\\widetilde{Z}^n_s(\\widetilde{\\omega })-\\widetilde{Z}_s(\\widetilde{\\omega })|\\,ds+n\\int _{t-\\frac{1}{n}}^t|\\widetilde{Z}_s(\\widetilde{\\omega })-\\widetilde{Z}_t(\\widetilde{\\omega })|\\,ds\\\\\\le &~\\sup _{t-\\frac{1}{n}\\le s\\le t}|\\widetilde{Z}^n_s(\\widetilde{\\omega })-\\widetilde{Z}_s(\\widetilde{\\omega })|+\\sup _{t-\\frac{1}{n}\\le s\\le t}|\\widetilde{Z}_s(\\widetilde{\\omega })-\\widetilde{Z}_t(\\widetilde{\\omega })|\\\\\\rightarrow &~ 0.\\end{split}$ Then (REF ) and right-continuity of the path yield that $\\mathbb {Q}\\left(\\widetilde{X}_{t}=\\widetilde{Y}_{t}+\\int _0^{t}c(s)\\,d\\widetilde{Z}^{}_s,~t\\in [0,T+\\epsilon ]\\right)=1.$ The desired result can be obtained by the same proof as Proposition REF .", "Remark 4.5 In the above proof, the local uniform convergence near a continuous point is necessary.", "As stated in Proposition REF , this is a direct consequence of the convergence in the $M_1$ topology.", "Local uniform convergence cannot be guaranteed in the Meyer-Zheng topology.", "For Meyer-Zheng topology, we only know that convergence is equivalent to convergence in Lebesgue measure but we do not have uniform convergence in general.", "We are now ready to state and prove the main result of this section.", "Theorem 4.6 Suppose $\\mathcal {A}_1$ -$\\mathcal {A}_6$ hold.", "Then $\\mathbb {P}^*$ is a relaxed solution to the MFG (REF ).", "For each $\\mathbb {P}\\in \\mathcal {R}(\\mu ^*)$ such that $J(\\mu ^*,\\mathbb {P})<\\infty $ , on an extension $(\\widetilde{\\Omega },\\widetilde{\\mathcal {F}},\\lbrace \\widetilde{\\mathcal {F}}_t,t\\in \\mathbb {R}\\rbrace ,\\widetilde{\\mathbb {P}})$ we have, $d\\widetilde{X}_t=\\int _Ub(t,\\widetilde{X}_t,\\mu ^*_t,u)\\,\\widetilde{Q}_t(du)dt+\\int _U\\sigma (t,\\widetilde{X}_t,\\mu ^*_t,u)\\,\\widetilde{M}(du,dt)+c(t)\\,d\\widetilde{Z}_t,$ and $\\mathbb {P}=\\widetilde{\\mathbb {P}}\\circ (\\widetilde{X},\\widetilde{Q},\\widetilde{Z})^{-1}$ .", "Let $\\widetilde{Z}^{[n]}_t=n\\int _{t-1/n}^t\\widetilde{Z}_s\\,ds$ .", "By the Lipschitz continuity of the coefficient $b$ and $\\sigma $ , there exists a unique strong solution $X^n$ to the following SDE on $(\\widetilde{\\Omega },\\widetilde{\\mathcal {F}},\\lbrace \\widetilde{\\mathcal {F}}_t,t\\in \\mathbb {R}\\rbrace ,\\widetilde{\\mathbb {P}})$ : $dX^n_t=\\int _Ub(t,X^n_t,\\mu ^{[n]}_t,u)\\,\\widetilde{Q}_t(du)dt+\\int _U\\sigma (t,X^n_t,\\mu ^{[n]}_t,u)\\,\\widetilde{M}(du,dt)+c(t)\\,d\\widetilde{Z}^{[n]}_t.$ For each $n$ , set $\\mathbb {P}^n=\\widetilde{\\mathbb {P}}\\circ (X^n,\\widetilde{Q},\\widetilde{Z})^{-1}$ .", "It is easy to check that $\\mathbb {P}^{n}\\in \\mathcal {R}^{[n]}(\\mu ^{[n]})$ .", "Standard estimates yield, $\\begin{split}E^{\\widetilde{\\mathbb {P}}}\\int _0^T |X^n_t-\\widetilde{X}_t|^2\\,dt\\le &~ CE^{\\widetilde{\\mathbb {P}}}\\int _0^T \\left|\\int _0^tc(s)\\,d\\widetilde{Z}^{[n]}_s-\\int _0^tc(s)\\,d\\widetilde{Z}_s\\right|^2\\,dt\\\\&+~CE^{\\widetilde{\\mathbb {P}}}\\int _0^T\\left(1+L(W_p(\\mu ^{[n]}_t,\\delta _0),W_p(\\mu ^*_t,\\delta _0))\\right)^2\\mathcal {W}_p(\\mu ^{[n]}_t,\\mu ^*_t)^2\\,dt.\\end{split}$ $\\widetilde{Z}^{[n]}\\rightarrow \\widetilde{Z}$ in $M_1$ a.s. implies $E^{\\widetilde{\\mathbb {P}}}\\int _0^T \\left|\\int _0^tc(s)\\,d\\widetilde{Z}^{[n]}_s-\\int _0^tc(s)\\,d\\widetilde{Z}_s\\right|^2\\,dt\\rightarrow 0.$ By the same arguments leading to (REF ) in the proof of Lemma REF , $E^{\\widetilde{\\mathbb {P}}}\\int _0^T\\left(1+L(W_p(\\mu ^{[n]}_t,\\delta _0),W_p(\\mu ^*_t,\\delta _0))\\right)^2\\mathcal {W}_p(\\mu ^{[n]}_t,\\mu ^*_t)^2\\,dt\\rightarrow 0.$ This yields, $\\lim _{n\\rightarrow \\infty }E^{\\widetilde{\\mathbb {P}}}\\int _0^T |X^n_t-\\widetilde{X}_t|^2\\,dt=0.$ Hence, up to a subsequence, dominated convergence implies $\\begin{split}\\lim _{n\\rightarrow \\infty }J^{[n]}(\\mu ^{[n]},\\mathbb {P}^n) =&~\\lim _{n\\rightarrow \\infty }E^{\\widetilde{\\mathbb {P}}}\\left[\\int _0^T \\int _U f(t,X^n_t,\\mu ^{[n]}_t,u)\\,\\widetilde{Q}_t(du)dt\\right] \\\\= & E^{\\widetilde{\\mathbb {P}}}\\left[\\int _0^T\\int _U f(t,X_t,\\mu ^*_t,u)\\,\\widetilde{Q}_t(du)dt\\right] \\\\= & J(\\mu ^*,\\mathbb {P}).\\end{split}$ Moreover, by Lemma REF , $\\lim _{n\\rightarrow \\infty }J^{[n]}(\\mu ^{[n]},\\mathbb {P}^{[n]})= J(\\mu ^*,\\mathbb {P}^*).$ Altogether, this yields, $J(\\mu ^*,\\mathbb {P})=\\lim _{n\\rightarrow \\infty }J^{[n]}(\\mu ^{[n]},\\mathbb {P}^n)\\ge \\lim _{n\\rightarrow \\infty }J^{[n]}(\\mu ^{[n]},\\mathbb {P}^{[n]})= J(\\mu ^*,\\mathbb {P}^*).$" ], [ "Approximating a given solutions to MFGs with singular controls", "In this subsection, we show how to approximate a given solution to a MFG with singular controls of the form (REF ) introduced in the previous subsection by a sequence of admissible control rules of MFGs with only regular controls.", "Let $\\mathbb {P}^*$ be any solution to the MFG (REF ).", "Since $(\\Omega ,\\lbrace \\mathcal {F}_t,t\\in \\mathbb {R}\\rbrace ,\\mathbb {P}^*,X,Q,Z)$ satisfies the associated martingale problem, there exists a tuple $(\\widehat{X},\\widehat{Q},\\widehat{Z},M)$ defined on some extension $(\\widehat{\\Omega },\\lbrace \\widehat{\\mathcal {F}}_t,t\\in \\mathbb {R}\\rbrace ,{\\mathbb {Q}})$ of the canonical path space, such that $\\mathbb {P}^*\\circ (X,Q,Z)^{-1}=\\mathbb {Q}\\circ (\\widehat{X},\\widehat{Q},\\widehat{Z})^{-1}$ and $\\mathbb {Q}\\left(\\widehat{X}_{\\cdot }=\\int _0^{\\cdot }\\int _U b(s,\\widehat{X}_s,\\mu ^*_s,u)\\,\\widehat{Q}_s(du)ds+\\int _0^{\\cdot }\\int _U\\sigma (s,\\widehat{X},\\mu ^*_s,u)\\,M(du,ds)+\\int _0^{\\cdot }c(s)\\,d\\widehat{Z}_s\\right)=1.$ Let $X^{[n]}$ be the unique strong solution of the SDE $dX^{[n]}_t=\\int _U b(t,X^{[n]}_t,\\mu ^{[n]}_t,u)\\,\\widehat{Q}_t(du)dt+\\int _U\\sigma (t,X^{[n]}_t,\\mu ^{[n]}_t,u)\\,M(du,dt)+c(t)\\,d\\widehat{Z}^{[n]}_t,$ where $\\widehat{Z}^{[n]}$ is defined by (REF ) and $\\mu ^{[n]}$ is any sequence satisfying $\\mu ^{[n]}\\rightarrow \\mu ^*$ in $\\mathcal {W}_{p,(\\widetilde{\\mathcal {D}}(\\mathbb {R}),d_{M_1})}$ .", "One checks immediately that $\\mathbb {P}^{[n]}:=\\mathbb {Q}\\circ (X^{[n]},\\widehat{Q}, \\widehat{Z})^{-1}\\in \\mathcal {R}^{[n]}(\\mu ^{[n]}).$ Our goal is to show that the sequence $\\lbrace \\mathbb {P}^{[n]}\\rbrace _{n \\ge 1}$ converges to $\\mathbb {P}^*$ in $\\mathcal {W}_p$ along some subsequence, which relies on the following lemma.", "Its proof uses the notion of a parameter representation of the thin graph of a function $x \\in {\\cal D}(0,T)$ introduced in Appendix .", "Proposition 4.7 On some probability space $(\\Omega ,\\mathcal {F},\\lbrace \\mathcal {F}_t,t\\ge 0\\rbrace ,\\mathbb {P})$ , let $X^n$ and $X$ be the unique strong solution to SDE, $dX^n_t=\\int _U b(t,X^n_t,\\mu ^{n}_t,u)\\,Q_t(du)dt+\\int _U\\sigma (t,X^n_t,\\mu ^n_t,u)\\,M(du,dt)+\\,dZ^n_t, ~t\\in [0,\\widetilde{T}]$ respectively, $dX_t=\\int _Ub(t,X_t,\\mu _t,u)\\,Q_t(du)dt+\\int _U\\sigma (t,X_t,\\mu _t,u)\\,M(du,dt)+\\,dZ_t, ~t\\in [0,\\widetilde{T}]$ where $\\widetilde{T}$ is a fixed positive constant, $b$ and $\\sigma $ satisfy $\\mathcal {A}_1$ and $\\mathcal {A}_5$ .", "If $Z^n\\rightarrow Z$ in $(\\mathcal {A}^m(0,\\widetilde{T}),d_{M_1})$ a.s. and $\\mu ^n\\rightarrow \\mu $ in $\\mathcal {W}_{p,(\\mathcal {D}(0,\\widetilde{T}),d_{M_1})}$ , then $\\lim _{n\\rightarrow \\infty } E^{\\mathbb {P}}d_{M_1}(X^n,X)=0.$ By the a.s. convergence of $Z^{n}$ to $Z$ in $M_1$ , there exists $\\underline{\\Omega }\\subseteq \\Omega $ with full measure such that $d_{M_1}(Z^n(\\omega ),Z(\\omega ))\\rightarrow 0$ for each $\\omega \\in \\underline{\\Omega }$ .", "Furthermore, by Proposition REF (2), for each $\\omega \\in \\underline{\\Omega }$ , there exist parameter representations $(u(\\omega ),r(\\omega ))\\in \\Pi _{Z(\\omega )}$ and $(u_n(\\omega ),r_n(\\omega ))\\in \\Pi _{Z^{n}(\\omega )}$ of $Z(\\omega )$ and $Z^n(\\omega )$ $(n \\in \\mathbb {N})$ , respectively, such that $\\Vert u_n(\\omega )-u(\\omega )\\Vert \\rightarrow 0\\textrm { and }\\Vert r_n(\\omega )-r(\\omega )\\Vert \\rightarrow 0.$ Parameter representations with the desired convergence properties are constructed in, e.g., [28]; see also [28].", "A careful inspection of [28] shows that the constructions of $(u(\\omega ),r(\\omega ))$ and $(u_n(\\omega ),r_n(\\omega ))$ only use measurable operations.", "As a result the mappings $(u(\\cdot ),r(\\cdot ))$ and $(u_n(\\cdot ),r_n(\\cdot ))$ are measurable.", "We now construct parameter representations $(u_{X^n}(\\omega ),r_{X^n}(\\omega ))$ and $(u_{X}(\\omega ),r_{X}(\\omega ))$ of $X^n(\\omega )$ and $X(\\omega )$ , respectively.", "Since $X(\\omega )$ (resp.", "$X^n(\\omega )$ ) jumps at the same time as $Z(\\omega )$ (resp.", "$Z^{n}(\\omega )$ ), we can choose $r_{X}(\\omega )=r(\\omega ),~~r_{X^n}(\\omega )=r_n(\\omega ).$ In the following, we will drop the dependence on $\\omega \\in \\underline{\\Omega }$ , if there is no confusion.", "By [28], parameter representations of $X^n$ and $X$ in terms of the parameter representations of $Z^n$ and $Z$ are given by, respectively, $u_{X^n}(t)=\\int _0^{r_n(t)}\\int _U b(s,X^n_s,\\mu ^{n}_{s},u)\\,Q_s(du)ds+\\int _0^{r_n(t)}\\int _U \\sigma (s,X^n_s,\\mu ^n_s,u)\\,M(du,ds)+u_n(t),$ and $u_{X}(t)=\\int _0^{r(t)} \\int _Ub(s,X_s,\\mu _s,u)\\,Q_s(du)ds+\\int _0^{r(t)}\\int _U \\sigma (s,X_s,\\mu _s,u)\\,M(du,ds)+u(t).$ Hence, by the Lipschitz property of $b$ and $\\sigma $ and BDG's inequality, we get, $\\begin{split}&~E\\sup _{0\\le t\\le \\widetilde{T}}|u_{X^n}(t)-u_{X}(t)|\\\\\\le &~CE\\left(\\int _0^{\\widetilde{T}}|{X^n}(s)-{X}(s)|^2\\,ds\\right)^{\\frac{1}{2}}\\\\&~+C\\left(\\int _0^{\\widetilde{T}}\\left(1+L(\\mathcal {W}_p(\\mu ^n_{s},\\delta _0),\\mathcal {W}_p(\\mu _{s},\\delta _0))\\right)^2\\mathcal {W}^2_p(\\mu ^n_{s},\\mu _{s})\\,ds\\right)^{\\frac{1}{2}}+CE\\sup _{0\\le t\\le \\widetilde{T}}|r_n(t)-r(t)|\\\\&~+E\\sup _{0\\le t\\le \\widetilde{T}}\\left|\\int _0^{r_n(t)}\\int _U \\sigma (s,X_s,\\mu _s,u)\\,M(du,ds)-\\int _0^{r(t)}\\int _U \\sigma (s,X_s,\\mu _s,u)\\,M(du,d{s})\\right|\\\\&~+E\\sup _{0\\le t\\le \\widetilde{T}}\\left|u_n(t)-u(t)\\right|.\\end{split}$ The same argument as in the proof of Theorem REF yields that the first two terms on the right hand side of (REF ) converge to 0 while the last three terms converge to 0 due to (REF ).", "Thus, $\\lim _{n\\rightarrow \\infty } E\\sup _{0\\le t\\le \\widetilde{T}}|u_{X^n}(t)-u_{X}(t)|=0.$ Corollary 4.8 Under the assumptions of Proposition REF , along a subsequence $\\mathbb {P}^{[n]}\\rightarrow \\mathbb {P}^*$ in $\\mathcal {W}_p.$ For each $\\widetilde{\\epsilon }>0$ , we extend the equations (REF ) and (REF ) by $\\widehat{X}_s=\\int _{-\\widetilde{\\epsilon }}^s\\int _U \\widetilde{b}(t,\\widehat{X}_t,\\mu ^*_t,u)\\,\\widehat{Q}_t(du)dt+\\int _{-\\widetilde{\\epsilon }}^s\\int _U \\widetilde{\\sigma }(t,\\widehat{X}_t,\\mu ^*_t,u)\\,M(du,dt)+\\int _{-\\widetilde{\\epsilon }}^s\\widetilde{c}(t)\\,d\\widehat{Z}_t,$ respectively, $X^{[n]}_s=\\int _{-\\widetilde{\\epsilon }}^s\\int _U \\widetilde{b}(t,X^{[n]}_t,\\mu ^{[n]}_t,u)\\,\\widehat{Q}_t(du)dt+\\int _{-\\widetilde{\\epsilon }}^s\\int _U\\widetilde{\\sigma }(t,X^{[n]}_t,\\mu ^{[n]}_t,u)\\,M(du,dt)+\\int _{-\\widetilde{\\epsilon }}^s\\widetilde{c}(t)\\,d\\widehat{Z}^{[n]}_t,$ where $\\widetilde{b}(s,\\cdot )=b(s,\\cdot ), ~\\widetilde{\\sigma }(s,\\cdot )=\\sigma (s,\\cdot ),~\\widetilde{c}(s)=c(s)\\textrm { when }s\\ge 0;~ \\widetilde{b}(s,\\cdot )=0, ~\\widetilde{\\sigma }(s,\\cdot )=0,~\\widetilde{c}(s)=c(0)\\textrm { when }s<0.$ Moreover, we have that $\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}(t)\\,d\\widehat{Z}^{[n]}_t=\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^+(t)\\,d\\widehat{Z}^{[n]}_t-\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^-(t)\\,d\\widehat{Z}^{[n]}_t,$ where a.s. in $({\\mathcal {A}}^m(-\\widetilde{\\epsilon },T+\\epsilon ),d_{M_1})$ , $\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^+(t)\\,d\\widehat{Z}^{[n]}_t\\rightarrow \\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^+(t)\\,d\\widehat{Z}^{}_t\\quad \\mbox{and} \\quad \\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^-(t)\\,d\\widehat{Z}^{[n]}_t\\rightarrow \\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^-(t)\\,d\\widehat{Z}^{}_t.$ Since $\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^+(t)\\,d\\widehat{Z}^{}_t$ and $\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}^-(t)\\,d\\widehat{Z}^{}_t$ never jump at the same time, Proposition REF implies that $\\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}(t)\\,d\\widehat{Z}^{[n]}_t\\rightarrow \\int _{-\\widetilde{\\epsilon }}^{\\cdot }\\widetilde{c}(t)\\,d\\widehat{Z}^{}_t$ a.s. in $({\\mathcal {A}}^m(-\\widetilde{\\epsilon },T+\\epsilon ),d_{M_1})$ .", "Hence, by Proposition REF , $E^{\\mathbb {Q}}d_{M_1}(X^{[n]},\\widehat{X})\\rightarrow 0.$ Hence, up to a subsequence, $d_{M_1}(X^{[n]},\\widehat{X})\\rightarrow 0 \\mbox{ in } \\mathcal {D}(-\\widetilde{\\epsilon },T+\\epsilon ); \\quad \\mathbb {Q}\\mbox{-a.s.},$ which implies the same convergence holds in $\\widetilde{\\mathcal {D}}_{0,T+\\epsilon }(\\mathbb {R})$ .", "For any nonnegative continuous function $\\phi $ satisfying $\\phi (x,q,z)\\le C(1+d_{M_1}(x,0)^p+\\mathcal {W}_p^p(q/T,\\delta _0)+d_{M_1}(z,0)^p),$ the uniform integrability of $d_{M_1}(X^{[n]},0)^p$ , $\\mathcal {W}_p^p(\\widehat{Q}/T,\\delta _0)$ and $d_{M_1}(\\widehat{Z},0)^p$ yields $E^{\\mathbb {Q}}\\phi (X^{[n]},\\widehat{Q},\\widehat{Z})\\rightarrow E^{\\mathbb {Q}}\\phi (\\widehat{X},\\widehat{Q},\\widehat{Z}).$ This implies $\\mathbb {Q}\\circ (X^{[n]},\\widehat{Q},\\widehat{Z})^{-1}\\rightarrow \\mathbb {Q}\\circ (\\widehat{X},\\widehat{Q},\\widehat{Z})^{-1}$ in $\\mathcal {W}_{p,\\Omega }$ by [29], that is, $\\mathbb {P}^{[n]}\\rightarrow \\mathbb {P}^*$ in $\\mathcal {W}_{p,\\Omega }$ ." ], [ "Wasserstein distance and representation of martingales", "Definition A.1 Let $(E,\\varrho )$ be a metric space.", "Denote by $\\mathcal {P}_p(E)$ the class of all probability measures on $E$ with finite moment of $p$ -th order.", "The $p$ -th Wasserstein metric on $\\mathcal {P}_p(E)$ is defined by: $\\mathcal {W}_{p,(E,\\varrho )}(\\mathbb {P}_1,\\mathbb {P}_2)=\\inf \\left\\lbrace \\left(\\int _{E\\times E}\\varrho (x,y)^p\\,\\gamma (dx,dy)\\right)^{\\frac{1}{p}}:\\gamma (dx,E)=\\mathbb {P}_1(dx),\\gamma (E,dy)=\\mathbb {P}_2(dy)\\right\\rbrace .$ The set $\\mathcal {P}_p(E)$ endowed with the Wasserstein distance is denoted by $\\mathcal {W}_{p,(E,\\varrho )}$ or $\\mathcal {W}_{p,E}$ or $\\mathcal {W}_{p}$ if there is no risk of confusion about the underlying state space or distance.", "It is well known [24] that for every continuous square integrable martingale $m$ with quadratic variation process $\\int _0^{\\cdot }\\int _U a(s,u)\\,v_s(du)ds$ , where $a=\\sigma \\sigma ^{\\top }$ and $\\sigma $ is a bounded measurable function and $v$ is $\\mathcal {P}(U)$ valued stochastic process, on some extension of the original probability space, there exists a martingale measure $M$ with intensity $v_s(du)ds$ such that $m_{\\cdot }=\\int _0^{\\cdot }\\int _U \\sigma (t,u)\\,M(du,dt)$ .", "This directly leads to the following proposition, which is frequently used in the main text.", "Proposition A.2 The existence of solution $\\mathbb {P}$ to the martingale problem (REF ) is equivalent to the existence of the weak solution to the following SDE $d\\bar{X}_t=\\int _U b(t,\\bar{X}_t,\\mu _t,u)\\,\\bar{Q}_s(du)ds+\\int _U \\sigma (t,\\bar{X}_t,\\mu _t,u)\\,\\bar{M}(du,dt)+c(t)\\,d\\bar{Z}_t,$ where $\\bar{X}$ , $\\bar{M}$ and $\\bar{Z}$ are defined on some extension $(\\bar{\\Omega },\\bar{\\mathcal {F}},\\bar{\\mathbb {P}})$ and $\\bar{M}$ is a martingale measure with intensity $\\bar{Q}$ .", "Moreover, the two solutions are related by $\\mathbb {P}=\\bar{\\mathbb {P}}\\circ (\\bar{X},\\bar{Q},\\bar{Z})^{-1}$ ." ], [ "Strong $M_1$ Topology in Skorokhod Space", "In this section, we summarise some definitions and properties about strong Skorokhod $M_1$ topology.", "For more details, please refer to Chapter 3, 11 and 12 in [30].", "Note that in [30] two $M_1$ topologies are introduced, the strong one and the weak one.", "In this paper, we only apply the strong one.", "So without abuse of terminologies, we just take $M_1$ topology for short.", "For $x\\in \\mathcal {D}(0,T)$ , denote by $Disc(x)$ the set of discontinuous points of $x$ .", "Note that on $[0,T]$ , $Disc(x)$ is at most countable.", "Define the thin graph of $x$ as $G_x=\\lbrace (z,t)\\in \\mathbb {R}^d\\times [0,T]:z\\in [x_{t-},x_t]\\rbrace ,$ where $x_{t-}$ is the left limit of $x$ at $t$ and $[a,b]$ means the line segment between $a$ and $b$ , i.e., $[a,b]=\\lbrace \\alpha a+(1-\\alpha )b:0\\le \\alpha \\le 1\\rbrace $ .", "On the thin graph, we define an order relation.", "For each pair $(z_i,t_i)\\in G_x$ , $i=1,2$ , $(z_1,t_1)\\le (z_2,t_2)$ if either of the following holds: (1) $t_1<t_2$ ; (2) $t_1=t_2$ and $|z_1-x_{t_1-}|<|z_2-x_{t_2-}|$ .", "Now we define the parameter representation, on which the $M_1$ topology depends.", "The mapping pair $(u,r)$ is called a parameter representation if $(u,r):[0,1]\\rightarrow G_x$ , which is continuous and nondecreasing w.r.t.", "the order relation defined above.", "Denote by $\\Pi _x$ all the parameter representations of $x$ .", "Let $d_{M_1}(x_1,x_2)=\\inf _{(u_i,r_i)\\in \\Pi _{x_i},i=1,2} ||u_1-u_2||\\vee ||r_1-r_2||.$ It can be shown that $d_{M_1}$ is a metric on $\\mathcal {D}(0,T)$ such that $\\mathcal {D}(0,T)$ is a Polish space.", "The topology induced by $d_{M_1}$ is called $M_1$ topology.", "For each $t\\in [0,T]$ and $\\delta >0$ , the oscillation function around $t$ is defined as $\\bar{v}(x,t,\\delta )=\\sup _{0\\vee (t-\\delta )\\le t_1\\le t_2 \\le (t+\\delta )\\wedge T}|x_{t_1}-x_{t_2}|,$ and the so called strong $M_1$ oscillation function is defined as $w_s(x,t,\\delta )=\\sup _{0\\vee (t-\\delta )\\le t_1< t_2<t_3\\le (t+\\delta )\\wedge T}|x_{t_2}-[x_{t_1},x_{t_3}]|,$ where $|x_{t_2}-[x_{t_1},x_{t_3}]|$ is the distance from $x_{t_2}$ to the line segment $[x_{t_1},x_{t_3}]$ .", "Moreover, $w_s(x,\\delta ):=\\sup _{0\\le t\\le T}w_s(x,t,\\delta ).$ Proposition B.1 The following statements about the characterization of $M_1$ convergence are equivalent, 1.", "$x^n\\rightarrow x$ in $M_1$ topology; 2. there exist $(u,r)\\in \\Pi _x$ and $(u^n,r^n)\\in \\Pi _{x^n}$ for each $n$ such that $\\lim \\limits _{n\\rightarrow \\infty }\\Vert u^n-u\\Vert \\vee \\Vert r^n-r\\Vert =0;$ 3.", "$x_n(t)\\rightarrow x(t)$ for each $t\\in [0,T]\\setminus Disc(x)$ including 0 and $T$ , and $\\lim _{\\delta \\rightarrow 0}\\overline{\\lim }_{n\\rightarrow \\infty }w_s(x^n,\\delta )=0.$ Moreover, each one of the above three items implies the local uniform convergence of $x^n$ to $x$ at each continuous point of $x$ , that is, for each $t\\notin Disc(x)$ , there holds $\\lim _{\\delta \\rightarrow 0}\\limsup _{n\\rightarrow \\infty }\\sup _{t-\\delta \\le s\\le t+\\delta }|x_n(s)-x(s)|=0.$ Remark B.2 (1) When restricted to $\\mathcal {C}(0,T)$ , the uniform topology is equivalent to the $M_1$ topology.", "Indeed, when $x^n\\in \\mathcal {C}(0,T)$ and $\\Vert x^n-x\\Vert \\rightarrow 0$ , then $x\\in \\mathcal {C}(0,T)$ .", "For any $r:[0,1]\\rightarrow [0,T]$ , define $u^n(t):=x^n_{r(t)},\\qquad u(t):=x_{r(t)}.$ Thus, $(u^n,r)$ and $(u,r)$ can serve as a parameter representation of $x^n$ and $x$ , respectively.", "Moreover, it holds that $\\Vert u^n-u\\Vert \\rightarrow 0$ .", "So we have $x^n\\rightarrow x$ in $M_1$ by Proposition REF (2).", "On the other hand, when $x^n,x\\in \\mathcal {C}(0,T)$ and $x^n\\rightarrow x$ in $M_1$ , by Proposition REF (2), there exist parameter representations $(u^n,r^n)$ and $(u,r)$ of $x^n$ and $x$ , respectively, such that $u^n(t)=x^n_{r^n(t)}, ~u(t)=x_{r(t)},~\\Vert u^n-u\\Vert \\rightarrow 0\\textrm { and }\\Vert r^n-r\\Vert \\rightarrow 0.$ To show that $\\Vert x^n-x\\Vert \\rightarrow 0$ , it is sufficient to prove that $t^n\\rightarrow t$ implies $|x^n_{t^n}-x_t|\\rightarrow 0$ .", "Let $r^n(s^n)=t^n$ and $r(s)=t$ .", "Then we have $|r(s^n)-r(s)|\\le |r^n(s^n)-r(s^n)|+|r^n(s^n)-r(s)|\\rightarrow 0.$ So we get $|x^n_{t^n}-x_t|=|u^n(s^n)-u(s)|\\le |u^n(s^n)-u(s^n)|+|u(s^n)-u(s)|=|u^n(s^n)-u(s^n)|+|x_{r(s^n)}-x_{r(s)}|\\rightarrow 0.$ (2) Proposition REF (3) implies that $(\\mathcal {D}(0,T),d_{M_1})$ convergence is stronger than $L^{\\alpha }[0,T]$ convergence, for any $\\alpha >0$ .", "In fact, if $x^n\\rightarrow x$ in $M_1$ , then $x^n_t\\rightarrow x_t$ for a.e.", "$t\\in [0,T]$ , due to Proposition REF (3).", "Moreover, $|x^n_t-x_t|^{\\alpha }\\,\\le 2^{\\alpha }\\left(d_{M_1}^{\\alpha }(x^n,0)+d_{M_1}^{\\alpha }(x,0)\\right)\\rightarrow 2^{\\alpha +1}d_{M_1}(x,0)<\\infty .$ Thus, the assertion follows from dominated convergence.", "Proposition B.3 A subset $A$ of $(\\mathcal {D}(0,T), d_{M_1})$ is relatively compact w.r.t.", "$M_1$ topology if and only if $\\sup _{x\\in A}||x||<\\infty $ and $\\lim _{\\delta \\downarrow 0}\\sup _{x\\in A}w^{\\prime }_s(x,\\delta )=0,$ where $w^{\\prime }_s(x,\\delta )=w_s(x,\\delta )\\vee \\bar{v}(x,0,\\delta )\\vee \\bar{v}(x,T,\\delta ).$ In [30], it is assumed that $x_{0-}=x_0$ , which implies there is no jump at the initial time.", "For singular control problems it is natural to admit jumps a the initial time.", "It is also implied by Proposition REF that the terminal time $T$ is a continuous point of $x\\in \\mathcal {D}(0,T)$ .", "This, too, is not appropriate for singular control problems.", "In order to adapt the relative compactness criteria stated in Proposition REF to functions with jumps at 0 and $T$ , we work on the extended state spaces $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ and $\\widetilde{\\mathcal {A}}(\\mathbb {R})$ .", "Convergence in $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ can be defined as convergence in $\\mathcal {D}(\\mathbb {R})$ , where a sequence $\\lbrace x^n, n \\ge 1\\rbrace $ converges to $x$ in $\\mathcal {D}(\\mathbb {R})$ if and only if the sequences $\\lbrace x^n|_{[a,b]}, n \\ge 1\\rbrace $ converge to $x|_{[a,b]}$ for all $a<b$ at which $x$ is continuous; see [30].", "Relative compactness of a sequence $\\lbrace x^n,n\\ge 1\\rbrace \\subseteq \\widetilde{\\mathcal {D}}(\\mathbb {R})$ is equivalent to that of the sequence $\\lbrace x^n|_{[a,b]},n\\ge 1\\rbrace \\subseteq \\mathcal {D}[a,b]$ for any $a<0$ and $b>T$ .", "Specifically, we have the following result.", "Proposition B.4 The sequence $\\lbrace x^n,n\\ge 1\\rbrace \\subseteq \\widetilde{\\mathcal {D}}(\\mathbb {R})$ is relatively compact if and only if $\\begin{split}\\sup _{n}||x_n||<\\infty \\quad \\mbox{and} \\quad \\lim _{\\delta \\downarrow 0}\\sup _{x\\in A}\\widetilde{w}_s(x,\\delta )=0,\\end{split}$ where the modified oscillation function $\\widetilde{w}_s$ is defined as $\\widetilde{w}_s(x,\\delta )={w}_s(x,\\delta )+\\sup _{0\\le s<t\\le \\delta }|x_s-[0,x_t]|.$ We notice that the modified oscillation function $\\widetilde{w}_s$ is defined in terms of the original oscillation function ${w}_s$ and the line segment (if it exists) between $0-$ and 0Due to the right-continuity of the elements in $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ there is no line segment between $T$ and $T+$ .. Corollary B.5 Let $A=\\lbrace z\\in \\widetilde{\\mathcal {A}}(\\mathbb {R}):z_T\\le K\\rbrace $ for some $K>0$ .", "Then $A$ is $(\\widetilde{\\mathcal {D}}(\\mathbb {R}),M_1)$ compact.", "This follows from Proposition REF as $w_s(z,t,\\delta )=0$ for each $z\\in A$ , $t \\in \\mathbb {R}$ and $\\delta > 0$ .", "Proposition B.6 A sequence of probability measures $\\lbrace \\mathbb {P}_n\\rbrace _{n\\ge 1}$ on $\\widetilde{\\mathcal {D}}(\\mathbb {R})$ is tight if and only if (1) for each $\\epsilon >0$ , there exists $c$ large enough such that $\\sup _n\\mathbb {P}_n(||x||>c)<\\epsilon ;$ (2) for each $\\epsilon >0$ and $\\eta >0$ , there exists $\\delta >0$ small enough such that $\\sup _n\\mathbb {P}_n(\\widetilde{w}_s(x,\\delta )\\ge \\eta )<\\epsilon .$ The following proposition shows that if two $M_1$ limits do not jump at the same time, then the $M_1$ convergence preserves by the addition operation.", "Proposition B.7 If $x^n\\rightarrow x$ and $y^n\\rightarrow y$ in $(\\mathcal {D}(0,T), d_{M_1})$ , and $Disc(x)\\cap Disc(y)={Ø}$ , then $x^n+y^n\\rightarrow x+y~in~M_1.$" ], [ "Sketch Proof of Proposition ", "It is sufficient to establish the equivalence of martingale problems in Definition REF and Proposition REF .", "Only the one-dimensional case is proved; the multi-dimensional case is similar.", "Proposition REF $\\Rightarrow $ Definition REF : Without loss of generality (see [23]), we can take $\\phi (y)=y,~y^2$ and following the proof of [23], we have that $M$ is a continuous martingale with the quadratic variation $\\langle M\\rangle _t=\\int _0^t\\int _U a(s,X_s,\\mu _s,u)\\,Q_s(du)ds,$ where $M_t=Y_t-\\int _0^t\\int _U b(s,X_s,\\mu _s,u)\\,Q_s(du)ds.$ By applying Itô's formula to $\\phi (X_t)$ and noting $X=Y+\\int _0 c(s)\\,dZ_s$ , the desired result follows from $\\begin{split}\\phi (X_t)=&\\phi (X_{0-})+\\int _0^t\\int _U\\phi ^{\\prime }(X_s)b(s,X_s,\\mu _s,u)\\,Q_s(du)ds+\\frac{1}{2}\\int _0^t\\int _U\\phi ^{\\prime \\prime }(X_s)a(s,X_s,\\mu _s,u)\\,Q_s(du)ds\\\\&+\\int _0^t\\phi ^{\\prime }(X_{s-})c(s)\\,dZ_s+\\sum _{0\\le s\\le t}\\left[\\phi (X_s)-\\phi (X_{s-})-\\phi ^{\\prime }(X_{s-})\\triangle X_s\\right]+\\int _0^t\\phi ^{\\prime }(X_s)\\,dM_s.\\end{split}$ Definition REF $\\Rightarrow $ Proposition REF : By Proposition REF , there exits $(\\overline{X},\\overline{Q},\\overline{Z})$ and a martingale measure $\\overline{M}$ with intensity $\\overline{Q}$ on some extension $(\\overline{\\Omega },\\overline{\\mathcal {F}},\\overline{\\mathbb {P}})$ , s.t.", "(REF ) holds and $\\mathbb {P}\\circ (X,Q,Z)^{-1}=\\overline{\\mathbb {P}}\\circ (\\overline{X},\\overline{Q},\\overline{Z})^{-1}$ .", "Let $\\overline{Y}_{\\cdot }=\\overline{X}_{\\cdot }-\\int _0^{\\cdot }c(s)\\,d\\overline{Z}_s.$ Then ${Y}_{\\cdot }:={X}_{\\cdot }-\\int _0^{\\cdot }c(s)\\,d{Z}_s\\overset{d}{=}\\overline{Y}.$ By applying Itô's formula to $\\phi (\\overline{Y}_t)$ , $\\phi (\\overline{Y}_t)-\\int _0^t\\int _U\\phi ^{\\prime }(\\overline{Y}_s)b(s,\\overline{X}_s,\\mu _s,u)\\,\\overline{Q}_s(du)ds-\\frac{1}{2}\\int _0^t\\int _U\\phi ^{\\prime \\prime }(\\overline{Y}_s)a(s,\\overline{X}_s,\\mu _s,u)\\,\\overline{Q}_s(du)ds$ is a martingale.", "Hence the following is also a martingale: $\\phi ({Y}_t)-\\int _0^t\\int _U\\phi ^{\\prime }({Y}_s)b(s,{X}_s,\\mu _s,u)\\,{Q}_s(du)ds-\\frac{1}{2}\\int _0^t\\int _U\\phi ^{\\prime \\prime }({Y}_s)a(s,{X}_s,\\mu _s,u)\\,{Q}_s(du)ds.$" ] ]
1612.05425
[ [ "Gr\\\"obner Bases of Neural Ideals" ], [ "Abstract The brain processes information about the environment via neural codes.", "The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes.", "Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code.", "On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gr\\\"obner basis with respect to that monomial order.", "How are these two types of generating sets -- canonical forms and Gr\\\"obner bases -- related?", "Our main result states that if the canonical form of a neural ideal is a Gr\\\"obner basis, then it is the universal Gr\\\"obner basis (that is, the union of all reduced Gr\\\"obner bases).", "Furthermore, we prove that this situation -- when the canonical form is a Gr\\\"obner basis -- occurs precisely when the universal Gr\\\"obner basis contains only pseudo-monomials (certain generalizations of monomials).", "Our results motivate two questions: (1)~When is the canonical form a Gr\\\"obner basis?", "(2)~When the universal Gr\\\"obner basis of a neural ideal is {\\em not} a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code?", "We give partial answers to both questions.", "Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice." ], [ "Introduction", "The brain is tasked with many important functions, but one of the least understood is how it builds an understanding of the world.", "Stimuli in one's environment are not experienced in isolation, but in relation to other stimuli.", "How does the brain represent this organization?", "Or, to quote from Curto, Itskov, Veliz-Cuba, and Youngs, “What can be inferred about the underlying stimulus space from neural activity alone?” [7].", "Curto et al.", "pursued this question for codes where each neuron has a region of stimulus space, called its receptive field, in which it fires at a high rate.", "They introduced algebraic objects that summarize neural-activity data, which are in the form of neural codes ($0/1$ -vectors where 1 means the corresponding neuron is active, and 0 means silence) [7].", "The neural ideal of a neural code is an ideal that contains the full combinatorial data of the code.", "The canonical form of a neural ideal is a generating set that is a minimal description of the receptive-field structure.", "Hence, the questions posed above have been investigated via the neural ideal or the canonical form [6], [7], [8], [10].", "As a complement to algebraic approaches, combinatorial and topological arguments are employed in related works [5], [11], [13].", "The aim of our work is to investigate, for the first time, how the canonical form is related to other generating sets of the neural ideal, namely, its Gröbner bases.", "This is a natural mathematical question, and additionally the answer could improve algorithms for computing the canonical form.", "Currently, there are two distinct methods to compute the canonical form of a neural ideal: the original method proposed in [7] and an iterative method introduced in [16].", "The former method requires the computation of primary decomposition of pseudo-monomial ideals.", "As a result, this method is rather inefficient.", "Even in dimension 5, one can find codes for which this algorithm takes hundreds or even thousands of seconds to terminate or halts due to lack of memory.", "The more recent iterative method relies entirely on basic polynomial arithmetic.", "This algorithm can efficiently compute canonical forms for codes in up to 10 dimensions; see [16].", "On the other hand, Gröbner basis computations are generally computationally expensive.", "Nevertheless, we take full advantage of tailored methods for Gröbner basis over Boolean rings [3].", "As we show later in Table REF , for small dimensions less than or equal to 8, Gröbner basis computations are faster than canonical form ones.", "For larger dimensions, we have observed that in general Gröbner basis computations are faster but the standard deviation on computational time is much larger.", "In dimension 9, the average time to compute a Gröbner basis is around 3 seconds, but there are codes for which that computation takes close to 10 hours to finish.", "Nevertheless, we believe that a thorough study of Gröbner basis of neural ideals is not only of theoretical interest, but it can lead to better procedures able to perform computations in larger dimensions.", "Indeed, among small codes, surprisingly many have canonical forms that are also Gröbner bases.", "Moreover, the iterative nature of the newer canonical form algorithm hints towards the ability to compute canonical forms and Gröbner bases of neural codes in large dimensions by `gluing' those of codes on small dimensions.", "Such decomposition results are a common theme in other areas of applied algebraic geometry [1], [9].", "The outline of this paper is as follows.", "Section  provides background on neural ideals, canonical forms, and Gröbner bases.", "In Section , we prove our main result: if the canonical form of a neural ideal is a Gröbner basis, then it is the universal Gröbner basis (Theorem REF ).", "We also prove a partial converse: if the universal Gröbner basis of a neural ideal contains only so-called pseudo-monomials, then it is the canonical form (Theorem REF ).", "Our results motivate other questions: When is the canonical form a Gröbner basis?", "If the universal Gröbner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code?", "Sections  and  provide some partial answers these questions.", "Finally, a discussion is in Section ." ], [ "Background", "This section introduces neural ideals and related topics, which were first defined by Curto, Itskov, Veliz-Cuba, and Youngs [7], and recalls some basics about Gröbner bases.", "We use the notation $[n]:=\\lbrace 1,2,\\dots , n\\rbrace $ ." ], [ "Neural codes and receptive fields", "A neural code (also known as a combinatorial code) on $n$ neurons is a set of binary firing patterns $C\\subset \\lbrace 0,1\\rbrace ^n$ , that is, a set of binary strings of neural activity.", "Note that neither timing nor rate of neural activity are recorded in a neural code.", "An element $c\\in C$ of a neural code is a codeword.", "Equivalently, a codeword is determined by the set of neurons that fire: $ \\operatorname{supp}(c):=\\lbrace i\\in [n] \\mid c_i=1\\rbrace \\subseteq [n]~.$ Thus, the entire code is identified with a set of subsets of co-firing neurons: $\\operatorname{supp}(C) =\\lbrace \\operatorname{supp}(c) \\mid c\\in C\\rbrace \\subseteq 2^{[n]}.$ In many areas of the brain, neurons are associated with receptive fields in a stimulus space.", "Of particular interest are the receptive fields of place cells, which are neurons that fire in response to an animal's location.", "More specifically, each place cell is associated with a place field, a convex region of the animal's physical environment where the place cell has a high firing rate [15].", "The discovery of place cells and related neurons (grid cells and head direction cells) won neuroscientists John O'Keefe, May Britt Moser, and Edvard Moser the 2014 Nobel Prize in Physiology and Medicine.", "Given a collection of sets $\\mathcal {U} = \\lbrace U_1,...,U_n\\rbrace $ in a stimulus space $X$ (here $U_i$ is the receptive field of neuron $i$ ), the receptive field code, denoted by $C(\\mathcal {U})$ , is: $C(\\mathcal {U})~:=~\\left\\lbrace c\\in \\lbrace 0,1\\rbrace ^n ~:~ \\left( \\bigcap _{i\\in \\operatorname{supp}(c)} U_i\\right) \\setminus \\left( \\bigcup _{j \\notin \\operatorname{supp}(c)} U_j \\right) \\ne \\emptyset \\right\\rbrace ~.$ As mentioned earlier, we often identify this code with the corresponding set of subsets of $[n]$ .", "Also, we use the following convention for the empty intersection: $\\bigcap _{i \\in \\emptyset } U_i := X$ .", "Example 2.1 Consider the sets $U_i$ in a stimulus space $X$ depicted in Figure REF .", "The corresponding receptive field code is $C(\\mathcal {U})=\\lbrace \\emptyset , 1,123, 13, 3 \\rbrace $ .", "Figure: Receptive fields U i U_i for which the code is C(𝒰)={∅,1,123,13,3}C(\\mathcal {U})=\\lbrace \\emptyset , 1,123, 13, 3 \\rbrace ." ], [ "The neural ideal and its canonical form", "A pseudo-monomial in $\\mathbb {F}_2[x_1,\\dots , x_n]$ is a polynomial of the form $f~=~ \\prod _{i\\in \\sigma } x_i\\prod _{j\\in \\tau }(1+x_j)~,$ where $\\sigma ,\\tau \\subseteq [n]$ with $\\sigma \\cap \\tau =\\emptyset $ .", "Every term in a pseudo-monomial $f= \\prod _{i\\in \\sigma } x_i\\prod _{j\\in \\tau }(1+x_j)$ divides its highest-degree term, $\\prod _{i\\in \\sigma \\cup \\tau } x_i$ .", "We will use this fact several times in this work.", "Each $v\\in \\lbrace 0,1\\rbrace ^n$ defines a pseudo-monomial $\\rho _v$ as follows: $\\rho _v~:=~\\prod _{i=1}^n(1-v_i-x_i)=\\prod _{\\lbrace i \\mid v_i=1\\rbrace }x_i\\prod _{\\lbrace j \\mid v_j=0\\rbrace }(1+x_j)=\\prod _{\\lbrace i\\in \\operatorname{supp}(v)\\rbrace }x_i\\prod _{\\lbrace j\\notin \\operatorname{supp}(v)\\rbrace }(1-x_j)~.$ Notice that $\\rho _v$ is the characteristic function for $v$ , that is, $\\rho _v(x)=1$ if and only if $x=v$ .", "Definition 2.2 Let $C \\subseteq \\lbrace 0,1\\rbrace ^n$ be a neural code.", "The neural ideal $J_C$ is the ideal in $\\mathbb {F}_2[x_1,\\dots , x_n]$ generated by all $\\rho _v$ for $v\\notin C$ : $J_C~:=~\\langle \\lbrace \\rho _v|v\\notin C\\rbrace \\rangle ~.$ It follows that the variety of the neural ideal is the code itself: $V(J_C)=C$ .", "The following lemma provides the algebraic version of the previous statement: Lemma 2.3 (Curto, Itskov, Veliz-Cuba, and Youngs [7]) Let $C \\subseteq \\lbrace 0,1\\rbrace ^n$ be a neural code.", "Then $ I(C) ~=~ J_C + \\langle x_i(1 + x_i) \\mid i \\in [n] \\rangle ~, $ where $I(C)$ is the ideal of the subset $C \\subseteq \\lbrace 0,1\\rbrace ^n$ .", "Note that the ideal generated by the Boolean relations $\\langle x_i(1 + x_i):i\\in [n]\\rangle $ is contained in $I(C)$ , regardless of the structure of $C$ .", "A pseudo-monomial $f$ in an ideal $J$ in $\\mathbb {F}_2[x_1,\\dots ,x_n]$ is minimal if there does not exist another pseudo-monomial $g\\in J$ , with $g \\ne f$ , such that $f=gh$ for some $h\\in \\mathbb {F}_2[x_1,\\dots ,x_n]$ .", "Definition 2.4 The canonical form of a neural ideal $J_C$ , denoted by ${\\rm CF}(J_C)$ , is the set of all minimal pseudo-monomials of $J_C$ .", "Algorithms for computing the canonical form were given in [7], [8], [16].", "In particular, [16] describes an iterative method to compute the canonical form that is significantly more efficient than the original method presented in [7].", "The canonical form ${\\rm CF}(J_C)$ is a particular generating set for the neural ideal $J_C$  [7].", "The main goal in this work is to compare ${\\rm CF}(J_C)$ to other generating sets of $J_C$ , namely, its Gröbner bases.", "Example 2.5 Returning to Example REF , the codewords $v$ that are not in $C(\\mathcal {U})= \\lbrace \\emptyset , 1,123, 13, 3 \\rbrace $ are 2, 12, and 23, so the neural ideal is $J_C=\\langle \\lbrace x_2(1+x_1)(1+x_3),~x_1x_2(1+x_3),~x_2x_3(1+x_1) \\rbrace \\rangle $ .", "The canonical form is ${\\rm CF}(J_{C(\\mathcal {U})} )=\\lbrace x_2(1+x_1),~x_2(1+x_3) \\rbrace $ .", "We will interpret these canonical-form polynomials in Example REF below." ], [ "Receptive-field relationships", "It turns out that we can interpret pseudo-monomials in $J_C$ (and thus in the canonical form) in terms of relationships among receptive fields.", "First we need the following notation: for any $\\sigma \\subseteq [n]$ , define: $x_\\sigma ~:=~\\prod _{i\\in \\sigma }x_i \\quad \\text{ and } \\quad U_\\sigma ~:=~ \\bigcap _{i\\in \\sigma } U_i~,$ where, by convention, the empty intersection is the entire space $X$ .", "Lemma 2.6 (Curto, Itskov, Veliz-Cuba, and Youngs [7]) Let $X$ be a stimulus space, let $\\mathcal {U}=\\lbrace U_i\\rbrace _{i=1}^n$ be a collection of sets in $X$ , and consider the receptive field code $C=C(\\mathcal {U})$ .", "Then for any pair of subsets $\\sigma ,\\tau \\subseteq [n]$ , $x_\\sigma \\prod _{i\\in \\tau }(1+x_i)\\in J_C \\iff U_\\sigma \\subseteq \\bigcup _{i\\in \\tau }U_i~.$ Thus, three types of receptive-field relationships (RF relationships) can be read off from pseudo-monomials in a neural ideal (e.g., those in the canonical form) [7]: Type 1: $x_\\sigma \\in J_C \\iff U_\\sigma = \\emptyset $ (where $\\sigma \\ne \\emptyset $ ).", "Type 2: $x_\\sigma \\prod _{i\\in \\tau }(1+ x_i)\\in J_C \\iff U_\\sigma \\subseteq \\bigcup _{i\\in \\tau }U_i$ (where $\\sigma , \\tau \\ne \\emptyset $ ).", "Type 3: $\\prod _{i\\in \\tau }(1+x_i)\\in J_C \\iff X\\subseteq \\bigcup _{i\\in \\tau }U_i$ (where $\\tau \\ne \\emptyset $ ), and thus $X = \\bigcup _{i\\in \\tau }U_i$ .", "Example 2.7 The canonical form in Example REF , which is $\\lbrace x_2(1+x_1),~x_2(1+x_3) \\rbrace $ , encodes two Type 2 relationships: $U_2 \\subseteq U_1$ and $U_2 \\subseteq U_3$ .", "Indeed, we can verify this in Figure REF .", "In this work, we reveal more types of RF relationships, which arise from non-pseudo-monomials.", "They often appear in Gröbner bases of neural ideals (see Section )." ], [ "Gröbner bases", "Here we recall some basics about Gröbner bases [2], [4], [12].", "Fix a monomial ordering $<$ of a polynomial ring $R=k[x_1,\\dots , x_n]$ over a field $k$ , and let $I$ be an ideal in $R$ .", "Let $LT_<(I)$ denote the ideal generated by all leading terms, with respect to the monomial ordering $<$ , of elements in $I$ .", "Definition 2.8 A Gröbner basis of $I$ , with respect to $<$ , is a finite subset of $I$ whose leading terms generate $LT_<(I)$ .", "One useful property of a Gröbner basis is that given a polynomial $f$ and a Gröbner basis $G$ , the remainder of $f$ when divided by the set of elements in $G$ is uniquely determined.", "A Gröbner basis is reduced if (1) every $f \\in G$ has leading coefficient 1, and (2) no term of any $f \\in G$ is divisible by the leading term of any $g \\in G$ for which $g\\ne f$ .", "For a given monomial ordering, the reduced Gröbner basis of an ideal is unique.", "Definition 2.9 A universal Gröbner basis of an ideal $I$ is a Gröbner basis that is a Gröbner basis with respect to every monomial ordering.", "The universal Gröbner basis of an ideal $I$ is the union of all the reduced Gröbner bases of $I$ .", "The universal Gröbner basis is an instance of a universal Gröbner basis, given that the set of all distinct reduced Gröbner bases of an ideal $I$ is finite [2].", "This fact is actually the main result of the theory of Gröbner fans first introduced in [14]." ], [ "Main Result", "In this section, we give the main result of our paper: if the canonical form is a Gröbner basis, then it is the universal Gröbner basis (Theorem REF ).", "Beyond being a natural expansion of some of Curto et al.", "'s results [7], our theorem is also of mathematical interest since there are few classes of ideals whose universal Gröbner bases are known.", "Indeed, such characterizations in general are known to be computationally difficult.", "Theorem 3.1 If the canonical form of a neural ideal $J_C$ is a Gröbner basis of $J_C$ with respect to some monomial ordering, then it is the universal Gröbner basis of $J_C$ .", "The proof of Theorem REF , which appears in Section REF , requires the following related results: Lemma 3.2 For a pseudo-monomial $f = x_{\\sigma } \\prod _{j \\in \\tau } (1+x_j)$ in $\\mathbb {F}_2[x_1, \\dots , x_n]$ , the leading term of $f$ with respect to any monomial ordering is its highest-degree term, $x_{\\sigma \\cup \\tau }$ .", "This follows from the fact that every term of $f$ divides $x_{\\sigma \\cup \\tau }$ , and two properties of a monomial ordering [4]: it is a well-ordering (so, $1<x_i$ ), and $x_{\\alpha } < x_{\\beta }$ implies $x_{\\alpha \\cup \\gamma } < x_{\\beta \\cup \\gamma }$ .", "Proposition 3.3 If the canonical form of a neural ideal $J_C$ is a Gröbner basis of $J_C$ with respect to some monomial ordering, then it is a universal Gröbner basis of $J_C$ .", "Let $G$ denote the canonical form, and assume that $G$ is a Gröbner basis with respect to some monomial ordering $<_1$ .", "Let $<_2$ denote another monomial ordering.", "As always, we have the containment $\\mathrm {LT}_{<_2}(G) \\subseteq \\mathrm {LT}_{<_2}(J_C)$ , which we must prove is an equality.", "Accordingly, let $f \\in J_C$ .", "We must show that $\\mathrm {LT}_{<_2}(f) \\in \\mathrm {LT}_{<_2}(G)$ .", "With respect to $<_1$ , the reduction of $f$ by $G$ is 0, so we can write $f$ as a polynomial combination of some of the $g_i\\in G$ in the following form: $ f~=~\\frac{\\mathrm {LT}_{<_1}(f)}{\\mathrm {LT}(g_{1})}g_{1}+\\frac{\\mathrm {LT}_{<_1}(r_1)}{\\mathrm {LT}(g_{2})}g_{2}+\\dots +\\frac{\\mathrm {LT}_{<_1}(r_{t-1})}{\\mathrm {LT}(g_{t})}g_{t}~=~h_1+\\dots +h_t~,$ where (for $i=1,\\dots , t$ ) we have $g_i \\in G$ , $h_i := \\frac{\\mathrm {LT}_{<_1}(r_{i-1})}{\\mathrm {LT}(g_{i})}g_{i}$ , $r_0:= f$ , and $r_i=f-h_1-\\dots -h_i$ is the remainder after the $i$ -th division of $f$ by $G$ .", "Note that in equation (REF ), the polynomial $g_i$ may appear multiple times, but this does not affect our arguments.", "By Lemma REF , the leading term of $g_i$ does not depend on the monomial ordering.", "Moreover, each $h_i$ is the product of a monomial and a pseudo-monomial, $g_i$ , so by a straightforward generalization of Lemma REF , the leading term of $h_i$ with respect to any monomial ordering is $\\mathrm {LT}_{<_1}(h_i)$ .", "Also note that when dividing by the Gröbner basis $G$ , $\\mathrm {LT}_{<_1}(r_i) <_1 \\mathrm {LT}_{<_1}(r_{i-1})$ so the $\\mathrm {LT}_{<_1}(r_{i})$ are distinct.", "This implies that the $\\mathrm {LT}_{<_1}(h_{i})$ are distinct since $\\mathrm {LT}_{<_1}(h_{i}) = \\mathrm {LT}_{<_1}(r_{i-1})$ .", "Hence, among the list of monomials $\\lbrace \\mathrm {LT}(h_{i})\\rbrace _{i=1}^t$ , there is a unique largest monomial with respect to $<_2$ , which we denote by $\\mathrm {LT}(h_{i^*})$ .", "Next, by examining the sum in (REF ), and noting that every term of $h_i$ divides the leading term of $h_i$ , we see that $\\mathrm {LT}_{<_2}(f)=\\mathrm {LT}(h_{i^*})$ .", "Thus, because $g_{i^*}$ divides $h_{i^*}$ , it follows that $\\mathrm {LT}(g_{i^*})$ divides $\\mathrm {LT}_{<_2}(f)$ , and so, $\\mathrm {LT}_{<_2}(f) \\in \\mathrm {LT}_{<_2}(G)$ .", "Thus, if the canonical form is a Gröbner basis with respect to some monomial ordering, then it is a Gröbner basis with respect to every monomial ordering." ], [ "Pseudo-monomials and hypercubes", "To prove our main result (Theorem REF ), we need to develop the connection between pseudo-monomials and hypercubes in the Boolean lattice.", "The Boolean lattice on $[n]$ is the power set $P([n]):=2^{[n]}$ , partially ordered by inclusion.", "Also, for $\\sigma \\subseteq [n]$ , we let $P(\\sigma )$ denote the power set of $\\sigma $ .", "The support of a monomial $\\prod _{i=1}^n x_i^{a_i}$ is the set $\\lbrace i \\in [n] \\mid a_i>0 \\rbrace $ .", "Definition 3.4 Let $f= x_{\\sigma } \\prod _{j \\in \\tau } (1+x_j)$ be a pseudo-monomial in $\\mathbb {F}_2[x_1, \\dots , x_n]$ .", "The hypercube of $f$, denoted by $H(f)$ , is the sublattice of the Boolean lattice on $[n]$ formed by the support of each term of $f$ .", "Remark 3.5 The hypercube of $f$ is the interval of the Boolean lattice from $\\sigma $ to $\\sigma \\cup \\tau $ : $H(f)=\\lbrace \\omega \\mid \\sigma \\subseteq \\omega \\subseteq \\sigma \\cup \\tau \\rbrace \\subseteq P([n])~,$ and thus its Hasse diagram is a hypercube (this justifies its name).", "This is because: $ f=x_{\\sigma } \\prod \\limits _{j \\in \\tau } (1+x_j) = \\sum \\limits _{\\lbrace \\theta \\mid \\theta \\subseteq \\tau \\rbrace } x_{\\sigma \\cup \\theta }~.$ Example 3.6 Let $f=x_1x_2(1+x_3)(1+x_4)=x_1x_2x_3x_4+x_1x_2x_3+x_1x_2x_4+x_1x_2$ .", "Figure REF shows part of the Hasse diagram of $P([4])$ , with the hypercube of $f$ indicated by circles and solid lines.", "Figure: Displayed is part of the Hasse diagram of the Boolean lattice P([4])P([4]).", "The hypercube of f=x 1 x 2 (1+x 3 )(1+x 4 )f=x_1x_2(1+x_3)(1+x_4) is indicated by circles and solid lines, and P([2])P([2]) is marked by dotted lines.", "If gg is a pseudo-monomial that divides ff, then its hypercube is contained ineither the hypercube of ff orone of the dashed-line squares “parallel\" to the hypercube of ff(see Example ).Via hypercubes, divisibility of pseudo-monomials has a nice geometric interpretation: Lemma 3.7 For pseudo-monomials $f=x_{\\sigma }\\prod _{j\\in \\tau }(1+x_j)$ and $g=x_{\\alpha }\\prod _{j\\in \\beta }(1+x_j)$ , the following are equivalent: $g|f$ , $\\alpha \\subseteq \\sigma $ and $\\beta \\subseteq \\tau $ , $H(g) \\subseteq P(\\sigma \\cup \\tau )$ and $ H(g) \\cap P(\\sigma ) = \\lbrace \\alpha \\rbrace $ , and $H(g) \\subseteq P(\\sigma \\cup \\tau )$ and $ \\left| H(g) \\cap P(\\sigma ) \\right| = 1 $ .", "The implication (1) $\\Leftarrow $ (2) is clear, and (1) $\\Rightarrow $ (2) follows from the fact that $\\mathbb {F}_2[x_1,\\dots ,x_n]$ is a unique factorization domain.", "For (2) $\\Rightarrow $ (3), assume that $\\alpha \\subseteq \\sigma $ and $\\beta \\subseteq \\tau $ .", "Then $H(g) \\subseteq P(\\alpha \\cup \\beta ) \\subseteq P(\\sigma \\cup \\tau )$ .", "So, we need only show that $H(g) \\cap P(\\sigma ) = \\lbrace \\alpha \\rbrace $ .", "To see this, we first recall: $ H(g) ~=~ \\lbrace \\alpha \\cup \\theta \\mid \\theta \\subseteq \\beta \\rbrace $ from Remark REF .", "Thus, $H(g) \\cap P(\\sigma )~=~\\lbrace \\alpha \\cup \\theta \\mid \\theta \\subseteq \\beta \\text{ and } \\theta \\subseteq \\sigma \\rbrace ~=~\\lbrace \\alpha \\rbrace ~,$ where the second equality follows from hypotheses: $\\alpha \\subseteq \\sigma $ and $\\sigma \\cap \\beta \\subseteq \\sigma \\cap \\tau = \\emptyset $ (because $\\beta \\subseteq \\tau $ ).", "(3) $\\Rightarrow $ (4) is clear, so we need only show (2) $\\Leftarrow $ (4).", "Accordingly, suppose $H(g) \\subseteq P(\\sigma \\cup \\tau )$ and $I:=H(g) \\cap P(\\sigma )$ consists of only one element.", "We claim that this element is $\\alpha $ .", "Indeed, let $\\omega \\in I$ (i.e., $\\omega \\in H(g)$ and $\\omega \\subseteq \\sigma $ ); then, $\\alpha $ also is in $I$ (because $\\alpha \\in H(g)$ and $\\alpha \\subseteq \\omega \\subseteq \\sigma $ ).", "So, $\\alpha = \\omega \\subseteq \\sigma $ .", "To complete the proof, we must show that $\\beta \\subseteq \\tau $ .", "To this end, let $k \\in \\beta $ .", "Then $\\alpha \\cup \\lbrace k\\rbrace $ is in $H(g)$ , by equation (REF ), so it is not in $P(\\sigma )$ (because $H(g) \\cap P(\\sigma )= \\lbrace \\alpha \\rbrace $ ).", "So, $k \\in ( \\beta \\setminus \\sigma )$ .", "Finally, $( \\beta \\setminus \\sigma ) \\subseteq \\tau $ , because $\\alpha \\cup \\beta \\subseteq \\sigma \\cup \\tau $ follows from the hypothesis $H(g) \\subseteq P(\\sigma \\cup \\tau )$ .", "So, $k \\in \\tau $ .", "Example 3.8 We return to the pseudo-monomial $f=x_1x_2(1+x_3)(1+x_4)$ , which we rewrite as $f=x_{\\sigma } \\prod _{j \\in \\tau } (1+x_j)$ , where $\\sigma =\\lbrace 1,2\\rbrace $ and $\\tau =\\lbrace 3,4\\rbrace $ .", "In Figure REF , $P(\\sigma )=P([2])$ is marked by the dotted line.", "According to Lemma REF , a pseudo-monomial $h$ divides $f$ if and only if the hypercube of $h$ satisfies two conditions: it includes a vertex from $P(\\sigma )$ , and it is contained within either the hypercube of $f$ or one of the dashed-line squares “parallel\" to the hypercube of $f$ in Figure REF ." ], [ "Multivariate division by pseudo-monomials", "The following result concerns reducing a given pseudo-monomial by a set of pseudo-monomials.", "Theorem 3.9 Consider a pseudo-monomial $f=x_{\\sigma }\\prod _{i\\in \\tau }(1+x_i)\\in \\mathbb {F}_2[x_1,\\dots ,x_n]$ , and let $G$ be a finite set of pseudo-monomials in $\\mathbb {F}_2[x_1,\\dots ,x_n]$ .", "If some remainder upon division of $f$ by $G$ is 0 for some monomial ordering, then there exists $g \\in G$ such that $g$ divides $f$ .", "Suppose that some remainder on division of $f$ by $G$ is 0: $ f~=~\\frac{\\mathrm {LT}(f)}{\\mathrm {LT}(g_{1})}g_{1}+\\frac{\\mathrm {LT}(r_1)}{\\mathrm {LT}(g_{2})}g_{2}+\\dots +\\frac{\\mathrm {LT}(r_{t-1})}{\\mathrm {LT}(g_{t})}g_{t}~=~h_1+\\dots +h_t~,$ where, as in the proof of Proposition REF , for $i=1,\\dots , t$ , we have $g_i \\in G$ , $h_i:= \\frac{\\mathrm {LT}(r_{i-1})}{\\mathrm {LT}(g_{i})}g_{i}$ , and $r_i=f-h_1-\\dots -h_i$ is the remainder after the $i$ -th division (and $r_0:= f$ ).", "Also, each term of $h_i$ divides the leading term of $h_i$ .", "By construction, $g_i | h_i$ .", "So, it suffices to show that there exists $i$ such that $h_i | f$ .", "We now claim that $\\mathrm {LT}(h_i)|\\mathrm {LT}(f)$ holds for all $i$ .", "We prove this claim by induction on $i$ .", "For the $i=1$ case, $\\mathrm {LT}(h_1)=\\mathrm {LT}(f)$ .", "If $i \\ge 2$ , then $\\mathrm {LT}(h_i)$ is the leading term of: $ r_{i-1} = f-h_1-\\dots - h_{i-1}~.$ We now examine the summands in (REF ).", "As $f$ is a pseudo-monomial, each term in $f$ divides $\\mathrm {LT}(f)$ , and the same holds for each remaining summand $h_i$ : as noted above, its terms divide $\\mathrm {LT}(h_i)$ , and thus (by induction hypothesis) divide $\\mathrm {LT}(f)$ .", "So, $\\mathrm {LT}(h_i)= \\mathrm {LT}(r_{i-1}) | \\mathrm {LT}(f)$ , proving our claim.", "We now assert that $h_i$ is a pseudo-monomial.", "To see this, recall that $h_i$ is the product of a monomial and a pseudo-monomial (namely, $g_i$ ), so we just need to show that its leading term is square-free.", "Indeed, this follows from two facts: $\\mathrm {LT}(h_i)| \\mathrm {LT}(f)$ and $f$ is a pseudo-monomial.", "Hence, $H(h_i) \\subseteq P(\\sigma \\cup \\tau )$ for every $i$ , because every term in $h_i$ divides $\\mathrm {LT}(h_i)$ which in turn divides $x_{\\sigma \\cup \\tau }=\\mathrm {LT}(f)$ .", "Thus, by Lemma REF , it is enough to show that $\\left| H(h_i) \\cap P(\\sigma ) \\right| = 1$ for some $i$ (because this would imply that $h_i|f$ ).", "The sum in (REF ) is over $\\mathbb {F}_2$ , so the polynomials $f, h_1, \\dots , h_t$ together must contain an even number of each term.", "We focus now on only those terms with support in $P(\\sigma )$ .", "The pseudo-monomial $f$ has only one such term (namely, $x_{\\sigma }$ ).", "Thus, some $h_{i^*}$ has an odd number of terms in $P(\\sigma )$ , i.e., $\\left| H(h_{i^*}) \\cap P(\\sigma ) \\right|$ is odd.", "On the other hand, both $H(h_{i^*})$ and $P(\\sigma ) $ are hypercubes in the Boolean lattice, so their intersection, if nonempty, also is a hypercube and thus has size $2^q$ for some $q \\ge 0$ .", "Hence, $q=0$ , so $\\left| H(h_{i^*}) \\cap P(\\sigma ) \\right|=1$ .", "This completes our proof." ], [ "Proof of Theorem ", "Theorem REF allows us to prove that when a canonical form is a Gröbner basis, it is reduced: Proposition 3.10 If the canonical form of a neural ideal $J_C$ is a Gröbner basis of $J_C$ , then it is a reduced Gröbner basis of $J_C$ .", "Suppose for contradiction that $\\mathrm {CF}(J_C)$ is a Gröbner basis, but not a reduced Gröbner basis.", "Then there exist $f, g \\in \\mathrm {CF}(J_C)$ , with $f \\ne g$ , such that $\\mathrm {LT}(g)$ divides some term of $f$ .", "Thus, $\\mathrm {LT}(g)$ divides $\\mathrm {LT}(f)$ (because every term in a pseudo-monomial divides the leading term).", "Thus, $\\mathrm {CF}(J_C)$ and $\\mathrm {CF}(J_C) \\setminus \\lbrace f\\rbrace $ both generate the same ideal of leading terms, and hence $\\mathrm {CF}(J_C) \\setminus \\lbrace f\\rbrace $ is also a Gröbner basis of $J_C$ .", "It follows that the remainder on division of $f$ by $\\mathrm {CF}(J_C) \\setminus \\lbrace f\\rbrace $ is 0, so by Theorem REF , there exists $h \\in \\mathrm {CF}(J_C) \\setminus \\lbrace f\\rbrace $ such that $h | f$ .", "Hence, $f$ is a non-minimal element of the canonical form, which is a contradiction.", "Now we can prove Theorem REF , which states that a canonical form that is a Gröbner basis is the universal Gröbner basis: Follows from Propositions REF and REF ." ], [ "Every pseudo-monomial in a reduced Gröbner basis is in the canonical form", "In this subsection, we prove the following partial converse of Theorem REF : if the universal Gröbner basis of a neural ideal consists of only pseudo-monomials, then it equals the canonical form (Theorem REF ).", "We first show that every pseudo-monomial in a reduced Gröbner basis is in the canonical form.", "Proposition 3.11 Let $J_C$ be a neural ideal.", "Let $G$ be a reduced Gröbner basis of $J_C$ .", "Then every pseudo-monomial in $G$ is in the canonical form of $J_C$ .", "Let $\\widehat{G}$ be the universal Gröbner basis of $J_C$ .", "Then every pseudo-monomial in $\\widehat{G}$ is in the canonical form of $J_C$ .", "Let $f$ be a pseudo-monomial in $G$ .", "Suppose that $f$ is not a minimal pseudo-monomial in $J_C$ : for some pseudo-monomial $h\\in J_C$ such that deg($h$ )<deg($f$ ), $h|f$ .", "Then for some $g\\in G$ , $\\mathrm {LT}(g)|\\mathrm {LT}(h)$ .", "Hence, $\\mathrm {LT}(g) | \\mathrm {LT}(f)$ (because $\\mathrm {LT}(h) | \\mathrm {LT}(f)$ ) and also $g\\ne f$ (because $\\deg (g) \\le \\deg (h) < \\deg (f)$ ).", "This is a contradiction: $f$ and $g$ cannot both be in a reduced Gröbner basis.", "Finally, (2) follows directly from (1).", "Theorem 3.12 Let $J_C$ be a neural ideal.", "The following are equivalent: the canonical form of $J_C$ is a Gröbner basis of $J_C$ , the canonical form of $J_C$ is the universal Gröbner basis of $J_C$ , and the universal Gröbner basis of $J_C$ consists of pseudo-monomials.", "The implication (1)$\\Rightarrow $ (2) is Theorem REF , and both (1)$\\Leftarrow $ (2) and (2)$\\Rightarrow $ (3) are clear.", "For (3)$\\Rightarrow $ (1), assume that the universal Gröbner basis $\\widehat{G}$ consists of pseudo-monomials.", "Then, by Proposition REF (2), $\\widehat{G}$ is contained in the canonical form of $J_C$ .", "Thus, the canonical form contains a Gröbner basis of $J_C$ (namely, $\\widehat{G}$ ) and hence is itself a Gröbner basis.", "Remark 3.13 Suppose we want to know whether a code's canonical form is a Gröbner basis.", "Theorem REF tells us how to do so without computing the canonical form: compute the universal Gröbner basis, and then check whether it contains only pseudo-monomials.", "See Example REF .", "Under certain conditions, e.g.", "small number of neurons, computing the Gröbner basis is more efficient than computing the canonical form, but is there some way to avoid computations entirely and yet still decide whether the canonical form is a Gröbner basis?", "In the next section, we give conditions under which we can resolve this decision problem quickly.", "Example 3.14 Consider the neural code $C = \\lbrace 0100,0101,0111\\rbrace $ .", "The universal Gröbner basis of $J_C$ is $\\widehat{G} = \\lbrace x_3(x_4 + 1) ,~ x_2 + 1,~ x_1\\rbrace $ , so it contains only pseudo-monomials.", "Thus, by Theorem REF , $\\widehat{G}$ is the canonical form.", "Example 3.15 Consider the neural code $C = \\lbrace 0101,1100,1110\\rbrace $ .", "The universal Gröbner basis of $J_C$ is $\\widehat{G} = \\lbrace x_4 x_3,~ x_3 (x_1 + 1),~ x_1 + x_4 + 1,~ x_2 + 1\\rbrace $ , which contains the non-pseudo-monomial $x_1+x_4+1$ .", "Thus, by Theorem REF , the canonical form is not a universal Gröbner basis of $J_C$ .", "Indeed, the canonical form is ${\\rm CF}(J_C)=\\lbrace x_3 (x_1 + 1),~ x_2 + 1,~ (x_4 + 1) (x_1 + 1),~ x_4 x_1,~ x_4 x_3\\rbrace $ , and, for a monomial ordering where $x_4>x_1$ , the leading term of the non-pseudo-monomial $x_1+x_4+1$ is $x_4$ , which is not divisible by any of the leading terms from the canonical form." ], [ "When is the canonical form a Gröbner basis?", "In this section we present some results that partially solve the question of when is the canonical form a Gröbner basis for the neural ideal.", "A complete answer to this question is not only of theoretical interest but perhaps also of practical relevance.", "Extensive computations suggest that, under certain conditions, Gröbner bases of neural ideals can be computed more efficiently than canonical forms.", "This is true for small neural codes.", "Moreover, the iterative nature of the newer canonical form algorithm hints towards the ability to compute canonical forms and Gröbner bases of neural codes in large dimensions by `gluing' those of codes on small dimensions.", "Such decomposition results are a common theme in other areas of applied algebraic geometry such as algebraic statistics and phylogenetic algebraic geometry [1], [9].", "Table REF displays a runtime comparison between the iterative canonical form algorithm described in [16] and a specialized Gröbner basis algorithm for Boolean rings implemented in SageMath based on the work in [3].", "We report the mean time (in seconds) of 100 randomly generated codes on $n$ neurons for $n = 4, \\dots , 8$ .", "More precisely, for each code, a number $m$ was chosen uniformly at random from $\\lbrace 1,\\dots , 2^n-1 \\rbrace $ and then $m$ codewords were chosen at random.", "These computations were performed on SageMath 7.2 running on a Macbook Pro with a 2.8 GHz Intel Core i7 processor and 16 GB of memory.", "Table: Runtime comparison of canonical form versus Gröbner basis computations.For codes on a larger number of neurons, our computations indicate that in general Gröbner bases computations are still more efficient than canonical form computations.", "However, even in the case of $n=9$ neurons we found codes whose Gröbner bases took over 6 hours to be computed.", "Proposition 4.1 Let $C$ be a neural code on $n$ neurons.", "If $|C| = 1$ or $|C| = 2^n-1$ , then the canonical form of $J_C$ is the universal Gröbner basis of $J_C$ .", "If $C = \\lbrace c\\rbrace $ , then Lemma REF implies that $J_C = \\langle x_1-c_1,~ x_2-c_2,~\\dots ,~ x_n-c_n\\rangle $ .", "When $|C| = 2^n-1$ , then by definition $J_C = \\langle \\rho _v \\rangle $ for the unique $v\\notin C$ .", "In either case, the indicated generating set is both the canonical form and the universal Gröbner basis of $J_C$ .", "A set of subsets $\\Delta \\subseteq 2^{[n]}$ is an (abstract) simplicial complex if $\\sigma \\in \\Delta $ and $\\tau \\subseteq \\sigma $ implies $\\tau \\in \\Delta $ .", "A neural code $C$ is a simplicial complex if its support $\\mathrm {supp}(C)$ is a simplicial complex.", "Proposition 4.2 If $C$ is a simplicial complex, then the canonical form of $J_C$ is the universal Gröbner basis of $J_C$ .", "If $C$ is a simplicial complex, then $J_C$ is a monomial ideal generated by the minimal Type 1 relationships (indeed, it is the Stanley-Reisner ideal of the simplicial complex $\\mathrm {supp}(C)$ ) [7].", "These minimal Type-1 relationships comprise the canonical form of $J_C$ , and also form the universal Gröbner basis of $J_C$ .", "The next result gives conditions that guarantee that the canonical form is not a Gröbner basis.", "Proposition 4.3 Let $\\mathcal {U}=\\lbrace U_i\\rbrace _{i=1}^n$ be a collection of sets in a stimulus space $X$ , and let $C=C(\\mathcal {U})$ denote the corresponding receptive field code.", "If one of the following conditions hold, then the canonical form of $J_C$ is not a Gröbner basis of $J_C$ : Two proper, nonempty receptive fields coincide: $\\emptyset \\ne U_i = U_j \\subsetneq X$ for some $i\\ne j \\in [n]$ .", "Two nonempty receptive fields are complementary: $U_i = X \\setminus U_j$ for some $i\\ne j \\in [n]$ with $U_i \\ne \\emptyset $ and $U_j \\ne \\emptyset $ .", "(1) Suppose $U_i, U_j \\in \\mathcal {U}$ are two sets with $\\emptyset \\ne U_i = U_j \\subsetneq X$ .", "By Lemma REF , both $f = x_i(x_j+1)$ and $g = x_j(x_i+1)$ are in $J_C$ .", "In fact, $f$ and $g$ are minimal pseudo-monomials in $J_C$ (because $\\emptyset \\ne U_i=U_j \\ne X$ ), so $f, g \\in \\mathrm {CF}(J_C)$ .", "Under any monomial ordering, $\\mathrm {LT}(f) = \\mathrm {LT}(g) = x_ix_j$ (by Lemma REF ), so the set $\\mathrm {CF}(J_C)$ is not reduced and thus cannot be a reduced Gröbner basis.", "Hence, by Proposition REF , $\\mathrm {CF}(J_C)$ cannot be a Gröbner basis.", "(2) Now assume that $U_i = X \\setminus U_j$ for some $i\\ne j \\in [n]$ , with $U_i \\ne \\emptyset $ and $U_j \\ne \\emptyset $ .", "Thus, $U_i\\cap U_j = \\emptyset $ and $U_i \\cup U_j = X$ , so Lemma REF implies that $f = x_ix_j$ and $g = (x_i+1)(x_j+1)$ are in $J_C$ .", "Now we proceed as in the previous paragraph: $f$ and $g$ are minimal pseudo-monomials in $\\mathrm {CF}(J_C)$ , and $\\mathrm {LT}(f) = \\mathrm {LT}(g) = x_ix_j$ , so, by Proposition REF , $\\mathrm {CF}(J_C)$ cannot be a Gröbner basis.", "The last result in this section concerns a class of codes that we call complement-complete.", "Definition 4.4 The complement of $c \\in \\lbrace 0,1\\rbrace ^n$ is the codeword $\\overline{c} \\in \\lbrace 0,1\\rbrace ^n$ defined by $\\overline{c}_i = 1$ if and only if $c_i = 0$ .", "A neural code $C$ is complement-complete if for all $c \\in C$ , then $\\overline{c}$ is also in $C$ .", "Example 4.5 The complement of the codeword $c_1=1000$ is $\\overline{c_1}=0111$ , and the complement of $c_2=1010$ is $\\overline{c_2}=0101$ .", "Thus, the code $C=\\lbrace 1000,0111,1010,0101\\rbrace $ is complement-complete.", "Definition 4.6 The complement of a pseudo-monomial $f = x_{\\sigma }\\prod _{i\\in \\tau }(1+x_i)$ is the pseudo-monomial $\\overline{f} = x_\\tau \\prod _{j\\in \\sigma }(1+x_j)$ .", "Lemma 4.7 Consider pseudo-monomials $f = x_{\\sigma }\\prod _{i\\in \\tau }(1+x_i)$ and $g = x_{\\sigma ^{\\prime }}\\prod _{i\\in \\tau ^{\\prime }}(1+x_i)$ .", "If $f$ divides $g$ , then $\\overline{f}$ divides $\\overline{g}$ .", "This follows from the fact that $f\\mid g$ if and only if $\\sigma ^{\\prime }\\subseteq \\sigma $ and $\\tau ^{\\prime } \\subseteq \\tau $ (Lemma REF ).", "Proposition 4.8 Let $C$ be a code on $n$ neurons, with $C \\subsetneq \\lbrace 0,1\\rbrace ^n$ .", "If $C$ is complement-complete, then the canonical form of $J_C$ is not a Gröbner basis of $J_C$ .", "Note that since $C \\ne \\lbrace 0,1\\rbrace ^n$ , $J_C$ is not trivial.", "We make the following claim: Claim: If $h$ is a pseudo-monomial in $J_C$ , then $\\overline{h}$ is also in $ J_C$ .", "To see this, let $S$ be the set of all degree-$n$ pseudo-monomials in $\\mathbb {F}_2[x_1,\\dots , x_n]$ that are multiples of $h$ (so, $S \\subseteq J_C$ ).", "Degree-$n$ pseudo-monomials in $\\mathbb {F}_2[x_1,\\dots , x_n]$ are characteristic functions $\\rho _v$ , so, every element of $S$ is some $\\rho _v$ , where $v \\notin C$ .", "Thus, every element of $\\overline{S}:=\\lbrace \\overline{f} \\mid f \\in S \\rbrace $ has the form $ \\overline{\\rho _v} = \\rho _{\\overline{v}} $ , where $v \\notin C$ , which is equivalent to $\\overline{v} \\notin C$ , as $C$ is complement-complete.", "So, $\\overline{S} \\subseteq J_C$ .", "Next, let $s \\in S$ , that is, $s = hq$ for some pseudo-monomial $q$ .", "Then $h\\overline{q}$ is also in $S$ .", "Since $\\gcd (q,\\overline{q}) = 1$ , it follows that $h = \\gcd (hq, h \\overline{q})$ , so $h = \\gcd \\lbrace S\\rbrace $ .", "Thus, $\\overline{h} = \\gcd \\lbrace \\overline{S}\\rbrace $ , so $\\overline{h} \\in J_C$ (because $\\overline{S} \\subseteq J_C$ ), which proves the claim.", "Now let $f \\in CF(J_C)$ .", "By the claim, $\\overline{f}$ is in $J_C$ , and now we assert that, like $f$ , the pseudo-monomial $\\overline{f}$ is in $CF(J_C)$ .", "Indeed, if a pseudo-monomial $d$ in $J_C$ divides $\\overline{f}$ , then by Lemma REF , the pseudo-monomial $\\overline{d}$ divides $f$ .", "Also, $\\overline{d} \\in J_C$ (by the claim), so $\\overline{d}=f$ (because $f$ is minimal), and thus $d = \\overline{f}$ .", "Hence, $\\overline{f}$ is minimal, and so $\\overline{f}$ is also in $CF(J_C)$ .", "Thus, $CF(J_C)$ contains two polynomials ($f$ and $\\overline{f}$ ) with the same leading term, and so is not a reduced Gröbner basis, and thus (by Proposition REF ) is not a Gröbner basis of $J_C$ .", "Example 4.9 Consider again the complement-complete code $C=\\lbrace 1000,0111,1010,0101\\rbrace $ from Example REF .", "The canonical form is $CF(J_C)=\\lbrace (x_1+1)(x_2+1),~(x_1+1)(x_4+1),~x_1x_2,~x_2(x_4+1),~x_1x_4,~x_4(x_2+1)\\rbrace $ .", "Note that $CF(J_C)$ is itself complement-complete; for example, $f=x_2(x_4+1)$ and $\\overline{f}=x_4(x_2+1)$ are both in $CF(J_C)$ .", "Also, we can show directly that $CF(J_C)$ is not a Gröbner basis, which is consistent with Proposition REF : with respect to any monomial ordering, the leading term of $f+\\overline{f}=x_2+x_4$ is not divisible by any of the leading terms in $CF(J_C)$ ." ], [ "New receptive-field relationships", "We saw earlier that if the universal Gröbner basis of a neural ideal consists of only pseudo-monomials, then it equals the canonical form (Theorem REF ).", "When this is not the case, there are non-pseudo-monomial elements in the universal Gröbner basis, so it is natural to ask what they tell us about the receptive fields of the code.", "In other words, what types of RF relationships, besides those of Types 1–3 (Lemma REF ), appear in Gröbner bases?", "Here we give a partial answer: Theorem 5.1 Let $\\mathcal {U}=\\lbrace U_i\\rbrace _{i=1}^n$ be a collection of sets in a stimulus space $X$ .", "Let $C=C(\\mathcal {U})$ denote the corresponding receptive field code, and let $J_C$ denote the neural ideal.", "Then for any subsets $\\sigma _1,\\sigma _2,\\tau _1,\\tau _2 \\subseteq [n]$ , and $m$ indices $1 \\le i_1 < i_2 < \\dots < i_m \\le n$ , with $m \\ge 2$ , we have RF relationships as follows: Type 4: $x_{\\sigma _1} \\prod _{i \\in \\tau _1} (1 + x_i) + x_{\\sigma _2} \\prod _{j \\in \\tau _2} (1 + x_j) \\in J_C$  $\\Rightarrow $   $U_{\\sigma _1}\\cap \\left( \\bigcap _{i\\in \\tau _1}U_i^c \\right) = U_{\\sigma _2}\\cap \\left( \\bigcap _{j\\in \\tau _2}U_j^c\\right)$ .", "Type 5: $x_{i_1}+ \\dots +x_{i_m} \\in J_C$  $\\Rightarrow $   $U_{i_k}\\subseteq \\bigcup _{j\\in [m] \\setminus \\lbrace k\\rbrace }U_{i_j}$ for all $k=1, \\dots , m$ , and if, additionally, $m$ is odd, then $\\bigcap _{k=1}^m U_{i_k}=\\emptyset $ .", "Type 6: $x_{i_1}+ \\dots +x_{i_m}+1 \\in J_C$  $\\Rightarrow $   $\\bigcup _{k=1}^m U_{i_k}=X$ .", "Throughout the proof, for $p \\in X$ , we let $c(p)$ denote the corresponding codeword in $C$ .", "Type 4.", "Let $f_1:=x_{\\sigma _1} \\prod _{i \\in \\tau _1} (1+x_i)$ , and let $f_2:=x_{\\sigma _2} \\prod _{j \\in \\tau _2} (1+x_j)$ .", "Also, let $W_1 := U_{\\sigma _1} \\cap \\left( \\bigcap _{i\\in \\tau _1}U_i^c \\right)$ , and let $W_2 := U_{\\sigma _2} \\cap \\left( \\bigcap _{j\\in \\tau _2}U_j^c \\right)$ .", "By symmetry, we need only show that $W_1 \\subseteq W_2$ .", "To this end, let $p \\in W_1$ (so, $c(p) \\in C$ ).", "First, because $f_1+f_2 \\in J_C$ and $V(J_C)=C$ , it follows that $f_1(c(p))=f_2(c(p))$ .", "Next, for $i=1,2$ , we have $p \\in W_i$ if and only if $f_i(c(p))=1$ .", "Thus, $p \\in W_2$ .", "Type 5.", "Let $g:= x_{i_1}+\\cdots +x_{i_m}$ .", "By symmetry, we need only show that $U_{i_1} \\subseteq \\bigcup _{l=2}^m U_{i_l}$ .", "To this end, let $ p \\in U_{i_1}$ (so, $c(p)_{i_1}=1$ ).", "Then $g \\in J_C$ implies the following equality in $\\mathbb {F}_2$ : $ 0 ~=~ g(c(p)) ~=~ c(p)_{i_1} + c(p)_{i_2} + \\dots + c(p)_{i_m} ~=~ 1 + c(p)_{i_2} + \\dots + c(p)_{i_m}~.$ Thus, for some $k \\ge 2$ , we have $c(p)_{i_k}=1$ , i.e., $p \\in U_{i_k}$ .", "Hence, $p \\in \\bigcup _{l=2}^m U_{i_l}$ .", "Now assume, additionally, that $m$ is odd.", "Suppose, for contradiction, that there exists $q \\in \\bigcap _{k=1}^m U_{i_k}$ .", "Then, like the sum (REF ) above, we have $0=g(c(q))=1+\\dots +1=m$ , which contradicts the hypothesis that $m$ is odd.", "So, $\\bigcap _{k=1}^m U_{i_k} = \\emptyset $ .", "Type 6.", "Let $h:=x_{i_1}+ \\dots +x_{i_m}+1$ .", "Let $p \\in X$ (so, $c(p) \\in C$ ).", "We must show that $p \\in \\bigcup _{k=1}^m U_{i_k}$ .", "Because $h \\in J_C$ , we have $0 = h(c(p)) = c(p)_{i_1} + \\dots + c(p)_{i_m} + 1$ .", "Thus, for some $k \\in [m]$ , we have $c(p)_{i_k}=1$ , i.e., $p \\in U_{i_k}$ .", "Hence, $p \\in \\bigcup _{k=1}^m U_{i_k}$ .", "Remark 5.2 Like the earlier RF relationships (those of Types 1–3 from Lemma REF ), some of our new ones (Types 4–6) are containments and some are equalities.", "Example 5.3 Recall the code $C = \\lbrace 0101,1100,1110\\rbrace $ , from Example REF , for which the universal Gröbner basis of $J_C$ is $\\widehat{G} = \\lbrace x_4 x_3,~ x_3 (x_1 + 1),~ x_1 + x_4 + 1,~ x_2 + 1\\rbrace $ .", "The polynomial $x_1+x_4+1$ encodes a Type 6 relationship: $U_1\\cup U_4=X$ .", "Also, the polynomial $x_2+1$ encodes a Type 3 relationship: $U_2=X$ , which together gives us $U_1\\cup U_4=U_2$ .", "The canonical form also contains the polynomial $x_1x_4$ , which encodes a Type 1 relationship: $U_1\\cap U_4=\\emptyset $ .", "We conclude that $U_1 \\dot{\\cup }U_4=U_2$ .", "Example 5.4 Consider the code $C=\\lbrace 00,11\\rbrace $ .", "The universal Gröbner basis of $C$ is $\\widehat{G} = \\lbrace x_1(1+x_1),~ x_1+x_2, ~x_2(1+x_2) \\rbrace $ .", "The polynomial $x_1+x_2$ encodes a Type 4 relationship: $U_1=U_2$ .", "(The polynomial $x_1+x_2$ also encodes Type 5 relationships.)", "This points to one of the advantages of our new RF relationships: we can read off some set equalities more quickly than from the canonical form.", "Indeed, the canonical form is ${\\rm CF}(J_C)=\\lbrace x_1(1+x_2),~x_2(1+x_1) \\rbrace $ , in which the Type 2 relationships are $U_1 \\subseteq U_2$ and $U_2 \\subseteq U_1$ – and only from there do we infer the equality $U_1=U_2$ ." ], [ "Discussion", "In this work, we proved that if a code's canonical form is a Gröbner basis of the neural ideal, then it is the universal Gröbner basis.", "Additionally, we gave conditions that guarantee or preclude this situation, and found three new types of receptive-field relationships that arise in neural ideals.", "Going forward, there are natural extensions to pursue: Give a complete characterization of codes for which the canonical form is a Gröbner basis.", "Beyond those of Types 1–6, what other receptive-field relationships can be read off from a Gröbner basis, and what do they tell us about a code?", "Solutions to these problems would help us extract information about the receptive-field structure directly from the neural code.", "Finally, we expect that our results can be used to improve canonical-form algorithms.", "Indeed, our experiments indicate that under certain conditions, Gröbner bases can be computed more efficiently than canonical forms.", "Moreover, every pseudo-monomial in the universal Gröbner basis of a neural ideal is in the canonical form – so, that subset of the canonical form can be obtained directly from the universal Gröbner basis.", "And, in the case when the universal Gröbner basis contains only pseudo-monomials, then we can conclude immediately that the basis is in fact the canonical form.", "Moreover, we hope to develop decomposition results to build canonical forms and Gröbner basis of codes in large dimensions by `gluing' those of codes in smaller dimensions." ], [ "Acknowledgments", "DM, RK, and EP conducted this research as part of the 2015 Pacific Undergraduate Research Experience in Mathematics Interns Program funded by the NSF (DMS-1045147 and DMS-1045082) and the NSA (H98230-14-1- 0131), in which RG and LG served as mentors and KP was a GTA.", "JL conducted this research as part of the 2016 NSF-funded REU in the Department of Mathematics at Texas A&M University (DMS-1460766), in which AS served as mentor and KP was a GTA.", "The authors thank Ihmar Aldana, Carina Curto, Vladimir Itskov, and Ola Sobieska for helpful discussions.", "LG was supported by the Simons Foundation Collaboration grant 282241.", "AS was supported by the NSF (DMS-1312473/DMS-1513364).", "The authors thank an anonymous referee for helpful comments which improved this work." ] ]
1612.05660
[ [ "Typical-medium, multiple-scattering theory for disordered systems with\n Anderson localization" ], [ "Abstract The typical medium dynamical cluster approximation (TMDCA) is reformulated in the language of multiple scattering theory to make possible first principles calculations of the electronic structure of substitutionally disordered alloys including the effect of Anderson localization.", "The TMDCA allows for a systematic inclusion of non-local multi-site correlations and at same time provides an order parameter, the typical density of states, for the Anderson localization transition.", "The relation between the dynamical cluster approximation and the multiple scattering theory is analyzed, and is illustrated for a tight-binding model." ], [ "Introduction", "During the last 50 years the study of the electronic [1] and phononic [2] properties of substitutionally disordered alloys has been a very active field of research [3], [4].", "Among the many theoretical methods proposed, the Green's function approach has proved to be particularly useful and convenient for calculating various physical quantities [5].", "One of the most successful and comprehensive schemes for the computation of the ensemble-averaged Green's function is the Coherent Potential Approximation (CPA) [6], [7], see also Refs. el.kr.74,ziman79.", "Today it is known that the CPA provides the exact solution for non-interacting fermions with diagonal (local) disorder on any lattice in the limit $Z \\rightarrow \\infty $ , where $Z$ is the coordination number, provided the appropriate quantum scaling of the hopping amplitude is employed [8].", "It is also possible to investigate interacting disordered electrons by combining the dynamical mean-field theory (DMFT) [9], [10], [11], [12], [13], [14], [15] with the CPA [16], [17], [18].", "Using the CPA, the single-particle excitations in quench-disordered [1], [2], [4], [5], [6], [7], [3], [19] systems can be computed.", "In crystalline materials, quenched disorder manifests itself as randomly embedded impurities or alloy disorder.", "Therefore the CPA is frequently employed in the calculation of the electronic structure of these systems.", "The CPA introduces an effective crystalline medium in which a spatially fluctuating random potential is replaced by a purely local, but energy-dependent potential.", "The effective potential is determined in such a way that the configurationally averaged Green's function is equal to the Green's function of the effective medium.", "The CPA was also reformulated in the framework of the multiple scattering theory [20] and combined with the Korringa-Kohn-Rostoker (KKR) basis [21], [22], [23] or linear muffin-tin orbital (LMTO) basis [24] sets.", "It has been used to calculate bulk properties [25], thermodynamic properties [26], [27], [28], the phase stability [29], [30], [31], [32], magnetic properties [33], [34], [35], the surface electronic structure [36], [37], [38], [32], segregation [39], [40], and other alloy characteristics.", "For decades attempts have been made to overcome the main shortcomings of the CPA by incorporating the missing nonlocal physics, e.g., by the molecular CPA [41], [5] and the dynamical cluster approximation (DCA) [42], [43] in model Hamiltonian calculations.", "Regarding the extension of the CPA for electronic structure calculations the KKR-Non-Local Coherent Potential Approximation (KKR-NLCPA) [44] and the KKR-DCA approach [45] were proposed, both implementing the DCA coarse graining [43].", "While the DCA and the KKR-NLCPA are able to include some nonlocal effects, they cannot describe the divergent behavior at the Anderson localization transition [46], [47], [48], [49], [50], [51].", "This is due to the fact that these effective medium theories employ the arithmetically averaged density of states (ADOS), which does not became critical at the Anderson transition, and hence cannot serve as an order parameter.", "As pointed out by Anderson [46] one should instead determine the most probable (“typical”) value of the local density of states (LDOS), which is given by the global maximum of the full probability distribution function of the LDOS.", "In the case of a disordered system near the localization transition the LDOS fluctuates strongly, such that the corresponding probability distribution function possesses long tails.", "Indeed it has been demonstrated that the probability distribution function of the LDOS has very different properties in the metallic and insulating phase, respectively.", "[52], [53] In particular, for weak disorder when states are extended, the probability distribution is Gaussian.", "By contrast, for strong disorder the probability distribution is asymmetric and is given by a log-normal distribution as obtained using analytic, field-theoretical approaches [54] for one-dimensional systems, or numerically exact calculations for three-dimensional lattices [52], [55], [56], [57]; see also Ref. PhysRevB.89.081107.", "The typical value of the LDOS is then determined by the geometric average [58].", "In an attempt to develop an order parameter formalism for Anderson localization, Dobrosavljević and collaborators [59], [60], [61], formulated an effective mean-field theory, the typical medium theory (TMT), which accounts for such changes in the probability distribution function.", "In particular, the TMT uses the geometrically averaged, i.e., typical, DOS in its self-consistency approach.", "The typical DOS vanishes at the localization transition.", "Thus, it can serve as an order parameter for Anderson localization.", "Despite this success, the TMT suffers from some of the same drawbacks as the CPA, i.e., it is still a local theory and hence does not include the crucial nonlocal quantum backscattering effects.", "Therefore the TMT can only provide a qualitative description of the Anderson localization transition.", "Recently, some of us proposed the typical-medium dynamical cluster approximation (TMDCA) [53], which extends the single-site TMT to a finite cluster and thereby allows for a systematic inclusion of nonlocal multisite correlations.", "We demonstrated that the TMDCA overcomes the shortcomings of the TMT and is able to qualitatively and quantitatively describe Anderson localization.", "We also extended this method to models with off-diagonal disorder[62], multiband systems [63], and interactions [64].", "Until now, the TMDCA has only been applied to model Hamiltonians [53], [63], [65].", "To incorporate this formalism directly into first-principles methods, the TMDCA should be reformulated such that access to the Green's function is provided in a language appropriate for calculations within density functional theory (DFT).", "This is best realized in the framework of multiple scattering theory.", "Therefore the purpose of the present study is to fill this gap and extend the TMDCA to the multiple scattering formalism.", "The advantage of this approach is its ability to treat both diagonal and off-diagonal disorder while requiring only small matrices due to the fast convergence of the scattering operators in angular momentum space [66].", "In addition, we outline an alternative approach to first-principles downfolding, model-based approaches[63] which is able to identify the Anderson localization transition in real materials.", "In particular, we show how the KKR-NLCPA can be extended to incorporate the typical medium formalism and demonstrate this in the case of a tight-binding model.", "The discussion of a material-specific implementation and the inclusion of electronic interactions will be postponed to future work.", "Section  describes the model Hamiltonian.", "Section REF is devoted to a detailed discussion of the approximations for the ensemble-averaged Green's function and, in particular, to a discussion of the formal equivalence of the DCA [43] and the KKR-NLCPA as implemented in Refs. Rowlands2003,Biava.", "These approximations are then applied to the tight-binding model of Sec.", ", and the correspondence between physical quantities calculated in the DCA and KKR-NLCPA is discussed.", "Section  addresses the typical medium formulation within the multiple scattering approach: namely, we discuss the general algorithm and the formulas for calculating the typical density of states.", "Numerical results for the tight-binding model are presented in Section , and Section  contains a conclusion of the paper.", "We consider the Anderson model of non-interacting electrons on a cubic lattice subject to a random diagonal potential, described by the Hamiltonian $ H=-\\sum _{\\langle i j \\rangle }W_{ij}(c_{i}^{\\dagger }c_{j}+h.c.", ")+\\sum _{i}V_i n_{i}\\,.$ Here the operator $c_{i}^\\dagger $ ($c_{i}$ ) creates (annihilates) an electron on site $i$ , and $n_{i} = c_{i}^\\dagger c_{i}$ is the number operator.", "The first term describes the hopping of electrons between nearest-neighbor sites $\\langle i,j\\rangle $ with the tight-binding hopping amplitude $W_{ij}=W$ .", "We set $4W = 1$ as the energy unit.", "The disorder is modeled through the energies $V_i$ of the local orbitals which are taken to be independent quenched random variables distributed according to some specified probability distribution $P(V_i)$ .", "In the following, for illustrative purposes, we use a binary alloy distribution which has a bimodal disorder distribution $P(V_i)=c_A\\delta (V_i-V_A)+(1-c_A)\\delta (V_i-V_B)$ , corresponding to a crystal randomly composed of $A (B)$ atoms at energy $V_A$ $(V_B)$ with concentration $c_A$ $(c_B)$ ." ], [ "Formalism", "The main difficulty in dealing with disordered systems is the absence of translational invariance.", "In order to use approaches like the DCA one therefore has to average the free energy and its functional derivatives (such as the Green's functions) over the possible disorder configurations.", "This is often justified since many experimental measurements, like in transport, spectroscopy and optical probes, tend to average over relatively large volumes and, thereby, over many local disorder configurations.", "Since this approach is computationally expensive, especially for large systems, one employs effective medium theories.", "These approaches are based on the idea that a heterogeneous medium can be replaced by an effective equivalent homogeneous medium.", "The problem is then reduced to finding a suitable representation for measured quantities, which in many cases is not provided by the average but by the full probability distribution.", "In this paper we demonstrate explicitly that the DCA and the KKR-NLCPA, when applied to a disordered, non-interacting tight-binding Hamiltonian, are equivalent.", "This fact, although having been observed in numerical calculations [67], has not been proven up to now.", "One goal of this paper is therefore to provide such a proof.", "Furthermore, the TMT has thus far not been formulated in the multiple scattering language.", "The latter provides a theoretical framework for the construction of the typical medium formalism within the DFT.", "In order to see how the ideas of the TMDCA can be incorporated into a multiple scattering formalism (MS-TMDCA), we first briefly summarize the equations and concepts of the DCA approach, followed by the KKR-NLCPA procedure.", "Before proceeding with the formalism, let us note the conceptual relation between the DCA and the KKR-NLCPA.", "The DCA method can be successfully applied to strongly correlated electrons on a lattice with or without disorder [68], to account for non-local dynamic correlations that capture spin and charge fluctuations in addition to configurational fluctuations.", "It should be mentioned that the KKR-NLCPA only accounts for disordered, non-interacting systems, i.e., the self-energy is associated with non-local correlations only due to configurational fluctuations.", "In this way the KKR-NLCPA can be viewed as the non-interacting limit of the DCA [43].", "The derivations given below closely follow Ref.", "mjarrell01a, but we shall not use diagrammatic perturbation theory.", "Instead we focus on the formal connection between the DCA and KKR-NLCPA.", "Namely, both employ the DCA mapping of a lattice with $N$ sites onto a periodic cluster of size $N_c= L_c^D$ ($L_c$ is the linear dimension of the cluster, and $D$ is the dimensionality of the system) embedded in a self-consistently determined host.", "Conceptually, this may be performed by partitioning the lattice with $N$ sites into clusters containing $N_c$ sites.", "In reciprocal space this is equivalent to the division of the Brillouin zone (BZ) of the underlying lattice into $N_c$ cells of size $(2\\pi /L_c)^D$ , centered at the reciprocal sub-lattice vectors ${\\bf K}$ .", "The lattice momenta within a given cell are denoted by $\\tilde{\\bf k}$ .", "In the self-consistency loop, the Green's function is coarse-grained (averaged) over the momenta $\\tilde{\\mathbf {k}}$ surrounding the cluster momentum $\\bf K$ .", "The clusters are subject to periodic boundary conditions, which allows one to use the usual lattice Fourier transform.", "Averaged cluster quantities possess translation invariance and can be taken from the reciprocal space into the real space (or vice versa) using the Fourier transform.", "In this approximation, correlations within the cluster are treated accurately up to a range $\\xi \\le L_c$ , while the physics on longer length scales is described at the static mean-field level.", "In the limit $N_c=1$ the purely local CPA is recovered.", "By increasing the cluster size, the DCA systematically interpolates between the single-site and the exact result while remaining in the thermodynamic limit.", "Here we consider the simple model described by Eq.", "(REF ) in $3D$ , with a cluster size $N_c=38$ .", "The notation in the present paper uses the following convention: an underscore denotes a matrix in the cluster real space, superscripts $I,J$ indicate specific elements of real-space matrices, an overbar represents quantities of the effective medium and the course-grained lattice quantities, an argument “$V$ ” denotes quantities calculated on the cluster for a particular disorder configuration, arguments $\\mathbf {K}$ and $\\mathbf {K}^{\\prime }$ indicate that the corresponding quantities are coarse-grained or calculated in the cluster reciprocal space, a subscript $l$ denotes local quantities, the description “effective medium” denotes the homogeneous, or translationally invariant, problem on the lattice or cluster.", "We use the short-hand notation $\\langle ...\\rangle =\\int dV_i P(V_i) (...)$ for disorder averaging.", "However, it is important to note that it is not necessary, nor even desired, to generate all disorder configurations.", "This would cause the algorithm to scale as $2^{N_c}$ for the binary alloy model described above.", "Rather, in the disorder-averaging procedure we generate the configurations stochastically and assume that the average restores the full point and space group symmetry of the lattice.", "Then, by averaging over these symmetries, we effectively generate more disorder configurations.", "As we will see below, the resulting algorithm scales as $N_c^3$ ." ], [ " A) DCA algorithm I: Reciprocal-space self-energy formulation", "To solve the disorder problem defined by Eq.", "(REF ), we first use the Dynamical Cluster Approximation (DCA), an effective medium cluster approximation in which the random potential of the Hamiltonian of Eq.", "(REF ) is replaced by the effective medium, defined by, as yet unknown, homogeneous self-energy $\\bar{\\Sigma }({\\mathbf {k}, \\omega })$ .", "In local (single-site) approximations, such as the Coherent Potential Approximation (CPA), all non-local corrections are neglected and the self-energy is a local quantity, i.e., in the CPA we approximate the lattice self-energy by a local self-energy $\\bar{\\Sigma }({\\mathbf {k}, \\omega }) = \\bar{\\Sigma }(\\omega )$ with only a frequency dependence.", "To include non-local correlation effects in the DCA, the effective-medium lattice self-energy is approximated by a constant within each DCA cell in momentum space [42], [69], $\\bar{\\Sigma }({\\mathbf {k}, \\omega }) = \\bar{\\Sigma }({\\mathbf {K}, \\omega })$ .", "The corresponding effective-medium lattice Green's function is then given by $G(\\mathbf {k},\\omega )=\\frac{1}{w-\\epsilon (\\mathbf {k})+\\mu -\\bar{\\Sigma }(\\mathbf {K},\\omega )},$ where $\\mathbf {k}=\\mathbf {K}+\\tilde{\\mathbf {k}}$ is the lattice reciprocal space vector, $\\mathbf {K}$ is the DCA cluster vector, $\\tilde{\\mathbf {k}}$ is cluster momenta within each DCA cell, $\\epsilon (\\mathbf {k})=-2W\\left(\\cos k_x+\\cos k_y +\\cos k_z \\right)$ is the band dispersion, and $\\mu $ is the chemical potential which is set to zero for a particle-hole symmetric case.", "To determine the DCA effective-medium self-energy $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ , we must solve a $N_c$ -site cluster problem.", "Following Ref.", "Jarrell.92 the effective-medium cluster Green's function is given by $\\bar{G}_{cl}(\\mathbf {K},\\omega )=\\bar{g}_{cl}(\\mathbf {K},\\omega )+\\bar{g}_{cl}(\\mathbf {K},\\omega )\\bar{\\Delta }(\\mathbf {K},\\omega )\\bar{G}_{cl}(\\mathbf {K},\\omega )\\,.$ Here $\\bar{\\Delta }(\\mathbf {K},\\omega )$ is the hybridization function (the effective-medium “bath”), which is obtained from integrating out all but the cluster degrees of freedom, and which describes the coupling between the cluster and the rest of the effective medium.", "We also introduce $\\bar{g}_{cl}$ , which is an “isolated” (with no hybridization to the effective-medium bath) cluster Green's function, and is defined as $\\bar{g}_{cl}(\\mathbf {K},\\omega )=\\frac{1}{\\omega -\\bar{\\Sigma }(\\mathbf {K},\\omega )-\\bar{\\epsilon }(\\mathbf {K})}\\,.$ Here $\\bar{\\epsilon }(\\mathbf {K})$ is the coarse-grained dispersion averaged over $\\tilde{\\mathbf {k}}$ points within the cell centered on $\\mathbf {K}$ $\\bar{\\epsilon }(\\mathbf {K}) = \\frac{N_{c}}{N}\\sum _{\\tilde{\\mathbf {k}}} \\epsilon _{\\mathbf {K}+\\tilde{\\mathbf {k}}}$ For the CPA formalism, which corresponds to the $N_c=1$ limit of the DCA, all cluster quantities are independent of momentum $\\mathbf {K}$ , and $\\bar{\\epsilon }(\\mathbf {K})$ is equal to a constant which here we set to zero.", "These formulas may be used to define a DCA algorithm where the effective-medium self-energy $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ is required to give an exact description of the original random medium.", "Therefore, to determine the yet-unknown $\\bar{\\Sigma } (\\mathbf {K},\\omega )$ and $\\bar{\\Delta }(\\mathbf {K},\\omega )$ , we replace the effective-medium potential $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ on the cluster by the random potential $V_I$ .", "We then demand that upon averaging the scattering caused by the random potential $V_I$ vanishes identically in the effective medium.", "This construction defines the self-consistency condition which determines $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ .", "As a first step, the cluster Green's function will be calculated.", "We first Fourier transform the cluster Green's function of Eq.", "(REF ) into the cluster real space.", "This is simplified by the fact that the Green's function of the effective medium is translationally invariant.", "Hence, the Green's function between sites $(I,J)$ belonging to the cluster in the effective medium is $\\bar{G}^{IJ}_{cl}(\\omega )= \\bar{g}_{cl}^{IJ}(\\omega )+\\sum _{K,L} \\bar{g}_{cl}^{IK}(\\omega )\\bar{\\Delta }^{KL}(\\omega )\\bar{G}^{LJ}_{cl}(\\omega )\\,.$ Equivalently, Eq.", "(REF ) can be written (in real space) in the matrix form $\\bar{\\underline{g}}_{cl}(\\omega )=\\left(\\omega \\mathbb {I} -\\bar{\\underline{\\Sigma }}(\\omega )-\\bar{\\underline{W}}) \\right)^{-1}\\,$ where $\\bar{W}^{IJ}=\\sum _{\\mathbf {K}}\\bar{\\epsilon }(\\mathbf {K})e^{i\\mathbf {K}(\\bf {R}_I-\\bf {R}_J)}$ is the Fourier transform of the cluster coarse-grained dispersion.", "Now, to consider the real disorder cluster problem embedded in the effective medium, everywhere on the cluster we replace the effective potential $\\underline{\\bar{\\Sigma }}(\\omega )$ of Eq.", "(REF ) by the random potential $V_I$ .", "The corresponding disorder cluster Green's function is then given by $G_{cl}^{I J}(\\omega ,V) &=& g_{cl}^{I J}(\\omega ,V) \\\\&+& \\sum _{K L}g_{cl}^{I K}(\\omega ,V)\\bar{\\Delta }^{K L}(\\omega )G_{cl}^{K J}(\\omega ,V)\\,,\\nonumber $ where the isolated cluster Green's function $\\underline{g}_{cl}(\\omega ,V)$ for a given disorder configuration $V$ is given by $\\underline{g}_{cl}(\\omega ,V)=\\left(\\omega \\mathbb {I}- \\underline{V} - \\underline{\\bar{W}}\\right)^{-1}\\,.$ Since the hybridization function $\\underline{\\bar{\\Delta }}(\\omega )$ is disorder independent, it represents the same hybridization function of the effective medium of Eq.", "(REF ).", "Notice, that the above Green's function can be rewritten in terms of the cluster-excluded (cavity) Green's function $ {\\mathcal {G}}^{I J}(\\omega )$ , conventionally used in the DCA community [12], [68], as $\\underline{G}_{cl}(\\omega ,V)=\\left(\\underline{{ \\mathcal {G} } }^{-1}(\\omega )- \\underline{V}\\right)^{-1} \\,,$ with $\\underline{{\\mathcal {G}}}^{-1}(\\omega )=\\left(\\omega \\mathbb {I} - \\underline{\\bar{W}}-\\underline{\\bar{\\Delta }}(\\omega )\\right)\\,.$ Due to this inverse, this algorithm and the subsequent algorithms below, scale with cluster size as $N_c^3$ .", "In the next step we impose the DCA self-consistency condition, which requires that the disorder-averaged cluster Green's function is equal to the effective medium cluster Green's function, with $\\left< G^{IJ}_{cl}(\\omega ,V)\\right> =\\bar{G }^{IJ}_{cl}(\\omega )\\,.$ Here $\\left< ...\\right>$ denotes averaging over disorder configurations on the cluster.", "Or, equivalently in momentum space, we write it as $\\bar{G}_{cl}(\\mathbf {K},\\omega )=FT(\\left<G^{IJ}_{cl}(\\omega ,V)\\right>),$ where FT stands for the Fourier Transform of the disorder-averaged cluster Green's function to the cluster reciprocal space.", "This condition is then used to update the self-energy, with $\\bar{\\Sigma }(\\mathbf {K},\\omega ) = \\omega - \\bar{\\epsilon }(\\mathbf {K}) - {\\bar{\\Delta }}(\\mathbf {K},\\omega ) -\\bar{G}_{cl}^{-1}(\\mathbf {K},\\omega )\\,,$ which is then used to calculate the coarse-grained lattice Green's function $\\bar{G}(\\mathbf {K},\\omega )$ .", "$\\bar{G}(\\mathbf {K},\\omega )=\\frac{N_{c}}{N}\\sum _{\\tilde{k}}\\frac{1}{\\omega +\\mu -\\epsilon _{\\mathbf {K}+\\tilde{\\mathbf {k}}}-\\bar{\\Sigma }(\\mathbf {K},\\omega )} \\,.$ At convergence this is identical to the effective medium cluster Green's function $G_{cl}(\\mathbf {K},\\omega )$ .", "Given the self-energy $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ through Eq.", "(REF ) one defines the new hybridization function $\\bar{\\Delta }(\\mathbf {K},\\omega )$ as: ${\\bar{\\Delta }}(\\mathbf {K},\\omega ) = \\omega - \\bar{\\epsilon }(\\mathbf {K}) - \\bar{\\Sigma }(\\mathbf {K},\\omega ) - \\bar{G}^{-1}(\\mathbf {K},\\omega )\\,,$ which closes the DCA loop.", "Figure: The DCA self-consistency loop.", "The arrows correspond to thesteps taken in the DCA algorithm described in Subsection .The numerical self-consistency loop is diagrammatically shown in Fig.", "REF , and below we describe the DCA iterative procedure: 1) First, a guess for the cluster self-energy $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ is made (usually set to zero) and the lattice coarse-grained Green's function $\\bar{G}(\\mathbf {K},\\omega )$ is calculated using Eq.", "(REF ).", "2) The effective-medium hybridization function $\\bar{\\Delta }(\\mathbf {K},\\omega )$ is constructed by solving Eq.", "(REF ) and Eq.", "(REF ), i.e., $\\bar{\\Delta }(\\mathbf {K},\\omega )=\\omega -\\bar{\\epsilon }(\\mathbf {K})-\\bar{\\Sigma }(\\mathbf {K},\\omega )-\\bar{G}^{-1}(\\mathbf {K},\\omega )$ .", "Since the cluster problem is solved in real space, we Fourier transform the obtained hybridization $\\bar{\\Delta }(\\mathbf {K},\\omega )$ to real space of the cluster.", "3) Next, the cluster problem is solved in real space and the disorder-averaged cluster Green's function $\\langle G_{cl}(\\omega ,V)\\rangle ^{IJ}=\\langle (\\omega \\mathbb {I}-\\underline{V}-\\underline{\\bar{W}}-\\underline{\\bar{\\Delta }}(\\omega ))^{-1}\\rangle $ is calculated.", "4) Once the cluster problem is solved, we construct a new cluster self-energy, $\\bar{\\Sigma }(\\mathbf {K},\\omega )=\\omega -\\bar{\\epsilon }(\\mathbf {K})-\\bar{\\Delta }(\\mathbf {K},\\omega )-\\bar{G}_{cl}^{-1}(\\mathbf {K},\\omega )$ , using the Fourier Transform of the disorder-averaged cluster Green's function to the cluster reciprocal space, i.e., $\\bar{G}_{cl}(\\mathbf {K},\\omega )=FT(\\langle G_{cl}(\\omega ,V)^{IJ}\\rangle )$ .", "The self-consistent procedure is repeated until $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ converges to the desired accuracy.", "We note that an equivalent self-consistency loop can be constructed by using the cluster-excluded Green's function $ \\underline{\\mathcal {G}}(\\omega )$ .", "[43] This can be done by noting that $\\bar{\\Sigma }(\\mathbf {K},\\omega ) = {\\mathcal {G}}^{-1}(\\mathbf {K},\\omega ) - \\bar{G}_{cl}^{-1}(\\mathbf {K},\\omega )$ .", "While in the DCA formalism the non-local contribution to the self-energy, obtained for $N_c>1$ , is encoded explicitly in the $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ , in the KKR-NLCPA formalism the cluster extensions involve a separate analysis of the local and non-local contributions.", "To provide a better connection between these methods, in the following we present an alternative DCA self-consistency analysis which involves an explicit separation of the local and non-local components of the self-energy.", "To this end, we introduce $\\tilde{\\alpha } (\\mathbf {K},\\omega )=\\bar{\\Sigma }(\\mathbf {K},\\omega )-\\bar{\\Sigma }_l(\\omega ),$ which defines the non-local corrections to the self-energy (the subscript $l$ is used to emphasize the local, momentum independent, quantity).", "For the CPA (i.e.", "in the $N_c$ =1 limit) this non-local contribution vanishes since the self-energy is purely local $\\bar{\\Sigma }(\\mathbf {K}, \\omega )\\rightarrow \\bar{\\Sigma }(\\omega )$ ).", "Using this definition, we rewrite the effective-medium lattice Green's function of Eq.", "(REF ) as $\\bar{G}(\\mathbf {k},\\omega )=\\frac{1}{\\bar{g}_l^{-1}(w)-\\tilde{\\alpha } (\\mathbf {K},\\omega )+\\mu -\\epsilon (\\tilde{\\mathbf {k}}+\\mathbf {K})},$ where we introduce the locator Green's function $\\bar{g}_l(\\omega )$ [5], defined as $\\bar{g}_l(\\omega )=\\frac{1}{\\omega -\\bar{\\Sigma }_l(\\omega )}\\,.$ Similarly, by applying the decomposition of the self-energy into local and non-local parts, with $\\bar{\\Sigma }(\\mathbf {K},\\omega )=\\tilde{\\alpha }(\\mathbf {K})+\\bar{\\Sigma }_l(\\omega )$ , we rewrite the effective-medium cluster Green's function of Eq.", "(REF ) and Eq.", "(REF ) as $\\bar{G}_{cl}^{-1}(\\mathbf {K},\\omega )=\\bar{g}_{cl}^{-1}(\\mathbf {K},\\omega )-\\bar{\\Delta }(\\mathbf {K},\\omega ),$ with the isolated cluster Green's function $\\bar{g}_{cl}^{-1}(\\mathbf {K},\\omega )=\\bar{g}_l^{-1}(\\omega )-\\tilde{\\alpha }(\\mathbf {K},\\omega )-\\bar{\\epsilon }(\\mathbf {K})\\,.$ Next, to find the yet unknown $\\tilde{\\alpha }(\\mathbf {K},\\omega )$ and $\\bar{g}_l(\\omega )$ we consider the disordered cluster embedded in such an effective medium.", "Given a disorder configuration $V_I$ we construct the corresponding cluster Green's function $\\underline{G}_{cl}(V,\\omega )=\\left( \\underline{g}_{l}^{-1}(V,\\omega )-\\bar{\\underline{W}}-\\bar{\\underline{\\Delta }}(\\omega ) \\right) ^{-1},$ where in Eq.", "REF we replaced the effective-medium potential $\\bar{\\Sigma }^{IJ}(\\omega )$ with the random-disorder potential $V_I$ for the locator Green's function, i.e.", "$\\underline{g}_{l}(V,\\omega )=(\\omega \\mathbb {I}-\\underline{V})^{-1}\\,.$ We then impose the DCA self-consistency condition, requiring that the disorder-average cluster Green's function and the effective-medium cluster Green's function are equal, i.e., $\\left<G^{IJ}_{cl}(\\omega ,V)\\right>=\\bar{G }^{IJ}_{cl}(\\omega )\\,.$ This allows us to define a new cluster self-energy, as in Eq.", "(REF ).", "Finally, we complete the self-consistency loop by calculating the lattice coarse-grained Green's function $\\bar{G}(\\mathbf {K},\\omega )=\\frac{N_c}{N}\\sum _{\\tilde{\\mathbf {k}}}\\frac{1}{\\bar{g}_l^{-1}(w)-\\tilde{\\alpha } (\\mathbf {K},\\omega )+\\mu -\\epsilon (\\tilde{\\mathbf {k}}+\\mathbf {K})}.$ This DCA-II algorithm is close in spirit to the KKR-NLCPA scheme with an explicit separation of the local and non-local contributions of self-energy.", "It is, of course, equivalent to the DCA-I algorithm described in the previous subsection.", "In Fig.", "REF we present the corresponding self-consistent loop of the computational procedure of the DCA-II algorithm.", "The steps of the iterative procedure are described as follows: Figure: An alternative DCA-II self-consistency algorithm with separate local and non-local contributions to the self-energy.1) We compute the lattice coarse grained Green's function $\\bar{G}(\\mathbf {K},\\omega )$ using Eq.", "(REF ), by making a guess for $\\bar{g}_l^{-1}(\\omega )-\\tilde{\\alpha }(\\omega )$ based on the values of the local and non-local self-energy components.", "If nothing is known a priori, the guess $\\bar{\\Sigma }_l(\\omega )=0$ and $\\tilde{\\alpha }=0$ , with resulting $\\bar{g}_l^{-1}(\\omega )-\\tilde{\\alpha }=\\omega $ may serve as the starting point.", "2) We then construct the effective-medium hybridization function $\\bar{\\Delta }(\\mathbf {K},\\omega )$ by solving Eq.", "(REF ) and Eq.", "(REF ), i.e., $\\bar{\\Delta }(\\mathbf {K},\\omega )=\\bar{g}_l^{-1}(\\omega )-\\tilde{\\alpha }(\\mathbf {K})-\\bar{\\epsilon }(\\mathbf {K})-\\bar{G}^{-1}(\\mathbf {K},\\omega )$ .", "Since the cluster problem is solved in real space we Fourier transform the obtained hybridization function, $\\bar{\\Delta }(\\mathbf {K},\\omega )$ , to the real space cluster.", "3) In the next step, we solve the cluster problem and calculate the disorder-averaged cluster Green's function $\\langle \\underline{G}_{cl}(\\omega ,V)\\rangle =\\langle ( \\omega \\mathbb {I}-\\underline{V}-\\underline{\\bar{W}}-\\underline{\\bar{\\Delta }}(\\omega ))^{-1}\\rangle $ .", "4) Once the cluster problem is solved, the disorder-averaged cluster Green's function $\\langle \\underline{G}_{cl} \\rangle $ is used to construct an isolated cluster Green's function $\\bar{\\underline{g}}_{cl}^{-1}=\\langle \\underline{G}_{cl}(V,\\omega )\\rangle ^{-1}+\\bar{\\underline{\\Delta }}(\\omega )$ , which we then use to get a new $\\tilde{\\underline{\\alpha }}(\\omega )=\\bar{\\underline{g}}_l^{-1}(\\omega )-\\bar{\\underline{g}}_{cl}^{-1}-\\bar{\\underline{W}}$ .", "Here $\\bar{\\underline{g}}_l$ is a local component of $\\bar{\\underline{g}}_{cl}$ .", "Notice, that in practice we combine these two steps in momentum space and instead calculate $\\bar{g}_l^{-1}(\\omega )-\\tilde{\\alpha }(\\mathbf {K},\\omega )=FT(\\langle G_{cl}^{IJ}(V,\\omega )\\rangle )^{-1}+\\bar{\\Delta }(\\mathbf {K},\\omega )+\\bar{\\epsilon }(\\mathbf {K})$ , where $\\bar{G}_{cl}(\\mathbf {K},\\omega )=FT(\\langle G_{cl}^{IJ}(\\omega ,V)\\rangle )$ .", "5) We repeat the self-consistent procedure through steps 1-5 until convergence is obtained." ], [ " KKR-NLCPA algorithm I ", "In this section we present the details of the KKR-NLCPA [45], [44] formalism applied to the tight-binding Hamiltonian.", "Essentially, the KKR-NLCPA is the static limit (i.e., with no inelastic scattering) of the DCA as derived by Jarrell and Krishnamurthy [43].", "To demonstrate this, in this subsection we present the “KKR-NLCPA algorithm I” which is an alternative to Refs.", "Biava,Rowlands2003 in a spirit which is very similar to the original DCA scheme.", "The derivation of the KKR-NLCPA makes use of the multiple scattering formulation in which the central quantity is the effective-medium cluster scattering path operator $ \\bar{\\tau } (\\mathbf {K},\\omega )$ rather than the cluster effective-medium Green's function $\\bar{G}(\\mathbf {K},\\omega )$ used in the DCA.", "The KKR-NLCPA is also an effective-medium method, where the original disorder problem of Eq.", "(REF ) is replaced by the effective-medium problem, such that the lattice effective scattering path operator $\\bar{\\tau }(\\mathbf {k},\\omega )$ is given as $\\bar{\\tau }(\\mathbf {k},\\omega )=\\frac{1}{\\bar{t}^{-1}(\\mathbf {k},\\omega )-G_0^{^{\\prime }}(\\mathbf {k},\\omega )}\\,.$ Here $\\bar{t}^{-1}(\\mathbf {k},\\omega )$ is the yet unknown homogeneous effective scattering $t$ -matrix and $G_0^{^{\\prime }}(\\mathbf {k},\\omega )$ is free space structure constant  [5], which for tight-binding Hamiltonian corresponds to a bare Green's function $G_0(\\mathbf {k},\\omega )$ .", "To determine the $t$ -matrix we use the DCA-like cluster embedding scheme, where the lattice t-matrix, is approximated by a cluster t-matrix with $\\bar{t}(\\mathbf {k},\\omega )=\\bar{t}(\\mathbf {K},\\omega )$ and is obtained by solving a cluster embedded in an effective-medium, with the effective medium cluster scattering path operator defined as $\\bar{\\tau }_{cl}(\\mathbf {K},\\omega )=\\bar{t}_{cl}(\\mathbf {K},\\omega )+\\bar{t}_{cl}(\\mathbf {K},\\omega )\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )\\bar{\\tau }_{cl}(\\mathbf {K},\\omega )\\,.$ Here $\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ arises from integrating out all but the cluster degrees of freedom and corresponds to the hybridization function $\\Delta (\\mathbf {K},\\omega )$ in the DCA scheme.", "In the KKR literature this quantity is referred as the “effective-medium renormalized interactor” (see for example Refs. ziman79,agonis92).", "Here we also define $\\bar{t}_{cl}(\\mathbf {K},\\omega )$ the isolated cluster t-matrix (with no hybridization to the effective medium) as $\\bar{t}_{cl}(\\mathbf {K},\\omega )=\\frac{1}{\\bar{t}^{-1}(\\mathbf {K},\\omega )-\\bar{G}_{0}^{^{\\prime }}(\\mathbf {K},\\omega )}\\,,$ Due to its explicit $\\mathbf {K}$ -dependence, $t_{cl}(\\mathbf {K},\\omega )$ takes into account non-local correlations up to the cluster size which are missing in the local KKR-CPA analysis for $N_c=1$ .", "We note, that for $N_c=1$ this quantity becomes local with $\\bar{t}_{cl}({\\mathbf {K},\\omega })\\rightarrow \\bar{t}_l(\\omega )$ , where the subscript $l$ indicates “local quantity”.", "Since the effective medium is translationally invariant, we Fourier transform Eq.", "(REF ) to the cluster real space, i.e., ${\\bar{\\tau }}^{IJ}_{cl}(\\omega )=\\bar{t}_{cl}^{IJ}(\\omega )+\\sum _{K,L}\\bar{t}_{cl}^{IK}(\\omega )\\bar{\\Delta }^{^{\\prime }KL}(\\omega )\\bar{\\tau }^{LJ}(\\omega )\\,.$ Equivalently, Eq.", "(REF ) in real space can be rewritten as $\\bar{\\underline{t}}_{cl}(\\omega )=\\left( \\bar{\\underline{t}}^{-1} (\\omega )-\\bar{\\underline{G}}_{0}^{^{\\prime }}(\\omega ) \\right)^{-1} \\,.$ To determine the effective-medium quantities we next introduce the impurity cluster, with the disorder placed on each cluster site, embedded in the effective medium.", "To do this, we replace the effective-medium quantities in Eq.", "(REF ) and Eq.", "(REF ) by their disorder-dependent counterparts.", "Note that $\\underline{\\bar{\\Delta }}^{\\prime }(\\omega )$ is independent of the disorder configuration of the cluster since it describes the effective medium.", "Hence it is not changed under the substitution of an effective cluster with the disorder-dependent cluster problem.", "Thus, we obtain an expression for the cluster path operator for a given disorder configuration $\\tau _{cl}^{I J}(\\omega ,V) &=& t_{cl}^{I J}(\\omega ,V) \\\\&+&\\sum _{K,L}t^{I K}_{cl}(\\omega ,V)\\bar{\\Delta }^{^{\\prime }K L}(\\omega )\\tau _{cl}^{L J}(\\omega ,V)\\nonumber \\,,$ where the cluster $t$ -matrix for a given disorder $V$ is ${\\underline{t}}_{cl}(\\omega ,V)=\\left( {\\underline{t}}^{-1} (V)-\\bar{\\underline{G}}_{0}^{^{\\prime }}(\\omega ) \\right)^{-1} ,$ where for the tight-binding model $\\underline{t}^{-1}=\\underline{V}^{-1}$ .", "With this transformation, the cluster path operator for a given disorder configuration reads $\\underline{\\tau }_{cl}(\\omega ,V)=\\left(\\underline{t}^{-1}(V)-\\underline{\\bar{G}}_{0}^{^{\\prime }}-\\underline{\\bar{\\Delta }}^{^{\\prime }}(\\omega )\\right)^{-1}\\,.$ In the next step, we impose the self-consistency condition, which requires that, when placing the cluster in the effective medium, no additional scattering is produced on average, i.e., $\\left<\\tau _{cl}(\\omega ,V)\\right>^{I J}=\\bar{\\tau }^{I J}_{cl}(\\omega ).$ We then close the self-consistency loop by calculating the coarse-grained lattice scattering path operator given as $\\bar{\\tau }(\\mathbf {K}, \\omega )=\\frac{N_c}{N}\\sum _{\\tilde{k}}\\frac{1}{\\bar{t}^{-1}(\\mathbf {K},\\omega )-G_0^{^{\\prime }}(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )},$ where $\\bar{t}^{-1}(\\mathbf {K},\\omega )=FT(\\left<\\tau _{cl}(\\omega ,V)\\right>^{I J})+\\bar{G}_{0}^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ is obtained from Eq.", "(REF ) and Eq.", "(REF ).", "Figure: The self-consistency loop of the KKR-NLCPA algorithm I presented in Subsection .The self-consistency loop used in this “KKR-NLCPA algorithm I\" is illustrated in Fig.", "REF and the iterative procedures are described as follows: 1) First we make a guess for the unknown cluster $\\bar{t}(\\mathbf {K},\\omega )$ scattering t-matrix (usually set to zero) and calculate the lattice coarse-grained scattering path operator $\\bar{\\tau }(\\mathbf {K},\\omega )$ using Eq.", "(REF ).", "2) Then we construct the effective-medium cluster renormalized interactor function $\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ by solving Eq.", "(REF ) and Eq.", "(REF ), i.e., $\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )=\\bar{t}^{-1}(\\mathbf {K},\\omega )-\\bar{\\tau }(\\mathbf {K},\\omega )^{-1}-\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )$ .", "3) Since the cluster problem is solved in real space, we Fourier transform the obtained cluster interactor $\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ to real space.", "We then solve the cluster problem and calculate the disorder-average scattering path operator $\\langle {\\underline{\\tau }}_{cl}(\\omega ,V)\\rangle =\\left\\langle \\left(\\underline{t}(V)^{-1}-\\underline{\\bar{G}}_0^{^{\\prime }}-\\underline{\\bar{\\Delta }}^{^{\\prime }}(\\omega )\\right)^{-1}\\right\\rangle $ .", "4) Once the cluster problem is solved, we Fourier transform the obtained disorder-averaged path operator to the cluster reciprocal space with $\\bar{\\tau }_{cl}(\\mathbf {K},\\omega )=FT(\\langle \\tau _{cl}(\\omega ,V)^{IJ}\\rangle )$ and use it to construct new cluster $\\bar{t}(\\mathbf {K},\\omega )$ scattering matrix, i.e., $\\bar{t}(\\mathbf {K},\\omega )=\\bar{\\tau }_{cl}^{-1}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )$ .", "5) We repeat steps 1-4 of the self-consistent procedure until $\\bar{t}(\\mathbf {K},\\omega )$ converges to the desired accuracy." ], [ " KKR-NLCPA algorithm II: Local and non-local $t$ -matrix contributions", "The KKR-NLCPA algorithm of Refs.", "Biava,Rowlands2003 with explicit separation of local and non-local contributions to the t-matrix is an alternative way of constructing the self-consistency.", "Here we show that this formalism can be derived from the one presented in the previous subsection “KKR-NLCPA algorithm I\".", "As discussed above, to explicitly distinguish between the local and non-local contributions to the scattering $t$ -matrix, we define a quantity $\\bar{\\alpha }(\\mathbf {K},\\omega )=\\bar{t}_l^{-1}(\\omega )-\\bar{t} ^{-1}(\\mathbf {K}, \\omega ).$ Here $\\bar{t}_{l}(\\omega )$ is the local component of the scattering $t$ -matrix, i.e., the single-site scattering matrix (we use a subscript $l$ to emphasize that $ \\bar{t}_{l}(\\omega )$ is a local quantity).", "Although $\\bar{\\alpha }(\\mathbf {K},\\omega )$ describes the non-local contributions to the scattering $t$ -matrix (in the same way as $\\tilde{\\alpha } (\\mathbf {K},\\omega )$ describes non-local corrections to the DCA self-energy), in the KKR-NLCPA literature it is called the correction to the free propagator [44].", "Also note, that in Ref.", "Biava this quantity is denoted as $\\delta G(\\mathbf {K},\\omega )$ .", "Using Eq.", "(REF ) and making the DCA approximation with $\\bar{t}(\\mathbf {k},\\omega )=\\bar{t}(\\mathbf {K},\\omega )$ , we rewrite the effective-medium lattice scattering path operator of Eq.", "(REF ) as $\\bar{\\tau }(\\mathbf {k},\\omega )=\\frac{1}{\\bar{t}^{-1}_l(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )-G_0^{^{\\prime }}(\\mathbf {k},\\omega )}.$ with the lattice vector $\\mathbf {k}=\\mathbf {K}+\\tilde{\\mathbf {k}}$ .", "To find the yet unknown $\\bar{t}^{-1}_l(\\omega )$ and $\\bar{\\alpha }(\\mathbf {K},\\omega )$ , we consider the cluster path operator $\\bar{\\tau }_{cl}(\\mathbf {K},\\omega )$ of Eq.", "(REF ), with $\\tau _{cl}^{-1}(\\mathbf {K},\\omega )=\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )-\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )-\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega ),$ where we used the fact that Eq.", "(REF ) can be rewritten as $t_{cl}^{-1}(\\mathbf {K},\\omega )=\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )-\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )\\,.$ This corresponds to Eq.", "(5) of Ref. Rowlands2003.", "Next, to determine the effective-medium quantities we solve the cluster, with the disorder placed on each cluster site, embedded in the effective medium.", "Replacing $\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )$ in Eq.", "(REF ) with the disorder-dependent $t^{-1}(V)$ in Eq.", "(REF ), we again obtain Eq.", "(REF ).", "In the next step, we impose a self-consistency condition, which requires that placing a cluster in the effective medium on average does not produce additional scattering, i.e., $\\left<\\tau _{cl}(\\omega ,V)\\right>^{I J}=\\bar{\\tau }^{I J}_{cl}(\\omega ).$ To complete the self-consistency loop, we coarse-grain the scattering path operator of Eq.", "(REF ) $\\bar{\\tau }(\\mathbf {K},\\omega )=\\frac{N_c}{N}\\sum _{\\tilde{\\mathbf {k}}} \\frac{1}{\\bar{t}^{-1}_l(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )-G_0^{^{\\prime }}(\\mathbf {k},\\omega )},$ where, $t^{-1}_l(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )=FT(\\left<\\tau _{cl}(\\omega ,V)\\right>^{I J})^{-1}+G_0^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ , according to Eq.", "(REF ) and Eq.", "(REF ).", "After self-consistency is achieved, we can calculate the cluster coarse-grained Green's function using Eq.", "(REF ): $\\bar{G}(\\mathbf {K},\\omega )&=& \\frac{N_c}{N}\\sum _{\\tilde{\\mathbf {k}}}( G_0(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega ) \\nonumber \\\\&+& G_0(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )\\bar{\\tau }(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )G_0(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )),$ where $G_0(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )$ the bare lattice Green's function $ G_{0}(\\mathbf {K}+\\tilde{\\mathbf {k}})=(\\omega -{\\epsilon }_{\\mathbf {K}+\\tilde{\\mathbf {k}}}+i\\eta )^{-1}$ and $\\bar{\\tau }(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega ) = {\\bar{t}_l(\\omega )}^{-1}-\\bar{\\alpha }(\\mathbf {K},\\omega )- G_{0}(\\mathbf {K}+\\tilde{\\mathbf {k}},\\omega )$ .", "In the tight-binding Hamiltonian, the bare Green's function $G_0$ and the structure constant $G_0^{^{\\prime }}$ are identical.", "Nevertheless we still use separate symbols to distinguish the physical meaning of the quantities.", "The self-consistency loop for the KKR-NLCPA algorithm II is presented in Fig.", "REF .", "It is easy to see that, using $\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )=\\bar{t}^{-1}(\\mathbf {K},\\omega )$ , both KKR-NLCPA algorithm I and KKR-NLCPA algorithm II are equivalent.", "Figure: The KKR-NLCPA algorithm II self-consistency loop applied to the tight binding model." ], [ "Relation between DCA and KKR-NLCPA quantities", "The application of the DCA and the KKR-NLCPA formalism to the tight-binding model discussed in Sec.", "REF and Sec.", "REF , shows that there exist formal analogies between quantities and self-consistency equations used in these two approaches.", "For example, the effective cluster locator of the DCA approach, $g_{cl}(K,\\omega )$ , plays the role of the cluster $t$ -matrix $t_{cl}(K,\\omega )$ in the KKR-NLCPA formalism; the coarse-grained hopping matrix $\\bar{W}^{IJ} $ is replaced by the free-space structure constant $\\bar{G}_0^{\\prime IJ}$ ; and the non-local contributions to the self-energy $\\tilde{\\alpha }(K,\\omega )$ correspond to the non-local contribution to the scattering $t$ -matrix $\\bar{\\alpha } (K,\\omega )$ .", "For completeness, in Table.", "REF , we summarize the one-to-one correspondence between the DCA and the KKR-NLCPA equations.", "In the following we explicitly show that quantities in the DCA are related to those in the KKR-NLCPA.", "In doing so we establish the formal equivalence of the two methods when applied to a tight-binding model.", "Using the obtained relationship, we construct an alternative Multiple-Scattering DCA (MS-DCA) algorithm which is a Green's function based multiple-scattering algorithm which allows one to calculate the disorder-averaged Green's function instead of the scattering path operator in self-consistent KKR-NLCPA loop.", "This step is necessary for the further implementation of the typical medium analysis in the multiple-scattering formalism." ], [ "DCA hybridization function and the NLCPA cluster renormalized interactor", "First, we show the relationship between the DCA hybridization function $\\bar{\\Delta }(\\mathbf {K},\\omega )$ and the KKR-NLCPA renormalized interactor $\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ .", "To establish this, we calculate the disorder-averaged Green's function of the cluster using the scattering-path operator $\\left<\\underline{G}_{cl}(\\omega )\\right>&=&\\left<\\left(\\underline{\\mathcal {G}}^{-1}(\\omega )-\\underline{V}\\right)^{-1}\\right>\\\\ \\nonumber &=& \\underline{\\mathcal {G}}(\\omega )+\\underline{\\mathcal {G}}(\\omega ) \\left<\\underline{\\tau }_{cl}(\\omega ,V)\\right> \\underline{\\mathcal {G}}(\\omega )\\,.$ Here the cluster scattering path operator matrix is given by $\\underline{\\tau }_{cl}(\\omega ,V)=\\left( \\underline{V}^{-1}-\\underline{\\mathcal {G}}(\\omega ) \\right)^{-1}\\,.$ Further, recalling Eq.", "(REF ) of the NLCPA procedure, the cluster path operator is $\\underline{\\tau }_{cl}(\\omega ,V) =\\left(\\underline{t}^{-1}(V)-\\underline{\\bar{G}}_{0}^{^{\\prime }}(\\omega )-\\underline{\\bar{\\Delta }}^{^{\\prime }} (\\omega ) \\right)^{-1}\\,.$ Comparing Eqs.", "(REF ) and (REF ), we find that the cluster-excluded Green's function $\\underline{\\mathcal {G}}(\\omega )$ satisfies the relation $\\underline{\\mathcal {G}} (\\omega ) =\\underline{\\bar{G}}^{^{\\prime }}_{0}(R)+\\underline{\\bar{\\Delta }}^{^{\\prime }}(\\omega )\\,.$ Next, in order to find the relation between the DCA hybridization function and the KKR-NLCPA interactor, we employ the expression for the cluster-excluded Green's function $\\underline{{\\mathcal {G}}}^{-1}(\\omega )=\\left(\\omega \\mathbb {I} - \\underline{\\bar{W}}-\\underline{\\bar{\\Delta }}(\\omega )\\right)$ .", "Thereby, the NLCPA cluster renormalized interactor $\\bar{\\Delta }^{^{\\prime }}$ is found to be related to the DCA hybridization function $\\bar{\\Delta }$ as $\\underline{\\bar{\\Delta }}^{^{\\prime }}(\\omega )=(\\omega \\mathbb {I} -\\underline{\\bar{W}}-\\underline{\\bar{\\Delta }}(\\omega ))^{-1}-\\underline{\\bar{G}}_{0}^{^{\\prime }}\\,.$ Using this relationship, we now obtain an expression for the cluster Green's function which can be calculated directly in the KKR-NLCPA self-consistency scheme, i.e., $\\left<\\underline{G_{cl}}(\\omega )\\right>&=&\\left<\\left(\\underline{\\mathcal {G}}^{-1}-\\underline{V}\\right)^{-1}\\right>\\\\ \\nonumber &=& \\left<\\left(\\left(\\underline{\\bar{\\Delta }}^{^{\\prime }}(\\omega )+\\underline{\\bar{G}}_{0}^{^{\\prime }}(\\omega )\\right)^{-1}-\\underline{V}\\right)^{-1}\\right>\\,.$ Figure: Multiple-Scattering DCA: a Green's function based multiple scattering algorithm where we calculate the disorder-averaged Green's function instead of the scattering-path operator." ], [ "DCA self-energy $\\Sigma (\\mathbf {K},\\omega )$ and the NLCPA\neffective corrections {{formula:c6cb768e-6990-4b93-a9b2-74678ef6de8c}}", "Here we establish the relationship between the DCA self-energy and the non-local $t$ -matrix corrections $\\bar{\\alpha }(\\mathbf {K},\\omega )$ of the KKR-NLCPA.", "In the latter, the lattice Green's function is given by $G(\\mathbf {k},\\omega )&=&G_0(\\mathbf {k},\\omega )+G_0(\\mathbf {k},\\omega )\\bar{\\tau }(\\mathbf {k},\\omega )G_0(\\mathbf {k},\\omega ) \\\\ \\nonumber &=&G_0(\\mathbf {k},\\omega )+G_0(\\mathbf {k},\\omega ) \\times \\\\ \\nonumber & & \\frac{1}{\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )-G_0(\\mathbf {k},\\omega )}G_0(\\mathbf {k},\\omega ) \\,.$ which can also be written as $G(\\mathbf {k},\\omega )=\\frac{1}{G_0^{-1}(\\mathbf {k},\\omega )-\\frac{1}{\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )}}\\,.$ At the same time, in the DCA scheme the lattice Green's function is given by $G(\\mathbf {k},\\omega )=\\frac{1}{G_0^{-1}(k,\\omega )-\\bar{\\Sigma }(\\mathbf {K},\\omega )}\\,.$ Comparing Eqs.", "(REF ) and (REF ), we find $\\bar{\\Sigma }^{-1}(\\mathbf {K},\\omega ) =\\bar{t}_l^{-1}(\\omega ) - \\bar{\\alpha }(\\mathbf {K},\\omega )\\,.$ This relationship shows how to obtain the self-energy in the KKR-NLCPA analysis." ], [ "Multiple-Scattering DCA algorithms", "Using the obtained relationships between the DCA and KKR-NLCPA for the hybridization function and the self-energy, we now construct an alternative MS-DCA self-consistency loop where the disorder-averaged cluster Green's function instead of the scattering-path operator is calculated directly in the self-consistency.", "The diagram with the MS-DCA algorithm is shown in Fig  REF , where, in addition to Eq.", "(REF ), Eq.", "(REF ) and Eq.", "(REF ), we also use the Dyson's equation with $\\Sigma (\\mathbf {K},\\omega )&=&\\mathcal {G}^{-1}(\\mathbf {K},\\omega )-G_{cl}^{-1}(\\mathbf {K},\\omega )\\nonumber \\\\&=&\\left(\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )\\right)^{-1}-G_{cl}^{-1}(\\mathbf {K},\\omega )$ As shown in Sec.", "below, this algorithm may be adapted to a typical medium approach.", "Table: The correspondence between quantities appearing in the DCA and the KKR-NLCPA formalism.While the DCA incorporates spatial correlations which are missing in the CPA, the average DOS calculated in the DCA [53] is not critical at the Anderson transition, [70], [71] and hence cannot be used as an order parameter.", "To identify Anderson localized states, one has to calculate the typical density of states (TDOS).", "Indeed, it has been demonstrated that the TDOS vanishes for localized states, and hence, can be used as a proper order parameter (Refs.", "Vlad2003 and PhysRevB.89.081107).", "In the typical medium theory the self-consistency involves the TDOS which vanishes continuously as the strength of the disorder increases towards the critical point (Refs.", "Vlad2003 and PhysRevB.89.081107).", "As a consequence, there is an additional step needed to calculate this order parameter.", "In the following, we show how to apply the typical medium analysis to the tight-binding model within the multiple scattering approach (MS-TMDCA).", "We have already shown in Fig.", "REF how to calculate the average Green's function in the self-consistency loop.", "To incorporate the typical medium analysis in the KKR-NLCPA, we modify the effective medium by replacing the algebraically averaged Green's function by its typical, i.e., geometrically averaged, counterpart.", "Thereby translational invariance is restored by the disorder average everywhere in the distribution of the density of states, including the average and typical values.", "To construct the typical Green's function, we use the same ansatz as in Ref. PhysRevB.89.081107).", "For each cluster configuration, we obtain the cluster density of states diagonal in the wave number, $\\rho _{cl}(\\mathbf {K},\\omega ,V)=-\\Im G_{cl} (\\mathbf {K},\\mathbf {K},\\omega ,V)/\\pi $ , assuming that the off-diagonal contributions vanish.", "We also calculate the LDOS on the cluster $\\rho _{cl}^{II}(\\omega ,V)=-\\Im G_{cl}^{II} (\\omega ,V)/\\pi $ .", "Then, we calculate the typical (geometrically averaged) density of states as $\\bar{\\rho }_{typ}(\\mathbf {K},\\omega ) &=& \\exp \\left(\\frac{1}{N_c} \\sum _{I=1}^{N_c} \\left\\langle \\ln \\rho _{cl}^{II} (\\omega ,V)\\right\\rangle \\right) \\nonumber \\\\&\\times & \\left\\langle \\frac{\\rho _{cl}(\\mathbf {K},\\omega ,V)}{\\frac{1}{N_c} \\sum _{I} \\rho _{cl}^{II} (\\omega ,V)} \\right\\rangle \\,.$ Here, as in the TMDCA, the local part of the cluster-momentum-resolved typical density of states is separated and treated with geometrical averaging over the disorder configurations, to avoid self-averaging as the cluster size increases.", "The form of the proposed typical density of states of Eq.", "() recovers the local TMT at $N_c=1$ limit and reduces to the DCA scheme at weak disorder strength.", "By using the Hilbert transform, we obtain the typical Green's function from Eq.", "(REF ) as $\\bar{G}_{typ}(\\mathbf {K},\\omega )=\\int d\\omega ^{\\prime } \\frac{\\bar{\\rho }_{typ}(\\mathbf {K},\\omega ^{\\prime })}{\\omega -\\omega ^{\\prime }}\\,,$ which replaces the average Green's function in the self-consistency loop, Fig.", "REF .", "The resulting algorithm for the MS-TMDCA applied to the tight-binding model is shown in Fig.", "REF .", "Figure: MS-TMDCA: a Green's function based typical medium multiple scattering algorithm.As discussed previously[65], the form of Eq.", "(REF ) is not unique.", "To be able to describe the transition it should contain the order parameter (the first term in the product), which vanishes at the transition.", "A possible, but different form reads [65] $\\bar{G}_{typ}(\\mathbf {K},\\omega ) &=& \\exp \\left(\\frac{1}{N_c} \\sum _{I=1}^{N_c} \\left\\langle \\ln \\rho _{cl}^{II} (\\omega ,V)\\right\\rangle \\right) \\nonumber \\\\&\\times & \\left\\langle \\frac{G_{cl}(\\mathbf {K},\\omega ,V)}{\\frac{1}{N_c} \\sum _{I} \\rho _{cl}^{II} (\\omega ,V)} \\right\\rangle \\,.$ It dispenses with the need for the Hilbert transform so that results at different frequencies are independent by calculating the typical $\\bar{G}_{typ}$ directly, at the expense of not recovering the TMT when $N_c=1$ .", "Nevertheless, it quickly converges to produce the same results as Eq.", "(REF ) and Eq.", "(REF ) for moderate cluster sizes, including $N_c=38$ , but will not be further employed in this discussion.", "It is important to note that the typical Green's function, $G_{typ}(\\mathbf {K},\\omega )$ , is used only to calculate the effective medium of the cluster problem.", "This geometrically averaged reference system carries no physical meaning other than the order parameter.", "Since experimental measurements of the single-particle spectra (which also determine the Fermi level), and of transport, two-particle spectra, etc.", "involve averages over large regions, these experiments are described by arithmetically averaged Green's functions obtained from functional derivatives of the arithmetically averaged free energy.", "In Tab.", "REF we summarize the naming conventions and acronyms for the different algorithms discussed in this manuscript.", "Here, in each abbreviation, we use the letters TM to indicate a typical medium approach, the letters DCA for a Green's function based approach, and NLCPA a scattering path operator ($\\tau $ ) based approach.", "Table: Acronyms of the algorithms used." ], [ "Numerical results for the tight-binding model", "In this section, we will present the results obtained using the KKR-NLCPA formalism as compared with the standard DCA and the typical medium DCA using the tight-binding model of Sec.", "." ], [ "Effective medium with arithmetic average: KKR-NLCPA vs. DCA", "As a starting point, we discuss results from benchmarking the KKR-NLCPA formalism with the DCA.", "As shown in Eq.", "(REF ) in Subsection REF , the DCA cluster excluded Green's function is related to the KKR-NLCPA cluster renormalized interactor through $\\mathcal {G}(\\mathbf {K},\\omega )=\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ .", "We use this relationship in the construction of the multiple-scattering algorithms which employ the arithmetically averaged and typical Green's functions.", "Figure: Comparison of the imaginary parts of the cluster excluded Green's function 𝒢(𝐊,ω)\\mathcal {G}(\\mathbf {K},\\omega ) from the DCA procedure and G ¯ 0 ' (𝐊,ω)+Δ ¯ ' (𝐊,ω)\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega ) calculated in the KKR-NLCPA procedure for N c =38N_c=38 at disorder strength, V A =0.7_A=0.7 and concentration, c A =0.5_A=0.5.", "The solid lines depict the DCA results while the dash lines are the corresponding KKR-NLCPA and their associated momenta 𝐊\\mathbf {K} correspond to each of the eight distinct cells obtained using the point-group symmetry of the cluster.To demonstrate this numerically, we show in Fig.", "REF a comparison of the imaginary parts of $\\mathcal {G}(\\mathbf {K},\\omega )=\\left(\\omega -\\bar{\\epsilon }(\\mathbf {K})-\\bar{\\Delta }(\\mathbf {K},\\omega )\\right)^{-1}$ from the standard DCA and $\\bar{G}_0^{^{\\prime }}(\\mathbf {K},\\omega )+\\bar{\\Delta }^{^{\\prime }}(\\mathbf {K},\\omega )$ calculated from the KKR-NLCPA algorithm shown in Fig.", "REF for the binary alloy disorder configurations for the concentration $c_A=0.5$ at disorder strength $V_A=0.7$ for the cluster size $N_c=38$ .", "In Fig.", "REF , the solid lines are the DCA results while the dash lines are the corresponding KKR-NLCPA.", "As evident from the plots, the two formalisms agree with each other within our numerical accuracy.", "This shows that the averaged medium KKR-NLCPA/DCA using the tight-binding model adequately reproduces the required behavior as in the DCA.", "Figure: A numerical comparison of the local self-energy 1 N c ∑ 𝐊 Σ ¯(𝐊,ω)\\frac{1}{N_c}\\sum _\\mathbf {K}\\bar{\\Sigma }(\\mathbf {K},\\omega ) obtained from the DCA procedureand the local KKR-NLCPA self-energy 1 N c ∑ 𝐊 [t l -1 (ω)-α ¯(𝐊)] -1 \\frac{1}{N_c}\\sum _\\mathbf {K}[t_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K})]^{-1}.", "Here, solid lines are the DCA results anddash lines are their KKR-NLCPA counterparts.To perform a further numerical check of these expressions we compare the DCA self-energy $\\bar{\\Sigma }(\\mathbf {K},\\omega )$ to the KKR-NLCPA (Fig.", "REF ) $[\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K},\\omega )]^{-1}$ .", "The results for the imaginary part of the local cluster average self-energy $\\Im \\bar{\\Sigma }(\\omega )=\\frac{1}{N_c}\\sum _{\\mathbf {K}}\\Im \\bar{\\Sigma }(\\mathbf {K},\\omega )$ is shown in Fig.", "REF and that for the imaginary part of the non-local momentum-resolved $\\Im \\bar{\\Sigma }(\\mathbf {K},\\omega )$ is shown in Fig.REF .", "In Fig.", "REF , the labels A–D and their associated momenta $\\mathbf {K}$ correspond to each of the four distinct cells obtained using the point-group and particle-hole symmetry ($\\bar{\\rho }(\\mathbf {K},\\omega ) = \\bar{\\rho }(\\mathbf {Q}-\\mathbf {K},-\\omega )$ , with $\\mathbf {Q}= (\\pi ,\\pi ,\\pi )$ ) of the cluster.", "For the local self-energy, Fig.", "REF , the two disorder strengths, $V_A=0.2$ and 0.7, correspond to the weak disorder and the band split regimes, respectively.", "Figure: A numerical comparison of the 𝐊\\mathbf {K}-resolved self-energy Σ ¯(𝐊,ω)\\bar{\\Sigma }(\\mathbf {K},\\omega ) obtained fromthe DCA procedure and the KKR-NLCPA self-energy [t ¯ l -1 (ω)-α ¯(𝐊)] -1 [\\bar{t}_l^{-1}(\\omega )-\\bar{\\alpha }(\\mathbf {K})]^{-1} at V A =0.7V_A=0.7 and c A =0.5c_A=0.5.We observe again that two procedure are numerically equivalent.", "Here, the solid lines are the DCAresults while the dashed lines depict their KKR-NLCPA counterparts.", "Note, the labels A–D and theirassociated momenta 𝐊\\mathbf {K} depict each of the four distinct cells obtained using the point-group andparticle-hole symmetry (ρ ¯(𝐊,ω)=ρ ¯(𝐐-𝐊,-ω)\\bar{\\rho }(\\mathbf {K},\\omega ) = \\bar{\\rho }(\\mathbf {Q}-\\mathbf {K},-\\omega ), with 𝐐=(π,π,π)\\mathbf {Q}= (\\pi ,\\pi ,\\pi )) of thecluster.In both cases, a good agreement between the two formalisms is observed.", "Similarly, as seen from Fig.", "REF , we find a very good agreement between the non-local self-energies calculated within the DCA and KKR-NLCPA formalisms, respectively.", "Here we present results for the large disorder $V_A=0.7$ strength only, since at low disorder the non-local effects are negligible.", "Figure: The arithmetically averaged local density of states (ADOS)obtained from the DCA and KKR-NLCPA calculations.", "We present data for the binary alloy disorder with c A =0.5c_A=0.5, forcluster size N c =38N_c=38 at small disorder V A =0.2V_A=0.2 and larger disorder V A =0.7V_A=0.7.The results show that the two procedures are equivalent when applied to the tight-binding model.To conclude our benchmarking of the multiple scattering KKR-NLCPA approach with the DCA, we show in Fig.", "REF a plot of the arithmetically averaged density of states (ADOS) obtained using the DCA algorithm I of Fig.", "REF and the KKR-NLCPA of Fig.", "REF .", "This again is a good quantity to check the reliability of the developed KKR-NLCPA formalism within the tight-binding model.", "The data are for $c_A=0.5$ at weak disorder $V_A=0.2$ and strong disorder $V_A=0.7$ using a finite cluster $N_c=38$ .", "The results clearly demonstrate that the two procedures are equivalent when applied to the tight-binding model." ], [ "Typical medium: MS-TMDCA vs. TMDCA", "In the typical medium analysis, the effective medium is characterized by a geometrically averaged typical density of states.", "To demonstrate that the proposed MS-TMDCA formalism captures correctly the Anderson localization transition, we compare our results of the typical density of states obtained from the MS-TMDCA procedure described in Fig.", "REF with the ones obtained using the TMDCA scheme [53].", "As shown in Fig.", "REF for a finite cluster size, $N_c=38$ at disorder strength, $V_A=0.7$ Figure: ADOS calculated in DCA and KKR-NLCPA procedures and TDOS calculated in MS-TMDCA and TMDCA procedures.", "States within the mobility edge marked by arrows are extended.the ADOS from the standard DCA (of Fig.", "REF ) and the KKR-NLCPA (of Fig.", "REF ) are the same (black line).", "The results from the TMDCA of Ref.", "Ekuma2015 and the MS-TMDCA of Fig.", "REF are also nearly indistinguishable.", "Here the TDOS is finite for the extended states (inside the mobility edge marked by arrows), and vanishes for the localized states at the top and bottom of the band and at the band center.", "By contrast, the ADOS remains finite even for localized states.", "With increasing $V_A$ it is possible to drive also the ADOS to zero at the band center due to the alloy disorder band splitting [72].", "We note that the above formalism applies to the single-band case.", "However, realistic systems often have many bands.", "Thus, in electronic structure calculations based on the multiple-scattering typical-medium dynamical cluster approximation formalism the above quantities are matrices in angular momentum space.", "This will require some modifications to the proposed MS-TMDCA scheme.", "In particular, for the ansatz Eq.", "(REF ) for the multi-orbital TMDCA formalism one may perform a geometric average for the diagonal terms $l=m$ and a linear average for the off-diagonal $l\\ne m$ components [63].", "The second ansatz, Eq.", "(REF ) requires less modification in that only a linear average of the Green function matrix occurs in the numerator, which appears to improve the numerical stability of the algorithm." ], [ "Conclusion", "In this paper we detail the construction of the typical medium approach in the framework of the multiple-scattering formalism.", "The constructed Multiple Scattering Typical Medium DCA (MS-TMDCA) formalism is a reformulation of the typical medium DCA [53] which has been successfully applied to model-Hamiltonian systems to quantitatively study and detect Anderson localization.", "Being motivated by the need for the development of appropriate numerical tools to study strong disorder effects in first principle calculations, we first provide a detailed comparison of two major effective medium algorithms used in the model Hamiltonian DCA [43] community and the real material multiple-scattering KKR-NLCPA community [45], [44], [67].", "To provide a bridge between the DCA and multiple-scattering approaches, we demonstrate explicitly that these two approaches when applied to a tight-binding Hamiltonian are equivalent.", "We pay particular attention to the non-trivial relations between the key quantities in the DCA and KKR-NLCPA methods.", "These relationship were used to construct various self-consistency procedures, including the MS-DCA which is a generalization of the DCA using the multiple-scattering language.", "Because the MS-DCA calculates the disorder-averaged Green's function rather than the scattering path operator used in the KKR-NLCPA, it can be readily generalized into a typical medium approach such as the MS-TMDCA (a multiple scattering algorithm).", "As an application of the MS-TMDCA formalism developed here we solve a tight-binding model on a cubic lattice with $N_c=38$ sites for various values of binary disorder strength.", "To check the validity of our mapping of the TMDCA to the multiple scattering approach, we compare numerical results for the density of states and the self-energy calculated within the DCA-based and the multiple scattering-based approaches (both for the algebraically averaged and geometrically averaged components).", "We find that the results from both approaches are numerically the same, affirming the successful mapping of the DCA-based formalism to a MS based formalism.", "The construction and application of the MS-TMDCA formalism to the tight-binding model presented in this paper is the first step towards an extension of the typical medium methodology into an ab-initio theory of disordered materials which is sensitive to Anderson localization.", "The material-specific implementation of this approach will be our next step.", "We thank J. Moreno for useful conversations.", "This work (H.T., Y.Z., and M.J.) was supported by the National Science Foundation under the NSF EPSCoR Cooperative Agreement No.", "EPS-1003897 with additional support from the Louisiana Board of Regents.", "LC thanks L. Vitos for valuable discussions.", "LC and DV acknowledge funding by the Deutsche Forschungsgesellschaft (DFG) through the Transregional Collaborative Research center TRR80." ] ]
1612.05611
[ [ "The Challenges of a Public Data Release: behind the scenes of SDSS DR13" ], [ "Abstract The Sloan Digitial Sky Surveys (SDSS) have been collecting imaging and spectoscopic data since 1998.", "These data as well as their derived data products are made publicly available through regular data releases, of which the 13th took place summer 2016.", "Although public data releases can be challenging to manage, they signficantly increase the impact of a survey, both scientifically and educationally." ], [ "Introduction", "The Sloan Digital Sky Survey (SDSS) started observing in 1998, with the goal to image the Northern hemisphere in five optical wavelength bands [11], using the dedicated 2.5m Sloan Foundation Telescope [5] at Apache Point Observatory (APO), New Mexico, US.", "Since then, SDSS has hosted several dedicated surveys, adding optical, infrared and integral-field spectroscopy to its orignal imaging survey [4].", "The current SDSS-IV (Blanton et al.", "in prep) consists of three spectroscopic surveys: i) APOGEE-2 (Apache Point Observatory Galaxy Evolution Experiment), obtaining infrared spectra of stars to unravel the chemical and dynamical formation history of the Milky Way (Majewski et al.", "in prep), ii) MaNGA (Mapping Nearby Galaxies at APO), an integral-field spectroscopic survey to study the formation, growth and evolution of galaxies [2], and iii) eBOSS (extended Baryon Oscillation Spectroscopic Survey), measuring redshifts of $\\sim $ 1.5 million galaxies and quasars from optical spectra to map the structure of the Universe and determine its expansion history [3].", "In 2017, SDSS will for the first time observe from the Southern hemisphere at Las Campanas Observatory (LCO), Chile, as part of APOGEE-2.", "The SDSS collaboration includes more than 50 member institutes spread over 4 continents, with close to 1000 scientists registered as members.", "SDSS has from its beginnings been dedicated to making its data publicly available through data releases.", "The first early data release was made public in 2002 [9], and in summer 2016 the Thirteenth SDSS data release (DR13) became publicly available [1].", "These public data releases have signficantly widened the impact of the SDSS beyond its collaborations: more than 7000 papers have been published based on SDSS data.", "The 2015 National Research Council report “Optimizing the U.S.", "Ground-Based Optical and Infrared Astronomy System” notes that within this US system the SDSS has a publication rate three times higher than any other telescope.", "In addition, SDSS data has been incorporated in various educational (e.g., Voyageshttp://voyages.sdss.org) and public engagement activities [7].", "In the field of data science, SDSS public data releases have been cited as the most influential data source ahead of any other source in any field of science [8].", "In this proceeding we discuss the mechanisms of an SDSS public data release, with a focus on the latest release - SDSS-IV's DR13." ], [ "Data and Data Products", "The main components of a data release are the data products.", "The raw SDSS data is transferred daily from APO to the Center for High Performance Computing (CHPC) at the University of Utah, where it is reduced by dedicated data reduction pipelines.", "Each survey within SDSS-IV has a survey data team that is responsible for this part of the process.", "Once the data is reduced and the relevant data products have been vetted by the survey data teams, they are copied to the Science Archive Server (SAS)http://data.sdss.org/sas where they remain as proprietary data for the SDSS-IV collaboration, until they are incorporated in a public data release (average time scale $\\sim $ 1.5 year).", "The SAS offers a file-based system that allows for low-level bulk data access and off line analysis.", "The SAS can also be accessed through a specialised web application, to visualise spectra.", "In addition, the Catalog Archive Server (CAS)http://skyserver.sdss.org, hosted at the Johns Hopkins University, offers Web browser-based graphical and SQL query interfaces to the catalog data that permit casual synchronous data retrieval [10] as well as batch-mode server-side analysis [6] with advanced capabilities like a server-side personal SQL database, data upload and data sharing via groups.", "There is also a command-line tool to submit CasJobs queries, and as of DR13, a jupyter notebook interface to retrieve catalog data from SkyServer and CasJobs within a docker container.", "Both the SAS and the CAS offer public data access, with the SAS hosting 267 TB of data products, as of DR13.", "All data releases are cumulative, in that they also contain data products from previous data releases.", "By accessing the latest SDSS data release, users will therefore always have access to all available SDSS public data." ], [ "Documentation", "Making data publicly available is a necessary but not sufficient condition for generating impact: in addition, the data has to be accessible.", "Users need to know where to find the data, and how to work with the data.", "Documentation is essential, and each SDSS data product therefore has a detailed data model.", "Each data release is also described in detail in an accompaning data release paper [1].", "However, a public data release serves many different audiences.", "Although for astronomers familiar with the SDSS data format the data model may be sufficient, novel users and students require more documentation.", "The SDSS websitehttp://www.sdss.org/ therefore offers a portal with more information on how to access the data (including links to the SAS, CAS and web application), as well as background information, links to technical papers, examples and tutorials.", "As teachers and instructors also use SDSS data in their classrooms, the website also offers links to education and public engagement activities.", "Finally, users can e-mail the SDSS helpdesk if they have any questions or encounter any problems when working with SDSS data." ], [ "Data Release Management", "The SDSS-IV data team is led by the Science Archive Scientist, the Catalog Archive Scientist and the Data Release Coordinator.", "The latter is repsonsible for managing the data release and keeping it on schedule.", "Planning for a public data release typically starts one year in advance.", "First the data products are generated by the survey data teams, and vetted for quality control by the survey science teams.", "Once the data products are finalised, catalogs are generated and loaded into the CAS.", "Most of the documenation is written during a one-week documentation workshop (”DocuFeest”), $\\sim $ four months before the data release.", "Several members of the data team, the survey teams and the education and public engagement team meet to generate most of the website content, as well as a comprehensive draft of the data release paper.", "Descriptions and tutorials for new data products have to be added into the documenation hierarchy: for DR13 this meant a redesign of the website to include the new MaNGA survey.", "The website and paper are finalised in the weeks after the DocuFeest.", "Meeting in person has proved crucial for efficient writing of documentation, as both the website and paper require input from all teams across SDSS-IV, and communication is key.", "As the date for the data release approaches, a plan is drafted for the announcement of the data release.", "SDSS-IV has a strong social media presence, with a general and survey-specific Twitter accounts, a Facebook page, and a blog aimed at the general public.", "For DR13, we issued a Facebook announcement and tweeted in almost all languages spoken within the collaboration (thanks to a team-wide effort to translate the tweets).", "A description of the content of DR13 was given in a multilingual blog post.", "Announcements were also issued over mailing lists aimed at professional astronomers." ], [ "Conclusion", "The impact of an astronomical survey is set by the reach of its data distribution system.", "If the data does not reach the astronomers for their research projects to make new discoveries, the teachers to teach their students how to work with astronomical data, and the general public to increase their awareness of astronomy and science, then impact will be limited to a small core survey team.", "A successful public data release, especially one aimed at a variety of end users, therefore needs to ensure that the data is not only freely available, but also clearly documented and in accessible formats.", "Managing a public data release is challenging: it involves keeping track of a myriad of different data products and their documenation, which includes data models, descriptions and tutorials, all aimed at different user audiences.", "Quality control of both the data products and the documentation is crucial.", "Good communication between different teams within the collaboration is key, given that the data release requires input from everyone.", "But despite the challenges and work involved, public data releases are worthwhile: they increase the impact of the survey, allow for more science output beyond the collaboration, and provide accessible data products not only for astronomical research, but also for education and public engagement.", "AW acknowledges support of a Leverhulme Trust Early Career Fellowship.", "The authors would like to thank the Center for Data Science at New York University for their hospitality during DocuFeest 2016.", "Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions.", "SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah.", "The SDSS web site is www.sdss.org.", "SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University." ] ]
1612.05668
[ [ "Discrete Multichannel Scattering with step-like potential" ], [ "Abstract We study direct and inverse scattering problem for systems of interacting particles, having web-like structure.", "Such systems consist of a finite number of semi-infinite chains attached to the central part formed by a finite number of particles.", "We assume that the semi-infinite channels are homogeneous at infinity, but the limit values of the coefficients may vary from one chain to another." ], [ "Introduction", "The aim of this article is to study the direct and inverse problems for small oscillations near equilibrium position for a system of particles $\\mathcal {A}=\\lbrace \\alpha , \\beta , \\ldots \\rbrace $ which interact with each other and perhaps with an external field.", "The interaction is described by the matrix $\\mathcal {L}= \\left( L(\\alpha , \\beta ) \\right)_{\\alpha ,\\beta \\in \\mathcal {A}}.$ We say that the particles $\\alpha $ and $\\beta $ interact with each other if $L(\\alpha , \\beta ) \\ne 0$ and we assume that each particle interacts with at most a finite number of its neighbours: $\\# \\lbrace \\beta , L(\\alpha , \\beta )\\ne 0 \\rbrace < \\infty $ for each $\\alpha \\in \\mathcal {A}$ .", "Our the system has a \"web-like\" structure: it includes a finite set of \"channels\", i.e.", "semi-infinite chains of particles attached to a \"central part\" formed by a finite number of interacting particles.", "Given a set of particles $\\mathcal {X}$ we denote by $\\mathcal {M}(\\mathcal {X})$ and $l^2(\\mathcal {X})$ the spaces of all functions on $\\mathcal {X}$ and square summable functions on $\\mathcal {X}$ respectively.", "The matrix $\\mathcal {L}$ is related to the Hessian matrix of the potential energy near equilibrium position, so we always assume that all $L(\\alpha ,\\beta )$ are real and $\\mathcal {L}> 0,$ here $\\mathcal {L}$ is considered as an operator in $l^2(\\mathcal {A})$ .", "In what follows we do not distinguish between matrices and the corresponding linear operators.", "After separation of variables one arrives to the spectral problem $\\mathcal {L}\\xi = \\lambda \\xi ; \\ \\xi =\\lbrace \\xi (\\alpha )\\rbrace \\in l^2(\\mathcal {A}).$ which can be considered as a discrete version of spectral problems for quantum graphs, see [3], [6], [8], [17] as well as later articles [9], [10], [11].", "The web-like structure of the system allows us to treat the spectral problem (REF ) as a scattering problem: the points of continuos spectra correspond to frequences of incoming and outcoming waves which are propagating along the channels; the points of discrete spectra correspond to proper oscillations of the system.", "The spectral data includes transmission and reflection coefficients: we consider waves incoming along one of the channels and observe (at infinite ends of the channels) how do such waves come through the system.", "The direct problem is to determine the spectral data through the characteristics of the system.", "The inverse problem deals with recovering of characteristics of the system from the spectral data.", "This is a classical setting of the scattering problem, see e.g.", "[16].", "This work is a continuation of [12] which considers the case of the same wave propagation speed along all channels.", "Now we assume that each channel has its own speed.", "The problem then becomes more complicated: different points of continuous spectra may have different multiplicities; we do not have a single scattering matrix for the whole spectra; besides the generalized eigenfunctions may exponentially decay along some of the channels.", "These eigenfunctions need to be treated specially, it does not make sense to observe the phase of a wave which decays exponentially at inifinity, just their amplitudes can be taken into account.", "Respectively, for such waves, only absolute values of the transmission coefficients can be included to spectral data.", "As in [12] the problem can be reduced to a system of difference Schrödinger equations on semi-infinite discrete string with the initial conditions related to the way the channels are attached to the central part.", "In our setting the potentials have different limits for different equations in this system.", "In the classical case of infinite string this corresponds to a step-like potential, such problem for continuous case has been studied for example in [2], [4], [7], for the Jacobi operators the case of step-like quasi-periodic potential has been considered in [5].", "We modify techniques of these articles, especially those in [4], in order to make it applicable for the graph setting.", "The scattering data are determined as the set of scattering coefficients, eigenvalues, and also as normalization constants of the corresponding eigenfunctions.", "These scattering data are associated with the data which can be measured in an experiment at infinite ends of channels.", "We solve the direct problem, i.e, description of scattering data from physical characteristic of the system, and (under additional assumptions) the inverse problem, i.e.", "reconstruction the characteristics of the channels from given spectral data, that is, we reconstruct the matrix $\\mathcal {L}$ on the channels.", "In order to make this reconstruction we reduce the problem to a discrete analog of Marchenko equation, which is known to have unique solution and actually admits numeric implementation.", "In this article we do not discuss possibility of reconstruction of data which is related to the central part of the graph.", "In order to have such reconstruction, one needs to demand additional sparsity conditions of the central part.", "We refer the reader to the recent book [15], which contains examples of such conditions.", "The article is organised as follows: in the next section we describe the system, derive the boundary condition which allows us to treat the problem as a system of equations on a string.", "In section we consider the characteristics of the channels, introduce the Jost solutions and also describe the spectral data related to the continuos spectra.", "In section we collect some known results, see e.g.", "[13], [14], [18], as well as some new ones about solutions of finite difference equations.", "These results will be used in the sequel.", "In particular in section we construct the special solutions, i.e.", "solutions which correspond to wave incoming along one of the channels.", "Using these solutions we study the structure of discrete spectra, this is done in section .", "As it was already mentioned, no scattering matrix can be defined for the whole spectra, however the scattering coefficients possess some symmetry which plays the crucial role in our construction.", "This symmetry is described in section .", "In section we return to discrete spectra and connect each eigenvalue of the problem with normalized matrix of eigenfucntions which gives the energies, completing the set of spectral data.", "In section we collect all previous results and finally obtain equations of the inverse scattering problem.", "Section contains some concluding remarks.", "Acknowledgments.", "We thank V. A. Marchenko, who suggested the problem, and also for numerous fruitful discussions." ], [ "Geometry of the system and the boundary condition", "We consider the systems $\\mathcal {A}$ which have a \"web-like\" structure.", "Namely $\\mathcal {A}=\\mathcal {A}_0\\cup \\mathcal {A}_1$ where $\\mathcal {A}_1$ is a central part, and $\\mathcal {A}_0$ is a union of a finite number of semi-infinite channels.", "For such web-like systems the inverse spectral problem can be treated as an inverse scattering problems: sending a wave along one of the channels and observing how does it pass through the system we are trying to reconstruct the characteristics of the system, i.e.", "the values $L(\\alpha ,\\beta )$ .", "Definition A sequence of particles $\\sigma =\\lbrace \\alpha (p)\\rbrace _{p=0}^\\infty $ is a channel if, for $p>0$ the particle $\\alpha (p)$ interacts with the particles $\\alpha (p-1)$ and $\\alpha (p+1)$ only (and, perhaps with the external field), while $\\alpha (0)$ interacts with $\\alpha (1)$ and some other particles in $\\mathcal {A}$ which do not belong to $\\sigma $ .", "We will use the following notation: - the set of all channels is $\\mathcal {C}$ , the channels will be denoted by $\\sigma $ , $\\nu $ , $\\gamma $ etc.", "; - the particles in $\\sigma \\in \\mathcal {C}$ are $\\sigma (0)$ , $\\sigma (1)$ , $\\sigma (2)$ , $\\ldots $ , the point $\\sigma (0)$ is called the attachment point of $\\sigma $ ; - $\\Gamma := \\lbrace \\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}}$ , $\\mathcal {A}_0:=\\cup _{\\sigma \\in \\mathcal {C}} \\cup _{k=1}^\\infty {\\sigma (k)}$ , $\\mathcal {A}_1:= \\mathcal {A}\\setminus \\mathcal {A}_0$ .", "- For $\\sigma \\in \\mathcal {C}$ , $k=1,2,\\ldots $ we also denote $-{\\mathfrak {b}}_\\sigma (k-1)=L(\\sigma (k-1), \\sigma (k)),\\qquad {\\mathfrak {a}}_\\sigma (k)=L(\\sigma (k), \\sigma (k)),$ so equation (REF ) on the channel $\\sigma $ takes the form: $-{\\mathfrak {b}}_\\sigma (k-1)\\xi (\\sigma (k-1)) +{\\mathfrak {a}}_\\sigma (k)\\xi (\\sigma (k))-{\\mathfrak {b}}_\\sigma (k)\\xi (\\sigma (k+1)) = \\lambda \\xi (\\sigma (k)).$ We assume that the number of particles in the central part as well as the number of channels is finite: $M:=\\sharp \\mathcal {A}_1<\\infty ,\\qquad \\sharp \\mathcal {C}<\\infty $ Also (for simplicity) we assume $\\sigma (0)\\ne \\nu (0)$ , $\\sigma ,\\nu \\in \\mathcal {C}$ , $\\sigma \\ne \\nu $" ], [ "Boundary conditions", "Let $\\xi \\in \\mathcal {M}(\\mathcal {A})$ be a solution to (REF ).", "Then it meets (REF ) and also, for each $\\alpha \\in \\mathcal {A}_1$ , $\\lambda \\xi (\\alpha ) - \\sum _{\\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta ) \\xi (\\beta ) = \\sum _{\\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta ) \\xi (\\beta )$ The only pairs $(\\alpha ,\\beta )\\in \\mathcal {A}_1\\times \\mathcal {A}_0$ for which $L(\\alpha ,\\beta )\\ne 0$ are of the form $(\\sigma (0),\\sigma (1))$ , $\\sigma \\in \\mathcal {C}$ , so with account of (REF ) this relation can be written as $\\lambda \\xi (\\alpha ) - \\sum _{\\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta ) \\xi (\\beta ) ={\\left\\lbrace \\begin{array}{ll}-{\\mathfrak {b}}_\\nu (0)\\xi (\\nu (1)), & \\alpha =\\nu (0)\\in \\Gamma , \\\\0, & \\alpha \\in \\mathcal {A}_1\\setminus \\Gamma \\end{array}\\right.", "}$ Consider the matrix $\\mathcal {L}_1= \\left( L(\\alpha ,\\beta )\\right)_{\\alpha ,\\beta \\in \\mathcal {A}_1}.$ Being a truncation of $\\mathcal {L}$ , the matrix $\\mathcal {L}_1$ also is strictly positive.", "Let $0<\\lambda _1\\le \\lambda _2 \\le \\ \\ldots \\le \\lambda _M$ and $p_1, \\ \\ldots , p_M$ be its eigenvalues and the corresponding normalised eigenvectors.", "We may choose $p_j$ 's to be real-valued.", "For $\\lambda \\notin \\lbrace \\lambda _j\\rbrace _{j=1}^M$ the operator $\\mathcal {L}_1-\\lambda I$ is invertible and $\\left( \\mathcal {L}_1-\\lambda I\\right)^{-1}= R(\\lambda )=\\left( r(\\alpha ,\\beta ;\\lambda ) \\right)_{\\alpha ,\\beta \\in \\mathcal {A}_1}, \\quad r(\\alpha ,\\beta ;\\lambda ) =\\sum _{j=1}^M \\frac{p_j(\\alpha )p_j(\\beta )}{\\lambda _j-\\lambda } .$ Relation (REF ) can be than written as $\\xi (\\alpha )= \\sum _{\\nu \\in \\mathcal {C}} r(\\alpha , \\nu (0); \\lambda ) {\\mathfrak {b}}_\\nu (0)\\xi (\\nu (1)); \\ \\alpha \\in \\mathcal {A}_1.$ In particular for $\\alpha =\\sigma (0)$ , $\\sigma \\in \\mathcal {C}$ we obtain $\\xi (\\sigma (0))= \\sum _{\\nu \\in \\mathcal {C}} r(\\sigma (0), \\nu (0); \\lambda ) {\\mathfrak {b}}_\\nu (0)\\xi (\\nu (1)).$ After introducing vector notations $\\vec{\\xi }(k):= (\\xi _\\sigma (k))_{\\sigma \\in \\mathcal {C}}, \\ k=0,1, \\ \\ldots , \\ \\mathcal {R}(\\lambda )=(r(\\sigma (0), \\nu (0); \\lambda ))_{\\sigma ,\\nu \\in \\mathcal {C}}$ we can rewrite (REF ) as $\\vec{\\xi }(0)=\\mathcal {R}(\\lambda ){\\mathbf {\\rm {diag}}}\\lbrace {\\mathfrak {b}}_\\nu (0)\\rbrace _{\\nu \\in \\mathcal {C}} \\ \\vec{\\xi }(1).$ This relation connects the values of solution $\\xi $ on $\\mathcal {A}_0$ and on $\\Gamma $ .", "It plays the role of the boundary condition for vector-valued scattering problem.", "The following statement holds.", "Theorem 2.1 ( See theorem 1 in [12]) Let $\\lambda \\notin \\lbrace \\lambda _j\\rbrace _{j=1}^M$ .", "A function $\\xi \\in \\mathcal {M}(\\mathcal {A}_0\\cup \\Gamma )$ admits prolongation to a function $\\xi \\in \\mathcal {M}(\\mathcal {A})$ satisfying (REF ) if and only if it satisfies (REF ) and also the boundary condition (REF ).", "The prolongation is unique and can be defined by (REF )." ], [ "Jost solutions ", "We assume that the channels are asymptotically homogeneous at infinity.", "Namely, for each $\\sigma \\in \\mathcal {C}$ there exist $b_\\sigma $ and $a_\\sigma $ such that ${\\mathfrak {a}}_\\sigma (k) \\rightarrow a_\\sigma , \\ {\\mathfrak {b}}_\\sigma (k) \\rightarrow b_\\sigma \\ \\text{as} \\ k\\rightarrow \\infty .$ Moreover $\\sum _{k=1}^\\infty k \\lbrace |{\\mathfrak {a}}_\\sigma (k)-a_\\sigma | + |{\\mathfrak {b}}_\\sigma (k)-b_\\sigma |\\rbrace < \\infty .$ This relation provides existence of Jost solutions on each channel $\\sigma \\in \\mathcal {C}$ .", "It is well known, see e.g.", "[15], [18], that for each $\\sigma \\in \\mathcal {C}$ there is a family $\\lbrace e_\\sigma (k, \\theta )\\rbrace _{k=0}^\\infty $ of functions holomorphic inside the open disk $, continuousup to the boundary and, for each $ and $k\\ge 1$ , $-{\\mathfrak {b}}_\\sigma (k-1)e_\\sigma (k-1,\\theta ) +{\\mathfrak {a}}_\\sigma (k)e_\\sigma (k,\\theta )-{\\mathfrak {b}}_\\sigma (k)e_\\sigma (k+1,\\theta ) = \\lambda _\\sigma (\\theta )e_\\sigma (k,\\theta ),$ where $\\lambda _\\sigma (\\theta )=a_\\sigma -b_\\sigma \\left( \\theta +\\theta ^{-1}\\right).$ In addition $e_\\sigma (k, \\theta )= \\theta ^k (1+o(1)) \\ \\text{as} \\ k \\rightarrow \\infty $ uniformly with respect to $\\theta \\in \\overline{}The functions $ e(k, )$ admit the representation\\begin{equation}e_\\sigma (k,\\theta )= c_\\sigma (k)\\sum _{m\\ge k} a_\\sigma (k,m) \\theta ^m, \\ k=0,1,\\ldots .\\end{equation}Here\\begin{equation*}c_\\sigma (k)=\\prod _{p=k}^\\infty \\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (p)}, \\ a_\\sigma (k,k)=1 \\ \\text{and} \\ \\lim _{k\\rightarrow \\infty } \\sum _{m\\ge k+1} |a_\\sigma (k,m)| =0\\end{equation*}For $ k1$ the coefficients $ a(k)$, $ b(k)$ can be expressed through the coefficients of the functions $ e(k, )$\\begin{gather*}\\frac{{\\mathfrak {a}}_\\sigma (k)-a_\\sigma }{b_\\sigma }= a_\\sigma (k-1,k)-a_\\sigma (k,k+1); \\\\\\frac{{\\mathfrak {b}}_\\sigma ^2(k)}{b_\\sigma ^2} = \\frac{{\\mathfrak {a}}_\\sigma (k)-a_\\sigma }{b_\\sigma } a_\\sigma (k,k+1)+a_\\sigma (k,k+2)-a_\\sigma (k-1,k+1)+ 1.\\end{gather*}These relations can be obtained by substituting representation (\\ref {eq:13}) in (\\ref {eq:10}) and then comparing the coefficientswith the same powers of $$.\\\\ \\\\In this article we assume that the coefficients $ a(k),b(k)$ approach their limit values faster than (\\ref {eq:09}).", "Namely we assume that, for some $ >0$,\\begin{equation*}\\sum _{k=1}^\\infty (1+\\epsilon )^k\\lbrace |{\\mathfrak {a}}_\\sigma (k)-a_\\sigma | + |{\\mathfrak {b}}_\\sigma (k)-b_\\sigma |\\rbrace < \\infty .\\end{equation*}Under this condition the functions $ e(k,)$ admit holomorphic prolongation to some vicinity of the unit disk $ D$, see e.g.", "chapter 10 in \\cite {Te}.", "This allows us to avoid extra technicalities which are necessary in the general case.\\subsection {Spectra, scattering data corresponding to the continuous spectra}$ As the parameter $\\theta $ runs through the unit circle $, the corresponding value $ ()$ defined by (\\ref {eq:11}) runs through the segment\\begin{equation*}I_\\sigma = [a_\\sigma -2b_\\sigma , a_\\sigma +2b_\\sigma ].\\end{equation*}We assume for the simplicity that $ C(a-2b, a+2b) $ is a connected set and choose $ a,bR$ so that\\begin{equation*}I=\\cup _{\\sigma \\in \\mathcal {C}} I_\\sigma = [a-2b,a+2b]\\end{equation*}We need the Zhukovskii mappings $ W, W: : $W : \\omega \\mapsto a-b(\\omega +\\omega ^{-1}), \\ W_\\sigma : \\theta \\mapsto a_\\sigma -b_\\sigma (\\theta +\\theta ^{-1}), \\\\\\theta _\\sigma (\\omega )= W_\\sigma ^{-1} \\circ W (\\omega ).$ Further we choose the set $J_\\sigma \\subset [-1,1]$ , so that $\\theta _\\sigma $ maps $ onto$ :=J$ and let\\begin{equation*}T_\\sigma ^+= \\theta _\\sigma ^{-1} (, \\ T_\\sigma ^-= \\theta _\\sigma ^{-1}(J_\\sigma )= T_\\sigma ^+.\\end{equation*}The set $ J$ consists of one or two segments attached to the points $ 1$.", "\\\\The functions $ e(k,())$ are holomorphic in $ .", "Moreover, for $\\omega \\in T_\\sigma ^+$ , the functions $e(k,\\theta _\\sigma (\\omega )^{-1})$ are well defined.", "By a special solution which correspond to $\\sigma \\in \\mathcal {C}$ we mean the function $\\psi ^\\sigma (\\alpha ,\\omega )$ , $\\omega \\in , $ A$with the following properties:\\\\ \\\\\\medskip 1.", "For each $ A0$ the function $ (, )$ is holomorphic in a vicinity of $$, except, perhaps a finite set $ O$ where it has poles.\\\\ \\\\\\medskip 2.", "For each $O$, the function $ (, )M(A)$ meets the equation\\begin{equation*}\\mathcal {L}\\psi ^\\sigma ( \\cdot ,\\omega ) = \\lambda \\psi ^\\sigma (\\cdot , \\omega ), \\ \\lambda =a+b(\\omega +\\omega ^{-1})\\end{equation*}\\medskip 3.", "For $$ there exist functions $ s()$, meromorhpic in a vicinity of $$ and a function $ s()$ continuous on $ T+$ such thatthe function $$ has the following representation along the channels:\\begin{equation}\\psi ^\\sigma (\\gamma (k), \\omega )={\\left\\lbrace \\begin{array}{ll}e_\\sigma (k, \\theta _\\sigma (\\omega )^{-1}) + s_{\\sigma \\sigma }(\\omega )e_\\sigma (k, \\theta _\\sigma (\\omega )), & \\gamma =\\sigma ; \\\\s_{\\gamma \\sigma }(\\omega )e_\\gamma (k, \\theta _\\gamma (\\omega )), & \\gamma \\ne \\sigma \\end{array}\\right.", "}, \\ \\omega \\in T^+_\\sigma ,\\end{equation}and for $ $\\psi ^\\sigma (\\gamma (k), \\omega )={\\left\\lbrace \\begin{array}{ll}\\theta _\\sigma (\\omega )^{-k} (1+o(1)), \\ \\mbox{as} \\ k\\rightarrow \\infty , & \\gamma =\\sigma \\\\s_{ \\gamma \\sigma }(\\omega )e_\\gamma (k,\\theta _\\gamma (\\omega )), & \\gamma \\ne \\sigma .\\end{array}\\right.", "}, \\ \\omega \\in \\overline{\\setminus T^+_\\sigma .", "}The {\\em scattering coefficients}, s_{\\sigma \\gamma }(\\omega ) are the elements of the scattering matrix, correspondingto the problem (\\ref {eq:01}).", "We mention that so far they are defined on different arcs T_\\sigma ^+\\subset .", "Insection \\ref {sec:Construction} we will prove existence and uniqueness of such special solutions for each \\sigma \\in \\mathcal {C}.\\\\The solution \\psi ^\\sigma corresponds to the wave incoming along the channel \\sigma and distributing through the whole system.This wave is well-defined if \\omega \\in T^+_\\sigma (respectively \\lambda \\in I_\\sigma ), for these values the equation (\\ref {eq:02}) along the channel \\sigma admits two independent bounded solutions corresponding to in- and out-coming waves.", "For\\omega \\in T^+_\\sigma \\cap T^+_\\gamma the wave incoming along \\sigma generates an outcoming wave along thechannel \\gamma , this wave can be observed at infinity.", "For \\omega \\in T^+_\\sigma \\cap T^-_\\gamma the waveincoming along \\sigma generates an exponentially decaying wave \\gamma , observation of phase of such wave atinfinity is virtually impossible, one can measure the absolute values only.", "These reasoning explain the definitionof the spectral data.$ Definition 3.1 By continuous spectral data corresponding to the channel $\\sigma \\in \\mathcal {C}$ we mean the set of functions $\\mathcal {S}_\\sigma = \\lbrace s_{\\gamma \\sigma }(\\omega ):\\text{ }\\omega \\in T_\\sigma ^+\\cap T_\\gamma ^+;\\text{ }\\gamma \\in \\mathcal {C}\\rbrace \\cup \\lbrace |s_{ \\gamma \\sigma }(\\omega )|:\\text{ }\\omega \\in T_\\sigma ^+\\cap T_\\gamma ^-; \\text{ } \\gamma \\in \\mathcal {C}\\rbrace $ By full continuous spectral data we mean $\\mathcal {S}= \\cup _{\\sigma \\in \\mathcal {C}} \\mathcal {S}_\\sigma .$ Later in sections and we will discuss discrete spectrum, which corresponds to eigenfunctions of the operator $\\mathcal {L}:l^2(\\mathcal {A})\\rightarrow l^2(\\mathcal {A})$ ." ], [ "Properties of solutions of finite difference equations", "We need to establish some properties of solutions to (REF ) as well as solutions to difference equations along the channels.", "In this section we collect known (see e.g.", "[13], [14], [18] ) properties of solutions of the finite-difference equation as well as some new statements related to solutions of (REF ) $-{\\mathfrak {b}}(k-1)x(k-1)+{\\mathfrak {a}}(k)x(k)-{\\mathfrak {b}}(k)x(k+1)=\\lambda x(k), \\quad k=1, 2, \\ldots \\,.$ with real coefficients ${\\mathfrak {a}}(k)$ , ${\\mathfrak {b}}(k)$ .", "We also assume that ${\\mathfrak {b}}(k)>0$ in order that the Jost solutions will be well defined, see e.g.", "[18].", "Given two functions $x=x(k)$ , $y=y(k)$ on the integers we define their Wronskian $\\lbrace x,y\\rbrace $ as $\\lbrace x,y\\rbrace (k)= x(k){y(k+1)}- x(k+1){y(k)}, \\quad k=0,1,\\ldots \\,.$ 1.", "Let $x$ , $y$ be solutions to (REF ).", "Then, for all $N$ , ${\\mathfrak {b}}(N)\\lbrace x,y\\rbrace (N)-{\\mathfrak {b}}(0)\\lbrace x,y\\rbrace (0)=0$ and ${\\mathfrak {b}}(N)\\lbrace x,\\overline{x}\\rbrace (N)-{\\mathfrak {b}}(0)\\lbrace x,\\overline{x}\\rbrace (0)=(\\lambda - \\bar{\\lambda })\\sum _{k=1}^N |x(k)|^2.$ If in addition $\\lambda =\\lambda (\\omega )$ and $x=x(k,\\omega )$ are differentiable functions of a parameter $\\omega $ , then ${\\mathfrak {b}}(N)\\lbrace \\dot{x},\\overline{x}\\rbrace (N)-{\\mathfrak {b}}(0)\\lbrace \\dot{x},\\overline{x}\\rbrace (0)=\\dot{\\lambda }\\sum _{k=1}^N|x(k)|^2 +(\\lambda -\\bar{\\lambda })\\sum _{k=1}^N\\dot{x}(k)\\overline{x(k)} \\, ,$ here and in what follows the dot denotes derivative with respect to $\\omega $ .", "If in addition $\\sum _{k=1}^\\infty k( |{\\mathfrak {b}}(k)-b|+|{\\mathfrak {a}}(k)-a|) < \\infty ,$ and $e(\\theta )=e(k,\\theta )$ are the corresponding Jost solutions, relation (REF ) yields ${\\mathfrak {b}}(0)\\lbrace e(\\theta ), \\overline{e}(\\theta ) \\rbrace (0) ={\\left\\lbrace \\begin{array}{ll}b(\\overline{\\theta }-\\theta ), & |\\theta |=1; \\\\b (\\overline{\\theta }-\\theta ) (|\\theta |^{-2}-1)\\sum _{k=1}^\\infty |e(k,\\theta )|^2, & |\\theta |<1.\\end{array}\\right.", "}$ Lemma 4.1 Let $\\omega \\in (-1,1)$ and $\\sigma \\in \\mathcal {C}$ be such that $\\theta _\\sigma (\\omega )\\in (-1,1)$ .", "Then $-{\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{e}_\\sigma ,(\\theta _\\sigma (\\omega )) e_\\sigma (\\theta _\\sigma (\\omega ))\\rbrace _\\sigma (0)= \\dot{\\lambda }\\sum _{k=1}^\\infty e_\\sigma (k,\\theta _\\sigma (\\omega ))^2.$ This is an immediate consequence of (REF ).", "2.", "We will consider Wronskians, which correspond to various channels $\\sigma \\in \\mathcal {C}$ .", "Given two functions $\\xi ,\\eta \\in \\mathcal {M}(\\mathcal {A})$ and $\\sigma \\in \\mathcal {C}$ we denote $\\lbrace \\xi ,\\eta \\rbrace _\\sigma (k):=\\lbrace \\xi (\\sigma (k))\\eta (\\sigma (k+1))-\\xi (\\sigma (k+1))\\eta (\\sigma (k))\\rbrace $ Remark If both $\\xi $ and $\\eta $ are solutions of (REF ) it follows from (REF ) that the quantity ${\\mathfrak {b}}_\\sigma (k)\\lbrace \\xi ,\\eta \\rbrace _\\sigma (k)$ depends on $\\sigma $ only.", "Lemma 4.2 Let $\\xi ,\\eta \\in \\mathcal {M}(\\mathcal {A})$ be solutions to (REF ).", "Then $\\sum _{\\sigma \\in \\mathcal {C}} {\\mathfrak {b}}_\\sigma (0)\\lbrace \\xi ,\\eta \\rbrace _\\sigma (0) = 0$ For $\\alpha \\in \\mathcal {A}_1$ we have $\\sum _{\\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta )\\xi (\\beta ) \\ + \\ \\sum _{\\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta )\\xi (\\beta ) = \\lambda \\xi (\\alpha ), \\\\\\sum _{\\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta )\\eta (\\beta ) \\ + \\ \\sum _{\\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta )\\eta (\\beta ) = \\lambda \\eta (\\alpha ),$ We multiply these relations by $\\eta (\\alpha )$ and $\\xi (\\alpha )$ respectively.", "Summation with respect to $\\alpha \\in \\mathcal {A}_1$ gives $\\sum _{\\alpha , \\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta )\\xi (\\beta )\\eta (\\alpha ) \\ + \\ \\sum _{\\alpha \\in \\mathcal {A}_1, \\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta )\\xi (\\beta )\\eta (\\alpha )= \\lambda \\sum _{\\alpha \\in \\mathcal {A}_1}\\xi (\\alpha )\\eta (\\alpha ), \\\\\\sum _{\\alpha , \\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta )\\eta (\\beta )\\xi (\\alpha ) \\ + \\ \\sum _{\\alpha \\in \\mathcal {A}_1,\\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta )\\xi (\\alpha )\\eta (\\beta )= \\lambda \\sum _{\\alpha \\in \\mathcal {A}_1}\\xi (\\alpha )\\eta (\\alpha ).$ The right-hand sides in these relations coincide as well as the first terms in the left-hand sides (since $L(\\alpha ,\\beta )=L(\\beta ,\\alpha )$ ).", "Therefore $\\sum _{\\alpha \\in \\mathcal {A}_1, \\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta ) \\left( \\xi (\\beta )\\eta (\\alpha ) - \\xi (\\alpha )\\eta (\\beta ) \\right)=0.$ This proves the lemma since the only option for $L(\\alpha ,\\beta )\\ne 0$ for $\\alpha \\in \\mathcal {A}_1$ , $\\beta \\in \\mathcal {A}_0$ is $\\alpha =\\sigma (0)$ , $\\beta =\\sigma (1)$ for some $\\sigma \\in \\mathcal {C}$ and in this case $L(\\sigma (1),\\sigma (0))=-{\\mathfrak {b}}_{\\sigma } (0)$ .", "3.", "We need a special statement in order to calculate the energy of eigenfunctions of the operator $\\mathcal {L}$ .", "Lemma 4.3 Let a function $\\xi (\\alpha )=\\xi (\\alpha ,\\omega )\\in \\mathcal {M}(\\mathcal {A})$ be differentiable with respect to $\\omega $ in a neighborhood of $\\hat{\\omega }\\in \\overline{, \\Im \\lambda (\\hat{\\omega })=0 and \\xi (\\alpha ) satisfy the equation\\begin{equation}\\lambda (\\omega )\\xi (\\alpha ,\\omega )= \\sum _{\\beta \\in \\mathcal {A}}L(\\alpha ,\\beta )\\xi (\\beta ,\\omega ).\\end{equation}Let also a function \\eta (\\alpha ) meet this equation for \\omega =\\hat{\\omega }.Then\\begin{equation*}\\dot{\\lambda }(\\hat{\\omega })\\sum _{\\alpha \\in \\mathcal {A}_1}\\xi (\\alpha ,\\hat{\\omega })\\overline{\\eta (\\alpha )}=\\sum _{\\sigma \\in \\mathcal {C}}{\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\xi }(\\hat{\\omega }), \\overline{\\eta }\\rbrace _\\sigma (0).\\end{equation*}In particular\\begin{equation*}\\dot{\\lambda }(\\hat{\\omega })\\sum _{\\alpha \\in \\mathcal {A}_1}|\\xi (\\alpha ,(\\hat{\\omega }))|^2=\\sum _{\\sigma \\in \\mathcal {C}}{\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\xi }(\\hat{\\omega }), \\overline{\\xi (\\hat{\\omega })}\\rbrace _\\sigma (0).\\end{equation*}}$ The statement uses the same idea as in lemma 2.1 in [12], we repeat here the construction: Differentiate () with respect to $\\omega $ : $\\dot{\\lambda }(\\omega )\\xi (\\alpha ,\\omega )+ {\\lambda }(\\omega )\\dot{\\xi }(\\alpha ,\\omega )= \\sum _{\\beta \\in \\mathcal {A}}L(\\alpha ,\\beta )\\dot{\\xi }(\\beta ,\\omega ).$ Besides $\\lambda (\\hat{\\omega })\\overline{\\eta (\\alpha )}= \\overline{ \\lambda (\\hat{\\omega })\\eta (\\alpha )}=\\sum _{\\beta \\in \\mathcal {A}}L(\\alpha ,\\beta ) \\overline{\\eta (\\beta )}.$ Combining these equations we obtain $\\dot{\\lambda }(\\hat{\\omega }) {\\xi }(\\alpha ,\\omega )\\overline{\\eta (\\alpha )} = \\sum _{\\beta \\in \\mathcal {A}}L(\\alpha ,\\beta ) [ \\dot{\\xi }(\\beta , \\hat{\\omega }) \\overline{\\eta (\\alpha ) } - \\dot{\\xi }(\\alpha , \\hat{\\omega }) \\overline{\\eta (\\beta )}],$ and $\\dot{\\lambda }(\\hat{\\omega })\\sum _{\\alpha \\in \\mathcal {A}_1} {\\xi }(\\alpha ,\\omega )\\overline{\\eta (\\alpha )}= \\sum _{\\alpha \\in \\mathcal {A}_1}\\sum _{\\beta \\in \\mathcal {A}_1}L(\\alpha ,\\beta ) [ \\dot{\\xi }(\\beta , \\hat{\\omega }) \\overline{\\eta (\\alpha ) } - \\dot{\\xi }(\\alpha , \\hat{\\omega }) \\overline{\\eta (\\beta )}] + \\\\\\sum _{\\alpha \\in \\mathcal {A}_1}\\sum _{\\beta \\in \\mathcal {A}_0}L(\\alpha ,\\beta ) [ \\dot{\\xi }(\\beta , \\hat{\\omega }) \\overline{\\eta (\\alpha ) } - \\dot{\\xi }(\\alpha , \\hat{\\omega }) \\overline{\\eta (\\beta )}].$ The first summand in the right-hand side of this relation vanishes because it is anti-symmetric in $\\alpha $ and $\\beta $ .", "In the second summand the only non-zero coefficient $L(\\alpha ,\\beta )$ appears in the case $\\alpha =\\sigma (0)$ , $\\beta =\\sigma (1)$ for some $\\sigma \\in \\mathcal {C}$ and $L(\\sigma (0),\\sigma (1))=-{\\mathfrak {b}}_\\sigma (0)$ ." ], [ "Construction of the special solutions", "Consider the diagonal matrices $B:={\\mathbf {\\rm {diag}}}\\lbrace b_\\sigma \\rbrace _{\\sigma \\in \\mathcal {C}}, \\ B(0):={\\mathbf {\\rm {diag}}}\\lbrace {\\mathfrak {b}}_\\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}},$ and also the matrix-functions in $\\bar{\\begin{equation*}\\mathcal {E}(k,\\omega ):={\\mathbf {\\rm {diag}}}\\lbrace e(k,\\theta _\\sigma (\\omega ))\\rbrace _{\\sigma \\in \\mathcal {C}}, \\ P(k,\\omega ):= {\\mathbf {\\rm {diag}}}\\lbrace p_\\sigma (k,\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}},\\end{equation*}here \\lbrace p_\\sigma (k,\\omega )\\rbrace _{k=0}^\\infty satisfy the equation on the channels:\\begin{equation*}-{\\mathfrak {b}}_\\sigma (k-1)p_\\sigma (k-1,\\omega )+{\\mathfrak {a}}_\\sigma (k)p_\\sigma (k,\\omega )-{\\mathfrak {b}}_\\sigma (k)p_\\sigma (k+1,\\omega )=\\lambda (\\omega )p_\\sigma (k,\\omega )\\end{equation*}for k=1,2,\\dots , \\sigma \\in \\mathcal {C} and with boundary conditions\\begin{equation}p_\\sigma (0,\\omega )=1, \\ p_\\sigma (1,\\omega )=0,\\quad \\sigma \\in \\mathcal {C}\\end{equation}The functions \\mathcal {E}(k,\\omega ) and P(k,\\omega ) are holomorphic in a neighborhood of \\overline{, except, perhaps zero, where P(k,\\omega ) may have poles.", "\\\\Moreover, it is well known, see \\cite {MS,Te}, that the Jost solutions form a fundamental system of solutions of the finite-difference equation (\\ref {eq:30}).", "If |\\theta |=1 and |\\theta |\\ne 1, it follows from (\\ref {eq:36}) that e(\\theta ), \\bar{e}(\\theta )=e(\\theta ^{-1}) are independent solutions of (\\ref {eq:30}) and any other solution \\lbrace x(k,\\theta )\\rbrace _{k\\ge 0} can be expressed as x(k,\\theta )=m(\\theta )e(k,\\theta )+n(\\theta )e(k,\\theta ^{-1}), with m(\\theta ),n(\\theta ) independent of k. Thus, for \\omega \\in T_\\sigma ^+ the function p_\\sigma (k,\\omega ) may be expressed in terms of the corresponding Jost solutions:\\begin{equation}p_\\sigma (k,\\omega )= \\frac{{\\mathfrak {b}}_\\sigma (0)}{b_\\sigma }\\frac{e_\\sigma (k,\\theta _\\sigma (\\omega ))e_\\sigma (1,\\theta _\\sigma (\\omega )^{-1}) - e_\\sigma (k,\\theta _\\sigma (\\omega )^{-1})e_\\sigma (1,\\theta _\\sigma (\\omega ))}{\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )}.\\end{equation}\\\\ \\\\In order to construct the special solutions we need auxiliary operator-function T(\\omega ): l^2(\\mathcal {C}) \\rightarrow l^2(\\mathcal {C}) defined as\\begin{equation}T(\\omega )= \\mathcal {E}(0,\\omega ) - \\mathcal {R}(\\lambda (\\omega )) B(0) \\mathcal {E}(1, \\omega ), \\ \\omega \\in \\end{equation}This function is holomorphic in a vicinity of \\overline{\\mathbb {D}}, except the set \\mathcal {O} of poles of the function \\mathcal {R}(\\lambda (\\omega )).\\medskip }Denote\\begin{equation*}\\Delta _\\sigma (\\omega ) ={\\left\\lbrace \\begin{array}{ll}b_\\sigma , & |\\theta _\\sigma (\\omega )|=1; \\\\b_\\sigma (|\\theta _\\sigma (\\omega )|^{-2}-1)\\sum _{k=1}^\\infty |e_\\sigma (k,\\theta _\\sigma (\\omega ))|^2, & |\\theta _\\sigma (\\omega )|<1,\\end{array}\\right.", "}\\end{equation*}and\\begin{equation}\\Delta (\\omega )={\\mathbf {\\rm {diag}}}\\lbrace \\Delta _\\sigma (\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}}, \\quad \\Phi (\\omega )={\\mathbf {\\rm {diag}}}\\lbrace \\bar{\\theta }_\\sigma (\\omega )-\\theta _\\sigma (\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}}.\\end{equation}The operator \\Delta (\\omega ) is positive uniformly with respect \\omega \\in \\overline{ {\\mathbb {D}}}, i.e.,for some C>0,\\begin{equation*}\\langle \\Delta (\\omega ){\\mathbf {x}, \\mathbf {x}} \\rangle \\ge C \\Vert {\\mathbf {x} }\\Vert ^2,\\quad {\\mathbf {x}}\\in l^2(\\mathcal {C}), \\, \\omega \\in \\overline{{\\mathbb {D}}}.\\end{equation*}Besides relation (\\ref {eq:36}) now reads\\begin{equation}B(0)\\left\\lbrace \\mathcal {E}(0,\\omega )\\mathcal {E}(1,\\omega )^*-\\mathcal {E}(0,\\omega )^*\\mathcal {E}(1,\\omega ) \\right\\rbrace =\\Phi (\\omega )\\Delta (\\omega ),\\end{equation}}\\begin{Lemma}(See lemma 3.1 in \\cite {LM})The following inequality holds for all \\theta \\in {\\mathbb {D}}\\setminus \\mathcal {O}, {\\mathbf {x}}=(x_\\sigma )_{\\sigma \\in \\mathcal {C}}\\in l^2(\\mathcal {C})\\begin{equation}|\\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}}\\rangle |\\ge |\\Im \\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}}\\rangle | \\ge C\\sum _{\\sigma \\in \\mathcal {C}} |\\bar{\\theta }_\\sigma (\\omega )-\\theta _\\sigma (\\omega )||x_\\sigma |^2.\\end{equation}\\end{Lemma}{\\begin{xmlelement*}{proof}We have\\begin{equation*}2\\Im \\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}} \\rangle =\\langle [ \\mathcal {E}(1,\\omega )^*B(0)T(\\omega )- T(\\omega )^*B(0)\\mathcal {E}(1,\\omega )]{\\mathbf {x}}, {\\mathbf {x}} \\rangle \\end{equation*}and, by (\\ref {eq:54}),\\begin{multline*}\\mathcal {E}(1,\\omega )^*B(0)T(\\omega )- T(\\omega )^*B(0)\\mathcal {E}(1,\\omega )=B(0)(\\mathcal {E}(0,\\omega )\\mathcal {E}(1, \\omega )^*-\\mathcal {E}(0, \\omega )^*\\mathcal {E}(1,\\omega ))+ \\\\(B(0)\\mathcal {E}(1,\\omega ))^*(\\mathcal {R}(\\lambda (\\omega ))^*-\\mathcal {R}(\\lambda (\\omega )))(B(0)\\mathcal {E}(1,\\omega ))\\end{multline*}The lemma now follows from (\\ref {eq:56}) - (\\ref {eq:58}) and from the fact that\\begin{equation*}\\mathcal {R}(\\lambda (\\omega ))^*-\\mathcal {R}(\\lambda (\\omega ))=b (\\bar{\\omega }- \\omega )\\underbrace{(|\\omega |^{-2}-1) \\left(\\sum _{l=1}^M \\frac{p_l(\\sigma (0))p_l(\\nu (0))}{|\\lambda _l - \\lambda (\\omega )|^2}\\right)_{\\sigma , \\nu \\in \\mathcal {C}}}_{\\Delta _1(\\omega )},\\end{equation*}here \\Delta _1(\\theta ) is a non-negative operator.\\end{xmlelement*}}$ Corollary 5.1 Operators $T(\\omega )$ are invertible for all non-real $\\omega $ in the open disk ${{\\mathbb {D}}}\\setminus \\lbrace \\mathcal {O}\\cup \\lbrace \\pm 1\\rbrace \\rbrace $ .", "Indeed, fix an $\\omega \\in [-1,1]$ and denote $\\delta (\\omega )=\\inf _\\sigma \\lbrace |\\Im \\theta _\\sigma (\\omega )| \\rbrace >0$ .", "Relation () now reads $\\Vert T(\\omega )\\mathbf {x}\\Vert >C\\delta (\\omega )\\Vert \\mathbf {x}\\Vert ,$ which yields invertibility of $T(\\omega )$ .", "Since $T(\\omega )$ is an analytic function, we can now claim that $T(\\omega )$ is invertible in a vicinity of $\\overline{\\mathbb {D}}$ , except perhaps at a finite set of points.", "Consider the analytic matrix functions : $D(\\omega ):= {\\mathbf {\\rm {diag}}}\\lbrace \\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )\\rbrace $ and $U(k,\\omega )=\\left[ -P(k,\\omega )+\\mathcal {E}(k,\\omega ) T(\\omega )^{-1} \\right]B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}.$ Lemma 5.1 Consider the vector function on $\\mathcal {A}_0\\cup \\Gamma $ $\\psi ^\\sigma (k,\\omega )= (\\psi ^\\sigma (\\gamma (k),\\omega ))_{\\gamma \\in \\mathcal {C}}:= U(k,\\omega ) {\\mathbf {n}}_\\sigma .$ This function satisfies the boundary condition (REF ) and, for $\\omega \\in T^+_\\sigma $ , admits representation () along the channels.", "Thus by (REF ) it may be prolongated to a special solution of (REF ).", "Representation () for $\\omega \\in T^+_\\sigma $ is just a consequence of (REF ) and ().", "In order to prove that $\\psi ^\\sigma (k,\\omega )$ meets the boundary condition for $\\omega \\in T^+_\\sigma $ , we prove that this condition is met by the whole matrix-function $U(k,\\omega )$ : $U(0,\\omega )=\\mathcal {R}(\\lambda (\\omega ))B(0)U(1,\\omega ),\\quad \\ \\omega \\in $ This implies that $\\psi ^\\sigma (k,\\omega )$ also meets the boundary condition.", "Relation (REF ) is straightforward: after factoring out the inessential factor $B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}$ in the definition (REF ) it becomes $-I+\\mathcal {E}(0,\\omega )T(\\omega )^{-1}=\\mathcal {R}(\\lambda (\\omega ))B(0)\\mathcal {E}(1,\\omega )T(\\omega )^{-1},$ which is just the definition of $T(\\omega )$ .", "So far the special solution is not defined at a point $\\omega _0\\in T^+_\\sigma $ in case $\\omega _0$ is a pole of $U(k,\\omega )$ .", "This case will be considered in the following sections: in section we prove that all poles of $U(k,\\omega )$ are simple, later in section we show that.", "if $\\omega _0\\in T_\\sigma ^+$ , the function $U(k,\\omega ){\\mathbf {n}}_\\sigma $ is continuos at $\\omega _0$ even if $\\omega _0$ is a pole of $U(k,\\omega )$ .", "In particular $\\text{Res}_{\\omega _0}u_{\\nu \\sigma }(\\omega )=0,\\quad \\nu \\in \\mathcal {C}$" ], [ "Singularities of $U(k,\\omega )$ .", "The matrix function $U(k,\\omega )$ is analytic in a vicinity of $\\bar{, in particular it has a finite number of poles which belong to \\overline{.\\begin{Lemma}There is a finite set \\Omega \\subset \\overline{ such that all poles of the matrix function U(k,\\omega ) \\in \\overline{ belong to\\Omega \\cup \\lbrace 0\\rbrace .", "In the origin U(k,\\omega ) has pole of order k, all poles in \\Omega are simple.", "}}{\\begin{xmlelement*}{proof}The proof follows the pattern of Lemma 4.1 in \\cite {LM}.We rewrite (\\ref {eq:64}) as\\begin{multline}U(k,\\omega )= -P(k,\\omega )B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} + \\\\\\mathcal {E}(k,\\omega ) T(\\omega )^{-1} B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}.\\end{multline}Therefore all poles of U(k,\\omega ) in \\overline{ belong to the finite set \\Omega \\cup \\lbrace 0\\rbrace where \\Omega includes all poles of T(\\omega )^{-1} and \\mathcal {E}(1,\\omega )^{-1} in \\overline{ .\\\\That at the origin the functions U(0,\\omega ) and U(1,\\omega ) have poles of order 0 and 1 respectively followsfrom the initial conditions for P.For k\\ge 2 the main contribution to singularity of U(k,\\omega ) at zero comes from the first term in theright-hand side of (\\ref {Eq:23a01}) because P(k,\\omega ) is a polynomial of degree k-2 with respect to\\lambda =a-b(\\omega +\\omega ^{-1}).\\\\It now suffices to study the singularities of the second term in the right-hand side in (\\ref {Eq:23a01}).", "Let \\Omega be the set of all such singularities in \\overline{\\mathbb {D}}.These singularities comes from the poles of T(\\omega )^{-1}, that is, when det(T(\\omega ))=0 and also from the zeros of det(\\mathcal {E}(1,\\omega ))=\\prod _{\\sigma \\in \\mathcal {C}}e_\\sigma (1,\\theta _\\sigma (\\omega )).\\\\Actually U(k,\\omega ) cannot have poles outside (-1,1).", "This is due to the poles of T(\\omega )^{-1} are located in (-1,1) and e_\\sigma (1,\\theta _\\sigma (\\omega ))=0 only if \\theta _\\sigma (\\omega )\\in (-1,1), see (\\ref {eq:36}), and thus \\omega \\in (-1,1).\\\\Let \\hat{\\omega }\\in (-1,0)\\cup (0,1).We are going to use (\\ref {eq:59}) as \\omega approaches \\hat{\\omega }.", "We have|\\theta _\\sigma (\\omega )-\\theta _\\sigma (\\bar{\\omega })|\\asymp |\\omega -\\bar{\\omega }|, so for any \\mathbf {y}\\in l^2(\\mathcal {C}) relation (\\ref {eq:59}) with\\mathbf {x}= T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y}gives\\begin{multline*}| \\langle \\mathcal {E}(1,\\omega )^*\\mathcal {E}(1,\\omega )^{-1}BD(\\omega )\\mathbf {y}, T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\rangle |\\ge \\\\C |\\omega -\\bar{\\omega }| \\Vert T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\Vert ^2.\\end{multline*}Since \\mathcal {E}(1,\\omega )^*\\mathcal {E}(1,\\omega )^{-1} is unitary and also | \\mbox{det}D(\\omega )| stays bounded from below near\\hat{\\omega }, the Schwartz inequality gives\\Vert T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\Vert \\le \\mbox{Const} \\frac{\\Vert \\mathbf {y}\\Vert }{|\\omega -\\bar{\\omega }|} \\ \\mbox{as} \\ \\omega \\rightarrow \\hat{\\omega },this is possible only in case \\hat{\\omega } is a simple pole of T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} .\\\\In case when \\hat{\\omega }\\in , the reasoning goes in a similar way, it suffices to let \\omega approach \\hat{\\omega } in a way that |\\theta _\\sigma (\\omega )-\\hat{\\theta }_\\sigma (\\omega )| \\asymp |\\omega -\\hat{\\omega }|.", "}}\\section {Relation for the scattering coefficients}\\end{xmlelement*}The special solution \\psi ^\\sigma (\\alpha ,\\omega ) is now well-defined for all \\omega \\in \\Omega .", "Thusthe non-diagonal scattering coefficients s_{\\sigma \\gamma }(\\omega )\\sigma \\ne \\gamma are also well-defied for all \\omega \\in \\Omega .", "Together with \\psi ^\\sigma (\\omega ) they are analytic in a vicinity of \\overline{.\\\\ \\\\The scattering coefficients s_{\\sigma ,\\sigma }(\\omega ) are so far well-defined for \\omega \\in T_\\sigma ^+ only.For \\omega \\in T_\\sigma ^- the corresponding value \\theta _\\sigma (\\omega ) belongs to the unit disk.", "Onecan construct (similarly to how this is done in Theorem 1.4.1 in \\cite {AM} for the continuous case)a real-valuedsolution e_\\sigma ^{(1)}(k,\\theta _\\sigma ) of(\\ref {eq:10}) such that\\begin{equation}e_\\sigma ^{(1)}(k,\\theta _\\sigma )=\\theta _\\sigma ^{-k} (1+o(1)), \\ k\\rightarrow \\infty .\\end{equation}This choice is not unique since by adding any multiple of e_\\sigma we obtain a solution which still meets this relation.", "However we fix some choice of functionse_\\sigma ^{(1)}(k,\\theta _\\sigma ).", "The functions p_\\sigma can be represented as\\begin{equation}p_\\sigma (k,\\omega )= \\frac{{\\mathfrak {b}}_\\sigma (0)}{b_\\sigma }\\frac{e_\\sigma (k,\\theta _\\sigma (\\omega ))e_\\sigma ^{(1)}(1,\\theta _\\sigma (\\omega )) - e_\\sigma ^{(1)}(k,\\theta _\\sigma (\\omega ))e_\\sigma (1,\\theta _\\sigma (\\omega ))}{\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )},\\end{equation}which yields\\begin{equation*}\\psi ^{\\sigma }(\\sigma (k),\\omega )=\\theta _\\sigma (\\omega )^{-k}(1+o(1)), \\ k\\rightarrow \\infty \\end{equation*}as it should be for a special solution.", "}}\\begin{Lemma}For each \\sigma ,\\gamma \\in \\mathcal {C}, \\sigma \\ne \\gamma we have\\begin{equation}b_\\gamma (\\theta _\\gamma (\\omega )^{-1}-\\theta _\\gamma (\\omega ))s_{\\gamma \\sigma }(\\omega )= b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )) s_{\\sigma \\gamma }(\\omega ), \\ \\omega \\in \\overline{\\setminus \\Omega }In addition the scattering coefficients s_{\\gamma ,\\sigma }(\\omega ) and s_{\\sigma ,\\gamma }(\\omega ) are continuous up to \\Omega \\end{equation}{\\begin{xmlelement*}{proof}Relation (\\ref {eq:67}) follows from (\\ref {eq:38}) for \\xi =\\psi ^\\sigma (\\omega ), \\eta =\\psi ^\\gamma (\\omega )if one takes into account that according (\\ref {eq:23}) and (\\ref {eq:23a}) we have\\begin{equation*}{\\mathfrak {b}}_\\nu (0)\\lbrace \\psi ^\\sigma ,\\psi ^\\gamma \\rbrace _{\\nu }(0)= {\\left\\lbrace \\begin{array}{ll}-b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega ))s_{\\sigma \\gamma }(\\omega ), & \\nu =\\sigma ; \\\\b_\\gamma (\\theta _\\gamma (\\omega )^{-1}-\\theta _\\gamma (\\omega ))s_{\\gamma \\sigma }(\\omega ), & \\nu =\\gamma ; \\\\0, & \\nu \\ne \\sigma ,\\gamma .\\end{array}\\right.", "}\\end{equation*}Continuity of the scattering coefficients s_{\\gamma \\sigma }(\\omega ) follows from the definition of U(k,\\omega ) and the fact that it has finitely many poles contained in \\Omega .\\end{xmlelement*}}\\end{Lemma}\\end{Lemma}\\begin{Cor}Let \\hat{\\omega }\\in \\overline{\\mathbb {D}}, then\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu }(k,\\omega )={\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\nu \\sigma }(k,\\omega )=0,\\quad \\hat{\\omega }\\in T^{+}_\\sigma \\cup T_\\nu ^+,\\quad \\nu \\in \\mathcal {C}\\end{equation*}\\end{Cor}{\\begin{xmlelement*}{proof}Denote\\begin{equation*}A_{\\nu \\sigma }=\\text{det}(\\tilde{T}_{\\nu \\sigma }(\\omega )),\\quad \\omega \\in \\overline{\\mathbb {D}},\\quad \\nu ,\\sigma \\in \\mathcal {C},\\end{equation*}where \\tilde{T}_{\\nu \\sigma }(\\omega ) denotes the matrix which comes from T(\\omega ) once we have removed the \\sigma -column and the \\nu -row.", "\\\\Let now \\omega _0\\in , \\lambda (\\omega _0), be a regular point of \\mathcal {R}(\\lambda (\\omega )) and \\det (T(\\omega _0))=0.", "Then there exists \\mathbf {x}=\\lbrace x_\\sigma \\rbrace \\in l^2(\\mathcal {C}) such that T(\\omega _0)\\mathbf {x}=0, thus the vector function \\vec{\\xi }(k)= \\mathcal {E}(k,\\omega _0)\\mathbf {x} satisfies the equations (\\ref {eq:02}) on the channels as well as the boundary conditions (\\ref {eq:08b}).", "Hence it can be prolongated to a solution of the whole problem (\\ref {eq:01}).", "\\\\For \\mathbf {x}\\in ł^2(\\mathcal {C}), denote \\mbox{supp} \\ \\mathbf {x}=\\lbrace \\sigma \\in \\mathcal {C};\\text{ } x_\\sigma \\ne 0 \\rbrace and let \\xi =\\lbrace \\xi (\\alpha )\\rbrace _{\\alpha \\in \\mathcal {A}} be a solution to (\\ref {eq:01}) with \\lambda =\\lambda (\\omega _0) which is obtained by prolongation of\\vec{\\xi }(k).", "Then \\eta (\\alpha )= \\bar{\\xi }(\\alpha ) also solves this problem.", "Relation (\\ref {eq:38}) yields0= \\sum _{\\sigma \\in \\mbox{supp} \\ \\mathbf {x}} {\\mathfrak {b}}_{\\sigma }(0)\\lbrace \\xi ,\\eta \\rbrace _{\\sigma }(0)=\\sum _{\\sigma \\in \\mbox{supp} \\ \\mathbf {x}} b_\\sigma |x_\\sigma |^2 (\\bar{\\theta }_\\sigma (\\omega _0)- {\\theta }_\\sigma (\\omega _0))Therefore \\omega _0\\in T_\\sigma ^- for each \\sigma \\in \\mbox{supp} \\ \\mathbf {x}.", "Thus A_{\\sigma \\nu }(\\omega _0)=0, for \\omega _0\\in T_\\nu ^+, \\sigma \\in \\mathcal {C}.\\\\Since the poles of U(k,\\omega ) are simple and (\\ref {eq:36}) implies that e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0 if \\omega \\in T_\\nu ^+, we obtain{\\mathbf {\\rm {Res}}}_{\\omega _0} u_{\\nu \\sigma }(k,\\omega )=0A simple application of lemma \\ref {L:6} gives us {\\mathbf {\\rm {Res}}}_{\\omega _0} u_{\\sigma \\nu }(k,\\omega )=0 \\end{xmlelement*}}}}$ Corollary 6.1 Let $\\omega \\in T_\\sigma ^+\\cup T_\\nu ^+$ , $\\sigma ,\\nu \\in \\mathcal {C}$ and $\\sigma \\ne \\nu $ .", "Then $s_{\\nu \\sigma }(\\omega ^{-1})=\\overline{s_{\\nu \\sigma }}(\\omega )$ In particular $|s_{\\nu \\sigma (\\omega )}|^2=s_{\\nu \\sigma }(\\omega )s_{\\nu \\sigma }(\\omega ^{-1})$ We have $u_{\\nu \\sigma }(k,\\omega )=e_\\nu (k,\\theta _\\nu (\\omega ))s_{\\nu \\sigma }(\\omega )$ and by construction of the matrix $U(k,\\omega )$ , we know that $u_{\\nu \\sigma }(k,\\omega ^{-1})=\\overline{u_{\\nu \\sigma }}(k,\\omega ),\\quad \\omega \\in T_\\sigma ^+\\cup T_\\nu ^+$ and the corollary follows." ], [ "Discrete spectra of the operator $\\mathcal {L}$ ", "Lemma 7.1 Let $\\hat{\\omega }\\in \\bar{\\setminus \\lbrace 0\\rbrace be a pole of U(k,\\omega ).", "Then \\lambda (\\hat{\\omega })=a-b(\\hat{\\omega }+\\hat{\\omega }^{-1})is an eigenvalue of (\\ref {eq:01}).", "If in addition \\hat{\\omega }\\in T_\\sigma ^+, then all elements u_{\\sigma \\nu }(k,\\omega ), \\nu \\in \\mathcal {C} areregular at \\hat{\\omega }.", "}{\\begin{xmlelement*}{proof}Let \\hat{\\omega }\\in \\bar{\\setminus \\lbrace 0\\rbrace be a (simple) pole of U(k,\\omega ).", "Denote\\begin{equation*}a_\\nu (\\omega )= \\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\theta _\\nu ( \\omega )-\\theta _\\nu ( \\omega )^{-1}} (\\omega -\\hat{\\omega }), \\ \\nu \\in \\mathcal {C}; \\quad A(\\omega )= {\\mathbf {\\rm {diag}}}\\lbrace a_\\nu (\\omega )\\rbrace _{\\nu \\in \\mathcal {C}}.\\end{equation*}We then have\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} U(k,\\omega )=\\lim _{\\omega \\rightarrow \\hat{\\omega }} U(k,\\omega )A(\\omega ).\\end{equation*}For each \\nu \\in \\mathcal {C} and for each \\omega \\ne \\hat{\\omega } the \\nu -th column of U(k,\\omega )A(\\omega )\\vec{\\phi }^\\nu (k,\\omega )= \\left( \\phi ^\\nu (\\sigma (k),\\omega ) \\right)_{\\sigma \\in \\mathcal {C}}=U(k,\\omega )A(\\omega ) \\mathbf {n}_\\nu can be prolongated into \\mathcal {A}_1 to a solution of (\\ref {eq:01}) with \\lambda =\\lambda (\\omega ) according (\\ref {eq:07}):\\begin{equation*}\\phi ^\\nu (\\alpha , \\omega )= \\sum _{\\gamma \\in \\mathcal {C}} r(\\alpha , \\gamma (0); \\lambda ) {\\mathfrak {b}}_\\gamma (0)\\phi ^\\nu (\\gamma (1),\\omega ); \\ \\alpha \\in \\mathcal {A}_1\\end{equation*}Since U(k,\\omega ) has a simple pole at \\hat{\\omega }, there exists the limit\\begin{equation*}\\vec{\\phi }^\\nu (k, \\hat{\\omega })=\\lim _{\\omega \\rightarrow \\hat{\\omega }} \\vec{\\phi }^\\nu (k, \\omega )\\end{equation*}\\medskip }{\\bf Claim} {\\em The vector \\vec{\\phi }^\\nu (k,\\hat{\\omega }) also can be prolongated to a solution of the problem(\\ref {eq:01})}.\\\\In case \\lambda (\\hat{\\omega }) is not a pole of \\mathcal {R}(\\lambda (\\omega )), the prolongation is straightfoward.", "By (\\ref {eq:07}), if \\lambda (\\hat{\\omega }) is at the same time an eigenvalue of \\mathcal {L}_1 one can apply the same reasonings as in lemma 4.3 in \\cite {LM}.", "We omit the details.\\\\ \\\\Let now\\begin{equation*}T(\\omega )^{-1}=(\\tau _{\\sigma \\nu }(\\omega ))_{\\sigma ,\\nu \\in \\mathcal {C}},\\end{equation*}and denote\\begin{equation}h_{\\sigma \\nu }(\\omega )= -\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1, \\theta _\\nu (\\omega ))}(\\omega -\\hat{\\omega })\\tau _{\\sigma \\nu }(\\omega ).\\end{equation}It follows from (\\ref {eq:64}) that for \\sigma \\ne \\nu we have\\begin{equation}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu } (k,\\cdot )= \\phi ^\\nu (\\sigma (k),\\hat{\\omega }) = e_\\sigma (k,\\hat{\\omega })h_{\\sigma \\nu }(\\hat{\\omega })\\end{equation}here we denote {\\mathfrak {m}}_{\\sigma \\nu }(\\hat{\\omega })=h_{\\sigma \\nu }(\\hat{\\omega }).\\\\ \\\\If e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0, representation (\\ref {Eq:33}) is valid for \\sigma =\\nu as well.Assume that e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))= 0.", "Then\\begin{equation}p_\\nu (k,\\hat{\\omega })e_\\nu (0,\\theta _\\nu (\\hat{\\omega }))=e_\\nu (k, \\theta _\\nu (\\hat{\\omega })),\\end{equation}because the expressions in both sides satisfy the same recurrence equation and the same initial conditions, and also it follows from lemma \\ref {le:02a}that \\dot{e}_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0 because \\ e_\\nu (1,\\theta _\\sigma (\\hat{\\omega }) ) and \\dot{e}_\\nu (1,\\theta _\\sigma (\\hat{\\omega }) ) cannot vanish simultaneously.Thus we again obtain (\\ref {Eq:33}), yet now\\begin{equation*}m_{\\nu \\nu }(\\hat{\\omega })= \\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu (\\hat{\\omega }),\\dot{e}_\\nu (\\hat{\\omega })\\rbrace _\\nu (0)}+h_{\\nu \\nu }(\\hat{\\omega }).\\end{equation*}Representation (\\ref {Eq:33}) is now valid for all \\nu ,\\sigma \\in \\mathcal {C}.", "\\\\ \\\\If \\hat{\\omega }\\in it follows from (\\ref {Eq:33}) that \\phi ^\\nu (\\alpha ,\\hat{\\omega }) is an eigenfunction of \\mathcal {L} with \\lambda (\\hat{\\omega }) as eigenvalue.", "If \\hat{\\omega }\\in , we may have \\theta _\\sigma (\\hat{\\omega })\\in , i.e., \\hat{\\omega }\\in T^+_\\sigma for some \\sigma \\in \\mathcal {C}.", "It follows from corollary \\ref {Rem:51} that for such \\sigma we have m_{\\sigma \\nu }(\\hat{\\omega })=0, in particular\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu }(k, \\cdot )=\\phi ^\\nu (\\sigma (k), \\hat{\\omega })=0, \\ \\hat{\\omega }\\in \\theta _\\sigma (\\hat{\\omega }) \\in \\end{equation*}and, again \\phi ^\\nu (\\alpha ,\\hat{\\omega }) is an eigenfunction of \\mathcal {L} with \\lambda (\\hat{\\omega }) as eigenvalue.\\end{xmlelement*}}$ Consider the matrix ${\\mathfrak {m}}(\\hat{\\omega })= \\left( m_{\\sigma \\nu }(\\hat{\\omega }) \\right)_{\\sigma ,\\nu \\in \\mathcal {C}}$ Properties of ${\\mathfrak {m}}(\\hat{\\omega })$ are summarized in the statement below Lemma 7.2 Let $\\hat{\\omega }\\in \\Omega $ .", "Then ${\\mathbf {\\rm {Res}}}U(k,\\hat{\\omega })= \\mathcal {E}(k, \\hat{\\omega }){\\mathfrak {m}}(\\hat{\\omega }), \\ k=0,1,2, \\ldots .$ The diagonal elements $m(\\nu ,\\nu ; \\hat{\\omega })$ satisfy $\\Vert \\phi ^\\nu (\\hat{\\omega })\\Vert ^2=-\\frac{b_\\nu (1-\\theta _\\nu (\\hat{\\omega })^{-2})}{b(1- \\bar{\\hat{\\omega }}^{-2})}\\theta _\\nu (\\hat{\\omega })m_{\\nu \\nu }(\\hat{\\omega }),$ where $\\phi ^\\nu =\\phi ^\\nu (\\alpha , \\hat{\\omega })$ is the eigenvector of $\\mathcal {L}$ , corresponding to the eigenvalue $\\lambda (\\hat{\\omega })$ and such that $\\phi ^\\nu (\\sigma (k), \\hat{\\omega }) = e_\\sigma (k,\\theta _\\sigma (\\hat{\\omega })) m(\\sigma ,\\nu ;\\hat{\\omega }) \\ \\sigma \\in \\mathcal {C}, \\ k\\ge 0.$ Relations (REF ) and (REF ) are already established in lemma REF .", "It remains to prove (REF ).", "We apply Lemma REF with $\\xi (\\alpha )= \\phi ^\\nu (\\alpha ,\\hat{\\omega }) $ : $\\dot{\\lambda }(\\hat{\\omega })\\sum _{\\alpha \\in \\mathcal {A}_1} |\\phi ^\\nu (\\alpha ,\\hat{\\omega })|^2=\\sum _{\\sigma \\in \\mathcal {C}}{\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\sigma (0)$ and calculate the Wronskians in the right-hand side of this equality.", "We then have $\\phi ^\\nu (\\sigma (k),\\omega ) = p_\\sigma (k,\\omega )h_\\nu (\\omega )\\delta _{\\sigma ,\\nu }+e_\\sigma (k,\\omega )h_{\\sigma \\nu }(\\omega ),$ with $h_\\nu (\\omega )=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1, \\theta _\\nu (\\omega ))}(\\omega -\\hat{\\omega }),$ and $h_{\\sigma \\nu }$ is already defined in ().", "Let $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ .", "Then, for all $\\sigma ,\\nu \\in \\mathcal {C}$ , $h_\\nu (\\hat{\\omega })=0, \\ \\phi ^\\nu (\\sigma (k),\\hat{\\omega })= e_\\sigma (k,\\theta _\\sigma (\\hat{\\omega }))h_{\\sigma \\nu }(\\hat{\\omega }),$ and $m_{\\sigma \\nu }(\\hat{\\omega })=h_{\\sigma \\nu }(\\hat{\\omega }), \\ {\\mathfrak {m}}(\\hat{\\omega })=\\left( m_{\\sigma \\nu }(\\hat{\\omega })\\right)_{\\sigma ,\\nu \\in \\mathcal {C}}$ We use (REF ), (REF ), (REF ), (REF ) and that $\\dot{p}_k(\\omega )=0$ for $k=0,1$ as follows from (): $\\dot{\\phi }^\\nu (\\sigma (k),\\omega ) = p_\\sigma (k,\\omega )\\dot{h}_\\nu (\\omega )\\delta _{\\sigma ,\\nu }+\\dot{e}_\\sigma (k,\\omega )h_{\\sigma \\nu }(\\omega )+e_\\sigma (k,\\omega )\\dot{h}_{\\sigma \\nu }(\\omega ), \\ k=0,1.$ Besides $\\dot{h}_\\nu (\\hat{\\omega })=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))}.$ Since $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ , we also have $h_\\nu (\\hat{\\omega })=0$ and, according to (REF ) and (), ${\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\sigma (0)=b_\\nu (\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1})\\overline{m_{\\sigma \\nu }(\\hat{\\omega })} \\delta _{\\sigma ,\\nu } + \\\\{\\mathfrak {b}}_\\sigma (0) \\lbrace \\dot{e}_\\sigma (\\hat{\\omega }), e_\\sigma (\\theta _\\sigma (\\hat{\\omega }))\\rbrace _\\sigma (0) |m_{\\sigma \\nu }(\\hat{\\omega })|^2 = \\\\b_\\nu (\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1})\\overline{m_{\\sigma \\nu }(\\hat{\\omega })} \\delta _{\\sigma ,\\nu } -\\dot{\\lambda }(\\hat{\\omega })\\sum _{k=1}^\\infty |\\phi ^\\nu (\\sigma (k), \\theta _\\sigma (\\hat{\\omega }))|^2.$ We can now return to (REF ) in order to obtain normalization condition (REF ) for the matrix ${\\mathfrak {m}}$ : In the case $e_\\nu (1,\\hat{\\omega })=0$ representation (REF ) is still valid, yet $h_\\nu (\\hat{\\omega })=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu ,\\dot{e}_\\nu \\rbrace _\\nu (0)}e_\\nu (0,\\hat{\\omega })\\ne 0.$ Taking () into account we again obtain (), yet now $m_{\\sigma \\nu }(\\hat{\\omega })= \\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu (\\hat{\\omega }),\\dot{e}_\\nu (\\hat{\\omega })\\rbrace _\\nu (0)}\\delta _{\\sigma ,\\nu }+h_{\\sigma \\nu }(\\hat{\\omega }).$ Relation (REF ) is still valid and for $\\nu \\ne \\sigma $ we arrive to relation (REF ).", "For $\\sigma =\\nu $ we have ${\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\nu (0)={\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{e}_\\nu (\\hat{\\omega })h_2(\\hat{\\omega }),\\overline{e_\\nu (\\hat{\\omega })}\\rbrace _\\nu (0) \\overline{m_{\\nu \\nu }(\\hat{\\omega })}= \\\\{\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{e}_\\nu (\\hat{\\omega }),\\overline{e_\\nu (\\hat{\\omega })}\\rbrace _\\nu (0) |m(\\nu ,\\nu ;\\omega )|^2-\\ {b_\\nu } ({\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}})\\overline{m_{\\nu \\nu }(\\hat{\\omega })},$ and we again arrive to (REF ).", "Now one can complete the proof in the same way as if $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ .", "Remark The matrix $\\overline{{\\mathfrak {m}}(\\hat{\\omega })}$ corresponds to the point of discrete spectra $\\lambda (\\hat{\\omega })$ .", "Its columns are normalized eigenfunctions.", "This normalizations is defined by relation REF and is therefore unique.", "We will see in the next section that $m_{\\nu \\nu }(\\hat{\\omega })$ , i.e.", "the energies of the normalized eigenfunctions, are the quantities which participate in the equations for the inverse scattering problem" ], [ "In order to obtain equations of the inverse scattering problem we introduce the matrix function $\\Delta _l(\\omega ):= {\\mathbf {\\rm {diag}}}\\left\\lbrace \\theta _\\nu (\\omega )^{l-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }\\right\\rbrace _{\\nu \\in \\mathcal {C}}, \\ l\\in {\\mathbb {Z}}$ and consider the integral ${\\mathcal {J}}(l,k) = \\left( j_{\\nu \\sigma }(l,k) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}= \\frac{1}{2\\pi I} \\int __l(\\omega ) U(k,\\omega ) d\\omega .$ Since $ may contain (simple) poles of $ U(k,) $ this integral as well as all integrals in this section is considered in principal value.", "We will calculate this integral in two ways: through the residues, this would correspond to the contribution of the discrete spectra, and through the scattering coefficients, this would correspond to the contribution of the continuos spectra.\\\\ \\\\Comparing two different expressions for $ Jl,k$ leads one to the equations of the inverse scattering problem.$ Let as before $U(k,\\omega )= (u_{\\nu \\sigma }(k,\\omega ))_{\\nu ,\\sigma }$ .", "Then $j_{\\nu \\sigma }(l,k)= \\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}u_{\\nu \\sigma }(k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega = \\\\\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}\\psi ^\\sigma _\\nu (k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ here $\\psi ^\\nu (\\cdot , \\omega )$ is the special solution, defined in (REF ).", "Assume first that $\\sigma \\ne \\nu $ .", "Then, according to () and (REF ) $j_{\\nu \\sigma }(l,k) =\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}e_\\nu (k,\\theta _\\nu (\\omega ))s_{\\sigma \\nu }(\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ Consider the function $q_{\\nu \\sigma }(n)=\\frac{1}{2\\pi I}\\int _s̰_{\\sigma \\nu }(\\omega ) \\theta _\\nu (\\omega )^{n-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega .$ Relation () now yields $j_{\\nu \\sigma }(l,k)=c_\\nu (k)\\sum _{m\\ge k} q_{\\nu \\sigma }(m+l)a_\\nu (k,m),$ this is the desired expression.", "Remark The functions $q_{\\nu ,\\sigma }$ cannot be determined from the spectral data generally speaking.", "However relation (REF ) determines the structure of the equation of the inverse scattering problem.", "Later in section REF we will use this structure to get rid of the functions which cannot be observed from the spectral data.", "Let now $\\nu =\\sigma $ and $T(\\omega )^{-1}= (\\tau _{\\sigma \\nu }(\\omega ))_{\\sigma ,\\nu }$ .", "Relations (REF ) (REF ) yield $j_{\\sigma \\sigma }(k,l)= \\\\ \\frac{1}{2\\pi I} \\int _\\lbrace \\bigl [-p_\\sigma (k,\\omega ) + e_\\sigma (k,\\theta _\\sigma )\\tau _{\\sigma \\sigma }(\\omega )\\bigr ] \\frac{b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma )}{{\\mathfrak {b}}_\\sigma (0)e_\\sigma (1,\\theta _\\sigma )} \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma (\\omega )}{d\\omega } \\Bigr \\rbrace d\\omega = \\\\\\frac{1}{2\\pi I} \\Bigl ( \\int _{T_\\sigma ^+} + \\int _{T_\\sigma ^-}\\Bigr ) \\Bigl \\lbrace \\ \\cdot \\ \\Bigr \\rbrace = j_{\\sigma \\sigma }^+(k,l) + j_{\\sigma \\sigma }^-(k,l).$ As $\\omega $ runs over $T_\\sigma ^+$ the function $\\theta _\\sigma (\\omega )$ runs over the whole $.", "Let $ ()$ be the inverse function.Relations (\\ref {Eq:02a01}) together with (\\ref {eq:23}) yield\\begin{equation*}j_{\\sigma \\sigma }^+(k,l) = \\frac{1}{2\\pi I} \\int _[ e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )\\bigr ] \\theta _\\sigma ^{l-1} d\\theta _\\sigma .\\end{equation*}Besides it follows from corollary \\ref {Rem:51} that $ s()$ is bounded on $ +$.\\\\Consider the Fourier series of $ s(())$:\\begin{equation*}s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))=\\sum _{n=-\\infty }^\\infty \\tilde{s}_\\sigma (n) \\theta ^{-n}; \\ \\tilde{s}_\\sigma (n) = \\frac{1}{2 \\pi I} \\int _s̰_{\\sigma \\sigma }(\\omega (\\theta _\\sigma )) \\theta _\\sigma ^{n-1} d \\theta _\\sigma .\\end{equation*}Together with (\\ref {eq:13}) this yields\\begin{multline*}e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )= \\\\c_\\sigma (k) \\sum _m \\bigl \\lbrace a_\\sigma (k,m) + \\sum _n a_\\sigma (k,n) \\tilde{s}_\\sigma (m+n) \\bigr \\rbrace \\theta _\\sigma ^{-m},\\end{multline*}and finally\\begin{equation}j_{\\sigma \\sigma }^+(k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\tilde{s}_\\sigma (l+n)\\right] .\\end{equation}$ We now study $j^-_{\\sigma \\sigma }(k,l)$ .", "Denote $a_1(k,\\omega )= - p_\\sigma (k,\\omega ) \\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma );$ $a_2(\\omega )=\\tau _{\\sigma \\sigma }(\\omega )\\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma ).$ We then have $\\psi ^\\sigma _\\sigma (k,\\omega )= a_1(k,\\omega )+e_\\sigma (k,\\theta _\\sigma (\\omega ))a_2(\\omega ),$ and $j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_1(k,\\omega ) \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega + \\\\\\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_2( \\omega ) e_\\sigma (k,\\theta _\\sigma (\\omega ))\\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega .$ As $\\omega $ runs over $T_\\sigma ^-$ , the function $\\theta _\\sigma (\\omega )$ runs twice in the opposite directions over $J_\\sigma =\\theta _\\sigma (T_\\sigma ^-)\\subset {\\mathbb {R}}$ and the values $\\theta _\\sigma (\\omega )$ and $\\theta _\\sigma (\\omega ^{-1})$ coinside.", "Respectively $e_\\sigma (k,\\theta _\\sigma (\\omega )), p_\\sigma (k,\\omega ) \\in {\\mathbb {R}}$ and $e_\\sigma (k,\\theta _\\sigma (\\omega ))=e_\\sigma (k,\\theta _\\sigma (\\omega ^{-1}))$ , $ p_\\sigma (k,\\omega ) = p_\\sigma (k,\\omega ^{-1}) $ , thus $a_1(k,\\omega )=a_1(k,\\omega ^{-1})$ .", "Besides $ a_2(\\omega ) = \\overline{a_2(\\omega ^{-1})}$ , since $\\psi ^\\sigma (k,\\omega )= \\overline{\\psi ^\\sigma (k,\\omega ^{-1})}$ for $\\omega \\in .\\\\ \\\\Therefore\\begin{equation}j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{\\Gamma _\\sigma } [a_2(\\omega )-a_2(\\omega ^{-1}) ] e_\\sigma (k,\\theta _\\sigma )\\theta _\\sigma ^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega ,\\end{equation}where $ =T-+$.\\\\ \\\\The function $ a1(k,)$ satisfies equation (\\ref {eq:02}) on the channel $$ andrelations (\\ref {eq:66a}), (\\ref {eq:66b}) yield\\begin{equation*}a_1(k,\\omega )= \\theta _\\sigma (\\omega )^{-k}(1+o(1)).\\end{equation*}Therefore$ ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)= b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega ))[ a_2(\\omega )-a_2(\\omega ^{-1})].$ On the other hand by (REF ) ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)=- \\sum _{\\nu \\in \\mathcal {C}\\setminus \\lbrace \\sigma \\rbrace }{\\mathfrak {b}}_\\nu (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0).$ We have $\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)=0, \\ \\omega \\in T_\\nu ^-.$ For $\\omega \\in T_\\nu ^+\\cap T_\\sigma ^-$ we obtain ${\\mathfrak {b}}_\\nu (0)\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)= \\\\{\\mathfrak {b}}_\\nu (0) \\lbrace s_{\\nu \\sigma }(\\omega )e_\\nu (\\theta _\\nu (\\omega )), s_{\\nu \\sigma }(\\omega ^{-1})e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) \\\\|s_{\\nu \\sigma }(\\omega )|^2 {\\mathfrak {b}}_\\nu (0)\\lbrace e_\\nu (\\theta _\\nu (\\omega )), e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) = \\\\|s_{\\nu \\sigma }(\\omega )|^2 b_\\nu (\\theta _\\nu (\\omega )^{-1}-\\theta _\\nu (\\omega )) =- |s_{\\sigma \\nu }(\\omega )|^2 \\frac{ ( b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma ))^2}{b_\\nu (\\theta _\\nu ^{-1}-\\theta _\\nu )}.$ Now, for $\\theta \\in J_\\sigma $ , we define $\\omega _\\sigma (\\theta )=W^{-1}\\circ W_\\sigma (\\theta )$ , here the functions $W$ , $W_\\sigma $ are defined by (REF ) and the branch is chosen so that $\\omega _\\sigma (\\theta )\\in \\Gamma _\\sigma $ .", "Let $N_\\sigma (\\theta )= \\lbrace \\nu ; \\ \\omega _\\sigma (\\theta )\\in T_\\nu ^+ \\rbrace , \\ \\theta \\in J_\\sigma .$ Denote $\\Phi _\\sigma (\\theta )= \\sum _{\\nu \\in N_\\sigma (\\theta )}|s_{\\sigma \\nu }(\\omega _\\sigma )|^2 \\left(\\frac{ b_\\sigma (\\theta ^{-1}-\\theta ) }{b_\\nu (\\theta _\\nu (\\omega _\\sigma )^{-1}-\\theta _\\nu (\\omega _\\sigma ))} \\right), \\ \\omega _\\sigma (\\theta )\\in T_\\sigma ^-.$ We finally obtain $a_2(\\omega )-a_2 (\\omega ^{-1})= \\Phi _\\sigma (\\theta _\\sigma (\\omega ))$ this function is expressed via the spectral data.", "Remark All functions $|s_{\\nu \\sigma }(\\omega )|$ participating in (REF ) are well defined and continuous.", "In addition, $s_{\\nu \\sigma }(\\omega ^{-1})=\\overline{s_{\\nu \\sigma }}(\\omega )$ for $\\omega \\in \\ T_\\nu ^+$ .", "Relation () now takes the form $j_{\\sigma \\sigma }^-(k,l)= \\frac{ 1 }{2\\pi I} \\int _{J_\\sigma } e_\\sigma (k,\\theta ) \\Phi _\\sigma (\\theta ) \\theta ^{l-1} d\\theta ,$ and with account () we obtain $j_{\\sigma \\sigma }^-(k,l)= c_\\sigma (k)\\sum _n a_\\sigma (k,n) q_{\\sigma \\sigma }(n+l),$ where $q_{\\sigma \\sigma }(n)= \\frac{1 }{2\\pi I} \\int _{J_\\sigma } \\Phi _\\sigma (\\theta )\\theta ^{n-1} d\\theta .$ Combining (REF ), (), and (REF ) we finally obtain $j_{\\sigma \\sigma } (k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\left( \\tilde{s}_\\sigma (l+n)+q_{\\sigma \\sigma }(n+l) \\right)\\right],$ the functions $\\tilde{s}_\\sigma (\\cdot )$ and $q_{\\sigma ,\\sigma }(\\cdot )$ are defined through the scattering data." ], [ "Let now $\\Theta (\\omega )= {\\mathbf {\\rm {diag}}}\\lbrace \\theta _\\sigma (\\omega ) \\rbrace _{\\sigma \\in \\mathcal {C}}.$ We use the representation () for $\\mathcal {E}(k,\\omega )$ , $\\mathcal {E}(k,\\omega )=C(k)\\sum _{m=-\\infty }^\\infty A(k,m)\\Theta (\\omega )^m.$ The coefficients $C(k)$ and $A(k,m)$ are diagonal matrices $C(k)={\\mathbf {\\rm {diag}}}\\lbrace c_\\sigma (k)\\rbrace _{\\sigma \\in \\mathcal {C}}, \\ A(k,m)={\\mathbf {\\rm {diag}}}\\lbrace a_\\sigma (k,m)\\rbrace _{\\sigma \\in \\mathcal {C}},$ and also $A(k,m)=0$ for $m<k$ .", "It follows from (REF ) and (REF ) that ${\\mathcal {J}}(l,k)=\\frac{1}{2\\pi I}\\int __l(\\omega )U(k,\\omega )d\\omega = \\\\C(k) \\left\\lbrace A(k,l)+\\sum _{l=-\\infty }^\\infty A(k,m) Z(l+m) \\right\\rbrace ,$ here the integral is taken as a principal value and the matrix $Z(n)=\\left( z_{\\nu ,\\sigma }(n) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}$ is given by the relations $z_{\\nu \\sigma }(n)=q_{\\nu \\sigma }(n), \\ \\nu \\ne \\sigma , \\\\z_{\\sigma \\sigma }(n)=\\tilde{s}_\\sigma (n)+q_{\\sigma \\sigma }(n),$ here $\\tilde{s}_\\sigma (n)$ are the Fourier coefficients of the reflection coefficient $s_{\\sigma \\sigma }$ with respect to $\\theta _\\sigma $ and the function $q_{\\nu \\sigma }$ is defined in (REF ) and (REF ).", "Let now $\\Omega \\subset \\bar{ be the set of all poles of U(k,\\omega ).", "It follows from (\\ref {eq:64}) that \\overline{U(k,\\omega )}=U(k,\\bar{\\omega }), thus for each \\hat{\\omega }\\in \\Omega \\cap the point \\bar{\\hat{\\omega }} also belongs to \\Omega .", "Moreover, for such \\hat{\\omega } we have \\overline{\\text{Res}_{\\hat{\\omega }}U(k,\\omega )}=\\text{Res}_{\\bar{\\hat{\\omega }}}U(k,\\omega ).Denote \\tilde{\\Omega }= \\lbrace \\hat{\\omega }\\in \\Omega : |\\hat{\\omega }| < 1\\rbrace \\cup \\lbrace \\hat{\\omega }\\in \\Omega : \\Im \\hat{\\omega }>0\\rbrace .", "We use (\\ref {Eq:39}) and (\\ref {Eq:01}):}$ ${\\mathcal {J}}(k,l)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} \\Delta _l(\\hat{\\omega })\\mathcal {E}(k,\\hat{\\omega }){\\mathfrak {m}}(\\hat{\\omega })=C(k)\\sum _mA(k,m)M(m+l),$ where $M(n)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} {\\mathbf {\\rm {diag}}}\\left\\lbrace \\frac{d\\theta _\\sigma }{d\\omega }(\\hat{\\omega }) \\right\\rbrace _{\\sigma \\in \\mathcal {C}}\\Theta (\\hat{\\omega })^{n-1}\\Re {\\mathfrak {m}}(\\hat{\\omega }).$ By compare this to (REF ) and taking into account that $A(k,m)=0$ for $m<k$ and $A(k,k)=I$ , we obtain a system of equations $A(k,m)+\\sum _{s=k}^\\infty A(k,s)F(s+m)=0,\\quad m=k+1,k+2, \\ldots \\,$ where $F(n)=Z(n)-M(n).$ Since $A(k,k)=I$ , this relation can be written as $F(k+m)+A(k,m)+\\sum _{s=k+1}^\\infty A(k,s)F(s+m)=0, \\quad k=1,2, \\ldots ,\\quad m>k.$ The matrices $A(k,m)$ are diagonal.", "The diagonal elements of $F(n)$ , can be expressed through the spectral data.", "The non-diagonal elements of the matrices $F(n)$ vanish, as follows from the lemma below.", "Lemma 8.1 The matrices $F(n)$ are diagonal for all $n\\ge 1$ : $F(n)={\\mathbf {\\rm {diag}}}\\lbrace f_\\sigma (n)\\rbrace _{\\sigma \\in \\mathcal {C}}$ .", "and their diagonal elements $f_\\sigma (n)$ are determined by $a_\\sigma (k,m)$ , $m>k\\ge \\Big [\\frac{n-1}{2}\\Big ]$ only.", "This statement is proved in [12], (Lemma 5.1) and we omit the proof.", "Theorem 8.1 The following properties hold for the systems under consideration 1) Equations (REF ) split into a system of independent scalar equations $f_\\nu (k+m)+a_\\nu (k,m)+\\sum _{s=k+1}^\\infty a_\\nu (k,s)f_\\nu (s+m)=0, \\,m\\ge k+1\\ge 1,$ here $f_\\nu (n)$ is defined in (REF ), (REF ), ().", "2) For each $k\\ge 0$ equations (REF ) has unique solutions $a_\\sigma (k,m)$ .", "The first statement follows directly from relation (REF ) which defines $F(n)$ and Lemma REF .", "It follows from the same lemma that the functions $f_\\nu (n)$ , $n\\ge 1$ are uniquely defined by the coefficients ${\\mathfrak {b}}_\\nu (k)$ , ${\\mathfrak {a}}_\\nu (k)$ , $k\\ge 1$ .", "Therefore (REF ) coincide with equations for inverse scattering problem for equation (REF ) with boundary condition $\\xi (\\nu (0))=0$ .", "It is well-known (see for example [14]) that the later has unique solution." ], [ "Concluding remarks", "In case that the continuos spectra $I=\\cup _{\\sigma \\in \\mathcal {C}}[a_\\sigma -2b_\\sigma ,a_\\sigma +2b_\\sigma ]$ splits into a number of disjoints intervals, one can repeat the procedure separately for each connected component of $I$ .", "If, say, $I^{(0)}$ is a connected component of $I$ and $\\sigma $ is a channel corresponding to this component, then each wave incoming along $\\sigma $ generates decaying waves in all channels which correspond to other connected components of $I$ .", "We omit the details.", "So far we have discussed reconstruction of the part $\\mathcal {A}_0$ of the whole system, that is, the channels.", "This information is, generally speaking, insufficient for reconstruction the whole matrix $\\mathcal {L}$ .", "However if the matrix $\\mathcal {L}_1$ corresponding to the \"central\" part of the system is sufficiently sparse and also we know the matrix $B(0)=\\text{diag}\\lbrace {\\mathfrak {b}}_\\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}}$ which realizes connections between the channels and the central part of the system, the whole matrix $\\mathcal {L}$ can be recovered from the scattering data.", "We refer the reader to Chapter 11 in [15], where statemets of such type are obtained." ], [ "Construction of the special solutions", "Consider the diagonal matrices $B:={\\mathbf {\\rm {diag}}}\\lbrace b_\\sigma \\rbrace _{\\sigma \\in \\mathcal {C}}, \\ B(0):={\\mathbf {\\rm {diag}}}\\lbrace {\\mathfrak {b}}_\\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}},$ and also the matrix-functions in $\\bar{\\begin{equation*}\\mathcal {E}(k,\\omega ):={\\mathbf {\\rm {diag}}}\\lbrace e(k,\\theta _\\sigma (\\omega ))\\rbrace _{\\sigma \\in \\mathcal {C}}, \\ P(k,\\omega ):= {\\mathbf {\\rm {diag}}}\\lbrace p_\\sigma (k,\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}},\\end{equation*}here \\lbrace p_\\sigma (k,\\omega )\\rbrace _{k=0}^\\infty satisfy the equation on the channels:\\begin{equation*}-{\\mathfrak {b}}_\\sigma (k-1)p_\\sigma (k-1,\\omega )+{\\mathfrak {a}}_\\sigma (k)p_\\sigma (k,\\omega )-{\\mathfrak {b}}_\\sigma (k)p_\\sigma (k+1,\\omega )=\\lambda (\\omega )p_\\sigma (k,\\omega )\\end{equation*}for k=1,2,\\dots , \\sigma \\in \\mathcal {C} and with boundary conditions\\begin{equation}p_\\sigma (0,\\omega )=1, \\ p_\\sigma (1,\\omega )=0,\\quad \\sigma \\in \\mathcal {C}\\end{equation}The functions \\mathcal {E}(k,\\omega ) and P(k,\\omega ) are holomorphic in a neighborhood of \\overline{, except, perhaps zero, where P(k,\\omega ) may have poles.", "\\\\Moreover, it is well known, see \\cite {MS,Te}, that the Jost solutions form a fundamental system of solutions of the finite-difference equation (\\ref {eq:30}).", "If |\\theta |=1 and |\\theta |\\ne 1, it follows from (\\ref {eq:36}) that e(\\theta ), \\bar{e}(\\theta )=e(\\theta ^{-1}) are independent solutions of (\\ref {eq:30}) and any other solution \\lbrace x(k,\\theta )\\rbrace _{k\\ge 0} can be expressed as x(k,\\theta )=m(\\theta )e(k,\\theta )+n(\\theta )e(k,\\theta ^{-1}), with m(\\theta ),n(\\theta ) independent of k. Thus, for \\omega \\in T_\\sigma ^+ the function p_\\sigma (k,\\omega ) may be expressed in terms of the corresponding Jost solutions:\\begin{equation}p_\\sigma (k,\\omega )= \\frac{{\\mathfrak {b}}_\\sigma (0)}{b_\\sigma }\\frac{e_\\sigma (k,\\theta _\\sigma (\\omega ))e_\\sigma (1,\\theta _\\sigma (\\omega )^{-1}) - e_\\sigma (k,\\theta _\\sigma (\\omega )^{-1})e_\\sigma (1,\\theta _\\sigma (\\omega ))}{\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )}.\\end{equation}\\\\ \\\\In order to construct the special solutions we need auxiliary operator-function T(\\omega ): l^2(\\mathcal {C}) \\rightarrow l^2(\\mathcal {C}) defined as\\begin{equation}T(\\omega )= \\mathcal {E}(0,\\omega ) - \\mathcal {R}(\\lambda (\\omega )) B(0) \\mathcal {E}(1, \\omega ), \\ \\omega \\in \\end{equation}This function is holomorphic in a vicinity of \\overline{\\mathbb {D}}, except the set \\mathcal {O} of poles of the function \\mathcal {R}(\\lambda (\\omega )).\\medskip }Denote\\begin{equation*}\\Delta _\\sigma (\\omega ) ={\\left\\lbrace \\begin{array}{ll}b_\\sigma , & |\\theta _\\sigma (\\omega )|=1; \\\\b_\\sigma (|\\theta _\\sigma (\\omega )|^{-2}-1)\\sum _{k=1}^\\infty |e_\\sigma (k,\\theta _\\sigma (\\omega ))|^2, & |\\theta _\\sigma (\\omega )|<1,\\end{array}\\right.", "}\\end{equation*}and\\begin{equation}\\Delta (\\omega )={\\mathbf {\\rm {diag}}}\\lbrace \\Delta _\\sigma (\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}}, \\quad \\Phi (\\omega )={\\mathbf {\\rm {diag}}}\\lbrace \\bar{\\theta }_\\sigma (\\omega )-\\theta _\\sigma (\\omega )\\rbrace _{\\sigma \\in \\mathcal {C}}.\\end{equation}The operator \\Delta (\\omega ) is positive uniformly with respect \\omega \\in \\overline{ {\\mathbb {D}}}, i.e.,for some C>0,\\begin{equation*}\\langle \\Delta (\\omega ){\\mathbf {x}, \\mathbf {x}} \\rangle \\ge C \\Vert {\\mathbf {x} }\\Vert ^2,\\quad {\\mathbf {x}}\\in l^2(\\mathcal {C}), \\, \\omega \\in \\overline{{\\mathbb {D}}}.\\end{equation*}Besides relation (\\ref {eq:36}) now reads\\begin{equation}B(0)\\left\\lbrace \\mathcal {E}(0,\\omega )\\mathcal {E}(1,\\omega )^*-\\mathcal {E}(0,\\omega )^*\\mathcal {E}(1,\\omega ) \\right\\rbrace =\\Phi (\\omega )\\Delta (\\omega ),\\end{equation}}\\begin{Lemma}(See lemma 3.1 in \\cite {LM})The following inequality holds for all \\theta \\in {\\mathbb {D}}\\setminus \\mathcal {O}, {\\mathbf {x}}=(x_\\sigma )_{\\sigma \\in \\mathcal {C}}\\in l^2(\\mathcal {C})\\begin{equation}|\\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}}\\rangle |\\ge |\\Im \\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}}\\rangle | \\ge C\\sum _{\\sigma \\in \\mathcal {C}} |\\bar{\\theta }_\\sigma (\\omega )-\\theta _\\sigma (\\omega )||x_\\sigma |^2.\\end{equation}\\end{Lemma}{\\begin{xmlelement*}{proof}We have\\begin{equation*}2\\Im \\langle \\mathcal {E}(1,\\omega )^*B(0)T(\\omega ){\\mathbf {x}}, {\\mathbf {x}} \\rangle =\\langle [ \\mathcal {E}(1,\\omega )^*B(0)T(\\omega )- T(\\omega )^*B(0)\\mathcal {E}(1,\\omega )]{\\mathbf {x}}, {\\mathbf {x}} \\rangle \\end{equation*}and, by (\\ref {eq:54}),\\begin{multline*}\\mathcal {E}(1,\\omega )^*B(0)T(\\omega )- T(\\omega )^*B(0)\\mathcal {E}(1,\\omega )=B(0)(\\mathcal {E}(0,\\omega )\\mathcal {E}(1, \\omega )^*-\\mathcal {E}(0, \\omega )^*\\mathcal {E}(1,\\omega ))+ \\\\(B(0)\\mathcal {E}(1,\\omega ))^*(\\mathcal {R}(\\lambda (\\omega ))^*-\\mathcal {R}(\\lambda (\\omega )))(B(0)\\mathcal {E}(1,\\omega ))\\end{multline*}The lemma now follows from (\\ref {eq:56}) - (\\ref {eq:58}) and from the fact that\\begin{equation*}\\mathcal {R}(\\lambda (\\omega ))^*-\\mathcal {R}(\\lambda (\\omega ))=b (\\bar{\\omega }- \\omega )\\underbrace{(|\\omega |^{-2}-1) \\left(\\sum _{l=1}^M \\frac{p_l(\\sigma (0))p_l(\\nu (0))}{|\\lambda _l - \\lambda (\\omega )|^2}\\right)_{\\sigma , \\nu \\in \\mathcal {C}}}_{\\Delta _1(\\omega )},\\end{equation*}here \\Delta _1(\\theta ) is a non-negative operator.\\end{xmlelement*}}$ Corollary 5.1 Operators $T(\\omega )$ are invertible for all non-real $\\omega $ in the open disk ${{\\mathbb {D}}}\\setminus \\lbrace \\mathcal {O}\\cup \\lbrace \\pm 1\\rbrace \\rbrace $ .", "Indeed, fix an $\\omega \\in [-1,1]$ and denote $\\delta (\\omega )=\\inf _\\sigma \\lbrace |\\Im \\theta _\\sigma (\\omega )| \\rbrace >0$ .", "Relation () now reads $\\Vert T(\\omega )\\mathbf {x}\\Vert >C\\delta (\\omega )\\Vert \\mathbf {x}\\Vert ,$ which yields invertibility of $T(\\omega )$ .", "Since $T(\\omega )$ is an analytic function, we can now claim that $T(\\omega )$ is invertible in a vicinity of $\\overline{\\mathbb {D}}$ , except perhaps at a finite set of points.", "Consider the analytic matrix functions : $D(\\omega ):= {\\mathbf {\\rm {diag}}}\\lbrace \\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )\\rbrace $ and $U(k,\\omega )=\\left[ -P(k,\\omega )+\\mathcal {E}(k,\\omega ) T(\\omega )^{-1} \\right]B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}.$ Lemma 5.1 Consider the vector function on $\\mathcal {A}_0\\cup \\Gamma $ $\\psi ^\\sigma (k,\\omega )= (\\psi ^\\sigma (\\gamma (k),\\omega ))_{\\gamma \\in \\mathcal {C}}:= U(k,\\omega ) {\\mathbf {n}}_\\sigma .$ This function satisfies the boundary condition (REF ) and, for $\\omega \\in T^+_\\sigma $ , admits representation () along the channels.", "Thus by (REF ) it may be prolongated to a special solution of (REF ).", "Representation () for $\\omega \\in T^+_\\sigma $ is just a consequence of (REF ) and ().", "In order to prove that $\\psi ^\\sigma (k,\\omega )$ meets the boundary condition for $\\omega \\in T^+_\\sigma $ , we prove that this condition is met by the whole matrix-function $U(k,\\omega )$ : $U(0,\\omega )=\\mathcal {R}(\\lambda (\\omega ))B(0)U(1,\\omega ),\\quad \\ \\omega \\in $ This implies that $\\psi ^\\sigma (k,\\omega )$ also meets the boundary condition.", "Relation (REF ) is straightforward: after factoring out the inessential factor $B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}$ in the definition (REF ) it becomes $-I+\\mathcal {E}(0,\\omega )T(\\omega )^{-1}=\\mathcal {R}(\\lambda (\\omega ))B(0)\\mathcal {E}(1,\\omega )T(\\omega )^{-1},$ which is just the definition of $T(\\omega )$ .", "So far the special solution is not defined at a point $\\omega _0\\in T^+_\\sigma $ in case $\\omega _0$ is a pole of $U(k,\\omega )$ .", "This case will be considered in the following sections: in section we prove that all poles of $U(k,\\omega )$ are simple, later in section we show that.", "if $\\omega _0\\in T_\\sigma ^+$ , the function $U(k,\\omega ){\\mathbf {n}}_\\sigma $ is continuos at $\\omega _0$ even if $\\omega _0$ is a pole of $U(k,\\omega )$ .", "In particular $\\text{Res}_{\\omega _0}u_{\\nu \\sigma }(\\omega )=0,\\quad \\nu \\in \\mathcal {C}$" ], [ "Singularities of $U(k,\\omega )$ .", "The matrix function $U(k,\\omega )$ is analytic in a vicinity of $\\bar{, in particular it has a finite number of poles which belong to \\overline{.\\begin{Lemma}There is a finite set \\Omega \\subset \\overline{ such that all poles of the matrix function U(k,\\omega ) \\in \\overline{ belong to\\Omega \\cup \\lbrace 0\\rbrace .", "In the origin U(k,\\omega ) has pole of order k, all poles in \\Omega are simple.", "}}{\\begin{xmlelement*}{proof}The proof follows the pattern of Lemma 4.1 in \\cite {LM}.We rewrite (\\ref {eq:64}) as\\begin{multline}U(k,\\omega )= -P(k,\\omega )B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} + \\\\\\mathcal {E}(k,\\omega ) T(\\omega )^{-1} B B(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1}.\\end{multline}Therefore all poles of U(k,\\omega ) in \\overline{ belong to the finite set \\Omega \\cup \\lbrace 0\\rbrace where \\Omega includes all poles of T(\\omega )^{-1} and \\mathcal {E}(1,\\omega )^{-1} in \\overline{ .\\\\That at the origin the functions U(0,\\omega ) and U(1,\\omega ) have poles of order 0 and 1 respectively followsfrom the initial conditions for P.For k\\ge 2 the main contribution to singularity of U(k,\\omega ) at zero comes from the first term in theright-hand side of (\\ref {Eq:23a01}) because P(k,\\omega ) is a polynomial of degree k-2 with respect to\\lambda =a-b(\\omega +\\omega ^{-1}).\\\\It now suffices to study the singularities of the second term in the right-hand side in (\\ref {Eq:23a01}).", "Let \\Omega be the set of all such singularities in \\overline{\\mathbb {D}}.These singularities comes from the poles of T(\\omega )^{-1}, that is, when det(T(\\omega ))=0 and also from the zeros of det(\\mathcal {E}(1,\\omega ))=\\prod _{\\sigma \\in \\mathcal {C}}e_\\sigma (1,\\theta _\\sigma (\\omega )).\\\\Actually U(k,\\omega ) cannot have poles outside (-1,1).", "This is due to the poles of T(\\omega )^{-1} are located in (-1,1) and e_\\sigma (1,\\theta _\\sigma (\\omega ))=0 only if \\theta _\\sigma (\\omega )\\in (-1,1), see (\\ref {eq:36}), and thus \\omega \\in (-1,1).\\\\Let \\hat{\\omega }\\in (-1,0)\\cup (0,1).We are going to use (\\ref {eq:59}) as \\omega approaches \\hat{\\omega }.", "We have|\\theta _\\sigma (\\omega )-\\theta _\\sigma (\\bar{\\omega })|\\asymp |\\omega -\\bar{\\omega }|, so for any \\mathbf {y}\\in l^2(\\mathcal {C}) relation (\\ref {eq:59}) with\\mathbf {x}= T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y}gives\\begin{multline*}| \\langle \\mathcal {E}(1,\\omega )^*\\mathcal {E}(1,\\omega )^{-1}BD(\\omega )\\mathbf {y}, T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\rangle |\\ge \\\\C |\\omega -\\bar{\\omega }| \\Vert T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\Vert ^2.\\end{multline*}Since \\mathcal {E}(1,\\omega )^*\\mathcal {E}(1,\\omega )^{-1} is unitary and also | \\mbox{det}D(\\omega )| stays bounded from below near\\hat{\\omega }, the Schwartz inequality gives\\Vert T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} \\mathbf {y} \\Vert \\le \\mbox{Const} \\frac{\\Vert \\mathbf {y}\\Vert }{|\\omega -\\bar{\\omega }|} \\ \\mbox{as} \\ \\omega \\rightarrow \\hat{\\omega },this is possible only in case \\hat{\\omega } is a simple pole of T(\\omega )^{-1}BB(0)^{-1}D(\\omega )\\mathcal {E}(1,\\omega )^{-1} .\\\\In case when \\hat{\\omega }\\in , the reasoning goes in a similar way, it suffices to let \\omega approach \\hat{\\omega } in a way that |\\theta _\\sigma (\\omega )-\\hat{\\theta }_\\sigma (\\omega )| \\asymp |\\omega -\\hat{\\omega }|.", "}}\\section {Relation for the scattering coefficients}\\end{xmlelement*}The special solution \\psi ^\\sigma (\\alpha ,\\omega ) is now well-defined for all \\omega \\in \\Omega .", "Thusthe non-diagonal scattering coefficients s_{\\sigma \\gamma }(\\omega )\\sigma \\ne \\gamma are also well-defied for all \\omega \\in \\Omega .", "Together with \\psi ^\\sigma (\\omega ) they are analytic in a vicinity of \\overline{.\\\\ \\\\The scattering coefficients s_{\\sigma ,\\sigma }(\\omega ) are so far well-defined for \\omega \\in T_\\sigma ^+ only.For \\omega \\in T_\\sigma ^- the corresponding value \\theta _\\sigma (\\omega ) belongs to the unit disk.", "Onecan construct (similarly to how this is done in Theorem 1.4.1 in \\cite {AM} for the continuous case)a real-valuedsolution e_\\sigma ^{(1)}(k,\\theta _\\sigma ) of(\\ref {eq:10}) such that\\begin{equation}e_\\sigma ^{(1)}(k,\\theta _\\sigma )=\\theta _\\sigma ^{-k} (1+o(1)), \\ k\\rightarrow \\infty .\\end{equation}This choice is not unique since by adding any multiple of e_\\sigma we obtain a solution which still meets this relation.", "However we fix some choice of functionse_\\sigma ^{(1)}(k,\\theta _\\sigma ).", "The functions p_\\sigma can be represented as\\begin{equation}p_\\sigma (k,\\omega )= \\frac{{\\mathfrak {b}}_\\sigma (0)}{b_\\sigma }\\frac{e_\\sigma (k,\\theta _\\sigma (\\omega ))e_\\sigma ^{(1)}(1,\\theta _\\sigma (\\omega )) - e_\\sigma ^{(1)}(k,\\theta _\\sigma (\\omega ))e_\\sigma (1,\\theta _\\sigma (\\omega ))}{\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )},\\end{equation}which yields\\begin{equation*}\\psi ^{\\sigma }(\\sigma (k),\\omega )=\\theta _\\sigma (\\omega )^{-k}(1+o(1)), \\ k\\rightarrow \\infty \\end{equation*}as it should be for a special solution.", "}}\\begin{Lemma}For each \\sigma ,\\gamma \\in \\mathcal {C}, \\sigma \\ne \\gamma we have\\begin{equation}b_\\gamma (\\theta _\\gamma (\\omega )^{-1}-\\theta _\\gamma (\\omega ))s_{\\gamma \\sigma }(\\omega )= b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega )) s_{\\sigma \\gamma }(\\omega ), \\ \\omega \\in \\overline{\\setminus \\Omega }In addition the scattering coefficients s_{\\gamma ,\\sigma }(\\omega ) and s_{\\sigma ,\\gamma }(\\omega ) are continuous up to \\Omega \\end{equation}{\\begin{xmlelement*}{proof}Relation (\\ref {eq:67}) follows from (\\ref {eq:38}) for \\xi =\\psi ^\\sigma (\\omega ), \\eta =\\psi ^\\gamma (\\omega )if one takes into account that according (\\ref {eq:23}) and (\\ref {eq:23a}) we have\\begin{equation*}{\\mathfrak {b}}_\\nu (0)\\lbrace \\psi ^\\sigma ,\\psi ^\\gamma \\rbrace _{\\nu }(0)= {\\left\\lbrace \\begin{array}{ll}-b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega ))s_{\\sigma \\gamma }(\\omega ), & \\nu =\\sigma ; \\\\b_\\gamma (\\theta _\\gamma (\\omega )^{-1}-\\theta _\\gamma (\\omega ))s_{\\gamma \\sigma }(\\omega ), & \\nu =\\gamma ; \\\\0, & \\nu \\ne \\sigma ,\\gamma .\\end{array}\\right.", "}\\end{equation*}Continuity of the scattering coefficients s_{\\gamma \\sigma }(\\omega ) follows from the definition of U(k,\\omega ) and the fact that it has finitely many poles contained in \\Omega .\\end{xmlelement*}}\\end{Lemma}\\end{Lemma}\\begin{Cor}Let \\hat{\\omega }\\in \\overline{\\mathbb {D}}, then\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu }(k,\\omega )={\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\nu \\sigma }(k,\\omega )=0,\\quad \\hat{\\omega }\\in T^{+}_\\sigma \\cup T_\\nu ^+,\\quad \\nu \\in \\mathcal {C}\\end{equation*}\\end{Cor}{\\begin{xmlelement*}{proof}Denote\\begin{equation*}A_{\\nu \\sigma }=\\text{det}(\\tilde{T}_{\\nu \\sigma }(\\omega )),\\quad \\omega \\in \\overline{\\mathbb {D}},\\quad \\nu ,\\sigma \\in \\mathcal {C},\\end{equation*}where \\tilde{T}_{\\nu \\sigma }(\\omega ) denotes the matrix which comes from T(\\omega ) once we have removed the \\sigma -column and the \\nu -row.", "\\\\Let now \\omega _0\\in , \\lambda (\\omega _0), be a regular point of \\mathcal {R}(\\lambda (\\omega )) and \\det (T(\\omega _0))=0.", "Then there exists \\mathbf {x}=\\lbrace x_\\sigma \\rbrace \\in l^2(\\mathcal {C}) such that T(\\omega _0)\\mathbf {x}=0, thus the vector function \\vec{\\xi }(k)= \\mathcal {E}(k,\\omega _0)\\mathbf {x} satisfies the equations (\\ref {eq:02}) on the channels as well as the boundary conditions (\\ref {eq:08b}).", "Hence it can be prolongated to a solution of the whole problem (\\ref {eq:01}).", "\\\\For \\mathbf {x}\\in ł^2(\\mathcal {C}), denote \\mbox{supp} \\ \\mathbf {x}=\\lbrace \\sigma \\in \\mathcal {C};\\text{ } x_\\sigma \\ne 0 \\rbrace and let \\xi =\\lbrace \\xi (\\alpha )\\rbrace _{\\alpha \\in \\mathcal {A}} be a solution to (\\ref {eq:01}) with \\lambda =\\lambda (\\omega _0) which is obtained by prolongation of\\vec{\\xi }(k).", "Then \\eta (\\alpha )= \\bar{\\xi }(\\alpha ) also solves this problem.", "Relation (\\ref {eq:38}) yields0= \\sum _{\\sigma \\in \\mbox{supp} \\ \\mathbf {x}} {\\mathfrak {b}}_{\\sigma }(0)\\lbrace \\xi ,\\eta \\rbrace _{\\sigma }(0)=\\sum _{\\sigma \\in \\mbox{supp} \\ \\mathbf {x}} b_\\sigma |x_\\sigma |^2 (\\bar{\\theta }_\\sigma (\\omega _0)- {\\theta }_\\sigma (\\omega _0))Therefore \\omega _0\\in T_\\sigma ^- for each \\sigma \\in \\mbox{supp} \\ \\mathbf {x}.", "Thus A_{\\sigma \\nu }(\\omega _0)=0, for \\omega _0\\in T_\\nu ^+, \\sigma \\in \\mathcal {C}.\\\\Since the poles of U(k,\\omega ) are simple and (\\ref {eq:36}) implies that e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0 if \\omega \\in T_\\nu ^+, we obtain{\\mathbf {\\rm {Res}}}_{\\omega _0} u_{\\nu \\sigma }(k,\\omega )=0A simple application of lemma \\ref {L:6} gives us {\\mathbf {\\rm {Res}}}_{\\omega _0} u_{\\sigma \\nu }(k,\\omega )=0 \\end{xmlelement*}}}}$ Corollary 6.1 Let $\\omega \\in T_\\sigma ^+\\cup T_\\nu ^+$ , $\\sigma ,\\nu \\in \\mathcal {C}$ and $\\sigma \\ne \\nu $ .", "Then $s_{\\nu \\sigma }(\\omega ^{-1})=\\overline{s_{\\nu \\sigma }}(\\omega )$ In particular $|s_{\\nu \\sigma (\\omega )}|^2=s_{\\nu \\sigma }(\\omega )s_{\\nu \\sigma }(\\omega ^{-1})$ We have $u_{\\nu \\sigma }(k,\\omega )=e_\\nu (k,\\theta _\\nu (\\omega ))s_{\\nu \\sigma }(\\omega )$ and by construction of the matrix $U(k,\\omega )$ , we know that $u_{\\nu \\sigma }(k,\\omega ^{-1})=\\overline{u_{\\nu \\sigma }}(k,\\omega ),\\quad \\omega \\in T_\\sigma ^+\\cup T_\\nu ^+$ and the corollary follows." ], [ "Discrete spectra of the operator $\\mathcal {L}$ ", "Lemma 7.1 Let $\\hat{\\omega }\\in \\bar{\\setminus \\lbrace 0\\rbrace be a pole of U(k,\\omega ).", "Then \\lambda (\\hat{\\omega })=a-b(\\hat{\\omega }+\\hat{\\omega }^{-1})is an eigenvalue of (\\ref {eq:01}).", "If in addition \\hat{\\omega }\\in T_\\sigma ^+, then all elements u_{\\sigma \\nu }(k,\\omega ), \\nu \\in \\mathcal {C} areregular at \\hat{\\omega }.", "}{\\begin{xmlelement*}{proof}Let \\hat{\\omega }\\in \\bar{\\setminus \\lbrace 0\\rbrace be a (simple) pole of U(k,\\omega ).", "Denote\\begin{equation*}a_\\nu (\\omega )= \\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\theta _\\nu ( \\omega )-\\theta _\\nu ( \\omega )^{-1}} (\\omega -\\hat{\\omega }), \\ \\nu \\in \\mathcal {C}; \\quad A(\\omega )= {\\mathbf {\\rm {diag}}}\\lbrace a_\\nu (\\omega )\\rbrace _{\\nu \\in \\mathcal {C}}.\\end{equation*}We then have\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} U(k,\\omega )=\\lim _{\\omega \\rightarrow \\hat{\\omega }} U(k,\\omega )A(\\omega ).\\end{equation*}For each \\nu \\in \\mathcal {C} and for each \\omega \\ne \\hat{\\omega } the \\nu -th column of U(k,\\omega )A(\\omega )\\vec{\\phi }^\\nu (k,\\omega )= \\left( \\phi ^\\nu (\\sigma (k),\\omega ) \\right)_{\\sigma \\in \\mathcal {C}}=U(k,\\omega )A(\\omega ) \\mathbf {n}_\\nu can be prolongated into \\mathcal {A}_1 to a solution of (\\ref {eq:01}) with \\lambda =\\lambda (\\omega ) according (\\ref {eq:07}):\\begin{equation*}\\phi ^\\nu (\\alpha , \\omega )= \\sum _{\\gamma \\in \\mathcal {C}} r(\\alpha , \\gamma (0); \\lambda ) {\\mathfrak {b}}_\\gamma (0)\\phi ^\\nu (\\gamma (1),\\omega ); \\ \\alpha \\in \\mathcal {A}_1\\end{equation*}Since U(k,\\omega ) has a simple pole at \\hat{\\omega }, there exists the limit\\begin{equation*}\\vec{\\phi }^\\nu (k, \\hat{\\omega })=\\lim _{\\omega \\rightarrow \\hat{\\omega }} \\vec{\\phi }^\\nu (k, \\omega )\\end{equation*}\\medskip }{\\bf Claim} {\\em The vector \\vec{\\phi }^\\nu (k,\\hat{\\omega }) also can be prolongated to a solution of the problem(\\ref {eq:01})}.\\\\In case \\lambda (\\hat{\\omega }) is not a pole of \\mathcal {R}(\\lambda (\\omega )), the prolongation is straightfoward.", "By (\\ref {eq:07}), if \\lambda (\\hat{\\omega }) is at the same time an eigenvalue of \\mathcal {L}_1 one can apply the same reasonings as in lemma 4.3 in \\cite {LM}.", "We omit the details.\\\\ \\\\Let now\\begin{equation*}T(\\omega )^{-1}=(\\tau _{\\sigma \\nu }(\\omega ))_{\\sigma ,\\nu \\in \\mathcal {C}},\\end{equation*}and denote\\begin{equation}h_{\\sigma \\nu }(\\omega )= -\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1, \\theta _\\nu (\\omega ))}(\\omega -\\hat{\\omega })\\tau _{\\sigma \\nu }(\\omega ).\\end{equation}It follows from (\\ref {eq:64}) that for \\sigma \\ne \\nu we have\\begin{equation}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu } (k,\\cdot )= \\phi ^\\nu (\\sigma (k),\\hat{\\omega }) = e_\\sigma (k,\\hat{\\omega })h_{\\sigma \\nu }(\\hat{\\omega })\\end{equation}here we denote {\\mathfrak {m}}_{\\sigma \\nu }(\\hat{\\omega })=h_{\\sigma \\nu }(\\hat{\\omega }).\\\\ \\\\If e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0, representation (\\ref {Eq:33}) is valid for \\sigma =\\nu as well.Assume that e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))= 0.", "Then\\begin{equation}p_\\nu (k,\\hat{\\omega })e_\\nu (0,\\theta _\\nu (\\hat{\\omega }))=e_\\nu (k, \\theta _\\nu (\\hat{\\omega })),\\end{equation}because the expressions in both sides satisfy the same recurrence equation and the same initial conditions, and also it follows from lemma \\ref {le:02a}that \\dot{e}_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0 because \\ e_\\nu (1,\\theta _\\sigma (\\hat{\\omega }) ) and \\dot{e}_\\nu (1,\\theta _\\sigma (\\hat{\\omega }) ) cannot vanish simultaneously.Thus we again obtain (\\ref {Eq:33}), yet now\\begin{equation*}m_{\\nu \\nu }(\\hat{\\omega })= \\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu (\\hat{\\omega }),\\dot{e}_\\nu (\\hat{\\omega })\\rbrace _\\nu (0)}+h_{\\nu \\nu }(\\hat{\\omega }).\\end{equation*}Representation (\\ref {Eq:33}) is now valid for all \\nu ,\\sigma \\in \\mathcal {C}.", "\\\\ \\\\If \\hat{\\omega }\\in it follows from (\\ref {Eq:33}) that \\phi ^\\nu (\\alpha ,\\hat{\\omega }) is an eigenfunction of \\mathcal {L} with \\lambda (\\hat{\\omega }) as eigenvalue.", "If \\hat{\\omega }\\in , we may have \\theta _\\sigma (\\hat{\\omega })\\in , i.e., \\hat{\\omega }\\in T^+_\\sigma for some \\sigma \\in \\mathcal {C}.", "It follows from corollary \\ref {Rem:51} that for such \\sigma we have m_{\\sigma \\nu }(\\hat{\\omega })=0, in particular\\begin{equation*}{\\mathbf {\\rm {Res}}}_{\\hat{\\omega }} u_{\\sigma \\nu }(k, \\cdot )=\\phi ^\\nu (\\sigma (k), \\hat{\\omega })=0, \\ \\hat{\\omega }\\in \\theta _\\sigma (\\hat{\\omega }) \\in \\end{equation*}and, again \\phi ^\\nu (\\alpha ,\\hat{\\omega }) is an eigenfunction of \\mathcal {L} with \\lambda (\\hat{\\omega }) as eigenvalue.\\end{xmlelement*}}$ Consider the matrix ${\\mathfrak {m}}(\\hat{\\omega })= \\left( m_{\\sigma \\nu }(\\hat{\\omega }) \\right)_{\\sigma ,\\nu \\in \\mathcal {C}}$ Properties of ${\\mathfrak {m}}(\\hat{\\omega })$ are summarized in the statement below Lemma 7.2 Let $\\hat{\\omega }\\in \\Omega $ .", "Then ${\\mathbf {\\rm {Res}}}U(k,\\hat{\\omega })= \\mathcal {E}(k, \\hat{\\omega }){\\mathfrak {m}}(\\hat{\\omega }), \\ k=0,1,2, \\ldots .$ The diagonal elements $m(\\nu ,\\nu ; \\hat{\\omega })$ satisfy $\\Vert \\phi ^\\nu (\\hat{\\omega })\\Vert ^2=-\\frac{b_\\nu (1-\\theta _\\nu (\\hat{\\omega })^{-2})}{b(1- \\bar{\\hat{\\omega }}^{-2})}\\theta _\\nu (\\hat{\\omega })m_{\\nu \\nu }(\\hat{\\omega }),$ where $\\phi ^\\nu =\\phi ^\\nu (\\alpha , \\hat{\\omega })$ is the eigenvector of $\\mathcal {L}$ , corresponding to the eigenvalue $\\lambda (\\hat{\\omega })$ and such that $\\phi ^\\nu (\\sigma (k), \\hat{\\omega }) = e_\\sigma (k,\\theta _\\sigma (\\hat{\\omega })) m(\\sigma ,\\nu ;\\hat{\\omega }) \\ \\sigma \\in \\mathcal {C}, \\ k\\ge 0.$ Relations (REF ) and (REF ) are already established in lemma REF .", "It remains to prove (REF ).", "We apply Lemma REF with $\\xi (\\alpha )= \\phi ^\\nu (\\alpha ,\\hat{\\omega }) $ : $\\dot{\\lambda }(\\hat{\\omega })\\sum _{\\alpha \\in \\mathcal {A}_1} |\\phi ^\\nu (\\alpha ,\\hat{\\omega })|^2=\\sum _{\\sigma \\in \\mathcal {C}}{\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\sigma (0)$ and calculate the Wronskians in the right-hand side of this equality.", "We then have $\\phi ^\\nu (\\sigma (k),\\omega ) = p_\\sigma (k,\\omega )h_\\nu (\\omega )\\delta _{\\sigma ,\\nu }+e_\\sigma (k,\\omega )h_{\\sigma \\nu }(\\omega ),$ with $h_\\nu (\\omega )=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1, \\theta _\\nu (\\omega ))}(\\omega -\\hat{\\omega }),$ and $h_{\\sigma \\nu }$ is already defined in ().", "Let $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ .", "Then, for all $\\sigma ,\\nu \\in \\mathcal {C}$ , $h_\\nu (\\hat{\\omega })=0, \\ \\phi ^\\nu (\\sigma (k),\\hat{\\omega })= e_\\sigma (k,\\theta _\\sigma (\\hat{\\omega }))h_{\\sigma \\nu }(\\hat{\\omega }),$ and $m_{\\sigma \\nu }(\\hat{\\omega })=h_{\\sigma \\nu }(\\hat{\\omega }), \\ {\\mathfrak {m}}(\\hat{\\omega })=\\left( m_{\\sigma \\nu }(\\hat{\\omega })\\right)_{\\sigma ,\\nu \\in \\mathcal {C}}$ We use (REF ), (REF ), (REF ), (REF ) and that $\\dot{p}_k(\\omega )=0$ for $k=0,1$ as follows from (): $\\dot{\\phi }^\\nu (\\sigma (k),\\omega ) = p_\\sigma (k,\\omega )\\dot{h}_\\nu (\\omega )\\delta _{\\sigma ,\\nu }+\\dot{e}_\\sigma (k,\\omega )h_{\\sigma \\nu }(\\omega )+e_\\sigma (k,\\omega )\\dot{h}_{\\sigma \\nu }(\\omega ), \\ k=0,1.$ Besides $\\dot{h}_\\nu (\\hat{\\omega })=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))}.$ Since $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ , we also have $h_\\nu (\\hat{\\omega })=0$ and, according to (REF ) and (), ${\\mathfrak {b}}_\\sigma (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\sigma (0)=b_\\nu (\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1})\\overline{m_{\\sigma \\nu }(\\hat{\\omega })} \\delta _{\\sigma ,\\nu } + \\\\{\\mathfrak {b}}_\\sigma (0) \\lbrace \\dot{e}_\\sigma (\\hat{\\omega }), e_\\sigma (\\theta _\\sigma (\\hat{\\omega }))\\rbrace _\\sigma (0) |m_{\\sigma \\nu }(\\hat{\\omega })|^2 = \\\\b_\\nu (\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1})\\overline{m_{\\sigma \\nu }(\\hat{\\omega })} \\delta _{\\sigma ,\\nu } -\\dot{\\lambda }(\\hat{\\omega })\\sum _{k=1}^\\infty |\\phi ^\\nu (\\sigma (k), \\theta _\\sigma (\\hat{\\omega }))|^2.$ We can now return to (REF ) in order to obtain normalization condition (REF ) for the matrix ${\\mathfrak {m}}$ : In the case $e_\\nu (1,\\hat{\\omega })=0$ representation (REF ) is still valid, yet $h_\\nu (\\hat{\\omega })=\\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu ,\\dot{e}_\\nu \\rbrace _\\nu (0)}e_\\nu (0,\\hat{\\omega })\\ne 0.$ Taking () into account we again obtain (), yet now $m_{\\sigma \\nu }(\\hat{\\omega })= \\frac{b_\\nu }{{\\mathfrak {b}}_\\nu (0)}\\frac{\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}}{\\lbrace e_\\nu (\\hat{\\omega }),\\dot{e}_\\nu (\\hat{\\omega })\\rbrace _\\nu (0)}\\delta _{\\sigma ,\\nu }+h_{\\sigma \\nu }(\\hat{\\omega }).$ Relation (REF ) is still valid and for $\\nu \\ne \\sigma $ we arrive to relation (REF ).", "For $\\sigma =\\nu $ we have ${\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{\\phi }^\\nu (\\hat{\\omega }), \\overline{\\phi ^\\nu (\\hat{\\omega })} \\rbrace _\\nu (0)={\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{e}_\\nu (\\hat{\\omega })h_2(\\hat{\\omega }),\\overline{e_\\nu (\\hat{\\omega })}\\rbrace _\\nu (0) \\overline{m_{\\nu \\nu }(\\hat{\\omega })}= \\\\{\\mathfrak {b}}_\\nu (0)\\lbrace \\dot{e}_\\nu (\\hat{\\omega }),\\overline{e_\\nu (\\hat{\\omega })}\\rbrace _\\nu (0) |m(\\nu ,\\nu ;\\omega )|^2-\\ {b_\\nu } ({\\theta _\\nu (\\hat{\\omega })-\\theta _\\nu (\\hat{\\omega })^{-1}})\\overline{m_{\\nu \\nu }(\\hat{\\omega })},$ and we again arrive to (REF ).", "Now one can complete the proof in the same way as if $e_\\nu (1,\\theta _\\nu (\\hat{\\omega }))\\ne 0$ .", "Remark The matrix $\\overline{{\\mathfrak {m}}(\\hat{\\omega })}$ corresponds to the point of discrete spectra $\\lambda (\\hat{\\omega })$ .", "Its columns are normalized eigenfunctions.", "This normalizations is defined by relation REF and is therefore unique.", "We will see in the next section that $m_{\\nu \\nu }(\\hat{\\omega })$ , i.e.", "the energies of the normalized eigenfunctions, are the quantities which participate in the equations for the inverse scattering problem" ], [ "In order to obtain equations of the inverse scattering problem we introduce the matrix function $\\Delta _l(\\omega ):= {\\mathbf {\\rm {diag}}}\\left\\lbrace \\theta _\\nu (\\omega )^{l-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }\\right\\rbrace _{\\nu \\in \\mathcal {C}}, \\ l\\in {\\mathbb {Z}}$ and consider the integral ${\\mathcal {J}}(l,k) = \\left( j_{\\nu \\sigma }(l,k) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}= \\frac{1}{2\\pi I} \\int __l(\\omega ) U(k,\\omega ) d\\omega .$ Since $ may contain (simple) poles of $ U(k,) $ this integral as well as all integrals in this section is considered in principal value.", "We will calculate this integral in two ways: through the residues, this would correspond to the contribution of the discrete spectra, and through the scattering coefficients, this would correspond to the contribution of the continuos spectra.\\\\ \\\\Comparing two different expressions for $ Jl,k$ leads one to the equations of the inverse scattering problem.$ Let as before $U(k,\\omega )= (u_{\\nu \\sigma }(k,\\omega ))_{\\nu ,\\sigma }$ .", "Then $j_{\\nu \\sigma }(l,k)= \\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}u_{\\nu \\sigma }(k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega = \\\\\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}\\psi ^\\sigma _\\nu (k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ here $\\psi ^\\nu (\\cdot , \\omega )$ is the special solution, defined in (REF ).", "Assume first that $\\sigma \\ne \\nu $ .", "Then, according to () and (REF ) $j_{\\nu \\sigma }(l,k) =\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}e_\\nu (k,\\theta _\\nu (\\omega ))s_{\\sigma \\nu }(\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ Consider the function $q_{\\nu \\sigma }(n)=\\frac{1}{2\\pi I}\\int _s̰_{\\sigma \\nu }(\\omega ) \\theta _\\nu (\\omega )^{n-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega .$ Relation () now yields $j_{\\nu \\sigma }(l,k)=c_\\nu (k)\\sum _{m\\ge k} q_{\\nu \\sigma }(m+l)a_\\nu (k,m),$ this is the desired expression.", "Remark The functions $q_{\\nu ,\\sigma }$ cannot be determined from the spectral data generally speaking.", "However relation (REF ) determines the structure of the equation of the inverse scattering problem.", "Later in section REF we will use this structure to get rid of the functions which cannot be observed from the spectral data.", "Let now $\\nu =\\sigma $ and $T(\\omega )^{-1}= (\\tau _{\\sigma \\nu }(\\omega ))_{\\sigma ,\\nu }$ .", "Relations (REF ) (REF ) yield $j_{\\sigma \\sigma }(k,l)= \\\\ \\frac{1}{2\\pi I} \\int _\\lbrace \\bigl [-p_\\sigma (k,\\omega ) + e_\\sigma (k,\\theta _\\sigma )\\tau _{\\sigma \\sigma }(\\omega )\\bigr ] \\frac{b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma )}{{\\mathfrak {b}}_\\sigma (0)e_\\sigma (1,\\theta _\\sigma )} \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma (\\omega )}{d\\omega } \\Bigr \\rbrace d\\omega = \\\\\\frac{1}{2\\pi I} \\Bigl ( \\int _{T_\\sigma ^+} + \\int _{T_\\sigma ^-}\\Bigr ) \\Bigl \\lbrace \\ \\cdot \\ \\Bigr \\rbrace = j_{\\sigma \\sigma }^+(k,l) + j_{\\sigma \\sigma }^-(k,l).$ As $\\omega $ runs over $T_\\sigma ^+$ the function $\\theta _\\sigma (\\omega )$ runs over the whole $.", "Let $ ()$ be the inverse function.Relations (\\ref {Eq:02a01}) together with (\\ref {eq:23}) yield\\begin{equation*}j_{\\sigma \\sigma }^+(k,l) = \\frac{1}{2\\pi I} \\int _[ e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )\\bigr ] \\theta _\\sigma ^{l-1} d\\theta _\\sigma .\\end{equation*}Besides it follows from corollary \\ref {Rem:51} that $ s()$ is bounded on $ +$.\\\\Consider the Fourier series of $ s(())$:\\begin{equation*}s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))=\\sum _{n=-\\infty }^\\infty \\tilde{s}_\\sigma (n) \\theta ^{-n}; \\ \\tilde{s}_\\sigma (n) = \\frac{1}{2 \\pi I} \\int _s̰_{\\sigma \\sigma }(\\omega (\\theta _\\sigma )) \\theta _\\sigma ^{n-1} d \\theta _\\sigma .\\end{equation*}Together with (\\ref {eq:13}) this yields\\begin{multline*}e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )= \\\\c_\\sigma (k) \\sum _m \\bigl \\lbrace a_\\sigma (k,m) + \\sum _n a_\\sigma (k,n) \\tilde{s}_\\sigma (m+n) \\bigr \\rbrace \\theta _\\sigma ^{-m},\\end{multline*}and finally\\begin{equation}j_{\\sigma \\sigma }^+(k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\tilde{s}_\\sigma (l+n)\\right] .\\end{equation}$ We now study $j^-_{\\sigma \\sigma }(k,l)$ .", "Denote $a_1(k,\\omega )= - p_\\sigma (k,\\omega ) \\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma );$ $a_2(\\omega )=\\tau _{\\sigma \\sigma }(\\omega )\\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma ).$ We then have $\\psi ^\\sigma _\\sigma (k,\\omega )= a_1(k,\\omega )+e_\\sigma (k,\\theta _\\sigma (\\omega ))a_2(\\omega ),$ and $j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_1(k,\\omega ) \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega + \\\\\\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_2( \\omega ) e_\\sigma (k,\\theta _\\sigma (\\omega ))\\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega .$ As $\\omega $ runs over $T_\\sigma ^-$ , the function $\\theta _\\sigma (\\omega )$ runs twice in the opposite directions over $J_\\sigma =\\theta _\\sigma (T_\\sigma ^-)\\subset {\\mathbb {R}}$ and the values $\\theta _\\sigma (\\omega )$ and $\\theta _\\sigma (\\omega ^{-1})$ coinside.", "Respectively $e_\\sigma (k,\\theta _\\sigma (\\omega )), p_\\sigma (k,\\omega ) \\in {\\mathbb {R}}$ and $e_\\sigma (k,\\theta _\\sigma (\\omega ))=e_\\sigma (k,\\theta _\\sigma (\\omega ^{-1}))$ , $ p_\\sigma (k,\\omega ) = p_\\sigma (k,\\omega ^{-1}) $ , thus $a_1(k,\\omega )=a_1(k,\\omega ^{-1})$ .", "Besides $ a_2(\\omega ) = \\overline{a_2(\\omega ^{-1})}$ , since $\\psi ^\\sigma (k,\\omega )= \\overline{\\psi ^\\sigma (k,\\omega ^{-1})}$ for $\\omega \\in .\\\\ \\\\Therefore\\begin{equation}j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{\\Gamma _\\sigma } [a_2(\\omega )-a_2(\\omega ^{-1}) ] e_\\sigma (k,\\theta _\\sigma )\\theta _\\sigma ^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega ,\\end{equation}where $ =T-+$.\\\\ \\\\The function $ a1(k,)$ satisfies equation (\\ref {eq:02}) on the channel $$ andrelations (\\ref {eq:66a}), (\\ref {eq:66b}) yield\\begin{equation*}a_1(k,\\omega )= \\theta _\\sigma (\\omega )^{-k}(1+o(1)).\\end{equation*}Therefore$ ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)= b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega ))[ a_2(\\omega )-a_2(\\omega ^{-1})].$ On the other hand by (REF ) ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)=- \\sum _{\\nu \\in \\mathcal {C}\\setminus \\lbrace \\sigma \\rbrace }{\\mathfrak {b}}_\\nu (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0).$ We have $\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)=0, \\ \\omega \\in T_\\nu ^-.$ For $\\omega \\in T_\\nu ^+\\cap T_\\sigma ^-$ we obtain ${\\mathfrak {b}}_\\nu (0)\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)= \\\\{\\mathfrak {b}}_\\nu (0) \\lbrace s_{\\nu \\sigma }(\\omega )e_\\nu (\\theta _\\nu (\\omega )), s_{\\nu \\sigma }(\\omega ^{-1})e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) \\\\|s_{\\nu \\sigma }(\\omega )|^2 {\\mathfrak {b}}_\\nu (0)\\lbrace e_\\nu (\\theta _\\nu (\\omega )), e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) = \\\\|s_{\\nu \\sigma }(\\omega )|^2 b_\\nu (\\theta _\\nu (\\omega )^{-1}-\\theta _\\nu (\\omega )) =- |s_{\\sigma \\nu }(\\omega )|^2 \\frac{ ( b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma ))^2}{b_\\nu (\\theta _\\nu ^{-1}-\\theta _\\nu )}.$ Now, for $\\theta \\in J_\\sigma $ , we define $\\omega _\\sigma (\\theta )=W^{-1}\\circ W_\\sigma (\\theta )$ , here the functions $W$ , $W_\\sigma $ are defined by (REF ) and the branch is chosen so that $\\omega _\\sigma (\\theta )\\in \\Gamma _\\sigma $ .", "Let $N_\\sigma (\\theta )= \\lbrace \\nu ; \\ \\omega _\\sigma (\\theta )\\in T_\\nu ^+ \\rbrace , \\ \\theta \\in J_\\sigma .$ Denote $\\Phi _\\sigma (\\theta )= \\sum _{\\nu \\in N_\\sigma (\\theta )}|s_{\\sigma \\nu }(\\omega _\\sigma )|^2 \\left(\\frac{ b_\\sigma (\\theta ^{-1}-\\theta ) }{b_\\nu (\\theta _\\nu (\\omega _\\sigma )^{-1}-\\theta _\\nu (\\omega _\\sigma ))} \\right), \\ \\omega _\\sigma (\\theta )\\in T_\\sigma ^-.$ We finally obtain $a_2(\\omega )-a_2 (\\omega ^{-1})= \\Phi _\\sigma (\\theta _\\sigma (\\omega ))$ this function is expressed via the spectral data.", "Remark All functions $|s_{\\nu \\sigma }(\\omega )|$ participating in (REF ) are well defined and continuous.", "In addition, $s_{\\nu \\sigma }(\\omega ^{-1})=\\overline{s_{\\nu \\sigma }}(\\omega )$ for $\\omega \\in \\ T_\\nu ^+$ .", "Relation () now takes the form $j_{\\sigma \\sigma }^-(k,l)= \\frac{ 1 }{2\\pi I} \\int _{J_\\sigma } e_\\sigma (k,\\theta ) \\Phi _\\sigma (\\theta ) \\theta ^{l-1} d\\theta ,$ and with account () we obtain $j_{\\sigma \\sigma }^-(k,l)= c_\\sigma (k)\\sum _n a_\\sigma (k,n) q_{\\sigma \\sigma }(n+l),$ where $q_{\\sigma \\sigma }(n)= \\frac{1 }{2\\pi I} \\int _{J_\\sigma } \\Phi _\\sigma (\\theta )\\theta ^{n-1} d\\theta .$ Combining (REF ), (), and (REF ) we finally obtain $j_{\\sigma \\sigma } (k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\left( \\tilde{s}_\\sigma (l+n)+q_{\\sigma \\sigma }(n+l) \\right)\\right],$ the functions $\\tilde{s}_\\sigma (\\cdot )$ and $q_{\\sigma ,\\sigma }(\\cdot )$ are defined through the scattering data." ], [ "Let now $\\Theta (\\omega )= {\\mathbf {\\rm {diag}}}\\lbrace \\theta _\\sigma (\\omega ) \\rbrace _{\\sigma \\in \\mathcal {C}}.$ We use the representation () for $\\mathcal {E}(k,\\omega )$ , $\\mathcal {E}(k,\\omega )=C(k)\\sum _{m=-\\infty }^\\infty A(k,m)\\Theta (\\omega )^m.$ The coefficients $C(k)$ and $A(k,m)$ are diagonal matrices $C(k)={\\mathbf {\\rm {diag}}}\\lbrace c_\\sigma (k)\\rbrace _{\\sigma \\in \\mathcal {C}}, \\ A(k,m)={\\mathbf {\\rm {diag}}}\\lbrace a_\\sigma (k,m)\\rbrace _{\\sigma \\in \\mathcal {C}},$ and also $A(k,m)=0$ for $m<k$ .", "It follows from (REF ) and (REF ) that ${\\mathcal {J}}(l,k)=\\frac{1}{2\\pi I}\\int __l(\\omega )U(k,\\omega )d\\omega = \\\\C(k) \\left\\lbrace A(k,l)+\\sum _{l=-\\infty }^\\infty A(k,m) Z(l+m) \\right\\rbrace ,$ here the integral is taken as a principal value and the matrix $Z(n)=\\left( z_{\\nu ,\\sigma }(n) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}$ is given by the relations $z_{\\nu \\sigma }(n)=q_{\\nu \\sigma }(n), \\ \\nu \\ne \\sigma , \\\\z_{\\sigma \\sigma }(n)=\\tilde{s}_\\sigma (n)+q_{\\sigma \\sigma }(n),$ here $\\tilde{s}_\\sigma (n)$ are the Fourier coefficients of the reflection coefficient $s_{\\sigma \\sigma }$ with respect to $\\theta _\\sigma $ and the function $q_{\\nu \\sigma }$ is defined in (REF ) and (REF ).", "Let now $\\Omega \\subset \\bar{ be the set of all poles of U(k,\\omega ).", "It follows from (\\ref {eq:64}) that \\overline{U(k,\\omega )}=U(k,\\bar{\\omega }), thus for each \\hat{\\omega }\\in \\Omega \\cap the point \\bar{\\hat{\\omega }} also belongs to \\Omega .", "Moreover, for such \\hat{\\omega } we have \\overline{\\text{Res}_{\\hat{\\omega }}U(k,\\omega )}=\\text{Res}_{\\bar{\\hat{\\omega }}}U(k,\\omega ).Denote \\tilde{\\Omega }= \\lbrace \\hat{\\omega }\\in \\Omega : |\\hat{\\omega }| < 1\\rbrace \\cup \\lbrace \\hat{\\omega }\\in \\Omega : \\Im \\hat{\\omega }>0\\rbrace .", "We use (\\ref {Eq:39}) and (\\ref {Eq:01}):}$ ${\\mathcal {J}}(k,l)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} \\Delta _l(\\hat{\\omega })\\mathcal {E}(k,\\hat{\\omega }){\\mathfrak {m}}(\\hat{\\omega })=C(k)\\sum _mA(k,m)M(m+l),$ where $M(n)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} {\\mathbf {\\rm {diag}}}\\left\\lbrace \\frac{d\\theta _\\sigma }{d\\omega }(\\hat{\\omega }) \\right\\rbrace _{\\sigma \\in \\mathcal {C}}\\Theta (\\hat{\\omega })^{n-1}\\Re {\\mathfrak {m}}(\\hat{\\omega }).$ By compare this to (REF ) and taking into account that $A(k,m)=0$ for $m<k$ and $A(k,k)=I$ , we obtain a system of equations $A(k,m)+\\sum _{s=k}^\\infty A(k,s)F(s+m)=0,\\quad m=k+1,k+2, \\ldots \\,$ where $F(n)=Z(n)-M(n).$ Since $A(k,k)=I$ , this relation can be written as $F(k+m)+A(k,m)+\\sum _{s=k+1}^\\infty A(k,s)F(s+m)=0, \\quad k=1,2, \\ldots ,\\quad m>k.$ The matrices $A(k,m)$ are diagonal.", "The diagonal elements of $F(n)$ , can be expressed through the spectral data.", "The non-diagonal elements of the matrices $F(n)$ vanish, as follows from the lemma below.", "Lemma 8.1 The matrices $F(n)$ are diagonal for all $n\\ge 1$ : $F(n)={\\mathbf {\\rm {diag}}}\\lbrace f_\\sigma (n)\\rbrace _{\\sigma \\in \\mathcal {C}}$ .", "and their diagonal elements $f_\\sigma (n)$ are determined by $a_\\sigma (k,m)$ , $m>k\\ge \\Big [\\frac{n-1}{2}\\Big ]$ only.", "This statement is proved in [12], (Lemma 5.1) and we omit the proof.", "Theorem 8.1 The following properties hold for the systems under consideration 1) Equations (REF ) split into a system of independent scalar equations $f_\\nu (k+m)+a_\\nu (k,m)+\\sum _{s=k+1}^\\infty a_\\nu (k,s)f_\\nu (s+m)=0, \\,m\\ge k+1\\ge 1,$ here $f_\\nu (n)$ is defined in (REF ), (REF ), ().", "2) For each $k\\ge 0$ equations (REF ) has unique solutions $a_\\sigma (k,m)$ .", "The first statement follows directly from relation (REF ) which defines $F(n)$ and Lemma REF .", "It follows from the same lemma that the functions $f_\\nu (n)$ , $n\\ge 1$ are uniquely defined by the coefficients ${\\mathfrak {b}}_\\nu (k)$ , ${\\mathfrak {a}}_\\nu (k)$ , $k\\ge 1$ .", "Therefore (REF ) coincide with equations for inverse scattering problem for equation (REF ) with boundary condition $\\xi (\\nu (0))=0$ .", "It is well-known (see for example [14]) that the later has unique solution." ], [ "Concluding remarks", "In case that the continuos spectra $I=\\cup _{\\sigma \\in \\mathcal {C}}[a_\\sigma -2b_\\sigma ,a_\\sigma +2b_\\sigma ]$ splits into a number of disjoints intervals, one can repeat the procedure separately for each connected component of $I$ .", "If, say, $I^{(0)}$ is a connected component of $I$ and $\\sigma $ is a channel corresponding to this component, then each wave incoming along $\\sigma $ generates decaying waves in all channels which correspond to other connected components of $I$ .", "We omit the details.", "So far we have discussed reconstruction of the part $\\mathcal {A}_0$ of the whole system, that is, the channels.", "This information is, generally speaking, insufficient for reconstruction the whole matrix $\\mathcal {L}$ .", "However if the matrix $\\mathcal {L}_1$ corresponding to the \"central\" part of the system is sufficiently sparse and also we know the matrix $B(0)=\\text{diag}\\lbrace {\\mathfrak {b}}_\\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}}$ which realizes connections between the channels and the central part of the system, the whole matrix $\\mathcal {L}$ can be recovered from the scattering data.", "We refer the reader to Chapter 11 in [15], where statemets of such type are obtained." ], [ "In order to obtain equations of the inverse scattering problem we introduce the matrix function $\\Delta _l(\\omega ):= {\\mathbf {\\rm {diag}}}\\left\\lbrace \\theta _\\nu (\\omega )^{l-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }\\right\\rbrace _{\\nu \\in \\mathcal {C}}, \\ l\\in {\\mathbb {Z}}$ and consider the integral ${\\mathcal {J}}(l,k) = \\left( j_{\\nu \\sigma }(l,k) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}= \\frac{1}{2\\pi I} \\int __l(\\omega ) U(k,\\omega ) d\\omega .$ Since $ may contain (simple) poles of $ U(k,) $ this integral as well as all integrals in this section is considered in principal value.", "We will calculate this integral in two ways: through the residues, this would correspond to the contribution of the discrete spectra, and through the scattering coefficients, this would correspond to the contribution of the continuos spectra.\\\\ \\\\Comparing two different expressions for $ Jl,k$ leads one to the equations of the inverse scattering problem.$ Let as before $U(k,\\omega )= (u_{\\nu \\sigma }(k,\\omega ))_{\\nu ,\\sigma }$ .", "Then $j_{\\nu \\sigma }(l,k)= \\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}u_{\\nu \\sigma }(k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega = \\\\\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}\\psi ^\\sigma _\\nu (k,\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ here $\\psi ^\\nu (\\cdot , \\omega )$ is the special solution, defined in (REF ).", "Assume first that $\\sigma \\ne \\nu $ .", "Then, according to () and (REF ) $j_{\\nu \\sigma }(l,k) =\\frac{1}{2\\pi I}\\int __\\nu (\\omega )^{l-1}e_\\nu (k,\\theta _\\nu (\\omega ))s_{\\sigma \\nu }(\\omega )\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega ,$ Consider the function $q_{\\nu \\sigma }(n)=\\frac{1}{2\\pi I}\\int _s̰_{\\sigma \\nu }(\\omega ) \\theta _\\nu (\\omega )^{n-1}\\frac{d\\theta _\\nu (\\omega )}{d\\omega }d\\omega .$ Relation () now yields $j_{\\nu \\sigma }(l,k)=c_\\nu (k)\\sum _{m\\ge k} q_{\\nu \\sigma }(m+l)a_\\nu (k,m),$ this is the desired expression.", "Remark The functions $q_{\\nu ,\\sigma }$ cannot be determined from the spectral data generally speaking.", "However relation (REF ) determines the structure of the equation of the inverse scattering problem.", "Later in section REF we will use this structure to get rid of the functions which cannot be observed from the spectral data.", "Let now $\\nu =\\sigma $ and $T(\\omega )^{-1}= (\\tau _{\\sigma \\nu }(\\omega ))_{\\sigma ,\\nu }$ .", "Relations (REF ) (REF ) yield $j_{\\sigma \\sigma }(k,l)= \\\\ \\frac{1}{2\\pi I} \\int _\\lbrace \\bigl [-p_\\sigma (k,\\omega ) + e_\\sigma (k,\\theta _\\sigma )\\tau _{\\sigma \\sigma }(\\omega )\\bigr ] \\frac{b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma )}{{\\mathfrak {b}}_\\sigma (0)e_\\sigma (1,\\theta _\\sigma )} \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma (\\omega )}{d\\omega } \\Bigr \\rbrace d\\omega = \\\\\\frac{1}{2\\pi I} \\Bigl ( \\int _{T_\\sigma ^+} + \\int _{T_\\sigma ^-}\\Bigr ) \\Bigl \\lbrace \\ \\cdot \\ \\Bigr \\rbrace = j_{\\sigma \\sigma }^+(k,l) + j_{\\sigma \\sigma }^-(k,l).$ As $\\omega $ runs over $T_\\sigma ^+$ the function $\\theta _\\sigma (\\omega )$ runs over the whole $.", "Let $ ()$ be the inverse function.Relations (\\ref {Eq:02a01}) together with (\\ref {eq:23}) yield\\begin{equation*}j_{\\sigma \\sigma }^+(k,l) = \\frac{1}{2\\pi I} \\int _[ e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )\\bigr ] \\theta _\\sigma ^{l-1} d\\theta _\\sigma .\\end{equation*}Besides it follows from corollary \\ref {Rem:51} that $ s()$ is bounded on $ +$.\\\\Consider the Fourier series of $ s(())$:\\begin{equation*}s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))=\\sum _{n=-\\infty }^\\infty \\tilde{s}_\\sigma (n) \\theta ^{-n}; \\ \\tilde{s}_\\sigma (n) = \\frac{1}{2 \\pi I} \\int _s̰_{\\sigma \\sigma }(\\omega (\\theta _\\sigma )) \\theta _\\sigma ^{n-1} d \\theta _\\sigma .\\end{equation*}Together with (\\ref {eq:13}) this yields\\begin{multline*}e_\\sigma (k, \\theta _\\sigma ^{-1}) + s_{\\sigma \\sigma }(\\omega (\\theta _\\sigma ))e_\\sigma (k, \\theta _\\sigma )= \\\\c_\\sigma (k) \\sum _m \\bigl \\lbrace a_\\sigma (k,m) + \\sum _n a_\\sigma (k,n) \\tilde{s}_\\sigma (m+n) \\bigr \\rbrace \\theta _\\sigma ^{-m},\\end{multline*}and finally\\begin{equation}j_{\\sigma \\sigma }^+(k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\tilde{s}_\\sigma (l+n)\\right] .\\end{equation}$ We now study $j^-_{\\sigma \\sigma }(k,l)$ .", "Denote $a_1(k,\\omega )= - p_\\sigma (k,\\omega ) \\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma );$ $a_2(\\omega )=\\tau _{\\sigma \\sigma }(\\omega )\\frac{b_\\sigma }{{\\mathfrak {b}}_\\sigma (0)}e_\\sigma (1,\\theta _\\sigma )^{-1}(\\theta _\\sigma ^{-1}-\\theta _\\sigma ).$ We then have $\\psi ^\\sigma _\\sigma (k,\\omega )= a_1(k,\\omega )+e_\\sigma (k,\\theta _\\sigma (\\omega ))a_2(\\omega ),$ and $j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_1(k,\\omega ) \\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega + \\\\\\frac{1}{2 \\pi I} \\int _{T_\\sigma ^-} a_2( \\omega ) e_\\sigma (k,\\theta _\\sigma (\\omega ))\\theta _\\sigma (\\omega )^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega .$ As $\\omega $ runs over $T_\\sigma ^-$ , the function $\\theta _\\sigma (\\omega )$ runs twice in the opposite directions over $J_\\sigma =\\theta _\\sigma (T_\\sigma ^-)\\subset {\\mathbb {R}}$ and the values $\\theta _\\sigma (\\omega )$ and $\\theta _\\sigma (\\omega ^{-1})$ coinside.", "Respectively $e_\\sigma (k,\\theta _\\sigma (\\omega )), p_\\sigma (k,\\omega ) \\in {\\mathbb {R}}$ and $e_\\sigma (k,\\theta _\\sigma (\\omega ))=e_\\sigma (k,\\theta _\\sigma (\\omega ^{-1}))$ , $ p_\\sigma (k,\\omega ) = p_\\sigma (k,\\omega ^{-1}) $ , thus $a_1(k,\\omega )=a_1(k,\\omega ^{-1})$ .", "Besides $ a_2(\\omega ) = \\overline{a_2(\\omega ^{-1})}$ , since $\\psi ^\\sigma (k,\\omega )= \\overline{\\psi ^\\sigma (k,\\omega ^{-1})}$ for $\\omega \\in .\\\\ \\\\Therefore\\begin{equation}j_{\\sigma \\sigma }^-(k,l)= \\frac{1}{2 \\pi I} \\int _{\\Gamma _\\sigma } [a_2(\\omega )-a_2(\\omega ^{-1}) ] e_\\sigma (k,\\theta _\\sigma )\\theta _\\sigma ^{l-1}\\frac{d\\theta _\\sigma }{d\\omega } d\\omega ,\\end{equation}where $ =T-+$.\\\\ \\\\The function $ a1(k,)$ satisfies equation (\\ref {eq:02}) on the channel $$ andrelations (\\ref {eq:66a}), (\\ref {eq:66b}) yield\\begin{equation*}a_1(k,\\omega )= \\theta _\\sigma (\\omega )^{-k}(1+o(1)).\\end{equation*}Therefore$ ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)= b_\\sigma (\\theta _\\sigma (\\omega )^{-1}-\\theta _\\sigma (\\omega ))[ a_2(\\omega )-a_2(\\omega ^{-1})].$ On the other hand by (REF ) ${\\mathfrak {b}}_\\sigma (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\sigma (0)=- \\sum _{\\nu \\in \\mathcal {C}\\setminus \\lbrace \\sigma \\rbrace }{\\mathfrak {b}}_\\nu (0) \\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0).$ We have $\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)=0, \\ \\omega \\in T_\\nu ^-.$ For $\\omega \\in T_\\nu ^+\\cap T_\\sigma ^-$ we obtain ${\\mathfrak {b}}_\\nu (0)\\lbrace \\psi ^\\sigma (\\omega ),\\psi ^\\sigma (\\omega ^{-1})\\rbrace _\\nu (0)= \\\\{\\mathfrak {b}}_\\nu (0) \\lbrace s_{\\nu \\sigma }(\\omega )e_\\nu (\\theta _\\nu (\\omega )), s_{\\nu \\sigma }(\\omega ^{-1})e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) \\\\|s_{\\nu \\sigma }(\\omega )|^2 {\\mathfrak {b}}_\\nu (0)\\lbrace e_\\nu (\\theta _\\nu (\\omega )), e_\\nu (\\theta _\\nu (\\omega )^{-1}) \\rbrace _\\nu (0) = \\\\|s_{\\nu \\sigma }(\\omega )|^2 b_\\nu (\\theta _\\nu (\\omega )^{-1}-\\theta _\\nu (\\omega )) =- |s_{\\sigma \\nu }(\\omega )|^2 \\frac{ ( b_\\sigma (\\theta _\\sigma ^{-1}-\\theta _\\sigma ))^2}{b_\\nu (\\theta _\\nu ^{-1}-\\theta _\\nu )}.$ Now, for $\\theta \\in J_\\sigma $ , we define $\\omega _\\sigma (\\theta )=W^{-1}\\circ W_\\sigma (\\theta )$ , here the functions $W$ , $W_\\sigma $ are defined by (REF ) and the branch is chosen so that $\\omega _\\sigma (\\theta )\\in \\Gamma _\\sigma $ .", "Let $N_\\sigma (\\theta )= \\lbrace \\nu ; \\ \\omega _\\sigma (\\theta )\\in T_\\nu ^+ \\rbrace , \\ \\theta \\in J_\\sigma .$ Denote $\\Phi _\\sigma (\\theta )= \\sum _{\\nu \\in N_\\sigma (\\theta )}|s_{\\sigma \\nu }(\\omega _\\sigma )|^2 \\left(\\frac{ b_\\sigma (\\theta ^{-1}-\\theta ) }{b_\\nu (\\theta _\\nu (\\omega _\\sigma )^{-1}-\\theta _\\nu (\\omega _\\sigma ))} \\right), \\ \\omega _\\sigma (\\theta )\\in T_\\sigma ^-.$ We finally obtain $a_2(\\omega )-a_2 (\\omega ^{-1})= \\Phi _\\sigma (\\theta _\\sigma (\\omega ))$ this function is expressed via the spectral data.", "Remark All functions $|s_{\\nu \\sigma }(\\omega )|$ participating in (REF ) are well defined and continuous.", "In addition, $s_{\\nu \\sigma }(\\omega ^{-1})=\\overline{s_{\\nu \\sigma }}(\\omega )$ for $\\omega \\in \\ T_\\nu ^+$ .", "Relation () now takes the form $j_{\\sigma \\sigma }^-(k,l)= \\frac{ 1 }{2\\pi I} \\int _{J_\\sigma } e_\\sigma (k,\\theta ) \\Phi _\\sigma (\\theta ) \\theta ^{l-1} d\\theta ,$ and with account () we obtain $j_{\\sigma \\sigma }^-(k,l)= c_\\sigma (k)\\sum _n a_\\sigma (k,n) q_{\\sigma \\sigma }(n+l),$ where $q_{\\sigma \\sigma }(n)= \\frac{1 }{2\\pi I} \\int _{J_\\sigma } \\Phi _\\sigma (\\theta )\\theta ^{n-1} d\\theta .$ Combining (REF ), (), and (REF ) we finally obtain $j_{\\sigma \\sigma } (k,l) = c_\\sigma (k) \\left[a_\\sigma (k,l)+\\sum _{n=-\\infty }^\\infty a_\\sigma (k,n) \\left( \\tilde{s}_\\sigma (l+n)+q_{\\sigma \\sigma }(n+l) \\right)\\right],$ the functions $\\tilde{s}_\\sigma (\\cdot )$ and $q_{\\sigma ,\\sigma }(\\cdot )$ are defined through the scattering data." ], [ "Let now $\\Theta (\\omega )= {\\mathbf {\\rm {diag}}}\\lbrace \\theta _\\sigma (\\omega ) \\rbrace _{\\sigma \\in \\mathcal {C}}.$ We use the representation () for $\\mathcal {E}(k,\\omega )$ , $\\mathcal {E}(k,\\omega )=C(k)\\sum _{m=-\\infty }^\\infty A(k,m)\\Theta (\\omega )^m.$ The coefficients $C(k)$ and $A(k,m)$ are diagonal matrices $C(k)={\\mathbf {\\rm {diag}}}\\lbrace c_\\sigma (k)\\rbrace _{\\sigma \\in \\mathcal {C}}, \\ A(k,m)={\\mathbf {\\rm {diag}}}\\lbrace a_\\sigma (k,m)\\rbrace _{\\sigma \\in \\mathcal {C}},$ and also $A(k,m)=0$ for $m<k$ .", "It follows from (REF ) and (REF ) that ${\\mathcal {J}}(l,k)=\\frac{1}{2\\pi I}\\int __l(\\omega )U(k,\\omega )d\\omega = \\\\C(k) \\left\\lbrace A(k,l)+\\sum _{l=-\\infty }^\\infty A(k,m) Z(l+m) \\right\\rbrace ,$ here the integral is taken as a principal value and the matrix $Z(n)=\\left( z_{\\nu ,\\sigma }(n) \\right)_{\\nu ,\\sigma \\in \\mathcal {C}}$ is given by the relations $z_{\\nu \\sigma }(n)=q_{\\nu \\sigma }(n), \\ \\nu \\ne \\sigma , \\\\z_{\\sigma \\sigma }(n)=\\tilde{s}_\\sigma (n)+q_{\\sigma \\sigma }(n),$ here $\\tilde{s}_\\sigma (n)$ are the Fourier coefficients of the reflection coefficient $s_{\\sigma \\sigma }$ with respect to $\\theta _\\sigma $ and the function $q_{\\nu \\sigma }$ is defined in (REF ) and (REF ).", "Let now $\\Omega \\subset \\bar{ be the set of all poles of U(k,\\omega ).", "It follows from (\\ref {eq:64}) that \\overline{U(k,\\omega )}=U(k,\\bar{\\omega }), thus for each \\hat{\\omega }\\in \\Omega \\cap the point \\bar{\\hat{\\omega }} also belongs to \\Omega .", "Moreover, for such \\hat{\\omega } we have \\overline{\\text{Res}_{\\hat{\\omega }}U(k,\\omega )}=\\text{Res}_{\\bar{\\hat{\\omega }}}U(k,\\omega ).Denote \\tilde{\\Omega }= \\lbrace \\hat{\\omega }\\in \\Omega : |\\hat{\\omega }| < 1\\rbrace \\cup \\lbrace \\hat{\\omega }\\in \\Omega : \\Im \\hat{\\omega }>0\\rbrace .", "We use (\\ref {Eq:39}) and (\\ref {Eq:01}):}$ ${\\mathcal {J}}(k,l)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} \\Delta _l(\\hat{\\omega })\\mathcal {E}(k,\\hat{\\omega }){\\mathfrak {m}}(\\hat{\\omega })=C(k)\\sum _mA(k,m)M(m+l),$ where $M(n)=\\sum _{\\hat{\\omega }\\in \\tilde{\\Omega }} {\\mathbf {\\rm {diag}}}\\left\\lbrace \\frac{d\\theta _\\sigma }{d\\omega }(\\hat{\\omega }) \\right\\rbrace _{\\sigma \\in \\mathcal {C}}\\Theta (\\hat{\\omega })^{n-1}\\Re {\\mathfrak {m}}(\\hat{\\omega }).$ By compare this to (REF ) and taking into account that $A(k,m)=0$ for $m<k$ and $A(k,k)=I$ , we obtain a system of equations $A(k,m)+\\sum _{s=k}^\\infty A(k,s)F(s+m)=0,\\quad m=k+1,k+2, \\ldots \\,$ where $F(n)=Z(n)-M(n).$ Since $A(k,k)=I$ , this relation can be written as $F(k+m)+A(k,m)+\\sum _{s=k+1}^\\infty A(k,s)F(s+m)=0, \\quad k=1,2, \\ldots ,\\quad m>k.$ The matrices $A(k,m)$ are diagonal.", "The diagonal elements of $F(n)$ , can be expressed through the spectral data.", "The non-diagonal elements of the matrices $F(n)$ vanish, as follows from the lemma below.", "Lemma 8.1 The matrices $F(n)$ are diagonal for all $n\\ge 1$ : $F(n)={\\mathbf {\\rm {diag}}}\\lbrace f_\\sigma (n)\\rbrace _{\\sigma \\in \\mathcal {C}}$ .", "and their diagonal elements $f_\\sigma (n)$ are determined by $a_\\sigma (k,m)$ , $m>k\\ge \\Big [\\frac{n-1}{2}\\Big ]$ only.", "This statement is proved in [12], (Lemma 5.1) and we omit the proof.", "Theorem 8.1 The following properties hold for the systems under consideration 1) Equations (REF ) split into a system of independent scalar equations $f_\\nu (k+m)+a_\\nu (k,m)+\\sum _{s=k+1}^\\infty a_\\nu (k,s)f_\\nu (s+m)=0, \\,m\\ge k+1\\ge 1,$ here $f_\\nu (n)$ is defined in (REF ), (REF ), ().", "2) For each $k\\ge 0$ equations (REF ) has unique solutions $a_\\sigma (k,m)$ .", "The first statement follows directly from relation (REF ) which defines $F(n)$ and Lemma REF .", "It follows from the same lemma that the functions $f_\\nu (n)$ , $n\\ge 1$ are uniquely defined by the coefficients ${\\mathfrak {b}}_\\nu (k)$ , ${\\mathfrak {a}}_\\nu (k)$ , $k\\ge 1$ .", "Therefore (REF ) coincide with equations for inverse scattering problem for equation (REF ) with boundary condition $\\xi (\\nu (0))=0$ .", "It is well-known (see for example [14]) that the later has unique solution." ], [ "Concluding remarks", "In case that the continuos spectra $I=\\cup _{\\sigma \\in \\mathcal {C}}[a_\\sigma -2b_\\sigma ,a_\\sigma +2b_\\sigma ]$ splits into a number of disjoints intervals, one can repeat the procedure separately for each connected component of $I$ .", "If, say, $I^{(0)}$ is a connected component of $I$ and $\\sigma $ is a channel corresponding to this component, then each wave incoming along $\\sigma $ generates decaying waves in all channels which correspond to other connected components of $I$ .", "We omit the details.", "So far we have discussed reconstruction of the part $\\mathcal {A}_0$ of the whole system, that is, the channels.", "This information is, generally speaking, insufficient for reconstruction the whole matrix $\\mathcal {L}$ .", "However if the matrix $\\mathcal {L}_1$ corresponding to the \"central\" part of the system is sufficiently sparse and also we know the matrix $B(0)=\\text{diag}\\lbrace {\\mathfrak {b}}_\\sigma (0)\\rbrace _{\\sigma \\in \\mathcal {C}}$ which realizes connections between the channels and the central part of the system, the whole matrix $\\mathcal {L}$ can be recovered from the scattering data.", "We refer the reader to Chapter 11 in [15], where statemets of such type are obtained." ] ]
1612.05528
[ [ "Does a band structure affect sphaleron processes?" ], [ "Abstract Inspired by a recent work of Tye and Wong, we examine an effect of a band structure on baryon number preservation criteria requisite for successful electroweak baryogenesis.", "Action of a reduced model is fully constructed including a time component of the gauge field that is missing in the original work.", "The band structure is estimated more precisely in wider energy range based on WKB framework with three connection formulas to find that the band structure has little effect on the criteria at around 100 GeV temperature.", "We also address an issue of suppression factors peculiar to the $(B+L)$-changing process in high-energy collisions at zero temperature." ], [ "ACFI-T16-34 Does a band structure affect sphaleron processes?", "Koichi Funakubo$^1$ [email protected] Kaori Fuyuto$^{2}$ [email protected] Eibun Senaha$^3$ [email protected] $^1$ Department of Physics, Saga University, Saga 840-8502 Japan and $^2$ Amherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts Amherst, MA 01003, USA $^3$ Department of Physics, National Taiwan University, Taipei 10617, Taiwan Inspired by a recent work of Tye and Wong, we examine an effect of a band structure on baryon number preservation criteria requisite for successful electroweak baryogenesis.", "Action of a reduced model is fully constructed including a time component of the gauge field that is missing in the original work.", "The band structure is estimated more precisely in wider energy range based on WKB framework with three connection formulas to find that the band structure has little effect on the criteria at around 100 GeV temperature.", "We also address an issue of suppression factors peculiar to the $(B+L)$ -changing process in high-energy collisions at zero temperature.", "It is known that conservation of baryon $(B)$ and lepton number $(L)$ is nonperturbatively violated by the chiral anomaly in electroweak theories [1].", "Experimental verification of such a violation is important not only for understanding the nonperturbative nature of the quantum field theory but also for phenomenology in particle physics and cosmology.", "In particular, existence of such an anomalous process has expanded the possibilities of baryogenesis in the early Universe: electroweak baryogenesis (EWBG) [2], leptogenesis [3], and baryogenesis via neutrino oscillations [4], etc.", "It is shown by 't Hooft that the $(B+L)$ -changing vacuum transition is suppressed by $e^{-S_{\\rm instanton}}\\simeq 10^{-162}$ , where $S_{\\rm instanton}$ is the instanton action.", "Later, the analysis was generalized to the process in high-energy collisions and found that the above exponential suppression can get alleviated with increasing energy  [5], [6].", "However, the perturbative calculation using an instanton-like configuration is no longer valid as the energy approaches to a sphaleron energy ($E_{\\rm sph}$ ) which corresponds to the height of the barrier separating topologically different vacua.", "A nonperturbative analysis in a simplified model indicates that the exponential suppression still persists even at very high energy [7].", "Recently, Tye and Wong revisited the possibility of the $(B+L)$ -changing process at high energies in the framework of a reduced model [8].", "In their analysis, the reduced model is quantized so that the energy spectrum has a band structure owing to the periodic potential, and the wave functions are given by Bloch waves.", "It is demonstrated that the band width gets wider with increasing energy, and the energy spectrum becomes almost continuous above $E_{\\rm sph}$ , and therefore the transition amplitude is free from the exponential suppression when the energy is larger than $E_{\\rm sph}$ .", "Such a description is missing in the past work.", "After their claim, discussions on the detectability of the $(B+L)$ -changing process at colliders or in high-energy cosmic rays have been revived [9].", "While the $(B+L)$ -changing process is unsuppressed at higher temperature than the electroweak phase transition, its probability below that temperature should be so small to satisfy the baryon-number preservation criteria (BNPC) for a successful EWBG scenario.", "It is a natural question to what extent the above band structure description can change the BNPC.", "Such an analysis is enormously important for a test of EWBG at the Large Hadron Collider (LHC) since the BNPC determines the minimal value of strength of the first-order electroweak phase transition, so the collider signatures in the Higgs sector.", "In this Letter, we investigate the effect of the band structure on the BNPC.", "As is done in Refs.", "[11], [10], [12], [13], [8], the reduced model is constructed by regarding the noncontractible loop parameter [14] as a dynamical variable.", "Unlike the work of [11], [8], we adopt a fully gauge-invariant approach in which the time component of the gauge field $A_0$ is also taken into account for the construction.", "Furthermore, we evaluate the band structure in a more precise manner utilizing the WKB method with three connection formulas depending on energy.", "With those improvements, we compare our results with those in Ref.", "[8] and quantify the impact of the band structure on EWBG.", "We also point out a problem of an exponential suppression factor besides the tunneling factor in the $(B+L)$ -changing process in high-energy collisions at zero temperature, which is not properly argued in Ref. [8].", "We consider the standard model (SM) for an illustrative purpose.", "The generalization to models beyond the SM is straightforward.", "Since the effect of a $U(1)_Y$ contribution on the sphaleron energy is a few %, we neglect it.", "The starting point is the following $SU(2)_L$ gauge-Higgs system: ${\\cal L}=-\\frac{1}{4}F^a_{\\mu \\nu }F^{a\\mu \\nu }+(D_{\\mu }\\Phi )^{\\dagger }D^{\\mu }\\Phi -\\lambda \\left(\\Phi ^{\\dagger }\\Phi -\\frac{v^2}{2}\\right)^2,$ where $D_{\\mu }=\\partial _{\\mu }+ig_2A_{\\mu }$ with $g_2$ being the gauge coupling.", "$F^{a}_{\\mu \\nu }$ is the field strength tensor and $\\Phi $ the $SU(2)_L$ doublet Higgs field.", "The vacuum expectation value of the Higgs field is denoted as $v(\\simeq 246~{\\rm GeV})$ .", "With the Manton's ansatz [14], the gauge and Higgs fields are cast into the form $A_i(\\mu ,r,\\theta ,\\phi )=&\\frac{i}{g_2}f(r)\\partial _iU(\\mu ,\\theta ,\\phi )U^{-1}(\\mu ,\\theta ,\\phi ), \\\\{\\Phi }(\\mu ,r,\\theta ,\\phi )=&\\frac{v}{\\sqrt{2}}\\bigg [(1-h(r))\\begin{pmatrix}0\\\\e^{-i\\mu }\\cos \\mu \\end{pmatrix}\\nonumber \\\\&\\hspace{28.45274pt}+h(r)U(\\mu ,\\theta , \\phi )\\begin{pmatrix}0\\\\1\\end{pmatrix}\\bigg ],$ with $&U(\\mu ,\\theta ,\\phi )\\nonumber \\\\&=\\begin{pmatrix}e^{i\\mu }(\\cos \\mu -i\\sin \\mu \\cos \\theta ) & e^{i\\phi }\\sin \\mu \\sin \\theta \\\\-e^{-i\\phi }\\sin \\mu \\sin \\theta & e^{-i\\mu }(\\cos \\mu +i\\sin \\mu \\cos \\theta ),\\end{pmatrix}$ with a noncontractible loop parameter $\\mu $ which runs from 0 to $\\pi $ .", "Note that Eqs.", "(REF ) and () are reduced to the vacuum configurations for $\\mu =0$ and $\\pi $ while the sphaleron for $\\mu =\\pi /2$ .", "Our aim is to study the transition from one vacuum to another along the least energy path such that the sphaleron configuration is realized at the maximal point.", "For this purpose, $\\mu $ is promoted to the dynamical value $\\mu (t)$ as proposed in Ref. [11].", "In contrast to the previous studies [11], [8], we construct the reduced model in a fully gauge-invariant way, where $A_0=ifU^{-1}\\partial _0U/g_2$ .", "Note that the Manton's ansatz with the $A_0=0$ gauge causes an unwanted divergence arising from the $D\\Phi $ term in asymptotic region $r\\rightarrow \\infty $ due to lack of the full gauge invariance, and therefore some prescriptions are needed [11], [8].", "After deriving the profile functions $f(r)$ and $h(r)$ by solving the equations of motion for the sphaleron, one obtains the classical action as $S[\\mu ]=g_2v\\int dt~\\bigg [\\frac{M(\\mu )}{2}\\left(\\frac{d}{dt}\\frac{\\mu (t)}{g_2v}\\right)^2-V(\\mu )\\bigg ],$ where $M(\\mu )&=\\frac{4\\pi }{g^2_2}\\left(\\alpha _0+\\alpha _1\\cos ^2\\mu +\\alpha _2\\cos ^4\\mu \\right),\\\\V(\\mu )&=\\frac{4\\pi }{g^2_2}\\sin ^2\\mu \\left(\\beta _1+\\beta _2\\sin ^2\\mu \\right).$ The coefficients of $\\alpha $ 's and $\\beta $ 's are found to be $\\alpha _0&=19.42,\\quad \\alpha _1=-1.937, \\quad \\alpha _2=-2.656,\\nonumber \\\\\\beta _1&=1.313,\\quad \\beta _2=0.603,$ The sphaleron mass and potential in units of mass are, respectively, given by $M_{\\rm sph} &= g_2vM\\left(\\frac{\\pi }{2}\\right) \\simeq 92.0~{\\rm TeV}, \\\\E_{\\rm sph}&=g_2vV\\left(\\frac{\\pi }{2}\\right)\\simeq 9.08~{\\rm TeV}.$ Note that $M_{\\rm sph}$ depends on the normalization of the kinetic term in Eq.", "(REF ) which is different from that in Ref.", "[8] by a factor of 4.", "With the same normalization, $M_{\\rm sph}=23.0$ TeV in our case and $M_{\\rm sph}=17.1$ TeV in Ref.", "[8], which implies that the $A_0$ contribution to $M_{\\rm sph}$ is not negligible.", "As will be discussed below, number of the bands can change according to the size of $M_{\\rm sph}$ .", "When a classical Hamiltonian is quantized, an ambiguity arises from an operator ordering.", "In our analysis, we adopt $\\mathcal {H}(\\mu ,p)&=g_2v\\left[\\hat{p}\\frac{1}{2M(\\hat{\\mu })}\\hat{p}+V(\\hat{\\mu }) \\right],$ where $\\hat{p}$ is the momentum conjugate operator that satisfies $[\\hat{\\mu },~\\hat{p}]=i$ .", "With this Hamiltonian, we solve the Schrödinger equation ${\\cal H}\\psi (\\mu )={\\cal E}\\psi (\\mu )$ , where ${\\cal E}=E/g_2v$ .", "Since the potential is a periodic function of $\\mu $ , the eigenfunction $\\psi (\\mu )$ is given by Bloch wave and energy spectrum has the band structure.", "The band edges are determined, in the WKB approximation, as solutions to $\\cos (\\Phi ({\\cal E}))=\\pm \\sqrt{T({\\cal E})}, $ where $T({\\cal E})$ is a transmission coefficient at ${\\cal E}$ and $\\Phi ({\\cal E})$ is defined by $\\Phi ({\\cal E}) = \\int ^{a({\\cal E})}_{b({\\cal E})}d\\mu \\,\\sqrt{2M(\\mu )({\\cal E}-V(\\mu ))},$ where $a({\\cal E})$ and $b({\\cal E})$ are the turning points determined by ${\\cal E}=V(\\mu )$ for ${\\cal E}<V(\\pi /2)$ , while $a({\\cal E})=\\pi /2$ and $b({\\cal E})=-\\pi /2$ for ${\\cal E}\\ge V(\\pi /2)$  [15].", "We denote the lower and upper edges of the $n$ -th band as ${\\cal E}_{-,n}$ and ${\\cal E}_{+,n}$ , respectively, so its band width is given by $\\Delta {\\cal E}_n={\\cal E}_{+,n}-{\\cal E}_{-,n}$ .", "Depending on the energy, three kinds of connection formulas are properly used: linear potential, parabolic potential and over-barrier approximations (for a review, see, e.g., Ref. [16]).", "Detailed calculations will be given in Ref. [17].", "To verify the validity of this method, we also applied it to an eigenvalue problem with a sine-type potential, where the band edges are known to be the characteristic values of the Mathieu function, and then confirmed that the above method sufficiently works well.", "Table REF shows the band structure in the reduced model.", "The number of the bands below $E_{\\rm sph}$ is 158 in our case while 148 in Ref.", "[8], which is attributed to the different values of $M_{\\rm sph}$ .", "While the band center energies are more or less the same as those in Ref.", "[8], significant differences exist in the band widths at $E\\simeq \\mathcal {O}(100)$ GeV.", "However, such a difference does not give any impact on the results we are concern with.", "One can see that the band widths become wider as the energy grows, while the band structure persists above $E_{\\rm sph}$ even though the band gaps gets smaller.", "Table: The band structure of the reduced model is listed.The lower edge and band width are shown, respectively.The sphaleron energy is about 9.08 TeV.Now we scrutinize how much the band structure affects EWBG.", "The typical energy scale of EWBG is a critical temperature $T_C$ at which the Higgs potential has degenerate minima if the electroweak phase transition is of first order, which is about $\\mathcal {O}(100)$ GeV in most cases.", "The decay rate of a false vacuum at finite temperatures is defined by [18] $\\Gamma _A(T)=\\frac{1}{Z_0(T)}\\int ^{\\infty }_0dE~J(E)e^{-E/T}, $ where $Z_0(T)=[2\\sinh (\\omega _0/(2T))]^{-1}$ is a partition function of a harmonic oscillator with an angular frequency $\\omega _0=g_2v\\sqrt{V^{\\prime \\prime }(0)/M(0)}$ and $J(E)$ is a probability current, which is cast into the form $J(E)=T(E)/2\\pi $ .", "The WKB approximation in high temperature regime of the field theoretic expression of (REF ) is used to estimate the $(B+L)$ -changing rate in electroweak theories [19].", "We numerically calculate $\\Gamma _A(T)$ for a single energy barrier in the region of $0\\le \\mu \\le \\pi $ .", "In the current framework of the periodic potential , we newly define the transition rate as $\\Gamma (T)=\\frac{1}{Z_0(T)}\\int ^{\\infty }_0dE~\\frac{\\eta (E)}{2\\pi }e^{-E/T},$ where $\\eta (E)$ is the density of states that embodies the band effect: $\\eta (E)=1$ for the conducting band and $\\eta (E)=0$ for the band gap.", "It should be noted that although what is really needed is the thermal average of the transition probability from $\\mu =0$ to $\\mu =\\pi $ , the definition (REF ) corresponds to setting the probability to unity for a state in one of the energy bands.", "Therefore, the above naive definition of the rate provides an overestimated result.", "We defer the more precise estimate to future work.", "Figure: Γ(T)\\Gamma (T) and Γ A (T)\\Gamma _A(T) are shown against temperature,where Γ(T)\\Gamma (T) is the transition rate taking the band effect into account whileΓ A (T)\\Gamma _A(T) is the ordinary transition rate.The numerical values of $\\Gamma (T)$ and $\\Gamma _A(T)$ are plotted in Fig.", "REF .", "Now we study the effect of the band structure on the BNPC, $\\Gamma (T) < H(T)$ , as needed for successful EWBG, where $H(T)\\propto T^2/m_{\\rm Pl}$ with $m_{\\rm Pl}$ being the Planck mass is the Hubble parameter at $T$ .", "Although the enhancement due to the band structure is sizable at temperatures below 30 GeV, $\\Gamma (T)$ is still too small to affect the BNPC quantitatively, since $H(T)$ decreases more slowly than $\\Gamma (T)$ for lower temperatures.", "On the other hand, the enhancement appears very small near the electroweak phase transition temperature.", "To quantify the effect, we denote $R(T) = \\Gamma (T)/\\Gamma _A(T)$ , where $\\Gamma _A(T)$ corresponds to the $(B+L$ )-changing rate so far used.", "The modified BNPC is expressed as [20] $\\frac{v(T)}{T}>\\frac{g_2}{4\\pi (\\beta _1+\\beta _2)}\\bigg [42.97+\\log {\\cal N}+\\log R(T)+\\cdots \\bigg ],$ where ${\\cal N}$ denotes the translational and rotational zero-mode factors of the fluctuations about the sphaleron, which may amount to about 10% correction to the leading constant term (see, e.g., Ref. [20]).", "With $R(T)$ obtained above, one finds that $\\log R(T=100~{\\rm GeV})\\simeq 0.05$ , so the band structure has little effect on the criteria (REF ) .", "It is an interesting question whether $(B+L)$ -changing process is visible at colliders such as LHC and Future Circular Collider-$hh$ etc.", "Tye and Wong claim that it might be possible thanks to the unsuppressed transition probability between the topologically inequivalent vacua.", "However, it is known that the creation of the classical configurations from the high-energy collision of the two particles suffers from another type of exponential suppression other than the tunneling suppression discussed above.", "In Refs.", "[12], [13], the $(B+L)$ -changing scattering amplitude is formulated by use of the complete set of the coherent state, which is dominated by the bounce configuration describing a transition from one vacuum to another.", "It is shown that, in the leading order of the WKB approximation, the overlap between the coherent state and the $n$ -particle state produces a multiplicative factor $\\mathbf {k}_1^2e^{-\\pi |k_1|/m_W}\\mathbf {k}_2^2e^{-\\pi |k_2|/m_W}\\cdots \\mathbf {k}_n^2e^{-\\pi |k_n|/m_W}$ with $k_i$ being a spatial momentum of the particle $i$ .", "Therefore, the overlap between the coherent state and the in-state composed of two particles whose total momentum is $E_{\\rm sph}$ yields a suppression factor $\\sim e^{-\\pi E_{\\rm sph}/m_W}\\simeq 10^{-155}$ , rendering $(B+L)$ -changing process unobservably small.", "We note in passing that existence of the suppression can also be shown using the complete set of the field eigenstates instead of that of the coherent states.", "Since each factor $\\mathbf {k}^2e^{-\\pi |k|/m_W}$ has a sharp peak at $|\\mathbf {k}|\\sim m_W$ , the above overlap suppression might be circumvented if the incoming two particles get scattered into multiple $W$ bosons where each has a momentum close to the $W$ boson mass, and then couple to the sphaleron all together.", "For this process to occur, about 80 $W$ bosons must be produced by the initial state scatterings, which would have a phase space suppression factor $\\sim (1/(4\\pi )^2)^{80}\\simeq 10^{-176}$ , again preventing the $(B+L)$ -changing process from being visible in high-energy collisions.", "Those types of the suppressions still remain regardless of the band structure, which is not properly discussed in Ref.", "[8] (see also Ref. [21]).", "It should be emphasized that the above types of the suppressions do not exist at temperatures higher than the weak scale, since so many particles whose momenta are of order $m_W$ are populated to have sizable overlap with the classical configuration.", "We have constructed the reduced model along the noncontractible loop connecting the classical vacua with different $B+L$ in a gauge-invariant manner, and investigated the energy eigenstates of the mode in detail.", "Based on the results, we have scrutinized to what extent the band structure can affect the EWBG scenario.", "Our findings show that the BNPC is not virtually altered for $T\\le T_C$ .", "Before closing, a few comments are in order.", "First of all, we make a remark on the formulation of the $(B+L)$ -changing process.", "The energy eigenstates obtained here are not those of $B+L$ or $\\mu $ .", "If an in-state of a high-energy collision is an eigenstate of $\\mu $ or a state localized at some $\\mu $ , it should be expressed as a superposition of the energy eigenstates.", "Such a state should be peaked at some energy in the asymptotic region where $B+L$ is almost conserved.", "The correct $(B+L)$ -changing rate must be formulated by taking into account this situation.", "Secondly, we have not taken thermal corrections into account in our analysis.", "In order to improve our results quantitatively, the reduced model should be constructed starting from the field theoretic model with thermal corrections, as done in [20] to improve the precision of the BNPC.", "These issues will be dealt with elsewhere [17].", "K. Funakubo is supported by JSPS KAKENHI Grant Number JP15K05057.", "K. Fuyuto is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists No. 15J01079.", "E.S.", "is supported in part by the Ministry of Science and Technology of R. O. C. under Grant No.", "MOST 104-2811-M-008-011." ] ]
1612.05431
[ [ "Theoretical aspects of charged Lepton Flavour Violation" ], [ "Abstract If observed, charged lepton flavour violation is a clear sign of new physics - beyond the Standard Model minimally extended to accommodate neutrino oscillation data.", "We briefly review several extensions of the Standard Model which could potentially give rise to observable signals, also emphasising the r\\^ole of charged lepton flavour violation in probing such new physics models." ], [ "Introduction", "Of the three observations which cannot be explained by the Standard Model (SM) - and which also include the baryon asymmetry of the Universe (BAU) and dark matter (DM) - neutrino oscillations provided the first evidence of new physics.", "Interestingly, among the several models successfully accounting for and explaining $\\nu $ -data, many offer the possibility to further address the BAU via leptogenesis, succeed in putting forward viable DM candidates, or even ease certain theoretical puzzles of the SM, as is the case of the flavour problem.", "Leptonic mixings and massive neutrinos offer a true gateway to many new experimental signals or deviations from SM predictions in the lepton sector; among others, these include charged lepton flavour violation (cLFV).", "The most minimal extension of the SM allowing to accommodate $\\nu $ oscillation data consists in the addition of right-handed neutrinos ($\\nu _R$ ) while preserving the total lepton number, thus giving rise to massive Dirac neutral leptons.", "In such a framework, individual lepton numbers are violated (as encoded in the $U_{\\rm {PMNS}}$ matrix), and cLFV transitions such as $\\mu \\rightarrow e \\gamma $ can occur, being mediated by $W^\\pm $ bosons and massive neutrinos; however, and due to the tiny values of light neutrino masses, the associated rate is extremely small, BR($\\mu \\rightarrow e \\gamma $ )$ \\sim \\mathcal {O}(10^{-55})$ , lying beyond the reach of any future experiment.", "Thus, the observation of a cLFV process would necessarily imply the existence of new physics degrees of freedom (beyond minimal extensions via massive Dirac neutrinos).", "A comprehensive review of the experimental status of a vast array of cLFV observables was presented at this Conference by W. Ootani [1].", "Whether or not charged and neutral LFV are related, or equivalently if cLFV arises from the mechanism of $\\nu $ -mass generation, and the question of which is the New Physics model that can be at the origin of these phenomena, have constituted the starting point to extensive studies.", "Here we very briefly review the prospects for cLFV observables of some appealing and well motivated SM extensions." ], [ "cLFV and New Physics models", "Interpreting experimental data on cLFV - be it in the form of a possible measurement or improved bounds - requires an underlying theoretical framework: new physics models can lead to “observable” cLFV introducing new sources of lepton flavour violation, as well as new operators at the origin of the flavour violating transitions and decays.", "A first, model-independent approach consists in parametrising cLFV interactions by means of higher-order non-renormalisable (dimension $d>4$ ) operators.", "The new low-energy effective Lagrangian can be written as $\\mathcal {L}^\\text{eff} = \\mathcal {L}^\\text{SM} + \\sum _{n \\ge 1}\\mathcal {C}_{ij}^{4+n} \\, \\Lambda ^{-n} \\, \\mathcal {O}_{ij}^{4+n}$ , in which $\\Lambda $ denotes the scale of new physics, and $\\mathcal {C}$ , $\\mathcal {O}$ the effective couplings and operators, with the former corresponding to complex matrices in flavour space.", "Contrary to the unique dimension-five Weinberg operator (common to all models with Majorana neutrino masses), there exists a large number of dimension-six operators, whose low-energy effects include cLFV.", "Regarding the cLFV dimension-six operators, these can be loosely classified as dipole, four-fermion and scalar/vector operators.", "In order to constrain the new physics scale and the amount of flavour violation thus introduced, the cLFV observables can be cast in terms of combinations of $\\mathcal {C}_{ij}^{6}$ and $\\Lambda ^{-2}$ ; simple, natural hypothesis on one allow to infer constraints on the other.", "Table REF collects some bounds on the scale of new physics (derived under the hypothesis of natural, $\\mathcal {O}(1)$ , effective couplings) and on the size of the new effective couplings (inferred for a choice $\\Lambda =1$  TeV).", "Table: Bounds on the effective couplings and lower bounds on thescale Λ\\Lambda (TeV), following the hypotheses described on the text;the last column refers to the observable leading to the most stringentbounds.", "Adapted from .Despite its appeal for leading to a generic evaluation of the new physics contributions to a given cLFV observable, and thus to model-independent constraints, there are several limitations to the effective approach.", "These include taking “natural” values for the couplings, assuming the dominance of a single operator when constraining a given process and the uniqueness of the new physics scale; the latter should be kept in mind when weighing the impact of the thus derived constraints on new physics.", "A second phenomenological approach consists in considering specific new physics models or theories, and evaluating the corresponding impact for a given class of cLFV processes.", "As extensively explored in the literature, cLFV might be a powerful test of new physics realisations, probing scales beyond collider reach, offering valuable hints on properties and parameters of a given model, and allowing to disentangle (and ultimately disfavour) between candidate models.", "A short, non-comprehensive list of examples has been presented at this Conference; below we highlight a small subset.", "By studying the correlation of (high- and low-energy) cLFV observables as predicted in the framework of generic, flavour violating, supersymmetric (SUSY) extensions of the SM, one can derive useful information on the nature of the dominant operator at work (dipole vs. scalar); for instance, this is also the case of littlest Higgs models, which can be efficiently tested via cLFV, as they lead to very distinctive patterns for ratios of cLFV observables [3].", "In the case of specific holographic composite Higgs models, it has been shown that although the predictions for most cLFV observables lie below experimental reach, current bounds on BR($\\mu \\rightarrow e \\gamma $ ) allow to constrain the size of boundary kinetic terms, and thus infer information on otherwise unreachable fundamental parameters of the model (cf.", "left panel of Fig.", "REF ) [4].", "A particularly interesting and rich example of “geometrical cLFV” is that of realisations of extra-dimensional Randall-Sundrum (RS) models, with anarchic ($d=5$ ) Yukawa couplings [5]: current bounds from $\\mu -e$ transitions and decays already constrain the scale of new physics to lie beyond LHC reach ($T_{\\text{KK}} \\gtrsim 4$  TeV, corresponding to having the first Kaluza-Klein excitations $m_\\text{KK}^\\text{1st} \\gtrsim 10$  TeV); future bounds should allow to exclude generic anarchic RS models up to 8 TeV (and correspondingly first excitations $m_\\text{KK}^\\text{1st}\\gtrsim 20$  TeV).", "Further examples of the constraining power of cLFV include (simple) SM extensions such as multi-Higgs doublet models, leptoquark constructions, additional $Z^\\prime $ bosons, etc.. Increasing the symmetry content of the model - be it in the form of a gauge or flavour symmetry - reduces its arbitrariness, thus rendering the model easier to test and possibly falsify.", "For instance, Left-Right symmetric models, which in addition to exhibiting a strong interplay of cLFV and lepton number violating observables, lead to very distinctive correlations of observables (see, for example, [6]), and can thus be easily falsifiable.", "Extended gauge groups, and in particular grand unified theories (GUTs), in addition to possibly incorporating a mechanism of neutrino mass generation, lead to scenarios of strong predictivity for cLFV, as illustrated on the right panel of Fig.", "REF  [7].", "The latter consists of a supersymmetrisation of an SO(10) type II seesaw model - as can be seen, data on any two observables would readily allow to exclude the model.", "Figure: On the left, BR(μ→eγ\\mu \\rightarrow e \\gamma ) as a function of the size ofboundary kinetic terms in the framework of Holographic CompositeHiggs models(from ); on theright panel, correlation between different cLFV observables in for anSO(10) type II SUSY seesaw model (from )." ], [ "cLFV from seesaw realisations", "Although cLFV need not arise from the mechanism of $\\nu $ mass generation, models in which this is indeed the case - such as the different seesaw realisations - are particularly appealing and well-motivated frameworks.", "Whether or not a given mechanism of neutrino mass generation does have an impact regarding cLFV stems from having non-negligble flavour violating couplings (e.g., the Yukawa couplings) provided that the rates are not suppressed by excessively heavy propagator masses.", "While “standard” high-scale seesaws do accommodate neutrino data with natural values of the neutrino Yukawa couplings, the typical scale of the mediators (close to the GUT scale) leads to a very strong suppression of the different cLFV rates.", "On the other hand, low-scale seesaws, or the embedding of the seesaw in larger frameworks (as is the case of the SUSY seesaw), are associated with a rich phenomenology, with a strong impact regarding cLFV.", "In low-scale seesaws (as is the case of low-scale type I seesaw, inverse and linear seesaw realisations, ...), the new “heavy” states do not fully decouple; their non-negligible mixings with the light (active) neutrinos lead to the non-unitarity of the left-handed lepton mixing matrix ($U_\\text{PMNS} \\rightarrow \\tilde{U}_\\text{PMNS}$ ), and thus to having modified neutral and charged lepton currents.", "The latter are at the origin of potentially abundant experimental/observational signatures, which have been intensively searched for in recent years; negative results have allowed to derive strong constraints on the parameter space of the new degrees of freedom (see [8], [9] for comprehensive discussions of cLFV in low-scale seesaws).", "A very appealing example of such low-scale models are Inverse Seesaw (ISS) realisations: other than right-handed neutrinos, further sterile states are added; in the case of a (3,3) ISS realisation, three copies of each are present.", "The masses of the light active neutrinos are given by a modified seesaw relation, $m_{\\nu _i} \\approx (Y_\\nu v)^2M_R^{-2} \\mu _X$ , where $\\mu _X$ is the only source of lepton number violation in the model.", "By taking small values of $\\mu _X$ , one can naturally accommodate the smallness of active neutrino masses for large Yukawa couplings and a comparatively low seesaw scale ($M_R$ lying close to the TeV scale).", "The spectrum contains, in addition to the light states, three heavier (mostly sterile) pseudo-Dirac pairs, whose masses are given by $m_{N} \\approx M_R \\pm \\mu _X$ .", "The (3,3) ISS opens the door to a very rich phenomenology, which includes abundant cLFV signatures, both at low- and at high-energies (see, for example, [10], [12], [11]).", "To illustrate the potential impact regarding high-intensity facilities, the left panel of Fig.", "REF displays the prospects for $\\mu - e$ conversion, as well as the Coulomb enhanced decay of a muonic atom (both for the case of Aluminium targets), as a function of the average mass of the heavier states, $<m_{4-9}>$ .", "Although CR($\\mu - e$ , Al) is in general associated to larger rates, for sterile states above the TeV, both observables are expected to be well within reach of the COMET experience (horizontal lines respectively denoting the sensitivity of Phase I and II), or of the Mu2e experiment.", "At higher energies (for example, in the case of a future circular collider, as FCC-ee), one can also explore cLFV in the decay of heavier states, as for instance in $Z \\rightarrow \\ell _i \\ell _j$ .", "In the ISS (3,3) realisation, especially in the “large” sterile mass regime, the cLFV $Z$ decays exhibit a strong correlation with cLFV 3-body decays (since the latter are dominated by the $Z$ -penguin contribution).", "The prospects for a (3,3) ISS realisation, for the case of $\\mu -\\tau $ flavour violation, are shown in the right panel of Fig.", "REF .", "Not only can one expect to have BR($Z \\rightarrow \\tau \\mu $ ) within FCC-ee reach, but this observable does allow to probe $\\mu -\\tau $ flavour violation well beyond the sensitivity of a future SuperB factory (large values of BR($\\tau \\rightarrow 3 \\mu $ ) are precluded in this realisation due to the violation of other cLFV bounds).", "Figure: On the left panel, BR(μ - e - →e - e - \\mu ^- e^- \\rightarrow e^- e^-, Al) - cyan - and CR(μ-e\\mu -e, Al)- blue - as a function of the average mass of the heavier, mostly sterilestates, in a (3,3) ISS realisation.", "Horizontal lines denote futureexperimental sensitivities (from ).On the right, BR(Z→τμZ \\rightarrow \\tau \\mu ) vs. BR(τ→3μ\\tau \\rightarrow 3 \\mu ) in a (3,3)ISS realisation.", "Vertical lines denote future experimentalsensitivities while the horizontal ones correspond to the prospects ofa GigaZ facility and of the FCC-ee (from ).In both cases, grey points are phenomenologically excluded.At the LHC, searches for heavy ISS mediators relying on cLFV signatures can be carried; as recently proposed, a significant number of events (after cuts) could be expected from the channel $q q^\\prime \\rightarrow \\tau \\mu +2$ jets (no missing $E^T$ ) [13].", "Another rich and well-motivated framework leading to observable cLFV is that of the SUSY seesaw (a high-scale seesaw embedded in the context of otherwise flavour conserving SUSY models).", "In the case of a type I SUSY seesaw [14], sizeable neutrino Yukawa couplings (as characteristic of a high-scale seesaw) and the possibility of new, not excessively heavy mediators (the SUSY partners), open the door to large contributions to cLFV observables.", "Having a unique source of flavour violation implies that the observables exhibit a high degree of correlation; such a synergy can be explored, allowing to put the seesaw hypothesis to the test and possibly hinting on certain parameters.", "The complementarity of two low-energy observables is depicted on the left panel of Fig.", "REF : BR($\\mu \\rightarrow e \\gamma $ ) vs. BR($\\tau \\rightarrow \\mu \\gamma $ ), for different seesaw scales (and for distinct values of the then unknown Chooz angle) [15].", "The determination of these two observables, in association with the discovery of SUSY, would allow to infer information on the seesaw scale $M_R$ , or then readily disfavour the SUSY seesaw as the source of cLFV.", "The potential of exploring the interplay of high-intensity (for instance $\\mu \\rightarrow e \\gamma $ and $\\mu -e$ conversion) and collider observables (for example, the splittings between the first and second generation charged slepton masses, $\\Delta m_{\\tilde{\\ell }}$ ) is summarised on the right panel of Fig.", "REF : “isolated” cLFV manifestations (i.e., outside the coloured regions) would allow to disfavour the SUSY seesaw hypothesis as the (unique) underlying source of lepton flavour violation, while “compatible” ones would strenghten it, furthermore hinting on the seesaw scale [16].", "Figure: On the left, correlation between BR(μ→eγ\\mu \\rightarrow e \\gamma ) and BR(τ→μγ\\tau \\rightarrow \\mu \\gamma ) in a type I SUSY seesaw for differentseesaw scales (from );on the right panel, 1 st 1^\\text{st} and 2 nd 2^\\text{nd} generationcharged slepton mass splittings vs. BR(μ→eγ\\mu \\rightarrow e \\gamma ), with CR(μ-e\\mu -e,Ti) on secondary y-axis in a type I SUSY seesaw, for different valuesof the heaviestright-handed neutrino mass M R 3 =10 13,14,15 M_{R_3} = 10^{13,14,15} GeV (M R 1 ,R 2 =10 10,11 M_{R_1,R_2} =10^{10,11} GeV) and for a flavour conserving modified mSUGRA benchmark(from )." ], [ "Outlook", "As of today, we have firm evidence that flavour is violated in the quark sector, as well as in the neutral lepton one.", "In the absence of a fundamental principle preventing it, there is no apparent reason for Nature to conserve charged lepton flavours.", "By itself, any observation of a cLFV process would constitute a clear signal of new physics - beyond the SM extended via massive (Dirac) neutrinos.", "As we aimed at illustrating in the present brief review, cLFV observables could provide valuable (indirect) information on the underlying new physics model, and certainly contribute to at least disfavour several realisations.", "The current (and planned) experimental programme, with numerous observables being searched for in a large array of high-intensity and high-energy experiments (see [1]) renders cLFV a privileged laboratory to search for new physics.", "AMT is gratefull to the Organisers of “Neutrino 2016” for the invitation and support.", "Part of the work here summarised was done within the framework of the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements No 690575 and No 674896." ] ]
1612.05561
[ [ "Effective Non-oscillatory Regularized L$_1$ Finite Elements for Particle\n Transport Simulations" ], [ "Abstract In this work, we present a novel regularized L$_1$ (RL$_1$) finite element spatial discretization scheme for radiation transport problems.", "We review the recently developed least-squares finite element method in nuclear applications.", "We then derive an L$_1$ finite element by minimizing the L$_1$ norm of the transport residual.", "To ensure the stability on incident boundary, we newly develop a consistent L$_1$ boundary condition (BC).", "The numerical tests demonstrate such a method effectively prevents the oscillations which would occur to least-squares finite element when discontinuity exists such as void and absorber.", "Further, the RL$_1$ method is accurate in problems with scattering." ], [ "Introduction", "Neutral particle transport problems are governed by the linear Boltzmann transport equation, a first-order, hyperbolic equation for the phase space density of particles.", "The streaming operator in the transport equation is a linear advection operator.", "Consequently, problems where particles travel long distances between scattering interactions, so-called streaming-dominated problems, e.g.", "void and strong absorber problems, can have discontinuous solutions, a fact that spatial discretization schemes should respect.", "On the other hand, in regions where particles travel very short distances between scattering interactions, the transport equation asymptotically limits to a diffusion equation.", "There has been much research into methods that are asymptotic preserving for this limit [1], [2].", "At present the most widely used spatial discretization scheme that can handle both discontinuous solutions in the streaming dominated case and preserves the asymptotic diffusion limit is the discontinuous finite element method (DFEM)[3], [4].", "DFEM is widely used despite the notorious disadvantage of requiring additional degrees of freedom compared with the continuous finite element method (CFEM) that is commonly used in elliptic and parabolic problems.", "Another approach to solve transport problems recasts the transport equation into a symmetric, second-order form.", "This can be done by deriving even-parity equations and self-adjoint angular flux equations[5], [6], [7], [8], [9] or by using the least-squares finite element method.", "The least-squares finite element method seeks solutions in a finite element space to minimize the squared residual of the transport equation.", "The minimizer of the squared residual is the solution to the weak form of a symmetrized transport equation.", "Thereafter, CFEM can be used to solve this transport equation.", "Recent work has successfully applied least-squares finite elements to a variety of transport applications[10], [11], [12], [13], [14], [15], [16], [17], [18].", "There remain a number of open problems regarding least-squares finite elements.", "For instance, using an under-resolved mesh in problems with strong absorbers can induce oscillations and negative particle densities[19], [20], [21], [22].", "These negative densities, in addition to being non-physical, yield nonsensical reaction rates, which are usually the primary quantity of interest for particle transport simulations.", "Moreover, in multi-D situations, void or near-void situations can also induce negativity in the particle density due to oscillations caused by the existence of discontinuities in the solution to the continuum equations.", "These oscillations exist even in solutions to the first-order transport with DFEM[22] when piecewise linear or high-degree basis functions are used.", "Even without oscillations, void regions can induce low accuracy of the least-squares method without specific corrections [12].", "Yet, in real-world problems, like radiation shielding problems or remote sensing, those situations are usual and inevitable.", "For the least-squares (LS) method, one cause of the oscillations is that the L$_2$ norm of transport residual overestimates the contribution from large residual components.", "Like least-squares fitting in data analysis, when trying to fit every single data point, the fitting principle would overweight the contribution from large-error points, leading to erroneous and oscillatory results[23], [24].", "One of the remedies is to develop finite elements that minimize the residual of the transport equation in other norms, such as L$_1$ .", "This is difficult because the L$_1$ norm is non-smooth and typically requires the solution of nonlinear equations.", "Previously, Jiang developed the iteratively reweighted least-squares (IRLS) method for linear advection[24].", "The IRLS method uses a power of the inverse residual as the weighting function for a least-squares method.", "For simplicity, IRLS uses an approximation to the residual in the weighting function and assumes the weighting function to be constant in each spatial cell.", "The rationale behind this choice is that such a method would approximate the L$_1$ norm-induced method.", "It was shown that for discontinuous boundary data, IRLS is extremely accurate in the interior of the domain.", "However, Lowrie and Roe [25] demonstrated that IRLS does not necessarily propagate information correctly from boundary so that if the incident boundary condition (BC) is smooth, IRLS can give erroneous results.", "In particular, large gradients of the solution can be mistakenly treated as discontinuities.", "Furthermore, Lowrie and Roe demonstrated that IRLS is not an L$_1$ method.", "In a more recent work, by developing efficient nonlinear solving techniques, Guermond approximated the L$_1$ solution for several problems in fluid dynamics based on Newton's method[26].", "The L$_1$ method is demonstrated to be accurate and stable in problems where least-squares has difficulty and oscillations and accurately treats smooth solutions in contrast to Lowrie's findings about IRLS.", "Yet, Guermond's implementation of L$_1$ does not have a continuum form, making it difficult to apply to problems of particle transport where strong interactions could dominate advection.", "It also lacks a theory for consistent boundary conditions, which is required for many transport problems.", "We wish to have a method with stability property of the L$_1$ method, as demonstrated by Guermond, with an implementation that builds on the technology recently developed for least-squares finite elements.", "Therefore, we introduce a regularized L$_1$ method for solving particle transport problems.", "Such a method is designed to be a regularized version of L$_1$ method in the sense that L$_1$ method will be used only when the pointwise residual becomes larger than certain criteria otherwise the least-squares method is used.", "Additionally, the scheme is designed to be compatible with source iteration and acceleration techniques like diffusion synthetic acceleration (DSA)[27] that are de rigeur for efficiently solving transport problem.", "We also develop a consistent L$_1$ BC which, as we will demonstrate, is necessary to obtain accurate solutions.", "The remainder of this paper is organized as follows: in Sec.", ", we derive the LS weak formulation solving one-speed steady-state radiation transport equation; we then derive an L$_1$ and further a regularized L$_1$ (RL$_1$ ) finite element method in a continuum form in Sec.", ".", "Therein, we further derive the consistent L$_1$ and RL$_1$ boundary weak formulation to prevent oscillations on incident boundaries.", "We briefly discuss the details of the implementation.", "Thereafter, we demonstrate the method efficacy in Sec.", ".", "We conclude the work in Sec.", ".", "Given that we are interested in spatial discretization of the transport equation, we will restrict ourselves in this work to steady state, energy-independent transport problems, with isotropic scattering and volumetric fixed source.", "Extending our method with time and energy dependence and anisotropic scattering is relatively straightforward using existing methods.", "The steady, mono-energetic transport equation is given by [28], [29], [6]: $\\vec{\\Omega }\\cdot \\nabla \\psi (\\vec{r},\\vec{\\Omega })+\\sigma _\\mathrm {t}(\\vec{r})\\psi (\\vec{r},\\vec{\\Omega })=\\frac{\\sigma _\\mathrm {s}\\phi (\\vec{r})}{4\\pi }+\\frac{{Q}(\\vec{r})}{4\\pi }, \\qquad \\vec{r}\\in \\mathcal {D}$ where $\\vec{\\Omega }\\in \\mathbb {S}_2$ is the directional vector on the unit sphere corresponding to the direction of travel for particles; $\\psi $ is the angular flux with units of particles per unit area per steradian per second; $\\sigma _\\mathrm {s}$ and $\\sigma _\\mathrm {t}$ are respectively the scattering and total cross sections with units of inverse length; $\\phi $ is the scalar flux defined as $\\phi (\\vec{r})=\\int \\limits _{{4\\pi }}d\\Omega \\ \\psi (\\vec{r},\\vec{\\Omega });$ $Q$ is the isotropic volumetric fixed source.", "The boundary conditions for Eq.", "(REF ) specify $\\psi $ for incoming directions on the boundary: $ \\psi (\\vec{r}, \\vec{\\Omega }) = \\psi ^\\mathrm {inc}(\\vec{r},\\vec{\\Omega }), \\qquad \\vec{r}\\in \\partial \\mathcal {D},\\quad \\vec{\\Omega }\\cdot \\hat{n} < 0,$ where $\\hat{n}(\\vec{r})$ is the unit outward normal on the boundary of the domain.", "For notational simplicity, we will also use the operator form as the following: $\\mathcal {L}\\psi =q_\\mathrm {s},$ where $\\mathcal {L}$ is the streaming plus removal operator $ \\mathcal {L}\\equiv \\vec{\\Omega }\\cdot \\nabla +\\sigma _\\mathrm {t},$ and the total (fixed plus scattering) source is $q_\\mathrm {s}\\equiv \\frac{\\sigma _\\mathrm {s}\\phi (\\vec{r})}{4\\pi }+\\frac{{Q}(\\vec{r})}{4\\pi }.$ We will discretize the angular component of the transport equation using the discrete ordinates (S$_N$ ) method [7] where we use a quadrature set for the angular space, $\\lbrace w_n, \\vec{\\Omega }_n\\rbrace $ , to obtain $N$ equations of the form $ \\mathcal {L}_n \\psi _n = q_\\mathrm {s},$ where $\\psi _n = \\psi (\\vec{r}, \\vec{\\Omega }_n),$ $ \\mathcal {L}_n \\equiv \\vec{\\Omega }_n \\cdot \\nabla +\\sigma _\\mathrm {t},$ and $ \\phi (\\vec{r}) \\approx \\sum _{n=1}^N w_n \\psi _n(\\vec{r}).$ On the boundary we write $\\psi _n(\\vec{r}) = \\psi ^\\mathrm {inc}(\\vec{r}, \\vec{\\Omega }_n)$ for $\\vec{r}\\in \\partial \\mathcal {D}$ and $\\vec{\\Omega }_n \\cdot \\hat{n} < 0$ .", "This choice of angular discretization will not affect the derivation of our spatial discretization.", "Therefore, we will drop the $n$ subscripts in the following sections.", "For example, we could use our regularized L$_1$ method with the spherical harmonics treatment of the angular variable." ], [ "Interior Weak Form of the Least-Squares Discretization", "To derive a LS discretization of the transport equation, we begin with defining the L$_2$ norm of transport residual $R=\\mathcal {L}\\psi -q_\\mathrm {s}$ away from the boundary of the domain: ${\\Gamma }_{\\mathrm {L}_2}(\\psi )=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ R(\\psi )^2,$ where we have restricted $\\psi $ to belong to a finite element space $\\mathcal {V}$ .", "We next introduce an arbitrarily small perturbation $\\epsilon v$ , where $\\epsilon >0 $ is an arbitrarily small number and $v$ is a weight function in the finite element space $\\mathcal {V}$ .", "Then we obtain the perturbed functional: ${\\Gamma }_{\\mathrm {L}_2}(\\psi +\\epsilon v)=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\left(R(\\psi +\\epsilon v)\\right)^2.$ To minimize the functional, we expect the first derivative to be zero in order to find the stationary point in the finite element space [30], i.e.", "$\\frac{\\partial {\\Gamma }_{\\mathrm {L}_2}}{\\partial \\epsilon }\\bigg |_{\\epsilon =0}=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\mathcal {L}v\\left(\\mathcal {L}\\psi -q_\\mathrm {s}\\right)=0.$ Therefore, after rearranging, we have the weak formulation for the interior: find $\\psi \\in \\mathcal {V}$ such that for any $v \\in \\mathcal {V}$ $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\mathcal {L}v\\mathcal {L}\\psi =\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\mathcal {L}vq_\\mathrm {s}.$" ], [ "Boundary condition and complete weak formulation", "It is straightforward as well to obtain a weak form for the BC for the LS method.", "Similar to Eq.", "(REF ), we define the functional for the BC measured by the L$_2$ norm: ${\\Gamma }_{\\mathrm {b,L}_2}=\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\left(\\psi -\\psi ^\\mathrm {inc}_{{}}\\right)^2,$ where $\\lambda $ is a cross section related multiplier and defined as: $\\lambda ={\\left\\lbrace \\begin{array}{ll}\\sigma _\\mathrm {t},&\\sigma _\\mathrm {t}>0,\\\\1.0,&\\mathrm {otherwise}.\\end{array}\\right.", "}$ For the non-void situation, the LS weak form is globally conservative[15], [6] with the choice in Eq.", "(REF ).", "The first choice of $\\lambda $ in void is somewhat arbitrary, but has been observed to be adequate.", "With the same procedure as in Sec.", "REF , we arrive at the boundary weak form: $\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|v\\psi =\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|v\\psi ^\\mathrm {inc}_{{}}.$ Combining this result with Eq.", "(REF ), we reach the complete LS weak form: $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\mathcal {L}v\\mathcal {L}\\psi +\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|v\\psi =\\nonumber \\\\\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\mathcal {L}vq_\\mathrm {s}+\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|v\\psi ^\\mathrm {inc}_{{}}.$" ], [ "Smoothed L$_1$ norm and L{{formula:09e0018b-9d8f-4081-b22e-f9961b55a26b}} finite element method", "Similar to the LS method, we begin by defining the L$_1$ norm of the transport residual: ${\\Gamma }_{\\mathrm {L}_1}(\\psi )=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ |R(\\psi )|$ A suitable finite element method would be developed by minimizing the functional above.", "With the procedure introduced in Sec.", "REF , we obtain a perturbed functional: ${\\Gamma }_{\\mathrm {L}_1}(\\psi +\\epsilon v)=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ |R(\\psi +\\epsilon v)|,\\ v\\in \\mathcal {V},\\ \\epsilon >0$ However, the functional ${\\Gamma }_{\\mathrm {L}_1}$ is not differentiable at the points where $R=0$ .", "To find the fixed point of Eq.", "(REF ) we develop an approximate L$_1$ norm, as in [20], [19].", "For a small number $\\zeta $ we approximate $|R|\\approx \\sqrt{R^2+\\zeta ^2}.$ Figure REF illustrates the effect of the approximation for different values of $\\zeta $ .", "With decreasing $\\zeta $ , $\\sqrt{x^2+\\zeta ^2}$ converges to $|x|$ rapidly.", "Figure: x 2 +ζ 2 \\sqrt{x^2+\\zeta ^2} vs |x||x| for different ζ\\zeta values.Introducing Eq.", "(REF ) into (REF ), we obtain a differentiable approximation to the perturbed L$_1$ norm functional: ${\\Gamma }_{\\mathrm {L}_1}(\\psi +\\epsilon v)\\approx \\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\sqrt{R(\\psi +\\epsilon v)^2+\\zeta ^2}.$ To minimize the convex functional, in the finite element space $\\mathcal {V}$ , we need to find its stationary point where the derivative is zero: $\\frac{\\partial {\\Gamma }_{\\mathrm {L}_1}}{\\partial \\epsilon }\\bigg |_{\\epsilon =0}=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}v(\\mathcal {L}\\psi -q_\\mathrm {s})}{\\sqrt{R^2+\\zeta ^2}}=0.$ Taking the limit $\\zeta \\rightarrow 0$ , the smoothed L$_1$ expression limits to the L$_1$ weak form as $\\sqrt{R^2+\\zeta ^2}\\rightarrow |R|$ for $|R| \\ne 0$ : $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}v(\\mathcal {L}\\psi -q_\\mathrm {s})}{|R|}=0,$ or equivalently, $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}v\\mathcal {L}\\psi }{|R|}=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}vq_\\mathrm {s}}{|R|}.$ This is equivalent to the weak form for the LS finite element with a weighting function that depends on the solution.", "Clearly, this weak formulation is nonlinear because the evaluation of $|R|$ requires the solution $|\\psi |$ ." ], [ "An L$_1$ BC", "The naïve BC for the L$_1$ method would use BC from the LS weak form.", "However, we have observed that the application of this BC causes stability problems on the incident boundaries.", "A hypothesis is that the norms measuring residuals on the boundary and the interior should be consistent.", "We can derive a regularized L$_1$ BC similar to the approach used for the interior functional.", "Namely, we write ${\\Gamma }_{b,\\mathrm {L}_1}=\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|.$ A boundary weak form as the following can be achieved through similar minimization process as above: $\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{v\\psi }{\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|}=\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{v\\psi ^\\mathrm {inc}_{{}}}{\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|}.$" ], [ "L$_1$ and regularized L{{formula:bd495e53-2860-4502-aa24-4bbb7c26cacd}} weak forms", "Whence we have the complete L$_1$ finite element weak formulations: $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}v\\mathcal {L}\\psi }{|R|}+\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{v\\psi }{\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|}\\nonumber \\\\=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\mathcal {L}vq_\\mathrm {s}}{|R|}+\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{v\\psi ^\\mathrm {inc}_{{}}}{\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|}$ Due to the assumption of letting $\\zeta $ vanish, the weak form above is an exact L$_1$ finite element formulation.", "However, solving such a weak form can be extremely challenging, especially when residual in the problem varies by several orders of magnitude and when the residual vanishes in certain regions.", "Instead, we propose a regularized L$_1$ formulation using a factor $\\theta >0$ : $\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\theta \\mathcal {L}v\\mathcal {L}\\psi }{\\max (\\theta ,|R|)}+\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{\\theta v\\psi }{\\max (\\theta ,\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|)}\\nonumber \\\\=\\int \\limits _{4\\pi }d\\Omega \\ \\int \\limits _\\mathcal {D}d\\vec{r}\\ \\frac{\\theta \\mathcal {L}vq_\\mathrm {s}}{\\max (\\theta ,|R|)}+\\int \\limits _{\\vec{n}\\cdot \\vec{\\Omega }<0}d\\Omega \\ \\int \\limits _{{\\partial \\mathcal {D}}}ds\\ \\lambda \\left|\\vec{n}\\cdot \\vec{\\Omega }\\right|\\frac{\\theta v\\psi ^\\mathrm {inc}_{{}}}{\\max (\\theta ,\\left|\\psi -\\psi ^\\mathrm {inc}_{{}}\\right|)}.$ This regularization is such that in regions with moderate to large residuals, i.e.", "$|R|>\\theta $ , the regularization factor $\\frac{\\theta }{\\max (\\theta ,|R|)} \\rightarrow \\frac{\\theta }{|R|}$ and the L$_1$ weak form is used.", "On the other hand, when the residual is small, the regularization factor goes to 1 and the least-squares weak form is used.", "Numerical experiments indicate that the solution is largely insensitive to the choice of $\\theta $ .", "The effects of varying it will be illustrated in Sec.", "REF .", "Normally, we choose $\\theta $ to be around $0.01|R|_\\mathrm {max}$ , where $|R|_\\mathrm {max}$ denotes the maximum absolute residual of all direction and space.", "Smaller $\\theta $ can be used, yet, we observe the efficiency of the linear solver is degraded without a concomitant gain in solution accuracy.", "In our implementation the residuals are evaluated on each spatial quadrature point.", "This is one of the drawbacks of this method in that it requires storing or performing on-the-fly calculations of the point wise residuals." ], [ "Nonlinear solution method", "As RL$_1$ is nonlinear, an appropriate nonlinear scheme is necessary in order to solve Eq.", "(REF ).", "We will use a Picard iteration scheme that evaluates the residual using the estimate of $\\psi $ from the previous iteration.", "The scheme is initialized using the unweighted least-squares solution.", "The steps in the solution are Calculate pointwise residuals for Nonlinear Iteration (NI) $l$ from NI $l-1$ ; Update the weak form Eq.", "(REF ); Solve Eq.", "(REF ) Given a nonlinear tolerance $tol$ , check nonlinear convergence $e=\\frac{\\Vert \\phi ^l-\\phi ^{l-1}\\Vert }{\\Vert \\phi ^l\\Vert }$ : If $e<tol$ , stop.", "else, go to Step 1.", "In the solution of Eq.", "(REF ) source iteration with diffusion synthetic acceleration (DSA) is utilized[7], [14].", "Though RL$_1$ , as well as LS, does not have consistent low order diffusion acceleration scheme[14], we have found that DSA is still effective.", "Developing a consistent DSA scheme for LS and RL$_1$ should be the target of future work.", "All numerical results below used finite element solutions carried out with the C++ Open source library deal.II[31].", "In this section, we present four 2D test problems with bilinear finite elements on rectangular meshes.", "We first investigate the behavior of RL$_1$ in void with discontinuous incident BC." ], [ "Void problem", "As mentioned above transport problems in voids can contain numerical artifacts such as oscillations and negative solutions due to the discontinuous nature of the analytic solution [21], [19], [20], [22].", "A demonstration of these phenomena can be seen in the solution to a beam problem at a grazing angle entering a void from the boundary.", "The solution to this problem is discontinuous with solution being zero outside the volume within the view of the beam.", "Schemes such as LS will have Gibbs oscillations due to the discontinuity.", "Figure REF shows the LS and RL$_1$ solution for this a 2-D void with where the domain is a 0.5$\\times $ 0.5 cm square.", "Along part of the boundary from $x=0.1$ to $0.3$ cm there is a unit incident angular flux at the angle $\\vec{\\Omega }=(1/\\sqrt{3},1/\\sqrt{3},1/\\sqrt{3})$ .", "The unit incident BC is applied only on part of the boundary.", "The LS solution oscillates near the discontinuity, and has an overshoot of 7%.", "The RL$_1$   solution, on the other hand, is monotone and non-negative.", "Figure: LS and RL 1 _1 method comparison in void transport problem with a grazing, incident flux on the boundary.", "100×\\times 100 cells are usedThis problem can also demonstrate the necessity of having a boundary condition based on L$_1$ .", "In Figure REF we compare the analytic, LS, and RL$_1$   solutions along the boundary at $y=0$ .", "In the RL$_1$   solution we use the L$_1$ boundary condition we derived above, and the standard LS boundary condition.", "As seen in the figure, the LS solution does over and under shoot the analytic solution.", "When the RL$_1$ method is used with the LS boundary condition, the result is a sharp oscillation near the discontinuity.", "These oscillations go away when the L$_1$ -based boundary condition is used.", "Figure: LS and RL 1 _1 with different boundary conditions on the void problem.The results in the previous two figures were performed with $\\theta /|R|_\\mathrm {max}=0.01$ .", "In Figure REF we show the line out of the solution $y=0.2$ cm with differing values of $\\theta /|R|_\\mathrm {max}$ .", "With a large value of $\\theta $ (the one with $\\theta /|R|_\\mathrm {max}=0.1$ ), RL$_1$ is able to effectively damp the oscillations around the discontinuous solution, though the solution does become slightly negative.", "The results with $\\theta /|R|_\\mathrm {max}=0.01$ and $0.0001$ are nearly indistinguishable in the figure.", "For this reason we will use $\\theta /|R|_\\mathrm {max}=0.01$ for the remainder of this work.", "Figure: Solution to the void problem at y=0.2y=0.2 cm for different values of θ/|R| max \\theta /|R|_\\mathrm {max}.As the mesh is uniformly refined we observe that the errors in the solution to the void problem decrease as the mesh size $h$ to the one-half power for both methods, as illustrated in Figure REF , yet, the constant is smaller for the RL$_1$ solutions.", "The converged RL$_1$ solutions have the smallest error.", "Typically, a converged solution requires around 40 nonlinear iterations with $tol=1\\times 10^{-4}$ used in this work.", "But only a few nonlinear iterations can provide a noticeable improvement over LS.", "Figure: Void problem convergence results." ], [ "Pure absorber problem", "The second test is with the same geometry as in the previous problem with the void replaced by an absorber with $\\sigma _\\mathrm {t}=1.0$ cm$^{-1}$ .", "Additionally, the unit incident angular flux is imposed through the whole bottom boundary with the same incident angle.", "Figures REF and REF present the angular flux for this problem.", "In the pseudo-color plots in Figure REF we observe the oscillations and negative values in the LS solution; these artifacts do not appear in the RL$_1$ solution.", "Figure REF compares RL$_1$ and LS line-outs along y=0.2 cm with the analytic solution.", "Indeed, neither of these methods resolves the discontinuity.", "However, the RL$_1$ solution is monotonic, in contrast to LS's oscillatory flux as a result of Gibbs phenomenon.", "We examine the L$_1$ error of scalar fluxes as well.", "As the convergence tests in Figure REF shows, we found both LS and RL$_1$ have a convergence order near one half, with RL$_1$ having lower error magnitudes as expected.", "Figure: Angular flux distributions in incident convergence test.Figure: Angular flux line-out and scalar flux errors for the pure absorber test problem." ], [ "Smooth boundary problem", "In this test, we still use the same material and geometry configurations as in Sec.", "REF except that the we use a smooth boundary condition specified at a grazing direction, $\\Omega _x=0.8688,~\\Omega _y=0.3599,\\Omega _z=0.3599$ .", "The boundary condition for this angle is $\\psi ^\\mathrm {inc}_{{}}={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1}{2}+\\frac{1}{2}\\cos \\left(2\\pi \\frac{x-0.2}{0.2}\\right),&x\\in (0.1,~0.3)\\ \\mathrm {cm},\\ y=0\\ \\mathrm {cm},\\\\0, &\\mathrm {otherwise}.\\end{array}\\right.", "}$ Figure: Angular flux distributions for the grazing incident.Figure REF illustrates the spatial distribution of angular flux for both methods.", "LS gives a negative angular flux even when the incident boundary condition is smooth in space.", "As the line-out plots presented in Figs.", "REF (incident boundary) and REF (outgoing boundary), with converged RL$_1$ , we see agreement with the analytic solution except for slight smearing on the outgoing boundary in Figure REF .", "Moreover, without convergence but with only one or two iterations of residual weighting, RL$_1$ can still deliver acceptable results.", "On the other hand, LS solution presents negative solutions.", "Moreover, RL$_1$ , in contrast to IRLS[24], [25], treats the smooth incident data correctly and propagates the information correctly throughout the domain.", "Figure: Boundary results comparison for smooth boundary problem." ], [ "Ackroyd test", "Finally, we present the results for the Ackroyd problem[32], [17].", "This is a heterogeneous problem that has a high-scattering ratio central block and outer shield ($\\sigma _\\mathrm {t}=0.2,\\sigma _\\mathrm {s}=0.19$ ).", "There is a void between the block and the shield.", "See Fig.", "REF for a schematic of a quarter of the problem geometry.", "The whole problem is symmetric about x-axis and y-axis.", "A reference scalar flux from an LS calculation using 320$\\times $ 320 cells with a Gauss-Chebyshev level-symmetric-like S$_{8}$ quadrature[33] is presented in Figure REF .", "We examine the solution along a line that crosses the void in Figure REF and another one on the boundary in Figure REF .", "In both cases, the coarse mesh (32$\\times $ 32) RL$_1$ solution agrees reasonably well with the fine mesh LS results, while the LS coarse-mesh solution has large and noticeable errors.", "When refining to 128$\\times $ 128, RL$_1$ agrees with the reference, while LS still has large errors.", "Figure: Ackroyd problem layout and reference solution.Figure: Ackroyd problem line-out plots." ], [ "Concluding remarks and future work", "In this work, we developed an effectively non-oscillatory regularized L$_1$ finite element scheme for solving radiation transport problems.", "Starting from minimizing the L$_1$ norm of transport residual, we derived a continuum form of L$_1$ finite element method along with a consistent nonlinear L$_1$ BC.", "The resulting continuum form is a weighted LS problem where the weight is the inverse of the absolute value of the residual.", "We regularize this form by switching the weight to unity when the residual is small.", "Numerical tests with void, pure absorber and heterogeneous problems demonstrate the efficacy and accuracy of our methodology.", "There exist opportunities to improve our method.", "For one, our technique for solving the resulting nonlinear equations is based on fixed-point iteration.", "The universal efficiency of such a scheme is to be investigated.", "Other possibilities, such as Jacobian-free Newton Krylov method[34], could improve the efficiency of our scheme.", "Furthermore, our scheme inherits the properties of LS in that it is only conservative in the limit of the mesh size going to zero.", "Other authors have presented ways to ameliorate this issue (such as SAAF weighting for problems without void) and other void treatments [35], [36].", "Another possibility is to use a high-order low-order formalism [37], [38] to solve the transport equation, where RL$_1$ is used for the high-order solve." ], [ "Acknowledgements", "W. Zheng is thankful to Dr. Wolfgang Bangerth from Colorado State University for suggestions on the methodology derivation.", "Also, appreciation goes to Dr. Milan Hanus from Texas A&M University for fruitful discussions and helpful proofreading.", "This project is funded by Department of Energy NEUP research grant from Battelle Energy Alliance, LLC- Idaho National Laboratory, Contract No: C12-00281." ] ]
1612.05581
[ [ "Atomic Gas in Debris Discs" ], [ "Abstract We have conducted a search for optical circumstellar absorption lines in the spectra of 16 debris disc host stars.", "None of the stars in our sample showed signs of emission line activity in either H$_{\\alpha}$, Ca II or Na I, confirming their more evolved nature.", "Four stars were found to exhibit narrow absorption features near the cores of the photospheric Ca II and Na I D lines (when Na I D data were available).", "We analyse the characteristics of these spectral features to determine whether they are of circumstellar or interstellar origins.", "The strongest evidence for circumstellar gas is seen in the spectrum of HD110058, which is known to host a debris disc observed close to edge-on.", "This is consistent with a recent ALMA detection of molecular gas in this debris disc, which shows many similarities to the $\\beta$ Pictoris system." ], [ "Introduction", "Debris discs represent the final stage of the planet formation process.", "The dust observed in these systems is thought to be replenished through collisions between solid bodies [66], and are therefore expected to be gas depleted.", "However, a few debris discs show the presence of small amounts of circumstellar gas.", "Whether this gas is remnant of the early disc stages (primordial) or of secondary origin (e.g.", "brought to the gas phase by photodesorption or by cometary activity) is a matter of contemporary debate [17], [47], [33], [34], [43], [20], [40].", "Circumstellar gas was first discovered around $\\beta $  Pictoris over 40 years ago when [55] noticed the presence of narrow Ca ii absorption lines located at the centre of $\\beta $  Pic photospheric lines.", "Subsequent to the first imaging of the debris disc around $\\beta $  Pic by [56], [22] observed similar features both in Ca ii and Na i.", "The radial velocities of the absorptions coincided with the radial velocity of the star, and were attributed to circumstellar gas orbiting the star.", "Variable absorption features have been indentified, attributed to the evaporation of solid kilometer-sized bodies falling into the star [3], [9], [62], [31], [65].", "Over a dozen other debris discs with circumstellar gas absorption features have been indentified with this method, such as HR 10 [35], HD 32297 [52], 49 Ceti [44], HD 172555 [32], HD 21620, HD 110411, HD 145964, HD 183324 [64], and $\\phi $  Leo [19].", "Detecting gas in discs using absorption lines is difficult since the gas disc must be favorably oriented close to edge-on in order to be detected.", "On the other hand, the advantage of this method is that it is sensitive to much lower gas column densities compared to direct observations of gas emission.", "In this work we study the optical spectra of 16 debris disc host stars in order to search for gas-bearing $\\beta $ Pictoris-like systems.", "One of the main difficulties of this technique is to rule out interstellar (IS) absorption features, which may be very similar in profiles and strength to circumstellar lines.", "This requires careful analysis of the line characteristics [36], [11], [52], [64]." ], [ "Target sample", "The sample consists of 16 debris disc host stars taken from the catalogue of [42] having spectral types ranging from B8 to F7 [23].", "The sample was selected in terms of their low dust fractional luminosities, with all stars having L$_{\\rm IR}$ /L$_{\\rm \\star }$ values less than those investigated by [18] being selected in order to choose true debris discs.", "Dust fractional luminosities were computed by fitting a black-body to the observed spectral energy distributions (SED).", "The stellar SEDs were approximated using templates from the library of stellar atmospheres by [7], for models with $log g =4.0$ and solar metallicity.", "Figure REF shows the results from the SED fitting routine, where both the observed SED and the fitted stellar + dust blackbody model model are plotted.", "L$_{\\rm IR}$ /L$_{\\rm \\star }$ values in our sample are found to range between 1.8$\\times 10^{-3}$ and 7.5$\\times 10^{-6}$ .", "We estimate our method to be accurate to factors of 2-3 by comparing our $L_{\\rm IR}$ /$L_{*}$ results with values already published in the literature [59], [45], [21].", "Based on their dust fractional luminosities, these systems are expected to be in a more advanced evolutionary stage than the sample studied by [18].", "All stars have Hipparcos distances available [61].", "Spectral standards of three spectral types were also observed in order to allow for spectral classification and removal of telluric absorption lines.", "Table REF summarizes the properties of our sample, where we have listed the properties of our observed target sample along with those of the 3 observed spectral standards (HR 5558, HR 5670 and HR 6045)." ], [ "AAT/UCLES Observations", "Optical Echelle spectroscopy of the 16 debris disc host stars was obtained using the UCL Echelle Spectrogaph (UCLES) at the 3.9 meter Anglo-Australian Telescope (AAT).", "All observations were taken on the night of May 20th 2000.", "A log of the observations is given in Table REF , including the slit-width and seeing values.", "The 31.6 g/mm grating was used in conjunction with the MIT/LL CCD.", "Observations were carried out with two wavelength settings in order to cover the blue region of the spectra between 3834-5440 $\\rm {Å}$ , and between 5550-9900 $\\rm {Å}$ in the red.", "The red observations were optimized for the study of H$_{\\alpha }$ , He i and the Na i D-lines, whilst the blue exposures covered the Ca ii K and H lines as well as several metallic lines in the 4000-5000 $\\rm {Å}$ region.", "The spectral resolution, estimated from measuring the $FWHM$ of Thorium-Argon arc lines, was 0.115 $\\rm {Å}$ ($\\sim 8.8~$ km s$^{-1}$ ) at Ca ii K (3933.663 $\\rm {Å}$ ), corresponding to a resolving power of $R=34200$ .", "At the location of the Na i D-lines, in the red part of the spectrum (5889.951 $\\rm {Å}$ ), the spectral resolution and resolving power were found to be 0.171 $\\rm {Å}$ and $R=34600$ respectively.", "Table: Debris disc host stars observed along with spectral standardsused for spectral classification.", "Spectral types are from MichiganSpectral Catalogue and distances are derived fromparallax measures listed in the Hipparcos catalogue.", "B-V colours are taken from the Simbaddatabase.", "The L IR _{\\rm IR}/L ☆ _{\\star } values were derived using the25 μ\\mu m and 60 μ\\mu m IRAS excess fluxes.", "L ☆ _{\\star } is theluminosity of the stellar model, normalised tothe 12 μ\\mu m flux.", "L IR _{\\rm IR} is the bolometric luminosity of theblack-body that fits the infrared excesses.Figure: Dust black-body fits to the infrared excesses used in thederivation of L IR _{\\rm IR}/L ☆ _{\\rm \\star } values for stars withdifferent spectral types in our sample.", "The observed IRASfluxes, as listed in , are plotted in blue with errorbars denoting the 1σ\\sigma photometric errors.", "The dotted line inred corresponds to the reference stellar SED from, normalised to the 12 μ\\mu m flux.", "The dashedline represents the fitted black-body and the solid line representsthe resulting SED (SED ☆ _{\\rm \\small \\star }+SED Dust Black body _{\\rm \\small DustBlack body }) that provides a satisfactory fit to the data.", "Thestellar temperatures characteristic of the corresponding stellarmodel from the library and the temperature ofthe fitted blackbody are indicated in each plot.Table: Log of UCLES observations.Overscan correction, dark subtraction and flat-fielding were performed using the Image Reduction and Analysis Facility (IRAF http://www.iraf.noao.edu) package developed by the National Optical Astronomy Observatory (NOAO).", "When more than one exposure for the same star was available, the different frames were coadded and the average taken.", "At this stage the `crreject' task was used in order to remove cosmic rays.", "Echelle spectra were extracted following standard echelle reduction procedures using the different tasks of the Echelle package in IRAF.", "These include order tracing, extracting and blaze correcting each order.", "Wavelength calibration was performed using reference arc lines from a Thorium-Argon lamp.", "Each order was wavelength calibrated individually, producing a dispersion accuracy always better than 0.004 $\\rm {Å}$ ." ], [ "Magellan/MIKE Observations", "Follow up observations of the target HD 110058 were obtained using the Magellan Inamori Kyocera Echelle (MIKE) spectrograph [2], mounted on the Magellan II telescope.", "In order to characterize the distribution of interstellar material in the surroundings of HD 110058, three stars located in the same direction and within a small range of distances from the main target were observed.", "All observations were carried out in the night of 2008 July 4 under clear sky conditions and with seeing between 0.9 and 1.2 arcseconds.", "The summary of the observations together with the target properties are shown in Table REF .", "MIKE allows to obtain both the red and blue side of the spectrum simultaneously, covering form 3350 to 9150 $Å$ .", "The 0.7x5  arcsec$^2$ slit was used which provided spectral resolutions of 42000 and 32000 in the blue and red parts of the spectra, respectively (as measured by fitting the FWHM of the Thorium-Argon lamp lines).", "All targets are bright (V$<$ 8), so signal-to-noise ratios (SN) higher than 15 can be obtained in both the blue and red arms in only a few minutes of total integration time (listed in the last two columns of Table REF ).", "Data reduction was performed using the CarPy MIKE data reduction package [30], which performs the standard steps for echelle reduction i.e.", "overscan subtraction, order tracing, sky subtraction, extraction, wavelength calibration.", "The pipeline also combines multiple frames of the same target and to produce final combined spectra.", "Table: Summary of MIKE observations.", "Spectral types are from theMichigan Spectral Catalog.", "Distances are derived from Hipparcos parallaxes.", "Column 6 denotes the angular separation fromthe direction of HD 110058." ], [ "Spectral classification and stellar parameters", "In order to check the accuracy of previous spectral classifications of our sources, several regions of the blue parts of the acquired spectra were compared to those of the observed standards (as listed in Table REF ) and also to archive spectra from the UVES Paranal Observatory Project [1].", "The latter provides a library of high-Resolution spectra of stars across the Hertzsprung-Russell Diagram at a resolution of $R\\sim 80000$ , which were degraded to the resolution of our UCLES spectra.", "Several regions of the 4000-5000 $\\rm {Å}$ part of the observed spectra of the stars were compared by eye to the spectra of the nearest spectral types.", "In some cases the template spectra were artificially broadened in order to match the rotational velocity of the star to be classified.", "No discrepancies between the Michigan spectral types of the target stars and their corresponding spectral templates were found, leading us to conclude that all of our stars have their previous Michigan Spectral Catalog classification confirmed.", "In cases when the Michigan Spectral Catalog had an uncertainty of the order of 1 or 2 spectral sub-classes, our method was able to discriminate and associate the star with one of those spectral types, as listed in Table REF .", "In the case of the five F-type stars (and the A9 star HD 99211), whose spectra are rich in metallic species, radial velocities (V$_{\\odot }$ ) were derived by cross-correlating the observed spectra with template spectra for stars of similar spectral type from the UVESPOP database, for which radial velocities had already been measured.", "The FXCORR task in IRAF was used for this purpose, which allows one to cross-correlate several echelle orders simultaneously, providing a very accurate determination of V$_{\\odot }$ .", "This is reflected in the small error estimates presented in Table REF , which correspond to the standard deviation of the velocities obtained in the different orders.", "For the case of the late B-type and early A-type stars, these stars have intrinsically fewer metallic lines that can be used for radial velocity estimations.", "Therefore the most prominent photospheric lines of Ca ii, Mg  ii and Fe i were used for performing Gaussian fitting to the centres of the photospheric lines.", "The heliocentric V$_{\\odot }$ values listed in Table REF for these stars correspond to the mean obtained using the different lines.", "The Mg ii doublet at 4481.13 $\\rm {Å}$ and 4481.33  $\\rm {Å}$ is unresolved in the case of the B- and early A-type stars, and therefore was not used for radial velocity determination.", "Earth radial velocity corrections were applied to each spectral range, in order to obtain heliocentric velocities (obtained using the Radial Velocity (RV) Starlink package).", "Projected rotational velocities ($v\\,\\sin {i}$ ) were derived using the STAROT package within DIPSO, which allows to artificially broaden the spectrum of a stellar template to match the spectra of the observed star.", "Template spectra with previously measured $v\\,\\sin {i}$ values for each spectral type were taken from the UVESPOP database, by selecting the stars with the lowest rotational velocities.", "Artificial broadening was then applied until a satisfactory fit to the observed spectra was achieved.", "Derived rotational velocities for our target stars are presented in column 3 of Table REF .", "Table: Measured parameters of the debris disc host stars.", "Column 2 confirms theprevious spectral classification for all of our targets.", "Columns 3and 4 list the projected rotational velocities (vsiniv\\,\\sin {i}) and theheliocentric radial velocities (V ⊙ _{\\odot }), respectively.", "Theuncertainties quoted for V ⊙ _{\\odot } and vsiniv\\,\\sin {i} correspond tothe dispersion obtained when deriving the quantities using thedifferent lines." ], [ "Evidence for circumstellar gas", "Of the 16 targets observed, 4 are found to exhibit narrow absorption features located near the centre of the photospheric Ca ii H & K lines - these stars are HD 61950, HD 75416, HD 110058 and HD 166841.", "Similar absorption features are seen in the Na i D lines, with the exception of HD 61950 and HD 75416, for which only Ca ii data were available.", "No narrow absorption components can be seen in either the Ca ii K or Na i D line profiles of the remaining stars of our sample.", "In addition, a narrow feature was also detected in the Ca ii H & K spectrum of the spectral standard HR 5558.", "Figure REF shows the H & K regions of HD 61950 and HD 75416 in the heliocentric velocity frame, with the radial velocities of the stars marked by a vertical line.", "Figure REF shows the Na i D and the Ca ii H& K lines of HD 110058, HD 166841 and HR 5558 in the heliocentric velocity frame.", "These features are too narrow to be photospheric in nature and therefore they must be produced by absorption of stellar light by either an interstellar or circumstellar gas cloud.", "There is a good correspondence between the radial velocities of the components from the different species, as is expected if both the calcium and sodium are located in the same cloud.", "For both HD 110058 and HD 166841 there is a strong absorption component that coincides, within the errors, with the radial velocity of the star.", "HD 61950 and HD 75416 have components that partially overlap the stellar radial velocity, while in the case of HR 5558 the velocity of the absorption feature is totally dissimilar to that of the star and is deemed to be interstellar in origin (see REF ).", "Figure: Heliocentric velocity plots of the narrow Ca ii H(dotted line) and K (solid line) absorption components present in thespectra of HD 61950 and of HD 75416.", "The vertical lines denote theradial velocity of the star.", "The H line spectra have been shiftedvertically for plotting purposes.Figure: Heliocentric velocity plots of narrow absorption featuresseen towards HD 110058 in both the Ca ii H & K lines (top panel) and in theNa i D lines (bottom panel).", "The solid lines correspond tothe Ca ii K and Na i D 2 _2 lines.", "The Na i regionof the spectra was divided by the spectrum of the B8 spectralstandard HR 6045 in order to remove telluric absorption features.Figure: Same as Figure  for HD 166841Figure: Same as Figure  for spectral standard HR 5558The fact that both the radial velocity of the star and of the absorbing cloud are similar is not a sufficient condition to rule out that the absorption lines detected are of interstellar origin.", "There is not a unique criterion that may be used in order to discriminate between an interstellar or circumstellar origin for absorption features, instead it is common to refer to a set of criteria that can be used together in order to resolve the nature of the absorption features [18].", "To rule out an interstellar origin for an absorption feature one has to consider: Is the radial velocity of the narrow absorption feature the same as the stellar velocity?", "Is the radial velocity of the narrow absorption consistent or not with known interstellar velocities in the same direction?", "Is the observed $N$ (Ca ii)/$N$ (Na i) column density ratio consistent with circumstellar (usually$>$ 1) or interstellar (usually$<$ 1) values?", "Is the absorption observed towards stars in the similar line of sight ?", "The $N$ (Ca ii)/$N$ (Na i) must, however, be treated with care as low density clouds or shocked shells in the interstellar medium have also been found to exhibit $N$ (Ca ii)/$N$ (Na i$>$ 1 [8], [57].", "When multiple epoch observations are available, one can also study variability of the absorption features, as circumstellar lines often vary with time [4], [51], [64].", "However not all circumstellar lines are variable, as for instance in $\\beta $ Pic the strongest component is stable and corresponds to the main disc absorption [4].", "We measured the equivalent widths and radial velocities of the narrow absorption features on the normalized spectra by using the Emission Line Fitting (ELF) Starlink routine, which fits Gaussian profiles to the absorption features and returns the equivalent width and centre of these unsaturated components.", "Table REF presents the derived equivalent widths and radial velocities of the narrow Ca ii and Na i absorption features present in the spectra of our targets.", "Table: Equivalent widths and heliocentric velocities of the narrowabsorption lines observed in the spectra of the target stars.", "Theequivalent widths are in mÅ\\rm {Å}.", "For stars where two entries arelisted, these correspond to separate velocity components." ], [ "Cloud modeling", "The Voigt Absorption Profile/Interstellar Dabbler software for modeling interstellar absorption lines [25], was used to estimate the column density ($N$ ), radial velocity ($v$ ) and the velocity dispersion ($b$ ) of the material causing the observed absorption in the Ca ii K and Na i D$_2$ lines.", "VAPID assumes a Gaussian line-of-sight velocity distribution for each absorbing cloud, and uses Voigt functions and least-squares optimization to estimate cloud parameters of the many cloud components required to reproduce the observed spectra.", "Uncertainties in the resulting parameters are estimated using a Monte Carlo method to derives confidence intervals in the parameter space.", "Oscillator strengths of 0.635 and 0.631 were assumed for the Ca ii K and Na i D$_2$ lines respectively [48].", "The best-fit model parameters that were found to reproduce successfully the data are presented in Table REF , while models versus data are plotted in Figures REF and REF .", "Table: Derived line profile parameters for narrow Ca K and Na D 2 _2 velocity components derived from VAPID modeling.", "Theparameter errors reported represent the 1σ\\sigma dispersion ineach parameter (i.e., single-parameter 68% confidenceintervals).", "The last column gives the NN(Ca ii)/NN(Na i) ratio.Figure: Cloud models for the Ca ii K absorption features seentowards HD 61950 (top) and HD 75416 (bottom).", "The solid line showsthe model that gives a satisfactory to the data (dotted line).", "Thevertical line represents the heliocentric velocity of thestar.Figure: Cloud models for the Ca ii K (left panel) andNa i D 2 _2 (right panel) absorption features seen towardsHD 110058, HD 166841 and HR 5558.", "The solid line shows the model thatgives a satisfactory fit to the data (dotted line).", "The vertical linerepresents the heliocentric velocity of the star." ], [ "Velocity Projections of known IS clouds/shells", "All the stars in this study lie in the Fourth Galactic Quadrant.", "This region is almost entirely occupied by the Scorpius-Centaurus association, extending between $\\sim 290^{\\rm o}-360^{\\rm o}$ in longitude and being the closest OB association to the Sun [5], [14].", "Three subgroups have been identified to reside within this association: the Upper Centaurus-Lupus (UCL), Lower Centaurus-Crux (LCC) and the Upper Scorpius (US) shells [15] .", "These subgroups have been studied extensively, and the interstellar medium in their proximity has been modeled as expanding spherical shells of gas, whose physical parameters such as shell centre, radius and expansion velocity have been derived by [14], [15] and [16].", "The location of our target stars with respect to the UCL, LCC and US shells are plotted in Figure .", "In addition to the UCL, LCC and US shells, one must also consider the structure of the local interstellar medium including the Local Interstellar Cloud [53].", "The LIC is a warm, low density interstellar cloud which is itself located within a hot Local Cavity in the interstellar medium which extends to approximately 50 pc [63].", "The LIC is thought to move in the direction $l=186^{\\rm o}\\pm 3$ , $b=-16^{\\rm o}\\pm 3$ , with a heliocentric velocity of $v=26\\pm 1$ km s$^{-1}$ [39].", "The Local Cavity is thought to contain several small clouds located within a few tens of parsecs of the Sun, with characteristics similar to the LIC.", "One such possible cloud is the `G' cloud, which appears to move at $v=29\\pm 1$ km s$^{-1}$ in the direction $l=184.5^{\\rm o}\\pm 2.3$ and $b=-20.5^{\\rm o}\\pm 1.8$ [38]." ], [ "The LIC and G clouds", "Given an interstellar cloud moving with velocity $v_w$ in a direction $(l_w,b_w)$ , the line of sight velocity component of the cloud in a direction $(l,b)$ is given by [12]: $\\frac{v}{v_w}=\\sin {b}\\cdot \\sin {b_w}+\\cos {b}\\cdot \\cos {b_w}\\cdot \\cos {(l-l_w)},$ where the terms on the right-hand side correspond to the cosine of the angle between $(l,b)$ and $(l_w,b_w)$ ." ], [ "The UCL, LCC and US shells", "For an expanding shell model the following equation can be used to determine the velocity component for a point $(l,b)$ on the surface of a sphere of radius $r_s$ , centred at a distance R from the Sun at $(l_o,b_o)$ : $v=\\pm \\frac{v_o}{r_{s}}\\sqrt{R^{2}(\\cos {\\theta }^{2}-1)+r{_s}^2},$ where $\\cos {\\theta }$ is given by the term in the right-hand side of Equation REF (substituting $(l_w,b_w)$ for $(l_o,b_o)$ ), $\\theta $ is the angle between $(l,b)$ and $(l_o,b_o)$ and $v_o$ is the expansion velocity with respect to the Local Standard of Rest (LSR) or V$_{\\rm LSR}$ .", "Conversion from LSR to Heliocentric velocities was performed using the RV Starlink package.", "We used equations REF and REF to calculate the projected velocity components for the different interstellar clouds and shells in the direction of our targets with detected absorption, which are presented in Table REF .", "Table: Heliocentric Velocity Projections of known interstellar clouds/shells in the direction of the stars presenting narrow absorption features." ], [ "HD 61950", "HD 61950 is a B8 dwarf that shows a distinctive two-component absorption feature seen in both the Ca H and K lines.", "The absorption components are centred at +8.3 and +18.1 km s$^{-1}$ , lying near the base of the broad photospheric line.", "The red-most component, at +18.1$\\pm 0.7$ km s$^{-1}$ coincides within the errors with the star's heliocentric velocity (20$\\pm 5$ km s$^{-1}$ ), although there is quite a large uncertainty on the latter due to the lack of prominent metallic lines on the spectrum of this B-type star.", "The Ca ii equivalent width ratio suggests that for both narrow components the line is unsaturated, as the derived K:H EW ratio is $\\sim 2$ in both cases (a Ca ii K:H ratio of 2 is expected in the case of unsaturated lines, as the oscillator strength of the H-line is half that of the K-line).", "Given that HD 61950 is the most distant star in our sample (d$=362$  pc), it is more probable that these features are of interstellar origin.", "The velocity projections of known clouds and shells predicts that the UCL shell should contribute an absorption feature at +9.4 km s$^{-1}$ (Table REF ), that coincides extremely well with the main absorption component seen at +8.3 km s$^{-1}$ .", "Comparison with Ca ii K observations toward stars in similar directions suggests that both components in the spectrum of HD 61950 are interstellar.", "The LSR velocities of the two narrow Ca ii absorption components seen toward HD 61950 (l=281$^{\\rm o}$ , b=-21$^{\\rm o}$ , d=362 pc) correspond to -3.7 km s$^{-1}$ and +5.9 km s$^{-1}$ .", "[28] detected two Ca ii K absorption components toward HD 76131 (l=273$^{\\rm o}$ , b=-7$^{\\rm o}$ , d=453 pc) at LSR velocities of -2 km s$^{-1}$ and +8 km s$^{-1}$ , with EWs of 110 m$\\rm Å$ and 21 m$\\rm Å$ respectively.", "These two velocity components are also detected in the spectra of HD 67536 (l=276$^{\\rm o}$ , b=-16$^{\\rm o}$ , d=450 pc), at LSR velocities of -4 km s$^{-1}$ and +5.3 km s$^{-1}$ with EW of 16 m$\\rm Å$ and 11 m$\\rm Å$ respectively.", "The LSR velocities of the two components detected toward HD 61950 coincide with the range of velocities of the two interstellar clouds reported by [28] are detected.", "It is therefore very likely that both absorption features seen toward HD 61950 are interstellar in nature." ], [ "HD 75416", "HD 75416 ($\\eta $  Cha) lies at a distance of 97 pc and is also a B8V star.", "It is the brightest member of the 8 Myr old $\\eta $  Cha cluster [41], and is the only debris disk in this young, disk rich, stellar association [54].", "Gaussian modeling of the photospheric lines indicate that the star is a fast rotator ($v\\,\\sin {i}\\sim $ 290 km s$^{-1}$ ), as was already noted by [54], and that the stellar heliocentric velocity is +15 $\\pm 5$  km s$^{-1}$ (in agreement with the V$_{\\odot }=14\\pm 10$  km s$^{-1}$ value derived by [13]).", "A very weak absorption feature is detected at the centre of the photospheric Ca ii K profile.", "Three separate components at heliocentric velocities of -10.2, +6.9 and +18.5 km s$^{-1}$ , were required in order to satisfactorily fit the observed profile (Figure REF ).", "There is no evidence of corresponding features in the intrinsically weaker Ca ii H line.", "The cloud models imply very low calcium column densities, with the strongest components being the ones at -10.2 and +6.9 km s$^{-1}$ .", "The velocity component observed at +6.9 km s$^{-1}$ could be related to the UCL shell (velocity projection of +4.3 km s$^{-1}$ ), however no known clouds/shells can account for the 18.5 km s$^{-1}$ component.", "This feature is interesting because, despite being very weak (only 1.3 m$\\rm Å$ equivalent width), it lies close to the stellar velocity.", "However, the lack of additional information regarding this absorption feature makes it difficult to draw any conclusions regarding the nature of this feature.", "Additional observations of the Na i D region could help in resolving the origin of these features" ], [ "HD 110058", "HD 110058 is an A0V star located at a distance of $107_{-8}^{+10}$  pc, in the direction of the LCC shell [61].", "HD 110058 was first identified by [42] as a debris disc host star.", "The disc was undetected at 1350 $\\mu $ m with SCUBA by [60], which allowed them to estimate an upper limit to the dust mass of less than $5\\times 10^{-6}\\,M_{\\odot }$ .", "[60] estimate the fractional disc luminosity of $1.89\\times 10^{-3}$ , in good agreement with our estimate of $2.0\\times 10^{-3}$ .", "This is very similar to the dust fractional luminosity of $\\beta $  Pic 0-3$,\\endcsname {moor2011}, and comparable to those ofother A-type star debris disc around which circumstellar gas has beendetected\\cite {Hughes08,moor2011,Dent14,moor2015,Lieman2016,marino2016}.$ Near-infrared Very Large Telescope/SPHERE imaging has revealed that, similarly to $\\beta $  Pic, the debris disc around HD 110058 is seen very close to edge-on [29].", "Recent ALMA observations confirm the near to edge-on inclination of the disc, and also show a 5-$\\sigma $ detection of carbon monoxide towards the system [40].", "There is some discrepancy in the literature regarding HD 110058's radial velocity.", "[45] measured the radial velocity of HD 110058 to be +21.7$\\pm 1.3$  km s$^{-1}$ , while [13] estimate a radial velocity of 5$\\pm 1$  km s$^{-1}$ .", "Our measurement lies somewhere in between the estimates of [45] and [13].", "A strong, sharp, absorption component is detected in both the Calcium and Sodium regions at the very base of the photospheric line.", "Modeling of the absorption feature indicates that there are actually two velocity components, centred near -1 km s$^{-1}$ and +12 km s$^{-1}$ respectively.", "None of the known interstellar clouds/shells appear to have projections at these velocities, and so cannot account for the observed features.", "The component near +12 km s$^{-1}$ , lies right at the stellar radial velocity of +12$\\pm 3$  km s$^{-1}$ we derived, which would argue for a circumstellar nature for this feature.", "It is interesting to note that in the case of the +12 km s$^{-1}$ component, the equivalent widths of the D$_1$ and D$_2$ lines are equal within the errors, suggesting saturation.", "The same effect is seen with regards to the H and K calcium lines.", "This could be explained by the presence of clumpy intervening material, similar to what has been modeled for $\\beta $ Pictoris [37].", "For both velocity components, the Ca ii/Na i ratios that we derive are close to unity, and are the largest found in our sample.", "[51] notes however that is is hard to differentiate a circumstellar from interstellar origin based on the abundance ratios alone, since a wide range of Ca ii/Na i ratios are detected even locally.", "In their search for atomic absorption in nearby debris discs, [51] find that their best candidates for circumstellar absorption have Ca ii/Na i ratios between 3.9 and 46.", "The coincidence between the radial velocities of the narrow absorption feature and the star, the lack of known interstellar clouds/shells that can account for absorptions at these velocities, together with the Ca ii/Na i ratio of $\\sim 1$ strongly argue for a circumstellar nature for the component seen at +12 km s$^{-1}$ .", "The fact that the component seen at $\\sim $ 12 km s$^{-1}$ is saturated while the one at $\\sim $ -1 km s$^{-1}$ is not, indicates that they have different column densities, suggesting that they may be different in nature.", "The MIKE observations of HD 110058 and of reference stars in the vecinity of HD 110058 confirm that the feature at a lower velocity is interstellar.", "The feature is observed in all stars (Figure ), and its depth increases with distance as expected if the absorption is caused by interstellar material.", "On the other hand, the absorption feature at $\\sim $ 12 km s$^{-1}$ is detected only towards HD 110058.", "Therefore we conclude that this absorption feature is indeed circumstellar." ], [ "HD 166841", "HD 166841 is B9V star located 214 pc away from the Sun in the direction of the centre of the UCL shell.", "The star was first identified as a debris disc host by [42], but no further studies of this object have been carried out.", "The value of the projected rotational velocity that we derive (245 $\\pm 8$  km s$^{-1}$ ) is consistent with the star being of late B-type.", "Despite being a quite distant star, the E(B-V) value we estimate assuming the normal colours of a B9V star is quite low (E(B-V)=0.02).", "A strong narrow absorption feature is easily seen right at the centre of the photospheric Ca ii and Na i lines.", "Both the calcium and sodium absorption features are found to be located at V$_{\\odot }$ =0$\\pm 1$  km s$^{-1}$ , coinciding extremely well with the radial velocity of the star.", "The derived equivalent widths of the different component suggest that the lines are not saturated and cloud modeling indicates that Ca ii/Na i column density ratio is $\\sim 0.3$ .", "Although the velocity projections of known clouds/shells cannot be responsible for the observed absorption feature, we investigated the presence of similar absorption components in the spectra of stars in similar directions.", "The LSR velocity of the component seen toward HD 166841 (l=326$^{\\rm o}$ , b=-22$^{\\rm o}$ , d=214 pc) is -1.8 km s$^{-1}$ .", "[28] reports +2.4$\\pm 0.1$  km s$^{-1}$ Ca ii K component on the direction of HD 142758 (l=325$^{\\rm o}$ , b=-4$^{\\rm o}$ , d=4000 pc) with 65 m$\\rm Å$ equivalent width.", "[28] also detected two Ca ii K absorption features in the spectrum of HD 143448 (l=324$^{\\rm o}$ , b=6$^{\\rm o}$ , d=520 pc), at -3.6$\\pm 0.1$  km s$^{-1}$ (EW=18 m$\\rm Å$ ) and +2.0$\\pm 0.6$  km s$^{-1}$ (EW=59 m$\\rm Å$ ), both referred to the LSR.", "These two stars are located at a higher Galactic latitude than HD 166841 and are more distant, but it is worth noting that the first component seen toward HD 143448 is similar within the errors to the one seen toward HD 166841, both in equivalent width and radial velocity.", "Despite being located right at the centre of the photospheric lines, there is not enough supporting evidence to attribute this feature to a circumstellar origin.", "The star is quite distant and there is some evidence of a correlated velocity structure between the feature seen toward HD 166841 and those towards stars in nearby lines of sight.", "One argument that could favor a circumstellar nature for the feature seen toward HD 166841 is the fact that when interstellar absorption components are detected toward field stars, generally more than one component is detected, whereas in the case of HD 166841 only one distinctive narrow absorption is observed.", "Observations of stars with lines of sight closer to HD 166841 than the stars quoted above could help to clarify the nature of HD 166841's narrow absorption feature." ], [ "HR 5558", "HR 5558 is an A0V spectral standard located 76 pc away from the Sun.", "We serendipitously detected a narrow absorption feature noticeable in the Na i D lines (EW of 20 m$\\rm Å$ and 33 m$\\rm Å$ in the D$_1$ and D$_2$ lines respectively), and less evident in the Ca ii K line.", "The feature was not detected in the Ca ii H line.", "There is no correspondence of the radial velocity of the absorption feature with the radial velocity of the star, suggesting that the absorption is of interstellar origin.", "This is supported by the low Ca ii/Na i ratio derived.", "Given the diagnostics above, plus the fact that HR 5558 is not known to be a debris disc host star, we conclude that the absorption feature seen toward this star is interstellar in origin." ], [ "Conclusions", "We have conducted a search for optical circumstellar absorption lines in the spectra of 16 debris disc host stars.", "We found no evidence of emission line activity, confirming their more evolved and quiescent evolutionary state.", "Four stars show narrow absorption features close to the centre the photospheric Ca ii and Na i D lines.", "In addition similar absorption features were detected in the spectrum of one our spectral standards.", "Of the four stars showing narrow absorption features, two are younger than 17 Myr (the other two stars do not have available ages in the literature).", "This is consistent with the findings of [64] in which they note that the stars that exhibit circumstellar gas activity are significantly younger than quiescent systems.", "We also find that the systems towards which absorption is detected are all fast rotators.", "This has been discussed as a possible proxy for detecting gas absorption in debris disks, since it can be indicative of a system seen close to edge-on [64].", "With the exception of HD 176638 none of the stars with $v\\,\\sin {i}$ <140 km s$^{-1}$ show signatures of gas absorption.", "We also note that gas absorption is only detected toward A- or late B-type stars, in agreement with previous detections of exocomet bearing systems [52], [64], [19].", "The features detected toward HD 61950 and HR 5558 can be almost unambiguously attributed to interstellar absorption, while in the case of HD 75416 and HD 166841 the association is less clear.", "The lack of known interstellar absorption features within the velocity ranges of the lines seen in both HD 75416 and HD 166841 (which coincide with the stellar velocities) leaves open the question of the nature of these features.", "HD 75416 ($\\eta $  Cha) is known to host an evolved protoplanetary disk, possibly transitioning into the debris disk phase [54].", "Re-observation of these two sources in order to search for variability, together with observations of stars in adjacent lines of sight, could help to disentangle the origin of these features.", "The non-detection of atomic gas absorption towards the CO-rich HD 181237 debris disc system is consistent with this system being seen close to pole-on [43].", "The most compelling evidence for a circumstellar gas is seen in one of the two velocity components of the absorption seen in the spectrum of HD 110058 (at +12 km s$^{-1}$ ).", "The good agreement with the stellar velocity, the Ca ii/Na i column density ratio close to unity, the lack of known interstellar clouds or shells at the correct velocities in HD 110058's direction, and the fact that the disc is observed close to edge-on strongly suggest that the absorption is caused by atomic gas present in HD 110058's disc.", "The additional MIKE data provides compelling evidence that the component at +12 km s$^{-1}$ arises in the circumstellar environment of HD 110058.", "The detection of optical absorption gas features towards HD 110058, adds to the many similiarities with the $\\beta $ Pic system (age, dust fractional luminosity, orientation, and presence of both atomic and molecurlar gas).", "In the latter, the gas is believed to be of secondary origin, i.e.", "produced by the release of atomic and molecular species by volatile-rich bodies [17], [34].", "Future studies of this new $\\beta $ Pic-like system could provide further information on the origins of gas in debris discs." ], [ "Acknowledgements", "A.S.H.", "carried out part of this work while being funded by the PPARC GeminiFundacion Andes UK/Chile studentship program.", "A.S.H.", "thanks Rafael Brahm for useful discussions on calibration of MIKE data.", "The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. SC acknowledges support from Millennium Science Initiative, Chilean Ministry of Economy: Nucleus P10-022-F." ] ]
1612.05465
[ [ "Interlayer excitons and Band Alignment in MoS$_2$/hBN/WSe$_2$ van der\n Waals Heterostructures" ], [ "Abstract Van der Waals heterostructures (vdWH) provide an ideal playground for exploring light-matter interactions at the atomic scale.", "In particular, structures with a type-II band alignment can yield detailed insight into free carrier-to-photon conversion processes, which are central to e.g.", "solar cells and light emitting diodes.", "An important first step in describing such processes is to obtain the energies of the interlayer exciton states existing at the interface.", "Here we present a general first-principles method to compute the electronic quasi-particle (QP) band structure and excitonic binding energies of incommensurate vdWHs.", "The method combines our quantum electrostatic heterostructure (QEH) model for obtaining the dielectric function with the many-body GW approximation and a generalized 2D Mott-Wannier exciton model.", "We calculate the level alignment together with intra and interlayer exciton binding energies of bilayer MoS$_2$/WSe$_2$ with and without intercalated hBN layers, finding excellent agreement with experimental photoluminescence spectra.", "Comparison to density functional theory calculations demonstrate the crucial role of self-energy and electron-hole interaction effects." ], [ "Methods", "All the ab-initio calculation in this work are performed with GPAW[59], [60].", "The band structures of the twisted bilayers were calculated at the DFT level with an LDA exchange correlation functional and double-zeta polarized atomic orbitals as a basis set.", "The HSE06 and LDA calculations for the monolayers were performed using a plane wave basis set with a cut off energy of 500 eV and $18\\times 18$ k-point grids.", "We find it useful to elaborate on the color scheme choice for the twisted bilayer band structure in fig:bandsrotation.", "Such choice should be understood by considering that the two layers have different primitive cells and that the unfolding procedure has to be performed separately.", "Indeed, the unfolding for a given layer, say MoS$_2$ , not only will project the eigenstates belonging onto MoS$_2$ to their primitive cell but also the WSe$_2$ ones.", "As the latter should rather be projected onto the WSe$_2$ primitive cell, we “hide” them by choosing a color scheme that goes from blue to white and decreases the size of the markers when going from states localized completely in MoS$_2$ to states localized completely in WSe$_2$ .", "The same argument applies to the unfolding to the WSe$_2$ cell, but a red color scheme is applied instead.", "For lattice matched heterostructures modeled in a minimal unit cell, it was checked that the atomic orbital basis yields the same band structure as well converged plane wave calculations.", "For the calculation of dielectric properties of van der Waals heterostructures we utilized dielectric building blocks available in Ref. database2015.", "Specifically the response function of each building block was calculated on a plane-waves basis with 100 eV cut-off energy and $100\\times 100$ k-point mesh.", "In order to avoid spurious interaction from artificial replica in the out-of-plane direction, a truncated Coulomb interaction with 20 Å  of vacuum is used.", "The interlayer distance between the layers are taken as average of the interlayer distance in their respective bulk form, specifically $d_{\\textrm {MoS}_2/\\textrm {WSe}_2}=6.51$ Å, $d_{\\textrm {MoS}_2/\\textrm {hBN}}=5.08$ Å , $d_{\\textrm {hBN}/\\textrm {hBN}}=3.2$ Å  and $d_{\\textrm {WSe}_2/\\textrm {hBN}}=5.28$ Å.", "The monolayer G$_0$ W$_0$ calculations have been performed employing a new efficient technique[61] that overcomes the problem of slow convergence of the band structures with k-point grid and yields well converged band gaps with $18\\times 18$ k-points (rather than $40\\times 40$ using standard approaches).", "We used an energy cut-off of 150 eV for the dielectric function and sum over empty states.", "The G$_0$ W$_0$ band energies were extrapolated as $1/N_G$ to the infinite plane wave limit.", "The Mott-Wannier equation in eq:MW was solved on a radial logarithmic grid ensuring numerical convergence of exciton energies up to $0.002$ eV." ], [ "Acknowledgement", "The authors acknowledge support the Center for Nanostructured Graphene (CNG), which is sponsored by the Danish National Research Foundation, Project DNRF58.", "The authors declare no competing financial interests." ], [ "Effect of Hybridization and Charge-Transfer", "In the main text we argued that the use of supercells is essential for a good description of the band structure of mismatched bilayers and the main differences between the bands of the isolated layers and the bilayers are consequence of charge transfer and hybrization separately.", "In this section of the supporting information we prove these arguments for the case of MoS$_2/$ WSe$_2$ based structures.", "We start out by demonstrating that using a supercell is unavoidable if an accurate band structure of the MoS$_2/$ WSe$_2$ is needed.", "This is shown in fig.", "REF , where we plot the band structure of the strained bilayer and isolated monolayers in panel (a) and the corresponding unstrained structures in panel (b).", "Figure: Panel (a) electronic bands for strained bilayer MoS 2 /_2/WSe 2 _2 (circles) and isolated monolayers (continuous line).", "Panel (b) the same as in (a) but for the unstrained structures.", "For the unstrained bilayer an alignment angle of ∼16.1 ∘ \\sim 16.1^{\\circ } is used.", "In the strained structures the lattice parameter is the average of the lattice parameter of isolated MoS 2 _2 and WSe 2 _2.", "The figure shows that the effect of charge transfer can be inferred from a simpler strained calculation.In both panels, the bands belonging to the isolated layers are drawn with continuous lines while the ones for the bilayers are drawn with circles.", "The unstrained bilayer is constructed using a supercell and an alignment angle of $\\sim 16.1$ as described in the main text, whereas for the strained bilayer we use a unit cell with the lattice parameter equal to the average of the lattice parameter of the isolated monolayers.", "From the figure it is evident that straining the layers has a considerable effect both on the curvature of the bands and on their positioning with respect to vacuum.", "We thus conclude that accurate band structures cannot be obtained without employing supercells.", "However, it is still possible to extract information about charge transfer and hybridization from the strained calculations.", "Indeed, we can see from fig.", "REF that the relative difference between isolated layers and bilayer, in both panels, are practically the same.", "Figure: Left panel: bands for strained MoS 2 /_2/WSe 2 _2 bilayer (circles) and constituent monolayers (continuous lines).", "Here the charge transfer effect manifests as constant shift in opposite direction for MoS 2 _2 (in blue) and WSe 2 _2 (in red).", "Right panel bands for the same systems but for the bilayer the Hartree potential is not updated self-consistently.", "Keeping the Hartree potential fixed to the one of the isolated layers prevent charge transfer and therefore bilayer and monolayers bands coincide as long as hybrization is negligible.", "Hybridization is present around Γ\\Gamma .Based on this consideration we proceed the analysis of charge transfer and hybridization using the strained calculations, which are computationally more feasible.", "As explained in the main text, we observe a constant shift in energy, upwards for MoS$_2$ and downwards for WSe$_2$ , and a wavevector dependent variation around $\\Gamma $ when comparing the isolated layers to the bilayer.", "While the effect of hybridization is a direct consequence of the mixing among wavefunctions of the two layers, charge transfer results from the rearrangement of the electrons at the bilayer interface due to the difference in band gap centers of the two materials.", "From a DFT calculation point of view, it is the self-consistent procedure, in particular the change in the Hartree potential in each loop, that allows the rearrangement of the electrons once the two materials are put together.", "This means that performing a non-self-consistent DFT calculation starting from the self-consistent ground state density of the isolated layers, would not allow for the update of the Hartree potential and consequent electrons rearrangement.", "The fully self-consistent band structures (reported for comparison from fig.", "REF (a)) and the non self-consistent ones are shown in fig.", "REF , left and right panel respectively.", "In panel (b), the isolated layers bands are now exactly on top of the bilayer ones throughout most of the Brillouin zone and therefore it should be now clear that the rigid shift of the bands was a signature of charge transfer.", "Furthermore the alteration of the bands around the $\\Gamma $ point has not disappeared.", "This is exactly what we expected considering that hybrization is a result of the overlap of the wavefunctions which is accounted for in the non-self-consistent calculation." ], [ "Validity of the QEH correction on G$_0$ W{{formula:98149d39-555d-4dd0-aa31-bf4a0f143931}} band structure", "To check the validity of our G$_0\\Delta $ W$_0$ approach we perform G$_0$ W$_0$ calculations for strained MoS$_2$ and WSe$_2$ isolated layers and MoS$_2/$ WSe$_2$ bilayers.", "The choice of strained structure is obviously imposed by the feasibility of a G$_0$ W$_0$ calculation for the bilayers.", "For the following calculation plane-wave mode has been used.", "In the left panel of fig.", "REF we report the bands for the strained MoS$_2/$ WSe$_2$ bilayer from a G$_0$ W$_0$ calculation (continuous black line) and the G$_0$ W$_0$ approach.", "Figure: Comparison of the G 0 Δ_0\\Delta W 0 _0 method to G 0 _0W 0 _0 calculation for the lattice-matched bilayer.", "Left panel: reference calculation.", "Central panel: G 0 Δ_0\\Delta W 0 _0 bands are shifted upwards for MoS 2 _2 and downwards for WSe 2 _2 to account for the charge transfer.", "The values of the shifts are extracted from the comparison between the isolated layers and bilayer LDA calculations.", "Right panel: the layer separation is artificially increased by 3Å.", "With the extra distance we expect the effect of charge transfer and hybridization to be completely negligible.The isolated layer bands are reported in all the panels as reference.", "As usual blue is used for MoS 2 _2 and red for WSe 2 _2.The G$_0$ W$_0$ bands for the isolated layers are also shown as reference.", "As expected from the extra screening that each layer provides to the other, the intra and inter layer gaps are reduced compared to the isolated layer ones and such an effect is grasped both from the full G$_0$ W$_0$ calculation and the G$_0\\Delta $ W$_0$ method.", "However, the agreement between G$_0$ W$_0$ and G$_0\\Delta $ W$_0$ is not striking.", "This is because the effect of charge transfer is still present at the G$_0$ W$_0$ level, since the Hartree potential generated by the charge rearrangement at the interface is the same as the DFT one.", "Only self-consistency, indeed, could relieve this problem.", "To prove that charge transfer is still there and that it is an effect inherited from the starting LDA calculation, we evaluate the layer dependent energy shift at the $K$ -point by comparing isolated layers and bilayers bands at the LDA level and then add these shifts to the G$_0\\Delta $ W$_0$ bands.", "The results are shown in the central panel of fig.", "REF .", "The agreement is nearly perfect and it supports our argument on the importance of charge transfer.", "As a side note, we mention that the effect of charge transfer using plane-wave mode, as opposed to LCAO, is a bit lower, namely we get an increase in interlayer gap of $0.11$ eV compared to the $0.21$ eV reported in the main text.", "As a further proof of the validity of the G$_0\\Delta $ W$_0$ method we repeat the G$_0$ W$_0$ for the bilayer adding 3 Å  to interlayer distance between MoS$_2$ and WSe$_2$ .", "This guarantees that charge transfer and hybridization effects are negligible.", "Screening effects, on the other hand, are still appreciable being the Coulomb coupling between the layers long range.", "The bands for such a system are shown in the right panel of fig.", "REF and it is clear that the G$_0\\Delta $ W$_0$ does a good job." ] ]
1612.05736
[ [ "H12CN and H13CN excitation analysis in the circumstellar outflow of R\n Scl" ], [ "Abstract Abridged.", "The 12CO/13CO ratio in the circumstellar envelope (CSE) of asymptotic giant branch (AGB) stars has been extensively used as the tracer of the photospheric 12C/13C ratio.", "However, spatially-resolved ALMA observations of R Scl, a carbon rich AGB star, have shown that the 12CO/13CO ratio is not consistent over the entire CSE.", "Hence, it can not necessarily be used as a tracer of the 12C/13C ratio.", "The most likely hypothesis to explain the observed discrepancy between the 12CO/13CO and 12C/13C ratios is CO isotopologue selective photodissociation by UV radiation.", "Unlike the CO isotopologue ratio, the HCN isotopologue ratio is not affected by UV radiation.", "Therefore, HCN isotopologue ratios can be used as the tracer of the atomic C ratio in UV irradiated regions.", "We have performed a detailed non-LTE excitation analysis of circumstellar H12CN and H13CN line emission around R Scl, observed with ALMA and APEX, using a radiative transfer code, ALI.", "The spatial extent of the molecular distribution for both isotopologues is constrained based on the spatially resolved H13CN(4-3) ALMA observations.", "We find fractional abundances of H12CN/H2 = (5.0 +\\- 2.0) x 10^{-5} and H13CN/H2 = (1.9 +\\- 0.4) x 10^{-6} in the inner wind (r < (2.0 +\\- 0.25) x 10^{15} cm) of R Scl.", "The derived circumstellar isotopologue ratio of H12CN/H13CN = 26.3 +\\- 11.9 is consistent with the photospheric ratio of 12C/13C ~ 19 \\pm 6.", "We show that the circumstellar H12CN/H13CN ratio traces the photospheric 12C/13C ratio.", "These results support the previously proposed explanation that CO isotopologue selective-shielding is the main factor responsible for the observed discrepancy between 12C/13C and 12CO/13CO ratios in the inner CSE of R Scl.", "This indicates that UV radiation impacts on the CO isotopologue ratio." ], [ "Introduction", "During late stellar evolutionary phases, stars produce almost all heavy elements in the universe through nucleosynthesis.", "In the last phase of evolution of low- and intermediate-mass stars (0.8-8 $M_{\\odot }$ ), heavy elements that are gradually built up in the inner layers are dredged up to the surface and are injected into the interstellar medium through intense stellar winds.", "These stars lose up to 80 percent of their initial mass typically at rates of $10^{-8} - 10^{-4}$ $M_{\\odot }$ yr$^{-1}$ during the asymptotic giant branch (AGB) phase.", "As a result, a circumstellar envelope (CSE) of gas and dust forms around the central star, [19], [20].", "AGB stars can be classified by the elemental C/O ratio: C/O > 1 the carbon rich C-type stars, C/O$\\sim $ 1 the S-type stars and C/O < 1 the oxygen rich M-type stars.", "Molecular emission lines from CSEs are excellent probes of the physical and chemical properties of the CSE and the central star.", "Observations of CO rotational transitions provide the most reliable measurements of the physical parameters of the CSE such as the mass-loss rate, density structure, expansion-velocity profile, kinetic temperature, and spatial extent [34], [47], [42], [5], [9].", "Observations of other abundant molecules set strong constraints on the chemical networks active in the CSE [39], [4], [31], [30], [10], [41].", "The study of isotopic ratios of evolved stars provides important information on the stellar evolutionary phases and chemical enrichment of the interstellar medium.", "The photospheric $^{12}$ C/$^{13}$ C ratio is a good tracer of the stellar nucleosynthesis.", "From an observational point of view, a direct estimate of the $^{12}$ C/$^{13}$ C ratio is very challenging.", "Hence, observing isotopologues of circumstellar carbon-bearing molecules are widely used to trace the elemental isotopic carbon ratio.", "Carbon monoxide, as the most abundant C-bearing molecule, has been extensively used to extract the carbon isotope ratio [18], [17], [47], [33], Assuming a solitary AGB star, known processes that may affect the chemical composition of the CSE are shocks due to stellar pulsations [7] and chromospheric activity in the inner wind ($\\sim r < 2.5 \\times 10^{14}$ cm), gas-dust interaction in the intermediate wind ($\\sim 2.5\\times 10^{14}<r<5 \\times 10^{15}$ cm), and photodissociation by the interstellar UV-radiation, associated photo-induced chemistry, and chemical fractionation processes, in the outer wind ($\\sim r>5\\times 10^{15}$ cm), [11].", "In the case of a clumpy envelope structure, the interstellar UV-radiation can penetrate the entire envelope and affect the chemistry in the inner wind [1].", "The chemical processes in the inner and intermediate wind are not expected to affect the isotopologues abundance ratios.", "However, differences in self-shielding against UV-radiation do cause different photodissociation rates of molecules with sharp discrete absorption bands such as isotopologues of H$_2$ , CO, C$_2$ H$_2$ , NO, [25].", "Thus, isotopologue selective photodissociation by UV radiation can change the isotopologue abundance ratios in the UV irradiated regions [45], [51].", "Moreover, ion- molecule charge-exchange reactions in cold regions may also affect the isotopologue abundance ratios [53].", "Photodissociation of the molecular gas in the CSEs is thought to be dominated by the interstellar radiation field (ISRF) from the outside.", "ISRF is the only UV radiation field which has been considered in the modelling of AGB CSEs.", "However, previous studies of UV spectra indicate the presence of a chromosphere in the outer atmosphere of carbon stars [21], [13].", "In binary systems, active binary companions can emit UV-radiation from the inside as well [44], [40].", "A recent search for UV emission from AGB stars has revealed that about 180 AGB stars, $\\sim $ 50 $\\%$ of the AGB stars observed with Galaxy Evolution Explorer (GALEX), have detectable near- or far-UV emission (Montez et al., in prep), supporting the possible existence of internal sources of UV-radiation.", "Nowadays, high spatial resolution Atacama Large Millimeter/submillimeter Array (ALMA) data enable us to study the impact of UV radiation sources on the isotopologue abundance ratios in the CSEs of evolved stars more accurately.", "In this paper, we derive the H$^{12}$ CN/H$^{13}$ CN ratio and compare it with previously reported $^{12}$ CO/$^{13}$ CO and $^{12}$ C/$^{13}$ C ratios to probe the effect of UV radiation on the CSE of R Scl.", "We present the physical characteristics of R Scl in Section 2.", "New spectral line observations of R Scl are presented in Section 3.", "The excitation analysis of H$^{12}$ CN and H$^{13}$ CN is explained in Section 4.", "In Section 5, we present the results.", "Finally, we discuss our results and draw conclusion in Sections 6 and 7, respectively." ], [ "R Sculptoris", "R Scl is a carbon-type AGB star at a distance of approximately 370 pc derived using K-band period-luminosity relationships [23], [54].", "It is a semi-regular variable with a pulsation period of 370 days.", "The stellar velocity $v_{LSR}^{\\ast }=$ ${-19}$ km s$^{-1}$ is determined from molecular line observations.", "A detached shell of gas and dust with a width of $2^{\\prime \\prime }$ $\\pm $ $1^{\\prime \\prime }$ was created during a recent thermal pulse at a distance of $19.5^{\\prime \\prime }$ $\\pm $ $0.5^{\\prime \\prime }$ from the central star [28], [29].", "This shell has been extensively observed in CO and dust scattered stellar light in the optical [15], [16], [38], [28], [29].", "A spiral structure in the CSE of R Scl induced by a binary companion was revealed by $^{12}$ CO($J$ =3-2) ALMA observations [28].", "Moreover, a recent study of the physical properties of the detached shell by [32] show that the previously assumed detached shell around R Scl is filled with gas and dust.", "High spatial resolution ALMA observations of $^{12}$ CO and $^{13}$ CO have allowed us to separate the detached-shell emission from the extended emission of the CSE [28], [52].", "These observations reveal a discrepancy between the circumstellar $^{12}$ CO/$^{13}$ CO and the photospheric $^{12}$ C/$^{13}$ C ratios, [52], hereafter V13.", "They measure an intensity ratio of $^{12}$ CO/$^{13}$ CO$>$ 60 in the inner wind.", "Using detailed radiative transfer modelling, they show that this implies a carbon isotope ratio that is not consistent with the photospheric $^{12}$ C/$^{13}$ C$\\sim $ 19 $\\pm $ 6 reported by [24].", "At the same time, they measure the intensity ratio, which varies from 1.5 to 40 in the detached shell with the average intensity ratio of $^{12}$ CO/$^{13}$ CO$\\sim $ 19 which, again taking into account radiative transfer, is still consistent with the photospheric $^{12}$ C/$^{13}$ C ratio.", "Therefore, the circumstellar $^{12}$ CO/$^{13}$ CO ratio from the inner parts of the CSE, provided by the high-resolution interferometric observations, does not necessarily measure the $^{12}$ C/$^{13}$ C ratio.", "It has been suggested in V13 that the lack of $^{13}$ CO in the recent mass loss might be due to the extra photodissociation of $^{13}$ CO by internal UV radiation from the binary companion or chromospheric activity, while the more abundant $^{12}$ CO would be self-shielded.", "To confirm isotopologue selective photodissociation of CO as the reason for the observed discrepancy between the aforementioned CO isotopologue and C isotope ratios in R Scl, we compare the H$^{12}$ CN/H$^{13}$ CN and $^{12}$ C/$^{13}$ C ratios.", "Both CO and HCN have line and continuum absorption bands in UV, respectively.", "Thus, CO isotopologues are photodissociated in well-defined bands, while HCN isotopologues are photodissociated via continuum.", "This implies that both HCN isotopologues would be equally affected by the UV-radiation, whereas CO isotopologues have different rates of photodissociation because of isotopologue self-shielding.", "Hence, this comparison would confirm the selective photodissociation of CO as the main reason for changing the CO isotopologue abundance ratio through the CSE of R Scl." ], [ "Single-dish data", "Single-dish observations of H$^{12}$ CN($J$ =2-1) and H$^{13}$ CN($J$ =2-1, 3-2) emission lines towards R Scl were performed using the APEX 12 m telescope, located on Llano Chajnantor in northern Chile, in July 2015.", "We used the SEPIA/band 5 and the SHeFI-APEX1 receivers.", "The observations were made in a beam switching mode.", "The antenna main-beam efficiency, $\\eta _{\\rm mb}$ , the full-width half-power beam width, $\\theta _{\\rm mb}$ , and the excitation energy of the upper transition level, $E_{up}$ , at the observational frequencies for H$^{12}$ CN and H$^{13}$ CN are presented in Table REF .", "The data reduction was done using XSXS is a package developed by P. Bergman to reduce and analyse single-dish spectra.", "It is publicly available from ftp://yggdrasil.oso.chalmers.se.", "A first-order polynomial was subtracted from the spectrum to remove the baseline.", "The measured antenna temperature was converted to the main-beam temperature using $T_{\\rm mb} = T^{\\ast }_{\\rm A}/\\eta _{\\rm mb}$ .", "In addition to the new data presented here, we have also used previously published single-dish observations of H$^{12}$ CN($J$ =1-0, 3-2) made with the Swedish-ESO Submillimetre Telescope (SEST) [35] and H$^{12}$ CN($J$ =4-3) observed with the Heinrich Hertz Submillimeter Telescope (HHT) [2], which are summarised in Table  REF ." ], [ "Interferometer data", "The ALMA observations of H$^{13}$ CN($J$ =4-3) were made on 14 Dec 2013, 25 Dec, 26 Apr 2014, and 24 Jul 2015 using ALMA band 7 (275-373 GHz).", "Figure REF shows the H$^{13}$ CN($J$ =4-3) integrated flux density over the velocity channels, a zero-moment map.", "The primary flux calibration was done using Uranus and bootstrapped to the gain calibrator J0143-3206 (0.27 Jy beam$^{-1}$ ) and J0106-4034 (0.23 Jy beam$^{-1}$ ).", "Based on the calibrator fluxes, the absolute flux has an uncertainty of around 10 $\\%$ .", "The data reduction was done with the Common Astronomy Software Application (CASA).", "More details of the data reduction will come in Maercker et al.", "in prep.", "The tasks \"imsmooth\" and \"immoments\" in CASA were used to smooth the image to $0.13^{\\prime \\prime } \\times 0.13^{\\prime \\prime }$ resolution and to integrate over velocity channels, respectively.", "Figure: Zero-moment H 13 ^{13}CN(4-3) map of R Scl observed with ALMA.", "The ALMA beam size is shown in the bottom left corner.Table: Observations of HCN towards the circumstellar envelope of R scl." ], [ "Spectroscopic treatment of HCN", "The molecule HCN is linear and polyatomic, with three vibrational modes: the H-C stretching mode $\\nu _1$ at 3 $\\mu $ m, the bending mode $\\nu _2$ at 14 $\\mu $ m, and the C-N stretching mode $\\nu _3$ at 5 $\\mu $ m. In our modelling, we take into account the $\\nu _1$ =1 and the $\\nu _2$ =1 states, while we neglect the $\\nu _3$ mode since this includes transitions that are about 300 times weaker than $\\nu _1$ =1 [3].", "The nuclear spin (or electric quadrupole moment) of the nitrogen nucleus leads to a splitting of the rotational levels into three hyperfine components.", "Furthermore, a splitting of the bending mode occurs due to rotation when the molecule is bending and rotating simultaneously.", "This is referred to as $l$ -type doubling.", "This interaction between the rotational and bending angular momenta results in $l$ -doubling of the bending mode into two levels $01^{1c}0$ and $01^{1d}0$ .", "In our modelling, the excitation analysis includes 126 energy levels.", "Hyperfine splitting of the rotational levels for $J$ =1 levels are included.", "In each of the vibrational levels, rotational levels up to $J$ =29 are considered, and the $l$ -type doubling for $\\nu _2$ =1 transitions are also included.", "For excitation analysis, the same treatment as [46], [8] with small adjustments was used.", "The HCN vibrational states are mainly radiatively excited [26], [49].", "In our modelling, the radiation arises from the central star, which is assumed to be a black body, and from the dust grains, which are distributed through the CSE." ], [ "Radiative transfer model", "To determine the HCN isotopologue abundances and constrain their distributions in the CSE, a non-local thermodynamic equilibrium (LTE) radiative transfer code based on the accelerated lambda iteration (ALI) method was used.", "The ALI method is described in detail by [43].", "The code has been implemented by [31] and was previously used by [30], [8].", "The CSE around R Scl is assumed to be spherically symmetric and is formed due to a constant mass loss rate $\\dot{M}\\sim 2\\times 10^{-7} $ $M_{\\odot }$ yr$^{-1}$ [56].", "The inner radius is assumed to be located at $r_{\\rm in} = 10^{14}$ cm.", "The H$_2$ number density is calculated assuming a constant mass loss rate, as in [48].", "The gas-expansion profiles is assumed to be: $v_{\\rm exp}\\left(r\\right) = v_{\\rm min} + \\left( v_\\infty - v_{\\rm min} \\right) \\left( 1 - \\frac{r_{\\rm in}}{r} \\right)^{\\rm b},$ where $v_\\infty $ is the terminal expansion velocity and $v_{\\rm min}$ is the minimum velocity at $r_{\\rm in}$ , which is taken as the sound speed 3 km s$^{-1}$ and b determines the shape of the radial velocity profile.", "A detailed discussion on the determination of the two free parameters $v_\\infty $ and b based on the shape of line profiles at radial offset positions from the ALMA observations is presented in Sect.", "REF .", "The radial distribution of the dust temperature is derived based on SED modelling (Maercker, private communication) to be a power-law given by: $T(r) = T_0 \\left(\\frac{r_0}{r}\\right)^{0.38},$ where $T_0$ = 1500 K and $r_0=7\\times 10^{13}$ cm are the dust condensation temperature and radius.", "Since ALI does not solve the energy balance equation, the same temperature profile as the dust temperature was used to describe the kinetic temperature of the gas.", "We also ran models using the gas temperature profile derived by W04, which led to no significant change in the results.", "The HCN fractional molecular abundance relative to H$_2$ $(n_{\\rm HCN}/n_{\\rm H2})$ is assumed to have a gaussian distribution: $f(r) = f_0 exp\\left(-\\left(\\frac{r}{R_e}\\right)^2\\right),$ where $f_0$ denotes the initial fractional abundance and $R_e$ is the $e$ -folding radius for HCN, the radius at which the abundance has dropped to 1/$e$ (37%).", "The stellar parameters are presented in Table REF .", "Table: The stellar parameters that are used in the radiative transfer modelling of the H 12 ^{12}CN and H 13 ^{13}CN isotopologues around R Scl." ], [ "Radial expansion velocity profile", "We use the spatially resolved ALMA observations of H$^{13}$ CN(4-3) to constrain two free parameters, b and $v_\\infty $ , in the gas radial expansion velocity profile, Eq.REF .", "We extracted intensities at a series of offset positions sampling every independent beam from the ALMA observations and the corresponding intensities from the modelling results.", "A series of models with b and $v_\\infty $ changing in the ranges $0.4 < b < 8.5$ and $8.5 < v_\\infty < 13$ were run to get good fits to the line shape of spectra at all positions.", "The model with b = 2.5 and $v_\\infty $ =10 km s$^{-1}$ leads to the best fits to the line shape of the H$^{13}$ CN(4-3) spectra at the offset positions from the centre.", "The velocity profiles with b = 0.4, 2.5 and 8.5 values are shown in Fig.", "REF .", "Models with b < 2.5 did not reproduce the shape of H$^{13}$ CN(4-3) line profiles in the inner part of the envelope (r $\\leqslant $ 0.2$ ^{\\prime \\prime } $ ); they predicted double-peaked line profiles contrary to the observed spectra.", "On the other hand, models with large b values reproduce narrow line shapes, which are also not consistent with the observations.", "To illustrate this, we plot the H$^{13}$ CN(4-3) intensity profiles at the centre of the star from ALMA observations and from three models with b = 0.4, 2.5 and 8.5 in Fig.", "REF .", "There is red-shifted excess emission in the ALMA H$^{13}$ CN(4-3) that can not be reproduced by our spherically symmetric model.", "The models with b = 0.4 and 8.5 can predict the total intensity of the H$^{13}$ CN(4-3) pretty well, but they fail to reproduce the line shapes at the inner part of the envelope (r$\\leqslant $ 0.2$ ^{\\prime \\prime } $ ).", "Without having spatially resolved observations, it is not possible to find out such effects and precisely constrain the gas expansion velocity profile.", "It should be noted that the wind-expansion velocity of R Scl likely changes over time [32].", "The gas expansion velocity profile used here hence mostly describes the change in terminal expansion velocity, rather than the acceleration due to radiation pressure on dust grains.", "Typical values for the exponent in accelerated-wind profiles are approx.", "1.5 for M-type stars [27] and even less for carbon stars.", "The fact that we derive a larger exponent confirms that the wind-expansion velocity of R Scl has been declining since the formation of the shell.", "Figure: Comparison of the H 13 ^{13}CN(4-3) intensity profile at the centre of the star from ALMA observations and three models with b = 0.4, 2.5 and 8.5.", "The model with b = 2.5 reasonably fits the line spectra, while models with b = 8.5 and 0.4 are not consistent with the observations at the centre of the star." ], [ "Radial distributions of HCN isotopologues", "The molecular photodissociation by UV-radiation is the dominant process controlling HCN survival throughout the CSE.", "Consequently, it is the most important factor in determining the size of the HCN molecular envelope in the CSE.", "Since both HCN isotopologues are equally affected by UV radiation, the same molecular distribution is expected for both H$^{12}$ CN and H$^{13}$ CN isotopologues.", "To estimate $R_e$ , we use the spatially resolved ALMA observations of H$^{13}$ CN(4-3) which strongly constrains the $e$ -folding radius.", "We ran 102 models with the fixed parameters detailed in Table REF and simultaneously varying $f_0$ and $R_e$ over the ranges $6\\times 10^{-7} < f_0 < 3 \\times 10^{-6}$ and $1\\times 10^{15} < R_e < 6\\times 10^{15}$ cm (6 values for $f_0$ and 17 values for $R_e$ ).", "To select the model with the best $R_e$ , a reduced $\\chi ^2$ statistic is used.", "The reduced chi-square is defined as $\\chi ^2_{red} = \\chi ^2 / (N-2)$ , where 2 is the number of free parameters in our modelling.", "In $\\chi ^2_{red}$ calculation, we only use the average intensities at the radial offset positions (9 positions) from four directions from the centre of the star which are shown in Fig.", "REF .", "The model with the minimum $\\chi ^2_{red}$ has the $e$ -folding radius $R_e = (2.0 \\pm 0.25) \\times 10^{15}$ cm.", "The cited error is for 1$\\sigma $ uncertainty derived from the $\\chi ^2_{red}$ distribution.", "Figure: Comparison of the integrated ALMA intensities of H 13 ^{13}CN(4-3) at radial offset points in the CSE of R Scl towards the west, east, north and south from the centre of the star with the best-fitting model which constrains the HCN molecular distributions at R e =2×10 15 cmR_e = 2 \\times 10^{15} cm.", "Error-bars on the observational points show 10%\\% uncertainty on the flux calibration.Figure: Two χ red 2 \\chi ^2_{red} maps calculated using the average intensities at radial offset points from H 13 ^{13}CN(4-3) ALMA observations (blue), and the total intensities of H 13 ^{13}CN(2-1, 3-2,4-3) (black).", "The min χ red 2 \\chi ^2_{red} values and the number of observational constraints that are used in the calculations are written.", "Contours are plotted at the 1,2,3 and 4 σ\\sigma standard deviation levels from the minimum χ red 2 \\chi ^2_{red}.", "The best-fitting model for H 13 ^{13}CN is shown by a black X." ], [ "The H$^{13}$ CN abundance", "To estimate the H$^{13}$ CN initial value $f_0$ , we use the total intensities of all three observed lines H$^{13}$ CN(2-1, 3-2, 4-3) in $\\chi ^2$ calculation.", "The uncertainty of the observed lines in $\\chi ^2$ calculation, $\\sigma $ , is assumed to be 20$\\%$ for H$^{13}$ CN(2-1) APEX data, 10$\\%$ for H$^{13}$ CN(4-3) ALMA data and 100$\\%$ for H$^{13}$ CN(3-2) undetected line.", "The H$^{13}$ CN(3-2) undetected line was only used to put a limit on the adjustable parameters.", "Since various rotational transitions come from different regions of the envelope, changing the parameters in the model has a different effect on these transitions.", "The $\\chi ^2$ map is shown in Fig.", "REF , accompanied with the $\\chi ^2_{red}$ map derived using H$^{13}$ CN(4-3) observations to constrain the $R_e$ .", "The best-fitting model is chosen among the six models with $R_e = 2.0 \\times 10^{15}$ cm and $6\\times 10^{-7} < f_0 < 3 \\times 10^{-6}$ , which are shown with a red line in the map.", "The best model with the minimum $\\chi ^2$ has the fractional abundance $f_0 = (1.9 \\pm 0.5) \\times 10^{-6}$ .", "The H$^{13}$ CN spectra accompanied with the best-fitting model are presented in Fig.", "REF .", "We also compare the average intensities at the radial offset positions from ALMA observations and the best-fitting model results in Fig.", "REF .", "Our excitation analysis shows differences between the kinetic and the excitation temperatures, meaning that all observed lines are formed under non-LTE conditions.", "The maximum tangential optical depth varies from $\\sim $ 0.09 up to 0.7 for the $J=2-1$ to $J=4-3$ lines, indicating that all lines are optically thin.", "Figure: Comparison between the average of integrated intensities of the H 13 ^{13}CN(4-3) at radial offset positions with best-fitting model which has values of f 0 =1.9×10 -6 f_0 = 1.9 \\times 10^{-6} and R e =2.0×10 15 R_e = 2.0 \\times 10^{15} cm.", "The grey region shows 1 σ\\sigma confidence level for R e R_e." ], [ "The H$^{12}$ CN abundance", "To estimate the H$^{12}$ CN fractional abundance $f_0$ , we ran 11 models with $R_e = 2.0 \\times 10^{15}$ cm and $f_0$ varying in the range of $1\\times 10^{-5} <f_0< 6.5\\times 10^{-5}$ .", "Figure REF shows the H$^{12}$ CN spectra overlaid with the best-fitting model with the minimum $\\chi ^2_{red} = 0.07$ .", "This model has a fractional abundance of $f_0 = (5.0 \\pm 2)\\times 10^{-5}$ .", "In the $\\chi ^2$ calculation, we consider the integrated intensity of the H$^{12}$ CN(2-1, 3-2, 4-3) lines.", "Since there is evidence of maser emission in the H$^{12}$ CN($J$ =1-0) line [36], [37], [26], [12], [50], [49], we do not include in the $\\chi ^2$ statistic.", "The difference between the kinetic and the excitation temperatures for H$^{12}$ CN indicates that all observed lines are formed under non-LTE conditions.", "The maximum tangential optical depth varies from $\\sim $ 8 up to 30 for $J=1-0$ to $J=4-3$ lines, indicating that all lines are optically thick.", "Figure: Line emission of H 12 ^{12}CN towards R Scl (black) overlaid with the best-fitting model (blue).", "Molecular transitions and the telescope used to get data are written in each panel.", "Second peaks in J=J=2-1, 3-2, 4-3 transitions are due to maser emission in the (01 1c 1^{1c}0) vibrational state." ], [ "Comparison of the best-fitting criteria based on single-dish and interferometric observations", "Comparison between two $\\chi ^2$ maps in Fig.", "REF shows that the best-fitting criteria based on the high-resolution interferometric and low-resolution single-dish observations are different.", "The single-dish observations require a larger $e$ -folding radius with less fractional abundance, while interferometric observations require a more compact envelope with higher abundance.", "This illustrates the importance of spatially resolved images of even a subset of the transitions used in molecular modelling.", "For comparison, a model selected only on the single-dish spectra is presented in the Appendix.", "The H$^{12}$ CN modelling for R Scl by W04 also implements the different best-fitting criteria based on the single-dish and interferometric observations.", "They require a more compact envelope with less abundance to fit the spatially-resolved H$^{12}$ CN(1-0) ATCA observations, while they can not fit the single-dish spectra using the same condition (see figures 7 and 8 in W04).", "A possible explanation for this discrepancy might be the spherically symmetric molecular distribution considered in our models, especially considering R Scl is known to have a binary companion whose influence gives rise to the spiral pattern observed in CO [28].", "Constraining this asymmetry in the inner wind is, with the currently available observations, not yet possible.", "However, while the asymmetry could affect the derived abundances, it does not affect our determination of the isotopologue ratio, which is the main aim of this work." ], [ "Asymmetry in the CSE ", "Our modelling is based on assuming a spherically symmetric envelope around R Scl.", "This is a first order approximation, and the east-west asymmetry in the CSE as seen in Fig.", "REF possibly due to the binary companion is not taken into account.", "Indeed, we are not able to precisely constrain the additional free parameters required to fully describe the physical condition such as the molecular distribution profiles from the available data.", "To find out the effect of this asymmetry in determining the adjustable parameters, we calculate the $\\chi ^2_{red}$ statistic in four directions from the central star.", "As seen in Fig.", "REF , $\\chi ^2_{red}$ in west, north and south lead to approximately the same range of adjustable parameters, while the contour map from the east shows a larger $R_e$ , which is expected to be due to the elongation of the molecular gas towards the east as seen also in Fig.", "REF .", "The east-west elongation of the molecular distribution could also be an explanation of the underestimate of the lower-$J$ HCN emission.", "Assuming that the lower-$J$ transitions are predominantly excited in the more extended aspherical region, a spherical symmetric model will either overestimate the lower-$J$ transitions when adopting the large $R_e$ found in the east-west direction, or underestimate them when adopting a smaller average $R_e$ .", "Figure: Comparison between four χ red 2 \\chi ^2_{red} maps derived using the intensities of H 13 ^{13}CN(4-3) at radial offset positions towards the east, west, north and south.", "The min χ red 2 \\chi ^2_{red} values and the number of observational constraints that are used in the calculations are written in each panel." ], [ "Comparison with previous studies", "We compared our results with those of previous studies.", "The H$^{12}$ CN circumstellar abundance value of $(5.0 \\pm 2)\\times 10^{-5}$ reported here for R Scl is higher than previously reported values of $8.1 \\times 10^{-6}$ and $1.2 \\times 10^{-5}$ by [36] and W04, respectively.", "It should be noted that [36] derived their abundance through a different method (see equation 3 in [36]).", "However, our result is consistent with the median value of the H$^{12}$ CN fractional abundances $f_0$ = 3.0 $\\times 10^{-5}$ for a sample of 25 carbon stars studied by [49] and the average value of $f_0$ = 4.6 $\\times 10^{-5}$ for a sample of five carbon stars studied by [26].", "The derived H$^{12}$ CN abundance here is also in good agreement with results of LTE stellar atmosphere models $f$ =$(1-5) \\times 10^{-5}$ reported for carbon-type stars by [55], [6].", "The $e$ -folding radius of $R_e$ = (2.0 $\\pm $ 0.25) $\\times 10^{15}$ cm derived here is smaller than the values reported by W04 and [36] ($r = 1.5\\times 10^{16}$ cm and $r = 3.6\\times 10^{15}$ cm, respectively).", "However, the result reported in [36] is not based on solving the radiative transfer equation (see equation 6 in [36]).", "The excitation analysis of W04 considered a fixed $f_{\\rm HCN}$ throughout the envelope out to the outer radius $r = 1.5\\times 10^{16}$ cm to fit the multi-transition spectra.", "However this radius is not consistent with the $J$ =1-0 ATCA intensity map, which requires a smaller envelope of $ r= 5.5 \\times 10^{15}$ cm and a higher photospheric abundance.", "We speculate that the difference between the two envelope sizes derived based on the interferometric observations of H$^{12}$ CN(1-0) by W04 ($\\sim $ 360 AU) and H$^{13}$ CN(4-3) here ($\\sim $ 130 AU) could be due to the influence of the binary companion at $\\sim $ 60 AU.", "As seen in CO in the outer envelope, the binary interaction causes a spiral density pattern that will alter the physical conditions in the outflow.", "The relative magnitude of the density pattern grows rapidly beyond the binary companion [22] and the east-west HCN extension could be the observational indication of the start of the density pattern.", "The wave might have a stronger effect on the excitation of H$^{12}$ CN(1-0) than that of H13CN(4-3)." ], [ "Comparison of the isotopologue ratios", "We have derived a ratio of circumstellar H$^{12}$ CN/H$^{13}$ CN = 26.3 $\\pm $ 11.9 for R Scl which is consistent with the photospheric atomic carbon ratio of $^{12}$ C/$^{13}$ C $\\sim $ 19 $\\pm $ 6 reported by [24].", "The derived isotopologue ratio is not affected by the limitation in the modelling (e.g.", "considering the effects of the binary companion on the physical condition), since the abundances are equally affected by these limitations.", "The CN molecule is also photodissociated in the continuum [14].", "Hence, the $^{12}$ CN/$^{13}$ CN isotopologue ratio is expected to follow the $^{12}$ C/$^{13}$ C isotopic ratio.", "An intensity ratio of $^{12}$ CN(1-0)/$^{13}$ CN(1-0) $\\sim $ 24 for R Scl reported by [35] is consistent with the photospheric $^{12}$ C/$^{13}$ C and the H$^{12}$ CN/H$^{13}$ CN ratio reported here.", "At the same time, the authors of V13 find an average value of $^{12}$ CO/$^{13}$ CO $\\sim $ 19 in the detached shell, consistent with the atomic carbon photospheric estimates, whereas they derive a lower limit of $^{12}$ CO/$^{13}$ CO > 60 for the present-day mass loss.", "Since H$^{12}$ CN/H$^{13}$ CN and $^{12}$ CN/$^{13}$ CN ratios are consistent with the $^{12}$ C/$^{13}$ C ratio, the most probable scenario of explaining the unexpectedly high $^{12}$ CO/$^{13}$ CO ratio in the inner CSE of R Scl is CO isotopologue selective-shielding.", "The extra photodissociation of less abundant $^{13}$ CO is most likely by the internal UV radiation as proposed by V13.", "In addition, [35] have reported an intensity ratio of $^{12}$ CN(1-0)/H$^{12}$ CN(1-0) $\\sim 4.3$ which is higher by a factor of two from the average value $2.0\\pm 0.7 $ reported for C-type stars.", "Assuming that HCN photodissociation is solely responsible for producing CN in the circumstellar environment, the ratio reported by [35] also supports the extra UV radiation in the CSE of R Scl.", "Consequently, taking isotopologue self-shielding of molecules which are dissociated in discrete bands in the UV, e.g.", "$^{12}$ CO/$^{13}$ CO, into account is very important in the determination of the isotopologue ratios.", "Moreover, the isotopologue ratios of these molecules are not always a reliable tracer of the atomic carbon isotope ratio $^{12}$ C/$^{13}$ C in UV irradiated regions.", "The discrepancy between the two ratios can be considerable in optically thick regions where the isotopologue self-shielding has more impact." ], [ "Conclusion", "We have performed a detailed non-LTE excitation analysis of H$^{12}$ CN and H$^{13}$ CN in the CSE of R Scl.", "The derived H$^{12}$ CN/H$^{13}$ CN = 26.3 $\\pm $ 11.9 is consistent with the photospheric ratio of $^{12}$ C/$^{13}$ C $\\sim $ 19 $\\pm $ 6 reported by [24].", "It is clearly shown that constraining the molecular distribution size, the fractional abundance and the radial expansion velocity profile in the CSE requires high-resolution spatially-resolved observations.", "Our results show that the circumstellar H$^{12}$ CN/H$^{13}$ CN ratio is a more reliable tracer of the photospheric $^{12}$ C/$^{13}$ C ratio than the circumstellar $^{12}$ CO/$^{13}$ CO ratio in the UV irradiated region.", "These results also support the isotopologue selective-shielding of CO as the reason for the lacking of $^{13}$ CO in the inner CSE of R Scl as previously claimed by V13.", "The extra photodissociation of $^{13}$ CO is most likely due to the internal radiation either from the binary companion or chromospheric activity.", "This indicates the important role of internal UV radiation as well as the ISRF on the chemical composition of the CSEs.", "Thus, should be considered in the chemical-physical modelling of the CSEs around evolved stars.", "In a more general context, the most abundant molecules in astrophysical regions that control the chemistry, for example H$_2$ , CO, N$_2$ , C$_2$ H$_2$ , have sharp discrete absorption bands in UV [25], which leads to isotopic fractionation in the UV irradiated regions.", "Thus, considering the isotopologue selective-shielding is very important in astrochemical models of AGB envelopes and other irradiated environments.", "We suggest that the comparison between the photospheric $^{12}$ C/$^{13}$ C ratio and the circumstellar $^{12}$ CO/$^{13}$ CO and H$^{12}$ CN/H$^{13}$ CN ratios can indirectly trace the internal embedded UV sources, which are difficult to observe directly, in the CSEs of evolved stars.", "This comparison cannot distinguish between the contributions from different potential UV sources such as ISRF, the chromospheric activity and active binary companions.", "This requires spatially-resolved observations of the photodissociated products such as atomic carbon.", "This paper makes use of ALMA data from project No.", "ADS/ JAO.ALMA$\\#$ 2012.1.00097.S.", "ALMA is a partnership of ESO (representing its member states), the NSF (USA) and NINS (Japan), together with the NRC (Canada), NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile.", "The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.", "This work is supported by ERC consolidator grant 614264.", "M.M.", "has received funding from the People Programme (Marie Curie Actions) of the EU’s FP7 (FP7/2007-2013) under REA grant agreement No.", "623898.11." ], [ "An example model which fits the total intensities of H$^{13}$ CN(2-1, 4-3) spectra very well, but is not consistent with the integrated intensities at radial offset positions from ALMA observations, Figs.", "REF and REF .", "This model has a radial expansion velocity profile with values b = 0.5 and $V_\\infty = 8.5$ .", "As it was mentioned in Sect.", "REF , the models with b < 2.5 give a double peak shape intensity profiles in the inner part ($r\\leqslant 0.2^{\\prime \\prime }$ ).", "Fig.", "REF shows the intensity profile at the central point of the star from the model and ALMA observations.", "This model has a larger abundance of $f_0 = 8.0 \\times 10^{-7}$ and a smaller radius of $R_e = 5.0 \\times 10^{15}$ cm compared to our best model which was discussed in Sect.", ".", "Thus, the best-fitting criteria based on the single-dish observations requires a smaller envelope and larger abundance.", "This model shows the importance of using spatially resolved interferometric data in constraining the adjustable parameters in the modelling of the CSE.", "Figure: Comparison of the ALMA integrated intensities of H 13 ^{13}CN(4-3) at radial offset points in the CSE of R Scl towards the west, east, north and south from the centre of the star with the model results.", "Error-bars on the observational points show 10%\\% uncertainty on the flux calibration.Figure: Comparison of the H 13 ^{13}CN(4-3) line emission towards R Scl at the centre of the star from ALMA observations and a model with fast accelerating wind (b = 0.5)." ] ]
1612.05573
[ [ "Dark Matter Interpretation of the Fermi-LAT Observation Toward the\n Galactic Center" ], [ "Abstract The center of the Milky Way is predicted to be the brightest region of gamma-rays generated by self-annihilating dark matter particles.", "Excess emission about the Galactic center above predictions made for standard astrophysical processes has been observed in gamma-ray data collected by the Fermi Large Area Telescope.", "It is well described by the square of an NFW dark matter density distribution.", "Although other interpretations for the excess are plausible, the possibility that it arises from annihilating dark matter is valid.", "In this paper, we characterize the excess emission as annihilating dark matter in the framework of an effective field theory.", "We consider the possibility that the annihilation process is mediated by either pseudo-scalar or vector interactions and constrain the coupling strength of these interactions by fitting to the Fermi Large Area Telescope data for energies 1-100 GeV in the 15 x 15 degree region about the Galactic center using self-consistently derived interstellar emission models and point source lists for the region.", "The excess persists and its spectral characteristics favor a dark matter particle with a mass in the range approximately from 50 to 190 (10 to 90) GeV and annihilation cross section approximately from 1E-26 to 4E-25 (6E-27 to 2E-25) cm^3/s for pseudo-scalar (vector) interactions.", "We map these intervals into the corresponding WIMP-neutron scattering cross sections and find that the allowed range lies well below current and projected direct detection constraints for pseudo-scalar interactions, but are typically ruled out for vector interactions." ], [ "Introduction", "Despite the overwhelming evidence from astrophysics and cosmology that roughly 80$\\%$ of the matter in our Universe is in the form of dark, non-baryonic particles, how this so-called dark matter (DM) fits with the Standard Model (SM) of particle physics is currently unknown.", "Determining the nature of DM is one of the most pressing questions in the physical sciences, and a wide array of experiments are underway which hope to shed light on its identity by observing its interactions with the better understood particles of the SM.", "Indirect detection is one of the promising avenues to elucidate the nature of DM.", "This method attempts to detect and discriminate the SM particles produced by DM particle annihilations (or decays) from those produced by conventional astrophysical processes.", "$\\gamma $ -rays of $\\sim $  GeV energies are a particularly effective messenger because they propagate unhindered on galactic scales, and thus can be effectively traced back along the direction of their origin.", "In recent years, the Fermi Large Area Telescope (Fermi-LAT) has mapped out the $\\gamma $ -ray sky with the highest sensitivity of space-borne detectors to date, leading to the current best limits on the annihilation cross section for $\\sim 100$  GeV DM annihilations that result in $\\gamma $ -rays.", "Numerical simulations of galaxy formation offer clues as to where DM annihilation is expected to shine the most brightly.", "The simulations typically predict a large concentration of DM close to the Galactic center (GC), which smoothly falls off with Galactocentric radius.", "They also predict localized over-densities of DM, some of which correspond to dwarf spheroidal satellite galaxies.", "Both targets provide complementary regions of interest for DM searches.", "The DM related emission from the dwarf galaxies is expected to be of lower intensity, but to be relatively free of standard astrophysical backgrounds.", "Searches for $\\gamma $ -ray emission from dwarf satellites of the Milky Way have so far shown no convincing signal of DM annihilation [1], [2].", "In contrast, the GC is expected to produce a higher intensity annihilation signal.", "However, the region about the GC is strongly confused because of the intense interstellar emission and numerous discrete sources of $\\gamma $ -rays that are summed along and through the line-of-sight toward the GC.", "The estimation of these fore-/background contributions pose a significant challenge for detection of DM annihilation at the GC.", "There seems to be an excess of $\\gamma $ -rays from the direction of the GC, above the expectations from astrophysics.", "This feature was first observed by Goodenough and Hooper [3], [4], and its general features, a spatial morphology remarkably consistent with predictions for a DM annihilation signal and a spectrum that peaks at a few GeV, persist in more recent analyses [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].", "The Fermi-LAT collaboration has released its own analysis [17] of the $\\gamma $ -rays from the direction of the inner galaxy based on specialized interstellar emission models (IEMs) for estimating the fore-/background emissions, and enabling the analysis to make the first separation of the $\\gamma $ -ray emission of the $\\sim 1$  kpc region about the GC from the rest of the Galaxy.", "Even with these IEMs, which represent the most sophisticated modeling to date, the excess persists.", "However, its spectral properties are strongly dependent on the assumed IEM, making it challenging to conclusively identify its origin.", "As a result, it remains unclear whether this signal arises from DM annihilation rather than from a currently unknown contribution from astrophysics such as a large population of milli-second pulsars, cosmic-ray (CR) proton or electron outbursts, additional cosmic ray sources, and/or emission from a stellar over-density in the Galactic bulge [18], [11], [19], [20], [21], [22], [16], [23].", "An interesting development is the use of statistical tools which indicate that GeV photons from the direction of the inner galaxy region show significantly more clustering than would be expected from Poisson noise from smooth components [24], [25], [26], [27].", "However, it remains difficult with the current models to disentangle whether this feature represents a property of the excess itself, or unmodelled variation in the background components [28].", "While it is clearly premature to claim that the GeV excess represents a confirmed signal of DM annihilation, in this paper we extract the properties of the excess under the assumption that it does.", "We make simultaneous fits to the parameters of generic, realistic particle physics model of DM annihilation together with those defining the broad characterization of the possible fore-/backgrounds determined using the methodology of Ref [17].", "As a result, we can compare with the expectations for such models from direct searches for DM and colliders, finding that the null results of those searches play a significant role in shaping the allowed parameter space.", "Our work is organized as follows.", "In Section , we very briefly review the methodology of the Fermi-LAT analysis [17] to formulate realistic IEMs, which crucially define the fore- and backgrounds as well as the astrophysical contributions from the GC itself.", "This is followed in Section  by a revisitation of some of the most important morphological and spectral features of the signal: its centroid and whether there is evidence for two separate components with distinct morphologies and spectra.", "In Section , we define realistic flexible DM models described by effective field theories (EFTs), and perform a maximum likelihood (ML) fit to determine the ranges of their parameters capable of describing the excess together with the IEM parameters.", "We compare the ML regions of those models to direct and collider searches for DM in Section .", "Section  contains our conclusions and outlook.", "The analysis presented in this paper employs the same data as used by Ref [17]: front converting events corresponding to the P7REP_CLEAN_V15 selection  [29], in the energy range 1-100 GeV, and with zenith angles less than 100$^\\circ $ .", "Exposure maps and the PSF for the pointing history of the observations were produced using the Fermi–LAT ScienceTools package (version 09-34-02)Available at http://fermi.gsfc.nasa.gov/ssc/data/analysis.", "Events are selected from approximately 62 months of data, from 2008-08-11 until 2013-10-15.", "We note that for high statistics analyses such as the one presented here a notable difference is not expected in the results obtained with the P7REP_CLEAN_V15 data processing and those processed using Pass 8 [30]; this is confirmed by several previous analyses [26], [16], [31]." ], [ "Interstellar Emission Models", "The interstellar emission is the largest contribution to the $\\gamma $ -ray emission toward and through the line-of-sight toward the GC.", "To separate the contribution by the Galaxy between our location and the inner 1 kpc region about the GC, and that on the other side of the GC, specialized IEMs (four in total) were developed for the Ref [17] analysis.", "The methodology employed templates calculated using the well-known GALPROP CR propagation modeling codeA description of the GALPROP code is available at http://galprop.stanford.edu that were scaled to the data outside of the inner $15^\\circ \\times 15^\\circ $ region about the GC.", "Under the assumption of Galactocentric azimuthal symmetry, these IEMs were used to estimate the fore-/background emission over the $15^\\circ \\times 15^\\circ $ region, enabling the separation.", "Employing this prescriptive methodology ensures that minimal biases are introduced when fitting to the inner region.", "In addition, point source lists were developed for each IEM with the properties of the individual point sources obtained in a combined ML fit over the $15^\\circ \\times 15^\\circ $ region.", "The construction of each IEM and its associated point-source list/model is a critical improvement over earlier works because the residual emission is strongly dependent on modeling both the over the region self-consistently.", "The four distinct IEMs from Ref [17] are labeled: Pulsars, intensity-scaled Pulsars, index-scaled OB stars, intensity-scaled OB stars, index-scaled The IEMs differ in the assumed distribution of the sources of CRs as tracing either the distributions of pulsars or OB stars; and in the procedure employed to scale the $\\gamma $ -ray intensity of the fore-/background components outside of the  $15^\\circ \\times 15^\\circ $  region to the data, either by scaling the normalization of the model templates for intensity-scaled IEMs, or scaling the normalization and spectral index (the latter only for gas-related templates interior to the solar circle) for the index-scaled IEMs.", "Notably, it was found that the data are compatible with a contribution from $\\gamma $ -rays from DM annihilation, and that the agreement between the data and the model significantly improves for all four IEMs when an additional component with a DM annihilation morphology is included in the fit." ], [ "Analysis Procedure", "We employ the procedure developed by the Fermi–LAT Collaboration in [17], which performs a ML fit of a model consisting of one of the four IEMs and its corresponding list of point sources to the data in the $15^\\circ \\times 15^\\circ $  region.", "For each model, we include a DM annihilation contribution (described below) and perform the fit using the gtlike package of the Fermi–LAT ScienceTools.", "The results of the fit are the coefficients of the interstellar emission components from within the the innermost $\\sim $ 1 kpc, as well as those describing the DM model under consideration.", "All point sources with a test statistic (defined as in [32]) $TS > 9$ are included in the model.", "Their fluxes and spectra are determined by iterative fits, with each iteration freeing the spectral parameters for a subset of point sources in order of decreasing TS.", "The DM spatial distribution used in this paper is described in this section.", "Because [17] tested spatial templates fixed at the position of Sgr A* we investigate the possibility of an offset from this location by refitting the DM spatial distribution and scanning the ML grid about the GC.", "If a large offset is found, it might challenge a DM interpretation of the excess.", "For some IEMs the DM spectrum obtained by [17] extended beyond 10 GeV, but a dedicated study of the spatial distribution $>10$  GeV was not made; this is also investigated in this section." ], [ "Dark Matter Component", "The results of numerical simulations for galaxy formation can broadly be described by the Navarro, Frenk, and White (NFW) profile [33]: $ \\rho (r) = \\rho _0 \\left( \\frac{r}{R_s}\\right)^{-\\gamma }\\left(1 + \\frac{r}{R_s}\\right)^{\\gamma - 3}$ For this analysis, we use a scale radius $R_s$ = 20 kpc, and $\\rho _0$ corresponding to a local DM density $\\rho _\\odot $ = 0.4 $\\mathrm {GeV/cm^3}$ .", "Two values for the inner slope $\\gamma $ of the DM distribution are considered, $\\gamma $ = 1, 1.2.", "The more cuspy distribution $\\gamma $ = 1.2 is motivated by the possibility of halo contraction due to the influence of baryons, which are typically not included in the simulations [34].", "The square of the NFW distribution is used as a template for DM annihilation, and we refer to it as the “NFW profile” (for $\\gamma =1$ ) or “NFW-c” (for $\\gamma $ = 1.2)." ], [ "NFW Centroid", "The centroid of the Milky Way DM halo is conventionally centered at the location of Sgr A*.", "Because a large offset from this location might disfavor a DM interpretation, we verify that the centroid of the excess is sufficiently close.", "An offset between the centroid of the DM halo and Sgr A* as large as approximately 2$^{\\circ }$ is consistent with numerical DM simulations, with the largest offsets tending to correlate with flatter central profiles [35], [36].", "An offset in the centroid position was previously reported in [13], [37], while other studies of the GC excess have found it to be consistent with Sgr A*.", "We investigate the centroid position of the excess by scanning the ML for different locations near Sgr A*, for each of the four IEMs.", "A power-law with exponential cut-off is employed for the spectral model, following [17].", "The scan is performed by making the ML fit following Sec.", "with the DM template centered at each point of a grid with spacing $0.2^\\circ $ centered on Sag A*.", "The results of the scan are shown in Fig.", "REF , where the color scale shows the $\\mathrm {2 \\Delta log L}$ as a function of Galactic latitude and longitude.", "The intersections of the dotted grid lines correspond to the points where the likelihood is evaluated.", "The circle indicates the position of Sgr A*, and the triangle is the most likely position of the centroid for that IEM.", "We find that the centroid position is offset from Sgr A* for all four IEMs, with the Pulsars, index-scaled model displaying the largest offset, both in longitude (0.6$^{\\circ }$ ) and latitude (0.2$^{\\circ }$ ).", "The other three models prefer an offset only in longitude (within 0.4$^{\\circ }$ up to the grid accuracy).", "Based on the scan, Sgr A* is not favored as the location of the NFW centroid for all four IEMs, however its position is roughly consistent with a DM interpretation for the GC excess and imperfections in the IEMs could plausibly introduce an offset.", "We therefore assume for the remainder of this paper that the DM distribution is centered at Sgr A*." ], [ "Multiple Component Fit", "Whether the high-energy tail ($>10$  GeV) of the GeV excess spectrum is related to that at lower energies remains an open issue.", "In [17], the excess emission above 10 GeV is most prominent in the intensity-scaled IEMs.", "For the index-scaled variants however, it is largely attributed to interstellar emission (see also [10]).", "The origin of the $> 10$  GeV excess has been previously investigated by several studies.", "In [28], the excess emission above 10 GeV is found to cut off in the innermost few degrees about the GC (unlike the excess at a few GeV) and therefore to have a different spatial morphology; secondary emission from unresolved millisecond pulsars is proposed as an interpretation.", "In [37], the excess emission above 10 GeV is found to have a similar radial profile as the peak emission.", "Ref [37] also discusses the interplay with the Fermi Bubbles, although the bubble morphology close to the Galactic plane is uncertain.", "Here we investigate the morphology of the $>10$  GeV excess emission present for the Pulsars and OB stars, intensity-scaled IEMs.", "We perform a ML fit over the 1-100 GeV energy range with two components to model the GC excess: an NFW template; and a second component that has either an NFW, gas, or a 2D gaussian (with half-width, half maximum of $1^\\circ $ , $2^\\circ $ , $5^\\circ $ , or $10^\\circ $ ) morphology.", "These are the same templates that were employed by [17].", "Six template combinations for the two intensity-scaled models are therefore tested.", "The spectrum for each template is modeled as a power law with an exponential cutoff function.", "The ML fit is performed iteratively, as described in section , and the results are shown in Tables REF and REF for the Pulsars and OB stars, intensity-scaled IEM, respectively.", "The NFW + NFW combination is favored over all of the others considered, for both IEMs.", "In Fig.", "REF the differential fluxes integrated over the $15^\\circ \\times 15^\\circ $  region for the two component fits, along with the fractional residuals, are shown for the Pulsars, intensity-scaled model.", "The contribution to the flux from each of the two spatial components and the IEM are shown, with the IEM broken down into the contributions from inverse Compton (IC), $\\pi ^0$ emission from the inner $\\sim $ 1 kpc (“ring 1\" in the legend), and from the point sources.", "For each of the six combinations we consider, the low energy excess is better described by an NFW morphology.", "The more peaked 2D gaussian templates ($1^\\circ $ and $2^\\circ $ ) have spectra that peak in the few GeV energy range and cutoff at higher energies.", "Note that their contribution is always well below the contribution assigned to the NFW template.", "On the other hand, the spectra for the broader 2D gaussian templates ($5^\\circ $ and $10^\\circ $ ) are more prominent at higher energies, suggesting that the high-energy tail of the GeV excess is consistent with an extended component in the region.", "The NFW morphology, which is peaked towards the GC and broadly extended in the region, is better suited to model the excess emission over the full energy range compared to the other options we have considered.", "However, due to the limitations of the IEMs together with the limited statistics at the higher energies, it is difficult to conclude decisively whether or not the high-energy tail is a true feature of the GC excess.", "Given the current preference for a single NFW morphology for both low and high energy components, we include the full energy range when comparing with the DM scenarios in Section  below." ], [ "Dark Matter Interpretation", "In this section we fit the parameters of particle physics models of DM, together with the parameters describing the fore-/backgrounds, extracting a comprehensive DM interpretation of the GC excess.", "As described in more detail below, we employ a parameterization of the DM particle physics model which allows for distinct annihilation rates into up-type quarks, down-type quarks, and leptons.", "Our parametrization has more flexibility than the often-considered annihilation into a single channel of SM particles and, in this sense, is better able to capture a wider array of realistic particle physics models for DM annihilation than those typically used in indirect searches." ], [ "EFT Description of Dark Matter Interactions", "We consider two representative EFTs that describe the DM interactions with the SM fermions.", "These theories form part of a universal set of operators to which any theory of DM flows at low energies, well below the masses of the particles responsible for communicating between the SM and the dark matter [38], [39], [40], [41], [42], [43].", "Such models have previously been considered to describe the GC excess [44], [45].", "More generalized constructions are employed here, and their parameters are fit together with the IEM parameters as described in Section .", "Of course, models with light mediators are also interesting, and worthy of investigation in their own right [46], [47], [48], [49], [50], [51], [52], [53].", "We leave exploration of such theories for future work.", "Both of our considered EFTs are chosen such that they mediate $s$ -wave (velocity-unsuppressed) annihilation, because a $p$ -wave annihilation mechanism would require such strong interactions to overcome the innate $v^2 \\sim 10^{-4}$ suppression that it is likely to already be ruled out by direct and/or collider searches.", "We further restrict them to follow the principle of minimal flavor violation (MFV) [54], such that the most stringent constraints from flavor-violating observables are mitigated by small Yukawa interactions.", "We consider models containing either pseudo-scalar or vector Lorentz structures described by Lagrangians ${\\cal L}_{\\rm ps}$ and ${\\cal L}_{\\rm vec}$ (respectively, in the fermion mass basis), ${\\cal L}_{\\rm ps} & = & \\overline{\\chi } \\gamma _5 \\chi \\times \\\\ & &\\sum _i \\left\\lbrace \\frac{m_{u_i}}{\\Lambda _u^3} ~\\overline{u}_i \\gamma _5 u_i+ \\frac{m_{d_i}}{\\Lambda _d^3} ~\\overline{d}_i \\gamma _5 d_i+ \\frac{m_{\\ell _i}}{\\Lambda _\\ell ^3} ~\\overline{\\ell }_i \\gamma _5 \\ell _i \\right\\rbrace , \\nonumber \\\\{\\cal L}_{\\rm vec} & = & \\overline{\\chi } \\gamma ^\\mu \\chi \\times \\\\ & &\\sum _i \\left\\lbrace \\frac{1}{\\Lambda _u^2} \\overline{u}_i \\gamma _\\mu u_i+ \\frac{1}{\\Lambda _d^2} \\overline{d}_i \\gamma _\\mu d_i+ \\frac{1}{\\Lambda _\\ell ^2} \\overline{\\ell }_i \\gamma _\\mu \\ell _i \\right\\rbrace , \\nonumber $ where $i=1,2,3$ is the sum over fermion flavor with the indicated relative weighting of $m_{f_i}$ $(1)$ for the pseudo-scalar (vector) interaction types, as dictated by the leading terms consistent with MFV.", "The $\\Lambda _{u,d,\\ell }$ are parameters with dimensions of energy which specify the separate interaction strengths between the DM and up-type quarks, down-type quarks, and charged leptons.", "Together with the DM mass, $m_\\chi $ , these coefficients specify the point in parameter space for the DM model.", "They represent generalizations (in that they allow the couplings of up-type and down-type quarks and leptons to vary independently) of the commonly considered interactions D4 and D5 used in DM searches via direct detection and at colliders [41]." ], [ "$\\gamma $ -ray Flux from Dark Matter Annihilation", "The interactions in both the pseudo-scalar and vector models defined in Eqs.", "(REF ,) lead to cross sections for a pair of DM particles to annihilate $\\chi \\overline{\\chi } \\rightarrow f \\overline{f}$ (where $f$ is any SM fermion): $\\langle \\sigma _f v \\rangle _{\\rm ps} &=& \\frac{N_f m_f^2 m_\\chi ^2}{\\Lambda _f^6 \\pi } \\sqrt{ 1 - \\frac{m_f^2}{m_\\chi ^2} } + {\\cal O}(v^2), \\\\\\langle \\sigma _f v \\rangle _{\\rm vec} &=& \\frac{N_f (2 m_\\chi ^2 + m_f^2)}{\\Lambda _f^4 \\pi } \\sqrt{ 1 - \\frac{m_f^2}{m_\\chi ^2} } + {\\cal O}(v^2),$ where $\\langle \\cdot \\rangle $ indicates averaging over the DM velocity profile, $N_f = 3$ (1) for quarks (leptons) counts their color degrees of freedom, and $\\Lambda _f$ is the appropriate $\\Lambda _{u,d,\\ell }$ for the fermion under consideration.", "The inclusive cross section for annihilation into up-type quarks, down-type quarks, and charged leptons is the sum of the individual cross sections for all three flavors of each fermion type, and the total cross section $\\langle \\sigma v \\rangle $ is the sum of the three inclusive cross sections.", "In presenting results, we typically trade the three parameters $\\Lambda _{u,d,\\ell }$ for $\\langle \\sigma v \\rangle $ and the fractional cross sections $f_u$ , $f_d$ , and $f_\\ell $ (with $f_u + f_d + f_\\ell = 1$ ).", "It is easy to map these back into the $\\Lambda _{u,d,\\ell }$ parameters using the appropriate single channel cross section from Eqs.", "(REF ) and ().", "Figure: Likelihood (2Δ logL \\mathrm {2 \\Delta logL}) as a function of the DM mass for thepseudo-scalar interaction model with NFW-c morphology.", "Results are shown for all four IEMs, as indicated.The $\\gamma $ -ray intensity and spectrum from DM annihilation is constructed by summing over all of the annihilation channels: $\\frac{dN_\\gamma }{dE} = \\sum _f\\frac{ \\langle \\sigma _f v \\rangle }{4 \\pi \\eta ~ m_\\chi ^2} \\frac{dN^f_\\gamma }{dE} \\times \\int _{\\Delta \\Omega } d\\Omega ^\\prime \\int _{los} ds \\ \\rho ^2 (r(s, \\psi )),$ where ${dN^f_\\gamma }/dE$ is the number of $\\gamma $ rays per annihilation into the $f \\overline{f}$ channel, generated from the PPPC 4 DM ID package [55] based on fits to Pythia 8.1 [56], and $\\eta $ = 2(4) for Majorana (Dirac) DM.", "The integral is the J-factor, obtained by integrating the DM density $\\rho ^2(\\mathbf {x})$ corresponding to either an NFW or NFW-c distribution, Eq.", "(REF ), over the line of sight ($los$ ) in direction $\\psi $ .", "To determine the preferred DM model parameters for each IEM, we fix the DM mass in the range from 10 – 250 GeV in 10 GeV increments.", "For each mass hypothesis the analysis procedure of Section  determines the fitted values of the DM model parameters $f_u$ , $f_d$ , and $f_\\ell $ , along with the coefficients of the interstellar emission components from within the innermost $\\sim $ 1 kpc and point sources, as usual.", "We repeat this scan for both NFW and NFW-c annihilation morphologies and for both the pseudo-scalar and vector models described above.", "We find that the DM component is detected with high statistical significance for all IEMs, and for pseudo-scalar as well as vector interactions.", "The likelihood values for pseudo-scalar interactions are summarized in Table REF .", "Table: Likelihood ( log L\\mathrm {log~L}) values for all IEMs for pseudo-scalar interactions and for NFW and NFW-c templates." ], [ "Results for Pseudo-scalar Interactions", "In Fig.", "REF , we display the likelihood profile as a function of the DM mass for each of the IEMs for the NFW-c annihilation morphology.", "The results for the NFW morphology are qualitatively similar.", "Each of the four IEMs shows a clear preference for particular DM masses, but there is considerable variation between them, with the index-scaled models favoring a mass around $\\sim $ 50 GeV, while the intensity-scaled models favor higher masses $\\sim $ 200 GeV.", "The results are consistent with the results obtained by [17], where the spectrum of the GC excess for the index-scaled IEMs displays a lower energy cutoff compared to the intensity-scaled IEMs.", "The spectra we consider here correspond to motivated DM scenarios, in contrast with the simpler assumptions made for the spectral model by [17].", "Figure: Flux fraction for annihilation into up-type quarks, down-type quarks, and charged leptons,for the pseudo-scalar interaction model with NFW-c morphology.", "Results are shown for all four IEMs, as indicated.In Fig.", "REF , we present the ML fractions into the three annihilation channels as a function of the DM mass, for each of the IEMs with the NFW-c annihilation morphology.", "These also vary considerably from one IEM to another, and are characterized by one channel or another typically dominating at any given DM mass hypothesis: charged leptons at lower masses $\\sim 10-20$  GeV; down-type quarks in the range $\\sim 50 -170$ GeV; and up-type quarks above 180 GeV and at lower masses $\\sim 20-40$  GeV.", "The lepton flux declines steeply above $\\sim $ 20 GeV, and its contribution to the flux is smaller for the index-scaled models (Pulsars in particular) compared to the intensity-scaled ones.", "This reflects in part the lower energy cutoff of the GC excess spectrum for the index-scaled models and the harder $\\gamma $ -ray spectra produced by charged leptons compared to quarks.", "Also of note is the sharp transition from annihilation into down-type quarks to up-type quarks at the top mass threshold, $\\sim 175$  GeV.", "This follows because the pseudo-scalar model annihilations are dominated by the heaviest quark kinematically accessible, and top quarks produced close to at rest decay into $\\sim 60$  GeV bottom quarks, corresponding to the ML region at $m_\\chi \\sim 50$  GeV.", "The best-fit DM mass for the Pulsars (OB stars) index-scaled IEM is $\\mathrm {50_{-10}^{+10}}$ GeV ($\\mathrm {70_{-10}^{+15}}$ GeV), and in both cases annihilation is predominantly into bottom quarksThe grid spacing is taken into account in the quoted uncertainties on the DM mass..", "These results are compatible with the findings of previous studies [57], [58] interpreting the spectrum of the excess as presented in Ref. [17].", "The intensity-scaled IEMs favor higher DM masses, $\\mathrm {180_{-5}^{+15}}$ GeV and $\\mathrm {190_{-15}^{+25}}$ GeV, for the Pulsars and OB stars variants, respectively, and primarily favor annihilation into top quarks.", "We note that the likelihood profile for the OB stars, intensity-scaled IEM is rather flat around the minimum, which yields a higher uncertainty in the best-fit DM mass, compared to the other IEMs.", "The uncertainties on the flux fractions into up-type and down-type quarks in this mass range are also somewhat larger.", "Figure: Differential fluxes (broken down into components, as indicated)integrated over the 15 ∘ ×15 ∘ 15^\\circ \\times 15^\\circ  region and corresponding fractional residuals for pseudo-scalar interactions and for the four IEMs.The differential fluxes for the ML model (and the data points) are shown for each IEM in Fig.", "REF .", "Individual model components are displayed separately, including the contribution to the DM flux from each annihilation final state, as well as their sum.", "The contribution from each DM annihilation channel illustrates the fact that the integrated DM flux originates primarily from annihilations into quarks with the harder spectrum from annihilation into leptons becoming important at higher energies, particularly for the intensity-scaled IEMs.", "The $\\gamma $ -ray emission correlated with gas from the innermost $\\sim $ 1 kpc is sub-dominant in the region.", "Fig.", "REF also shows the fractional residuals as a function of energy.", "The agreement between data and model is at the level of a few % or better up to $\\sim 30$  GeV for all IEMs, and is generally worse at higher energies for all but the Pulsars, index-scaled IEM.", "It is plausible that the energy cutoff at the DM mass in the annihilation spectrum limits its ability to describe the excess at the higher energies while simultaneously providing a good fit to the data in the few GeV range.", "We note that the fractional residuals based on realistic DM models including up-type, down-type, and lepton final states generally improve (for the same number of free parameters) over the results in [17] based on a power law with exponential cutoff spectrum.", "Figure: Residuals (data – model) in three energy bands, for the four IEMs.", "The rows correspond to the range 1 - 1.6 GeV (top), 1.6 - 10 GeV (center), and 10 - 100 GeV (bottom).", "The columns, going from left to right are: Pulsars, index-scaled; Pulsars, intensity-scaled;OB stars, index-scaled; OB stars, intensity-scaled.Residual count (data-model) maps are shown in Fig.", "REF for the energy bands $1-1.6$ , $1.6-10$ , and $10-100$  GeV, for each IEM.", "Structured excesses and deficits remain that may be attributed to imperfect modeling of the interstellar emission.", "Because of this, we do not rule out the DM models corresponding to IEMs with larger fractional residuals as these discrepancies might be explained by limitations in the IEMs.", "There is better agreement with the data when the DM spectrum is modeled with power law functions in 10 independent energy bins as done in [17]; perhaps unsurprising given the larger number of free parameters for the spectral model.", "Figure: Differential flux integrated over the 15 ∘ ×15 ∘ 15^\\circ \\times 15^\\circ  region for the DM component for pseudo-scalar interactions, NFW and NFW-c profiles, for all four IEMs, as indicated.", "The bands represent the fit uncertainties on the normalization.The differential flux from the total DM annihilation component for both profiles (NFW, NFW-c) and all four IEMs are summarized in Fig.", "REF .", "The bands represent the 1$\\sigma $ fit uncertainty on the flux summing the up-type, down-type, and lepton final states.", "For the index-scaled variants of the IEMs, the spectrum peaks at a few GeV, while for the intensity-scaled counterparts the peak shifts to higher energies.", "This is consistent with the requirement that the high energy tail in the spectrum for the intensity-scaled IEMs, predominantly from annihilations into leptons, has to cutoff at the same energy (corresponding to the DM mass) as the contribution to the flux from annihilations into up-type and down-type quarks, which dominate the DM flux at lower energies.", "Finally, we note that the flux for NFW-c profile is smaller compared to the NFW profile.", "As a consequence, a simple rescaling based on $J-$ factors when comparing fit results obtained with different profiles is not accurate, as the flux assigned to the DM component has a dependence on the specific morphology.", "Figure: Masses and cross sections for pseudo-scalar interaction models(including one and two sigma uncertainties as the tick marks) for NFW and NFW-c DM profiles,and the four IEMs, as indicated.", "Also shown are the cross sections saturating the standard thermal relic density (grey dashed line)and the Fermi–LAT 95%95\\% C.L.", "bounds from dwarf spheroidal galaxies, assuming 100%100\\% annihilation into bb ¯b \\overline{b}.We translate the DM template flux for each IEM into the inclusive annihilation cross section, with the results shown in Fig.", "REF .", "Also shown for comparison is the $\\langle \\sigma v \\rangle $ predicting saturation the measured DM relic density for a standard cosmology [59].", "The results for the index-scaled models are comparable to those found in most of the earlier studies of the GeV excess [4], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].", "The intensity-scaled models however are consistent with larger DM masses and cross sections, as first discussed in [57], based on the spectra from [17]." ], [ "Results for Vector Interactions", "The analysis for the vector-type DM interactions proceeds very similarly to the analysis of the pseudo-scalar interactions described above.", "For each IEM and both NFW and NFW-c morphologies, the DM mass is scanned and the couplings to up-type quarks, down-type quarks, and charged leptons is fit.", "The results are presented in Figs.", "REF and REF , respectively, for each IEM with the NFW-c profile (the results for the NFW profile are qualitatively similar.)", "Similarly to pseudo-scalar interactions, lower DM masses are favored by the index-scaled IEMs, compared to the intensity-scaled.", "However, in general, lower DM masses are favored for the vector interaction models than for the pseudo-scalar ones for the same IEM.", "In addition, because the coupling to SM fermions is assumed to be flavor-universal for the vector interaction model, there is no sharp transition in behavior at the top quark mass.", "For the Pulsars, index-scaled IEM, there are two close-to-degenerate minima in the likelihood profile, with the lower mass dominated by annihilations into leptonsFor annihilations into leptons, secondary $\\gamma $ -ray emission via IC processes is neglected.", "Note that for DM masses $\\lesssim 10$  GeV, IC photons are mainly produced at energies $< 1$  GeV [60], [61]..", "The fitted values of $\\langle \\sigma v \\rangle $ and the DM mass for each of the IEMs and DM profiles are shown in Fig.", "REF ." ], [ "Comparison with Other Searches", "As seen in sections REF and  REF , DM interpretations of the GC excess cover a broad range of masses ($\\sim 10 - 200$ GeV) and $\\langle \\sigma v \\rangle $ , depending on the IEM, DM profile, and interaction type.", "One crucial avenue toward exploring a DM hypothesis for the excess is to compare the regions of parameter space best describing the excess with the results from other searches for DM.", "Null results of such searches can sharpen the target parameter space or even exclude candidate explanations, whereas positive results could strengthen a DM interpretation of the excess and better define the characteristics of candidate models." ], [ "Indirect Searches", "For masses in the range $10 - 200$  GeV, the strongest constraints from indirect detection are generally from Fermi–LAT observations of dwarf spheroidal galaxies [1].", "These limits appear to constrain the region relevant for explanations of the GC excess, but are derived from less theoretically motivated DM annihilation models where the DM annihilates into one species of SM fermion at a time.", "As such, they do not precisely apply to the models considered here, although similar conclusions are likely.", "The bound based on the assumption of $100\\%$ annihilation into $b \\overline{b}$ , corrected to account for Dirac (rather than Majorana) DM particles, is shown on Figures REF and REF for reference.", "The dwarf spheroidal bounds for annihilations into leptons are not displayed in these figures.", "Although they would in principle be more pertinent to constrain our low mass, vector interaction results, they are still not adequate as the final state channel we consider here is an equal weight mixture of $e^+e^-$ , $\\mu ^+\\mu ^-$ , $\\tau ^+\\tau ^-$ and therefore not directly comparable.", "The limitations in the IEMs, modeling uncertainties in the dwarf halos [62], [63], [64], [65], modifications to the particle physics model for DM [66], and large uncertainties in the J-factor for the GC [67], all widen the relative uncertainties when confronting the parameters describing the GC excess with the limits from observations of dwarf spheroidal galaxies.", "Because of this, care must be taken when contrasting these limits with a DM interpretation of the GC excess.", "The particle physics models under consideration also lead to annihilations producing anti-matter, such as positrons or anti-protons.", "Positrons in particular show excess production compared to naive expectations [68], [69], leading to limits which do not significantly constrain the parameters for the GC excess [70].", "Recently, Ref.", "[71] performed a detailed analysis of the anti-proton spectrum measured by AMS-02 [72], and also found an indication for an excess component roughly consistent with the parameter space describing a DM interpretation of the GC excess (see also [73] for a less optimistic view).", "The interpretation of CR anti-matter measurements is complicated by propagation, energy losses, and other modeling uncertainties related to particle fragmentation, as well as the spatial distribution of astrophysical sources.", "Consequently, the interpretation of these data in terms of DM is unclear." ], [ "Direct Searches", "Coupling to quarks implies coupling to hadrons, and thus is bounded from direct searches for DM scattering with heavy nuclei.", "Models with pseudo-scalar interactions map onto a scattering cross section which is both suppressed by the small velocities of DM in the Galactic halo and are also spin-dependent.", "As a result, the expectation is that the constraints from direct searches result in mild constraints.", "In contrast, vector interactions lead to velocity-unsuppressed spin-independent scattering and are strongly constrained by direct searches.", "For the vector models, which contribute to the spin-independent cross section $\\sigma _{\\rm SI}$ , we follow the usual convention mapping onto this quantity defined at zero relative velocity.", "For pseudo-scalar interactions, we compute the integrated cross section for DM scattering with a nucleon by integrating over the recoil energy of the nucleus and the velocity of the DM, which we assume follows a Maxwellian distribution, using techniques developed in [74], [75], [76], [77], (specifically using the code presented in Ref. [76]).", "This integrated cross section should be distinguished from usual spin-dependent cross section $\\sigma _{\\rm SD}$ , defined at zero velocity scattering, and is a more appropriate measure of scattering which is strongly velocity-dependent.", "Figure: ML points for the pseudo-scalar models, for each IEM and profile considered, as indicated, mapped into the plane of the DM mass and the integrated cross section,as described in the text.", "Also shown are current constraints from LUX (upper shaded region)and projections from XENONnT, LZ, and Darwin (dashed and dotted lines).", "The lower shaded region indicates the neutrino floor.Figure: ML points for the vector models, for each IEM and profile considered, as indicated, mapped into the plane of the DM mass and σ SI \\sigma _{\\rm SI},as described in the text.", "Also shown are current constraints from LUX (upper shaded region)and projections from XENON1T (dashed line).", "The lower shaded region indicates the neutrino floor.In Figs.", "REF and  REF , we show the ML points for the pseudo-scalar and vector models mapped into the WIMP-neutron spin-dependent integrated cross section, respectively, for each IEM and both NFW and NFW-c. For comparison, the limits from the LUX search for DM scattering with Xenon are presented [78], also mapped into $\\sigma _{\\rm SI}$ or the integrated cross section for spin-dependent scattering with neutrons.", "For the vector models, the limits from LUX easily exclude all of the ML points except for the point with dark matter masses around 10 GeV which annihilates predominantly into leptons for the Pulsars, index-scaled IEM with NFW-c profile, which has sufficiently small coupling to quarks that the scattering with nuclei is highly suppressed.", "For the pseudo-scalar models, the predictions for the ML points lie well below below the LUX bounds, with the lower mass points potentially probed long-term by Darwin [79], while the higher mass points are slightly above the neutrino floor [80] and out of the reach of these experiments.", "These results illustrate the importance of the IEM modeling and its influence on characterization of the putative signal, which can lead to drastic differences in the expectations from complementary searches." ], [ "Collider Searches", "Searches at the Large Hadron Collider (LHC) are more model-dependent, and can be classified based on the masses and couplings of the particles mediating the interaction.", "When such particles are heavy compared to the typical collider energies, they can be described by the same EFTs employed in this paper.", "The results of searches in this regime are typically not competitive with direct searches except at masses far below those of interest to describe the GC excess [81], [82].", "For lighter mediating particles, the limits depend sensitively on the specific couplings to the DM as well as to the SM fermions.", "In particular, for values of the cross sections similar to what has been found in past characterizations of the GeV excess, cases where a pseudo-scalar mediator's coupling to DM is significantly weaker than the coupling to quarks are mildly constrained by LHC data, and the opposite limit is essentially unconstrained [83].", "Given the wide range of parameter space (which is even larger for the specialized IEM analysis considered here), it seems possible that the LHC could eventually hope to observe an excess consistent with a pseudo-scalar mediator interpretation if parameters are favorable.", "Similar remarks apply to the vector mediator models, although all but the Pulsars, index-scaled IEM with NFW-c profile are already excluded by direct detection experiments.", "This latter model is consistent with vanishing coupling to quarks, and thus is unlikely to be excluded by searches at the LHC." ], [ "Summary", "The excess of $\\sim $  GeV $\\gamma $ -rays from the direction of the GC is an indication that there is something in the $\\gamma $ -ray sky beyond our current knowledge.", "Whether this source ultimately proves to originate from DM annihilation or from a more conventional astrophysical source still remains to be determined, and is likely to require further experimental input.", "As part of this process, we have examined key aspects of the putative signal using the specialized IEMs, developed by the Fermi–LAT Collaboration [17].", "Our goal in characterizing potential DM explanations is to explore the implications from complementary searches, which can rule out or favor a DM interpretation.", "Our results illustrate the impact of interstellar emission modeling on the extracted characteristics of the excess and highlight the need for improved modeling to capture a more realistic range of possibilities.", "As far as the gross characteristics of the excess are concerned, we find an offset of $\\sim 0.5^\\circ $ of the excess centroid from Sgr A* for all four IEMs considered.", "We further find no significant evidence that the tail of the excess has a different spatial morphology than the few GeV bump, with both high energy and low energy components favoring an NFW morphology compared to the other morphologies we have considered.", "We also consider flexible and realistic particle physics models for DM interacting with up-type quarks, down-type quarks, and charged leptons, for two separate interaction types (pseudo-scalar and vector) leading to $s$ -wave annihilation.", "These theories are described by EFTs, valid when the momentum transfer is small compared to the masses of the particles mediating the interactions – to describe annihilation, this implies the mediators are heavier than the DM itself.", "We find that the choice of IEM has a large impact on the preferred DM mass, annihilation cross section, and primary annihilation channel.", "In particular, we identify regions with higher masses and annihilation predominantly into top quarks.", "Comparing the ML points in parameter space with direct and collider searches, we find that all of the vector models aside from one at DM mass $\\sim 10$  GeV and annihilating into leptons are ruled out by null results from the LUX experiment.", "The pseudo-scalar models predict spin-dependent and velocity-dependent scattering with nuclei at a rate far below the current sensitivity, but in some cases within the grasp of future planned experiments.", "It would be interesting, but beyond the scope of this work, to extend our analysis beyond the EFT limit to the case of models where the DM can annihilate directly into the mediator particles themselves.", "The GeV excess is a compelling hint that there is more to learn about the Galaxy.", "It is likely to take a combined effort of observation and interpretation to unravel its nature." ], [ "Acknowledgements", "The authors are pleased to acknowledge conversations with D. Finkbeiner, D. Hooper, M. Kaplinghat, T. Slatyer, and C. Weniger.", "The work of CK and SM is supported in part by Department of Energy grant DE-SC0014431.", "The work of TMPT and PT is supported in part by National Science Foundation grants PHY-1316792 and PHY-1620638.", "GALPROP development is partially funded via NASA grants NNX09AC15G, NNX10AE78G, and NNX13AC47G." ] ]
1612.05687
[ [ "Production of $e^{+}e^{-}$ from U+U collisions at $\\sqrt{s_{NN}}$ = 193\n GeV and Au+Au collisions at $\\sqrt{s_{NN}}$ = 19.6, 27, 39, 62.4, and 200 GeV\n as measured by STAR" ], [ "Abstract We present STAR's measurement of the $e^{+}e^{-}$ continuum as a function of centrality, invariant mass, and transverse momentum for U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV.", "Also reported are the acceptance-corrected $e^{+}e^{-}$ invariant mass spectra for minimum-bias Au+Au collisions at $\\sqrt{s_{NN}}$ = 27, 39, and 62.4 GeV and U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV.", "The connection between the integrated $e^{+}e^{-}$ excess yields normalized by charge particle multiplicity ($dN_{ch}/dy$) at mid-rapidity and the lifetime of the fireball is discussed." ], [ "Introduction", "Relativistic heavy-ion collisions produced by the Relativistic Heavy Ion Collider (RHIC) are capable of generating a hot, dense, strongly interacting medium.", "In order to study this medium, electromagnetic probes, such as $e^{+}e^{-}$ pairs, lend themselves as a natural choice.", "The $e^{+}e^{-}$ are generated at all stages of the collision, interact electromagnetically, and thus, are able to traverse the strongly interacting medium with minimal effects on their final state while preserving information imprinted on them by their parent(s).", "Pairs with an invariant mass (M$_{ee}$ ) less than $\\sim 1.2$ GeV/c$^{2}$ are of particular interest because it contains production from the $\\rho $ meson.", "The $\\rho $ has been suggested to have its spectral function modified by the medium [1] and is in agreement with measurements reported in [2], [3], [5], [4], [6], [7].", "The in-medium modification of the $\\rho $ spectral function is considered a possible link to chiral symmetry restoration [8].", "Moreover, it has been suggested that the integrated yield of $e^{+}e^{-}$ production within the low mass region (LMR, M$_{ee}$ $\\lesssim $ 1.2 GeV/c$^{2}$ ) can be related to the lifetime of the fireball [9].", "The versatility of RHIC enables a systematic study of the $e^{+}e^{-}$ continuum.", "The Beam Energy Scan Program has enabled the Solenoidal Tracker At RHIC (STAR) to collect enough statistics to study $e^{+}e^{-}$ production from Au+Au collisions at $\\sqrt{s_{NN}}$ = 19.6, 27, 39, and 62.4 GeV while the total baryon density remained approximately constant, which the $\\rho $ spectral function depends on.", "Here, the $e^{+}e^{-}$ continuum is studied as a function of $\\sqrt{s_{NN}}$ and presented in the context of the fireball lifetime.", "Additionally, RHIC's versatility is evident from the different colliding species that it can accelerate.", "By switching from a Au+Au collision system at $\\sqrt{s_{NN}}$ = 200 GeV to a U+U collision system at $\\sqrt{s_{NN}}$ = 193 GeV, the collision energy density is expected in most central collisions to be up to 20% higher than Au+Au collisions [10] while the collision energy remains within a couple percent.", "If the energy density is increased, one may expect a longer fireball lifetime, and in turn, a larger $e^{+}e^{-}$ yield in the LMR.", "Here, the $e^{+}e^{-}$ continuum from U+U collisions is studied and compared to the continuum from Au+Au collisions.", "This paper presents a systematic study of the acceptance-corrected $e^{+}e^{-}$ production measured by STAR as a function of $\\sqrt{s_{NN}}$ = 27, 39, and 62.4 GeV.", "Furthermore, STAR measurements of the $e^{+}e^{-}$ continuum produced by U+U collisions are presented.", "Figure: (Color online) (Left) β -1 ~\\beta ^{-1} as a function of momentum p for all tracks originating from the collision vertex.", "The red lines depict the selection criteria used to reject slower hadrons while keeping electrons.", "(Middle) The TPC's electron identification as a function of momentum after TOF velocity selection.", "The red lines depict the selection criteria used to select a pure electron sample.", "The bulb beneath the selected area are pions.", "(Right) The electron purity as a function of momentum where the green (online) bands represent systematic uncertainty and the solid gray region represents the region where the absolute yield of electrons and overlapping hadrons have been extrapolated." ], [ "Data Sample and Analysis", "STAR has collected 70M, 130M, and 67M events from the top-80% most central Au+Au collisions at $\\sqrt{s_{NN}}$ = 27, 39, and 62.4 GeV, respectively, and 270M events from the top-80% most central U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV, where the top-80% represents STAR's minimum bias selection.", "The Time Projection Chamber (TPC) [11] and Time of Flight (TOF) [12] detectors are used to identify electrons and positrons within the STAR acceptance ($p_{T}^{e}$ $>$ 0.2 GeV/c, $|\\eta ^{e}|$ $<$ 1, and $|y_{ee}|$ $<$ 1).", "The TPC provides the particle identification and tracking via ionization energy loss (dE/dx) while the TOF is used to reject the slower hadrons then enhancing the TPC's particle identification and purity.", "This is depicted in Fig.", "REF , where the left panel demonstrates the TOF velocity selection criteria (between the red lines), the middle panel shows the electron sample selection based on the TPC electron identification (n$\\sigma _{e}$ ) after the TOF velocity rejection of slower hadrons.", "The right panel illustrates the purity of the selected electrons as a function of momentum, the average purity for each data sample is at the level of 95% or greater.", "Selected electrons (and positrons) are combined to form the $e^{+}e^{-}$ foreground.", "This contains background from the combinatorial background and correlated pairs (e.g.", "jets and double Dalitz decays).", "To estimate and remove the background, a geometric mean with a charge-acceptance correction is subtracted from the uncorrected $e^{+}e^{-}$ distribution as a function of the pair invariant mass and pair transverse momentum ($p_{T}^{ee}$ ).", "After subtraction, the continuum is then corrected for efficiency and acceptance losses.", "Details on the methods used may be found here [13].", "Shown in the top panel of Fig.", "REF is the $e^{+}e^{-}$ continuum as a function of invariant mass for minimum bias U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV.", "Known hadronic contributions to the invariant mass spectrum are then modeled and compared to the data.", "Contributions are modeled from $\\pi ^{0}$ , $\\eta $ , $\\eta $ ', $\\omega $ , $\\phi $ , J/$\\psi $ , $c\\bar{c}$ , Drell-Yan, and $b\\bar{b}$ , where Drell-Yan and $b\\bar{b}$ are only modeled for U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV and Au+Au collisions at $\\sqrt{s_{NN}}$ = 200 GeV.", "The top panel in Fig.", "REF also shows the known hadronic contributions as perforated lines and the cocktail sum of their contributions shown as the solid line.", "Figure: (Color online) (Top) The corrected e + e - e^{+}e^{-} invariant mass spectrum from U+U collisions (black markers).", "The perforated lines represent known hadronic contributions and the cocktail (solid line) is a summation of these contributions.", "(Bottom) The data over cocktail ratio as a function of invariant mass.", "The red line is the ratio of Rapp et al.", ", calculations plus the cocktail to the cocktail.", "The systematic uncertainties are shown as the shaded areas and the statistical uncertainties as the error bars.", "The yellow region along the dotted line at data/cocktail = 1 represents the cocktail uncertainty.Figures REF and REF are the $e^{+}e^{-}$ invariant mass spectrum for different $p_{T}^{ee}$ and centrality ranges, respectively.", "For reference, the all-inclusive distribution from Fig.", "REF is shown as the bottom distribution in each figure.", "Figure: (Color online) The e + e - e^{+}e^{-} invariant mass spectrum (markers) for different p T ee p_{T}^{ee} ranges.", "The systematic uncertainties are represented by the shaded regions and the statistical uncertainties are represented by the error bars.", "The hadronic cocktail is shown for each p T ee p_{T}^{ee} range (solid line).Figure: (Color online) The e + e - e^{+}e^{-} invariant mass spectrum (markers) for different centrality ranges.", "The systematic uncertainties are represented by the shaded regions and the statistical uncertainties are represented by the error bars.", "The hadronic cocktail is shown for each centrality range (solid line)." ], [ "Results and Discussion", "Figures REF , REF , and REF exhibit a difference between the invariant mass spectrum and the hadronic cocktail without $\\rho $ contributions, or excess.", "A model by Rapp et al.", "that incorporates broadening of the $\\rho $ spectral function (HG_med) and QGP thermal radiation is in agreement with this observation.", "This is demonstrated by overlaying the ratio of the model calculations plus the cocktail to the cocktail and comparing to the data over cocktail ratio in the bottom panel of Fig.", "REF .", "This has previously been shown for Au+Au collisions at $\\sqrt{s_{NN}}$ = 19.6, 27, 39, 62.4, and 200 GeV [5], [6], [14].", "Taking a step further, the excess is corrected for STAR's kinematic acceptance.", "The acceptance-corrected excess at mid-rapidity has been normalized by the charge particle multiplicity at mid-rapidity ($dN_{ch}/dy$ ) to cancel volume effects and is presented in Fig.", "REF for U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV and Au+Au collisions at $\\sqrt{s_{NN}}$ = 27, 39, 62.4, and 200 GeV.", "In the same figure, we compare our results with model calculations from Rapp et al.", "that incorporate thermal radiation from the QGP and a broadened $\\rho $ spectral function.", "Figure: (Color online) The acceptance-corrected e + e - e^{+}e^{-} invariant mass excess yield normalized by dN ch /dydN_{ch}/dy.", "Data from minimum bias Au+Au collisions (black markers) are shown in all four panels at s NN \\sqrt{s_{NN}} = 27, 39, 62.4, and 200 GeV, starting from the top and reading from left to right, respectively.", "Data from minimum bias U+U collisions (red markers) at s NN \\sqrt{s_{NN}} = 193 GeV is shown in the lower right panel.", "Systematic uncertainties are the shaded areas and statistical uncertainties are the error bars.", "Calculations from Rapp et al.", ", are shown in each panel for contributions from the hadronic medium (dashed blue curve), QGP (dashed pink curve), and their sum (solid red curve).", "In the lower right panel, the Rapp et al.", "calculations are for U+U collisions.To study the possible connection between the lifetime and excess yield, the $e^{+}e^{-}$ excess yield has been integrated from 0.4 to 0.75 GeV/c$^{2}$ , normalized by $dN_{ch}/dy$ , and plotted in Fig.", "REF as a function of $dN_{ch}/dy$ .", "On the same figure, lifetime calculations from Rapp et al.", "[9], [15] are plotted for each corresponding yield measurement as a bar while the trend for Au+Au at $\\sqrt{s_{NN}}$ = 200 GeV centrality calculations is plotted as a dashed line.", "There is an increase in normalized yields at higher collision energies with respect to the lower energies, and at $\\sqrt{s_{NN}}$ = 200 GeV there is an increase in normalized yields in more central collisions with respect to peripheral collisions.", "The expected lifetime in model calculations also has an increasing trend from peripheral to central collisions [6].", "Figure: (Color online) (Left y-axis) The integrated acceptance-corrected e + e - e^{+}e^{-} excess yield normalized by dN ch /dydN_{ch}/dy as a function of dN ch /dydN_{ch}/dy, where data markers represent each point.", "Statistical uncertainties are represented by the error bars and the systematic uncertainties are represented by the shaded regions.", "(Right y-axis) The lifetime of the fireball calculations by Rapp et al.", ", as a function of dN ch /dydN_{ch}/dy, where the calculations are represented by the solid bars that have been offset in the +x-direction and the red dashed curve are lifetime calculations for Au+Au at s NN \\sqrt{s_{NN}} = 200 GeV." ], [ "Summary and Outlook", "We have presented STAR measurements of the $e^{+}e^{-}$ invariant mass for U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV along with the acceptance-corrected excess $e^{+}e^{-}$ yields for minimum bias U+U collisions at $\\sqrt{s_{NN}}$ = 193 GeV and minimum bias Au+Au collisions at $\\sqrt{s_{NN}}$ = 27, 39, 62.4, and 200 GeV.", "Comparisons between these measurements and model calculations, which include broadening of the $\\rho $ spectral function and QGP thermal radiation, have been shown to be in agreement.", "Also, we reported the normalized integrated excess yields as a function of $dN_{ch}/dy$ for various collision systems.", "These measurements are consistent with theoretical calculations that indicate a longer fireball lifetime for collisions that are central and have a higher $\\sqrt{s_{NN}}$ .", "Measurements made during the Beam Energy Scan Program were made at approximately the same total baryon density.", "At $\\sqrt{s_{NN}}$ $<$ 20 GeV, the total baryon density rises as $\\sqrt{s_{NN}}$ decreases.", "Models such as [1] suggest that the $\\rho $ spectral function is dependent upon the total baryon density; hence, if the total baryon density increases, the $\\rho $ yield is expected to increase too.", "To test this relation and distinguish between models, STAR will take advantage of RHIC's second Beam Energy Scan Program [16] where Au+Au will be collided at $\\sqrt{s_{NN}}$ = 7.7, 9.1, 11.5, 14.5, and 19.6 GeV.", "It is expected that the number of events recorded will provide statistical uncertainties similar to the statistical uncertainties quoted in STAR's Au+Au measurement at $\\sqrt{s_{NN}}$ = 200 GeV [5].", "In addition to the increased statistics, the inner sector of the TPC will be replaced to provide additional tracking points [17] and a complimentary forward Time of Flight (eTOF) detector will be installed [18].", "The TPC upgrade will lead to additional reduction in the statistical and systematic uncertainties, while the eTOF will lead to an extension in the rapidity reach and the ability to measure the $e^{+}e^{-}$ dependence on rapidity." ] ]
1612.05484
[ [ "Phase rotation symmetry and the topology of oriented scattering networks" ], [ "Abstract We investigate the topological properties of dynamical states evolving on periodic oriented graphs.", "This evolution, that encodes the scattering processes occurring at the nodes of the graph, is described by a single-step global operator, in the spirit of the Ho-Chalker model.", "When the successive scattering events follow a cyclic sequence, the corresponding scattering network can be equivalently described by a discrete time-periodic unitary evolution, in line with Floquet systems.", "Such systems may present anomalous topological phases where all the first Chern numbers are vanishing, but where protected edge states appear in a finite geometry.", "To investigate the origin of such anomalous phases, we introduce the phase rotation symmetry, a generalization of usual symmetries which only occurs in unitary systems (as opposed to Hamiltonian systems).", "Equipped with this new tool, we explore a possible explanation of the pervasiveness of anomalous phases in scattering network models, and we define bulk topological invariants suited to both equivalent descriptions of the network model, which fully capture the topology of the system.", "We finally show that the two invariants coincide, again through a phase rotation symmetry arising from the particular structure of the network model." ], [ "Introduction", "Topological insulators are remarkable materials where the particular topology of the bulk states leads to protected degrees of freedom with exceptional properties at the boundary of the system.", "For example, such edge states may provide a unidirectional propagation of waves, and are robust against various perturbations.", "In this context, periodically driven (Floquet) dynamical systems have been shown to exhibit specific anomalous topological properties with no equivalent in equilibrium systems [1], [2].", "This anomalous behavior manifests itself by the existence of boundary states in finite geometry despite the vanishing of the topological index which usually accounts for all topological properties in equilibrium systems.", "More precisely, the first Chern number associated with the bands of the Bloch Hamiltonian that effectively describes the stroboscopic dynamics vanishes in this case.", "The existence of these anomalous boundary states can instead be associated with a topological property of the full bulk evolution operator $U(t)$ , which, unlike the effective Hamiltonian, accounts for the entire evolution at all times during one driving period [2].", "This behavior can be generalized to a more general class of time-dependent dynamical systems.", "For linear systems, the evolution operator is generated by the Hamiltonian $H(t)$ of the system through an equation of motion ${\\rm i}\\partial _t U(t) = H(t) U(t)$ with initial condition $U(0) = \\text{Id}$ , which is formally solved by the time-ordered exponential $U(t) = \\lim _{N \\rightarrow \\infty } {\\rm e}^{-{\\rm i}t/N \\, H(N t/N)} \\cdots {\\rm e}^{-{\\rm i}t/N \\, H(n t/N)} \\cdots {\\rm e}^{-{\\rm i}t/N \\, H(t/N)}.$ Namely, $U(t)$ results from an infinite product of infinitesimal free evolutions governed by instantaneous Hamiltonians $H(n t/N)$ .", "As the Hamiltonians at different times generically do not commute, the evolution operator $U(t)$ can be cumbersome to manipulate.", "However, it is often convenient to alternatively consider evolutions composed of a finite sequence of step operations described by unitary step operators $U_n$ , so that after $N$ operations the evolution operator reads $U = U_N U_{N-1}\\dots U_1 \\, .$ Such a stepwise dynamics suitably describes the effective discrete-time evolution of various experimental systems such as, in two dimensions, arrays of evanescently coupled optical waveguides with sufficiently sharp bending [3], [4] and atomic discrete-time quantum walks, where the operators $U_n$ may consist of coin or shift operations applied to a spin-$1/2$ quantum state trapped in an optical lattice [5].", "Periodically driven systems include both evolutions generated by a time-periodic Hamiltonian $H(t)=H(t+T)$ and stepwise evolutions where the sequence of operations is repeated periodically.", "In both cases, the Floquet operator of the evolution can be defined, respectively by $U_\\text{F}= U(T)$ and by $U_\\text{F}= U$ .", "Despite their lack of explicit time dependence, stepwise evolutions were predicted to host anomalous topological chiral edge states in two dimensions, showing that the sequence structure (REF ) is enough to engineer such topological phases [1], [2], [6], [7], [8], [9], [10].", "An important physical example was revealed by Liang, Pasek and Chong [7], [8] who described spatially periodic arrays of coupled photonic resonators in terms of unitary scattering matrices that locally encode the transmission and reflection coefficients of the optical signal between resonators, in order to go beyond the effective tight-binding description.", "Within this framework, the system can be seen as an oriented scattering network similar to that introduced by Chalker and Coddington to describe the Hall plateau transition [11], [12], which consists in links over which a directed current flows in one direction connecting nodes where incoming currents are scattered into outgoing currents, as represented in figure REF .", "Notably, the unidirectionality of the links plays a role similar to that of time as it forces the currents to cross the nodes in a given order that is fixed by the connectivity of the network.", "This behavior can originate from various physical mechanisms that explicitly break time-reversal symmetry, such as a perpendicular magnetic field like in the original Chalker-Coddington model [11], [12] or a flow of the propagation medium like in the array of acoustic circulators recently proposed by Khanikaev et al.", "[13] and Souslov et al. [14].", "When time-reversal symmetry is preserved, as it is in most photonic systems, it is fair to use similar one-way oriented networks to describe one of the two spin copies of the system, provided that certain spin-flip processes can be neglected [7], [15], [16], [17].", "Due to this particular resurgence of an effective time, a fruitful analogy between scattering networks and Floquet dynamics was envisioned [18], [19], [8], an important consequence of which is the discovery of anomalous chiral edge states in such systems, while there is remarkably no external periodic driving as it would be in a Floquet system.", "The efficiency of this approach motivated two recent microwave experiments that probed the existence of these anomalous topological edge states [15], [16].", "Figure: Example of an oriented network.", "The direction of propagation along the links is represented by an arrow.", "The nodes represent the unitary scattering events between incoming and outgoing amplitudes.Due to the unitarity of scattering events, the number of incoming links is equal to the number of outgoing links at each node.Despite the accumulation of theoretical and experimental results on such systems, several questions remain open.", "First, the entire network is described by a unitary scattering matrix, the Ho-Chalker evolution operator [12], which takes into account all the scattering events at the same time.", "In this picture, there is no Floquet dynamics, and the relation between both descriptions is not clear.", "A second issue is that even in the Floquet picture, a bulk topological characterization of network models is not available.", "The question of the characterization of the bulk topology of such systems is particularly crucial in the case of anomalous phases, where the first Chern numbers of the bands all vanish.", "Moreover, the way to engineer such phases remains an open question.", "Generically, bands of a two-dimensional gapped system where time-reversal symmetry is broken have a non-vanishing first Chern number.", "We therefore expect that an additional mechanism imposes their vanishing in certain conditions.", "To answer this set of questions, we introduce in section  a new symmetry specific to unitary systems, the phase rotation symmetry, and show how this property constrains the value of the first Chern numbers associated to the spectral projectors of a gapped unitary operator.", "In particular, a strong version of the phase rotation symmetry ensures the vanishing of first Chern numbers, a necessary condition to obtain anomalous topological phases.", "In oriented scattering networks, this phase rotation symmetry subtly enters at two different levels.", "First, it relates the evolution operator of certain networks to that of a Floquet-like system and allows for the definition of topological invariants.", "More precisely, a particular class of cyclic oriented networks is introduced in section , where the particular structure and connectivity of the scattering network constrains its evolution operator to possess a particular phase rotation symmetry, which we call a structure constraint.", "This observation leads to several important results as it enables us to understand the structure of the evolution operator spectrum of the network model.", "Due to this insight, we are able to directly define a bulk topological invariant characterizing the system.", "The structure constraint also enables us to explicit the relationship between (cyclic) oriented network models and Floquet stepwise dynamics, and to define another bulk topological invariant for such dynamics.", "Indeed, both topological invariants are related and equivalent, as we finally show in section .", "The second role of the phase rotation symmetry in scattering networks is to provide an interpretation of the vanishing of first Chern numbers which is found in specific networks [7], [8].", "At particular points of the phase diagram, another phase rotation symmetry, stronger than the structure constraint, may exist and enforce this vanishing.", "This allows us to propose a qualitative way to identify, in real space, whether a given oriented network may exhibit a vanishing first Chern number phase, which is developed in section .", "We consider systems described by a unitary evolution operator $U(t)$ .", "This evolution may be derived from the microscopic description of the system, or rather be an effective description of the relevant degrees of freedom.", "We focus on situations where it is sensible to concentrate on the evolution operator $U=U(T)$ after some finite time $T$ .", "Time-periodic dynamics provide the most common example of such a situation, as the evolution operator after one period $U(T)$ (here called the Floquet operator) describes the evolution of the system on long time scales.", "As we shall see in the next section, there are other cases where such a description is relevant ; this is in particular the case of oriented scattering networks, the study of which constitute the bulk of section .", "In a crystal, discrete space periodicity enables to block-diagonalize the evolution operator into a family of Bloch evolution operators $U(k)$ which are finite matrices, and are labeled by a quasi-momentum $k$ living on a $d$ -dimensional torus called the Brillouin torus (we will only consider the case $d=2$ here).", "The spectrum of the evolution operator $U$ is called its phase spectrum.", "Its eigenstates $\\mathinner {|{\\psi _n(k)}\\rangle }$ satisfy the eigenvalue equation $U(k) \\mathinner {|{\\psi _n(k)}\\rangle } = {\\rm e}^{-{\\rm i}\\varepsilon _n(k)} \\mathinner {|{\\psi _n(k)}\\rangle }$ where the eigenphases $\\varepsilon _n(k)$ , which constitute the phase spectrum, are confined on the unit circle in the complex plane.", "The minus sign in (REF ) is arbitrary; it is chosen here for the analogy between $U$ and the evolution operator.", "Generically, the phase spectrum displays phase bands separated from each other by phase gaps, as illustrated in figure REF .", "Each band corresponds to a family of orthogonal projectors $k \\mapsto P(k)$ , which describe the spectral projector on the corresponding arc in the unit circle.", "Figure: Phase spectrum.", "Illustration of a phase spectrum with four bands and four gaps." ], [ "The phase rotation symmetry ", "Unitary systems share the particularity to have a periodic spectrum.", "This allows us to consider a rotation of those spectra by an angle $\\zeta $ , corresponding to the transformation ${\\rm e}^{-{\\rm i}\\varepsilon } \\rightarrow {\\rm e}^{-{\\rm i}(\\varepsilon + \\zeta )}$ , as depicted in figure REF .", "Figure: The phase rotation.", "On the level of the spectrum, a phase rotation of angle ζ\\zeta maps eigenvalues e -iε {\\rm e}^{-{\\rm i}\\varepsilon } to e -i(ε+ζ) {\\rm e}^{-{\\rm i}(\\varepsilon +\\zeta )}.We consider situations where the phase spectrum is invariant under such a phase rotation.", "Although a symmetry of the phase spectrum can be accidental, this situation is not typical and instead we consider situations where the invariance of the phase spectrum under such a phase rotation is associated to a phase rotation symmetry of the form ${Z}U {Z}^{-1} = {\\rm e}^{{\\rm i}\\zeta } U$ where ${Z}$ is a unitary phase rotation operator acting on the states of the Hilbert space.", "The phase rotation symmetry (REF ) is the evidence of a redundancy in the description of the system.", "Indeed, if $\\mathinner {|{\\psi }\\rangle }$ is an eigenstate of $U$ with eigenvalue ${\\rm e}^{-{\\rm i}\\varepsilon }$ , then ${Z}\\mathinner {|{\\psi }\\rangle }$ is also an eigenstate of $U$ , with the eigenvalue ${\\rm e}^{-{\\rm i}(\\varepsilon +\\zeta )}$ ; more generally, ${Z}^m \\mathinner {|{\\psi }\\rangle }$ (with $m$ an integer) is an eigenstate of $U$ with eigenvalue ${\\rm e}^{-{\\rm i}(\\varepsilon +m\\zeta )}$ .", "Crucially, such a symmetry has no equivalent in Hamiltonian systems, as it would correspond to an unphysical energy translation $E \\rightarrow E + \\Delta E$ .", "In contrast, it can arise in unitary systems as the phase spectrum lies on a circle.", "When $\\zeta /2\\pi $ is irrational, the irrational rotation of the Floquet spectrum ensures that it is fully gapless.", "On the other hand, when $\\zeta /2\\pi = m/M$ is a rational, where $m/M$ is an irreducible fraction, a phase is mapped to itself applying the phase rotation $M$ times.", "Phases being defined modulo $2\\pi $ it is sufficient to consider $0 \\le \\zeta \\le 2\\pi $ , so we can set $m=1$ without loss of generality.", "As we are interested in gapped unitary operators, we will focus on cases where $\\zeta = 2\\pi /M$ where $M$ is an integer.", "The phase rotation symmetry (REF ) then reads ${Z}U {Z}^{-1} = {\\rm e}^{{\\rm i}2\\pi /M} U \\ .$ In practice, it is more convenient to use the Bloch version of this symmetry.", "Assuming that the operator ${Z}$ is local in space (i.e.", "it does not couple different unit cells), (REF ) straightforwardly translates as ${Z}U(k) {Z}^{-1} = {\\rm e}^{{\\rm i}2\\pi /M} U(k)$ where $U(k)$ is the Fourier transform of $U$ .", "As the variable $k$ is not affected by the phase rotation, we will omit it when the meaning is clear.", "Let us assume that $U$ has a gap around ${\\rm e}^{-{\\rm i}\\eta }$ .", "Then, due to the phase rotation symmetry (REF ), there is also a gap around ${\\rm e}^{-{\\rm i}(\\eta + 2\\pi /M)}$ .", "A fundamental domain $F$ for the phase rotation symmetry is then defined by the interval between these two values, so that it represents the shorter arc that links ${\\rm e}^{-{\\rm i}\\eta }$ and ${\\rm e}^{-{\\rm i}(\\eta + 2\\pi /M)}$ on the unit circle (see figure REF ).", "The fundamental domain $F$ plays a role similar to that of a unit cell: starting from the part of the spectrum over the arc $F$ , the whole spectrum is recovered by $M$ successive applications of the phase rotation of an angle $2\\pi /M$ (for eigenvalues) and of the unitary operator ${Z}$ (for eigenvectors) as illustrated in figure REF .", "Phase rotation symmetry allows one to reduce the description of the system by removing its redundancy, essentially by keeping only the eigenstates in one fundamental domain.", "As an example, this reduction procedure will be carried out explicitly in the case where ${Z}^M = \\text{Id}$ during the study of oriented scattering networks in section , and it will allow us to account for the topological properties of such systems.", "Figure: Examples of phase rotation invariant spectra.", "The spectra are invariant under a rotation of 2π/M2\\pi /M, with (a) M=2M=2 and (b,c) M=3M=3.", "In all cases, a fundamental domain FF for the symmetry can be chosen.", "For our purposes, the most convenient choice is an interval of length 2π/M2\\pi /M with both ends lying in a spectral gap.", "In each case, a possible choice of fundamental domain is represented in purple.Notice that (REF ) implies that ${Z}U^M {Z}^{-1} = U^M,$ which has the usual form of an actual symmetry (with an equivalent in Hamiltonian systems) for $U^M$ .", "Equation (REF ) means that the system recovers a symmetry represented by the operator ${Z}$ after $M$ successive identical evolutions.", "Besides, the $M^{\\text{th}}$ power of the phase rotation operator is also a symmetry of $U$ , as ${Z}^{M} U {Z}^{-M} = U.$ In general, this symmetry can be arbitrary.", "When ${Z}^M$ is scalar and $U$ is gapped, the phase rotation operator assumes the particular form ${Z}\\simeq \\text{diag}(1,{\\rm e}^{{\\rm i}2\\pi /M}, {\\rm e}^{{\\rm i}2\\pi \\times 2/M}, \\dots , {\\rm e}^{{\\rm i}2 \\pi (M-1)/M}) \\otimes \\text{Id}= {Z}_0$ in an adequate basis, which emphasizes its cyclic behavior (see appendix )." ], [ "Topological states and the phase rotation symmetry", "As we have seen, phase rotation symmetry enables to reduce the degrees of freedom in the description of the system.", "Another important consequence of this symmetry is to impose particular constraints on the topological properties of the system.", "As we shall see, a crucial consequence of the phase rotation symmetry is that the spectral projector over one fundamental domain has a vanishing first Chern number.", "For concreteness, we focus on two dimensional crystals in the following.", "Each band of the evolution operator $U$ carries a first Chern number, which is computed from the projector family $k \\mapsto P(k)$ as $C_1(P) = \\frac{{\\rm i}}{2\\pi } \\int \\operatorname{tr}P {\\rm d}P \\wedge {\\rm d}P \\, .$ Let us recall two important properties of the first Chern number which will be useful later.", "First, it is invariant under conjugation by a constant unitary operator $\\mathcal {U}$ , $C_1(\\mathcal {U} P \\mathcal {U}^{-1}) = C_1(P).$ Moreover, it is additive: if $P$ and $Q$ are mutually orthogonal projector families (so $P Q = 0 = Q P$ ), then $C_1(P+Q) = C_1(P) + C_1(Q).$ A nonvanishing first Chern number signals a nontrivial bulk topology of the system, which manifests itself in the appearance of robust chiral edge states at the boundary of a finite sample.", "When $U$ corresponds to a time-independent Hamiltonian evolution, the first Chern numbers fully characterize the bulk topological properties of the system (at least in the Altland-Zirnbauer symmetry class A).", "In general, however, this is not the case: there are the so-called anomalous topological phases which display a nontrivial topology despite having vanishing first Chern numbers [1], [2].", "Such topological properties are instead captured by taking into account the full time-dependent evolution in the bulk [2], and not only the bulk evolution operator after a finite amount of time." ], [ "Consequences of the phase rotation symmetry", "Let us denote by $\\Pi $ the spectral projector on states with eigenvalues ${\\rm e}^{-{\\rm i}\\varepsilon } \\in F$ , i.e.", "on a fundamental domain.", "The spectral projector $\\Pi _m$ on the $m$ -th rotated fundamental domain ${\\rm e}^{-{\\rm i}2\\pi m /M} F$ is then obtained by the action of ${Z}$ as $\\Pi _m={Z}^{m} \\Pi {Z}^{-m}$ .", "Due to the invariance of the first Chern number under conjugation by a constant unitary operator (REF ), all the rotated fundamental domains have the same first Chern number $C_1(\\Pi _m) = C_1({Z}^m \\Pi {Z}^{-m}) = C_1(\\Pi ).$ Second, as these projectors sum to identity $\\sum _{m=0}^{M-1} {Z}^{m} \\Pi {Z}^{-m} = \\text{Id}$ and due to the additivity of the first Chern number (REF ), we infer that $C_1(\\Pi ) = 0.$ As a consequence, the first Chern number of the spectral projector on any rotated fundamental domain vanishes.", "This is one of the main results of this paper.", "In general, the projector $\\Pi $ on a fundamental domain $F$ of the phase rotation symmetry does not correspond to a single band, as there may be phase gaps inside of $F$ (see figure REF ).", "In the particular situation where $\\Pi $ does correspond to a single bandA single band does not necessarily correspond to a single state.", "The projector $\\Pi $ may have a rank higher than one, provided that the corresponding eigenstates of $U$ are degenerate (at least at some point of the Brillouin torus)., we say that the evolution operator is endowed with a strong phase rotation symmetry.", "It follows from the previous discussion that in this situation, the first Chern numbers of each band in the spectrum of $U(T)$ vanish.", "As a consequence, the corresponding phase is either topologically trivial or anomalous.", "This observation is particularly interesting as it provides an explanation to the prevalence of anomalous topological states in certain contexts.", "When time-reversal symmetry is broken, we typically expect the appearance of nonvanishing first Chern numbers, at least when the corresponding phase does not include a time-reversal invariant point.", "However, there are systems where only anomalous phases appear (a concrete example is discussed in section REF ): this surprising behavior is explained by the existence of a phase rotation symmetry (at least at particular points of the phase diagram) which prevents nonvanishing first Chern numbers from appearing, despite the breaking of time-reversal symmetry.", "The propagation of waves in a time-reversal breaking metamaterial can be described by an oriented scattering network composed of unitary scattering matrices (the nodes) connected to each other by oriented links.", "The dynamics of waves in the network model are then described by a unitary evolution operator which contains all the vertex scattering matrices as well as the connectivity of the network.", "Oriented networks models were originally introduced by Chalker and Coddington to describe the Hall plateau transition [11], [12].", "In a semi-classical picture, electronic wave packets in a disordered two-dimensional electron gas under strong magnetic field follow the equipotentials of the smooth disorder potential, in a direction fixed by the magnetic field.", "The quantum Hall transition essentially arises when the equipotentials of the disorder percolate; however, near the transition, the relevant equipotentials approach the saddle points of the disorder potential and become closer and closer.", "Hence, wave packets can tunnel from an equipotential to another giving rise to quantum percolation [11] (see also [20] for a pedagogical introduction).", "This process is described by scattering matrices, one per saddle point, within the Chalker–Coddington model [11] that distorts the equipotentials into a periodic square lattice of such scattering matrices connected by incoming and outgoing directed links, the so-called L-lattice.", "Remarkably, this oriented network model captures most of the essential features of the Hall plateau transition.", "In the original model [11], random phases are added on each link to take into account the Aharanov–Bohm phase accumulated on the closed disorder equipotentials of various sizes.", "A fully space-periodic oriented network, without such random phases, was introduced by Ho and Chalker [12], who showed that a Dirac equation emerged from an expansion of a discrete evolution operator of the scattering network model.", "More recently, Liang, Pasek and Chong [7], [8] introduced a similar formalism to investigate the properties of an array of coupled photonic resonators beyond tight-binding descriptions.", "In such a system, the coupling between resonators is described by unitary scattering matrices that encode the transmission and reflection coefficients of the optical signal, rather than by an effective tight-binding Hamiltonian.", "The same formalism was also applied to sound waves in arrays of acoustic circulators by Khanikaev et al. [13].", "In both situations, the light or sound waves in the system are described by a huge scattering matrix, which can be understood as the evolution operator of the system.", "Notably, robust chiral edge states appear in a finite system, precisely in the phase gap(s) of the bulk scattering matrix.", "This is not a surprise in light of the connection with the quantum Hall effect.", "What is more surprising is that Liang, Pasek and Chong unveiled that photonic arrays support anomalous topological states similar to that described by Rudner et al.", "in periodically driven systems [2], despite the lack of explicit time dependence of the system.", "The existence of such anomalous topological states appears to be a fundamental property of unitary systems, as it crucially depends on the periodicity of the phase spectrum; such a behavior may in principle emerge whenever a unitary description of the system is adopted [9].", "In contrast, they are not captured in an effective tight-binding description [7], [17].", "As we have seen in Section REF , the pervasiveness of anomalous phases can be attributed to the existence of particular constraints, like a phase rotation symmetry.", "This is indeed the case in oriented scattering networks, where anomalous phases can be tracked down to the existence of particular symmetric points of the phase diagram where a phase rotation symmetry is present.", "As we shall see, such phase rotation symmetries can further be interpreted in terms of classical loop configurations of the oriented network.", "This interpretation is particularly powerful as it allows one to design anomalous phases in a straightforward way.", "In a potentially topological anomalous system, first Chern numbers are not sufficient to distinguish the possible topological phases (as they are always zero), and more precise bulk invariants are required.", "Crucially, the unidirectionality of the links plays a similar role to that of time as it forces the wave packets to visit the vertices in a given order which is determined by the connectivity of the network.", "In the following, we define a class of scattering networks, cyclic oriented networks, where it is possible to map the network model to a (stepwise) time-dependent system to study its properties.", "This mapping is allowed by the existence of a structure constraint which encodes the particular connectivity of the cyclic oriented network.", "On the level of the evolution operator describing the entire scattering network, this constraint manifests itself as a phase rotation symmetry, which allows for the definition of bulk topological invariants that fully characterize the network model." ], [ "Oriented scattering network models", "In general, oriented scattering network models consist of a directed graph, composed of a set of vertices (or nodes) representing scattering matrices, which are connected to each other by directed edges (or links) over which flows a directed current [20].", "At each vertex $v$ , the number $b_v$ of incoming links is equal to the number of outgoing links to guarantee the unitarity of scattering events, which are described by a scattering matrix $S_v \\in U(b_v)$ , which relates the incoming amplitudes $c^{\\text{in}}_e$ on each incoming edge $e$ to the outgoing amplitudes $c^{\\text{out}}_f$ on each outgoing edge $f$ by $c^{\\text{out}}_f = (S_v)_{f e} \\, c^{\\text{in}}_e.$ Here, we will only consider spatially periodic graphs.", "There may be several scattering matrices in a unit cell, but for simplicity we will further assume that all scattering matrices have the same size $b$ , i.e.", "that each vertex is connected to the same number of links.", "The most simple nontrivial situations is $b=2$ , where matrices are $U(2)$ rotations, and it is usually possible to reduce any network model to this situation [20], [21].", "While network models can be used in any space dimension, we shall focus on two-dimensional systems.", "Waves in such a spatially periodic network are described by a unitary Bloch scattering matrix.", "In the bulk, Bloch reduction gives a matrix $S(k_x,k_y)$ from which one can hope to extract topological invariants.", "In a finite cylinder geometry, a bigger matrix $S_{\\text{cylinder}}(k_y)$ (whose size depends on the height of the cylinder) describes both the bulk and the edge states of the finite system.", "In both cases, we obtain a periodic phase spectrum : as usually in topological systems, the bulk phase gaps host the chiral anomalous edge states that appear in a finite geometry." ], [ "An archetypal example: the L-lattice", "One of the simplest examples of oriented scattering networks is the L-lattice, which was introduced by Chalker and Coddington [11].", "We illustrate the main focal points of our analysis on this example, namely (i) the definition of bulk topological invariants that fully characterize the network model and (ii) the existence of special points of the phase diagram where a strong version of the phase rotation symmetry ensures the vanishing of the first Chern numbers, allowing only for anomalous topological phases.", "Figure: L-lattice.", "(a) The L-lattice as a square Bravais lattice with basis vectors e x e_x and e y e_y, with its four inequivalent links and two inequivalent nodes.", "A unit cell is enhanced in red and detailed in (b).The L-lattice is an oriented network model on a square Bravais lattice with two inequivalent vertices and four inequivalent oriented links per unit cell (which somehow ressembles two L letters connected together).", "More precisely, the unit cell is composed of two vertices $U_1$ and $U_2$ and of four inequivalent oriented links $(a_1,b_1,a_2,b_2)$ connecting the vertices, as represented in figure REF .", "The unitary matrices $U_j \\in U(2)$ encode how amplitudes on their two incoming links are scattered into their two outgoing links, as $\\begin{pmatrix}a_2(x,y,t+T) \\\\b_2(x,y,t+T)\\end{pmatrix}=U_1 \\,\\begin{pmatrix}a_1(x,y,t) \\\\b_1(x,y,t)\\end{pmatrix}$ and $\\begin{pmatrix}a_1(x,y+1,t+T) \\\\b_1(x-1,y,t+T)\\end{pmatrix}=U_2 \\,\\begin{pmatrix}a_2(x,y,t) \\\\b_2(x-1,y+1,t)\\end{pmatrix}.$ For simplicity, we choose the parametrization $U_j= \\begin{pmatrix}\\cos {\\theta _j}& \\sin {\\theta _j} \\\\-\\sin {\\theta _j} & \\cos {\\theta _j}\\end{pmatrix}$ of the vertex scattering matrices, where the parameters $\\theta _j$ control the transmission and reflection at each vertex.", "Complex phases can be introduced but will not change the properties we discuss here, namely the existence of two distinct topological phases both with vanishing first Chern numbers [7].", "Moreover, for convenience and to compare with [7], we focus on the situation where both angles are controlled by a single parameter $\\theta = \\theta _2 = \\pi /2 - \\theta _1$ .", "A state $\\mathinner {|{\\psi }\\rangle }$ of the system is given by a set of amplitudes $\\lbrace a_1(x,y)$ , $b_1(x,y)$ , $a_2(x,y)$ , $b_2(x,y)\\rbrace $ for all positions $(x,y)$ in the square Bravais lattice.", "Following Ho and Chalker [12], we consider the discrete evolution operator $\\mathcal {S}$ that describes the evolution of a state $\\mathinner {|{\\psi }\\rangle }$ after its amplitude on each link has been scattered at the nodes of the network.", "In other words, this operator effectively describes the scattering processes at all the nodes simultaneously.", "When focusing on the stationary bulk states, we can assume translation invariance and Fourier transform both the stationary states and the evolution operator into their block-diagonal Bloch version.", "The Bloch version of the Ho-Chalker evolution operator reads $\\mathcal {S}(k) =\\begin{pmatrix}0 & U_2(k) \\\\U_1(k) & 0\\end{pmatrix}$ in the Bloch basis $(a_1(k),b_1(k),a_2(k),b_2(k))$ , where $k$ is in the two-dimensional Brillouin zone.", "For the choice of unit cell shown in figure REF (b), the two unitary blocks are given by $U_1(k) = \\begin{pmatrix}\\sin {\\theta } & \\cos {\\theta } \\\\-\\cos {\\theta } & \\sin {\\theta }\\end{pmatrix}\\qquad \\text{and}\\qquad U_2(k) = \\begin{pmatrix}\\cos {\\theta }\\, {\\rm e}^{-{\\rm i}k_y}& \\sin {\\theta }\\,{\\rm e}^{-{\\rm i}k_x} \\\\-\\sin {\\theta }\\, {\\rm e}^{{\\rm i}k_x}& \\cos {\\theta }\\, {\\rm e}^{{\\rm i}k_y}\\end{pmatrix}.$ The block-antidiagonal form (REF ) of the evolution operator is reminiscent of the cyclic structure of the oriented network: as $a_1$ and $b_1$ are oriented from $U_2$ to $U_1$ , whereas $a_2$ and $b_2$ are oriented from $U_1$ to $U_2$ , a wave packet traveling in the network will always encounter a succession $U_1 \\rightarrow U_2 \\rightarrow U_1 \\rightarrow U_2 \\rightarrow \\cdots $ of nodes (and will never, for example, come across two successive $U_1$ nodes).", "It is convenient to reframe this particular block-antidiagonal form in terms of the structure constraint $D \\mathcal {S}(k) D^{-1} = - \\mathcal {S}(k)\\qquad \\text{where}\\qquad D = \\begin{pmatrix}\\text{Id}& 0 \\\\ 0 & - \\text{Id}\\end{pmatrix}$ where $\\text{Id}$ is here the two-by-two identity matrix.", "We recognize a particular case of the phase rotation symmetry (REF ) with ${Z}=D$ and $M=2$ .", "As we shall see in section REF , such a structure constraint can be generalized to a whole class of network models.", "(On first sight, this particular case may looks like a chiral symmetry, but this is not the case as $\\mathcal {S}$ is an evolution operator and not a Hamiltonian.)", "The well-known phase diagram of the L-lattice [12], [7] with respect to the parameter $\\theta $ is represented in figure REF .", "Due to the form of matrices $U_j$ , it is $\\pi $ -periodic with respect to $\\theta $ , and we can restrict the discussion to a range of that length.", "The phase spectrum of $\\mathcal {S}(k)$ consists in four bands that touch at the critical value $\\theta _c=\\pi /4$ , and this critical point separates two phases where the four bands are well-defined (i.e.", "separated by gaps), which we call phases I and II.", "Notably, such phases are topologically inequivalent, a smoking gun evidence of which is the existence of robust chiral edge states at an interface between them (see figure REF ).", "Following a longstanding analogy between network models and Floquet stepwise evolutions [18], [19], Liang, Pasek and Chong [7], [8] studied the topology of network models by focusing on the Floquet operator $U_{\\text{F}}(k)=U_2(k)U_1(k)$ which represents a sequence of two steps, in contrast with the Ho-Chalker evolution operator $\\mathcal {S}(k)$ that accounts for the different scattering processes simultaneously.", "The equivalence between both points of view is rooted into the existence of the structure constraint (REF ).", "Due to this phase rotation symmetry, the description of the system from the point of view of the Ho-Chalker evolution operator $\\mathcal {S}(k)$ is redundant, and its spectrum reduced to a fundamental domain is directly related to the (entire) spectrum of $U_{\\text{F}}(k)$ .", "The structure constraint enables to define bulk invariants that characterize the network model: for each bulk gap ${\\rm e}^{-{\\rm i}\\eta }$ of the Ho-Chalker evolution operator $k\\mapsto \\mathcal {S}(k)$ , there is a bulk invariant $W_{\\eta }^{\\text{HC}}[\\mathcal {S}] \\in \\mathbb {Z}$ which essentially accounts for the number of edge states appearing in the bulk gap ${\\rm e}^{-{\\rm i}\\eta }$ when an interface is considered.", "We defer the definition of such invariants to the section , but we will now discuss their essential properties.", "The redundancy expressed by the phase rotation symmetry (REF ) is translated at the level of such invariants by the identity $W_{\\eta }^{\\text{HC}}[\\mathcal {S}] = W_{\\eta +2\\pi /M}^{\\text{HC}}[\\mathcal {S}]$ where $M=2$ in the case of the L-lattice.", "Crucially, this invariant is relative to a reference evolution which has to be chosen arbitrarily.", "For the unit cell in figure REF , we obtain $W_{0}^{\\text{HC}}[\\mathcal {S}_{\\text{I}}]={1}$ and $W_{\\pi /2}^{\\text{HC}}[\\mathcal {S}_{\\text{I}}]={1}$ in phase I and $W_{0}^{\\text{HC}}[\\mathcal {S}_{\\text{II}}]={0}$ and $W_{\\pi /2}^{\\text{HC}}[\\mathcal {S}_{\\text{II}}]={0}$ for phase II.", "A different choice of unit cell leads to different values for the invariants (see table REF for an example, and section REF for a more detailed discussion), yet the differences between invariants do not depend on particular choices.", "Usually, only such differences carry a physical meaning; for example, their variation at an interface is expected to give the algebraic number of chiral edge states (counted with chirality) in the corresponding bulk gap.", "Particular physical situations may however naturally select only one unit cell.", "Figure: Two possible unit cells of the L-lattice.Table: Relative invariants for the L-lattice.", "The values of the invariants are given for two choices of a unit cell (a) and (b), for the two phases I and II, and for the two gaps η=0\\eta =0 and η=π\\eta =\\pi .", "We observe that the values do not coincide when the unit cell changes, but that the difference between two phases W η HC [𝒮 I ]-W η HC [𝒮 II ]W_{\\eta }^{\\text{HC}}[\\mathcal {S}_{\\text{I}}] - W_{\\eta }^{\\text{HC}}[\\mathcal {S}_{\\text{II}}] is invariant with respect to the choice of the unit cell, as it is expected for a physically observable quantity.", "The chosen unit cells are represented in figure , and a more detailed account of the choice of the reference evolution is explained in the general case, in section .The code used to compute the phase spectra and the topological invariants is available in Supplemental Materials at URL https://arxiv.org/src/1612.05769/anc." ], [ "Classical loop configurations and anomalous phases", "When the scattering matrices $U_j$ correspond to full reflection or full transmission, they do not split an incoming wave packet.", "In this situation, they describes a classical or ballistic propagation (as opposed to a wave-like propagation).", "In the L-lattice, such a behavior arises at two special points of the phase diagram (do not confuse with the phase spectrum), when $\\theta =0$ or $\\theta =\\pi /2$ (see figure REF ).", "Here, we observe that the network is composed only of small loops, and the corresponding point of the phase diagram is therefore called a classical loop configuration.", "Notably, such loops rotate clockwise in phase I and counter-clockwise in phase II.", "Away from the classical configurations, the network model can be understood as a superposition of more complicated loop configurations, where the loops now extend over several unit cells.", "The direction of rotation of such loops is preserved all over the gapped phase, and the transition at $\\theta =\\pi /4$ between clockwise and counter-clockwise phases is marked by a percolation of the possible trajectories, which allows for a path through the entire system.", "Notably, a strong version of the phase rotation symmetry is satisfied at the points at the classical loop configurations, which ensures that the band structure at those points is either trivial or anomalous, a property which extends to the entire gapped phase, as topological invariants cannot change unless a gap closes.", "In these two situations, a unitary operator ${Z}_\\theta $ can be found so that ${Z}_\\theta \\mathcal {S}(k) {Z}^{-1}_\\theta ={\\rm i}\\mathcal {S}(k)$ with ${Z}_{0} = \\begin{pmatrix}-\\sigma _z & 0 \\\\0 & {\\rm i}\\sigma _z\\end{pmatrix}\\qquad \\text{and}\\qquad {Z}_{\\pi /2}= \\begin{pmatrix}-\\sigma _z & 0 \\\\0 & -{\\rm i}\\sigma _z\\end{pmatrix}$ meaning that there is only one band in the fundamental domain of the phase rotation symmetry.", "As shown in section REF , this directly implies the vanishing of the first Chern number of each band.", "In the section , we will see that such classical loop configurations provide, along with phase rotation symmetry, a valuable tool to design anomalous phases in network models.", "Figure: Phase diagram and loops configurations of the L-Lattice.", "(Do not confuse with the phase spectrum of figures and .)", "The L-lattice hosts two gapped phases with a transition at θ=π/4\\theta =\\pi /4.", "When varying θ\\theta , each of these phases can be continuously deformed into lattices of clockwise (θ=0\\theta =0) or anti-clockwise (θ=π/2\\theta =\\pi /2) loops, that both satisfy a phase rotation symmetry with one band in the fundamental domain.Figure: Interfaces of the L-lattice.We consider interfaces between the two phases of the L-lattice in a cylinder geometry (the system has periodic boundary conditions in the xx direction and is infinite in the yy direction).", "This allows one to (i) avoid potential ambiguities due to the relative character of the invariant and (ii) confirm that the existence of chiral edge states is indeed due to the bulk topology, and not merely from the oriented nature of the links.Remarkably, the two chiral edge states (one at each interface) are found to have different group velocities, which is consistent with the simple intuitive sketch in (a) where one of the two channels (in red) can flow easily rather than the other one (in blue) is forced to propagate in pilgrimage, resulting in a decreasing of its velocity along the yy axis compared to that of the other boundary state.", "(a) Interfaces between two networks with respectively θ=0\\theta =0 and θ=π/2\\theta =\\pi /2.", "The system in periodic in both direction and finite in the xx direction.", "At the two interfaces, edge states with different velocity, in sign and amplitude, arise.", "(b) Eigenvalues of the corresponding Ho-Chalker evolution operator with θ≈0\\theta \\approx 0 and θ≈π/2\\theta \\approx \\pi /2 for clarity.", "Bulk states are represented in green, the fast boundary state in red and the slow boundary state in blue.The code used to compute the phase spectra and the topological invariants is available in Supplemental Materials at URL https://arxiv.org/src/1612.05769/anc.In the following, we first generalize this set of observations to a more general class of scattering networks, cyclic oriented networks (section REF ).", "Their precise definition allows us to elucidate the correspondence between the Ho-Chalker-like description and the reduced Floquet-like description (section REF ), which sets the ground for a proper definition of bulk topological invariants for this class of network models (section ).", "As a byproduct, we also propose a standard way to define topological invariants for a stepwise (or discrete time) evolution.", "The orientation of the links of the L-lattice is such that a wave packet traveling on the network will encounter the nodes $U_1$ and $U_2$ in a cyclic way during its evolution, namely, in a periodic sequence of the form $\\cdots \\rightarrow U_2 \\rightarrow U_1 \\rightarrow U_2 \\rightarrow U_1 \\rightarrow \\cdots $ (there are, for example, no $U_1 \\rightarrow U_1$ in this sequence).", "From the point of view of the wave packet, the situation is similar to a stepwise evolution periodic in time, similar to a Floquet dynamics with a (Bloch)-Floquet operator $U_{\\text{F}}=U_2(k)U_1(k)$ .", "As we shall see, there is indeed a mapping between a particular class of network models that generalize the L-lattice and stepwise Floquet evolutions.", "A cyclic oriented network is a (space-periodic) oriented network where any path along the directed edges is constrained to travel through a periodic sequence of the nodes, always in the same order $\\cdots \\rightarrow U_s \\rightarrow U_1 \\rightarrow U_{2} \\rightarrow \\cdots \\rightarrow U_{s-1} \\rightarrow U_s \\rightarrow U_1 \\rightarrow \\cdots $ , where $U_j \\in U(b)$ describes the scattering events at the corresponding node.", "A unit cell of such a network consists in $s$ nodes and $b\\times s$ oriented links (in the examples, we will always consider $b={2}$ ).", "As we shall see, such a network model can be mapped to a time-periodic stepwise evolution composed of $s$ unitary operations $U_n \\in U(b)$ .", "Let us denote by $a_n, b_n, c_n, \\dots $ the incoming amplitudes at the node $U_n$ , and by $a_{n+1}, b_{n+1}, c_{n+1}, \\dots $ the outgoing amplitudes at the same node (which are, on the cyclic network, the incoming amplitudes on the next node $U_{n+1}$ ).", "In reciprocal space, the Ho-Chalker evolution operator of such a network then reads $\\mathcal {S}(k)=\\begin{pmatrix}0 & 0 & \\cdots & U_s(k) \\\\U_1(k) & 0 & \\cdots & 0 \\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & U_{s-1}(k) & 0\\end{pmatrix} \\qquad \\in U(b\\times s)$ in the Bloch basis $(a_1(k),b_1(k), a_2(k), b_2(k), \\dots a_s(k),b_s(k))$ .", "As for the L-lattice, the form of $\\mathcal {S}(k)$ in this well-chosen basis is reminiscent of the cyclic structure of the oriented network.", "We interpret it as stemming from the existence of a structure constraint $D \\mathcal {S}(k) D^{-1} = {\\rm e}^{{\\rm i}2\\pi /s} \\mathcal {S}(k)$ where $D$ is the block-diagonal unitary matrix that reads $D = \\text{diag}(1,{\\rm e}^{{\\rm i}2\\pi /s}, {\\rm e}^{{\\rm i}4 \\pi /s}, \\dots , {\\rm e}^{{\\rm i}2(s-1)\\pi /s}) \\otimes \\text{Id}_{b} \\in U(b\\times s)$ in the same basis as $\\mathcal {S}(k)$ , which is the standard phase rotation operator (REF ) that satisfies $D^s=\\text{Id}$ .", "Although the explicit expressions (REF ) and (REF ) for the Ho-Chalker evolution operator and its symmetry might depend on the basis and unit cell choices, they will be modified in a covariant way so that constraint (REF ) will always be preserved.", "Cyclic oriented networks with a given number $s$ of non-equivalent nodes and $b$ of incoming links per node define an equivalence class of networks models (where the connectivity of the underlying graph is fixed).", "The structure constraint (REF ) implements the restriction to this equivalence class at the level of the Ho-Chalker evolution operators $\\mathcal {S}(k)\\in U(b\\times s)$ in Bloch representation, and evolutions that preserve equation (REF ) therefore stay in the corresponding class.", "Indeed, the structure constraint is a particular case of the phase rotation symmetry (REF ) where ${Z}=D$ and with $M=s$ , and the cyclic form (REF ) of the evolution operator highlights the reduction in the number of degrees of freedom enabled by the existence of the phase rotation symmetryThe number of degrees of freedom is reduced from $s^2 \\, {b}^2$ for a generic unitary matrix to $s \\, {b}^2$ when it is taken into account.. As a consequence, the spectrum of $\\mathcal {S}$ is redundant: more precisely, it is obtained by $s-1$ successive rotations of the spectrum contained in a fundamental domain of length $2 \\pi /s$ .", "Moreover, the total first Chern number of the bands of $\\mathcal {S}(k)$ in such a fundamental domain vanishes.", "This set of properties will allow us to develop a mapping between the network model and a stepwise Floquet evolution.", "To do so, the first step is to relate the spectrum of the Ho-Chalker evolution operator $\\mathcal {S}$ to the spectrum of an associated Floquet evolution operator." ], [ "Two points of view: simultaneous steps and sequence of steps", "The particular form (REF ) of the Ho-Chalker evolution operator $\\mathcal {S}$ imposed by the structure constraint (REF ) implies that its $s$ -th power $\\mathcal {S}^s$ is block-diagonal and reads $\\mathcal {S}^s = \\text{diag} ( U_{\\text{F}}^{(1)}, U_{\\text{F}}^{(2)}, \\cdots , U_{\\text{F}}^{(s)} )$ in the same basis as equation (REF ), where $U_{\\text{F}}^{(n)} \\in U(b)$ denotes the cyclic permutation of the Floquet operator starting at step $n$ , namely $U_{\\text{F}}^{(n)} = U_{n-1} \\cdots U_2 U_1 U_s \\cdots U_{n+1} U_n \\, .$ The restriction to a fundamental domain of the spectrum of the Ho-Chalker operator $\\mathcal {S}$ is identical to the spectrum of the Floquet operators $U_\\text{F}^{(n)} \\in U(b)$ , up to a constant scaling factor, as illustrated in figure REF .", "In this sense, $\\mathcal {S}$ can be reduced to the smaller-dimensional operator $U_\\text{F}^{(n)}$ .", "The eigenstates of the $U_\\text{F}^{(n)}$ can be obtained from the eigenstates of $\\mathcal {S}$ .", "The converse is not fully possible without the knowledge of the matrices $U_j(k)$ , but we will see that the first Chern numbers of the bands of any of the $U_\\text{F}^{(n)}$ (for any given $n$ ) entirely determine the ones of $\\mathcal {S}$ .", "Let $\\mathinner {|{\\psi }\\rangle }$ be an eigenstate of $\\mathcal {S}$ with eigenvalue $\\lambda $ , so that $\\mathcal {S}\\mathinner {|{\\psi }\\rangle } = \\lambda \\mathinner {|{\\psi }\\rangle }$ , and thus $\\mathcal {S}^s \\mathinner {|{\\psi }\\rangle } = \\lambda ^s \\mathinner {|{\\psi }\\rangle }$ .", "Decomposing the vector $\\mathinner {|{\\psi }\\rangle }$ into $s$ smaller vectors $\\mathinner {|{\\varphi ^{(r)}}\\rangle }$ as $\\mathinner {|{\\psi }\\rangle } = \\left( \\mathinner {|{\\varphi ^{(1)}}\\rangle }, \\cdots , \\mathinner {|{\\varphi ^{(s)}}\\rangle } \\right)^T$ it follows from (REF ) that $U_n \\mathinner {|{\\varphi ^{(n)}}\\rangle } = \\lambda \\mathinner {|{\\varphi ^{(n-1)}}\\rangle }$ and we infer from equation (REF ) the eigenvalue equation for the Floquet operators $U_{\\text{F}}^{(n)} \\mathinner {|{\\varphi ^{(n)}}\\rangle } = \\lambda ^s \\mathinner {|{\\varphi ^{(n)}}\\rangle } .$ Importantly, the phase spectrum of $U_{\\text{F}}^{(n)}$ does not depend on $n$ , meaning that the Floquet spectrum is invariant under a change of the origin of time, as expected.", "This construction can be applied to the set of $b\\times s/s=b$ eigenvectors $\\mathinner {|{\\psi _j}\\rangle }$ of $\\mathcal {S}$ with eigenvalues $\\lambda _j$ in the fundamental domain $F$ to obtain two linearly independent eigenstates $\\mathinner {|{\\varphi _j^{(n)}}\\rangle }$ of $U_{\\text{F}}^{(n)}$ .", "As a consequence, we have on the one hand $\\mathcal {S}= \\sum _{r=0}^{s-1} \\sum _{j=1}^{b} {\\rm e}^{-{\\rm i}2 \\pi r/s} \\lambda _j D^{r} \\mathinner {|{\\psi _j}\\rangle }\\!\\mathinner {\\langle {\\psi _j}|} D^{-r}$ and on the other hand $U_{\\text{F}}^{(n)} = \\sum _{j=1}^{b} \\lambda _j^s \\mathinner {|{\\varphi _j^{(n)}}\\rangle }\\!\\mathinner {\\langle {\\varphi _j^{(n)}}|}$ where the correspondence between $\\mathinner {|{\\psi _j}\\rangle }$ and $\\mathinner {|{\\varphi _j^{(n)}}\\rangle }$ is given by (REF ) and illustrated in figure REF .", "Indeed, the complete correspondence between the Ho-Chalker description and the Floquet description involves, on one side, the Ho-Chalker evolution operator $\\mathcal {S}(k)$ and, on the other side, the stepwise Floquet evolution with steps $(U_1, \\dots , U_{s})$ [as opposed to only the Floquet operator $U_{\\text{F}}^{(n)}$ , from which it is not possible to reconstruct $\\mathcal {S}(k)$ entirely].", "In particular, both points of view allow for a complete topological characterization of the system.", "However, we have seen that the phase spectrum of the Floquet operator $U_{\\text{F}}^{(n)}$ is enough to reconstruct the phase spectrum of $\\mathcal {S}$ , and we will see in the next paragraph that this is also true for the first Chern numbers of their bands.", "Figure: Relation between the spectra of 𝒮\\mathcal {S} and U F (n) U_\\text{F}^{(n)}.", "The spectrum of 𝒮\\mathcal {S} restricted to a fundamental domain FF of the phase rotation constraint corresponding to the structure constraint is in direct correspondence with the (full) spectra of all blocks U F (n) U_\\text{F}^{(n)} of the repeated evolution operator 𝒮 s \\mathcal {S}^s.", "By phase rotation symmetry DD, the first Chern number on the fundamental domain FF is zero, and thus the two bands of FF ave opposite first Chern numbers." ], [ "Consequences on the first Chern numbers", "We have seen that the spectra of the Ho-Chalker operator $\\mathcal {S}(k)$ and of the Floquet operator $U_{\\text{F}}^{(n)}$ are in direct correspondence, and can be obtained from one another, possibly up to a constant phase.", "In addition, the first Chern numbers of their bands are also in direct correspondence.", "More precisely, let $P^{\\text{F}}_{\\eta , \\eta ^{\\prime }}$ be the projector on states between the gaps $\\eta $ and $\\eta ^{\\prime }$ of $U_{\\text{F}}^{(n)}$ , and let $P^{\\text{HC}}_{\\eta /s, \\eta ^{\\prime }/s}$ be the projector on states between the gaps $\\eta /s$ and $\\eta ^{\\prime }/s$ of $\\mathcal {S}$ .", "Then, $C_1(P^{\\text{F}}_{\\eta , \\eta ^{\\prime }}) = C_1(P^{\\text{HC}}_{\\eta /s, \\eta ^{\\prime }/s}).$ Although a more direct proof could be devised, we infer this identity from our results on the complete topological characterization of network models which is discussed in the last section, and in particular from equation (REF ) (which is proven in appendix ) and the relation (REF ).", "A particular but typical situation arises when all bands are well-defined and composed of only one state.", "Then, the spectrum of $\\mathcal {S}$ is composed of $b\\times s$ bands separated from each other by $b\\times s$ gaps.", "Due to the phase rotation symmetry, it is sufficient to consider the $b$ bands in a fundamental domain described by projectors $P[\\psi _j] = \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|}$ , with $j = 1, \\dots , b$ .", "The Floquet operator $U_{\\text{F}}^{(n)}$ has also $b$ bands corresponding to projectors $P[\\varphi _j^{(n)}] = \\mathinner {|{\\varphi _j^{(n)}}\\rangle }\\mathinner {\\langle {\\varphi _j^{(n)}}|}$ , and equation (REF ) simplifies into $C_1\\left(P[\\psi _j]\\right) = C_1(P[\\varphi _j^{(n)}]).$ This illustrates that the first Chern number of a band $j$ of a generalized Ho-Chalker operator is simply obtained from any of its associated Floquet operators $U_{\\text{F}}^{(n)}(k)$ , as sketched in figure REF .", "Equation (REF ) is of practical importance, since $U_{\\text{F}}^{(n)}$ has a smaller dimension than that of $\\mathcal {S}$ .", "To obtain a vanishing first Chern number phase (where $C_1(P[\\psi _j])=0$ for all of the $b\\times s$ bands of $\\mathcal {S}$ ), it is therefore enough to show that the Floquet operator $U_{\\text{F}}^{(n)}$ has a vanishing first Chern number phase (where $C_1(P[\\varphi _j^{(n)}])=0$ ).", "This is far easier, as we have to deal with $U(b)$ matrices instead of larger $U(b\\times s)$ matrices.", "As we have seen in section REF , this is achieved when $U_{\\text{F}}^{(n)}$ is endowed with a (strong) phase rotation symmetry (REF ), with only one band in the fundamental domain, that is to say, when there is a unitary operator ${Z}\\in U(b)$ such that ${Z}U_{\\text{F}}^{(n)}(k) {Z}^{-1} = {\\rm e}^{{\\rm i}2\\pi /b} U_{\\text{F}}^{(n)}(k) \\ .$ For $b=2$ , such a constraint is similar to the phase shift property pointed out by Asbóth and Edge in two-dimensional discrete-time quantum walks [10]." ], [ "A procedure to identify vanishing first Chern number phases in network models", "The Ho-Chalker evolution operators $\\mathcal {S}$ of cyclic oriented networks always have a phase rotation symmetry with ${Z}=D$ , so that there are $b$ bands in the fundamental domain (in this section, we will always consider situations where $b= 2$ ).", "This allows one to reduce the dimension of the problem and map it onto a Floquet dynamics.", "However, this does not guarantee the vanishing of the first Chern numbers.", "To obtain anomalous phases where all first Chern numbers vanish, an extra condition has to be found on the Floquet operator, such as equation (REF ), where another phase rotation symmetry applies to $U_{\\text{F}}^{(n)}$ .", "This approach is tantamount to the one consisting in directly finding out a stronger phase rotation symmetry for the Ho-Chalker evolution operator $\\mathcal {S}$ with only one band in the fundamental domain, as discussed in section REF on an example.", "Yet, it is usually convenient to work in the Floquet point of view where smaller matrices are involved, as discussed in section REF .", "In this section, we introduce a simple qualitative method to establish whether a cyclic oriented network has a vanishing first Chern number phase or not.", "Our analysis lies on two points.", "First, we identify the possible classical loops configurations, as we did for the L-lattice (see section REF ).", "These configurations are obtained by considering the possible loops in the unit cell when the nodes are either fully transmitting or fully reflecting.", "Intuitively, these configurations ensure that the phase being described is gapped, since the amplitude of a state cannot escape from a loop to propagate in the network.", "Second, we associate a Floquet operator $U_{\\text{F}}^{(n)}$ to each classical loop configuration (as the the first Chern numbers do not depend on the choice of the starting node, we can arbitrarily choose one of them).", "Importantly, the Floquet operator is a product of either diagonal or anti-diagonal step operators $U_n$ , because of the classical loop structure, and it is therefore itself either diagonal or anti-diagonal.", "Depending on its form, one can possibly conclude about the existence of a phase rotation symmetry (REF ) for the Floquet operator by easily exhibiting a suitable phase rotation operator ${Z}$ .", "In particular, if $U_{\\text{F}}$ is anti-diagonal, then equation (REF ) is always satisfied with ${Z}=\\sigma _z$ , which guaranties the vanishing of the first Chern number of bands of the Floquet operator, and therefore the vanishing of the first Chern number of the bands of the Ho-Chalker evolution operator.", "Let us now apply this analysis to concrete cyclic oriented networks." ], [ "The L-lattice ($s=2$ )", "The two loops configurations of the L-lattice have already been discussed in section REF (see figure REF ), where we exhibited a rotation phase symmetric operator for the Ho-Chalker evolution operator.", "As discussed above, one can equivalently consider any of the associated Floquet operator $U_{\\text{F}}^{(n)}$ .", "In phase I, a loop corresponds to the sequence $a_1 \\rightarrow b_2 \\rightarrow b_1, \\rightarrow a_2 \\rightarrow a_1$ meaning that $U_1$ is anti-diagonal (it changes $a$ to $b$ and $b$ to $a$ ) and $U_2$ is diagonal.", "Their product is thus anti-diagonal, and the first Chern numbers therefore vanish.", "In phase II, a loop corresponds to the sequence $a_1 \\rightarrow a_2 \\rightarrow b_1 \\rightarrow b_2 \\rightarrow a_1$ , meaning that $U_1$ is diagonal and $U_2$ is anti-diagonal.", "Again, their product is anti-diagonal and the first Chern numbers vanish.", "This is of course in agreement with the analysis of the Ho-Chalker operator done in section REF .", "This reasoning can now be applied to cyclic oriented networks beyond the L-lattice." ], [ "The oriented Kagome lattice ($s=3$ )", "The cyclic oriented network with $s=3$ corresponds to a Kagome lattice shown in figure REF (a).", "In this case, the unit cell is composed of $s=3$ inequivalent nodes $U_j$ ($j \\in [1,s]$ ) and $2s=6$ inequivalent oriented links denoted by $(a_j, b_j)$ [see figure REF (b)].", "Note that the oriented Kagome lattice has been considered to describe arrays of optical coupled resonators arranged in a honeycomb lattice by Pasek and Chong [8].", "Figure: Oriented Kagome lattice.", "(a) Oriented Kagome lattice with 6 inequivalent links and 3 inequivalent nodes per unit cell enhanced in red and detailed in (b).This oriented network allows for different possible loops configurations.", "Let us identify some of those which necessarily correspond to a vanishing first Chern number phase.", "Following the method discussed above, we select loops such that the product of the three $U_j$ 's is anti-diagonal.", "As previously, $U_j$ is anti-diagonal if it changes $a\\leftrightarrow b$ and is diagonal otherwise.", "Figure: Examples of loops configurations in the oriented Kagome lattice (a) and (b) Loops configurations that display a vanishing first Chern number phase.", "The phase spectra (d) and (e) for these configurations in a ribbon geometry confirm this result, and reveal that (a) corresponds to a trivial topological phase (with no edge state in the gaps) whereas (b) corresponds to an anomalous topological one (with one chiral edge state per edge in each gap).", "The configuration (c) displays no loop so that only the first Chern number on a fundamental domain is constrained to vanish.", "The phase spectrum on a strip geometry (f) confirms this result.With this in mind, it is clear that the two configurations represented in figures  REF (a) and (b) correspond to a vanishing first Chern number phase.", "Indeed, for the loops sketched in figure  REF (a), $U_1$ is anti-diagonal whereas $U_2$ and $U_3$ are diagonal, and for the loops shown in figure REF (b), all the $U_j$ 's are anti-diagonal.", "These results are confirmed by a direct diagonalization of the phase spectrum in a strip geometry which exhibits, for each of these two configurations, an equal (algebraic) number of edge states in each gap [0 in the spectrum (d) and 1 per edge in the spectrum (e) of the figure REF ], as expected for a vanishing first Chern number phase.", "In contrast, if one considers a case where any incoming amplitude to a node is partially scattered onto each outgoing link, then this does not correspond to a loops configuration and $U_{\\text{F}}$ is neither diagonal nor diagonal [see figure REF (c)].", "Thus, provided such a phase is gapped, the first Chern numbers may not vanish, as shown in figure REF (f).", "This analysis gives one an insight on the control of the first Chern number.", "However, to discriminate a topologically trivial phase [figure REF (d)] from an anomalous topological one (with zero first Chern number and edge states [figure REF (e)]), it is still required to compute the topological invariants defined in section REF ." ], [ "Topological characterization of cyclic scattering network models", "We now want to fully characterize cyclic scattering network models, and in particular to account for anomalous phases.", "Both the Ho-Chalker and the Floquet points of view provide ways to define proper bulk topological invariants, which crucially depend on the existence of the structure constraint (REF ).", "In both cases, we interpolate the discrete evolution to a continuous one, in order to use the tools of homotopy theory.", "A first step in this direction is to properly define topological invariants for a stepwise evolution; the section REF is devoted to this task.", "Equipped with this tool, it is possible to actually define topological invariants for the network model in section REF .", "In the Ho-Chalker point of view, we require the interpolation to always satisfy the structure constraint of the oriented network to be able to define meaningful invariants.", "No such requirement is necessary in the Floquet point of view, as the reduction to the stepwise dynamics already takes the structure constraint into account.", "Indeed, both points of view are equivalent, and the corresponding invariants can be related one to another." ], [ "Topological characterization of a stepwise evolution", "A topological characterization of periodically driven systems was proposed by Rudner, Lindner, Berg, and Levin [2] for systems without specific symmetries in two dimensions.", "A topological invariant $W_{\\eta }$ can be assigned to each spectral gap $\\eta $ of the Bloch-Floquet operator $U(t=T,k)$ , thus remarkably establishing a new bulk-boundary correspondence for periodically driven systems [2].", "This index $W_{\\eta }[U]$ is defined as the degree (or winding number) of a periodized evolution operator $V_{\\eta }(t,k)$ built from the full evolution operator $U(t,k)$ .", "This method is applicable both to Floquet systems actually periodically driven in time, and to lattices of evanescently coupled light waveguides when the paraxial direction of propagation plays somehow the role of time [22].", "In contrast, stepwise evolutions like (REF ) are not continuous maps, as opposed to the usual evolution operator (REF ).", "Hence, the index $W$ is not directly applicable to such evolutions.", "In the following, we propose a systematic procedure to extend Rudner et al.", "'s [2] $W$ index to discrete evolutions.", "In order to do so, an interpolating continuous-time evolution must be associated to any stepwise evolution $U = U_{N} \\cdots U_{1}$ .", "The main idea is that when the stepwise evolution is correctly specified, one can assume that each step $U_{n}$ is generated by a time-independent Hamiltonian $H_{n}$ , so that $U_{n} = {\\rm e}^{-{\\rm i}\\tau _n H_{n}}$ for some duration $\\tau _n$ .", "The description of the stepwise evolution does not contain more information.", "A similar method, where a Hamiltonian realizing the discrete-time quantum walk is explicitly constructed, was described by Asbóth and Edge [10].", "In this paragraph, we review the construction by Rudner et al.", "[2] of a topological invariant $W_{\\eta }$ for unitary evolutions, and define all the tools required for the upcoming construction.", "We start with a continuous unitary evolution $U(t,k)$ from an origin time $t=0$ to a finite time $t=T$ , and assume that $U(T)$ is gapped.", "It is then convenient to define a time-independent effective Hamiltonian $H^{\\text{eff}}(k)$ .", "In the context of Floquet theory of periodically driven systems, such an effective Hamiltonian would generate the stroboscopic evolution at discrete times $U(n T) = [U(T)]^{n}$ .", "Namely, we want that $U(T,k) = {\\rm e}^{-{\\rm i}\\, T H^{\\text{eff}}(k)}$ .", "A crucial point of reference [2] is that the effective Hamiltonian is not unique, as it is defined as a logarithm of the Floquet operator.", "More precisely, the branch cut $\\eta $ of the logarithm must be chosen in a spectral gap of the Floquet operator $U(T)$ , to define (e.g., by spectral decomposition) $H^{\\text{eff}}_{\\eta }(k) = \\frac{{\\rm i}}{T} \\log _{-\\eta } U(T,k)$ where the complex logarithm with branch cut along an ray with angle $-\\eta \\in \\mathbb {R}$ is defined as $\\log _{-\\eta }({\\rm e}^{{\\rm i}\\varphi }) = {\\rm i}\\varphi \\quad \\text{for}\\quad - \\eta - 2 \\pi < \\varphi < - \\eta .$ The periodized evolution operator is then defined asAlthough different from the original one from [2], this definition is equivalent and leads to the same topological invariant, see [23].", "$V_{\\eta }(t,k) = U(k,t) {\\rm e}^{{\\rm i}t H^{\\text{eff}}_{\\eta }(k)}.$ Finally, as $V_{\\eta }$ is periodic both in time and on the Brillouin zone $\\rm {BZ}$ , the bulk topological index is defined as its degree or winding number $W_{\\eta }[U] \\equiv \\operatorname{deg}(V_\\eta ) \\in \\mathbb {Z},$ where the degree of a periodic map is formally defined as $\\operatorname{deg}(V_\\eta ) \\equiv \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left(V_\\eta \\right)^* \\chi \\qquad \\text{where}\\qquad \\left(V_\\eta \\right)^* \\chi = \\operatorname{tr}\\left[ (V_\\eta ^{-1} {\\rm d}V_\\eta )^3\\right].$ When $\\eta $ and $\\eta ^{\\prime }$ are in the same spectral gap of $U(T)$ (called the quasi-energy spectrum in the context of periodically driven systems), then $W_{\\eta }[U] = W_{\\eta ^{\\prime }}[U]$ (and indeed, $W_{\\eta +2\\pi }[U]=W_{\\eta }[U]$ ).", "When there are several gaps in the phase spectrum, however, there are as many topological invariants defined for the unitary evolution.", "Remarkably, the interface between two driven systems with bulk evolution operators $U_{\\text{left}}$ and $U_{\\text{right}}$ carries $n_{\\text{es}}(\\eta )$ topologically protected chiral edge states (counted algebraically with chirality) in the gap of quasi-energy $\\eta $ with [2] $n_{\\text{es}}(\\eta ) = W_{\\eta }[U_{\\text{left}}] - W_{\\eta }[U_{\\text{right}}].$ At equilibrium, the first Chern numbers of energy bands are sufficient to characterize the topology of quantum Hall like systems.", "In a periodically driven system, the quasi-energy bands can also carry a nonzero first Chern number.", "Rudner et al.", "[2] showed that even though the data of all the first Chern numbers is not sufficient to fully characterize the topology of Floquet states, they are still significant, and give the variation in the $W$ invariant between the gaps above and below the band.", "More precisely, let $- 2\\pi < \\eta _1, \\eta _2 < 0$ be two quasi-energies and $P_{\\eta _1,\\eta _2}(k)$ the spectral projector on states with quasi-energy between $\\eta _1$ and $\\eta _2$ , i.e.", "on eigenstates with eigenvalues ${\\rm e}^{-{\\rm i}\\eta }$ in the arc joining ${\\rm e}^{-{\\rm i}\\eta _1}$ and ${\\rm e}^{-{\\rm i}\\eta _2}$ clockwise on the circle $U(1)$ .", "The difference between the gap invariants $W$ is then related to the first Chern number $C_1$ of the quasi-energy band in between by $W_{\\eta _2}[U] - W_{\\eta _1}[U] = - C_{1}(P_{\\eta _1,\\eta _2}).$" ], [ "Interpolating a discrete-time evolution", "In a stepwise evolution, one only knows the evolution operator at several discrete times.", "We would like to interpolate such data to a physically relevant continuous evolution.", "As such an interpolation is not unique, we need a specification to construct it in a unique way, or at least to get equivalent interpolations from the point of view of topological properties.", "The main idea is that the choice of the interpolation should not add or remove anything to the topology.", "Roughly speaking, each step $U_i$ should be interpolated from the identity $\\text{Id}$ through an evolution of the form $\\mathcal {U}_{\\text{int}} = {\\rm e}^{-{\\rm i}t H_i}$ for some effective Hamiltonian $H_i$ .", "The choice of an interpolation where the phases grow linearly is, however, a natural choice in this context [24], as it necessarily corresponds to a trivial evolution when all first Chern numbers of $H_i$ vanish.", "In this situation all such interpolations are actually equivalent.", "In the general case where the step operators may carry nonzero first Chern numbers, we must assume that each step operator stems from a constant Hamiltonian, and moreover that the evolution was sufficiently short with respect to the characteristic time scales of the Hamiltonian.", "This hypothesis is necessary to accurately interpolate the discrete-time evolution, as it ensures that the gap $\\eta = -\\pi $ of the step operator is trivial.", "Without this additional information, there are not enough data to unambiguously reconstruct the evolution (essentially because it is not sufficiently discretized to capture all the physical information of the system).", "We first consider the evolution operator generated by a (known) constant Hamiltonian, i.e.", "to do the one-step version in a situation where the result is known.", "A time-independent Hamiltonian $H$ generates a step evolution operator $U_{\\text{F}} = U_1 = {\\rm e}^{-{\\rm i}T H}$ .", "In this case, we already know that the correct interpolation is indeed $U(t) = {\\rm e}^{-{\\rm i}t H}.$ To express it in terms of the step operator $U_1$ only, we use the effective Hamiltonian, defined in (REF ), corresponding to $U_1$ and with logarithm branch cut $\\eta = -\\pi $ .", "We have $H^{\\text{eff}}_{\\eta =-\\pi }[{\\rm e}^{-{\\rm i}H T}] = H$ at least for a small enoughTo be precise, the maximum energy in absolute value $\\max | \\sigma (H) |$ of $H$ should be smaller than $\\pi /T$ , or $h/2T$ if we restore the Planck constant.", "In this way, there is always a gap around the eigenvalue ${\\rm e}^{-{\\rm i}\\pi }$ .", "$T$ .", "We may then interpolate the evolution ending with the step operator $U_1$ by $\\mathcal {U}_{\\text{int}}[U_1](t) \\equiv \\exp \\left( -{\\rm i}t H^{\\text{eff}}_{-\\pi }[U_1] \\right).$ and we see that this formula immediately generalizes to any step operator $U_1$ , even without knowing some underlying time dynamics.", "Besides this definition being natural as the choice of cut, $\\eta = -\\pi $ ensures that when $U_1 = {\\rm e}^{-{\\rm i}T H}$ (for $T$ small enough) we recover $\\mathcal {U}_{\\text{int}}[U_1](t) = {\\rm e}^{-{\\rm i}t H}.$ This is obviously true in $t=T$ whatever the choice of $\\eta $ is, but it is not necessarily valid for intermediate times.", "Besides with this choice, $W_{\\eta }[\\mathcal {U}_{\\text{int}}[U_1]] = \\operatorname{deg}[\\exp \\left( -{\\rm i}t H^{\\text{eff}}_{-\\pi }[U_1] \\right) \\cdot \\exp \\left( {\\rm i}t H^{\\text{eff}}_{\\eta }[U_1] \\right)] = -C_1(P_{-\\pi ,\\eta })$ correctly accounts for the topology of the time-independent system (e.g.", "for $\\eta =0$ it gives, up to the usual sign, the first Chern number of the valence band).", "We now move on to the general case.", "When there are several steps in the stepwise evolution, a natural interpolation of the full evolution consists in concatenating the one-step interpolations (see figure REF ).", "Thus for each step operator $U_i$ involved in a stepwise evolution, an explicit interpolation is given by definition (REF ), as long as this operator is gapped around $-\\pi $ , so that it coincides with time evolution when $U_j$ actually comes from a Hamiltonian dynamics.", "When $U_j$ is gapped, but not at phase $-\\pi $ , a constant ($k$ independent) rotation of the phase spectrum can be factored out of the step operator.", "When the step operator has vanishing first Chern numbers, all interpolations by constant effective Hamiltonians are topologically equivalent, so any choice of such a rotation is acceptable.", "On the other hand, when the step operator has nonvanishing first Chern numbers, we must provide additional data for the interpolation to be unique, and the hypothesis of a trivial gap around $-\\pi $ is a natural and sufficient choice.", "The critical case where the step operator $U_j$ is gapless is discussed in Section REF .", "To be precise, let us consider a time-periodic stepwise evolution of period $T$ , namely, the data of several times $t_1, \\dots ,t_s$ and $t_s = T$ and corresponding unitary operators $U_1,\\dots ,U_s$ .", "We assume that such operators are gapped at phase $-\\pi $ .", "The evolution operator is only defined at discrete times $p T + t_j$ for $1 \\le j < s$ and $p \\in \\mathbb {N}$ as $U(p T + t_j) = U_{j} U_{j-1} \\cdots U_1 \\; (U_{\\text{F}}^{(1)})^{p}\\qquad \\text{and}\\qquad U(0) = \\text{Id}$ where $U_{\\text{F}}^{(1)} = U_{s} U_{s-1} \\cdots U_1.$ Indeed, $U_{\\text{F}}^{(1)}=U(T)$ .", "We interpolate this evolution as $U(t) = \\mathcal {U}_{\\text{int}}[U_j]\\left( \\displaystyle \\frac{t-t_{j-1}}{t_j-t_{j-1}} \\; T \\right) U_{j-1} \\cdots U_1 \\qquad \\text{for $t_{j-1} \\le t \\le t_j$}$ where by convention $t_0 = 0$ and $t_{s}=T$ (indeed, the first step is simply the correctly rescaled $\\mathcal {U}_{\\text{int}}[U_1]$ ).", "We can then extend the previous notation by setting $\\mathcal {U}_{\\text{int}}[U_1, \\dots , U_s] = U(t),$ where it is implied that $U_1, \\dots , U_s$ are in the right order.", "Note that the choice of times $t_j$ for the interpolation is completely arbitrary and will not influence its topological properties.", "Figure: Construction of the interpolation of a stepwise evolution.", "The evolution is constituted of ss time-ordered step operators U j U_j.", "A time t j t_j is attributed to each step and a continuous map is systematically built to interpolate between two successive steps.", "It follows a continuous map t→U(t)t\\rightarrow U(t) from Id\\text{Id} to U F (1) U_\\text{F}^{(1)}.Equivalently, we may consider the piecewise constant Hamiltonian $H(t) = \\mathcal {H}^{\\text{eff}}_{-\\pi }[U_j] \\quad \\text{for $t_{j-1} \\le t \\le t_j$}$ with $t_0=0$ and $t_s=T$ , and consider the corresponding evolution operator." ], [ "Topological invariants for discrete evolutions", "The previous construction allows us to define the topological invariant associated with the stepwise evolution as the topological invariant associated with this continuous-time evolution as $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}] = W_{\\eta }\\big [ \\mathcal {U}_{\\text{int}}[U_1, \\dots , U_s] \\big ].$ Note that functionally, $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}]$ depends on the entire sequence of steps $(U_1,...,U_s)$ and not only on their product $U_{\\text{F}}^{(1)}$ .", "This invariant is indeed associated with an ordered sequence of steps, but is invariant under any circular permutation of the step sequence.", "Such permutations correspond to the Floquet operators $U_{\\text{F}}^{(n)} = U_{n-1} \\cdots U_2 U_1 U_s \\cdots U_{n+1} U_n$ defined in (REF ), and one has $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}] = W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(n)}] \\qquad \\forall \\, n \\in {1,\\ldots ,s} \\, .$ This property simply means that the choice of origin of time (i.e.", "the first step) is not relevant for a periodic system, similarly to Floquet systems with a continuous time evolution.", "We, however, give an independent and explicit proof of this property in appendix , based on the fact that $W^{\\text{SWE}}_{\\eta }$ is a degree computation [see equation (REF )] and hence invariant under homotopy (i.e.", "under smooth deformations).", "We show that the two periodized interpolations are homotopic, so their degrees coincide.", "Although it was devised with oriented scattering networks in mind, this construction is applicable to any stepwise evolutions like discrete-time quantum walks." ], [ "Topological invariants for cyclic network models", "Equipped with tools to define topological invariants for stepwise evolutions, we can now move on the case of cyclic scattering networks.", "Physically, the effective Floquet stepwise evolution essentially describes the evolution from the point of view of one wave packet traveling on the network, whereas the Ho-Chalker point of view consists in studying the global evolution of the entire network." ], [ "The Floquet point of view", "As we have seen in section REF , a cyclic oriented network model can be mapped to a time-periodic stepwise evolution with steps $(U_1, U_2, \\dots , U_{s})$ , where $k \\mapsto U_j(k) \\in U(b)$ are maps from the Brillouin zone to the unitary group.", "In this Floquet point of view, such step operators are multiplied (in the right order) to obtain the Floquet operator $U_{\\text{F}}^{(n)} = U_{n-1} \\cdots U_2 U_1 U_s \\cdots U_{n+1} U_n$ associated to the network model.", "Hence we simply apply the results of section REF to define as a topological invariant of the network the quantity $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}] \\in \\mathbb {Z}$ defined in equation REF which, as we said before, does actually not depend on the choice of the first step." ], [ "The Ho-Chalker point of view", "We would like now to apply the same reasoning for the Ho-Chalker evolution operator $\\mathcal {S}(k) \\in U(b\\times s)$ .", "It is still possible to interpolate from the identity to $\\mathcal {S}(k)$ .", "However, this method does not take into account the nature of the scattering network, which is expressed by the structure constraint (REF ), and is likely to fail.", "A strong evidence in this direction is that we do not expect to observe anomalous topological phases (with vanishing first Chern numbers) in this situation, as the interpolation effectively reproduces the effect of a constant Hamiltonian $\\mathcal {H}^{\\text{eff}}_{-\\pi }[\\mathcal {S}]$ , and equation (REF ) tells that the degree of a single-step evolution only captures first Chern numbers.", "As it is clear from the examples that such phases do exist (see section REF ), this construction fails to capture the full topology of the network model.", "To fully take into account the structure of the cyclic oriented network, we need a somehow more complex interpolation.", "Starting from the structure constraint (REF ) for $\\mathcal {S}$ , which encodes the particular relations between the different links of the network, we propose instead the following interpolation: $\\mathcal {U}_{\\text{int,HC}}[\\mathcal {S}](t) =\\begin{pmatrix}0 & 0 & \\cdots & \\mathcal {U}_{\\text{int}}[U_s](t) \\\\\\mathcal {U}_{\\text{int}}[U_1](t) & 0 & \\cdots & 0 \\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & \\mathcal {U}_{\\text{int}}[U_{s-1}](t) & 0\\end{pmatrix} \\qquad \\in U(b\\times s).$ which has to be compared with (REF ).", "Crucially, the structure constraint (REF ) is satisfied all along this interpolation, namely, for every $t \\in [0,T]$ , we have $D \\, \\mathcal {U}_{\\text{int,HC}}[\\mathcal {S}](t) \\, D^{-1} = {\\rm e}^{{\\rm i}2\\pi /s} \\, \\mathcal {U}_{\\text{int,HC}}[\\mathcal {S}](t).$ We then define $W_{\\eta }^{\\text{HC}}[\\mathcal {S}] = W_\\eta \\big [\\mathcal {U}_{\\text{int,HC}}[\\mathcal {S}]\\big ] \\in \\mathbb {Z}$ as the invariant for the $\\mathcal {S}$ matrix describing the one-step evolution of a cyclic oriented network." ], [ "Relation between the topological invariants: equivalence between both points of view", "So far we have defined two different topological invariants for a cyclic oriented network: (1) $W_\\eta ^\\text{HC}$ is associated to the one-step Ho-Chalker evolution operator $\\mathcal {S}$ describing the network model and (2) $W_\\eta ^{\\text{SWE}}$ is associated to the stepwise Floquet evolution constituted of $s$ steps $(U_{1}, \\dots , U_{s})$ , the product of which is a Floquet operator $U_{\\text{F}}^{(n)}$ .", "We expect that such invariants are related, especially in view of the relation (REF ) between $\\mathcal {S}^s$ and the $U_{\\text{F}}^{(n)}$ .", "Because of the structure constraint (REF ), the spectrum of $\\mathcal {S}$ is redundant and can be fully deduced from a fundamental domain $F$ (as illustrated in figure REF ).", "This property translates simply on the invariant $W_\\eta ^\\text{HC}$ , which are also partially redundant.", "Namely, from equation (REF ) we have $W_{\\eta + \\frac{2\\pi }{s}}^\\text{HC}[\\mathcal {S}] = W_{\\eta }^\\text{HC}[\\mathcal {S}] - C_1(P_{\\eta ,\\eta + \\frac{2\\pi }{s}}) = W_{\\eta }^\\text{HC}[\\mathcal {S}]$ since the first Chern number on a fundamental domain vanishes, $C_1(P_{\\eta ,\\eta + \\frac{2\\pi }{s}}) = C_1(\\Pi ) = 0$ , see equation (REF ).", "As a consequence, $W_{\\eta }^\\text{HC}[\\mathcal {S}]$ is $2\\pi /s$ -periodic and the system is fully characterized by computing the $W$ invariant over a fundamental domain only.Note that this applies for any phase rotation symmetric systems, and not only cyclic oriented networks.", "We are now able to state the main result of this section, that is, $W_{\\eta /s}^\\text{HC}[\\mathcal {S}] = W_{\\eta }^{\\text{SWE}}[U_{\\text{F}}^{(n)}].", "$ On the left-hand side, we know from (REF ) and from the previous paragraph that the invariant is defined for $\\eta /s$ without ambiguity, so that the previous formula is still $2\\pi $ -periodic in $\\eta $ .", "On the right hand side, we know [see (REF ) and appendix ] that the invariants associated to any $U_{\\text{F}}^{(n)}$ with $n \\in 1, \\dots , s$ are all equal.", "The previous equality can be interpreted as follows: the topological informationIn particular, note that the $W$ 's also contain the first Chern numbers of the different bands.", "from the one-step evolution of an oriented network ruled by $\\mathcal {S}\\in U(b\\times s)$ is fully equivalent to the one from the $s$ -step stepwise Floquet evolution with steps $(U_1, \\dots , U_s)$ with $U_j \\in U(b)$ leading to the Floquet operator $U_{\\text{F}}^{(1)} \\in U(b)$ or any of its cyclic permutations $U_{\\text{F}}^{(n)}$ that appear in the block diagonal operator $\\mathcal {S}^s$ .", "From the topological point of view, these are two equivalent descriptions of the same problem.", "The identity (REF ) is proven by direct computation of both invariants, which are proven equal through the relation between $\\mathcal {S}^s$ and $U_{\\text{F}}^{(n)}$ .", "The actual proof is quite technical and therefore postponed to appendix , but we encourage the reader to have a glimpse at it." ], [ "The relative nature of the invariants", "As we have seen in the example of the L-lattice in section REF , the invariants for cyclic oriented networks are actually relative invariants, in a way which is very similar (and formally equivalent for $s=2$ ) to the standard chiral symmetric (class AIII) topological insulators, which are very clearly discussed in reference [25].", "More precisely, such invariants are relative to reference evolution which satisfies the structure constraint.", "There is indeed a large gauge freedom in this choice: starting from a given reference evolution $U_{\\text{ref}}$ , the conjugation by any change of basis matrix $M(k)$ commuting with $D$ gives another equally valid reference evolution $M(k) U_{\\text{ref}} M^{-1}(k)$ .", "The relative invariants with respect to such evolutions are indeed generally not equal.", "This relative character of the topological invariants is particularly clear in the Ho-Chalker point of view, where the reference evolution is indeed chosen as the constant Bloch evolution operator $U_{\\text{ref}}(t,k) =\\begin{pmatrix}0 & 0 & \\cdots & \\text{Id}\\\\\\text{Id}& 0 & \\cdots & 0 \\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & \\mathcal {\\text{Id}}& 0\\end{pmatrix}$ satisfying the structure constraint and from which the interpolation starts.", "Though it may not be as obvious, the invariant defined in the Floquet point of view is indeed also relative.", "Notably, the choice as a reference of the Bloch evolution operator $U_{\\text{ref}}(t,k)$ defined in equation (REF ) makes the invariants depend on the choice of the unit cell, as we have observed in the case of the L-lattice.", "This is because the Bloch representation as a $k$ -dependent matrix $U_{\\text{ref}}(t,k)$ (like in equation (REF )) of the operator $U_{\\text{ref}}(t)$ (acting on the Hilbert space) actually depends on the choice of the unit cell [26].", "Importantly, the difference of topological indices at an interface and for a given gap is well defined and unambiguous." ], [ "Gapless steps and their ambiguities ", "Let us illustrate a possible ambiguity of our definitions in the case of the L-lattice (see section REF ).", "In classical loop configurations, at $\\theta =0$ and $\\theta =\\pi $ , we observe that the step operator $U_2(k)$ is gapless.", "In principle, this prevents from defining a proper topological invariant.", "On the other hand, this situation should only arise in critical situations when (a part of) the nodes behave classically (as perfectly reflecting or transmitting elements).", "Such situations could either arise at the middle of a phase or at a transition point.", "In the first case (an example of which is provided by the L-lattice), we expect that the limit values of the invariants on all sides of the critical manifold should agree: in this case, the common limit value can be taken as the value of the invariant at that point.", "On the other hand, when the critical point marks a phase transition, we do not expect a well-defined topological invariant, and the various limits are indeed expected to disagree." ], [ "Conclusion", "In this paper, we have introduced the phase rotation symmetry, a new symmetry specific to dynamical systems described by a unitary operator, which has no equivalent in Hamiltonian systems.", "We then illustrated its power on our main subject, the description and the topology of oriented scattering network models.", "As we have seen, the phase rotation symmetry is the signal of a redundancy in the description of the system, and allows one to reduce and simplify this description, but also to better understand the internal structure of the system.", "We introduced a particular class of cyclic scattering networks where wave packets always encounter the same cycle of nodes.", "At the level of their evolution operator, such network models are characterized by a particular phase rotation symmetry, which is always present, and which we call a structure constraint.", "Two important consequences stem from the existence of this particular phase rotation symmetry.", "The first one is that the description of the cyclic network model can be reduced to a particular form, and then fully mapped into a stepwise Floquet dynamics.", "This mapping justifies and expresses the relationship between scattering networks and Floquet dynamics found in several works [12], [18], [19], [8].", "A second consequence is that the structure constraint allows one to define bulk topological invariants which fully account for the topology of the network model, and in particular for topological anomalous phases.", "Even though such phases were already experimentally observed [15], [16], such an invariant was not known until now.", "Notably, the topological invariants can be defined directly from the constrained evolution operator of the network model, or from the corresponding Floquet dynamics: both points of view indeed coincide, but they may be equally useful in different situations.", "Scattering network models notably allow for topological anomalous phases, where the system is topologically nontrivial despite the vanishing of all first Chern numbers.", "Such phases are particularly interesting, and we may want to design them.", "Although the phase rotation symmetry alone is not sufficient to ensure that a phase is topologically anomalous, a strong version of this property is actually enough to ensure that all first Chern numbers vanish, which is of clear interest to the design of anomalous phases.", "An example of a procedure to identify anomalous phases in network models based on phase rotation symmetries in classical loop configurations is also proposed, which can also be applied to engineer anomalous Floquet phases in other kinds of systems.", "Whilst it was mainly used to study scattering networks, the phase rotation symmetry is a new item in the toolbox of (potentially topological) general unitary evolutions, where it can be applied both as a reduction procedure and as a design principle.", "Several generalizations of the phase rotation symmetry can be imagined, for example where the symmetry operator is antiunitary, i.e.", "where ${Z}\\overline{U} {Z}^{-1} = {\\rm e}^{{\\rm i}\\zeta } U.$ It is also possible to consider more exotic generalizations: the antiunitary phase rotation constraint can be reformulated as ${Z}U {Z}^{-1} = {\\rm e}^{{\\rm i}\\zeta } \\overline{U}$ (with different ${Z}$ and $\\zeta $ ), and we can consider other constraints where $\\overline{U}$ is replaced by, e.g., $U^{-1}$ (which corresponds to an actual chiral symmetry when $\\zeta =0$ ), $U^{T}$ or $U^{\\alpha }$ .", "Some of such generalizations seem to appear in stepwise evolutions.", "Indeed, another very simple generalization, which should be physically relevant, consists of including the possibility of a nontrivial action on the Brillouin zone, where, for example, a constraint like ${Z}U(k) {Z}^{-1} = {\\rm e}^{{\\rm i}\\zeta } U(-k)$ could be considered.", "This assortment of examples aims at highlighting that the world of unitary dynamical evolutions is far richer than its Hamiltonian counterpart: new kinds of effective constraints or symmetries can emerge, the phase rotation symmetry being the prime example of such.", "Whether such constraints deserve or not to be named symmetries depends on the context and on the meaning we attribute to the word.", "For instance, the phase rotation symmetry does indicate a redundancy in the description, which may be broken in other physical situations.", "As such, we believe that this label is indeed relevant in the context of the effective description of wave propagation.", "The structure constraint of cyclic scattering networks seems to protect the topological phase, in the same way than standard symmetries are necessary for symmetry-protected topological phases to exist.", "This statement can serve as an interpretation of the fact that our approach to characterize the topology of oriented scattering networks only covers the particular class of cyclic network models.", "Other kinds of (spatially periodic) network models exist, which can also display anomalous topological states, yet evade our characterization.", "A topological characterization of such systems based on the same principle should be possible, but requires further analysis.", "Another open question involves the effects of a structure-constraint-breaking defects or disorder on the topological phases.", "Physically, such imperfections are not necessarily present in experimental realizations, but they may arise quite naturally, and we expect they should at some point spoil the topology; the question is to what extend they may be tolerated while still keeping protected edge states.", "This work was supported by the French Agence Nationale de la Recherche (ANR) under Grant TopoDyn (ANR-14-ACHN-0031).", "The work of C.T.", "was supported by the PRIN project Mathematical problems in kinetic theory and applications (prot.", "2012AZS52J)." ], [ "The standard phase rotation operator ", "Let us consider a phase rotation operator ${Z}$ for the evolution operator $U$ , such that ${Z}U {Z}^{-1} = {\\rm e}^{{\\rm i}2 \\pi /M} U.$ In general, the phase rotation operator has no special form.", "In this appendix, we show that when the $M^{\\text{th}}$ power of ${Z}$ is scalarIn particular, this is necessarily the case in an irreducible representation space, where all symmetries, including ${Z}^M$ , are scalar., that is to say ${Z}^M = {\\rm e}^{{\\rm i}\\phi } \\, \\text{Id}$ and the evolution operator $U$ is gapped, then the phase rotation operator ${Z}$ assumes the standard form ${Z}\\simeq {Z}_0 \\equiv \\text{diag}(1,{\\rm e}^{{\\rm i}2\\pi /M}, {\\rm e}^{{\\rm i}4\\pi /M}, \\dots , {\\rm e}^{{\\rm i}2 \\pi (M-1)/M}) \\otimes \\text{Id}_{b} \\in U(b\\times M)$ in an adequate basis.", "The standard phase rotation operator emphasizes the cyclic nature of the phase rotation symmetry (fundamental domains are simply rotated by the action of the operator ${Z}_0$ ).", "When ${Z}$ is a phase rotation symmetry of $U$ (namely ${Z}U {Z}^{-1} = {\\rm e}^{{\\rm i}2 \\pi /M} U$ ) and $R$ is a symmetry of $U$ (namely $R U R^{-1} = U$ ), then ${Z}R$ and $R {Z}$ are both phase rotation symmetries of $U$ .", "Hence, many phase rotations symmetries can be constructed from the standard phase rotation operator ${Z}_0$ , when it is a phase rotation symmetry of $U$ , which may not be reduced to the standard form.", "In general, it may also happen that the operator ${Z}= {Z}_0 R$ is a phase rotation symmetry, while ${Z}_0$ is not.", "Let us now prove the preceding statement.", "First, we redefine the phase rotation operator so that ${Z}^M = \\text{Id}$ by replacing ${Z}$ with ${\\rm e}^{-{\\rm i}\\phi /M} {Z}$ .", "Let then $F$ be a fundamental domain for the phase rotation symmetry, chosen to have its ends in a gap of $U$ (this is possible because we assumed that $U$ is gapped, as explained in section REF ).", "Let $\\psi _1, \\dots , \\psi _{b}$ be the eigenstates of $U$ with eigenvalue in $F$ .", "Because of the phase rotation symmetry, the family $(\\psi _i, \\dots , \\psi _{b}, {Z}\\psi _1, \\dots , {Z}\\psi _{b}, \\dots , {Z}^{M-1} \\psi _1, \\dots , {Z}^{M-1} \\psi _{b})$ is a basis.", "In this basis, ${Z}$ is block-diagonal, and assumes the form ${Z}\\simeq B \\otimes \\text{Id}_{b}$ where $B$ reads $B \\simeq \\begin{pmatrix}0 & 0 & \\cdots & 1 \\\\1 & 0 & \\cdots & 0 \\\\\\vdots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 1 & 0\\end{pmatrix}$ As $B$ is a $M \\times M$ circulant matrix, it is diagonalizable and its eigenvalues are the $M^{\\text{th}}$ roots of unity.", "Hence, ${Z}$ is then diagonalized as ${Z}\\simeq \\text{diag}(1,{\\rm e}^{{\\rm i}2\\pi /M}, {\\rm e}^{{\\rm i}2\\pi \\times 2/M}, \\dots , {\\rm e}^{{\\rm i}2 \\pi (M-1)/M}) \\otimes \\text{Id}_{b}$ which concludes the proof." ], [ "Proof of the equality between the SWE-invariants of all circular permutations $U_{\\text{F}}^{(n)}$ ", "We start by proving identity (REF ) for $n=2$ , namely that $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}] = W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(2)}]$ , in order not to overload the explicit expressions.", "The generalization to any $n$ is straightforward, as discussed in the end of the paragraph.", "The choice of times $t_j$ in interpolation (REF ) is completely arbitrary and does not change the value invariant: they actually do not appear in the computation, as it can be seen in expression (REF ), for example.", "Thus, from now on, we can choose the natural and regular time-step: $t_j = jT/s$ , so from the definitions (REF ) and (REF ) the computation of $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}]$ is reduced to the degree of the map $V^{(1)}_\\eta (t,k) = \\left\\lbrace \\begin{array}{lcl}\\mathcal {U}_{\\rm int}[U_1]\\big ( t s \\big ) {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & & 0 \\le t \\le \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_j]\\big ( (t-(j-1)\\frac{_T}{^s})s \\big )U_{j-1} \\ldots U_1 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & & (j-1)\\dfrac{_T}{^s} \\le t \\le j \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_s]\\big ( (t-(s-1)\\frac{_T}{^s})s \\big )U_{s-1} \\ldots U_1 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & & (s-1)\\dfrac{_T}{^s} \\le t \\le T\\end{array}\\right.$ which is defined piecewise for $j\\in \\lbrace 1,\\ldots s\\rbrace $ , and where we dropped the $k$ dependency on the right hand side.", "Similarly, the computation of $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(2)}]$ is reduced to the degree of the map $V^{(2)}_\\eta (t,k) = \\left\\lbrace \\begin{array}{lcl}\\mathcal {U}_{\\rm int}[U_2]\\big ( t s \\big ) {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(2)}]} & & 0 \\le t \\le \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_{j+1}]\\big ( (t-(j-1)\\frac{_T}{^s})s \\big )U_{j} \\ldots U_2 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(2)}]} & & (j-1)\\dfrac{_T}{^s} \\le t \\le j \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_1]\\big ( (t-(s-1)\\frac{_T}{^s})s \\big )U_{s} \\ldots U_2 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(2)}]} & & (s-1)\\dfrac{_T}{^s} \\le t \\le T\\end{array}\\right.$ In order to show that the degrees of these two maps are equal, we will use the homotopy invariance of the degree [2].", "Consider the following smooth deformation $\\widetilde{V}(r;t,k) = V^{(2)}_\\eta (t,k) \\mathcal {U}_{\\rm int}[U_1](rT,k) {\\rm e}^{{\\rm i}r \\frac{T}{s} H^{\\rm eff}_\\eta [U_{\\text{F}}^{(1)}](k)}$ where $r \\in [0,1]$ is a deformation parameter.", "Obviously one has $\\widetilde{V}(0;t,k) = V^{(2)}_\\eta (t,k)$ .", "The expression at $r=1$ is somehow close to $V^{(1)}_\\eta $ since $\\mathcal {U}_{\\rm int}[U_1](T,k)=U_1$ .", "We deduce from (REF ) that $U_{\\text{F}}^{(2)} = U_1 U_{\\text{F}}^{(1)} U_1^{-1}\\qquad \\Rightarrow \\qquad {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(2)}]} = U_1 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} U_1^{-1}$ which follows from the spectral decomposition of definition (REF ).", "Hence, $\\widetilde{V}(1;t,k) = \\left\\lbrace \\begin{array}{ll}\\mathcal {U}_{\\rm int}[U_2]\\big ( t s \\big ) U_1 {\\rm e}^{{\\rm i}(t+\\frac{T}{s}) H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & 0 \\le t \\le \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_{j+1}]\\big ( (t-(j-1)\\frac{_T}{^s})s \\big )U_{j} \\ldots U_2 U_1 {\\rm e}^{{\\rm i}(t+\\frac{T}{s}) H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & (j-1)\\dfrac{_T}{^s} \\le t \\le j \\dfrac{_T}{^s} \\\\\\mathcal {U}_{\\rm int}[U_1]\\big ( (t-(s-1)\\frac{_T}{^s})s \\big )U_{s} \\ldots U_2 U_1 {\\rm e}^{{\\rm i}(t+\\frac{T}{s}) H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} & (s-1)\\dfrac{_T}{^s} \\le t \\le T\\end{array}\\right.$ which looks like $V_\\eta ^{(1)}$ but somewhat shifted in time.", "By homotopy invariance $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(2)}] = \\operatorname{deg}\\big ( V_\\eta ^{(2)}\\big ) = \\operatorname{deg}\\big ( \\widetilde{V}(0;\\cdot )\\big ) = \\operatorname{deg}\\big ( \\widetilde{V}(1;\\cdot )\\big )$ and the degree integral formula (REF ) can be decomposed in pieces corresponding to the different steps: $&24 \\pi ^2 \\operatorname{deg}\\big ( \\widetilde{V}(1;\\cdot )\\big ) \\cr & = \\sum _{j=1}^{s-1} \\int _{\\rm {BZ}} \\int _{(j-1)T/s}^{jT/s} \\Big (\\mathcal {U}_{\\rm int}[U_{j+1}]\\big ( (t-(j-1)\\frac{_T}{^s})s \\big ) U_j \\ldots U_1 {\\rm e}^{{\\rm i}(t+\\frac{T}{s}) H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} \\Big )^*\\chi \\cr & \\qquad + \\int _{\\rm {BZ}} \\int _{(s-1)T/s}^{T} \\Big (\\mathcal {U}_{\\rm int}[U_{1}]\\big ( (t-(s-1)\\frac{_T}{^s})s \\big ) {\\rm e}^{{\\rm i}(t-(s-1)\\frac{T}{s}) H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} \\Big )^*\\chi $ where $\\rm {BZ}$ is the Brillouin zone and where in the last part we have used the fact that $U_s \\ldots U_1 = U_{\\text{F}}^{(1)} = {\\rm e}^{-{\\rm i}T \\mathcal {H}_\\eta ^{\\rm eff}[U_{\\text{F}}^{(1)}]}.$ Then by a change of variable $t \\mapsto t-T/s$ for the first term, and $t \\mapsto t-(s-1)T/s$ for the second, we end up by reordering the terms as $&24 \\pi ^2 \\operatorname{deg}\\big ( \\widetilde{V}(1;\\cdot )\\big ) \\cr & = \\sum _{j=1}^{s} \\int _{\\rm {BZ}} \\int _{(j-1)T/s}^{jT/s} \\Big (\\mathcal {U}_{\\rm int}[U_{j}]\\big ( (t-(j-1)\\frac{_T}{^s})s \\big ) U_{j-1} \\ldots U_1 {\\rm e}^{{\\rm i}t H_{\\eta }^{\\rm eff}[U_{\\text{F}}^{(1)}]} \\Big )^*\\chi \\cr &=24 \\pi ^2 \\operatorname{deg}\\big ( V_\\eta ^{(1)}\\big ) = 24 \\pi ^2 W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}]$ where the empty product $U_{j-1} \\ldots U_1$ is the identity for $j=1$ .", "This concludes the proof.$\\square $ The generalization to $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(1)}] = W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(n)}] $ for any $n$ is straightforward by noticing that $U_{\\text{F}}^{(n)} = (U_{n-1}\\ldots U_1) \\, U_{\\text{F}}^{(1)} \\, (U_{n-1}\\ldots U_1)^{-1}$ and by defining the corresponding homotopy $\\widetilde{V}(r;t,k) = V^{(n)}_\\eta (t,k) \\mathcal {U}_{\\rm int}[U_{n-1} \\ldots U_1](rT,k) {\\rm e}^{{\\rm i}r (n-1) \\frac{T}{s} H^{\\rm eff}_\\eta [U_{\\text{F}}^{(1)}](k)}.$" ], [ "Proof of the identity (", "The proof of the identity (REF ) between the one-step invariant $W_{\\eta /s}^\\text{HC}[\\mathcal {S}]$ in the Ho-Chalker point of view and the stepwise invariant $W^{\\text{SWE}}_{\\eta }[U_{\\text{F}}^{(n)}]$ in the Floquet point of view is done by direct computation of each invariant, and using the fact that $\\mathcal {S}^s$ is related to the $U_{\\text{F}}^{(n)}$ .", "First we start with the one-step evolution invariant $W_{\\eta }^\\text{HC}[\\mathcal {S}] = \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( \\mathcal {U}_{\\rm {int},D}[\\mathcal {S}] {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [\\mathcal {S}]} \\right)^* \\chi $ see (REF ) and (REF ), where $V^*\\chi = \\operatorname{tr}( (V^{-1} {\\rm d}V)^3)$ .", "Then, using identity $(AB)^*\\chi = A^*\\chi + B^\\star \\chi - 3 {\\rm d}\\operatorname{tr}(A^{-1} {\\rm d}A {\\rm d}B \\, B^{-1})$ (see e.g.", "appendix A of reference [23]), we get $W_{\\eta }^\\text{HC}[\\mathcal {S}] = \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( \\mathcal {U}_{\\rm {int},D}[\\mathcal {S}] \\right)^* \\chi + \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [\\mathcal {S}]} \\right)^* \\chi $ Indeed by Stokes formula the third term is vanishing since it is reduced to an integration over the boundaries $t=0$ and $t=T$ of $[0,T]\\times \\rm {BZ}$ .", "At $t=0$ the two maps are constant ($k$ -independent) and at $t=T$ we get $\\mathcal {U}_{\\rm {int},\\text{HC}}[\\mathcal {S}](T) = \\mathcal {S}$ whereas ${\\rm e}^{{\\rm i}T H^{\\rm eff}_\\eta [\\mathcal {S}]} = \\mathcal {S}^{-1}$ , leading to $\\operatorname{tr}(\\mathcal {S}^{-1} {\\rm d}\\mathcal {S}{\\rm d}(\\mathcal {S}^{-1}) \\mathcal {S}) = - \\operatorname{tr}(\\mathcal {S}^{-1} {\\rm d}\\mathcal {S})^2 = 0$ by antisymmetry.", "Note that even if the two quantities in the latter equation are not integers anymore, they will however respectively coincide with some terms coming from the computation of $W$ for the SWE.", "Before that the first part can already be improved by noticing that $\\mathcal {U}^{-1}_{\\rm {int},\\text{HC}}[\\mathcal {S}] {\\rm d}\\mathcal {U}_{\\rm {int},\\text{HC}}[\\mathcal {S}] & = \\text{diag} \\left( \\mathcal {U}^{-1}_{\\rm {int}}[U_1] {\\rm d}\\mathcal {U}_{\\rm {int}}[U_1], \\quad \\ldots \\quad , \\, \\mathcal {U}^{-1}_{\\rm {int}}[U_s] {\\rm d}\\mathcal {U}_{\\rm {int}}[U_s] \\right) \\cr & = \\text{diag} \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_{-\\pi }[U_1]} {\\rm d}{\\rm e}^{-{\\rm i}t H^{\\rm eff}_{-\\pi }[U_1]} ,\\quad \\ldots \\quad , \\,{\\rm e}^{{\\rm i}t H^{\\rm eff}_{-\\pi }[U_s]} {\\rm e}^{-{\\rm i}t H^{\\rm eff}_{-\\pi }[U_s]} \\right)$ see (REF ) and (REF ), so that finally $W_{\\eta }^\\text{HC}[\\mathcal {S}] = \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\sum _{n=1}^s \\left({\\rm e}^{-{\\rm i}t H^{\\rm eff}_{-\\pi }[U_n] } \\right)^* \\chi + \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [\\mathcal {S}]} \\right)^* \\chi $ On the other hand, the invariant of the SWE is computed similarly: from definitions (REF ) and (REF ), separating and rescaling the time of each step $t_{j-1} \\le t \\le t_j$ in the integral by a change of variables $t^{\\prime }=(t-t_{j-1})/(t_j-t_{j-1})$ , one has the following decomposition $W_{\\eta }^{\\rm SWE}[U_{\\text{F}}^{(1)}] = &\\, \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( \\mathcal {U}_{\\rm {int}}[U_1] \\right)^* \\chi + \\left( \\mathcal {U}_{\\rm {int}}[U_2]U_1 \\right)^* \\chi +\\dots + \\left( \\mathcal {U}_{\\rm {int}}[U_s] U_{s-1} \\ldots U_1 \\right)^* \\chi \\cr & + \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [U_{\\text{F}}^{(1)}]} \\right)^* \\chi $ Then using again identity (REF ), the fact that $U_n^*\\chi =0$ since $\\chi $ is a 3-form and $U_n$ only depends on the two-dimensional variable $k$ and not on $t$ , and the boundary values of $\\mathcal {U}_{\\rm {int}}[U_n] = \\text{Id}, \\, U_n$ at $t=0,\\, T$ respectively, we get $W_{\\eta }^{\\rm SWE}[U_{\\text{F}}^{(1)}] = &\\, \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\sum _{n=1}^s \\left( {\\rm e}^{-{\\rm i}t H^{\\rm eff}_{-\\pi }[U_n]} \\right)^* \\chi + \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [U_{\\text{F}}^{(1)}]} \\right)^* \\chi \\cr & - \\frac{1}{8\\pi ^2} \\int _{\\rm {BZ}} \\sum _{n=2}^s \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right)$ We see similarities between (REF ) and the latter equation, however it involves the particular choice of $U_{\\text{F}}^{(1)}$ whereas the first one involves $\\mathcal {S}$ that is in some sense more symmetric.", "Hence to see the equality between the two invariants we use the following trick: since all the $W_{\\eta }^{\\rm SWE}[U_{\\text{F}}^{(j)}]$ are all equal from appendix , we can symmetrize the previous quantity as $W_{\\eta }^{\\rm SWE}[U_{\\text{F}}^{(1)}] = & \\,\\dfrac{1}{s} \\sum _{j=1}^s W_{\\eta }^{\\rm SWE}[U_{\\text{F}}^{(j)}]\\cr = & \\, \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\sum _{n=1}^s \\left( {\\rm e}^{-{\\rm i}t H^{\\rm eff}_{-\\pi }[U_n]} \\right)^* \\chi + \\dfrac{1}{s} \\sum _{j=1}^s \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [U_{\\text{F}}^{(j)}]} \\right)^* \\chi \\cr & - \\frac{1}{8\\pi ^2} \\int _{\\rm {BZ}} \\dfrac{1}{s} \\left(\\sum _{n=2}^s \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right) + \\circlearrowleft \\right).$ Indeed the first term is already symmetric and then remain unchanged, whereas the two other terms appear symmetrized, where $\\circlearrowleft $ corresponds to all the possible cyclic permutations.", "For example, when $s = 3$ the last term is simply equal to $&\\dfrac{1}{3} \\left(\\sum _{n=2}^3 \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right) + \\circlearrowleft \\right) \\cr & = \\dfrac{1}{3} \\Big ( U_2^{-1} {\\rm d}U_2 {\\rm d}(U_1) U_1^{-1} + U_3 {\\rm d}U_3 {\\rm d}(U_2 U_1) (U_2 U_1)^{-1} \\cr & \\hspace{28.45274pt} + U_3^{-1} {\\rm d}U_3 {\\rm d}(U_2) U_2^{-1} + U_1 {\\rm d}U_1 {\\rm d}(U_3 U_2) (U_3 U_2)^{-1} \\cr & \\hspace{28.45274pt} + U_1^{-1} {\\rm d}U_1 {\\rm d}(U_3) U_3^{-1} + U_2 {\\rm d}U_2 {\\rm d}(U_1 U_3) (U_1 U_3)^{-1} \\Big )$ each line corresponding to one of the cyclic permutations of $(3,2,1)$ , namely $(1,3,2)$ and $(2,1,3)$ .", "Coming back to the general case and comparing (REF ) with (REF ), we see that the identity between the two invariants holds if and only if we have the following equality $&\\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_{\\eta /s}[\\mathcal {S}]} \\right)^* \\chi \\cr &= \\, \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\dfrac{1}{s} \\sum _{j=1}^s \\left( {\\rm e}^{{\\rm i}t H^{\\rm eff}_\\eta [U_{\\text{F}}^{(j)}]} \\right)^* \\chi \\cr &\\hspace{28.45274pt} - \\frac{1}{8\\pi ^2} \\int _{\\rm {BZ}} \\dfrac{1}{s} \\left(\\sum _{n=2}^s \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right) + \\circlearrowleft \\right)$ Note the difference of parameters in the effective Hamiltonians, coming from (REF ).", "This equality will be proved using the spectral decompositions of $\\mathcal {S}$ and $\\mathcal {S}^s$ .", "The spectral decomposition of $\\mathcal {S}$ is $\\mathcal {S}= \\sum _{r=0}^{s-1} \\sum _{j=1}^{b} {\\rm e}^{-{\\rm i}2 \\pi r/s} \\lambda _j D^{r} \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ due to the structure constraint (REF ), see equation (REF ).", "Hence, $H_\\eta ^{\\rm eff}[\\mathcal {S}] ={\\rm i}\\sum _{r=0}^{s-1} \\sum _{j=1}^{b} \\ln _{-\\eta }(\\lambda _j) D^r \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}+ 2 \\pi \\sum _{r=1}^{s-1} \\sum _{j=1}^{b} \\frac{r}{s} D^r \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ Besides, the $s$ -th power of $\\mathcal {S}$ reads $\\mathcal {S}^{s} = \\sum _{r=0}^{s-1} \\sum _{j=1}^{b} \\lambda _j^{s} D^{r} \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ so its effective Hamiltonian is $H_\\eta ^{\\rm eff}[\\mathcal {S}^s] = &\\, {\\rm i}\\sum _{r=0}^{s-1} \\sum _{j=1}^{b} s \\ln _{-\\eta }(\\lambda _j) D^r \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ The two effective Hamiltonians are indeed not equal for the same branch cut.", "However, if $\\lambda ^s = {\\rm e}^{{\\rm i}\\varphi }$ with $-\\eta - 2 \\pi < \\varphi < -\\eta $ , then $-\\frac{\\eta }{s} - 2 \\pi < -\\frac{\\eta }{s} - \\frac{2\\pi }{s} < \\frac{\\varphi }{s} < - \\frac{\\eta }{s}$ from which we deduce (using the definition (REF ) of the logarithm) that $H_\\eta ^{\\rm eff}[\\mathcal {S}^s] = s H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] + 2 \\pi \\sum _{r=1}^{s-1} \\sum _{j=1}^{b} r D^r \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ On top of that, since $\\mathcal {S}^s$ is block diagonal, we immediately deduce its effective Hamiltonian in terms of the $U_{\\text{F}}^{(n)}$ from (REF ), so we finally get $\\text{diag}\\left( H_\\eta ^{\\rm eff}[U_{\\text{F}}^{(1)}], \\ldots , H_\\eta ^{\\rm eff}[U_{\\text{F}}^{(s)}] \\right)= s H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] + 2 \\pi \\sum _{r=1}^{s-1} \\sum _{j=1}^{b} r D^r \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ We now wish to take the exponential of ${\\rm i}t$ times this equality, and to compute the 3-form $\\chi $ on the result.", "The two terms on the right hand side commute because they are both decomposed on the mutually orthogonal projectors $D^{r} \\mathinner {|{\\psi _j}\\rangle }\\mathinner {\\langle {\\psi _j}|} D^{-r}$ , and the left hand side is block diagonal so $\\sum _{n=1}^s \\left( {\\rm e}^{{\\rm i}t H_\\eta ^{\\rm eff}[U_{\\text{F}}^{(n)}] }\\right)^*\\chi = \\left( \\Big ({\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\Big )^s \\prod _{r=1}^{s-1} D^r {\\rm e}^{{\\rm i}2 \\pi r t \\Pi }D^{-r}\\right)^*\\chi $ where $\\Pi \\equiv \\mathinner {|{\\psi _1}\\rangle }\\!\\mathinner {\\langle {\\psi _1}|} + \\cdots + \\mathinner {|{\\psi _{b}}\\rangle }\\!\\mathinner {\\langle {\\psi _{b}}|}$ is the projector on the fundamental domain $F$ of $\\mathcal {S}$ as explained in section REF .", "Using again identity (REF ) we get $&\\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( \\Big ({\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\Big )^s \\prod _{r=1}^{s-1} D^r {\\rm e}^{{\\rm i}2 \\pi r t \\Pi }D^{-r}\\right)^*\\chi \\cr & = \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}}\\left( \\Big ({\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\Big )^s\\right)^*\\chi + \\sum _{r=1}^{s-1} \\left( D^r {\\rm e}^{{\\rm i}2 \\pi r t \\Pi }D^{-r}\\right)^*\\chi + 0$ where the 0 comes from the fact that ${\\rm e}^{-{\\rm i}2 \\pi r t \\Pi } = \\text{Id}$ both at $t=0$ and 1.", "First, using identity (A13) of [27] and equation (REF ), we have $\\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\left( D^r {\\rm e}^{{\\rm i}2 \\pi r t \\Pi }D^{-r}\\right)^*\\chi = - r \\, C_1(D^r \\Pi D^{-r}) = - r \\, C_1(\\Pi ) = 0$ for every $r=1,\\ldots ,s-1$ , so that $\\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}} \\sum _{n=1}^s \\left( {\\rm e}^{{\\rm i}t H_\\eta ^{\\rm eff}[U_{\\text{F}}^{(n)}] }\\right)^*\\chi = \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}}\\left( \\Big ({\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\Big )^s\\right)^*\\chi $ Then, by induction on $s$ of identity (REF ), and with the fact that ${\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] } = \\mathcal {S}^{-1}$ at $t=T$ we get $&\\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}}\\left( \\Big ({\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\Big )^s\\right)^*\\chi \\cr & = s \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}}\\left( {\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\right)^*\\chi + \\frac{1}{8\\pi ^2} \\int _{\\rm {BZ}} \\sum _{k=1}^{s-1} \\operatorname{tr}\\Big (\\mathcal {S}^{-1}{\\rm d}\\mathcal {S}{\\rm d}(\\mathcal {S}^k) \\mathcal {S}^{-k} \\big )$ Finally, because of the specific form of $\\mathcal {S}$ given by (REF ), ${\\rm d}(\\mathcal {S}^k) \\mathcal {S}^{-k}$ is always block-diagonal for any $k$ , with blocks of the form ${\\rm d}(U_n \\ldots U_{n-k+1}) (U_n \\ldots U_{n-k+1})^{-1}$ and all the corresponding cyclic permutations.", "From which we infer $\\sum _{k=1}^{s-1} \\operatorname{tr}\\Big (\\mathcal {S}^{-1}{\\rm d}\\mathcal {S}{\\rm d}(\\mathcal {S}^k) \\mathcal {S}^{-k} \\big ) = \\left(\\sum _{n=2}^s \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right) + \\circlearrowleft \\right)$ Putting all together the last three equations, we get $\\frac{1}{24\\pi ^2}& \\int _{[0,T]\\times \\rm {BZ}} \\sum _{n=1}^s \\left( {\\rm e}^{{\\rm i}t H_\\eta ^{\\rm eff}[U_{\\text{F}}^{(n)}] }\\right)^*\\chi \\cr &= s \\frac{1}{24\\pi ^2} \\int _{[0,T]\\times \\rm {BZ}}\\left( {\\rm e}^{{\\rm i}t H_{\\eta /s}^{\\rm eff}[\\mathcal {S}] }\\right)^*\\chi \\cr &\\hspace{14.22636pt}+ \\frac{1}{8\\pi ^2} \\int _{\\rm {BZ}} \\left(\\sum _{n=2}^s \\operatorname{tr}\\left( U_n^{-1}{\\rm d}U_n {\\rm d}(U_{n-1} \\ldots U_1) (U_{n-1} \\ldots U_1)^{-1} \\right) + \\circlearrowleft \\right)$ which establishes the equality (REF ) and completes the proof of identity (REF ) between the two invariants.$\\square $" ] ]
1612.05769
[ [ "The signal of ill-defined CPT weakening entanglement in the $B_d$ system" ], [ "Abstract In the presence of quantum gravity fluctuations (space-time foam), the CPT operator may be ill-defined.", "Its perturbative treatment leads to a modification of the Einstein-Podolsky-Rosen correlation of the neutral meson system by adding an Entanglement-weakening term of the wrong exchange symmetry, the $\\omega$-effect.", "In the current paper we identify how to probe the complex $\\omega$ in the entangled $B_d$-system using Flavour(f)-CP(g) eigenstate decay channels: the connection between the Intensities for the two time-ordered decays (f, g) and (g, f) is lost.", "Appropriate observables are constructed allowing independent experimental determinations of Re($\\omega$) and Im($\\omega$), disentangled from CPT violation in the evolution Hamiltonian Re($\\theta$) and Im($\\theta$).", "2-$\\sigma$ tensions for both Re($\\theta$) and Im($\\omega$) are shown to be uncorrelated." ], [ "Introduction", "The physics of discrete symmetries in particle and nuclear physics has always been a fascinating subject, since the observation of CP Violation in the neutral Kaon system [1], which was a clear experimental surprise, and set the scene for subsequent precision tests of such discrete symmetries in other systems, including entangled neutral meson factories.", "Today CP violation in the $K$ and $B_d$ systems, as well as T violation with entangled $B_d$ 's [2], have been demonstrated experimentally to great accuracy.", "However, their combination CPT remains unbroken.", "This is believed to be due to one of the crucial theorems of modern physics, ensuring CPT Invariance of quantum field theory models that are Lorentz invariant, local (in their interactions) and unitary (that is they conserve probability) [3].", "This is basically a theorem of flat space-time.", "Quantum gravity or in general deviations from any of the three assumptions may lead to (independent) violations of CPT, which, if observed in nature, would undoubtedly constitute an indication of completely novel physics.", "Having mentioned quantum gravity, it is worth recalling a corollary by Wald [4], according to which a potential decoherence induced during observations in local scattering experiments in which the experimenter has no access to microscopic quantum gravity degrees of freedom, may lead to an effectively ill-defined CPT quantum mechanical operator.", "This observation prompted the authors of [5] to introduce a different observable for this kind of decoherence-induced CPT violation, termed $\\omega $ -effect.", "The $\\omega $ -effect is different from the situation where CPT violation is violated in the effective hamiltonian, parameterised by the complex $\\theta $ parameter.", "Among other possible sources, $\\theta $ can be due to, e.g., Lorentz violation [6], [7] as a result of propagation in some Lorentz violating space-time (or otherwise) backgrounds.", "In the latter case the quantum mechanical operator that implements CPT symmetry is well defined but simply does not commute with the hamiltonian.", "The $\\omega $ -effect, if observed, points to an observation of a phenomenon that is exclusively linked to ill-defined nature of the CPT operator, which to date is theoretically linked only to fundamental decoherence [4], independently of any violation of CPT in the hamiltonian.", "Recently, a study for separate direct evidence of T, CP, CPT symmetry violation was accomplished [8].", "It was based on the precise identification of genuine asymmetry parameters in the time evolution of intensities between the two decays in a B-Factory of entangled neutral $B_d$ -meson states.", "Their values were obtained from the BaBar measurements [2] of the different Flavour-CP eigenstate decay channels.", "The concept, put forward in [9], [10], uses the entangled character of the initial state as the crucial ingredient to (i) connect experimental double decay rates with specific meson transitions probabilities and (ii) identify the transformed transition to that taken as a reference [11], [12].", "Possible fake effects [8] were demonstrated to be well under control by measurements in the same experiment.", "The methodology, discussed in [9], [10], appears to be [13] crucially dependent on the assumed maximal entanglement between $B^0_d$ and $\\bar{B}^0_d$ , or between two orthogonal superpositions of them, as given by the Einstein-Podolsky-Rosen (EPR) correlation [14] imposed by their decay from the $\\Upsilon (4S)$ -state with C = -.", "The corresponding antisymmetric state of the system has two important implications: (i) the program of using Entanglement and the decays as filtering measurements to prepare and detect the meson states can be implemented at any time for the first decay, even in presence of mixing during the previous entangled evolution; (ii) the coefficients of the different time-dependent terms in the double decay rate intensities for the time-ordered decays to $(g,f)$ are related to those for the time-ordered decays to $(f,g)$ .", "The antisymmetry of the entangled state is kept for any two independent states of the neutral mesons, so its evolution leads to a trivial time dependence with definite symmetry under the combined exchange $(f,t_0; g,t_0+t)\\rightarrow (g,t_0-t;f,t_0)$ .", "As a consequence, the double decay rate intensity (see eq.", "(REF ) below) satisfies for the coefficients of its time dependence with $\\omega =0$ , $C_{h}[f,g]=C_{h}[{g},{f}],\\quad C_{c}[{f},{g}]=C_{c}[g,f]\\quad \\text{and}\\quad S_{c}[f,g]=-S_{c}[g,f],$ where the time-ordered decays $(f,g)$ and $(g,f)$ are, in general, not connected by any symmetry transformation.", "At this level, they can be considered as two different experimental ways of measuring the same quantity when $\\omega =0$ .", "In the application to definite Flavour or CP eigenstates decay products, the preparation by maximal entanglement of the initial state of a single neutral meson is usually referred to as “flavour tagging” $B^0_d$ , $\\bar{B}^0_d$ , or “CP tagging” $B_+$ , $B_-$ .", "The underlying assumption considers $B^0_d$ , $\\bar{B}^0_d$ as two states of the same field, in order to impose Bose statistics with charge conjugation C and permutation $\\mathcal {P}$ with C$\\mathcal {P}$ = +, and it may be invalidated if the CPT operator cannot be intrinsically well defined, as mentioned above.", "This latter circumstance may occur, for example, in the context of an extended class of quantum gravity models, where the structure of quantum space time at Planckian scales ($10^{-35}$ m) may actually be fuzzy, characterised by a “foamy” nature (space-time foam) [15], [16], [5].", "Let us emphasize once more that this kind of CPT breaking is different from an explicit CPT violation in the hamiltonian dynamics such that [CPT, H] $\\ne 0$ , as conventionally introduced, in the context of the Weisskopf-Wigner approach [17], [18], [19] for the neutral meson system, in the mass matrix.", "This last CPT violation does not invalidate the analysis followed in [8] and, in fact, genuine observables for CPT violation were found with their values obtained from experiment.", "However, the CPT breaking associated to “ill-defined” particle-antiparticle states modifies the EPR correlation, producing the aforementioned $\\omega $ -effect [5], [20], [21].", "Treating it in perturbation theory, in such a way that we still talk the language of $B^0_d$ , $\\bar{B}^0_d$ , the perturbed two-particle state will contain a component of the “wrong” symmetry at the instant of their production by the decay of $\\Upsilon (4S)$ : $|\\Psi _0\\rangle \\propto |B^0_d\\rangle |\\bar{B}^0_d\\rangle -|\\bar{B}^0_d\\rangle |B^0_d\\rangle +\\omega \\big [|B^0_d\\rangle |\\bar{B}^0_d\\rangle +|\\bar{B}^0_d\\rangle |B^0_d\\rangle \\big ] ,$ where $\\omega = |\\omega | e^{i\\Omega }$ is a complex CPT-breaking parameter [5], [20], associated with the non-identical particle nature of the neutral meson and antimeson states.", "The presence of an $\\omega $ -effect weakens the entanglement of the initial state (REF ), as follows from the fact that when $\\omega =\\pm 1$ the state simply reduces to a product state, whilst when $\\omega =0$ the state is fully entangled.", "We emphasize that the modification in eq.", "(REF ) is due to the loss of indistinguishability of $B^0_d$ and $\\bar{B}^0_d$ and not due to violation of symmetries in the production process.", "Evidently, the probabilities for the two states connected by a permutation are different due to the presence of $\\omega $ .", "This modification of the initial state vector has far-reaching consequences for the concept of meson tagging and for the relation of the time dependent intensities between the decays to time-ordered $(f,g)$ and $(g,f)$ channelAnother important aspect of the $\\omega $ effect is its dynamical generation during a decoherence evolution, in which the particle interacts with its gravitational environment, for instance.", "As discussed in [22], a time-dependent contribution to the $\\omega $ parameter may be generated in specific models of quantum decoherence, which could be present even if the initial state has an $\\omega $ =0.", "The relative magnitude of $\\text{Re}\\left(\\omega (t)\\right)$ and $\\text{Im}\\left(\\omega (t)\\right)$ in this case depends crucially on the decoherence space-time foam model used, but their generic form involves oscillatory dependences on time.", "Specifically, if one ignores conventional CPT violating effects, then the analysis of [22] has shown that the evolution of an entangled two-particle state contains in certain quantum space-time-foam models time-dependent $\\omega (t)$ parts, which to leading order in appropriate small quantities assume the form: $|\\psi \\rangle \\ni e^{-i ( \\lambda _0^{(1)} + \\lambda _0^{(2)})t} \\, \\omega (t) \\Big (|k, \\uparrow \\rangle ^{(1)} \\, |-k, \\uparrow \\rangle ^{(2)} - |k, \\downarrow \\rangle ^{(1)} \\, |-k, \\downarrow \\rangle ^{(2)}\\Big )~, \\quad \\omega (t) \\sim \\omega _0 \\, {\\rm sin}(2 |\\Delta E| \\, t)$ that is purely generated by the evolution with no $\\omega $ effect in the initial state $t=0$ .", "In the above formula, the superscripts $(i), i=1,2 $ refer to individual particles, $k$ is the momentum of the particle (assuming the decaying initial state to be at rest, for brevity), $\\Delta E $ is the energy difference between the appropriate single particle states, and the arrows denote the corresponding quantum numbers of a generic two state system, while $\\lambda _0$ are the energy eigenvalues.", "The parameter $\\omega _0$ in (REF ) is in general complex.", "In some concrete models of space-time quantum foam it could be purely imaginary [22].", "In the present work we shall consider only constant $\\omega $ in the initial state (REF ).", "We reserve details for the phenomenology of a time-dependent $\\omega $ -effect, generated during the evolution, for a future publication.", ".", "In what follows we will study the non-trivial time evolution of eq.", "(REF ), in the simplified but physically relevant case of a time independent $\\omega $ , in order to (i) establish the appearance of terms of the (previously forbidden) type $|B^0_d\\rangle |B^0_d\\rangle $ and $|\\bar{B}^0_d\\rangle |\\bar{B}^0_d\\rangle $ , and (ii) introduce a set of observables, which actually serve as a direct way for measuring $\\omega $ , based on the violation of the relations eq.", "(REF ), i.e.", "using as observables for $\\omega \\ne 0$ : $C_{h}^\\omega [f,g]-C_{h}^\\omega [g,f],\\quad C_{c}^\\omega [f,g]-C_{c}^\\omega [g,f]\\quad \\text{and}\\quad S_{c}^\\omega [f,g]+S_{c}^\\omega [g,f],$ and checking experimentally the robustness of the correlation between the two states assumed during the tagging.", "This paper demonstrates that the comparison between the double decay rate Intensities for time-ordered ($f=$ Flavour, $g=$ CP) eigenstate decay products and $(g,f)$ is sensitive to both $\\text{Re}\\left(\\omega \\right)$ and $\\text{Im}\\left(\\omega \\right)$ ." ], [ "Double decay rates, time dependent intensities", "The eigenstates of the effective hamiltonian $\\mathbf {H}$ areAs is commonplace, subindices “H” and “L” correspond to the heavy and light $B_d$ states.", "$\\begin{split}&\\mathbf {H}|B_H\\rangle =\\mu _H|B_H\\rangle ,\\quad |B_H\\rangle =p_H|B^0_d\\rangle +q_H|\\bar{B}^0_d\\rangle ,\\\\&\\mathbf {H}|B_L\\rangle =\\mu _L|B_L\\rangle ,\\quad \\ |B_L\\rangle =p_L|B^0_d\\rangle -q_L|\\bar{B}^0_d\\rangle .\\end{split}$ In terms of them $|\\Psi _0\\rangle \\propto |B_L\\rangle |B_H\\rangle -|B_H\\rangle |B_L\\rangle \\\\+\\omega \\Big \\lbrace \\theta \\big [|B_H\\rangle |B_L\\rangle +|B_L\\rangle |B_H\\rangle \\big ]+(1-\\theta )\\frac{p_L}{p_H}|B_H\\rangle |B_H\\rangle -(1+\\theta )\\frac{p_H}{p_L}|B_L\\rangle |B_L\\rangle \\Big \\rbrace ,$ where $\\theta $ is a CP and CPT violating complex parameter given by $\\theta =\\frac{\\mathbf {H}_{22}-\\mathbf {H}_{11}}{\\mu _H-\\mu _L}$ .", "The time evolution of two-meson flavour states is $\\begin{pmatrix}|\\mathrm {A}(t)\\rangle \\\\ |B^0_d(t)\\rangle |B^0_d(t)\\rangle \\\\ |\\mathrm {S}(t)\\rangle \\\\ |\\bar{B}^0_d(t)\\rangle |\\bar{B}^0_d(t)\\rangle \\end{pmatrix}=e^{-\\Gamma \\,t}e^{-i2M\\,t}\\begin{pmatrix}1 & \\begin{matrix} 0 & & & 0 & & & 0 \\end{matrix}\\\\\\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} & \\mathrm {C} + \\mathrm {E}_{[+]} e^{i\\Delta \\mu \\,t}+\\mathrm {E}_{[-]} e^{-i\\Delta \\mu \\,t}\\end{pmatrix}\\begin{pmatrix}|\\mathrm {A}\\rangle \\\\ |B^0_d\\rangle |B^0_d\\rangle \\\\ |\\mathrm {S}\\rangle \\\\ |\\bar{B}^0_d\\rangle |\\bar{B}^0_d\\rangle \\end{pmatrix},$ where $\\mu _H+\\mu _L=2M-i\\Gamma $ , $\\mu _H-\\mu _L=\\Delta \\mu =\\Delta M-i\\frac{\\Delta \\Gamma }{2}$ , $\\begin{matrix}|\\mathrm {A}(t)\\rangle =\\frac{1}{\\sqrt{2}}\\left[|B^0_d(t)\\rangle |\\bar{B}^0_d(t)\\rangle -|\\bar{B}^0_d(t)\\rangle |B^0_d(t)\\rangle \\right],\\\\|\\mathrm {S}(t)\\rangle =\\frac{1}{\\sqrt{2}}\\left[|B^0_d(t)\\rangle |\\bar{B}^0_d(t)\\rangle +|\\bar{B}^0_d(t)\\rangle |B^0_d(t)\\rangle \\right],\\end{matrix}$ and $|B^0_d(t)\\rangle =e^{-i\\mathbf {H}t}\\,|B^0_d\\rangle ,\\quad |\\bar{B}^0_d(t)\\rangle =e^{-i\\mathbf {H}t}\\,|\\bar{B}^0_d\\rangle .$ The $\\mathrm {C}$ , $\\mathrm {E}_{[\\pm ]}$ matrices are $\\mathrm {C}=\\begin{pmatrix}\\frac{1}{2}(1-\\theta ^2) & \\frac{1}{\\sqrt{2}}\\frac{q}{p}\\theta \\sqrt{1-\\theta ^2} & -\\frac{1}{2}\\frac{q^2}{p^2}(1-\\theta ^2)\\\\\\frac{1}{\\sqrt{2}}\\frac{p}{q}\\theta \\sqrt{1-\\theta ^2} & \\theta ^2 & -\\frac{1}{\\sqrt{2}}\\frac{q}{p}\\theta \\sqrt{1-\\theta ^2}\\\\-\\frac{1}{2}\\frac{p^2}{q^2}(1-\\theta ^2) & -\\frac{1}{\\sqrt{2}}\\frac{p}{q}\\theta \\sqrt{1-\\theta ^2} & \\frac{1}{2}(1-\\theta ^2)\\end{pmatrix},$ $\\mathrm {E}_{[+]}=\\begin{pmatrix}\\frac{1}{4}(1+\\theta )^2 & -\\frac{1}{2\\sqrt{2}}\\frac{q}{p}(1+\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{4}\\frac{q^2}{p^2}(1-\\theta ^2)\\\\-\\frac{1}{2\\sqrt{2}}\\frac{p}{q}(1+\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{2}(1-\\theta ^2) & -\\frac{1}{2\\sqrt{2}}\\frac{q}{p}(1-\\theta )\\sqrt{1-\\theta ^2}\\\\\\frac{1}{4}\\frac{p^2}{q^2}(1-\\theta ^2) & -\\frac{1}{2\\sqrt{2}}\\frac{p}{q}(1-\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{4}(1-\\theta )^2\\end{pmatrix},$ $\\mathrm {E}_{[-]}=\\begin{pmatrix}\\frac{1}{4}(1-\\theta )^2 & \\frac{1}{2\\sqrt{2}}\\frac{q}{p}(1-\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{4}\\frac{q^2}{p^2}(1-\\theta ^2)\\\\\\frac{1}{2\\sqrt{2}}\\frac{p}{q}(1-\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{2}(1-\\theta ^2) & \\frac{1}{2\\sqrt{2}}\\frac{q}{p}(1+\\theta )\\sqrt{1-\\theta ^2}\\\\\\frac{1}{4}\\frac{p^2}{q^2}(1-\\theta ^2) & \\frac{1}{2\\sqrt{2}}\\frac{p}{q}(1+\\theta )\\sqrt{1-\\theta ^2} & \\frac{1}{4}(1+\\theta )^2\\end{pmatrix}.$ In eqs.", "(REF )-(REF ), $\\frac{q}{p}$ is the usual meson mixing quantity given by $\\frac{q^2}{p^2}=\\frac{\\mathbf {H}_{21}}{\\mathbf {H}_{12}}=\\frac{q_Hq_L}{p_Hp_L}$ .", "Before addressing actual observables, it is worth noting that, attending to eq.", "(REF ), it is clear that the presence of the symmetric state $|\\mathrm {S}\\rangle $ in eq.", "(REF ) induces the appearance of $|B^0_d\\rangle |B^0_d\\rangle $ and $|\\bar{B}^0_d\\rangle |\\bar{B}^0_d\\rangle $ states.", "The transition amplitude for the decay of the first state into $|f\\rangle $ at time $t_{0}$ , and then the second state into $|g\\rangle $ at time $t+t_{0}$ is $\\langle f,t_0;g,t+t_0|T|\\Psi _0\\rangle $ .", "Squaring and integrating over $t_{0}$ , the double decay rate $I(f,g;t)$ is obtained: $I(f,g;t)=\\int _0^\\infty \\!\\!\\!\\!\\!", "dt_0\\,|\\langle f,t_0;g,t+t_0|T|\\Psi _0\\rangle |^2\\,.$ Expanding to first order in $\\omega $ , $\\theta $ and taking $\\Delta \\Gamma =0$ , $I(f,g;t)$ has the following form for generic $f$ and $g$ decay channelsIn the notation of reference [8], $C_{h}[f,g]$ , $C_{c}[f,g]$ and $S_{c}[f,g]$ are the $\\omega \\rightarrow 0$ limit of (respectively) $C_{h}^\\omega [f,g]$ , $C_{c}^\\omega [f,g]$ and $S_{c}^\\omega [f,g]$ in eqs.", "(REF )-(REF ).", ": $I(f,g;t)=\\frac{\\langle \\Gamma _f\\rangle \\langle \\Gamma _g\\rangle }{\\Gamma }e^{-\\Gamma \\,t}\\big \\lbrace C_{h}^\\omega [f,g]+C_{c}^\\omega [f,g]\\cos (\\Delta Mt)+S_{c}^\\omega [f,g]\\sin (\\Delta Mt)\\big \\rbrace ,$ with $C_{h}^\\omega [f,g]=N_{[f,g]}\\Big [1-R_{f}R_{g}+\\text{Re}\\left(\\theta \\right)(C_{g}R_{f}+C_{f}R_{g})-\\text{Im}\\left(\\theta \\right)(S_{f}+S_{g})\\\\+\\frac{1}{1+(x/2)^2}\\big \\lbrace (2C_{f}+xS_{f})\\text{Re}\\left(\\omega \\right)+(xC_{f}-2S_{f})R_{g}\\text{Im}\\left(\\omega \\right)\\big \\rbrace \\Big ],$ $C_{c}^\\omega [f,g]=N_{[f,g]}\\Big [-(C_{f}C_{g}+S_{f}S_{g})-\\text{Re}\\left(\\theta \\right)(C_{g}R_{f}+C_{f}R_{g})+\\text{Im}\\left(\\theta \\right)(S_{f}+S_{g})\\\\+\\frac{1}{1+(x/2)^2}\\big \\lbrace -(2C_{g}+xS_{g})\\text{Re}\\left(\\omega \\right)+(-xC_{g}+2S_{g})R_{f}\\text{Im}\\left(\\omega \\right)\\big \\rbrace \\Big ],$ $S_{c}^\\omega [f,g]=N_{[f,g]}\\Big [(C_{g}S_{f}-C_{f}S_{g})+\\text{Re}\\left(\\theta \\right)(R_{g}S_{f}-R_{f}S_{g})+\\text{Im}\\left(\\theta \\right)(C_{f}-C_{g})\\\\+\\frac{1}{1+(x/2)^2}\\big \\lbrace (xC_{g}-2S_{g})\\text{Re}\\left(\\omega \\right)-(2C_{g}+xS_{g})R_{f}\\text{Im}\\left(\\omega \\right)\\big \\rbrace \\Big ],$ where, in terms of the decays amplitudes $\\langle f|T|\\bar{B}^0_d\\rangle \\equiv \\bar{A}_f$ and $\\langle f|T|B^0_d\\rangle \\equiv A_f$ , the following parameters are usedBy construction $C_{f}^2+S_{f}^2+R_{f}^2=1$ .", ": $\\lambda _{f}\\equiv \\frac{q}{p}\\frac{\\bar{A}_f}{A_f},\\quad C_{f}=\\frac{1-|\\lambda _{f}|^2}{1+|\\lambda _{f}|^2},\\quad R_{f}= \\frac{2\\text{Re}\\left(\\lambda _{f}\\right)}{1+|\\lambda _{f}|^2},\\quad S_{f}= \\frac{2\\text{Im}\\left(\\lambda _{f}\\right)}{1+|\\lambda _{f}|^2},$ $N_{[f,g]}=\\frac{1-\\delta ^2}{(1+|\\omega |^2)(1-\\delta C_{f})(1-\\delta C_{g})},\\ \\text{and }\\langle \\Gamma _f\\rangle =\\frac{|\\bar{A}_f|^2+|A_f|^2}{2}.$ In addition, $x=\\frac{\\Delta M}{\\Gamma }\\simeq 0.77$ and $\\delta =\\frac{1-|q/p|^2}{1+|q/p|^2}\\simeq 1-2\\times 10^{-3}$ .", "It is worth reminding that for flavour-specific decay channels $X+\\ell ^\\pm $ (“$\\ell ^\\pm $ ” for short in the following), we have $=\\pm 1$ , $R_{\\ell ^\\pm }=S_{\\ell ^\\pm }=0$ ." ], [ "Sensitivity to ${\\omega }$", "Coming back to the transition amplitude $\\langle f,t_0;g,t+t_0|T|\\Psi _0\\rangle $ , it has the following structure: $\\langle f,t_0;g,t+t_0|T|\\Psi _0\\rangle \\propto e^{(-iM-\\Gamma /2)(2t_0+t)}\\begin{pmatrix}[e^{-i\\Delta \\mu \\,t/2}\\mathcal {A}_{f}^{L}\\mathcal {A}_{g}^{H}-e^{i\\Delta \\mu \\,t/2}\\mathcal {A}_{f}^{H}\\mathcal {A}_{g}^{L}]\\\\+\\omega \\theta [e^{-i\\Delta \\mu \\,t/2}\\mathcal {A}_{f}^{L}\\mathcal {A}_{g}^{H}+e^{i\\Delta \\mu \\,t/2}\\mathcal {A}_{f}^{H}\\mathcal {A}_{g}^{L}]\\\\+\\omega (1-\\theta )\\frac{p_L}{p_H}e^{-i\\Delta \\mu \\,(t_0+t/2)}\\mathcal {A}_{f}^{H}\\mathcal {A}_{g}^{H}\\\\-\\omega (1+\\theta )\\frac{p_H}{p_L}e^{i\\Delta \\mu \\,(t_0+t/2)}\\mathcal {A}_{f}^{L}\\mathcal {A}_{g}^{L}\\end{pmatrix}.$ The prefactor $e^{(-iM-\\Gamma /2)(2t_0+t)}$ gives a global $e^{-2\\Gamma t_0}e^{-\\Gamma t}$ dependence in $|\\langle f,t_0;g,t+t_0|T|\\Psi _0\\rangle |^2$ .", "One can readily observe that the $\\omega $ -dependent terms, even for $\\theta =0$ (i.e.", "already for the leading $\\omega $ contribution), do introduce an additional non-trivial $t_0$ dependence.", "Ignoring that $e^{(-iM-\\Gamma /2)(2t_0+t)}$ prefactor, it is clear that combining the transformations $t\\mapsto -t$ and $f\\leftrightarrows g$ , the first contribution, the standard $\\omega =0$ one, just receives a $(-)$ sign.", "This implies that, in the absence of $\\omega $ , in the $t$ -dependence of $I(f,g;t)$ , $I(f,g;t)\\sim e^{-\\Gamma t}\\left(C_{h}[f,g]+C_{c}[f,g]\\cos (\\Delta Mt)+S_{c}[f,g]\\sin (\\Delta Mt)\\right)$ we necessarily have [8]: $C_{h}[f,g]=C_{h}[g,f]$ , $C_{c}[f,g]=C_{c}[g,f]$ and $S_{c}[f,g]=-S_{c}[g,f]$ .", "In the presence of $\\omega \\ne 0$ the situation changes drastically.", "From the remaining contributions in eq.", "(REF ), the ones induced by the evolution of the $\\omega $ -dependent term in eq.", "(REF ), the situation is more involved: the first one, proportional to $\\omega \\theta $ and $t_0$ -independent, is clearly invariant under the combination of $f\\leftrightarrows g$ and $t\\mapsto -t$ .", "The last two terms are separately invariant under $f\\leftrightarrows g$ , but have no well defined transformation under $t\\mapsto -t$ ; moreover, contrary to the previous contributions, they depend on $t_0$ , the time elapsed between production of the $B\\bar{B}$ pair and the first decayFor small $\\omega $ and $\\theta $ , these terms give the leading $\\omega $ contributions: the $t_0$ dependence integrated over in eq.", "(REF ) produces extra dilution factors $x/(1+(x/2)^2)^{-1}$ and $1/(1+(x/2)^2)^{-1}$ in eqs.", "(REF ) to (REF ); fortunately, in the $B_d$ system, they do not thwart significantly the sensitivity to $\\omega $ .. Out of those properties, the simple assignment of symmetry/antisymmetry under $f\\leftrightarrows g$ to the $t$ -even/$t$ -odd terms in $e^{\\Gamma \\,t}\\,I(f,g;t)$ , possible when $\\omega =0$ , does not apply when $\\omega \\ne 0$ .", "This simple remark provides the first understanding of the potential sensitivity to the presence of $\\omega \\ne 0$ : while in the absence of $\\omega $ , the measurement of intensities for decays into $f$ and $g$ with the two different orderings (i) first $f$ then $g$ and (ii) first $g$ then $f$ , provides two experimentally independent measurements of the same theoretical quantities, in the presence of $\\omega $ the situation has changed.", "Deviations from the standard $f\\leftrightarrows g$ symmetry properties are a gateway to probe for $\\omega $ .", "The BaBar collaboration performed separate analyses [2] for the two different time orderings of the two $B$ meson decays.", "Previous studies, like [21], exploited the use of two flavour specific decay channels to obtain bounds on $\\text{Re}\\left(\\omega \\right)$ through the appearance of $|B^0_d\\rangle |B^0_d\\rangle $ and $|\\bar{B}^0_d\\rangle |\\bar{B}^0_d\\rangle $ states for $t=0$ .", "Equation (REF ) shows that, using flavour specific channels alone, there is no sensitivity to $\\text{Im}\\left(\\omega \\right)$ : since $R_{\\ell ^\\pm }=0$ , the terms in $\\text{Im}\\left(\\omega \\right)$ would be absentEquation (REF ) gives the intensity $I(f,g;t)$ expanded up to linear order in $\\theta $ and $\\omega $ : the full result has indeed contributions that depend on $\\text{Im}\\left(\\omega \\right)$ and do not vanish when both $f$ and $g$ are flavour specific, but they have additional $\\omega $ and/or $\\theta $ suppressions which make them irrelevant.", "In any case, the actual fits in section are conducted using the full expressions.. Fortunately enough, besides addressing the two different time orderings, in [2], one decay is flavour specific (labelled $\\ell ^\\pm $ ), while the other is CP specific (decays into $J/\\Psi K_{S,L}$ , labelled $K_{S,L}$ for short): sensitivity to both $\\text{Re}\\left(\\omega \\right)$ and $\\text{Im}\\left(\\omega \\right)$ is thus expected." ], [ "Experimental observables", "In order to reduce experimental uncertainties in the different channels, the BaBar collaboration, in reference [2], fixed the constant term and measured the coefficients $\\mathrm {C}[f,g]$ and $\\mathrm {S}[f,g]$ of the decay intensity $\\mathbf {g}_{f,g}(t)\\propto e^{-\\Gamma \\, t}\\left\\lbrace 1+\\mathrm {C}[f,g]\\cos (\\Delta M\\,t)+\\mathrm {S}[f,g]\\sin (\\Delta M\\,t)\\right\\rbrace ,$ using for the $f$ and $g$ states one flavour specific channel, $X\\ell ^+\\nu $ or $X\\ell ^-\\bar{\\nu }$ , and one CP eigenstate, $J/\\Psi K_S$ or $J/\\Psi K_L$ .", "Obviously we should have $\\mathrm {C}[f,g]=\\frac{C_{c}^\\omega [f,g]}{C_{h}^\\omega [f,g]}\\quad \\text{and}\\quad \\mathrm {S}[f,g]=\\frac{S_{c}^\\omega [f,g]}{C_{h}^\\omega [f,g]},$ where one should remember that in the coefficients $\\mathrm {C}[f,g]$ and $\\mathrm {S}[f,g]$ , the ordering of $f$ and $g$ means that $f$ corresponds to the first (in time) decay product of the entangled state evolved in time, and $g$ corresponds to the second (in time) decay product.", "In the case under consideration, the flavour specific decays simplify significantly the expressions, which are, at linear order in $\\theta $ , $\\omega $ , $\\mathrm {C}[\\ell ^\\pm ,g]=\\mp C_{g}+\\text{Re}\\left(\\theta \\right)R_{g}(C_{g}\\mp 1)+\\text{Im}\\left(\\theta \\right)S_{g}(1\\mp C_{g})\\\\+\\frac{1}{1+(x/2)^2}\\left\\lbrace -xS_{g}\\text{Re}\\left(\\omega \\right)+xC_{g}R_{g}\\text{Im}\\left(\\omega \\right)\\right\\rbrace ,$ $\\mathrm {S}[\\ell ^\\pm ,g]=\\mp S_{g}+\\text{Re}\\left(\\theta \\right)S_{g}R_{g}+\\text{Im}\\left(\\theta \\right)(\\pm 1-C_{g}\\mp S_{g}^2)\\\\+\\frac{1}{1+(x/2)^2}\\left\\lbrace xC_{g}\\text{Re}\\left(\\omega \\right)+xS_{g}R_{g}\\text{Im}\\left(\\omega \\right)\\right\\rbrace .$ In the presence of $\\omega $ , the time ordering definite symmetry is not valid anymore and therefore it is relevant to write the completely different coefficients $\\mathrm {C}[f,\\ell ^\\pm ]=\\mp C_{f}+\\text{Re}\\left(\\theta \\right)R_{f}(C_{f}\\mp 1)+\\text{Im}\\left(\\theta \\right)S_{f}(1\\mp C_{f})\\\\+\\frac{1}{1+(x/2)^2}\\left\\lbrace \\pm (2(C_{f}^2-1)+xC_{f}S_{f}))\\text{Re}\\left(\\omega \\right)\\mp xR_{f}\\text{Im}\\left(\\omega \\right)\\right\\rbrace ,$ $\\mathrm {S}[f,\\ell ^\\pm ]=\\pm S_{f}-\\text{Re}\\left(\\theta \\right)S_{f}R_{f}+\\text{Im}\\left(\\theta \\right)(\\mp 1+C_{f}\\pm S_{f}^2)\\\\+\\frac{1}{1+(x/2)^2}\\left\\lbrace \\pm (x(1-S_{f}^2)-2C_{f}S_{f})\\text{Re}\\left(\\omega \\right)\\mp 2R_{f}\\text{Im}\\left(\\omega \\right)\\right\\rbrace .$ As anticipated, $\\mathrm {C}[\\ell ^\\pm ,g]-\\mathrm {C}[g,\\ell ^\\pm ]$ and $\\mathrm {S}[\\ell ^\\pm ,g]+\\mathrm {S}[g,\\ell ^\\pm ]$ are linear in $\\omega $ , and thus the fact that the BaBar collaboration distinguished the different decay time orderings in [2], now reveals crucial to disentangle the $\\omega $ effect: $\\mathrm {C}[\\ell ^\\pm ,g]-\\mathrm {C}[g,\\ell ^\\pm ]=\\frac{1}{1+(x/2)^2}\\\\ \\times \\left\\lbrace \\left[xS_{g}\\mp 2(C_{g}^2-1)\\mp xC_{g}S_{g}\\right]\\text{Re}\\left(\\omega \\right)+xR_{g}\\left[C_{g}\\pm 1\\right]\\text{Im}\\left(\\omega \\right)\\right\\rbrace ,$ $\\mathrm {S}[\\ell ^\\pm ,g]+\\mathrm {S}[g,\\ell ^\\pm ]=\\frac{1}{1+(x/2)^2}\\\\ \\times \\left\\lbrace \\left[xC_{g}\\pm x(1-S_{g}^2)\\mp 2C_{g}S_{g}\\right]\\text{Re}\\left(\\omega \\right)+R_{g}\\left[xS_{g}\\mp 2\\right]\\text{Im}\\left(\\omega \\right)\\right\\rbrace .$ These combinations are linearly sensitive not only to $\\text{Re}\\left(\\omega \\right)$ but also to $\\text{Im}\\left(\\omega \\right)$ when $R_{g}\\ne 0$ .", "The sensitivity to $\\text{Im}\\left(\\omega \\right)$ depends critically on the use of a CP eigenstate channel with large $R_{g}$ , as is the case with $J/\\Psi K_S$ and $J/\\Psi K_L$ ." ], [ "Results", "We are now ready to present the results obtained from a global fit to available BaBar experimental data, following the same statistical treatment as in reference [8].", "We use the sixteen experimental observables measured by BaBar in [2]: $\\mathrm {C}[\\ell ^\\pm ,K_{S,L}]$ , $\\mathrm {C}[K_{S,L},\\ell ^\\pm ]$ , $\\mathrm {S}[\\ell ^\\pm ,K_{S,L}]$ and $\\mathrm {S}[K_{S,L},\\ell ^\\pm ]$ .", "Taking into account full covariance information on statistical and systematic uncertainties, we perform a fit in terms of the set of parameters $\\lbrace \\text{Re}\\left(\\theta \\right)$ , $\\text{Im}\\left(\\theta \\right)$ , $\\text{Re}\\left(\\omega \\right)$ , $\\text{Im}\\left(\\omega \\right)$ , $$ , $S_{K_S}$ , $R_{K_S}$ , $$ , $S_{K_L}$ , $R_{K_L}\\rbrace $ with the known constraints $C_{f}^2+S_{f}^2+R_{f}^2=1$ .", "Therefore we generalize the corresponding fit presented in reference [8] to the actual situation where deviations from EPR entanglement are present due to the $\\omega $ -effect [5].", "A more restricted fit is also done in the case where no wrong sign flavour decays are allowed in the $B_d\\rightarrow J/\\Psi K$ decays, that is with $\\lambda _{K_S}+\\lambda _{K_L}=0$ .", "In table REF (I) we present the general result of the fit, whose most salient features are the following: Experimental data – more precisely the BaBar measurements in [2] – are sensitive for the first time to $\\text{Im}\\left(\\omega \\right)$ , revealing a tantalizing $2.4\\sigma $ deviation from $\\text{Im}\\left(\\omega \\right)=0$ .", "These observables are also sensitive to $\\text{Re}\\left(\\omega \\right)$ , but they do not show any significant deviation from $\\text{Re}\\left(\\omega \\right)=0$ , and the previous determination $\\text{Re}\\left(\\omega \\right)=(0.8\\pm 4.6)\\times 10^{-3}$ [21] – using semileptonic channels – is still better than the present one.", "The results of the fit for the CPT violating parameter $\\theta $ – in the evolution hamiltonian – are compatible with the previous determination in [8] and the one performed by the BaBar collaboration in reference [23].", "An exciting $2\\sigma $ effect in $\\text{Re}\\left(\\theta \\right)$ is still present.", "The parameters that measure the presence of wrong flavour decays in $B_d\\rightarrow J/\\Psi K$ , i.e.", "$-$ , $S_{K_S}+S_{K_L}$ and $R_{K_S}+R_{K_L}$ , do not show any significant deviation from zero and the results are consistent with [8].", "In the case of $S_{K_S}$ and $R_{K_S}$ we observe that they differ by more than $1\\sigma $ with respect to the determination in [8] without including the $\\omega $ effect.", "Should this persist in the future, it could affect the precise determination of the unitarity triangle angle $\\beta $ .", "In table REF (II) we present the results of the same fit with the additional requirement of not having wrong flavour decays, $\\lambda _{K_S}+\\lambda _{K_L}=0$ .", "No significant differences were noticed with respect to the conclusions discussed above for the general case.", "For completeness we show, when relevant, both analyses together in the same plots without further comments.", "Table: Summary of results.In figure REF is shown the result for the new parameters not previously considered in the analyses where EPR entangled initial states where assumed.", "A deviation of the complex number $\\omega $ from zero is found at 95% confidence level.", "This deviation comes essentially from $\\text{Im}\\left(\\omega \\right)$ and it represents a measurement of this parameter for the first time; the measurement of $\\text{Re}\\left(\\omega \\right)$ does not improve on the value obtained previously [21] from flavour specific decays.", "Figure: Imω\\text{Im}\\left(\\omega \\right) vs. Reω\\text{Re}\\left(\\omega \\right) in the general fit (blue regions with solid contours), and in the fit with λ K S +λ K L =0\\lambda _{K_S}+\\lambda _{K_L}=0 (red regions with dashed contours); darker to lighter regions correspond to two-dimensional 68%, 95% and 99% CL.", "Figures and obey the same colour coding for the two fits and the CL regions.The stability of the fitted value of the complex CPT violating parameter $\\theta $ is shown in figures REF and REF , where it is clear that the results for $\\text{Re}\\left(\\theta \\right)$ and $\\text{Im}\\left(\\theta \\right)$ do not change from the constrained case $\\omega =0$ to the general case with arbitrary $\\omega $ .", "Cross correlations among the different components of $\\theta $ and $\\omega $ are shown in figure REF .", "For example, figure REF shows the independence of $\\text{Im}\\left(\\omega \\right)$ and $\\text{Re}\\left(\\theta \\right)$ : furthermore one can see in that figure that the point $(0,0)$ in this projection is at more than $2.5\\sigma $ from the best fit values (or even at $3\\sigma $ in the $\\lambda _{K_S}+\\lambda _{K_L}=0$ constrained analysis).", "Figure: Restricted.Figure: Imω\\text{Im}\\left(\\omega \\right) vs. Imθ\\text{Im}\\left(\\theta \\right).Finally, in figure REF , one can see the near linear correlation among $\\text{Im}\\left(\\omega \\right)$ and $R_{K_S}$ .", "This explains why the presence of $\\omega $ affects both $R_{K_S}$ and $S_{K_S}$ .", "Figure: S K S S_{K_S} vs. Imωsign(R K S )\\text{Im}\\left(\\omega \\right)\\text{sign}(R_{K_S})." ], [ "Conclusions", "In the present article we have discussed the possibility of probing the entanglement-weakening CPT Violating parameter $\\omega $ , that potentially signifies the breakdown of CPT operation as a result of quantum decoherence of matter in some models of quantum gravity, by means of identifying appropriate asymmetry parameters in the time evolution of intensities (REF ) between the two decays in a B factory, based on observables that have already been used in previous studies [8] probing independently T, CP and CPT symmetries in the absence of $\\omega $ .", "In the current analysis we have included, simultaneously with the $\\omega $ , also the conventional CPT parameter $\\theta $ , already considered in [8], which parameterises CPT violation in the case of a well-defined CPT operator which however does not commute with the hamiltonian of the system, indicating a violation of CPT parameterised within the framework of effective field theories (e.g.", "due to Lorentz symmetry violation by a space-time background), in contrast to the parameter $\\omega $ that goes beyond that framework.", "As we have demonstrated in the present article the set of observables of the B system (REF ), (REF ), (REF ) and (REF ) allow for a simultaneous determination (bounds) of the CPT violating parameters $\\omega $ and $\\theta $ , which can thus be disentangled.", "The results obtained from the experimental data from the BaBar measurements [2] (see table REF (I)) are sensitive for the first time to $\\text{Im}\\left(\\omega \\right)$ , pointing towards a $2.4\\sigma $ deviation from $\\text{Im}\\left(\\omega \\right)=0$ , which we interpret as an upper bound.", "The observables (REF ),(REF ) are also sensitive to $\\text{Re}\\left(\\omega \\right)$ , but they do not show any significant deviation from $\\text{Re}\\left(\\omega \\right)=0$ , and in this sense they are inferior to the previous analyses [21] using equal sign semileptonic decay asymmetries of the B system, which yield $\\text{Re}\\left(\\omega \\right)=(0.8\\pm 4.6)\\times 10^{-3}$ .", "The results (REF ) and (REF ) also allow a fit to the CPT violating parameter $\\theta $ , and are compatible with the previous determination in [8] and the one performed by the BaBar collaboration in [23], pointing towards a $2\\sigma $ effect in $\\text{Re}\\left(\\theta \\right)$ , also interpreted as an upper bound for the corresponding parameter.", "Moreover, the parameters that measure the presence of wrong flavour decays in $B_d\\rightarrow J/\\Psi K$ , i.e.", "$-$ , $S_{K_S}+S_{K_L}$ and $R_{K_S}+R_{K_L}$ , do not show any significant deviation from zero and the results are consistent with [8].", "In the case of $S_{K_S}$ and $R_{K_S}$ we observe that they differ by more than $1\\sigma $ with respect to the determination in [8] without including the $\\omega $ effect.", "Should this persist in the future, it could affect the precise determination of the unitarity triangle angle $\\beta $ .", "Before closing we stress once more that a quantum-gravity-decoherence-induced CPT violating and entanglement-weakening parameter $\\omega $ may not only characterise the initial state of an entangled (neutral) meson system, but may also be generated as a result of a decoherening time evolution that goes beyond the local effective field theory framework [22].", "A full analysis of that case will appear in a forthcoming publication." ], [ "Acknowledgments", "JB, FJB and MN acknowledge financial support from the Spanish MINECO through Grants FPA2015-68318-R, FPA2014-54459-P and the Severo Ochoa Excellence Center Project SEV-2014-0398, and from Generalitat Valenciana through Grants PROMETEOII/2013/017 and PROMETEOII/2014/049.", "The work of NEM is supported in part by the U.K. Science and Technology Facilities Council (STFC) via the grants ST/L000326/1 and ST/P000258/1.", "MN acknowledges support from Fundação para a Ciência e a Tecnologia (FCT, Portugal) through postdoctoral grant SFRH/BPD/112999/2015 and through the project CFTP-FCT Unit 777 (UID/FIS/00777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU." ] ]
1612.05652
[ [ "On a Rogers-Ramanujan type identity from crystal base theory" ], [ "Abstract We refine and generalise a Rogers-Ramanujan type partition identity arising from crystal base theory.", "Our proof uses the variant of the method of weighted words recently introduced by the first author." ], [ "Introduction", "As combinatorial statements the Rogers-Ramanujan identities assert that for $i = 0$ or 1 and for all non-negative integers $n$ , the number of partitions of $n$ into parts differing by at least two and having at most $i$ ones is equal to the number of partitions of $n$ into parts congruent to $\\pm (2-i)$ modulo 5.", "As $q$ -series identities they say that $ \\sum _{n \\ge 0} \\frac{q^{n^2+ (1-i)n}}{(q;q)_n} = \\frac{1}{(q^{2-i};q^5)_{\\infty }(q^{3+i};q^5)_{\\infty }},$ where for $n \\in \\mathbb {N} \\cup \\lbrace \\infty \\rbrace $ we have $(a;q)_n := \\prod _{k=0}^{n-1} (1-aq^k).$ A Lie-theoretic interpretation and proof of these identities were given by Lepowsky and Wilson [21], [22].", "Up to a factor of $(-q;q)_{\\infty }$ , the right-hand side of (REF ) is the principally specialised Weyl-Kac character formula for level 3 standard $A_1^{(1)}$ -modules [19], [20], while the product of this factor and the left-hand side corresponds to bases constructed from vertex operators.", "The vertex operator approach of Lepowsky and Wilson was subsequently extended by many authors to treat level $k$ and/or other affine Lie algebras, beginning a fruitful interaction between Lie theory and partition theory.", "For some examples of vertex operator constructions leading to partition identities, see [8], [9], [23], [24], [25], [30], and for some combinatorial approaches to such partition identities we refer to [1], [7], [11], [16].", "In [28], Primc observed that the difference conditions in certain vertex operator constructions correspond to energy functions of perfect crystals, and in [27] he studied partition identities of the Rogers-Ramanujan type coming from crystal base theory.", "(For other early examples of the study of Rogers-Ramanujan type identities from the point of view of crystal bases, see [17], [26], [29].)", "Here the Weyl-Kac character formula again gives the partitions defined by congruence conditions, while the crystal base character formula of Kang, Kashiwara, Misra, Miwa, Nakashima and Nakayashiki [18] ensures the correspondence with partitions defined by difference conditions.", "In this paper we will be concerned with the following partition identity of Primc.", "Consider partitions $(\\lambda _1, \\lambda _2,\\dots )$ into parts of four colours $a,b,c,d$ , with the order $1_{a} < 1_{b} < 1_{c} <1_{d} <2_{a} < 2_{b} <2_{c} < 2_{d} < \\cdots ,$ where $k_z$ denotes the part $k$ of colour $z$ for $k \\in \\mathbb {N}$ and $z \\in \\lbrace a,b,c,d\\rbrace $ .", "Let the minimal difference between consecutive parts of colour $x$ and $y$ be given by the entry $(x,y)$ in the matrix $ D={\\begin{bordermatrix }\\text{} & a & b & c & d \\\\a & 2&1&2&2 \\\\b &1&0&1&1 \\\\c &0&1&0&2 \\\\d&0&1&0&2\\end{bordermatrix }}.$ Then $\\sum _{\\lambda } q^{\\sum _{k \\ge 1}\\big ( (2k-1)A_k(\\lambda ) + 2k (B_k(\\lambda ) + C_k(\\lambda )) + (2k+1)D_k(\\lambda )\\big )} = \\frac{1}{(q)_{\\infty }},$ where the sum is over the coloured partitions $\\lambda $ satisfying the difference conditions given by (REF ) and where $A_k(\\lambda )$ (resp.", "$B_k(\\lambda ), C_k(\\lambda ), D_k(\\lambda )$ ) denotes the number of parts $k$ of colour $a$ (resp.", "$b, c ,d$ ) in $\\lambda $ .", "In other words, if the coloured integers in (REF ) are transformed by $ \\begin{aligned}k_{a} &\\rightarrow 2k-1,\\\\k_{b} &\\rightarrow 2k,\\\\k_{c} &\\rightarrow 2k,\\\\k_{d} &\\rightarrow 2k+1,\\end{aligned}$ the generating function for the resulting coloured partitions with the difference conditions inherited from (REF ) is equal to the generating function for ordinary partitionsThis was actually stated with a question mark by Primc, who was unsure of the application of the crystal base formula of [18] to the case of the $A_1^{(1)}$ -crystal whose energy matrix is (REF ).", "We are indebted to K. Misra for pointing out that this case is covered by Section 1.2 of [18], rendering Primc's question mark unnecessary.. Our main result is a generalisation and refinement of Primc's identity.", "Theorem 1.1 Let $A(n;k,\\ell ,m)$ denote the number of four-coloured partitions of $n$ with the ordering (REF ) and matrix of difference conditions (REF ), having $k$ parts coloured $a$ , $\\ell $ parts coloured $c$ and $m$ parts coloured $d$ .", "Then $\\sum _{n,k,\\ell ,m \\ge 0} A(n;k,\\ell ,m) q^n a^k c^{\\ell } d^m = \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q;q)_{\\infty }(cq;q^2)_{\\infty }}.$ Under the dilations $ \\begin{aligned}q &\\rightarrow q^2,\\\\a &\\rightarrow aq^{-1},\\\\d &\\rightarrow dq,\\end{aligned}$ the integers in (REF ) are transformed by $\\begin{aligned}k_a &\\rightarrow (2k-1)_a,\\\\k_b &\\rightarrow 2k_b,\\\\k_c &\\rightarrow 2k_c,\\\\k_d &\\rightarrow (2k+1)_d,\\end{aligned}$ their order becomes $1_a < 2_b <2_c < 3_d <3_a <4_b <4_c<5_d < \\cdots $ and the matrix $D$ in (REF ) becomes $D_2={\\begin{bordermatrix }\\text{} & a & b & c & d \\\\a & 4&1&3&2 \\\\b &3&0&2&1 \\\\c &1&2&0&3 \\\\d&2&3&1&4\\end{bordermatrix }}.$ Considering the $a$ -parts and $c$ -parts together coloured red and the $b$ -parts and the $d$ -parts together coloured green, this gives the following refinement of Primc's identity in terms of two-coloured partitions.", "Corollary 1.2 Let $\\mathcal {P}_2$ denote the set of partitions where parts may appear in two colours, say red and green, and let $c(\\lambda _i)$ denote the colour of a part $\\lambda _i$ .", "Let $A_2(n;k,\\ell ,m)$ denote the number of partitions $(\\lambda _1, \\lambda _2,\\dots )$ of $n$ in $\\mathcal {P}_2$ having $k$ odd red parts, $\\ell $ even red parts, and $m$ odd green parts, such that no part is a green 1 and $\\lambda _i - \\lambda _{i+1} \\ge {\\left\\lbrace \\begin{array}{ll}1,& \\text{if $\\lambda _i$ is odd and $c(\\lambda _i) \\ne c(\\lambda _{i+1})$}, \\\\2,& \\text{if $\\lambda _i$ is even and $c(\\lambda _i) \\ne c(\\lambda _{i+1})$}, \\\\3,& \\text{if $\\lambda _i$ is odd and $c(\\lambda _i) = c(\\lambda _{i+1})$}.\\end{array}\\right.", "}$ Then $ \\sum _{n,k,\\ell ,m \\ge 0} A_2(n;k,\\ell ,m) q^n a^k c^{\\ell } d^m = \\frac{(-aq;q^4)_{\\infty }(-dq^3;q^4)_{\\infty }}{(q^2;q^2)_{\\infty }(cq^2;q^4)_{\\infty }}.$ In other words, if $B_2(n;k,\\ell ,m)$ denotes the number of partitions of $n$ in $\\mathcal {P}_2$ such that odd parts are distinct and only parts congruent to 2 modulo 4 may be green, having $k$ parts congruent to 1 modulo 4, $\\ell $ green parts, and $m$ parts congruent to 3 modulo 4, then $A_2(n;k,\\ell ,m) = B_2(n;k,\\ell ,m).$ One recovers Primc's identity by setting $a=c=d=1$ , as the dilations in (REF ) correspond to (REF ) and the infinite product in (REF ) becomes $\\frac{(-q;q^4)_{\\infty }(-q^3;q^4)_{\\infty }}{(q^2;q^2)_{\\infty }(q^2;q^4)_{\\infty }}=\\frac{1}{(q;q)_{\\infty }}.$ Another nice application of Theorem REF is the dilation $q &\\rightarrow q^4,\\\\a &\\rightarrow aq^{-3},\\\\c &\\rightarrow cq^{-2},\\\\d &\\rightarrow dq^3,$ where the ordering of integers (REF ) becomes $1_a < 2_c < 4_b < 5_a < 6_c < 7_d < 8_b < 9_a < \\cdots ,$ the matrix $D$ in (REF ) becomes $D_4={\\begin{bordermatrix }\\text{} & a & b & c & d \\\\a & 8&1&7&2 \\\\b &7&0&6&1 \\\\c &1&2&0&3 \\\\d&6&7&5&8\\end{bordermatrix }},$ and we obtain the following partition identity.", "Corollary 1.3 Let $A_4(n;k,\\ell ,m)$ denote the number of partitions $\\lambda = (\\lambda _1,\\lambda _2,\\dots )$ of $n$ with $k,\\ell $ , and $m$ parts congruent to 1, 2, and 3 modulo 4, respectively, with no part equal to 3, such that $\\lambda _i - \\lambda _{i+1} \\ge 5$ if ($i$ ) $\\lambda _ i \\equiv 3 \\pmod {4}$ or if ($ii$ ) $\\lambda _i \\equiv 0,1 \\pmod {4}$ and $\\lambda _{i+1} \\equiv 1,2 \\pmod {4}$ .", "Then $\\sum _{n,k,\\ell ,m \\ge 0} A_4(n;k,\\ell ,m) q^n a^k c^{\\ell } d^m = \\frac{(-aq;q^8)_{\\infty }(-dq^7;q^8)_{\\infty }}{(q^4;q^4)_{\\infty }(cq^2;q^8)_{\\infty }}.$ In other words, if $B_4(n;k,\\ell ,m)$ denotes the number of partitions of $n$ into even parts not congruent to 6 modulo 8 and distinct odd parts congruent to $\\pm 1$ modulo 8, with $k,\\ell $ , and $m$ parts congruent to $1,2$ , and 7 modulo 8, respectively, then $A_4(n;k,\\ell ,m) = B_4(n;k,\\ell ,m).$ The proof of Theorem REF relies on the variant of the method of weighted words recently introduced by the first author [14], [16].", "The difference with the original method of Alladi and Gordon [2] is that instead of using the minimal partitions and $q$ -series identities, we use recurrences and $q$ -difference equations (with colours) coming from the difference conditions in (REF ) and we solve them directly.", "This is presented in the next section, and in Section 3 we give some examples and another application of Theorem REF ." ], [ "Idea of the proof", "To prove Theorem REF , we proceed as follows.", "Define $G_k=G_k (q;a,c,d)$ (resp.", "$E_k=E_k (q;a,c,d)$ ) to be the generating function for coloured partitions satisfying the difference conditions from (REF ) with the added condition that the largest part is at most (resp.", "equal to) $k$ .", "Then we want to find $\\lim _{k \\rightarrow \\infty } G_k$ , which is the generating function for all partitions with difference conditions, as there is no more restriction on the size of the largest part.", "We start by using (REF ) to give simple recurrence equations relating the $G_k$ 's and the $E_k$ 's.", "Then we combine them to obtain a big recurrence equation involving only $G_{k_d}$ 's.", "This is done in Section REF .", "Then we use the technique consisting of going back and forth from $q$ -difference equations to recurrences introduced by the first author [13], [14], [15], and conclude using Appell's comparison theorem.", "This is done in Section REF ." ], [ "Recurrences and $q$ -difference equations", "We use combinatorial reasoning on the largest part of partitions to state some recurrences.", "We have the following identities: Lemma 2.1 For all $k \\ge 1,$ we have $G_{k_{d}}-G_{k_{c}}=E_{k_{d}}&=dq^k \\left(E_{k_{c}}+E_{k_{a}}+G_{(k-1)_{c}}\\right), \\\\G_{k_{c}}-G_{k_{b}}=E_{k_{c}}&=cq^k \\left(E_{k_{c}}+E_{k_{a}}+G_{(k-1)_{c}}\\right), \\\\G_{k_{b}}-G_{k_{a}}=E_{k_{b}}&=q^k \\left(E_{k_{b}}+G_{(k-1)_{d}}\\right), \\\\G_{k_{a}}-G_{(k-1)_{d}}=E_{k_{a}}&=aq^k \\left(E_{(k-1)_{b}}+G_{(k-2)_{d}}\\right),$ with the initial conditions $E_{0_{a}}&=E_{0_{c}}=E_{0_{d}}=0,\\\\ E_{0_{b}}&=1,\\\\ G_{{-1}_d}&=G_{0_{a}}=0,\\\\ G_{0_{b}}&=G_{0_{c}}=G_{0_{d}}=1,$ Proof: We give details only for (REF ).", "The other identities follow in a similar manner.", "The first equality $G_{k_{d}}-G_{k_{c}} = E_{k_{d}}$ follows directly from the definitions.", "Next, in a partition counted by $E_{k_{d}}$ we remove the largest part of size $k$ and colour $d$ , giving the factor $dq^k$ .", "An examination of the difference conditions in (REF ) shows that in the partition remaining the largest part could be $k_c$ , $k_a$ , or a part at most $(k-1)_c$ .", "This corresponds to the terms $E_{k_{c}}+E_{k_{a}}+G_{(k-1)_{c}}$ .", "$\\Box $ The recurrences (REF )-() completely characterise the coloured partitions with difference conditions of Theorem REF .", "Next we give a recurrence equation involving only $G_{k_{d}}$ 's.", "Proposition 2.2 For all $k \\ge 3$ we have $\\begin{aligned}(1-cq^k)G_{k_{d}}&= \\frac{1-cq^{2k}}{1-q^k}G_{(k-1)_{d}}\\\\&+ \\frac{aq^k+dq^k+adq^{2k}}{1-q^{k-1}}G_{(k-2)_{d}} +\\frac{adq^{2k-1}}{1-q^{k-2}}G_{(k-3)_{d}},\\end{aligned}$ with the initial conditions $G_{0_d} &= 1,\\\\G_{1_d} &= \\frac{q}{1-q} + \\frac{(1+aq)(1+dq)}{1-cq},\\\\G_{2_d} &= \\frac{q^3}{(1-q)(1-q^2)} + \\frac{(1+aq)(1+dq)(1-q^3)}{(1-q)(1-q^2)(1-cq)}.$ Proof: To find the correct initial conditions, we use Lemma REF .", "Now let us prove (REF ).", "We first observe that $G_{k_{b}} = G_{(k-1)_{d}} +E_{k_{a}} +E_{k_{b}}.$ By Equation (), it is clear that for all $k$ , $E_{k_{b}}= \\frac{q^k}{1-q^k} G_{(k-1)_{d}}.$ Now substituting this with $k$ replaced by $k-1$ into Equation (), we get $E_{k_{a}}= \\frac{aq^{k}}{1-q^{k-1}} G_{(k-2)_{d}}.$ Thus combining Equations (REF ), (REF ) and (REF ), we obtain $G_{k_{b}}= \\frac{1}{1-q^k} G_{(k-1)_{d}}+\\frac{aq^k}{1-q^{k-1}} G_{(k-2)_{d}}.$ Let us now turn to $E_{k_{c}}$ .", "By Equation (), we have $E_{k_{c}}= \\frac{cq^k}{1-cq^k} \\left( E_{k_{a}}+G_{(k-1)_{c}} \\right).$ Substituting (REF ), we obtain $E_{k_{c}}= \\frac{cq^k}{1-cq^k} \\left( \\frac{aq^{k}}{1-q^{k-1}} G_{(k-2)_{d}}+G_{(k-1)_{c}}\\right).$ Finally, by Equations (REF ) and () and the initial conditions, for all $k$ , we have $d E_{k_{c}} = c E_{k_{d}}.$ Combining that with (REF ), we obtain that for all $k$ , $E_{k_{d}}= \\frac{dq^k}{1-cq^k} \\left( \\frac{aq^{k}}{1-q^{k-1}} G_{(k-2)_{d}}+G_{(k-1)_{c}} \\right).$ Using Equations (REF ), (REF ), (REF ) and the fact that $G_{k_{d}} = G_{k_{b}}+E_{k_{c}} +E_{k_{d}},$ we obtain $G_{k_{d}} &= \\frac{1}{1-q^k} G_{(k-1)_{d}} +\\frac{aq^k}{1-q^{k-1}} G_{(k-2)_{d}} \\\\&+ \\frac{(c+d)q^k}{1-cq^k} \\left( \\frac{aq^{k}}{1-q^{k-1}} G_{(k-2)_{d}}+G_{(k-1)_{c}} \\right).$ Rearranging gives an expression for $G_{(k-1)_{c}}$ in terms of $G_{k_{d}}$ 's.", "$G_{(k-1)_{c}} = \\frac{1-cq^k}{(c+d)q^k} \\left( G_{k_{d}} - \\frac{1}{1-q^{k}} G_{(k-1)_{d}}- \\frac{aq^{k}(1+dq^k)}{(1-q^{k-1})(1-cq^k)} G_{(k-2)_{d}}\\right).$ Substituting this into (REF ) and simplifying leads to $E_{k_{c}} = \\frac{c}{c+d} G_{k_{d}}- \\frac{c}{(c+d)(1-q^{k})} G_{(k-1)_{d}}- \\frac{acq^{k}}{(c+d)(1-q^{k-1})} G_{(k-2)_{d}}.$ On the other hand, using (REF ), (REF ) and the fact that $E_{k_{c}} = G_{k_{c}}- G_{k_{b}},$ we obtain $\\begin{aligned}E_{k_{c}} &= \\frac{1-cq^{k+1}}{(c+d)q^{k+1}} G_{(k+1)_{d}} - \\frac{1-cq^{k+1}}{(c+d)q^{k+1}(1-q^{k+1})} G_{k_{d}} \\\\&- \\frac{a+c+d+adq^{k+1}}{(c+d)(1-q^{k})} G_{(k-1)_{d}}- \\frac{aq^k}{1-q^{k-1}} G_{(k-2)_{d}}.\\end{aligned}$ Equating (REF ) and (REF ) and replacing $k$ by $k-1$ yields the desired recurrence equation.", "$\\Box $ Finding $ \\lim _{k \\rightarrow \\infty } G_{k}(q;a,c,d)$ To finish the proof we wish to calculate $\\lim _{k \\rightarrow \\infty } G_{k_d}(q;a,c,d)$ , where the $G_{k_d}$ 's satisfy the recurrence of order 3 in (REF ).", "This is in constrast to classical partition identities, where the recurrence/$q$ -difference equation is typically of order 1 (see for example [3], [4], [5], [6]).", "The problem of treating higher order recurrences/$q$ -difference equations has recently come up in work of the first author on overpartition identities [12], [13], [14], [15], and her method applies here as well.", "Specifically, we transform the recurrence of order 3 into a simple one of order 2, and then, as in the classical case, we apply Appell's comparison theorem [10] to find the desired limit.", "In the conclusion we sketch an alternative method suggested by the referee, which gives a $q$ -hypergeometric generating function for $G_{k_d}(q;a,c,d)$ from which the limit also follows.", "For all $k \\ge 0$ , let us define $H_k := \\frac{G_{k_{d}}(q)}{1-q^{k+1}}.$ Thus $(H_k)$ satisfies the following recurrence equation for $k \\ge 0$ : $(1-cq^k-q^{k+1}+cq^{2k+1})H_k = (1-cq^{2k})H_{k-1}+ (aq^k+dq^k+adq^{2k})H_{k-2} +adq^{2k-1}H_{k-3},$ To obtain the correct values of $H_k$ for all $k \\ge 0$ using Equation (REF ), we define the initial values $H_{-1}=1$ and $H_{k}=0$ for all $k \\le -2$ .", "We now define $f(x):= \\sum _{k \\ge 0} H_{k-1} x^k,$ and convert Equation (REF ) into a $q$ -difference equation on $f$ : $(1-x)f(x)= (1+\\frac{c}{q}+ax^2q+dx^2q)f(xq)-(1+xq)(\\frac{c}{q}-adx^2q^2)f(xq^2),$ together with the initial conditions $f(0)&= H_{-1} = 1,\\\\ f^{\\prime }(0) &= H_{0}=\\frac{1}{1-q}.$ This is a $q$ -difference equation of order 2, which is still not obvious to solve.", "But we make another transformation to obtain a very simple recurrence of order 2.", "Define $g(x):= \\frac{f(x)}{(-x;q)_{\\infty }}.$ We obtain: $(1-x^2)g(x)= (1+\\frac{c}{q}+ax^2q+dx^2q)g(xq)-(\\frac{c}{q}-adx^2q^2)g(xq^2),$ and $g(0)&=f(0) = 1,\\\\ g^{\\prime }(0) &= f^{\\prime }(0) - \\frac{f(0)}{1-q}=\\frac{1}{1-q} - \\frac{1}{1-q}=0.$ Finally let us define $(a_n)$ as $\\sum _{n \\ge 0} a_n x^n := g(x).$ Then $(a_n)$ satisfies the recurrence equation $\\left(1-q^n -cq^{n-1}+cq^{2n-1}\\right)a_n = \\left(1+aq^{n-1}+dq^{n-1}+adq^{2n-2}\\right)a_{n-2},$ which simplifies as $a_n = \\frac{\\left(1+aq^{n-1}\\right)\\left(1+dq^{n-1}\\right)}{\\left(1-q^n\\right)\\left(1-cq^{n-1}\\right)} a_{n-2},$ and the initial conditions $a_0&=g(0) = 1,\\\\ a_1 &= g^{\\prime }(0)=0.$ Thus for all $n \\ge 0$ , we have $a_{2n} = \\frac{(-aq;q^2)_n(-dq;q^2)_n}{(q^2;q^2)_n(cq;q^2)_n}a_0 = \\frac{(-aq;q^2)_n(-dq;q^2)_n}{(q^2;q^2)_n(cq;q^2)_n},$ and $a_{2n+1} = \\frac{(-aq^2;q^2)_n(-dq^2;q^2)_n}{(q^3;q^2)_n(cq^2;q^2)_n}a_1 = 0.$ We now conclude using Appell's comparison theorem.", "We have $\\lim _{k \\rightarrow \\infty } G_k(q;a,c,d) &= \\lim _{k \\rightarrow \\infty } H_k\\\\&= \\lim _{x \\rightarrow 1^-} (1-x) \\sum _{k \\ge 0} H_{k-1} x^k\\\\&= \\lim _{x \\rightarrow 1^-} (1-x) f(x)\\\\&= \\lim _{x \\rightarrow 1^-} (1-x) g(x) \\prod _{k \\ge 0} (1+xq^k)\\\\&= (-q;q)_{\\infty } \\lim _{x \\rightarrow 1^-} (1-x^2) \\sum _{n \\ge 0} a_{2n} x^{2n}\\\\&= (-q;q)_{\\infty } \\lim _{n \\rightarrow \\infty } a_{2n}\\\\&= \\frac{(-q;q)_{\\infty }(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q^2;q^2)_{\\infty }(cq;q^2)_{\\infty }}\\\\&= \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q;q)_{\\infty }(cq;q^2)_{\\infty }}.$ We used Appell's theorem on the second line, and on the sixth with $x$ replaced by $x^2$ .", "Examples and further results We begin this section by illustrating Corollaries REF and REF .", "First, the eleven two-coloured partitions of 6 satisfying the difference conditions in Corollary REF and having no green 1 are the following, where green parts are marked with a prime: ${\\begin{array}{c}(6), (6^{\\prime }), (5,1), (5^{\\prime },1), (4,2), (4^{\\prime },2), (4,2^{\\prime }), (4^{\\prime },2^{\\prime }), \\\\(3^{\\prime },2,1), (2,2,2), (2^{\\prime },2^{\\prime },2^{\\prime }).\\end{array}}$ On the other hand, the eleven two-coloured partitions with distinct odd parts where only parts 2 modulo 4 can be green are ${\\begin{array}{c}(6), (6^{\\prime }), (5,1), (4,2), (4, 2^{\\prime }), (3,2,1), (3,2^{\\prime },1), \\\\(2,2,2), (2,2,2^{\\prime }), (2,2^{\\prime },2^{\\prime }), (2^{\\prime },2^{\\prime },2^{\\prime }).\\end{array}}$ One may then easily verify that $A_2(6;k,\\ell ,m) = B_2(6;k,\\ell ,m)$ for a given choice of $(k,\\ell ,m)$ .", "For example, $A_2(6;1,0,1) = B_2(6;1,0,1) = 1$ , the relevant partitions being $(5^{\\prime },1)$ and $(3,2,1)$ , respectively.", "Next, the thirteen partitions of 14 satisfying the difference conditions in Corollary REF and having no part equal to 3 are ${\\begin{array}{c}(14), (13,1), (12,2), (11,2,1), (10,4), (10,2,2), (9,2,2,1), \\\\(8,2,2,2), (7,2,2,2,1), (6,6,2), (6,4,4), (6,2,2,2,2), (2,2,2,2,2,2,2),\\end{array}}$ while the thirteen partitions of 14 satisfying the congruence conditions are ${\\begin{array}{c}(12,2), (10,4), (10,2,2), (9,4,1), (9,2,2,1), (8,4,2), (8,2,2,2), \\\\(7,4,2,1), (7,2,2,2,1), (4,4,4,2), (4,4,2,2,2), (4,2,2,2,2,2), (2,2,2,2,2,2,2).\\end{array}}$ Again, one easily verifies that $A_4(13;k,\\ell ,m) = B_4(13;k,\\ell ,m)$ for a given choice of $(k,\\ell ,m)$ .", "We close with one more application of Theorem REF .", "Here parts divisible by 3 may appear in two kinds.", "Performing the dilation $q &\\rightarrow q^3,\\\\a &\\rightarrow aq^{-1},\\\\c &\\rightarrow 1,\\\\d &\\rightarrow dq,$ the ordering of integers (REF ) becomes $2_a < 3_b < 3_c < 4_d < 5_a < 6_b < 6_c < 7_d < 8_a < 9_b < 9_c < \\cdots $ and the matrix $D$ in (REF ) becomes $D_3={\\begin{bordermatrix }\\text{} & a & b & c & d \\\\a & 6&2&5&4 \\\\b &4&0&3&2 \\\\c &1&3&0&5 \\\\d&2&4&1&6\\end{bordermatrix }}.$ Letting $b$ -parts and $c$ -parts be ordinary and primed multiples of 3, respectively, we obtain the following partition identity.", "Corollary 3.1 Let $\\mathcal {P}_3$ denote the set of partitions where parts divisible by 3 may appear in two kinds, say ordinary and primed.", "Let $A_3(n;k,m)$ denote the number of partitions of $n$ in $\\mathcal {P}_3$ with $k$ and $m$ parts congruent to 2 and 1 modulo 3, respectively, such that $\\lambda _i \\ne 1$ and $\\lambda _i - \\lambda _{i+1} \\ge {\\left\\lbrace \\begin{array}{ll}3, &\\text{if $(\\lambda _i,\\lambda _{i+1}) \\pmod {3} \\subset (\\lbrace 0,2\\rbrace ,\\lbrace 0^{\\prime },2\\rbrace )$ or $(\\lbrace 0^{\\prime },1\\rbrace ,\\lbrace 0,1\\rbrace )$}, \\\\4, &\\text{if $3 \\nmid \\lambda _i, \\lambda _{i+1}$ and $\\lambda _i - \\lambda _{i+1} \\lnot \\equiv 2 \\pmod {3}$}.\\end{array}\\right.", "}$ Then $ \\sum _{n,k,m \\ge 0} A_3(n;k,m)q^n a^k d^{m} = \\frac{(-aq^2;q^6)_{\\infty } (-dq^4;q^6)_{\\infty } (-q^3;q^3)_{\\infty }}{(q^3;q^3)_{\\infty }}.$ In other words, if $B_3(n;k,m)$ denotes the number of partitions of $n$ in $\\mathcal {P}_3$ with $k$ and $m$ parts congruent to 2 and 4 modulo 6, respectively, such that primed multiples of 3 may not repeat, then $A_3(n;k,m) = B_3(n;k,m).$ Note that the generating function in (REF ) differs only slightly from the infinite product appearing in the Alladi-Andrews-Gordon generalisation of Capparelli's identity [1], $(-aq^2;q^6)_{\\infty } (-bq^4;q^6)_{\\infty } (-q^3;q^3)_{\\infty }.$ Concluding Remarks The referee has kindly pointed out that the recurrence in Proposition REF can be used to give a $q$ -hypergeometric generating function for $G_{k_d}$ , $ G_{k_d} = (1-q^{k+1}) \\sum _{i=0}^{\\lfloor (k+1)/2 \\rfloor } \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_i(cq;q^2)_i}.$ The idea is to recursively define sequences $g_k^{(i)}$ and $h_k^{(i)}$ with $g_k^{(0)} = G_{k_d}$ by $h_k^{(i)} &:= \\lim _{c \\rightarrow \\infty } g_k^{(i)}, \\\\g_{k}^{(i+1)} &:= (1-cq^{2i+1}) (g_k^{(i)} - h_k^{(i)}),$ where the existence of $h_k^{(i)}$ follows from the recurrence (plus initial conditions) for $g_k^{(i)}$ .", "At each step one uses the recurrence for the $g_k^{(i)}$ and formula for the $h_k^{(i)}$ to find a recurrence for the $g_k^{(i+1)}$ and a formula for the $h_k^{(i+1)}$ .", "The recurrence for $g_k^{(i)}$ is the following: $\\left( 1-cq^k \\right) g_k^{(i)} &= \\frac{1-cq^{2k}}{1-q^k} g_{k-1}^{(i)} + \\frac{aq^k+dq^k+adq^{2k}}{1-q^{k-1}} g_{k-2}^{(i)}\\\\&+ \\frac{adq^{2k-1}}{1-q^{k-2}} g_{k-3}^{(i)} + \\frac{q^{\\binom{k-2i+2}{2}}\\left(1-cq^{2i-1} \\right)(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_{i-1}},$ where $1/(q;q)_n =0$ for $n <0$ .", "From this one can deduce the recurrence for $h_k^{(i)}$ : $h_k^{(i)} = \\frac{q^k}{1-q^k}h_{k-1}^{(i)} + \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_{i-1}}.$ The result is $h_k^{(i)} = (1-q^{k+1}) \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_i}.$ We leave the details to the interested reader.", "Since $G_{k_d} = g_k^{(0)} = \\sum _{i \\ge 0} \\frac{h_k^{(i)}}{(cq;q^2)_i},$ we obtain (REF ).", "Note that if we replace $k$ by $2k-1+\\delta $ for $\\delta = 0,1$ and use the fact that $\\sum _{i=0}^{\\infty } \\frac{q^{\\binom{2i+\\delta }{2}}}{(q;q)_{2i+\\delta }} = (-q;q)_{\\infty },$ we have $\\lim _{k \\rightarrow \\infty } g_{2k-1+\\delta }^{(0)} &= \\lim _{k \\rightarrow \\infty } (1-q^{2k+\\delta }) \\sum _{i=0}^{k} \\frac{q^{\\binom{2i+\\delta }{2}}(-aq;q^2)_{k-i}(-dq;q^2)_{k-i}}{(q;q)_{2i+\\delta }(q^2;q^2)_{k-i}(cq;q^2)_{k-i}} \\\\&= \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q^2;q^2)_{\\infty }(cq;q^2)_{\\infty }} \\sum _{i=0}^{\\infty } \\frac{q^{\\binom{2i+\\delta }{2}}}{(q;q)_{2i+\\delta }} \\\\&= \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q;q)_{\\infty }(cq;q^2)_{\\infty }},$ in agreement with Section 2.3.", "This kind of argument differs from the usual approach and should be kept in mind for future studies of partition identities.", "Acknowledgements We thank the referee for a careful reading of the paper and the many valuable suggestions for its improvement, especially for clarifying the early work on vertex operators and crystal bases and for detailing an alternative approach to Section 2.3." ], [ "Examples and further results", "We begin this section by illustrating Corollaries REF and REF .", "First, the eleven two-coloured partitions of 6 satisfying the difference conditions in Corollary REF and having no green 1 are the following, where green parts are marked with a prime: ${\\begin{array}{c}(6), (6^{\\prime }), (5,1), (5^{\\prime },1), (4,2), (4^{\\prime },2), (4,2^{\\prime }), (4^{\\prime },2^{\\prime }), \\\\(3^{\\prime },2,1), (2,2,2), (2^{\\prime },2^{\\prime },2^{\\prime }).\\end{array}}$ On the other hand, the eleven two-coloured partitions with distinct odd parts where only parts 2 modulo 4 can be green are ${\\begin{array}{c}(6), (6^{\\prime }), (5,1), (4,2), (4, 2^{\\prime }), (3,2,1), (3,2^{\\prime },1), \\\\(2,2,2), (2,2,2^{\\prime }), (2,2^{\\prime },2^{\\prime }), (2^{\\prime },2^{\\prime },2^{\\prime }).\\end{array}}$ One may then easily verify that $A_2(6;k,\\ell ,m) = B_2(6;k,\\ell ,m)$ for a given choice of $(k,\\ell ,m)$ .", "For example, $A_2(6;1,0,1) = B_2(6;1,0,1) = 1$ , the relevant partitions being $(5^{\\prime },1)$ and $(3,2,1)$ , respectively.", "Next, the thirteen partitions of 14 satisfying the difference conditions in Corollary REF and having no part equal to 3 are ${\\begin{array}{c}(14), (13,1), (12,2), (11,2,1), (10,4), (10,2,2), (9,2,2,1), \\\\(8,2,2,2), (7,2,2,2,1), (6,6,2), (6,4,4), (6,2,2,2,2), (2,2,2,2,2,2,2),\\end{array}}$ while the thirteen partitions of 14 satisfying the congruence conditions are ${\\begin{array}{c}(12,2), (10,4), (10,2,2), (9,4,1), (9,2,2,1), (8,4,2), (8,2,2,2), \\\\(7,4,2,1), (7,2,2,2,1), (4,4,4,2), (4,4,2,2,2), (4,2,2,2,2,2), (2,2,2,2,2,2,2).\\end{array}}$ Again, one easily verifies that $A_4(13;k,\\ell ,m) = B_4(13;k,\\ell ,m)$ for a given choice of $(k,\\ell ,m)$ .", "We close with one more application of Theorem REF .", "Here parts divisible by 3 may appear in two kinds.", "Performing the dilation $q &\\rightarrow q^3,\\\\a &\\rightarrow aq^{-1},\\\\c &\\rightarrow 1,\\\\d &\\rightarrow dq,$ the ordering of integers (REF ) becomes $2_a < 3_b < 3_c < 4_d < 5_a < 6_b < 6_c < 7_d < 8_a < 9_b < 9_c < \\cdots $ and the matrix $D$ in (REF ) becomes $D_3={\\begin{bordermatrix }\\text{} & a & b & c & d \\\\a & 6&2&5&4 \\\\b &4&0&3&2 \\\\c &1&3&0&5 \\\\d&2&4&1&6\\end{bordermatrix }}.$ Letting $b$ -parts and $c$ -parts be ordinary and primed multiples of 3, respectively, we obtain the following partition identity.", "Corollary 3.1 Let $\\mathcal {P}_3$ denote the set of partitions where parts divisible by 3 may appear in two kinds, say ordinary and primed.", "Let $A_3(n;k,m)$ denote the number of partitions of $n$ in $\\mathcal {P}_3$ with $k$ and $m$ parts congruent to 2 and 1 modulo 3, respectively, such that $\\lambda _i \\ne 1$ and $\\lambda _i - \\lambda _{i+1} \\ge {\\left\\lbrace \\begin{array}{ll}3, &\\text{if $(\\lambda _i,\\lambda _{i+1}) \\pmod {3} \\subset (\\lbrace 0,2\\rbrace ,\\lbrace 0^{\\prime },2\\rbrace )$ or $(\\lbrace 0^{\\prime },1\\rbrace ,\\lbrace 0,1\\rbrace )$}, \\\\4, &\\text{if $3 \\nmid \\lambda _i, \\lambda _{i+1}$ and $\\lambda _i - \\lambda _{i+1} \\lnot \\equiv 2 \\pmod {3}$}.\\end{array}\\right.", "}$ Then $ \\sum _{n,k,m \\ge 0} A_3(n;k,m)q^n a^k d^{m} = \\frac{(-aq^2;q^6)_{\\infty } (-dq^4;q^6)_{\\infty } (-q^3;q^3)_{\\infty }}{(q^3;q^3)_{\\infty }}.$ In other words, if $B_3(n;k,m)$ denotes the number of partitions of $n$ in $\\mathcal {P}_3$ with $k$ and $m$ parts congruent to 2 and 4 modulo 6, respectively, such that primed multiples of 3 may not repeat, then $A_3(n;k,m) = B_3(n;k,m).$ Note that the generating function in (REF ) differs only slightly from the infinite product appearing in the Alladi-Andrews-Gordon generalisation of Capparelli's identity [1], $(-aq^2;q^6)_{\\infty } (-bq^4;q^6)_{\\infty } (-q^3;q^3)_{\\infty }.$" ], [ "Concluding Remarks", "The referee has kindly pointed out that the recurrence in Proposition REF can be used to give a $q$ -hypergeometric generating function for $G_{k_d}$ , $ G_{k_d} = (1-q^{k+1}) \\sum _{i=0}^{\\lfloor (k+1)/2 \\rfloor } \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_i(cq;q^2)_i}.$ The idea is to recursively define sequences $g_k^{(i)}$ and $h_k^{(i)}$ with $g_k^{(0)} = G_{k_d}$ by $h_k^{(i)} &:= \\lim _{c \\rightarrow \\infty } g_k^{(i)}, \\\\g_{k}^{(i+1)} &:= (1-cq^{2i+1}) (g_k^{(i)} - h_k^{(i)}),$ where the existence of $h_k^{(i)}$ follows from the recurrence (plus initial conditions) for $g_k^{(i)}$ .", "At each step one uses the recurrence for the $g_k^{(i)}$ and formula for the $h_k^{(i)}$ to find a recurrence for the $g_k^{(i+1)}$ and a formula for the $h_k^{(i+1)}$ .", "The recurrence for $g_k^{(i)}$ is the following: $\\left( 1-cq^k \\right) g_k^{(i)} &= \\frac{1-cq^{2k}}{1-q^k} g_{k-1}^{(i)} + \\frac{aq^k+dq^k+adq^{2k}}{1-q^{k-1}} g_{k-2}^{(i)}\\\\&+ \\frac{adq^{2k-1}}{1-q^{k-2}} g_{k-3}^{(i)} + \\frac{q^{\\binom{k-2i+2}{2}}\\left(1-cq^{2i-1} \\right)(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_{i-1}},$ where $1/(q;q)_n =0$ for $n <0$ .", "From this one can deduce the recurrence for $h_k^{(i)}$ : $h_k^{(i)} = \\frac{q^k}{1-q^k}h_{k-1}^{(i)} + \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_{i-1}}.$ The result is $h_k^{(i)} = (1-q^{k+1}) \\frac{q^{\\binom{k-2i+1}{2}}(-aq;q^2)_i(-dq;q^2)_i}{(q;q)_{k-2i+1}(q^2;q^2)_i}.$ We leave the details to the interested reader.", "Since $G_{k_d} = g_k^{(0)} = \\sum _{i \\ge 0} \\frac{h_k^{(i)}}{(cq;q^2)_i},$ we obtain (REF ).", "Note that if we replace $k$ by $2k-1+\\delta $ for $\\delta = 0,1$ and use the fact that $\\sum _{i=0}^{\\infty } \\frac{q^{\\binom{2i+\\delta }{2}}}{(q;q)_{2i+\\delta }} = (-q;q)_{\\infty },$ we have $\\lim _{k \\rightarrow \\infty } g_{2k-1+\\delta }^{(0)} &= \\lim _{k \\rightarrow \\infty } (1-q^{2k+\\delta }) \\sum _{i=0}^{k} \\frac{q^{\\binom{2i+\\delta }{2}}(-aq;q^2)_{k-i}(-dq;q^2)_{k-i}}{(q;q)_{2i+\\delta }(q^2;q^2)_{k-i}(cq;q^2)_{k-i}} \\\\&= \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q^2;q^2)_{\\infty }(cq;q^2)_{\\infty }} \\sum _{i=0}^{\\infty } \\frac{q^{\\binom{2i+\\delta }{2}}}{(q;q)_{2i+\\delta }} \\\\&= \\frac{(-aq;q^2)_{\\infty }(-dq;q^2)_{\\infty }}{(q;q)_{\\infty }(cq;q^2)_{\\infty }},$ in agreement with Section 2.3.", "This kind of argument differs from the usual approach and should be kept in mind for future studies of partition identities." ], [ "Acknowledgements", "We thank the referee for a careful reading of the paper and the many valuable suggestions for its improvement, especially for clarifying the early work on vertex operators and crystal bases and for detailing an alternative approach to Section 2.3." ] ]
1612.05423
[ [ "Models, networks and algorithmic complexity" ], [ "Abstract I aim to show that models, classification or generating functions, invariances and datasets are algorithmically equivalent concepts once properly defined, and provide some concrete examples of them.", "I then show that a) neural networks (NNs) of different kinds can be seen to implement models, b) that perturbations of inputs and nodes in NNs trained to optimally implement simple models propagate strongly, c) that there is a framework in which recurrent, deep and shallow networks can be seen to fall into a descriptive power hierarchy in agreement with notions from the theory of recursive functions.", "The motivation for these definitions and following analysis lies in the context of cognitive neuroscience, and in particular in Ruffini (2016), where the concept of model is used extensively, as is the concept of algorithmic complexity." ], [ "Abstract", "I aim to show that models, classification or generating functions, invariances and datasets are algorithmically equivalent concepts once properly defined, and provide some concrete examples of them.", "I then show that a) neural networks (NNs) of different kinds can be seen to implement models, b) that perturbations of inputs and nodes in NNs trained to optimally implement simple models propagate strongly, c) that there is a framework in which recurrent, deep and shallow networks can be seen to fall into a descriptive power hierarchy in agreement with notions from the theory of recursive functions.", "The motivation for these definitions and following analysis lies in the context of cognitive neuroscience, and in particular in [17], where the concept of model is used extensively, as is the concept of algorithmic complexity." ], [ "Models", "Let us first define formally the notion of model as in [17] in the context of computation: Definition 1 The optimal model or k-model of a dataset is the shortest program that generates (or, equivalently, compresses) the dataset efficiently (i.e., succinctly in the Kolmogorov sense).", "A model is any program that generates the dataset, optimal or not.", "In general, we don't have access to optimal models.", "Some examples of models are: A Lempel-Ziv-Welch (LZW) compressed version of a file is an implementation of a model of the data in the file (running on the programming environment that can carry out the decompression).", "The implementation may be a poor one if the data does not contain regularities that are in the form of substring repetition (e.g., as in the digits of $\\pi $ , which can be generated by a simple algorithm but without repetition regularities).", "A program that provided with initial condition inputs, generates dynamical data for some physical system (e.g., positions and velocities of particles in a gas).", "Any physics model encoded in equations (e.g., Maxwell's equations) and associated methods (e.g., calculus) which can procedurally be computed given some initial/boundary conditions to generate data.", "The following, as we will discuss, are also equivalent to models: A pattern recognition program.", "E.g., a feedforward neural network (NN) trained to recognize images of hands, whether it is shallow (one layer, SNN) or deep (multiple layers, DNN).", "Such networks essentially provide an implementation to compute a function.", "A recurrent NN (RNN) classifying inputs by going into an attractor (a Hopfield network).", "An LSTM network trained to classify sequences into labels, for example for speech identification of a given word.", "Again, there are invariances (who says the words, or how) encoded by the network, and again we can think of the network as encoding the model for a spoken word.", "In the human brain and in machine learning we talk about neural networks, and, in general of recurrent neural networks.", "For example, learned sequences of motor activity are encoded in many vertebrate brains by “complex spatio-temporal patterns of neural activity” [10].", "We also point out that in general the Kolmogorov Complexity of a string is uncomputable due to the halting problem of Turing machines [7].", "However, there are ways to deal with this limitation in practice.", "For instance, if we limit computation to programming using primitive recursive functions (PR), which are described by programs that include for-loops but not while loopsSee the BlooP and FlooP languages as discussed in [11], then all such programs halt.", "Algorithmic complexity limited to PR is therefore computable.", "For example, feedforward networks are PR, and one may in practice seek to find the simplest (shortest as a program) such feedforward NN that computes a PR function.", "Of course, it may be that if one uses a more general (complete) language, a shorter program can be produced for the same PR function." ], [ "Compressing a disordered stacked dataset", "In the discussion below we will refer often to binary (two class) classification of images to make the discussion concrete, although the reasoning is more general.", "We will use the symbols $\\mathbb {B}=\\lbrace 0,1\\rbrace $ (a set with two elements) and $\\mathcal {X}\\equiv \\mathbb {B}^{n} $ for image space (with $n$ proportional to the number of pixels in an image).", "A (digital) class function is a map from $f(x): \\mathcal {X}\\rightarrow \\mathbb {B}$ (e.g., from images to binary numbers, as in binary classification of images).", "The discussion is entirely framed in finite, discrete sets.", "We restrict the discussion throughout to discrete images, both in pixel and in digitization and function values.", "Image space is thus large but finite.", "If we consider 24 bit/pixel images of 1000 by 1000 pixels, with size $ |\\mathcal {X}| = 2^n$ , with $n=24\\cdot 10 ^6$ .", "This is a large number, but the set of binary class functions $f(x) \\in \\mathcal {F}: \\mathcal {X}\\rightarrow \\mathbb {B}$ is much largerWe will use $\\log $ to refer to $\\log _{2}$ througout., $\\log |\\mathcal {F}| = |\\mathcal {X}| $ .", "In practice, we can assume that the number of possible images is much smaller, however, because many images are never encountered.", "To be concrete, here we will refer to a stacked dataset $D_{f}$ : the dataset generated from all hand images, which we imagine as a random stack of images $x \\in \\mathcal {X}$ (a datacube).", "Thus, the dataset is generated using the same rule (“hand”, which we imagine as a program that takes some inputs and generates an image of a hand) with varying parameters $\\theta \\in \\mathcal {P}= \\mathbb {B}^{m}$ at each iteration plus a varying background for each image, which we assume belongs to a space of possible backgrounds $\\mathcal {B}$ .", "We call the set of “hand” images $ \\mathcal {H}\\subset \\mathcal {P} \\times \\mathcal {B}$ (not all combinations of background and hand realizations are possible, they have to fit).", "It is a finite set of size $| \\mathcal {H}| < |\\mathcal {P}| \\times | \\mathcal {B} | $ .", "Although this type of dataset is not the most general, it is fitting for the discussion of neural networks below.", "We will also assume that the stack is disordered, to make compression more difficultHowever, it is possible that the optimal compressing program will identify the underlying group structure of image generation and use that to order the images and then compress them better.", "This concept has been proposed by Barbour as the mechanism behind the emergence for the parameter of time, for instance, if we imagine the images to be snapshots of a dynamical system [1]..", "The stacked dataset, a string of bits, has length $l(D_{f})=|\\mathcal {H}|\\, \\log |\\mathcal {X}|$ .", "We assume throughout that $|\\mathcal {X}| >>|\\mathcal {H}|>>1.$" ], [ "Models, class function and the generating function", "Neural networks encode functions.", "Eg., a feedforward network maps inputs to outputs.", "In what sense is a neural network or a function a model of the dataset?", "As defined, models are mathematical objects (programs) which unpack into a dataset.", "From optimal model to generating function to class function: Let us first see how a k-model can be used to define a classification function.", "To be concrete, let us imagine the staked dataset generated from all the images of a rotating and flexing hand we can generate with a few parameters (rotations, shape changes), a large, random stack of each image ($x \\in \\mathcal {X}$ ).", "Since we generated the data simply, we know we can compress it into a program in which part of the program describes what is invariant (generating function), and what is changing (shape, perspective).", "More formally, the program can be thought as consisting of three parts.", "One part encodes what a hand is in algorithmic terms through an image generating function of some parameters $x=g(\\theta )$ , with $\\theta \\in \\mathcal {P}$ : $g(\\theta ): \\mathcal {P} \\longrightarrow \\mathcal {X}.$ The second part is a sequence $\\theta _{i}$ where the hand model parameters are changed and has no structure.", "The last part describes the background, which we assume also has no structure.", "The optimal model (a compressing program) must thus encode a hand image generating function $x_{\\theta }=g(\\theta )$ .", "A suboptimal model need not do this (e.g., in the worse case it may say “print the following bits in the dataset: ..... ”).", "If the model is optimal, the first part will be the algorithmically simplest program that encodes the generating function, and presumably unique.", "To find it we could for example carry out all possible reordering of images in the dataset, and compare the models that our compressor produces for each case.", "Part of the model will be invariant to such operations—the one corresponding to the generating function.", "The make the example a bit more concrete, the model may look like this:   def Model:       def HandImageCreator(theta)         # Inputs parameter theta which defines position,         # rotation, shape or other hand features.", "# Generates image of hand with white background \t# Provides a list of the background pixel IDs \t [ ... ] \treturn image, backgroundPixels      # we need a list of thetas in some order    P= CreateParameterSet [...]      for each theta in P:        handImage, backgroundPixels = HandImageCreator(theta)          B= CreatePossibleBackgroundsList(backgroundPixels)          for b in B:    \t        \t\thandImage.addBackground(b)\t\t             \tplace handImage in Stack \t    return Stack We could actually create a finite listing of all possible parameter-background combinations and simply iterate over them or over a permutation of this list (this model is PR).", "An useful concept to specify the generating function is that of the Kolmogorov Structure function, $\\Phi _k$ of the dataset [7].", "This function is $\\Phi _k(D_{f})= \\log |S|$ where $S$ is the smallest set that 1) contains the dataset as an element and 2) can be described with no more than $k$ bits.", "The function equals the length of the dataset for $k=0$ ($\\Phi _0(D_{f})=l$ ) and 0 when the $k = \\mathcal {K} [D_{f}]$ ($\\Phi _{\\mathcal {K} [D_{f}]}(D_{f})= 0$ ), since there is only one set that contains $D_{f}$ ).", "As we add more bits, we constrain further the size of $S$ , which will decrease rapidly while regularities are encoded.", "At some point, the rate of decrease of $\\log |S|$ will be of 1 per bit, which means we are then encoding the incompressible part of the dataset—in this case constraining the random list of parameters and background.", "Thus, the program at the critical length $k^{*}$ [7] is the one that makes use of the generating function, and may look like this “Iterate over all possible images and keep those which have maximal correlation with the output of the generating function for some parameter $\\theta $ with due care for masking.", "Create an element per permutation of this stack.” Here we assume that the generating function provides images of a hand and a mask for the white background—a list of background pixels to compute the correlation with the masked image.", "The set $S^*$ defined using $k^{*}$ bits is called the “algorithmic sufficient statistic for the dataset”, and satisfies $\\mathcal {K}(x)+c=\\mathcal {K}(S^{*})+\\log |S^{*}| = k^{*}+\\log |S^{*}|$ .", "From the generating function we can construct a classifier of hands (class function).", "We can write a program that given an image $x$ scans over all allowed parameter vales (including backgrounds $b$ ) and outputs 1 if there exits a value such that $g(\\theta ,b)=x$ , or zero otherwise.", "We can also carry out the search with a fixed $b_{0}$ by allowing some error, $|| g(\\theta , b_{0}) - x || < \\epsilon $ .", "From class function to model: To illustrate now how a classification or class function can be used to create a model, let us begin with a simple type, a function defined by a feedforward network such as the ones used for image classification.", "Such a network encodes a function $f(x)$ that when input an image of hand $x$ outputs a value of 1, and 0 otherwise, $f(x): \\mathcal {X}\\longrightarrow \\mathbb {B}.$ Geometrically, this function is an invariant over the manifold of hand images embedded in image space—a point which we will return to later.", "With the class function we can create a model to generate datasets of images of hands: Model 1 Find the set of points $ \\mathcal {H}=\\lbrace x \\in \\mathcal {X}| \\, f(x)=1 \\rbrace ,$ and list them (or a specified subset) in some order.", "In general, there will be many solutions and therefore elements in $ \\mathcal {H}$ , which brings to light the meaning of compression.", "There will be many such points.", "For simplicity, assume we can simply list them (in some random order).", "Thus, we can use $f(x)$ and a number (a parameter) to select an element in the list to unpack the function into an image of a hand.", "We can also create a generating function from the class function in a similar way: use the class function to generate all images of hands in some order and use that list as the generating function table from integers (parameter space) to images.", "In this way, we can talk about some types of models (models of stacked datasets) as functions represented/encoded by neural networks or other machine learning systems.", "The same reasoning can be applied to recurrent neural network (RNN) trained for some similar task: e.g., once given an input, the trained network will orbit around dynamical attractor the encodes that “memory” or classification output.", "The interesting point is that the attractor (or the output of a subset of nodes mapped to a single output here for binary functions) that the system enters is invariant under a set of transformation of the inputs, exactly as we discussed before.", "We note that in this example, class as well as generating function are primitive recursive, since the input space is finite.", "Figure: Left: Input (image) space 𝒳\\mathcal {X}, the subspace of hand images ℋ={x∈𝒳|f(x)=1} \\mathcal {H}=\\lbrace x \\in \\mathcal {X}| \\, f(x)=1 \\rbrace and a example of automorphism (arrows) leaving the class one set ℋ \\mathcal {H} invariant.", "Right: a sample element x∈ℋ⊂𝒳x\\in \\mathcal {H}\\subset \\mathcal {X}." ], [ "Algorithmic complexity of a class function", "We have a notion of algorithmic complexity for strings or datasets, the Kolmogorov complexity, i.e., the length of the shortest (Turing) program capable of generating the string in some universal language [7].", "We would like to extend this notion to functions in the present context.", "This motivates the following definitions.", "Let an ordering of a countable set $S$ be a function $x_{i}: \\mathbb {Z} \\rightarrow S$ .", "Definition 2 The algorithmic complexity ${\\mathcal {K}}[f]$ of a binary valued function $f(x)$ is the algorithmic complexity of the ordered stacked dataset $S_{ \\mathcal {H}}\\equiv [ \\lbrace x \\in \\mathcal {X}| \\: f(x)=1\\rbrace ] = [ \\mathcal {H}]$ (a string or list) given its ordering, $\\mathcal {K}[{f }]= \\mathcal {K}[D_{f } | ordering] \\approx k_{f}^{*}$ where $k^{*}$ is the length shortest program capable of generating all hand images (the hard part) and then a set of all possible stacks from permutations of the images (easy).", "Thus, only $k_{f}^{*} $ bits are needed to specify the class function, or, equivalently, the generating function.", "If we did not know the order, we would need $\\log |\\mathcal {H}| !", "\\approx |\\mathcal {H}| \\log |\\mathcal {H}| $ bits to specify it: the algorithmic complexity of the dataset is $\\mathcal {K}[{D_{f} }] = k_{f}^{*} + \\log |\\mathcal {H}| !", "\\approx k_{f}^{*} + |\\mathcal {H}| \\log |\\mathcal {H}| $ .", "This is large, but much smaller that the length of the dataset, $\\mathcal {K}[{D_{f} }] << l(D_{f})=|\\mathcal {H}|\\, \\log |\\mathcal {X}|$ .", "Ignoring the program length $k^{*}$ (which we assume is small compared to the other quantities), compression results from us being able to use $\\log |\\mathcal {H}| $ bits to specify a hand image, instead of the full $ \\log |\\mathcal {X}| $ , and the compression ratio is $\\rho = \\log |\\mathcal {H}| / \\log |\\mathcal {X}|$ ." ], [ "The group of invariances of the class function", "A class function defines an equivalence relation (i.e., images in the same class are equivalent).", "This motivates the following definition (Figure REF ).", "Definition 3 Given a class function $f(x): \\mathcal {X}\\rightarrow \\mathbb {B}$ , we define the set of invariant transformations of the function (or more simply invariances) to be automorphisms of the domain space $T:\\mathcal {X}\\rightarrow \\mathcal {X}$ such that $\\forall x \\in \\mathcal {H}$ we have $T(x) \\in \\mathcal {H}=\\lbrace x \\in \\mathcal {X}| \\, f(x)=1 \\rbrace $ .", "The set of all such automorphisms forms a group, $\\mathcal {G}_{f}$ , the group of invariances of the class function, which acts on input (image) space.", "Examples of such automorphisms are permutations of the elements of particular class in $\\mathcal {X}$ that leave the other one alone (permutation on $\\mathcal {H}$ x Identity on $\\bar{\\mathcal {H}}$ ).", "More generally, $\\mathcal {G}_{f}$ contains as elements all permutations that leave elements in the same class.", "Also, $\\mathcal {G}_{f}$ is a big set, with as many elements as permutations of elements in $\\mathcal {H}$ multiplied by those of its complement, $G_{f} \\sim S_{\\mathcal {H}} \\times S_{\\bar{\\mathcal {H}}}$ That $\\mathcal {G}_{f}$ forms a group is easy to see: it has an identity, composition of invariances is an invariance, each invariance has an inverse, and composition is associative.", "It is immediate that the group of invariances is equivalent to the class function as well (up to labeling).", "Theorem 1 The group of invariances of a class function $f(x): \\mathcal {X}\\rightarrow \\mathbb {B}$ uniquely determines the class function up to labeling.", "Proof: by definition, $f(x)=f(x^{\\prime })$ if and only if there exists $T\\in \\mathcal {G}_{f}$ such that $T(x) =x^{\\prime }$ .", "Thus, the set of invariances partitions $\\mathcal {X}$ into subsets which can be assigned a unique, distinct label (the equivalence class), reproducing the class function.", "Similar reasoning applies to an $q$ -class (multiclass) functions.", "Each class has associated a group of invariances of the set of images.", "Thus, learning a class is equivalent to learning the associated function invariance group (up to label, i.e., up to $\\log q!$ bits, with $q$ the number of classes).", "Thus class functions are equivalent to subgroups of invariances or permutations, so by Cayley's theorem, to some group (every group is isomorphic to a group permutations).", "For instance, the group of rotations acts on image space an appears as a homeomorphism from the rotation group to the group of permutations of the hand image space (a permutation representation of rotations).", "We can see that in this framework, function, invariance group and dataset algorithmic complexities are essentially equivalent notions.", "It is the structure of the group of invariances that is related to the depth of the associated generating function, which can be seen as using the group actions to iterate from image to image.", "For instance, rotations of images (group action) can sequentially generate many of the images in the stack.", "This is a “deep”, recursive operation (there is a Lie group involved), and one that a classifier can exploit." ], [ "The efficiency of implementation of a class function by a neural network", "Although neural networks can be seen to implement functions (or models, group of invariances or datasets), they may be inefficient implementations.", "We can imagine, for instance, a network with dead-end or even disconnected nodes.", "A neural network may be defined, essentially, by a set of connectivity matrices and biases (we assume the node activation function to be constant).", "This motivates the definition of implementation length.", "Definition 4 The implementation length of a neural network, $l(\\mathcal {N})$ , is the minimal program length required to specify the architecture and parameter values of $\\mathcal {N}$ (the algorithmic complexity of its architecture).", "We note that implementation length is directly related to the number of parameters in the networks's architecture, and hence to the amount of data it will need for training.", "We can now define the concept of implementation efficiency of a function (or model) by a network, which quantifies how succinct is the implementation of the class function.", "Definition 5 Let network $\\mathcal {N}_{f}$ implement a function $f(x)$ .", "The efficiency of the network is defined by the ratio of the algorithmic complexity of the function and the minimal description length of the network implementation, which we can assume lies between 0 (inefficient) and 1 (optimal) after some normalization: $\\mathcal {E}[\\mathcal {N}_{f}] = { {\\mathcal {K}}[f] \\over l(\\mathcal {N}_f)}.$" ], [ "The entropy of a class function; Deep and shallow functions", "Can we define the entropy of a class function?", "We know that the stacked dataset must contain at least $|\\mathcal {H}| \\log |\\mathcal {H}|$ bits, and no more than $|\\mathcal {H}| \\log |\\mathcal {X}| = l(D_{f})$ .", "We recall here that entropy rate is closely related to Lempel-Ziv compression [7].", "Definition 6 Let $h_{D_{f}}$ be the entropy rate of the dataset.", "The total (Shannon) information or entropy of the dataset/model/function is $H[D_{f}]=h_{D_{f}}\\, | \\mathcal {H}|$ .", "We say a classification function is deep when its algorithmic complexity is much smaller than its entropy, i.e., $K[D_{f}] <<H[D_{f}] \\approx l_{LZW}[D_{f}] \\approx |\\mathcal {H}| \\, h_{D_{f}}$ .", "Intuitively, a function is deep if it is doing a lot (generating entropyOne is tempted to make use here, or a connection to, ideas in the mathematics of fractals, such as fractal dimension, for a metric that relates to this idea of generating entropy or “filling” output space.)", "despite being algorithmically simple.", "Consider the following description lengths for the dataset, starting from the initial long one: $d &=& l(D_{f}) = |\\mathcal {H}| \\log |\\mathcal {X}| \\\\d_{lzw}&=& l(D_{f}) \\, h_{D_{f}} = h_{D_{f}}\\, |\\mathcal {H}| \\log |\\mathcal {X}| \\\\d_{\\mathcal {K}}&=& \\mathcal {K}[D_{f}] = k^{*} + |\\mathcal {H}| \\log |\\mathcal {H}|$ We note that it is known that for most, but not all datasets, $d_{lzw} \\approx d_{\\mathcal {K}}$ [7].", "Examples of deep generating functions from $\\mathcal {P} = \\mathbb {B} ^{m} \\rightarrow \\mathcal {X}= \\mathbb {B} ^{n}$ are “For a given ordered input of $m$ bits, move $2^{m} * n$ positions to to the right in the binary expansion of $\\pi $ and extract the following $n$ bits ” using algorithms from Ramanujan's work [3].", "Or, “using an elementary Cellular Automaton such as Rule 110 [19] with dimension $n$ and some random initial condition, carry out $2^{m}$ steps and return that line of $n$ -bits.” In both cases, many iterations are needed.", "The functions are highly recursive (basically performing the same operation over and over), with at least as many recursions as there are “images” in the stack ($ |\\mathcal {H}|$ ).", "The role of recursion in computation is also discussed by Bennett [2], who proposed the concept of logical depth of a string at a significance level $s$ as the minimal number of steps $t$ required to compute it with $s$ bits more than the minimum program length.", "On the other extreme, we may talk of “shallow” functions, which require the direct use of a look-up table to output data.", "An example would be a function to describe Chaitin's constant $\\Omega $ or other truly incompressible datasets.", "As we discuss below, a shallow NN is appropriate for such a task, but deep functions benefit from the availability of the for-loops offered by multilayer or recurrent networks." ], [ "Shallow vs. Deep vs. Recurrent networks", "A neural network can be seen as implementing a program to compute a function as a sequence of steps in a special programming language.", "In this language, each layer represents a set of parallel computational steps in a sequence of several possible.", "Each step uses the same basic operations.", "As we saw, the function thus encoded provides the means for compression of datasets which can then be represented efficiently by the equation $f(x)=1$ .", "A Shallow network is a network of the form [14] $\\mathcal {N} (x) = \\sum _{k=1}^{N} a_{k}\\, \\sigma ( \\langle w_{k} , x \\rangle + b_{k} ) .$ A deep network can be recursively defined by $h^{1}=x$ and $N (x) = h^{L+1}$ , where $h^{l+1} = \\sum _{k=1}^{N_{l}} a_{k}^{l} \\, \\sigma ( \\langle w_{k}^{l} , h^{l}\\rangle + b_{k}^{l} )$ for $l=1,..,L$ , which includes a SNN as a special case with only one layer.", "An RNN has the form $h^{t+1} = \\sum _{k=1}^{N} a_{k} \\, \\sigma ( \\langle w_{k} , h^{t}\\rangle + \\langle v_{k} , x^{t}\\rangle + b_{k} )$ where $N (x^{t}) = \\sigma _{o} (\\langle w_{o}, h^{t} \\rangle +b_{o}).$ Feedforward NNs have been shown to be able to approximate any multivariate primitive recursive function [8], [12].", "Moreover, if the function to be approximated is compositional or recursive, e.g., $f(x_1, á á á , x_8) = h_3 \\left(h_{21} \\left(h_{11} \\left(x_1, x_2\\right), h_{12}\\left(x_3, x_4\\right)\\right), h_{22}\\left(h_{13}\\left(x_5, x_6\\right), h_{14}\\left(x_7, x_8\\right)\\right) \\right)$ (which is best visualized as a compositional tree), then it is known that a hierarchical, compositional network performs better (will need less training data) [14].", "This is a way of saying that if the data we want the NN to learn is in some sense simple (compositional or recursive), then simpler structure networks (hierarchical) will perform better (given the same amount of training data) than shallow networks (SNNs), because they can achieve the same accuracy with simpler architectures (fewer nodes or “complexity”).", "Thus, although shallow networks can approximate any function, they are limited as a programming language in terms of their efficiency.", "Compositionally (depth) provides NNs more programming power.", "Intuitively, a random function can be efficiently encoded by a shallow network, since all that can be done is provide the function table, which is essentially what a SNN will do.", "If a dataset has low algorithmic complexity (allows for a short description) yet its model/function has high entropy, it is fairly intuitive that generally speaking deep networks will provide more succinct, and hence easier to train network structures than shallow ones, which are limited to one step computations (smaller programming language repertoire) and essentially to producing function tables [15].", "However, a DNN is again a limited type of system as a programming language: the number of steps it can carry out are fixed once and for all.", "In essence, a DNN can only carry out a fixed, finite number of for-loop steps and can be seen as the equivalent to primitive recursive functions (see, e.g., [19]).", "It is not capable of universal computation, as its depth is fixed.", "Once trained, it computes a primitive recursive function: once its weights are determined, is deterministic w.r.t its inputs.", "Recurrent neural networks (RNNs), on the other hand, are known to be Turing complete [18], [9].", "Recurrence enables universal modeling.", "In an RNN, outputs depend on inputs and its initial state.", "Thus, it can respond to the same input in different ways, depending on its state (program).", "In computation theory terms, it includes the analog of the while loop, which extends their reach and makes them universal.", "This gives it a much larger repertoire, including memory, and puts it the class of $\\mu $ -recursive functions or universal Turing machines.", "Thus, although all NNs are universal with regard to primitive recursive functions, RNNs, then DNNs and last SNNs should form an efficient encoding hierarchy and, hence, a hierarchy in terms of training set requirements (since the architecture of the network determines how many trainable parameters it has).", "We formalize this in the following conjecture, which is in the same spirit as the one in [14]: Conjecture 1 (Network Hierarchy) The implementation efficiency of deep functions (with $\\mathcal {K}[f] << \\mathcal {H}[f]$ ) by networks is highest for recurrent, then deep, then shallow networks, i.e., $\\mathcal {E}[\\mathcal {N}^{recurrent}_{f}] \\ge \\mathcal {E}[\\mathcal {N}^{deep}_{f}] \\ge \\mathcal {E}[\\mathcal {N}^{shallow}_{f}]$ Furthermore, the inequalities are stronger the larger the ratio of entropy to algorithmic complexity of the function, $ \\mathcal {R}_{f}= \\mathcal {H}[f] / \\mathcal {K}[f] $ .", "We sketch a possible proof: SNNs and DNNs are equivalent computationally to PR functions, a subset of the language used in $\\mu $ -recursive functions.", "RNNs are universal.", "This is already known [18].", "Although we deal here with functions that are computable by SNNs (using lookup tables, since the input and output space is finite), access restricted to a subset of a programming language can only lead to longer algorithmic descriptions and therefore longer implementation lengths and heavier architectures (more parameters to train).", "This is immediate as well.", "In particular, not having access to for-loops (SNNs), or only to a fixed finite number of them (as in DNNs) is a disadvantage.", "We can take the viewpoint of the generating function angle (equivalently to class function or model).", "We have $|\\mathcal {H}| \\log |\\mathcal {H}|$ bits of information as inputs (generating function parameters) to expand into the dataset, which has $|\\mathcal {H}| \\log |\\mathcal {X}|$ random looking bits.", "This is to be achieved by iteration (repeated calculation).", "In order to generate randomly looking sequences, recursion, reuse of prior calculations must be employed (for-loop steps)—copying inputs many times will not generate entropy.", "Deep functions will benefit greatly from DNN architectures.", "A shallow function computing an algorithmically random string will require a shallow network with essentially as many nodes as the size of $|\\mathcal {X}|$ (to function as a lookup table).", "In the case of a deep (simple) function, a DNN or an RNN, depending on how deep the function is, will require fewer nodes (and training points), because this architecture can exploit and represent the simplicity of the underlying function.", "Paralleling the reasoning in [14], in this case we would need $\\log |\\mathcal {X}|$ nodes (one per input), but it would be interesting to getter a better estimate on the number of nodes as a function of the size of both input and hand-space, and the algorithmic complexity of the model." ], [ "Modeling in the Solomonoff prior", "With regard to the practical implication of this conjecture, we can extend it making reference to the Solomonoff priorAlternatively, we can argue the relevance of entropic strings from a anthropic algorithmic principle: cognitive system can one model deep functions.", "[7].", "We recall here the (Solomonoff) algorithmic or universal (un-normalized) probability $P_\\mathcal {U}(x)$ of a string $x$ .", "This is the probability that a given string $x$ could be generated by a random program.", "An important result is that this is given by $ P_\\mathcal {U}(x)\\approx 2^{-K_\\mathcal {U}(x)}$ [13].", "Thus, short programs contribute most to the probability of observing a given data string and, furthermore, the probability of a given string to be produced by a random program is dominated by its Kolmogorov complexity.", "We can hypothesize that such is the case in the real world: most datasets, models or functions in the real world will be short but entropic (simple programs are know to produce entropic datasets [19]).", "Conjecture 2 The implementation efficiency of networks is higher for deep than shallow networks in the Solomonoff universal prior.", "That is, for most dataset derived functions (with high universal probability), $\\mathcal {E}[\\mathcal {N}^{r}_{f}] \\ge \\mathcal {E}[\\mathcal {N}^{d}_{f}] \\ge \\mathcal {E}[\\mathcal {N}^{s}_{f}]$ with the probability function defined by the Solomonoff prior.", "Furthermore, the inequalities are stronger the larger the ratio of entropy to algorithmic complexity of the function, $ \\mathcal {R}_{f}= \\mathcal {H}[f] / \\mathcal {K}[f] $ .", "The proof rests on showing that most simple functions are also entropic." ], [ "Perturbation of an optimal, well trained network", "Here we wish to consider a scenario in which a) one class is much smaller than the other (l$|\\mathcal {H}|/|\\mathcal {X}| <<1$ ) and b) the smaller class is very large still compared to the class function program length (the function is deep).", "For instance, let us consider the simplest trained NN with perfect performance in identifying hand images.", "To train this NN, suppose that we have used a short, simple program to generate a large number of hand images by transformation of a template.", "For simplicity, let us consider only 3D translations and rotations of the hand.", "Those involve 6 degrees of freedom.", "Allowing for 64 bits per parameter, we have some $2^{64\\cdot 6} \\sim 10^{116}$ possible hand images, to which we need to add backgrounds using a random number generator (another big space).", "This may seem big, but it is small compared to the input space $2^{24\\cdot 10^{6}}$ in monochrome 24 bit images.", "Perturbation of inputs to the NN: Let us suppose, then, that we have found a perfectly performing, efficient network $D$ for our binary classification problem.", "What can we say about its connectivity?", "For comparison, let $D_{r}$ be a network with the same architecture but random weights.", "If we input an arbitrary image of a hand $h \\in \\mathcal {X}$ to $D$ it will certainly output 1 (for hand, $D(h)=1$ , while the outcome with the second network will be random.", "Now suppose that we perturb the hand image.", "Since the space of hand images is much smaller than the space of not-hands (low relative entropy of $f(x)$ ), the effect on $D$ will be to change its output to 0 with high probability, $D(h+\\delta h)=0$ , while the effect on the random network will be random.", "We can summarize this by averaging over the space of perturbations, $\\langle D(h) - D(h+\\delta ) \\rangle =1, \\:\\: \\:\\: \\langle D_{r}(h) - D_{r}(h+\\delta ) \\rangle =0$ Moreover, if we ask the same question with “not-hand” input images $\\bar{h} \\in \\mathcal {X}$ , a perturbation will not have much effect in either case.", "In the first case, this is simply because the subspace of not-hand images is much bigger than the subspace of hands.", "A perturbation of “not-hand” will, with probability near one, leave the image in the same class.", "In the case of the random network, the outcome will be random, and the change again average out.", "$\\langle D(\\bar{h}) - D(\\bar{h}+\\delta ) \\rangle =0, \\:\\: \\:\\: \\langle D_{r}(\\bar{h}) - D_{r}(\\bar{h}+\\delta ) \\rangle =0$ So we have two similarly sized NNs, and their response to perturbation is very different depending on the class of the image they are processing.", "We can summarize this in the following theorem: Theorem 2 In an efficient, well trained NN encoding a class function in which one class is much smaller than the other, a perturbation of the NN input when fed by an example in the smaller class will propagate efficiently with high probability, but not in a randomly parameterized NN with the same architecture or for activations in the other, larger class.", "Perturbation of nodes in the NN: What about perturbations of a node in the network?", "An efficient but shallow NN implementing a deep function with a large range of size $|\\mathcal {X}|$ will use many nodes, basically as many as the size of $|\\mathcal {X}|$ (to see this one can refer to the work in [14], with $\\epsilon $ seen as scale parameter to discretize the input space into hypercubes of size $\\epsilon ^{1/m}$ ).", "Perturbation of the activation function or value of a node will only affect the output for some input values, but not all the input space.", "For an efficient DNN, on the other hand, each node will have to play a crucial role in the calculation of all the values in the function table.", "A perturbation of the node will have a big impact on the entire function.", "The impact on a performing deep efficient network will again be to take to the larger class.", "If it did not, we could simply remove that node and get an equally performing simpler network.", "Perturbations will always lead to classification into the larger class space.", "Theorem 3 In an efficient, well trained NN encoding a class function in which one class is much smaller than the other, a perturbation of a node NN when fed by an example in the smaller class will propagate efficiently with high probability if the network is deep, but not in a shallow NN, or randomly parameterized NN with the same architecture or for activations in the other, larger class.", "This potentially establishes a link between the perturbation complexity index (PCI) measured by perturbing human cortical neuronal networks by TMS [6] and algorithmic complexity.", "Furthermore, perturbations in such a network will appear to be decorrelated at different locations due to the non-linear nature of signal transmission in NNs.", "This provides the “information” aspect and a potential explanation for hard to compress multichannel EEG streams using LZW in the PCI [6].", "Finally, we note that although hierarchical networks represent a subset of all multivariate functions [14], it would appear that RNNs can bypass this limitation, as they are universal, by affording compositional DNNs of infinite depth." ], [ "Discussion", "What mechanisms are in place to evolve simple programs or efficient networks?", "A network is actually a computer program, and it may be a good or a poor implementation of the model, so we may ask how long or even how compressible this program is as a string.", "Such program may contain much underused, inefficient or even irrelevant code, while maintaining great classification performance and in fact implementing a simple model.", "For example, the program, in general Turing machine terms, could have lengthy code to write and erase a billion random digits before outputting the class label.", "In a network we could have orphaned nodes that don't contribute to the final computation.", "So the program or network architecture, as a string, may actually be huge and incompressible (algorithmically very complex).", "How such an inefficient program has come to be found during training among all possible others of that size is an interesting question.", "We can think of this aspect of the problem from the point of view of evolutionary programming.", "Let us suppose we set up a problem with a simple program as the solution.", "We can imagine we have setup an evolutionary system where programs are bred to perform well on the task, starting from small programs that aggregate to larger ones at each generation.", "It is fairly clear that in such an organic-growth type of search, the large, inefficient but performing program will never be found, since smaller solutions are at hand, and the space of programs becomes huge quickly with program size.", "From this growth structure, hierarchies and power laws will emerge naturally [16].", "Since we may know (because we defined it) the underlying model to be very simple, searching for solutions from short to longer programs will speed up the process significantly.", "We can conjecture that as real neural networks develop in brains during learning, they must implement simplicity principles in the process, such as sparsity.", "One mechanism is a synaptic cap growth model, which seems to force in a natural way the emergence of sparse networks [10].", "Of course, sparsity is one way to approach the $\\mathcal {K}$ minimum.", "Another example is [5], where it is shown that one way to create robust networks with high memory capacity (attractors or learned sequences) is by a) sparsity, b) more numerous, stronger bidirectional connections than random networks.", "In machine learning, e.g., from statistical learning theory to Echo State Networks, sparsity is important and normally used.", "Robustness of such networks is also related to good generalization [4] and hence sparsity or, probably more generally, Kolmogorov simplicity.", "And in terms of resources, simplicity is great: less memory and computation will be needed.", "Scarcity, competition are both helpful in this context, since they will lead to some form of simplicity.", "This may be the reason why evolutionary search is successful.", "It provides a gradient towards simple programs.", "Memory, power/energy and computation time as limited resources will undoubtedly be important for model selection in competitive worlds.", "Why is the universe simple?", "Is the answer anthropic — i.e., “If it wasn't simple, there would not be a discussion” — or can we do better?", "Regardless of the answer, assuming there is simplicity exists is a valid starting point (as generations of physicists can attest).", "Acknowledgements: This work partially supported by the FET Open Luminous project (this project has received funding from the European Union's Horizon 2020 research and innovation programme H2020-FETOPEN-2014-2015-RIA under agreement No.", "686764).", "The author is very grateful to Tomaso Poggio for his lectures (9.520 Statistical Learning Theory and Applications - MIT) and inspiring discussions on the topics of network/function compositionally, depth and their links to simplicity." ] ]
1612.05627
[ [ "Metallicity-dependendent kinematics and morphology of the Milky Way\n bulge" ], [ "Abstract We use N-body chemo-dynamic simulations to study the coupling between morphology, kinematics and metallicity of the bar/bulge region of our Galaxy.", "We make qualitative comparisons of our results with available observations and find very good agreement.", "We conclude that this region is complex, since it comprises several stellar components with different properties -- i.e.", "a boxy/peanut bulge, thin and thick disc components, and, to lesser extents, a disky pseudobulge, a stellar halo and a small classical bulge -- all cohabiting in dynamical equilibrium.", "Our models show strong links between kinematics and metallicity, or morphology and metallicity, as already suggested by a number of recent observations.", "We discuss and explain these links." ], [ "Introduction", "Observations have recently revealed an interesting coupling between kinematics and metallicity of stars in the Milky Way (MW) bar/bulge region.", "[6] (her Fig.", "4, see also [4]), reviewing the link between metallicity and kinematics, collected data from a number of sources and plotted the velocity dispersion ($\\sigma $ ) versus the absolute value of the latitude ($b$ ).", "She found that for low metallicity stars, $\\sigma $ shows very little, if any, trend with $|b|$ , while for high metallicity stars $\\sigma $ clearly decreases with increasing $|b|$ .", "[17] (hereafter N13b; see the lower panels of their figure 6) binned the ARGOS data by metallicity, and plotted the velocity dispersion as a function of longitude $l$ .", "This revealed that the higher metallicity stars (0$<$ [Fe/H]) have two clear trends: First, stars at $l$ =0$^{\\circ }$ have high velocity dispersions which decrease with increasing $|l|$ , and, second, stars at low latitude ($|b|$ =5$^{\\circ }$ ) have a larger velocity dispersion than stars at high latitude ($|b|$ =10$^{\\circ }$ ).", "They also showed that the lower metallicity stars (-1.0$<$ [Fe/H]$<$ -0.5) have higher velocity dispersions than the higher metallicity stars and also that the velocity dispersion depends only little on longitude or latitude.", "Just after this letter was first submitted, [24] ([24], hereafter Z16) presented APOGEE data for the MW bulge.", "Given the relevance of these data to our results we added a posteriori a short discussion of them.", "We present here a theoretical study of this interesting chemo-kinematic coupling.", "Previous simulation studies of metallicity in the bar/bulge region [7], [14], [9] used pure N-body simulations and thus included neither gas nor star formation.", "Nevertheless, by assuming an initial metallicity radial distribution, they were able to study its redistribution due to bar formation and evolution.", "Here, we avoid such a short-cut, and use a coupled chemical-kinematical-morphological approach, based on an N-body simulation obtained with a code including both gas and star formation, coupled to a chemical evolution code which follows the distribution of the chemical elements as a function of time and location in the galaxy.", "The results of this letter were presented in two meetings on our Galaxy, one in Paris (19 - 23/09/2016)http://www.iap.fr/vie_scientifique/ateliers/MilkyWay_Workshop/2016/ and then click on program and the other in Canberra (21 - 25/11/2016)https://www.aao.gov.au/files/conferences/Lia_GASP16_0.pdf.", "Our disc galaxy formation model has been described in detail by [3] ([3], hereafter A16).", "We will present it here only very briefly.", "We assume that the galaxy we model underwent a major merger about 8 – 10 Gyr ago and start our simulations with two spherical protogalaxies composed solely of dark matter and gas, while stars form all through the simulation.", "These two protogalaxies are set on an orbit bringing them to a collision.", "The stars born before the merging undergo violent relaxation and form a spheroidal, centrally concentrated object.", "The stars born during the merging are strongly shuffled by the quickly varying potential and form mainly a thick disc or an extended stellar halo.", "Up to this point the evolution is merger driven.", "After the merger settles down the evolution becomes secular, and the thin disc starts forming from the gas accreted from the gaseous halo.", "Some of it may thicken and contribute to the thick disc, but, as gas accretion continues all through the simulation, a thin disc of relatively younger stars is always present." ], [ "Code", "The code describing the dynamical evolution is based on gadget3 (a Tree SPH code, [20], [21]) and is described in A16 and in Rodionov et al.", "(subm.).", "The mass of the baryonic particles is $10^4 M_\\odot $ , and that of the dark matter ones $4 \\times 10^4 M_\\odot $ , with 10 and 17.5 million particles in each of these components, respectively.", "The softening is 25 pc.", "For the chemical part we adopt the Single Stellar Population (SSP) formalism, where a single simulation particle represents a population of stars of the same age.", "Within that “stellar particle\", stars are distributed according to the IMF of [11].", "They release their ejecta after a finite, mass dependent, lifetime.", "Yields for 12 selected elements are metallicity dependent and are taken from [18] for single stars and from [10] for SNIa (see Appendix C in [12] for details).", "A more detailed description will be given elsewhere (Rodionov et al., in prep.).", "There is one crucial free parameter in the present application of our code, namely the metallicity at the beginning of our simulation.", "If we had a cosmological simulation we would have started with zero metals.", "However, in order to fully follow the dynamics and evolution of the bar/bulge region we need to have a dynamical simulation which does not start at the time of the Big Bang and the formation of the large scale structure.", "It just starts from the progenitors of the last major merger and we need to assume that some of the elements are already formed.", "In this letter we use an initial metallicity of [Fe/H]=-1 [23].", "More on different choices will be included in Rodionov et al.", "(in prep.", ")." ], [ "Model", "Simulation mdf732, described in some detail in A16, has properties that make it a reasonable choice for a qualitative model of the bar/bulge region of our Galaxy.", "In particular it has a classical bulge with only 9 – 12% of the total stellar mass and a bar of roughly the correct size, with a boxy/peanut inner part.", "We thus reran it using 27.5 million particles to enhance the signal-to-noise, because we need to split the data by metallicity, $b$ and $l$ .", "We use here the snapshot of this new simulation at time $t$ =10 Gyr, but we made sure that other late times in the simulation around this one gave very similar results.", "As described in A16, there are two times which can be considered as landmark times for the disc galaxy formation.", "The first is the beginning of the merging period, or merging time, $t_{bm}$ , which is the time beyond which the distance between the centres of the two protogalaxies becomes and stays less than 1 kpc (A16).", "The second time is $t_{bd}$ , the time at which the thin disc starts forming.", "The values of these times for our simulations are 1.4 and 2.2 Gyr, respectively (A16).", "Although we do not claim to have a full model of our Galaxy, our model is a sufficiently reasonable approximation to use for the present qualitative comparison.", "Here we follow [16] and consider only stars within a cylindrical radius of 3.5 kpc from the centre, so as to concentrate on the bar/bulge region.", "Except for Fig.", "REF and its discussion, we use everywhere the geometry of our Galaxy, placing the bar major axis at 27$^{\\circ }$ to the line from the Sun to the Galactic centre and assuming that the Sun is on the equatorial plane at a distance of 8 kpc from the centre.", "In all the following analysis we use three metallicity bins, namely low metallicity (-1.0$<$ [Fe/H]$<$ -0.5, hereafter LM), intermediate (-0.5$<$ [Fe/H]$<$ 0, hereafter IM) and high metallicity (0$<$ [Fe/H]$<$ 0.5, hereafter HM) bins, with one exception, namely in Sect.", "REF , where the high metallicity bin is defined as 0$<$ [Fe/H], to follow N13b.", "Furthermore, in Sects.", "REF and REF we add a yet higher metallicity bin with 0.5$<$ [Fe/H]$<$ 1 (hereafter HHM)." ], [ "Comparison to the Babusiaux plot", "To show the link between metallicity and kinematics, [6] plotted the dispersion of the line-of-sight velocity as a function of latitude and constrains her plot to stars around $l$ =0$^{\\circ }$ and from either the LM, or the HM metallicity ranges, i.e.", "neglecting all stars with IM metallicities.", "We repeated this for our simulation and three values of the longitude and give the results in the left panel of Fig.", "REF .", "We find that, for LM stars, $\\sigma $ takes higher values than for HM ones.", "Furthermore, for LM stars $\\sigma $ varies little with $|b|$ or $|l|$ , while for HM stars it decreases considerably with increasing $|b|$ .", "Note also that this decrease is strongest for $l$ =0$^{\\circ }$ , and weakens with increasing $|l|$ .", "We thus conclude that there is an excellent qualitative agreement between our simulations and observations (and even a reasonable quantitative one, as can be seen by comparing our figure with figure 4 of [6])." ], [ "Comparison to ARGOS and APOGEE results", "Fig.", "REF shows the velocity dispersion as a function of $l$ for three metallicity bins, i.e.", "from left to right, stars with 0$<$ [Fe/H], IM and LM stars respectively, i.e.", "the same bins as in N13b, so that it can be directly compared to the lower panels of figure 6 of that paper.", "LM stars have in general higher values of $\\sigma $ than HM ones.", "For HM stars, $\\sigma $ is much larger at small $|b|$ than at large $|b|$ .", "This difference with $|b|$ is very strong for l=0$^{\\circ }$ and decreases with increasing $|l|$ .", "For small values of $|b|$ , and still for HM stars, there is also a dependence on $l$ , the velocity dispersion being higher at the centre and decreasing with increasing $|l|$ , as expected.", "This decrease is stronger for low than for high $|b|$ .", "These trends can also be found for LM stars, but to a much less pronounced.", "The IM stars have an intermediate behaviour, as expected.", "We thus reproduce well the main observational characteristics found by N13b.", "In the lowest metallicity bin, however, the $\\sigma $ stays considerably flatter with $l$ than in N13b.", "This discrepancy is even stronger when we compare with the Z16 data.", "Indeed, the Z16 data give results qualitatively in agreement with [17], but quantitatively showing a smaller difference between the kinematics of the various populations.", "This could well be understood by the different selection criteria of the N13b and Z16 samples (inclusion or not of foreground stars, choice of metallicity bins, latitudes, etc.).", "Nevertheless, the effect of the various selection criteria on the data is well beyond the scope of this letter and will be left for future work.", "However, this difference between the two observational data sets helped us focus on the initial [Fe/H] used our simulations (Sect.", "REF ) and we made a new calculation, starting the chemical evolution from zero metals initially.", "This produced a considerably stronger decrease of $\\sigma $ with $|l|$ , although always less than the corresponding decrease in the higher metallicity bins.", "We can thus achieve a good qualitative agreement.", "It is difficult, however, to obtain a quantitative agreement and to find, even approximately, what the best initial metallicity value is.", "We thus again limit ourselves to a qualitative agreement.", "We conclude that there is very good qualitative agreement between our simulations and observations, and we stress that this is found directly from the simulations, without having to add by hand any further stars, implying that our scenario of the galaxy formation history gives a right mix of the various types of stars.", "In the middle panel of Fig.", "REF we plot the mean star formation time after binning stars by metallicity and longitude.", "We see clearly that the stars in the low metallicity bin were born preferentially between, on average, one or two Gyr from the beginning of the simulation, i.e.", "around $t_{bm}$ (1.4 Gyr), while those in the HM bin were born considerably later, between, on average, 4 and 7 Gyr ago.", "Note also that there is not much dependence of these birth times on $|b|$ .", "The right panel of Fig.", "REF shows the star formation history of stars within the inner 3.5 kpc (SFH$_{3.5}$ ), i.e.", "the histogram of the stellar mass born at a given time.", "This is done separately for our three metallicity bins, i.e.", "LM, IM, HM and the yet higher metallicity HHM bin.", "We note that the LM and HM have SFH time distributions of quite different forms.", "The former were born much before the latter, in good agreement with what was shown in the middle panel of Fig.", "REF .", "Furthermore, the birth times of the LM stars are quite concentrated around $t_{bm}$ (1.4 Gyr), suggesting that these star formations are due to a starburst, presumably from strong inflow of gas during the merging.", "On the other hand, those of the HM stars are very spread out, starting from 7 Gyr ago and continuing up to the end of the simulation.", "The IM stars are intermediate, as expected." ], [ "Spatial distribution of metallicity-defined populations", "To understand the spatial distribution of populations with different metallicities we viewed them side-on, i.e.", "edge-on with the line of sight in the equatorial plane and along the bar minor axis (Fig.", "REF ).", "In this figure we wish to view the bar and its B/P structure in an optimal manner and not to compare with observations.", "We thus exclude stars from the foreground and background disc component, using a cut-off of $|\\Delta y|$$<$ 0.5 kpc, where $y$ is the distance of the `star' from the plane perpendicular to the line of sight and through the centre of the galaxy.", "On the contrary, in Fig.", "REF we view the bar in $l$ , $b$ coordinates, i.e.", "as viewed from the Sun, and introduce only the cut-off of $R$$<$ 3.5 kpc.", "We then split, both for Figs.", "REF and REF all stars in four metallicity bins – LM, IM, HM and HHM.", "The difference between the distributions corresponding to different metallicity ranges is striking.", "The lowest metallicities are distributed in a spheroidal-like shape flattened in the vertical direction, and including a low density disc, which has a thick and smooth outline in the $z$ direction.", "They do not show any, or very little, X-shaped structure, in good agreement with observations [15], [22], [19], [13].", "The IM stars show a clear X-shape embedded in a spheroidal-like or boxy-like part, implying that the stars in this bin come from two quite different populations and components.", "The HM bin has only the X-shape and a clear underlying disc component.", "These two panels show that the X-shape is of the off-centred type (OX, [8]), i.e.", "that the branches of the X join only in pairs, two on either side of the centre ($>$ —-$<$ ), contrary to the centred X (CX), where all four branches join in the centre ($>$$<$ ).", "This is the case for about half of the [8] sample of X-shaped bulges.", "Finally, the distribution of the material in the last metallicity bin (HHM) is centrally concentrated and its vertical extent is small compared to its horizontal extent.", "Although this definitely needs further study, it is tempting to associate it with a discy pseudo-bulge (see e.g.", "[2] and [1] for definitions and reviews).", "These density distributions show that the bar/bulge region is complex, with many co-existing stellar populations, while the ratio of their mass is a function of location.", "This also is in good agreement with observations ." ], [ "Summary and discussion", "We use numerical simulations that follow the formation of a disc galaxy subsequent to a major merger of two protogalaxies with extended gaseous haloes.", "In a previous paper (A16) we showed that such events could form realistic disc galaxies.", "We thus apply such a model to the central bar/bulge region of our Galaxy.", "Metallicity distribution functions [16] imply the existence of more than one stellar population in this region.", "We thus analysed, as in observations, separately three populations, with metallicities -1.$<$ [Fe/H]$<$ -0.5 (LM), -0.5$<$ [Fe/H]$<$ 0 (IM) and 0$<$ [Fe/H]$<$ 0.5 (HM), respectively.", "As in observations, we limited ourselves to the inner 3.5 kpc so as to concentrate on the bar/bulge region.", "We compared the velocity dispersions of the LM and HM populations (IM stars have an intermediate behaviour) and found them quite different.", "This implies that they have quite different kinematics.", "LM stars have large velocity dispersions, which, furthermore, show little, or no, dependence on either latitude or longitude.", "HM stars located near the equatorial plane have roughly the same velocity dispersion as the LM ones, but this dispersion decreases substantially with increasing $|l|$ , or b.", "The decrease with $b$ is stronger for smaller longitudes.", "We thus find excellent qualitative agreement between our simulations results and those of spectroscopic observations, concerning the kinematics, the metallicities and their link.", "This argues that the properties we are discussing must be generic to disc galaxies with boxy/peanut/X bulges, because our model, although it gives a reasonable fit to a number of MW quantities and properties, was not specifically built for this purpose.", "We also followed the ages and the SFH$_{3.5}$ separately for each of our metallicity defined populations.", "We find that the LM stars were born early on and in a relatively narrow time range centred roughly on $t_{bm}$ (1.4 Gyr).", "On the other hand, the HM stars are born later, up to considerably later, and over a much broader time range which extends to the end of the simulation.", "Finally we inspected the morphology of each of these populations separately.", "We found that the LM population makes a flattened spheroidal-like object, which could be considered as a compound of the thick disc, a stellar halo and (whenever present) a small classical bulge.", "The IM and HM populations show clearly an X-shape when viewed side-on and are the two only populations to do so.", "A clear conclusion from all our results is that the central region of our Galaxy, and therefore of a number of external barred galaxies, is very complex, which can be understood by realising that a number of components are cohabiting in this area.", "These components have widely different morphological, kinematical and chemical properties and the properties of any region depend on the relative density of these components in it.", "Such a mix may not be obvious to disentangle quantitatively, but is a natural consequence of the dynamical processes governing the formation of the MW.", "In A16 we had decomposed the baryons as a function of their birth time in five different populations.", "Using morphology, radial projected density profiles and circularity parameters, we found that stars born in the individual protogalaxies, i.e.", "before the merging, undergo violent relaxation during the merger and end up mainly in a triaxial classical bulge and a stellar halo.", "Stars born during the merging period contribute partly to a similar though more extended spheroid and partly to a thick disc and the bar that forms from it.", "Stars born in the beginning of this period mainly contribute to the classical bulge and only little to the thick disc.", "On the other hand, when we consider stars born at times nearer to the end of the merging period, it is the thick disc that is the main contributor.", "At times after the end of the merging period (more precisely for times after $t_{bd}$ ) the gas accreted from the gaseous halo forms a thin disc and its substructures such as spirals, or a bar, or a boxy/peanut bulge.", "This picture is of course a very simplified description of how the stellar populations form, because for example stars born after $t_{bd}$ (2.2 Gyr) in the thin disc could become members of the thick disc after being perturbed by small dark or luminous satellites, or by spirals.", "Furthermore, material from the gaseous halo will fall in at all times, and not only after $t_{bd}$ .", "Thus the boundaries set by the landmark times are not sharp and the whole evolution can be seen as a continuous sequence from an all-spheroid to an all thin disc formation, with strong changes at the landmark times.", "The results we find here are well compatible with this evolution picture proposed in A16.", "We find that the LM stars are old, and are born in a relative short time range around the merging time.", "They have larger velocity dispersions than HM stars, compatible with them being stars partly in a spheroid and partly in a thick disc.", "Most important, their 2D projected density distribution has the right shape for such components.", "The HM stars were born on average much later than the LM ones, and over a much more extended range of times, extending all the way to the present time.", "They generally have smaller velocity dispersions, as would be expected for disc stars.", "For $b$ =0$^{\\circ }$ , and in general up to roughly $|b|$ =4$^{\\circ }$ , their velocity dispersion decreases with increasing $|l|$ (left panel of Fig.", "REF , as expected for a disc.", "Most important, the corresponding 2D projected density distribution is that of a disc population, some of it having undergone the bar instability and then formed a clear X-shape.", "Thus in the bar/bulge region we find a classical bulge component, a thin and a thick disc component, together with the corresponding bar and boxy/peanut/X bulge, and, most probably, a discy bulge, although to affirm the latter we need yet higher resolution simulations.", "The ratio of masses of these components vary from one barred galaxy to another, so the one we have here may not correspond accurately to that of the MW.", "It is important to stress that none of the components is strictly confined to a given bracket, be it in time, or age, or metallicity.", "Thus at any location there is mix of populations.", "This cohabitation explains the links of metallicity with kinematics (as found in observations) and also with morphology, as stressed here.", "Regions that are dominated by the spheroid population will have both spheroid kinematics and spheroid metallicities, while regions which are dominated by disc population will have both disc kinematics and disc metallicities.", "This makes the link between kinematics and metallicities obvious, since we can not have e.g.", "spheroid kinematics and disc metallicities in the same component.", "Thus such links should be found also in external galaxies having a B/P structure viewed edge-on." ], [ "Acknowledgements", "We thank A. Bosma, C. Chiappini, K. Freeman, M. Ness and G. Zasowski for stimulating and useful discussions and the referee for a constructive report.", "We acknowledge financial support from the CNES (Centre National d'Etudes Spatiales - France), as well as HPC resources from GENCI - TGCC/CINES x2016047665 (program SINGSONG) and from the Mesocentre of Aix-Marseille Université (program DIFOMER)." ] ]
1612.05643
[ [ "A $QQ\\to QQ$ planar doublebox in canonical form" ], [ "Abstract We consider a planar doublebox with four massive external momenta and two massive internal propagators.", "We derive the system of differential equations for the relevant master integrals, cast it in canonical form, write it as a $d\\log$ form and solve it in terms of iterated integrals up to depth four." ], [ "Introduction", "Differential equations are a powerful tool for solving master integrals [1], [2], [3], [4], [5], [6], [7].", "This method received considerable boost by the observation that the system of differential equations can be often put in a canonical form, by a suitable choice of master integrals exhibiting uniform transcendentality [8] (see also [9] and the review [10]).", "In such a form the solution of the differential equations becomes much simpler and lands on Chen iterated integrals [11] which in many cases evaluate to harmonic [12] and Goncharov polylogarithms [13].", "This technique has proven efficient and has been applied in a number of contexts [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25].", "In this short note, we apply this method to a planar doublebox topology with four external massive momenta $p_1^2=p_2^2=p_3^2=p_4^2=-m^2$ and two internal massive propagators, all with the same mass $m$ .", "The integral is depicted in Figure REF .", "More explicitly, we consider the class of two-loop integrals given by the following propagators $g_{a_1,\\dots ,a_9}=\\frac{e^{2\\epsilon \\gamma _{E}}}{(\\pi ^{d/2})^2}\\, \\int d^{d} k_1\\, d^{d} k_2\\, \\frac{P_4^{-a_4}P_6^{-a_6}}{P_1^{a_1}\\, P_2^{a_2}\\, P_3^{a_3}\\, P_5^{a_5}\\, P_7^{a_7}\\, P_8^{a_8}\\, P_9^{a_9}}$ where $d=4-2\\epsilon $ and $\\begin{array}{ccccc}P_1 = (k_1-p_1)^2 & \\quad & P_2= k_1^2+m^2 & \\quad & P_3= (k_1+p_2)^2 \\\\P_4= (k_1+p_2+p_3)^2+m^2 & \\quad & P_5 = (k_2-p_1)^2 & \\quad & P_6= k_2^2+m^2\\\\P_7= (k_2+p_2)^2 & \\quad & P_8= (k_2+p_2+p_3)^2+m^2 & \\quad & P_9=(k_1-k_2)^2\\end{array}$ where we work in the Euclidean and we assume that $a_4\\le 0$ and $a_6\\le 0$ , namely these correspond to two irreducible numerators of the doublebox.", "Figure: Momentum labels of the propagators.", "The figure does not include the irreducible numerators P 4 P_4 and P 6 P_6.", "The thick internal lines are the massive propagators.This kind of integrals arises (among others) in $2\\rightarrow 2$ scattering of massive particles which can exchange massless non-abelian gauge bosons, such as massive quarks in QCD.", "Such a topology can also be considered as an extension of that studied in [8] (albeit with a different labelling of propagators and momenta), where the roles of two numerators/propagators have been exchanged.", "Indeed, several master integrals relevant in this model were considered in literature before, mostly in the context of Bhabha scattering, using again the differential equations approach or evaluation via Mellin-Barnes representation [26], [27], [28], [29], [30], [31].", "On the contrary, to the best of our knowledge, the seven propagators topologies presented in this note are new in literature.", "The problem has three quadratic invariants, $s\\equiv (p_1+p_2)^2$ , $t\\equiv (p_2+p_3)^2$ and $m^2$ , giving rise to two adimensional ratios.", "In the following it proves convenient to express the integrals in terms of the dimensionless variables $\\frac{s}{m^2} = \\frac{(1-x)^2}{x} \\qquad \\qquad \\frac{t}{m^2} = \\frac{(1-y)^2}{y}$ which are commonly used for massive particles scattering and rationalize typical square roots arising in this context.", "In the rest of the article we provide the details regarding the system of differential equations and the basis of master integrals of uniform transcendentality that allows to write it in canonical form.", "Then we cast the latter in $d\\log $ form depending on letters chosen from an alphabet of 16.", "Finally we sketch the solution of the differential equations in terms of iterated integrals, most of which are expressed in terms of Goncharov and harmonic polylogarithms (whose definition we review in Appendix ), and the procedure to fix the integration constants.", "The results up to depth four are collected in electronic format in the ancillary file results.m." ], [ "Basis of master integrals", "The system possesses 25 master integrals, which can be put in canonical form as follows: $f_1 &= \\epsilon ^2\\, g_{0,2,0,0,0,0,0,2,0} \\\\f_2 &= m^2\\, \\frac{\\epsilon ^2 (4 \\epsilon +1) g_{0,1,0,0,2,0,0,0,2}}{\\epsilon +1} \\\\f_3 &= s\\, \\epsilon ^2\\, g_{0,2,0,0,2,0,1,0,0} \\\\f_4 &= s\\, \\epsilon ^2\\, g_{2,0,0,0,0,0,2,0,1} \\\\f_5 &= \\frac{1}{2} \\sqrt{t} \\sqrt{4 m^2+t}\\, \\epsilon ^2\\, \\left(2 g_{0,2,0,0,0,0,0,1,2}+g_{0,2,0,0,0,0,0,2,1}\\right) \\\\f_6 &= t\\, \\epsilon ^2\\, g_{0,2,0,0,0,0,0,2,1} \\\\f_7 &= s^2\\, \\epsilon ^2\\, g_{2,0,1,0,2,0,1,0,0} \\\\f_8 &= \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, g_{0,2,0,0,1,0,1,1,0} \\\\f_9 &= \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, g_{0,2,0,0,1,0,1,0,1} \\\\f_{10} &= m^2\\, \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^2\\, g_{0,3,0,0,1,0,1,0,1} \\\\f_{11} &= s\\, \\epsilon ^2 \\left(\\frac{3}{2}\\, \\epsilon \\, g_{0,2,0,0,1,0,1,0,1}+m^2\\, g_{0,2,0,0,2,0,1,0,1}-m^2\\, g_{0,3,0,0,1,0,1,0,1}\\right) \\\\f_{12} &= 2\\, \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, g_{0,1,1,0,2,0,0,0,1} \\\\f_{13} &= 2\\, \\sqrt{t} \\sqrt{4 m^2+t}\\, \\epsilon ^3\\, g_{0,2,0,0,1,0,0,1,1} \\\\f_{14} &= s^{3/2}\\, \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, g_{1,1,1,0,2,0,1,0,0} \\\\f_{15} &= \\sqrt{s} \\sqrt{4 m^6+s (t+m^2)^2}\\, \\epsilon ^3\\, g_{0,2,0,0,1,0,1,1,1} \\\\f_{16} &= \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, \\left(g_{0,2,0,0,1,-1,1,1,1}-m^2\\, g_{0,2,0,0,1,0,1,1,1}\\right) \\\\f_{17} &= s\\, \\sqrt{t} \\sqrt{4 m^2+t}\\, \\epsilon ^2\\, \\left(m^2\\, g_{0,3,0,0,1,0,1,1,1}-\\epsilon \\, g_{0,2,0,0,1,0,1,1,1}\\right) \\\\f_{18} &= 4\\, \\sqrt{s+t} \\sqrt{4 m^2+s+t}\\, \\epsilon ^4\\, g_{1,1,0,0,0,0,1,1,1} \\\\f_{19} &= 2\\, s\\, \\sqrt{t} \\sqrt{4 m^2+t}\\, \\epsilon ^3\\, g_{1,1,0,0,0,0,1,1,2} \\\\f_{20} &= 2\\, m^2\\, \\sqrt{s} \\sqrt{4 m^2+s}\\, \\epsilon ^3\\, g_{1,2,0,0,0,0,1,1,1} \\\\f_{21} &= \\frac{1}{2} s\\, \\epsilon ^2\\, \\big [ 2\\, m^2 \\left(m^2\\, g_{1,2,0,0,0,0,1,2,1}-2 \\epsilon \\, g_{1,2,0,0,0,0,1,1,1}\\right)-\\left(2 m^2+t\\right) \\epsilon \\, g_{1,1,0,0,0,0,1,1,2}\\big ] \\\\f_{22} &= s \\left(4 m^2+s\\right)\\, \\epsilon ^4\\, g_{1,1,1,0,1,0,1,1,0} \\\\f_{23} &= s^2\\, \\sqrt{t} \\sqrt{4 m^2+t}\\, \\epsilon ^4\\, g_{1,1,1,0,1,0,1,1,1} \\\\f_{24} &= s^{3/2}\\, \\sqrt{4 m^2+s}\\, \\epsilon ^4\\, g_{1,1,1,-1,1,0,1,1,1} \\\\f_{25} &= \\frac{1}{2} s\\, \\epsilon ^2\\, \\Big [-4 m^2\\, g_{0,3,0,0,1,0,1,0,1}+\\frac{g_{2,0,0,0,0,0,2,0,1}}{1-2 \\epsilon }+2 \\epsilon \\big (-4 g_{0,1,1,0,2,0,0,0,1}+\\\\ &+2 g_{0,2,0,0,1,-1,1,1,1}+g_{0,2,0,0,1,0,1,0,1}-g_{0,2,0,0,1,0,1,1,0}-2 m^2\\, g_{0,2,0,0,1,0,1,1,1}+ \\nonumber \\\\& +s\\, g_{1,1,1,0,2,0,1,0,0}+2 m^2\\, g_{1,2,0,0,0,0,1,1,1}+\\frac{s\\, g_{2,0,1,0,2,0,1,0,0}}{2 \\epsilon -1} + \\nonumber \\\\& + \\epsilon \\, \\left(g_{1,1,1,-1,1,-1,1,1,1}-2 g_{1,1,0,0,0,0,1,1,1}+s\\, g_{1,1,1,-1,1,0,1,1,1}-t\\, g_{1,1,1,0,1,0,1,1,0}\\right)\\big )\\Big ] \\nonumber $ where the integrals are all normalized with a common overall factor $(m^2)^{2\\epsilon }$ in order to guarantee the correct dimensions in dimensional regularization.", "The integral topologies are depicted in figure REF .", "Figure: Master integral topologies.We performed the relevant reductions via IBP identities [32], [33], [34], [35] by use of FIRE [36], [37], [38] and LiteRed [39], [40]." ], [ "The alphabet", "After performing the change of variables (REF ), the differential equations for the system can be written as a $d\\log $ form $d f = \\epsilon \\, d A\\, f \\qquad ,\\qquad A = \\sum _{i}\\, M_i\\, \\log (\\text{letter})$ where the letters are chosen from an alphabet that includes the set of [15] $\\text{letter}\\in & \\bigg \\lbrace x, 1\\pm x, y, 1\\pm y, x+y, 1+ x y, x+y-4x y+x^2 y+ x y^2, \\nonumber \\\\&\\quad \\frac{1+Q}{1-Q}, \\frac{1+x+(1-x)Q}{1+x-(1-x)Q}, \\frac{1+y+(1-y)Q}{1+y-(1-y)Q} \\bigg \\rbrace \\cup \\dots $ with the addition of four more letters, that we define $\\dots \\cup \\bigg \\lbrace \\frac{4+v+\\beta }{4+v-\\beta },\\frac{\\beta \\beta _v+4+3 v}{\\beta \\beta _v-4-3 v},\\frac{\\beta \\beta _u+4+v \\beta _u^2}{\\beta \\beta _u-4-v \\beta _u^2},\\frac{\\left((4+v) \\beta _u+\\beta \\right) \\left(\\beta +(4+3 v) \\beta _u\\right)}{\\left((4+v) \\beta _u-\\beta \\right) \\left(\\beta -(4+3 v) \\beta _u\\right)} \\bigg \\rbrace $ We have borrowed the notation of [15] and [41] as follows $Q\\equiv \\sqrt{\\frac{(x+y)(1+x y)}{x+y-4x y+x^2 y+ x y^2}}$ and $& \\displaystyle u\\equiv \\frac{s}{4m^2} \\qquad & \\qquad v\\equiv \\frac{t}{4m^2}\\\\& \\beta _u \\equiv \\sqrt{1+u} \\qquad & \\qquad \\beta _v \\equiv \\sqrt{1+v}\\\\& \\beta _{uv} \\equiv \\sqrt{1+u+v} \\qquad & \\qquad \\beta \\equiv \\sqrt{u\\, v^2 + (4+v)^2}$ The matrices $M$ are given in electronic form in the ancillary file results.m.", "They are sparse matrices with non-vanishing elements which, as required, do not depend on $x$ , $y$ nor $\\epsilon $ ." ], [ "The solution up to order 4", "The differential equations for integrals $f_{15}-f_{17}$ and $f_{23}-f_{25}$ have nontrivial dependence on the letters (REF ).", "Integral $f_{15}$ develops a dependence on them at order 3; $f_{16}$ , $f_{17}$ , $f_{24}$ and $f_{25}$ at order 4 and $f_{23}$ at order 5.", "All other integrals and those mentioned above up to those orders are expressible in terms of Goncharov polylogarithms $G$ , which in some cases reduce to harmonic polylogarithms $H$ .", "The definitions of these functions are recalled in appendix for completeness.", "The solution can be determined straightforwardly in a recursive manner, order by order in $\\epsilon $ .", "Given the expansion of the vector solution $f=f^{(0)}+\\epsilon f^{(1)}+\\epsilon ^2 f^{(2)}+...$ and using (REF ) we have to recursively solve the equation $df^{(n\\!+\\!1)}=dA f^{(n)}$ .", "We can integrate it in two steps by starting from an arbitrary base point and integrating on a straight horizontal line in the $(x,y)$ plane taking $y$ fixed, followed by an integration on a straight vertical line taking $x$ fixed.", "This is why the analytical results we obtain can be written in terms of harmonic polylogarithms with argument $x$ and Goncharov polylogarithms with argument $y$ with possible $x$ -dependent indexes.", "Since we take the starting point to be arbitrary, this procedure fixes the solution order by order up to an integration constant unknown at each order.", "The latter can be fixed imposing physical conditions on the analytic structure of the result (requiring that only branch cuts associated to the cuts of the integrals are present) or comparing the results to analytic evaluations of the integrals in some limit, e.g.", "$y\\rightarrow 1$ , which is regular.", "We have benefited from the tools of the HPL.m package [42], [43], for taking such limits.", "For instance, the result for $f_{23}$ reads up to order 4 $f_{23}^{(4)} &= 8 H_0(x) G_{0,0,0}(y)+16 H_1(x) G_{0,0,0}(y)+\\frac{4}{3} \\pi ^2 G_{-1,0}(y)-\\frac{2}{3} \\pi ^2 G_{0,0}(y)+ \\nonumber \\\\& +\\frac{4}{3} \\pi ^2 G_{1,0}(y)+8 G_{-1,0,0,0}(y)-8 G_{0,0,-1,0}(y)-8 G_{0,0,1,0}(y)+8 G_{1,0,0,0}(y)+ \\nonumber \\\\&+\\frac{4}{3} \\pi ^2 G_0(y) H_0(x)+\\frac{8}{3} \\pi ^2 G_0(y) H_1(x)-\\frac{4 \\pi ^4}{45}$ where the integration constant has been fixed using the information that the master integral has a vanishing limit for $y\\rightarrow 1$ .", "Similarly, the result for integral $f_{16}$ , at order 3 reads $f_{16}^{(3)} &= -G_0(y) H_{0,0}(x)-2 G_0(y) H_{0,1}(x)+H_0(x) G_{-\\frac{1}{x},0}(y)+2 H_1(x) G_{-\\frac{1}{x},0}(y)+ \\nonumber \\\\&-H_0(x) G_{-x,0}(y)-2 H_1(x) G_{-x,0}(y)+H_{0,0}(x) G_{-\\frac{1}{x}}(y)+H_{0,0}(x) G_{-x}(y)+ \\nonumber \\\\&+2 H_{0,1}(x) G_{-\\frac{1}{x}}(y)+2 H_{0,1}(x) G_{-x}(y)+2 G_{-\\frac{1}{x},-1,0}(y)-G_{-\\frac{1}{x},0,0}(y)-2 G_{-x,-1,0}(y)+ \\nonumber \\\\&+G_{-x,0,0}(y)-2 H_{-1,0,0}(x)-4 H_{-1,0,1}(x)-H_{0,1,0}(x)-2 H_{0,1,1}(x)-H_{1,0,0}(x)+ \\nonumber \\\\&-2 H_{1,0,1}(x)+\\frac{5}{6} \\pi ^2 G_{-\\frac{1}{x}}(y)+\\frac{1}{2} \\pi ^2 G_{-x}(y)-\\frac{2}{3} \\pi ^2 G_0(y)-\\frac{4}{3} \\pi ^2 H_{-1}(x)+\\frac{1}{3} \\pi ^2 H_0(x)+ \\nonumber \\\\&-\\frac{2}{3} \\pi ^2 H_1(x)-4 \\zeta (3)$ Such expressions are collected in electronic form in the ancillary file results.m.", "They have been successfully checked against already available results in literature and numerical integration using FIESTA [44], [45].", "In particular, they are amenable of fast and precise numerical evaluation, for instance using GiNaC [46], [47].", "For the integrals with a dependence on the letters (REF ), one can use their $d\\log $ form, immediately available from (REF ) and integrate it from a base point in the plane $(x,y)$ to given values of the Mandelstam variables.", "We performed various consistency checks of this against numerical evaluations.", "For these we found a pedestrian numerical integration with Mathematica's NIntegrate sufficient.", "Nevertheless, it would be interesting to ascertain whether an expression in terms of Goncharov polylogarithms could be found for these integrals, which would allow for a much more efficient numerical evaluation.", "All integrals are regular in $y\\rightarrow 1$ and can be expressed in terms of harmonic polylogarithms, even those depending on the letters (REF ) as they all degenerate to combinations of the $\\lbrace x,1\\pm x\\rbrace $ alphabet in this limit.", "We provide these limits as well in the ancillary file, expressed in terms of HPL's.", "Some of them can be easily simplified and give rise to extremely concise answers.", "For instance, integrals $f_{24}$ and $f_{25}$ reads in this limit $f_{24}^{(4)} &\\underset{y\\rightarrow 1}{\\longrightarrow } 2 \\text{Li}_4 x + \\frac{\\log ^4 x}{24} + 2\\zeta (2) \\log ^2 x + 28 \\zeta (4) \\\\f_{25}^{(4)} &\\underset{y\\rightarrow 1}{\\longrightarrow } \\frac{1}{3}\\, \\log ^4\\left(\\tfrac{s}{m^2}\\right) - \\zeta (2)\\log ^2\\left(\\tfrac{s}{m^2}\\right) + \\frac{32}{3} \\zeta (3) \\log \\left(\\tfrac{s}{m^2}\\right) - 7 \\zeta (4)$ where in the last result we restored the dependence on $\\tfrac{s}{m^2}$ instead of $x$ for convenience." ], [ "Comments", "In this note we have been able to write a differential equation in the canonical form for the set of master integrals associated to an on-shell planar doublebox Feynman integral with two internal massive propagators and massive external momenta (see Figure REF ).", "This integral is a building block for the NNLO computation of massive quark-quark scattering amplitudes with full dependence on the mass.", "While some of the subtopologies of this system had already been computed, to our knowledge, the full seven propagator doublebox is a novel result.", "The particular differential equation we found for this system can be put in $d\\log $ form such that the solution of each integral can be written in terms of iterated integrals of those forms.", "For most of the integrals, we have been able to write them up to depth four with the use of harmonic and Goncharov polylogarithms, whose analytic properties and extensions are well known and are amenable of fast numerical evaluation.", "It would be interesting to extend this analysis to the other topologies relevant for the computation of the NNLO massive quark-quark scattering (some of them also needed for Bhabha scattering with finite electron mass).", "Among others, doubleboxes with different massive internal propagators routing would be needed.", "It would be interesting to determine if a differential equation in the canonical form for those systems exists and if they could be expressed in terms of generalized polylogarithms or elliptic functions kick in." ], [ "Acknowledgements", "We thank Andi Brandhuber, Johannes Henn, Andreas von Manteuffel for very useful comments.", "MB particularly thanks Joe Hayling and Rodolfo Panerai for extra CPU's.", "The work of MB was supported in part by the Science and Technology Facilities Council Consolidated Grant ST/L000415/1 String theory, gauge theory & duality." ], [ "Goncharov and harmonic polylogarithms", "The results of this paper are mostly expressed in terms of Goncharov and harmonic polylogarithms.", "In this appendix we review their definition.", "Goncharov polylogarithms [13] are defined recursively as follows $G_{a_1,\\dots a_n} (y) = \\int _0^y \\frac{dt}{t-a_{1}}\\, G_{a_{2}, \\dots ,a_{n}} (t)$ where $G_{a_1} (y) = \\int _0^y \\frac{dt}{t-a_{1}} \\,, \\qquad a_{1} \\ne 0$ and for $a_{1}=0$ , the definition reads $G_{\\underbrace{0,\\dots ,0}_{n}}(y) = 1/n!", "\\log ^n y$ .", "Harmonic polylogarithms [12] $H_{a_1,a_2,\\dots ,a_n}(x)$ , with indices $a_i \\in \\lbrace 1,0,-1\\rbrace $ , are defined recursively as follows $H_{a_1,a_2,\\dots ,a_n}(x) = \\int _0^x f_{a_1}(t)\\, H_{a_2,\\dots ,a_n}(t)\\, d t$ where $& f_{\\pm 1}(x)=\\frac{1}{1 \\mp x} \\qquad f_0(x)=\\frac{1}{x}\\nonumber \\\\& H_{\\pm 1}(x)= \\mp \\log (1\\mp x)\\qquad H_0 (x)= \\log x$ and at least one of the indices $a_i$ is non-zero.", "When all indices are vanishing the definition reads $H_{\\underbrace{0,0,\\ldots ,0}_{n}}(x) = \\frac{1}{n!", "}\\log ^n x$" ] ]
1612.05609
[ [ "A Blast Wave Model With Viscous Corrections" ], [ "Abstract Hadronic observables in the final stage of heavy ion collision can be described well by fluid dynamics or blast wave parameterizations.", "We improve existing blast wave models by adding shear viscous corrections to the particle distributions in the Navier-Stokes approximation.", "The specific shear viscosity $\\eta/s$ of a hadron gas at the freeze-out temperature is a new parameter in this model.", "We extract the blast wave parameters with viscous corrections from experimental data which leads to constraints on the specific shear viscosity at kinetic freeze-out.", "Preliminary results show $\\eta/s$ is rather small." ], [ "Introduction", "In the initial stage of heavy-ion collisions, a hot and dense fireball is created.", "The fireball cools rapidly and expands into the surrounding vacuum.", "This process continues until it reaches a final freeze-out stage when hadrons decouple from each other.", "Hadronic observables in the final stage of the heavy-ion collisions can be described well by fluid dynamics or blast wave parameterizations.", "In this contribution, we construct a blast wave model with viscous corrections by calculating the viscous stress tensor from the parameterized flow field in the Navier-Stokes approximation, similar to the models developed in [1][2]." ], [ "Viscous Corrections In Fluid Dynamics And Blast Wave", "Fluid dynamics models the time evolution of the nuclear matter in heavy ion collisions, determined by initial conditions and the equation of state.", "When the system becomes dilute enough, kinetic freeze-out happens at a temperature $T=T_{\\mathrm {fo}}$ .", "At the kinetic freeze-out fluid cells are translated into particles through the Cooper-Frye formula $E\\frac{d^3N}{dp^3}=\\frac{g}{(2\\pi )^3}\\int _\\sigma f(p,T)p^{\\mu }d\\sigma _{\\mu }\\, .$ Shear viscous corrections, and in particular the specific shear viscosity $\\eta /s$ will enter both the fluid dynamic equations of motion and the Cooper-Frye formula at freeze-out.", "Extractions of $\\eta /s$ from data using viscous fluid dynamics thus conflate the effects of shear viscosity on the dynamical evolution and on freeze-out.", "This is to some extent also true in calculations that switch to a hadronic transport model for freeze-out.", "Typical values extracted for $\\eta /s$ from fluid dynamics or fluid dynamics plus hadronic transport are $(1-2)/4\\pi $ [3], averaged over all temperatures and including freeze-out in the averaging.", "Efforts are on the way to pin down the temperature evolution of $\\eta /s$ during the dynamical evolution [4].", "In contrast, the blast wave model fits the flow field and shape of the fireball in the transverse plane.", "It becomes then sensitive to $\\eta /s$ at kinetic freeze-out only." ], [ "Viscous Blast Wave Parameterization", "The hadron spectrum at freeze-out is given by Eq.", "(1) above where $f(p,T)$ is the distribution function of the given hadron species, $g$ is its degeneracy and $\\sigma ^\\mu $ is the normal vector on the freeze-out hypersurface.", "In equilibrium $f(p,T)=f_0(p,T)$ and with shear viscous corrections $f(p,T)=f_0(p,T)+\\delta f$ , where $f_0(p,T)=\\frac{1}{e^{p\\cdot u/T}\\pm 1}$ $\\delta f=\\frac{1}{2}\\frac{p_\\mu p_\\nu }{T^2}\\frac{\\pi ^{\\mu \\nu }}{e+p}f_0(p,T)$ and $\\pi ^{\\mu \\nu }$ is the shear stress tensor [5][6].", "In the first-order (Navier-Stokes) approximation, $\\pi ^{\\mu \\nu }$ is given by $\\pi ^{\\mu \\nu }=2\\eta \\langle \\partial ^{\\mu }u^{\\nu }\\rangle $ we plug Eq.", "(4) into Eq.", "(3) and find $\\delta f=\\frac{1}{T^3}\\frac{\\eta }{s}p_{\\mu }p_{\\nu }\\langle \\partial ^{\\mu }u^{\\nu }\\rangle f_0(p,T)\\, .$ We have neglected a Bose/Fermi enhancement/suppression factor since we will not work at very small momenta $p\\lesssim T$ .", "We see that the viscous correction at freeze out is directly proportional to $\\eta /s$ .", "We choose constant $\\tau =\\sqrt{t^2-z^2}$ as the freeze-out hypersurface.", "We follow [7] and parameterize the blast wave as follows.", "The coordinates are given by $x^{\\mu }=(t,x,y,z)=(\\tau \\cosh \\eta ,\\rho R_x\\cos \\phi ,\\rho R_y\\sin \\phi ,\\tau \\sinh \\eta )$ where $\\rho =\\sqrt{\\frac{x^2}{R_x^2}+\\frac{y^2}{R_y^2}}$ is the normalized radial coordinate.", "The hadron momentum is $p^{\\mu }=(m_T\\cosh Y,p_T\\cos \\theta ,p_T\\sin \\theta ,m_T\\sinh Y)$ where $m_T=\\sqrt{m^2+p_T^2}$ is the transverse mass and $Y$ is momentum rapidity.", "We will work with hadrons at $Y=0$ here.", "The flow velocity can be written as [5][7] $u^{\\mu }=(\\cosh \\eta _L\\cosh \\eta _T,\\sinh \\eta _T\\cos \\phi _b,\\sinh \\eta _T\\sin \\phi _b,\\sinh \\eta _L\\cosh \\eta _T)$ where $\\eta _L$ is given by the longitudinal velocity $v_L=\\tanh \\eta _L$ , $\\eta _T$ is given by the transverse velocity $v_T=\\tanh \\eta _T$ .", "For longitudinal flow, we choose $v_L=\\frac{z}{t}$ , thus $\\eta _L=\\eta $ , which enforces boost invariance.", "For transverse flow, we use the parameterization $v_T=\\alpha \\rho ^n, \\qquad \\alpha =\\alpha _0+\\alpha _2\\cos 2\\phi _b$ where $\\alpha $ is the surface velocity, $\\alpha _0$ is the average surface velocity, $\\alpha _2$ is an elliptic deformation of the flow field and $n$ is a power term.", "Let us summarize the parameters in this blast wave: freeze-out temperature $T$ , surface velocity $\\alpha _0$ , velocity profile power $n$ , velocity deformation $\\alpha _2$ , ratio of event plane radii $R_y/R_x$ and specific viscosity $\\eta /s$ .", "We are left to calculate the Navier-Stokes shear stress tensor given our velocity field.", "In detail this tensor is $\\pi ^{\\mu \\nu }=2\\eta \\langle \\partial ^{\\mu }u^{\\nu }\\rangle $ where [6] $\\langle \\partial ^{\\mu }u^{\\nu }\\rangle \\equiv [\\frac{1}{2}(\\Delta _{\\sigma }^{\\mu } \\Delta _{\\nu }^{\\tau })-\\frac{1}{3}\\Delta ^{\\mu \\nu }\\Delta _{\\sigma \\tau }]\\partial ^{\\sigma }u^{\\tau }, \\qquad \\Delta ^{\\mu \\nu }=g^{\\mu \\nu }-u^\\mu u^\\nu \\, .$ The computation of the Navier-Stokes shear tensor is now straight forward but results in long expressions.", "We show one derivative as an example $\\partial _2u^1=\\frac{\\partial u^1}{\\partial y}=n\\sinh \\eta _T\\cosh ^2\\eta _T\\frac{\\sin \\phi }{\\rho R_y}\\cos \\phi _b-\\sinh \\eta _T\\frac{\\tan ^2\\phi _b}{(1+\\tan ^2\\phi _b)^{3/2}}\\frac{1}{\\rho R_y\\sin \\phi }\\, .$" ], [ "Extraction Of Freeze Out Shear Viscosity From Data", "To extract the blast wave parameters from experimental data, we use a Bayesian package from the Models and Data Analysis Initiative (MADAI) project [4][8].", "We first calculate training points in parameter space given a prior range for each parameter.", "The package then uses a Gaussian process emulator to estimate spectra at random parameter values.", "Finally it will do a likelihood analysis and give the maximum likelihood parameters.", "The data used here is from the ALICE collaboration for Pb+Pb collisions at 2.76 TeV in the 40$\\%$ -50$\\%$ centrality bin [9][10].", "Our fit ranges are 0.325-2.75 GeV/$c$ , 0.225-1.95 GeV/$c$ and 0.425-1.25 GeV/$c$ for proton, kaon and pion respectively.", "We use both the transverse momentum spectra (without their absolute normalization) and elliptic flow $v_2$ to constrain the blast wave parameters.", "We set $\\tau $ =9 fm/$c$ , $R_xR_y$ =12 fm$^2$ and assume $\\partial _\\tau u^\\mu $ =0.", "Figure: Likelihood analysis of our fit.", "The x and y axes show our chosen prior ranges for all parameters.", "The plots on the diagonal show likelihood distributions and the off diagonal plots show correlations.The likelihood analysis given by the MADAI package is shown in Fig.", "1.", "The best parameter values are $T$ =0.119 GeV, $\\alpha _0$ =0.838$c$ , $n$ =1.07, $R_y/R_x$ =1.31, $\\alpha _2$ =0.0372$c$ , $\\eta /s$ =1.13/4$\\pi $ .", "We can check the quality of our fit results by calculating the transverse momentum spectra, see Fig.", "2 and elliptic flow $v_2$ , see Fig.", "3 for the best parameter values.", "The $p_T$ spectra with the best fit parameters describe the data rather well.", "Elliptic flow is generally described satisfactorily as well, but there is slight tension between the kaons and pions.", "This can be ameliorated by allowing pions to have a separate (lower) freeze-out temperature which improves the fits (not shown here) and leads to a very similar value of the specific shear viscosity.", "Figure: Transverse momentum spectra for protons (left), kaons (center) and pions (right).", "Solid line: viscous blast wave result with best fit parameters.", "Data: ALICE collaboration .Figure: Same as Fig.", "2 for elliptic flow v 2 v_2.", "Data: ALICE collaboration ." ], [ "Conclusions", "We have developed a viscous blast wave model.", "We have used this model to extract $\\eta /s$ at kinetic freeze-out in a complimentary way.", "The preliminary results show $\\eta /s$ is rather small.", "In the future, we will improve our statistical analysis and work on more centrality bins.", "We may also use the viscous blast wave for other projects like quark recombination.", "This work was supported by the U.S. National Science Foundation." ] ]
1612.05629
[ [ "Sojourn times and fixation dynamics in multi-player games with\n fluctuating environments" ], [ "Abstract We study evolutionary multi-player games in finite populations, subject to fluctuating environments.", "The population undergoes a birth-death process with absorbing states, and the environment follows a Markovian process, resulting in a fluctuating payoff matrix for the evolutionary game.", "Our focus is on the fixation or extinction of a single mutant in a population of wildtypes.", "We show that the nonlinear nature of fitnesses in multi-player games gives rise to an intricate interplay of selection, genetic drift and environmental fluctuations.", "This generates effects not seen in simpler two-player games.", "To analyse trajectories towards fixation we analytically calculate sojourn times for general birth-death processes in populations of two types of individuals and in fluctuating environments." ], [ "Introduction", "Models of evolutionary dynamics frequently involve randomness, and the timing of birth, death and mutation events is statistical.", "The modelling framework most commonly used in evolutionary dynamics, game theory, epidemiology and population dynamics is that of a Markovian birth-death process, see e.g.", "[1], [2].", "These processes capture so-called intrinsic stochasticity (or demographic noise) in finite populations.", "More traditional approaches (see e.g.", "[3], [4]), based on deterministic rate equations, neglect all randomness and are valid formally only in the limit of infinite populations.", "Deterministic models are analysed relatively easily using tools from nonlinear dynamics, however they do not capture fluctuation-driven phenomena such as fixation and extinction.", "These features can only be characterised within a stochastic model, and the analysis is frequently based on methods from non-equilibrium statistical mechanics [5], [6].", "Often the relative growth rates or interaction coefficients between the different types of individuals themselves fluctuate in time, as environmental conditions change.", "Fluctuating environments lead to a second layer of stochasticity, in addition to the intrinsic stochasticity in finite populations.", "Examples of fluctuating environments include variation of external conditions (e.g., temperature, pH, presence or absence of nutrients), or targeted intervention, such as phases of antibiotic treatment, see e.g.", "[7], [8], [9], [10], [11].", "Biological examples include the dynamics of persister cells in bacterial populations [9], [12], and the emergence of phenotype switching [13].", "Theoretical work exists to study such systems [7], [13], [14], [15], but often disregards intrinsic stochasticity or focuses on growth in continuously varying environments.", "We are mostly interested in the dynamics of fixation in fluctuating environments, which is not accounted for in the deterministic approach.", "Previous work [16] addressed the case of two-player evolutionary dynamics in switching environments with discrete states, and an analytical framework was constructed to calculate fixation probabilities and times.", "The analysis focused on games with linear payoffs and two-player interaction.", "Even for this relatively simple case, unexpected behaviour was observed due to the environmental switching; for example fixation probabilities of an invading mutant may be maximised for intermediate switching rates of the environmental process.", "The purpose of this work is to study fixation of mutants in evolutionary multi-player games in fluctuating environments.", "In multi-player games each individual interacts with more than one other individual in each instance of the game, and the resulting payoff or fitness functions become non-linear in the composition of the population.", "This can generate multiple non-trivial fixed points of the resulting replicator equations [17]; these are the equilibria of selection at which different species coexist and have the same fitness.", "These equilibrium points shape the dynamics and outcome of evolution in the presence of demographic noise.", "We are interested in the interplay between selection, random drift and environmental fluctuations in this non-linear scenario.", "We approach this in a general setting and in the context of a stylised minimal model.", "While we do not aim to model an immediate application, we believe our approach is the most suitable to identify the main phenomena and principal mechanisms.", "On the other hand it is clear that the scenario of multi-player interaction is of importance for a number of applications, and potentially more prevalent in the real world than two-player games.", "The latter are often analysed as an approximation as they are mathematically easier to handle.", "Applications of multi-player games have extensively been surveyed for example in [18], [19].", "These include modelling in ecology (e.g.", "biological markets and auctions [20], [21], [22]), population genetics (e.g dynamics of the Medea allele, and `playing the field' in the context of the sex-ratio game [23], [24]), and in the social sciences (e.g.", "public good games [25], [26]).", "Multi-player games in finite populations have of course been investigated in the literature (see e.g.", "[27], [28]).", "However, most existing studies focus on one fixed multi-player game, whereas our aim is to analyse cases in which the game is shaped by an external fluctuating environment.", "In order to do this, we develop a theory for the analytical calculation of sojourn times for arbitrary two-species birth-death processes in randomly switching environments.", "We then apply it to understand fixation dynamics in fluctuating three-player games, and study the success of an invading mutant in an existing population of wildtypes.", "The remainder of this paper is organised as follows.", "In Sec.", "we define the components of the model, in particular the birth-death dynamics, the environmental process and the setup of a multi-player game.", "We introduce sojourn times for general birth-death processes in switching environments in Sec.", "; these can be calculated analytically, full mathematical details can be found in the Supplement.", "We then study fixation phenomena in three-player games in Sec.", ", before we present out conclusions and an outlook in Sec.", ".", "Further technical details of the model setup and the analysis can be found in the Supplement." ], [ "Evolutionary dynamics and environmental switching", "We consider a population of $N$ individuals.", "The size of the population does not vary in time.", "Each individual can be of one of two types, $A$ or $B$ ; we will occasionally refer to these as `species'.", "The population is assumed to be well mixed, i.e., every individual in the population can interact with any other individual.", "The state of the population is therefore fully specified by the number of individuals of type $A$ , which we write as $i\\in \\lbrace 0,\\dots ,N\\rbrace $ .", "The number of individuals of type $B$ is $N-i$ .", "Evolution occurs through a discrete-time birth-death process, governed by transition probabilities $\\omega _i^\\pm $ from state $i$ to states $i\\pm 1$ , respectively.", "More precisely, $\\omega _i^+$ is the probability to find the population in state $i+1$ in the next time step, if it is currently in state $i$ .", "A similar definition applies for $\\omega _i^-$ , and $1-\\omega _i^+-\\omega _i^-$ is the probability for the population to remain in $i$ .", "To model environmental influence, we assume that the rates of the birth-death process within the population depend on a discrete state $\\sigma $ of the environment.", "We write $\\omega _{i,\\sigma }^\\pm $ for the probability to transition from $i$ to $i\\pm 1$ if the environment is in state $\\sigma $ .", "In principle, much of our theory applies to arbitrary number of environmental states, although our analysis of multi-player games focuses on the case of two such states.", "Crucially, we assume that the process of the environmental variable is Markovian and that its transition rates do not depend on the state, $i$ , of the population.", "We write $\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}$ for the probability that the state of the environment changes to $\\sigma ^{\\prime }$ in the next time step, if it is currently in state $\\sigma $ .", "Finally, we assume that the states of both the population and the environment can change in any one time step (i.e., the two processes are not mutually exclusive).", "In our model the dynamics of the population occurs via a standard Moran process [29], [2], [1] $\\omega _{i,\\sigma }^+&=&\\frac{i(N-i)}{N^2}\\frac{f_A^\\sigma (i)}{f^\\sigma (i)}, \\nonumber \\\\\\omega _{i,\\sigma }^-&=&\\frac{i(N-i)}{N^2}\\frac{f_B^\\sigma (i)}{f^\\sigma (i)},$ where $f_A^\\sigma (i)$ denotes the reproductive fitness of species $A$ in a population with $i$ individuals of type $A$ and when the environment is in state $\\sigma $ .", "The quantity $f^\\sigma (i)=\\frac{i f_A^\\sigma (i)+(N-i) f_B^\\sigma (i)}{N}$ is the mean fitness in the population if the environment is in state $\\sigma $ .", "The states $i=0$ and $i=N$ are absorbing states of the population dynamics for all environmental states, i.e.", "$\\omega _{i=0,\\sigma }^+=\\omega _{i=N,\\sigma }^-=0$ .", "The above birth and death rates depend on the state of the environment, $\\sigma $ , through the fitnesses $f_A^\\sigma (i)$ and $f_B^\\sigma (i)$ .", "These in turn are derived from payoffs in a multi-player game via the common exponential mapping [1] $f_A^\\sigma (i)=e^{\\beta \\pi _A^\\sigma (i)}, f_B^\\sigma (i)=e^{\\beta \\pi _B^\\sigma (i)}$ , where $\\beta \\ge 0$ denotes the intensity of selection, and where $\\pi _A^\\sigma (i)$ and $\\pi _B^\\sigma (i)$ are the payoffs to individuals of either type in the multi-player game in environment $\\sigma $ ." ], [ "Multi-player games", "We assume that individual-based interactions take place between $n$ players, where $n$ is a fixed integer, $2\\le n\\le N$ .", "We will write $a_{j,\\sigma }$ for the payoff to an individual of type $A$ if they face $j$ other individuals of type $A$ and $n-1-j$ individuals of type $B$ in such an encounter of $n$ players.", "The payoff to a player of type $B$ is $b_{j,\\sigma }$ is they play against $j$ individuals of type $A$ , and $n-1-j$ (other) players of type $B$ .", "These payoffs depend on the environmental state $\\sigma $ .", "This information is summarised in the following payoff matrix (cf.", "[17]) $\\begin{tabular}{c|ccccc}& A\\cdots A & A\\cdots AB & \\cdots & AB\\cdots B& B\\cdots B \\\\\\hline A & a_{n-1,\\sigma } & a_{n-2,\\sigma } &\\cdots & a_{1,\\sigma } & a_{0,\\sigma }\\\\B & b_{n-1,\\sigma } & b_{n-2,\\sigma } &\\cdots & b_{1,\\sigma } & b_{0,\\sigma },\\\\\\end{tabular}$ and we have $\\pi _A^\\sigma (i)=\\sum _{j=0}^{n-1} a_{j,\\sigma } H^A(i,j)$ , and $\\pi _B^\\sigma (i)=\\sum _{j=0}^{n-1} a_{j,\\sigma } H^B(i,j)$ .", "The quantity $H^A(i,j)$ is the probability that precisely $j$ individuals among the $n-1$ opponents of a player of type $A$ are of type $A$ as well, in a population with $i$ type $A$ individuals in total.", "Similarly, $H^B(i,j)$ is the probability for an individual of type $B$ to face precisely $j$ individuals of type $A$ , and $n-1-j$ players of type $B$ .", "Detailed expressions can for example be found in [17].", "In order to analyse the evolutionary dynamics of games in finite populations, it is often useful to first consider the limit of an infinite population.", "Stochastic effects are suppressed in this limit, but often the qualitative behaviour of the stochastic system is determined by the structure of the underlying deterministic flow.", "Writing $x = <i>/N$ , where angular brackets denote an ensemble average, and keeping the environment fixed for the time being, one finds the following dynamics in the limit $N\\rightarrow \\infty $ : $\\dot{x} = \\omega _\\sigma ^+(x)-\\omega _\\sigma ^-(x),$ where $\\omega _\\sigma ^\\pm (x)$ is obtained from $\\omega _{i,\\sigma }^\\pm $ by obvious substitutions.", "Eq.", "(REF ) can formally be derived from the leading order of an expansion in the inverse system size, or equivalently from the Kramers-Moyal expansion of the relevant master equation; see [5], [6] for details.", "We will mostly be interested in the fixed points of Eq.", "(REF ).", "The rates $\\omega _\\sigma ^{\\pm }(x)$ contain a factor of $x(1-x)$ .", "This is because evolutionary events affecting the state of the population require two individuals of the different types (one selected for reproduction, the other for removal).", "As a consequence, one always has the trivial fixed points at $x=0$ and $x=1$ .", "Further fixed points can exist at locations $x^\\star $ ($0\\le x^\\star \\le 1$ ) for which $\\pi _A^\\sigma (x^\\star )=\\pi _B^\\sigma (x^\\star )$ .", "The underpinning deterministic flows of multi-player games can then, qualitatively, be classified according to the number and stability of these internal fixed points.", "Below we will focus on three-player games with a designated deterministic structure.", "We choose the values of the payoff matrix elements, $a_{j,\\sigma }$ and $b_{j,\\sigma }$ , so as to have fixed points in specific locations; a prescription for doing this is given for general $n$ -player games in the Supplement.", "Figure: Deterministic flow in the two environments, we plot x ˙\\dot{x} as a function of xx, see text for details.", "The internal fixed points are located at x ☆ =0.3x^\\star =0.3 and x ☆ =0.7x^\\star =0.7.", "The environment σ=1\\sigma = 1 is shown on the left and σ=-1\\sigma = -1 is shown on the right.", "The fixed points have opposite stability in the two environments.", "The method of choosing payoff matrix elements such that fixed points are at designated locations is described in the Supplement.A typical example of the resulting deterministic flow can be found in Fig.", "REF , where plot $\\dot{x}$ as a function of $x$ .", "We show a case of a game with (non-trivial) fixed points at $x^\\star =0.3$ and $x^\\star =0.7$ , but in which the deterministic flow has opposite directions in the two environments.", "In Fig.", "REF and in all subsequent figures below, we have always set $b_{j,\\sigma } =1$ ($j=0,1,2$ ) in both environments $(\\sigma =\\pm 1)$ .", "Effectively, this is an overall normalisation of payoff, individuals of the wildype $B$ always have $\\pi _B\\equiv 1$ , irrespective of the composition of the population.", "The remaining pay-off matrix elements, $a_{j,\\sigma }$ are then calculated as described in the Supplement.", "We use $\\beta =1/2$ in all sections except Sec.", "REF ." ], [ "Fixation probability and fixation times", "We will now turn to the dynamics in finite populations, and consider situations in which one single mutant is placed in a population of $N-1$ wildtype individuals, i.e., the initial condition is $i=1$ .", "While the deterministic flow of the dynamics in the limit of infinite populations may have internal fixed points, the outcome of evolution in a finite population will necessarily be extinction or fixation of the invading mutant; through genetic drift the population will eventually reach one of the absorbing states, $i=0$ (extinction of the mutant) or $i=N$ (fixation), and no further dynamics occurs.", "To analyse these processes we study the probability that the mutants fixates (as opposed to going extinct), and the mean time that they take to do so.", "The fixation probabilities and fixation times in switching environments have already been calculated in [16], and we briefly summarise the results here.", "Writing $\\varphi _{i,\\sigma }$ for the probability that the system reaches fixation ($i = N$ ), given that it currently contains $i$ individuals of type $A$ and is in environment $\\sigma $ , one has $\\varphi _{i,\\sigma }=\\sum _{\\sigma ^{\\prime }}\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}\\left[\\omega _{i,\\sigma ^{\\prime }}^+ \\varphi _{i+1,\\sigma ^{\\prime }}+\\omega _{i,\\sigma ^{\\prime }}^- \\varphi _{i-1,\\sigma ^{\\prime }}+(1-\\omega _{i,\\sigma ^{\\prime }}^+-\\omega _{i,\\sigma ^{\\prime }}^-)\\varphi _{i,\\sigma ^{\\prime }}\\right],$ subject to the boundary conditions $\\varphi _{0,\\sigma }=0$ and $\\varphi _{N,\\sigma }=1$ for all $\\sigma $ .", "Fixation times can be obtained from a similar relation.", "The unconditional fixation time $t_{i,\\sigma }$ is the time until absorption (either at $i=0$ or at $i=N$ ) if the population starts in state $i$ and if the initial state of the environment is $\\sigma $ .", "Similarly, we can introduce the conditional fixation time for a given initial condition; this is an ensemble average of fixation time for trajectories that result in the fixation of the mutant.", "A procedure for solving Eqs.", "(REF ) and its analogue for the conditional and unconditional fixation times in switching environments is described in detail in [16]." ], [ "Sojourn times", "We next introduce an additional tool for the analysis of population dynamics in switching environments, and describe the calculation of mean sojourn times.", "The concept of sojourn times is well known in the context of birth-death processes with absorbing states [30]; they describe the mean time spent in each state before absorption occurs.", "More precisely, we write $t_{i,\\sigma ;j}$ for the mean time the population spends in state $j$ before absorption, if it is started in state $i$ and if the initial state of the environment is $\\sigma $ .", "We stress that no requirement is made for the environment to be in a specific state when the population `sojourns' in $j$ .", "Additionally, for the purposes of unconditional sojourn times, we do not specify whether the dynamics ends in the state with 0 or $N$ mutants.", "The end-point is relevant for so-called conditional sojourn times though; we write $t_{i,\\sigma ;j}^\\star $ for the mean time the population spends in state $j$ , if started at $i$ and with initial environmental state $\\sigma $ , conditioned on fixation of the mutant.", "In simulations this object is measured as follows: Run a large number of independent realisations, all started from a population with $i$ mutants and $N-i$ wildtypes, and in environment $\\sigma $ .", "Then run each of these samples until the mutant has either gone extinct or reached fixation.", "Only take into account the runs in which fixation occurs.", "In this sample of trajectories measure the average time the population has spent in state $j$ .", "As part of this work we have developed a method for the calculation of conditional and unconditional sojourn times for general birth-death processes with two species and in switching environments.", "The calculation is broadly based on backward equation techniques [5], [16], and relies on `return probabilities', that is the likelihood that the population returns to state $i$ at a later time if it is started there.", "The key novelty here is that the dynamics of the enviroment needs to be accounted for as well.", "The calculation is lengthy, and we relegate the full mathematical details to the Supplement.", "The unconditional and conditional sojourn times discussed in the next section for specific games and environmental processes have been computed from Eqs.", "(S27) and (S34) respectively." ], [ "Dynamics of three-player games", "Using this analytical approach to calculate the sojourn times in switching environments (see Supplement), we can now examine the specific case of three-player games.", "We notice interesting behaviour in the three-player games which is not found in the two-player case; a minimum in the conditional fixation times with respect to the switching parameters for example.", "The sojourn times allow us to investigate where the system spends its time and to understand in more detail the mechanistic origin of the observed behaviour.", "Figure: Conditional fixation time as a function of the switching parameters TT and δ + \\delta ^+.", "Starting environment is σ=1\\sigma = 1 in the left-hand panel and σ=-1\\sigma = -1 on the right.", "The fixed points of the system are at x ☆ =0.3x^\\star = 0.3 and x ☆ =0.7x^\\star = 0.7.", "Filled white dots in each panel indicate the point at which the conditional fixation time is minimal." ], [ "Notation", "We focus on systems in which the environment can take two different states, $\\sigma =\\pm 1$ .", "In [16] results for the case of two-player games were described as a function of the switching probabilities of the environmental state (per time step), $p^+=\\mu _{+\\rightarrow -}$ and $p^-=\\mu _{-\\rightarrow +}$ .", "In order to better reflect the characteristic properties of the switching process we introduce $T = 1/p^+ + 1/p^-$ and $\\delta ^+ = p^-/(p^-+p^+)$ , and use these quantities in place of the parameters $p^\\pm $ .", "The parameter $T$ can be interpreted as the average length of time that it takes for the environment to switch from one state to the other, and then back again.", "We will refer to $T$ as the `cycle time' or `switching period', keeping in mind though that the switching between the environments is stochastic so that there are no strictly periodic cycles.", "The parameter $\\delta ^+$ is the average proportion of time spent in the environment $\\sigma = 1$ .", "The proportion of time spent in environment $\\sigma =-1$ is given by $\\delta ^-=p^+/(p^++p^-)=1-\\delta ^+$ .", "In order for $0\\le p^\\pm \\le 1$ , we require $T\\ge 2$ and $1/T \\le \\delta ^+ \\le 1 - 1/T$ .", "These conditions are easily understood, keeping in mind the discrete-time nature of the dynamics.", "The typical cycle must be at least two time steps long, and the minimum proportion of time spent in either environment is one time step, i.e.", "a fraction $1/T$ of the total (average) cycle.", "We are most interested in the dynamics of the system when the internal fixed points are neither very close to one another, nor to the absorbing boundaries.", "Cases where the fixed points are close to one another or to the boundaries exhibit behaviour similar to games where there are only one or no internal fixed points.", "Novel behaviour is most clearly observed when the internal fixed points are sufficiently isolated, and so we focus on this regime.", "In the following we will mostly use an example in which the two internal fixed points are located at $x^\\star =0.3$ and $x^\\star = 0.7$ , and with deterministic flow as shown in Fig.", "REF .", "We have tested other cases and have observed similar behaviour.", "Unless specified otherwise we always use a population size of $N=50$ , and start with one single mutant $i=1$ ." ], [ "Conditional fixation time", "The conditional fixation time is obtained using the formalism of [16], and is shown as a function of the switching parameters $T$ and $\\delta ^+$ in Fig.", "REF .", "The data in the figure reveals that there is a particular set of switching parameters $T$ and $\\delta ^+$ which minimises the conditional fixation time, as indicated by filled circles in Fig.", "REF .", "That is, we have a minimum with respect to $T$ for fixed $\\delta ^+$ and vice versa.", "In order to investigate the details of the dynamics leading to this effect, it is useful to discuss the conditional sojourn times, to gain an understanding of where the population spends its time on the way to fixation.", "These are obtained using the theory detailed in the Supplement, results are shown in Fig.", "REF .", "Figure: Re-scaled conditional sojourn time as a function of the switching parameter δ + \\delta ^+ and the position x=i/Nx = i/N.", "Specifically we plot the fraction of time spent in each state, i.e., the sojourn times have been normalised to sum to unity for fixed δ + \\delta ^+.", "Starting environment σ=1\\sigma = 1 (left) and σ=-1\\sigma = -1 (right).", "We have fixed T=100T = 100.", "When δ + ∼1\\delta ^+ \\sim 1 or δ + ∼0\\delta ^+ \\sim 0 the system tends to loiter around the stable fixed point in the σ=1\\sigma = 1 and σ=-1\\sigma = -1 environments respectively.We first focus on the effects of varying the model parameters $\\delta ^+$ and $\\delta ^-$ , which reflect the proportion of time spent in each of the two environmental states.", "As seen in Fig.", "REF successful trajectories (i.e.", "those in which the mutant reaches fixation) spend most of their time around the stable fixed point of the `dominant' environment, when either $\\delta ^+\\gg \\delta ^-$ , or vice versa.", "For example, when $\\delta ^+$ is close to one, most time is spent near the stable fixed point $x^\\star = 0.3$ of environment $\\sigma =1$ .", "This illustrates that successful trajectories will loiter around that the relevant stable fixed point fixed point if one environmental state is significantly more frequent than the other.", "However, as the parameters $\\delta ^+$ and $\\delta ^-$ are moved away from the extremes, the time the system spends in the different states, $i=1,\\dots ,N=1$ , is more evenly distributed, and the population has a lower propensity to get trapped near fixed points.", "At the value of $\\delta ^+$ which corresponds to the minimum in the conditional fixation time (see Fig.", "REF ), the conditional sojourn times are fairly evenly spread across $i$ (Fig.", "REF ).", "Trajectories then do not loiter around any of the internal fixed points, and successful runs reach fixation relatively quickly.", "Figure: Re-scaled conditional sojourn time as a function of the switching parameter TT and the position x=i/Nx = i/N.", "Normalisation is as in the previous figure.", "Starting environment σ=1\\sigma = 1 (left) and starting environment σ=-1\\sigma = -1 (right).", "We have fixed δ + =0.5\\delta ^+ = 0.5.We next turn to the effects of varying the time scale of the environmental switching time, i.e.", "the role of the parameter $T$ .", "When the cycle period $T$ is small, the switching process between the environments is fast and so the initial state of the environment at the start of the dynamics does not have any significant effect.", "This is confirmed in Fig.", "REF , where we show the conditional sojourn time as a function of $i/N$ and $T$ for fixed $\\delta ^+=\\delta ^-=1/2$ .", "The two panels in the figure show the data for starts in the two different environments, and the sojourn times are seen to be quantitatively similar across the two panels when $T$ is small ($T\\lesssim 10^3$ in our example).", "The sojourn time is then relatively constant across different states $i$ of the population, indicating that time is spent fairly evenly; we note again that the data in the figure is for $\\delta ^+ = 0.5$ .", "This indicates that the population does not have sufficient time to settle near either of the fixed points, due to the fast switching dynamics.", "While each fixed point is attractive in one environment and a repeller in the other, these effects `average' out under fast environmental switching.", "For longer duration of the switching cycle $T$ however, the system begins to spend longer stretches of time in either environment, and so is able to spend more time at either fixed point.", "The conditional sojourn times begin to peak around the locations of the fixed points for $T \\gtrsim 10^3$ , see Fig.", "REF .", "If the cycle is longer still ($T\\gtrsim 10^4$ ) most of the dynamics occurs in the starting environment, and so the population will spend most of its time around the stable fixed point in that environment.", "This is where we begin to see a marked difference between the two panels in Fig.", "REF .", "For time scales above $T\\approx 10^5$ the environment effectively never switches state before fixation is reached; this corresponds to the regime in Fig.", "REF in which the conditional fixation time reaches a value which is characteristic of the starting environment and independent of $\\delta ^+$ .", "The value of the cycle period $T\\sim 10^3$ which minimises the conditional fixation time in Fig.", "REF corresponds roughly to the point at which the environment is neither switching so often that the trajectories are constantly changing direction, nor switching so little that the system loiters around a fixed point.", "This can be seen in Fig.", "REF as the value of $T$ where the sojourn times begin to stop being spread equally across all values of $i/N$ , and start to peak around both fixed points." ], [ "Probability for a single mutant to reach fixation", "The probability that a single mutant reaches fixation is shown in Fig.", "REF as a function of the switching period, $T$ , and the fraction of time spent in environment $\\sigma =1$ .", "The two panels show data for starts in either of the environmental states.", "The figure reveals that there is no combination of $T$ and $\\delta ^+$ , which would extremise the fixation probability in the same way as the conditional fixation times.", "The main conclusion we draw from the figure, is, quite simply, that the likelihood that the mutant fixates is the greater the higher the proportion of time that the system spends in environment $\\sigma =+1$ , corresponding to the flow depicted in the left-hand panel of Fig.", "REF .", "This is the case for starts in either of the two environmental states.", "We also note that the fixation probability increases with the cycle period $T$ if the start occurs in environment $\\sigma = 1$ , but that it decreases with $T$ for starts in $\\sigma = -1$ .", "Figure: Unconditional sojourn time as a function of the switching parameter δ + \\delta ^+ and the position x=i/Nx = i/N.", "Starting environment σ=1\\sigma = 1 (left) and σ=-1\\sigma = -1 (right).", "Data is for fixed T=100T = 100.", "The system can be seen to spend less time around the initial position at i=1i = 1 as δ + \\delta ^+ is increased, indicating that the system leaves the starting position more easily when δ + \\delta ^+ is higher, as one might expect.", "This correlates with the increase in fixation probability with δ + \\delta ^+ in Fig.", ".", "The qualitative behaviour is roughly the same in either environment.Figure: Unconditional sojourn times as a function of the switching parameter TT and the position x=i/Nx = i/N.", "Starting environment σ=1\\sigma = 1 (left) and σ=-1\\sigma = -1 (right), for fixed δ + =0.5\\delta ^+ = 0.5.", "As TT is increased, the system spends more time in the starting environment.", "In the left-hand panel, the system moves away from the starting position more easily with increasing TT, which corresponds to the increase with TT of the fixation probability in the left-hand panel of Fig.", ".", "The opposite is true in the right-hand panel.", "The fixation probability can be seen to decrease with TT in the right-hand panel of Fig.", ".We now comment on the mechanisms generating these effects.", "The system always begins with one mutant, very close to the absorbing boundary at $i=0$ .", "One might expect, therefore, that the propensity of the system to move away from the initial position, and from extinction, would have a great impact on the over all fixation probability.", "The unconditional sojourn times shown in Fig.", "REF confirm that the system spends less of its trajectory near the starting position as $\\delta ^+$ is increased.", "This correlates with the increase in fixation probability with $\\delta ^+$ in Fig.", "REF .", "In Fig.", "REF we show the unconditional sojourn time as a function of the cycle period $T$ .", "As this time scale $T$ is increased, the initial state of the environment becomes more relevant, as the first switch occurs later on average.", "This reduces the probability of immediate extinction on starts in the environment $\\sigma = 1$ for which the flow is away from the $i=0$ boundary.", "This correlates with the increase in fixation probability with $T$ in the left-hand panel of Fig.", "REF .", "If the initial state of the environment is $\\sigma =-1$ the argument is reversed.", "In this environment the gradient of selection is towards $i=0$ for small mutant numbers; a prolonged initial time spent in this environment hence reduces the likelihood that the mutant is successful, as indicated by the decrease of fixation probability in the right-hand panel of Fig.", "REF ." ], [ "Fixation in finite time", "The combination of Figs.", "REF and REF reveals an intriguing observation.", "The fixation time, conditioned on fixation of the mutant, is minimal for a certain combination of $T$ and $\\delta ^+$ , as indicated in Fig.", "REF .", "On the other hand, no such extremum is found for the fixation probability in Fig.", "REF .", "The fixation probability measures the likelihood for the mutant to be successful eventually – including at long times.", "This indicates an interesting balance of two effects: if evolution is allowed to run indefinitely the mutant has the highest chance of success when $T$ and $\\delta ^+$ are large, and the starting environment is $\\sigma = 1$ .", "If however, we are interested in fixation on moderate time scales, the mutant does better near the point of minimal conditional fixation time in Fig.", "REF .", "In order to investigate this further we focus on a case with relatively strong selection, $\\beta =1$ .", "The corresponding conditional fixation times and fixation probabilities are shown as functions of $\\delta ^+$ and $T$ in Fig.", "REF .", "Again a minumum in fixation time is found, but no extremum of the fixation probability.", "We introduce $Q_{i,\\sigma }(t)$ as the probability that the mutant has reached fixation $t$ time steps (or sooner) after the system is started with $i$ mutants and in environmental state $\\sigma $ .", "We then have the discrete-time backward master equation (see e.g.", "[5], [6], [30] for general references) $Q_{i,\\sigma }(t+1)=\\sum _{\\sigma ^{\\prime }}\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}\\left[\\omega _{i,\\sigma }^+ Q_{i+1,\\sigma ^{\\prime }}(t)+\\omega _{i,\\sigma }^- Q_{i-1,\\sigma ^{\\prime }}(t)+(1-\\omega _{i,\\sigma }^+-\\omega _{i,\\sigma }^-)Q_{i,\\sigma ^{\\prime }}(t)\\right].$ The initial conditions are $Q_{i,\\sigma }(t=0)=\\delta _{i,N}$ for all $\\sigma $ , and allow us to numerically obtain the $Q_{i,\\sigma }(t)$ by iterating Eq.", "(REF ) forward.", "Results are summarised in Fig.", "REF .", "In the left-hand panel we show the fraction of samples, started with one single mutant, in which the mutant has reached fixation by time $t=2000$ , a maximum is discernible near $\\delta ^+=0.7$ and just below $T\\approx 10^3$ .", "This demonstrates that the total fraction of samples that reach fixation by a finite time $t$ can have a local maximum at a location in the $\\delta ^+-T$ plane, even when the eventual fixation probability does not have such a maximum (cf.", "left-hand panel of Fig.", "REF ).", "This is clarified further in the right-hand panel of Fig.", "REF , where we show $Q_{1,\\sigma =1}(t)$ as a function of time, $t$ , for three different choices of the cycle period (and at a fixed value of $\\delta ^+=0.7$ ).", "These correspond to the points marked in the left-hand panel of Fig.", "REF .", "While more trajectories reach fixation eventually for $T=500$ , the proportion that have reached fixation at times $t\\approx 1000-5000$ is higher for the other two choices of $T$ shown in the figure ($T=50$ and $T=5000$ ).", "Figure: Fixation probability (left) and conditional fixation time (right) as a function of the switching parameters δ + \\delta ^+ and TT for the case β=1\\beta =1.", "In both cases, the starting environment is σ=1\\sigma = 1 and the fixed points are located at x ☆ =0.3x^\\star =0.3 and x ☆ =0.7x^\\star =0.7.", "The behaviour is qualitatively the same as in Figs.", "and .Figure: Left: Proportion of trajectories which have reached state i=Ni=N by time t=2000t=2000.", "Right: Fraction of trajectories which have reached i=Ni=N by time tt.", "We fix δ + =0.7\\delta ^+=0.7 and show three different choices of TT in the right-hand panel; these are indicated by the crosses in the left-hand panel.", "All data is for β=1\\beta =1, starting at i=1i=1 and in environment σ=1\\sigma =1." ], [ "Conclusions and outlook", "In summary, we have analysed the dynamics of fluctuating multi-player games in finite populations.", "Fluctuations of the payoff matrix are taken to originate from changes of an environmental state affecting the relative success of mutants and resident wildtypes.", "These environmental fluctuations could for example represent availability or absence of nutrients, the application of treatment in the context of bacterial populations or variations in other external conditions.", "Our analysis focuses on a stylised model with interaction between multiple individuals – modelled as a multi-player game – but representing general nonlinear payoff structures.", "This adds complexity relative to the two-player case studied in [16].", "In this earlier work fitness differences between mutants and wildtypes are linear in their relative frequencies, and so at most one internal equilibrium point is permitted by the resulting replicator equations.", "In this work we address higher-order interaction leading to more complex frequency-dependent fitness landscapes, with multiple non-trivial selection balance points.", "Specifically, we focus on games which allow two internal fixed points and on the regime in which these are well separated from each other and from the states at which mutants have reached fixation or gone extinct.", "We have then analysed in detail the likelihood for an invading mutant to reach fixation in the resident population.", "We have also studied the dynamics leading to fixation, in particular conditional fixation times, and the time spent in each state of the population on the path to fixation, the so-called sojourn times [30].", "We have presented a comprehensive theory to calculate these analytically for multi-player games in finite populations subject to fluctuating environments with discrete states.", "Our analysis indicates that the time scales and detailed dynamics of the environmental switching process can affect the success of the invading mutant in several ways.", "For example, we find that the conditional fixation time can have a local minimum in the space of all (Markovian) switching processes, indicating that the mutant succeeds quickly under those circumstances, if it reaches fixation.", "At the same time the fixation probability appears to exhibit monotonous behaviour in the time-scale of the switching process and fraction of time spent in either of two environments.", "This indicates an interesting balance of two effects: a propensity to reach fixation quickly and the overall fixation probability at long times.", "A more detailed analysis of the dynamics, based on a backward-master equation approach, reveals that these effects may be in competition with each other.", "There are switching dynamics for which there is a pronounced tendency to reach fixation in the early stages of the dynamics, but other parameters of the environmental dynamics may lead to a higher chance of fixation eventually.", "In broader terms our findings contribute to constructing a more general theory of population dynamics with selection, random genetic drift and environmental fluctuations.", "While we have focused on a selected set of three-player games, the calculation of sojourn times in switching environments is applicable to general birth-death processes with fluctuating discrete environmental states.", "Hamilton has characterised the complexity of multi-player games as `sea-sickness' [17].", "We believe our work is a contribution to reducing this discomfort in dealing with multi-player dynamics and towards an understanding of the outcomes of evolutionary processes with fluctuating non-linear interaction." ], [ "Acknowledgements", "We thank Pete(r) Ashcroft for discussions.", "JWB acknowledges a studentship by the Engineering and Physical Sciences Research Council (EPSRC, UK).", "Supplementary Material" ], [ "Construction of multi-player games", "We briefly describe the construction of multi-player games with specified locations of internal fixed points in the deterministic limit.", "In this limit the dynamics are given by $\\dot{x} = \\omega _\\sigma ^+(x)-\\omega _\\sigma ^-(x).$ The subscript $\\sigma $ is not relevant for the argument that follows, so we omit it for the remainder of this section of the Supplement.", "Given that each of the rates contains a factor $x(1-x)$ there are always two trivial fixed points $x^\\star =0$ and $x^\\star =1$ .", "The remaining fixed points of Eq.", "(REF ) are the – non-trivial – roots of $\\omega ^+(x)-\\omega ^-(x)=0$ .", "This is equivalent to $\\pi _A(x)=\\pi _B(x),$ where the latter payoffs are given by $\\pi _A(x)&=&\\sum _{j=0}^{n-1} a_j \\left(\\begin{array}{c} n-1 \\\\ j \\end{array}\\right) x^j (1-x)^{n-j-1} \\nonumber \\\\\\pi _B(x)&=&\\sum _{j=0}^{n-1} b_j \\left(\\begin{array}{c} n-1 \\\\ j \\end{array}\\right) x^j (1-x)^{n-j-1}$ in an $n$ -player game.", "We note that $\\left(\\begin{array}{c} n-1 \\\\ j \\end{array}\\right) x^j (1-x)^{n-j-1}$ is the probability for a given player to face $j$ opponents of type $A$ and $n-1-j$ opponents of type $B$ in an $n$ -player game, if the fraction of type $A$ -players in the population is $x$ ." ], [ "General construction", "The construction of the game with specified non-trivial fixed points $x_1^\\star ,\\dots , x_{n-1}^\\star $ is equivalent to finding coefficients $c_j\\equiv a_j-b_j$ , $j=0,1,\\dots ,n-1$ , such that $\\sum _{j=0}^{n-1} c_j \\left(\\begin{array}{c} n-1 \\\\ j \\end{array}\\right) x^j (1-x)^{n-j-1}=0$ for $x\\in \\lbrace x_1,\\dots ,x_{n-1}\\rbrace $ .", "We construct a solution by induction.", "The problem is straightforward for $n=2$ , in which case it reduces to finding $c_0$ and $c_1$ such that $c_0(1-x_1^\\star )+c_1x_1^\\star =0$ .", "One finds $\\frac{c_{1}}{c_{0}} = 1 - \\frac{1}{x_1^\\star } .$ Suppose now that we have constructed $c_0,\\dots ,c_{n-1}$ such that $x_1^\\star ,\\dots ,x_{n-1}^\\star $ are the roots of $\\sum _{j=0}^{n-1} c_j \\left(\\begin{array}{c} n-1 \\\\ j \\end{array}\\right) x^j (1-x)^{n-j-1}=0.$ We can introduce a new root at $x_n^\\star $ by multiplying by $1- x + \\left(1-\\frac{1}{x_n^\\star }\\right)x$ .", "One then obtains $&&\\sum _{j=0}^{n-1}\\left[c_{j}\\left(\\begin{array}{c} n-1\\\\ j\\end{array}\\right) + \\left(1-\\frac{1}{x_n^\\star }\\right) c_{j-1}\\left(\\begin{array}{c} n-1\\\\ j-1 \\end{array}\\right)\\right]x^j\\left(1-x\\right)^{n-j}\\nonumber \\\\&&+ c_{0}\\left(1-x\\right)^n+ c_{n-1}\\left(1-\\frac{1}{x_n^\\star }\\right)x^n = 0 .$ This can be written in the form $\\sum _{j=0}^{n} c_j^{\\prime } \\left(\\begin{array}{c} n \\\\ j \\end{array}\\right) x^j (1-x)^{n-j}=0.$ with $c_0^{\\prime } &=& c_0, \\nonumber \\\\c_n^{\\prime } &=& c_{n-1}\\left(1-\\frac{1}{x_n^\\star }\\right), \\nonumber \\\\c_{j}^{\\prime } &=& \\frac{1}{\\left(\\begin{array}{c} n\\\\ j\\end{array}\\right)}\\left[c_{j}\\left(\\begin{array}{c} n-1\\\\ j\\end{array}\\right)+c_{j-1}\\left(1-\\frac{1}{x_n^*}\\right)\\left(\\begin{array}{c} n-1\\\\ j-1\\end{array}\\right)\\right] .$ This completes the inductive construction." ], [ "Three-player games", "The resulting relations are relatively compact for three-player games.", "We will have internal fixed points at $x_1^\\star $ and $x_2^\\star $ if we choose $\\frac{c_{1,\\sigma } }{c_{0,\\sigma } }= \\frac{1}{2} \\left[ \\left(1-\\frac{1}{x_1^\\star } \\right) + \\left(1-\\frac{1}{x_2^\\star }\\right) \\right],$ and $\\frac{c_{2,\\sigma } }{c_{0,\\sigma } } = \\left( 1-\\frac{1}{x_1^\\star }\\right)\\left(1 - \\frac{1}{x_2^\\star }\\right).$ We have here included the subscript $\\sigma $ to indicate the dependence of the game on the environmental state.", "The coefficient $c_{0,\\sigma }$ can be chosen arbitrarily, its sign determines the direction of the flow at $x=0$ , and hence the stability of $x_1^\\star $ and $x_2^\\star $ , respectively.", "In the main paper we use $c_{0, \\sigma =1} = 1$ and $c_{0, \\sigma =-1} = -1$ ." ], [ "Initial steps of the calculation", "In order to compute sojourn times we introduce the quantity $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}=\\mbox{Prob}\\left(\\begin{array}{p{7cm} | p{5cm}} There exists a time t\\ge t_0 at which the system reaches state j, and when it does so for the first time the environment is in state \\sigma ^{\\prime }.", "& The population is started in state i and the environent in \\sigma at initial time t_0.", "\\end{array}\\right).$ We note that the initial time $t_0$ is obviously immaterial, as the dynamics is Markovian.", "One then has $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}=\\sum _{\\sigma ^{\\prime \\prime }} \\mu _{\\sigma \\rightarrow \\sigma ^{\\prime \\prime }} \\left[ \\omega _{i,\\sigma }^+ \\varphi _{i+1,\\sigma ^{\\prime \\prime };j,\\sigma ^{\\prime }}+\\omega _{i,\\sigma }^- \\varphi _{i-1,\\sigma ^{\\prime \\prime };j,\\sigma ^{\\prime }}+(1-\\omega _{i,\\sigma }^+-\\omega _{i,\\sigma }^-) \\varphi _{i,\\sigma ^{\\prime \\prime };j,\\sigma ^{\\prime }}\\right],$ with the following boundary conditions $\\varphi _{0,\\sigma ;j,\\sigma ^{\\prime }}=\\delta _{0,j}\\delta _{\\sigma ,\\sigma ^{\\prime }}, ~\\varphi _{N,\\sigma ;j,\\sigma ^{\\prime }}=\\delta _{N,j}\\delta _{\\sigma ,\\sigma ^{\\prime }}, ~\\varphi _{j,\\sigma ;j,\\sigma ^{\\prime }}=\\delta _{\\sigma ,\\sigma ^{\\prime }}.$ The first and second of these reflect the fact that the states $i=0$ and $i=N$ are absorbing, so once the population is in state $i=0$ or $i=N$ it remains there at all future times.", "In the third relation in Eq.", "(REF ) we have $i=j$ so that trajectories, started at $(i,\\sigma )$ , will reach state $j$ immediately at $t=t_0$ ; they then only contribute to the above probability if $\\sigma ^{\\prime }=\\sigma $ .", "For a fixed $j$ Eqs.", "(REF ) impose constraints on $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}$ at $i=0$ , $i=N$ and at $i=j$ .", "Focusing on a given value of $j$ it is hence convenient to treat the cases $i<j$ and $i>j$ separately.", "To proceed we introduce the following quantities, $\\psi _{i,\\sigma ;j,\\sigma ^{\\prime }}=\\sum _{\\sigma ^{\\prime \\prime }} \\mu _{\\sigma \\rightarrow \\sigma ^{\\prime \\prime }}\\varphi _{i,\\sigma ^{\\prime \\prime };j,\\sigma ^{\\prime }}.$ Focusing first on $j>i$ , we define $\\nu _{i,\\sigma ^{\\prime };j,\\sigma ^{\\prime }}=\\psi _{i,\\sigma ;j,\\sigma ^{\\prime }}-\\psi _{i-1,\\sigma ;j,\\sigma ^{\\prime }},$ and we then arrive at the following after substitution in Eq.", "(REF ), $\\nu _{i+1,\\sigma ;j,\\sigma ^{\\prime }}=\\gamma _{i,\\sigma }\\nu _{i,\\sigma ;j,\\sigma ^{\\prime }}+\\frac{1}{\\omega _{i,\\sigma }^+}\\left[\\left(\\underline{\\underline{\\mu }}^{-1}-{\\rm 1\\!\\!I}\\right)\\sum _{k=1}^j \\underline{\\nu }_{k,\\sigma ;j}\\right]_{\\sigma ^{\\prime }}.$ We have used vector and matrix notation for convenience.", "Indices run over the states of the environment; for example the components of $\\underline{\\nu }_{k,\\sigma ;j}$ correspond to the index $\\sigma ^{\\prime }$ , and $\\underline{\\underline{\\mu }}$ has entries $\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}$ .", "We then use the condition $\\sum _{i=1}^j\\nu _{i,\\sigma ;j,\\sigma ^{\\prime }}=\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}$ to determine $\\nu _{i,\\sigma ;j,\\sigma ^{\\prime }}$ .", "The probabilities $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}$ can then be found using $\\underline{\\varphi }_{i,\\sigma ;j} = \\underline{\\underline{\\mu }}^{-1} \\sum _{k=1}^i\\underline{\\nu }_{k,\\sigma ;j} .$ Similarly, for $j<i$ we write $\\lambda _{i,\\sigma ;j,\\sigma } = \\psi _{i,\\sigma ;j,\\sigma } - \\psi _{i+1,\\sigma ;j,\\sigma }$ , and find $\\lambda _{i-1,\\sigma ;j,\\sigma ^{\\prime }}=\\frac{1}{\\gamma _{i,\\sigma }}\\lambda _{i,\\sigma ;j,\\sigma ^{\\prime }}+\\frac{1}{\\omega _{i,\\sigma }^-}\\left[\\left(\\underline{\\underline{\\mu }}^{-1}-{\\rm 1\\!\\!I}\\right)\\displaystyle \\sum _{k=i}^{N-1} \\underline{\\lambda }_{k,\\sigma ;j}\\right]_{\\sigma ^{\\prime }},$ One then proceeds using $\\displaystyle \\sum _{i=j}^{N-1}\\lambda _{i,\\sigma ;j,\\sigma ^{\\prime }} = \\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}$ , and $\\underline{\\varphi }_{i,\\sigma ;j} = \\underline{\\underline{\\mu }}^{-1} \\displaystyle \\sum _{k=i}^{N-1}\\underline{\\lambda }_{k,\\sigma ;j}$ .", "Now we have at our disposal a means by which to calculate the complete set of probabilities $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}$ .", "Using these probabilities, one can then compute the unconditional and conditional sojourn times." ], [ "Calculation of unconditional sojourn times", "As the next step we compute $r_{i,\\sigma ;\\sigma ^{\\prime }}=\\mbox{Prob}\\left(\\begin{array}{p{6.5cm}|p{5cm}} There exists a time t>t_0 at which the population returns to state i, and when it does so for the first time the environment is in state \\sigma ^{\\prime }.", "& The population is started in state i and the environent in \\sigma at initial time t_0.", "\\end{array}\\right).$ We stress the requirement that $t$ be strictly greater than $t_0$ , marking a difference compare to the above definition of $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}$ , where we only require $t\\ge t_0$ .", "Hence, $r_{i,\\sigma ;\\sigma ^{\\prime }}$ is in general distinct from $\\varphi _{i,\\sigma ;i,\\sigma ^{\\prime }} = \\delta _{\\sigma , \\sigma ^{\\prime }}$ .", "We have $r_{i,\\sigma ;\\sigma ^{\\prime }} = \\mu _{\\sigma \\rightarrow \\sigma ^{\\prime }}\\left(1-\\omega _{i,\\sigma }^+-\\omega _{i,\\sigma }^-\\right)+\\sum _{\\sigma ^{\\prime \\prime }}\\mu _{\\sigma \\rightarrow \\sigma ^{\\prime \\prime }}\\left(\\omega _{i,\\sigma }^+\\varphi _{i+1,\\sigma ^{\\prime \\prime };i,\\sigma ^{\\prime }}+\\omega _{i,\\sigma }^-\\varphi _{i-1,\\sigma ^{\\prime \\prime };i,\\sigma ^{\\prime }}\\right).$ We can now turn to sojourn times.", "Consider a trajectory which begins in state $\\left(i,\\sigma \\right)$ .", "The probability of spending a total of $t$ time steps in a particular state $j$ , irrespective of the state the environment is in at that time, is then given by $q_t(j|i,\\sigma ) = \\sum _{\\sigma _{1}...\\sigma _{t-1}} \\varphi _{i,\\sigma ;j,\\sigma _{1}} r_{j,\\sigma _{1};\\sigma _{2}}r_{j,\\sigma _{2};\\sigma _{3}}...r_{j,\\sigma _{t-2};\\sigma _{t-1}}\\left(1-\\sum _{\\sigma _t}r_{j,\\sigma _{t-1};\\sigma _{t}}\\right) .$ The trajectory first has to reach state $j$ , as indicated by $\\varphi _{i,\\sigma ;j,\\sigma _{1}}$ , it then has to `return' $t$ times [in the sense of Eq.", "(REF )], indicated by the factors $r_{j,\\sigma _{1};\\sigma _{2}}r_{j,\\sigma _{2};\\sigma _{3}}...r_{j,\\sigma _{t-2};\\sigma _{t-1}}$ , and it must then not return to $j$ again, see the factor $1-\\sum _{\\sigma _t}r_{j,\\sigma _{t-1};\\sigma _{t}}$ .", "This can be written in a more compact matrix notation $\\underline{q_t}(j|i) = \\underline{\\underline{\\varphi _{ij}}}\\left(\\underline{\\underline{r_{j}}}\\right)^{t-1}\\underline{x_j},$ where $\\left(\\underline{x_{j}}\\right)_{\\sigma } = 1 - \\sum _{\\sigma ^{\\prime }}r_{j,\\sigma ;\\sigma ^{\\prime }} .$ The unconditional sojourn time is then the first moment of the distribution over $t$ defined by $q_t(j|i,\\sigma )$ , $t_{i,\\sigma ;j} = \\sum _{t=1}^\\infty tq_{t}\\left(j|i\\sigma \\right)=\\left(\\underline{\\underline{\\varphi _{ij}}}\\left[\\sum _{t=1}^\\infty t\\left(\\underline{\\underline{r_{j}}}\\right)^{t-1}\\right]\\underline{x_j}\\right)_{\\sigma }.$ Letting $\\underline{\\underline{{\\rm 1\\!\\!I}}} - \\underline{\\underline{r_{j}}} = \\underline{\\underline{S_{j}}} $ , one can evaluate the series to find $t_{i,\\sigma ;j} = \\left(\\underline{\\underline{\\varphi _{ij}}}\\left(\\underline{\\underline{S_{j}^{-1}}}\\right)^2\\underline{x_j}\\right)_{\\sigma }.$ Therefore, one can calculate the unconditional Sojourn times once the probabilities $\\underline{\\underline{\\varphi _{ij}}}$ have been obtained as described above." ], [ "Conditional sojourn times", "The conditional Sojourn times can be calculated in a similar way.", "We introduce the following shorthand $q_{t}^*\\left(j|i,\\sigma \\right) = \\mbox{Prob}\\left(\\begin{array}{p{5cm}|p{5cm}} The population spends exactly t steps at j before absorption.", "& The starting point is (i, \\sigma ), and the mutant reaches fixation.\\end{array}\\right) .$ Using Bayes' theorem we have $q_{t}^*\\left(j|i,\\sigma \\right) = \\frac{\\mbox{Prob}\\left(\\mbox{spends $t$ steps at $j$ and reaches fixation}~ |~ \\mbox{starts at $(i, \\sigma )$}\\right)}{\\mbox{Prob}\\left(\\mbox{reaches fixation}~ |~ \\mbox{starts at $(i, \\sigma )$}\\right)} .$ We also note that $\\mbox{Prob}\\left(\\mbox{reaches fixation}~ |~ \\mbox{starts at $(i,\\sigma )$}\\right) = \\varphi _{i,\\sigma } = \\sum _{\\sigma ^{\\prime }} \\varphi _{i,\\sigma ; N , \\sigma ^{\\prime }} .$ Hence, similar to the unconditional case, $q_{t}^*\\left(j|i,\\sigma \\right) \\sum _{\\sigma ^{\\prime }} \\varphi _{i,\\sigma ; N , \\sigma ^{\\prime }}= \\sum _{\\sigma _{1}...\\sigma _{t-1}} \\varphi _{i,\\sigma ;j,\\sigma _{1}} r_{j,\\sigma _{1};\\sigma _{2}}r_{j,\\sigma _{2};\\sigma _{3}}...r_{j,\\sigma _{t-2};\\sigma _{t-1}}\\left(\\underline{y_{j}}\\right)_{\\sigma _{t-1}},$ where $\\left(\\underline{y_{j}}\\right)_{\\sigma } = \\omega _{j, \\sigma }^+ \\sum _{\\sigma _{t}}\\mu _{\\sigma \\rightarrow \\sigma _{t}}\\left(1 - \\sum _{\\sigma ^{\\prime }} \\varphi _{j+1,\\sigma _{t} ; j, \\sigma ^{\\prime }}\\right).$ Using more compact notation we have $q_{t}^*\\left(j|i,\\sigma \\right) = \\frac{\\left(\\underline{\\underline{\\varphi _{ij}}}\\left(\\underline{\\underline{r_{j}}}\\right)^{t-1}\\underline{y_j},\\right)}{\\displaystyle \\sum _{\\sigma ^{\\prime }} \\varphi _{i,\\sigma ; N , \\sigma ^{\\prime }}},$ Using a procedure similar to the unconditional case, we obtain $t_{i,\\sigma ;j}^* = \\sum _{t=1}^\\infty tq_{t}^*\\left(j|i,\\sigma \\right) = \\frac{\\left(\\underline{\\underline{\\varphi _{ij}}}\\left(\\underline{\\underline{S_{j}^{-1}}}\\right)^2\\underline{y_j}\\right)_{\\sigma }}{\\displaystyle \\sum _{\\sigma ^{\\prime }} \\varphi _{i,\\sigma ; N , \\sigma ^{\\prime }}} .$ These conditional Sojourn times can also be calculated, given the complete set of probabilities $\\varphi _{i,\\sigma ;j,\\sigma ^{\\prime }}$ ." ] ]
1612.05530
[ [ "Cluster Realization of $U_q(\\mathfrak{g})$ and Factorization of the\n Universal $R$-Matrix" ], [ "Abstract For each simple Lie algebra $\\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\\mathfrak{g})$ into certain quantum torus algebra $D_\\mathfrak{g}$ via the positive representations of split real quantum group.", "The quivers corresponding to $D_\\mathfrak{g}$ is obtained from amalgamation of two basic quivers, where each of them is mutation equivalent to the cluster structure of the moduli space of framed $G$-local system on a disk with 3 marked points when $G$ is of classical type.", "We derive a factorization of the universal $R$-matrix into quantum dilogarithms of cluster variables, and show that conjugation by the $R$-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk." ], [ "Introduction", "For any finite dimensional complex simple Lie algebra $\\mathfrak {g}$ , Drinfeld [5] and Jimbo [24] associated to it a remarkable Hopf algebra $\\mathcal {U}_q(\\mathfrak {g})$ known as quantum group, which is certain deformation of the universal enveloping algebra.", "To better understand the structure of $\\mathcal {U}_q(\\mathfrak {g})$ , a very natural problem is to find certain embeddings into simpler algebras.", "In [13], [14], through the generalization of Gelfand-Tsetlin representations, embeddings of the whole quantum group $\\mathcal {U}_q(\\mathfrak {g})$ into certain rational functions $\\mathbb {C}(\\mathbf {T}_q)$ of quantum torus have been found.", "Another well-known result is provided by Feigin's homomorphism [1], [37] which embeds the positive Borel part $\\mathcal {U}_q(\\mathfrak {b}_+)$ of $\\mathcal {U}_q(\\mathfrak {g})$ directly into a quantum torus algebra $\\mathbb {C}[\\mathbf {T}_q]$ .", "However, the explicit extension of Feigin's map to the whole quantum group, i.e.", "given by polynomial embeddings of $\\mathcal {U}_q(\\mathfrak {g})$ into certain quantum torus algebra, appears to be much more subtle.", "While the case for $\\mathcal {U}_q(\\mathfrak {sl}_n)$ is known previously [31], the case for general types has only been solved recently with the introduction of positive representations of split real quantum groups." ], [ "Quantum group embeddings via positive representations", "The notion of positive representations was introduced in [11] as a new research program devoted to the representation theory of split real quantum groups $\\mathcal {U}_{q}(\\mathfrak {g}_\\mathbb {R})$ and its modular double $\\mathcal {U}_{q\\widetilde{q}}(\\mathfrak {g}_\\mathbb {R})$ introduced in [6], [7], in the regime where $|q|=1$ .", "It is motivated by the simplest case $\\mathcal {U}_{q\\widetilde{q}}(\\mathfrak {sl}(2,\\mathbb {R}))$ which has been studied extensively by Teschner et al.", "[3], [35], [36] from the point of view of non-compact conformal field theory.", "Explicit construction of the positive representations $\\mathcal {P}_\\lambda $ of $\\mathcal {U}_{q\\widetilde{q}}(\\mathfrak {g}_\\mathbb {R})$ associated to a simple Lie algebra $\\mathfrak {g}$ has been obtained for the simply-laced case in [11], [17], [18] and non-simply-laced case in [20], where the generators of the quantum groups are realized by positive essentially self-adjoint operators acting on certain Hilbert spaces.", "As a consequence of the construction, if one forgets the real structure of such representations, one can express the generators in terms of Laurent polynomials of certain $q$ -commuting variables, and we obtain a full embedding of quantum groups $\\mathcal {U}_q(\\mathfrak {g})\\hookrightarrow \\mathbb {C}[\\mathbf {T}_q]$ into certain quantum torus algebra, thus solving the long-standing problem of generalizing the Feigin's homomorphism.", "The construction of the positive representations of $\\mathcal {U}_q(\\mathfrak {g}_\\mathbb {R})$ relies heavily on Lusztig's total positivity of reductive groups and is closely related to the structure of the quantum principal affine space $\\mathcal {O}_q[G/N]$ .", "Its harmonic analysis on $L^2(G_{q\\widetilde{q}}^+(\\mathbb {R}))$ through the Gauss-Lusztig decomposition [19], [22] also involves the structure of the coordinate ring $\\mathcal {O}_q[G]$ and the double Bruhat cell $\\mathcal {O}_q[G^{w_0,w_0}]$ .", "Therefore the theory of positive representations is long considered to have a strong connection to the theory of quantum cluster algebra [2] in which these objects represent [12], [15].", "In particular both theories share a similar positivity phenomenon under some mutation operations, where for example the generators of $\\mathcal {U}_q(\\mathfrak {g}_\\mathbb {R})$ are always represented as Laurent polynomials of positive operators with positive $q$ -integral coefficients, thus naturally acting on $\\mathcal {P}_\\lambda $ as positive self-adjoint operators.", "In a recent work of [39], Schrader and Shapiro found explicitly an embedding of $\\mathcal {U}_q(\\mathfrak {sl}_n)$ into certain quantum torus algebra $\\mathcal {D}_{\\mathfrak {sl}_n}$ , which arises from quantizing the Fock and Goncharov's construction of the cluster coordinates on the moduli spaces of framed $PGL_n$ -local systems on the punctured disk with two marked points, where the structure can be nicely summarized into certain quiver diagrams given by $n$ -triangulations [9].", "It turns out that their construction fit nicely into the framework of positive representations, and one can carry over the explicit constructions of $\\mathcal {P}_\\lambda $ and obtain a new quantum torus algebra embedding for arbitrary type of $\\mathcal {U}_q(\\mathfrak {g})$ .", "Our first main result (Theorem REF ) states that there is an embedding of algebra $\\mathcal {U}_q(\\mathfrak {g})\\hookrightarrow \\mathcal {D}_\\mathfrak {g}/\\sim $ into a quantum torus algebra (modulo some central elements), which can be represented by some quiver diagrams associated to $\\mathcal {D}_\\mathfrak {g}$ .", "The embeddings of the generators of $\\mathcal {U}_q(\\mathfrak {g})$ can then be expressed explicitly by certain paths on the quiver.", "In particular, the previously rather ad hoc explicit expressions, especially in the exceptional types, can now be visualized in a very simple manner (see Figure REF - Figure REF ).", "Furthermore, a change of words of the reduced expression of the longest element $w_0\\in W$ of the Weyl group, which induces unitary equivalences of the positive representations, correspond to certain quiver mutations and hence quantum cluster mutations of $\\mathcal {D}_\\mathfrak {g}$ .", "This makes the connection between positive representations and the theory of (quantum) cluster algebra much more explicit.", "It strongly suggests that in fact we have an embedding into the global functions on the corresponding cluster $\\mathcal {X}$ -variety $\\mathcal {U}_q(\\mathfrak {g})\\hookrightarrow \\bigcap _\\mathbf {i}\\mathcal {D}_\\mathfrak {g}^\\mathbf {i}/\\sim $ associated to all the seed equivalence class of $\\mathcal {D}_\\mathfrak {g}$ , where the generators of $\\mathcal {U}_q(\\mathfrak {g})$ stay polynomial in any cluster.", "However this requires a separate proof and will be considered in future works.", "Finally, the proof of injectivity of the embedding in type $A_n$ by [39] involves explicitly some combinatorial hive-type conditions related to the work of Knutson-Tao [34].", "It will be interesting to see the analogues of such combinatorics coming from other types of quantum groups from our construction using positive representations." ], [ "Basic quivers and framed $G$ -local systems", "The quiver corresponding to $\\mathcal {D}_\\mathfrak {g}$ is naturally associated to the triangulation of a punctured disk with two marked points.", "It can be constructed by gluing (amalgamating) two copies of “basic quivers\" $Q$ associated to a triangle.", "It turns out that the basic quiver is mutation equivalent to the quiver giving a (classical) cluster algebra structure on the moduli space of framed $G$ -local system, or the configuration space $Conf_3\\mathcal {A}_G$ of triples of principal flags, recently discovered for classical types [25] and type $G_2$ [26].", "Both constructions require the use of elementary quivers associated to simple reflections of the longest element $w_0$ .", "In particular, by providing a different construction than the ones in [25], [26], the description of $Q$ in this paper may allow us to construct quantum higher Teichmüller theory in full generality in a representation theoretical setting of quantum groups, where the quiver describes the coordinates of the framed $G$ -local system and their Poisson structure, and hence also the quantization of these coordinates.", "The uniqueness of $Q$ can also potentially be used to solve the series of conjectures proposed in [26].", "We also expect that such geometric description of the basic quivers will let us better understand the geometric construction of another quantum group embedding via the Grothendieck-Springer resolution proposed by [38], which turns out to be quite hard to write down explicitly.", "The basic quiver plays an important role in the description of the universal $R$ -matrix realized as half-Dehn twist, which we will described next." ], [ "Universal $R$ -matrix as half-Dehn twist", "Using the quantum cluster embedding (REF ), our second main result (Theorem REF and Corollary REF ) of the paper gives the factorization of the reduced $R$ matrix into products of quantum dilogarithms such that the arguments are given by monomials of the quantum cluster variables $X_i\\in \\mathcal {D}_\\mathfrak {g}$ associated to the chosen reduced expression of $w_0$ , and the factorization is invariant under the change of reduced expression.", "This result generalizes the factorization of [39] in type $A_n$ for a specific choice of $w_0$ , and the earlier well-known result for $\\mathcal {U}_q(\\mathfrak {sl}_2)$ by Faddeev [7].", "It is different from the usual multiplicative formula discovered independently by Kirillov-Reshetikhin [32] and Levendorskii-Soibelman [27], [28], which is further extended to the superalgebra case in [33].", "Since each factor is expressed in terms of quantum cluster variables, in fact it can be viewed as a sequence of quiver mutations on two copies of the $\\mathcal {D}_\\mathfrak {g}$ -quiver associated to a disk of two punctures and two marked points.", "Our final main result of the paper (Theorem REF ) shows that the conjugation by the universal $R$ -matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in the twice punctured disk.", "This factorization can also be split into 4 blocks such that each block corresponds to a flip of triangulations of the twice punctured disk, where the basic quiver $Q$ associated to each triangle is being mutated to a different configuration.", "In the case of $\\mathcal {U}_q(\\mathfrak {sl}_2)$ , such identification of the factorization of the $R$ -matrix appears in quantum Teichmüller theory [30] as an element of the mapping class group, and the corresponding factorization is also used to re-derive Kashaev's knot invariant [16].", "For general Hopf algebra $\\mathcal {A}$ , Kashaev has constructed an embedding $\\phi :\\mathcal {D}(\\mathcal {A})\\longrightarrow H(A)\\otimes H(A)^{op}$ of the Drinfeld's double $\\mathcal {D}(\\mathcal {A})$ into a tensor square of the Heisenberg double $H(\\mathcal {A})$ , and the image of the universal $R$ -matrix can be similarly decomposed into a product of 4 variants of the $S$ -tensors [29]: $\\phi ^{\\otimes 2}(\\mathcal {R})=S_{14}^{\\prime \\prime }S_{13}\\widetilde{S}_{24}S_{23}^{\\prime }\\in (H(A)\\otimes H(A)^{op})^{\\otimes 2}.$ This has been utilized for example to construct new quantum invariant for “colored triangulations\" of topological spaces recently [40].", "Although the two factorizations are realized on different tensor spaces, we believe there is a strong connection between the two different factorizations, where $\\mathcal {A}$ is identified with the Borel part of $\\mathcal {U}_q(\\mathfrak {g})$ , and it will be interesting to find an explicit relationship between them.", "We hope that the factorization in this paper opens up a new class of invariants which can be explicitly constructed." ], [ "Generalization to the split real setting", "The embeddings of quantum groups as well as the factorization of $R$ -matrix in this paper is treated in a formal algebraic setting.", "However, as the construction comes from the positive representations of split real quantum groups, it is natural to conclude that the theory developed in this paper can be generalized to the split real setting.", "For example, the monomials of the embedding constructed out of the quantum cluster variables $X_i\\in \\mathcal {D}_\\mathfrak {g}$ are all manifestly positive self-adjoint if we put back in the split real form.", "In particular, throughout the paper, we use the correspondence (see Remark REF for more details): $\\Psi ^q(x)\\sim g_b^*(x)$ to identify both the compact and non-compact quantum dilogarithm functions.", "This suggests that in fact all the quiver mutations and $R$ -matrix decomposition work in the split real setting.", "In this case the non-compact version is well-defined as the quantum cluster variables are positive self-adjoint, therefore the formal power series manipulations can be replaced by actions of unitary operators.", "Furthermore, Faddeev's modular double can be easily recovered by applying the transcendental relations [7], [11] to the quantum cluster variables: $\\widetilde{X_i}:=X_i^{\\frac{1}{b_i}},$ and the simple analytic version of the Langlands duality [20] interchanging the long and short roots can then be easily recovered as well (this is made more explicit in the quiver diagrams of type $B_n, C_n$ and $G_2$ ).", "The perspectives of the applications of such phenomenon in the split real case look very promising, and will be explored elsewhere." ], [ "Outline of the paper", "The paper is organized as follows.", "In Section , we fix the convention used throughout the paper and recall the definition of quantum group $\\mathcal {U}_q(\\mathfrak {g})$ .", "In Section , we recall the definition and properties of quantum torus algebra, the associated quivers, and their cluster structure.", "In Section , we recall the construction of the positive representations of split real quantum groups, and define the new quantum torus algebra $\\mathcal {D}_\\mathfrak {g}$ in which $\\mathcal {U}_q(\\mathfrak {g})$ embeds.", "In Section we construct explicitly the $\\mathcal {D}_\\mathfrak {g}$ -quiver associated to the algebra $\\mathcal {D}_\\mathfrak {g}$ using the elementary quivers, and in Section we give an explicit embedding of $\\mathcal {U}_q(\\mathfrak {g})$ for all each simple types of $\\mathfrak {g}$ , where the generators are represented by certain paths on the quivers.", "In Section we discuss the quiver mutations associated to a change of reduced expression of the longest element $w_0\\in W$ of the Weyl group, and we use this to show in Section that the basic quiver associated to a triangle of a triangulation is uniquely defined.", "In Section we recall the definition of universal $R$ -matrix, and using the quantum group embedding, we give a factorization formula of $R$ , which is proved in Section .", "Finally in Section , we show that the factorization of $R$ can be realized as half-Dehn twist of a twice punctured disk with two marked points, where the basic quiver associated to each triangle is mutated to certain new configurations, and we give explicitly its sequence of mutations." ], [ "Notations and definitions of $\\mathcal {U}_q(\\mathfrak {g})$", "In order to fix the convention we use throughout the paper, we follow the notations used in [20], [21] for the root systems and recall the definition of the quantum group $\\mathcal {U}_q(\\mathfrak {g})$ , where $\\mathfrak {g}$ is of general type [4], as well as the Drinfeld's double $\\mathfrak {D}_\\mathfrak {g}$ of the Borel part.", "Definition 2.1 Let $I$ denote the set of nodes of the Dynkin diagram of $\\mathfrak {g}$ where $|I|=n=rank(\\mathfrak {g}).$ Let $w_0\\in W$ be the longest element of the Weyl group of $\\mathfrak {g}$ , and let $N:=l(w_0)=\\dim \\mathfrak {n}_-$ be its length, which is also the dimension of the unipotent subgroup $\\mathfrak {n}_-$ of $\\mathfrak {g}$ .", "We call a sequence $\\mathbf {i}=(i_1,...,i_N)\\in I^N$ a reduced word of $w_0$ if $w_0=s_{i_1}...s_{i_N}$ is a reduced expression, where $s_{i_k}$ are the simple reflections of the root spaceWe will sometimes omit the commas in $\\mathbf {i}$ for typesetting purpose.. We denote by $\\mathfrak {R}$ the set of all reduced words of $w_0$ .", "We let $n_i^\\mathbf {i}\\in \\mathbb {Z}_{>0}$ be the number of letters $i$ appearing in $\\mathbf {i}$ .", "We will write $n_i:=n_i^\\mathbf {i}$ if no confusion arises.", "If we have another reduced word $\\mathbf {i}^{\\prime }\\in \\mathfrak {R}$ , we will sometimes write $\\mathbf {i}^{\\prime }=(i_1^{\\prime },...,i_N^{\\prime })$ and $n_i^{\\prime }:=n_i^{\\mathbf {i}^{\\prime }}$ .", "Clearly we have $\\sum _{i=1}^n n_i=N.$ Definition 2.2 We index the nodes of the Dynkin diagrams as follow, where black nodes correspond to short roots, and white nodes correspond to long roots.", "Type $A_n$ : [scale=.4] [xshift=0 cm,thick] (0 cm, 0) circle (.3 cm); in 1,...,5 [xshift=cm,thick] (cm,0) circle (.3cm); [dotted,thick] (8.3 cm,0) – +(1.4 cm,0); in 0.15,...,3.15 [xshift=cm,thick] (cm,0) – +(1.4 cm,0); in 1,...,5 t (2*-2,-1) $$ ; t (10,-1)$n$ ; Type $B_n$ : [scale=.4] [xshift=0 cm,thick,fill=black] (0 cm, 0) circle (.3 cm); in 1,...,5 [xshift=cm,thick] (cm,0) circle (.3cm); [dotted,thick] (8.3 cm,0) – +(1.4 cm,0); in 1.15,...,3.15 [xshift=cm,thick] (cm,0) – +(1.4 cm,0); [thick] (0.3 cm, .1 cm) – +(1.4 cm,0); [thick] (0.3 cm, -.1 cm) – +(1.4 cm,0); in 1,...,5 t (2*-2,-1) $$ ; t (10,-1)$n$ ; Type $C_n$ : [scale=.4] [xshift=0 cm,thick] (0 cm, 0) circle (.3 cm); in 1,...,5 [xshift=cm,thick,fill=black] (cm,0) circle (.3cm); [dotted,thick] (8.3 cm,0) – +(1.4 cm,0); in 1.15,...,3.15 [xshift=cm,thick] (cm,0) – +(1.4 cm,0); [thick] (0.3 cm, .1 cm) – +(1.4 cm,0); [thick] (0.3 cm, -.1 cm) – +(1.4 cm,0); in 1,...,5 t (2*-2,-1) $$ ; t (10,-1)$n$ ; Type $D_n$ : [scale=.4] [xshift=0 cm,thick] (0 cm, 1) circle (.3 cm); [xshift=0 cm,thick] (0 cm, -1) circle (.3 cm); in 1,...,5 [xshift=cm,thick] (cm,0) circle (.3cm); [dotted,thick] (8.3 cm,0) – +(1.4 cm,0); [xshift=0.25 cm] (0 cm,1) – +(1.4 cm,-1); [xshift=0.25 cm] (0 cm,-1) – +(1.4 cm,1); in 1.15,...,3.15 [xshift=cm,thick] (cm,0) – +(1.4 cm,0); in 2,...,5 t (2*-2,-1) $$ ; t (10,-1)$n-1$ ; t (-1,-1)0; t (-1,1)1; Type $E_n$ : [scale=.4] [xshift=0 cm,thick] (0 cm, 0) circle (.3 cm); in 1,...,4 [xshift=cm,thick] (cm,0) circle (.3cm); in 0.15,...,2.15 [xshift=cm,thick] (cm,0) – +(1.4 cm,0); [xshift=3.15cm, dotted, thick] (3.15cm,0) –+(1.4 cm,0); in 1,...,4 t (2*-2,1) $$ ; t (8,1) $n-1$ ; [xshift=0 cm,thick] (4 cm, -2) circle (.3 cm); [xshift=0 cm] (4 cm,-0.25) – +(0 cm,-1.5); t (4,-3)0; Type $F_4$ : [scale=.4] [thick] (-2 cm ,0) circle (.3 cm); t (-2,-1) 1; [thick] (0 ,0) circle (.3 cm); t (0,-1) 2; [thick,fill=black] (2 cm,0) circle (.3 cm); t (2,-1) 3; [thick,fill=black] (4 cm,0) circle (.3 cm); t (4,-1) 4; [thick] (15: 3mm) – +(1.5 cm, 0); [xshift=-2 cm,thick] (0: 3 mm) – +(1.4 cm, 0); [thick] (-15: 3 mm) – +(1.5 cm, 0); [xshift=2 cm,thick] (0: 3 mm) – +(1.4 cm, 0); Type $G_2$ : [scale=.4] [thick] (0 ,0) circle (.3 cm); t (0,-1) 1; [thick,fill=black] (2 cm,0) circle (.3 cm); t (2,-1) 2; [thick] (30: 3mm) – +(1.5 cm, 0); [thick] (0: 3 mm) – +(1.5 cm, 0); [thick] (-30: 3 mm) – +(1.5 cm, 0); Definition 2.3 Let $q$ be a formal parameter.", "Let $\\lbrace \\alpha _i\\rbrace _{i\\in I}$ be the set of positive simple roots.", "Let $(-,-)$ be the inner product of the root lattice, and we define $a_{ij}:=\\frac{2(\\alpha _i,\\alpha _j)}{(\\alpha _i,\\alpha _i)},$ such that $A:=(a_{ij})$ is the Cartan matrix.", "We normalize $(-,-)$ as follows: we choose the symmetrization factors (also called the multipliers) $d_i:=\\frac{1}{2}(\\alpha _i,\\alpha _i)=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}1&\\mbox{$i$ is long root or in the simply-laced case,}\\\\\\frac{1}{2}&\\mbox{$i$ is short root in type $B,C,F$,}\\\\\\frac{1}{3}&\\mbox{$i$ is short root in type $G_2$,} \\\\ \\end{array}\\right.$ and $(\\alpha _i,\\alpha _j)=-1$ when $i,j$ are adjacent in the Dynkin diagram, such that $d_ia_{ij}=d_ja_{ji}.$ We then define $q_i:=q^{d_i},$ which we will also write as $q_l&:=q,\\\\q_s&:=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}q^{\\frac{1}{2}}&\\mbox{$\\mathfrak {g}$ is of type $B_n, C_n, F_4$},\\\\q^{\\frac{1}{3}}&\\mbox{$\\mathfrak {g}$ is of type $G_2$}, \\\\ \\end{array}\\right.$ for the $q$ parameters corresponding to long and short roots respectively.", "Definition 2.4 Let $A=(a_{ij})$ denote the Cartan matrix.", "We define $\\mathfrak {D}_\\mathfrak {g}$ to be the $\\mathbb {C}(q_s)$ -algebra generated by the elements $\\lbrace E_i, F_i,K_i^{\\pm 1}, K_i^{\\prime \\pm 1}| i\\in I\\rbrace $ subject to the following relations (we will omit the relations involving $K_i^{-1}, {K_i^{\\prime }}^{-1}$ below for simplicity): $K_iE_j&=q_i^{a_{ij}}E_jK_i, &K_iF_j&=q_i^{-a_{ij}}F_jK_i,\\\\K_i^{\\prime }E_j&=q_i^{-a_{ij}}E_jK_i^{\\prime }, &K_i^{\\prime }F_j&=q_i^{a_{ij}}F_jK_i^{\\prime },\\\\K_iK_j&=K_jK_i, &K_i^{\\prime }K_j^{\\prime }&=K_j^{\\prime }K_i^{\\prime }, &K_iK_j^{\\prime } = K_j^{\\prime }K_i,\\\\&&[E_i,F_j]&= \\delta _{ij}\\frac{K_i-K_i^{\\prime }}{q_i-q_i^{-1}},$ together with the Serre relations for $i\\ne j$ : $\\sum _{k=0}^{1-a_{ij}}(-1)^k\\frac{[1-a_{ij}]_{q_i}!}{[1-a_{ij}-k]_{q_i}![k]_{q_i}!", "}E_i^{k}E_jE_i^{1-a_{ij}-k}&=&0,\\\\\\sum _{k=0}^{1-a_{ij}}(-1)^k\\frac{[1-a_{ij}]_{q_i}!}{[1-a_{ij}-k]_{q_i}![k]_{q_i}!", "}F_i^{k}F_jF_i^{1-a_{ij}-k}&=&0,$ where $[k]_q:=\\frac{q^k-q^{-k}}{q-q^{-1}}$ is the $q$ -number and $[n]_q!", ":=\\prod _{k=1}^n [k]_q$ the $q$ -factorial.", "The algebra $\\mathfrak {D}_\\mathfrak {g}$ is a Hopf algebra with comultiplication $\\Delta (E_i)=&1\\otimes E_i+E_i\\otimes K_i,&\\Delta (K_i)&=K_i\\otimes K_i,\\\\\\Delta (F_i)=&F_i\\otimes 1+K_i^{\\prime }\\otimes F_i,&\\Delta (K_i^{\\prime })&=K_i^{\\prime }\\otimes K_i^{\\prime },$ the counit $\\epsilon (E_i)&=\\epsilon (F_i)=0, & \\epsilon (K_i)&=\\epsilon (K_i^{\\prime })=1,\\\\$ and antipode $S(E_i)&=-K_i^{-1}E_i, &S(K_i)&=K_i^{-1},\\\\S(F_i)&=-F_iK_i, &S(K_i^{\\prime })&=(K_i^{\\prime })^{-1}.$ Definition 2.5 The quantum group $\\mathcal {U}_q(\\mathfrak {g})$ is defined as the quotient $\\mathcal {U}_g(\\mathfrak {g}):=\\mathfrak {D}_\\mathfrak {g}/\\langle K_iK_i^{\\prime }=1|i\\in I\\rangle ,$ and it inherits a well-defined Hopf algebra structure from $\\mathfrak {D}_\\mathfrak {g}$ .", "Remark 2.6 $\\mathfrak {D}_\\mathfrak {g}$ is the Drinfeld's double of the quantum Borel subalgebra $\\mathcal {U}_q(\\mathfrak {b})$ generated by $E_i$ and $K_i$ .", "Definition 2.7 We define the rescaled generators $\\mathbf {e}_i :=\\left(\\frac{\\sqrt{-1}}{q_i-q_i^{-1}}\\right)^{-1}E_i,&&\\mathbf {f}_i :=\\left(\\frac{\\sqrt{-1}}{q_i-q_i^{-1}}\\right)^{-1}F_i.$ By abuse of notation, we will also denote by $\\mathfrak {D}_\\mathfrak {g}$ the $\\mathbb {C}(q_s)$ -algebra generated by $\\lbrace \\mathbf {e}_i, \\mathbf {f}_i, K_i, K_i^{\\prime }|i\\in I\\rbrace $ and the corresponding quotient by $\\mathcal {U}_q(\\mathfrak {g})$ .", "The generators satisfy all the defining relations above except (REF ) which is modified to be $[\\mathbf {e}_i, \\mathbf {f}_j]=\\delta _{ij} (q_i-q_i^{-1})(K_i^{\\prime }-K_i).$" ], [ "Quantum cluster $\\mathcal {X}$ -tori", "We recall the definition of the quantum cluster $\\mathcal {X}$ -tori following [9], [39] and their properties that are needed, as well as some notations and modification that fit the needs of this paper." ], [ "Quantum torus algebra and quivers", "Definition 3.1 (Quantum torus algebra) A seed $\\mathbf {i}$ is a triple $(I, I_0, B, D)$ where $I$ is a finite set, $I_0\\subset I$ is a subset called the frozen subset, $B=(b_{ij})_{i,j\\in I}$ a skew-symmetrizable $\\mathbb {Q}$ -valued matrix called the exchange matrix, and $D=diag(d_i)_{i\\in I}$ is a diagonal matrix such that $DB=B^TD$ is skew-symmetric.", "Let $q$ be a formal parameter.", "We define the quantum torus algebra $\\mathcal {X}_{\\mathbf {i}}$ associated to the seed $\\mathbf {i}$ to be an associative algebra over $\\mathbb {C}(q^d)$ , where $d=\\min _{i\\in I}(d_i)$ , defined by generators $X_i^{\\pm 1}, i\\in I$ subject to the relations $X_iX_j=q_i^{-2b_{ij}}X_jX_i,\\;\\;\\;\\;\\;\\;i,j\\in I$ where $q_i:=q^{d_i}$ .", "The generators $X_i$ are called the quantum cluster variables, and they are said to be frozen if $i\\in I_0$ .", "We call $d_i$ the multipliers of the variables $X_i$ .", "We denote by $\\mathbf {T}_\\mathbf {i}$ the non-commutative field of fraction of $\\mathcal {X}_\\mathbf {i}$ .", "The structure of the quantum torus algebra $\\mathcal {X}_{\\mathbf {i}}$ associated to a seed $\\mathbf {i}$ can be conveniently encoded in a quiver: Definition 3.2 (Quiver associated to $\\mathbf {i}$ ) We associate to each seed $\\mathbf {i}$ a generalized quiver $Q^\\mathbf {i}=(Q_0,w)$ with vertices $Q_0$ labeled by $I$ , and for each pair $i,j\\in Q_0$ a weight $w_{ij}:=d_ib_{ij}=-w_{ji}.$ We will draw arrows from $i\\xrightarrow{} j$ if $w_{ij}>0$ .", "We will call an isomorphism $\\pi :S\\simeq Q_0$ from a finite set $S$ an external label of the quiver $Q$ .", "We will use squares to denote frozen nodes $i\\in I_0$ and circles otherwise.", "In the sequel, when $q_i=q_s$ or $q_l$ given by Definition REF , we will distinguish the arrows by thick or thin arrows instead of writing the weights.", "We will also use dashed lines to denote arrows with half the weights, which only occurs between frozen nodes.", "Figure: Arrows between nodes and their algebraic meaning.We introduce the following notations which will be useful throughout the paper: Definition 3.3 We denote by $X_{i_1^{m_1},...,i_n^{m_n}}:=q^C X_{i_1}^{m_1}... X_{i_n}^{m_n},$ where $C$ is the unique rational number such that $q^C X_{i_1}^{m_1}... X_{i_n}^{m_n}=q^{-C} X_{i_n}^{m_n}... X_{i_1}^{m_1}.$ Explicitly, if $X_iX_j=q^{c_{ij}}X_jX_i$ , then $C=-\\frac{1}{2}\\sum _{p>q} m_p m_q c_{i_p}c_{i_q}.$ If we introduce a $*$ -structure such that $q^*=q^{-1}$ and $X_i^*=X_i$ (and positive), then the expression $X_{i_1^{m_1},...,i_n^{m_n}}$ is also (positive) self-adjoint.", "We also denote by $X(i_1,..., i_n):=\\sum _{k=1}^n X_{i_1,..., i_k}.$ Definition 3.4 A permutation of a seed $\\sigma : \\mathbf {i}\\longrightarrow \\mathbf {i}^{\\prime }$ is a bijection $\\sigma :I\\longrightarrow I^{\\prime }$ such that $\\sigma (I_0)&=I_0^{\\prime },\\\\b^{\\prime }_{ij}&= b_{\\sigma (i)\\sigma (j)},\\\\d^{\\prime }_i&=d_{\\sigma (i)}.$ It induces an isomorphism $\\sigma ^*:\\mathbf {T}_{\\mathbf {i}^{\\prime }}\\longrightarrow \\mathbf {T}_{\\mathbf {i}}$ by $\\sigma ^*(\\widehat{X}_{\\sigma (i)}):=X_i,$ where $\\widehat{X}_{\\sigma (i)}$ denotes the quantum cluster variables of $\\mathbf {T}_{\\mathbf {i}^{\\prime }}$ ." ], [ "Quantum cluster mutation", "Next we define the cluster mutations of a seed and its quiver, and the quantum cluster mutations for the algebra.", "Definition 3.5 (Cluster mutation) Given a pair of seeds $\\mathbf {i}=(I,I_0,B, D)$ , $\\mathbf {i}^{\\prime }=(I^{\\prime },I_0^{\\prime },B^{\\prime },D^{\\prime })$ and an element $k\\in I\\setminus I_0$ , a cluster mutation in direction $k$ is an isomorphism $\\mu _k:\\mathbf {i}\\longrightarrow \\mathbf {i}^{\\prime }$ such that $\\mu _k(I_0)=I_0^{\\prime }$ , $b^{\\prime }_{\\mu _k(i),\\mu _k(j)} &= \\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}-b_{ij}&\\mbox{if $i=k$ or $j=k$},\\\\ b_{ij}+\\frac{b_{ik}|b_{kj}|+|b_{ik}|b_{kj}}{2}&\\mbox{otherwise}, \\\\ \\end{array}\\right.\\\\d^{\\prime }_{\\mu _k(i)}&=d_i.$ Then the quiver mutation $Q^\\mathbf {i}\\longrightarrow Q^{\\mathbf {i}^{\\prime }}$ corresponding to the mutation $\\mu _k$ can be performed by: (1) reverse all the arrows incident to the vertex $k$ ; (2) for each pair of arrows $i\\xrightarrow{} k$ and $k\\xrightarrow{}j$ associate the arrow $i\\xrightarrow{}j$ .", "(3) delete any arrows with weight $w_{ij}=0$ .", "Definition 3.6 (Quantum cluster mutation) The cluster mutation in direction $k$ , $\\mu _k:\\mathbf {i}\\longrightarrow \\mathbf {i}^{\\prime }$ , induces an isomorphism $\\mu _k^q:\\mathbf {T}_{\\mathbf {i}^{\\prime }}\\longrightarrow \\mathbf {T}_{\\mathbf {i}}$ called the quantum cluster mutation, defined by $\\mu _k^q(\\widehat{X}_i)=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}X_k^{-1}&\\mbox{if $i=k$},\\\\ \\displaystyle X_i\\prod _{r=1}^{|b_{ki}|}(1+q_i^{2r-1}X_k)&\\mbox{if $i\\ne k$ and $b_{ki}\\le 0$},\\\\\\displaystyle X_i\\prod _{r=1}^{b_{ki}}(1+q_i^{2r-1}X_k^{-1})^{-1}&\\mbox{if $i\\ne k$ and $b_{ki}\\ge 0$}, \\\\ \\end{array}\\right.$ where we denote by $\\widehat{X}_i$ the quantum cluster variables corresponding to $\\mathcal {X}_{\\mathbf {i}^{\\prime }}$ with exchange matrix $B^{\\prime }$ such that $b^{\\prime }_{ki}=-b_{ki}$ for every $i\\in I$ .", "The quantum cluster mutation $\\mu _k^q$ can be written as a composition of two homomorphisms $\\mu _k^q=\\mu _k^\\#\\circ \\mu _k^{\\prime },$ where $\\mu _k^{\\prime }:\\mathbf {T}_{\\mathbf {i}^{\\prime }}\\longrightarrow \\mathbf {T}_\\mathbf {i}$ is a monomial transformation defined by $\\mu _k^{\\prime }(\\widehat{X}_i):=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}X_k^{-1}&\\mbox{if $i=k$},\\\\ X_i&\\mbox{if $i\\ne k$ and $b_{ki}\\le 0$},\\\\ q_i^{b_{ik}b_{ki}}X_iX_k^{b_{ik}}&\\mbox{if $i\\ne k$ and $b_{ki}\\ge 0$}, \\\\ \\end{array}\\right.$ and $\\mu _k^\\#:\\mathbf {T}_\\mathbf {i}\\longrightarrow \\mathbf {T}_\\mathbf {i}$ is a conjugation by the quantum dilogarithm function $\\mu _k^\\#:=Ad_{\\Psi ^{q_k}(X_k)},$ where $\\Psi ^{q}:=\\prod _{r=0}^\\infty (1+q^{2r+1}x)^{-1}.$ In the remaining of the paper, however, we will use the notation $g_{b_k}(x)&:=\\Psi ^{q_k}(x)^{-1},\\\\g_{b_k}^*(x)&:=g_{b_k}^{-1}(x)=\\Psi ^{q_k}(x)$ instead, in accordance to the universal $R$ -operator formula given in [21].", "The various identities of $g_b(x)$ that are needed in this paper are summarized in Appendix .", "Remark 3.7 We remark that $g_b(x)$ , where $q=e^{\\pi \\sqrt{-1}b^2}$ , is the notation for the non-compact quantum dilogarithm, which plays a central role in the theory of positive representation, various quantum Teichmüller theories [9], [30] and non-rational conformal field theories [3], [35], [36].", "It is composed by two commuting copies, associated to the so-called Faddeev's modular double, of the compact quantum dilogarithm $\\Psi ^q(x)$ [7], [8], and it is a unitary operator when $x$ is positive self-adjoint.", "In this paper however, we are only interested in the formal algebraic theory, hence one may consider only the compact part and think of the correspondence $g_b(x)\\sim \\Psi ^q(x)^{-1}=\\prod _{r=0}^\\infty (1+q_k^{2r+1}x) = Exp_{q^{-2}}\\left(-\\frac{u}{q-q^{-1}}\\right),$ where $Exp_q(x)&:=\\sum _{k\\ge 0} \\frac{x^k}{(k)_q!", "},\\\\(k)_q&:=\\frac{1-q^k}{1-q}.$ The use of the notation $g_b(x)$ suggests that the theory of the current paper can be naturally applied to the case of the non-compact split real setting, where all the algebraic relations are satisfied, and naturally the positivity and self-adjointness of the operators are automatically taken care into account, which makes the choice extremely natural.", "The following version of the useful Lemma from [39] is rewritten in the notation of the current paper: Lemma 3.8 Let $\\mu _{i_1}, ... ,\\mu _{i_k}$ be a sequence of mutation, and denote the intermediate seeds by $\\mathbf {i}_j:=\\mu _{i_j}...\\mu _{i_1}(\\mathbf {i})$ .", "Then the induced quantum cluster mutation $\\mu _{i_1}^q ...\\mu _{i_k}^q: \\mathbf {T}_{\\mathbf {i}_k}\\longrightarrow \\mathbf {T}_{\\mathbf {i}}$ can be written as $\\mu _{i_1}^q ...\\mu _{i_k}^q=\\Phi _k\\circ M_k,$ where $M_k:\\mathbf {T}_{\\mathbf {i}_k}\\longrightarrow \\mathbf {T}_{\\mathbf {i}}$ and $\\Phi _k:\\mathbf {T}_\\mathbf {i}\\longrightarrow \\mathbf {T}_\\mathbf {i}$ with $M_k&:= \\mu _{i_1}^{\\prime }\\mu _{i_2}^{\\prime }... \\mu _{i_k}^{\\prime },\\\\\\Phi _k&:= Ad_{g_{b_{i_1}}^*(X_{i_1})}Ad_{g_{b_{i_2}}^*(\\mu _{i_1}^{\\prime }(X_{i_2}^{\\mathbf {i}_1}))}... Ad_{g_{b_{i_k}}^*(\\mu _{i_1}^{\\prime }...\\mu _{i_{k-1}}^{\\prime }(X_{i_k}^{\\mathbf {i}_{k-1}}))},$ and $X^{\\mathbf {i}}_i$ denotes the corresponding quantum cluster variables of the algebra $\\mathcal {X}_\\mathbf {i}$ ." ], [ "Amalgamation", "We also recall the procedure of amalgamation of two quivers [10]: Definition 3.9 Let $Q_1:=Q^{\\mathbf {i}_1}$ and $Q_2:=Q^{\\mathbf {i}_2}$ be a pair of quivers associated to the seed $\\mathbf {i}_1=(I^1,I_0^1,B^1,D^1), \\mathbf {i}_2=(I^2,I_0^2,B^2,D^2)$ and with edge weights $w^1, w^2$ respectively, and let $J_1\\subset I_0^1, J_2\\subset I_0^2$ be certain subsets of the frozen nodes of $Q_1$ and $Q_2$ respectively.", "Assume there exists a bijection $\\phi :J_1\\longrightarrow J_2$ such that $d_{\\phi (i)}=d_i$ for $i\\in J_1$ .", "Then the amalgamation of $Q_1$ and $Q_2$ along $\\phi $ is a new quiver $Q$ constructed as follows: (1) The vertices of $Q$ are given by $Q_1\\cup _\\phi Q_2$ by identifying vertices $i\\in Q_1$ and $\\phi (i)\\in Q_2$ and assigned with the same weight $d_i$ , (2) The frozen nodes of $Q$ are given by $(I_0^1\\setminus J_1)\\sqcup (I_0^2\\setminus J_2)$ , i.e.", "we “defroze\" the vertices that are glued.", "(3) The weights $w$ of the edges of $Q$ are given by $w_{ij}=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}0&\\mbox{if $i\\in I^k\\setminus J_k$ and $j\\in I^{2-k}\\setminus J_{2-k}$ for $k=1,2$},\\\\w^k_{ij}&\\mbox{if $i\\in I^k\\setminus J_k$ or $j\\in I^k\\setminus J_k$ for $k=1,2$},\\\\w^1_{ij}+w^2_{\\phi (i)\\phi (j)}&\\mbox{if $i,j\\in J_1$}.", "\\\\ \\end{array}\\right.$ Amalgamation of a pair of quiver induces an embedding $\\mathcal {X}\\longrightarrow \\mathcal {X}_1\\otimes \\mathcal {X}_2$ of the corresponding quantum cluster $\\mathcal {X}$ -tori by $X_i\\mapsto \\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}X_i\\otimes 1&\\mbox{if $i\\in Q_1\\setminus J_1$},\\\\1\\otimes X_i&\\mbox{if $i\\in Q_2\\setminus J_2$},\\\\X_i\\otimes X_{\\phi (i)}&\\mbox{otherwise}.", "\\\\ \\end{array}\\right.$ Visually it is just gluing two quivers together along the chosen frozen nodes, such that the weights of the corresponding arrows among those nodes are added." ], [ "From positive representations to quantum group embedding", "In this section, we introduce the notion of positive representations, and show that one can construct from it the quantum group embedding into certain quantum torus algebra." ], [ "Positive representations $\\mathcal {P}_\\lambda $ of {{formula:5d5a4322-59fd-4105-a8a8-7e29c6f08373}}", "In [11], [18], [20], a special class of representations called the positive representations is constructed for split real quantum groups $\\mathcal {U}_q(\\mathfrak {g}_\\mathbb {R})$ (and its modular double, which is not needed in this paper).", "Here $\\mathcal {U}_q(\\mathfrak {g}_\\mathbb {R})$ is defined to be $\\mathcal {U}_q(\\mathfrak {g})$ endowed with the star structure $\\mathbf {e}_i^*=\\mathbf {e}_i, \\;\\;\\;\\;\\;\\;\\mathbf {f}_i^*=\\mathbf {f}_i, \\;\\;\\;\\;\\;\\;K_i^*=K_i,$ and necessarily $|q_i|=1$ for every $i\\in I$ , whence we let $q_i=e^{\\pi \\sqrt{-1} b_i^2}\\in \\mathbb {C}$ for $b_i\\in \\mathbb {R}$ .", "We assume the $q_i$ 's are not root of unity for simplicity.", "Theorem 4.1 (Positive representations) There exists a family of irreducible representations $\\mathcal {P}_{\\lambda }$ of $\\mathcal {U}_{q}(\\mathfrak {g}_\\mathbb {R})$ parametrized by the $\\mathbb {R}_+$ -span of the cone of positive weights $\\lambda \\in P_\\mathbb {R}^+\\subset \\mathfrak {h}_\\mathbb {R}^*$ , or equivalently by $\\lambda :=(\\lambda _1,...,\\lambda _n)\\in \\mathbb {R}_+^n$ where $n=rank(\\mathfrak {g})$ , such that For each reduced word $\\mathbf {i}\\in \\mathfrak {R}$ , the generators $\\mathbf {e}_i,\\mathbf {f}_i,K_i$ are represented by positive essentially self-adjoint operators acting on $L^2(\\mathbb {R}^N)$ , Each generators can be represented by monomials generated by the positive operators $\\lbrace e^{\\pm \\pi b_i x_i}, e^{\\pm 2\\pi b_i p_i}\\rbrace _{i=1,..., N},$ where $p_i=\\frac{1}{2\\pi \\sqrt{-1}}\\frac{\\partial }{\\partial x_i}$ are the momentum operators such that $[p_i,x_i]=\\frac{1}{2\\pi \\sqrt{-1}}$ , and each monomials are positive essentially self-adjoint.", "There exists a unitary equivalence $\\Phi $ between positive representations corresponding to different reduced words, hence the representation does not depend on the choice of reduced expression of $w_0$ .", "In the theory of positive representations of split real quantum groups, the representation carries a real structure and the operators are represented by unbounded positive operators on certain Hilbert spaces.", "However in this paper, we will only be dealing with the representation formally, so all the generators and relations are treated on the algebraic level only.", "Hence if we define formally $X_i^{\\pm 1}=e^{\\pm \\pi b_i x_i},\\;\\;\\;\\;\\;\\;Y_i^{\\pm 1} = e^{\\pm 2\\pi b_i p_i},$ then algebraically we have for $i=1,..., N$ : $X_iY_i&=q_i Y_i X_i,\\\\X_iY_j&=Y_jX_i,\\;\\;\\;\\;\\;\\;i\\ne j.\\nonumber $ As a corollary, if we just consider the quantum torus algebra $\\mathbb {C}[\\mathbf {T}_q]$ generated by the elements $\\langle X_i^{\\pm 1}, Y_i^{\\pm 1}\\rangle _{i=1,...,N}$ subjected to (REF ), we obtain Corollary 4.2 The positive representations give an embedding of $\\mathcal {U}_q(\\mathfrak {g})$ into $\\mathbb {C}[\\mathbf {T}_q]$ , generalizing the Feigin's homomorphism $\\mathcal {U}_q(\\mathfrak {b})\\longrightarrow \\mathbb {C}[\\mathbf {T}_q]$ .", "Remark 4.3 In [18], [20], we showed that one can shift the generators $\\mathbf {e}_i, \\mathbf {f}_i$ by some appropriate $K_i$ factors such that the modified quantum group embeds into the “true” quantum torus algebra $\\mathbb {C}\\langle X_i^{\\pm 1}, Y_i^{\\pm 1}\\rangle $ with the relations $X_i Y_i=q_i^2 Y_i X_i$ instead." ], [ "Explicit construction of $\\mathcal {P}_\\lambda $", "The positive representations $\\mathcal {P}_\\lambda $ was computed explicitly for all types of $\\mathfrak {g}$ .", "Let us first recall some notations used in [18], [20].", "Definition 4.4 We denote by $p_u=\\frac{1}{2\\pi \\sqrt{-1}}\\frac{\\partial }{\\partial u}$ and $e(u)&:=e^{\\pi bu},\\;\\;\\;\\;\\;\\;[u]:=q^\\frac{1}{2}e(u)+q^{-\\frac{1}{2}}e(-u),$ so that whenever $[p,u]=\\frac{1}{2\\pi \\sqrt{-1}}$ , $[u]e(-2p)&:=(q^\\frac{1}{2}e^{\\pi bu}+q^{-\\frac{1}{2}}e^{-\\pi bu})e^{-2\\pi bp} \\\\&= e^{\\pi bu-2\\pi b p}+e^{-2\\pi b u-2\\pi b p}\\\\&= e(u-2p)+e(-u-2p)$ is self-adjoint.", "Definition 4.5 By abuse of notation, we denote by $[u_s+u_l]e(-2p_s-2p_l):=e^{\\pi b_s(-u_s-2p_s)+\\pi b_l(-u_l-2p_l)}+e^{\\pi b_s(u_s-2p_s)+\\pi b_l(u_l-2p_l)},$ where $u_s$ (resp.", "$u_l$ ) is a linear combination of the variables corresponding to short roots (resp.", "long roots).", "The parameters $\\lambda _i$ are also considered in both cases.", "Similarly $p_s$ (resp.", "$p_l$ ) are linear combinations of the momentum variables corresponding to the short roots (resp.", "long roots).", "This applies to all simple $\\mathfrak {g}$ , with the convention given in Definition REF .", "Definition 4.6 (Notation) Let $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ be a reduced word for $w_0$ .", "We associate to $\\mathbf {i}$ a set of $N$ variables indexed in two ways: $u_{i}^k$ denotes the $k$ -th variables from the leftThis differs from the previous notation used in [18], [20] where the variables read from the right.", "This version will be more convenient in this paper.", "in $\\mathbf {i}$ corresponding to the root index $i$ .", "$v_j$ denotes the $j$ -th variable from the left in $\\mathbf {i}$ , i.e.", "corresponding to $i_j$ , and $i_j$ is the root index corresponding to $v_j$ .", "We denote the corresponding momentum operators as $p_i^k$ and $p_j$ respectively if no confusion arises.", "$v(i,k)$ denotes the index such that $u_i^k= v_{v(i,k)}$ .", "Example 4.7 For type $A_3$ , let $\\mathbf {i}=(1,2,1,3,2,1)$ .", "Then the 6 variables are ordered as: $(u_1^1, u_2^1, u_1^2, u_3^1, u_2^2, u_1^3) = (v_1,v_2,v_3,v_4,v_5,v_6).$ Now we can summarize the construction of the positive representations as follows: Theorem 4.8 Given a reduced word $\\mathbf {i}\\in \\mathfrak {R}$ , the positive representation $\\mathcal {P}_\\lambda \\simeq L^2(\\mathbb {R}^N)$ of $\\mathcal {U}_q(\\mathfrak {g}_\\mathbb {R})$ is parametrized by $\\lambda =(\\lambda _i)\\in \\mathbb {R}_{\\ge 0}^{n}$ , and the generators are represented in the form $\\mathbf {f}_i &= F_i^1+F_i^2+...+F_i^{n_i},\\\\K_i &= e(-2\\lambda _i-\\sum _{j=1}^N a_{i_j,i}v_j),$ where $F_i^k &= \\left[-\\sum _{j=1}^{v(i,k)}a_{i_j,i}v_j+u_i^k -2\\lambda _1\\right]e(2p_i^k)\\\\&= e\\left(-\\sum _{j=1}^{v(i,k)}a_{i_j,i}v_j+u_i^k -2\\lambda _1+2p_i^k\\right)+e\\left(\\sum _{j=1}^{v(i,k)}a_{i_j,i}v_j-u_i^k -2\\lambda _1+2p_i^k\\right)\\nonumber \\\\&=:F_i^{k,-}+F_i^{k,+}$ splitting according to Definition REF .", "The representation of $\\mathbf {e}_j$ is explicitly written case by case.", "In general, if $j=i_N$ , then $\\mathbf {e}_{j} = [v_N]e(-p_N),$ Otherwise $\\mathbf {e}_j=\\Phi \\circ [v_N]e(-2p_N)\\circ \\Phi ^{-1},$ where we recall $\\Phi $ is the unitary transformation, expressed in terms of quantum dilogarithms, that relates $\\mathcal {P}_\\lambda $ to another representations corresponding to a reduced word $\\mathbf {i}^{\\prime }\\in \\mathfrak {R}$ with $i_N^{\\prime }=j$ .", "Theorem 4.9 Each generator $\\mathbf {e}_j$ is expressed as a polynomial in $X_i^{\\pm 1}$ and $Y_i^{\\pm 1}$ (cf.", "(REF )), with a unique initial term of the form $[u_i^k]e(-2p_i^k +... )$ for some index $i,k$ .", "One determines this initial term by applying the transformation (REF ) and tracing the changes of the corresponding initial term by the following rules: if $j=i_N$ , then from (REF ) the initial term is $[v_{i_N}]$ by definition.", "If we have a change of word $(...i,j,i,...)\\longleftrightarrow (...,j,i,j,...)$ , inducing a change of variables $(..., u_i^{k}, u_j^{l}, u_i^{k+1},...) \\longleftrightarrow (..., u_j^{l}, u_i^{k}, u_j^{l+1},... ),$ then the initial term changes from $[u_i^k]\\longleftrightarrow [u_j^{l+1}]$ .", "If we have a change of word $(...,i,j,i,j,...)\\longleftrightarrow (...,j,i,j,i,...)$ , inducing a change of variables $(..., u_i^{k}, u_j^{l}, u_i^{k+1}, u_j^{l+1}...) \\longleftrightarrow (..., u_j^{l}, u_i^{k}, u_j^{l+1}, u_i^{k+1}... ),$ then the initial term changes from $[u_j^l]\\longleftrightarrow [u_j^{l+1}]$ .", "In type $G_2$ , for the change of word $(2,1,2,1,2,1)\\longleftrightarrow (1,2,1,2,1,2)$ , the initial term for $\\mathbf {e}_1$ is $[u_1^3]\\longleftrightarrow [u_1^1]$ and initial term for $\\mathbf {e}_2$ is $[u_2^1]\\longleftrightarrow [u_2^3]$ .", "From the explicit expression (REF ), we have Proposition 4.10 If we write $\\mathbf {f}_i$ as $\\mathbf {f}_i=F_i^{n_i,-}+F_i^{n_i-1,-}... F_i^{1,-}+ F_i^{1,+}+ F_i^{2,+}+...+F_i^{n_i,+},$ then each term $q_i^{-2}$ -commute with all the terms on the right, and each term $q_i^{-2}$ -commute with $K_i^{-1}$ .", "Remark 4.11 Feigin's homomorphism $U_q(\\mathfrak {b}_-)\\longrightarrow \\mathbb {C}[\\mathbf {T}_q]$ is given by the expression of $K_i$ and half of $\\mathbf {f}_i$ : $\\mathbf {f}_i^{\\prime }:= F_i^{1,+}+ F_i^{2,+}+...+F_i^{n_i,+}$ only, so the expression of Theorem REF is really a “double\" of Feigin's homomorphism." ], [ "Embedding of $\\mathcal {U}_q(\\mathfrak {g})$ into quantum torus algebra {{formula:5c6d1631-21a0-4c4f-8f4d-fd4ffb708bbc}}", "Now we are ready to construct the quantum torus algebra $\\mathcal {D}_\\mathfrak {g}$ in the favor of [39] that will provide a clear description of the embedding of the generators of the quantum group $\\mathcal {U}_q(\\mathfrak {g})$ .", "Definition 4.12 Define $2N+2n$ variables indexed by $S=\\lbrace f_i^{-n_i},..., f_i^{n_i}\\rbrace _{i\\in I}\\cup \\lbrace e_i^0\\rbrace _{i\\in I}\\simeq \\lbrace 1,..., 2N+2n\\rbrace $ as follows: For each $i\\in I$ , we take the consecutive “ratio” of the monomial terms of $\\mathbf {f}_i$ as: $X_{f_i^k}=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll} F_i^{n_i,-}& k=-n_i,\\\\q_iF_i^{k,-} (F_i^{k-1,-})^{-1}&n<0,\\\\q_i F_i^{1,+}(F_i^{1,-})^{-1}&k=0,\\\\q_i F_i^{k+1,+}(F_i^{k,+})^{-1}&k>0,\\\\q_i K_i^{-1}(F_i^{n_i,+})^{-1}& k=n_i.", "\\\\ \\end{array}\\right.$ Let the initial term of $\\mathbf {e}_i$ be $[v_n]e(-2p_n)&=e(v_n-2p_n)+e(-v_n-2p_n)\\\\&=:E_i^-+E_i^+$ as in Theorem REF .", "Then we define $X_{e_i^0}=q_i E_i^+(E_i^-)^{-1}= e(-2v_n).$ We note that each $X_k$ are self-adjoint.", "Moreover, since all $X_k$ are expressed formally as a monomial, we have $X_jX_k=q_j^{-2b_{jk}}X_kX_j$ for some skew-symmetrizable exchange matrix $B=(b_{jk})$ and $q_j:=q_i$ if $j=f_i^k$ or $e_i^0$ .", "By abuse of notation, we will use the same variables for the definition below: Definition 4.13 We define the quantum torus algebra $\\mathcal {D}_\\mathfrak {g}$ to be the algebra generated by the elements $X_{f_i^{-n_i}},..., X_{f_i^{n_i}}, X_{e_i^0},\\;\\;\\;\\;\\;\\;i=1,..., n$ subject to the relations (REF ).", "The corresponding $\\mathcal {D}_\\mathfrak {g}$ -quiver is associated to the seed $(S, S_0, B, D)$ where $D=diag(d_i)$ and the frozen nodes are $S_0=\\lbrace f_i^{-n_i}\\rbrace _{i\\in I}\\cup \\lbrace f_i^{n_i}\\rbrace _{i\\in I}$ .", "Now we can state our first main result of the paper.", "Theorem 4.14 We have an embedding of algebra $\\iota :\\mathfrak {D}_\\mathfrak {g}\\hookrightarrow \\mathcal {D}_\\mathfrak {g},$ which induces an embedding of the quantum group into a quotient of $\\mathcal {D}_\\mathfrak {g}$ $\\mathcal {U}_q(\\mathfrak {g})\\hookrightarrow \\mathcal {D}_\\mathfrak {g}/\\langle \\iota (K_i)\\iota (K_i)^{\\prime }=1\\rangle .$ By construction from (REF ), we can write (cf.", "Definition REF ) $\\mathbf {f}_i&=X_{f_i^{-n_i}}+q_i X_{f_i^{-n_i}}X_{f_i^{-n_i+1}}+... + q_i^{n_i-1}X_{f_i^{-n_i}}...X_{f_i^{n_i-1}}\\\\&=X(f_i^{-n_i}, f_i^{-n_i+1},..., f_i^{n_i-1})\\\\K_i^{\\prime } &= X_{f_i^{-n_i},..., f_i^{n_i}}.$ Given a reduced word $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ , if $i=i_N$ , then one computes explicitly $\\mathbf {e}_i &= [u_i^1]e(-2p_i^1) \\\\&= e^{\\pi b_i u_i^1-2\\pi b_i p_i^1}+e^{\\pi b_i u_i^1-2\\pi b_i p_i^1}\\\\&=X_{f_i^{n_i}}+q_iX_{f_i^{n_i}}X_{e_i^0}\\\\&=X(f_i^{n_i}, e_i^0),\\\\K_i &=X_{f_i^{n_i}, e_i^0, f_i^{-n_i}}.$ Otherwise, from the construction of positive representation, each mutation of the reduced expression of $w_0$ correspond to a unitary transformation $\\Phi $ given by the quantum dilogarithm function with an argument given by a consecutive difference of the $F_i^n$ 's in the corresponding mutated quiver.", "(This is described in detail in Section .)", "Hence $\\mathbf {e}_i$ will be expressed as a sum of monomials, each of which is expressed as a product of $X_{f_i^n}$ and the ratios between the initial term, which is given by $X_{e_i^0}$ .", "The explicit expression is given in the next section.", "Furthermore, the unitary transformation $\\Phi $ has the properties that for any reduced word $\\mathbf {i}\\in \\mathfrak {R}$ , if $\\mathbf {e}_j$ is expressed as $\\mathbf {e}_i=X(i_1,..., i_k)=X_{i_1}+... + X_{i_1,..., i_k},$ then the leading term $X_{i_1}=X_{f_i^{n_i}}$ and the ending term satisfies $X_{i_1,...,i_k}X_{f_i^{-n_i}}=q_i^{-2}X_{f_i^{-n_i}}X_{i_1,...,i_k}.$ The unitary transformation $\\Phi $ , while inducing a change of variables given by a linear transformation, will keep $K_i$ as a monomial.", "Hence from the relation $[\\mathbf {e}_i, \\mathbf {f}_i]=(q_i-q_i^{-1})(K_i^{\\prime }-K_i),$ we see that the term $ X_{i_1,..., i_k, f_i^{-n_i}}$ does not vanish.", "Since we already have $K_i^{\\prime }=X_{f_i^{-n_i},..., f_i^{n_i}}$ , we must have $K_i = X_{i_1,..., i_k, f_i^{-n_i}},$ hence giving the desired homomorphism of $\\mathfrak {D}_\\mathfrak {g}$ into $\\mathcal {D}_\\mathfrak {g}$ .", "Since the positive representation is a faithful irreducible representation coming from the quantization of the induced representation of the left regular representation, the homomorphism $\\iota $ is an embedding.", "More explicitly, by choosing the parameters $\\lambda $ such that $2\\sqrt{-1}\\lambda _i\\in Q_i+b_i\\mathbb {N}, \\;\\;\\;\\;\\;\\;Q_i:=b_i+b_i^{-1},$ we recover every finite dimensional highest weight irreducible representation for the compact quantum group $\\mathcal {U}_q(\\mathfrak {g})$ .", "This fact has been utilized for example to calculate the eigenvalues of the positive Casimir operators [23].", "In particular all the PBW basis cannot be identically zero in the representation, hence the homomorphism $\\iota $ is indeed an embedding.", "Finally, we note that the algebra $\\mathcal {D}_\\mathfrak {g}$ has a center generated by $\\iota (K_i)\\iota (K_i^{\\prime })$ .", "Hence taking the quotient with $\\iota (K_i)\\iota (K_i^{\\prime })=1, i\\in I$ we obtain the desired embedding of $\\mathcal {U}_q(\\mathfrak {g})$ as well.", "Remark 4.15 Note that by the Cartan involution, one can also define another embedding $\\iota ^w: \\mathfrak {D}_\\mathfrak {g}&\\longrightarrow \\mathcal {D}_\\mathfrak {g},\\\\\\mathbf {e}_i &\\mapsto \\iota (\\mathbf {f}_i),\\\\\\mathbf {f}_i &\\mapsto \\iota (\\mathbf {e}_i),\\\\K_i &\\mapsto \\iota (K_i^{\\prime }),\\\\K_i^{\\prime } &\\mapsto \\iota (K_i).$ This interchanges the expressions of the explicit embeddings of $\\mathbf {e}_i$ and $\\mathbf {f}_i$ in the quantum torus algebra $\\mathcal {D}_\\mathfrak {g}$ .", "Remark 4.16 In [39], the proof of the injectivity of $\\iota $ is explicitly checked on the PBW basis.", "The expression relating the PBW exponents to those of the $q$ -tori generators turns out to involve some combinatorial hive-type conditions from the work of Knutson-Tao [34].", "It will be interesting to see explicitly analogues of such combinatorics in other types." ], [ "Construction of the $\\mathcal {D}_\\mathfrak {g}$ -quiver", "Let us now describe the explicit construction of the quiver associated to the $\\mathcal {D}_\\mathfrak {g}$ algebra in more details." ], [ "Relation among cluster variables", "First we have the obvious relations.", "Lemma 5.1 $X_{f_i^0}$ and $X_{e_i^0}$ mutually commute with each other.", "By Definition REF , the formal expression of all the $X_{f_i^0}$ 's and $X_{e_i^0}$ 's do not contain any momentum operators $e(2p)$ , hence they commute with each other.", "Next we have the following observation: Lemma 5.2 Recall that $F_i^{k,\\pm }$ is defined in (REF ).", "We have $F_i^{k,\\pm } F_j^{l,\\pm } = q_i^{\\mp a_{ij}}F_j^{l,\\pm }F_i^{k,\\pm }$ if $v(i,k) < v(j,l)$ .", "Let us consider the $+$ case, while the $-$ case is similar.", "By definition, if $v(i,k)<v(j,l)$ , then there are no terms of $u_j^l$ appearing in $F_i^{k,+}$ , hence $e(2p_j^l)$ in $F_j^{l,+}$ commutes with everything in $F_i^{k,+}$ , while $e(2p_i^k)$ from $F_i^{k,+}$ $q$ -commutes with $e(...a_{ij}u_i^k+...)$ from $F_j^{l,+}$ giving the factor $q_i^{-a_{ij}}$ .", "Thus one can derive the commutation relation directly between the variables $X_{f_i^k}$ and $X_{f_j^l}$ .", "First, by definition we have $X_{f_i^k}X_{f_i^l}=q_i^{-2} X_{f_i^l}X_{f_i^k}$ whenever $l=k+1$ and commute otherwise.", "Corollary 5.3 Assume $i\\ne j, k,l\\ge 0$ and $v(i,k)<v(j,l)$ , we have: $X_{f_i^k}X_{f_j^l} = q_i^C X_{f_j^l}X_{f_i^k},$ where $C=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}2a_{ij}& v(i,k)<v(j,l)<v(i,k+1)<v(j,l+1),\\\\0 & v(i,k)<v(j,l)<v(j,l+1)<v(i,k+1),\\\\0& v(i,k)<v(i,k+1)<v(j,l)<v(j,l+1),\\\\a_{ij}& \\mbox{$k=n_i$ and $l=n_j$}, \\\\ \\end{array}\\right.$ where in the inequalities we let the boundaries be $v(i,0)=-\\infty $ and $v(i,n_i):=+\\infty $ .", "Next, we observe that by construction, the cluster variables $X_{f_i^k}$ and $X_{f_j^l}$ with $k,l\\le 0$ have exactly the commutation relations opposite to (REF ), while if $k,l\\ne 0$ have different signs they commute with each other.", "Finally, the cluster variables $X_{e_i^0}$ is given by $e(-2u_j^k)$ where $[u_j^k]$ is the initial term of the positive representation of $\\mathbf {e}_i$ which is determined explicitly.", "Then we have $X_{e_i^0}X_{f_j^k}&=q_i^2X_{f_j^k}X_{e_i^0},\\\\X_{e_i^0}X_{f_j^{k-1}}&=q_i^{-2}X_{f_j^{k-1}}X_{e_i^0},\\;\\;\\;\\;\\;\\;k\\ne 1.$ Combining the above relations among $X_k$ , this completes the description of the $\\mathcal {D}_\\mathfrak {g}$ -quiver." ], [ "Elementary quiver associated to simple reflections", "With the above observations, a more conceptual way of constructing the $\\mathcal {D}_\\mathfrak {g}$ quiver motivated by [26] is as follows.", "We define the following quiver: Definition 5.4 The elementary quiver $Q_i^k$ associated to the $v(i,k)$ -th simple reflection $s_i$ in $w_0$ , i.e.", "corresponding to the variable $u_i^k$ , is constructed by the frozen nodes [every node/.style=inner sep=0, minimum size=0.6cm, thick, x=0.5cm, y=0.5cm] draw] (i) at(0,0) $f_i^{k-1}$ ; draw] (j) at (6,0) $f_i^k$ ; [thick, ->](i) to (j); t (3,0.3) $d_i$ ; and for every $j$ connected to $i$ in the Dynkin diagram we have in addition [every node/.style=inner sep=0, minimum size=0.6cm, thick, x=0.6cm, y=0.6cm] draw] (i) at(0,0) $f_i^{k-1}$ ; draw] (j) at (6,0) $f_i^k$ ; draw] (k) at (3,3) $f_j^l$ ; [thick, ->](j) to (k); [thick, ->](k) to (i); t (5.5,1.5) $\\frac{|d_ia_{ij}|}{2}$ ; t (0.5,1.5) $\\frac{|d_ia_{ij}|}{2}$ ; for the unique $l$ with $v(j,l)<v(i,k)<v(j,l+1)$ , and the nodes $f_i^a$ have weight $d_i$ .", "In particular, according to the convention from Definition REF , in the last figure if either $i$ or $j$ is long, $X_{f_i^k}X_{f_j^l}=q^{-1}X_{f_j^l}X_{f_i^k}$ and we will denote both arrows by (thick) dashed arrows, while if both $i$ and $j$ are short, $X_{f_i^k}X_{f_j^l}=q_s^{-1}X_{f_j^l}X_{f_i^k}$ and we will denote both arrows by thin dashed arrows.", "For example, in type $A_3$ , an elementary quiver associated to $s_2$ is drawn as in Figure REF , where we indicate the corresponding simple reflection, and indicate with dashed arrows the weights $\\frac{|d_2a_{2k}|}{2}=\\frac{1}{2}$ .", "Figure: Elementary quiver in type A 3 A_3.Proposition 5.5 The subquiver of the $\\mathcal {D}_\\mathfrak {g}$ -quiver generated by $f_i^n$ with $n\\ge 0$ corresponding to the reduced word $(i_1,...,i_N)\\in \\mathfrak {R}$ is given by amalgamation of the elementary quivers $Q_{s_i}^k$ along vertices with the same indices, modulo the arrows between $f_i^0$ 's.", "The subquiver of the $\\mathcal {D}_\\mathfrak {g}$ -quiver generated by $f_i^n$ for every $n$ corresponding to the reduced word $(i_1,...,i_N)\\in \\mathfrak {R}$ is obtained by amalgamation of the elementary quivers $Q_{s_i}^{\\pm k}$ along vertices with the same indices, where $Q_{s_i}^{-k}$ are the elementary quivers corresponding to the opposite words $(i_N,...,i_1)$ of the opposite nodes $f_i^{-n}$ .", "This follows directly from Corollary REF .", "The $\\mathcal {D}_\\mathfrak {g}$ quiver is then obtained by the above amalgamation together with the quivers connecting the nodes $e_i^0$ ." ], [ "Explicit embedding of $\\mathcal {U}_q(\\mathfrak {g})$", "In the previous section, we constructed the $\\mathcal {D}_\\mathfrak {g}$ -quivers as amalgamation of the elementary quivers $Q_i^k$ together with the arrows joining the nodes $e_i^0$ .", "In particular, they can be presented in a way that is symmetric along a vertical axis, where the arrows are flipped over.", "It turns out that this can be expressed as an amalgamation of a pair of basic quivers associated to $\\mathfrak {g}$ , and that these basic quivers are mutation equivalent to the cluster structure of framed $G$ -local system associated to the disk with 3 marked points, recently discovered by [25], [26].", "We will determine and describe the basic quivers in Section .", "By Theorem REF , the action of $K_i$ (resp.", "$K_i^{\\prime }$ ) are obtained by multiplying $X_{f_i^{-n_i}}$ (resp.", "$X_{f_i^{n_i}})$ to the last term of $\\mathbf {e}_i$ (resp.", "$\\mathbf {f}_i$ ), hence we will omit it from the description below for simplicity.", "Definition 6.1 ($E_i$ and $F_i$ -path) Since $\\mathbf {f}_i=X(f_i^{-n_i},..., f_i^{n_i-1})$ , we will call the path of the quiver given by the nodes $f_i^{-n_i}\\longrightarrow f_i^{-n_i+1}\\longrightarrow ...\\longrightarrow f_i^{n_i-1}\\longrightarrow f_i^{n_i}$ an $F_i$ -path.", "On the other hand, if $\\mathbf {e}_i=X(m_1,m_2,..., m_k)$ (or similar variants in type $C_n$ , $E_8$ and $F_4$ ), we call the path of the quiver given by the nodes $m_1\\longrightarrow m_2,...\\longrightarrow m_k\\longrightarrow f_i^{-n_i}$ an $E_i$ -path.", "From the relations $[\\mathbf {e}_i,\\mathbf {f}_i]=(q_i-q_i^{-1})(K_i^{\\prime }-K_i)$ , one can derive the fact that the path always begin with $m_1 = f_i^{n_i}$ .", "Remark 6.2 In [39], the $E_i$ -paths and $F_i$ -paths are also known as the $V_i$ -paths and $\\Lambda _i$ -paths respectively in type $A_n$ , which describe the corresponding shapes of the paths, see Figure REF ." ], [ "Toy example: Type $A_2$", "In this section, we illustrate the method of recovering the $\\mathcal {D}_\\mathfrak {g}$ -quiver.", "Let $\\mathfrak {g}$ be of type $A_2$ , and recall the notation from Definition REF .", "For simplicity we label the variables $(u_1^1,u_2^1,u_1^2)$ below by $(u,v,w)$ .", "Proposition 6.3 [18] The positive representation $\\mathcal {P}_\\lambda $ of $\\mathcal {U}_{q\\widetilde{q}}(\\mathfrak {sl}(3,\\mathbb {R}))$ with parameters $\\lambda =(\\lambda _1,\\lambda _2)\\in \\mathbb {R}_+^2$ , corresponding to the reduced word $\\mathbf {i}=(1,2,1)$ acting on $f(u,v,w)\\in L^2(\\mathbb {R}^3)$ , is given by $\\mathbf {e}_1=&[w]e(-2p_w),\\\\\\mathbf {e}_2=&[u]e(-2p_u-2p_v+2p_w)+[v-w]e(-2p_v),\\\\\\mathbf {f}_1=&[-u-2\\lambda _2]e(2p_u)+[-2u+v-w-2\\lambda _2]e(2p_w),\\\\\\mathbf {f}_2=&[u-v-2\\lambda _1]e(2p_v),\\\\K_1=&e(-2u+v-2w-2\\lambda _1),\\\\K_2=&e(u-2v+w-2\\lambda _2).$ In the expanded form, we have $\\mathbf {e}_1=&e(w-2p_w)+e(-w-2p_2),\\\\\\mathbf {e}_2=&e(u-2p_u-2p_v+2p_w)+e(v-w-2p_v)+e(-v+w-2p_v)+e(-u-2p_u-2p_v+2p_w),\\\\\\mathbf {f}_1=&e(-2u+v-w-2\\lambda _2+2p_w)+e(-u-2\\lambda _2+2p_u)+e(u+2\\lambda _2+2p_u)+e(2u-v+w+2\\lambda _2+2p_w),\\\\\\mathbf {f}_2=&e(u-v-2\\lambda _1+2p_v)+e(-u+v+2\\lambda _1+2p_v).$ We recover the following cluster variables following Definition REF by taking successive ratios of the $\\mathbf {f}_i$ generators: $X_{f_1^{-2}}&=e(-2u+v-w-2\\lambda _2+2p_w),\\\\X_{f_1^{-1}}&=e(u-v+w+2p_u-2p_w),\\\\X_{f_1^0}&=e(2u+4\\lambda _2),\\\\X_{f_1^1}&=e(u-v+w-2p_u+2p_w),\\\\X_{f_1^2}&=e(w-2p_w),\\\\X_{f_2^{-1}}&=e(u-v-2\\lambda _1+2p_v),\\\\X_{f_2^0}&=e(-2u+2v+4\\lambda _1),\\\\X_{f_2^1}&=e(v-w-2p_v),$ and taking the ratio of the initial terms of the $\\mathbf {e}_i$ generators (i.e.", "the first and last terms in the expanded form) we have $X_{e_1^0}&=e(-2w),\\\\X_{e_2^0}&=e(-2u).$ Hence treating the operators as formal algebraic variables, from their commutation relations we recover the quiver describing the quantum torus algebra $D_{\\mathfrak {sl}_3}$ in Figure REF .", "Figure: A 2 A_2-quiver, with the E i E_i-paths colored in red.Using the notation from Definition REF , we see that the $F_i$ -path is expressed as $V$ -shaped paths in the quiver diagram.", "$\\mathbf {f}_1&=X_{f_1^{-2}}+X_{f_1^{-2},f_1^{-1}}+X_{f_1^{-2},f_1^{-1},f_1^0}+X_{f_1^{-2},f_1^{-1},f_1^0,f_1^2}\\\\&=X(f_1^{-2}, f_1^{-1}, f_1^0, f_1^1),\\\\\\mathbf {f}_2&=X_{f_2^{-1}}+X_{f_2^{-1},f_2^0}\\\\&=X(f_2^{-1}, f_2^0),\\\\K_1^{\\prime }&=X_{f_1^{-2}, f_1^{-1}, f_1^0, f_1^1, f_1^2},\\\\K_2^{\\prime }&=X_{f_2^{-1},f_2^0, f_2^1}.$ Since the exponents of the variables $\\lbrace X_{f_i^{k}}\\rbrace _{k\\ne 0}$ and $X_{e_i^0}$ for $i=1,2$ forms a complete basis of the linear space spanned by $\\langle u,v,w,p_u, p_v, p_w,\\lambda _1,\\lambda _2\\rangle $ , one can solve for the $\\mathbf {e}_i$ action in terms of these cluster variables.", "As a result, we obtain $\\mathbf {e}_1&=X_{f_1^2}+X_{f_1^2, e_1^0}\\\\&=X(f_1^2, e_1^0),\\\\\\mathbf {e}_2&=X_{f_2^1}+X_{f_2^1,f_1^1}+X_{f_2^1,f_1^1,e_2^0}+X_{f_2^1,f_1^1,e_2^0,f_1^{-1}}\\\\&=X(f_2^1,f_1^1,e_2^0,f_1^{-1}),\\\\K_1&=X_{f_1^2,e_1^0,f_1^{-2}},\\\\K_2&=X_{f_2^1,f_1^1,e_2^0,f_1^{-2},f_2^{-1}},$ which gives the $E_i$ -path (highlighted in red) as $\\Lambda $ -shaped paths in the quiver diagram as desired.", "Let us now turn to the general cases." ], [ "Type $A_n$", "The quiver associated to type $A_n$ and the quantum group embedding $\\mathcal {U}_q(\\mathfrak {sl}_{n+1})$ is fully described in [39].", "Let us choose the reduced word $\\mathbf {i}= (1\\; 21\\; 321\\;4321... n\\; (n-1)\\;...1).$ Then $n_i=n+1-i$ .", "Using the explicit expression of the positive representations from [18] in type $A_n$ , we have $\\mathbf {f}_i&=X(f_i^{-n+i-1},..., f_i^{n-i}),\\\\\\mathbf {e}_i&=X(f_i^{n-i+1}, f_{i-1}^{n-i+1},..., f_1^{n-i+1}, e_i^0, f_1^{-n+i-1},..., f_i^{-n+i-1}).$ Here the initial terms are given by $X_{e_i^0} = e(-2u_1^{n+1-i}).$ The quiver is shown in Figure REF .", "We see that the $F_i$ -path follows a $\\Lambda $ -shaped path, while $E_i$ -path follows a $V$ -shaped path in the quiver (highlighted in red).", "The quiver can obviously be generalized to arbitrary rank.", "Figure: A 5 A_5-quiver, with the E i E_i-paths colored in red." ], [ "Type $B_n$", "Using the explicit expression of the positive representations from [20], we choose the reduced word $\\mathbf {i}= (1212\\; 32123\\; 4321234... n\\; (n-1)\\;...1...\\;(n-1)\\; n).$ Here recall that 1 is short and all other roots are long.", "Then $n_1=n$ and $n_i = 2n+2-2i$ .", "$\\mathbf {f}_1&=X(f_{1}^{-n},..., f_1^{n-1}),\\\\\\mathbf {f}_i&=X(f_i^{-2n+2i-2},..., f_i^{2n-2i+1})& i\\ge 2,\\\\\\mathbf {e}_1&=X(f_1^n, f_2^{2n-3}, f_1^{n-1}, f_2^{2n-5},..., f_2^1, f_1^1, e_1^0, f_1^{-1}, f_2^{-1},..., f_2^{-2n+3}),\\\\\\mathbf {e}_i&=X(f_i^{2n-2i+2}, f_{i+1}^{2n-2i-1}, f_i^{2n-2i}, f_{i+1}^{2n-2i-3},..., f_{i+1}^1, f_i^2, e_i^0, f_i^{-2}, f_{i+1}^{-1},..., f_{i+1}^{-2n+2i+1})& i\\ge 2.$ The initial terms are given by $X_{e_i^0}=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}e(-2u_1^1)&i=1,\\\\ e(-2u_i^2)&i>1.", "\\\\ \\end{array}\\right.$ The quiver is shown in Figure REF .", "Both the $E_i$ -path and $F_i$ -path follow a zig-zag shaped path in the quiver.", "Moreover, the quiver can naturally be generalized to arbitrary rank.", "Figure: B 5 B_5-quiver, with the E i E_i-paths colored in red." ], [ "Type $C_n$", "We choose the same word as type $B_n$ : $\\mathbf {i}= (1212\\; 32123\\; 4321234... n\\; (n-1)\\;...1...\\;(n-1)\\; n).$ Here 1 is long and all other roots are short.", "Then the expression for $\\mathbf {f}_i$ is the sameAlthough the algebraic expressions are the same, the $q$ -commuting relations are not due to different long and short roots.", "as type $B_n$ , while $\\mathbf {e}_i$ are the same for $i\\ge 2$ , but modification is made to $\\mathbf {e}_1$ : $\\mathbf {e}_1&=X(f_1^n, *f_2^{2n-3}, f_1^{n-1}, *f_2^{2n-5},...,*f_2^1, f_1^1, e_1^0, f_1^{-1}, *f_2^{-1},..., *f_2^{-2n+3}),$ where $X(...,a, *b,...)$ means adding the extra factors as follows: $&...+X_{...}+X_{...,a}+[2]_{q_s} X_{...,a,b}+X_{...,a,b^2}+X_{...,a,b^2,...}+...\\\\&=...+X_{...}+(X_{...,a}^{\\frac{1}{2}}+X_{...,a,b^2}^{\\frac{1}{2}})^2+X_{...,a,b^2,...}+...\\;.\\nonumber $ The initial terms are same as type $B_n$ : $X_{e_i^0}=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}e(-2u_1^1)&i=1,\\\\ e(-2u_i^2)&i>1.", "\\\\ \\end{array}\\right.$ The quiver is shown in Figure REF .", "We see that the quiver is exactly the same as type $B_n$ case, except that the weights of the arrows are modified.", "Furthermore, the $E_i$ -path for $\\mathbf {e}_1$ now “stops” at certain vertices (corresponding to (REF )), which we highlighted in red.", "Figure: C 5 C_5-quiver, with the E i E_i-paths and the repeated nodes of 𝐞 1 \\mathbf {e}_1 colored in red." ], [ "Type $D_n$", "We choose the word corresponding to splitting of type $B_{n-1}$ : $\\mathbf {i}=(012012\\;320123\\;43201234... (n-1)\\;...2012...\\;(n-1)),$ where 0 and 1 are the splitting nodes that are paired.", "Then $n_0=n_1=n-1$ and $n_i=2n-2i$ for $i\\ge 2$ : $\\mathbf {f}_0&=X(f_0^{-n+1},..., f_0^{n-2}),\\\\\\mathbf {f}_1&=X(f_1^{-n+1},..., f_1^{n-2}),\\\\\\mathbf {f}_i&=X(f_i^{-2n+2i},..., f_i^{2n-2i-1}) \\;\\;\\;\\;\\;\\;i\\ge 2,$ and $\\mathbf {e}_0&=X(f_0^{n-1}, f_2^{2n-5}, f_1^{n-2}, f_2^{2n-7}, f_0^{n-3}, f_2^{2n-9}, f_1^{n-4}, f_2^{2n-11},..., f_2^1, f_{\\epsilon }^1, e_0^0, f_{\\epsilon }^{-1}, f_2^{-1},..., f_2^{-2n+5}),\\\\\\mathbf {e}_1&=X(f_1^{n-1}, f_2^{2n-5}, f_0^{n-2}, f_2^{2n-7}, f_1^{n-3}, f_2^{2n-9}, f_0^{n-4}, f_2^{2n-11},..., f_2^1, f_{1-\\epsilon }^1, e_1^0, f_{1-\\epsilon }^{-1}, f_2^{-1},..., f_2^{-2n+3}),\\\\\\mathbf {e}_i&=X(f_i^{2n-2i}, f_{i+1}^{2n-2i-3}, f_i^{2n-2i-2}, f_{i+1}^{2n-2i-5},..., f_{i+1}^1, f_i^2, e_i^0, f_i^{-2}, f_{i+1}^{-1},..., f_{i+1}^{-2n+2i-1})\\;\\;\\;\\;\\;\\;i\\ge 2,$ where $\\epsilon =n \\mbox{(mod 2)}\\in \\lbrace 0,1\\rbrace $ .", "The initial terms are given by $X_{e_i^0}=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}e(-2u_i^1)&i=0,1,n\\mbox{ is even,}\\\\e(-2u_{1-i}^1)&i=0,1,n\\mbox{ is odd,}\\\\ e(-2u_i^2)&i>1.", "\\\\ \\end{array}\\right.$ The quiver is shown in Figure REF .", "Note that the action of $E_i$ and $F_i$ are the same as type $B_{n-1}$ for $i\\ne 0,1$ .", "Furthermore, it follows naturally that the $B_{n-1}$ -quiver comes from a folding of the $D_n$ -quiver, with the weights of the arrows appropriately adjusted.", "In the quiver, we highlight the $E_0$ -path in red, where we see that it alternates between root 0 and root 1, while the $E_1$ -path interchanges 0 and 1.", "Figure: D 6 D_6-quiver, with the E i E_i-path of 𝐞 0 \\mathbf {e}_0 colored in red." ], [ "Type $E_n$", "For type $E_n$ , we let 0 be the extra node (cf.", "Definition REF ).", "The explicit expression of the positive representations for the generators $\\mathbf {f}_i$ and $K_i^{\\prime }$ is given by Theorem REF and Theorem REF , while the expression for the generators $\\mathbf {e}_i$ is given in the Appendix of the author's Ph.D. Thesis [17] Here we choose $\\mathbf {i}$ to begin with 343 instead of 434 for technical convenience..", "The explicit expression is, however, rather ad hoc.", "Using the procedure describe in the beginning of this section, we can solve for the cluster variables $X_i$ to rewrite the expression as certain $E$ -paths on some quiver diagrams, which is a lot easier to visualize.", "Interestingly, we see that unlike type $A$ to $D$ , most of the actions of $\\mathbf {e}_i$ actually pass through rows corresponding to other roots throughout the whole quiver.", "As before, we only list the representations of $\\mathbf {e}_i$ and $\\mathbf {f}_i$ , while again the representation of the $K_i$ and $K_i^{\\prime }$ variables are expressed as the product of the last term of $\\mathbf {e}_i$ and $\\mathbf {f}_i$ with $X_{f_i^{-n_i}}$ and $X_{f_i^{n_i}}$ respectively." ], [ "Type $E_6$", "Following [17], we choose the longest word to be $\\mathbf {i}= (3\\;43\\;034\\; 230432\\;12340321\\;5432103243054321),$ which comes from the embedding of Dynkin diagram $A_1\\subset A_2\\subset A_3\\subset D_4\\subset D_5\\subset E_6$ by successively adding the nodes 3,4,0,2,1,5 to the diagram.", "Then the $\\mathbf {f}_i$ variables are expressed as $\\mathbf {f}_1&=X(f_1^{-4},...,f_1^3),\\\\\\mathbf {f}_2&=X(f_2^{-7},...,f_2^6),\\\\\\mathbf {f}_3&=X(f_3^{-11},..., f_3^{10}),\\\\\\mathbf {f}_4&=X(f_4^{-7},..., f_4^6),\\\\\\mathbf {f}_5&=X(f_5^{-2},f_5^{-1},f_5^0,f_5^1),\\\\\\mathbf {f}_0&=X(f_0^{-5},..., f_0^{4}),$ while the $\\mathbf {e}_i$ variables are expressed as certain paths on the quiver: $\\mathbf {e}_1&=X(f_1^4,e_1^0),\\\\\\mathbf {e}_2&=X(f_2^7, f_1^{3}, f_2^{5}, f_3^{8}, f_0^{3}, f_3^{6}, f_4^{3},f_3^{4}, f_0^{1}, f_3^{2}, e_2^0, f_3^{-2},f_0^{-1}, f_3^{-4}, f_4^{-3}, f_3^{-6}, f_0^{-3}, f_3^{-8}, f_2^{-5},f_1^{-3}),\\\\\\mathbf {e}_3&=X(f_3^{11}, f_2^{6}, f_3^{9}, f_4^{5}, f_3^{7}, f_2^{3}, f_3^{5}, f_2^{1}, f_3^{3}, f_4^{1}, f_3^{1}, e_3^0, f_3^{-1}, f_4^{-1}, f_3^{-3}, f_2^{-1}, f_3^{-5}, f_2^{-3}, f_3^{-7}, f_4^{-5}, f_3^{-9}, f_2^{-6})\\\\\\mathbf {e}_4&=X(f_4^7, f_3^{10}, f_0^{4}, f_3^{8}, f_2^{4}, f_1^{1}, f_2^{2},e_4^0, f_2^{-2}, f_1^{-1}, f_2^{-4}, f_3^{-8}, f_0^{-4}, f_3^{-10}),\\\\\\mathbf {e}_5&=X(f_5^2, f_4^{6}, f_3^{9}, f_2^{5}, f_1^{2}, e_5^0, f_1^{-2},f_2^{-5}, f_3^{-9}, f_4^{-6}),\\\\\\mathbf {e}_0&=X(f_0^{5}, f_3^{10}, f_4^{6}, f_5^{1}, f_4^{4}, f_3^{6}, f_0^{2},f_3^{4}, f_4^{2}, e_0^0, f_4^{-2}, f_3^{-4}, f_0^{-2}, f_3^{-6},f_4^{-4}, f_5^{-1}, f_4^{-6}, f_3^{-10}).$ The initial terms are given by $&X_{e_1^0}=e(-2u_1^4),&& X_{e_2^0}=e(-2u_3^2),&& X_{e_3^0}=e(-2u_3^1),\\\\&X_{e_4^0}=e(-2u_2^2),&& X_{e_5^0}=e(-2u_1^2),&& X_{e_0^0}=e(-2u_4^2).$ The quiver is shown in Figure REF , where the labeling of each row is given by $f_i^{-n_i},..., f_i^{n_i}$ , hence each $F_i$ -path is represented as a horizontal path.", "The different $E_i$ -paths, starting from $f_i^{n_i}$ and ending at $f_i^{-n_i}$ , are shown in different colors.", "Figure: E 6 E_6-quiver, with the E i E_i-paths colored in different colors." ], [ "Type $E_7$", "Following [17], we choose the longest word to be $\\mathbf {i}=(3\\;43\\;034\\;230432\\;12340321\\;5432103243054321\\;654320345612345034230123456),$ which comes from the embedding of Dynkin diagram $A_1\\subset A_2\\subset A_3\\subset D_4\\subset D_5\\subset E_6\\subset E_7$ by successively adding the nodes 3,4,0,2,1,5,6 to the diagram.", "Then the $\\mathbf {f}_i$ variables are expressed as $\\mathbf {f}_1&=X(f_1^{-6},..., f_1^5),\\\\\\mathbf {f}_2&=X(f_2^{-11},..., f_2^{10}),\\\\\\mathbf {f}_3&=X(f_3^{-17},..., f_3^{16}),\\\\\\mathbf {f}_4&=X(f_4^{-12},..., f_4^{11}),\\\\\\mathbf {f}_5&=X(f_5^{-6},..., f_5^5),\\\\\\mathbf {f}_6&=X(f_6^{-3},..., f_6^2),\\\\\\mathbf {f}_0&=X(f_0^{-8},..., f_0^7),$ while the $\\mathbf {e}_i$ variables are expressed as certain paths on the quiver: $\\mathbf {e}_1=&X(f_1^{7}, f_2^{10}, f_3^{15}, f_4^{10}, f_5^{4}, f_6^{1}, f_5^{2},f_4^{6}, f_3^{9}, f_2^{5}, f_1^{3}, e_1^0, f_1^{-1}, f_2^{-5},f_3^{-9}, f_4^{-6}, f_5^{-2}, f_6^{-1}, f_5^{-4}, f_4^{-10},f_3^{-15}, f_2^{-10}),\\\\\\mathbf {e}_2=&X(f_2^{11}, f_3^{16}, f_0^{7}, f_3^{14}, f_4^{9}, f_5^{3}, f_4^{7},f_3^{10}, f_0^{4}, f_3^{8}, f_2^{4}, f_1^{2}, f_2^{2}, e_2^0, \\\\&\\;\\;\\;\\;\\;\\;f_2^{-2}, f_1^{0}, f_2^{-4}, f_3^{-8}, f_0^{-4}, f_3^{-10}, f_4^{-7},f_5^{-3}, f_4^{-9}, f_3^{-14}, f_0^{-7}, f_3^{-16}),\\\\\\mathbf {e}_3=&X(f_3^{17}, f_4^{11}, f_3^{15}, f_2^{9}, f_3^{13}, f_4^{8}, f_3^{11},f_2^{6}, f_3^{9}, f_4^{5}, f_3^{7}, f_2^{3}, f_3^{5}, f_2^{1},f_3^{3}, f_4^{1}, f_3^{1}, e_3^0, \\\\&\\;\\;\\;\\;\\;\\;f_3^{-1}, f_4^{-1}, f_3^{-3}, f_2^{-1}, f_3^{-5},f_2^{-3}, f_3^{-7}, f_4^{-5}, f_3^{-9}, f_2^{-6}, f_3^{-11},f_4^{-8}, f_3^{-13}, f_2^{-9}, f_3^{-15}, f_4^{-11}),\\\\\\mathbf {e}_4=&X(f_4^{12}, f_5^{5}, f_4^{10}, f_3^{14}, f_0^{6}, f_3^{12}, f_2^{7},f_1^{4}, f_2^{5}, f_3^{8}, f_0^{3}, f_3^{6}, f_4^{3}, f_3^{4},f_0^{1}, f_3^{2}, e_4^0, \\\\&\\;\\;\\;\\;\\;\\;f_3^{-2}, f_0^{-1},f_3^{-4}, f_4^{-3}, f_3^{-6}, f_0^{-3}, f_3^{-8}, f_2^{-5}, f_1^{-2},f_2^{-7}, f_3^{-12}, f_0^{-6}, f_3^{-14}, f_4^{-10}, f_5^{-5}),\\\\\\mathbf {e}_5=&X(f_5^{6}, f_6^{2}, f_5^{4}, f_4^{9}, f_3^{13}, f_2^{8}, f_1^{4},e_5^0, f_1^{-4}, f_2^{-8}, f_3^{-13}, f_4^{-9}, f_5^{-4},f_6^{-2}),\\\\\\mathbf {e}_6=&X(f_6^3, e_6^0),\\\\\\mathbf {e}_0=&X(f_0^{8}, f_3^{16}, f_2^{10}, f_1^{6}, f_2^{8}, f_3^{12}, f_0^{5},f_3^{10}, f_4^{6}, f_5^{1}, f_4^{4}, f_3^{6}, f_0^{2}, f_3^{4},f_4^{2}, e_0^0,\\\\&\\;\\;\\;\\;\\;\\;f_4^{-2}, f_3^{-4}, f_0^{-2}, f_3^{-6}, f_4^{-4},f_5^{-1}, f_4^{-6}, f_3^{-10}, f_0^{-5}, f_3^{-12}, f_2^{-8},f_1^{-4}, f_2^{-10}, f_3^{-16}).$ The initial terms are given by $&X_{e_1^0}=e(-2u_1^2),&& X_{e_2^0}=e(-2u_2^2),&& X_{e_3^0}=e(-2u_3^1),&&X_{e_4^0}=e(-2u_3^2),\\\\& X_{e_5^0}=e(-2u_1^4),&& X_{e_6^0}=e(-2u_6^3), && X_{e_0^0}=e(-2u_4^2).$ The quiver is shown in Figure REF , again the labeling of each row is given by $f_i^{-n_i},..., f_i^{n_i}$ , hence each $F_i$ -path is represented as a horizontal path.", "The different $E_i$ -paths, starting from $f_i^{n_i}$ and ending at $f_i^{-n_i}$ , are shown in different colors.", "Figure: E 7 E_7-quiver, with the E i E_i-paths colored in different colors." ], [ "Type $E_8$", "Following [17], we choose the longest word to be $\\mathbf {i}=&(3\\;43\\;034\\;230432\\;12340321\\;5432103243054321\\;654320345612345034230123456\\\\&765432103243546503423012345676543203456123450342301234567),$ which comes from the embedding of Dynkin diagram $A_1\\subset A_2\\subset A_3\\subset D_4\\subset D_5\\subset E_6\\subset E_7\\subset E_8$ by successively adding the nodes 3,4,0,2,1,5,6,7 to the diagram.", "Then the $\\mathbf {f}_i$ variables are expressed as $\\mathbf {f}_1&=X(f_1^{-10},..., f_1^9),\\\\\\mathbf {f}_2&=X(f_2^{-19},..., f_2^{18}),\\\\\\mathbf {f}_3&=X(f_3^{-29},..., f_3^{28}),\\\\\\mathbf {f}_4&=X(f_4^{-22},..., f_4^{21}),\\\\\\mathbf {f}_5&=X(f_5^{-14},..., f_5^{13}),\\\\\\mathbf {f}_6&=X(f_6^{-9},..., f_6^8),\\\\\\mathbf {f}_7&=X(f_7^{-3},..., f_7^2),\\\\\\mathbf {f}_0&=X(f_0^{-14},..., f_0^{13}),$ while the $\\mathbf {e}_i$ variables are expressed as certain paths on the quiver: $\\mathbf {e}_1=&X(f_1^{10}, f_2^{18}, f_3^{27}, f_4^{20}, f_5^{12}, f_6^{7}, f_5^{10},f_6^{5}, f_5^{8}, f_4^{14}, f_3^{19}, f_2^{12}, f_1^{6}, f_2^{10},f_3^{15}, f_4^{10}, f_5^{4}, f_6^{1}, f_5^{2}, f_4^{6}, f_3^{9},f_2^{5}, f_1^{2}, e_1^0, \\\\&\\;\\;\\;\\;\\;\\;f_1^{-2}, f_2^{-5}, f_3^{-9}, f_4^{-6},f_5^{-2}, f_6^{-1}, f_5^{-4}, f_4^{-10}, f_3^{-15}, f_2^{-10},f_1^{-6}, f_2^{-12}, f_3^{-19}, f_4^{-14}, f_5^{-8}, f_6^{-5},f_5^{-10}, f_6^{-7}, f_5^{-12}, \\\\&\\;\\;\\;\\;\\;\\;f_4^{-20}, f_3^{-27}, f_2^{-18}),\\\\\\mathbf {e}_2=&X(f_2^{19}, f_3^{28}, f_0^{13}, f_3^{26}, f_4^{19}, f_5^{11}, f_4^{17},f_5^{9}, f_4^{15}, f_3^{20}, f_0^{9}, f_3^{18}, f_2^{11}, f_3^{16},f_0^{7}, f_3^{14}, f_4^{9}, f_5^{3}, f_4^{7}, f_3^{10}, f_0^{4},f_3^{8}, f_2^{4}, f_1^{1}, f_2^{2}, e_2^0,\\\\&\\;\\;\\;\\;\\;\\;f_2^{-2}, f_1^{-1}, f_2^{-4}, f_3^{-8}, f_0^{-4}, f_3^{-10}, f_4^{-7}, f_5^{-3},f_4^{-9}, f_3^{-14}, f_0^{-7}, f_3^{-16}, f_2^{-11}, f_3^{-18},f_0^{-9}, f_3^{-20}, f_4^{-15}, f_5^{-9}, f_4^{-17}, f_5^{-11},\\\\&\\;\\;\\;\\;\\;\\;f_4^{-19}, f_3^{-26}, f_0^{-13}, f_3^{-28}),\\\\\\mathbf {e}_3=&X(f_3^{29}, f_4^{21}, f_3^{27}, f_2^{17}, f_3^{25}, f_4^{18}, f_3^{23},f_4^{16}, f_3^{21}, f_2^{13}, f_3^{19}, f_4^{13}, f_3^{17},f_4^{11}, f_3^{15}, f_2^{9}, f_3^{13}, f_4^{8}, f_3^{11}, f_2^{6},f_3^{9}, f_4^{5}, f_3^{7}, f_2^{3}, f_3^{5}, f_2^{1}, f_3^{3},f_4^{1}, f_3^{1}, e_3^0,\\\\&\\;\\;\\;\\;\\;\\;f_3^{-1}, f_4^{-1}, f_3^{-3}, f_2^{-1}, f_3^{-5}, f_2^{-3},f_3^{-7}, f_4^{-5}, f_3^{-9}, f_2^{-6}, f_3^{-11}, f_4^{-8},f_3^{-13}, f_2^{-9}, f_3^{-15}, f_4^{-11}, f_3^{-17}, f_4^{-13},f_3^{-19}, f_2^{-13}, f_3^{-21}, f_4^{-16}, \\\\&\\;\\;\\;\\;\\;\\;f_3^{-23},f_4^{-18}, f_3^{-25}, f_2^{-17}, f_3^{-27}, f_4^{-21}),\\\\\\mathbf {e}_4=&X(f_4^{22}, f_5^{13}, f_4^{20}, f_3^{26}, f_0^{12}, f_3^{24}, f_2^{15},f_3^{22}, f_0^{10}, f_3^{20}, f_4^{14}, f_5^{7}, f_4^{12}, f_5^{5},f_4^{10}, f_3^{14}, f_0^{6}, f_3^{12}, f_2^{7}, f_1^{3}, f_2^{5},f_3^{8}, f_0^{3}, f_3^{6}, f_4^{3}, f_3^{4}, f_0^{1}, f_3^{2},e_4^0, \\\\&\\;\\;\\;\\;\\;\\;f_3^{-2}, f_0^{-1}, f_3^{-4}, f_4^{-3},f_3^{-6}, f_0^{-3}, f_3^{-8}, f_2^{-5}, f_1^{-3}, f_2^{-7},f_3^{-12}, f_0^{-6}, f_3^{-14}, f_4^{-10}, f_5^{-5}, f_4^{-12},f_5^{-7}, f_4^{-14}, f_3^{-20}, f_0^{-10}, f_3^{-22}, \\\\&\\;\\;\\;\\;\\;\\;f_2^{-15}, f_3^{-24}, f_0^{-12}, f_3^{-26}, f_4^{-20}, f_5^{-13}),\\\\\\mathbf {e}_5=&X(f_5^{14}, f_6^{8}, f_5^{12}, f_4^{19}, f_3^{25}, f_2^{16}, f_1^{8},f_2^{14}, f_3^{21}, f_4^{15}, f_5^{8}, f_6^{4}, f_5^{6}, f_6^{2},f_5^{4}, f_4^{9}, f_3^{13}, f_2^{8}, f_1^{4}, e_5^0,\\\\&\\;\\;\\;\\;\\;\\;f_1^{-4},f_2^{-8}, f_3^{-13}, f_4^{-9}, f_5^{-4}, f_6^{-2}, f_5^{-6},f_6^{-4}, f_5^{-8}, f_4^{-15}, f_3^{-21}, f_2^{-14}, f_1^{-8},f_2^{-16}, f_3^{-25}, f_4^{-19}, f_5^{-12}, f_6^{-8}),\\\\\\mathbf {e}_6=&X(f_6^{9}, f_7^{2}, f_6^{6}, f_6^{7}, f_5^{10}, f_5^{11}, f_4^{17},f_4^{18}, f_3^{23}, f_3^{24}, [f_0^{11}, f_2^{15}, f_0^{11}], f_3^{22}, f_3^{23},f_4^{16}, f_4^{17}, f_5^{9}, f_5^{10}, f_6^{5}, f_6^{6}, f_7^{1},f_6^{3}, e_6^0, \\\\&\\;\\;\\;\\;\\;\\;f_6^{-3}, f_7^{-1}, f_6^{-6}, f_6^{-5}, f_5^{-10},f_5^{-9}, f_4^{-17}, f_4^{-16}, f_3^{-23}, f_3^{-22}, [f_0^{-11}, f_2^{-15},f_0^{-11}], f_3^{-24}, f_3^{-23}, f_4^{-18}, f_4^{-17}, \\\\&\\;\\;\\;\\;\\;\\;f_5^{-11}, f_5^{-10}, f_6^{-7}, f_6^{-6}, f_7^{-2}),\\\\\\mathbf {e}_7=&X(f_7^3,e_7^0),\\\\\\mathbf {e}_0=&X(f_0^{14}, f_3^{28}, f_2^{18}, f_1^{9}, f_2^{16}, f_3^{24}, f_0^{11},f_3^{22}, f_2^{14}, f_1^{7}, f_2^{12}, f_3^{18}, f_0^{8}, f_3^{16},f_2^{10}, f_1^{5}, f_2^{8}, f_3^{12}, f_0^{5}, f_3^{10}, f_4^{6},f_5^{1}, f_4^{4}, f_3^{6}, f_0^{2}, f_3^{4}, f_4^{2}, e_0^0,\\\\&\\;\\;\\;\\;\\;\\;f_4^{-2}, f_3^{-4}, f_0^{-2}, f_3^{-6}, f_4^{-4}, f_5^{-1}, f_4^{-6},f_3^{-10}, f_0^{-5}, f_3^{-12}, f_2^{-8}, f_1^{-5}, f_2^{-10},f_3^{-16}, f_0^{-8}, f_3^{-18}, f_2^{-12}, f_1^{-7}, f_2^{-14},f_3^{-22}, f_0^{-11},\\\\&\\;\\;\\;\\;\\;\\;f_3^{-24}, f_2^{-16}, f_1^{-9}, f_2^{-18}, f_3^{-28}).$ Here for the action of $\\mathbf {e}_6$ , the path corresponding to $...[A,B,A]...$ is split as: $...+X_{...}+ X_{..., A}+X_{..., B}+X_{..., A,B} + X_{..., A, B, ...}+...\\;.$ We see that the path for $\\mathbf {e}_6$ is special in the sense that it revisited certain nodes twice.", "The same phenomenon also appear in type $F_4$ below.", "Finally the initial terms are given by $&X_{e_1^0}=e(-2u_1^2),&& X_{e_2^0}=e(-2u_2^2),&& X_{e_3^0}=e(-2u_3^1),&&X_{e_4^0}=e(-2u_3^2),\\\\& X_{e_5^0}=e(-2u_1^4),&& X_{e_6^0}=e(-2u_6^3), && X_{e_7^0}=e(-2u_7^3), &&X_{e_0^0}=e(-2u_4^2).$ The $E_8$ -quiver is shown in Figure REF , where we have highlighted the different $E_i$ -paths of the $\\mathbf {e}_i$ generators except $\\mathbf {e}_6$ .", "For the special case of $\\mathbf {e}_6$ , we highlight it separately in Figure REF .", "Figure: E 8 E_8-quiver, with the E i E_i-paths (except E 6 E_6) colored in different colors.Figure: E 8 E_8-quiver, with the E 6 E_6-paths colored in red." ], [ "Type $F_4$", "The explicit expression for type $F_4$ positive representations can be found in [20], where we choose $\\mathbf {i}=(3232\\;12321\\;432312343213234),$ where 1,2 is long, 3,4 is short, corresponding to the embedding of the Dynkin diagram: $B_2\\subset B_3\\subset F_4.$ Then the $\\mathbf {f}_i$ variables are expressed as $\\mathbf {f}_1&=X(f_1^{-4},..., f_1^3),\\\\\\mathbf {f}_2&=X(f_2^{-8},..., f_2^7),\\\\\\mathbf {f}_3&=X(f_3^{-9},..., f_3^8),\\\\\\mathbf {f}_4&=X(f_4^{-3},..., f_4^2),$ while the $\\mathbf {e}_i$ variables are expressed as certain paths on the quiver: $\\mathbf {e}_1=&X(f_1^{4}, f_2^{7}, *f_3^{7}, f_2^{6}, *f_3^{5}, f_2^{5}, f_1^{2}, e_1^0,f_1^{-2}, f_2^{-5}, *f_3^{-5}, f_2^{-6}, *f_3^{-7}, f_2^{-7}),\\\\\\mathbf {e}_2=&X(f_2^{8}, *f_3^{8}, f_2^{7}, f_1^{3}, f_2^{5}, *f_3^{4}, f_2^{4},f_1^{1}, f_2^{2}, e_2^0, f_2^{-2}, f_1^{-1}, f_2^{-4}, *f_3^{-4},f_2^{-5}, f_1^{-3}, f_2^{-7}, *f_3^{-8}),\\\\\\mathbf {e}_3=&X(f_3^{9}, f_4^{2}, f_3^{6}, f_3^{7}, f_2^{6}, f_3^{5}, f_3^{6},f_4^{1}, f_3^{3}, f_2^{3}, f_3^{2}, f_2^{1}, f_3^{1}, e_3^0, f_3^{-1},f_2^{-1}, f_3^{-2}, f_2^{-3}, f_3^{-3}, f_4^{-1}, f_3^{-6},f_3^{-5}, f_2^{-6},\\\\&\\;\\;\\;\\;\\;\\;f_3^{-7}, f_3^{-6}, f_4^{-2}),\\\\\\mathbf {e}_4=&X(f_4^3, e_4^0),$ where we recall from type $C_n$ that $X(...,a, *b, ...)$ corresponds to the extra factors as follows: $&...+X_{...}+X_{...,a}+[2]_{q_s} X_{...,a,b}+X_{...,a,b^2}+X_{...,a,b^2,...}+...\\\\&=...+X_{...}+(X_{...,a}^{\\frac{1}{2}}+X_{...,a,b^2}^{\\frac{1}{2}})^2+X_{...,a,b^2,...}+...\\;.$ The initial terms are given by $&X_{e_1^0}=e(-2u_1^2),&&X_{e_2^0}=e(-2u_2^2),&& X_{e_3^0}=e(-2u_3^1),&& X_{e_4^0}=e(-2u_4^3).$ The quiver is shown in Figure REF , where the repeated nodes $*$ are highlighted.", "We note that the $E_1$ and $E_3$ paths overlapped a little bit.", "Figure: F 4 F_4-quiver, with the E i E_i-paths colored in different colors." ], [ "Type $G_2$", "The explicit expression for type $G_2$ positive representations can be found in [20].", "We choose $\\mathbf {i}=(2,1,2,1,2,1)$ .", "Then we have $\\mathbf {f}_1&=X(f_1^{-3},..., f_1^2),\\\\\\mathbf {f}_2&=X(f_2^{-3},...,f_2^2),\\\\\\mathbf {e}_1&=X(f_1^3, e_1^0),\\\\\\mathbf {e}_2&=X(f_2^3, f_1^2, *f_2^2, f_1^1,f_2^1,e_2^0, f_2^{-1},f_1^{-1},*f_2^{-2}, f_1^{-2}),$ where again $X(...,a, *b, ...)$ corresponds to the extra factors: $&...+X_{...}+X_{...,a}+[2]_{q_s} X_{...,a,b}+X_{...,a,b^2}+X_{...,a,b^2,...}+...\\;.$ The inital terms are given by $X_{e_1^0}=e(-2u_1^3), \\;\\;\\;\\;\\;\\;X_{e_2^0}=e(-2u_2^1).$ The quiver is shown in Figure REF .", "Figure: G 2 G_2-quiver, with the E i E_i-paths colored in red." ], [ "Quiver mutations for different choice of $w_0$", "Recall from the construction of the positive representations that a change of reduced expression of $w_0$ corresponds to a unitary transformation $\\Phi $ (cf.", "(REF )).", "This is expressed in terms of conjugation by quantum dilogarithms, followed by a linear transformation.", "As we have seen in Section , conjugation by the quantum dilogarithms naturally correspond to mutations of the quiver diagram.", "In this section we will describe the corresponding mutation associated to a change of words.", "In particular, by extending the mutations below to the full quiver, we obtain an alternate proof of Theorem REF for the rules of finding the initial term $X_{e_j^0}$ of the generators $\\mathbf {e}_j$ ." ], [ "Quiver mutation in simply-laced case", "First we note that if $a_{ij}=0$ , i.e.", "$s_is_j=s_js_i$ , there is no mutation or change of variables occurring.", "That is, swapping the reflections does not affect the quiver diagram at all.", "In the simply-laced case, the unitary transformation $\\Phi $ corresponding to the change of words $w_0=...s_is_js_i... \\longleftrightarrow ... s_j s_i s_j ...$ is expressed in terms of conjugation by a single quantum dilogarithm.", "Consider the following amalgamation $Q$ of elementary quivers corresponding to $s_is_js_i$ , where we exclude the nodes outside the root indices $i$ and $j$ : Figure: The s i s j s i s_is_js_i quiver.This corresponds to the representation of the $\\mathbf {f}_i$ generators in the full quiver $\\mathcal {D}_\\mathfrak {g}$ as $\\mathbf {f}_i &= .... + X_{...f_i^{k-1}}+ X_{... f_i^{k-1}, f_i^k} + X_{... f_i^{k-1}, f_i^k, f_i^{k+1}} + ...\\\\\\mathbf {f}_j &= .... + X_{...f_j^{l-1}}+ X_{... f_j^{l-1}, f_j^l} + ...$ Then the mutation corresponding to the unitary transformation $\\Phi $ giving the change of words $s_is_js_i\\longleftrightarrow s_js_is_j$ is given by mutation at $f_i^k$ , followed by a renaming of variables, where we have defined a new external labeling for the mutated quiver $\\widehat{Q}$ by the rules: $\\hat{f}_i^t&:= f_i^{t+1}\\;\\;\\;\\;\\;\\;t\\ge k,\\\\\\hat{f}_j^t&:= f_j^{t-1} \\;\\;\\;\\;\\;\\;t\\ge l+1,\\\\\\hat{f}_j^l&:=f_i^k$ and stays the same otherwise.", "Figure: After mutation at f i k f_i^k.Figure: Rearranging and renaming the quiver.In the representation level, a change of words corresponds to a unitary transformation $\\Phi $ by the conjugation of the quantum dilogarithm $g_b(X_{f_i^k})$ : $Ad_{g_b(X_{f_i^k})}\\cdot \\mathbf {f}_i &= ... + X_{...f_i^{k-1}}+ X_{... f_i^{k-1}, f_i^k, f_i^{k+1}} + ...\\\\&=\\mu _{f_i^k}^{\\prime }(... +X_{...f_{i}^{k-1}}+ X_{... f_{i}^{k-1}, f_{i}^{k+1}} + ...)\\\\&=\\mu _{f_i^k}^{\\prime }(... +X_{...\\hat{f}_{i}^{k-1}}+ X_{... \\hat{f}_{i}^{k-1}, \\hat{f}_i^k} + ...),\\\\Ad_{g_b(X_{f_i^k})}\\cdot \\mathbf {f}_j &= ... + X_{...f_j^{l-1}, f_i^k}+ X_{... f_j^{l-1}} + X_{... f_j^{l-1}, f_j^l} + ...\\\\&=\\mu _{f_i^k}^{\\prime }(...+X_{...f_j^{l-1}} + X_{...f_j^{l-1}, f_i^k}+X_{... f_j^{l-1}, f_i^k, f_j^l} +...)\\\\&=\\mu _{f_i^k}^{\\prime }(...+X_{...\\hat{f}_j^{l-1}} +X_{...\\hat{f}_j^{l-1}, \\hat{f}_j^l}+X_{... \\hat{f}_j^{l-1}, \\hat{f}_j^l, \\hat{f}_j^{l+1}} +...).$ Hence using $\\mu _k^q=Ad_{g_b^*(X_j)}\\circ \\mu _k^{\\prime }$ , we have $\\mathbf {f}_i &= ...+\\widehat{X}_{...\\hat{f}_{i}^{k-1}}+ \\widehat{X}_{... \\hat{f}_{i}^{k-1}, \\hat{f}_i^k}+...,\\\\\\mathbf {f}_j &= ...+\\widehat{X}_{...\\hat{f}_j^{l-1}} + \\widehat{X}_{...\\hat{f}_j^{l-1}, \\hat{f}_j^l}+\\widehat{X}_{... \\hat{f}_j^{l-1}, \\hat{f}_j^l, \\hat{f}_j^{l+1}} +...,$ where we denote the mutated cluster variables by $\\widehat{X}_j:=\\mu _{f_i^k}^q(X_j)$ associated to the mutated quiver $\\widehat{Q}$ , and we see that the representation of the $\\mathbf {f}_i$ generators are invariant under the quiver mutation.", "When we take into account the whole quiver $\\mathcal {D}_\\mathfrak {g}$ , we see that the nodes precisely come in pair.", "Hence we have Corollary 7.1 The cluster embedding $\\iota : \\mathfrak {D}_\\mathfrak {g}\\longrightarrow \\mathcal {D}_\\mathfrak {g}$ corresponding to $\\mathbf {i}=(...iji...)$ and $\\mathbf {i}^{\\prime }=(...jij...)$ is related by quiver mutations at the pair of nodes $\\lbrace f_i^k, f_i^{-k}\\rbrace $ (the order does not matter)." ], [ "Quiver mutation in doubly-laced case", "Following the notation above, we consider the following amalgamation of quiver corresponding to $s_is_js_is_j$ where the root $i$ is long and $j$ is short.", "All the arrows are thick except the two corresponding to $s_j$ .", "Figure: The s i s j s i s j s_is_js_is_j quiver.The unitary transformation $\\Phi $ of the positive representations corresponding to the change of words $s_is_js_is_j \\longleftrightarrow s_js_is_js_i$ is expressed as 3 pairs of quantum dilogarithm transformations [18].", "The mutation corresponding to $\\Phi $ is then given by mutation at $f_j^l, f_i^k, f_j^l$ , with the weights $d_i$ of each nodes taken into account.", "Figure: After mutation at f j l f_j^l.Figure: After mutation at f i k f_i^k.Figure: After second mutation at f j l f_j^l.No renaming of the variables is necessary after the last step, and again we have expressed the generators $\\mathbf {f}_i$ in terms of the mutated cluster variables $\\widehat{X}_j$ associated to the mutated quiver.", "Similarly as before, for the full quiver we have Corollary 7.2 The cluster embedding $\\iota : \\mathfrak {D}_\\mathfrak {g}\\longrightarrow \\mathcal {D}_\\mathfrak {g}$ corresponding to $\\mathbf {i}=(...ijij...)$ and $\\mathbf {i}^{\\prime }=(...jiji...)$ is related by quiver mutations at the pair of nodes $\\lbrace f_j^l,f_j^{-l}\\rbrace $ , $\\lbrace f_i^k,f_i^{-k}\\rbrace $ and $\\lbrace f_j^l,f_j^{-l}\\rbrace $ ." ], [ "Quiver mutation in type $G_2$", "We consider the following amalgamation of quiver corresponding to $s_2s_1s_2s_1s_2s_1$ where the root 1 is long and 2 is short.", "All the arrows are thick except the three corresponding to $s_2$ .", "Figure: The s 2 s 1 s 2 s 1 s 2 s 1 s_2s_1s_2s_1s_2s_1 quiver.In [18], we found that the unitary transformation $\\Phi $ changing the words $s_2s_1s_2s_1s_2s_1 \\longleftrightarrow s_1s_2s_1s_2s_1s_2$ is given by conjugations by 11 quantum dilogarithms.", "This corresponds to the following sequence of mutations (starting from the left): $f_1^2, f_1^1,f_2^2,f_1^2, f_2^2,f_2^1,f_2^2, f_1^2, f_1^1, f_2^2, f_1^2$ Figure: After mutation at f 1 2 ,f 1 1 ,f 2 2 ,f 1 2 f_1^2, f_1^1,f_2^2,f_1^2.Figure: After mutation at f 2 2 ,f 2 1 ,f 2 2 f_2^2,f_2^1,f_2^2.Figure: After mutation again at f 1 2 ,f 1 1 ,f 2 2 ,f 1 2 f_1^2, f_1^1, f_2^2, f_1^2.Figure: Rearranging the quiver.We see that we have to permute the index: $\\hat{f}_2^1:=f_2^2,\\;\\;\\;\\;\\;\\;\\hat{f}_2^2:=f_2^1$ at the end.", "Similarly as before, this expresses the generators $\\mathbf {f}_i$ in terms of the mutated cluster variables $\\widehat{X}_j$ , and for the full quiver we have Corollary 7.3 The cluster embedding $\\iota : \\mathfrak {D}_\\mathfrak {g}\\longrightarrow \\mathcal {D}_\\mathfrak {g}$ corresponding to $\\mathbf {i}=(2,1,2,1,2,1)$ and $\\mathbf {i}^{\\prime }=(1,2,1,2,1,2)$ is related by quiver mutations at the pair of nodes $&\\lbrace f_1^2,f_1^{-2}\\rbrace ,\\lbrace f_1^1,f_1^{-1}\\rbrace ,\\lbrace f_2^2,f_2^{-2}\\rbrace ,\\lbrace f_1^2,f_1^{-2}\\rbrace ,\\lbrace f_2^2,f_2^{-2}\\rbrace , \\lbrace f_2^1,f_2^{-1}\\rbrace \\\\& \\lbrace f_2^2,f_2^{-2}\\rbrace ,\\lbrace f_1^2, f_1^{-2}\\rbrace , \\lbrace f_1^1,f_1^{-1}\\rbrace ,\\lbrace f_2^2, f_2^{-2}\\rbrace ,\\lbrace f_1^2, f_1^{-2}\\rbrace .$ Remark 7.4 In [26], it is also found that the above change of words can be realized by 12 mutations coming from a more natural geometric consideration: $(f_1^1,f_1^2,f_2^2,f_1^2),(f_2^2,f_2^1,f_1^1,f_2^2),(f_1^1,f_1^2,f_2^2,f_1^2),$ where the groups correspond to the permutations (12)(23)(12) of the vertices of the triangle where the quiver is attached in the framed $G$ -local system.", "The end result differs from the above quiver by an additional permutation of $f_1^1$ and $f_1^2$ , but such difference will not play a role in this paper." ], [ "Basic quivers", "In Section , we obtain explicitly the $\\mathcal {D}_\\mathfrak {g}$ -quiver corresponding to the embedding of the quantum group $\\mathcal {U}_q(\\mathfrak {g})$ associated to the reduced expression of the longest element $w_0=s_{i_1}...s_{i_N}$ .", "By their symmetric presentations, we observe that the $\\mathcal {D}_\\mathfrak {g}$ -quiver is given by amalgamation of some quivers $Q$ and $\\widetilde{Q}$ where $\\widetilde{Q}$ is obtained by a mirror image of $Q$ along the vertical axis, together with flipping all the arrows.", "More precisely, let us arrange the quiver $Q$ so that its frozen vertices $\\lbrace e_i^0, f_i^0, f_i^{n_i}\\rbrace _{i\\in I}$ are fixed on a triangle $ABC$ as shown in Figure REF .", "Let $\\widetilde{Q}$ be the mirror image of $Q$ with frozen vertices $\\lbrace e_i^0, f_i^0, f_i^{-n_i}\\rbrace _{i\\in I}$ fixed on a triangle $A^{\\prime }B^{\\prime }C^{\\prime }$ , but such that all arrows are flipped (i.e.", "with the same indexing, it has the exchange matrix $-B$ instead).", "Then the $\\mathcal {D}_\\mathfrak {g}$ -quiver is obtained by amalgamating $Q$ and $\\widetilde{Q}$ along the frozen vertices at $\\lbrace e_i^0, f_i^0\\rbrace _{i\\in I}$ .", "We will call such external labeling of the basic quivers $Q$ and $\\widetilde{Q}$ the standard form.", "Figure: Amalgamating the quivers QQ and Q ˜\\widetilde{Q} in standard form.We note that there are some freedom of choices of the quivers $Q$ , namely, we can choose arbitrary arrows among the nodes $e_i^0$ and $f_i^0$ .", "In order to fix the ambiguity, we first note that such amalgamation of two triangles give a triangulation of the disk with one puncture and two marked points: Figure: Triangulation of a disk with one puncture and two marked points.Therefore in order to realize the embedding naturally as associated to triangulations of such surface, the quiver $Q$ associated to the triangle $ABC$ , should be mutation equivalent to the quiver $\\widetilde{Q}$ associated to triangle $B^{\\prime }A^{\\prime }C^{\\prime }$ in this clockwise order.", "Let $\\mathcal {M}$ be the mutation sequence reversing the reduced word $(i_1,...,i_N)\\longrightarrow (i_N,...,i_1)$ .", "If $Q^{\\prime }$ is the subquiver of $Q$ with nodes $\\lbrace f_i^k\\rbrace $ , then $\\mathcal {M}(Q^{\\prime })$ is naturally given by a mirror image of $Q^{\\prime }$ with all the arrows flipped, or in terms of Figure REF , the triangle is flipped from $ABC$ to $B^{\\prime }A^{\\prime }C^{\\prime }$ .", "It turns out that we have to identify the frozen nodes with its Dynkin involution $\\theta :I\\longrightarrow I,$ where by definition, the longest element acts on simple roots as $w_0\\cdot \\alpha _i=-\\alpha _{\\theta (i)}.$ Hence if we let $\\mathcal {M}_\\theta $ be the mutation sequence changing the reduced word $\\mathcal {M}_\\theta :(i_1,...,i_N)\\longrightarrow (\\theta (i_1),...,\\theta (i_N)),$ then we naturally also want to identify $Q$ with $\\mathcal {M}_\\theta (Q)$ .", "With these observations, we made the following definition.", "Definition 8.1 A basic quiver $Q$ for $\\mathcal {D}_\\mathfrak {g}$ corresponding to the word $w_0=s_{i_1}...s_{i_N}$ is a quiver associated to the triangle $ABC$ such that the amalgamation of $Q$ and $\\widetilde{Q}$ gives the $\\mathcal {D}_\\mathfrak {g}$ -quiver, $\\mathcal {M}(Q)$ is identical to the quiver $\\widetilde{Q}$ , where the frozen nodes $\\lbrace f_i^0, f_i^{n_i}, e_i^0\\rbrace $ of $\\mathcal {M}(Q)$ is identified with the frozen nodes $\\lbrace f_i^{-n_i}, f_i^0, e_{\\theta (i)}^0\\rbrace $ of $\\widetilde{Q}$ .", "$Q$ is identical to the quiver $\\mathcal {M}_\\theta (Q)$ , where the frozen nodes $\\lbrace f_i^0, f_i^{n_i},e_i^0\\rbrace $ of $Q$ is identified with the frozen nodes $\\lbrace f_{\\theta (i)}^0, f_{\\theta (i)}^{n_i}, e_{\\theta (i)}^0\\rbrace $ of $\\mathcal {M}_\\theta (Q)$ .", "Note that when $\\theta =id$ , the third condition is trivial.", "Theorem 8.2 For each $\\mathfrak {g}$ of simple Lie type, there exists a unique basic quiver $Q$ .", "We need to solve for the required relations among the nodes $\\lbrace e_i^0, f_i^0\\rbrace $ that satisfies the definition of a basic quiver.", "First of all, by the construction of the elementary quivers in Section REF , we can naturally determine the arrows between $\\lbrace f_i^0\\rbrace $ already by reading $w_0$ from the left.", "Furthermore, Theorem REF implies that any quiver mutations preserve the relations below whenever the initial term $X_{e_i^0}=e(-2u_j^a)$ for $a>1$ in the quantum group embedding: Figure: NO_CAPTIONHence for the basic quiver, we see that in order for $\\mathcal {M}(Q)$ to be identical to $\\widetilde{Q}$ , we must also have the above quiver for $a=1$ , and this establishes the arrows between $\\lbrace e_i^0\\rbrace $ and $\\lbrace f_i^0\\rbrace $ .", "Therefore it remains to determine the arrows among the nodes $\\lbrace e_i^0\\rbrace $ .", "Since quiver mutation is a bijection, it suffices to construct the basic quiver for some reduced word $\\mathbf {i}\\in \\mathfrak {R}$ .", "Hence throughout the proof, we will use the same reduced word for $\\mathbf {i}$ in Section for each type of $\\mathfrak {g}$ in the construction of the $\\mathcal {D}_\\mathfrak {g}$ -quiver.", "Let $Q_+$ denote the subquiver of $\\mathcal {D}_\\mathfrak {g}$ containing the nodes $\\lbrace f_i^n\\rbrace _{n\\ge 0, i\\in I}$ and $\\lbrace e_i^0\\rbrace _{i\\in I}$ .", "The mutation sequences $\\mathcal {M}$ and $\\mathcal {M}_\\theta $ are obtained by recursively bringing the required index to the right of $\\mathbf {i}$ using the swapping $s_is_j=s_js_i$ , or the Coxeter moves.", "Type $A_n$ .", "This is a very special case as the change of words $\\mathbf {i}=(121321...n,(n-1),...1)\\longleftrightarrow (123... n, 123... (n-1),... 123121)$ does not require any Coxeter moves, but only swapping between commuting reflections.", "Therefore $\\mathcal {M}=id$ , and the definition of basic quiver requires that $Q$ associated to $ABC$ is the same as $\\widetilde{Q}$ associated to $B^{\\prime }A^{\\prime }C^{\\prime }$ , where the order of $\\lbrace e_i^0\\rbrace $ is reversed.", "In particular it says that the sides $\\overrightarrow{AC}$ and $\\overrightarrow{BC}$ of $Q$ is the same as the sides $\\overrightarrow{B^{\\prime }C^{\\prime }}$ and $\\overrightarrow{A^{\\prime }C^{\\prime }}$ of $\\widetilde{Q}$ , which by definition is just the sides $\\overrightarrow{CB}$ and $\\overrightarrow{CA}$ of $Q$ .", "This uniquely determines the arrows among the nodes $\\lbrace f_i^0\\rbrace $ , and between the nodes $\\lbrace e_i^0\\rbrace $ and $\\lbrace f_i^0\\rbrace $ .", "The Dynkin involution is given by $\\theta (i):=n+1-i.$ By considering the mutation $\\mathcal {M}_\\theta $ of the Dynkin involution $(121321... n,(n-1),... 1) \\longrightarrow (n,(n-1),n,(n-2),(n-1),n,.... 123...n)$ which in a sense is just flipping the diagram upside-down, we observe that the arrows between consecutive $\\lbrace e_i^0\\rbrace $ are mutated once whenever there is a change of word $(...121...)\\longleftrightarrow (...212...)$ .", "The arrows among $\\lbrace e_i^0\\rbrace $ are chosen such that $Q=\\mathcal {M}_\\theta (Q)$ .", "The end result forces $Q$ to have a magical $\\mathbb {Z}_3$ symmetry, and we recover the well-known basic quiver for type $A_n$ associated to $n$ -triangulation first studied by [9].", "More precisely, the basic quiver $Q$ is obtained by attaching to $Q_+$ the additional arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type A n A_n.Figure: Basic quiver in type A n A_n with ℤ 3 \\mathbb {Z}_3 symmetry.Type $B_n$ and $C_n$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=(1212\\; 32123\\;...n(n-1)...1...(n-1)n)$ consists of $\\frac{2}{3}n(n-1)(n-2)$ simply-laced mutations, and $\\frac{1}{2}n(n-1)$ doubly-laced mutations.", "Recall from Section REF that each doubly-laced mutation corresponds to 3 quiver mutations.", "Hence the change of words $\\mathcal {M}$ corresponds to $\\frac{2}{3}n(n-1)(n-2)+\\frac{3}{2}n(n-1)=\\frac{1}{6}n(n-1)(4n+1)$ quiver mutations.", "By comparing $\\mathcal {M}(Q)$ with $\\widetilde{Q}$ , we found that the basic quiver is obtained by adjoining $Q_+$ the following arrows in type $B_n$ : Figure: Additional arrows attaching to Q + Q_+ to give QQ in type B n B_n.and the following arrows in type $C_n$ : Figure: Additional arrows attaching to Q + Q_+ to give QQ in type C n C_n.Type $D_n$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=(012012\\; 320123\\;...(n-1)...2012...(n-1))$ consists of $\\frac{2}{3}n(n-1)(n-2)$ mutations.", "When $n$ is even, we have $\\theta =id$ , and the condition $\\mathcal {M}(Q)=\\widetilde{Q}$ uniquely determines the arrows among the frozen nodes.", "Otherwise when $n$ is odd, the condition $\\mathcal {M}(Q)=\\widetilde{Q}$ uniquely determines the arrows among the frozen nodes except $e_0^0$ and $e_1^0$ .", "In this case the Dynkin involution is given by $\\theta (i)=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}1-i&i=0,1,\\\\i&otherwise, \\\\ \\end{array}\\right.$ hence we see that from our choice of $w_0$ , $\\mathcal {M}_\\theta $ is trivial.", "This means that we cannot have arrows between $e_0^0$ and $e_1^0$ .", "The resulting basic quiver $Q$ is then obtained by taking $Q_+$ and adjoining the following arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type D n D_n.Type $E_6$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=(3\\;43\\;034\\; 230432\\;12340321\\;5432103243054321)$ consists of 78 mutations.", "The Dynkin involution is given by $\\theta (i)=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}6-i&i>0,\\\\0&i=0, \\\\ \\end{array}\\right.$ whence the change of words $\\mathcal {M}_\\theta $ consists of 42 mutations.", "After comparing the quiver, we found that the basic quiver $Q$ is obtained by taking $Q_+$ and adjoining the following arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type E 6 E_6.Type $E_7$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=(3\\;43\\;034\\;230432\\;12340321\\;5432103243054321\\;654320345612345034230123456)$ consists of 336 mutations.", "By comparing $\\mathcal {M}(Q)$ and $\\widetilde{Q}$ , the basic quiver $Q$ is found to be obtained by taking $Q_+$ and adjoining the following arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type E 7 E_7.Type $E_8$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=&(3\\;43\\;034\\;230432\\;12340321\\;5432103243054321\\;654320345612345034230123456\\\\&765432103243546503423012345676543203456123450342301234567)$ consists of 1120 mutations.", "By comparing $\\mathcal {M}(Q)$ and $\\widetilde{Q}$ , the basic quiver $Q$ is found to be obtained by taking $Q_+$ and adjoining the following arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type E 8 E_8.Type $F_4$ .", "The change of words $\\mathcal {M}$ for $\\mathbf {i}=(3232\\;12321\\;432312343213234)$ consists of 32 simply-laced mutations and 18 doubly-laced mutations, hence it corresponds to $32+3\\times 18=86$ quiver mutations.", "The resulting basic quiver is obtained by adjoining to $Q_+$ the following arrows: Figure: Additional arrows attaching to Q + Q_+ to give QQ in type F 4 F_4.Type $G_2$ .", "Finally, the change of words $\\mathcal {M}$ for $\\mathbf {i}=(2,1,2,1,2,1)\\longrightarrow (1,2,1,2,1,2)$ is described in Section REF , which consists of 11 or 12 quiver mutations.", "The basic quiver $Q$ can be presented as follows for the two cases in Figure .", "Figure: Basic quiver QQ for w 0 =s 2 s 1 s 2 s 1 s 2 s 1 w_0=s_2s_1s_2s_1s_2s_1.Figure: Basic quiver for w 0 =s 1 s 2 s 1 s 2 s 1 s 2 w_0=s_1s_2s_1s_2s_1s_2, i.e.", "ℳ(Q)\\mathcal {M}(Q).This is identical to the $G_2$ quiver found in [26].", "In particular, it demonstrates the Langland's duality of the change of short and long roots as a change of weights of the quivers in the diagram.", "Also as mentioned before, $\\mathcal {M}(Q)$ is a mirror image of $Q$ with all arrows flipped as desired.", "This completes the proof of the Theorem.", "Corollary 8.3 The basic quiver is mutation equivalent to the $Conf_3 \\mathcal {A}_G$ quiver for $G$ of type $A_n,B_n,C_n,D_n,G_2$ described in [25], [26].", "The $Conf_3 \\mathcal {A}_G$ quivers described in [25], [26] correspond to the words $w_0=w_c^\\frac{h}{2}$ where $w_c$ is the Coxeter element and $h$ the Coxeter number.", "Hence one can check directly the mutation sequence from the change of words of $w_0$ and compare the quivers.", "Remark 8.4 As we can see, the arrows joining the nodes $\\lbrace e_i^0\\rbrace $ and the arrows joining the nodes $\\lbrace f_{\\theta (i)}^0\\rbrace $ turns out to be opposite to each other.", "This should reflect some internal symmetries of the moduli spaces $Conf_3 \\mathcal {A}_G$ of the configurations of triples of principal flags, and one should be able to find a more conceptual way to fix the basic quiver.", "We believe that this uniqueness theorem can be used to solve the series of conjectures regarding the uniqueness of the cluster structure proposed in Section 3 of [26].", "The mutation $\\mathcal {M}$ corresponds to the transposition interchanging the sides $AC$ and $BC$ of the triangle where $\\lbrace f_i^0\\rbrace $ and $\\lbrace f_i^{n_i}\\rbrace $ are attached respectively (cf.", "Figure REF ).", "On the other hand, in order to realize $S_3$ symmetry, we also want a mutation sequence corresponding to transposition of sides $AB$ and $AC$ , where $\\lbrace e_{\\theta (i)}^0\\rbrace $ and $\\lbrace f_i^0\\rbrace $ are attached respectively.", "Also note that $f_i^{n_i}=:e_i^{-m_i}$ in the quantum group embedding.", "Hence such mutation should correspond to the longest Lusztig's transformation (see Definition REF ): $T_{i_1}T_{i_2}...T_{i_N}(\\mathbf {e}_i)&= q_i\\mathbf {f}_{\\theta (i)}K_{\\theta (i)}^{-1}\\\\T_{i_1}T_{i_2}...T_{i_N}(\\mathbf {f}_i)&= q_i\\mathbf {e}_{i}K_{i}\\\\$ In [21], we showed that these transformations $T_i$ are represented by certain unitary transformation given by conjugation of the Weyl elements.", "Hence we conjecture that Conjecture 8.5 The Lusztig's isomorphisms $T_i$ are represented by a sequence of quiver mutations.", "This will also give a representation theoretic meaning of the mutation sequences found explicitly for type $A_n,B_n,C_n,D_n$ [25] and $G_2$ [26], as well as proving the conjecture of $S_3$ symmetry regarding type $E_n$ and $F_4$ proposed in [26].", "Remark 8.6 In the product formula of $R$ described in the next section, the transformations $T_i$ generate the split-real version of the so-called quantum Weyl group introduced in [27], which is a byproduct of the representation theory of the quantized algebra of functions on $G$ , and is based on a choice of “good generators\" for certain representations of the quantized enveloping algebra.", "Through this conjecture, it will be interesting to recast the concept of quantum Weyl group into the language of cluster transformations.", "We thank Yan Soibelman for the remarks." ], [ "Factorization of the $R$ -matrix", "In this section, we will prove a factorization formula for the universal $R$ matrix such that it is expressed in terms of products of quantum dilogarithms, with the arguments given by monomials of the quantum cluster variables.", "This generalizes the factorization in type $A_n$ found in [39], which in turn is a generalization of the factorization given in [7] for $\\mathcal {U}_q(\\mathfrak {sl}_2)$ , where it has been used to construct new continuous braided tensor category of representations of $\\mathcal {U}_q(\\mathfrak {sl}(2,\\mathbb {R}))$ [3], [35], [36]." ], [ "Positive Lusztig's isomorphism", "First we recall the positive version of Lusztig's isomorphism giving the expression of non-simple root generators: Definition 9.1 [21] We define the “positive version\" of Lusztig's isomorphism on the simple generators by: $T_i(K_j)&:=K_jK_i^{-a_{ij}},\\\\T_i(\\mathbf {e}_i)&:=q_i f_iK_i^{-1},\\\\T_i(\\mathbf {e}_j)&:=(-1)^{a_{ij}}\\left[\\left[\\mathbf {e}_i,...[\\mathbf {e}_i,\\mathbf {e}_j]_{q_i^{\\frac{a_{ij}}{2}}}\\right]_{q_i^{\\frac{a_{ij}+2}{2}}}...\\right]_{q_i^{\\frac{-a_{ij}-2}{2}}}\\prod _{k=1}^{-a_{ij}}(q_i^k-q_i^{-k})^{-1},\\\\T_i(\\mathbf {f}_i)&:=q_i \\mathbf {e}_i K_i,\\\\T_i(\\mathbf {f}_j)&:=(-1)^{a_{ij}}\\left[\\left[\\mathbf {f}_i,...[\\mathbf {f}_i,\\mathbf {f}_j]_{q_i^{\\frac{a_{ij}}{2}}}\\right]_{q_i^{\\frac{a_{ij}+2}{2}}}...\\right]_{q_i^{\\frac{-a_{ij}-2}{2}}}\\prod _{k=1}^{-a_{ij}}(q_i^k-q_i^{-k})^{-1},$ where $[X,Y]_q:=qXY-q^{-1}YX.$ Proposition 9.2 [21] Let $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ be a reduced word.", "Let $\\mathbf {e}_{\\alpha _k}:=T_{i_1}T_{i_2}...T_{i_{k-1}}(\\mathbf {e}_{i_k})$ and similarly for $\\mathbf {f}_{\\alpha _k}$ .", "Then $\\mathbf {e}_{\\alpha _k}$ and $\\mathbf {f}_{\\alpha _k}$ are positive self-adjoint operators under the positive representation $\\mathcal {P}_\\lambda $ for every $k=1,...,N$ ." ], [ "Coproduct of $\\mathfrak {D}_\\mathfrak {g}$ and the {{formula:0ad84b2f-56cb-4320-9d12-23a797680dbf}} -quiver", "The coalgebra structure of $\\mathcal {U}_q(\\mathfrak {g})$ can naturally be represented by amalgamation of two $\\mathcal {D}_\\mathfrak {g}$ quivers, associated to triangulations of a disk with two punctures and two marked points on the boundary: Figure: Triangulation of a disk with two punctures and two marked points.Definition 9.3 The $\\mathcal {Z}_\\mathfrak {g}$ quiver is obtained by amalgamating two $\\mathcal {D}_\\mathfrak {g}$ -quivers, where the frozen nodes $f_i^{n_i}$ of the first quiver is identified with $f_i^{-n_i}$ of the second quiver.", "For simplicity, we will denote the vertices of the second $\\mathcal {D}_\\mathfrak {g}$ -quiver by $\\lbrace {f^{\\prime }}_i^{-n_i}... {f^{\\prime }}_i^{n_i}, {e^{\\prime }}_i^0\\rbrace _{i\\in I}$ such that $f_i^{n_i}={f^{\\prime }}_i^{-n_i}$ in $Z_\\mathfrak {g}$ .", "We will also denote by $\\mathcal {Z}_\\mathfrak {g}$ the corresponding quantum torus algebra.", "Then one can easily observe the following Proposition 9.4 We have an embedding $(\\iota \\otimes \\iota ) \\circ \\Delta : \\mathfrak {D}_\\mathfrak {g}\\longrightarrow \\mathcal {Z}_\\mathfrak {g}\\subset \\mathcal {D}_\\mathfrak {g}\\otimes \\mathcal {D}_\\mathfrak {g}$ where the coproduct $\\Delta (\\mathbf {e}_i)$ (resp.", "$\\Delta (\\mathbf {f}_i)$ ) can be represented in the $Z_\\mathfrak {g}$ -quiver by concatenating the $E_i$ -path (resp.", "$F_i$ -path) of the two $\\mathcal {D}_\\mathfrak {g}$ quivers and ignoring the last vertex.", "The coproduct $\\Delta (K_i)$ (resp.", "$\\Delta (K_i^{\\prime })$ ) is given by the product of the monomials along the $E_i$ -paths of $\\Delta (\\mathbf {e}_i)$ (resp.", "$F_i$ -paths of $\\Delta (\\mathbf {f}_i)$ ).", "The iterated coproduct $\\Delta ^n(X), X\\in \\mathfrak {D}_\\mathfrak {g}$ can be obtained by amalgamating $n+1$ copies of $\\mathcal {D}_\\mathfrak {g}$ in the same way.", "We will consider $\\Delta (\\mathbf {f}_i)$ , where the other statements are similar.", "Recall that $\\Delta (\\mathbf {f}_i)=\\mathbf {f}_i\\otimes 1+K_i^{\\prime }\\otimes \\mathbf {f}_i.$ Then the first half of the $F_i$ -path in $\\mathcal {Z}_\\mathfrak {g}$ is the $F_i$ -path in the first $\\mathcal {D}_\\mathfrak {g}$ quiver, which gives the polynomial $\\mathbf {f}_i\\otimes 1$ .", "On the other hand, the second half of the $F_i$ -path in $\\mathcal {Z}_\\mathfrak {g}$ is obtained by multiplying the $F_i$ -path in the second copy of $\\mathcal {D}_\\mathfrak {g}$ quiver, and the product of the first half of the $F_i$ -path, which by definition represents $K_i^{\\prime }$ .", "Hence combining it gives $K_i^{\\prime }\\otimes \\mathbf {f}_i$ , and hence the concatenation of the $F_i$ -path represents $\\Delta (\\mathbf {f}_i)$ as desired." ], [ "Standard description of the universal $R$ -matrix", "Recall that the universal $R$ -matrix of the quantum group $\\mathcal {U}_q(\\mathfrak {g})$ is an element in certain completion of the tensor square $\\mathcal {R}\\in \\mathcal {U}_q(\\mathfrak {g})\\widehat{\\otimes } \\mathcal {U}_q(\\mathfrak {g})$ and it gives the braiding relation: $\\mathcal {R}\\Delta (X)=\\Delta ^{op}(X)\\mathcal {R},\\;\\;\\;\\;\\;\\;X\\in \\mathcal {U}_q(\\mathfrak {g})$ In [21], a natural expression of $R$ in the split real case is constructed.", "Given a reduced word $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ , We have the well-known decomposition $\\mathcal {R}=\\mathcal {K}\\overline{R}$ Here the Cartan part is given by $\\mathcal {K}=\\prod _{ij} q_i^{(A^{-1})_{ij} H_i\\otimes H_j}$ where $A$ is the Cartan matrix, and formally we write $K_i=:q_i^{H_i}$ .", "The reduced $R$ -matrix is given by $\\overline{\\mathcal {R}}=\\prod _{k=1}^N{}^{op} g_{b_{i_k}}(\\mathbf {e}_{\\alpha _k}\\otimes \\mathbf {f}_{\\alpha _k})$ where $\\mathbf {e}_{\\alpha _k}=T_{i_1}T_{i_2}...T_{i_{k-1}}\\mathbf {e}_{i_k}$ and similarly for $\\mathbf {f}_{\\alpha _k}$ .", "The product $\\Pi ^{op}$ is taken with $k=1$ from the right.", "Note that (see Remark REF ) if we write $g_{b}(x)=Exp_{q^{-2}}(-\\frac{x}{q-q^{-1}})$ , then (REF ) coincides with the well-known formula [32], [33], [27], [28].", "Also $\\mathcal {R}$ naturally extends to $\\mathfrak {D}_\\mathfrak {g}$ by replacing $H_i\\otimes H_j$ in $\\mathcal {K}$ with $-H_i\\otimes H_j^{\\prime }$ instead, where $K_j^{\\prime }=:q_j^{H_j^{\\prime }}$ .", "The action of the Cartan part on $\\mathfrak {D}_\\mathfrak {g}$ is easy to describe (see Section ).", "In particular, it describes a monomial transformation on the quantum torus algebra $\\mathcal {X}_\\mathbf {i}$ , where $X_{f_i^{-n_i}}$ and $X_{f_i^{n_i}}$ on both $\\mathcal {D}_\\mathfrak {g}$ components of the $Z_\\mathfrak {g}$ -quiver is modified, and this does not change the underlying quiver.", "Hence we will focus on studying the reduced $R$ -matrix, which corresponds to certain quiver mutations." ], [ "First factorization of the reduced $R$ -matrix", "Now we can state our second main result of the paper.", "Under the embedding $\\iota \\otimes \\iota :\\mathfrak {D}_\\mathfrak {g}\\otimes \\mathfrak {D}_\\mathfrak {g}\\longrightarrow \\mathcal {D}_\\mathfrak {g}\\otimes \\mathcal {D}_\\mathfrak {g}$ , we have the following factorization of the reduced $R$ -matrix, which generalizes the case of $\\mathcal {U}_q(\\mathfrak {sl}_2)$ first described by Faddeev [7], as well as the type $A_n$ case by [39].", "Theorem 9.5 Let $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ be a reduced word.", "Let us rewrite the embedding of $\\mathbf {f}_i$ from Proposition REF as $\\mathbf {f}_i&=F_i^{n_i,-}+...+F_i^{1,-}+F_i^{1,+}+...F_i^{n_i,+}\\\\&=\\sum _{i_k=i} X_k^- + \\sum _{i_k=i} X_k^+,\\;\\;\\;\\;\\;\\;k=1,...,N,$ where $X_{v(i,k)}^\\pm := F_i^{k,\\pm }.$ Then under the embedding $\\iota \\otimes \\iota $ , the reduced $R$ matrix factorization is given by $\\overline{R}=&g_{b_{i_N}}(\\mathbf {e}_{i_N}\\otimes X_N^+)...g_{b_{i_2}}(\\mathbf {e}_{i_2}\\otimes X_2^+)g_{b_{i_1}}(\\mathbf {e}_{i_1}\\otimes X_1^+)\\cdot \\nonumber \\\\&g_{b_{i_1}}(\\mathbf {e}_{i_1}\\otimes X_1^-)g_{b_{i_2}}(\\mathbf {e}_{i_2}\\otimes X_2^-)... g_{b_{i_N}}(\\mathbf {e}_{i_N}\\otimes X_N^-).$ We will prove the Theorem in Section .", "Since $\\mathbf {e}_i=\\Phi [u]e(-2p)\\Phi ^*$ for some unitary transformation by (REF ), and $[u]e(-2p)=e^{\\pi b_i(u-2p)}+e^{\\pi b_i(-u-2p)},$ each $\\mathbf {e}_i$ can also be split into $\\mathbf {e}_i=\\mathbf {e}_i^-+\\mathbf {e}_i^+$ such that $\\mathbf {e}_i^-\\mathbf {e}_i^+=q_i^{-2} \\mathbf {e}_i^+\\mathbf {e}_i^-,$ where $\\mathbf {e}_i^\\pm := \\Phi e^{\\pi b_i(\\mp u-2p)} \\Phi ^*.$ Then we have Corollary 9.6 Under the embedding $\\iota \\otimes \\iota $ , the reduced $R$ matrix can also be factorized as $\\overline{R}=R_4\\cdot R_3\\cdot R_2\\cdot R_1$ where $R_4=&g_{b_{i_N}}(\\mathbf {e}_{i_N}^+\\otimes X_N^+)... g_{b_{i_2}}(\\mathbf {e}_{i_2}^+\\otimes X_2^+)g_{b_{i_1}}(\\mathbf {e}_{i_1}^+\\otimes X_1^+),\\\\R_3=&g_{b_{i_N}}(\\mathbf {e}_{i_N}^-\\otimes X_N^+)... g_{b_{i_2}}(\\mathbf {e}_{i_2}^-\\otimes X_2^+)g_{b_{i_1}}(\\mathbf {e}_{i_1}^-\\otimes X_1^+),\\\\R_2=&g_{b_{i_1}}(\\mathbf {e}_{i_1}^+\\otimes X_1^-)g_{b_{i_2}}(\\mathbf {e}_{i_2}^+\\otimes X_2^-)... g_{b_{i_N}}(\\mathbf {e}_{i_N}^+\\otimes X_N^-),\\\\R_1=&g_{b_{i_1}}(\\mathbf {e}_{i_1}^-\\otimes X_1^-)g_{b_{i_2}}(\\mathbf {e}_{i_2}^-\\otimes X_2^-)... g_{b_{i_N}}(\\mathbf {e}_{i_N}^-\\otimes X_N^-).$ Note that from the remark above, $g_{b_{i_n}}(\\mathbf {e}_{i_n}\\otimes X_n) = g_{b_{i_n}}(\\mathbf {e}_{i_n}^+\\otimes X_n)g_{b_{i_n}}(\\mathbf {e}_{i_n}^-\\otimes X_n).$ Hence it suffices to show that we can arrange all the $\\mathbf {e}_i^+$ to the left hand side of $\\mathbf {e}_j^-$ in $R_1$ and $R_2$ (similarly for $R_3$ and $R_4$ to the right).", "This is equivalent to the statement: $\\left[\\mathbf {e}_{i_n}^-\\otimes X_n^-, \\mathbf {e}_{i_m}^+\\otimes X_m^+\\right] = 0, \\;\\;\\;\\;\\;\\;n>m.$ T2his follows from Lemma REF and $\\mathbf {e}_i^+\\mathbf {e}_j^- = q_i^{a_{ij}} \\mathbf {e}_j^-\\mathbf {e}_i^+$ by conjugating it to the rank 2 case.", "This simplies the proof of [39] as well as generalizing it to arbitrary type." ], [ "Full factorization of the reduced $R$ matrix", "In order to realize the $R$ matrix factorization as certain quiver mutation sequences, we have to decompose the terms $g_{b_i}(\\mathbf {e}_i^\\pm \\otimes X_N^\\pm )$ in the decomposition in Corollary REF .", "In other words, we have to decompose $g_{b_i}(\\mathbf {e}_i)$ .", "Then for the monomial terms that we obtain after the decomposition, we compare it with Lemma REF in order to obtain the mutation sequence.", "Proposition 9.7 For every generators $\\mathbf {e}_i\\in \\mathcal {U}_q(\\mathfrak {g})$ , consider the explicit embedding given in Section for the chosen reduced word $\\mathbf {i}$ .", "Then we can decompose $g_{b_i}(\\mathbf {e}_i^\\pm )$ into products of the form $g_{b_i}(\\mathbf {e}_i^\\pm ) = \\prod g_b(X_{...}),$ (in type $G_2$ we also need $g_b^*$ ) where each argument is given by certain cluster monomials $X_{...}$ .", "It suffices to consider $g_{b_i}(\\mathbf {e}_i^-)$ , while the decomposition for $g_{b_i}(\\mathbf {e}_i^+)$ is just a reflection.", "For the generators $\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}\\mathbf {e}_i& A_n,B_n, D_n,E_n,\\\\\\mathbf {e}_i,i\\ne 1& C_n,\\\\\\mathbf {e}_4&F_4,\\\\\\mathbf {e}_2&G_2, \\\\ \\end{array}\\right.$ let us write the embedding of the generators of $\\mathbf {e}_i^-$ as a sum of monomials in the form $\\mathbf {e}_i^-=E_i^0+E_i^1+...+E_i^{m_i},$ where $E_i^0=X_{f_i^{n_i}}$ and ends before the next term $E_i^{m_i+1} = X_{f_i^{n_i},..., e_i^0}$ .", "Note that $\\mathbf {e}_4^-$ in type $F_4$ and $\\mathbf {e}_2^-$ in type $G_2$ is just a monomial, hence the statement is trivial.", "For all generators except $\\mathbf {e}_1$ in type $B_n$ and $\\mathbf {e}_6$ in type $E_8$ , we have $E_i^kE_i^l = q^2 E_i^l E_i^k$ for $k>l$ , hence we can apply () to obtain $g_{b_i}(\\mathbf {e}_i^-)=g_{b_i}(E_i^{m_i})...g_{b_i}(E_i^1)g_{b_i}(E_i^0).$ For $\\mathbf {e}_1$ in type $B_n$ , since $E_1^{2n+1}E_1^{2n}=q^2 E_1^{2n}E_1^{2n+1}$ , using (REF ) we obtain $g_{b_s}(\\mathbf {e}_1^-)=g_{b_s}(E_i^{m_i})...g_{b_s}(E_i^3)g_b(qE_i^2E_i^3)g_{b_s}(E_i^2)g_{b_s}(E_i^1)g_{b}(qE_i^0E_i^1)g_{b_s}(E_i^0).$ For $\\mathbf {e}_6$ in type $E_8$ , the path comes in blocks as follows $\\mathbf {e}_6^-=&X_{f_6^9}+(X_{f_6^9,f_7^2}+X_{f_6^9,f_7^2,f_6^6}) + ... (X_{...f_4^{18}}+X_{...f_3^{23}})\\\\&+(X_{...f_3^{24}}+X_{...,f_3^{24},f_0^{11}})+(X_{...,f_3^{24},f_2^{15}}+X_{...,f_3^{24},f_2^{15},f_0^{11}})\\\\&++(X_{... f_3^{22}}+X_{...f_3^{23}})+... + (X_{...f_6^5}+X_{...f_6^6})+X_{...f_7^1}+X_{...f_6^3}.$ One can check that each block $q^{-2}$ -commutes with all the blocks to the right of it, and within each block the two terms also $q^{-2}$ -commute with each other.", "Hence apply repeatedly () we arrive at the decomposition of the same form as others.", "For the long generators $\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}\\mathbf {e}_1&C_n,\\\\\\mathbf {e}_1,\\mathbf {e}_2&F_4, \\\\ \\end{array}\\right.$ let us write the embedding of the generators $\\mathbf {e}_i^-$ as $\\mathbf {e}_i^-&=E_i^0+E_i^1+...+E_i^k+[2]_{q_s}(q^2E_i^kE_i^{k+1})^{\\frac{1}{2}}+E_i^{k+1}+...\\\\&=E_i^0+E_i^1+...+((E_i^k)^{1/2}+(E_i^{k+1})^{1/2})^2+...$ whenever we have the double term $(...a,*b,...)$ appear in the $E_i$ path such that $E_i^k:=X_{...,a}, \\;\\;\\;\\;\\;\\;(qE_i^kE_i^{k+1})^{1/2}=X_{...a,b}, \\;\\;\\;\\;\\;\\;E_i^{k+1}:=X_{...,a,b^2}.$ Then each block $q^{-2}$ -commutes with the terms on the right, and since $E_i^{k+1}E_i^k=q^4 E_i^kE_i^{k+1}$ , by (REF ), we have $g_b(\\mathbf {e}_i^-)=...g_b(E_i^{k+1})g_{b_s}((qE_i^kE_i^{k+1})^{\\frac{1}{2}})g_b(E_i^k)...g_b(E_i^1)g_b(E_i^0)$ whenever the double term appears in the $E_i$ -path.", "The remaining two special cases are as follows: The generator $\\mathbf {e}_3$ of type $F_4$ is given by: $\\mathbf {e}_3^-&=X_{f_3^9}+(X_{f_3^9,f_4^2}+X_{f_3^9,f_4^2,f_3^6})+(X_{...f_3^7}+X_{...f_2^6})\\\\&+(X_{...f_3^5}+X_{...f_3^6})+((X_{...f_4^1}+X_{...f_3^3}))+((X_{...f_2^3}+X_{...f_3^2}))+X_{...f_2^1}+X_{...f_3^1},$ where each blocks $q_s^{-2}$ commute with all the blocks to the right of it.", "Within each block, the terms inside the single brackets $q_s^{-2}$ commute, while the terms inside double brackets $q^{-2}$ commute.", "Hence by () and (REF ) we can decompose $g_{b_s}(\\mathbf {e}_3^-)$ .", "For the generator $\\mathbf {e}_2$ in type $G_2$ , we have to involve conjugations, which gives $&g_{b_s}(\\mathbf {e}_2^-)\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1}+X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1}+X_{f_2^3,f_1^2,(f_2^2)^2}+[2]_{q_s}X_{f_2^3,f_1^2,f_2^2}+X_{f_2^3,f_1^2}+X_{f_2^3})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1}+X_{f_2^3,f_1^2,(f_2^2)^2}+[2]_{q_s}X_{f_2^3,f_1^2,f_2^2}+X_{f_2^3,f_1^2}+X_{f_2^3})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_b^*(X_{f_1^1})g_{b_s}(X_{f_2^3,f_1^2,(f_2^2)^2}+[2]_{q_s}X_{f_2^3,f_1^2,f_2^2}+X_{f_2^3,f_1^2}+X_{f_2^3,f_1^2,f_1^1}+X_{f_2^3})g_b(X_{f_1^1})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_b^*(X_{f_1^1})g_b^*(X_{f_1^1,f_1^2})g_{b_s}(X_{f_2^3,f_1^2,(f_2^2)^2}+[2]_{q_s}X_{f_2^3,f_1^2,f_2^2}+X_{f_2^3,f_1^2}+X_{f_2^3})g_b(X_{f_1^1,f_1^2})g_b(X_{f_1^1})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_b^*(X_{f_1^1})g_b^*(X_{f_1^1,f_1^2})g_{b_s}^*(X_{f_2^2})g_{b_s}(X_{f_2^3,f_1^2}+X_{f_2^3}+X_{f_2^3,f_2^2})g_{b_s}(X_{f_2^2})g_b(X_{f_1^1,f_1^2})g_b(X_{f_1^1})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_b^*(X_{f_1^1})g_b^*(X_{f_1^1,f_1^2})g_{b_s}^*(X_{f_2^2})g_b^*(X_{f_1^2})g_{b_s}(X_{f_2^3}+X_{f_2^3,f_2^2})g_b(X_{f_1^2})g_{b_s}(X_{f_2^2})g_b(X_{f_1^1,f_1^2})g_b(X_{f_1^1})\\\\&=g_{b_s}(X_{f_2^2,f_1^2,(f_2^2)^2,f_1^1,f_2^1})g_b^*(X_{f_1^1})g_b^*(X_{f_1^1,f_1^2})g_{b_s}^*(X_{f_2^2})g_b^*(X_{f_1^2})g_{b_s}(X_{f_2^3})g_{b_s}(X_{f_2^3,f_2^2})g_b(X_{f_1^2})g_{b_s}(X_{f_2^2})g_b(X_{f_1^1,f_1^2})g_b(X_{f_1^1}).$" ], [ "Universal $R$ -matrix as half-Dehn twist", "Finally we state the final main result of the paper.", "Consider the $\\mathcal {Z}_\\mathfrak {g}$ -quiver associated to the triangulation of the disk with two marked points $A,C$ and two punctures $B,D$ as before, where the basic quiver $Q$ and its mirror image $\\widetilde{Q}$ are put onto the triangles according to Section , and we label the nodes using the standard form.", "Let $P$ be the permutation $P(X\\otimes Y):=Y\\otimes X.$ Note that $P\\circ Ad_\\mathcal {R}$ acts as identity on the coalgebra structure, hence it naturally corresponds to an automorphism of seed $\\mathbf {i}\\longrightarrow \\mathbf {i}$ .", "Theorem 10.1 We have $P\\circ Ad_{\\mathcal {R}}=(\\mu _{i_1}^q...\\mu _{i_T}^q\\circ \\sigma ^*)^{-1}= (\\sigma ^*)^{-1}\\circ \\mu _{i_T}^q...\\mu _{i_1}^q$ for some mutation sequence $\\mu _{i_T}...\\mu _{i_1}:\\mathbf {i}\\longrightarrow \\mathbf {i}^{\\prime }$ realizing the half-Dehn twist, and $\\sigma :\\mathbf {i}^{\\prime }\\simeq \\mathbf {i}$ is a permutation of the quiver returning to the original seed.", "More precisely, recall from Lemma REF that $\\mu _{i_1}^q...\\mu _{i_T}^q=\\Phi _T \\circ M_T.$ Then we have $Ad_{\\overline{R}}&=\\Phi _T^{-1},\\\\P\\circ Ad_{\\mathcal {K}}&=(\\sigma ^*)^{-1}\\circ M_T^{-1}.$ The factors $R_1,R_2,R_3,R_4$ in (REF ) correspond to the sequences of quiver mutations realizing the 4 flips of triangulations giving the half-Dehn twist as follows: Figure: Half-Dehn twist.Remark 10.2 We note that the mutation sequence is not unique, for example, using (REF ) one can replace 2 $g_b$ 's with 3 $g_b$ 's, thus giving the same mutation (with different permutation index at the end) but with a longer sequence.", "In terms of the quiver associated to the triangulations, the 4 flips are realized as follows.", "Let us write $\\mu _{R_i}$ for the sequence of quiver mutations (starting from a standard form) corresponding to $R_i$ , and $\\sigma _i$ the permutation bringing the labeling of the basic quivers back to the standard form.", "Then we have the following configuration: Figure: Flip of triangulations corresponding to R 1 R_1.Figure: Flip of triangulations corresponding to R 2 R_2.Figure: Flip of triangulations corresponding to R 3 R_3.Figure: Flip of triangulations corresponding to R 4 R_4.After the 4 flips, the quiver comes back to the original configuration with $B\\longleftrightarrow D$ , $Q\\longleftrightarrow Q^{\\prime }, \\widetilde{Q}\\longleftrightarrow \\widetilde{Q^{\\prime }}$ , and we have $\\sigma =\\sigma _4\\circ \\sigma _3\\circ \\sigma _2\\circ \\sigma _1.$ Figure: Half-Dehn twist of the quiver 𝒵 𝔤 \\mathcal {Z}_\\mathfrak {g}.We observe that for each flip, one can think of the quiver mutation as rotating both basic quivers clockwise by 30 degree, and then stack the right quiver on top of the left one.", "In the next subsection, we will show how to obtain such mutation sequence." ], [ "Explicit mutation sequence for the Half-Dehn twist", "By the symmetry of the decomposition (REF ) as well as the mutation configurations, we can see that $R_2$ and $R_3$ commute, and the mutation sequence corresponding to $R_4$ and $R_3$ in some sense are just “mirror images\" to those of $R_1$ and $R_2$ respectively.", "Using the explicit decomposition from Proposition REF , we arrive at the following more precise description of the quiver mutation giving the half-Dehn twist: Proposition 10.3 The mutation sequence is a mirrored palindrome: $\\mu _{R_4}&=(\\rho \\sigma _1)_* (\\mu _{R_1}^{-1}),& \\sigma _4&=\\rho _*(\\sigma _1^{-1}),\\\\\\mu _{R_3}&=(\\rho \\sigma _2)_* (\\mu _{R_2}^{-1}),& \\sigma _3&=\\rho _*(\\sigma _2^{-1}),$ where $\\rho $ is the permutation given by the reflection $f_i^k\\longleftrightarrow f_i^{-k}$ : $\\rho : \\lbrace f_i^k, e_i^0, {f^{\\prime }}_i^k, {e^{\\prime }}_i^0\\rbrace \\longleftrightarrow \\lbrace f_i^{-k}, e_i^0, {f^{\\prime }}_i^{-k}, {e^{\\prime }}_i^0\\rbrace ,$ and for a permutation $\\pi $ we denote by $\\pi _*(X):=\\pi \\circ X \\circ \\pi ^{-1}$ .", "Hence below we will only study the mutation sequence corresponding to $R_1$ and $R_2$ .", "To describe the mutation sequence, let us define the following notation: Definition 10.4 Let $S=(s_0,...,s_n),T=(t_0,...,t_m)$ be two sequences (of the nodes of some quiver).", "If $s_n$ and $t_1$ denote the same node in the quiver, then we define a new sequence of length $n+m+1$ : $\\langle S- T\\rangle :=(s_0,...,s_n=t_0,...,t_m)$ to be the concatenation of the two sequences, and it is indexed from $-n$ to $m$ such that $\\langle S-T\\rangle _0= s_n = t_0.$ If $\\mathcal {P}$ is a sequence constructed in this way, then we define its flip to be $\\mathcal {F}(\\mathcal {P}):=\\langle T-S\\rangle $ whenever $t_m=s_0$ in some other quiver in which this sequence is indexing.", "Definition 10.5 If $\\mu _T:=\\mu _{j_M}...\\mu _{j_1}$ is a mutation sequence, we alternatively write it as $\\mu _T=:\\lbrace j_1\\longrightarrow j_2\\longrightarrow ...\\longrightarrow j_M\\rbrace .$ Then given a sequence $\\mathcal {P}$ , we denote the $k$ -shifted mutation subsequence of length $m$ by $\\mathcal {P}[k,m]:=\\lbrace \\mathcal {P}_{1-k}\\longrightarrow \\mathcal {P}_{2-k}\\longrightarrow ...\\longrightarrow \\mathcal {P}_{m-k}\\rbrace .$ Definition 10.6 We define the sequences $\\mathcal {P}_{E_i}^{Q}&:=(f_i^{n_i},..., e_i^0),\\\\\\mathcal {P}_{E_i}^{\\widetilde{Q}}&:=(e_i^0,..., f_i^{-n_i}),\\\\\\mathcal {P}_{F_i}^{Q}&:=(f_i^{n_i},..., f_i^0),\\\\\\mathcal {P}_{F_i}^{\\widetilde{Q}}&:=(f_i^0,..., f_i^{-n_i})$ to be the $E_i$ and (reverse of) $F_i$ paths of the quiver $Q$ and $\\widetilde{Q}$ respectively.", "Similarly we use ${}^{\\prime }$ to denote the corresponding paths in the second quiver $1\\otimes \\mathcal {D}_\\mathfrak {g}\\subset \\mathcal {Z}_\\mathfrak {g}$ .", "Finally, given a reduced word $\\mathbf {i}$ , we denote by $\\mathbf {i}^{\\prime }$ the reversed word, and recall that (cf.", "Definition REF ) $v^{\\prime }(i,k)=m$ if $i_m$ is the $k$ -th appearance of the root index $i$ from the left of $\\mathbf {i}^{\\prime }$ , i.e.", "right of $\\mathbf {i}$ ." ], [ "Toy Example: Type $A_2$", "To demonstrate the procedure, let us first look at the toy example in type $A_2$ in detail using the notation of our paper.", "This has also been worked out in detail in [39] with slightly different notations.", "Figure: 𝒵 A 2 \\mathcal {Z}_{A_2}-quiver.Recall the embedding of type $A_2$ from Section REF .", "First of all, the $g_b(\\mathbf {e}_i)$ can be easily decomposed using () with $g_b(\\mathbf {e}_i)=g_b(\\mathbf {e}_i^+)g_b(\\mathbf {e}_i^-)$ as $g_b(\\mathbf {e}_1^+)&=g_b(X_{f_1^2,e_1^0}),\\\\g_b(\\mathbf {e}_1^-)&=g_b(X_{f_1^2}),\\\\g_b(\\mathbf {e}_2^+)&=g_b(X_{f_2^1,f_1^1,e_2^0,f_1^{-1}})g_b(X_{f_2^1,f_1^1,e_2^0}),\\\\g_b(\\mathbf {e}_2^-)&=g_b(X_{f_2^1,f_1^1})g_b(X_{f_2^1}).$ Hence by Corollary REF , the reduced $R$ matrix decomposed as: $R_4&=g_b(X_{f_1^2,e_1^0}\\otimes X_{f_1^{-2},f_1^{-1},f_1^0,f_1^2})g_b(X_{f_2^1,f_1^2,e_2^0,f_1^{-1}}\\otimes X_{f_2^{-1},f_2^0})\\\\&\\;\\;\\;\\;\\;\\;g_b(X_{f_2^1,f_1^2,e_2^0}\\otimes X_{f_2^{-1},f_2^0})g_b(X_{f_1^2,e_1^0}\\otimes X_{f_1^{-2},f_1^{-1},f_1^0}).\\\\R_3&=g_b(X_{f_1^2}\\otimes X_{f_1^{-2},f_1^{-1}})g_b(X_{f_2^1,f_1^1}\\otimes X_{f_2^{-1}})g_b(X_{f_2^1}\\otimes X_{f_2^{-1}})g_b(X_{f_1^2}\\otimes X_{f_1^{-2}}).\\\\R_2&=g_b(X_{f_1^2,e_1^0}\\otimes X_{f_1^{-2},f_1^{-1}})g_b(X_{f_2^1,f_1^2,e_2^0,f_1^{-1}}\\otimes X_{f_2^{-1}})g_b(X_{f_2^1,f_1^2,e_2^0}\\otimes X_{f_2^{-1}})g_b(X_{f_1^2,e_1^0}\\otimes X_{f_1^{-2}}).\\\\R_1&=g_b(X_{f_1^2}\\otimes X_{f_1^{-2},f_1^{-1}})g_b(X_{f_2^1,f_1^1}\\otimes X_{f_2^{-1}})g_b(X_{f_2^1}\\otimes X_{f_2^{-1}})g_b(X_{f_1^2}\\otimes X_{f_1^{-2}}).$ Then we calculate term by term the corresponding mutation sequence (recall that $f_i^{n_i}$ is glued to ${f^{\\prime }}_i^{-n_i}$ ): $X_{f_1^2}\\otimes X_{f_1^{-2}}&\\sim \\mu _{f_1^2}\\\\X_{f_2^1}\\otimes X_{f_2^{-1}} = \\mu ^{\\prime }_{f_1^2}(X^{\\mu }_{f_2^1}\\otimes X^{\\mu }_{f_2^{-1}}))&\\sim \\mu _{f_2^1}\\\\X_{f_2^1,f_1^1}\\otimes X_{f_2^{-1}}=\\mu ^{\\prime }_{f_1^2}\\mu ^{\\prime }_{f_{2^1}}(X^{\\mu ^2}_{f_1^1}\\otimes 1)&\\sim \\mu _{f_1^1}\\\\X_{f_1^2}\\otimes X_{f_1^{-2},f_1^{-1}}=\\mu ^{\\prime }_{f_1^2}\\mu ^{\\prime }_{f_2^1}\\mu ^{\\prime }_{f_1^1}(1\\otimes X^{\\mu ^3}_{f_1^{-1}})&\\sim \\mu _{{f^{\\prime }}_1^{-1}}\\\\\\cdots &\\sim \\cdots $ and so on, where we denoted by $X^{\\mu ^n}$ the corresponding mutated quantum cluster variables after $n$ mutations (but we do not change the labels).", "Then we obtain: $\\mu _{R_1}&=\\mu _{{f^{\\prime }}_1^{-1}}\\mu _{f_1^1}\\mu _{{f^{\\prime }}_2^{-1}}\\mu _{{f^{\\prime }}_1^{-2}}, &\\sigma _1&=({f^{\\prime }}_2^0, {f^{\\prime }}_2^{-1}, f_1^1, e_2^0)(e_1^0, {f^{\\prime }}_1^{-2}, {f^{\\prime }}_1^{-1}, {f^{\\prime }}_1^0),\\\\\\mu _{R_2}&=\\mu _{{f^{\\prime }}_1^{-1}}\\mu _{f_1^{-1}}\\mu _{{f^{\\prime }}_2^{-1}}\\mu _{{f^{\\prime }}_1^{-2}}, &\\sigma _2&=({f^{\\prime }}_2^0,{f^{\\prime }}_2^{-1}, {f}_1^{-1}, f_2^{-1})(f_1^{-2},{f^{\\prime }}_1^{-2},{f^{\\prime }}_1^{-1},{f^{\\prime }}_1^0),\\\\\\mu _{R_3}&=\\mu _{{f^{\\prime }}_1^1}\\mu _{f_1^1}\\mu _{{f^{\\prime }}_2^0}\\mu _{{f^{\\prime }}_1^0},&\\sigma _3&=({f^{\\prime }}_2^1,{f^{\\prime }}_2^0,f_1^1,e_2^0)(e_1^0,{f^{\\prime }}_1^0,{f^{\\prime }}_1^1,{f^{\\prime }}_1^2),\\\\\\mu _{R_4}&=\\mu _{{f^{\\prime }}_1^1}\\mu _{f_1^{-1}}\\mu _{{f^{\\prime }}_2^0}\\mu _{{f^{\\prime }}_1^0},&\\sigma _4&=({f^{\\prime }}_2^1,{f^{\\prime }}_2^0,f_1^{-1},f_2^{-1})(f_1^{-2},{f^{\\prime }}_1^0,{f^{\\prime }}_1^1,{f^{\\prime }}_1^2).$ Note that $\\sigma _i$ are given by shifting along the concatenation of the $F_i$ path in the right quiver and $E_i$ path in the left quiver, and that the mutation corresponding to $R_3$ and $R_4$ are the mirror reflections of $R_2$ and $R_1$ satisfying Proposition REF .", "We display the configurations in Figure REF , omitting $R_3$ and $R_4$ .", "Also recall that $Q=\\widetilde{Q}$ in type $A_n$ due to the $S_3$ symmetry, hence in fact under this identification all 4 flips are identical.", "Figure: The flipping of triangle μ R 1 \\mu _{R_1} of the basic quivers, before changing the index back to standard form.Figure: The flipping of triangle μ R 2 \\mu _{R_2} of the basic quivers, before changing the index back to standard form." ], [ "Type $A_n$", "Let $\\mathbf {i}=(1213214321...n...1)$ be the usual reduced word.", "To study $R_1$ and $R_2$ , let $\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}:= \\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{E_i}^Q\\rangle ,\\;\\;\\;\\;\\;\\;\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}:=\\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{E_i}^{\\widetilde{Q}}\\rangle $ be the concatenation of the $F_i, E_i$ path of the right and left quiver respectively.", "Then the mutation sequences $\\mu _{R_j}$ , $j=1,2$ , are given by $\\mu _{R_j}:=\\lbrace \\mathcal {P}_1^j\\longrightarrow \\mathcal {P}_2^j\\longrightarrow ...\\mathcal {P}_N^j\\rbrace $ where for $v^{\\prime }(i,k)=m$ , $\\mathcal {P}_m^j$ are the $k$ -shifted subsequences $\\mathcal {P}_m^1=\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}[k,i],\\;\\;\\;\\;\\;\\;\\mathcal {P}_m^2=\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}[k,i].$ Let $\\mathcal {P}_i^{{}_Q^{\\widetilde{Q}^{\\prime }}}:= \\mathcal {F}(\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }})=\\langle \\mathcal {P}_{E_i}^Q-\\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}\\rangle ,\\;\\;\\;\\;\\;\\;\\mathcal {P}_i^{{}_{\\widetilde{Q}}^{\\widetilde{Q}^{\\prime }}}:=\\mathcal {F}(\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }})=\\langle \\mathcal {P}_{E_i}^{\\widetilde{Q}}-\\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}\\rangle $ be the concatenation of the $E_i, F_i$ path of the bottom and top quiver respectively after the flip of triangulation.", "The permutations $\\sigma _1,\\sigma _2$ are then defined by renaming the corresponding sequence: $\\sigma _1:\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}\\mapsto \\mathcal {P}_i^{{}_Q^{\\widetilde{Q}^{\\prime }}},\\;\\;\\;\\;\\;\\;\\sigma _2:\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}\\mapsto \\mathcal {P}_i^{{}_{\\widetilde{Q}}^{\\widetilde{Q}^{\\prime }}}$ For example, in type $A_3$ , we have $\\mathbf {i}=(1,2,1,3,2,1)$ and $\\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_1^0,{f^{\\prime }}_1^{-1},{f^{\\prime }}_1^{-2},{f^{\\prime }}_1^{-3}=f_1^3,e_1^0),\\\\\\mathcal {P}_2^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_2^0,{f^{\\prime }}_2^{-1},{f^{\\prime }}_2^{-2}=f_2^2,f_1^2,e_2^0),\\\\\\mathcal {P}_3^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_3^0,{f^{\\prime }}_3^{-1}=f_3^1,f_2^1,f_1^1,e_1^0),$ and hence the mutation sequence giving the first flip of triangulations is $\\mu _{R_1}&=\\lbrace f_1^3\\longrightarrow f_1^2\\longrightarrow f_2^2\\longrightarrow f_1^1\\longrightarrow f_2^1\\longrightarrow f_3^1\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow {f^{\\prime }}_2^{-1}\\longrightarrow f_2^2\\longrightarrow {f^{\\prime }}_1^{-1}\\rbrace .$" ], [ "Type $B_n$ and {{formula:f92a9642-78b8-46e4-bd97-c9516498e254}}", "Let the reduced word $\\mathbf {i}$ be as in (REF ).", "It turns out that type $B_n$ and type $C_n$ have identical mutation sequences.", "Define the sequence $\\mathcal {S}:&=(f_1^n,f_1^{n-1},f_1^{n-2},...,f_1^1,e_1^0),\\\\\\mathcal {S}^{\\prime }:&=(e_1^0,f_1^1,f_1^2,...,f_1^n),$ and let $\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}&:=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}\\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{E_i}^Q\\rangle & i\\ne 1,\\\\(\\mathcal {P}_{F_1}^{\\widetilde{Q}^{\\prime }}-\\mathcal {S}\\rangle & i=1, \\\\ \\end{array}\\right.\\\\\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}&:= \\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}\\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{E_i}^{\\widetilde{Q}}\\rangle &i\\ne 1,\\\\\\langle \\mathcal {P}_{F_1}^{\\widetilde{Q}^{\\prime }}-\\mathcal {S}^{\\prime }\\rangle &i=1.", "\\\\ \\end{array}\\right.$ Then the mutation sequences $\\mu _{R_j}$ , $j=1,2$ , are given by $\\mu _{R_j}:=\\lbrace \\mathcal {P}_1^j\\longrightarrow \\mathcal {P}_2^j\\longrightarrow ...\\mathcal {P}_N^j\\rbrace $ For $i\\ne 1$ and $v^{\\prime }(i,k)=m$ , $\\mathcal {P}_m^j$ are the $k$ -shifted subsequences $\\mathcal {P}_m^1=\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}[k,m_i],\\;\\;\\;\\;\\;\\;\\mathcal {P}_m^2=\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}[k,m_i],$ where $m_i=2(n-i)+1$ is the length of the $E_i$ path.", "For $i=1$ and $v^{\\prime }(1,k)=m$ , let $f_1^n\\longrightarrow f_2^{2n-3}\\longrightarrow f_1^{n-1}\\longrightarrow f_2^{2n-5}\\longrightarrow f_1^{n-2}\\longrightarrow ...\\longrightarrow f_1^2\\longrightarrow f_2^1\\longrightarrow f_1^1$ be the $E_i$ path of $\\mathbf {e}_1$ in $Q$ (ignore the double count in type $C_n$ ).", "Then $\\mathcal {P}_m^1=&\\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(1-k)\\longrightarrow f_2^{2n-3}\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(1-k)\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(2-k)\\longrightarrow f_2^{2n-5}\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(2-k) ...\\\\&...\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(n-k-1)\\longrightarrow f_2^1\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(n-k-1)\\longrightarrow \\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}(n-k).$ Let $e_1^0\\longrightarrow f_1^{-1}\\longrightarrow f_2^{-1}\\longrightarrow f_1^{-2}\\longrightarrow ...\\longrightarrow f_1^{-(n-1)}\\longrightarrow f_2^{-(2n-3)}\\longrightarrow f_1^{-n}$ be the $E_i$ path of $\\mathbf {e}_1$ in $\\widetilde{Q}$ .", "Then $\\mathcal {P}_m^2=&\\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(1-k)\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(2-k)\\longrightarrow f_2^{-1}\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(2-k)\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(3-k)\\longrightarrow f_2^{-3}\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(3-k) ...\\\\&...\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(n-k)\\longrightarrow f_2^{-(2n-3)}\\longrightarrow \\mathcal {P}_1^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}(n-k).$ Let $\\mathcal {P}_i^{{}_Q^{\\widetilde{Q}^{\\prime }}}:= \\mathcal {F}(\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}),\\;\\;\\;\\;\\;\\;\\mathcal {P}_i^{{}_{\\widetilde{Q}}^{\\widetilde{Q}^{\\prime }}}&:= \\mathcal {F}(\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }})$ Then the permutations $\\sigma _1,\\sigma _2$ are again defined by renaming the corresponding sequence: $\\sigma _1:\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}\\mapsto \\mathcal {P}_i^{{}_Q^{\\widetilde{Q}^{\\prime }}},\\;\\;\\;\\;\\;\\;\\sigma _2:\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}\\mapsto \\mathcal {P}_i^{{}_{\\widetilde{Q}}^{\\widetilde{Q}^{\\prime }}}$ For example, in type $B_3$ , we have $\\mathbf {i}=(1,2,1,2,3,2,1,2,3)$ and $\\mathcal {P}_1^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_1^0,{f^{\\prime }}_1^{-1},{f^{\\prime }}_1^{-2},{f^{\\prime }}_1^{-3}=f_1^3,f_1^2,f_1^1,e_1^0),\\\\\\mathcal {P}_2^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_2^0,{f^{\\prime }}_2^{-1},{f^{\\prime }}_2^{-2},{f^{\\prime }}_2^{-3},{f^{\\prime }}_2^{-4}=f_2^4,f_3^1,f_2^2,e_2^0),\\\\\\mathcal {P}_3^{Q\\widetilde{Q}^{\\prime }}&: ({f^{\\prime }}_3^0,{f^{\\prime }}_3^{-1},{f^{\\prime }}_3^{-2}=f_3^2,e_3^0),$ and hence the mutation sequence giving the first flip of triangulations is (spacing according to $\\mathbf {i}^{\\prime }$ ): $\\mu _{R_1}=\\lbrace &f_3^2\\longrightarrow \\\\&f_2^4\\longrightarrow f_3^1\\longrightarrow f_2^2\\longrightarrow \\\\&f_1^3 \\longrightarrow f_2^3\\longrightarrow f_1^3\\longrightarrow f_1^2\\longrightarrow f_2^1\\longrightarrow f_1^2\\longrightarrow f_1^1\\longrightarrow \\\\&{f^{\\prime }}_2^{-3}\\longrightarrow f_2^4\\longrightarrow f_3^1\\longrightarrow \\\\& {f^{\\prime }}_3^{-1}\\longrightarrow \\\\& {f^{\\prime }}_2^{-2}\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow f_2^4\\longrightarrow \\\\& {f^{\\prime }}_1^{-2}\\longrightarrow f_2^3\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow f_1^3\\longrightarrow f_2^1\\longrightarrow f_1^3\\longrightarrow f_1^2\\longrightarrow \\\\& {f^{\\prime }}_2^{-1}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow \\\\& {f^{\\prime }}_1^{-1}\\longrightarrow f_2^3\\longrightarrow {f^{\\prime }}_1^{-1}\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow f_2^1\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow {f^{\\prime }}_1^{-3}\\rbrace .$" ], [ "Type $D_n$", "The description of the $D_n$ mutation sequences is a lot more complicated.", "Let the reduced word $\\mathbf {i}$ be as in (REF ).", "For $i\\ne 0,1$ , define as before $\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}=\\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{E_i}^{Q}\\rangle .$ Let $\\overline{n}:=n\\mbox{ (mod 3)}\\in \\lbrace 0,1,2\\rbrace $ and define the following sequences, which are constructed by repeating in blocks of 4: $\\mathcal {S}_1&=(X_1, ...,{f^{\\prime }}_0^{-3k+\\overline{n}-2}, {f^{\\prime }}_0^{-3k-\\overline{n}-1},{f^{\\prime }}_1^{-3k-\\overline{n}-1}, {f^{\\prime }}_1^{-3k-\\overline{n}},...,{f^{\\prime }}_1^{-n+1}),\\\\\\mathcal {S}_2&=(X_2, ...,{f^{\\prime }}_0^{-3k-\\overline{n}-1}, {f^{\\prime }}_0^{-3k-\\overline{n}},{f^{\\prime }}_1^{-3k-\\overline{n}}, {f^{\\prime }}_1^{-3k-\\overline{n}+1},...,{f^{\\prime }}_0^{-n+1}),\\\\\\mathcal {S}_0&=(X_0, ...,{f^{\\prime }}_0^{-3k-\\overline{n}}, {f^{\\prime }}_0^{-3k-\\overline{n}+1},{f^{\\prime }}_1^{-3k-\\overline{n}+1}, {f^{\\prime }}_1^{-3k-\\overline{n}+2},...,{f^{\\prime }}_0^{-n+1}),$ where the starting terms are given by $X_i=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}{f^{\\prime }}_1^0&\\overline{n}=i,\\\\{f^{\\prime }}_0^0&\\mbox{otherwise}.", "\\\\ \\end{array}\\right.$ Let $\\mathcal {T}_0=(f_0^{n-1}, f_1^{n-1},f_1^{n-2},f_0^{n-2},f_0^{n-3},f_1^{n-3},..., f_\\epsilon ^1,f_{1-\\epsilon }^1,e_1^0)$ where $\\epsilon :=n\\mbox{ (mod 2)}\\in \\lbrace 0,1\\rbrace $ .", "Let $\\mathcal {T}_1=\\mathcal {P}_{E_1^\\#}^{Q}$ denote the $E_1$ path in $Q$ , but with the last term $e_1^0$ replaced by $e_0^0$ , and let $\\mathcal {T}_2=\\mathcal {P}_{E_0}^{Q}$ .", "Finally, we define $\\mathcal {U}_j:=\\langle \\mathcal {S}_j-\\mathcal {T}_j\\rangle , \\;\\;\\;\\;\\;\\;j=0,1,2.$ Then the mutation sequence for $R_1$ is given by $\\mu _{R_1}:=\\lbrace \\mathcal {P}_1\\longrightarrow \\mathcal {P}_2\\longrightarrow ...\\longrightarrow \\mathcal {P}_N\\rbrace .$ For $i\\ne 0,1$ and $v^{\\prime }(i,k)=m$ , we have as before $\\mathcal {P}_m=\\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}[k,m_i]$ where $m_i=2(n-i)-1$ is the length of the $E_i$ path.", "For $i=0,1$ and $v^{\\prime }(i,k)=m$ , we have $\\mathcal {P}_m&=\\mathcal {U}_{\\overline{k-1+i}}[K_k^i, 2n-3]$ where $K_k^0=(0,2,3,4,6,7,8,10,11,12...)$ and $K_k^1=(0,1,2,4,5,6,8,9,10...)$ .", "Then the permutation is given by $\\sigma _1: \\mathcal {P}_i^{Q\\widetilde{Q}^{\\prime }}\\mapsto \\mathcal {P}_i^{{}_Q^{\\widetilde{Q}^{\\prime }}},\\;\\;\\;\\;\\;\\;i\\ne 0,1$ and $\\sigma _1: \\mathcal {U}_j\\mapsto \\langle \\mathcal {T}_{\\overline{j+1-n}}-\\mathcal {S}_{\\overline{j+1}}\\rangle ,\\;\\;\\;\\;\\;\\;j=0,1,2.$ The second flip $R_2$ is described similarly, where all the sequences $\\mathcal {T}_j$ are reversed and the root indexes $0\\longleftrightarrow 1$ interchanged, and $\\mathcal {S}_j$ are replaced by $\\mathcal {S}_{\\overline{j-1}}$ .", "For example, in type $D_4$ , we have $\\mathbf {i}=(0,1,2,0,1,2,3,2,0,1,2,3)$ , and $\\mathcal {P}_2^{Q\\widetilde{Q}^{\\prime }}&=({f^{\\prime }}_2^0,{f^{\\prime }}_2^{-1},{f^{\\prime }}_2^{-2},{f^{\\prime }}_2^{-3}, {f^{\\prime }}_2^{-4}=f_2^4,f_3^1,f_2^2,e_2^0),\\\\\\mathcal {P}_3^{Q\\widetilde{Q}^{\\prime }}&=({f^{\\prime }}_3^0,{f^{\\prime }}_3^{-1},{f^{\\prime }}_3^{-2}=f_3^2, e_3^0),\\\\\\mathcal {U}_1&=({f^{\\prime }}_1^0,{f^{\\prime }}_0^{-1},{f^{\\prime }}_0^{-2},{f^{\\prime }}_1^{-2},{f^{\\prime }}_1^{-3}=f_1^3,f_2^3,f_0^2,f_2^1,f_1^1,e_0^0),\\\\\\mathcal {U}_2&=({f^{\\prime }}_0^0,{f^{\\prime }}_0^{-1},{f^{\\prime }}_1^{-1},{f^{\\prime }}_1^{-2},{f^{\\prime }}_0^{-3}=f_0^3,f_2^3,f_1^2,f_2^1,f_0^1,e_0^0),\\\\\\mathcal {U}_0&=({f^{\\prime }}_0^0,{f^{\\prime }}_1^0,{f^{\\prime }}_1^{-1},{f^{\\prime }}_0^{-2},{f^{\\prime }}_0^{-3}=f_0^3,f_1^3,f_1^2,f_0^2,f_0^1,f_1^1,e_1^0),$ and hence the mutation sequence giving the first flip of triangulations is (spacing according to $\\mathbf {i}^{\\prime }$ ): $\\mu _{R_1}=\\lbrace & f_3^2\\longrightarrow \\\\&f_2^4\\longrightarrow f_3^1\\longrightarrow f_2^2\\longrightarrow \\\\&f_1^3\\longrightarrow f_2^3\\longrightarrow f_0^2\\longrightarrow f_2^1\\longrightarrow f_1^1\\longrightarrow \\\\&f_0^3\\longrightarrow f_1^3\\longrightarrow f_1^2\\longrightarrow f_0^2\\longrightarrow f_0^1\\longrightarrow \\\\&{f^{\\prime }}_2^{-3}\\longrightarrow f_2^4\\longrightarrow f_3^1\\longrightarrow \\\\&{f^{\\prime }}_3^{-1}\\longrightarrow \\\\&{f^{\\prime }}_2^{-2}\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow f_2^4\\longrightarrow \\\\&{f^{\\prime }}_1^{-2}\\longrightarrow f_0^3\\longrightarrow f_2^3\\longrightarrow f_1^2\\longrightarrow f_2^1\\longrightarrow \\\\&{f^{\\prime }}_0^{-2}\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow f_1^3\\longrightarrow f_2^3\\longrightarrow f_0^2\\longrightarrow \\\\&{f^{\\prime }}_2^{-1}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow \\\\&{f^{\\prime }}_1^{-1}\\longrightarrow {f^{\\prime }}_0^{-2}\\longrightarrow f_0^3\\longrightarrow f_1^3\\longrightarrow f_1^2\\longrightarrow \\\\&{f^{\\prime }}_0^{-1}\\longrightarrow {f^{\\prime }}_1^{-1}\\longrightarrow {f^{\\prime }}_1^{-2}\\longrightarrow f_0^3\\longrightarrow f_2^3\\rbrace .$" ], [ "Exceptional types", "The mutation sequences can be worked out in the exception type, but there are no apparent patterns, so we will not present here.", "We know that the reduced $R$ matrix corresponds to $T=4\\prod _{i=1}^n n_i \\mathcal {E}_i$ mutations, where $\\mathcal {E}_i$ is the number of factors in the $g_b(\\mathbf {e}_i)$ decomposition as in Proposition REF .", "One check explicitly that indeed the mutation sequences give the half-Dehn twist.", "Combining with the classical types, we have Proposition 10.7 The half-Dehn twist can be represented by $T$ quiver mutations, where $T=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}\\frac{2}{3}n(n+1)(n+2)&\\mbox{Type $A_n$}\\\\\\frac{4}{3}n(4n^2-1)&\\mbox{Type $B_n$ and $C_n$}\\\\\\frac{4}{3}n(n-1)(4n-5)&\\mbox{Type $D_n$}\\\\1196&\\mbox{Type $E_6$}\\\\3464&\\mbox{Type $E_7$}\\\\12064&\\mbox{Type $E_8$}\\\\976&\\mbox{Type $F_4$}\\\\144&\\mbox{Type $G_2$} \\\\ \\end{array}\\right.$ and each flip of triangulations are given by $\\frac{T}{4}$ quiver mutations.", "In type $G_2$ , from the factorization of $g_{b_s}(\\mathbf {e}_2)$ , we see that it involves the factor $g_{b_s}^*(X_{...})$ .", "We use the fact that $\\mu _k^q&=Ad_{g_b^*(X_k)}\\circ \\mu _k^{\\prime },\\\\&=Ad_{g_b(X_k^{-1})}\\circ \\mu _k^{\\prime \\prime },$ where $\\mu _k^{\\prime \\prime }$ is the same as $\\mu _k$ but with $b_{ij}\\longrightarrow b_{ji}$ inverted in the formula.", "With slight modification of Lemma REF , we obtain a mutation sequence $\\mu _{R_1}$ of length 36 given by $\\mu _{R_1}=\\lbrace &{f^{\\prime }}_1^{-3}\\longrightarrow f_1^1\\longrightarrow f_1^2\\longrightarrow f_2^2\\longrightarrow f_1^1\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow f_2^2\\longrightarrow f_1^1\\longrightarrow f_2^3\\longrightarrow f_1^2\\longrightarrow f_1^1\\longrightarrow f_2^1\\longrightarrow \\\\&{f^{\\prime }}_1^{-2}\\longrightarrow f_1^1\\longrightarrow f_1^2\\longrightarrow f_2^3\\longrightarrow f_1^1\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow f_2^3\\longrightarrow f_1^1\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow f_1^2\\longrightarrow f_1^1\\longrightarrow f_2^2\\longrightarrow \\\\&{f^{\\prime }}_1^{-1}\\longrightarrow f_1^1\\longrightarrow f_1^2\\longrightarrow {f^{\\prime }}_2^2\\longrightarrow f_1^1\\longrightarrow {f^{\\prime }}_2^{-1}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow f_1^1\\longrightarrow {f^{\\prime }}_2^{-1}\\longrightarrow f_1^2\\longrightarrow f_1^1\\longrightarrow f_2^3\\rbrace ,$ where $f_i^{n_i}$ and ${f^{\\prime }}_i^{-n_i}$ are identified.", "The basic quiver (cf.", "Figure REF ) can be attached to a triangle as in Figure REF .", "Then the mutation $\\mu _{R_1}$ appears as in Figure REF , and we can determine $\\sigma _1$ to be: $\\sigma _1:\\langle \\mathcal {S}_i-\\mathcal {T}_i\\rangle \\mapsto \\langle \\mathcal {T}_i-\\mathcal {S}_i\\rangle ,\\;\\;\\;\\;\\;\\;i=1,2,$ where $\\mathcal {S}_1=\\mathcal {P}_{F_1}^{\\widetilde{Q}^{\\prime }}&=({f^{\\prime }}_1^0,{f^{\\prime }}_1^{-1},{f^{\\prime }}_1^{-2},{f^{\\prime }}_1^{-3}),& \\mathcal {T}_1&=(f_1^3,e_1^0),\\\\\\mathcal {S}_2=\\mathcal {P}_{F_2}^{\\widetilde{Q}^{\\prime }}&=({f^{\\prime }}_2^0,{f^{\\prime }}_2^{-1},{f^{\\prime }}_2^{-2},{f^{\\prime }}_2^{-3}),&\\mathcal {T}_2&=(f_2^3,f_2^2,f_2^1,e_2^0).$ Similarly, the description for $\\mu _{R_2}$ is given by $\\mu _{R_2}=\\lbrace &{f^{\\prime }}_1^{-3}\\longrightarrow {f^{\\prime }}_2^{-3}\\longrightarrow f_1^{-2}\\longrightarrow f_1^{-1}\\longrightarrow f_2^{-2}\\longrightarrow f_1^{-2}\\longrightarrow f_2^{-1}\\longrightarrow f_2^{-2}\\longrightarrow f_1^{-2}\\longrightarrow f_2^{-1}\\longrightarrow f_1^{-1}\\longrightarrow f_1^{-2}\\longrightarrow \\\\&{f^{\\prime }}_1^{-2}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow f_1^{-2}\\longrightarrow f_1^{-1}\\longrightarrow f_2^{-1}\\longrightarrow f_1^{-2}\\longrightarrow e_2^0\\longrightarrow f_2^{-1}\\longrightarrow f_1^{-2}\\longrightarrow e_2^0\\longrightarrow f_1^{-1}\\longrightarrow f_1^{-2}\\longrightarrow \\\\&{f^{\\prime }}_1^{-1}\\longrightarrow {f^{\\prime }}_2^{-1}\\longrightarrow f_1^{-2}\\longrightarrow f_1^{-1}\\longrightarrow e_2^0\\longrightarrow f_1^{-2}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow e_2^0\\longrightarrow f_1^{-2}\\longrightarrow {f^{\\prime }}_2^{-2}\\longrightarrow f_1^{-1}\\longrightarrow f_1^{-2}\\rbrace ,$ where $e_i^0$ and ${f^{\\prime }}_i^{-n_i}$ are identified.", "The permutation is then given by $\\sigma _2:\\langle \\mathcal {S}_i-\\mathcal {T}^{\\prime }_i\\rangle \\mapsto \\langle \\mathcal {T}^{\\prime }_i-\\mathcal {S}_i\\rangle ,\\;\\;\\;\\;\\;\\;i=1,2,$ where $\\mathcal {S}_i$ is the same as before, while $\\mathcal {T}^{\\prime }_1&=(e_1^0,f_1^{-3}),\\;\\;\\;\\;\\;\\;\\mathcal {T}^{\\prime }_2=(e_2^0,f_2^{-1},f_2^{-2},f_2^{-3}).$ Figure: Basic quiver in type G 2 G_2 attached to a triangle.Figure: The flipping of triangle μ R 1 \\mu _{R_1} of the basic quivers in type G 2 G_2, before changing the index back to standard form.", "The basic quivers are stacked according to Figure .Figure: The flipping of triangle μ R 2 \\mu _{R_2} of the basic quivers in type G 2 G_2, before changing the index back to standard form.", "The basic quivers are stacked according to Figure ." ], [ "Alternative factorization of the reduced $R$ matrix", "From Remark REF , one can use the Cartan involution and replace the first factor $\\mathbf {e}_i\\otimes 1$ in the reduced $R$ matrix with the embedding by the $F_i$ paths.", "Then the embedding $\\iota ^w\\otimes \\iota $ induces a very simple factorization of the reduced $R$ matrix, where $g_b(\\mathbf {e}_i^-) &= g_b(F_i^{1,-})...g_b(F_i^{n_i,-}),\\\\g_b(\\mathbf {e}_i^+) &= g_b(F_i^{n_i,+})...g_b(F_i^{1,+}),$ and hence by Corollary REF , Corollary 10.8 Under the embedding $\\iota ^w\\otimes \\iota $ , the reduced $R$ matrix factorizes as $\\overline{R}=R_4\\cdot R_3\\cdot R_2\\cdot R_1,$ where $R_1&=\\prod _{k=1}^{N}{}^{op}\\prod _{j=1}^{n_{i_k}}{}^{op} g_b(F_{i_k}^{j,+}\\otimes X_k^+),\\\\R_2&=\\prod _{k=1}^{N}{}^{op}\\prod _{j=1}^{n_{i_k}} g_b(F_{i_k}^{j,-}\\otimes X_k^+),\\\\R_3&=\\prod _{k=1}^{N}\\prod _{j=1}^{n_{i_k}}{}^{op} g_b(F_{i_k}^{j,+}\\otimes X_k^-),\\\\R_4&=\\prod _{k=1}^{N}\\prod _{j=1}^{n_{i_k}} g_b(F_{i_k}^{j,-}\\otimes X_k^-).$ and recall that $\\Pi ^{op}$ means multiplying from the right.", "The embedding $\\iota ^w\\otimes \\iota $ corresponds to a new quiver $\\widetilde{\\mathcal {Z}_\\mathfrak {g}}$ , which is another amalgamation of the two quivers $\\mathcal {D}_\\mathfrak {g}$ , where the nodes $\\lbrace f_{i}^{n_i}\\rbrace $ of the first quiver are glued to $\\lbrace {f^{\\prime }}_i^{n_i}\\rbrace $ of the second quiver instead (see Figure REF below).", "Then one can describe for every type of $\\mathfrak {g}$ the mutation sequence giving the flip of triangulations on $\\widetilde{\\mathcal {Z}_\\mathfrak {g}}$ easily: Proposition 10.9 Let $\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}:=\\langle \\mathcal {P}_{F_i}^{\\widetilde{Q}^{\\prime }}-\\mathcal {P}_{F_i}^{\\widetilde{Q}}\\rangle $ be the concatenation of the $F_i$ -paths in the corresponding subquivers of $\\widetilde{\\mathcal {Z}_\\mathfrak {g}}$ .", "Then the mutation sequence giving the flip of triangulation is $\\mu _{R_1}=\\lbrace \\mathcal {P}_1\\longrightarrow \\mathcal {P}_2\\longrightarrow ...\\mathcal {P}_N\\rbrace ,$ where as before if $i_m$ is the $k$ -th appearance of the root index $i$ from the right of $\\mathbf {i}$ , then $\\mathcal {P}_m=\\mathcal {P}_i^{\\widetilde{Q}\\widetilde{Q}^{\\prime }}[k,n_i].$ When $\\mathbf {i}$ corresponds to the Coxeter element of the Weyl group, $w_0=w_c^{h/2}$ , this coincides with the mutation sequence of the flip of triangulations (where two quivers mirrored to each other are glued) described in [25] in the classical type.", "Hence this construction generalizes those of [25], and at the same time provides a representation theoretic meaning of the sequences giving the flip of triangulations described there.", "Figure: Half-Dehn twist of the quiver 𝒵 𝔤 ˜\\widetilde{\\mathcal {Z}_\\mathfrak {g}}.Although the description of the $R$ matrix factorization is very nice, we see that after 4 flips it does not return to the original quiver, but rather a mirror image with all the arrows flipped.", "A full Dehn twist, however, return us to the original configuration.", "If Conjecture REF is true, which gives a quiver mutation equivalence between $\\iota $ and $\\iota ^w$ (with Dynkin involution), this will relate such nice presentation of the $R$ matrix factorization to the canonical one found in the main theorem." ], [ "Proof of Theorem ", "Let $\\widetilde{R}$ denote the right hand side of (REF ).", "The strategy is to show that $\\mathcal {K}\\widetilde{R}$ also gives the braiding relations (REF ) as well.", "First of all, we have $Ad_{\\mathcal {K}}(1\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes K_i^{\\prime }) &= K_i\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes 1,\\\\Ad_{\\mathcal {K}}(\\mathbf {f}_i\\otimes 1+K_i\\otimes \\mathbf {f}_i) &= \\mathbf {f}_i\\otimes K_i^{\\prime }+1\\otimes \\mathbf {f}_i,\\\\Ad_{\\mathcal {K}} \\Delta (K_i) &=K_i\\otimes K_i.$ Hence in order to prove the braiding relations, it suffices to show $\\widetilde{R}\\Delta (\\mathbf {e}_i)&=(1\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes K_i^{\\prime }),\\\\\\widetilde{R}\\Delta (\\mathbf {f}_i)&=(\\mathbf {f}_i\\otimes 1+K_i\\otimes \\mathbf {f}_i),\\\\\\widetilde{R}\\Delta (K_i)&=K_i\\otimes K_i,$ where the last one is trivial.", "We begin with several Lemmas: Lemma 11.1 For any $\\mathfrak {sl}_2$ triple $(\\mathbf {e}, \\mathbf {f}, K, K^{\\prime })$ , and any self-adjoint element $X$ , we have $Ad_{g_b(\\mathbf {e}\\otimes X)} (\\mathbf {f}\\otimes 1+K^{\\prime }\\otimes X) = \\mathbf {f}\\otimes 1+K\\otimes X.$ This is a well-known result by considering the formal power series expansion of $g_b$ (recall that we restrict ourselves to the compact case, but it holds for the non-compact case as well).", "Recall $g_b(u)=Exp_{q^{-2}}(-\\frac{u}{q-q^{-1}})=\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(n-1)}u^n}{(q^n-q^{-n})...(q-q^{-1})},$ and that we have $\\mathbf {e}^n\\mathbf {f}- \\mathbf {f}\\mathbf {e}^n =(q^{n}-q^{-n})(q^{n-1}K^{\\prime }-q^{1-n}K)\\mathbf {e}^{n-1}..$ Hence we can work out $&g_b(\\mathbf {e}\\otimes X)(\\mathbf {f}\\otimes 1+K^{\\prime }\\otimes X)- (\\mathbf {f}\\otimes 1+K\\otimes X)g_b(\\mathbf {e}\\otimes X)\\\\=&(\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(n-1)}\\mathbf {e}^n}{(q^n-q^{-n})...(q-q^{-1})}\\otimes X^n) (\\mathbf {f}\\otimes 1+K^{\\prime }\\otimes X)\\\\&-(\\mathbf {f}\\otimes 1+K\\otimes X)\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(n-1)}\\mathbf {e}^n}{(q^n-q^{-n})...(q-q^{-1})}\\otimes X^n\\\\=&(\\mathbf {f}\\otimes 1)\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(n-1)}\\mathbf {e}^n}{(q^n-q^{-n})...(q-q^{-1})}\\otimes X^n \\\\&+(K^{\\prime }\\otimes X)\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(3+n)}\\mathbf {e}^n}{(q^n-q^{-n})...(q-q^{-1})}\\otimes X^n \\\\&-\\sum _{n\\ge 0}(q^{n}K^{\\prime }-q^{-n}K)\\frac{(-1)^nq^{\\frac{1}{2}n(n+1)}\\mathbf {e}^{n}}{(q^{n}-q^{-n})...(q-q^{-1})}\\otimes X^{n+1} \\\\&-(\\mathbf {f}\\otimes 1+K\\otimes X)\\sum _{n\\ge 0}\\frac{(-1)^nq^{\\frac{1}{2}n(n-1)}\\mathbf {e}^n}{(q^n-q^{-n})...(q-q^{-1})}\\otimes X^n\\\\=&0.$ For simplicity, let us define $Y_i^k:=\\left\\lbrace \\begin{array}{llllllllllllllllllllllllllllllllllllllllllll}F_i^{n_i+1-k,-}& k\\le n_i,\\\\ F_i^{k-n_i,+}&k>n_i, \\\\ \\end{array}\\right.$ such that $\\mathbf {f}_i = Y_i^1+Y_i^2+...+Y_i^{2n_i}$ .", "Lemma 11.2 We have for $1\\le k\\le 2n_i$ , $&Ad_{g_b(\\mathbf {e}_i \\otimes Y_i^k)}(\\mathbf {f}_i\\otimes 1+K_i\\otimes \\sum _{l=1}^{k-1} Y_i^l+K_i^{\\prime }\\otimes \\sum _{l=k}^{2n_i} Y_i^l)\\\\&=\\mathbf {f}_i\\otimes 1+K_i\\otimes \\sum _{l=1}^{k} Y_i^l+K_i^{\\prime }\\otimes \\sum _{l=k+1}^{2n_i} Y_i^l$ and invariant under $Ad_{g_b(\\mathbf {e}_j\\otimes Y_j^l)}$ for $j\\ne i$ if $Y_j^l$ comes after $Y_i^{k-1}$ and before $Y_i^k$ in the decomposition (REF ).", "We observe that by Lemma REF , $Ad_{g_b(\\mathbf {e}_i\\otimes Y_i^k)}(K_i\\otimes Y_i^l)&=K_i\\otimes Y_i^l\\;\\;\\;\\;\\;\\;l<k,\\\\Ad_{g_b(\\mathbf {e}_i\\otimes Y_i^k)}(K_i^{\\prime }\\otimes Y_i^l)&=K_i^{\\prime }\\otimes Y_i^l\\;\\;\\;\\;\\;\\;l>k.$ Hence we only care about the term $(\\mathbf {f}_i\\otimes 1 + K_i^{\\prime }\\otimes Y_i^k)$ .", "By Lemma REF $Ad_{g_b(\\mathbf {e}_i\\otimes Y_i^k)}(\\mathbf {f}_i\\otimes 1+K_i^{\\prime }\\otimes Y_i^k)= \\mathbf {f}_i\\otimes 1+K_i\\otimes Y_i^k$ and we are done.", "Finally, again by Lemma REF it is easy to check that $\\mathbf {e}_j\\otimes Y_j^l$ commute with $K_i\\otimes \\sum _{l=1}^{k} Y_i^l+K_i^{\\prime }\\otimes \\sum _{l=k+1}^{2n_i} Y_i^l$ whenever $Y_j^l$ comes after $Y_i^{k-1}$ and before $Y_i^k$ in the decomposition (REF ).", "Lemma 11.3 For the reduced word $\\mathbf {i}=(i_1,...,i_N)\\in \\mathfrak {R}$ , if $i_N=i$ , then $\\mathcal {K}\\widetilde{R}\\Delta (\\mathbf {e}_i) = \\Delta ^{op}(\\mathbf {e}_i)\\mathcal {K}\\widetilde{R}.$ Note that if $i_N=i$ , then $X_N^\\pm =X_{f_i^{\\pm n_i}}$ , $\\mathbf {e}_i=X_{f_i^{n_i}}+X_{f_i^{n_i},e_i^0}$ and $K_i=X_{f_i^{n_i},e_i^0,f_i^{-n_i}}$ .", "We have $X_{f_i^{- n_i}}X_{e_i^0} = q_i^2 X_{e_i^0}X_{f_i^{- n_i}}$ Hence $&Ad_{g_b(\\mathbf {e}_i\\otimes X_N^-)} (1\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes K_i)\\\\&=Ad_{g_b(\\mathbf {e}_i\\otimes X_{f_i^{-n_i}})} (1\\otimes X_{f_i^{n_i}}+1\\otimes X_{f_i^{n_i},e_i^0}+e_i\\otimes K_i)\\\\&=1\\otimes X_{f_i^{n_i}}+(1\\otimes X_{f_i^{n_i},e_i^0}+\\mathbf {e}_i\\otimes K_i)(1+q_i\\mathbf {e}_i\\otimes X_{f_i^{-n_i}}X_{e_i^0})^{-1}\\\\&=1\\otimes X_{f_i^{n_i}}+1\\otimes X_{f_i^{n_i},e_i^0}(1+q_ie\\beta _i\\otimes X_{f_i^{-n_i}})(1+q_i\\mathbf {e}_i\\otimes X_{f_i^{-n_i}}X_{e_i^0})^{-1}\\\\&=1\\otimes \\mathbf {e}_i.$ One then check directly that $1\\otimes \\mathbf {e}_i$ commutes with all the factors $\\mathbf {e}_j\\otimes X_k^\\pm $ for every $j,k$ , except the last term $\\mathbf {e}_i\\otimes X_N^+$ , where we have the reverse of the above: $Ad_{g_b(\\mathbf {e}_i\\otimes X_N^+)}(1\\otimes \\mathbf {e}_i) = 1\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes K_i^{\\prime },$ and hence $\\mathcal {K}\\widetilde{R}\\Delta (\\mathbf {e}_i)=\\mathcal {K}(1\\otimes \\mathbf {e}_i+\\mathbf {e}_i\\otimes K_i^{\\prime })\\widetilde{R} = \\Delta ^{op}(\\mathbf {e}_i)\\mathcal {K}\\widetilde{R}$ as required.", "In general, we use the fact that the decomposition of $\\widetilde{R}$ is invariant under the change of words $\\mathcal {M}$ .", "Let $\\widehat{F}_i^k$ denote the representation of $\\mathbf {f}_i$ using the mutated cluster variables $\\widehat{X_i}:=\\mathcal {M}(X_i)$ under the change of words $\\mathcal {M}$ (cf.", "Section ) Lemma 11.4 　 (1) For the change of words $\\mathcal {M}: (... iji...)\\longleftrightarrow (...jij...)$ we have $g_b(\\mathbf {e}_1\\otimes F_1^{k+1,\\pm })g_b(\\mathbf {e}_2\\otimes F_2^{l,\\pm })g_b(\\mathbf {e}_1\\otimes F_1^{k,\\pm })=g_b(\\mathbf {e}_2\\otimes \\widehat{F}_2^{l+1,\\pm })g_b(\\mathbf {e}_1\\otimes \\widehat{F}_1^{k,\\pm })g_b(\\mathbf {e}_2\\otimes \\widehat{F}_2^{l,\\pm })$ for $v(i,k)<v(j,l)<v(i,k+1)$ .", "(2) For the change of words $\\mathcal {M}: (... ijij...)\\longleftrightarrow (...jiji...)$ where $i$ is short and $j$ is long, we have $&g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k+1,\\pm })g_b(\\mathbf {e}_j\\otimes F_j^{l+1,\\pm })g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,\\pm })g_b(\\mathbf {e}_j\\otimes F_j^{l,\\pm })\\\\&=g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l+1,\\pm })g_{b_s}(\\mathbf {e}_i\\otimes \\widehat{F}_i^{k+1,\\pm })g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l,\\pm })g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,\\pm })$ for $v(j,l)<v(i,k)<v(j,l+1)<v(i,k+1)$ .", "We will prove the $+$ case, while the $-$ case is similar.", "Proof of (1).", "In the simply-laced case, recall that we have $F_i^{k+1,+}F_i^{k,+}=q^2 F_i^{k,+}F_i^{k+1,+},$ $F_i^{k,+}F_j^{l,+}=q^{-1}F_j^{l,+}F_i^{k,+}.$ Hence $\\frac{[\\mathbf {e}_j\\otimes F_j^{l,+}, \\mathbf {e}_i\\otimes F_i^{k,+}]}{q-q^{-1}} &= \\mathbf {e}_j\\mathbf {e}_i\\otimes F_j^{l,+}F_i^{k,+} -\\mathbf {e}_i\\mathbf {e}_j\\otimes F_i^{k,+}F_j^{l,+}\\\\&=\\frac{\\mathbf {e}_j\\mathbf {e}_i\\otimes -q^{-1}\\mathbf {e}_j\\mathbf {e}_i}{q-q^{-1}}\\otimes F_j^{l,+}F_i^{k,+}\\\\&=\\mathbf {e}_{ij}\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+}$ where $\\mathbf {e}_{ij}=T_i(\\mathbf {e}_j)$ is given by the Lusztig's isomorphism.", "Hence using (REF ), we have $&g_b(\\mathbf {e}_i\\otimes F_i^{k+1,+})g_b(\\mathbf {e}_j\\otimes F_j^{l,+})g_b(\\mathbf {e}_i\\otimes F_i^{k,+})\\\\&=g_b(\\mathbf {e}_i\\otimes F_i^{k+1,+})g_b(\\mathbf {e}_i\\otimes F_i^{k,+})g_b(\\mathbf {e}_{ij}\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})g_b(\\mathbf {e}_j\\otimes F_j^{l,+})\\\\&=g_b(\\mathbf {e}_i\\otimes (F_i^{k+1,+}+F_i^{k,+}))g_b(\\mathbf {e}_{ij}\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})g_b(\\mathbf {e}_j\\otimes F_j^{l,+}).$ Similarly, we have $&g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l+1,+})g_b(\\mathbf {e}_i\\otimes \\widehat{F}_i^{k,+})g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l,+})\\\\&=g_b(\\mathbf {e}_i\\otimes \\widehat{F}_i^{k,+})g_b(\\mathbf {e}_{ij}\\otimes q^{-1/2}\\widehat{F}_j^{l+1,+}\\widehat{F}_i^{k,+})g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l+1,+})g_b(\\mathbf {e}_j\\otimes \\widehat{F}_j^{l,+})\\\\&=g_b(\\mathbf {e}_i\\otimes \\widehat{F}_i^{k,+})g_b(\\mathbf {e}_{ij}\\otimes q^{-1/2}\\widehat{F}_j^{l+1,+}\\widehat{F}_i^{k,+})g_b(\\mathbf {e}_j\\otimes (\\widehat{F}_j^{l+1,+}+\\widehat{F}_j^{l,+})).$ If we write down the quantum cluster variables as $F_i^k &= X_1, &F_i^{k+1}&=X_{1,2}, &F_j^l&=X_3,\\\\\\widehat{F}_i^k &= \\widehat{X_1}, &\\widehat{F_j}^l&=\\widehat{X_3}, &\\widehat{F_j}^{l+1}&=\\widehat{X_{3,4}},$ then we have $\\widehat{X_1}&=X_1(1+qX_2),\\\\\\widehat{X_3}&=X_3(1+qX_2^{-1})^{-1},\\\\\\widehat{X_4}&=X_2^{-1},$ and one can see that $F_i^{k+1,+}+F_i^{k,+}&=\\widehat{F}_i^{k,+},\\\\F_j^{l,+}F_i^{k,+}&=\\widehat{F}_j^{l+1,+}\\widehat{F}_i^{k,+},\\\\F_j^{l,+}&=\\widehat{F}_j^{l+1,+}+\\widehat{F}_j^{l,+}$ as required.", "Proof of (2).", "We have $F_i^{k,+}F_j^{l,+}=q^{-1}F_j^{l,+}F_i^{k,+}$ whenever $v(j,l)<v(i,k)$ .", "Let $v&=\\mathbf {e}_i\\otimes F_i^{k,+},\\\\u&=\\mathbf {e}_j\\otimes F_j^{l,+},\\\\\\frac{c}{[2]_{q_s}}&=\\frac{[u,v]}{q-q^{-1}}=\\frac{q^{1/2}\\mathbf {e}_j\\mathbf {e}_i-q^{-1/2}\\mathbf {e}_i\\mathbf {e}_j}{q-q^{-1}}\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+}=\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+},\\\\d&=\\frac{q_s^{-1}cv-q_s vc}{q-q^{-1}}=\\frac{\\mathbf {e}_Y\\mathbf {e}_i-\\mathbf {e}_i\\mathbf {e}_Y}{q_s-q_s^{-1}}\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2=\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2,$ where $\\mathbf {e}_X:=T_i(\\mathbf {e}_j)$ and $\\mathbf {e}_Y:=T_iT_j(\\mathbf {e}_i)$ are given by the Lusztig's isomorphism.", "We have $\\mathbf {e}_Y\\mathbf {e}_X&=q\\mathbf {e}_X\\mathbf {e}_Y,\\\\\\mathbf {e}_X\\mathbf {e}_i&=q\\mathbf {e}_i\\mathbf {e}_X,\\\\\\mathbf {e}_j\\mathbf {e}_Y&=q\\mathbf {e}_Y\\mathbf {e}_j, \\\\\\frac{[\\mathbf {e}_j,\\mathbf {e}_X]}{q-q^{-1}}&=\\mathbf {e}_Y^2,$ and hence $u,c,d,v$ satisfies the condition for (REF ).", "Applying (REF ) repeatedly and rearranging, we have (we underline the terms to be transformed): $&\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k+1,+})g_b(\\mathbf {e}_j\\otimes F_j^{l+1,+})}\\;\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,+})g_b(\\mathbf {e}_j\\otimes F_j^{l,+})}\\\\=_(\\ref {g1212})&g_b(\\mathbf {e}_j\\otimes F_j^{l+1,+})g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l+1,+}F_i^{k+1,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2)\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k+1,+})}\\\\&\\underline{g_b(\\mathbf {e}_j\\otimes F_j^{l,+})}g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2)g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,+})\\\\=_(\\ref {g1212})&g_b(\\mathbf {e}_j\\otimes F_j^{l+1,+})g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l+1,+}F_i^{k+1,+})\\underline{g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2)g_b(\\mathbf {e}_j\\otimes F_j^{l,+})}\\\\&g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k+1,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k+1,+})^2)\\\\&\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k+1,+})g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})}g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2)g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,+})\\\\=_(\\ref {gvu})&\\underline{g_b(\\mathbf {e}_j\\otimes F_j^{l+1,+})}g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l+1,+}F_i^{k+1,+})\\underline{g_b(\\mathbf {e}_j\\otimes F_j^{l,+})}g_b(\\mathbf {e}_Y^2\\otimes F_j^{l+1,+}F_j^{l,+}(F_i^{k+1,+})^2)\\\\&g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2)g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k+1,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k+1,+})^2)\\\\&g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})g_{b_s}(\\mathbf {e}_X\\otimes F_j^{l,+}F_i^{k,+}F_i^{k+1,+})\\\\&\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k+1,+})}g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2)\\underline{g_{b_s}(\\mathbf {e}_i\\otimes F_i^{k,+})}\\\\=_(\\ref {guv})&g_b(\\mathbf {e}_j\\otimes (F_j^{l+1,+}+F_j^{l,+}))\\\\&\\underline{g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l+1,+}F_i^{k+1,+})g_b(\\mathbf {e}_Y^2\\otimes F_j^{l+1,+}F_j^{l,+}(F_i^{k+1,+})^2)g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k+1,+})}\\\\&g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2)\\\\&\\underline{g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k+1,+})^2)g_{b_s}(\\mathbf {e}_X\\otimes F_j^{l,+}F_i^{k,+}F_i^{k+1,+})g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k,+})^2)}\\\\&g_{b_s}(\\mathbf {e}_i\\otimes (F_i^{k+1,+}+F_i^{k,+}))\\\\=_(\\ref {gb2})&g_b(\\mathbf {e}_j\\otimes (F_j^{l+1,+}+F_j^{l,+}))\\underline{g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}(F_j^{l+1,+}+F_j^{l,+})F_i^{k+1,+})g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}F_j^{l,+}F_i^{k,+})}\\\\&\\underline{g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2)g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l,+}(F_i^{k+1,+}+F_i^{k,+})^2)}g_{b_s}(\\mathbf {e}_i\\otimes (F_i^{k+1,+}+F_i^{k,+}))\\\\=_(\\ref {guv})&g_b(\\mathbf {e}_j\\otimes (F_j^{l+1,+}+F_j^{l,+}))g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}(F_j^{l+1,+}+F_j^{l,+})F_i^{k+1,+}+q^{-1/2}F_j^{l,+}F_i^{k,+})\\\\&g_b(\\mathbf {e}_X\\otimes q^{-1}F_j^{l+1,+}(F_i^{k+1,+})^2+q^{-1}F_j^{l,+}(F_i^{k+1,+}+F_i^{k,+})^2)g_{b_s}(\\mathbf {e}_i\\otimes (F_i^{k+1,+}+F_i^{k,+})),$ where in the last line, we observe that the terms $q^2$ commute, hence we can apply (REF ).", "On the other hand, by applying $(\\ref {g1212})$ once, we have $&g_b(\\mathbf {e}_j\\otimes \\widehat{F}_2^{l+1,+})g_{b_s}(\\mathbf {e}_i\\otimes \\widehat{F}_1^{k+1,+})g_b(\\mathbf {e}_j\\otimes \\widehat{F}_2^{l,+})g_{b_s}(\\mathbf {e}_i\\otimes \\widehat{F}_1^{k,+})\\\\=&g_b(\\mathbf {e}_j\\otimes (\\widehat{F}_2^{l+1,+}+\\widehat{F}_2^{l,+}))g_{b_s}(\\mathbf {e}_Y\\otimes q^{-1/2}\\widehat{F}_2^{l,+}\\widehat{F}_1^{k+1,+})\\\\&g_b(\\mathbf {e}_X\\otimes q^{-1}\\widehat{F}_2^{l,+}(\\widehat{F}_1^{k+1,+})^2)g_{b_s}(\\mathbf {e}_i\\otimes (\\widehat{F}_1^{k+1,+}+\\widehat{F}_1^{k,+})).$ To compare, again we write out the quantum cluster variables as $F_i^{k,+} &= X_1, &F_i^{k+1,+} &= X_{1,2},&F_j^{l,+} &= X_{3},&F_j^{l+1,+} &= X_{3,4},\\\\\\widehat{F}_i^{k,+} &= \\widehat{X}_1, &\\widehat{F}_i^{k+1,+} &= \\widehat{X}_{1,2},&\\widehat{F}_j^{l,+} &= \\widehat{X}_{3},&\\widehat{F}_j^{l+1,+} &= \\widehat{X}_{3,4}.$ Recall that we need to do mutation three times according to Section REF , which gives at the end $\\widehat{X}_1&=D_2^{-1}X_{1,2,4},\\\\\\widehat{X}_2&=X_{2,4}^{-1}D_1,\\\\\\widehat{X}_3&=D_1^{-1}X_3 D_3,\\\\\\widehat{X}_4&=D_3^{-1}X_4,$ where $D_1&=(1+q_s X_2)(1+q_s^3 X_2)+q X_{2^2,4},\\\\D_2&=(1+q_s X_2+q_s X_{2,4}),\\\\D_3&=(1+q_s X_2+q_s X_{2,4})(1+q_s^3X_2+q_s^3X_{2,4}).$ Now we can check directly that $F_j^{l+1,+}+F_j^{l,+}&=\\widehat{F}_2^{l+1,+}+\\widehat{F}_2^{l,+},\\\\F_j^{l+1,+}+F_j^{l,+})F_i^{k+1,+}+q^{-1/2}F_j^{l,+}F_i^{k,+}&=\\widehat{F}_2^{l,+}\\widehat{F}_1^{k+1,+},\\\\F_j^{l+1,+}(F_i^{k+1,+})^2+ F_j^{l,+}(F_i^{k+1,+}+F_i^{k,+})^2&=\\widehat{F}_2^{l,+}(\\widehat{F}_1^{k+1,+})^2,\\\\F_i^{k+1,+}+F_i^{k,+}&=\\widehat{F}_1^{k+1,+}+\\widehat{F}_1^{k,+}$ and this completes the proof.", "Remark 11.5 In type $G_2$ , using the mutation sequence that gives the half-Dehn twist from Section , one can conjugate the representation of $\\Delta (\\mathbf {e}_2)$ by (REF ) and check the braiding relation directly.", "Using the fact that the standard form of the universal $R$ matrix is invariant under the change of words, we conclude that the analogue of Lemma REF also holds in type $G_2$ .", "[Proof of Theorem REF ] First it is obvious that $\\mathcal {K}$ and $\\widetilde{R}$ commute with both $\\Delta (K_i)$ and $\\Delta (K_i^{\\prime })$ by direct calculation.", "As a consequence of Lemma REF , we have $\\mathcal {K}\\widetilde{R}\\Delta (\\mathbf {f}_i) = \\mathcal {K}(\\mathbf {f}_i\\otimes 1+K^{\\prime }\\otimes \\mathbf {f}_i)\\widetilde{R} = \\Delta ^{op}(\\mathbf {f}_i)\\mathcal {K}\\widetilde{R}$ as required.", "As a consequence of Lemma REF , we can choose freely the reduced word $\\mathbf {i}$ with any choice of index on the right of $\\mathbf {i}$ , and by Lemma REF , we obtain $\\mathcal {K}\\widetilde{R}\\Delta (\\mathbf {e}_i) = \\Delta ^{op}(\\mathbf {e}_i)\\mathcal {K}\\widetilde{R}$ for every root index $i$ , thus completing the proof of the braiding relations.", "Finally, recall that by the construction of the positive representations $\\mathcal {P}_\\lambda $ , one can choose appropriate discrete parameters $\\lambda $ and restrict it to give any irreducible highest weight finite dimensional representations of $\\mathcal {U}_q(\\mathfrak {g})$ [18].", "Then $\\mathcal {K}\\widetilde{R}$ satisfies the braiding (REF ) on every finite dimensional representations of $\\mathcal {U}_q(\\mathfrak {g})$ , and as a formal power series it has constant term equals 1, hence we conclude that $\\mathcal {K}\\widetilde{R}$ equals the universal $R$ matrix." ], [ "Acknowledgment", "tocsectionAcknowledgments I would like to thank Gus Schrader and Alexander Shapiro for stimulating discussions who inspired the constructions carried out in this work.", "I would also like to thank Masahito Yamazaki and Rei Inoue for valuable comments.", "This work is supported by JSPS KAKENHI Grant Numbers JP16K17571 and Top Global University Project, MEXT, Japan." ], [ "Quantum dilogarithm identities", "The compact quantum dilogarithm function is defined to be the infinite product $\\Psi ^q(x)=\\prod _{r=0}^\\infty (1+q^{2r+1}x) ^{-1},$ which is well defined for $0<q<1$ .", "In the split real case, where $q=e^{\\pi i b^2}$ with $0<b<1$ , the infinite product is not so well-behaved.", "To treat this case, the non-compact quantum dilogarithm $g_b(x)$ is composed of two commuting copies, associated to the so-called Faddeev's modular double, of the compact quantum dilogarithm $\\Psi ^q(x)$ [7], [8].", "It is a meromorphic function that can be represented as an integral expression: $g_b(x):=\\exp \\left(\\frac{1}{4}\\int _{\\mathbb {R}+i0} \\frac{x^{\\frac{t}{ib}}}{\\sinh (\\pi bt)\\sinh (\\pi b^{-1}t)}\\frac{dt}{t}\\right),$ such that by functional calculus, it is a unitary operator when $x$ is positive self-adjoint, and there is a $b$ -duality: $g_b(x)=g_{b^{-1}}(x^{\\frac{1}{b^2}}).$ In this paper however, we are only interested in the formal algebraic calculation, hence one may consider only the compact part and think about the correspondence in terms of formal power series $g_b(x)\\sim \\Psi ^q(x)^{-1}=\\prod _{r=0}^\\infty (1+q^{2r+1}x) = Exp_{q^{-2}}\\left(-\\frac{u}{q-q^{-1}}\\right),$ where $Exp_q(x)&:=\\sum _{k\\ge 0} \\frac{x^k}{(k)_q!", "},\\\\(k)_q&:=\\frac{1-q^k}{1-q}.$ In particular, we can rewrite the identities of $Exp_q(x)$ derived in [33] for the quantum dilogarithm function $g_b(x)$ that are needed in this paper.", "In particular, by writing in this way, the argument of $g_b(x)$ are all manifestly positive self-adjoint so that the identities are well-defined in the split real setting.", "We will be interested in two types of identities: the pentagon equation (PE) and the quantum exponential relation (QE), Simply-laced case.", "Let $u,v$ be self-adjoint variables.", "If $uv=q^2 vu$ , then we have the pentagon equation and the quantum exponential relation: $(PE): \\;\\;\\;\\;\\;\\;g_b(v)g_b(u)&=g_b(u)g_b(q^{-1}uv)g_b(v),\\\\(QE): \\;\\;\\;\\;\\;\\;g_b(u+v)&=g_b(u)g_b(v).$ Let again $u,v$ be self-adjoint and $c:=\\frac{[u,v]}{q-q^{-1}},$ such that $uc=q^2cu,\\;\\;\\;\\;\\;\\;cv=q^2vc.$ Then we have the generalized pentagon equation: $(PE):\\;\\;\\;\\;\\;\\;g_b(v)g_b(u)=g_b(u)g_b(c)g_b(v).$ in which (REF ) is a special case.", "Doubly-laced case.", "In the doubly-laced case we have $q_s=q^{1/2}$ .", "Let $u,v$ be self-adjoint variables, and let $c:=\\frac{[u,v]}{q_s-q_s^{-1}},\\;\\;\\;\\;\\;\\;d:=\\frac{q_s^{-1}cv-q_s vc}{q-q^{-1}},$ such that $uc=q^2 cu, \\;\\;\\;\\;\\;\\;cd=q^2 dc, \\;\\;\\;\\;\\;\\;dv=q^2 vd,\\;\\;\\;\\;\\;\\;\\frac{q^{-1}ud-qdu}{q-q^{-1}}=\\frac{c^2}{[2]_{q_s}^2}.$ We have $(PE):\\;\\;\\;\\;\\;\\;g_{b_s}(v)g_b(u)&=g_b(u)g_{b_s}(\\frac{c}{[2]_{q_s}})g_b(d)g_{b_s}(v),\\\\(QE):\\;\\;\\;\\;\\;\\;g_{b_s}(c+v)&=g_{b_s}(c)g_b([2]_{q_s}d)g_{b_s}(v).$ In particular if $uv=q^2vu$ and substitute $u\\mapsto quv^{-1}/[2]_{q_s}$ , we have: $g_{b_s}(u+v)&=g_{b_s}(u)g_b(q^{-1}uv)g_{b_s}(v),\\\\g_b((u+v)^2)&=g_b(u^2)g_{b_s}(q^{-1/2}uv)g_b(v^2).$ These two relations are related by the $b$ -duality (REF ).", "Triply-laced case.", "For completeness we also translate the type $G_2$ identity of [33] to $g_b(x)$ , which becomes more natural looking.", "Let $q_s=q^{1/3}$ , and let $u,v$ be self-adjoint.", "Define $c&:=\\frac{q_s^{-1}uv-q_svu}{q_s^2-q_s^{-2}},\\\\d&:=\\frac{q_s^{-2}cv-q_s^2vc}{q_s-q_s^{-1}},\\\\d^{\\prime }&:=\\frac{q_s^{-2}uc-q_s^2cu}{q_s-q_s^{-1}},$ such that these relations are satisfied: $ud^{\\prime }&=q^2d^{\\prime }u, & d^{\\prime }c&=q^2cd^{\\prime },& cd&=q^2dc,&dv&=q^2 vd,& \\\\c^2&=\\frac{q^{-1}ud-qdu}{q-q^{-1}},&c^2&=\\frac{q^{-1}d^{\\prime }v-qvd^{\\prime }}{q-q^{-1}}, &c^3&=\\frac{q^{-2}d^{\\prime }d-q^2 dd^{\\prime }}{q-q^{-1}}.$ Then we have $(QE):\\;\\;\\;\\;\\;\\;g_{b_s}(u+v)=&g_{b_s}(u)g_b(d^{\\prime })g_{b_s}(c)g_b(d)g_{b_s}(v).$ In particular if $uv=q^2vu=q_s^6vu$ , we have $g_{b_s}(u+v)&=g_{b_s}(u)g_b(q^{-2}u^2v)g_{b_s}(q^{-1}uv)g_b(q^{-2}uv^2)g_{b_s}(v),\\\\g_b((u+v)^3)&=g_b(u^3)g_{b_s}(q^{-2}u^2v)g_b(q^{-3}u^3v^3)g_{b_s}(q^{-2}uv^2)g_b(v^3),$ which are related by the $b$ -duality (REF ).", "On the other hand, let $\\mathbf {e}_1,\\mathbf {e}_2$ be the generators of $\\mathcal {U}_q(\\mathfrak {g}_{G_2})$ with $\\mathbf {e}_1$ long and $\\mathbf {e}_2$ short, and $\\zeta _1\\zeta _2=q^{-1}\\zeta _2\\zeta _1$ .", "Let the non-simple root generators be $\\mathbf {e}_W:=T_1(\\mathbf {e}_2)&=\\frac{[\\mathbf {e}_2,\\mathbf {e}_1]_{q_s^{3/2}}}{q_s^3-q_s^{-3}},\\\\\\mathbf {e}_X:=T_1T_2(\\mathbf {e}_1)&=\\frac{[\\mathbf {e}_Y,\\mathbf {e}_W]_{q_s^{-1/2}}}{q_s-q_s^{-1}},\\\\\\mathbf {e}_Y:=T_1T_2T_1(\\mathbf {e}_2)&=\\frac{[\\mathbf {e}_2,\\mathbf {e}_W]_{q_s^{1/2}}}{q_s^2-q_s^{-2}},\\\\\\mathbf {e}_Z:=T_1T_2T_1T_2(\\mathbf {e}_1)&=\\frac{[\\mathbf {e}_2,\\mathbf {e}_Y]_{q_s^{-1/2}}}{q_s-q_s^{-1}}.$ Then we have (PE): $&g_{b_s}(\\mathbf {e}_2\\otimes \\zeta _2)g_b(\\mathbf {e}_1\\otimes \\zeta _1)\\\\&=g_b(\\mathbf {e}_1\\otimes \\zeta _1)g_{b_s}(\\mathbf {e}_W\\otimes q^{1/2}\\zeta _1\\zeta _2)g_b(\\mathbf {e}_X\\otimes q^3\\zeta _1^2 \\zeta _2^3)g_{b_s}(\\mathbf {e}_Y\\otimes q\\zeta _1\\zeta _2^2)g_b(\\mathbf {e}_Z\\otimes q^{3/2}\\zeta _1\\zeta _2^3)g_{b_s}(\\mathbf {e}_2\\otimes \\zeta _2).\\nonumber $" ] ]
1612.05641
[ [ "Ultrafast demagnetization in bulk vs thin films: an ab-initio study" ], [ "Abstract We report on {\\it ab-initio} simulations of the quantum dynamics of electronic charge and spin when subjected to intense laser pulses.", "By performing separate calculations for a Ni thin film and bulk Ni, we conclude that surface effects have a dramatic influence on amplifying the laser induced demagnetization.", "We show that the reason for this amplification is due to increased spin-currents on the surface of the thin film.", "This enhancement is a direct consequence of the broken symmetry originating from the surface formation.", "We find that the underlying physics of demagnetization, during the early femtoseconds, for both bulk and thin film is dominated by spin-flips induced by spin-orbit coupling.", "After the first $\\sim 40$ fs this changes in that the dominant cause of demagnetization is the flow of spin-currents, which leads to stronger demagnetization in the film compared to that of the bulk." ], [ "Introduction", "Femtomagnetism[1], whereby the magnetic properties of a material are manipulated on the femtosecond timescale, was initiated by the experimental observation[2], [3], [4], [5] of ultrafast demagnetization of ferro-magnets subjected to an intense laser pulse.", "Due to the important technological implications of this phenomenon, e.g.", "in spintronics[6] or data storage[7], laser induced control of magnetism has since become a highly active field[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].", "As the devices utilizing ultra-short laser pulses to control magnetism hold the promise to reach the fastest possible electronic timescales, offering a speedup of several orders of magnitude over current state-of-the-art, magnetically operated devices, a large amount of work has gone into understanding the underlying physics of light-matter interactions.", "Among the most prominent suggested mechanisms are spin-orbit induced spin-dynamics[28], [29], [30], [31], all optical manipulation of spins[32], [33], [34], Elliott-Yafet scattering[35], [24], [36], [37], [38], [39], Coulomb Exchange scattering[25] and Super-diffusive spin transport[27], [40].", "Most of the theoretical work to-date deals with the effect of laser pulses on the bulk of the magnetic material.", "Research that deals with interface/surface are model calculations[26], [27] focusing only on a single aspect namely the diffusion of electrons across the interface.", "However, realistic devices contain surfaces, interfaces and bulk regions and a detailed understanding of the behaviour of these regions separately, under the influence of a laser pulse, is crucial[41], [42] to unraveling the complete physics of demagnetization.", "Experimentally it is a very challenging task to distinguish between contributions coming from various regions of the sample, but theoretically, by considering separate calculations for thin film and bulk of the same material, these effects can be disentangled.", "In the present work we use Ni thin film and bulk calculations to show that the underlying physics responsible for ultra-fast demagnetization in early femtoseconds is the same in both; spin-orbit induced spin-flips.", "Interestingly, we find that symmetry breaking originating from the formation of a surface (or interface) greatly enhances the demagnetization process.", "By explicitly treating spin and charge-currents in our simulations we find that the main reason behind this enhanced demagnetization in thin films is the presence of large spin-currents generated in these broken symmetry systems.", "In order to do this theoretical work we use the ab-initio method of time-dependent density functional theory (TDDFT)[43], [44], [45], [46], which is, in principle, an exact theory for studying light-matter interaction and makes no assumptions about the form of the electron dynamics.", "The study of dynamics of spins using TDDFT requires an extension where the magnetization density[47] is treated as an unconstrained vector field.", "Such an extension was recently performed[28], facilitating the present work.", "The calculations presented here are purely electronic in nature and include contributions like (a) spin- orbit induced spin-filps (b) restricted set of magnon excitations (by means of a super-cell calculations) and (c) spin-diffusion (or spin currents) to the process of demagnetization.", "Processes such as Elliott-Yafet scattering and electron-phonon (or lattice) induced spin-relaxation are ignored.", "A comparison with experiments then allows one to quantify the contribution of purely electronic processes to the physics of demagnetization and time scales at which other processes like Elliott-Yafet scattering become significant.", "In the present work we find that for times-scales below 120fs purely electronic processes dominate the physics of demagnetization." ], [ "Theoretical Aspects", "Within TDDFT[43] a Kohn-Sham(KS) Hamiltonian is used for time evolving the electronic wave-function.", "The scalar-relativistic KS Hamiltonian used in the present work reads: $& &\\hat{H}_{\\scriptscriptstyle \\rm S}=\\left[\\frac{1}{2}\\left( \\hat{p} +\\frac{1}{c}{\\bf A}_{\\rm ext}(t)\\right)^2 +v_{s}({\\bf r},t) \\right.", "\\\\ \\nonumber &+&\\left.\\frac{1}{2c} \\hat{\\sigma } \\cdot {\\bf B}_{s}({\\bf r},t) +\\frac{1}{4c^2} \\hat{\\sigma } \\cdot ({\\nabla }v_{s}({\\bf r},t) \\times [\\hat{p} + \\frac{1}{c} \\textbf {A}_{\\rm ext} (t) ])\\right],$ where c is the speed of light, $ \\hat{\\sigma }$ are the Pauli spin operators and ${\\bf A}_{\\rm ext}(t)$ is the vector potential representing an applied laser field.", "In the present work it is assumed that the wavelength of the applied laser pulse is much longer than the length of a unit cell.", "This assumption allows for the so-called dipole approximation and it implies that the spatial dependence of the vector potential ${\\bf A}_{\\rm ext}$ can be disregarded.", "The final term of Eq.", "(REF ) is the spin-orbit coupling term which is included in the most general form and thus automatically includes any derived forms, e.g.", "Rashba-Dresselhaus coupling.", "The KS effective potential $v_{s}({\\bf r},t) = v_{\\rm ext}({\\bf r},t)+v_{\\rm H}({\\bf r},t)+v_{\\rm xc}({\\bf r},t)$ is decomposed into the external potential $v_{\\rm ext}$ , the classical electrostatic Hartree potential $v_{\\rm H}$ and the exchange-correlation (XC) potential $v_{\\rm xc}$ .", "Similarly, the KS magnetic field is written as ${\\bf B}_{s}({\\bf r},t)={\\bf B}_{\\rm ext}(t)+{\\bf B}_{\\rm xc}({\\bf r},t)$ where ${\\bf B}_{\\rm ext}(t)$ is the magnetic field of the applied laser pulse plus possibly an additional magnetic field and ${\\bf B}_{\\rm xc}({\\bf r},t)$ is the XC magnetic field.", "In the present work we use the adiabatic local density approximation[48], [49] (ALSDA) for the XC functional.", "In order to analyze the contribution of each term in this Hamiltonian (REF ) to the dynamics of the magnetization ${\\bf M}(t)= \\int {\\bf m}({\\bf r},t) \\mathrm {d}^3 r = \\int \\langle \\hat{\\sigma } \\hat{n}({\\bf r},t)\\rangle \\mathrm {d}^3 r$ , where $\\hat{n}$ is the density operator and ${\\bf m}({\\bf r},t)$ is the magnetization density, we start by calculating the dynamics of the magnetization density using Ehrenfest's theorem: $\\frac{\\partial }{\\partial t} m_j(\\textbf {r}, t)= i \\langle [\\hat{H_s},\\hat{\\sigma }_j \\hat{n}({\\bf r},t)]\\rangle ,$ Substituting $\\hat{H}_s$ from Eq.", "REF into Eq.", "REF leads to $ \\nonumber \\frac{\\partial }{\\partial t} m_j(\\textbf {r}, t) =i \\Big [ &\\langle [ \\frac{1}{2}(\\hat{p} + \\frac{1}{c}\\textbf {A}_{\\rm ext}(t))^2, \\hat{\\sigma }_j \\hat{n}({\\bf r},t)] \\rangle + \\langle [\\frac{1}{2c} \\hat{\\sigma } \\cdot \\rm B_{\\scriptscriptstyle \\rm S}({\\bf r},t),\\hat{\\sigma }_j \\hat{n}({\\bf r},t)] \\rangle \\\\ &+ \\langle [\\frac{1}{4c^2} \\hat{\\sigma } \\cdot (\\nabla v_{\\scriptscriptstyle \\rm S}({\\bf r},t) \\times [\\hat{p} +\\frac{1}{c} \\textbf {A}_{\\rm ext}]),\\hat{\\sigma }_j \\hat{n}({\\bf r},t)] \\rangle \\Big ].$ Evaluating the commutators results in $ \\nonumber \\frac{\\partial }{\\partial t} {\\bf m}(\\textbf {r}, t) &=- \\nabla \\cdot \\overleftrightarrow{\\textbf {J}}({\\bf r},t) + \\frac{1}{c}[\\textbf {B}_s ({\\bf r},t) \\times \\textbf {m} ({\\bf r},t)] + \\frac{1}{4c^2} [\\nabla n({\\bf r},t) \\times \\nabla v_{\\scriptscriptstyle \\rm S}({\\bf r},t)] \\nonumber \\\\& \\hspace*{28.45274pt}+ \\frac{1}{2c^2} [\\overleftrightarrow{\\textbf {J}}^T({\\bf r},t)-Tr\\lbrace \\overleftrightarrow{\\textbf {J}}({\\bf r},t)\\rbrace ] \\cdot \\nabla v_{\\scriptscriptstyle \\rm S}({\\bf r},t),$ where $\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}\\!\\!", "(\\mathbf {r})=\\,\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}_p\\!\\!", "(\\mathbf {r})\\,+\\,\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}_d\\!\\!", "(\\mathbf {r})$ corresponds to the total spin-current tensor with paramagnetic component $\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}_p\\!\\!", "(\\mathbf {r})= \\langle \\hat{\\sigma } \\otimes \\frac{1}{2} \\lbrace \\hat{n}(\\mathbf {r}), \\hat{\\mathbf {p}}\\rbrace \\rangle $ and diamagnetic component $\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}_d\\!\\!", "(\\mathbf {r})= \\mathbf {m}(\\mathbf {r}) \\otimes \\frac{1}{c} \\mathbf {A}_{ext}$ .", "The change in the global moment, $\\partial _t \\textbf {M}(t)= \\int \\partial _t \\textbf {m}({\\bf r},t) \\mathrm {d}^3 r$ , can be evaluated by integrating Eq.", "REF ; the integral over $- \\nabla \\cdot \\overleftrightarrow{\\textbf {J}}({\\bf r},t)$ vanishes due to Gauss's law and the integral over $\\frac{1}{4c^2} [\\nabla n({\\bf r},t) \\times \\nabla v_{\\scriptscriptstyle \\rm S}({\\bf r},t)]$ vanishes upon integrating by parts.", "For ALSDA $\\textbf {B}_{\\rm xc} ({\\bf r},t) \\times \\textbf {m} ({\\bf r},t)=0$ and in the absence of any external magnetic field (i.e.", "B$_{\\rm ext}$ =0), the dynamics of the magnetization is given by: $ \\nonumber \\frac{\\partial }{\\partial t}\\mathbf {M}(t)&=& \\frac{1}{2c^2} \\int d^3r \\,[\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}\\hspace{-5.69046pt}\\phantom{|}^T(\\mathbf {r},t)- Tr\\lbrace \\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}(\\mathbf {r},t)\\rbrace ] \\cdot \\nabla v_s(\\mathbf {r},t)\\\\[0.3cm]&=& \\frac{1}{2c^2} \\int d^3r\\begin{bmatrix} \\hat{x} \\\\ \\hat{y} \\\\ \\hat{z} \\end{bmatrix} \\times \\begin{bmatrix}\\nabla v_s(\\mathbf {r},t) \\times \\mathbf {j}_x(\\mathbf {r},t) \\\\\\nabla v_s(\\mathbf {r},t) \\times \\mathbf {j}_y(\\mathbf {r},t) \\\\\\nabla v_s(\\mathbf {r},t) \\times \\mathbf {j}_z(\\mathbf {r},t) \\\\\\end{bmatrix}$ with $\\nonumber \\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}(\\mathbf {r},t) =\\begin{pmatrix} \\mathbf {j}_x^T(\\mathbf {r},t) \\\\ \\mathbf {j}_y^T(\\mathbf {r},t)\\\\ \\mathbf {j}_z^T(\\mathbf {r},t) \\end{pmatrix}$ Here each spin-current density describes the flow of the respective spin-component.", "The $z$ -component of Eq.", "REF reads: $\\frac{\\partial }{\\partial t}M_z(t) = \\frac{1}{2c^2} \\int d^3r \\,[\\nabla v_s(\\mathbf {r},t) \\times \\mathbf {j}_y(\\mathbf {r},t)]_x -[\\nabla v_s(\\mathbf {r},t) \\times \\mathbf {j}_x(\\mathbf {r},t)]_y$" ], [ "Results", "In order to distinguish the behaviour of spins on the surface (or interface) from those in the bulk of a sample subjected to an intense laser pulse, we study two cases in the present work; laser induced spin-dynamics in (a) bulk Ni and (b) a free standing film of Ni.", "Bulk Ni is easy to simulate in an electronic structure code which uses periodic boundary conditions (Elk code is used for all simulations[50]), but the computation of a thin film requires special care; in the present work we have used a 5 atomic layer thick film with a 5 layer thick vacuum.", "The $z$ -axis points in plane of the film and $y$ -axis out of the plane.", "The ground state of this thin film is ferromagnetic with the magnetization pointing along the $z$ -axis (in-plane of the film) (with a layer resolved moment of 0.71, 0.68 and 0.66 $\\mu _{\\rm B}$ starting from the layer adjacent to the vacuum).", "The pump laser pulse is then applied perpendicular onto this surface.", "At $t=0$ we begin the simulation from the ground-state for both systems (bulk and film) and the results for the relative moment $M_z(t)/M_z(t=0)$ as a function of time are shown in Fig.", "REF .", "In both cases, under the influence of the laser pulse of fluence 8.05 mJ/cm$^2$ , the system is first optically excited (i.e.", "electrons are excited to energetically higher lying states), followed by a global loss in the magnetic moment.", "The two cases differ in the amount of demagnetization; the bulk shows a small loss of moment ($\\sim 8\\%$ during the simulation) while for the thin film this loss is much larger ($\\sim 20\\%$ ).", "Furthermore the calculations done in the film geometry can be compared to realistic experiments, where the laser pulse is applied perpendicular to the surface of the sample and such a comparison is made in Fig.", "REF c. The results show that we only reach an agreement with the XMCD data of Ref.", "[11] for the first $\\sim $ 120fs beyond which the theoretical work saturates and experimental data continues to demagnetize.", "The reason behind this disagreement between theory and experiment is the fact that in the present work we have included electronic only contribution (like spin-flips, spin-current and magnons) to the process of demagnetization and electron-lattice, electron-phonon, other Elliott-Yafet like mechanisms and radiation losses are ignored.", "These results thus point to two important findings –(a) how significant and at what time scales is the contribution of the electronic processes to the total (experimental) demagnetization and (b) the time scales at which dissipative processes like Elliott-Yafet mechanism start to be significant.", "We find that, in the present case, around 120fs such processes start to contribute and become dominant for longer times.", "At this point, it is natural to ask if the underlying mechanism for ultrafast demagnetization in bulk differs from a thin film.", "To answer this we note that in a purely electronic simulation there are two distinct spin excitation processes that can lead to a loss in the moment: processes that lead to local moment loss like magnons or non-collective canting of spins between atoms and processes that lead to global moment loss like spin-flips (or Stoner like excitations).", "We find that in both cases, for the first $\\sim $ 120fs, the observed loss in magnetization along the $z$ -direction is not accompanied by an increase in magnetization in the $x$ or $y$ directions, indicating that the long-range non-collinearity plays very little role on these time scales and the dynamics of Fig.", "REF is dominated by spin-flip processes for both bulk and thin films.", "Furthermore, the term in the Hamiltonian (Eq.", "REF ) responsible for this magnetization dynamics is the spin-orbit coupling and setting this term to zero leads to no global demagnetization.", "Figure: The vector field 𝐣 x (t=120fs){\\bf j}_x(t=120fs) around the Ni atom.", "The choice of t=120t=120 fs is made based on the demagnetization being at its maximum.", "To easily visualize this vector field we show the streamlines in the left panel for bulk and in the right panel for Ni atom adjacent to the vacuum in the film.Despite the similarity in the underlying physics, the amount of demagnetization is markedly different in the two cases (film and bulk).", "The question then arises: what leads to this strong difference?", "From Eq.", "REF it is clear that the spin-current tensor, $\\stackrel{\\tiny {\\mbox{$\\leftrightarrow $}}}{\\mathbf {J}}$ , is the sole quantity responsible for demagnetization.", "Hence to understand the difference between the thin film and the bulk demagnetization, in Fig.", "REF , we plot the $x$ -spin-component of the spin-current, ${\\bf j}_x $ , which can be defined following Eq.", "REF as ${\\bf j}_x({\\bf r})=\\langle \\hat{\\sigma }_x \\otimes \\frac{1}{2} \\lbrace \\hat{n}(\\mathbf {r}), \\hat{\\mathbf {p}} + \\frac{1}{c} \\mathbf {A}_{ext}\\rbrace \\rangle = \\langle \\hat{\\sigma }_x \\otimes \\hat{{\\bf j}}\\rangle $ .", "This quantity can be interpreted as the vector field showing the flow of spins orientated along the $x$ -direction.", "As an aid to visualizing these currents, we can create streamlines by following the direction of the vector at each point with the strength of this flow shown in colour.", "Fig.", "REF thus shows how the spins pointing in the $x$ -direction are flowing (circulating) around the $y$ -axis.", "From these results it is clear that for both the bulk and the film the spin-current loops clockwise about the $y$ -axis, as is required by Eq.", "REF to cause a change in the moment.", "However, the magnitude of the spin-current in the case of the thin film is much larger than for the bulk and this enhanced spin-current then leads to larger demagnetization in the film.", "The reason behind this we find to be the broken symmetry due to surface/interface formation, which allows for large surface currents due to the presence of localized electronic wave-function (a natural consequence of the lower symmetry).", "The flow of this surface current towards the center of the film then causes large spin-currents every where in the film resulting in a significant demagnetization.", "Figure: Top Panel: The electric field of the applied laser pulse, with peak intensity 1×10 13 1\\times 10^{13} W/cm 2 ^2, FWHM of 17 fs, and fluence of 91.2791.27 mJ/cm 2 ^2.Lower panel: The dynamics of the zz-component of the magnetic moment for top (black) and central (red) layers of Ni film and for bulk Ni (blue).The shorter pulse leads to a significant increase of the observed demagnetization.", "Additionally, the demagnetization occurs much faster.To gather further insight into different behaviour of thin films and bulk it is instructive to compare the layer resolved magnetization of the film to that of the bulk.", "These results, for an ultra-short pulse with FWHM=17fs, are plotted in Fig.", "REF for the bulk, the top layer of the film and the very central layer of the film.", "From this data it is clear that during the first $\\sim $ 30 fs the demagnetization in the bulk and the central layer of the film are similar while the top layer of the film demagnetizes faster.", "Beyond 40fs the layers of the film continue to demagnetize while the bulk saturates.", "These results raise two interesting questions; why the demagnetization is almost the same for the bulk and the film in the first 30fs and why do the layers of the film continue to demagnetize whereas the bulk saturates beyond 40fs?", "One of the major reasons for these effects is that below $\\sim $ 30fs the spin-currents have almost the same magnitude for the bulk and the film, subsequently the broken symmetry allows for stronger currents in the film, while for the case of the bulk these currents stay small.", "In case of the film these spin-currents flow from one layer to another and continue to flow beyond the first 40fs, leading to further demagnetization of the layers of the film.", "This explains the observed physics of demagnetization in the two cases.", "It would be interesting to know the thickness at which the very central layer of the film start to behave like the bulk of Ni.", "But TDDFT, a state-of-the-art method to deal with light-matter interaction, comes at a very high computational cost.", "As a result, treating a film greater than a few atomic-layer thickness is not feasible with the resources available to us.", "Finally, it is crucial to mention that in Fig.", "REF we have used the predictive power of TDDFT to study the influence of ultra-short laser pulses on the spin dynamics.", "Presently there are no experiments reported which utilize such pulses, however the parameters are within the range of available technology.", "The effect of this ultra-short pulse is faster and much enhanced demagnetization; for a film $51\\%$ of the moment is lost while for bulk this value is $17\\%$ .", "The corresponding values for the long pulse (see Fig.", "REF ) are 20% for the film and 8% for the bulk." ], [ "Discussion", "By performing ab-initio simulations for laser pulse excited ferromagnetic thin films, we have demonstrated that thin films show enhanced spin-orbit mediated demagnetization compared to a bulk.", "We have further demonstrated that this enhancement in the demagnetization near the surface of a film is due to the broken symmetry which increases the rotating spin currents in the system.", "These calculations show the importance of treating the spin-orbit coupling as well as charge- and spin-currents at the same footing.", "From our calculations one can conclude that for a typical laser pump pulse (as currently used in experiments), demagnetization (caused by purely electronic processes) is strong within the first few atomic layers of a material but will then decrease to the smaller bulk value as we get deeper into the sample and it becomes more bulk-like.", "For longer duration pulses, such as that used in Fig.", "REF , for the first $\\sim $ 120fs purely electronic processes dominate the physics of demagnetization.", "Beyond first 120fs, these electronic mechanisms will exist in addition to dissipative mechanisms such as Elliott-Yafet, and it will require more ingenious experiments to disentangle and distinguish them.", "This early time purely electronic regime is of great importance for optimal control and device production at ultrashort timescales as it allows for the possibility of coherent control of the electrons." ], [ "Computational Details", "Considering future coherent control of spins by light, in the present work we explicitly concentrate on the electronic degrees of freedom during the early femtoseconds.", "Dissipative processes that induce decoherence such as radiation and phonons are not included (nuclei are kept fixed during the simulation).", "At this point it is important to mention that collective magnetic excitations (e.g.", "magnons) are also purely electronic in nature and have been included in the present work through usage of a large super-cell.", "However, we found that magnons only have a minor contribution to the spin-dynamics in the time scales studied in the present work[28].", "State-of-the art full potential linearized augmented plane wave (LAPW) method implemented within the Elk code[50] is used in the present work.", "The core electrons (with Eigenvalues below 95eV) are treated using the radial Dirac equation while higher lying electrons are treated using the scalar relativistic Hamiltonian in the presence of the spin-orbit coupling.", "To obtain the 2-component Pauli spinor states, the Hamiltonian containing only the scalar potential is diagonalized in the LAPW basis: this is the first-variational step.", "The scalar states thus obtained are then used as a basis to set up a second-variational Hamiltonian with spinor degrees of freedom[51].", "This is more efficient than simply using spinor LAPW functions, but care must be taken to ensure that there is a sufficient number of first-variational eigenstates for convergence of the second-variational problem.", "In the present work 300 empty states per k-point are used.", "For Ni a face centered cubic crystal structure with lattice spacing of $3.52$ Å  is used.", "A k-point grid of $8\\times 8\\times 8$ is used for the bulk and $1\\times 8\\times 8$ for the film calculations.", "A full geometry optimization for the Ni-film was performed, however, we found that for intense laser pulses, such as used in present work, geometry optimization does not change results significantly.", "The maximum augmented plane-wave cutoff for the orbitals was $3.5$ au$^{-1}$ corresponding to an energy cutoff of approximately 160 eV, and for the density and potential a value of 12 au$^{-1}$ was used.", "An angular momentum cutoff of 8 for orbitals and 7 for densities and potential was used.", "A time step of $0.002$ fs was used for time propagation[52].", "The initial state for all TDDFT calculations was the DFT ground-state, which is at a temperature of 0K.", "We use the adiabatic local density approximation[48], [49] for the XC functional.", "The film calculations required 900000 CPU hours, running on the HYDRA supercomputer at RZG Garching for 1 month." ], [ "Acknowledgements", "TM and SS would like to thank QUTIF-SPP for funding.", "PE and SS would like to acknowledge funding from DFG through SFB762 project." ] ]
1612.05663
[ [ "A new technique for infrared scintillation measurements" ], [ "Abstract We propose a new technique to measure the infrared scintillation light yield of rare earth (RE) doped crystals by comparing it to near UV-visible scintillation of a calibrated Pr:(Lu$_{0.75}$Y$_{0.25}$)$_{3}$Al$_5$O$_{12}$ sample.", "As an example, we apply this technique to provide the light yield in visible and infrared range up to \\SI{1700}{nm} of this crystal." ], [ "Introduction", "A new concept, all-optical particle radiation detector has been very recently proposed based on the mechanism of upconversion in RE-doped materials  [1].", "This process, in which low energy incident optical radiation (infrared light - IR) is converted into higher energy emitted photons (visible light), is efficiently accomplished by incorporating RE ions in inorganic matrices due to their $f$ -electrons configurations.", "In fact, ground state absorption (level 0) allows the rare earth ions to reach a metastable intermediate state (level 1), characterized by relatively long lifetimes ($\\approx $ ms), then another photon delivered by a pump laser tuned to the transition $1 \\rightarrow 2$ promotes the ion to a more energetic state (level 2).", "Radiative transition from this latter excited state back to the ground state is then observed by means of traditional detectors as photomultipliers (PMT) or photodiodes (PD).", "To date, this mechanism has been extensively applied for the development of lasers and optical devices [2], [3] but its applicability in the field of particle detection has not been deeply investigated.", "The visible light yield (LY) of this novel device depends on the upconversion efficiency and on the number of RE-ions excited into the metastable level 1 per energy unit of the particle.", "This latter quantity can be estimated by studying the LY in the IR band which very few articles in literature are concerned with [4].", "Actually, there is little interest in the RE-doped crystals infrared scintillation for their long decay lifetimes and for the low quantum efficiency of PMT in this spectral region.", "The aim of this work is to propose a method that allows us to tackle a systematic investigation of the IR LY in several different materials, composed of different matrices, dopants and concentrations.", "This method is based on the luminescence comparison with a reference Pr:(Lu$_{0.75}$ Y$_{0.25}$ )$_{3}$ Al$_5$ O$_{12}$ single crystal whose light yield in the near UV-visible is known.", "Moreover, this method is applied to this crystal, thereby yielding its IR LY." ], [ "Experimental setup", "The mixed lutetium-yttrium aluminum garnet sample has been grown via Czochralsky method at ITME, Warsaw and its preparation is described elsewhere [5].", "The interest in mixed Pr:LuYAG crystals is related to its much better performance in terms of light yield and energy resolution as compared to either Pr:LuAG or Pr:YAG.", "The $5\\times 5\\times 5$ mm$^3$ sample chosen for the present measurements has a reported light yield of 27000ph/MeV and a 5.3% energy resolution in the UV-VIS range [5].", "To investigate its light yield in the IR band, the Pr:LuYAG sample is excited by X-rays generated by an electron gun that can be operated both in continuous and in pulsed mode sweeping the electron beam at 100Hz frequency [6].", "The several intense electron beam impinges on a $\\sim $ 10-thick tantalum foil placed in front of the sample to make sure that the whole crystal is excited only by X-rays.", "The scintillation response of the crystal sample is measured using Si (mod.", "Hamamtsu S1337-1010BQ) and InGaAs (mod.", "Thorlabs DET20C) PDs.", "The small contribution to the PD signal of the X-rays that are not absorbed in the crystal and reach the PD can be estimated and then subtracted by covering the PD with an thin aluminum foil.", "In order to verify that the radioluminescence is determined by the Pr$^{3+}$ ions emission, the X-rays excitation spectra are compared with those obtained when the crystal is irradiated with a pulsed, frequency-quadrupled Nd:YAG laser (266nm).", "In fact, whereas the host matrix is transparent at this wavelength, the Pr$^{3+}$ ions are directly excited into $4f5d$ levels, thus allowing us to simulate the energy transfer process from the electron-hole recombination to the RE ions after the X-ray excitation." ], [ "Spectroscopic analysis", "We have recorded spectra of the Pr:LuYAG emission due to X-ray and UV excitation (Fig.", "REF ).", "The OceanOptics 650 RedTide and OceanOptics NIRQuest512 spectrometers were used for the 200–850nm and 900–1700nm regions, respectively.", "In both spectra one can clearly see emissions related to the same Pr$^{3+}$ energy levels.", "We observe a strong $4f5d \\rightarrow 4f^2$ emission in the range from 300 to 450nm and narrow lines in the visible/near-infrared region due to emission from levels $^{3}\\mathrm {P}_0$ , $^{1}\\mathrm {D}_2$ , $^{1}\\mathrm {G}_4$ .", "The spectra below 850nm are similar to those reported by previous authors in a Pr:LuAG crystal [7].", "The main near-infrared transitions that we identify are: $^1\\mathrm {D}_2 \\rightarrow $ $^{3}\\mathrm {F}_{3,4}$ , $^1\\mathrm {G}_4 \\rightarrow $ $^3\\mathrm {H}_4$ (900–1100nm) and $^1\\mathrm {D}_2 \\rightarrow $ $^1\\mathrm {G}_4$ (1400–1600nm).", "The emission from $^1\\mathrm {G}_4$ is expected to be partially non radiatively quenched, whereas the lower lying levels are strongly quenched in LuYAG matrix.", "As a consequence no mid-infrared emission is expected from these latter levels." ], [ "Method", "We measure the total number of charge carriers generated per X-ray pulse in the photodiode $n_e=Q_d R_0/G\\tau $ with $Q_d$ being the time integrated photodiode signal, $R_0 = $ 1 the impedance of oscilloscope, $G = {0.25}{mV/fC}$ and $\\tau \\approx $ 480s which are the gain and the time constant of the active integrator, respectively.", "If the PD quantum efficiency $\\eta $ is constant in the considered range, $n_e$ is also given by equation $n_e = E_{in} \\cdot \\mathrm {LY} \\cdot \\eta \\cdot \\dfrac{\\Delta \\Omega }{4\\pi } \\left( 1- R \\right)$ where $E_{in}$ is the energy deposited in the sample, $R$ the crystal reflectivity, ${\\Delta \\Omega }=A/ d^2$ is the solid angle subtended by the PD with area $A$ located at a distance $d$ from the crystal.", "Moreover, the measured charge per pulse $Q_{bs}$ is related through a proportionality constant $k$ to the energy released by X-ray pulse in the crystal.", "In fact, as shown in the Fig.", "REF , the measured luminescence $Q_d$ linearly depends on $Q_{bs}$ .", "It is then possible to obtain a general expression for the measured light yield in a definite wavelength range: $\\mathrm {LY} = \\dfrac{Q_d}{Q_{bs}}d^2 \\dfrac{4\\pi R_0}{kG\\tau } \\dfrac{1}{\\eta \\left( 1-R \\right)A}.$ As in the point source approximation the following expression holds true $\\dfrac{Q_{d}(x)}{Q_{bs}}= \\left(\\dfrac{Q_d}{Q_{bs}}d^2\\right) \\dfrac{1}{\\left( x_0 + x \\right)^2} = \\dfrac{a}{\\left( x_0 + x \\right)^2}$ the parameters $a=\\dfrac{Q_d}{Q_{bs}}d^2$ and $x_0$ can be obtained by a fit of data recorded at different relative distances $x$ of the photodiode from the scintillating crystal, as shown in Fig.", "REF .", "We observe that in this type of measurements it crucial to precisely know the efficiency of the X-ray generation process, related to the previously mentioned proportionality constant $k$ .", "The latter can be estimated if the X-ray energy is fully released in the sample and provided the LY and the $a$ parameter of a generic crystal in any wavelength range are known.", "As the sensitivity of the detector that has been used to measure the light yield of our LuYAG:Pr crystal reported in [5] peaks in the UV range, it is reasonable to expect that the reported light yield is mainly determined by $4f5d \\rightarrow 4f^2$ radiative transitions.", "In addition, due to fast integration time the slow components might have been underestimated.", "In order to select the photodiode signal component due to the UV scintillation, the previously described measurements are repeated at a fixed distance with optical longpass filters (Thorlabs, FGL and FEL filter sets) located in front of the photodiodes.", "The use of filters allows us to estimate the light yield in narrower bandwidths and with a better accuracy by taking into account the wavelength dependence of the photodiode's responsivity.", "As the energy of the X-rays can be assumed to be in the range of few tens of keV, the LY of our sample measured in [5] at 662keV has to be reduced by 10%, in agreement with previously published data [8], [9], [10].", "We report in Table REF the results of the light yield measurements in several optical ranges.", "Table: Measured light yield of the LuYAG:Pr crystal for different optical ranges." ], [ "Conclusions", "We have demonstrated a practical way to accurately estimate the infrared light yield of a RE-doped crystal by comparison with the near UV-visible luminescence of a reference crystal.", "The presented method allows us to make accurate LY measurements since it is based on the point-source approximation, as verified by performing measurements at several distances from the source, and because it is possible to vary the energy released per pulse in the crystal and to average over several measurements.", "Furthermore, the use of several optical longpass filters reduces the error due to the wavelength dependence of the photodiode quantum efficiency.", "The method has been applied to a (Lu$_{0.75}$ Y$_{0.25}$ )$_{3}$ Al$_5$ O$_{12}$ crystal whose emission in the UV range had been previously measured.", "In spite of its high LY in the UV, its emission in the near infrared band is limited to a few thousands of ph/MeV, probably due to the $^1\\mathrm {G}_{4}$ manifold quenching.", "Although the states of our interest $^3\\mathrm {H}_\\mathrm {J}$ and $^3\\mathrm {F}_\\mathrm {J}$ , characterized by ms-long radiative lifetimes, are efficiently populated by the relaxation of the higher manifolds, no mid-infrared emission is expected from them in this host matrix.", "The data reported in this work can be used to extend the LY measurements up to the mid-infrared band, provided that photodiodes with a lower band gap than InGaAs are used.", "In order to identify the most suitable crystals for the development of the upconversion-based detector, several RE-doped crystals are currently being investigated with the method described in this work.", "The preliminary results obtained for Nd:YAG and Tm:YAG are particularly encouraging for our aims and will soon be published." ], [ "Acknowledgments", "The growth of the crystals used in this research has been financed from the funds of the Polish National Science Centre granted on the basis of Decision no.", "DEC-2012/05/B/ST5/00324.", "The authors FC, AFB, CB, GC, and MG acknowledge the technical assistance of Mr. E. Berto and Mr. L. Barcellan.", "Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION" ] ]
1612.05507
[ [ "BSDEs with default jump" ], [ "Abstract We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process $(\\lambda_t)$.", "We give a priori estimates for these equations and prove comparison and strict comparison theorems.", "These results are generalized to drivers involving a singular process.", "The special case of a $\\lambda$-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale.", "We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a totally defaultable risky asset.", "The case of claims paying dividends is also studied via a singular process." ], [ "Introduction", "The aim of the present paper is to study BSDEs driven by a Brownian motion and a martingale measure associated with a default jump process with intensity process $\\lambda =(\\lambda _t)$ .", "The applications we have in mind are the pricing and hedging issues for contingent claims in an imperfect financial market with default.", "The theory on BSDEs driven by a Brownian motion and a Poisson random measure has been studied extensively by several authors (we refer e.g.", "to Barles, Buckdahn and Pardoux [3], Royer [21], Quenez and Sulem [20]).", "The present study relies on many arguments which are used in this literature.", "Nevertheless, the treatment of a default jump requires some specific arguments which are not straightforward and we present here a rigorous analysis of these BSDEs with default jump.", "To our knowledge, there are few results on non linear BSDEs with default jump.", "The papers [7] and [1] concern only the existence and the uniqueness of the solution, established under different assumptions (see Remark REF for more details).", "In our paper, we first provide some useful a priori estimates, from which the existence and uniqueness result directly follows.", "We also give a representation property of the solution when the driver is $\\lambda $ -linear, as well as comparison and strict comparison theorems in the general case.", "We moreover allow the driver of these equations to have some singular component, in the sense that the driver may be of the generalized form $g(t,\\omega ,y,z,k)dt + dD_t(\\omega ), $ where $D$ is a finite variation càdlàg process with square integrability conditions.", "This allows us to treat the case of dividends in our financial application.", "The paper is organized as follows: in Section , we present the theory of BSDEs with default jump.", "More precisely, in Section REF , we present the mathematical setup.", "In Section REF , we state some a priori estimates, from which we derive the existence and the uniqueness of the solution.", "In Section REF , we introduce the definition of a $\\lambda $ -linear driver, where $\\lambda $ refers to the intensity of the jump process, which generalizes the notion of a linear driver given in the literature on BSDEs to the case with default jump.", "When the driver is $\\lambda $ -linear, we provide an explicit solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale.", "In Section REF , we establish a comparison theorem, which holds under an appropriate assumption on the driver.", "We also prove a strict comparison theorem, which requires an additional assumption.", "Some interesting counterexamples of comparison theorems are given when the assumptions of these theorems are violated.", "We then turn to the application in Mathematical Finance in section .", "We consider a financial market with a defaultable risky asset and we study pricing and hedging issues for a European option paying a payoff $\\xi $ at the maturity $T$ and intermediate dividends modeled via a singular process $D$ .", "The case of a perfect market model is first studied via the theory of $\\lambda $ -linear BSDEs with default jump, while the case of imperfections, expressed via the nonlinearity of the wealth dynamics, is then treated by the theory of nonlinear BSDEs with generalized driver developed in Section .", "In this setting, the pricing system is expressed as a nonlinear expectation/evaluation $\\mathcal {E}^{g, \\cdot }: $ $(\\xi ,D) \\mapsto \\mathcal {E}^{^{g,D}}(\\xi ),$ induced by a nonlinear BSDE with default jump (solved under the primitive probability measure $P$ ) with generalized driver $g(t, \\cdot )dt+dD_t$ .", "Properties of consistency, monotonicity, convexity, non-arbitrage of this pricing system rely on the properties of the associated BSDE.", "As an illustrative example of market imperfections, we consider the case when the seller of the option is a large investor whose trading strategy may affect the market asset prices and the default intensity." ], [ "Probability setup", "Let $(\\Omega , \\mathcal {G},\\mathbb {P})$ be a complete probability space equipped with two stochastic processes: a unidimensional standard Brownian motion $W$ and a jump process $N$ defined by $N_t={\\bf 1}_{\\vartheta \\le t}$ for any $t\\in [0,T]$ , where $\\vartheta $ is a random variable which modelizes a default time.", "We assume that this default can appear at any time that is $P(\\vartheta \\ge t)>0$ for any $t\\ge 0$ .", "We denote by ${\\mathbb {G}}=\\lbrace \\mathcal {G}_t, t\\ge 0 \\rbrace $ the complete natural filtration of $W$ and $N$ .", "We suppose that $W$ is a ${\\mathbb {G}}$ -Brownian motion.", "Let $(\\Lambda _t)$ be the predictable compensator of the non decreasing process $(N_t)$ .", "Note that $(\\Lambda _{t \\wedge \\vartheta })$ is then the predictable compensator of $(N_{t \\wedge \\vartheta } )= (N_t)$ .", "By uniqueness of the predictable compensator, $\\Lambda _{t \\wedge \\vartheta } = \\Lambda _t$ , $t\\ge 0$ a.s. We assume that $\\Lambda $ is absolutely continuous w.r.t.", "Lebesgue's measure, so that there exists a nonnegative process $\\lambda $ , called the intensity process, such that $\\Lambda _t=\\int _0^t \\lambda _s ds$ , $t\\ge 0$ .", "Since $\\Lambda _{t \\wedge \\vartheta } = \\Lambda _t$ , $\\lambda $ vanishes after $\\vartheta $ .", "We denote by $M$ the compensated martingale which satisfies $M_t = N_t-\\int _0^t\\lambda _sds\\,.$ Let $T >0$ be the finite horizon.", "We introduce the following sets: ${\\mathcal {S}}^{2}$ is the set of ${\\mathbb {G}}$ -adapted RCLL processes $\\varphi $ such that $\\mathbb {E}[\\sup _{0\\le t \\le T} |\\varphi _t | ^2] < +\\infty $ .", "${\\mathcal {A}}^2$ is the set of real-valued non decreasing RCLL adapted processes $A$ with $A_0 = 0$ and $\\mathbb {E}(A^2_T) < \\infty $ .", "${\\mathbb {H}}^2$ is the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert Z\\Vert ^2:= \\mathbb {E}\\Big [\\int _0^T|Z_t|^2dt\\Big ]<\\infty \\,.$ ${\\mathbb {H}}^2_{\\lambda }$ is the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert U\\Vert _{\\lambda }^2:=\\mathbb {E}\\Big [\\int _0^T|U_t|^2\\lambda _tdt\\Big ]<\\infty \\,.$ In the following, ${\\mathcal {P}}$ denotes the ${\\mathbb {G}}$ -predictable $\\sigma $ -algebra on $\\Omega \\times [0,T]$ .", "Note that for each $U \\in {\\mathbb {H}}^2_{\\lambda }$ , we have $\\Vert U\\Vert _{\\lambda }^2=\\mathbb {E}\\Big [\\int _0^{T\\wedge \\vartheta }|U_t|^2\\lambda _tdt\\Big ] \\,$ because the ${\\mathbb {G}}$ -intensity $\\lambda $ vanishes after $\\vartheta $ .", "Moreover, we can suppose that for each $U$ in ${\\mathbb {H}}^2_{\\lambda }$ ($= L^2(\\Omega \\times [0,T], {\\mathcal {P}}, dP \\lambda _t dt)$ ), its representant, still denoted by $U$ , vanishes after $\\vartheta $ .", "Moreover, $\\mathcal {T}$ is the set of stopping times $\\tau $ such that $\\tau \\in [0,T]$ a.s.  and for each $S$ in $\\mathcal {T}$ , $\\mathcal {T}_{S}$ is the set of stopping times $\\tau $ such that $S \\le \\tau \\le T$ a.s. We recall the martingale representation theorem in this framework (see e.g.", "[16]): Lemma 2.1 Any ${\\mathbb {G}}$ -local martingale $m= (m_t)_{0\\le t \\le T}$ has the representation $m_t = m_0+\\int _0^t z_sdW_s+\\int _0^t l_sdM_s \\,, \\quad \\forall \\,t\\in [0,T] \\quad a.s. \\,,$ where $z= (z_t)_{0\\le t \\le T}$ and $l= (l_t)_{0\\le t \\le T}$ are predictable processes such that the two above stochastic integrals are well defined.", "If $m$ is a square integrable martingale, then $z \\in {\\mathbb {H}}^2$ and $l \\in {\\mathbb {H}}^2_{\\lambda }$ .", "We now introduce the following definitions.", "Definition 2.2 (Driver, $\\lambda $ -admissible driver) A function $g$ is said to be a driver if $g:\\Omega \\times [0,T] \\times {\\bf R}^3 \\rightarrow {\\bf R}$ ; $(\\omega , t,y, z, k) \\mapsto g(\\omega , t,y,z,k) $ is $ {\\mathcal {P}} \\otimes {\\mathcal {B}}({\\bf R}^3)- $ measurable, and such that $g(.,0,0,0) \\in {\\mathbb {H}}^2$ .", "A driver $g$ is called a $\\lambda $ -admissible driver if moreover there exists a constant $ C \\ge 0$ such that $dP \\otimes dt$ -a.s. , for each $(y, z, k)$ , $(y_1, z_1, k_1)$ , $(y_2, z_2, k_2)$ , $|g( t, y, z_1, k_1) - g( t, y, z_2, k_2)| \\le C (|y_1 - y_2|+ |z_1 - z_2| + \\sqrt{\\lambda }_t |k_1 - k_2 |).$ The positive real $C$ is called the $\\lambda $ -constant associated with driver $g$ .", "Note that condition (REF ) implies that for each $\\,t > \\vartheta $ , since $\\lambda _t=0$ , $g$ does depend on $k$ .", "In other terms, for each $(y,z,k)$ , we have: $g(t,y,z,k)= g(t,y,z,0), \\quad t > \\vartheta \\quad dP \\otimes dt-{\\rm a.s.}$ Definition 2.3 (BSDE with default jump) Let $g$ be a $\\lambda $ -admissible driver, let $\\xi \\in {L}^2({\\mathcal {G}_T})$ .", "A process $(Y, Z, K)$ in $ \\mathcal {S}^2 \\times {\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ is said to be a solution of the BSDE with default jump associated with terminal time $T$ , driver $g$ and terminal condition $\\xi $ if it satisfies: $-dY_t = g(t,Y_t, Z_t,K_t ) dt - Z_t dW_t - K_t dM_t; \\quad Y_T=\\xi .$" ], [ "First properties of BSDEs with a default jump", "For $\\beta >0$ , $\\phi \\in I\\!\\!H^{2} $ , and $l \\in I\\!\\!H_\\lambda ^{2}$ , we introduce the norms $\\Vert \\phi \\Vert _{\\beta }^2 := E[\\int _0^T e^{\\beta s} \\phi _s^2 ds], $ and $\\Vert k \\Vert _{\\lambda ,\\beta }^2 := E[\\int _0^T e^{\\beta s} k_s^2 \\lambda _s \\, ds] $ .", "We first show some a priori estimates for BSDEs with a default jump, from which we derive the existence and uniqueness of the solution." ], [ "A priori estimates for BSDEs with a default jump", "Proposition 2.4 Let $\\xi ^1$ , $\\xi ^2$ $\\in L^2({\\mathcal {G}}_T)$ .", "Let $g^1$ and $g^2$ be two $\\lambda $ -admissible drivers.", "For $i=1,2$ , let $(Y^i, Z^i ,K^i)$ be a solution of the BSDE associated with terminal time $T$ , driver $g^i$ and terminal conditions $\\xi ^i$ .", "Let $\\bar{\\xi }:= \\xi ^1 - \\xi ^2$ and $\\bar{g}(s): = g^1(s, Y^2_s, Z^2_s, K_s^2) - g^2(s, Y^2_s, Z^2_s, K_s^2)$ .", "For $s$ in $[0,T]$ , denote $\\bar{Y}_s := Y^1_s - Y^2_s, \\,\\,\\, \\bar{Z}_s := Z^1_s - Z^2_s$ , $\\bar{K}_s := K^1_s - K^2_s $ .", "Let $ \\eta , \\beta >0 $ be such that $\\beta \\ge \\frac{3}{\\eta } +2C $ and $\\eta \\le \\frac{1}{C^2}$ .", "For each $t \\in [0,T]$ , we then have $e^{\\beta t} (\\bar{Y}_t) ^2 \\le \\mathbb {E}[ e^{\\beta T} \\bar{\\xi }\\,^2 \\mid {\\mathcal {G}}_t ] +\\ \\eta \\,{\\mathbb {E}}[ \\int _t^T e^{\\beta s} \\bar{g}(s)^2 ds \\mid {\\mathcal {G}}_t ] \\;\\; \\text{ \\rm a .s.", "}\\,$ Moreover, $\\Vert \\bar{Y} \\Vert _\\beta ^2 \\le T [e^{\\beta T} \\mathbb {E}[\\bar{\\xi }\\,^2] + \\eta \\Vert \\bar{g} \\Vert _\\beta ^2].$ If $\\eta < \\frac{1}{C^2}$ , we have $\\Vert \\bar{Z} \\Vert _\\beta ^2 + \\Vert \\bar{K} \\Vert _{\\lambda ,\\beta }^2\\le \\frac{1}{1 - \\eta C^2} [e^{\\beta T} \\mathbb {E}[\\bar{\\xi }\\,^2] + \\eta \\Vert \\bar{g} \\Vert _\\beta ^2].$ By Itô's formula applied to the semimartingale $e^{\\beta s} \\bar{Y}_s$ between $t$ and $T$ , we get $e^{\\beta t} \\bar{Y}_t ^2 & + \\beta \\int _t^T e^{\\beta s} \\bar{Y}_s^2 ds +\\int _t^T e^{\\beta s} \\bar{Z}_s^2 ds + \\int _t^T e^{\\beta s} \\bar{K}_s^2 \\lambda _s ds\\nonumber \\\\&=e^{\\beta T} \\bar{Y}_T ^2 + 2 \\int _t^T e^{\\beta s} \\bar{Y}_s (g^1 (s, Y^1_s, Z^1_s, K^1_s) - g^2 (s, Y^2_s, Z^2_s, K^2_s)) ds \\nonumber \\\\&\\quad - 2 \\int _t^T e^{\\beta s} \\bar{Y}_s \\bar{Z}_s dW_s- \\int _t^T e^{\\beta s} (2 \\bar{Y}_{s^-} \\bar{K}_s + \\bar{K}_s^2) d M_s.", "$ Taking the conditional expectation given ${\\mathcal {G}}_t$ , we obtain $e^ {\\beta t} & \\bar{Y}_t^2+ E \\left[\\beta \\int _t^T e^{\\beta s} \\bar{Y}_s^2 ds + \\int _t^T e^{\\beta s} ( \\bar{Z}_s^2 + \\bar{K}_s^2 \\lambda _s) ds \\mid {\\mathcal {G}}_t \\right] \\nonumber \\\\& \\le E \\left[ e^{\\beta T} \\bar{Y}_T ^2 \\mid {\\mathcal {G}}_t\\right] + 2 E \\left[ \\int _t^T e^{\\beta s} \\bar{Y}_s (g^1 (s, Y^1_s, Z^1_s, K^1_s) - g^2 (s, Y^2_s, Z^2_s, K^2_s)) ds \\mid {\\mathcal {G}}_t\\right].$ Now, $ g^1(s,Y^1_s, Z^1_s, K^1_s) - g^2(s,Y^2_s,Z^2_s,K^2_s)= g^1(s,Y^1_s, Z^1_s, K^1_s) - g^1(s,Y^2_s,Z^2_s,K^2_s) + \\bar{g}_s$ .", "Since $g^1$ satisfies condition (REF ), we derive that $|g^1(s,Y^1_s, Z^1_s, K^1_s) - g^1(s,Y^2_s,Z^2_s,K^2_s)|\\le C|\\bar{Y}_s| + C|\\bar{Z}_s| + C |\\bar{K}_s| \\sqrt{\\lambda }_s + |\\bar{g}_s|.$ Note that, for all non negative numbers $\\lambda $ , $y$ , $z$ , $k$ , $g$ and $\\varepsilon >0$ , we have $ 2y (Cz + Ck \\sqrt{\\lambda }+ g) \\le \\frac{ y^2}{\\varepsilon ^2}+ \\varepsilon ^2(Cz+ Ck \\sqrt{\\lambda }+ g)^2 \\le \\frac{ y^2}{\\varepsilon ^2} + 3 \\varepsilon ^2(C^2 y^2+ C^2 k^2\\lambda +g^2)$ .", "Hence, $e^ {\\beta t} \\bar{Y}_t^2 + {\\mathbb {E}} \\left[\\beta \\int _t^T e^{\\beta s} \\bar{Y}_s^2 ds + \\int _t^T e^{\\beta s} ( \\bar{Z}_s^2 +\\bar{K}_s^2\\lambda _s) ds \\mid {\\mathcal {G}}_t \\right] \\le \\mathbb {E}\\left[ e^{\\beta T} \\bar{Y}_T ^2 \\mid {\\mathcal {G}}_t\\right] \\nonumber \\\\+ {\\mathbb {E}} \\left[ (2C+\\frac{ 1}{\\varepsilon ^2}) \\int _t^T e^{ \\beta s} \\bar{Y}_s^2 ds + 3C^2 \\varepsilon ^2 \\int _t^T e^{\\beta s} ( \\bar{Z}_s^2 + \\bar{K}_s^2\\lambda _s)ds+ 3 \\varepsilon ^2 \\int _t^T e^{\\beta s} \\bar{g}_s^2 ds \\mid {\\mathcal {G}}_t \\right].$ Let us make the change of variable $\\eta = 3 \\epsilon ^2$ .", "Then, for each $\\beta , \\eta >0$ chosen as in the proposition, these inequalities lead to (REF ).", "By integrating (REF ), we obtain (REF ).", "Using inequality (REF ), we derive (REF ).", "Remark 2.5 By classical results on the norms of semimartingales, one similarly shows that $\\Vert \\bar{Y} \\Vert _{S^2}$ $ \\le $ $ K \\left( \\mathbb {E}[\\bar{\\xi }\\,^2] + \\Vert \\bar{g} \\Vert _{I\\!\\!H^2}\\right)$ , where $K$ is a positive constant only depending on $T$ and $C$ ." ], [ "Existence and uniqueness result for BSDEs with a default jump", "Proposition 2.6 Let $g$ be a $\\lambda $ -admissible driver, let $\\xi \\in {L}^2({\\mathcal {G}_T})$ .", "There exists an unique solution $(Y, Z, K)$ in $ \\mathcal {S}^2 \\times {\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ of the BSDE (REF ).", "Remark 2.7 This result generalizes the existence and uniqueness result obtained in [7] under stronger assumptions.", "Indeed, suppose that $\\xi $ is $\\mathcal {G}_{\\vartheta \\vee T}$ -measurable (as in [7]) and that $g$ is replaced by $g {\\bf 1}_{t \\le \\tau }$ (which is a $\\lambda $ -admissible driver), then the solution $(Y, Z, K)$ of the associated BSDE (REF ) is equal to the solution of the BSDE with random terminal time $\\vartheta $ , driver $g$ and terminal condition $\\xi $ , considered in [7].", "Note that the boundedness assumption made on the default intensity process $(\\lambda _t)$ in [7] is not necessary to ensure this result.", "Proof.", "We show this result by using the a priori estimates given in Proposition REF .", "The arguments are classical and a short proof is given for completeness.", "Let us first consider the case when the driver $g(t)$ does not depend on the solution.", "By using the representation property of ${\\mathcal {G}}$ -martingales (Lemma REF ) together with classical computations, one can show that there exists a unique solution of the BSDE (REF ) associated with terminal condition $\\xi \\in L^2({\\mathcal {F}}_T)$ and driver process $g(t)$ $\\in $ $I\\!\\!H^2$ .", "Let us turn to the case with a general $\\lambda $ -admissible driver $g(t,y,z,k)$ .", "Denote by $I\\!\\!H_\\beta ^2$ the space $I\\!\\!H^2 \\times I\\!\\!H^2 \\times I\\!\\!H^2_\\lambda $ equipped with the norm $\\Vert Y, Z, K \\Vert _\\beta ^2 := \\Vert Y \\Vert _{\\beta }^2 + \\Vert Z \\Vert _{\\beta }^2 + \\Vert K \\Vert _{\\lambda ,\\beta }^2 $ .", "We define a mapping $\\Phi $ from $I\\!\\!H_\\beta ^2$ into itself as follows.", "Given $(U, V, L) \\in I\\!\\!H_\\beta ^2$ , let $(Y, Z,K) = \\Phi (U, V, L)$ be the the solution of the BSDE associated with driver $g^1(s) :=g(s, U_s , V_s, L_s)$ .", "Let us prove that the mapping $\\Phi $ is a contraction from ${I\\!\\!H}_\\beta ^2$ into ${I\\!\\!H}_\\beta ^2$ .", "Let $(U^{\\prime }, V^{\\prime }, L^{\\prime } )$ be another element of ${I\\!\\!H}_\\beta ^2$ and let $(Y^{\\prime }, Z^{\\prime },k^{\\prime }) :=\\Phi (U^{\\prime }, V^{\\prime }, L^{\\prime })$ , that is, the solution of the RBSDE associated with driver process $g(s, U^{\\prime }_s , V^{\\prime }_s, L^{\\prime }_s)$ .", "Set $\\bar{U} = U - U^{\\prime }$ , $\\bar{V} = V - V^{\\prime }$ , $\\bar{L} = L - L^{\\prime }$ , $\\bar{Y} = Y - Y^{\\prime }$ , $\\bar{Z} = Z-Z^{\\prime }$ , $\\bar{K} = K - K^{\\prime }$ .", "Let $\\Delta g_\\cdot := g(\\cdot , U, V, L) - g(\\cdot , U^{\\prime }, V^{\\prime }, L^{\\prime })$ .", "Using the estimates (REF ) and (REF ) with $\\lambda $ -constant equal to 0 (since the driver $g^1$ does not depend on the solution), we derive that for all $\\eta , \\beta >0$ such that $\\beta \\ge \\frac{3}{\\eta }$ , we have $\\Vert \\bar{Y}\\Vert _\\beta ^2 + \\Vert \\bar{Z}\\Vert _\\beta ^2 + \\Vert \\bar{K}\\Vert _{\\lambda , \\beta }^2 \\le \\eta (T+1) \\Vert \\Delta g \\Vert _\\beta ^2.$ Since the driver $g$ is $\\lambda $ -admissible with $\\lambda $ -constant $C$ , we get $\\Vert \\bar{Y}\\Vert _\\beta ^2 + \\Vert \\bar{Z}\\Vert _\\beta ^2 + \\Vert \\bar{K}\\Vert _{\\lambda , \\beta }^2 \\le \\eta (T+1) 3C^2 (\\Vert \\bar{U}\\Vert _\\beta ^2 + \\Vert \\bar{V}\\Vert _\\beta ^2 + \\Vert \\bar{L}\\Vert _{\\lambda , \\beta }^2),$ for all $\\eta , \\beta >0$ with $\\beta \\ge \\frac{3}{\\eta }$ .", "Choosing $\\eta = \\frac{1}{(T+1) 6C^2 }$ and $\\beta = \\frac{3}{\\eta }$ , we derive that $ \\Vert (\\overline{Y}, \\overline{Z}, \\overline{K}) \\Vert _\\beta ^2 \\le \\frac{1}{2}\\Vert (\\overline{U}, \\overline{V}, \\overline{K}) \\Vert _\\beta ^2.", "$ Hence, for $\\beta = 18 (T+1) C^2$ , $\\Phi $ is a contraction from ${I\\!\\!H}_\\beta ^2$ into ${I\\!\\!H}_\\beta ^2$ and thus admits a unique fixed point $(Y, Z, K)$ in the Banach space ${I\\!\\!H}_\\beta ^2$ , which is the solution of BSDE (REF ).", "$\\square $ By similar arguments, we have the following generalized result.", "Proposition 2.8 [BSDEs with default jump and “generalized driver\"] Let $g$ be a $\\lambda $ -admissible driver, let $\\xi \\in {L}^2({\\mathcal {G}_T})$ and let $D$ be a finite variational RCLL adapted process with square integrable total variation process.", "There exists an unique solution $(Y, Z, K)$ (also denoted by $(Y^D(T, \\xi ), Z^D(T, \\xi ), K^D(T, \\xi ))$ ) in $ \\mathcal {S}^2 \\times {\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ of the BSDE associated with “generalized driver\" $g(t, \\cdot )dt +dD_t$ and terminal condition $\\xi $ , that is $-dY_t = g(t,Y_t, Z_t,K_t ) dt + dD_t - Z_t dW_t - K_t dM_t; \\quad Y_T=\\xi .$ Remark 2.9 Let $D$ be a finite variational RCLL adapted process.", "Its associated total variation process is square integrable if and only if $D$ can be decomposed as follows: $D=A-A^{\\prime },$ with $A$ , $A^{\\prime } \\in \\mathcal {A}^2$ ." ], [ "$\\lambda $ -", "We introduce the notion of $\\lambda $ -linear BSDEs in our framework with default jump.", "Definition 2.10 ($\\lambda $ -linear driver) A driver $g$ is called $\\lambda $ -linear if it is of the form: $g(t,y,z,k)= \\varphi _t + \\delta _t y+ \\beta _t z+ \\gamma _t \\, k\\, \\lambda _t,$ where $(\\varphi _t) \\in {\\mathbb {H}}^2$ , and where $(\\delta _t)$ , $(\\beta _t)$ and $(\\gamma _t)$ are ${\\bf R}$ -valued predictable processes such that $(\\delta _t)$ , $(\\beta _t)$ and $(\\gamma _t\\sqrt{ \\lambda _t})$ are bounded.", "Remark 2.11 Note that a driver $g$ is $\\lambda $ -linear if and only if it is of the form: $g(t,y,z,k)= \\varphi _t + \\delta _t y+ \\beta _t z+ \\nu _t \\, k\\, \\sqrt{\\lambda _t},$ where $(\\varphi _t) \\in {\\mathbb {H}}^2$ , and where $(\\delta _t)$ , $(\\beta _t)$ and $(\\nu _t)$ are bounded ${\\bf R}$ -valued predictable processes.", "From this observation, it follows that a $\\lambda $ -linear driver is $\\lambda $ -admissible.", "We will now prove that the solution of a $\\lambda $ -linear BSDE, that is the solution of BSDE (REF ) associated with a $\\lambda $ -linear driver, can be written as a conditional expectation via an exponential semimartingale.", "We first show a preliminary result.", "Proposition 2.12 Let $(\\beta _s)$ and $(\\gamma _s)$ be two real-valued ${\\mathbb {G}}$ -predictable processes such that the random variable $\\int _0^T (\\beta _r ^2 + \\gamma _r ^2 \\lambda _r)\\,dr$ is bounded.", "Let $(\\zeta _s)$ be the process satisfying the forward SDE: $\\quad d \\zeta _{s} = \\zeta _{s^-} (\\beta _s d W_s +\\gamma _s dM_s)$ with $\\zeta _0 =1$ .", "The process $(\\zeta _s)$ satisfies the so-called Doléans-Dade formula, that is $\\zeta _s = \\exp \\lbrace \\int _0^s \\beta _r d W_r - \\frac{1}{2} \\int _0^s \\beta ^2_r dr \\rbrace \\exp \\lbrace - \\int _0^s \\gamma _r \\lambda _rdr \\rbrace (1 + \\gamma _{\\vartheta }{\\bf 1}_{ \\lbrace s \\ge \\vartheta \\rbrace }).$ For each $T >0$ , the process $(\\zeta _s)_{0\\le s\\le T}$ is a martingale and satisfies $ {\\mathbb {E}}[\\sup _{0 \\le s \\le T}\\zeta _s^p] < + \\infty $ , for all $p \\ge 2$ .", "Moreover, if $\\gamma _{\\vartheta } \\ge -1$ (resp.", "$> -1$ ) a.s. , then $\\zeta _s \\ge 0$ (resp.", "$>0$ ) a.s. for each $s \\in [0,T]$ .", "Proof.", "By definition, the process $(\\zeta _s)$ is a local martingale.", "Let $T>0$ .", "Let us show that $ {\\mathbb {E}}[\\sup _{0 \\le s \\le T}\\zeta _s^2] < + \\infty .$ By Itô's formula applied to $\\zeta _s^2$ , we get $d\\zeta _s^2 = 2 \\zeta _{s^-} d\\zeta _s + d[\\zeta ,\\zeta ]_s$ .", "We have $d[\\zeta ,\\zeta ]_s= \\zeta _{s^-}^2 \\beta _s^2 ds + \\zeta _{s^-}^2 \\gamma _s^2 dN_s.$ Using (REF ), we thus derive that $d\\zeta _s^2 = \\zeta _{s^-} ^2 [ 2 \\beta _s dW_s + (2 \\gamma _s + \\gamma _s^2) dM_s + (\\beta _s ^2 +\\gamma _s^2 \\lambda _s )ds ].$ It follows that $\\zeta ^2$ is an exponential semimartingale which can be written: $\\zeta _s^2 = \\eta _s \\exp \\lbrace \\int _0^s (\\beta _r ^2 + \\gamma _r ^2 \\lambda _r)\\,dr \\rbrace ,$ where $\\eta $ is the exponential local martingale satisfying $d \\eta _{s} = \\eta _{s^-} [ 2 \\beta _s dW_s+(2 \\gamma _s + \\gamma _s^2) dM_s ],$ with $\\eta _0 =1$ .", "By equality (REF ), the local martingale $\\eta $ is non negative.", "Hence, it is a supermartingale, which yields that ${\\mathbb {E}}[ \\eta _T] \\le 1$ .", "Now, by assumption, $\\int _0^T (\\beta _r ^2 + \\gamma _r ^2 \\lambda _r)\\,dr$ is bounded.", "By (REF ), it follows that ${\\mathbb {E}}[\\zeta _T^2] \\le {\\mathbb {E}} [\\eta _T]\\,K \\le K,$ where $K$ is a positive constant.", "By martingale inequalities, we derive that $ {\\mathbb {E}}[\\sup _{0 \\le s \\le T}\\zeta _s^2] < + \\infty .$ Hence, the process $(\\zeta _s)_{0\\le s\\le T}$ is a martingale.", "By an induction argument as in the proof of Proposition A.1 in [20], one can prove that the integrability property of $\\zeta $ holds for all integer $p\\ge 2$ .", "The last assertion follows from the Doléans-Dade formula.", "$\\square $ Remark 2.13 The inequality $\\gamma _{\\vartheta } \\ge -1$ a.s.  is equivalent to the inequality $\\gamma _t \\ge -1$ , $dt\\otimes dP \\text{ a.s.}$ .", "Indeed, we have ${\\mathbb {E}} [{\\bf 1}_{\\gamma _{\\vartheta } < -1}]$$ ={\\mathbb {E}} [\\int _0^{+ \\infty }{\\bf 1}_{\\gamma _{r} < -1}dN_r]$ $= {\\mathbb {E}} [\\int _0^{+ \\infty }{\\bf 1}_{\\gamma _{r} < -1}\\lambda _r dr]$ , because the process $(\\int _0^t \\lambda _r dr)$ is the ${\\mathbb {G}}$ -predictable compensator of the default jump process $N$ .", "Theorem 2.14 (Representation of the solution of a $\\lambda $ -linear BSDE) Let $\\xi $ $\\in $ $L^2({\\mathcal {G}}_T)$ .", "Let $g$ be a $\\lambda $ -linear driver of the form (REF ).", "Let $(Y, Z, K)$ be the solution in $S^{2} \\times I\\!\\!H^{2} \\times I\\!\\!H_{\\lambda }^{2}$ of the BSDE associated with driver $g$ and terminal condition $\\xi $ , that is $- dY_t = (\\varphi _t + \\delta _t Y_t + \\beta _t Z_t + \\gamma _t K_t \\lambda _t )dt - Z_t dW_t - K_t dM_t; \\;\\; Y_T = \\xi .$ For each $t$ $\\in $ $[0,T]$ , let $(\\Gamma _{t,s})_{s \\in [t,T]}$ (called the adjoint process) be the unique solution of the following forward SDE $d \\Gamma _{t,s} = \\displaystyle \\Gamma _{t,s^-} \\left[ \\delta _s ds + \\beta _s d W_s +\\gamma _s dM_s\\right] ; \\;\\;\\Gamma _{t,t} = 1.$ The process $(Y_t)$ satisfies $ Y_t \\,= \\,{\\mathbb {E}} \\,[ \\, \\Gamma _{t,T} \\,\\,\\xi + \\int _t^T \\Gamma _{t,s}\\, \\varphi _sds \\mid {\\mathcal {G}}_t \\, ], \\quad 0 \\le t \\le T,\\quad {\\rm a.s.}\\,$ Remark 2.15 The process $(\\Gamma _{t,s})_{s \\in [t,T]}$ , defined by (REF ), satisfies $\\Gamma _{t,s} = e^{ \\int _t^s \\delta _s ds} \\exp \\lbrace \\int _t^s \\beta _r d W_r - \\frac{1}{2} \\int _t^s \\beta ^2_r dr \\rbrace \\exp \\lbrace - \\int _t^s \\gamma _r \\lambda _rdr \\rbrace (1 + \\gamma _{\\vartheta }{\\bf 1}_{ \\lbrace s \\ge \\vartheta >t\\rbrace }).$ Note that the process $(e^{ \\int _t^s \\delta _s ds})_{t \\le s \\le T}$ is positive and bounded since $\\delta $ is bounded.", "Using Proposition REF , since $\\beta $ and $\\gamma \\sqrt{ \\lambda }$ are bounded, we derive that $ {\\mathbb {E}}[\\sup _{t \\le s \\le T}\\Gamma _{t,s}^2] < + \\infty .$ Moreover, if $\\gamma _{\\vartheta } \\ge -1$ (resp.", "$> -1$ ) a.s. , then $\\Gamma _{t,s} \\ge 0$ (resp.", "$>0$ ) a.s. for each $s \\in [t,T]$ .", "Proof.", "Fix $t$ $\\in $ $[0,T]$ .", "By applying the Itô product formula to $Y_s \\Gamma _{t,s}$ , we get $-d(Y_s \\Gamma _s) & =- Y_{s^-} d\\Gamma _s - \\Gamma _{s^-}dY_s - d [ Y, \\Gamma ]_s\\\\& = -Y_s \\Gamma _s \\delta _s ds + \\Gamma _{s} \\left[ \\varphi _s + \\delta _s Y_s + \\beta _s Z_s +\\gamma _s K_s \\lambda _s \\right]ds \\\\& \\quad - \\beta _s Z_s \\Gamma _s ds - \\Gamma _s \\gamma _s K_s \\lambda _s ds - \\Gamma _s(Y_s \\beta _s+ Z_s) dW_s- \\Gamma _{s^-}[K_s (1 +\\gamma _s )+Y_{s^-} \\gamma _s] dM_s,$ (where $\\Gamma _{t,s}$ is denoted by $\\Gamma _s)$ .", "Setting $dm_s = -\\Gamma _{t,s} (Y_s \\beta _s+ Z_s) dW_s- \\Gamma _{t,s^-} [K_s(1 +\\gamma _s) +Y_{s^-} \\gamma _s]dM_s,$ we get $-d(Y_s \\Gamma _s)=\\Gamma _s \\varphi _s ds - d m_s$ .", "By integrating between $t$ and $T$ , we obtain $Y_t=\\xi \\Gamma _{t,T} + \\int _t^T \\Gamma _{t,s} \\varphi _s ds - (m_T -m_t) \\quad {\\rm a.s.}$ By Remark REF , we have $(\\Gamma _{t,s})_{ t\\le s\\le T}$ $\\in $ $S^{2}$ .", "Moreover, $Y$ $\\in $ $S^{2}$ , $Z$ $\\in $ $I\\!\\!H^{2}$ , $K$ $\\in $ $I\\!\\!H^{2}_{\\lambda }$ , and $\\beta $ and $\\gamma $ are bounded.", "It follows that the local martingale $m= (m_s)_{t\\le s\\le T}$ is a martingale.", "Hence, by taking the conditional expectation in equality (REF ), we get equality (REF ).", "$\\square $ By similar arguments, we have the generalized representation result.", "Proposition 2.16 Let $\\xi \\in {L}^2({\\mathcal {G}_T})$ and let $D$ be a finite variational RCLL adapted process with square integrable total variation process.", "Let $(\\delta _t)$ , $(\\beta _t)$ and $(\\gamma _t)$ be ${\\bf R}$ -valued predictable processes such that $(\\delta _t)$ , $(\\beta _t)$ and $(\\gamma _t\\sqrt{ \\lambda _t})$ are bounded.", "Let $(Y,Z,K)$ be the solution of the BSDE: $- dY_t = ( \\delta _t Y_t + \\beta _t Z_t + \\gamma _t K_t \\lambda _t )dt + dD_t - Z_t dW_t - K_t dM_t; \\;\\; Y_T = \\xi .$ For each $t$ $\\in $ $[0,T]$ , we have $ Y_t \\,= \\,{\\mathbb {E}} \\,[ \\, \\Gamma _{t,T} \\,\\,\\xi + \\int _t^T \\Gamma _{t,s}\\, dD_s \\mid {\\mathcal {G}}_t \\, ] \\quad {\\rm a.s.}\\,,$ where $(\\Gamma _{t,s})_{s \\in [t,T]}$ is the adjoint process defined by (REF )." ], [ "Comparison theorems for BSDEs with a default jump", "We give here a comparison theorem and a strict comparison result for BSDEs with a default jump under additional assumptions on the driver.", "Theorem 2.17 (Comparison theorems for BSDEs with default jump) Let $\\xi _1$ and $\\xi _2$ $\\in $ $L^2({\\mathcal {G}}_T)$ .", "Let $g_1$ and $g_2$ be two $\\lambda $ -admissible drivers.", "For $i=1,2$ , let $(Y^i, Z^i, K^i)$ be a solution in $S^{2} \\times I\\!\\!H^{2} \\times I\\!\\!H_{\\lambda }^{2}$ of the BSDE $-dY^i_t = \\displaystyle g_i (t, Y^i_t, Z^i_t, K^i_t) dt - Z^i_t dW_t - K^i_t dM_t; \\quad Y^i_T = \\xi _i.$ (i) (Comparison theorem).", "Assume that there exists a predictable process $(\\gamma _t)$ with $(\\gamma _t\\sqrt{ \\lambda _t}) \\;\\text{ bounded} \\;\\; \\text{ and } \\quad \\gamma _t \\ge -1, \\quad dt\\otimes dP \\text{ a.s.}\\,$ such that $g_1(t, Y^2_t, Z^2_t, K^1_t) - g_1(t,Y^2_t,Z^2_t,K^2_t) \\ge \\gamma _t ( K^1_t- K^2_t) \\lambda _t , \\;\\; t \\in [0,T],\\; \\; dt\\otimes dP \\text{ a.s.}$ Suppose also that $\\xi _1 \\ge \\xi _2 \\text{ a.s. }\\quad {\\rm and} \\quad g_1 (t, Y^2_t, Z^2_t, K^2_t) \\ge g_2 (t, Y^2_t, Z^2_t, K^2_t), \\;\\; t \\in [0,T],\\; \\; dt\\otimes dP \\text{ a.s.}$ We then have $Y^1_t \\ge Y^2_t$ for all $t \\in [0,T]$ .", "(ii) (Strict Comparison Theorem).", "Suppose moreover that the second inequality in (REF ) is strict, that is $\\gamma _t > -1$ .", "If $Y_{t_0}=0$ , for some $t_0 \\in [0,T]$ , then the inequalities in (REF ) are equalities.", "Proof.", "Setting $\\bar{Y}_s = Y^1_s - Y^2_s$ ; $\\bar{Z}_s = Z^1_s - Z^2_s$ ; $\\bar{K}_s = K^1_s - K^2_s,$ we have $-d \\bar{Y}_s \\displaystyle = h_s ds - \\bar{Z}_s dW_s - \\bar{K}_s dM_s; \\quad \\bar{Y}_s = \\xi _1 - \\xi _2,$ where $h_s:=g_1 (s, Y^1_s, Z^1_s, K^1_s) - g_2(s, Y^2_s, Z^2_s, K^2_s)$ .", "Set $\\delta _s := (g_1(s,Y^1_{s^-}, Z^1_s, K^1_s) - g_1(s, Y^2_{s^-}, Z^1_s, K^1_s))/ \\bar{Y}_s$ if $\\bar{Y}_s \\ne 0$ , and 0 otherwise.", "Set $\\beta _s := (g_1(s,Y^2_{s^-}, Z^1_s, K^1_s) - g_1(s, Y^2_{s^-}, Z^2_s, K^1_s))/ \\bar{Z}_s$ if $\\bar{Z}_s \\ne 0$ , and 0 otherwise.", "By classical linearization techniques, we obtain $h_s = \\delta _s \\bar{Y}_s + \\beta _s \\bar{Z}_s+ g_1(s, Y^2_s, Z^2_s, K^1_s) - g_1(s,Y^2_s,Z^2_s,K^2_s) + \\varphi _s,$ where $\\varphi _s := g_1(s,Y^2_{s^-}, Z^2_s, K^2_s) - g_2(s, Y^2_{s^-}, Z^2_s, K^2_s)$ .", "Using the assumption (REF ), we get $\\,h_s \\ge \\delta _s \\bar{Y}_s + \\beta _s \\bar{Z}_s + \\gamma _s\\, \\,\\bar{K}_s \\lambda _s+\\varphi _s \\,\\, \\quad ds \\otimes dP-{\\rm a.s.}$ Since $g_1$ satisfies condition (REF ), the predictable processes $\\delta $ and $\\beta $ are bounded.", "Fix $t$ $\\in $ $[0,T]$ .", "Let $\\Gamma _{t,.", "}$ be the process defined by (REF ).", "Since $\\delta $ , $\\beta $ and $\\gamma \\sqrt{ \\lambda }$ are bounded, it follows from Remark REF that $\\Gamma _{t,.", "}$ $\\in $ $S^2$ .", "Also, since $\\gamma _s \\ge -1$ , we have $\\Gamma _{t,.}", "\\ge 0$ a.s.  By Itô's formula and similar computations as in the proof of Theorem REF , we derive that $-d(\\bar{Y}_s \\Gamma _{t,s}) & = \\Gamma _{t,s}(h_s - \\delta _s \\bar{Y}_s - \\beta _s \\bar{Z}_s -\\gamma _s \\,\\bar{K}_s \\,\\lambda _s)\\,ds - dm_s,$ where $m$ is a martingale (because $\\Gamma _{t,.", "}$ $\\in $ $S^2$ , $\\bar{Y}$ $\\in $ $S^{2}$ , $\\bar{Z}$ $\\in $ $I\\!\\!H^{2}$ , $\\bar{K}$ $\\in $ $I\\!\\!H^{2}_{\\lambda }$ and $\\beta $ , $\\gamma \\sqrt{ \\lambda }$ are bounded).", "Using the inequality (REF ) together with the non negativity of $\\Gamma $ , we thus get $-d(\\bar{Y}_s \\Gamma _{t,s})\\ge \\Gamma _{t,s} \\varphi _s ds - d m_s$ .", "By integrating between $t$ and $T$ and by taking the conditional expectation, we obtain $\\bar{Y}_t \\,\\ge \\,{\\mathbb {E}}\\, [\\, \\Gamma _{t,T} \\,(\\xi _1 - \\xi _2) + \\int _t^T\\Gamma _{t,s}\\, \\varphi _s ds \\mid {\\mathcal {G}}_t ],\\;\\; 0 \\le t \\le T,\\quad {\\rm a.s.}\\,$ By assumption (REF ), $\\varphi _s \\ge 0$ and $\\xi _1 - \\xi _2$ $\\ge 0$ , which, together with the non negativity of $\\Gamma _{t,T}$ , implies that $\\bar{Y}= Y^1 - Y^2 \\ge 0$ .", "The assertion (i) thus follows.", "Suppose now that $\\gamma _t > -1$ .", "By Remark REF , $\\Gamma _{t,T}>0$ a.s.", "The assertion (ii) thus follows from (REF ).", "$\\square $ We give here some counter-examples related to the comparison theorems for BSDEs with a default jump.", "Remark 2.18 Let us give an example which shows that in the case where assumption (REF ) is violated, that is when $\\gamma $ takes values $<-1$ with positive measure, then, even if the terminal condition is nonnegative, the solution $Y$ of the linear BSDE with default jump may take strictly negative values.", "Hence, in this case, the comparison theorem does not hold.", "Suppose that the process $\\lambda $ is bounded.", "Let $g$ be a $\\lambda $ -linear driver of the particular form $g(t,\\omega ,k)=\\gamma k \\lambda _t(\\omega ),$ where $\\gamma $ is a real constant (this corresponds to the driver of the $\\lambda $ -linear BSDE (REF ) with $\\delta _s= \\beta _s=\\varphi _s=0$ and $\\gamma _s= \\gamma $ ).", "At terminal time $T$ , the associated adjoint process $\\Gamma _{0,s}$ satisfies (see (REF ) and Remark REF ) : $\\Gamma _{0,T} = \\exp \\lbrace - \\int _0^T \\gamma \\lambda _rdr \\rbrace (1 + \\gamma {\\bf 1}_{ \\lbrace T \\ge \\vartheta \\rbrace })= \\exp \\lbrace - \\int _0^T \\gamma \\lambda _rdr \\rbrace (1 + \\gamma N_T),$ where the second equality follows from the definition of the default jump process $N$ .", "Let $Y$ be the solution of the BSDE associated with driver $g$ and terminal condition $\\xi :=N_T.$ The representation property of linear BSDEs with default jump (see (REF )) gives $Y_0=\\mathbb {E}[\\Gamma _{0,T} \\xi ]=\\mathbb {E}[\\Gamma _{0,T} N_T ].$ Hence, by $(\\ref {eqq})$ , we get $Y_0=\\mathbb {E}[\\Gamma _{0,T}N_T]=\\mathbb {E}[e^{-\\gamma \\int _0^T \\lambda _sds}(1+\\gamma N_T) N_T]=(1+\\gamma )\\mathbb {E}[e^{-\\gamma \\int _0^T \\lambda _sds}N_T],$ where for the last equality we have used the fact that $N_T=N_T^2$ .", "Equation (REF ) shows that when $\\gamma <-1$ , we have $Y_0<0$ although $\\xi \\ge 0$ a.s.", "This example also gives a counter-example for the strict comparison theorem by taking $\\gamma =-1$ .", "Indeed, in this case, the relation $(\\ref {eqrefff})$ at time 0 yields that $Y_0=0$ .", "Now, we have $\\mathbb {E}[\\xi ]= \\mathbb {E}[N_T]=1-P(\\vartheta > T).$ Hence, under the additional assumption $P(\\vartheta >T)<1$ , we get $\\mathbb {E}[\\xi ]>0$ , which implies that $P(\\xi >0)>0$ , even though $Y_0=0$ .", "Proposition 2.19 (Comparison theorems for BSDEs with “generalized driver\") Let $\\xi _1$ and $\\xi _2$ $\\in $ $L^2({\\mathcal {G}}_T)$ .", "Let $g_1$ and $g_2$ be two $\\lambda $ -admissible drivers.", "Let $D^1$ and $D^2 $ be finite variational RCLL adapted processes with square integrable total variation.", "Let $(Y^i, Z^i, K^i)$ be a solution in $S^{2} \\times I\\!\\!H^{2} \\times I\\!\\!H_{\\lambda }^{2}$ of the BSDE $-dY^i_t = \\displaystyle g_i (t, Y^i_t, Z^i_t, K^i_t) dt +dD^i_t- Z^i_t dW_t - K^i_t dM_t; \\quad Y^i_T = \\xi _i.$ (i) (Comparison theorem).", "Assume that there exists a predictable process $(\\gamma _t)$ satisfying (REF ) with (REF ) and that (REF ) holds.", "Moreover, suppose that the process $\\bar{D}:= D^1-D^2$ is non decreasing.", "We then have $Y^1_t \\ge Y^2_t$ for all $t \\in [0,T]$ .", "(ii) (Strict comparison theorem).", "Suppose moreover that $\\gamma _t > -1$ .", "If $Y_{t_0}=0$ , for some $t_0 \\in [0,T]$ , then the inequalities in (REF ) are equalities and $D^1-D^2$ is constant on the time interval $[t_0,T]$ .", "Proof.", "Using the same arguments and notation as above, we obtain: $\\bar{Y}_t \\,\\ge \\,{\\mathbb {E}}\\, [\\, \\Gamma _{t,T} \\,(\\xi _1 - \\xi _2) + \\int _t^T\\Gamma _{t,s}\\, (\\varphi _s ds + d\\bar{D}_s) \\mid {\\mathcal {G}}_t ],\\;\\; 0 \\le t \\le T,\\quad {\\rm a.s.}\\,$ Hence, $\\bar{Y}_t \\ge 0$ a.s. (ii) Suppose moreover that $Y_{t_0}=0$ a.s. and that $\\gamma _t > -1$ .", "Since $\\gamma _t > -1$ , we have $\\Gamma _{t,T}>0$ .", "We thus get $\\xi _1=\\xi _2$ a.s. and $\\varphi _t=0$ , $t \\in [t_0,T]$ $dt \\otimes dP$ -a.s. Set $\\tilde{D}_t:= \\int _{t_0,t} \\Gamma _{t_0,s}d \\bar{D}_s,$ for each $t \\in [t_0,T]$ .", "By assumption, $\\tilde{D}_T \\ge 0$ a.s. and $\\mathbb {E}[\\tilde{D}_T \\mid {\\mathcal {G}}_{t_0}]=0$ a.s.", "Hence $\\tilde{D}_T = 0$ a.s. Now, since $\\Gamma _{t_0,s}>0$ , $s\\ge t_0$ a.s. , we can write $\\bar{D}_T-\\bar{D}_{t_0}= \\int _{t_0,T} \\Gamma ^{-1}_{t_0,s}d \\tilde{D}_s$ .", "We thus get $\\bar{D}_T=\\bar{D}_{t_0}$ a.s. $\\square $" ], [ "Financial market with defaultable risky asset", "We consider a financial market which consists of one risk-free asset, whose price process $S^0$ satisfies $dS_t^{0}=S_t^{0} r_tdt$ , and two risky assets with price processes $S^{1},S^{2}$ evolving according to the equations: $dS_t^{1}&=&S_t^{1}[\\mu _t^1dt + \\sigma ^1_tdW_t]\\\\dS_t^{2}&=&S_{t^-}^{2} [\\mu ^2_tdt+\\sigma ^2_tdW_t-dM_t],$ where the process $(M_t)$ is given by (REF ).", "Note that the second risky asset is defaultable with total default.", "We have $S_t^2=0$ , $t \\ge \\vartheta $ a.s. All the processes $\\sigma ^1,\\sigma ^2,$ $r, \\mu ^1,\\mu ^2$ are predictable (that is ${\\mathcal {P}}$ -measurable).", "We set $\\sigma =(\\sigma ^1,\\sigma ^2)^{\\prime }$ .", "We suppose that $\\sigma ^1, \\sigma ^2 > 0$ , and $r$ , $\\sigma ^1,\\sigma ^2,$ ${(\\sigma ^1)}^{-1}$ , ${(\\sigma ^2)}^{-1}$ are bounded.", "Let us consider an investor who can invest in the three tradable assets.", "At time 0, he invests the amount $x \\ge 0$ in the three assets.", "For $i=1,2$ , we denote by $\\varphi _t^i$ the amount invested in the $i^{\\textit {th}}$ risky asset.", "Since after time $\\vartheta $ , the investor cannot invest his wealth in the defaultable asset (since its price is equal to 0), we have $\\varphi _t^2=0$ for each $t \\ge \\vartheta $ .", "A process $\\varphi = (\\varphi ^1, \\varphi ^2)^{\\prime }$ belonging to ${\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ is called a risky assets stategy.", "Let $C_t$ be the cumulated cash amount which has been withdrawn from the market portfolio between time 0 and time $t$ .", "The process $C$ belongs to ${\\mathcal {A}}^2$ , that is, $C$ is an RCLL adapted non decreasing process satisfying $C_0=0$ and $E[C_T^2] < + \\infty $ .", "The value of the associated portfolio (or wealth) at time $t$ is denoted by $V^{x, \\varphi , C}_t$ .", "The amount invested in the non risky asset at time $t$ is then given by $V^{x, \\varphi , C}_t - (\\varphi _t^1+ \\varphi _t^2)$ ." ], [ "Pricing of European options with dividends in a perfect market model", "In this section, we suppose that the market model is perfect.", "In this case, by the self financing condition, the wealth process $V^{x, \\varphi , C}$ (simply denoted by $V$ ) follows the dynamics: $dV_t & = r_t V_t+\\varphi _t^1 (\\mu ^1_t - r_t)+\\varphi _t^2(\\mu ^2_t - r_t)) dt - dC_t+(\\varphi _t^1 \\sigma ^1_t + \\varphi _t^2 \\sigma ^2_t) dW_t - \\varphi _t^2 dM_t\\\\& = \\left(r_t V_t+(\\varphi _t^1\\sigma _t^1+\\varphi _t^2\\sigma _t^2) \\theta _t^1- \\varphi _t^2 \\theta _t^2 \\lambda _t \\right) dt - dC_t+\\varphi _t ^{\\prime } \\sigma _t dW_t - \\varphi _t^2 dM_t,$ where $\\theta _t^1:=\\dfrac{\\mu _t^1-r_t}{\\sigma _t^1}$ , $\\theta _t^2:= - \\dfrac{\\mu _t^2-\\sigma _t^2 \\theta _t^1-r_t}{\\lambda _t }\\,{\\bf 1}_{\\lbrace t \\le \\vartheta \\rbrace }$ .", "Loosely speaking, $dC_t$ represents the amount withdrawn from the portfolio during the time period $[t, t + dt]$ .", "Suppose that the processes $\\theta ^1$ and $\\theta ^2 \\sqrt{\\lambda }$ are bounded.", "Let $T>0$ .", "Let $\\xi $ be a $\\mathcal {G}_T$ -measurable random variable belonging to ${L}^2$ , and let $D$ be a non decreasing process belonging to ${\\mathcal {A}}^2$ .", "We consider a European option with maturity $T$ , payoff $\\xi $ and cumulative dividend process $D$ .", "For each $t\\in [0,T]$ , $dD_t$ represents the dividend amount paid to the owner of the option between time $t$ and time $t+dt$ .", "The aim is to price this contingent claim.", "Let us consider a seller who wants to sell the option at time 0.", "With the amount he receives at time 0 from the buyer, he wants to be able to construct a portfolio which allows him to pay to the buyer the amount $\\xi $ at time $T$ and the intermediate dividends.", "By Proposition REF , there exists an unique process $(X, Z, K) \\in \\mathcal {S}^2 \\times {\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ solution of the following $\\lambda $ -linear BSDE: $- dX_t = \\displaystyle - (r_t X_t+Z_t\\theta _t^1+K_t \\theta _t^2 \\lambda _t) dt + dD_t- Z_t dW_t - K_t dM_t\\,; \\quad X_T=\\xi .$ Note that the driver of this BSDE is given for each $(\\omega , t,y,z,k)$ by $g(\\omega , t,y,z,k) = - r_t(\\omega ) y - z \\theta ^1_t (\\omega )- \\theta ^2_t (\\omega )\\lambda _t (\\omega )\\, k.$ Since by assumption, the coefficients $r, \\sigma ^2, \\theta ^1$ , $\\theta ^2 \\sqrt{\\lambda }$ are predictable and bounded, it follows that $g$ is a $\\lambda $ -linear driver (see Definition REF ).", "The solution $(X, Z, K)$ corresponds to the replicating portfolio.", "More precisely, the hedging risky assets stategy $\\varphi $ is such that $ {\\varphi _t}^{\\prime } \\sigma _t = Z_t \\;\\; ; \\;\\;- \\varphi _t^2 = K_t,$ where ${\\varphi _t}^{\\prime } \\sigma _t= {\\varphi ^1_t} \\sigma ^1_t + {\\varphi ^2_t} \\sigma ^2_t$ .", "Note that this defines a change of variables $\\Phi $ defined by: $\\Phi :{\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda } \\rightarrow {\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda };(Z, K) \\mapsto \\Phi (Z, K):= \\varphi $ , where $\\varphi = (\\varphi ^1, \\varphi ^2)$ is given by (REF ), which is equivalent to $ \\varphi _t^{2} = - {K_t} \\;\\; ; \\;\\;\\varphi _t^{1} = \\frac{Z_t - \\varphi _t^{2} \\sigma ^2_t }{\\sigma ^1_t}=\\frac{Z_t + \\sigma ^2_t K_t}{\\sigma ^1_t}.$ The process $D$ corresponds to the cumulated cash withdrawal.", "The process $X$ coincides with $V^{X_0, \\varphi , D}$ , the value of the portfolio associated with initial wealth $x=X_0$ , portfolio strategy $\\varphi $ and cumulated (dividend) cash withdrawal $D$ .", "From the seller's point of view, this portfolio is a hedging portfolio since, by investing the initial amount $X_0$ in the reference assets along the strategy $\\varphi $ , it allows him to pay the amount $\\xi $ to the buyer at time $T$ and the intermediate dividends.", "We derive that $X_0$ is the initial price of the option, called hedging price, and denoted by $X_0^D(\\xi )$ .", "Similarly, for each time $t \\in [0,T]$ , $X_t$ is the hedging price at time $t$ of the option, and is denoted by $X_t^D(\\xi )$ .", "Since the driver $g$ given by (REF ) is $\\lambda $ -linear, the representation property of the solution of a $\\lambda $ -linear BSDE (see Theorem REF ) yields $X^D_t(\\xi )=\\mathbb {E}[e^{-\\int _t ^T r_s ds} \\zeta _{t,T}\\xi + \\int _t^T e^{-\\int _t ^s r_u du} \\zeta _{t,s} dD_s\\,|\\,{\\mathcal {G}}_t],$ where $\\zeta $ satisfies $d\\zeta _{t,s}= \\zeta _{t,s^-} [-\\theta ^1_s dW_s - \\theta ^2_s dM_s]; \\quad \\zeta _{t,t}=1.$ This defines a linear price system $X$ : $(\\xi ,D) \\mapsto X^D(\\xi )$ .", "Suppose now that $\\theta ^2_t < 1$ , $0 \\le t \\le \\vartheta \\,$ $\\,dt \\otimes dP$ -a.s.  Moroever, by Proposition REF , the process $\\zeta _{0,.", "}$ is a square integrable positive martingale.", "By classical results, the probability measure with density $\\zeta _{0,T}$ on ${\\mathcal {G}}_T$ is the unique martingale probability measure, and $X$ corresponds to the classical free-arbitrage price system (see e.g.", "Proposition 7.9.11 in [16])." ], [ "Nonlinear pricing of European options with dividends in an imperfect market with default", "From now on, we assume that there are imperfections in the market which are taken into account via the nonlinearity of the dynamics of the wealth.", "More precisely, we suppose that the wealth process $V^{x, \\varphi ,C}_t$ (or simply $V_t$ ) associated with an initial wealth $x$ , a strategy $\\varphi =(\\varphi ^1, \\varphi ^2)$ in ${\\mathbb {H}}^2 \\times {\\mathbb {H}}^2_{\\lambda }$ and a cumulated withdrawal process $C$ satisfies the following dynamics: $-dV_t= g(t,V_t, {\\varphi _t}^{\\prime } \\sigma _t , - \\varphi _t^{2} ) dt - {\\varphi _t}^{\\prime } \\sigma _t dW_t+dC_t +\\varphi _t^{2} dM_t; \\; V_0=x,$ where $g$ is a nonlinear $\\lambda $ -admissible driver (see Definition REF ).", "Equivalently, setting $Z_t= {\\varphi _t}^{\\prime } \\sigma _t$ and $K_t= - \\varphi _t^2 $ , $-dV_t= g(t,V_t, Z_t,K_t ) dt - Z_t dW_t+dC_t - K_t dM_t ; \\; V_0=x.$ Note that in the special case of a perfect market, $g$ is given by (REF ).", "Let us consider a European option with maturity $T$ , terminal payoff $\\xi \\in {L}^2({\\mathcal {G}_T})$ and dividend process $D \\in \\mathcal {A}^2$ in this market model.", "Let $(X^D(T, \\xi ), Z^D(T, \\xi ), K^D(T, \\xi )),$ also denoted by $(X,Z,K)$ , be the solution of BSDE associated with terminal time $T$ , “generalized driver\" $g(\\cdot )dt+dD_t$ and terminal condition $\\xi $ , that is satisfying $-dX_t = g(t,X_t, Z_t,K_t ) dt + dD_t - Z_t dW_t - K_t dM_t; \\quad X_T=\\xi .$ The process $X=X^D(T, \\xi )$ is equal to the wealth process associated with initial value $x= X_0$ , strategy $\\varphi $ $= \\Phi (Z, K)$ (see (REF )) and cumulated amount $D$ of cash withdrawals that is $X= V^{X_0, \\varphi ,D}$ .", "Its initial value $X_0=X^D_0(T, \\xi )$ is thus a sensible price (at time 0) of the option for the seller since this amount allows him/her to construct a trading strategy $\\varphi $ , called hedging strategy, such that the value of the associated portfolio is equal to $\\xi $ at time $T$ .", "Moreover, the cash withdrawals perfectly replicate the dividends of the option.", "Similarly, $X_t=X^D_t(T, \\xi )$ is a sensible price for the seller at time $t$ .", "For each maturity $S\\in [0,T]$ and for each pair “payoff-dividend\" $(\\xi , D) \\in {L}^2({\\mathcal {G}_S}) \\times {\\mathcal {A}}^2,$ we define the $g$ -value process by ${\\mathcal {E}}_{t,S}^{^{g,D}} (\\xi ):= X^D_t(S, \\xi ), $ $t \\in [0,S]$ .", "Note that ${\\mathcal {E}}_{t,S}^{^{g,D}} (\\xi )$ can be defined on the whole interval $[0,T]$ by setting ${\\mathcal {E}}_{t,S}^{^{g,D}} (\\xi ):= {\\mathcal {E}}_{t,T}^{^{g^S,D^S}} (\\xi )$ for $t \\ge S$ , where $g^S(t,.", "):= g (t,.)", "{\\bf 1}_{t \\le S}$ and $D_t^S := D_{t \\wedge S}$ .", "This leads to a nonlinear pricing system ${\\mathcal {E}}^{^{g,\\cdot }}: (S,\\xi ,D) \\mapsto {\\mathcal {E}}^{^{g,D}}_{\\cdot ,S}(\\xi ).$ When there are no dividends, it reduces to the nonlinear pricing system ${\\mathcal {E}}^{^{g,0}}$ (usually denoted by ${\\mathcal {E}}^{^{g}}$ ), first introduced by El Karoui-Quenez ([15]) in a Brownian framework We now give some properties on this nonlinear pricing system $\\mathcal {E}^{^{g,\\cdot }}$ which generalize those given in [15] to the case with a default jump and dividends.", "$\\bullet $ Consistency.", "By the flow property for BSDEs, $\\mathcal {E}^{^{g,\\cdot }}$ is consistent.", "More precisely, let $S^{\\prime }\\in [0,T]$ , $\\xi \\in L^2(\\mathcal {G}_T)$ , $D\\in \\mathcal {A}^2,$ and let $S$ be a stopping time smaller than $S^{\\prime }$ .", "Then for each time $t$ smaller than $S$ , the $g$ -value of the option associated with payoff $\\xi $ , (cumulated) dividend process $D$ and maturity $S^{\\prime }$ coincides with the $g$ -value of the option associated with maturity $S$ , payoff $\\mathcal {E}_{S,T}^{^{g,D}}(\\xi )$ and dividend process $D$ , that is $\\mathcal {E}_{t,S^{\\prime }}^{^{g,D}}(\\xi )=\\mathcal {E}_{t,S}^{^{g,D}}(\\mathcal {E}_{S,S^{\\prime }}^{^{g,D}}(\\xi )) \\text{ a.s. }$ $\\bullet $ Zero-one law.", "If $g(t,0,0,0) = 0$ Note that when the market is perfect, $g$ is given by (REF ) and thus satisfies $g(t,0,0,0) = 0$ ., then the price of the European option with null payoff and no dividends is equal to 0.", "More precisely, $\\mathcal {E}^{^{g,\\cdot }}$ satisfies the Zero-one law property: for all maturity $S \\in [0,T]$ , for all payoff $\\xi \\in L^2(\\mathcal {G}_S),$ and cumulated dividend process $D \\in \\mathcal {A}^2$ , $\\mathcal {E}_{t,S}^{^{g,D^A}}({\\bf 1}_A \\xi )=\\ {\\bf 1}_A \\mathcal {E}_{t,S}^{^{g,D }}( \\xi )$ $a.s$ for $t \\le S$ , $A \\in {\\mathcal {G}}_t$ , and $\\xi $ $\\in $ $L^2({\\mathcal {G}}_S)$ , where $D^A$ is the process defined by $D^A_s:= (D_s -D_t){\\bf 1}_A{\\bf 1}_{s\\ge t}$ .", "Because of the presence of the default jump, the nonlinear pricing system $\\mathcal {E}^{^{g,\\cdot }}$ is not necessarily monotone with respect to $(\\xi ,D)$ .", "We introduce the following Assumption.", "Assumption 3.1 Assume that there exists a map $\\gamma : [0,T] \\times \\Omega \\times {\\bf R}^4 \\rightarrow {\\bf R}\\,; \\, (\\omega , t, y,z, k_1, k_2) \\mapsto \\gamma _t^{y,z,k_1,k_2}(\\omega )$ ${\\mathcal {P} } \\otimes {\\mathcal {B}}({\\bf R}^4) $ -measurable, satisfying $ dP\\otimes dt $ -a.s. , for each $(y,z, k_1, k_2)$ $\\in $ ${\\bf R}^4$ , $(\\gamma _t^{y,z,k_1,k_2}\\sqrt{ \\lambda _t}) \\;\\text{ bounded} \\;\\; \\text{ and } \\quad \\gamma _t^{y,z,k_1,k_2} \\ge -1,$ and $ g( t,y,z, k_1)- g(t,y,z, k_2) \\ge \\gamma _t^{y,z, k_1,k_2} (k_1 - k_2 ) \\lambda _t,$ Recall that $\\lambda $ vanishes after $\\vartheta $ and $g(t,\\cdot )$ does not depend on $k$ on $\\lbrace t >\\vartheta \\rbrace $ .", "Hence, the inequality (REF ) is always satisfied on $\\lbrace t >\\vartheta \\rbrace $ .", "Note that the above assumption holds e.g.", "if $g(t,\\cdot )$ is non decreasing with respect to $k$ , or if $g$ is ${\\mathcal {C}}^1$ in $k$ with $ \\partial _k g(t, \\cdot ) \\ge - \\lambda _t$ on $\\lbrace t \\le \\vartheta \\rbrace $ .", "In the case of a perfect market, it is satisfied when $\\theta ^2_t\\le 1$ .", "Before giving some additional properties (which hold under this Assumption), we introduce the following partial order relation, defined for each fixed time $S \\in [0,T]$ , on the set of pairs \"payoff-dividends\" by: for each $(\\xi ^1,D^1), (\\xi ^2,D^2) \\in L^2(\\mathcal {G}_S) \\times \\mathcal {A}^2$ by $(\\xi ^1,D^1) \\succ (\\xi ^2,D^2) \\,\\,\\quad \\text{ if } \\quad \\xi ^1 \\ge \\xi ^2\\,\\, {\\rm a.s.} \\text{ and } D^1-D^2 \\text{ is non decreasing.", "}$ Loosely speaking, the non decreasing property of $D^1-D^2$ corresponds to the fact that the instantaneous dividends paid between times $s$ and $s+ds$ corresponding to $D^1$ are greater or equal to the ones corresponding to $D^2$ , that is $dD^1_s \\ge dD^2_s$ .", "$\\bullet $ Monotonicity.", "Under Assumption REF , the nonlinear pricing system ${\\mathcal {E}}^{^{g,\\cdot }}$ is non decreasing with respect to the payoff and the dividend.", "More precisely, for all maturity $S \\in [0,T]$ , for all payoffs $\\xi _1, \\xi _2 \\in L^2(\\mathcal {G}_S),$ and cumulated dividend processes $D^1, D^2 \\in \\mathcal {A}^2,$ the following property holds: If $(\\xi ^1,D^1) \\succ (\\xi ^2,D^2)$ , then we have ${\\mathcal {E}}_{t,S}^{g,D^1}(\\xi _1) \\ge {\\mathcal {E}}_{t,S}^{g,D^2}(\\xi _2)$ ,   $t \\in [0,S]$ a.s.", "This property follows from the comparison theorem for BSDEs with “generalized drivers\" (Proposition REF (i)) applied to $g^1=g^2=g$ and $\\xi ^1$ , $\\xi ^2$ , $D^1$ , $D^2$ (Indeed, in this case, by Assumption REF , Assumption (REF ) holds with $\\gamma _t:= \\gamma _t^{Y^2_{t^-},Z_t^2,K^1_t,K^2_t}$ ).", "Using this comparison theorem, we also derive the following property: $\\bullet $ Convexity.", "Under Assumption REF , if $g$ is convex with respect to $(y,z,k)$ , then the nonlinear pricing system ${\\mathcal {E}}^{^{g,D}}$ is convex, that is, for any $\\alpha \\in [0,1]$ , $S \\in [0,T]$ , $\\xi _1, \\xi _2 \\in L^2(\\mathcal {G}_S), D^1, D^2 \\in \\mathcal {A}^2$ $\\mathcal {E}_{t,S}^{^{g,\\alpha D^1+(1-\\alpha ) D^2}}(\\alpha \\xi _1+(1-\\alpha ) \\xi _2) \\le \\alpha \\mathcal {E}_{t,S}^{^{g,D^1}}(\\xi _1)+(1-\\alpha )\\mathcal {E}_{t,S}^{^{g,D^2}}(\\xi _2), \\,\\,\\, \\, \\text{ for all } t \\in [0,S].$ $\\bullet $ Nonnegativity.", "Under Assumption REF , when $g(t,0,0,0) \\ge 0$ , the nonlinear pricing system ${\\mathcal {E}}^{^{g,\\cdot }}$ is nonnegative, that is, for each $S\\in [0,T]$ , for all non negative $\\xi \\in {L}^2({\\mathcal {G}_S})$ and all $D \\in \\mathcal {A}^2$ , we have ${\\mathcal {E}}^{^{g,D}}_{\\cdot , S} (\\xi )\\ge 0$ a.s.", "Moreover, under the additional assumption $\\gamma _t^{y,z,k_1,k_2}>-1$ in Assumption REF , using the strict comparison theorem (Proposition REF (ii)), we derive the following no arbitrage property: $\\bullet $ No arbitrage.", "Under Assumption REF with $\\gamma _t^{y,z,k_1,k_2}>-1$ , the nonlinear pricing system ${\\mathcal {E}}^{^{g,\\cdot }}$ satisfies the no arbitrage property: for all maturity $S \\in [0,T]$ , and for all payoffs $\\xi ^1, \\xi ^2 \\in L^2(\\mathcal {G}_S)$ , and cumulated dividend processes $D^1, D^2 \\in \\mathcal {A}^2$ , $t_0 \\in [0,S],$ and $A \\in \\mathcal {G}_{t_0}$ , Suppose that $(\\xi ^1,D^1) \\succ (\\xi ^2,D^2)$ , and ${\\mathcal {E}}_{t_0, S}^{^{g,D^1}}(\\xi _1)={\\mathcal {E}}_{t_0, S}^{^{g,D^2}}(\\xi _2)$ a.s. on $A \\in \\mathcal {G}_{t_0}$ .", "Then, $\\xi _1=\\xi _2$ a.s. on $A$ and $(D_t^1-D_t^2)_{t_0 \\le t \\le S}$ is a.s. constant on $A$ , that is $D^1_S - D^1_{t_0} = D^2_S - D^2_{t_0}$ a.s. on $A$ .", "In other words, the payoffs and the instantaneous dividends paid between $t_0$ and $S$ are equal a.s. on $A$ .", "The No arbitrage property also ensures that when $\\gamma _t^{y,z,k_1,k_2}>-1$ , the nonlinear pricing system $\\mathcal {E}^g$ is strictly monotone.", "Note that when the market is perfect, the condition $\\gamma _t^{y,z,k_1,k_2}>-1$ is satisfied when $\\theta ^2_t <1$ .", "Remark 3.2 Several authors have studied dynamic risk measures defined as the solutions of BSDEs (see e.g.", "[19], [4], [20]).", "In our framework with a default jump, given a $\\lambda $ -admissible driver, one can define a dynamic measure of risk $\\rho ^{g}$ as follows: for each $S \\in [0,T]$ and $\\xi \\in L^2({\\mathcal {G}}_S)$ , we set $\\rho ^{g}_\\cdot (\\xi , S) = -\\mathcal {E}^{g}_{\\cdot ,S} (\\xi ),$ where $\\mathcal {E}^{g}_{\\cdot ,S} (\\xi )$ denotes the solution of the BSDE associated with terminal condition $\\xi $ , terminal time $T$ and driver $g$ .", "Then, by the results of this section, the dynamic risk-measure $\\rho ^{g}$ satisfies analogous properties to the ones of the nonlinear pricing system $\\mathcal {E}^{g}_{\\cdot ,S} = \\mathcal {E}^{g,0}_{\\cdot ,S}$ (corresponding to the case with no dividends).", "We now introduce the definition of an $\\mathcal {E}^{^{g,D}}$ -supermartingale which generalizes the classical notion of $\\mathcal {E}^g$ -supermartingale.", "Definition 3.3 Let $D \\in \\mathcal {A}^2$ and $Y \\in \\mathcal {S}^2$ .", "The process $Y$ is said to be a $\\mathcal {E}^{^{g,D}}$ -supermartingale (resp.", "$\\mathcal {E}^{^{g,D}}$ -martingale) if ${\\mathcal {E}}_{\\sigma ,\\tau }^{^{g,D}}(Y_{\\tau }) \\le Y_{\\sigma }$ (resp.", "$= Y_{\\sigma }$ ) a.s. on $\\sigma \\le \\tau $ , for all $ \\sigma , \\tau \\in \\mathcal {T}_0$ .", "Proposition 3.4 For all $S\\in [0,T]$ , payoff $\\xi \\in {L}^2({\\mathcal {G}_S})$ and dividend process $D \\in {\\mathcal {A}}^2$ , the associated $g$ -value process ${\\mathcal {E}}_{\\cdot ,S}^{^{g,D}} (\\xi )$ is an $\\mathcal {E}^{^{g,D}}$ -martingale.", "Moreover, for all $x \\in \\mathbb {R}$ , portfolio strategy $\\varphi $ $\\in $ ${\\mathbb {H}}^2\\times {\\mathbb {H}}^2_{\\lambda }$ and cash withdrawal process $D \\in {\\mathcal {A}}^2$ , the associated wealth process $V^{x, \\varphi ,D}$ is an $\\mathcal {E}^{^{g,D}}$ -martingale.", "Proof.", "The first assertion follows from the consistency property of $\\mathcal {E}^{^{g,D}}$ .", "The second one is obtained by noting that $V^{x, \\varphi ,D}$ is the solution of the BSDE with “generalized driver\" $g(\\cdot )dt+dD_t$ , terminal time $T$ and terminal condition $V_T^{x, \\varphi ,D}$ .", "$\\square $ Example (Large investor seller) When the seller is a large trader, his hedging portfolio may affect the prices of the risky assets and the default probability.", "He may take into account these feedback effect in his market model as follows.", "In order to simplify the presentation, we consider the case when the seller's strategy affects only the default intensity.", "We are given a family of probability measures parametrized by $V$ and $\\varphi $ .", "More precisely, for each $V \\in \\mathcal {S}^2$ and $\\varphi \\in {\\mathbb {H}}^2$ , let $Q^{V, \\varphi }$ be the probability measure equivalent to $P$ , which admits $L^{V, \\varphi }$ as density with respect to $P$ , where $(L^{V, \\varphi })$ is the solution of the following SDE: $dL_t^{V, \\varphi }=L_{t^-} \\gamma (t,{V}_{t^-}, \\varphi _t)dM_t; \\quad L^{V,\\varphi }_0=1.$ Here, $\\gamma : (\\omega ,t ,y, \\varphi _1, \\varphi _2) \\mapsto \\gamma (\\omega , t ,y, \\varphi _1, \\varphi _2) $ is a ${\\mathcal {P}}\\otimes {\\mathcal {B}}({\\bf R}^3)/\\mathcal {B}({\\bf R})$ -measurable function defined on $ \\Omega \\times [0,T] \\times {\\bf R}^2$ with $\\gamma (t, \\cdot ) >-1$ , and such that $(\\gamma (t, \\cdot )\\sqrt{\\lambda _t})$ is uniformly bounded.", "Note that by Proposition REF , we have $L \\in {\\mathcal {S}}^2$ .", "By Girsanov's theorem, the process $W$ is a $Q^{V, \\varphi }$ -Brownian motion and the process $M^{V, \\varphi }$ defined as $M^{V, \\varphi }_t:= N_t- \\int _0^t \\lambda _s (1+ \\gamma (s,V_s, {\\varphi }_s))ds= M_t - \\int _0^t \\lambda _s \\gamma (s,V_s, {\\varphi }_s)ds$ is a $Q^{V, \\varphi }$ -martingale.", "Hence, under $Q^{V, \\varphi }$ , the ${\\mathbb {G}}$ -default intensity process is equal to $\\lambda _t (1+ \\gamma (t,V_t, {\\varphi }_t))$ .", "The process $\\gamma (t,V_t, {\\varphi }_t)$ represents the impact of the seller's strategy on the default intensity.", "The dynamics of the wealth process associated with an initial wealth $x$ and a risky assets stategy $\\varphi $ satisfy $dV_t = \\left(r_t V_t+(\\varphi _t^1\\sigma _t^1+\\varphi _t^2\\sigma _t^2) \\theta _t^1- \\varphi _t^2 \\theta _t^2 \\lambda _t \\right) dt - dC_t+\\varphi _t ^{\\prime } \\sigma _t dW_t - \\varphi _t^2 dM^{V, \\varphi }_t,$ Let us show that this model can be seen as a particular case of the model described above associated with an appropriate map $\\lambda $ -admissible driver $g$ .", "First, note that the dynamics of the wealth (REF ) can be written $dV_t = \\left(r_t V_t+(\\varphi _t^1\\sigma _t^1+\\varphi _t^2\\sigma _t^2) \\theta _t^1- \\varphi _t^2 \\theta _t^2 \\lambda _t + \\gamma (t,V_t, {\\varphi }_t) \\lambda _t \\varphi _t^2 \\right)dt - dC_t+ \\varphi _t ^{\\prime } \\sigma _t dW_t - \\varphi _t^2 dM_t,$ Equivalently, setting $Z_t= {\\varphi _t}^{\\prime } \\sigma _t$ and $K_t= - \\varphi _t^2 $ , $-dV_t= g(t,V_t, Z_t,K_t ) dt - Z_t dW_t+dC_t - K_t dM_t,$ where $g(t,y,z,k) = -r_t y - z \\theta ^1_t - \\theta ^2_t \\lambda _t k + \\gamma \\left(t,y, (\\sigma ^1_t)^{-1}(z + { \\sigma ^2_t} k), -k \\right)\\lambda _t k.$ We are thus led to the general model described above associated with this driver.", "This model can be easily generalized to the case when the coefficients $\\mu ^1$ , $\\sigma ^1$ , $\\mu ^2$ , $\\sigma ^2$ also depend on the hedging cost $V$ (equal to the price of the option) and on the hedging strategy $\\varphi ^2$ .", "The coefficients may also depend on $\\varphi = (\\varphi ^1, \\varphi ^2)$ , but in this case, we have to assume that the map $\\Psi :$ $(\\omega , t,y,\\varphi ) \\mapsto (z,k)$ with $z={\\varphi }^{\\prime } \\sigma _t(\\omega ,t,y,\\varphi )$ and $k=- \\varphi ^2$ is one to one with respect to $\\varphi $ , and such that its inverse $\\Psi ^{-1}_{\\varphi } $ is ${\\mathcal {P}}\\otimes {\\mathcal {B}} ({\\bf R}^3)$ -measurable.", "Appendix For $p \\ge 2$ , we introduce the spaces ${\\mathcal {S}}^{p}$ , ${\\mathbb {H}}^p$ and ${\\mathbb {H}}^p_{\\lambda }$ defined as follows.", "Let ${\\mathcal {S}}^{p}$ be the set of ${\\mathbb {G}}$ -adapted RCLL processes $\\varphi $ such that $\\mathbb {E}[\\sup _{0\\le t \\le T} |\\varphi _t | ^p] < +\\infty $ .", "Let ${\\mathbb {H}}^p$ be the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert Z\\Vert _p^p:= \\mathbb {E}\\Big [(\\int _0^T|Z_t|^2dt)^{p/2}\\Big ]<\\infty \\,.$ Let ${\\mathbb {H}}^p_{\\lambda }$ be the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert U\\Vert _{p,\\lambda }^p:=\\mathbb {E}\\Big [(\\int _0^T|U_t|^2\\lambda _tdt)^{p/2}\\Big ]<\\infty \\,.$ BSDEs with a default jump in $L^p$ Proposition A.1 Let $p\\ge 2$ and let $T >0$ .", "Let $g$ be a $\\lambda $ -admissible driver such that $g(t,0,0,0)$ $\\in $ $I\\!\\!H^{p}$ .", "Let $\\xi \\in {L}^p({\\mathcal {G}_T})$ .", "There exists a unique solution $(Y, Z, K)$ in $ \\mathcal {S}^p \\times {\\mathbb {H}}^p \\times {\\mathbb {H}}^p_{\\lambda }$ of the BSDE with default (REF ).", "Remark A.2 The above result still holds in the case when there is a ${\\mathbb {G}}$ -martingale representation theorem with respect to $W$ and $M$ , even if ${\\mathbb {G}}$ is not generated by $W$ and $M$ .", "Proof.", "The proof relies on the same arguments as in the proof of Proposition A.2 in [20] together with the arguments used in the proof of Proposition REF .", "$\\square $ BSDEs with a default jump and change of probability measure     Let $(\\beta _s)$ and $(\\gamma _s)$ be two real-valued ${\\mathbb {G}}$ -predictable processes such that $\\int _0^T (\\beta _r ^2 + \\gamma _r ^2 \\lambda _r)\\,dr$ is bounded.", "Let $(\\zeta _s)$ be the process satisfying the forward SDE: $\\quad d \\zeta _{s} = \\zeta _{s^-} (\\beta _s d W_s +\\gamma _s dM_s),$ with $\\zeta _0 =1$ .", "By Proposition REF , $\\zeta $ is a $p$ -integrable martingale, that is $\\zeta _T \\in L^p$ for all $p\\ge 1$ .", "We suppose that $\\gamma > -1$ , which implies that $\\zeta _s >0$ , $s \\in [0,T]$ a.s. Let $Q$ be the probability measure equivalent to $P$ which admits $\\zeta _{T}$ as density with respect to $P$ on ${\\mathcal {G}}_{T}$ .", "By Girsanov's theorem (see [16] Chapter 9.4 Corollary 4.5), the process $W^{\\beta }_t := W_t - \\int _0^t \\beta _s ds$ is a $Q$ -Brownian motion and the process $M^{\\gamma }$ defined as $M^{\\gamma }_t := M_t-\\int _0^t\\lambda _s\\gamma _s ds= N_t-\\int _0^t \\lambda _s(1+\\gamma _s) ds$ is a $Q$ -martingale.", "We now show a representation theorem for $(Q, {\\mathbb {G}})$ -local martingales with respect to $W^{\\beta }$ and $M^{\\gamma }$ .", "Proposition A.3 Let $m= (m_t)_{0\\le t \\le T}$ be a $(Q, {\\mathbb {G}})$ -local martingale.", "There exists a unique pair of predictable processes $(z_t, k_t)$ such that $m_t= m_0 +\\int _0^t z_s d W^{\\beta }_s + \\int _0^t k_s dM^{\\gamma }_s \\quad 0 \\le s \\le T\\quad {\\rm a.s.}$ Proof.", "Since $m$ is a $Q$ -local martingale, the process $\\bar{m}_t:=\\zeta _t m_t$ is a $P$ -local martingale.", "By the martingale representation theorem (Lemma REF ), there exists a unique pair of predictable processes $(Z,K)$ such that $\\bar{m}_t= \\bar{m}_0 +\\int _0^t Z_s d W_s + \\int _0^t K_s dM_s \\quad 0 \\le t \\le T\\quad {\\rm a.s.}$ Then, by applying Itô's formula to $m_t= \\bar{m}_t (\\zeta _t)^{-1}$ and by classical computations, one can derive the existence of $(z,k)$ satisfying $(\\ref {alpharepres})$ .", "$\\square $ From this result together with Proposition REF and Remark REF , we derive the following corollary.", "Corollary A.4 Let $p\\ge 2$ and let $T >0$ .", "Let $g$ be a $\\lambda $ -admissible driver such that $g(t,0,0,0)$ $\\in $ $I\\!\\!H^{p}_Q$ .", "Let $\\xi \\in {L}_Q^p({\\mathcal {G}_T})$ .", "There exists a unique solution $(Y, Z, K)$ in $ \\mathcal {S}_Q^p \\times {\\mathbb {H}}_Q^p \\times {\\mathbb {H}}^p_{Q, \\lambda }$ of the BSDE with default: $-dY_t = g(t,Y_t, Z_t,K_t ) dt - Z_t W^{\\beta }_t - K_t dM^{\\gamma }_t; \\quad Y_T=\\xi .$" ], [ "Appendix", "For $p \\ge 2$ , we introduce the spaces ${\\mathcal {S}}^{p}$ , ${\\mathbb {H}}^p$ and ${\\mathbb {H}}^p_{\\lambda }$ defined as follows.", "Let ${\\mathcal {S}}^{p}$ be the set of ${\\mathbb {G}}$ -adapted RCLL processes $\\varphi $ such that $\\mathbb {E}[\\sup _{0\\le t \\le T} |\\varphi _t | ^p] < +\\infty $ .", "Let ${\\mathbb {H}}^p$ be the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert Z\\Vert _p^p:= \\mathbb {E}\\Big [(\\int _0^T|Z_t|^2dt)^{p/2}\\Big ]<\\infty \\,.$ Let ${\\mathbb {H}}^p_{\\lambda }$ be the set of ${\\mathbb {G}}$ -predictable processes such that $\\Vert U\\Vert _{p,\\lambda }^p:=\\mathbb {E}\\Big [(\\int _0^T|U_t|^2\\lambda _tdt)^{p/2}\\Big ]<\\infty \\,.$" ], [ "BSDEs with a default jump in $L^p$", "Proposition A.1 Let $p\\ge 2$ and let $T >0$ .", "Let $g$ be a $\\lambda $ -admissible driver such that $g(t,0,0,0)$ $\\in $ $I\\!\\!H^{p}$ .", "Let $\\xi \\in {L}^p({\\mathcal {G}_T})$ .", "There exists a unique solution $(Y, Z, K)$ in $ \\mathcal {S}^p \\times {\\mathbb {H}}^p \\times {\\mathbb {H}}^p_{\\lambda }$ of the BSDE with default (REF ).", "Remark A.2 The above result still holds in the case when there is a ${\\mathbb {G}}$ -martingale representation theorem with respect to $W$ and $M$ , even if ${\\mathbb {G}}$ is not generated by $W$ and $M$ .", "Proof.", "The proof relies on the same arguments as in the proof of Proposition A.2 in [20] together with the arguments used in the proof of Proposition REF .", "$\\square $" ], [ "BSDEs with a default jump and change of probability measure", "Let $(\\beta _s)$ and $(\\gamma _s)$ be two real-valued ${\\mathbb {G}}$ -predictable processes such that $\\int _0^T (\\beta _r ^2 + \\gamma _r ^2 \\lambda _r)\\,dr$ is bounded.", "Let $(\\zeta _s)$ be the process satisfying the forward SDE: $\\quad d \\zeta _{s} = \\zeta _{s^-} (\\beta _s d W_s +\\gamma _s dM_s),$ with $\\zeta _0 =1$ .", "By Proposition REF , $\\zeta $ is a $p$ -integrable martingale, that is $\\zeta _T \\in L^p$ for all $p\\ge 1$ .", "We suppose that $\\gamma > -1$ , which implies that $\\zeta _s >0$ , $s \\in [0,T]$ a.s. Let $Q$ be the probability measure equivalent to $P$ which admits $\\zeta _{T}$ as density with respect to $P$ on ${\\mathcal {G}}_{T}$ .", "By Girsanov's theorem (see [16] Chapter 9.4 Corollary 4.5), the process $W^{\\beta }_t := W_t - \\int _0^t \\beta _s ds$ is a $Q$ -Brownian motion and the process $M^{\\gamma }$ defined as $M^{\\gamma }_t := M_t-\\int _0^t\\lambda _s\\gamma _s ds= N_t-\\int _0^t \\lambda _s(1+\\gamma _s) ds$ is a $Q$ -martingale.", "We now show a representation theorem for $(Q, {\\mathbb {G}})$ -local martingales with respect to $W^{\\beta }$ and $M^{\\gamma }$ .", "Proposition A.3 Let $m= (m_t)_{0\\le t \\le T}$ be a $(Q, {\\mathbb {G}})$ -local martingale.", "There exists a unique pair of predictable processes $(z_t, k_t)$ such that $m_t= m_0 +\\int _0^t z_s d W^{\\beta }_s + \\int _0^t k_s dM^{\\gamma }_s \\quad 0 \\le s \\le T\\quad {\\rm a.s.}$ Proof.", "Since $m$ is a $Q$ -local martingale, the process $\\bar{m}_t:=\\zeta _t m_t$ is a $P$ -local martingale.", "By the martingale representation theorem (Lemma REF ), there exists a unique pair of predictable processes $(Z,K)$ such that $\\bar{m}_t= \\bar{m}_0 +\\int _0^t Z_s d W_s + \\int _0^t K_s dM_s \\quad 0 \\le t \\le T\\quad {\\rm a.s.}$ Then, by applying Itô's formula to $m_t= \\bar{m}_t (\\zeta _t)^{-1}$ and by classical computations, one can derive the existence of $(z,k)$ satisfying $(\\ref {alpharepres})$ .", "$\\square $ From this result together with Proposition REF and Remark REF , we derive the following corollary.", "Corollary A.4 Let $p\\ge 2$ and let $T >0$ .", "Let $g$ be a $\\lambda $ -admissible driver such that $g(t,0,0,0)$ $\\in $ $I\\!\\!H^{p}_Q$ .", "Let $\\xi \\in {L}_Q^p({\\mathcal {G}_T})$ .", "There exists a unique solution $(Y, Z, K)$ in $ \\mathcal {S}_Q^p \\times {\\mathbb {H}}_Q^p \\times {\\mathbb {H}}^p_{Q, \\lambda }$ of the BSDE with default: $-dY_t = g(t,Y_t, Z_t,K_t ) dt - Z_t W^{\\beta }_t - K_t dM^{\\gamma }_t; \\quad Y_T=\\xi .$" ] ]
1612.05681
[ [ "An elementary proof for the Krull dimension of a polynomial ring" ], [ "Abstract This is an expository paper in which it is proved that, for every infinite field ${\\mathbf{F}}$, the polynomial ring ${\\mathbf{F}}[t_1,\\ldots, t_n]$ has Krull dimension $n$.", "The proof uses only \"high school algebra\" and the rudiments of undergraduate \"abstract algebra.\"" ], [ "Prime ideals and Krull dimension", "In this paper, a ring is a commutative ring with a multiplicative identity.", "A prime ideal in a ring $R$ is an ideal $ \\mathfrak {P}\\ne R$ such that, if $a,b \\in R$ and $ab \\in \\mathfrak {P}$ , then $a \\in \\mathfrak {P}$ or $b \\in \\mathfrak {P}$ .", "The ring $R$ is an integral domain if and only if $\\lbrace 0\\rbrace $ is a prime ideal in $R$ .", "In a unique factorization domain, a nonzero principal ideal $ \\mathfrak {P}$ is prime if and only if $ \\mathfrak {P}\\ne R$ and $ \\mathfrak {P}$ is generated by an irreducible element.", "For example, in the ring $\\mathbf {Z}$ , an ideal $ \\mathfrak {P}$ is prime if and only if $ \\mathfrak {P}= \\lbrace 0\\rbrace $ or $ \\mathfrak {P}= p\\mathbf {Z}$ for some prime number $p$ .", "An ideal chain of length $n$ in the ring $R$ is a strictly increasing sequence of $n+1$ ideals of $R$ .", "A prime ideal chain of length $n$ in $R$ is a strictly increasing sequence of $n+1$ prime ideals of $R$ .", "The Krull dimension of $R$ is the supremum of the lengths of prime ideal chains in $R$ .", "Eisenbud [3] wrote, “Arguably the most fundamental notion in geometry and topology is dimension.... [Its] ...algebraic analogue plays an equally fundamental role in commutative algebra and algebraic geometry.” We shall prove that, for every infinite field ${\\mathbf {F}}$ , the polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_n]$ has Krull dimension $n$ .", "Krull dimension has many applications.", "In algebraic geometry, if $S$ is a nonempty set of polynomials in ${\\mathbf {F}}[t_1,\\ldots , t_n]$ , then the variety (also called the algebraic set) $V$ determined by $S$ is the set of points in ${\\mathbf {F}}^n$ that are common zeros of the polynomials in $S$ : $V = V(S) = \\left\\lbrace (x_1 , \\ldots , x_n ) \\in {\\mathbf {F}}^n: f(x_1,\\ldots , x_n) = 0 \\text{ for all } f \\in S \\right\\rbrace .$ The vanishing ideal ${ \\mathfrak {I}}(V)$ is the set of polynomials that vanish on the variety $V$ : ${ \\mathfrak {I}}(V) = \\left\\lbrace f \\in {\\mathbf {F}}[t_1,\\ldots , t_n] : f(x_1,\\ldots , x_n) = 0 \\text{ for all } (x_1 , \\ldots , x_n ) \\in V \\right\\rbrace .$ The quotient ring ${\\mathbf {F}}[V] ={\\mathbf {F}}[t_1,\\ldots , t_n]/{ \\mathfrak {I}}(V)$ is called the coordinate ring of $V$ .", "One definition of the dimension of the variety $V$ is the Krull dimension of its coordinate ring ${\\mathbf {F}}[V]$ .", "For example, if $S = \\lbrace 0\\rbrace \\subseteq \\mathbf {F}[t_1,\\ldots , t_n]$ is the set whose only element is the zero polynomial, then $V = V( \\lbrace 0\\rbrace ) = \\mathbf {F}^n$ , and the vanishing ideal of $V$ is $ \\mathcal {I}(V) = \\mathcal {I}(\\mathbf {F}^n) = \\lbrace 0\\rbrace $ .", "We obtain the coordinate ring $\\mathbf {F}[V]= {\\mathbf {F}}[t_1,\\ldots , t_n]/{ \\mathfrak {I}}(V) \\cong {\\mathbf {F}}[t_1,\\ldots , t_n]$ and so the variety $\\mathbf {F}^n$ has dimension $n$ .", "Nathanson [5] explicitly computes the dimensions of some varieties generated by monomials.", "The proof that the Krull dimension of the polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_n]$ is $n$ uses only high school algebra and the rudiments of undergraduate abstract algebra.", "“High school algebra” means formal manipulations of polynomials.", "Results from high school algebra are collected and proved in Sections  and .", "“Rudimentary abstract algebra” means results whose proofs use only material found in standard undergraduate algebra texts.", "Section  contains results from abstract algebra.", "For other proofs, see Atiyah and Macdonald [1], Cox, Little, and O'Shea [2], and Kunz [4]." ], [ "A lower bound for the Krull dimension", "A polynomial in one variable $f = \\sum _{i =0}^d c_i t^i \\in R[t]$ has degree $d$ if $c_d \\ne 0$ .", "The leading term of $f$ is $c_dt^d$ , and the leading coefficient of $f$ is $c_d$ .", "The polynomial $f$ is monic if its leading coefficient is 1.", "Let $ \\mathbf {N} _0$ denote the set of nonnegative integers.", "For variables $t_1,\\ldots , t_n$ and for the $n$ -tuple $I = (i_1, i_2, \\ldots , i_n) \\in \\mathbf {N} _0^n$ , we define the monomial $t^I = t_1^{i_1}t_2^{i_2}\\cdots t_n^{i_n}.$ The degree of the monomial $t^I$ is $|I| = i_1+i_2+\\cdots + i_n$ .", "The degree of the variable $t_j$ in the monomial $t^I$ is $i_j$ .", "Let ${\\mathbf {F}}$ be a field.", "The polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_n]$ is a vector space over ${\\mathbf {F}}$ .", "A basis for this vector space is the set of monomials $\\left\\lbrace t^I: I \\in \\mathbf {N} _0^n \\right\\rbrace $ .", "Every nonzero polynomial $f \\in \\mathbf {F}[t_1,\\ldots , t_n]$ has a unique representation in the form $f = \\sum _{I \\in \\mathcal {I}} c_It^I$ where $ \\mathcal {I}$ is a nonempty finite subset of $ \\mathbf {N} _0^n$ and $c_I \\in {\\mathbf {F}}\\setminus \\lbrace 0\\rbrace $ for every $n$ -tuple $I \\in \\mathcal {I}$ .", "The degree of the polynomial $f$ is $\\max ( |I|: I \\in \\mathcal {I})$ , and the degree of the variable $t_j$ in $f$ is $\\max ( i_j: I \\in \\mathcal {I})$ .", "The degree of the zero polynomial is undefined.", "We write $S \\subseteq T$ is $S$ is a subset of $T$ , and $S \\subset T$ if $S$ is a proper subset of $T$ .", "Theorem 1 Let ${\\mathbf {F}}$ be a field and let $R = {\\mathbf {F}}[t_1,\\ldots , t_n]$ be the polynomial ring in $n$ variables with coefficients in $\\mathbf {F}$ .", "Let $ \\mathfrak {P}_0 = \\lbrace 0\\rbrace $ , and, for $k = 1,2, \\ldots , n$ , let $ \\mathfrak {P}_k$ be the ideal of $R$ generated by $\\lbrace t_1,t_2,\\ldots , t_k\\rbrace $ .", "Then $ \\mathfrak {P}_0 \\subset \\mathfrak {P}_1 \\subset \\mathfrak {P}_2 \\subset \\cdots \\subset \\mathfrak {P}_n$ is a strictly increasing chain of prime ideals in $R$ , and so the Krull dimension of $R$ is at least $n$ .", "The ideal $ \\mathfrak {P}_0 = \\lbrace 0\\rbrace $ is prime because $R$ is an integral domain.", "For $k \\in \\lbrace 1,\\ldots , n\\rbrace $ , every monomial in every nonzero polynomial in the ideal $ \\mathfrak {P}_k = \\left\\lbrace \\sum _{j=1}^k t_j f_j: f_j \\in R \\text{ for } j \\in \\lbrace 1,\\ldots , n\\rbrace \\right\\rbrace $ is divisible by $t_j$ for some $j \\in \\lbrace 1,\\ldots , k\\rbrace $ .", "For $k \\in \\lbrace 0,1,\\ldots , n-1\\rbrace $ , we have $t_{k+1} \\in \\mathfrak {P}_{k+1}$ but $t_{k+1} \\notin \\mathfrak {P}_k$ , and so (REF ) is a strictly increasing sequence of ideals.", "Moreover, $1 \\notin \\mathfrak {P}_n$ , and so $ \\mathfrak {P}_n$ is a proper ideal in $R$ .", "We shall prove that each ideal $ \\mathfrak {P}_k$ is prime.", "Let $k \\in \\lbrace 1,\\ldots , n\\rbrace $ .", "Consider the polynomial $f \\in R$ of the form (REF ).", "If $ \\mathcal {I}_1 = \\lbrace (i_1,\\ldots , i_n)\\in \\mathcal {I}: i_j \\ge 1 \\text{ for some } j \\in \\lbrace 1,\\ldots , k \\rbrace \\rbrace $ and $ \\mathcal {I}_2 = \\lbrace (i_1,\\ldots , i_n)\\in \\mathcal {I}: i_j = 0 \\text{ for all } j \\in \\lbrace 1,\\ldots , k\\rbrace \\rbrace $ then $f= f_1 + f_2$ , where $f_1 = \\sum _{I \\in \\mathcal {I}_1} c_I t^I \\in \\mathfrak {P}_k\\operatorname{\\qquad \\text{and}\\qquad }f_2 = \\sum _{I \\in \\mathcal {I}_2} c_I t^I \\in R.$ We have $f \\in \\mathfrak {P}_k$ if and only if $f_2 = 0$ .", "For example, let $n = 2$ and $f = t_1^3 + 2t_1^2t_2 + 4t_2^3\\in R = \\mathbf {F}[t_1,t_2].$ Choosing $k = 1$ , we obtain $ \\mathcal {I}_1 = \\lbrace (3,0), (2,1)\\rbrace $ and $ \\mathcal {I}_2 = \\lbrace (0,3) \\rbrace $ , and so $f_1 = t_1^3 + 2t_1^2t_2 \\in \\mathfrak {P}_1$ and $f_2 = 4 t_2^3 \\in R\\setminus \\mathfrak {P}_1$ .", "It follows that $f \\notin \\mathfrak {P}_1$ .", "Let $f \\in \\mathfrak {P}_k$ , and let $g$ and $h$ be polynomials in $R$ such that $f = gh$ .", "We write $g = g_1+g_2$ and $h = h_1 + h_2$ , where $g_1$ and $h_1$ are polynomials in $ \\mathfrak {P}_k$ , and $g_2$ and $h_2$ are polynomials in $R$ that are sums of monomials not divisible by $t_j$ for any $j \\in \\lbrace 1,\\ldots , k\\rbrace $ .", "Note that $g_2 h_2 \\in \\mathfrak {P}_k$ if and only if $g_2h_2=0$ if and only if $g_2 = 0$ or $h_2 = 0$ .", "We have $f = gh = (g_1+g_2) ( h_1 + h_2) = (g_1h_1+g_1h_2+g_2h_1) + g_2h_2.$ Because $ \\mathfrak {P}_k$ is an ideal, $g_1h_1+g_1h_2+g_2h_1\\in \\mathfrak {P}_k$ and so $g_2 h_2 = f - (g_1h_1+g_1h_2+g_2h_1) \\in \\mathfrak {P}_k.$ It follows that either $g_2 = 0$ and $g \\in \\mathfrak {P}_k$ , or $h_2 = 0$ and $h \\in \\mathfrak {P}_k$ .", "Therefore, $ \\mathfrak {P}_k$ is a prime ideal.", "The main result of this paper (Theorem REF ) is that the polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_n]$ has Krull dimension $n$ .", "Although this implies the maximality of the prime ideal chain (REF ), there is a nice direct proof of this result.", "Theorem 2 The prime ideal chain (REF ) is maximal.", "The first step is to prove that $ \\mathfrak {P}_n$ is a maximal ideal.", "The ideal $ \\mathfrak {P}_n$ consists of all polynomials with constant term 0.", "Let $g \\in R\\setminus \\mathfrak {P}_n$ , and let $ \\mathfrak {I}$ be the ideal of $R$ generated by the set $\\lbrace t_1,\\ldots , t_n, g \\rbrace $ .", "The polynomial $g$ has constant term $c \\ne 0$ , and $g-c \\in \\mathfrak {P}_n \\subseteq \\mathfrak {I}$ .", "The equation $c^{-1} (g - (g-c) ) = 1$ implies that 1 is in $ \\mathfrak {I}$ , and so $R = \\mathfrak {I}$ .", "Thus, the ideal $ \\mathfrak {P}_n$ is maximal.", "We shall prove that, for every integer $k \\in \\lbrace 1,2,\\ldots , n\\rbrace $ , if $ \\mathfrak {P}^{\\prime }$ is a prime ideal of $R$ such that $ \\mathfrak {P}_{k-1} \\subset \\mathfrak {P}^{\\prime } \\subseteq \\mathfrak {P}_k$ then $ \\mathfrak {P}^{\\prime } = \\mathfrak {P}_k$ .", "This implies that (REF ) is a maximal prime ideal chain.", "Let $k \\ge 1$ , and let $ \\mathfrak {P}$ ' be an ideal of $R$ satisfying (REF ).", "It follows that $ \\mathfrak {P}^{\\prime } \\setminus \\mathfrak {P}_{k-1} \\ne \\emptyset $ .", "Because $ \\mathfrak {P}^{\\prime } \\subseteq \\mathfrak {P}_k$ , every polynomial in $ \\mathfrak {P}^{\\prime } \\setminus \\mathfrak {P}_{k-1} $ contains at least one monomial of the form $t_k^{i_k} t_{k+1}^{i_{k+1}} \\cdots t_n^{i_n}$ with $i_k \\ge 1$ .", "Let ${\\ell }_k$ be the smallest positive integer $i_k$ such that a monomial of the form (REF ) occurs with a nonzero coefficient in some polynomial in $ \\mathfrak {P}^{\\prime } \\setminus \\mathfrak {P}_{k-1} $ .", "There exists a polynomial $f$ in $ \\mathfrak {P}^{\\prime } \\setminus \\mathfrak {P}_{k-1} $ that contains a monomial of the form  (REF ) with $i_k = \\ell _k$ .", "$t_k^{\\ell _k} t_{k+1}^{i_{k+1}} \\cdots t_n^{i_n}$ We write $f=f_1 + f_2$ , where $f_1 = \\sum _{\\begin{array}{c}I = (i_1,\\ldots , i_n) \\in \\mathbf {N} _0^k \\\\ i_j \\ge 1\\text{ for some } j \\in \\lbrace 1,\\ldots , k-1\\rbrace \\end{array}} c_I t^I\\in \\mathfrak {P}_{k-1}$ and $f_2 = \\sum _{\\begin{array}{c}I = (i_1,\\ldots , i_n) \\in \\mathbf {N} _0^k \\\\ i_j = 0\\text{ for all } j \\in \\lbrace 1,\\ldots , k-1\\rbrace \\\\ \\text{ and } i_k \\ge \\ell _k\\end{array}} c_I t^I = t_k^{{\\ell }_k} h$ for some nonzero polynomial $h \\in R$ .", "Because $f_2$ contains the monomial (REF ) in which the variable $t_k$ occurs with degree exactly $\\ell _k$ , the polynomial $h$ must contain a monomial not divisible by $t_i$ for all $i \\in \\lbrace 1,\\ldots , k-1, k\\rbrace $ , and so $h \\notin \\mathfrak {P}_k$ .", "It follows that $ t_k^{{\\ell }_k -1} h \\notin \\mathfrak {P}^{\\prime }$ .", "We have $f_2 = f - f_1 \\in \\mathfrak {P}^{\\prime }$ because $f \\in \\mathfrak {P}^{\\prime }$ and $f_1 \\in \\mathfrak {P}_{k-1} \\subseteq \\mathfrak {P}^{\\prime }$ .", "We factor $f_2$ as follows: $f_2 = t_k^{{\\ell }_k} h = t_k \\left( t_k^{{\\ell }_k -1} h \\right).$ Because $ t_k^{{\\ell }_k -1} h \\notin \\mathfrak {P}^{\\prime }$ and $ \\mathfrak {P}$ ' is a prime ideal, it follows that $t_k \\in \\mathfrak {P}^{\\prime }$ .", "Therefore, $ \\mathfrak {P}^{\\prime }$ contains $\\lbrace t_1,\\ldots , t_{k-1}, t_k \\rbrace $ , and so $ \\mathfrak {P}^{\\prime } = \\mathfrak {P}_k$ .", "This completes the proof." ], [ "Results from high school algebra", "Let $d$ be a nonnegative integer.", "A polynomial $f = \\sum _{I \\in \\mathcal {I}} c_It^I \\in \\mathbf {F}[t_1,\\ldots , t_n]$ is homogeneous of degree $d$ if all of its monomials have degree $d$ .", "Let $\\lambda \\in \\mathbf {F}$ and $a = (a_1,\\ldots , a_n) \\in \\mathbf {F}^n$ .", "If $t^I = t_1^{i_1}\\cdots t_n^{i_n}$ is a monomial of degree $d$ , then $(\\lambda a)^I = (\\lambda a_1)^{i_1}\\cdots (\\lambda a_n)^{i_n}= \\lambda ^{\\sum _{j=1}^n i_j} a_1^{i_1}\\cdots a_n^{i_n}= \\lambda ^d a^I.$ If $f = \\sum _{I \\in \\mathcal {I}} c_It^I$ is homogeneous of degree $d$ , then $f(\\lambda a_1,\\ldots , \\lambda a_n) = \\sum _{I \\in \\mathcal {I}} c_I (\\lambda a)^I= \\lambda ^d \\sum _{I \\in \\mathcal {I}} c_I a^I = \\lambda ^df( a_1,\\ldots , a_n).$ For example, the polynomial $f$ defined by (REF ) is homogeneous of degree 3, and $f(\\lambda a_1, \\lambda a_2)& = (\\lambda a_1)^3 + 2(\\lambda a_1)^2 (\\lambda a_2) + 4(\\lambda a_2)^3 \\\\& = \\lambda ^3 \\left( a_1^3 + 2 a_1^2 a_2 + 4 a_2^3 \\right)= \\lambda ^3 f( a_1, a_2).$ Let $ f = \\sum _{I \\in \\mathcal {I}} c_It^I$ be a nonzero polynomial of degree $d$ , and let $ \\mathcal {I}_d = \\lbrace I = (i_1,\\ldots , i_n) \\in \\mathcal {I}: |I| = \\sum _{j=1}^n i_j = d \\rbrace .$ Then $ \\mathcal {I}_d \\ne \\emptyset $ and $f_d = \\sum _{ I \\in \\mathcal {I}_d} c_It^I \\in {\\mathbf {F}}[t_1,\\ldots , t_n]$ is a homogeneous polynomial of degree $d$ .", "The real numbers $\\mathbf {R}$ , the complex numbers and the rational functions with real coefficients or with complex coefficients are infinite fields.", "An example of a finite field is ${\\mathbf {F}}_2 = \\lbrace 0, 1\\rbrace $ , with addition defined by $0+0=1+1=0$ and $1+0 = 0+1 = 1$ , and with multiplication defined by $0\\cdot 0 = 0\\cdot 1 = 1 \\cdot 0 = 0$ and $1 \\cdot 1 = 1$ .", "In the ring of polynomials $\\mathbf {F}_2[t_1,t_2]$ with coefficients in $\\mathbf {F}_2$ , the homogeneous polynomial $f(t_1,t_2) = t_1^2 + t_1t_2$ has the property that $f(a_1,1) = a_1^2 + a_1 = 0$ for all $a_1 \\in \\mathbf {F}_2$ .", "The following Lemma shows that this behavior is impossible for polynomials with coefficients in an infinite field.", "Lemma 1 Let ${\\mathbf {F}}$ be an infinite field, and let $f$ be a nonzero polynomial in ${\\mathbf {F}}[t_1,\\ldots , t_n]$ .", "There exist infinitely many points $(a_1,\\ldots , a_{n-1},a_n) \\in \\left( \\mathbf {F}\\setminus \\lbrace 0\\rbrace \\right)^n$ such that $f(a_1,\\ldots , a_{n-1},a_n) \\ne 0.$ If $f$ is homogeneous, then there exist infinitely many points $(a_1,\\ldots , a_{n-1}) \\in \\left( \\mathbf {F}\\setminus \\lbrace 0\\rbrace \\right)^{n-1}$ such that $f(a_1,\\ldots , a_{n-1},1) \\ne 0.$ By induction on the number $n$ of variables.", "If $n = 1$ , then $f(t_1)$ has only finitely many zeros.", "Because the field $\\mathbf {F}$ is infinite, there exist infinitely many $a_1\\in \\mathbf {F}\\setminus \\lbrace 0\\rbrace $ with $f(a_1) \\ne 0$ .", "If $f$ is homogeneous, then $f(t_1) = c_dt_1^d$ for some $d \\in \\mathbf {N} _0$ and $c_d \\in \\mathbf {F}\\setminus \\lbrace 0\\rbrace $ , and $f(1) = c_d \\ne 0$ .", "Let $n \\ge 2$ , and assume that the Lemma holds for polynomials in $n-1$ variables.", "A polynomial $f \\in {\\mathbf {F}}[t_1, \\ldots , t_{n-1}, t_n]$ can also be represented as a polynomial in the variable $t_n$ with coefficients in the polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_{n-1}]$ .", "Thus, there exist polynomials $f_0, f_1,\\ldots , f_d \\in {\\mathbf {F}}[t_1,\\ldots , t_{n-1}]$ such that $f_d \\ne 0$ and $f(t_1, \\ldots , t_{n-1}, t_n) = \\sum _{j=0}^d f_j(t_1,\\ldots , t_{n-1}) t_n^j.$ By the induction hypothesis, there exist infinitely many points $(a_1,\\ldots , a_{n-1}) \\in \\left( \\mathbf {F}\\setminus \\lbrace 0\\rbrace \\right)^{n-1}$ such that $f_d(a_1,\\ldots , a_{n-1}) \\ne 0.$ By (REF ) and (REF ), the polynomial $f(a_1, a_2,\\ldots , a_{n-1}, t_n) = \\sum _{j=0}^d f_j(a_1,\\ldots , a_{n-1}) t_n^j \\in {\\mathbf {F}}[t_n]$ has degree $d$ .", "Because a nonzero polynomial of degree $d$ has at most $d$ roots in ${\\mathbf {F}}$ , and because the field ${\\mathbf {F}}$ is infinite, there exist infinitely many elements $a_n \\in {\\mathbf {F}} \\setminus \\lbrace 0\\rbrace $ such that $f(a_1, a_2,\\ldots , a_{n-1}, a_n) \\ne 0.$ Let $f \\in {\\mathbf {F}}[t_1,\\ldots , t_{n-1}, t_n]$ be a homogeneous polynomial of degree $d$ , and let $(a_1,\\ldots , a_{n-1},a_n) \\in \\left( \\mathbf {F}\\setminus \\lbrace 0\\rbrace \\right)^n$ satisfy $f(a_1,\\ldots , a_{n-1},a_n) \\ne 0$ .", "Applying the homogeneity identity (REF ) with $\\lambda = a_n^{-1}$ , we obtain $f(a_n^{-1} a_1,\\ldots , a_n^{-1} a_{n-1}, 1) = a_n^{-d} f(a_1,\\ldots , a_{n-1},a_n) \\ne 0$ This completes the proof.", "We shall apply Lemma REF to prove an important result (Lemma REF ) that may appear “complicated” and “technical,” but is essential in Section  in the proof of the fundamental theorem about Krull dimension.", "Let $t_1,\\ldots , t_n, x_1,\\ldots , x_n$ be variables, and let $I = (i_1,\\ldots , i_n) \\in \\mathbf {N} _0^n$ .", "We consider the polynomials $(t_j+x_jt_n)^{i_j} \\in \\mathbf {F}[x_1,\\ldots , x_n,t_1,\\ldots , t_n]= \\mathbf {F}[x_1,\\ldots , x_n,t_1,\\ldots , t_{n-1}] [ t_n]$ for $j = 1, \\ldots , n$ .", "As a polynomial in $t_n$ , the leading term of $(t_j+x_jt_n)^{i_j}$ is $x_j^{i_j} t_n^{i_j}$ and the leading term of $(t_1+x_1t_n)^{i_1} (t_2+x_2t_n)^{i_2}\\cdots (t_{n-1}+x_{n-1}t_n)^{i_{n-1}} (x_nt_n)^{i_n}$ is $\\prod _{j=1}^n (x_jt_n)^{i_j}= \\left( \\prod _{j=1}^nx_j^{i_j} \\right) t_n^{\\sum _{j=1}^n i_j}= x^I t_n^{ |I| }.$ Let $g$ be a nonzero polynomial in ${\\mathbf {F}}[t_1,\\ldots , t_n]$ .", "If $g$ has degree $d$ in $t_n$ , then there exist unique polynomials $g_0,\\ldots , g_d \\in {\\mathbf {F}}[t_1,\\ldots , t_{n-1}]$ with $g_d \\ne 0$ such that $g(t_1,\\ldots , t_{n-1}, t_n)= \\sum _{j =0}^d g_j(t_1,\\ldots , t_{n-1}) t_n^j.$ The polynomial $g$ is monic in $t_n$ if $g_d =1$ , that is, if $g(t_1,\\ldots , t_{n-1}, t_n)= t_n^d + \\sum _{j =0}^{d-1} g_j(t_1,\\ldots , t_{n-1}) t_n^j.$ Lemma 2 Let ${\\mathbf {F}}$ be an infinite field, and let $f = \\sum _{I \\in \\mathcal {I}} c_It^I \\in {\\mathbf {F}}[t_1,\\ldots , t_n]$ be a nonzero polynomial.", "There exist $a_1,\\ldots , a_{n-1}, \\lambda \\in {\\mathbf {F}}$ with $\\lambda \\ne 0$ and a polynomial $g \\in {\\mathbf {F}}[t_1,\\ldots , t_n]$ such that $g$ is monic in $t_n$ and $g(t_1,\\ldots , t_{n-1}, t_n)= \\lambda ^{-1} f(t_1+a_1t_n, t_2+a_2t_n, \\ldots , t_{n-1}+a_{n-1}t_n, t_n).$ If $f$ has degree $d$ , then $ \\mathcal {I}_d = \\lbrace I = (i_1,\\ldots , i_n) \\in \\mathcal {I}: |I| = \\sum _{j=1}^n i_j = d \\rbrace \\ne \\emptyset $ and $f_d = \\sum _{ I \\in \\mathcal {I}_d} c_It^I \\in {\\mathbf {F}}[t_1,\\ldots , t_n]$ is a nonzero homogeneous polynomial of degree $d$ .", "We introduce additional variables $x_1,\\ldots , x_n$ , and consider $f_d(t_1+ x_1t_n, t_2+x_2t_n, \\ldots , t_{n-1}+x_{n-1}t_n, x_nt_n)$ as a polynomial in $t_n$ with coefficients in the ring ${\\mathbf {F}}[x_1,\\ldots , x_n, t_1,\\ldots , t_{n-1}]$ .", "Applying (REF ), we have $f_d(t_1+ & x_1t_n, t_2+x_2t_n, \\ldots , t_{n-1}+x_{n-1}t_n, x_nt_n) \\\\& = \\sum _{I = (i_1,\\ldots , i_n) \\in \\mathcal {I}_d } c_I \\prod _{j=1}^{n-1} (t_j+x_jt_n)^{i_j} \\ (x_nt_n)^{i_n} \\\\& = \\sum _{ I= (i_1,\\ldots , i_n) \\in \\mathcal {I}_d } c_I \\prod _{j=1}^n (x_j t_n)^{i_j}+ \\text{ lower order terms in $t_n$} \\\\& = \\left( \\sum _{ I \\in \\mathcal {I}_d } c_I x^I \\right) t_n^d + \\text{ lower order terms in $t_n$} \\\\& = f_d(x_1,\\ldots , x_{n-1}, x_n) t_n^d + \\text{ lower order terms in $t_n$}.$ By Lemma REF , there exist $a_1,\\ldots , a_{n-1} \\in {\\mathbf {F}}$ such that $\\lambda = f_d(a_1,\\ldots , a_{n-1},1) \\ne 0.$ It follows that $\\lambda ^{-1}f_d(t_1+ & a_1t_n, t_2+a_2t_n, \\ldots , t_{n-1}+a_{n-1}t_n, t_n) \\\\& = \\lambda ^{-1}f_d(a_1,\\ldots , a_{n-1}, 1) t_n^d + \\text{ lower order terms in $t_n$} \\\\& = t_n^d + \\text{ lower order terms in $t_n$}$ and so the polynomial $g(t_1, & \\ldots , t_{n-1}, t_n) \\\\& = \\lambda ^{-1}f(t_1+ a_1t_n, t_2+a_2t_n, \\ldots , t_{n-1}+a_{n-1}t_n, t_n) \\\\& = \\lambda ^{-1} f_d(t_1+ a_1t_n, \\ldots , t_{n-1}+a_{n-1}t_n, t_n) + \\text{ lower order terms in $t_n$} \\\\& = \\lambda ^{-1} f_d(a_1,\\ldots , a_{n-1}, 1) t_n^d + \\text{ lower order terms in $t_n$} \\\\& = t_n^d + \\text{ lower order terms in $t_n$}$ is monic in $t_n$ .", "This completes the proof." ], [ "Results from undergraduate algebra", "To obtain an upper bound for the Krull dimension of the polynomial ring, we need to study the image of a chain of ideals in a quotient ring.", "Lemma 3 If $ \\mathfrak {I}\\subset \\mathfrak {I}^{\\prime } \\subset \\mathfrak {I}^{\\prime \\prime }$ is an ideal chain in the ring $R$ , then $\\lbrace \\mathfrak {I}\\rbrace \\subset \\mathfrak {I}^{\\prime }/ \\mathfrak {I}\\subset \\mathfrak {I}^{\\prime \\prime } / \\mathfrak {I}$ is an ideal chain in the quotient ring $R/ \\mathfrak {I}$ .", "If $ \\mathfrak {I}^{\\prime }$ is a prime ideal in $R$ , then $ \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ is a prime ideal in $R/ \\mathfrak {I}$ .", "In the quotient ring $R/ \\mathfrak {I}$ , the set containing only the coset $0+ \\mathfrak {I}= \\mathfrak {I}$ is $\\lbrace \\mathfrak {I}\\rbrace $ .", "The sets $ \\mathfrak {I}^{\\prime }/ \\mathfrak {I}= \\lbrace a + \\mathfrak {I}: a \\in \\mathfrak {I}^{\\prime } \\rbrace $ and $ \\mathfrak {I}^{\\prime \\prime }/ \\mathfrak {I}= \\lbrace b+ \\mathfrak {I}: b \\in \\mathfrak {I}^{\\prime \\prime }\\rbrace $ are ideals in $R/ \\mathfrak {I}$ .", "We have $ \\mathfrak {I}^{\\prime }/ \\mathfrak {I}\\ne \\lbrace \\mathfrak {I}\\rbrace $ because $ \\mathfrak {I}^{\\prime } \\ne \\mathfrak {I}$ .", "Because $ \\mathfrak {I}^{\\prime }$ is a proper subset of $ \\mathfrak {I}^{\\prime \\prime }$ , there exists $b \\in \\mathfrak {I}^{\\prime \\prime }$ with $b \\notin \\mathfrak {I}^{\\prime }$ , and $b + \\mathfrak {I}\\in \\mathfrak {I}^{\\prime \\prime }/ \\mathfrak {I}$ .", "If $b + \\mathfrak {I}\\in \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ , then there exists $a \\in \\mathfrak {I}^{\\prime }$ such that $b+ \\mathfrak {I}= a+ \\mathfrak {I}$ , and so $b-a \\in \\mathfrak {I}\\subseteq \\mathfrak {I}^{\\prime }$ .", "It follows that $b = (b-a)+a \\in \\mathfrak {I}^{\\prime }$ , which is absurd.", "Therefore, $b + \\mathfrak {I}\\notin \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ and $ \\mathfrak {I}^{\\prime }/ \\mathfrak {I}\\ne \\mathfrak {I}^{\\prime \\prime }/ \\mathfrak {I}$ .", "Let $ \\mathfrak {I}^{\\prime }$ be a prime ideal in $R$ .", "If $a,b \\in R$ and $(a+ \\mathfrak {I})(b+ \\mathfrak {I}) \\in \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ then there exists $c \\in \\mathfrak {I}^{\\prime }$ such that $ab+ \\mathfrak {I}= c + \\mathfrak {I}$ and so $ab = c + x$ for some $x \\in \\mathfrak {I}\\subseteq \\mathfrak {I}^{\\prime }$ .", "Therefore, $ab \\in \\mathfrak {I}^{\\prime }$ .", "Because $ \\mathfrak {I}^{\\prime }$ is a prime ideal, we have $a \\in \\mathfrak {I}^{\\prime }$ or $b \\in \\mathfrak {I}^{\\prime }$ , and so $a+ \\mathfrak {I}\\in \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ or $b+ \\mathfrak {I}\\in \\mathfrak {I}^{\\prime }/ \\mathfrak {I}$ .", "This completes the proof.", "Theorem 3 If $ \\mathfrak {P}_1 \\subset \\mathfrak {P}_2 \\subset \\cdots \\subset \\mathfrak {P}_m$ is a prime ideal chain in the ring $R$ , then $\\lbrace \\mathfrak {P}_1 \\rbrace \\subset \\mathfrak {P}_2 / \\mathfrak {P}_1 \\subset \\cdots \\subset \\mathfrak {P}_m / \\mathfrak {P}_1$ is a prime ideal chain in the quotient ring $R/ \\mathfrak {P}_1$ .", "This follows immediately from Lemma REF .", "We also need some results about integral extensions of a ring.", "The ring $S$ is an extension ring of $R$ and the ring $R$ is a subing of $S$ if $R \\subseteq S$ and the multiplicative identity in $R$ is the multiplicative identity in $S$ .", "An element $a \\in S$ is integral over $R$ if there is a monic polynomial $f \\in R[t]$ of degree $d \\ge 1$ such that $f(a) = 0$ .", "The ring $S$ is an integral extension of $R$ if $S$ is an extension ring of $R$ and every element of $S$ is integral over $R$ .", "For example, the ring of Gaussian integers $\\mathbf {Z}[i] = \\lbrace a+bi:a,b \\in \\mathbf {Z}\\rbrace $ is an integral extension of $\\mathbf {Z}$ because $a+bi \\in \\mathbf {Z}[i] $ is a root of the monic quadratic polynomial $t^2 -2at + a^2+b^2 \\in \\mathbf {Z}[t]$ .", "The ring $R$ is an integral extension of itself because every element $a \\in R$ is a root of the monic linear polynomial $t-a \\in R[t]$ .", "Lemma 4 Let $S$ be an extension ring of $R$ .", "If $ \\mathfrak {P}$ is a prime ideal in $S$ , then $ \\mathfrak {P}\\cap R$ is a prime ideal in $R$ .", "Let $S$ be an integral domain that is an integral extension of the ring $R$ .", "If $ \\mathfrak {I}$ is a nonzero ideal in $S$ , then $ \\mathfrak {I}\\cap R$ is a nonzero ideal in $R$ .", "(i) If $ \\mathfrak {I}$ is an ideal in $S$ , then $ \\mathfrak {I}\\cap R$ is an ideal in $R$ .", "Let $ \\mathfrak {P}$ be a prime ideal in $S$ , and let $ \\mathfrak {p}= \\mathfrak {P}\\cap R$ .", "If $a,b \\in R$ and $ab \\in \\mathfrak {p}$ , then $ab \\in \\mathfrak {P}$ and so $a \\in \\mathfrak {P}$ or $b \\in \\mathfrak {P}$ .", "It follows that $a \\in \\mathfrak {p}$ or $b \\in \\mathfrak {p}$ , and so $ \\mathfrak {p}$ is a prime ideal in $R$ .", "(ii) Let $S$ be an integral domain that is an integral extension of the ring $R$ , and let $ \\mathfrak {I}$ be a nonzero ideal in $S$ .", "Let $a \\in \\mathfrak {I}$ , $a \\ne 0$ .", "Because $S$ is integral over $R$ , there is a monic polynomial $f$ of minimum degree $d$ $f = t^d + c_{d-1}t^{d-1} + \\cdots + c_1t + c_0 \\in R[t]$ such that $f(a) = a^d + c_{d-1}a^{d-1} + \\cdots + c_1a + c_0 = 0.$ If $c_0 = 0$ , then $\\left( a^{d-1} + c_{d-1}a^{d-2} + \\cdots + c_1\\right) a = 0.$ Because $S$ is an integral domain and $a \\ne 0$ , we obtain $a^{d-1} + c_{d-1}a^{d-2} + \\cdots + c_1 = 0$ and so $a$ is a root of a monic polynomial of degree $d-1$ .", "This contradicts the minimality of $d$ .", "Therefore, $c_0 \\ne 0$ .", "Because $a \\in \\mathfrak {I}$ and $ \\mathfrak {I}$ is an ideal, we have $a^d + c_{d-1}a^{d-1} + \\cdots + c_1a= \\left( a^{d-1} + c_{d-1}a^{d-2} + \\cdots + c_1\\right) a \\in \\mathfrak {I}$ and so $c_0 = -\\left( a^d + c_{d-1}a^{d-1} + \\cdots + c_1a \\right) \\in \\mathfrak {I}\\cap R.$ Thus, $ \\mathfrak {I}\\cap R \\ne \\lbrace 0\\rbrace $ .", "This completes the proof.", "Lemma 5 Let $S$ be a ring, and let $R$ be a subring of $S$ such that $S$ is integral over $R$ .", "If $ \\mathfrak {P}$ and $ \\mathfrak {I}$ are ideals in $S$ such that $ \\mathfrak {P}\\subset \\mathfrak {I}$ and $ \\mathfrak {P}$ is prime, then $ \\mathfrak {P}\\cap R \\ne \\mathfrak {I}\\cap R$ .", "The quotient ring $S/ \\mathfrak {P}$ is an integral domain because the ideal $ \\mathfrak {P}$ is prime.", "The ring $R/ \\mathfrak {P}$ is a subring of $S/ \\mathfrak {P}$ , and $( \\mathfrak {P}\\cap R)/ \\mathfrak {P}= \\lbrace \\mathfrak {P}\\rbrace $ .", "Thus, to prove that $ \\mathfrak {P}\\cap R \\ne \\mathfrak {I}\\cap R$ , it suffices to prove that $( \\mathfrak {I}\\cap R)/ \\mathfrak {P}\\ne \\lbrace \\mathfrak {P}\\rbrace $ .", "We prove first that $S/ \\mathfrak {P}$ is an integral extension of $R/ \\mathfrak {P}$ .", "Let $a \\in S$ , and consider the coset $a + \\mathfrak {P}$ .", "Because $a$ is integral over $R$ , there is a monic polynomial $f = t^d + \\sum _{i=0}^{d-1} c_i t^i \\in R[t]$ such that $f(a) = a^d + \\sum _{i=0}^{d-1} c_i a^i = 0.$ Defining the monic polynomial $\\tilde{f}= t^d + \\sum _{i=0}^{d-1} (c_i + \\mathfrak {P}) t^i \\in (R/ \\mathfrak {P}) [t]$ we obtain $\\tilde{f}( a + \\mathfrak {P})& = ( a + \\mathfrak {P})^d + \\sum _{i=0}^{d-1} (c_i + \\mathfrak {P})(a+ \\mathfrak {P})^i \\\\& = \\left( a^d + \\sum _{i=0}^{d-1} c_i a^i \\right) + \\mathfrak {P}\\\\& = f(a)+ \\mathfrak {P}= \\mathfrak {P}$ and so $a + \\mathfrak {P}$ is integral over the ring $R/ \\mathfrak {P}$ .", "Thus, $S/ \\mathfrak {P}$ is an integral extension of $R/ \\mathfrak {P}$ .", "The ideal $ \\mathfrak {I}/ \\mathfrak {P}$ is nonzero in $S/ \\mathfrak {P}$ because $ \\mathfrak {P}$ is a proper subset of $ \\mathfrak {I}$ .", "It follows from Lemma REF that $( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) $ is a nonzero ideal in $R/ \\mathfrak {P}$ , that is, $( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) \\ne \\lbrace \\mathfrak {P}\\rbrace $ .", "We shall prove that $( \\mathfrak {I}\\cap R)/ \\mathfrak {P}= ( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) $ .", "The inclusion $( \\mathfrak {I}\\cap R)/ \\mathfrak {P}\\subseteq ( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) $ is immediate.", "To prove the opposite inclusion, let $r+ \\mathfrak {P}\\in ( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) $ for some $r \\in R$ .", "There exists $a \\in \\mathfrak {I}$ such that $r+ \\mathfrak {P}= a+ \\mathfrak {P}$ , and so $r -a = b \\in \\mathfrak {P}\\subset \\mathfrak {I}$ .", "It follows that $r = a+ b \\in \\mathfrak {I}$ and so $r \\in \\mathfrak {I}\\cap R$ .", "Therefore, $( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) \\subseteq ( \\mathfrak {I}\\cap R)/ \\mathfrak {P}$ .", "We conclude that $( \\mathfrak {I}\\cap R) / \\mathfrak {P}= ( \\mathfrak {I}/ \\mathfrak {P}) \\cap (R/ \\mathfrak {P}) \\ne \\lbrace \\mathfrak {P}\\rbrace $ , and so $ \\mathfrak {P}\\cap R \\ne \\mathfrak {I}\\cap R$ .", "This completes the proof.", "Theorem 4 Let $S$ be an integral domain, and let $R$ be a subring of $S$ such that $S$ is integral over $R$ .", "If $ \\mathfrak {P}_0 \\subset \\mathfrak {P}_1 \\subset \\cdots \\subset \\mathfrak {P}_m$ is a prime ideal chain in $S$ , then $ \\mathfrak {P}_0 \\cap R \\ \\subset \\ \\mathfrak {P}_1 \\cap R \\ \\subset \\cdots \\subset \\ \\mathfrak {P}_m \\cap R$ is a prime ideal chain in $R$ .", "This follows immediately from Lemmas REF and REF .", "A minimal prime ideal in a ring $R$ is a nonzero prime ideal $ \\mathfrak {P}$ such that, if $ \\mathfrak {P}^{\\prime } $ is a prime ideal and $\\lbrace 0 \\rbrace \\subseteq \\mathfrak {P}^{\\prime } \\subseteq \\mathfrak {P}$ , then $ \\mathfrak {P}^{\\prime } = \\lbrace 0 \\rbrace $ or $ \\mathfrak {P}^{\\prime } = \\mathfrak {P}$ .", "In a field, the only prime ideal is $\\lbrace 0\\rbrace $ and there is no minimal prime ideal.", "Lemma 6 In a unique factorization domain $R$ , a prime ideal is minimal if and only if it is a principal ideal generated by an irreducible element.", "In a unique factorization domain, the principal ideal generated by an irreducible element is a nonzero prime ideal.", "Let $ \\mathfrak {P}$ be a minimal prime ideal in $R$ .", "Because $ \\mathfrak {P}\\ne \\lbrace 0\\rbrace $ and $ \\mathfrak {P}\\ne R$ , the ideal $ \\mathfrak {P}$ contains a nonzero element that is not a unit.", "This element is a product of irreducible elements.", "Because $ \\mathfrak {P}$ is a prime ideal, it contains at least one of these irreducible factors.", "If $a \\in \\mathfrak {P}$ and $a$ is irreducible, then $ \\mathfrak {P}$ contains the principal ideal $\\langle a \\rangle $ , which is a nonzero prime ideal.", "The minimality of $ \\mathfrak {P}$ implies that $ \\mathfrak {P}= \\langle a \\rangle $ .", "Thus, every minimal prime ideal is a principal ideal generated by an irreducible.", "Conversely, let $a$ be an irreducible element in $R$ , and consider the nonzero prime ideal $\\langle a \\rangle $ .", "If $ \\mathfrak {P}$ is a nonzero prime ideal contained in $\\langle a \\rangle $ , then $ \\mathfrak {P}$ contains a nonzero element that is not a unit.", "This element is a product of irreducibles, and, because the ideal $ \\mathfrak {P}$ is prime, it must contain at least one of these irreducible elements.", "Let $b$ be an irreducible element in $ \\mathfrak {P}$ , and let $\\langle b \\rangle $ be the principal ideal generated by $b$ .", "We have $\\langle b \\rangle \\subseteq \\mathfrak {P}\\subseteq \\langle a \\rangle $ and so $a$ divides $b$ .", "Because $a$ and $b$ are irreducible elements in $R$ , it follows that $a$ and $b$ are associates and so $\\langle a \\rangle = \\mathfrak {P}= \\langle b \\rangle $ .", "Therefore, $ \\langle a \\rangle $ is a minimal prime ideal.", "This completes the proof.", "Lemma 7 Let $S_0$ be an extension ring of $R$ .", "If there is a finite set $\\lbrace x_1,\\ldots , x_d\\rbrace \\subseteq S_0$ such that every element of $S_0$ is an $R$ -linear combination of elements of $\\lbrace x_1,\\ldots , x_d\\rbrace $ , that is, if $S_0 = \\left\\lbrace \\sum _{j=1}^d r_j x_j:r_j \\in R \\text{ for } j = 1,\\ldots , d \\right\\rbrace $ then $S_0$ is an integral extension of $R$ .", "Note that $S_0 \\ne \\lbrace 0\\rbrace $ because $1 \\in S_0$ , and so $ \\lbrace x_1,\\ldots , x_d\\rbrace \\ne \\lbrace 0\\rbrace $ .", "The Kronecker delta $\\delta _{i,j} $ is defined by $\\delta _{i,j} = 1$ if $i=j$ , and $\\delta _{i,j} = 0$ if $i \\ne j$ .", "Let $s \\in S_0$ .", "Because $S_0$ is a ring and $ \\lbrace x_1,\\ldots , x_d\\rbrace \\subseteq S_0$ , for all $i \\in \\lbrace 1,\\ldots , d \\rbrace $ we have $sx_i \\in S_0$ , and so there exist $r_{i,j} \\in R$ such that $sx_i = \\sum _{j=1}^d r_{i,j} x_j.$ Equivalently, $\\sum _{j=1}^d (\\delta _{i,j}s - r_{i,j} )x_j = 0$ and so the homogeneous system of linear equations $\\sum _{j=1}^d (\\delta _{i,j}s - r_{i,j} )t_j = 0 \\qquad \\text{for $i=1,\\ldots , d$.", "}$ has the nonzero solution $ \\lbrace x_1,\\ldots , x_d\\rbrace $ .", "This implies that the determinant of the matrix of coefficients of this system of linear equations is 0.", "This $d \\times d$ matrix is $\\left(\\begin{matrix}s - a_{1,1} & -a_{1,2} & \\cdots & -a_{1,d} \\\\-a_{2,1} & s - a_{2,2} & \\cdots & -a_{2,d} \\\\\\vdots & & & \\\\-a_{n,1} & -a_{n,2} & \\cdots & s - a_{d,d}\\end{matrix}\\right).$ and its determinant is a monic polynomial of degree $d$ in $s$ with coefficients in $R$ .", "Therefore, $s$ is integral over $R$ .", "This completes the proof.", "Lemma 8 Let $S$ be an integral domain, let $R$ be a subring of $S$ , and let $a \\in S$ .", "Let $R[a]$ be the smallest subring of $S$ that contains $R$ and $a$ .", "If $a$ is integral over $R$ , then the ring $R[a]$ is an integral extension of $R$ .", "Every element of $R[a]$ is a polynomial in $a$ with coefficients in $R$ , that is, an $R$ -linear combination of elements in the infinite set $\\lbrace a^i:i=0,1,2,\\ldots \\rbrace $ .", "Because $a$ is integral over $R$ , there is a monic polynomial $f \\in R[t]$ of degree $d$ such that $f(a) = 0$ .", "Rearranging this equation, we obtain $a^d = \\sum _{j=0}^{d-1} c_{d,j} a^j$ with $c_{d,j} \\in R$ for $j = 0,1,\\ldots , d-1$ .", "If $i \\ge d$ and $a^i = \\sum _{j=0}^{d-1} c_{i,j} a^j$ with $c_{i,j} \\in R$ for $j = 0,1,\\ldots , d-1$ , then $a^{i+1} & = a \\cdot a^i = a \\sum _{j=0}^{d-1} c_{i,j} a^j= \\sum _{j=0}^{d-2} c_{i,j} a^{j+1}+ c_{i,d-1} a^d \\\\& = \\sum _{j=1}^{d-1} c_{i,j-1} a^{j}+ c_{i,d-1} \\sum _{j=0}^{d-1} c_{i,j} a^j \\\\& = c_{i,d-1} c_{i,0} + \\sum _{j=1}^{d-1} ( c_{i,j-1} + c_{i,d-1} c_{i,j} ) a^j\\\\& = \\sum _{j=0}^{d-1} c_{i+1,j} a^j$ where $c_{i+1,0} = c_{i,d-1} c_{i,0}$ and $c_{i+1,j} = c_{i,j-1} + c_{i,d-1} c_{i,j} \\in R$ for $j=1,\\ldots , d-1$ .", "It follows by induction that every nonnegative power of $a$ can be written as an $R$ -linear combination of elements in the finite set $\\lbrace 1,a,a^2,\\ldots , a^{d-1} \\rbrace $ , and so every element in $R[a]$ is also an $R$ -linear combination of elements in the finite set $\\lbrace 1,a,a^2,\\ldots , a^{d-1} \\rbrace $ .", "By Lemma REF , the ring $R[a]$ is an integral extension of $R$ ." ], [ "An upper bound for the Krull dimension", "Theorem 5 Let ${\\mathbf {F}}$ be an infinite field.", "For every nonnegative integer $n$ , the Krull dimension of the polynomial ring ${\\mathbf {F}}[t_1,\\ldots , t_n]$ is $n$ .", "Let $m$ be the Krull dimension of the ring $R = {\\mathbf {F}}[t_1,\\ldots , t_n]$ .", "We have $m \\ge n$ by Theorem REF .", "We must prove that $m \\le n$ .", "The proof is by induction on $n$ .", "If $n = 0$ , then $R$ is the field ${\\mathbf {F}}$ , the only prime ideal in a field is $\\lbrace 0\\rbrace $ , there is no minimal prime ideal, and $R$ has Krull dimension 0.", "If $n = 1$ , then $R ={\\mathbf {F}}[t_1]$ is a principal ideal domain, and, consequently, a unique factorization domain.", "By Lemma REF , the nonzero prime ideals in $R$ are the principal ideals generated by irreducible polynomials, and every nonzero prime ideal in $R$ is minimal.", "Therefore, ${\\mathbf {F}}[t_1]$ has Krull dimension 1.", "Let $n \\ge 2$ , and let $R = {\\mathbf {F}}[t_1,\\ldots , t_{n-1}, t_n] = R^{\\prime }[t_n]$ , where $R^{\\prime } ={\\mathbf {F}}[t_1,\\ldots , t_{n-1}]$ .", "By the induction hypothesis, $R^{\\prime }$ has Krull dimension at most $n-1$ .", "Let $\\lbrace 0\\rbrace = \\mathfrak {P}_0 \\subset \\mathfrak {P}_1 \\subset \\cdots \\subset \\mathfrak {P}_m$ be a maximal chain of prime ideals in $R$ .", "The polynomial ring $R $ is a unique factorization domain, and the ideal $ \\mathfrak {P}_1$ is a minimal prime ideal in $R$ .", "By Lemma REF , $ \\mathfrak {P}_1$ is a principal ideal generated by an irreducible polynomial $f = f(t_1,\\ldots , t_n)$ .", "Here is the critical application of Lemma REF : There exist $a_1,\\ldots , a_{n-1}, \\lambda \\in {\\mathbf {F}}$ with $\\lambda \\ne 0$ such that $g = g(t_1,\\ldots , t_{n-1}, t_n)= \\lambda ^{-1} f(t_1+a_1t_n, t_2+a_2t_n, \\ldots , t_{n-1}+a_{n-1}t_n, t_n)$ is a polynomial that is monic in the variable $t_n$ with coefficients in $R^{\\prime }$ .", "We can represent $g$ in the form $g = \\tilde{g}(t_n) = t_n^d + \\sum _{i=0}^{d-1} c_i t_n^i \\in R^{\\prime }[t_n]$ for some positive integer $d$ and polynomials $c_0,c_1,\\ldots , c_{d-1} \\in R^{\\prime }$ .", "The function $\\varphi : R \\rightarrow R$ defined by $\\varphi (a) = a$ for $a \\in \\mathbf {F}$ and $\\varphi (t_j) = {\\left\\lbrace \\begin{array}{ll}t_j + a_j t_n & \\text{ if $j = 1, \\ldots , n-1$} \\\\t_n & \\text{ if $j=n$}\\end{array}\\right.", "}$ is a ring isomorphism.", "Because a ring isomorphism sends prime ideals to prime ideals, $\\lbrace 0\\rbrace = \\varphi ( \\mathfrak {P}_0) \\subset \\varphi ( \\mathfrak {P}_1) \\subset \\cdots \\subset \\varphi ( \\mathfrak {P}_m )$ is also a maximal chain of prime ideals in $R$ .", "We have $\\varphi (f(t_1,\\ldots , t_n) ) & = f(t_1+a_1t_n, t_2 + a_2 t_n, \\ldots , t_{n-1} + a_{n-1} t_n, t_n) \\\\& = \\lambda g(t_1,\\ldots , t_n)$ and so $\\varphi ( \\mathfrak {P}_1)$ is the principal ideal generated by $\\lambda g(t_1,\\ldots , t_n)$ .", "Because $\\lambda \\in \\mathbf {F}\\setminus \\lbrace 0\\rbrace $ is a unit in $R$ , the principal ideal $\\varphi ( \\mathfrak {P}_1)$ is also generated by $g(t_1,\\ldots , t_n)$ .", "Thus, we can assume that the minimal ideal $ \\mathfrak {P}_1$ in the prime ideal chain (REF ) is a principal ideal generated by a monic polynomial $g \\in R^{\\prime }[t_n]$ .", "The quotient ring $R/ \\mathfrak {P}_1$ is an integral domain because $ \\mathfrak {P}_1$ is a prime ideal.", "By Theorem REF , $\\lbrace \\mathfrak {P}_1 \\rbrace \\subset \\mathfrak {P}_2/ \\mathfrak {P}_1 \\subset \\cdots \\subset \\mathfrak {P}_m/ \\mathfrak {P}_1$ is a prime ideal chain in $R/ \\mathfrak {P}_1$ .", "Every coset in the quotient ring $R / \\mathfrak {P}_1$ is of the form $f + \\mathfrak {P}_1$ , where $f = \\sum _{i=0}^k f_i t_n^i \\in R^{\\prime }[t_n]$ and $f_i \\in R^{\\prime }$ for $i = 0,1,\\ldots , k$ .", "It follows that $f_i + \\mathfrak {P}_1 \\in R^{\\prime }/ \\mathfrak {P}_1$ , and so $f + \\mathfrak {P}_1 = \\left( \\sum _{i=0}^k f_i t_n^i \\right) + \\mathfrak {P}_1= \\sum _{i=0}^k (f_i + \\mathfrak {P}_1) (t_n + \\mathfrak {P}_1)^i\\in \\left( R^{\\prime }/ \\mathfrak {P}_1 \\right) [t_n + \\mathfrak {P}_1].$ This proves that $R / \\mathfrak {P}_1 \\subseteq \\left( R^{\\prime }/ \\mathfrak {P}_1 \\right) [t_n + \\mathfrak {P}_1]$ .", "Conversely, $ \\left( R^{\\prime }/ \\mathfrak {P}_1 \\right) [t_n + \\mathfrak {P}_1] \\subseteq R / \\mathfrak {P}_1$ and so $\\left( R^{\\prime }/ \\mathfrak {P}_1 \\right) [t_n + \\mathfrak {P}_1]= R / \\mathfrak {P}_1$ .", "Thus, we see that the quotient ring $R / \\mathfrak {P}_1$ is also the extension ring of $R^{\\prime } / \\mathfrak {P}_1$ that is generated by the coset $t_n + \\mathfrak {P}_1$ .", "From (REF ), we have $(t_n + \\mathfrak {P}_1)^d & + \\sum _{i=0}^{d-1} (c_i+ \\mathfrak {P}_1) (t_n + \\mathfrak {P}_1)^i \\\\& = \\left( t_n^d + \\sum _{i=0}^{d-1} c_i t_n^i \\right) + \\mathfrak {P}_1 \\\\& = \\tilde{g}(t_n) + \\mathfrak {P}_1 = g + \\mathfrak {P}_1= \\mathfrak {P}_1$ and so $t_n + \\mathfrak {P}_1$ is integral over $R^{\\prime } / \\mathfrak {P}_1$ .", "By Lemma REF , $R/ \\mathfrak {P}_1 = (R^{\\prime }/ \\mathfrak {P}_1) [ t_n + \\mathfrak {P}_1] $ is an integral extension of $R^{\\prime }/ \\mathfrak {P}_1$ .", "By Theorem REF , $\\lbrace \\mathfrak {P}_1 \\rbrace \\cap R^{\\prime }/ \\mathfrak {P}_1 \\subset \\left( \\mathfrak {P}_2/ \\mathfrak {P}_1 \\right) \\cap R^{\\prime }/ \\mathfrak {P}_1\\subset \\cdots \\subset \\left( \\mathfrak {P}_m/ \\mathfrak {P}_1 \\right) \\cap R^{\\prime }/ \\mathfrak {P}_1$ is a prime ideal chain of length $m-1$ in the ring $R^{\\prime }/ \\mathfrak {P}_1$ .", "The degree of $t_n$ in the polynomial $g$ is positive, and so the degree of $t_n$ in every nonzero polynomial in the principal ideal $ \\mathfrak {P}_1 = \\langle g \\rangle $ is positive.", "The degree of $t_n$ in every polynomial in $R^{\\prime } = \\mathbf {F}[t_1,\\ldots , t_{n-1}]$ is 0, and so $R^{\\prime } \\cap \\mathfrak {P}_1 = \\lbrace 0\\rbrace .$ This implies that the homomorphism $\\psi :R^{\\prime } \\rightarrow R/ \\mathfrak {P}_1$ defined by $\\psi (f) = f+ \\mathfrak {P}_1$ is one-to-one, and so $R^{\\prime } \\cong \\psi (R^{\\prime }) = R^{\\prime }/ \\mathfrak {P}_1.$ Applying the isomorphism $\\psi ^{-1}: R^{\\prime }/ \\mathfrak {P}_1 \\rightarrow R^{\\prime }$ to the maximal prime ideal chain (REF ) in $R^{\\prime }/ \\mathfrak {P}_1$ gives a prime ideal chain of length $m-1$ in $R^{\\prime }$ .", "The induction hypothesis implies that $m-1 \\le n-1$ .", "This completes the proof." ] ]
1612.05670
[ [ "Maximal lower bounds in the L\\\"owner order" ], [ "Abstract We show that the set of maximal lower bounds of two symmetric matrices with respect to the L\\\"owner order can be identified to the quotient set $O(p,q)/(O(p)\\times O(q))$.", "Here, $(p,q)$ denotes the inertia of the difference of the two matrices, $O(p)$ is the $p$-th orthogonal group, and $O(p,q)$ is the indefinite orthogonal group arising from a quadratic form with inertia $(p,q)$.", "We also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices.", "We exhibit a correspondence between the maximal lower bounds $C$ of two matrices $A,B$ and certain pairs of subspaces, describing the directions on which the quadratic form associated with $C$ is tangent to the one associated with $A$ or $B$.", "The present results refines a theorem from Kadison that characterizes the existence of the infimum of two symmetric matrices and a theorem from Moreland, Gudder and Ando on the existence of the positive semidefinite infimum of two positive semidefinite matrices." ], [ "Introduction", "The Löwner partial order is a basic notion in matrix theory [14], [6].", "It describes the pointwise ordering of real quadratic forms.", "These forms constitute an ordered vector space that is not a lattice, meaning that two quadratic forms may have two uncomparable minimal upper bounds, or dually, two uncomparable maximal lower bounds.", "A classical result by Kadison shows that it is an antilattice, meaning that greatest lower bounds exist only in trivial cases: Theorem 1.1 (Kadison, see [10]) Two symmetric matrices cannot have a greatest lower bound in the Löwner order unless they are comparable in this order.", "We refer to the work of Kalauch, Lemmens, and van Gaans [11] for a recent approach to Kadison's theorem and generalizations in the setting of Riesz spaces.", "Lower bounds of symmetric matrices have also been extensively studied in the setting of quantum observables [2], [15], [8], where the main motivation is the uniqueness of a positive semidefinite maximal lower bound.", "Moreland and Gudder have solved this problem in [15].", "Their result has been generalized to any pair of positive semidefinite bounded self-adjoint operators by Ando [2].", "His proof involved the notion of generalized short, which in the finite dimensional case is defined for positive semidefinite matrices $X,Y$ by $[Y]X = \\max \\lbrace Z \\mid 0 \\preceq Z \\preceq X,\\; \\operatorname{Im}Z \\subseteq \\operatorname{Im}Y \\rbrace \\,.$ Their results show that the uniqueness of a positive semidefinite maximal lower bound is decided by the comparability of such generalized shorts: Theorem 1.2 (Moreland and Gudder, Ando, see [15], [2]) Two positive semidefinite matrices $A$ and $B$ cannot have a unique positive semidefinite maximal lower bound unless the generalized shorts $[A]B$ and $[B]A$ are comparable.", "The aforementionned theorems raise the issue of characterizing the whole set of maximal lower bounds of two symmetric matrices $A$ and $B$ .", "Our first main result (Theorem REF ) shows that this set can be identified to the quotient space $O(p,q)/(O(p)\\times O(q))$ where $(p,q)$ denote the inertia of $A-B$ , $O(p)$ denotes the $p$ -th orthogonal group, and $O(p,q)$ is the indefinite orthogonal group arising from a quadratic form with inertia $(p,q)$ .", "It follows that the set of maximal lower bounds is of dimension $pq$ .", "When $p+q=n$ , the dimension of the set of maximal lower bounds is $p(n-p)$ which coincides with the dimension of the Grassmannian $\\operatorname{Gr}(n,p)$ .", "This suggests to look for a parametrization of maximal lower bounds by $p$ -dimensionnal subspaces of $\\operatorname{\\mathbb {R}}^n$ .", "Our second main result (Theorem REF ) leads to such a parametrization.", "We study more generally the following problem: given subspaces $\\mathcal {U}$ and $\\mathcal {V}$ , parametrize the set of maximal lower bounds $C$ of two symmetric matrices $A$ and $B$ that satisfy $\\mathcal {U}\\subseteq \\operatorname{Ker}(B-C)$ and $\\mathcal {V}\\subseteq \\operatorname{Ker}(A-C)$ .", "We give geometric conditions for the existence of solutions in terms of $\\mathcal {U}, \\mathcal {V}$ and the indefinite quadratic form $A-B$ , and, if these conditions are met, a parametrization of the set of solutions showing that this set is of dimension $(p-\\dim \\mathcal {U})(q-\\dim \\mathcal {V})$ .", "These results have a geometric consequence that will be dealt with in Section .", "When specialized to positive definite forms, the Löwner order corresponds to the inclusion order of ellipsoids, up to a reversal.", "We deduce from Theorem REF that given an ellipsoid $\\mathcal {E}_C$ minimally enclosing two ellipsoids $\\mathcal {E}_A, \\mathcal {E}_B$ , the set of tangency points of $\\mathcal {E}_C$ with $\\mathcal {E}_A$ (resp.", "$\\mathcal {E}_B$ ) spans the kernel of $A-C$ (resp.", "$B-C$ ).", "Moreover, the sum of these kernels must span the whole space.", "Although the present results are stated for real quadratic forms, they carry over to hermitian forms, up to immediate changes.", "Let us finally point out some applied motivations of the present work.", "The question of selecting a minimal ellipsoid containing two given ellipoids, or a maximal ellipsoid contained in their intersection, appears in a number of applied fields, including optimization [5], control [13], reachability analysis of dynamical systems [12], program verification [1], information geometry and mathematical morphology [3], [7].", "In most of the applications dealt with there, an important issue is to select effectively a remarkable minimal upper bound or maximal lower bound.", "For instance, selections arising from minimum or maximal volume considerations (Löwner's ellipsoids [4]) are frequently used.", "We expect the present characterization of the set of all maximal lower bounds to lead to more flexibility in some of these applications." ], [ "Notation", "In the sequel, $\\mathcal {M}_{p,q}$ denotes the set of $p \\times q$ (real) matrices, $\\mathcal {S}_n$ denotes the set of $n \\times n$ symmetric matrices and $A^T$ denotes the transpose of a matrix $A$ .", "The kernel of $A$ is denoted by $\\operatorname{Ker}A$ , its range by $\\operatorname{Im}A$ and its rank by $\\operatorname{rk}A$ .", "We denote the orthogonal complement of a subspace $V$ with respect to the standard scalar product by $V^\\perp $ .", "The group of $n \\times n$ invertible matrices is denoted by $\\mathrm {GL}_n$ .", "If $A\\in \\mathcal {S}_n$ , the inertia of $A$ is the triple $(p,q,r)$ , where $p$ (resp.", "$q,r$ ) is the number of positive (resp.", "negative, zero) eigenvalues of $A$ , counted with multiplicities.", "Given two square matrices $A,B$ , the direct sum of those matrices, denoted $A \\oplus B$ , is the block diagonal matrix with blocks $A$ and $B$ : $A \\oplus B =\\begin{pmatrix}A & \\\\& B\\end{pmatrix} \\,.$ We denote by $J_{p,q,r}$ the canonical bilinear form of inertia $(p,q,r)$ on $\\operatorname{\\mathbb {R}}^{p+q+r}$ .", "It is defined by $J_{p,q,r} (x,y)= \\sum _{i=1}^{p} x_iy_i - \\sum _{i=p+1}^{p+q} x_iy_i \\,,$ and the corresponding matrix in the canonical basis of $\\operatorname{\\mathbb {R}}^n$ is $I_p \\oplus (-I_q) \\oplus 0_r$ , where $I_n$ (resp.", "$0_n$ ) denote the identity matrix (resp.", "zero matrix) of size $n \\times n$ .", "When $r = 0$ , we use the notation $J_{p,q}$ and we denote by $\\mathcal {O}(p,q)$ the associated (indefinite) orthogonal group of square matrices $S$ such that $S J_{p,q} S^T = J_{p,q}$ .", "When $q=r=0$ , $\\mathcal {O}(p,q)$ becomes the standard orthogonal group $\\mathcal {O}(p)$ .", "The set of symmetric matrices is endowed with the Löwner order $\\preceq $ , which is such that $A \\preceq B \\iff \\forall x \\in \\operatorname{\\mathbb {R}}^n,\\; x^T Ax \\le x^TBx \\hspace{5.0pt}.$ We write $A \\prec B$ when $x^T Ax < x^TBx$ holds for all $x\\ne 0$ .", "The set of positive semidefinite matrices, denoted $\\mathcal {S}_n^+$ , is the set of matrices $A$ such that $A \\succeq 0$ .", "We say that the matrix $M$ is positive definite (resp.", "negative definite) over a subspace $\\mathcal {V}$ if $x^TMx > 0$ (resp.", "$x^TMx < 0$ ) holds for all nonzero vectors $x$ in $\\mathcal {V}$ .", "Given a positive semidefinite matrix $M$ , the square root of the matrix $M$ is the unique positive semidefinite matrix, denoted $M^{1/2}$ , such that $M^{1/2}M^{1/2} = M$ ." ], [ "Statement of the main theorem", "Our first main result is a parametrization of the set of maximal lower bounds of two symmetric matrices with respect to the Löwner order.", "It implies that this set is of dimension $pq$ and that it can be identified with $\\mathcal {O}(p,q)/ \\big (\\mathcal {O}(p) \\times \\mathcal {O}(q) \\big )$ , the quotient set of the indefinite orthogonal group $\\mathcal {O}(p,q)$ by the maximal compact subgroup $\\mathcal {O}(p) \\times \\mathcal {O}(q)$ .", "Theorem 3.1 Let $A,B,C\\in \\mathcal {S}_n$ be such that $C \\preceq A$ and $C \\preceq B$ , and let $(p,q,r)$ denote the inertia of $A-B$ .", "The following statements are equivalent: C is a maximal lower bound of $A$ and $B$ $\\operatorname{Ker}( A-C ) + \\operatorname{Ker}( B-C ) = \\operatorname{\\mathbb {R}}^n$ $\\operatorname{rk}( A-C ) = p$ and $\\operatorname{rk}( B-C ) = q$ For all $P \\in \\mathrm {GL}_n$ revealing the inertia of $A-B$ , i.e.", "such that $A-B = P J_{p,q,r}P^T$ , there exists a unique $M \\in \\mathcal {M}_{p,q}$ such that: $C & = A - PS( I_p \\oplus 0_{q} \\oplus 0_{r} ) S P^T\\quad \\text{with}\\\\S & =\\begin{pmatrix}( I_p + MM^T ) ^ {1/2} & M \\\\M^T & ( I_q + M^TM ) ^ {1/2}\\end{pmatrix} \\oplus 0_r \\hspace{5.0pt}.$ Remark 3.2 Assertion (REF ) can also be rewritten in terms of the matrix $B$ : $C = B - PS (0_p \\oplus I_q \\oplus 0_r )S P^T$ Remark 3.3 A similar theorem holds for minimal upper bounds, in which case (REF ) is unchanged, while (REF ) and (REF ) read : $\\operatorname{rk}( A-C ) = q$ and $\\operatorname{rk}(B-C ) = p$ , $C = A + PS(0_p \\oplus I_q \\oplus 0_r )SP^T=B +PS(I_p \\oplus 0_{q} \\oplus 0_{r} )S P^T$ Before proving Theorem REF , we draw two corollaries.", "Theorem REF , Corollary REF and Corollary REF are proved in Section REF .", "Corollary 3.4 Let $A,B \\in \\mathcal {S}_n$ , and let $(p,q,r)$ denote the inertia of $A-B$ .", "Then, the set of maximal lower bounds of $A$ and $B$ is homeomorphic to the quotient set ${\\raisebox {.2em}{\\mathcal {O}(p,q)}\\left\\bad.\\raisebox {-.2em}{\\big (\\mathcal {O}(p) \\times \\mathcal {O}(q)\\big )}\\right.}", "\\cong \\operatorname{\\mathbb {R}}^{pq} \\hspace{5.0pt}.$ Corollary 3.5 Let $A,B \\in \\mathcal {S}_n^+$ , and let $(p^{\\prime },q^{\\prime },r^{\\prime })$ denote the inertia of $B[A] - A[B]$ .", "The rank of a positive semidefinite maximal lower bound of $A,B$ cannot exceed $p^{\\prime }+q^{\\prime }+\\dim \\operatorname{Ker}(A-B)$ .", "Moreover, the set of positive semidefinite maximal lower bounds of $A$ and $B$ which have this rank is homeomorphic to the quotient set ${\\raisebox {.2em}{\\mathcal {O}(p^{\\prime },q^{\\prime })}\\left\\bad.\\raisebox {-.2em}{\\big (\\mathcal {O}(p^{\\prime }) \\times \\mathcal {O}(q^{\\prime })\\big )}\\right.}", "\\cong \\operatorname{\\mathbb {R}}^{p^{\\prime }q^{\\prime }} \\hspace{5.0pt}.$ We note that Kadison's result can be recovered as a special case of Corollary REF .", "Indeed, the existence of greatest lower bound of two matrices $A,B$ is equivalent to the existence of a unique maximal lower bound of these matrices, which, by Corollary REF , cannot happen unless $pq=0$ , meaning that $A\\preceq B$ or $B\\preceq A$ .", "The result from Moreland and Gudder can be recovered from Corollary REF in the same way.", "If two positive semidefinite matrices $A,B$ have a unique positive semidefinite maximal lower bound $C$ , then the uniqueness implies that $p^{\\prime }q^{\\prime } = 0$ , which means that $[B]A \\preceq [A]B$ or $[A]B \\preceq [B]A$ ." ], [ "Preliminary lemmas", "We present two results which will be useful in the proof of Theorem REF .", "Lemma 3.6 Let $P,Q \\in \\mathcal {S}_n$ .", "We have $\\operatorname{Ker}P + \\operatorname{Ker}Q = \\operatorname{\\mathbb {R}}^n \\Rightarrow \\operatorname{Ker}P \\cap \\operatorname{Ker}Q = \\operatorname{Ker}( P - Q )$ The inclusion $\\operatorname{Ker}P \\cap \\operatorname{Ker}Q \\subseteq \\operatorname{Ker}( P - Q )$ is trivial.", "Let $x \\in \\operatorname{Ker}( P-Q )$ and assume $x \\notin \\operatorname{Ker}P$ .", "Then $Px = Qx \\ne 0$ , and so $\\operatorname{Im}P \\cap \\operatorname{Im}Q \\ne \\lbrace 0 \\rbrace $ .", "Taking the orthogonal complement contradicts $\\operatorname{Ker}P + \\operatorname{Ker}Q = \\operatorname{\\mathbb {R}}^n$ .", "Lemma 3.7 (Polar decomposition of $\\mathcal {O}(p,q)$ , see [9]) For every $S \\in \\mathcal {O}(p,q)$ , there exists a unique triple $(M,U,V) \\in \\mathcal {M}_{p,q} \\times \\mathcal {O}(p) \\times \\mathcal {O}(q)$ such that: $S =\\begin{pmatrix}\\big ( I_p + MM^T \\big )^{1/2} & M \\\\M^T & \\big ( I_q + M^TM \\big )^{1/2}\\end{pmatrix}\\begin{pmatrix}U & \\\\& V\\end{pmatrix}\\hspace{5.0pt}.$" ], [ "Proof of Theorem ", "We now prove Theorem REF .", "We shall prove $(i) \\iff (ii) \\iff (iii) \\;\\;\\text{and}\\;\\; (ii) \\iff (iv) \\,.$" ], [ "$\\lnot (i) \\Rightarrow \\lnot (ii)$", "If $C$ is not a maximal lower bound of $A$ and $B$ , then there is some $C^* \\in \\mathcal {S}_n$ such that : $C \\precneqq C^* \\,, \\quad C^* \\preceq A \\,, \\quad C^* \\preceq B \\,.$ Let $x\\in \\operatorname{Ker}(A-C)$ .", "We have $0 \\preceq A-C^* \\preceq A-C$ , thus $x^T(A-C^*)x = 0$ and by nonnegativity, $Ax = C^*x$ .", "By assumption, $Ax = Cx$ , so that $x \\in \\operatorname{Ker}(C-C^*)$ .", "This shows that $\\operatorname{Ker}(A-C) \\subseteq \\operatorname{Ker}(C-C^*)$ .", "By symmetry, this is also true when changing $A$ into $B$ .", "As a consequence of this inclusion and the assumption $C \\ne C^*$ , we have: $\\operatorname{Ker}(A-C) + \\operatorname{Ker}(B-C) \\subseteq \\operatorname{Ker}(C-C^*) \\ne \\operatorname{\\mathbb {R}}^n \\,.$ If the sum of kernels is not equal to $\\operatorname{\\mathbb {R}}^n$ , then the set $\\operatorname{Ker}(A-C)^\\perp \\cap \\operatorname{Ker}(B-C)^\\perp $ is not $\\lbrace 0 \\rbrace $ .", "Let $u$ be a unit vector in $\\operatorname{Ker}(A-C)^\\perp \\cap \\operatorname{Ker}(B-C)^\\perp $ .", "Let $z \\in \\operatorname{\\mathbb {R}}^n$ .", "We write $z = x+y$ , with $x \\in \\operatorname{Ker}A-C$ and $y \\in \\operatorname{Ker}( A-C )^\\perp $ .", "Then we have, for $\\epsilon > 0$ , $z^T (A-C - \\epsilon uu^T)z = y^T(A-C-\\epsilon uu^T ) y\\,.$ The quadratic map $y \\mapsto y^T(A-C)y$ is positive definite on $\\operatorname{Ker}(A-C)^\\perp $ as the matrix $A-C$ is positive semidefinite, so the quadratic map $z \\mapsto z^T(A-C-\\epsilon uu^T)z$ is nonnegative over $\\operatorname{\\mathbb {R}}^n$ for $\\epsilon $ small enough.", "By symmetry, this is also true when changing $A$ into $B$ , so that for $\\epsilon $ small enough, we have $ A \\succeq C + \\epsilon uu^T\\,, B \\succeq C + \\epsilon uu^T \\,,$ and thus $C$ is not a maximal lower bound.", "We know from Lemma REF that $\\operatorname{Ker}(A-B) = \\operatorname{Ker}(A-C) \\cap \\operatorname{Ker}(B-C)$ .", "We choose $K_A$ to be a direct summand of $\\operatorname{Ker}(A-B)$ in $\\operatorname{Ker}(A-C)$ .", "Similarly, we choose $K_B$ a direct summand of $\\operatorname{Ker}(A-B)$ in $\\operatorname{Ker}(B-C)$ .", "Recall that $(p,q,r)$ denotes the inertia of $A-B$ .", "We have, $x^T(A-B)x = x^T(A-C)x > 0$ for all nonzero $x \\in K_B$ , so from the definition of the inertia, $\\dim K_B \\le p$ and by symmetry, $\\dim K_A \\le q$ .", "Those inequalities are equalities since $n = \\dim \\operatorname{\\mathbb {R}}^n = \\dim K_A + \\dim K_B + \\dim \\operatorname{Ker}( A-B ) \\le q + p + r = n \\,.$ We conclude using the rank-nullity theorem: since $\\operatorname{rk}(A-C) = n - \\dim \\operatorname{Ker}(A-C)$ , we have $\\operatorname{rk}(A-C) = n - q - r = p$ .", "Similarly, we have $\\operatorname{rk}(B-C) = q$ .", "We always have $\\operatorname{Ker}( A-C ) \\cap \\operatorname{Ker}( B-C ) \\subseteq \\operatorname{Ker}(A - B )$ , and so $\\dim \\big ( \\operatorname{Ker}( A-C ) + \\operatorname{Ker}( B-C ) \\big )& = 2n-p-q \\\\& - \\dim \\big ( \\operatorname{Ker}( A-C ) \\cap \\operatorname{Ker}( B-C ) \\big ) \\\\& \\ge 2n-p-q-(n-r) \\\\& = n \\,.$ Without loss of generality, we may assume that $P = I_n$ , so that $A-B = J_{p,q,r}$ .", "As before, we can write $\\operatorname{\\mathbb {R}}^n = K_A \\oplus K_B \\oplus \\operatorname{Ker}( A- B ) $ with $\\operatorname{Ker}(A-C) = K_A \\oplus \\operatorname{Ker}( A-B ) \\,, \\;\\operatorname{Ker}(B-C) = K_B \\oplus \\operatorname{Ker}( A-B ) \\,.$ We build a basis of $\\operatorname{\\mathbb {R}}^n$ respecting this subspace decomposition: we take a basis $\\mathcal {B}_A$ of $K_A$ , a basis $\\mathcal {B}_B$ of $K_B$ and $\\mathcal {B}_{A-B}$ of $\\operatorname{Ker}(A-B)$ , and our basis of $\\operatorname{\\mathbb {R}}^n$ is $\\big [ \\mathcal {B}_B \\,;\\, \\mathcal {B}_A \\,;\\, \\mathcal {B}_{A-B} \\big ]$ .", "In this basis, the matrices of the quadratic forms $A-C$ and $B-C$ are block-diagonal matrices: $A-C =(M_p \\oplus 0_q \\oplus 0_r)\\;\\;\\text{and}\\;\\;B-C = (0_p \\oplus M_q \\oplus 0_r )\\,,$ where the off-diagonal blocks are zero, because the matrices $M_p$ and $M_q$ (respectively of size $p\\times p$ and $q\\times q$ ) are positive definite.", "The matrix $\\Sigma = M_p^{1/2} \\oplus M_q^{1/2}$ is in the indefinite orthogonal group $\\mathcal {O}(p,q)$ , since $\\Sigma J_{p,q} \\Sigma ^T = J_{p,q}$ .", "By Lemma REF , there is a unique tuple $(M,U,V) \\in \\mathcal {M}_{p,q} \\times \\mathcal {O}(p) \\times \\mathcal {O}(q)$ such that : $\\Sigma =\\begin{pmatrix}( I_p + MM^T ) ^ {1/2} & M \\\\M^T & ( I_q + M^TM ) ^ {1/2}\\end{pmatrix}\\begin{pmatrix}U & \\\\& V\\end{pmatrix} \\,.$ The matrix $M$ does not depend on the choice of the matrix $\\Sigma $ : it is easily shown that all matrices $\\Xi $ such that $\\Xi (\\Xi ^T) =M_p \\oplus M_q$ only differ from $\\Sigma $ by a block-diagonal orthogonal-block matrix : $\\Xi = \\Sigma ( U^{\\prime } \\oplus V^{\\prime } )\\;,\\;U^{\\prime } \\in \\mathcal {O}(p), \\; V^{\\prime } \\in \\mathcal {O}(q) \\,,$ and any block-diagonal orthogonal-block matrix of this form multiplied on the right vanishes when computing $C$ .", "Indeed, if we denote $S =\\begin{pmatrix}( I_p + MM^T ) ^ {1/2} & M \\\\M^T & ( I_q + M^TM ) ^ {1/2}\\end{pmatrix}\\oplus 0_r\\; \\text{and} \\;W =U \\oplus V \\oplus 0_r\\,,$ we have $C = A - (SW)(I_p \\oplus 0_{q} \\oplus 0_r)(SW)^T= A - S(I_p \\oplus 0_{q} \\oplus 0_r)S\\,.$ Without loss of generality, we may again assume that $P = I_n$ .", "After change of basis with the invertible matrix $Q :=S^{-1} +( 0_{p} \\oplus 0_q \\oplus I_r )$ , we have $QAQ^T =(I_p \\oplus 0_q \\oplus 0_{r} )$ and $QBQ^T =( 0_p \\oplus I_q \\oplus 0_r )$ .", "The sum of the kernels of those matrices is $\\operatorname{\\mathbb {R}}^n$ , thus this is also the case for $A$ and $B$ .", "This concludes the proof of Theorem REF .", "$\\Box $ We have shown in the proof $(ii) \\Rightarrow (iv)$ of Theorem REF that we can associate to every matrix $\\Sigma \\in \\mathcal {O}(p,q)$ a maximal lower bound $C$ of $A$ and $B$ , via the continuous map $\\Phi $ from $\\mathcal {O}(p,q)$ to $\\mathcal {S}_n$ defined by: $\\Phi : \\Sigma \\mapsto A - P\\begin{pmatrix}\\Sigma & \\\\ & 0_r\\end{pmatrix}\\begin{pmatrix} I_p \\oplus 0_q & \\\\ & 0_{r} \\end{pmatrix}\\begin{pmatrix}\\Sigma & \\\\ & 0_r\\end{pmatrix}^TP^T \\,.$ Moreover, we have previously shown that two matrices $\\Sigma _1, \\Sigma _2 \\in \\mathcal {O}(p,q)$ produce the same maximal lower bound $C$ if and only if $\\Sigma _1 = \\Sigma _2(U \\oplus V)$ for some matrices $U \\in \\mathcal {O}(p), V\\in \\mathcal {O}(q)$ .", "This proves that the map $\\Phi $ is a bijection from $\\mathcal {O}(p,q)/ \\big (\\mathcal {O}(p) \\times \\mathcal {O}(q)\\big )$ to the set of maximal lower bounds of $A,B$ .", "By Lemma REF , the quotient set can be identified to $\\mathcal {M}_{p,q} \\cong \\operatorname{\\mathbb {R}}^{pq}$ by means of the continuous bijection $S$ defined by: $S : M \\mapsto \\begin{pmatrix}( I_p + MM^T ) ^ {1/2} & M \\\\M^T & ( I_q + M^TM ) ^ {1/2}\\end{pmatrix} \\,.$ It remains to show that the map $\\Phi \\circ S$ has a continuous inverse.", "We write $\\Phi \\circ S(M) = A - P\\bigg [\\begin{pmatrix}A(M) & B(M) \\\\B(M)^T & M^TM \\\\\\end{pmatrix}\\oplus 0_r \\bigg ]P^T \\,,$ with $A(M) := I + MM^T$ and $B(M) := (I+MM^T)^{1/2}M$ .", "The matrix $M$ can be recovered continuously with $M = A(M)^{-1/2} B(M)$ , since $A(M) \\succeq I$ cannot vanish.", "Before treating the general case, we shall prove the corollary when $A,B$ are positive definite.", "Note that, in this case, we have $[A]B = B$ and $[B]A = A$ .", "First, the inertias of the matrices $A-B$ and $B^{-1} - A^{-1}$ are the same: the matrices $A,B$ can be reduced simultaneously by an invertible congruence $X \\mapsto PXP^T$ to diagonal matrices with positive diagonal elements $a_i$ and $b_i$ .", "The fact that $a_i- b_i> 0$ is equivalent to $b_i^{-1} - a_i^{-1} >0$ shows that the inertias are identical.", "Also, note that $p^{\\prime }+q^{\\prime }+\\dim \\operatorname{Ker}(A-B) = n$ , so matrices that have this rank are invertible.", "Since the map $X \\mapsto X^{-1}$ is monotonically decreasing on the set of positive definite matrices, it is a (continuous) bijection between the set of minimal upper bounds of $A^{-1},B^{-1}$ (which are positive definite definite) and the set of positive definite maximal lower bounds of $A,B$ .", "By Corollary REF , the former set is homeomorphic to $\\mathcal {O}(p^{\\prime },q^{\\prime }) / \\big (\\mathcal {O}(p^{\\prime }) \\times \\mathcal {O}(q^{\\prime })\\big ) \\cong \\operatorname{\\mathbb {R}}^{p^{\\prime }q^{\\prime }}$ , so the same is true for the latter set.", "Now let $A,B$ denote positive semidefinite matrices.", "We may assume that $\\operatorname{Ker}(A-B) = \\lbrace 0\\rbrace $ , since it does not influence the structure of the set of maximal lower bounds of $A,B$ by Theorem REF , so that $\\operatorname{Ker}A \\cap \\operatorname{Ker}B = \\lbrace 0\\rbrace $ .", "Let $R_{A,B}$ denote the set $\\operatorname{Im}A \\cap \\operatorname{Im}B$ .", "We claim that there are direct summands $R_A$ and $R_B$ of $R_{A,B}$ in $\\operatorname{Im}A$ and $\\operatorname{Im}B$ respectively so that the matrices of the quadratic forms $A,B$ are block-diagonal in $\\operatorname{\\mathbb {R}}^n = R_A \\oplus R_{A,B} \\oplus R_B$ .", "Indeed, we have $\\operatorname{\\mathbb {R}}^n = \\operatorname{Ker}B \\oplus R_{A,B} \\oplus \\operatorname{Ker}A$ .", "In such a decomposition, the quadratic forms $A,B$ have matrices of the form $A = \\begin{pmatrix}A_{11} & A_{12} \\\\A_{12}^T & A_{22}\\end{pmatrix} \\oplus 0_b\\quad \\text{and}\\quad B = 0_a \\oplus \\begin{pmatrix}B_{22} & B_{23} \\\\B_{23}^T & B_{33}\\end{pmatrix} \\,.$ We define the matrix $U$ mapping $w = (x,y,z) \\in \\operatorname{Ker}B \\oplus R_{A,B} \\oplus \\operatorname{Ker}A$ to $Uw = (x -A_{11}^{-1} A_{12}y,y,z - B_{33}^{-1} B_{23}^Ty)$ .", "One can easily check that the subspaces $R_A$ and $R_B$ defined as the image of $\\operatorname{Ker}B$ and $\\operatorname{Ker}A$ respectively by $U$ satisfy the desired condition.", "Moreover, up to a transformation $X \\mapsto V^TXV$ with $V$ block-diagonal, we may assume that $A = I_a \\oplus S_A \\oplus 0_b$ and $B = 0_a \\oplus S_B \\oplus I_b$ , where $S_A,S_B$ denote positive definite matrices such that $S_A - S_B = J_{p^{\\prime },q^{\\prime }}$ .", "Note that the short $[B]A$ (resp.", "$[A]B$ ) is given by $0_a \\oplus S_A \\oplus 0_b$ (resp.", "$0_a \\oplus S_B \\oplus 0_b$ ).", "Let $C$ denote a positive semidefinite maximal lower bound of $A,B$ .", "Using the characterization in Theorem REF , $C$ is given in block form by $C =\\begin{pmatrix}-XX^T - YY^T & * & * \\\\* &* &* \\\\*& * &- Y^TY - W^TW\\end{pmatrix}$ with $M =\\begin{psmallmatrix}X & Y \\\\Z & W\\end{psmallmatrix}\\in \\mathcal {M}_{a+p^{\\prime },b+q^{\\prime }}$ .", "The fact that $C$ is positive semidefinite implies that $X,Y,W$ are zero matrices, so that $C = 0_a \\oplus S_C \\oplus 0_b$ , where the matrix $S_C$ is given by $S_C = S_A -\\begin{pmatrix}I_{p^{\\prime }} + ZZ^T & (I_{p^{\\prime }}+ZZ^T)^{1/2}Z \\\\Z^T(I_{p^{\\prime }}+ZZ^T)^{1/2} & Z^TZ\\end{pmatrix} \\,.$ By Theorem REF , $S_C$ is a (positive semidefinite) maximal lower bound of the matrices $S_A$ and $S_B$ .", "This concludes the proof since $S_A,S_B$ are positive definite." ], [ "Notation and preliminary lemma", "We first give some notation that will be useful in the sequel.", "We define the linear operators $\\pi _p$ , $\\pi _q$ and $\\pi _r$ , mapping respectively $\\operatorname{\\mathbb {R}}^n$ to $\\operatorname{\\mathbb {R}}^p$ , $\\operatorname{\\mathbb {R}}^q$ and $\\operatorname{\\mathbb {R}}^r$ , that select the first $p$ coordinates, the following $q$ and the last $r$ coordinates.", "Their matrices in the canonical basis of $\\operatorname{\\mathbb {R}}^n$ are $\\pi _p =\\begin{pmatrix}I_p & 0_{pq} & 0_{pr}\\end{pmatrix},\\qquad \\pi _q =\\begin{pmatrix}0_{qp} & I_q & 0_{qr}\\end{pmatrix},\\qquad \\pi _r =\\begin{pmatrix}0_{rp} & 0_{rq} & I_r\\end{pmatrix}\\,.$ We denote by $\\Vert \\cdot \\Vert $ the spectral norm (largest singular value) of a matrix.", "We define $\\mathcal {B}_{p,q}$ to be the open unit ball of $\\mathcal {M}_{p,q}$ with respect to this norm: $\\mathcal {B}_{p,q} := \\big \\lbrace X \\in \\mathcal {M}_{p,q} \\mid \\Vert X\\Vert < 1 \\big \\rbrace \\,.$ Lemma 4.1 The map $\\phi _{p,q}$ from $\\mathcal {M}_{p,q}$ to $\\mathcal {B}_{p,q}$ defined by : $\\phi _{p,q}(X)= \\big ( I_q + XX^T \\big ) ^{-1/2} X$ is a bijection, with inverse $\\psi _{p,q}(Y)=\\big ( I_q - YY^T \\big ) ^{-1/2} Y \\hspace{5.0pt}.$ Moreover, $\\phi _{p,q}(X) = X \\big (I_p + X^TX\\big ) ^{-1/2}\\quad \\text{and}\\quad \\big (\\phi _{p,q}(X)\\big )^T = \\phi _{q,p}(X^T)\\hspace{5.0pt}.$ Let $X=UDV^T$ denote the singular value decomposition of $X$ , so that $U,V$ are orthogonal matrices, and $D$ is a matrix consisting of a diagonal block and a zero block.", "Then, $\\phi _{p,q}(X)= U\\phi _{p,q}(D)V^T$ , and a similar property holds for the map $\\psi _{p,q}$ .", "Therefore, it suffices to check that $\\psi _{p,q} \\circ \\phi _{p,q} (X)= X$ when $X=D$ , which is straighforward.", "By symmetry, we obtain that $\\phi _{p,q} \\circ \\psi _{p,q} (Y)=Y$ holds for all $Y$ .", "The other properties are proved similarly." ], [ "Statement of the problem and the theorem", "As stated in Theorem REF , the kernels $\\operatorname{Ker}(A-C)$ and $\\operatorname{Ker}(B-C)$ are central to the characterization of maximal lower bounds.", "In the following, we investigate the problem of the selection of a maximal lower bound of two symmetric matrices where subspaces of those kernels have been predetermined.", "When $x^TCx > 0$ , the line $\\operatorname{\\mathbb {R}}x$ meets the surface $\\lbrace x \\mid x^TCx = 1 \\rbrace $ at two opposite points.", "Moreover, if $x \\in \\operatorname{Ker}(A-C)$ , the surfaces $\\lbrace x \\mid x^TCx = 1 \\rbrace $ and $\\lbrace x \\mid x^TAx = 1 \\rbrace $ are tangent at those points.", "When $x^TCx \\le 0$ , it may be interpreted as a tangency at $\\infty $ .", "For this reason, constraints on the kernels are called tangency constraints.", "Finally, following Theorem REF , the dimension of the kernel of $A-B$ does not influence the structure of the set of maximal lower bound of $A$ and $B$ .", "Thus, we assume that a reduction has been done and state Problem REF and Theorem REF accordingly.", "Problem 4.2 (Maximal lower bounds with tangency constraints) Let $A,B \\in \\mathcal {S}_n$ and let $(p,q,0)$ denote the inertia of $A-B$ .", "Let $\\mathcal {U},\\mathcal {V}$ be subspaces of $\\operatorname{\\mathbb {R}}^n$ .", "We wish to find in $C \\in \\mathcal {S}_n$ : ${\\left\\lbrace \\begin{array}{ll}C\\;\\text{is a maximal lower bound of}\\; A,B \\\\\\forall u \\in \\mathcal {U}, \\; Cu = Bu \\\\\\forall v \\in \\mathcal {V}, \\; Cv = Av \\\\\\end{array}\\right.", "}$ Our second main result gives conditions for Problem REF to have a solution and, if these conditions are met, a parametrization of the set of solutions.", "It shows that the set of solutions is of dimension $(p-\\dim \\mathcal {U})(q-\\dim \\mathcal {V})$ , so that the problem has a unique solution if and only if one of the subspaces has maximal dimension.", "Theorem 4.3 Problem REF has a solution if and only if $A-B$ is positive definite over $\\mathcal {U}$ $A-B$ is negative definite over $\\mathcal {V}$ $\\mathcal {U}$ and $\\mathcal {V}$ are orthogonal with respect to the indefinite scalar product $A-B$ If these conditions are met, then the set of solutions can be parametrized as in (REF ) of Theorem REF , with $M \\in \\phi _{p,q}^{-1} ( \\mathcal {B}_{p,q} \\cap \\mathcal {W} )$ where $\\mathcal {W}$ is the affine subspace of $\\mathcal {M}_{p,q}$ defined by $R \\in \\mathcal {W}\\iff {\\left\\lbrace \\begin{array}{ll}\\forall u \\in \\mathcal {U},\\, R^T\\pi _p(u) + \\pi _q(u) = 0 \\\\\\forall v \\in \\mathcal {V},\\, R\\pi _q(v) + \\pi _p(v) = 0\\end{array}\\right.}", "\\,.$ As soon as the conditions above are met, the intersection $\\mathcal {B}_{p,q} \\cap \\mathcal {W}$ is nonempty.", "The subspace $\\mathcal {W}$ has dimension $ (p-\\dim \\mathcal {U})(q-\\dim \\mathcal {V})$ , so that the solution is unique if and only if $\\dim \\mathcal {U}= p \\;\\text{or}\\; \\dim \\mathcal {V}= q \\,.$ Remark 4.4 When $\\mathcal {U}$ and $\\mathcal {V}$ have maximal dimension, $\\mathcal {V}$ is the orthogonal complement of $\\mathcal {U}$ with respect the indefinite form $A-B$ .", "Thus Theorem REF establishes a bijective correspondance between maximal lower bounds of $A,B$ and $p$ -dimensionnal subspaces over which the matrix $A-B$ is positive definite.", "In this way, the set of maximal lower bounds is parametrized by an open semi-algebraic subset of the Grassmannian $\\operatorname{Gr}(n,p)$ ." ], [ "Preliminary lemmas", "Before proving Theorem REF , we prove two useful results.", "First, Lemma REF shows that when the matrix $J_{p,q}$ is negative definite over a subspace $\\mathcal {V}$ , then there is a contractive mapping from the last $q$ coordinates of any vector of $\\mathcal {V}$ to its first $p$ coordinates.", "Lemma 4.5 Let $\\mathcal {V}$ be a subspace of $\\operatorname{\\mathbb {R}}^n$ over which $J_{p,q}$ is negative definite, with $p+q = n$ .", "There is a matrix $R \\in \\mathcal {M}_{p,q}$ with ${R} < 1$ such that : $\\forall x \\in \\mathcal {V}, \\; \\pi _p(x) = R \\pi _q(x) \\,.$ First, we show that the map $R_c$ from $\\operatorname{\\mathbb {R}}^q$ to $\\mathcal {V}$ defined by $R_c : z \\mapsto v \\quad \\text{s.t.", "}\\quad v \\in \\mathcal {V}\\quad \\text{and}\\quad \\pi _q(x) = v$ is well defined.", "Let $v,w \\in \\mathcal {V}$ such that $\\pi _q(v) = \\pi _q(w)$ .", "We have $v-w \\in \\mathcal {V}$ , thus $(v-w)^TJ_{p,q}(v-w) \\le 0$ .", "This is rewritten as $\\Vert \\pi _p(v) - \\pi _p(w) \\Vert _2 \\le \\Vert \\pi _q(v) - \\pi _q(w) \\Vert _2 = 0$ , hence $\\pi _p(v) = \\pi _q(w)$ , which implies $v=w$ .", "The map $R_c$ is linear as its inverse map is the restriction of the linear map $\\pi _q$ to $\\mathcal {V}$ .", "Then, we define the linear map $R$ from $\\operatorname{\\mathbb {R}}^q$ to $\\operatorname{\\mathbb {R}}^p$ by $R(x) := \\pi _p \\circ R_c (x) $ on $\\pi _q(\\mathcal {V})$ and we choose $R$ to be zero on ${\\pi _q(\\mathcal {V})}^\\perp $ .", "By definition of $R_c$ and $R$ , we have for all $x \\in \\operatorname{\\mathbb {R}}^n$ , $R \\pi _q(x) = \\pi _p(x)$ .", "The matrix $J_{p,q}$ is negative definite over $\\mathcal {V}$ , meaning that $\\Vert \\pi _p(x) \\Vert _2 < \\Vert \\pi _q(x) \\Vert _2$ when $x \\in \\mathcal {V}$ is nonzero.", "This implies that the map $R$ is a contraction on $\\pi _q(\\mathcal {V})$ : $\\Vert R \\pi _q(x) \\Vert _2 = \\Vert \\pi _p(x) \\Vert _2 < \\Vert \\pi _q(x) \\Vert _2 \\,.$ As $R$ is zero on $\\pi _q(\\mathcal {V})^\\perp $ , the map $R$ is a contraction on $\\operatorname{\\mathbb {R}}^q$ : ${R} < 1$ .", "Then, we solve Problem REF in the easiest case, when the subspaces are $\\mathcal {U}= \\lbrace 0\\rbrace $ and $\\mathcal {V}= \\operatorname{\\mathbb {R}}x$ , for $x \\in \\operatorname{\\mathbb {R}}^n$ .", "Since the proposition does not change if $r \\ne 0$ , we give its statement in the most general case.", "Proposition 4.6 Let $A,B \\in \\mathcal {S}_n$ and $x \\in \\operatorname{\\mathbb {R}}^n$ .", "Then, there exists a maximal lower bound $C$ of $A$ and $B$ such that $Ax = Cx$ if and only if $x^TAx < x^TBx$ or $Ax = Bx$ .", "Without loss of generality, we may assume that $A-B = J_{p,q,r}$ .", "($\\Rightarrow $ ): Assume that $C$ is a maximal lower bound of $A$ and $B$ such that $Ax = Cx$ .", "The constraint $Ax = Cx$ implies that $x^TAx = x^TCx \\le x^TBx$ holds.", "We shall thus show that if $x^TAx = x^TBx$ , then $Ax = Bx$ .", "Using Theorem REF , there is $M \\in \\mathcal {M}_{p,q}$ such that $C = A -\\begin{pmatrix}I+MM^T & (I+MM^T)^{1/2}M \\\\M^T(I+MM^T)^{1/2} & M^TM\\end{pmatrix}\\oplus 0_r\\,.$ The condition $Ax = Cx$ is rewritten as $\\phi _{p,q}(M)\\pi _q(x) = -\\pi _p(x) \\,.$ The function $\\phi _{p,q}$ maps the matrix $M$ to an element in the open ball $\\mathcal {B}_{p,q}$ , so ${\\phi _{p,q}(M)} < 1$ .", "It follows that $\\Vert \\pi _p(x)\\Vert _2 = \\Vert \\phi _{p,q}(M)\\pi _q(x) \\Vert _2 < \\Vert \\pi _q(x) \\Vert _2$ if $\\pi _q(x) \\ne 0$ .", "However, the assumption $x^T(A-B)x = 0$ implies $\\Vert \\pi _p(x) \\Vert _2 = \\Vert \\pi _q(x) \\Vert _2$ , thus $\\pi _q(x) = 0$ and $\\pi _p(x)= 0$ .", "We conclude with $(A-B)x = \\pi _p(x) - \\pi _q(x) = 0$ .", "($$ ): If $Ax = Bx$ , then, by Theorem REF , every maximal lower bound satisfies $Ax = Cx$ .", "If $x^T(A-B)x < 0$ , then $\\Vert \\pi _p(x) \\Vert _2 < \\Vert \\pi _q(x) \\Vert _2$ .", "It is easily seen that using $M = \\phi _{p,q}^{-1} \\Bigg ( \\frac{-\\pi _p(x)\\pi _q(x)^T}{{\\Vert \\pi _q(x)\\Vert _2}^2} \\Bigg )$ in the characterization in Theorem REF provides a solution satisfying $Ax = Cx$ ." ], [ "Feasibility ($\\Rightarrow $ )", "Given a solution $C$ to Problem REF , we have for $u \\in \\mathcal {U}$ , $u^T(A-B)u = u^T(A-C)u - u^T(B-C)u$ , where the first term is nonnegative and the second is zero.", "Hence $A-B$ is nonnegative over $\\mathcal {U}$ .", "For $v \\in \\mathcal {V}$ , the reverse holds and $A-B$ is nonpositive over $\\mathcal {V}$ .", "Moreover, if we have $x^T(A-B)x = 0$ for some $x \\in \\mathcal {U}\\cup \\mathcal {V}$ , then by Proposition REF , we have $(A-B)x = 0$ and $x=0$ as $A-B \\in GL_n$ .", "This shows that $A-B$ is positive definite over $\\mathcal {U}$ and negative definite over $\\mathcal {V}$ .", "Finally, for $u \\in \\mathcal {U}$ and $v \\in \\mathcal {V}$ , as $\\mathcal {U}\\subseteq \\operatorname{Ker}B-C$ and $\\mathcal {V}\\subseteq \\operatorname{Ker}A-C$ , we have $u^T(A-B)v = u^T(A-C)v - u^T(B-C)v = 0$ , so $\\mathcal {U}$ and $\\mathcal {V}$ are orthogonal with respect to $A-B$ .", "We will use the characterization (REF ) in Theorem REF to build a solution to Problem REF .", "Without loss of generality, we may assume that we work in a basis of $\\operatorname{\\mathbb {R}}^n$ revealing the inertia of $A-B = J_{p,q}$ .", "Furthermore, we may assume that $\\dim \\mathcal {U}= p$ and $\\dim \\mathcal {V}= q$ .", "If this is not the case, let $\\mathcal {U}_0$ denote a subspace of $\\big [ (A-B) \\cdot \\mathcal {U}\\big ]^\\perp $ over which $A-B$ is positive definite that has maximal dimension.", "Then let $\\mathcal {V}_0$ denote a subspace of $\\big [ (A-B) \\cdot \\mathcal {V}\\big ]^\\perp \\cap \\big [ (A-B) \\cdot (\\mathcal {U}\\oplus \\mathcal {U}_0)\\big ]^\\perp $ over which $A-B$ is negative definite that has maximal dimension.", "The subspaces $\\mathcal {U}\\oplus \\mathcal {U}_0$ and $\\mathcal {V}\\oplus \\mathcal {V}_0$ then satisfy the assumptions.", "We will prove that there is a matrix $R$ of size $p \\times q$ satisfying ${R} < 1$ and $u \\in \\mathcal {U}\\iff \\pi _q(u) = -R^T\\pi _p(u) \\,,\\quad v \\in \\mathcal {V}\\iff \\pi _p(v) = -R\\pi _q(v)\\,.$ The proof is done in two steps.", "First, we build a matrix $R$ satisfying the second and third equivalences using Lemma REF .", "The matrix $J_{p,q}$ is negative definite over $\\mathcal {V}$ , so that for all nonzero $x\\in \\mathcal {V}$ , we have ${\\pi _q(x)}_2^2 > {\\pi _p(x)}_2^2 \\ge 0$ , so $\\pi _q(x) \\ne 0$ and $\\pi _q(\\mathcal {V}) = \\operatorname{\\mathbb {R}}^q$ .", "Then, we use the orthogonality condition to show that the first equivalence holds.", "Let $u \\in \\mathcal {U}$ and $v \\in \\mathcal {V}$ .", "We have $\\pi _q(v)^T\\big (R^T\\pi _p(u) - \\pi _q(u)\\big ) & = \\pi _p(u)^T\\pi _p(v) - \\pi _q(u)^T\\pi _q(v) \\\\& = u^TJ_{p,q}v \\\\& = 0 \\,.$ Hence $R^T\\pi _p(u) - \\pi _q(u) \\in \\operatorname{\\mathbb {R}}^q$ is orthogonal to $\\pi _q(\\mathcal {V}) = \\operatorname{\\mathbb {R}}^q$ , and is thus zero.", "It now suffices to take $M = \\phi _{p,q}^{-1}(R)$ to build a solution to Problem REF using (REF ) in Theorem REF .", "Let $C$ be a solution of Problem REF .", "According to Theorem REF , we can associate with $C$ a unique $M \\in \\mathcal {M}_{p,q}$ .", "Given vectors $u \\in \\mathcal {U}$ and $v \\in \\mathcal {V}$ , the constraints $Av = Cv$ and $Bu = Cu$ can be rewritten as $\\phi _{p,q}(M)^T\\pi _p(u) = -\\pi _q(u) \\;\\;\\text{and}\\;\\; \\phi _{p,q}(M)\\pi _q(v) = -\\pi _p(v) \\,.$ Moreover, we have ${\\phi _{p,q}(M)} < 1$ , so that $\\phi _{p,q}(M) \\in \\mathcal {W} \\cap \\mathcal {B}_{p,q}$ .", "Conversely, one checks easily that any solution $R$ of (REF ) provides a solution, as long as $R \\in \\mathcal {B}_{p,q}$ .", "We have shown previously that as soon as the problem is feasible, the set $\\mathcal {W} \\cap \\mathcal {B}_{p,q}$ is nonempty.", "Let $R \\in \\mathcal {W} \\cap \\mathcal {B}_{p,q}$ .", "If $\\dim \\mathcal {U}\\ne p$ and $\\dim \\mathcal {V}\\ne q$ , since $\\dim \\pi _p(\\mathcal {U}) = \\dim \\mathcal {U}$ and $\\dim \\pi _q(\\mathcal {V}) = \\dim \\mathcal {V}$ , we can choose nonzero vectors $u_p \\in {\\pi _p(\\mathcal {U})}^\\perp $ and $v_q \\in {\\pi _q(\\mathcal {V})}^\\perp $ .", "The ball $\\mathcal {B}_{p,q}$ is an open set, thus for small enough positive $\\epsilon $ , the matrix $R^{\\prime } := R + \\epsilon u_p{v_q}^T$ is also in $\\mathcal {B}_{p,q}$ and satisfies the equations (REF ).", "The matrix $R^{\\prime }$ produces a different solution than $R$ since $R \\ne R^{\\prime }$ and $\\phi _{p,q}$ is a bijection, so that $\\dim \\mathcal {W} \\ge \\dim {\\pi _p(\\mathcal {U})}^\\perp \\times \\dim {\\pi _p(\\mathcal {V})}^\\perp = (p-\\dim \\mathcal {U})(q-\\dim \\mathcal {V})$ .", "If $R,R^{\\prime } \\in \\mathcal {W}$ are solutions of (REF ), then we have $\\forall u \\in \\mathcal {U},\\, (R-R^{\\prime })\\pi _q(u) = 0 \\qquad \\forall v \\in \\mathcal {V},\\, \\pi _p(v)^T(R-R^{\\prime }) = 0 \\,$ which yields the reverse inequality $\\dim \\mathcal {W} \\le \\dim {\\pi _p(\\mathcal {U})}^\\perp \\times \\dim {\\pi _p(\\mathcal {V})}^\\perp $ .", "$\\Box $" ], [ "Examples", "We recall the definition of ellipsoids, the equivalence between the inclusion of ellipsoids and the Löwner order and the algebraic counterpart of tangency between ellipsoids.", "Definition 5.1 We denote by $\\mathcal {Q}_A$ the quadric associated with the symmetric matrix $A$ , defined by: $\\mathcal {Q}_A = \\lbrace x \\in \\operatorname{\\mathbb {R}}^n \\mid x^TAx \\le 1 \\rbrace \\,.$ The set $\\mathcal {Q}_A$ is convex if and only if the matrix $A$ is positive semidefinite.", "Then, we call the set $\\mathcal {Q}_A$ an ellipsoid, and it will also be denoted by $\\mathcal {E}_A$ .", "The set $\\mathcal {E}_A$ is bounded if and only if the matrix $A$ is positive definite.", "Moreover, it always has a nonempty interior.", "The inclusion of the ellipsoid $\\mathcal {E}_A$ in the quadric $\\mathcal {Q}_B$ is equivalent to the positivity of the matrix $A-B$ , meaning that the inclusion of ellipsoids in quadrics and the ordering of the corresponding matrices is equivalent, up to reversal: $\\mathcal {E}_A \\subseteq \\mathcal {Q}_B \\iff B \\preceq A \\,.$ This also means that, given positive definite matrices $A,B$ , the quadric $\\mathcal {Q}_C$ associated with a maximal lower bound $C$ of $A$ and $B$ in the Löwner order is a minimal upper bound for the ellipsoids $\\mathcal {E}_A$ and $\\mathcal {E}_B$ , in the inclusion order.", "Remark 5.2 In the general case, $\\mathcal {Q}_A \\subseteq \\mathcal {Q}_B \\Rightarrow B \\preceq A \\,,$ as shown with $A = 2 \\oplus (-2)$ and $B = 1 \\oplus (-1)$ .", "For $(x,y)\\in \\mathcal {Q}_A$ , one clearly has $2x^2-2y^2 \\le 1 \\le 2$ , which implies $(x,y) \\in \\mathcal {Q}_B$ .", "However, we have $A-B = 1 \\oplus (-1) \\lnot \\succeq 0$ ." ], [ "In dimension 2: $\\mathcal {O}(1,1)/\\big ( \\mathcal {O}(1) \\times \\mathcal {O}(1) \\big )$", "This case arises whenever two symmetric matrices $A$ and $B$ of order 2 are not comparable.", "The maps $( X \\mapsto X+\\lambda I_n)_{\\lambda \\in \\operatorname{\\mathbb {R}}}$ and $( X \\mapsto U^TXU )_{U \\in \\mathrm {GL}_n}$ are all order-preserving isomorphisms.", "This implies that, given such an isomorphism $\\phi $ , the set of maximal lower bounds of $\\phi (A)$ and $\\phi (B)$ is exactly the image of the set of maximal lower bounds of $A$ and $B$ by the map $\\phi $ .", "Thus one can easily show that we may assume without loss of generality that $A = \\begin{pmatrix}2 & 0 \\\\ 0 & 1\\end{pmatrix}\\qquad B = \\begin{pmatrix}1 & 0 \\\\ 0 & 2\\end{pmatrix} \\,.$ We have the explicit description of the set of hyperbolic isometries $\\mathcal {O}(1,1)$ : $\\mathcal {O}(1,1) = \\Bigg \\lbrace \\begin{pmatrix}\\epsilon _1 \\operatorname{ch}\\theta & \\epsilon _2 \\operatorname{sh}\\theta \\\\\\epsilon _1 \\operatorname{sh}\\theta & \\epsilon _2 \\operatorname{ch}\\theta \\end{pmatrix}\\mid \\theta \\in \\operatorname{\\mathbb {R}},\\, \\epsilon _1,\\epsilon _2 \\in \\lbrace -1,1\\rbrace \\Bigg \\rbrace \\,.$ The quotient set $\\mathcal {O}(1,1)/\\big ( \\mathcal {O}(1) \\times \\mathcal {O}(1) \\big )$ is in this case equal to the classical set of hyperbolic rotations: $\\mathcal {O}(1,1)/\\big ( \\mathcal {O}(1) \\times \\mathcal {O}(1) \\big )= \\Bigg \\lbrace \\begin{pmatrix}\\operatorname{ch}\\theta & \\operatorname{sh}\\theta \\\\\\operatorname{sh}\\theta & \\operatorname{ch}\\theta \\end{pmatrix}\\mid \\theta \\in \\operatorname{\\mathbb {R}}\\Bigg \\rbrace \\,.$ Note that in this special case, the quotient set has a group structure.", "This gives us the parametrization of the minimal upper bounds $C_\\theta $ of $A$ and $B$ : $C_\\theta = \\begin{pmatrix}2 - \\operatorname{ch}^2 \\theta & \\operatorname{ch}\\theta \\operatorname{sh}\\theta \\\\\\operatorname{ch}\\theta \\operatorname{sh}\\theta & 2 - \\operatorname{ch}^2 \\theta \\end{pmatrix} \\,.$ Moreover, the tangency subspaces to $\\mathcal {E}_A$ and $\\mathcal {E}_B$ are respectively equal to $\\operatorname{\\mathbb {R}}\\begin{pmatrix} \\operatorname{sh}\\theta & -\\operatorname{ch}\\theta \\end{pmatrix}^T$ and $\\operatorname{\\mathbb {R}}\\begin{pmatrix} \\operatorname{ch}\\theta & -\\operatorname{sh}\\theta \\end{pmatrix}^T$ .", "This is depicted in Figure REF .", "Figure: Minimal quadrics 𝒬 θ \\mathcal {Q}_\\theta (in red) associated with ℰ A \\mathcal {E}_A and ℰ B \\mathcal {E}_B (in blue) for various values of θ\\theta ." ], [ "The quotient Lorentz set: $\\mathcal {O}(n,1)/\\big ( \\mathcal {O}(n) \\times \\mathcal {O}(1) \\big ), \\; n\\ge 2$ ", "Following Lemma REF , the set $\\mathcal {O}(n,1)/\\big ( \\mathcal {O}(n) \\times \\mathcal {O}(1) \\big )$ can be identified to $\\operatorname{\\mathbb {R}}^n$ via the bijection $\\phi := \\phi _{n,1}$ defined by $\\phi : {w}\\mapsto \\begin{pmatrix}(I_n + {w}{w}^T)^{1/2} & {w} \\\\{w}^T & \\sqrt{1+{w}^T{w}}\\end{pmatrix} \\,.$ In this case, when $pq = n > 1$ , the quotient set does not have a group structure.", "Let $(e_i)_{1\\le i\\le n}$ denote the canonical base of $\\operatorname{\\mathbb {R}}^n$ .", "The product $M := \\phi (e_1)\\phi (e_2)$ can be computed explicitly and it is not even symmetric: we have $M_{2,1} = 0$ whereas $M_{1,2} = 1$ .", "We shall illustrate the results of Theorem REF on an example with $p = 2$ and $q = 1$ , with the matrices $A = 2 \\oplus 2 \\oplus 1$ and $B = 1 \\oplus 1 \\oplus 2$ , so that $A-B = J_{2,1}$ .", "Theorem REF states that the set of maximal lower bounds of $A$ and $B$ , denoted $\\mathcal {I}_{A,B}$ , has dimension 2 and its elements $C_{w}$ are given, for ${w}\\in \\operatorname{\\mathbb {R}}^2$ by $C_{w} =A -\\begin{pmatrix}I_2 + {w}{w}^T & (I_2 + {w}{w}^T)^{1/2}{w} \\\\{w}^T(I_2 + {w}{w}^T)^{1/2} & {w}^T{w}\\end{pmatrix}\\,.$ For all ${w}\\in \\operatorname{\\mathbb {R}}^2$ , we also have $\\dim \\operatorname{Ker}(A-C_{w}) = 1$ and $\\dim \\operatorname{Ker}(B-C_{w}) = 2$ .", "Let $v = (x\\;0\\;z)^T$ denote some non-zero vector.", "We shall solve Problem REF in the cases where $(\\mathcal {U}, \\mathcal {V}) = (\\operatorname{\\mathbb {R}}v, \\lbrace 0\\rbrace )$ and $(\\mathcal {U}, \\mathcal {V}) = (\\lbrace 0\\rbrace , \\operatorname{\\mathbb {R}}v)$ ." ], [ "Case 1: $\\mathcal {U}= \\operatorname{\\mathbb {R}}v$ and {{formula:85b5478f-7176-4b56-a8b6-d45cb0647dd7}} .", "In this case, we have $p \\ne \\dim \\mathcal {U}$ and $q \\ne \\dim \\mathcal {V}$ , so by Theorem REF the set of solutions is not reduced to a point.", "The problem has a solution if and only if $x^2 > z^2$ , and the solutions are parametrized by the contractive elements of the affine subspace $\\mathcal {W}$ of $\\mathcal {M}_{2,1}$ defined by $ R \\in \\mathcal {W}$ if and only if $R^T (x\\,0)^T + z = 0$ .", "Denoting $r = -z/x$ , so that $|r| < 1$ , we have $\\mathcal {W}= \\big \\lbrace R_t := (r\\;t)^T \\mid t \\in \\operatorname{\\mathbb {R}}\\big \\rbrace \\,.$ Moreover, we have ${R_t}^2 = r^2+t^2$ so that, since $r^2 < 1$ , the set $\\mathcal {W}\\cap \\mathcal {B}_{p,q}$ is non-empty.", "Then, for $|t| < \\sqrt{1-r^2}$ , we recover the matrix ${w} = \\phi _{p,q}^{-1}(R_t) = (1-r^2-t^2)^{-1/2} \\begin{pmatrix}r \\\\ t\\end{pmatrix} \\,.$ Finally, we get the parametrization of the kernels: $\\operatorname{Ker}(A-C_{w}) & = \\operatorname{Span}\\Big \\lbrace \\begin{pmatrix}z & -tx & x\\end{pmatrix}^T \\Big \\rbrace \\,,\\\\\\operatorname{Ker}(B-C_{w}) & = \\operatorname{Span}\\Big \\lbrace \\begin{pmatrix}x & 0 & z\\end{pmatrix}^T\\;,\\;\\begin{pmatrix}txz & x^2+y^2 & -x^2t\\end{pmatrix}^T\\Big \\rbrace \\,.$ The set of solutions is parametrized by a single real parameter $t$ as expected from Theorem REF .", "In this case, we have $q = \\dim \\mathcal {V}$ , so the solution is unique.", "Indeed, the problem has a solution if and only if $x^2 < z^2$ and the affine subspace $\\mathcal {W}$ of $\\mathcal {M}_{2,1}$ is reduced to the point $R := (-x/z\\,,0)$ , which satisfies ${R} < 1$ .", "Figure REF depicts several minimal quadrics associated with the ellipsoids $\\mathcal {E}_A$ and $\\mathcal {E}_B$ .", "Figure: Minimal quadrics 𝒬 C w \\mathcal {Q}_{C_{w}} (in green) associated with ℰ A \\mathcal {E}_A and ℰ B \\mathcal {E}_B (in blue and red) for various values of w{w}.", "The black line shows the tangency points between the quadrics ℰ B \\mathcal {E}_B and 𝒬 C w \\mathcal {Q}_{C_{w}}.The author thanks Stéphane Gaubert for pointing out the problem of parametrization of maximal lower bounds.", "The author also thanks him with Xavier Allamigeon for many useful remarks and suggestions on previous versions of this work.", "Finally, the author thanks Peter Semrl for pointing out the link with Kadison's result." ] ]
1612.05664
[ [ "Bayesian Distributed Lag Interaction Models to Identify Perinatal\n Windows of Vulnerability in Children's Health" ], [ "Abstract Epidemiological research supports an association between maternal exposure to air pollution during pregnancy and adverse children's health outcomes.", "Advances in exposure assessment and statistics allow for estimation of both critical windows of vulnerability and exposure effect heterogeneity.", "Simultaneous estimation of windows of vulnerability and effect heterogeneity can be accomplished by fitting a distributed lag model (DLM) stratified by subgroup.", "However, this can provide an incomplete picture of how effects vary across subgroups because it does not allow for subgroups to have the same window but different within-window effects or to have different windows but the same within-window effect.", "Because the timing of some developmental processes are common across subpopulations of infants while for others the timing differs across subgroups, both scenarios are important to consider when evaluating health risks of prenatal exposures.", "We propose a new approach that partitions the DLM into a constrained functional predictor that estimates windows of vulnerability and a scalar effect representing the within-window effect directly.", "The proposed method allows for heterogeneity in only the window, only the within-window effect, or both.", "In a simulation study we show that a model assuming a shared component across groups results in lower bias and mean squared error for the estimated windows and effects when that component is in fact constant across groups.", "We apply the proposed method to estimate windows of vulnerability in the association between prenatal exposures to fine particulate matter and each of birth weight and asthma incidence, and estimate how these associations vary by sex and maternal obesity status, in a Boston-area prospective pre-birth cohort study." ], [ "Introduction", "A growing body of research supports an association between maternal exposure to air pollution during pregnancy and a variety of birth and children's health outcomes.", "Epidemiological studies have found that maternal exposure to ambient air pollution is associated with decreased birth weight as well as increased risk of preterm birth and respiratory disorders including asthma [22], [11], [20], [23], [18], [9], [5].", "The National Institute of Environmental Health Sciences (NIEHS) has identified both the estimation of windows of vulnerability and the identification of susceptible subpopulations as critical research directions in environmental health research [17].", "A critical methodological gap is the lack of available statistical methods to simultaneously identify windows of vulnerability and susceptible populations.", "Windows of vulnerability are time periods during which exposure to a toxin has an increased association with current or future health status [1], [27].", "Prenatal systems development is a multi-event process progressing sequentially from early gestation [10].", "The identification of windows of vulnerability, which in turn corresponds to sensitive stages of development, can inform our understanding of the underlying pathways through which an environmental exposure operates.", "Presumably, the window is defined by developmental specific events (i.e.", "gene expression changes, growth/cell density, vascularization etc.)", "that are transient and environmental exposure that is concurrent to these events.", "The distributed lag model (DLM) framework has a long history in air pollution research and was originally developed for time-series analysis where an outcome observed on a given day is jointly regressed on exposures over a previous time period [19], [30].", "Several recent studies have applied DLMs to estimate windows vulnerability during which air pollution has an elevated association with preterm birth [24], [3], decreased birth weight [25], childhood asthma [8], and disrupted neurodevelopment [6].", "A critical consideration in the estimation of windows of vulnerability is that the prenatal developmental process is not homogeneous across all subgroups of individuals.", "For example, females display earlier fetal breathing than males [2].", "Because developmental timing varies, it is natural to hypothesize that the association between prenatal exposure and health outcomes in a given subgroup may not only vary in the effect size but in the timing of the window of vulnerability.", "Therefore, when interest focuses on windows of vulnerability, there are at least four potential patterns of effect heterogeneity: 1) both the effect size and the timing of the window vary by subgroup; 2) only the effect size varies by subgroup; 3) only the timing of the window varies by subgroup; or 4) both the window and the effect size are the same for all subgroup.", "Existing methods only accommodate patterns (1) and (4), but not (2) and (3).", "A DLM can estimate a common window and effect size (pattern 4) or, when stratified by group, estimate lagged effects assuming that both the effect size and the exposure window vary by subgroup (pattern 1), [8], [6].", "The stratified DLM approach does not share information across groups relating to either the timing of the window or the effect size within the window.", "In the context of functional regression, [26] proposed a functional interaction model for gene-environment interactions with a time-varying environmental exposure.", "This approach shares information across groups (genotype) by assuming the same windows for each genotype but the effect is scaled by the number of major alleles (0, 1, or 2).", "Hence, this approach can estimate effects satisfying pattern 2 under the additional constraint that the effects are proportional to the number of major alleles.", "However, none of these approaches allow for the estimation across all four patterns of effect heterogeneity.", "In this paper we propose a new method that provides greater flexibility in characterizing effect heterogeneity when identifying windows of vulnerability is of interest.", "The proposed Bayesian distributed lag interaction model (BDLIM) partitions the distributed lag function from a standard DLM into a time component that identifies windows of vulnerability and a scale component that quantifies the magnitude of the effect within the window.", "The approach allows both the window and the scale components to either vary or stay constant across subgroups.", "As such, the BDLIM framework can estimate a model with any of the four effect heterogeneity patterns.", "To our knowledge, models that assume the window of vulnerability is the same across subgroups but the effect within the window varies across groups, and vice versa, have not previously been considered in the literature.", "The proposed approach allows the user to directly answer the question of whether effect heterogeneity manifests itself via changes in the window of vulnerability, the magnitude of the effect, or both, which in turn more directly answers the question of whether an environmental exposure affects the developmental process of an infant similarly across subgroups.", "In the BDLIM framework the time-varying weight function is treated as a functional predictor that is scaled by the scalar effect size.", "This partitioning requires that identifiability constraints be placed on the parameters of the weight function.", "Under certain assumptions about effect heterogeneity the model can be reparameterized to relax the identifiability constraints on the parameter space and reduces to a mixed effects model.", "In other cases, including the general linear model setting for discrete responses, we use a slice sampler to efficiently estimate the model from the constrained space.", "We make software available for BDLIM in the R package regimes (REGression In Multivariate Exposure Settings).", "We use BDLIM to estimate the association between fine particulate matter (PM$_{2.5}$ ) measured weekly over pregnancy and two outcomes–birth weight for gestational age (BWGA) $z$ -score and asthma incidence–in a prospective Boston-area pregnancy cohort.", "Following [13], we evaluate whether the association between PM$_{2.5}$ and BWGA $z$ -score varies by both sex and maternal obesity status.", "Following [8], we evaluate how effects on asthma incidence vary by sex." ], [ "The ACCESS Data", "We analyze data from the Asthma Coalition on Community, Environment, and Social Stress (ACCESS) project [28].", "ACCESS is a prospective, longitudinal study designed to investigate the effects of stress and other environmental factors, including air pollution, on asthma risk in a urban U.S. setting.", "The ACCESS cohort includes data on 997 mother-child pairs that were recruited between August 2002 and January 2007.", "The women were at least 18 years of age, spoke English or Spanish, and received prenatal care at one of two Boston, MA area hospitals or affiliated community health centers.", "To date, the ACCESS cohort has been used to study the relationship of air pollution exposures, maternal stress, and other risk factors with outcomes including asthma, wheeze, and birth weight [4], [8], [13].", "Like previous ACCESS studies, we limit the analysis to full-term ($\\ge $ 37 weeks), live births with complete exposure, outcome, and covariate data.", "For each child we consider BWGA $z$ -scores and maternal-reported and clinician-diagnosed asthma as outcomes.", "The data contain maternal and child covariate information including: maternal age at enrollment; race/ethnicity (black, hispanic, and white); maternal education (two categories less than high school and high school diploma or more); self-reported smoking during pregnancy; indicator of maternal pre-pregnancy obesity; infant sex; maternal atopy (ever self-reported doctor-diagnosed asthma, eczema, or hay fever); season of birth; a previously described maternal stress index [4]; and a previously described neighborhood disadvantage index [4].", "Maternal exposures of PM$_{2.5}$ were estimated with a hybrid land use regression model that incorporates satellite-derived aerosol optical depth measures [12].", "Each mother was assigned an average PM$_{2.5}$ exposure value for each week of pregnancy based on the predicted value at her address of residence.", "We limit our analysis to exposure during the first 37 weeks of pregnancy." ], [ "Approach", "Interest focuses on estimation of the association between time-varying exposure $X_i(t)$ , $t\\in \\mathcal {T}$ , and scalar outcome $Y_i$ while controlling for a vector of baseline covariates ${\\mathbf {Z}}_i$ .", "We denote individuals by $i=1,\\dots ,n$ .", "We parameterize the time-varying effect of exposure as $\\beta w(t)$ , where $w(t)$ is a continuous weight function that captures temporal variation in the association between $X(t)$ and $Y$ and $\\beta $ is the scalar effect size.", "The weight function $w(t)$ identifies windows of vulnerability in which the exposure effect is elevated relative to other time periods.", "To allow for heterogeneity among subgroups (e.g.", "infant sex) indexed by $j$ , we allow either or both of these quantities to vary across levels of $j$ , denoted $\\beta _j$ and $w_j(t)$ .", "Similar to a stratified DLM, this parameterization allows for group-level modification of both the window and effect size.", "In addition, this parameterization accommodates scenarios not yet considered whereby only the location of the window or only the magnitude of the effect, but not both, vary by group." ], [ "BDLIM for a single population", "In the BDLIM regression model for a single population with no effect heterogeneity we assume $E(Y_i)=\\mu _i$ and $g\\left(\\mu _i\\right)=\\alpha + \\beta \\int _{t\\in \\mathcal {T}} X_{i}(t)w(t)dt + {\\mathbf {Z}}_i^T{\\gamma },$ where $g(\\cdot )$ is a monotone link function, $\\alpha $ is the intercept, and ${\\gamma }$ is a vector of unknown regression coefficients for the covariates ${\\mathbf {Z}}$ .", "The model in (REF ) is similar to a functional linear model in which the total effect of exposure $X(t)$ on outcome $Y$ at time $t$ is $\\beta w(t)$ .", "We will refer to this model as BDLIM-n, where the “n\" indicates no heterogeneity between subgroups.", "For identifiability, we constrain the weight function such that $\\int _{t\\in \\mathcal {T}} w(t)^2 dt=1$ and $\\int _{t\\in \\mathcal {T}} w(t)dt\\ge 0$ .", "The weight function is allowed to be both positive and negative to account for exposures that are a toxicant during some time periods but a nutrient during others.", "The constraint $\\int _{t\\in \\mathcal {T}} w(t)dt\\ge 0$ assures that $\\beta $ reflects the direction of the cumulative effect, $\\beta \\int _{t\\in \\mathcal {T}} w(t)dt$ .", "The constraint $\\int _{t\\in \\mathcal {T}} w(t)^2 dt=1$ ensures that the magnitude of $\\beta $ (i.e.", "$|\\beta |$ ) is identifiable.", "To gain some insight into the BDLIM approach consider the case where $w(t)=1$ .", "In this case $\\int _{t\\in \\mathcal {T}} X_{i}(t)w(t)dt=\\bar{X}_i$ and (REF ) becomes a linear model with scalar exposure covariate equal to the mean exposure over the full pregnancy.", "However, once the weight function $w(t)$ varies with $t$ , the weighted exposure $\\int _{t\\in \\mathcal {T}} X_{i}(t)w(t)dt$ gives greater relative weight to some time windows.", "These up-weighted times are considered the windows of vulnerability." ], [ "BDLIM model for effect heterogeneity", "A key advantage of the BDLIM framework is the ability to estimate the three hypothesized patterns of heterogeneity where either $\\beta $ , $w(t)$ , or both $\\beta $ and $w(t)$ vary by group.", "When either $\\beta $ or $w(t)$ is constant across groups, BDLIM yields a more parsimonious model that results in more powerful tests of an interaction.", "Consider analysis of data for groups $j=1,\\dots ,J$ .", "When both the effect size and the window of vulnerability are group-specific, the BDLIM model is $g\\left(\\mu _i\\right)= \\alpha _{j_i} + \\beta _{j_i}\\int _{t\\in \\mathcal {T}} X_{i}(t)w_{j_i}(t)dt + {\\mathbf {Z}}_{i}^T{\\gamma },$ where the subscript $j_i$ denotes group $j$ to which individual $i$ belongs.", "We refer to this model as BDLIM-bw where “bw” indicates that both $\\beta $ and $w(t)$ vary across groups.", "The BDLIM framework extends to the new scenarios where only $\\beta $ or $w(t)$ varies by group.", "If it is hypothesized that groups share a common window, e.g.", "first trimester, but the groups are differentially susceptible within that window, the model is $g\\left(\\mu _i\\right)= \\alpha _{j_i} + \\beta _{j_i}\\int _{t\\in \\mathcal {T}} X_i(t)w(t)dt + {\\mathbf {Z}}_{i}^T{\\gamma },$ which we denote BDLIM-b.", "Alternatively, if the effect is the same across groups but the windows are different, perhaps shifted by a few weeks, then the model, denoted BDLIM-w, is $g\\left(\\mu _i\\right)= \\alpha _{j_i} + \\beta \\int _{t\\in \\mathcal {T}} X_{i}(t)w_{j_i}(t)dt + {\\mathbf {Z}}_i^T{\\gamma }.$ Both BDLIM-b and BDLIM-w estimate patterns of effect heterogeneity not previously addressed." ], [ "Parameterization of the functional components", "We assume a truncated basis function representation of both $X_i(t)$ and $w(t)$ .", "Common choices for the basis expansions include splines, wavelets, Fourier series, and principal components (PCs).", "We use the first $K$ PCs of the covariance matrix of $X(t)$ as the basis to represent both $X(t)$ and $w(t)$ for all groups.", "Here, $K$ is chosen to be the number of PCs that explain a prespecified proportion of the total variability in the exposures, for example 99% of the total variation.", "Hence, $X_i(t)=\\sum _{k=1}^{K}\\xi _{ik}\\psi _k(t)$ and $w(t)=\\sum _{k=1}^{K}\\theta _k\\psi _k(t)$ .", "In practice, we observe the exposures measured over a discrete grid, in our case $t=1,\\dots ,37$ weeks of pregnancy.", "Let ${\\mathbf {X}}$ be a $n\\times T$ matrix with row $i$ being the observed exposures for individual $i$ measured at times $t=1,\\dots ,T$ , ${\\mathbf {X}}_i=(X_{i1},\\dots ,X_{iT})$ .", "We use the covariance matrix of ${\\mathbf {X}}$ to estimate the basis functions $\\lbrace \\psi _k(t)\\rbrace _{k=1}^K$ .", "It is reasonable to expect that $w(t)$ is moderately smooth.", "However, the raw PCs of the covariance matrix of ${\\mathbf {X}}$ are potentially rough.", "To obtain a smooth orthonormal basis, we use fast covariance estimation (FACE) proposed by [29] to obtain the eigenfunctions of a smoothed covariance matrix, as implemented in the R package refund [7].", "There are several potential alternative approaches.", "In Appendix C of the Supplementary Materials we consider pre-smoothing the exposures and using the PCs of the covariance of the smoothed data.", "[14] discussed several methods for regularizing functional predictors that are sampled sparsely or on irregular grids.", "The orthonormal PC basis facilitates implementation of the constraints on $w(\\cdot )$ .", "When $w(t)=\\sum _{k=1}^{K}\\theta _k\\psi _k(t)$ , the constraint $\\int _{t\\in \\mathcal {T}} w(t)^2dt=1$ is satisfied if and only if $\\Vert {\\theta }\\Vert =1$ , where ${\\theta }=(\\theta _1,\\dots ,\\theta _k)^T$ .", "Additionally, $\\int _{t\\in \\mathcal {T}} w(t)dt\\ge 0$ holds for a set of observed times if and only if $\\mathbf {1}^T{\\Psi }{\\theta }\\ge 0$ where $\\mathbf {1}$ is a $T$ -vector of ones and ${\\Psi }$ is a $T\\times K$ matrix with row $t$ taking the values $[\\psi _1(t),\\dots ,\\psi _K(t)]$ .", "The resulting constrained parameter space on ${\\theta }$ is, therefore, defined by the surface of a unit $K$ -hemiball on one side of the hyperplane defined by $\\mathbf {1}^T{\\Psi }{\\theta }=0$ .", "In some cases, this constraint can be alleviated as discussed in Section REF .", "Using the PC representation for both $X(t)$ and $w(t)$ the model in (REF ) is $g(\\mu _i)=\\alpha +\\beta \\left({\\Psi }\\xi _i\\right)^T{\\Psi }{\\theta }+ {\\mathbf {Z}}_i^T{\\gamma },$ where ${\\theta }=\\left(\\theta _1,\\dots ,\\theta _T\\right)^T$ and $\\xi _i=(\\xi _{i1},\\dots ,\\xi _{iK})^T$ .", "Because ${\\Psi }$ is orthonormal, $ \\left({\\Psi }\\xi _i\\right)^T{\\Psi }{\\theta }=\\xi _i^T{\\theta }$ .", "The model in (REF ) is easily adapted for effect heterogeneity with group specific $\\beta _j$ and ${\\theta }_j$ ." ], [ "Reparameterization to remove constraints in the linear model ", "For the normal linear model, BDLIM-n and BDLIM-bw can be reparameterized to reduce the computational burden imposed by the constrained parameter space.", "For BDLIM-n, the unknown parameters ${\\theta }$ for the weight functions $w(t)$ are constrained to $\\Vert {\\theta }\\Vert =1$ and $\\mathbf {1}^T{\\Psi }{\\theta }\\ge 0$ ; however, $\\beta $ is unconstrained and ${\\theta }^*=\\beta {\\theta }$ is also unconstrained in $\\mathbb {R}^{K}$ .", "Importantly, $\\beta =\\Vert {\\theta }^*\\Vert \\times \\text{sign}(\\mathbf {1}^T{\\Psi }{\\theta }^*)$ , ${\\theta }={\\theta }^*\\Vert {\\theta }^*\\Vert ^{-1}$ , and $\\widehat{\\mathbf {w}}={\\Psi }\\widehat{{\\theta }}$ are each uniquely identified from ${\\theta }^*$ .", "This generalizes to the BDLIM-bw with group specific ${\\theta }^*_j$ .", "Hence, we reparameterize BDLIM-n and BDLIM-bw in terms of ${\\theta }^*$ and estimate the model with standard Markov chain Monte Carlo (MCMC) methods without any constraints on the parameters.", "Then the posterior sample of ${\\theta }^*$ can be deconvoluted into the posterior distributions of $\\beta $ and ${\\theta }$ by partitioning each MCMC draw.", "This approach is not applicable for the BDLIM-b and BDLIM-w (see Appendix A of the Supplementary Materials) and we sample directly from the constrained parameter space as described in Section REF ." ], [ "Prior specification and computation ", "We complete the model by assigning prior distributions to the unknown regression parameters.", "The prior for ${\\theta }$ is uniform over the surface of the unit $K$ -hemiball for ${\\theta }$ .", "The prior likelihood can be represented as proportional to a constrained mutlivariate standard normal, $\\pi ({\\theta })\\propto \\exp (-{\\theta }^T{\\theta }/2){1}\\lbrace \\Vert {\\theta }\\Vert =1\\rbrace {1}\\lbrace \\mathbf {1}^T{\\Psi }{\\theta }\\ge 0\\rbrace $ , where ${1}\\lbrace \\cdot \\rbrace $ is an indicator function.", "We assign normal priors to $\\beta $ and $\\gamma $ .", "When reparameterized, $\\beta $ becomes a scale parameter for ${\\theta }^*$ .", "Specifically, we assume that $\\beta \\sim \\text{N}(0,\\tau ^2)$ and let $\\kappa =\\beta ^2\\tau ^{-2}$ .", "Then ${\\theta }^*\\sim \\text{N}(0,\\kappa \\tau ^2{\\mathbf {I}})$ , where $\\tau $ is fixed and $\\kappa \\sim \\chi ^2_1$ .", "The resulting model has closed form full conditionals: ${\\theta }^*$ can be sampled from a multivariate normal and $\\kappa $ from a generalized inverse-Gaussian distribution with density function $f(\\kappa ;\\lambda ,\\chi ,\\psi )\\propto \\kappa ^{\\lambda -1} \\exp \\lbrace -(\\chi /\\kappa +\\psi \\kappa )/2\\rbrace $ , where $\\lambda =-(K-1)/2$ , $\\psi =1$ , and $\\chi =\\tau ^{-2}{\\theta }^{*T}{\\theta }^*$ .", "Finally, we assume a flat prior on the intercept $\\alpha $ and, for the linear model with residuals $\\epsilon _i\\sim \\text{N}(0,\\sigma ^2)$ , a gamma prior on the precision parameter $\\sigma ^{-2}$ .", "We use MCMC to simulate the posterior of the unknown parameters.", "For the BDLIM-n and BDLIM-bw in the linear model, all unknown parameters have simple conjugate forms and can be sampled via Gibbs sampler.", "For BDLIM-b, BDLIM-w, and models with a non-linear link function, we propose a slice sampling approach [16] based on the elliptical slice sampler proposed by [15] (see Appendix B of the Supplementary Materials for details)." ], [ "Summarizing the posterior of $w(t)$ ", "Summarizing the posterior distribution of $w(t)$ deserves special consideration in light of the constraint $\\int _{t\\in \\mathcal {T}} w(t)^2dt=1$ .", "It is typical to summarize the posterior distribution with the posterior mean.", "However, the posterior mean of ${\\theta }$ almost surely does not satisfy $\\Vert \\bar{{\\theta }}\\Vert =1$ and $\\int _{t\\in \\mathcal {T}} w(t)^2 dt= 1$ .", "To obtain a point estimate for $w(t)$ such that $\\int _{t\\in \\mathcal {T}} w(t)^2 dt=1$ and $\\int _{t\\in \\mathcal {T}} w(t) dt \\ge 0$ , we take the posterior mean of ${\\theta }$ in the topology of a $K$ -hemiball parameter space.", "We do this by taking the Bayes estimate with respect to the loss function $L({\\theta },\\widehat{{\\theta }}) = [({\\theta }-\\widehat{{\\theta }})^T({\\theta }-\\widehat{{\\theta }})]/{1}\\lbrace \\Vert \\widehat{{\\theta }}\\Vert =1\\rbrace $ .", "The resulting estimate $\\widehat{{\\theta }}$ is the posterior mean projected onto the $K$ -hemiball.", "That is, $\\widehat{{\\theta }}=\\bar{{\\theta }}\\Vert \\bar{{\\theta }}\\Vert ^{-1}$ where $\\bar{{\\theta }}$ is the posterior mean.", "Since each draw from the posterior satisfies $\\mathbf {1}^T{\\Psi }{\\theta }\\ge 0$ , it follows that $\\mathbf {1}^T{\\Psi }\\widehat{{\\theta }}\\ge 0$ .", "We identify windows of vulnerability as time periods where the pointwise 95% posterior intervals of $\\widehat{w}(t)$ do not contain 0.", "A drawback to this point estimate is that in the absence of an effect ($\\beta =0$ ) or when ${\\theta }$ is not well identified by the data, the posterior of ${\\theta }$ will reflect the prior.", "In this case the projected estimator can be erratic; however, the posterior interval will still reflect the uncertainty in $\\widehat{w}(t)$ .", "The effect $\\beta $ is interpretable even when a window is not identified.", "Because $\\int _{t\\in \\mathcal {T}} w(t)dt\\ge 0$ , the effect $\\beta $ and the cumulative effect $\\beta \\int _{t\\in \\mathcal {T}} w(t)dt$ share the same significance level, i.e.", "$\\Pr (\\beta >0 | D) = \\Pr (\\beta \\int _{t\\in \\mathcal {T}} w(t)dt >0 | D)$ .", "Hence, the $\\alpha $ -level posterior interval for the cumulative effect will not contain 0 if and only if the $\\alpha $ -level posterior interval for $\\beta $ does not contain 0 and, regardless of whether a window is identified, we can conclude that there is an overall effect." ], [ "Comparing models and identifying the pattern of heterogeneity", "We quantify the evidence in the data supporting each of the four potential patterns of effect heterogeneity with the mean log posterior predictive density (MLPPD).", "Specifically, for model $k$ (where $k=1,2,3,4$ indicates the four BDLIM variants) $\\text{MLPPD}_k = S^{-1}\\sum _{s=1}^S \\log \\Pr \\left(\\mathbf {Y}|\\zeta _k^{(s)}\\right)$ , where $\\zeta _k$ is the vector of all parameters for model $k$ and $s=1,\\dots ,S$ enumerates the draws from the simulated posterior distribution.", "We compute $\\widehat{\\text{MLPPD}}_k$ for each of the four effect heterogeneity models, then normalize as $\\widehat{P}_k= \\frac{\\exp \\left(\\widehat{\\text{MLPPD}}_k \\right)}{\\sum _{l=1}^4 \\exp \\left(\\widehat{\\text{MLPPD}}_l \\right)},$ In the simulation study presented in Section REF , we compare the performance of computed using the normalized MLPPD to identify the correct model to that of the deviance information criterion [21].", "Note that DIC is $-2\\text{MLPPD}+p_D$ , where $p_D$ is an additional penalty for model size.", "We show that normalized MLPPD is less likely to identify a misspecified pattern of effect heterogeneity." ], [ "Simulation overview", "We tested the performance of BDLIM with two simulation studies.", "Simulation A compares the BDLIM-n to a DLM when interest focuses on estimation of the effect of a time-varying exposure in a single group.", "Results suggest that the two methods perform similarly.", "We have relegated most of the details for this simulation to Appendix C of the Supplementary Materials.", "We also compared tuning choices for BDLIM, including using different numbers of PCs and using natural splines to pre-smooth the exposures instead of smoothing the covariance matrix of ${\\mathbf {X}}$ with FACE, in Appendix C of the Supplementary Materials.", "Simulation B highlights the advantage of BDLIM for subgroup analyses and effect heterogeneity estimation.", "For both simulations we used the observed exposures and covariates from the birth weight analysis of the ACCESS data.", "Using observed weekly air pollution levels during pregnancies ensures that the exposure data have realistic temporal trends and autocorrelations.", "Hence, the data contain $n=506$ individuals with exposures measured at 37 evenly spaced time points, ten binary covariates, three continuous covariates, and an intercept.", "For the second simulation we divided the data into two groups, 239 girls ($j=0$ ) and 267 boys ($j=1$ ).", "For each scenario we simulated 1000 datasets and analyzed them with BDLIM and DLM.", "For BDLIM we used 15 knots to estimate the covariance matrix.", "We used N$(0,10^2)$ priors on $\\beta $ and $\\gamma $ and used the first $K$ PCs that explain 99% of the variation in exposure.", "We assumed Gaussian errors and put a gamma$(0.01,0.01)$ prior on $\\sigma ^{-2}$ .", "For DLM we used natural cubic splines with flat priors on the regression coefficients.", "We fit the DLM with degrees of freedom ranging from three to ten and have presented results only from the best performing model." ], [ "Simulation A: Comparison to DLM with no effect heterogeneity", "This simulation compares BDLIM-n to DLM when there is no effect heterogeneity.", "Both models are correctly specified but use different parameterizations and basis functions.", "We used three scenarios each with data simulated from a different weight function: $w^1(t) &=& \\sqrt{\\frac{t^4(1-t)^{4}}{\\mathcal {B}(5,5)}} \\\\w^2(t) &=& \\frac{\\sin (t\\pi - \\pi /4)}{\\sqrt{(T^{-1}\\sum _{k\\in 1}^{T}\\sin (t_k\\pi - \\pi /4)^2)}}\\\\w^3(t) &=& 1,$ where $\\mathcal {B}(\\cdot ,\\cdot )$ is a beta function and $t$ is scaled to the unit interval.", "The superscripts identify the weight functions and correspond to the three scenarios in simulation A.", "The intercept and regression coefficients for the covariates are simulated as standard normals and we generated independent normal residuals with zero mean and standard deviation of six.", "Figure REF shows the weight functions and the first 100 estimated weight functions using BDLIM-n.", "Figure: Estimated weight functions w ^(t)\\widehat{w}(t) for simulation A.", "The grey lines show the estimated weight functions from BDLIM-n for the first 100 datasets.", "The thick black dashed line is the true weight functions.Both BDLIM-n and DLM can estimate the total time-varying effect $\\beta w(t)$ , while only BDLIM-n individually identifies $\\beta $ and $w(t)$ .", "For this reason we focused the comparison on estimation of $\\beta w(t)$ and the cumulative effect $37^{-1}\\sum _{t=1}^{37}\\beta w(t)$ .", "Supplemental Table 1 shows that the models performed similarly.", "The models had similar model fit as measured by DIC, had similar RMSE and coverage near 95% for the estimate of $\\beta w(t)$ .", "For the cumulative effect, $T^{-1}\\sum _{t=1}^T\\beta w(t)$ , we found that the bias and RMSE were similar for the methods and that both had posterior interval coverage near 95%.", "Hence, when there is no subgroup-specific analysis there is no information lost by using BDLIM instead of DLM." ], [ "Simulation B: Performance with effect heterogeneity", "The second simulation scenario compares BDLIM using the four parameterizations for exposure effect heterogeneity.", "The five simulation scenarios are as follows: B.1: $w_1(t)=w^1(t)$ , $\\beta _1=0.1$ , $w_2(t)=w^1(t)$ , $\\beta _2=0.1$ , no heterogeneity.", "B.2 : $w_1(t)=w^1(t)$ , $\\beta _1=0.1$ , $w_2(t)=w^1(t)$ , $\\beta _2=-0.2$ , heterogeneity in $\\beta $ only.", "B.3: $w_1(t)=w^1(t)$ , $\\beta _1=0.1$ , $w_2(t)=w^1(t)$ , $\\beta _2=0.0$ , one group with no effect.", "B.4: $w_1(t)=w^1(t)$ , $\\beta _1=0.1$ , $w_2(t)=w^2(t)$ , $\\beta _2=0.2$ , heterogeneity in $w(t)$ and $\\beta $ .", "B.5: $w_1(t)=w^1(t)$ , $\\beta _1=0.1$ , $w_2(t)=w^2(t)$ , $\\beta _2=0.1$ , heterogeneity in $w(t)$ only.", "For each simulation scenario, Table REF shows the model fits from each model using normalized MLPPD, as described in Section REF , and DIC.", "For each of the five scenarios, there is more than one correctly specified model because BDLIM-bw is correctly specified under any of the four forms of effect heterogeneity.", "Use of MLPPD to identify the best fitting model (indicated with a $*$ in Table REF ) selects one of the correctly specified models with very high probability (at least 95% for all scenarios) and almost never selects a misspecified model.", "In contrast, DIC selects the simplest, correctly specified model (shown in bold in Table REF ) at a higher rate but also identifies a misspecified model at a higher rate for scenarios B.4 and B.5.", "Therefore, we conclude that use of MLPPD is a more conservative choice in that it selects a misspecified model at a lower rate at the cost of less power to rule out the most general BDLIM-bw model.", "Table: Comparison of model fit with four BDLIM parameterizations for simulation B.", "The top panel show the mean MLPPD across the 1000 simulated datasets.", "The second panel shows the proportion of times each parameterization ranked as the best fitting model based on MLPPD.", "The third panel shows the average DIC for each parameterization.", "The fourth panel shows the proportion of times each model was selected as the best fitting model based on DIC.", "Numbers in bold indicate that that model is the simplest model that is correctly specified while an asterisk ( * ^*) indicates that the model is correctly specified.", "Note that multiple models can be correctly specified and the BDLIM-bw is always correctly specified.Table REF summarizes inference for $\\beta $ and $w(t)$ for the different BDLIM models.", "Overall, using the model that matches the pattern of heterogeneity (indicated by bold in Table REF ) in the data provides the best inference.", "This is most notable for scenario B.4 where both $\\beta _j$ and $w_j(t)$ vary by group and BDLIM-bw provides accurate inference while the other approaches have greater bias, larger RMSE, and lower coverage for both $\\beta $ and $w(t)$ .", "Similarly, for scenarios B.2 and B.3, BDLIM-b results in improved estimation of $w(t)$ by sharing information across groups.", "To a lesser extent, BDLIM-w yields $\\beta $ estimates with lower RMSE in scenario B.5.", "In summary, BDLIM-n is nearly identical to a standard distributed lag model when there is no heterogeneity across groups.", "When there is heterogeneity across groups, using BLDIM we can identify a correctly specified model with high probability.", "When that model is a reduced model, BDLIM results in improved inference of the weight function and effect size.", "Table: Simulation results for the inference on β\\beta and w(t)w(t) using BDLIM for simulation B.", "For each scenario 1000 datasets were fit.", "The table shows the bias, RMSE, and 95% credible interval coverage for β ^\\widehat{\\beta } and the RMSE and 95% credible interval coverage for w ^(t)\\widehat{w}(t).", "The bold rows indicate the model that was most frequently selected as the MLPPD model for that simulation scenario." ], [ "Impact of sex and maternal obesity on air pollution effects on birth weight", "We used BDLIM to estimate the association between PM$_{2.5}$ and BWGA $z$ -score.", "Following the analysis of [13], we estimated this association by child sex and maternal obesity status.", "Of the 506 children with complete data including BWGA $z$ -score, there were 155 females with non-obese mothers, 182 males with non-obese mothers, 84 females with obese mothers, and 85 males with obese mothers.", "[13] associated average PM$_{2.5}$ over the entire pregnancy with BWGA $z$ -score.", "Here, we estimate the association using the four variations of BDLIM to identify windows of vulnerability and use MLPPD to select the best fitting model.", "We assumed a normal linear model and used the same priors as described for the simulation in Section REF and set $K$ to explain 99% of the variance in $X(t)$ .", "The BDLIM-b model had the highest normalized MLPPD at 0.96.", "The other models were 0.02 for BDLIM-bw, 0.01 for BDLIM-w, and 0.01 for BDLIM-n.", "Hence, we present results from the BDLIM-b, which assumes a single weight functions $w(t)$ shared by all four groups but group specific $\\beta _j$ .", "Figure REF shows the estimated group-specific effects $\\widehat{\\beta }_j$ .", "The results are consistent with those reported by [13], with a negative association between PM$_{2.5}$ and BWGA $z$ -score among boys with obese mothers but not in the other groups.", "In addition, the posterior probability of a pairwise difference between boys with obese mothers and the other three groups range from 0.93 to 0.97.", "The posterior difference for the pairwise comparisons between the three non-significant groups range from 0.50 to 0.84 suggesting little evidence of differences between those groups.", "Further, because the model is saturated we can perform an ANOVA decomposition on the posterior sample to investigate main effects of sex and obesity as well as an interaction effect (see Appendix D of the Supplementary Materials for details).", "The estimated weight function $\\widehat{w}(t)$ (Figure REF ) shows a trend of increased vulnerability in the earlier part of pregnancy, approximately weeks 5 through 20.", "Although we do not identify a window with high probability, the estimated cumulative effect over the full pregnancy always has the same sign and significance level as $\\hat{\\beta }_j$ .", "For males infants with obese mothers we estimate a cumulative effect of -0.225 with 95% credible interval (-0.476, -0.001).", "Hence, the results are suggestive of a negative association between PM$_{2.5}$ exposures in early pregnancy and lower BWGA $z$ -score among boys with obese mothers.", "The estimated time-varying effects $\\beta _j w(t)$ are presented in Appendix D of the Supplementary Materials.", "Results from the BDLIM-bw model are presented there as well.", "Comparing the results using BDLIM-bw and BDLIM-b shows that the model with a shared weight function (BDLIM-b) is more suggestive of the location of the window of vulnerability and has, on average, 14% smaller posterior standard deviation for $\\widehat{\\beta }_j$ .", "Therefore, there is evidence that the magnitude of the effect varies across groups, but no evidence that the timing of the window varies across groups primarily because there is no effect in three of the four groups.", "Figure: Estimated group specific effect sizes β ^ j \\widehat{\\beta }_j (left panel) and weight function w ^(t)\\widehat{w}(t) (right panel) using the BDLIM-b model for the BWGA zz-score analysis." ], [ "Sex-specific effects of prenatal air pollution on asthma incidence", "We next used a logistic BDLIM to estimate sex-specific associations between PM$_{2.5}$ and childhood asthma incidence in the ACCESS cohort.", "[8] analyzed these data by stratifying by sex and applying a standard DLM to data in each stratum.", "Here we assess whether the magnitude of the effect, the timing of the effect, or both vary by sex.", "This analysis included data from 544 births with complete data including asthma.", "Again, BDLIM-b was the best fitting model for the asthma analysis with normalized MLPPD of 0.43.", "The other were 0.21 for BDLIM-bw, 0.20 for BDLIM-n, and 0.17 for BDLIM-w. We describe the results from the BDLIM-b model.", "Figure REF shows the estimated weight function $\\widehat{w}(t)$ and the estimated group-specific effects $\\widehat{\\beta }_j$ using BDLIM-b.", "Figure REF shows a positive and statistically significant association between PM$_{2.5}$ exposure and asthma incidence in boys but not in girls.", "The weight function (Figure REF ) identifies a window of vulnerability in weeks 13-21.", "Hence, PM$_{2.5}$ exposures during the weeks 13-21 were positively associated with asthma among boys.", "This is comparable to the widows identified by [8] which were 12-26 weeks for boys using a sex-stratified analysis and 14-20 weeks when testing sex differences.", "Web Figure 5 shows the estimated sex-specific odds ratios for a 10$\\mu g$ /m$^3$ increase in PM$_{2.5}$ , which peak around 1.25 for boys.", "The results using BDLIM-bw were very similar and are included in Appendix E of the Supplementary Materials; however, BDLIM-b yielded posterior standard deviations of $\\widehat{\\beta }_j$ that were 10% smaller than their BDLIM-bw counterparts.", "Figure: Estimated group specific effect sizes β ^ j \\widehat{\\beta }_j (left panel) and weight function w ^(t)\\widehat{w}(t) (right panel) using the BDLIM-b model on the asthma analysis." ], [ "Discussion", "In this paper we have proposed BDLIM as a new tool that can be used estimate effect heterogeneity in time-varying exposures.", "This addresses a critical methodological gap for simultaneous estimation of windows of vulnerability and identifying susceptible subpopulations.", "Specifically, BDLIM allows for estimation of effect heterogeneity when subgroups have a common window of vulnerability but different effects within the window (BDLIM-b) or when subgroups have the same effect size but in different windows (BDLIM-w).", "In these scenarios, the resulting estimates had reduced bias and RMSE, an advantage of pooling information across groups when the effects are not different.", "We demonstrated this advantage both in the simulation study and the data analysis.", "The proposed approach partitions the time-varying effect into two components.", "The first is a constrained functional predictor that captures the temporal variation in the effect and identifies windows of vulnerability.", "The second component is a scalar effect size that quantifies the effect within the window.", "In some situations the constraint on the parameters of the weight function can be removed by reparameterizing the model.", "In this case the scalar effect size becomes a scale parameter in the model with a generalized inverse-Gaussian full conditional, which allows for simulating the posterior with a Gibbs sampler.", "In other situations, including generalized linear models, we use a slice sampler to efficiently simulate the posterior of the contained parameters.", "The constraints of the weight function make summarizing the posterior of the weight function difficult because the posterior mean does not satisfy the constraints.", "We address this by using a point estimate that is the Bayes estimate with respect to a non-standard loss function specifically chosen to yield a posterior summary that satisfies the constraints.", "We then identify windows of vulnerability where the pointwise posterior interval does not contain zero.", "We analyzed data from the ACCESS cohort on the association between prenatal exposure to PM$_{2.5}$ and both birth weight and asthma incidence.", "In both cases the BDLIM analyses suggested a common window of vulnerability but different effect sizes within the window for both outcomes.", "Hence, in both analyses the model providing the best fit to the data could not have been estimated using existing methods.", "Our results identified a window of weeks 13-21 where PM$_{2.5}$ exposures were associated with increased asthma incidence in boys but not in girls.", "In the analysis of the birth weight data, there was strong evidence of a negative association between PM$_{2.5}$ in the earlier part of pregnancy and decreased BWGA $z$ -score among boys born to obese mothers.", "The proposed approach assumes that $\\beta $ and/or $w(t)$ are the same for all groups or different for all groups.", "When there are more than two groups it may be of interest to understand if only certain pairs of groups share one or more component.", "In the BWGA $z$ -score analysis we performed an a posteriori ANOVA decomposition to investigate if the main effect of child sex or maternal obesity status is important.", "This could be done because the model for $\\beta $ was saturated.", "Another simple approach would be to define groups based on preliminary analysis and rerun the model.", "In the BWGA $z$ -score analysis this would mean two groups: boys with obese mothers in one group and everyone else in another.", "A more sophisticated extension would be to extend the approach to directly model how covariates influence each component, such as modeling the weight functions as $w_j(t) = w_0(t) + w_{sex}(t) + w_{obese}(t)$ .", "Identifying susceptible populations and windows of vulnerability are critical areas of future research as highlighted in the NIEHS strategic plan [17].", "BDLIM provides an essential tool for simultaneous identification of susceptible populations and critical windows of vulnerability when estimating of the health effects of environmental exposures." ], [ "Supplementary Material", "Supplementary Material is available upon request." ], [ "Funding", "The ACCESS study has been supported by grants R01 ES010932, R01 ES013744; U01 HL072494, and R01 HL080674 (Wright RJ, PI).", "This work was supported by USEPA grant 834798 and NIH grants (ES020871, ES007142, CA134294, ES000002, P30 ES023515).", "This publication's contents are solely the responsibility of the grantee and do not necessarily represent the official views of the US EPA." ] ]
1612.05800
[ [ "Pointwise ergodic theorems in symmetric spaces of measurable functions" ], [ "Abstract For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages." ], [ "Introduction", "The celebrated Dunford-Schwartz and Wiener-Wintner-type ergodic theorems are two of the major themes of ergodic theory.", "Due to their fundamental roles, these theorems have been revisited ever since their first appearance.", "For instance, A. Garcia [10] gave an elegant self-contained proof of Dunford-Schwartz Theorem for $L^1- L^{\\infty }-$ contractions, and I. Assani [1], [2] extended Bourgain's Return Times theorem to $\\sigma -$ finite setting.", "In this article, among other results, we extend Dunford-Schwartz and Wiener-Wintner ergodic theorems to fully symmetric spaces of measurable functions.", "We begin by showing, in Section 3, that if one works in a space of real valued measurable functions, then the class of absolute contractions coincides with the class of Dunford-Schwartz operators, hence one can assume without loss of generality that the linear operator in question contracts $L^\\infty -$ norm.", "This helps us derive, in Section 4, Dunford-Schwartz pointwise ergodic theorem in any fully symmetric space of functions $E$ in an infinite measure space $\\Omega $ such that the characteristic function $\\chi _\\Omega \\notin E$ .", "Note that, as it is shown in Section 2 of the article, the class of such spaces $E$ is significantly wider than the class of $L^p-spaces$ , $1\\le p<\\infty $ .", "Section 5 is devoted to extension of weighted Dunford-Schwartz-type ergodic theorems to fully symmetric spaces $E$ with $\\chi _\\Omega \\notin E$ .", "In the last, Section 6, of the article we utilize Return Times theorem for $\\sigma -$ finite measure to show that Wiener-Wintner ergodic theorem holds in any fully symmetric space $E$ such that $\\chi _\\Omega \\notin E$ and with the set sequences $\\lbrace \\lambda ^k\\rbrace $ , $\\lambda \\in \\lbrace z\\in \\mathbb {C}: \\ |z|=1\\rbrace $ , extended to the set all bounded Besicovitch sequences." ], [ "Preliminaries", "Let $(\\Omega ,\\mu )$ be a complete measure space.", "Denote by $\\cal L^0$ ($\\cal L_h^0$ ) the linear space of equivalence classes of almost everywhere (a.e.)", "finite complex (respectively, real) valued measurable functions on $\\Omega $ .", "Let $\\chi _E$ be the characteristic function of a set $E\\subset \\Omega $ .", "Denote $\\mathbf {1} = \\chi _\\Omega $ .", "Given $1\\le p\\le \\infty $ , let $\\cal L^p\\subset \\cal L^0$ be the $L^p-$ space equipped with the standard norm $\\Vert \\cdot \\Vert _p$ .", "Assume that $(\\Omega ,\\mu )$ is $\\sigma $ -finite.", "If $f \\in \\cal L^1 + \\cal L^{\\infty }$ , then a non-increasing rearrangement of $f$ is defined as $\\mu _t(f)=\\inf \\lbrace \\lambda >0: \\ \\mu \\lbrace |f| > \\lambda \\rbrace \\le t\\rbrace , \\ \\ t>0$ (see [12]).", "A Banach space $(E, \\Vert \\cdot \\Vert _E)\\subset \\cal L^1 + \\cal L^{\\infty }$ is called symmetric (fully symmetric) if $f \\in E, \\ g \\in \\cal L^1 + \\cal L^{\\infty }, \\ \\mu _t(g)\\le \\mu _t(f) \\ \\ \\forall \\ t>0$ (respectively, $f \\in E, \\ g \\in \\cal L^1 + \\cal L^{\\infty }, \\ \\int \\limits _0^s\\mu _t(g)dt\\le \\int \\limits _0^s\\mu _t(f)dt \\ \\ \\forall \\ s>0\\ (\\text{writing } \\ g \\prec \\prec f))$ implies that $g \\in E$ and $\\Vert g\\Vert _E\\le \\Vert f\\Vert _E$ .", "Simple examples of fully symmetric spaces are $\\cal L^1\\cap \\cal L^{\\infty }$ with the norm $\\Vert f\\Vert _{\\cal L^1\\cap \\cal L^{\\infty }}=\\max \\left\\lbrace \\Vert f\\Vert _1, \\Vert f\\Vert _{\\infty } \\right\\rbrace $ and $\\cal L^1 + \\cal L^{\\infty }$ with the norm $\\Vert f\\Vert _{\\cal L^1 + \\cal L^{\\infty }}=\\inf \\left\\lbrace \\Vert g\\Vert _1+ \\Vert h\\Vert _{\\infty }: \\ f = g + h, \\ g \\in \\cal L^1, \\ h \\in \\cal L^{\\infty } \\right\\rbrace =\\int _0^1 \\mu _t(f) dt$ (see [12]).", "Denote by $E_+$ the set all nonnegative functions from a symmetric space $E$ .", "A symmetric space $(E, \\Vert \\cdot \\Vert _E)$ is said to possess Fatou property if conditions $\\lbrace f_{n}\\rbrace \\subset E_+, \\ \\ f_{n}\\le f_{n+1} \\ \\forall \\ n, \\ \\ \\sup _n \\Vert f_{n}\\Vert _E<\\infty $ imply that there exists $f=\\sup \\limits _{n}f_{n}\\in E_+$ and $\\Vert f\\Vert _E=\\sup \\limits _{n} \\Vert f_{n}\\Vert _E$ .", "It is know that if $E = E^{\\times \\times }$ , where $E^\\times = \\lbrace g \\in \\cal L^1 + \\cal L^{\\infty }: \\ \\Vert g\\Vert _{E^{\\times }}=\\sup \\limits _{\\Vert f\\Vert _E\\le 1}\\int _{\\Omega } |f\\cdot g| d\\mu < \\infty \\rbrace $ is the Köthe dual space of $E$ , then symmetric space $(E, \\Vert \\cdot \\Vert _E)$ has Fatou property (see, for example, [17]).", "Since $(\\cal L^1 + \\cal L^{\\infty })^{\\times \\times } = \\cal L^1 + \\cal L^{\\infty }$ and $(\\cal L^1\\cap \\cal L^{\\infty })^{\\times \\times } = \\cal L^1\\cap \\cal L^{\\infty }$ [5], the spaces $(\\cal L^1 + \\cal L^{\\infty }, \\Vert \\cdot \\Vert _{\\cal L^1 + \\cal L^{\\infty }})$ and $(\\cal L^1\\cap \\cal L^{\\infty },\\Vert \\cdot \\Vert _{\\cal L^1 \\cap \\cal L^{\\infty }})$ possess Fatou property.", "A sequence $ \\lbrace f_{n}\\rbrace \\subset \\cal L^0$ is said to converge to $ f \\in \\cal L^0$ in measure topology $t_\\mu $ if $f_{n} \\chi _E \\rightarrow f\\chi _E$ in measure $\\mu $ whenever $\\mu (E) < \\infty $ .", "It is clear that $f_{n} \\rightarrow f$ a.e.", "implies $f_{n} \\rightarrow f$ in $t_\\mu $ .", "Note that in the case $\\sigma -$ finite measure $\\mu $ the algebra $(\\cal L^0, t_\\mu )$ is a complete metrizable topological algebra.", "In what follows we rely on the fact that any symmetric space with Fatou property is fully symmetric and its unit ball is closed in $t_\\mu $ [11].", "Define $\\cal R_\\mu = \\lbrace f \\in \\cal L^1 + \\cal L^{\\infty }: \\ \\mu _t(f) \\rightarrow 0 \\text{ \\ as \\ } t\\rightarrow \\infty \\rbrace .$ By [12], $(\\cal R_\\mu , \\Vert \\cdot \\Vert _{\\cal L^1 + \\cal L^\\infty })$ is a symmetric space.", "In addition, $\\cal R_\\mu $ is the closure of $\\cal L^1\\cap \\cal L^{\\infty }$ in $\\cal L^1 + \\cal L^{\\infty }$ (see [12]).", "Furthermore, it follows from definitions of $\\cal R_\\mu $ and $\\Vert \\cdot \\Vert _{\\cal L^1 + \\cal L^{\\infty }}$ that if $f \\in \\cal R_\\mu , \\ g \\in L^1+L^{\\infty } \\text{ \\ and \\ } g \\prec \\prec f,$ then $g\\in \\cal R_\\mu $ and $\\Vert g\\Vert _{\\cal L^1 + \\cal L^{\\infty }} \\le \\Vert f\\Vert _{\\cal L^1 + \\cal L^{\\infty }}$ .", "Therefore $(\\cal R_\\mu , \\Vert \\cdot \\Vert _{\\cal L^1 + \\cal L^{\\infty }})$ is a fully symmetric space.", "It is clear that if $\\mu (\\Omega ) < \\infty $ , then $\\cal R_\\mu = \\cal L^1$ .", "Proposition 2.1 If $\\mu (\\Omega ) = \\infty $ , then a symmetric space $E \\subset \\cal L^1+\\cal L^{\\infty }$ is contained in $\\cal R_\\mu $ if and only if $\\mathbf {1}\\notin E$ .", "As $\\mu (\\Omega ) = \\infty $ , we have $\\mu _t(\\mathbf {1}) = 1$ for all $t > 0$ , hence $\\mathbf {1} \\notin \\cal R_\\tau $ .", "Therefore $ E$ is not contained in $\\cal R_\\mu $ whenever $\\mathbf {1} \\in E$ .", "Let $\\mathbf {1} \\notin E$ .", "If $f \\in E$ and $\\lim _{t\\rightarrow \\infty } \\mu _t(f) = \\alpha >0$ , then $\\mu _t(\\mathbf {1}) \\equiv 1 \\le \\frac{1}{\\alpha } \\mu _t(f),$ implying $\\mathbf {1} \\in E$ , a contradiction.", "Thus $\\mathbf {1} \\notin E$ entails $ E \\subset \\cal R_\\mu $ .", "To outline the scope of applications of what follows, we assume that $\\mu (\\Omega ) = \\infty $ and present some examples of fully symmetric spaces $E$ with $\\mathbf {1}\\notin E$ .", "1.", "Let $\\Phi $ be an Orlicz function, that is, $\\Phi :[0,\\infty )\\rightarrow [0,\\infty )$ is convex, continuous at 0 and such that $\\Phi (0)=0$ and $\\Phi (u)>0$ if $u\\ne 0$ .", "Let $L^\\Phi =L^\\Phi (\\Omega ,\\mu )=\\left\\lbrace f \\in \\cal L^0(\\Omega ,\\mu ): \\ \\int _{\\Omega } \\left(\\Phi \\left(\\frac{|f|}{a} \\right)\\right) d \\mu <\\infty \\text{ \\ for some \\ } a>0 \\right\\rbrace $ be the corresponding Orlicz space, and let $\\Vert f\\Vert _\\Phi =\\inf \\left\\lbrace a>0: \\int _{\\Omega } \\left(\\Phi \\left(\\frac{|f|}{a} \\right)\\right) d \\mu \\le 1 \\right\\rbrace $ be the Luxemburg norm in $L^\\Phi $ .", "It is well-known that $(L^\\Phi , \\Vert \\cdot \\Vert _\\Phi )$ is a fully symmetric space with Fatou property.", "Since $\\mu (\\Omega ) = \\infty $ , we have $\\int _{\\Omega } \\left(\\Phi \\left(\\frac{\\mathbf {1}}{a} \\right)\\right) d \\mu = \\infty $ for all $a>0$ , hence $\\mathbf {1} \\notin L^\\Phi $ .", "2.", "A symmetric space $(E, \\Vert \\cdot \\Vert _E)$ is said to have order continuous norm if $\\Vert f_{n}\\Vert _E\\downarrow 0 \\ \\ \\text{whenever} \\ \\ f_{n}\\in E_+ \\ \\ \\text{and} \\ \\ f_{n}\\downarrow 0.$ If $E$ is a symmetric space with order continuous norm, then $\\mu \\lbrace |f| > \\lambda \\rbrace < \\infty $ for all $f \\in E$ and $\\lambda > 0$ , so $E \\subset \\cal R_\\mu $ ; in particular, $\\mathbf {1} \\notin E$ .", "3.", "Let $\\varphi $ be an increasing concave function on $[0, \\infty )$ with $\\varphi (0) = 0$ and $\\varphi (t) > 0$ for some $t > 0$ , and let $\\Lambda _\\varphi =\\Lambda _\\varphi (\\Omega ,\\mu ) = \\left\\lbrace f \\in \\cal L^0(\\Omega ,\\mu ): \\ \\Vert f \\Vert _{\\Lambda _\\varphi } =\\int _0^{\\infty } \\mu _t(f) d \\varphi (t) < \\infty \\right\\rbrace ,$ the corresponding Lorentz space.", "It is well-known that $\\Lambda _\\varphi $ is a fully symmetric space with Fatou property; in addition, if $\\varphi (\\infty ) = \\infty $ , then $\\mathbf {1} \\notin \\Lambda _\\varphi $ .", "4.", "A Banach lattice $(E,\\Vert \\cdot \\Vert _E)$ is called $q$ -concave, $1 \\le q < \\infty $ , if there exists a constant $M>0$ such that $\\left(\\sum _{i=1}^n\\Vert x_i\\Vert ^q\\right)^{\\frac{1}{q}} \\le M \\left\\Vert \\left(\\sum _{i=1}^n|x_i|^q\\right)^{\\frac{1}{q}}\\right\\Vert _E$ for every finite set $\\lbrace x_i\\rbrace _{i=1}^n \\subset E$ .", "If a Banach lattice $(E,\\Vert \\cdot \\Vert _E)$ is a $q$ -concave for some $1 \\le q < \\infty $ , then there is no a sublattice of $E$ isomorphic to $l^{\\infty }$ , and the norm $\\Vert \\cdot \\Vert _E$ is order continuous [15].", "Therefore, if a fully symmetric function space $(E, \\Vert \\cdot \\Vert _{E})$ is $q$ -concave, then $\\mathbf {1}\\notin E$ .", "5.", "Let $(E(0,\\infty ), \\Vert \\cdot \\Vert _{E(0,\\infty )})$ be a fully symmetric space.", "If $s>0$ , let the bounded linear operator $D_s$ in $E(0,\\infty )$ be given by $D_s(f)(t) = f(t/s), \\ t > 0$ .", "The Boyd index $q_E$ is defined as $q_E=\\lim \\limits _{s \\rightarrow +0}\\frac{\\log s}{\\log \\Vert D_{s}\\Vert }.$ It is known that $1\\le q_E\\le \\infty $ [17].", "Since $\\Vert D_{s}\\Vert \\le \\max \\lbrace 1,s\\rbrace $ [17], $\\mathbf {1} \\in E(0,\\infty )$ would imply $D_{s}(\\mathbf {1}) =\\mathbf {1}$ and $\\Vert D_{s}\\Vert =1$ for all $s \\in (0,1)$ , hence $q_E = \\infty $ .", "Thus, if $q_E < \\infty $ , we have $\\mathbf {1}\\notin E(0,\\infty )$ .", "The next property of the fully symmetric space $\\cal R_\\mu $ is crucial.", "Proposition 2.2 For every $f\\in \\cal R_\\mu $ and $\\epsilon >0$ there exist $g_{\\epsilon }\\in \\cal L^1$ and $h_{\\epsilon }\\in \\cal L^{\\infty }$ such that $f=g_{\\epsilon }+h_{\\epsilon }$ and $\\Vert h_\\epsilon \\Vert _{\\infty }\\le \\epsilon $ .", "If $\\Omega _{\\epsilon }=\\lbrace |f|>\\epsilon \\rbrace , \\ \\ g_{\\epsilon }=f \\cdot \\chi _{\\Omega _{\\epsilon }}, \\ \\ h_{\\epsilon }=f \\cdot \\chi _{\\Omega \\setminus \\Omega _{\\epsilon }},$ then $\\Vert h_{\\epsilon }\\Vert _{\\infty }\\le \\epsilon $ .", "Besides, as $f\\in \\cal L^1+\\cal L^{\\infty }$ , we have $f=g_{\\epsilon }+h_{\\epsilon }=g+h$ for some $g\\in \\cal L^1$ , $h\\in \\cal L^{\\infty }$ .", "Then, since $f\\in \\cal R_\\mu $ , we have $\\mu (\\Omega _{\\epsilon })<\\infty $ , which implies that $g_{\\epsilon }=g\\cdot \\chi _{\\Omega _{\\epsilon }}+(h-h_{\\epsilon })\\cdot \\chi _{\\Omega _{\\epsilon }}\\in \\cal L^1.$" ], [ "Dunford-Schwartz operators and absolute contractions", "Definition 3.1 A linear operator $T: \\cal L^1 + \\cal L^{\\infty } \\rightarrow \\cal L^1 + \\cal L^{\\infty }$ is called a Dunford-Schwartz operator if $\\Vert T(f)\\Vert _1\\le \\Vert f\\Vert _1 \\ \\ \\forall \\ \\ f\\in L^1 \\text{ \\ \\ and \\ \\ } \\Vert T(f)\\Vert _{\\infty }\\le \\Vert f\\Vert _\\infty \\ \\ \\forall \\ f \\in \\cal L^{\\infty }.$ In what follows, we will write $T\\in DS$ ($T\\in DS^+$ ) to indicate that $T$ is a Dunford-Schwartz operator (respectively, a positive Dunford-Schwartz operator).", "It is clear that $\\Vert T\\Vert _{\\cal L^1 + \\cal L^{\\infty } \\rightarrow \\cal \\cal L^1 + \\cal L^{\\infty }} \\le 1$ for all $T\\in DS$ and, in addition, $Tf \\prec \\prec f$ for all $f \\in \\cal L^1 + \\cal L^{\\infty }$ [12].", "Therefore $T(E) \\subset E$ for every fully symmetric space $E$ and $\\Vert T\\Vert _{E \\rightarrow E} \\le 1$ (see [12]).", "In particular, $T(\\cal R_\\mu ) \\subset \\cal R_\\mu $ , and the restriction of $T$ on $\\cal R_\\mu $ is a linear contraction (also denoted by $T$ ).", "We say that a linear operator $T: \\cal L^1\\rightarrow \\cal L^1$ is an absolute contraction and write $T\\in AC$ if $\\Vert T(f)\\Vert _1\\le \\Vert f\\Vert _1 \\ \\ \\forall \\ f\\in \\cal L^1 \\text{ \\ and \\ } \\Vert T(f)\\Vert _{\\infty }\\le \\Vert f\\Vert _{\\infty } \\ \\ \\forall \\ f\\in \\cal L^1 \\cap \\cal L^{\\infty }.$ If $T\\in AC$ is positive, we will write $T\\in AC^+$ .", "Definition 3.2 A complete measure space $(\\Omega , \\cal A,\\mu )$ is called semifinite if every subset of $\\Omega $ of non-zero measure admits a subset of finite non-zero measure.", "A semifinite measure space $(\\Omega , \\cal A, \\mu )$ is said to have the direct sum property if the Boolean algebra $(\\cal A /\\sim )$ of equivalence classes of measurable sets is complete, that is, every subset of $(\\cal A /\\sim )$ has a least upper bound.", "Note that every $\\sigma -$ finite measure space has the direct sum property.", "A detailed account on measures with direct sum property is found in [7]; see also [14].", "Absolute contractions (or $\\cal L^1-\\cal L^{\\infty }-$ contractions) were considered in [10] and also in [13].", "It is clear that if $T \\in DS$ , then $T| \\cal L^1\\in AC$ .", "It turns out that if $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L^0_h$ and $(\\Omega ,\\mu )$ has the direct sum property, then $T\\in AC$ can be uniquely extended to a $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous $DS$ operator: Theorem 3.1 Let $(\\Omega ,\\mu )$ have the direct sum property and let $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L_h^0$ .", "Then for any $T\\in AC$ there exists a unique $S \\in DS$ such that $S| \\cal L^1 = T$ and $S| \\cal L^{\\infty }$ is $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous.", "Remark 3.1 In what follows, we will only need Theorem REF in the case of $\\sigma -$ finite measure.", "However, since this theorem, and Lemma REF below, are interesting results by themselves, we prove them in more general settings, namely, for a measure with the direct sum property and for a semifinite measure, respectively.", "In order to prove Theorem REF , we will need Lemma REF below.", "Let us denote $\\cal F=\\lbrace F\\subset \\Omega : \\ 0<\\mu (F)<\\infty \\rbrace ,$ and let $\\lbrace F_{\\alpha }\\rbrace $ be the directed set consisting of the elements of $\\cal F$ , ordered by inclusion.", "Lemma 3.1 If $(\\Omega ,\\mu )$ is semifinite, then $\\int \\chi _{F_{\\alpha }}g\\rightarrow \\int g$ for all $g\\in \\cal L^1$ .", "It is enough to prove the statement for $g\\ge 0$ .", "Then $\\int \\chi _{F_{\\alpha }}g\\le \\int g$ for every $\\alpha $ and, since $\\left\\lbrace \\int \\chi _{F_{\\alpha }}g \\right\\rbrace $ is an increasing net of real numbers, we have $\\lim _{\\alpha } \\int \\chi _{F_{\\alpha }}g=\\sup _{\\alpha }\\int \\chi _{F_{\\alpha }}g=s<\\infty .$ There exists a sequence $\\lbrace f_n\\rbrace \\subset \\cal F$ such that $\\lim _{n\\rightarrow \\infty } \\int \\chi _{F_n}g=s.$ Without loss of generality, we can assume that $F_n\\subset F_{n+1}$ for all $n \\ge 1$ .", "Let $E=\\cup F_n$ and assume that $g\\chi _{\\Omega \\setminus E}\\ne 0$ .", "Then there is $F\\in \\cal F$ such that $F\\subset \\Omega \\setminus E$ and $g\\chi _F>0$ , and we obtain $\\lim _{n\\rightarrow \\infty }\\int \\chi _{F_n\\cup F}g=\\int \\chi _{E\\cup F}g>\\int \\chi _Eg=\\lim _{n\\rightarrow \\infty }\\int \\chi _{F_n}g=s.$ This entails that $\\sup _{\\alpha }\\int \\chi _{F_{\\alpha }}g>s$ , a contradiction, so $g\\chi _{\\Omega \\setminus E}=0$ .", "Consequently, $s=\\lim _{n\\rightarrow \\infty } \\int \\chi _{F_n}g=\\int \\chi _Eg=\\int g,$ hence $\\int \\chi _{F_{\\alpha }}g\\rightarrow \\int g$ .", "Let us first establish Theorem REF for positive operators.", "Theorem 3.2 If $(\\Omega ,\\mu )$ , $\\cal L^{\\infty }$ , and $\\cal L^1$ are as in Theorem REF , then, given $T\\in AC^+$ , there exists a unique $S \\in DS^+$ such that $S| \\cal L^1 = T$ and $S|\\cal L^{\\infty }$ is $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous.", "Since $(\\Omega ,\\mu )$ has the direct sum property, Radon-Nikodym theorem is valid [7], hence $ (\\cal L^1)^* = \\cal L^{\\infty }$ , and the adjoint operator $T^*: \\cal L^{\\infty } \\rightarrow \\cal L^{\\infty }$ is defined.", "Besides, $\\Vert T^*\\Vert _{\\cal L^{\\infty }\\rightarrow \\cal L^{\\infty }} = \\Vert T\\Vert _{\\cal L^1 \\rightarrow \\cal L^1} \\le 1$ and $T^*$ is also $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous.", "Moreover, $\\int T^*(f)g=\\int fT(g)$ for all $f\\in \\cal L^{\\infty }$ , $g\\in \\cal L^1$ .", "In particular, it follows that the linear operator $T^*$ is positive.", "Now, if $f\\in \\cal L^{\\infty }_+\\cap \\cal L^1$ , then, with $\\chi _{F_{\\alpha }}$ as in Lemma REF , we have $0 \\le \\int T^*(f)\\chi _{F_{\\alpha }} = \\int f T(\\chi _{F_{\\alpha }}) \\le \\int f,$ for all $\\alpha $ , hence $T^*(f) \\in \\cal L^{\\infty }_+\\cap \\cal L^1$ .", "In addition, $\\begin{aligned}\\Vert T^*(f) \\Vert _1& = \\sup \\left\\lbrace \\left| \\int T^*(f)g \\right|:\\ g \\in \\cal L^{\\infty } = (\\cal L^1)^*,\\ \\Vert g\\Vert _{\\infty }\\le 1 \\right\\rbrace \\\\&\\le \\sup \\left\\lbrace \\int T^*(f)|g|:\\ g \\in \\cal L^{\\infty },\\ \\Vert g\\Vert _{\\infty }\\le 1 \\right\\rbrace \\\\&=\\sup \\left\\lbrace \\int T^*(f) g:\\ g \\in \\cal L^{\\infty }_+,\\ \\Vert g\\Vert _{\\infty }\\le 1 \\right\\rbrace .\\end{aligned}$ Since, by Lemma REF , for every $g \\in \\cal L^{\\infty }_+$ with $\\Vert g\\Vert _{\\infty }\\le 1$ we have $\\begin{aligned}\\int T^*(f) g&= \\lim _{\\alpha } \\int T^*(f) \\chi _{F_{\\alpha }}g \\\\& \\le \\sup \\left\\lbrace \\int T^*(f)h=\\int f T(h):\\ h \\in \\cal L^{\\infty }_+\\cap \\cal L^1,\\ \\Vert h\\Vert _{\\infty } \\le 1 \\right\\rbrace \\le \\int f,\\end{aligned}$ it follows that $\\Vert T^*(f)\\Vert _1\\le \\Vert f\\Vert _1$ whenever $f\\in \\cal L^{\\infty }_+\\cap \\cal L^1$ .", "Therefore $T^*$ is a positive linear $\\Vert \\cdot \\Vert _1-$ contraction on $\\cal L^{\\infty }\\cap \\cal L^1$ and, since $\\cal L^{\\infty }\\cap \\cal L^1$ is dense in $\\cal L^1$ , it uniquely extends to a $DS^+$ operator, which we also denote by $T^*$ .", "For the $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous adjoint operator $T^{**}: (\\cal L^{\\infty })^{\\ast } \\rightarrow (\\cal L^{\\infty })^{\\ast }$ and all $f, \\ g\\in \\cal L^{\\infty }\\cap \\cal L^1$ , we have $\\int T^{**}(f)g=\\int fT^*(g) = \\int T(f) g,$ hence $ T^{**}(f) = T(f)$ whenever $f \\in \\cal L^{\\infty }\\cap \\cal L^1$ .", "In the same way as $T^*$ , $T^{**}$ uniquely extends to a $DS^+$ operator (which we also denote by $T^{**}$ ) such that $T^{**}(g) = T(g)$ for every $g \\in \\cal L^1$ and $T^{**}|\\cal L^{\\infty }$ is $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous.", "Let $S\\in DS^+$ be another operator such that $S(g) = T(g)$ for every $g \\in \\cal L^1$ and $S|\\cal L^{\\infty }$ is $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ continuous.", "Given $f \\in \\cal L^{\\infty }$ and $g \\in \\cal L^1$ , it follows from Lemma REF that $ \\int g \\chi _{F_{\\alpha }}f \\rightarrow \\int gf$ , that is, $\\chi _{F_{\\alpha }}f \\rightarrow f$ in $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ topology, so $\\cal L^{\\infty }\\cap \\cal L^1$ is $\\sigma (\\cal L^{\\infty }, \\cal L^1)-$ dense in $\\cal L^{\\infty }$ .", "Therefore $S|\\cal L^{\\infty } = T^{**}|\\cal L^{\\infty }$ , which completes the proof.", "We shall recall now the following statement on the existence and properties of linear modulus of a bounded linear operator $T:\\cal L^1\\rightarrow \\cal L^1$ ($T:\\cal L^{\\infty } \\rightarrow \\cal L^{\\infty }$ ) (see [13]): Theorem 3.3 Let $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L_h^0$ .", "Then for any bounded linear operator $T:\\cal L^1\\rightarrow \\cal L^1$ ($T:\\cal L^{\\infty } \\rightarrow \\cal L^{\\infty }$ ) there exists a unique positive bounded linear operator $|T|:\\cal L^1\\rightarrow \\cal L^1$ (respectively, $|T|:\\cal L^{\\infty } \\rightarrow \\cal L^{\\infty }$ ) such that $ \\Vert \\ |T| \\ \\Vert = \\Vert T\\Vert $ ; $|T(f)|\\le |T|(|f|)$ for all $f\\in \\cal L^1$ (respectively, for all $f\\in \\cal L^{\\infty }$ ); $|T|(f )=\\sup \\lbrace |T(g)|: g\\in \\cal L^1, |g|\\le f\\rbrace $ for all $f\\in \\cal L^1_+$ (respectively, $|T|(f)=\\sup \\lbrace |T(g)|: g \\in \\cal L^{\\infty }, |g|\\le f\\rbrace $ for all $f \\in \\cal L^{\\infty }_+$ ).", "The operator $|T|$ is called the linear modulus of $T$ .", "In addition, $|T|$ satisfies the following properties [13]).", "Proposition 3.1 If $T \\in AC$ , then $|T^k(f)|\\le |T|^k(|f|)$ for every $f\\in \\cal L^1, \\ k=1,2, \\dots $ ; $\\Vert T\\Vert _{1,\\infty }:= \\sup \\lbrace \\Vert Tf\\Vert _{\\infty }: f \\in \\cal L^1 \\cap \\cal L^{\\infty }, \\ \\Vert f\\Vert _{\\infty } \\le 1 \\rbrace = \\Vert \\ |T| \\ \\Vert _{1,\\infty }$ ; $ |T^*| = |T|^*$ (in the case direct sum property of $(\\Omega ,\\cal A, \\mu )$ ).", "Here is a proof of Theorem REF .", "By virtue of Theorem REF (i) and Proposition REF (ii), $|T| \\in AC^+$ .", "The adjoint operators $T^*$ and $|T|^*$ are contractions in $\\cal L^{\\infty }$ such that $|T^*| = |T|^*$ , by Proposition REF (iii).", "As in proof of Theorem REF , $|T|^*$ uniquely extends to a $DS^+$ operator, which we also denote by $|T|^*$ .", "It follows from $\\Vert T^* f\\Vert _1 \\le \\Vert |T^*| (|f|)\\Vert _1 = \\Vert |T|^* (|f|)\\Vert _1 \\le \\Vert f\\Vert _1, \\ f \\in \\cal L^{\\infty } \\cap \\cal L^1,$ that $T^*$ is a $\\Vert \\cdot \\Vert _1-$ contraction in $\\cal L^{\\infty } \\cap \\cal L^1$ .", "Therefore, as in the proof of Theorem REF , $T^*$ and then $T^{**}$ can be uniquely extended to $DS$ operators.", "Repeating the argument at the end of proof of Theorem REF , we obtain the result.", "An important consequence of Theorem REF is that if $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L_h^0$ , then, given $T\\in AC$ , one can assume, without loss of generality, that $T\\in DS$ , so that $\\Vert T(f)\\Vert _{\\infty }\\le \\Vert f\\Vert _{\\infty }$ for every $f\\in \\cal L^{\\infty }$ .", "Remark 3.2 Let $E$ be a subspace of $\\cal L^1+\\cal L^\\infty $ such that $T(E)\\subset E$ if $T\\in DS$ .", "In what follows, if a.e.", "convergence of conventional or weighted ergodic averages holds for $T\\in DS$ and every $f\\in E_h$ , these averages also converge a.e.", "whenever $f\\in E$ and $T\\in DS^+$ because then $T(E_h)\\subset E_h$ .", "Therefore, we routinely assume that $T\\in DS$ if $\\cal L^{\\infty },\\cal L^1\\subset \\cal L_h^0$ and $T\\in DS^+$ in the general case." ], [ "Dunford-Schwartz pointwise ergodic theorem in fully symmetric spaces", "Dunford-Schwartz theorem on pointwise convergence of the ergodic averages $a_n(f)=\\frac{1}{n} \\sum _{k=0}^{n-1}T^k(f)$ for $T\\in DS$ acting in the $L^p-$ space, $1\\le p<\\infty $ , of real valued functions of an arbitrary measure space was established in [8]; see also [9].", "Note that, since the set on which the function $T^k(f)\\in \\cal L^p$ , $1\\le p<\\infty $ , $k=0,1,2,\\dots $ , does not equal to zero is a countable union of sets of finite measure, one can assume that the measure is $\\sigma -$ finite.", "In this section we prove Dunford-Schwartz pointwise ergodic theorem in $\\cal R_\\mu $ , Theorem REF , for real and complex valued (when $T\\in DS^+$ ) functions, arguably the most general version of the classical result.", "In view of Remark REF , Dunford-Schwartz pointwise ergodic theorem in $\\cal L^1$ can be stated as follows.", "Theorem 4.1 Let $(\\Omega , \\mu )$ be an arbitrary measure space.", "Assume that either $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L_h^0$ and $T\\in DS$ or $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L^0$ and $T\\in DS^+$ .", "Then the averages (REF ) converge a.e.", "for all $f\\in \\cal L^1$ .", "Below is an extension of Theorem REF to the fully symmetric space $\\cal R_\\mu $ .", "Theorem 4.2 Let $(\\Omega ,\\mu )$ be a measure space.", "Assume that either $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L_h^0$ and $T\\in DS$ or $\\cal L^{\\infty }, \\cal L^1\\subset \\cal L^0$ and $T\\in DS^+$ .", "If $f \\in \\cal R_\\mu $ , then the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in \\cal R_\\mu $ .", "Without loss of generality, assume that $f\\ge 0$ .", "By Proposition REF , given $k=1,2, \\dots $ , there are $0 \\le g_k\\in \\cal L^1$ and $0 \\le h_k\\in \\cal L^{\\infty }$ such that $f=g_k+h_k \\text{ \\ and \\ } \\Vert h_k\\Vert _{\\infty }\\le \\frac{1}{k}.$ Since $\\lbrace g_k\\rbrace \\subset \\cal L^1$ it follows from Theorem REF that the averages (REF ) converge a.e.", "for each $g_k $ .", "As $\\lbrace a_n(g_k)\\rbrace \\subset \\cal L_h^1$ , we can assume without loss of generality that $\\limsup _n a_n(g_k)(\\omega )=\\liminf _n a_n(g_k)(\\omega )$ for all $\\omega \\in \\Omega $ and every $k$ .", "Then, for a fixed $k$ and $\\omega \\in \\Omega $ , we have $0 \\le \\Delta (\\omega )=\\limsup _n a_n(f)(\\omega )-\\liminf _n a_n(f)(\\omega )=$ $=\\limsup _n a_n(h_k)(\\omega )-\\liminf _n a_n(h_k)(\\omega )\\le 2 \\sup _n | a_n(h_k)(\\omega )|,$ which together with $\\Vert a_n(h_k) \\Vert _{\\infty }\\le \\frac{1}{k}$ , $n=1,2,\\dots $ , implies that there exists $\\Omega _k \\in \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _k)=0$ such that $0 \\le \\Delta (\\omega )\\le \\frac{2}{k}$ whenever $\\omega \\in \\Omega _k$ .", "Then, letting $\\Omega _f=\\cap _k\\Omega _k$ , we obtain $\\mu (\\Omega \\setminus \\Omega _f)=0$ and $\\Delta (\\omega )\\le \\frac{2}{k}$ for all $\\omega \\in \\Omega _f$ and every $k$ .", "Therefore $\\Delta (\\omega )=0$ , hence $\\limsup _n a_n(f)(\\omega )=\\liminf _n a_n(f)(\\omega ), \\ \\ \\omega \\in \\Omega _f,$ and we conclude that the sequence $\\lbrace a_n(f)\\rbrace $ converges a.e.", "to a $\\mu -$ measurable function $\\widehat{f}$ on $\\Omega $ .", "Note that, since $\\cal L^0$ is complete in the measure topology, $\\widehat{f}$ cannot be infinite on a set of positive measure, hence $\\widehat{f}\\in \\cal L^0$ .", "Since $\\cal L^1+\\cal L^{\\infty }$ satisfies Fatou property, its unit ball is closed in measure topology $t_\\mu $ , and (REF ) implies that $\\widehat{f} \\in \\cal L^1+\\cal L^{\\infty }$ .", "Since $ a_n(f)\\rightarrow \\widehat{f}$ in $t_\\mu $ , it follows that $\\mu _t( a_n(f))\\rightarrow \\mu _t(\\widehat{f}) \\ \\ \\text{a.e.", "on} \\ \\ (0,\\infty )$ (see, for example, [12]).", "Besides, the inclusion $T \\in DS$ implies that $\\mu _t( a_n(f))\\prec \\prec \\mu _t( f)$ for everyl $n$ (see, for example, [12]).", "Utilizing Fatou property for $\\cal L^1(0,s)$ and the measure convergence $\\mu _t( a_n(f))\\rightarrow \\mu _t(\\widehat{f}) \\ \\ \\text{on} \\ \\ (0,s),$ we derive $\\int \\limits _0^s\\mu _t(\\widehat{f})dt \\le \\sup _{n\\ge 1} \\int \\limits _0^s\\mu _t(a_n(f))dt \\le \\int \\limits _0^s\\mu _t(\\widehat{f})dt$ for all $s>0$ , that is, $\\mu _t(\\widehat{f})\\prec \\prec \\mu _t( f)$ .", "Since $\\cal R_\\mu $ is a fully symmetric space and $f\\in \\cal R_\\mu $ , it follows that $\\widehat{f}\\in \\cal R_\\mu $ .", "Now we present a version of Theorem REF for a fully symmetric space $E \\subset \\cal R_\\mu $ .", "Theorem 4.3 Let $(\\Omega ,\\mu )$ be an infinite measure space, and let $E$ be a fully symmetric space with $\\mathbf {1}\\notin E$ .", "Assume that either $E \\subset \\cal L_h^0$ and $T\\in DS$ or $E \\subset \\cal L^0$ and $T\\in DS^+$ .", "Then for every $f \\in E$ the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in E$ .", "By Proposition REF , $E \\subset \\cal R_\\mu $ .", "Then, by Theorem REF , given $f \\in E$ , the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in \\cal R_\\mu $ .", "Since $E$ is a fully symmetric space, it follows as in Theorem REF that $\\widehat{f} \\in E$ .", "The next theorem implies that, in the model case $\\Omega =(0,\\infty )$ , if a symmetric space $E\\subset \\cal L^1+\\cal L^{\\infty }$ is such that $E\\setminus \\cal R_\\mu \\ne \\emptyset $ , then Dunford-Schwartz pointwise ergodic theorem does nod hold in $E$ .", "Theorem 4.4 Let $\\Omega =(0, \\infty )$ , and let $\\mu $ be Lebesgue measure.", "Then, given $f \\in (\\cal L^1+\\cal L^{\\infty }) \\setminus \\cal R_\\mu $ , there exists $T\\in DS$ such that the averages $a_n(\\mu _t(f))$ do not converge a.e.", "Since $f \\in (\\cal L^1+\\cal L^{\\infty }) \\setminus \\cal R_\\mu $ , it follows that $\\mu _t(f) \\ge \\epsilon $ for all $t >0$ and some $\\varepsilon >0$ .", "Without loss of generality we can assume that $\\mu _t(f) \\ge 1$ for all $t >0$ .", "Let $\\lbrace n_k\\rbrace _{k=1}^{\\infty }$ be an increasing sequence of integers with $n_0=0$ (the choice of this sequence is given below).", "Consider the function $\\varphi (t)=\\sum _{k=0}^{\\infty }\\left(\\chi _{[n_k,n_{k +1}-1)}(t) - \\chi _{[n_{k +1}-1,n_{k+1})}(t)\\right)$ and the operator $T$ in $\\cal L^1+\\cal L^{\\infty }$ defined by $T(f)(t) = \\varphi (t) f(t+1), \\ \\ f \\in \\cal L^1+\\cal L^{\\infty }.$ It is clear that $T \\in DS$ and $a_n(\\mu _t(f))==\\frac{1}{n} \\left(\\mu _{t}(f)+\\sum _{k=1}^{n-1}\\varphi (t)\\varphi (t+1)\\cdot \\cdots \\cdot \\varphi (t+k-1)\\mu _{t+k}(f)\\right).$ Show that the averages $a_n(\\mu _t(f))$ do not converge a.e.", "Fix $\\epsilon \\in (0,1)$ .", "If $t \\in (\\epsilon , 1)$ , then $a_{n_1}(\\mu _t(f)) =\\frac{1}{n_1}\\sum _{m=0}^{n_1-1}\\mu _{t+m}(f)\\ge 1.$ Next, since $a_{n_2}(\\mu _t(f)) =\\frac{1}{n_2}\\left(\\sum _{m=0}^{n_1-1}\\mu _{t+m}(f)-\\sum _{m=n_1}^{n_2-1}\\mu _{t+m}(f)\\right)\\le \\frac{1}{n_2}\\left(n_1\\mu _\\epsilon (f)-(n_2-n_1)\\right),$ one can choose big enough $n_2 $ so that $a_{n_2}(\\mu _t(f))<-\\frac{1}{2}$ for every $t\\in (\\epsilon ,1)$ .", "As $a_{n_3}(\\mu _t(f)) =\\frac{1}{n_3}\\left(\\sum _{m=0}^{n_1}\\mu _{t+m}(f)-\\sum _{m=n_1+1}^{n_2-1}\\mu _{t+m}(f)+\\sum _{m=n_2+1}^{n_3-1}\\mu _{t+m}(f)\\right),$ there exists $n_3$ such that $a_{n_3}(\\mu _t(f))>\\frac{1}{2}$ for every $t\\in (\\epsilon ,1)$ .", "Continuing this process, we construct a sequence of positive integers $n_1 < n_2 < ...$ such that for every $t \\in (\\epsilon , 1)$ , $a_{n_{2k-1}}(\\mu _t(f)) >\\frac{1}{2} \\text{ \\ \\ and \\ \\ }a_{n_{2k}}(\\mu _t(f)) <- \\frac{1}{2}, \\ \\ k=1,2,\\dots $ Thus, the averages $a_n(\\mu _t(f))$ diverge for each $t \\in (\\epsilon , 1)$ ." ], [ "Weigthed ergodic theorem for Dunford-Schwartz operators in fully symmetric spaces", "Let $(\\Omega ,\\mu )$ be a measure space.", "Given $T\\in DS$ , a bounded sequence $\\overline{\\beta }=\\lbrace \\beta _k\\rbrace _{k=0}^{\\infty }$ of complex numbers is called a good weight for $T$ if the sequence of weigthed ergodic averages $a_n(\\overline{\\beta }, f)=\\frac{1}{n} \\sum _{k=0}^{n-1} \\beta _k T^k(f)$ converges $\\mu -$ a.e.", "for every $f \\in \\cal L^1 $ (see, for example, [6]).", "In [3], [4], [6], [16], various classes of good weights for $T\\in DS$ in $L^p-$ spaces of real valued functions on finite and infinite measure spaces were studied.", "In particular, it follows from a general measure space extension of [3] given in [6] that bounded Besicovitch sequences are good weights for any $L^1-$ contraction with mean ergodic modulus, in particular, for Dunford-Schwartz operators, in an arbitrary measure space.", "The convergence also holds when the functions are not assumed to be real valued but $T\\in DS^+$ ; see Theorem REF below.", "Let $C_1$ be the unit circle in the field $\\mathbb {C}$ of complex numbers, and let $\\mathbb {Z}$ be the set of all integers.", "A function $P : \\mathbb {Z} \\rightarrow \\mathbb {C}$ is said to be a trigonometric polynomial if $P(k)=\\sum _{j=1}^{s} z_j\\lambda _j^k$ , $k\\in \\mathbb {Z}$ , for some $s\\in \\mathbb {N}$ , $\\lbrace z_j \\rbrace _1^s \\subset \\mathbb {C}$ , and $\\lbrace \\lambda _j \\rbrace _1^s \\subset \\mathbb {C}_1$ .", "A sequence $\\lbrace \\beta _k \\rbrace \\subset C$ is called a bounded Besicovitch sequence if (i) $| \\beta _k | \\le C < \\infty $ for all $k$ ; (ii) for every $\\epsilon >0$ there exists a trigonometric polynomial $P$ such that $\\limsup _n \\frac{1}{n} \\sum _{k=0}^{n-1} | \\beta _k - P(k) | < \\epsilon .$ From what was just noticed and taking into account Remark REF we have the following.", "Theorem 5.1 Assume that either $T\\in DS$ and $\\cal L^1\\subset \\cal L^0_h$ or $T\\in DS^+$ in $\\cal L^1\\subset \\cal L^0$ .", "Then any bounded Besicovitch sequence $\\overline{\\beta }=\\lbrace \\beta _k \\rbrace $ is a good weight for $T$ .", "Now we will show that if $\\overline{\\beta }$ is a good weight for $T\\in DS$ , then the averages (REF ) converge a.e.", "in any fully symmetric space $E$ with $\\mathbf {1}\\notin E$ .", "It should be pointed out that the boundedness of a sequence $\\overline{\\beta }$ implies that $a_n(\\overline{\\beta },f)(E) \\subset E \\text{ \\ \\ and \\ \\ } \\Vert a_n(\\overline{\\beta },f)\\Vert _{E \\rightarrow E} \\le \\sup _{k\\ge 0} | \\beta _k| < \\infty .$ for every fully symmetric space $E$ and $T\\in DS$ .", "As before, we begin with the fully symmetric space $\\cal R_\\mu $ .", "Theorem 5.2 Let $(\\Omega ,\\mu )$ be a measure space.", "Let $\\overline{\\beta }=\\lbrace \\beta _k\\rbrace _{k=0}^{\\infty }$ be a good weight for $T\\in DS$ .", "If $f \\in \\cal R_\\mu $ , then the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in \\cal R_\\mu $ .", "Without loss of generality assume that $(\\Omega ,\\mu )$ is $\\sigma -$ finite and $f\\ge 0$ .", "By Proposition REF , given $k=1,2, \\dots $ , there are $g_k\\in \\cal L^1$ and $h_k\\in \\cal L^{\\infty }$ such that $f=g_k+h_k \\text{ \\ and \\ } \\Vert h_k\\Vert _{\\infty }\\le \\frac{1}{k}.$ Since $\\lbrace g_k\\rbrace \\subset \\cal L^1$ and ${\\bf \\beta } =\\lbrace \\beta _k\\rbrace _{k=0}^{\\infty }$ is a good weight for $T$ the averages (REF ) converge a.e.", "for all $g_k $ .", "Thus we can assume without loss of generality that $\\limsup _n a_n(\\overline{\\beta },g_k)(\\omega )=\\liminf _n a_n(\\overline{\\beta },g_k)(\\omega )$ for all $\\omega \\in \\Omega $ and every $k$ .", "Then, for a fixed $k$ and $\\omega \\in \\Omega $ , we have $0 \\le \\Delta (\\omega )=\\limsup _n a_n(\\overline{\\beta },f)(\\omega )-\\liminf _n a_n(\\overline{\\beta },f)(\\omega )=$ $=\\limsup _n a_n(\\overline{\\beta },h_k)(\\omega )-\\liminf _n a_n(\\overline{\\beta },h_k)(\\omega )\\le 2 \\sup _n | a_n(\\overline{\\beta },h_k)(\\omega )|,$ which together with $\\Vert a_n(\\overline{\\beta },h_k) \\Vert _{\\infty }\\le \\frac{1}{k}$ , $n=1,2,\\dots $ , implies that there exists $\\Omega _k \\in \\cal A$ with $\\mu (\\Omega \\setminus \\Omega _k)=0$ such that $0 \\le \\Delta (\\omega )\\le \\frac{2}{k}$ whenever $\\omega \\in \\Omega _k$ .", "Then, letting $\\Omega _f=\\cap _k\\Omega _k$ , we obtain $\\mu (\\Omega \\setminus \\Omega _f)=0$ and $\\Delta (\\omega )\\le \\frac{2}{k}$ for all $\\omega \\in \\Omega _f$ and every $k$ .", "Therefore $\\Delta (\\omega )=0$ , hence $\\limsup _n a_n(\\overline{\\beta },f)(\\omega )=\\liminf _n a_n(\\overline{\\beta },f)(\\omega ), \\ \\ \\omega \\in \\Omega _f,$ and we conclude that the sequence $\\lbrace a_n(\\overline{\\beta },f)\\rbrace $ converges a.e.", "to a $\\mu -$ measurable function $\\widehat{f}$ on $\\Omega $ .", "Note that, since $\\cal L^0$ is complete in the measure topology $t_\\mu $ , the function $\\widehat{f}$ cannot be infinite on a set of positive measure, that is, $\\widehat{f}\\in \\cal L^0$ .", "Since $\\cal L^1+\\cal L^{\\infty }$ satisfies Fatou property, its unit ball is closed in measure, so (REF ) implies that $\\widehat{f} \\in \\cal L^1+\\cal L^{\\infty }$ .", "As $a_n(\\overline{\\beta }, f)\\rightarrow \\widehat{f}$ in $t_\\mu $ , it follows that $\\mu _t(a_n(\\overline{\\beta }, f))\\rightarrow \\mu _t(\\widehat{f}) \\ \\ \\text{a.e.", "on} \\ \\ (0,\\infty )$ (see, for example, [12]).", "Setting $M(\\overline{\\beta })= \\max \\lbrace 1,\\sup _{k\\ge 1} | \\beta _k|\\rbrace $ , we have $\\frac{1}{M(\\overline{\\beta })}(\\frac{1}{n} \\sum _{k=0}^{n-1} \\beta _k T^k) \\in DS$ , hence $\\mu _t(\\frac{1}{M(\\overline{\\beta })}a_n(\\overline{\\beta },f))\\prec \\prec \\mu _t( f)$ (see, for example, [12]).", "Using Fatou property for $\\cal L^1(0,s)$ and the measure convergence $\\mu _t\\left(\\frac{1}{M(\\overline{\\beta })}a_n(\\overline{\\beta },f)\\right)\\rightarrow \\mu _t\\left(\\frac{\\widehat{f}}{M(\\overline{\\beta })}\\right)$ on $(0,s)$ , we derive $\\int \\limits _0^s\\mu _t\\left(\\frac{\\widehat{f}}{M(\\overline{\\beta })}\\right)dt\\le \\sup _{n\\ge 1}\\int \\limits _0^s\\mu _t\\left(\\frac{1}{M(\\overline{\\beta })}a_n(\\overline{\\beta },f)\\right)dt \\le \\int \\limits _0^s\\mu _t\\left(\\frac{f}{M(\\overline{\\beta })}\\right)dt$ for all $s>0$ , that is, $\\mu _t(\\widehat{f})\\prec \\prec \\mu _t( f)$ .", "Since $\\cal R_\\mu $ is a fully symmetric space and $f\\in \\cal R_\\mu $ , it follows that $\\widehat{f}\\in \\cal R_\\mu $ .", "Since, by Proposition REF , $E \\subset \\cal R_\\mu $ for every symmetric space $E$ with $\\mathbf {1}\\notin E$ , utilizing Theorem REF and repeating the ending of its proof, we obtain the following.", "Theorem 5.3 Let $(\\Omega ,\\mu )$ be an infinite measure space, and let $T$ and $\\overline{\\beta }$ be as in Theorem REF .", "Assume that $E$ is a fully symmetric space with $\\mathbf {1}\\notin E$ .", "If $f \\in E$ , then the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in E$ .", "In view of Theorems REF and REF , we now have the following.", "Corollary 5.1 Let $(\\Omega ,\\mu )$ be an infinite measure space, and let $E$ be a fully symmetric space with $\\mathbf {1}\\notin E$ .", "Assume that either $E\\subset \\cal L^0_h$ and $T\\in DS$ or $E\\subset \\cal L^0$ and $T\\in DS^+$ .", "Then for every $f\\in E$ and a bounded Besicovitch sequence $\\lbrace \\beta _k \\rbrace $ the averages (REF ) converge a.e.", "to some $\\widehat{f} \\in E$ ." ], [ "Wiener-Wintner-type ergpdic theorems in fully symmetric spaces", "Assume that $(Y,\\nu )$ is a finite measure space and $\\phi :Y\\rightarrow Y$ a measure preserving transformation (m.p.t.).", "If $(\\Omega , \\mu )$ is a measure space, $\\tau :\\Omega \\rightarrow \\Omega $ a m.p.t., $f:\\Omega \\rightarrow \\mathbb {C}$ , and $g\\in L^1(Y)$ , denote $a_n(f,g)(\\omega ,y)=\\frac{1}{n}\\sum _{k=0}^{n-1}f(\\tau ^k\\omega )g(\\phi ^ky)$ Here is an extension of Bourgain's Return Times theorem to $\\sigma -$ finite measure [1].", "Theorem 6.1 Let $(\\Omega ,\\mu )$ be a $\\sigma -$ finite measure space, $\\tau : \\Omega \\rightarrow \\Omega $ be a m.p.t., and let $A\\subset \\Omega $ , $\\mu (A)<\\infty $ .", "Then there exists a set $\\Omega _A \\subset \\Omega $ such that $\\mu (\\Omega \\setminus \\Omega _A)=0$ and for any $(Y,\\nu ,\\phi )$ and $g\\in \\cal L^1(Y)$ the averages $a_n(\\chi _A,g)(\\omega ,y)=\\frac{1}{n}\\sum _{k=0}^{n-1}\\chi _A(\\tau ^k\\omega )g(\\phi ^ky)$ converge $\\nu -$ a.e.", "for all $\\omega \\in \\Omega _A$ .", "The next theorem is a version of Theorem REF where the functions $\\chi _A$ and $g\\in \\cal L^1(Y)$ are replaced by $f\\in \\cal L^1(\\Omega )$ and $g\\in \\cal L^\\infty (Y)$ , respectively.", "Theorem 6.2 Let $(\\Omega ,\\mu )$ be a measure space, $\\tau : \\Omega \\rightarrow \\Omega $ a m.p.t., and let $f\\in \\cal L^1(\\Omega )$ .", "Then there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that for any $(Y,\\nu ,\\phi )$ and $g\\in \\cal L^\\infty (Y)$ the averages (REF ) converge $\\nu -$ a.e.", "for all $\\omega \\in \\Omega _f$ .", "Assume first that $\\mu $ is $\\sigma -$ finite.", "Fix $f\\in \\cal L^1(\\Omega )$ .", "Then there exist $\\lbrace \\lambda _{m,i}\\rbrace \\subset \\mathbb {C}$ and $A_{m,i}\\subset \\Omega $ with $\\mu (A_{m,i})<\\infty $ , $m=1,2,\\dots $ , $1\\le i\\le l_m$ , such that $\\Vert f-f_m\\Vert _1\\rightarrow 0, \\text{ \\ where \\ } f_m=\\sum _{i=1}^{l_m}\\lambda _{m,i}\\chi _{A_{m,i}}.$ If $\\Omega _{m,j}=\\left\\lbrace \\omega \\in \\Omega : \\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|f-f_m|(\\tau ^k(\\omega )) >\\frac{1}{j} \\right\\rbrace ,$ then, due to the maximal ergodic inequality, we have $\\mu (\\Omega _{m,j})\\le j\\Vert f-f_m\\Vert _1,$ which implies that $\\mu (\\cap _m \\Omega _{m,j})=0$ for a fixed $j$ .", "Therefore, denoting $\\Omega _0=\\Omega \\setminus \\bigcup _j\\bigcap _m \\Omega _{m,j},$ we obtain $\\mu (\\Omega \\setminus \\Omega _0)=0$ .", "If $\\omega \\in \\Omega _0$ , then $\\omega \\notin \\Omega _{m_j,j}$ for every $j$ and some $m_j$ , and therefore $\\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|f-f_{m_j}|(\\tau ^k(\\omega ))\\le \\frac{1}{j} \\text{ \\ for all \\ } j \\text{ \\ and \\ } \\omega \\in \\Omega _0.$ Now, by Theorem REF , there exist $\\Omega _{j,i}\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _{j,i})=0$ such that for every $(Y,\\nu ,\\phi )$ and $g\\in \\cal L^\\infty (Y)$ the averages $\\frac{1}{n}\\sum _{k=0}^{n-1}\\chi _{A_{m_j,i}}(\\tau ^k(\\omega ))g(\\phi ^ky)$ converge $\\nu -$ a.e.", "for all $\\omega \\in \\Omega _{j,i}$ .", "Then, letting $\\Omega _f=\\bigcap \\limits _{j=1}^\\infty \\bigcap \\limits _{i=1}^{l_{m_j}}\\Omega _{j,i}\\bigcap \\Omega _0,$ we obtain $\\mu (\\Omega \\setminus \\Omega _f)=0$ .", "If we pick any $(Y,\\nu ,\\phi )$ and $g\\in \\cal L^\\infty (Y)$ , then the averages $a_n(f_{m_j},g)(\\omega ,y)$ converge $\\nu -$ a.e.", "for every $j$ and all $\\omega \\in \\Omega _f$ , and it follows that there are $Y_0\\subset Y$ with $\\nu (Y\\setminus Y_0)=0$ and $C>0$ such that $|g(\\phi ^ky)|\\le C$ for all $k$ and $y\\in Y_0$ and $\\liminf _n\\operatorname{Re}a_n(f_{m_j},g)(\\omega ,y)=\\limsup _n\\operatorname{Re}a_n(f_{m_j},g)(\\omega ,y),$ $\\liminf _n\\operatorname{Im}a_n(f_{m_j},g)(\\omega ,y)=\\limsup _n\\operatorname{Im}a_n(f_{m_j},g)(\\omega ,y),$ for all $y\\in Y_0$ , $k$ , and $\\omega \\in \\Omega _f$ .", "Let $\\omega \\in \\Omega _f$ and $y\\in Y_0$ .", "Given $k$ , taking into account (REF ), we have $\\Delta (\\omega ,y)=\\limsup _n\\operatorname{Re}a_n(f,g)(\\omega ,y)-\\liminf _n\\operatorname{Re}a_n(f,g)(\\omega ,y)=$ $\\limsup _n\\operatorname{Re}a_n(f-f_{m_j},g)(\\omega ,y)-\\liminf _n\\operatorname{Re}a_n(f-f_{m_j},g)(\\omega ,y)\\le $ $\\le 2\\sup _n a_n(|f-f_{m_j}|,|g|)(\\omega ,y)\\le 2C \\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|f-f_{m_j}|(\\tau ^k(\\omega ))\\le \\frac{2C}{j}.$ Therefore $\\Delta (\\omega ,y)=0$ .", "Similarly, $\\limsup _n\\operatorname{Im}a_n(f,g)(\\omega ,y)=\\liminf _n\\operatorname{Im}a_n(f,g)(\\omega ,y),$ and we conclude that the averages (REF ) converge $\\nu -$ a.e.", "and all $\\omega \\in \\Omega _f$ .", "If $\\mu $ is an arbitrary measure, we observe that, since $f\\in \\cal L^1(\\Omega )$ , the restriction of $\\mu $ on the set $\\lbrace \\omega \\in \\Omega : f(\\tau ^k(\\omega ))\\ne 0\\rbrace $ is $\\sigma -$ finite for each $k$ , which reduces the argument to the case of $\\sigma -$ finite measure space $(\\Omega , \\mu )$ .", "Now we extend Theorem REF to $\\cal R_\\mu =\\cal R_\\mu (\\Omega )$ .", "Theorem 6.3 Let $(\\Omega , \\mu )$ be a measure space, $\\tau : \\Omega \\rightarrow \\Omega $ be a m.p.t., and let $f\\in \\cal R_\\mu $ .", "Then there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that for any finite measure space $(Y,\\nu )$ , any measure preserving transformation $\\phi : Y\\rightarrow Y$ , and any $g\\in \\cal L^\\infty (Y)$ the averages (REF ) converge $\\nu -$ a.e.", "for all $\\omega \\in \\Omega _f$ .", "Due to Proposition REF , given a natural $m$ , there exists $f_m\\in \\cal L^1(\\Omega )$ and $h_m\\in \\cal L^\\infty (\\Omega )$ such that $f=f_m+h_m$ and $\\Vert h_m\\Vert _\\infty \\le \\frac{1}{m}$ .", "Then there is $\\Omega _0 \\subset \\Omega $ such that $\\mu (\\Omega \\setminus \\Omega _0)=0$ and $h_m(\\omega )\\le \\frac{1}{m}$ for all $m$ and $\\omega \\in \\Omega _0$ .", "By Theorem REF , as $\\lbrace f_m\\rbrace _{m=1}^\\infty \\subset \\cal L^1(\\Omega )$ , for every $m$ there is a set $\\Omega _m \\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _m)=0$ such that for every $(Y,\\nu ,\\phi )$ and $g\\in \\cal L^1(Y)$ the averages $a_n(f_m,g)(\\omega ,y)=\\frac{1}{n}\\sum _{k=0}^{n-1}f_m(\\tau ^k(\\omega ))g(\\phi ^k(y))$ converge $\\nu -$ a.e.", "for all $\\omega \\in \\Omega _m$ .", "Therefore, if $\\Omega _f=\\cap _{m=0}^\\infty \\Omega _m$ , then $\\mu (\\Omega \\setminus \\Omega _f)=0$ , $h_m(\\omega )\\le \\frac{1}{m}$ for all $m$ and $\\omega \\in \\Omega _f$ , and for every $(Y,\\nu ,\\phi )$ and $g\\in L^1(Y)$ , the averages (REF ) converge $\\nu -$ a.e.", "for all $m$ and $\\omega \\in \\Omega _f$ .", "Fix $\\omega \\in \\Omega _f$ , $(Y,\\nu ,\\phi )$ , $g\\in \\cal L^1(Y,\\nu )$ and show that the averages (REF ) converge $\\nu -$ a.e.", "As the averages (REF ) converge $\\nu -$ a.e.", "for each $m$ , there is a set $Y_1\\subset Y$ with $\\nu (Y\\setminus Y_1)=0$ such that the sequence (REF ) converges for every $m$ and $y\\in Y_1$ .", "Also, since the averages $\\frac{1}{n}\\sum _{k=0}^{n-1}|g|(\\phi ^k(y))$ converge $\\nu -$ a.e., there is a set $Y_2\\subset Y$ such that $\\nu (Y\\setminus Y_2)=0$ and the sequence $\\frac{1}{n}\\sum _{k=0}^{n-1}|g|(\\phi ^k(y))$ converges for all $y\\in Y_2$ .", "Then, letting $Y_0=Y_1\\cap Y_2$ , we conclude that $\\nu (Y\\setminus Y_0)=0$ , $\\frac{1}{n}\\sum _{k=0}^{n-1}|g|(\\phi ^ky)<\\infty $ , and the sequence (REF ) converges for all $m$ and $y\\in Y_0$ .", "Now, if $y\\in Y_0$ , we have $\\liminf _n a_n(f_m,g)(\\omega ,y)=\\limsup _n a_n(f_m,g)(\\omega ,y),$ which implies that, for every $m$ , $\\Delta (\\omega )=\\limsup _n a_n(f,g)(\\omega ,y)-\\liminf _na_n(f,g)(\\omega ,y)=$ $=\\limsup _n a_n(h_m,g)(\\omega ,y)-\\liminf _na_n(h_m,g)(\\omega ,y)\\le $ $\\le 2\\sup _n \\frac{1}{n}\\sum _{k=0}^{n-1}|h_m(\\tau ^k(\\omega ))|\\cdot |g(\\phi ^k(y))|\\le \\frac{2}{m}\\sup _n \\frac{1}{n}\\sum _{k=0}^{n-1}|g|(\\phi ^k(y)).$ Therefore $\\Delta (\\omega )=0$ for every $y\\in Y_0$ , that is, the averages (REF ) converge $\\nu -$ a.e.", "Letting in Theorem REF $Y=\\mathbb {C}_1=\\lbrace y\\in \\mathbb {C}: |y|=1\\rbrace $ with Lebesgue measure $\\nu $ , $\\phi _\\lambda (y)=\\lambda y$ , $y\\in Y$ , for a given $\\lambda \\in Y$ , and $g(y)=y$ whenever $y\\in Y$ , we obtain Wiener-Wintner theorem for $\\cal R_\\mu $ .", "Theorem 6.4 Let $(\\Omega , \\mu )$ be a measure space, $\\tau : \\Omega \\rightarrow \\Omega $ be a m.p.t.", "If $f\\in \\cal R_\\mu $ , then there is a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that the sequence $\\frac{1}{n} \\sum _{k=0}^{n-1} \\lambda ^kf(\\tau ^k(\\omega ))$ converges for all $\\omega \\in \\Omega _f$ and $\\lambda \\in \\mathbb {C}_1$ .", "Let $P(k)=\\sum _{j=1}^s z_j\\lambda _j^k, \\ k=0,1,2,\\dots $ be a trigonometric polynomial (see Section 5).", "Then, by linearity, Theorem REF implies the following.", "Corollary 6.1 If $(\\Omega ,\\mu )$ is a measure space, $\\tau : \\Omega \\rightarrow \\Omega $ is a m.p.t.", "and $f\\in \\cal R_\\mu $ , then there is a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that the sequence $a_n(\\lbrace P(k)\\rbrace ,f)(\\omega )=\\frac{1}{n}\\sum _{k=0}^{n-1}P(k)f(\\tau ^k(\\omega ))$ converges for every $\\omega \\in \\Omega _f$ and any trigonometric polynomial $\\lbrace P(k)\\rbrace $ .", "We will need the following.", "Proposition 6.1 If $(\\Omega ,\\mu )$ is a measure space, $\\tau : \\Omega \\rightarrow \\Omega $ is a m.p.t.", "and $f\\in \\cal L^1(\\Omega )\\cap \\cal L^\\infty (\\Omega )$ , then there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that the sequence $\\frac{1}{n} \\sum _{k=0}^{n-1}b_kf(\\tau ^k(\\omega ))$ converges for every $\\omega \\in \\Omega _f$ and any bounded Besicovitch sequence $\\lbrace b_k\\rbrace $ .", "By Corollary REF , there exists a set $\\Omega _{f,1}\\subset \\Omega $ , $\\mu (\\Omega \\setminus \\Omega _{f,1})=0$ , such that the sequence $\\frac{1}{n}\\sum _{k=0}^{n-1}P(k)f(\\tau ^k\\omega )$ converges for every $\\omega \\in \\Omega _{f,1}$ and any trigonometric polynomial $\\lbrace P(k)\\rbrace $ .", "Also, since $f\\in \\cal L^\\infty (\\Omega )$ , there is a set $\\Omega _{f,2}\\subset \\Omega $ , $\\mu (\\Omega \\setminus \\Omega _{f,2})=0$ , such that $|f(\\tau ^k\\omega )|\\le \\Vert f\\Vert _\\infty $ for every $k$ and $\\omega \\in \\Omega _{f,2}$ .", "If we set $\\Omega _f=\\Omega _{f,1}\\cap \\Omega _{f,2}$ , then $\\mu (\\Omega \\setminus \\Omega _f)=0$ .", "Now, let $\\omega \\in \\Omega _f$ , and let $\\lbrace b_k\\rbrace $ be a Besicovitch sequence.", "Fix $\\epsilon >0$ , and choose a trigonometric polynomial $P(k)$ to satisfy condition (REF ).", "Then we have $\\Delta (\\omega )=\\limsup _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f)(\\omega )-\\liminf _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f)(\\omega )=$ $=\\limsup _n\\operatorname{Re}a_n(\\lbrace b_k-P(k)\\rbrace ,f)(\\omega )-\\liminf _n\\operatorname{Re}a_n(\\lbrace b_k-P(k)\\rbrace ,f)(\\omega )\\le $ $\\le 2\\Vert f\\Vert _\\infty \\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|b_k-P(k)|<2\\Vert f\\Vert _\\infty \\epsilon $ for all sufficiently large $n$ .", "Therefore $\\Delta (\\omega )=0$ , and we conclude that the sequence $\\operatorname{Re}\\frac{1}{n} \\sum _{k=0}^{n-1}b_kf(\\tau ^k\\omega )$ converges.", "Similarly, we obtain convergence of the sequence $\\operatorname{Im} \\frac{1}{n} \\sum _{k=0}^{n-1}b_kf(\\tau ^k\\omega )$ , which completes the proof.", "Theorem 6.5 Let $(\\Omega ,\\mu )$ be a measure space.", "If $f\\in \\cal L^1(\\Omega )$ , then there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ , such that the sequence $a_n(\\lbrace b_k\\rbrace ,f)(\\omega )=\\frac{1}{n}\\sum _{k=0}^{n-1}b_kf(\\tau ^k\\omega )$ converges for every $\\omega \\in \\Omega _f$ and any bounded Besicovitch sequence $\\lbrace b_k\\rbrace $ .", "Let a sequence $\\lbrace f_m\\rbrace \\subset \\cal L^1(\\Omega )\\cap \\cal L^\\infty (\\Omega )$ be such that $\\Vert f-f_m\\Vert _1\\rightarrow 0$ .", "As in proof of Theorem REF , we construct a subsequence $\\lbrace f_{m_j}\\rbrace $ and a set $\\Omega _0\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _0)=0$ such that $\\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|f-f_{m_j}|(\\tau ^k\\omega )\\le \\frac{1}{j} \\text{ \\ for all \\ } j \\text{ \\ and \\ } \\omega \\in \\Omega _0.$ By Proposition REF , given $j$ , there is $\\Omega _j\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _j)=0$ such that the sequence $\\frac{1}{n}\\sum _{k=0}^{n-1}b_kf_{m_j}(\\tau ^k\\omega )$ converges for every $\\omega \\in \\Omega _j$ and any Besicovitch sequence $\\lbrace b_k\\rbrace $ .", "If we set $\\Omega _f=\\cap _{j=1}^\\infty \\Omega _j \\cap \\Omega _0$ , then $\\mu (\\Omega \\setminus \\Omega _f)=0$ , and for any $\\omega \\in \\Omega _f$ and any bounded Besicovitch sequence $\\lbrace b_k\\rbrace $ such that $\\sup _k|b_k|\\le C$ we have $\\Delta (\\omega )=\\limsup _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f)(\\omega )-\\liminf _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f)(\\omega )=$ $=\\limsup _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f-f_{m_j})(\\omega )-\\liminf _n\\operatorname{Re}a_n(\\lbrace b_k\\rbrace ,f-f_{m_j})(\\omega )\\le $ $\\le 2\\sup _n\\frac{1}{n} \\sum _{k=0}^{n-1}|b_k|\\cdot |f-f_{m_j}|(\\tau ^k\\omega )\\le \\frac{2C}{j}.$ Therefore $\\Delta (\\omega )=0$ , hence the sequence $\\operatorname{Re}\\frac{1}{n}\\sum _{k=0}^{n-1}b_kf(\\tau ^k\\omega )$ converges.", "Similarly, we derive convergence of the sequence $\\operatorname{Im}\\frac{1}{n}\\sum _{k=0}^{n-1}b_kf(\\tau ^k\\omega )$ , and the proof is complete.", "Taking into account that the sequence $\\lbrace b_k\\rbrace $ is bounded, we obtain, as in the proof of Theorem REF , the following extension of Wiener-Wintner theorem.", "Theorem 6.6 Let $(\\Omega ,\\mu )$ be a measure space, and let $\\tau :\\Omega \\rightarrow \\Omega $ be a m.p.t.", "Given $f\\in \\cal R_\\mu $ , then there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that the sequence (REF ) converges for every $\\omega \\in \\Omega _f$ and every bounded Besicovitch sequence $\\lbrace b_k\\rbrace $ .", "Finally, in view of Proposition REF , we have the following.", "Theorem 6.7 Let $(\\Omega ,\\mu )$ be an infinite measure space, and let $\\tau :\\Omega \\rightarrow \\Omega $ be a m.p.t.", "Assume that $E=E(\\Omega )$ is a fully symmetric space such that $\\mathbf {1}\\notin E$ .", "Then for every $f\\in E$ there exists a set $\\Omega _f\\subset \\Omega $ with $\\mu (\\Omega \\setminus \\Omega _f)=0$ such that the sequence (REF ) converges for every $\\omega \\in \\Omega _f$ and every bounded Besicovitch sequence $\\lbrace b_k\\rbrace $ ." ] ]
1612.05802
[ [ "Edge Momentum Transport by Neutrals: an Interpretive Numerical Framework" ], [ "Abstract Due to their high cross-field mobility, neutrals can contribute to momentum transport even at the low relative densities found inside the separatrix and they can generate intrinsic rotation.", "We use a charge-exchange dominated solution to the neutral kinetic equation, coupled to neoclassical ions, to evaluate the momentum transport due to neutrals.", "Numerical solutions to the drift-kinetic equation allow us to cover the full range of collisionality, including the intermediate levels typical of the tokamak edge.", "In the edge there are several processes likely to contribute to momentum transport in addition to neutrals.", "Therefore, we present here an interpretive framework that can evaluate the momentum transport through neutrals based on radial plasma profiles.", "We demonstrate its application by analysing the neutral angular momentum flux for an L-mode discharge in the ASDEX Upgrade tokamak.", "The magnitudes of the angular momentum fluxes we find here due to neutrals of up to $1{-}2\\;\\mathrm{N\\, m}$ are comparable to the net torque on the plasma from neutral beam injection, indicating the importance of neutrals for rotation in the edge." ], [ "Introduction", "Momentum transport in the tokamak edge is a crucial issue since rotation suppresses magnetohydrodynamic instabilities such as resistive wall modes [1] and flow shear in the edge suppresses turbulence, leading to high-confinement mode (H-mode) operation [2].", "Neutral particles are always present in the edge of the confined plasma volume, and despite their low relative density can contribute strongly to transport due to their high cross-field mobility, as has been demonstrated theoretically [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].", "The influence of neutrals on confinement has also been observed experimentally [13], [14], [15] and the poloidal location of gas-fuelling, inboard versus outboard, has been seen to be important for the low- to high-confinement transition threshold [16], [17], [18].", "It has been shown previously that when neutrals dominate the angular momentum transport, the radial electric field and hence toroidal rotation can be self-consistently calculated [10], [11], [12].", "It was also shown that the neutrals can generate intrinsic rotation, since a toroidal heat flux in the ions drives a radial flux of angular momentum through neutrals, as explained in [11].", "However, there will be other contributions, potentially of similar or greater magnitude, for instance from turbulence, ion orbit losses [19], [20], non-axisymmetric magnetic fields [21] or finite orbit width effects [22].", "It is therefore important to evaluate the momentum transport due to neutrals from experimental profiles, so that the magnitudes of the various effects can be compared.", "Analytical solutions [10], [11] rely on asymptotic ordering of the collisionality into the Pfirsch-Schlüter (high collisionality) or banana (low collisionality) regimes.", "However, the conditions typical in the edge plasma produce order unity collisionality, for example $\\nu _{\\mathrm {ii}}L_{\\Vert }/v_{T}\\sim 0.2{-}0.5$ for the profiles we show in Section , where $v_{T}$ is the thermal velocity, $\\nu _{\\mathrm {ii}}$ the ion-ion collision rate and $L_{\\Vert }=\\oint d\\theta \\,\\left(dl_{\\Vert }/d\\theta \\right)/2\\pi $ the parallel connection length.", "In order to relax this restriction numerical solutions of the drift-kinetic equation must be used.", "Recently, the radial electric field and toroidal rotation have been calculated numerically from the constraint that the angular momentum flux through neutrals vanishes in steady state in the absence of external torque [12] that is, as noted above, assuming that the neutrals dominate the momentum transport and also that the radial gradient of the neutrals is the dominant profile gradient.", "The latter assumption makes the calculation of the momentum flux local (otherwise it would depend on, for instance, the radial gradient of the toroidal rotation and hence on second derivatives of the density, temperature, etc.).", "Here, in order to avoid these assumptions, we introduce a new approach which takes the background plasma profiles, including the radial electric field, as given, allowing us to calculate the angular momentum flux through neutrals directly, without requiring that they are the only channel for momentum transport.", "This provides an interpretive tool that can be applied to experimental data in order to compare the magnitude of the neutral momentum transport to other mechanisms and to external sources of momentum such as neutral beam injection (NBI) heating.", "The interpretive approach is derived in Section and applied to an L-mode discharge from ASDEX Upgrade (AUG) in Section .", "We discuss the implications of the results in Section ." ], [ "Neutral momentum transport", "We outline here the calculation of the angular momentum flux through neutrals, which are coupled by charge exchange to kinetic ions.", "Integrated over a flux surface, the radial flux of toroidal angular momentum carried by the neutral population is $V^{\\prime }\\left\\langle R\\hat{\\zeta }\\cdot \\Pi _{\\mathrm {n}}\\cdot \\nabla \\psi \\right\\rangle $ where $\\Pi _{\\mathrm {n}}=m_{\\mathrm {i}}\\int d^{3}v\\,vvf_{\\mathrm {n}}$ is the stress tensor of the neutrals, $f_{\\mathrm {n}}$ is the distribution function of the neutrals, $2\\pi \\psi $ is the poloidal flux, $V$ is the volume enclosed by a flux surface, a prime denotes a derivative with respect to $\\psi $ , $R$ is the major radius, $\\hat{\\zeta }=\\nabla \\zeta /\\left|\\nabla \\zeta \\right|$ with $\\zeta $ the toroidal angle (increasing in the co-current direction), and $m_{\\mathrm {i}}$ is the mass of the ions or neutrals.", "Our solution to the kinetic equation for $f_{\\mathrm {n}}$ relies on several approximations.", "We take a short charge-exchange (CX) mean-free-path (MFP) expansion; although the MFP is not always short compared to profile length scales in the edge, it was found in [9] that the approximation is surprisingly accurate, compared to a full solution allowing arbitrary MFP for a special class of self-similar profiles.", "We neglect ionization, but this is most important for the determination of the neutral density profile, which we take as an input; ionization affects the momentum transport only by changing the effective collision rate [6].", "We use a simplified CX collision operator [4] that allows us to close the neutral kinetic equation in an efficient way [6], as described below.", "These approximations allow us to avoid the computational expense of Monte Carlo neutral codes, e.g.", "EIRENE [23], while retaining coupling to the kinetic ion distribution, rather than only a drifting Maxwellian as implemented in [23].", "The steady state neutral kinetic equation then takes the form [6] $v\\cdot \\nabla f_{\\mathrm {n}}=\\frac{1}{\\tau _{\\mathrm {CX}}}\\left(\\frac{n_{\\mathrm {n}}}{n_{\\mathrm {i}}}f_{\\mathrm {i}}-f_{\\mathrm {n}}\\right),$ where $n_{\\mathrm {n}}$ and $n_{\\mathrm {i}}$ are the neutral and ion densities and $f_{\\mathrm {i}}$ is the ion distribution function.", "$\\tau _{\\mathrm {CX}}^{-1}=n_{\\mathrm {i}}\\left\\langle \\sigma v\\right\\rangle _{\\mathrm {CX}}$ is the characteristic rate for charge-exchange interactions with $\\left\\langle \\sigma v\\right\\rangle _{\\mathrm {CX}}$ the thermal charge exchange rate, where $\\left\\langle \\sigma v\\right\\rangle _{\\mathrm {CX}}=4.21\\times 10^{-14}\\mathrm {\\; m^{3}\\, s^{-1}}$ for Deuterium ions and neutrals at $300\\;\\mathrm {eV}$ [24], which is a typical temperature value of the profiles we use in Section .", "Solving perturbatively for small $\\tau _{\\mathrm {CX}}v_{T}/L$ , where $v_{T}$ is the thermal speed and $L$ is a characteristic length scale of the background profiles, fn(0) =nnnifi, fn(1) =-CXv(nnnifi).", "$f_{\\mathrm {n}}^{(0)}$ contributes a term to the angular momentum flux which, being proportional to $\\Pi _{\\mathrm {i}}$ , is negligible at $\\mathcal {O}(\\delta )$ in the gyroradius expansion [25], where $\\delta =\\rho /L$ and $\\rho $ is the gyroradius.", "Thus we need keep only $f_{\\mathrm {n}}^{(1)}$ and so V'Rn =-miCXV'Rd3v (v)(v)v(nnnifi) -miCXdd(V'Rnnnid3v(v)(v)2fi), neglecting $\\nabla ^{2}\\psi $ and using the identities $\\left\\langle \\nabla \\cdot A\\right\\rangle =\\frac{1}{V^{\\prime }}\\frac{d}{d\\psi }\\left(V^{\\prime }\\left\\langle \\nabla \\psi \\cdot A\\right\\rangle \\right)$ for any $A$ [25] and $vv:\\nabla \\left(R\\hat{\\zeta }\\right)=0$This identity follows from the fact that $vv$ is a symmetric tensor, while $\\nabla \\left(R\\hat{\\zeta }\\right)$ is antisymmetric..", "Including the gyroradius correction, the ion distribution function at the particle position $r$ is fi(r) =fi,gc0(r)-e1Tifi,gc0(r)-fi,gc0(r)+gi(r), where $f_{\\mathrm {i,gc0}}$ is a Maxwellian, $\\rho =r-R_{\\mathrm {gc}}$ is the gyroradius vector with $R_{\\mathrm {gc}}$ the guiding centre position, $\\Phi _{1}=\\Phi -\\left\\langle \\Phi \\right\\rangle $ is the poloidally varying part of the electrostatic potential $\\Phi $ and 0, 1 subscripts refer to the order in $\\delta $ .", "The first two terms do not contribute to () as they are isotropic in $v$ .", "We obtain the non-adiabatic piece of the perturbed distribution function, $g_{\\mathrm {i}}=f_{\\mathrm {i},gc1}+e_{\\mathrm {i}}\\Phi _{1}f_{\\mathrm {i,gc0}}/T_{\\mathrm {i}}$ from numerical solutions of the first order drift kinetic equation using the perfect neoclassical solver (run here in radially-local mode) [26], which assume that the flow is subsonic, $V\\sim \\mathcal {O}(\\delta v_{T})$ .", "As noted in the introduction, this system was used to explore the effects of magnetic geometry and collisionality on intrinsic rotation driven by momentum transport through neutrals [12], [27].", "In the next section we introduce the interpretive framework which can be used to diagnose experimental results." ], [ "Interpretive modelling of momentum flux", "We have evaluated the angular momentum flux through neutrals for L-mode profiles from the AUG discharge #26601, shown in FIG.", "REF , in order to demonstrate the potential of our interpretive tool.", "This takes profiles of ion density $n_{\\mathrm {i}}$ and temperature $T_{\\mathrm {i}}$ , radial electric field $E_{r}$ and neutral density $n_{\\mathrm {n}}$ as inputs to calculate the angular momentum flux as described in the previous section.", "It is also possible to constrain the neoclassical solutions with the toroidal rotation profile of any ion species instead of $E_{r}$ .", "For the AUG case, CXRS measurements were performed on the $\\mathrm {He}^{2+}$ impurity to measure the ion temperature (assumed equal to the $\\mathrm {He}^{2+}$ temperature) and infer the radial electric field [28].", "Electron density was measured with the Thomson scattering, Lithium beam and interferometry diagnostics; we neglect the impurity density, taking $n_{\\mathrm {i}}=n_{\\mathrm {e}}$ .", "Profiles are plotted as functions of the normalized flux label $\\rho _{\\mathrm {pol}}=\\sqrt{\\psi /\\psi _{\\mathrm {sep}}}$ where $\\psi _{\\mathrm {sep}}$ is the value of $\\psi $ at the separatrix and we take $\\psi =0$ at the magnetic axis.", "To obtain neutral density profiles, we ran KN1D [29], extending the plasma profiles (taking $T_{\\mathrm {e}}=T_{\\mathrm {i}}$ ) into the scrape-off layer as exponentials, with decay lengths of 3 cm for the density and 0.5 cm for the temperature.", "KN1D's molecular pressure input was chosen to give a value of $n_{\\mathrm {n}}\\approx 10^{16}\\;\\mathrm {m^{-3}}$ at the separatrix, consistent with the low-field side neutral density modelled in [30].", "We use the output of KN1D to set the flux surface average of the neutral density $\\left\\langle n_{\\mathrm {n}}\\right\\rangle $ , which we keep fixed while choosing different poloidal profiles in FIGs.", "REF and REF .", "Within a drift-orbit width of the separatrix, the ion distribution function is likely to depart strongly from the conventional neoclassical prediction due to orbit loss effects [19], [20], which we do not consider here.", "We therefore exclude this region in the presentation of our results, as indicated by the shaded regions in the figures.", "Figure: Comparison of outward flux of toroidal angular momentum through neutralsfor different poloidal profile shapes: δ\\delta -function at outboardmidplane (blue, dotted), Gaussian with width π/5 rad \\pi /5\\;\\mathrm {rad}centred at outboard midplane (green, solid) and uniform (red, dashed).The total angular momentum flux from the NBI source is indicated asthe horizontal, dashed line.", "Positive values represent radially outwardflux of co-current angular momentum.", "Grey shaded area is within apoloidal gyroradius of the separatrix.For poloidally uniform neutrals, the outward flux of co-current toroidal angular momentum passing through a flux surface reaches nearly $-1\\;\\mathrm {N\\, m}$ , as shown in FIG.", "REF .", "The discharge had $1.1\\;\\mathrm {MW}$ of NBI heating with $39.8\\;\\mathrm {keV}$ particles; the momentum is deposited at a tangency radius $R_{\\mathrm {NBI}}=0.93\\;\\mathrm {m}$ and so the external momentum input from the NBI is $1.05\\;\\mathrm {N\\, m}$ (indicated as the dashed horizontal line in FIG.", "REF ), of a similar magnitude to the momentum flux carried by the neutrals.", "This gives the net outward flux of angular momentum carried by all mechanisms through our simulation domain at the plasma edge, assuming that the NBI momentum is predominantly deposited in the core plasma and provides the only source of angular momentum.", "The neutrals thus carry a significant angular momentum flux near the plasma edge and can be expected to play an important role in regulating the plasma rotation in L-mode.", "For comparison one can make a simple estimate of the angular momentum flux.", "Assume that the fast neutrals, those coming from a charge exchange reaction with a hot ion in the confined plasma, escape immediately from the confined region.", "Then the total momentum carried out of the plasma is $\\Gamma _{\\mathrm {n}}m_{\\mathrm {i}}\\bar{V}_{\\zeta }R_{\\mathrm {out}}\\approx 0.3\\;\\mathrm {N\\, m}$ where $\\Gamma _{\\mathrm {n}}\\approx 2\\times 10^{21}\\;\\mathrm {s}^{-1}$ is the total inward flux of neutrals (estimated from the flux density across the separatrix of $10^{20}\\;\\mathrm {m^{-2}\\, s^{-1}}$ simulated by KN1D), $\\bar{V}_{\\zeta }\\approx 20\\;\\mathrm {km}\\,\\mathrm {s}^{-1}$ is a typical toroidal rotation velocity and $R_{\\mathrm {out}}\\approx 2\\;\\mathrm {m}$ is a typical radius in the outboard edge region of the plasma.", "This estimate is similar in magnitude to our prediction but has the opposite sign; this is a consequence of the very different assumptions made.", "For this estimate each neutral experiences only one CX interaction and acts purely as a momentum sink.", "On the other hand in the short MFP model we use, each neutral undergoes several CX interactions.", "The direction of the flux (inward or outward) is then influenced by the directions of the gradients in both neutral density and plasma toroidal rotation, heat flux, etc.", "because neutrals exchange momentum between different flux surfaces.", "In particular there can be an inward pinch of momentum due to the density gradient of the neutrals; on a given flux surface there are more neutrals coming from the outside than the inside, all carrying the momentum picked up from a CX reaction with an ion.", "Note that, as we see in FIG.", "REF , the toroidal rotation does not change much across our domain; thus the momentum carried by each inward or outward going particle will be similar, so that this effect gives rise to an inward flux of angular momentum.", "Poloidal localization of the neutrals can enhance the angular momentum flux.", "In FIG.", "REF we compare the momentum flux for three different assumed poloidal profiles: uniform, Gaussian (with standard deviation $\\pi /5\\;\\mathrm {rad}$ ) or $\\delta $ -function.", "The localized profiles are centred at the outboard midplane.", "For ease of comparison we have used the same flux-surface averaged neutral density profile for all of the curves.", "In reality, the neutral distribution will be some combination of a roughly uniform piece from wall recycling and a localized part near the gas fuelling valve (represented by the Gaussian shape here), so the enhancement above the uniform level due to outboard localization could be expected to be more moderate than the limiting case shown.", "Previous work, both analytical [10], [11] and numerical [12], used $\\delta $ -function localized neutrals for convenience.", "We can see from FIG.", "REF that there is not a major difference between this approximation and the case of neutrals localized as a Gaussian, with both the trends and order of magnitude being correctly captured, although the sharper localization of the $\\delta $ -function does further enhance the momentum flux.", "Figure: Comparison of outward flux of toroidal angular momentum through neutralsfor different poloidal positions: inboard midplane (blue, dashed),outboard midplane (green, solid) and the nearest poloidal positionto the X-point (red, dotted).", "Poloidal profile of the neutrals isGaussian with width π/5 rad \\pi /5\\;\\mathrm {rad}.", "The total angular momentumflux from the NBI source is indicated as the horizontal, dashed line.Positive values represent radially outward flux of co-current angularmomentum.", "Grey shaded area is within a poloidal gyroradius of theseparatrix.The neutrals provide more effective transport of toroidal angular momentum when they are on the outboard side of the tokamak than on the inboard side or near the X-point, due to the larger major radius and poloidal magnetic field $B_{\\mathrm {p}}$ , as can be seen from (), noting that $\\left|\\nabla \\psi \\right|=RB_{\\mathrm {p}}$ .", "This is confirmed by FIG.", "REF which compares the momentum flux through Gaussian localized neutrals centred at the outboard midplane, inboard midplane and adjacent to the X-point, highlighting the importance of the poloidal location of the neutrals for the angular momentum flux that they carry." ], [ "Discussion", "We have demonstrated here that charge-exchanging neutrals can carry a significant angular momentum flux in the tokamak edge.", "The formalism developed in [10], [11] and first implemented numerically in [12] has been extended, to enable interpretive studies evaluating the momentum transport through neutrals for experimental profiles and equilibria.", "In this interpretive mode, the radial variation of the background profiles does not have to be neglected compared to the radial gradient of the neutral density.", "We model an L-mode discharge from AUG to demonstrate this capability and show that a significant flux of angular momentum, comparable in magnitude to the total momentum input from NBI, can be carried by the neutrals, motivating further application to experimental data in future.", "The approach taken here allows rapid experimentation with parameters and profiles and provides both qualitative insight and at least order of magnitude estimates of the momentum transport due to neutrals.", "Despite the limitations of the modelling described here, some conclusions are clear.", "The strength of the neutral momentum transport is much larger for neutrals located on the outboard side of the tokamak than on the inboard side.", "This is due both to the smaller moment of force at smaller major radius and to the smaller physical-space gradients of flux-function plasma profiles which have poloidally constant gradients in $\\psi $ -space, as $\\left|\\nabla \\psi \\right|=RB_{\\mathrm {p}}$ is smaller on the inboard side.", "Likewise near the X-point, where $B_{\\mathrm {p}}$ is small, the influence of the neutrals will be weak.", "Thus the influence of neutrals on the plasma rotation may be minimized by fuelling from the inboard side or X-point.", "On the other hand, if it is desired to drive intrinsic rotation using the neutral angular momentum flux [10], [11], [12], then this will compete more effectively with other momentum transport channels if the neutrals are located on the outboard side and may be able to generate substantial radial electric fields in H-mode pedestals.", "Intrinsic rotation is particularly important for future tokamaks such as ITER since the angular momentum source from NBI heating will be weaker in larger devices.", "The interesting case is that with steep temperature profiles in H-mode and although our modelling is restricted to L-mode plasmas, it is in the region where the edge transport barrier will form in H-mode that neutrals are most important.", "However, modelling steep temperature gradients is extremely challenging since the deviation of the bulk ion distribution from a Maxwellian distribution is not small, so that a non-linear collision operator is needed, and the steep gradients also necessitate radially-global solutions.", "State of the art numerical solutions of the neutral kinetic equation couple only to a drifting-Maxwellian plasma [23].", "However, the temperature gradient driven departure of the ions from a local Maxwellian distribution drives an angular momentum flux in the neutral distribution function [4], [11].", "Thus when the external torque is large, so that the rotation-driven momentum flux dominates, the drifting-Maxwellian description of the bulk plasma is sufficient, but when there are steep temperature gradients, or for intrinsic rotation where the external torque vanishes, a kinetic model for the plasma is needed.", "In the linear region we have considered, where standard neoclassical theory and modelling are valid, the transport is proportional to the gradients, so significantly stronger momentum transport through neutrals may be expected to arise in steep gradient regions, such as the H-mode pedestal.", "Quantitative evaluation, requiring the development of more sophisticated codes coupling fully kinetic neutrals to kinetic ions, remains a challenging subject for future research, which can build on existing progress in neoclassical pedestal modelling [31], [26], [32], [33]." ], [ "Acknowledgements", "The authors are grateful to Matt Landreman for advice and help with the perfect code and to Stuart Henderson for assistance with the ADAS database.", "We also thank Prof. Arne Kallenbach and Dr. Rachael McDermott for carefully reading the manuscript.", "This work was supported by the Framework grant for Strategic Energy Research (Dnr.", "2014-5392) and the International Career Grant (Dnr.", "330-2014-6313) from Vetenskapsrådet." ] ]
1612.05577
[ [ "Nonequilibrium photon production in partonic transport simulations" ], [ "Abstract We discuss the implementation of leading-order photon production in nonequilibrium partonic transport simulations.", "In this framework photons are produced by microscopic scatterings, where we include the exact matrix elements of Compton scattering, quark-antiquark annihilation, and bremsstrahlung processes.", "We show how the hard-thermal loop inspired screening of propagators has to be modified such that the microscopic production rate agrees well with the analytically known resummed leading-order rate.", "We model the complete quark-gluon plasma phase of heavy-ion collisions by using the partonic transport approach called the Boltzmann approach to multiparton scatterings (BAMPS), which solves the ultrarelativistic Boltzmann equation with Monte Carlo methods.", "We show photon spectra and elliptic flow of photons from BAMPS and discuss nonequilibrium effects.", "Due to the slow quark chemical equilibration in BAMPS, the yield is lower than the results from other groups; in turn we see a strong effect from scatterings of energetic jet-like partons with the medium.", "This nonequilibrium photon production can dominate the thermal emission, such that the spectra are harder and the photonic elliptic flow of the quark-gluon plasma becomes negative." ], [ "Introduction", "Photons have been used for decades as a valuable probe of the hot matter created in heavy-ion collisions.", "Such matter, as created, e.g., in Au + Au collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL or in Pb + Pb collisions at the Large Hadron Collider (LHC) at CERN, is highly dynamic, and temporarily the energy density is high enough that a so-called quark-gluon plasma (QGP) is formed [1], [2], [3], [4].", "Photons are emitted from the initial nucleon-nucleon contacts (prompt photons), during the subsequent QGP phase and the hot hadron gas (HG) phase (thermal photons and jet-medium photons), by the fragmentation of jets outside the fireball, and finally by the decay of long-lived resonances into real photons.", "The sum of all but the latter sources is called the direct photon contribution, and experiments have succeeded in separating decay from direct photons (ALICE experiment at the LHC [5], PHENIX experiment at RHIC [6], [7], [8]).", "The measurements extend down to transverse momenta $p_T=0.4\\ (0.9)~\\mathrm {GeV}$ for RHIC (LHC), and both find an exponential excess above $N_\\text{coll}$ -scaled prompt photons, which indicates a strong additional source, most likely the shining QGP and hot HG.", "The decay background subtraction is done via different methods, and improvements of the direct photon data are expected in the future.", "Recently, ALICE and PHENIX have measured elliptic and triangular flow of direct photons for several centrality classes (PHENIX, $\\sqrt{s}=200~\\mathrm {GeV}$ : $0\\%-20\\%,20\\%-40\\%,40\\%-60\\%$ [9], ALICE,$\\sqrt{s}=2.76~\\mathrm {TeV}$ : $0\\%-40\\%,$ [10]).", "Both experiments show unexpectedly large flow; however, the measurement is extremely challenging and error bars are still large.", "It is nearly impossible for experiments to disentangle the measured time-integrated photon spectra into their separate sources.", "Theoretical models however, compared with data, do not suffer from this problem.", "The ultimate goal is the explanation of the measured photon spectra by the correct combination of photon production mechanisms of hard and soft quantum chromo or electro dynamical (QCD or QED) interactions and a suitable spacetime evolution of the high-energy heavy-ion collision.", "It is furthermore desirable to explain the elliptic and triangular flow of photons in theoretical models.", "The explanation of elliptic flow for hadrons has required accurate modeling of the initial state and a correct treatment of the nearly hydrodynamic expansion of the medium with suitable viscosity [11], [12], [13], [14], [15].", "It is crucial to also describe the flow of photons; however, its physical picture is substantially different.", "Photons leave the fireball without any further scattering such that their flow originates solely from the production process.", "For now, the large elliptic flow of photons poses a formidable challenge for dynamical models, and the simultaneous description of the yield and the flow of direct photons remains an unsolved puzzle.", "Until now, popular descriptions of the spacetime evolution of heavy-ion collisions are given by fireball parametrizations [16], [17] or hydrodynamic simulations [18], [19], [20], [21], [22], [23], [24], [15].", "Photon spectra can be obtained from those models by folding the spacetime evolution of temperature $T$ and four-velocity $u^\\mu $ over analytically known photon production rates $R(T,u^\\mu )$ [25], [26], [27], [28], [16], [29], [30].", "Transport approaches, such as the Boltzmann approach to multiparton scatterings (BAMPS)  [31], parton-hadron-string dynamics (PHSD) [32], [33] or urqmd [34], [35] have two possibilities to study photon or dilepton production: ”coarse-graining\" of the particle ensemble [36] and obtaining a spacetime background which can be used in the same way as a hydrodynamic evolution as described above, or by using the microscopic cross sections for the desired photon production processes and generating photons within the transport framework directly.", "The latter method will be our choice in the Boltzmann approach to multiparton scatterings (BAMPS) [31], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], which is based on the numerical solution of the Boltzmann equation (BE).", "We show how tree-level and radiative scattering diagrams can be implemented in dynamical transport simulations to nearly reproduce full leading-order (LO) photon rates.", "Subsequently we compute results for the QGP phase of high-energy nuclear reactions.", "The physical difference of our approach compared to hydro, fireball, or coarse-graining approaches is the intrinsic nonequilibrium nature - high or low energetic jets and the non, nearly, or full thermal medium is treated equally.", "Furthermore, spacetime-dependent quark and gluon fugacitiesFor high-energy reactions the number of quarks and antiquarks is very similar, so that it makes sense to speak of an absolute quark fugacity defined as $\\lambda _{q}\\equiv n_{q+\\bar{q}}/n_{q+\\bar{q}}^{\\text{equilibrium}}$ with the density $n$ .", "influence the photon rates by default.", "As a main result, we claim that the photon yield of the QGP can be much smaller than previously thought, due to the small initial quark content of the fireball.", "Furthermore, the pre-equilibrium phase of the QGP does not contribute significantly to yield or elliptic flow of direct photons.", "Second, we show how important nonequilibrium photon production can be for the elliptic flow: energetic particles behave ”jet-like“, and make the elliptic flow for higher transverse momenta negative.", "These results provide necessary complementary aspects to hydrodynamic calculations, which in most cases does not include strong off-equilibrium dynamics.", "Recently, much work is done concerning alternative rates (see, e.g., [50]) or rather ignored effects, such as viscous corrections (e.g., [51]) or unknown sources (e.g., [52]).", "This paper is organized as follows: In Sec.", "we describe our theoretical and numerical transport framework.", "In Sec.", "we introduce our implementation of $2 \\leftrightarrow 2$ and $2 \\rightarrow 3$ (radiative) photon production processes, their comparisons with hard thermal loop (HTL) resummed rates and explain the handling of interference effects.", "We discuss the scaling behavior with quark fugacities in Sec.", "REF .", "In Sec.", "REF we show qualitatively in a box model how jet-medium interactions and a flowing thermal medium compete for elliptic flow and clarify the term jet-photon conversion in Sec.", "REF .", "Section  is devoted to results for transverse momentum spectra and elliptic flow of photons from the QGP phase and its physical implications.", "Finally, we conclude in Sec.", "and give an outlook on possible next steps.", "Our units are $\\hbar =c=k=1$ ; the spacetime metric is given by $g^{\\mu \\nu }=\\text{diag}(1,-1,-1,-1)$ .", "Greek indices run from 0 to 3." ], [ "The partonic cascade Boltzmann Approach to Multiparton Scatterings", "We simulate the partonic evolution of heavy-ion collisions by using the $(3+1)$ -dimensional transport approach (BAMPS) which solves the relativistic Boltzmann equation by Monte Carlo techniques [31], [53] for on-shell quarks and gluons by using perturbative QCD (pQCD) scattering matrix elements including $2\\leftrightarrow 2$ and $2\\leftrightarrow 3$ (radiative) processes.", "With the phase-space distribution function $f^i(x,k)\\equiv f_\\textbf {k}^i$ for particle species $i$ , the BE reads $k^\\mu \\frac{\\partial }{\\partial x^\\mu }f^i_\\textbf {k} =\\mathcal {C}^{2\\rightarrow 2}[f]+\\mathcal {C}^{2\\leftrightarrow 3}[f],$ where $\\mathcal {C}^{2\\rightarrow 2}[f]$ and $\\mathcal {C}^{2\\leftrightarrow 3}[f]$ are the elastic and inelastic collision terms.", "BAMPS uses the test particle method: The physical particle number is increased by an integer factor $N_\\text{test}$ ; however, all cross sections $\\sigma $ are simultaneously scaled down, $\\sigma \\rightarrow \\sigma /N_\\text{test}$ .", "This procedure increases the statistics but does not affect the physical results.", "Throughout this work, we include three flavors of light quarks, antiquarks, and gluons.", "All particles are on shell and massless (corresponding to an ideal equation of state) and carry physical electric charges and degeneracies.", "We neglect heavy quarks (see Refs.", "[45], [54], [55]) because their presence is subdominant for photon observables.", "Space is discretized in small cells with volume $\\Delta V$ and particles scatter and propagate within time steps $\\Delta t$ .", "Within each cell, the probability for binary scattering is $P_{22}=\\frac{\\sigma _{\\text{tot},22}(s)}{N_{\\text{test}}}v_{\\text{rel}}\\frac{\\Delta t}{\\Delta V},$ where $\\sigma _{\\text{tot},22}(s)$ is the (in general Mandelstam-$s$ -dependent) binary total cross section.", "For $2\\rightarrow 3$ particle scattering the probability is equivalently $P_{23}=\\frac{\\sigma _{\\text{tot},23}(s)}{N_{\\text{test}}}v_{\\text{rel}}\\frac{\\Delta t}{\\Delta V}.$ The inelastic $3\\rightarrow 2$ backreaction has a similar probability expressionWe do not include $3\\rightarrow 2$ processes involving photons, because these are subdominant processes.", "For gluon radiation it is implemented.. For massless particles, the relative velocity of the two incoming particle with four-momenta $p_{1,2}=(E_{1,2},\\vec{p}_{1,2})$ is $v_{\\text{rel}}=s/(2E_1E_2)$ .", "For binary collisions, the cross sections are obtained via tree-level pQCD matrix elements, where propagators are “screened” by a LO HTL Debye mass (for photon production, see Sec. ).", "For gluon radiation in $2\\rightarrow 3$ inelastic collisions we use the Gunion-Bertsch approximation for the matrix elements [56], which was further improved in Ref.", "[57], whereas for radiated photons we use the full QCD+QED matrix element.", "BAMPS features a running coupling $\\alpha _s(Q^2)$ , which is evaluated at the momentum transfer $Q^2$ of the respective scattering process [45].", "With this setup, the nuclear modification factor and elliptic flow in heavy-ion collisions could simultaneously be described in a former study [58].", "The framework allowed for several kinetic studies such as the determination of transport coefficients, heavy quarks, Mach cones, jet energy loss and momentum asymmetry, see e.g., Refs.", "[59], [47], [60], [61], [62], [63], [64], [45], [65], [66], [39], [67], [68], [69]." ], [ "Radiative cross sections", "The bremsstrahlung process $q+q\\rightarrow q+q+\\gamma $ is an important ingredient to the LO photon rate (more details in Sec.", "), thus in the following we give details regarding the evaluation of the total cross section.", "This is similar to the method for gluon radiation done earlier.", "Radiative processes (particles 1 + 2 $\\rightarrow $ 3 + 4 + 5) are described by the momentum labels $p_1,p_2,p_3,p_4$ and $p_5\\equiv k$ .", "All considered $2\\rightarrow 3$ processes have an internal gluon propagator with momentum $q$ .", "The rapidity in the center of momentum (CoM) frame is defined as $y=1/2 \\ln \\left[ (E+p_z)/(E-p_z)\\right]$ , where $y$ is the rapidity of the radiated photon, and its energy is $\\omega = k_\\perp \\cosh y$ .", "The energy of the outgoing particle 3 is $E_3=q_\\perp \\cosh y_3$ , with its rapidity being $y_3$ .", "Here, $k_\\perp , q_\\perp $ are the momentum components perpendicular to the $z$ axis in momentum space.", "The angle between $\\vec{q}_\\perp $ and $\\vec{k}_\\perp $ is $\\phi $ .", "The total cross section for radiative processes is defined as $\\sigma _{2\\rightarrow 3}&=\\frac{1}{2s}\\int \\frac{\\mathrm {d}^3 p_3}{(2\\pi )^32E_3}\\frac{\\mathrm {d}^3 p_4}{(2\\pi )^3 2E_4}\\frac{\\mathrm {d}^3 k }{(2\\pi )^3 2E_k}\\nonumber \\\\&\\quad \\times \\ (2\\pi )^4 \\delta ^{(4)}\\left(p_1+p_2-(p_3+p_4+k) \\right)\\left|\\mathcal {M}_{2\\rightarrow 3}\\right|^2\\nonumber \\\\&=\\frac{1}{256\\pi ^4}\\frac{1}{\\nu }\\frac{1}{s}\\int ^{s/4}_0 \\mathrm {d}q_\\perp ^2 \\int ^{s/4}_{k^2_{\\perp ,\\text{min}}} \\mathrm {d}k_T^2 \\int ^{y_{\\text{max}}}_{y_{\\text{min}}} \\mathrm {d}y \\int _0^\\pi \\mathrm {d}\\phi \\nonumber \\\\&\\quad \\times \\ \\left|\\mathcal {M}_{2\\rightarrow 3} \\right|^2 \\mathcal {J}\\left[s,q_\\perp ,k_\\perp ,\\phi ,y \\right],$ with a symmetry factor $\\nu =n!$ for $n$ identical final-state particles, the radiative matrix element $ \\left|\\mathcal {M}_{2\\rightarrow 3} \\right|^2$ and the Jacobian $\\mathcal {J}\\left[s,q_\\perp ,k_\\perp ,\\phi ,y \\right]=\\sum \\left\\lbrace \\left(\\frac{\\partial F}{\\partial y_3} \\right)^{-1} \\right\\rbrace ,$ where the sum is over the roots of $F&=(p_1+p_2-p_3-k)^2\\nonumber \\\\&=s-2\\sqrt{s}\\left(q_\\perp \\cosh y_3 + k_\\perp \\cosh y \\right)+ 2q_\\perp k_\\perp \\cos \\phi \\nonumber \\\\&\\ + 2q_\\perp k_\\perp \\left(\\cosh y_3 \\cosh y - \\sinh y_3 \\sinh y \\right).$ The lower integration limit $k^2_{\\perp ,\\text{min}}>0$ is further explained in Sec.", "REF .", "The limits in the rapidity of the outgoing photon $y_{\\text{max}},y_{\\text{min}}$ are functions of $k^2_{\\perp ,\\text{min}}$ , $k_\\perp $ and $s$ .", "For given coordinates $s, k_\\perp , q_\\perp , y, \\phi $ we can unambiguously obtain four-momenta in the CoM frame $p_1,p_2,p_3,p_4,k$ to get the value of the matrix element at this point without any approximation.", "The bremsstrahlung matrix element will be discussed in Sec.", "REF .", "More details regarding these kinematics can be found in Ref.", "[57]." ], [ "Photon production rate in partonic transport", "The emission rate of photons from an equilibrated quark-gluon plasma of temperature $T$ at leading-order $\\mathcal {O}\\left(e^2g^2T^4 \\right)$ was first determined in Refs.", "[70], [71].", "In nearly all phenomenological studies concerning photons in heavy-ion collisions, these rates are used and we will denote them as “AMY” rates.", "Because the full leading-order rate contains both $2\\leftrightarrow 2$ photon production (namely Compton-scattering and quark-antiquark annihilation) and higher-order processes, such as bremsstrahlung and inelastic pair annihilation including coherence effects, we implement the $2\\leftrightarrow 2$ processes and the $2\\rightarrow 3$ processes separately.", "We emphasize, that the advantage of transport simulations lies in the use of microscopic rates.", "We do not rely on thermal distributions of incoming partons, any pair of partons can produce a photon (given that the process is kinematically and diagrammatically allowed).", "However, in this section we use thermal distributions to show the validity of the total photon production rates in the transport framework by comparing to analytically known thermal rates." ], [ "$2\\leftrightarrow 2$ processes for photon production", "The authors of Refs.", "[72], [73] have computed the $2\\leftrightarrow 2$ contribution to the photon rate in the $k/T\\gg 1$ limit.", "This limit could be dropped in Refs.", "[70], [71].", "Essentially, in the so-called HTL improved rate the momentum transfer $t$ in photon production matrix elements is split up into a soft region $t<t^\\star $ and a hard region $t>t^\\star $ .", "The rate from the hard region is treated in a straightforward way by integrating the appropriate squared matrix elements $\\left|\\mathcal {M}_{\\text{Compton}}\\right|^2$ and $\\left|\\mathcal {M}_{\\text{Annihilation}}\\right|^2$ in the rate integral (see Appendix  ).", "The soft region, $t<t^\\star $ , is treated differently, by using effective HTL vertices and propagators in the corresponding loop diagrams.", "In the end, both soft and hard contributions are added and turn out to be independent of $t^\\star $ .", "In principle, the $t<t^\\star $ calculation of the dressed loop diagram corresponds to the kinetic ($t>t^\\star $ ) computation of the rate while making the propagators effectively massive, using a mass of order $gT$ .", "For the vertices, however, a similar interpretation would be difficult.", "Within the partonic transport model BAMPS, we deal only with vacuum matrix elements, essentially the same which are used in the $t>t^\\star $ region of Refs.", "[72], [73].", "The matrix element for Compton scattering $qg\\rightarrow q\\gamma $ reads $\\left|\\mathcal {M}_{\\text{Compton}}\\right|^2=\\frac{16}{3}\\pi ^2\\alpha \\alpha _s \\left(\\frac{s^2+st}{s^2} + \\frac{s^2+st}{u^2}\\right).$ The matrix element for quark-antiquark annihilation $q\\bar{q}\\rightarrow g\\gamma $ is $\\left|\\mathcal {M}_{\\text{Annihilation}}\\right|^2=\\frac{128}{9}\\pi ^2\\alpha \\alpha _s \\left(\\frac{tu}{t^2} + \\frac{tu}{u^2}\\right).$ where $s,t,u$ are the usual Mandelstam variables.", "To avoid a crude cut-off like $t^\\star $ in the momentum integration within BAMPS, we dress the quark propagators with a thermal mass $m_{D,q}\\sim gT$ , motivated by the HTL effective propagators.", "This screening of infrared divergencies naturally has a large effect on the total photon rate (and also the differential one), and must be carefully investigated, which is the purpose of this section.", "In BAMPS we use the formulas from Appendix  to compute the Debye mass from the given (in general nonequilibrium) distribution functions.", "We want to mention here the systematic uncertainty concerning the strong coupling entering the Debye mass.", "It can be fixed (e.g., $\\alpha _s=0.3$ ) or a running $\\alpha _s(Q^2)$ , where the scale $Q^2$ has to be specified.", "In the commonly used electric scale $Q=a2\\pi T$ , the prefactor $a$ is not clear, but $\\mathcal {O}(1)$ .", "In former versions of BAMPS, $Q^2$ was taken to be the momentum transfer of the specific process, $Q^2=s,t,u$ .", "Moreover, in Ref.", "[74] it was argued, that the coupling can be evaluated at the Debye mass itself, $m_D^2=\\frac{4\\pi }{3}\\alpha _s(m_D^2)\\left(N_c+\\frac{N_f}{2} \\right)T^2$ .", "To allow for comparison with other groups we set the coupling in this paper fix to $\\alpha _s=0.3$ for the photon production, unless otherwise stated.", "Note, that the procedure from this section and the following Secs.", "REF and REF is only strictly valid for fixed coupling.", "Recent hydrodynamical calculations of photon rates also keep the coupling fixed.", "Figure: The Born matrix element integrated with Boltzmann statistics (green dotted line).", "Reducing this rate by C stat =0.84C_{\\text{stat}}=0.84 (orange dashed line), the total rate RR equals the Born rate with quantum statistics, which equals approximately the elastic HTL improved rate, see panel (a).", "Having the screened matrix elements at hand, we then carry out the integration to obtain the total cross section and finally the photon spectra.", "These rates will by construction not be equal to the HTL improved rate, which is why we multiply the thermal masses by a real number $\\kappa $ .", "The propagators for the different channels read correspondingly, $\\frac{1}{t^2}&\\rightarrow \\frac{1}{(t-\\kappa m_{D,q}^2)^2},\\quad \\frac{1}{u^2}\\rightarrow \\frac{1}{(u-\\kappa m_{D,q}^2)^2},\\nonumber \\\\\\frac{1}{s^2}&\\rightarrow \\frac{1}{(s+\\kappa m_{D,q}^2)^2}.$ Table: The comparison of AMY with Born-photon rates for higher moments of the photon rate, using the fixed value of κ=2.45\\kappa =2.45.It is now our strategy to choose the value of $\\kappa $ in such a way that our simplified procedure leads to a rate that resembles the HTL improved rate closely (a similar procedure was done for heavy quark energy loss; e.g., in Ref. [75]).", "We do this by comparing the moments of the rate (where the $n$ th moment is defined as $\\int _0^\\infty \\mathrm {d}E E^n \\frac{\\mathrm {d}R}{\\mathrm {d}E }$ ).", "To this end we solve the integral in Eq.", "(REF ) numerically (as in Appendix A of [76]) first for quantum statistical distributions and screened matrix elements including the $\\kappa $ -factor (we call this “Born” rate), and compare the result to the HTL improved ($2 \\leftrightarrow 2$ ) rate from [70], [71].", "We adjust $\\kappa $ [which is of order $\\mathcal {O}(1)$ ] so that the total ratesThe total rate is the total number of photons emitted per volume per time (0th moment), $R=\\int _0^\\infty \\mathrm {d}E \\frac{\\mathrm {d}R}{\\mathrm {d}E}$ .", "$R$ are equal.", "The comparison is shown in Fig.", "REF , where we plot $\\mathrm {d}R/\\mathrm {d}E$ in both schemes.", "One observes that the Born rate (blue cross shaded area) has a slightly shifted peak when compared with the HTL improved rate.", "To get a handle on the quality of the comparison, we compare higher moments of the rate; the results are shown in Table REF .", "Note that the result for $\\kappa $ is rather insensitive to the numerical integration limits, as the integrand drops to zero for $E/T\\rightarrow 0,\\infty $ ." ], [ "Correction of the distribution functions", "Finally, we need to correct for the small effect of the distribution functions.", "In the present numerical study, we can only use Boltzmann (classical) statistics, in initial and final states.", "There is no Pauli blocking or Bose enhancement [77].", "That is why we will multiply the cross sections (equivalent to the rate) with a factor $C_{\\text{stat}}$ in BAMPS to get the correct number of photons even without quantum statistics.", "This factor does not alter the differential cross section, as it is an overall prefactor.", "Note that also the Debye mass follows the Boltzmann distribution, because it is dynamically computed from the simulation.", "To obtain $C_{\\text{stat}}$ , we solve Eq.", "(REF ) numerically with Boltzmann distributions and ignore Pauli blocking or Bose enhancement, but keep the fixed value of $\\kappa $ from the procedure aboveHere again, the Debye mass is in Boltzmann form.. Then we compare again to the HTL resummed $2\\leftrightarrow 2$ rate from Refs.", "[70], [71], which uses quantum statistics.", "The difference of both total rates is $C_{\\text{stat}}$ .", "The rates are shown in Fig.", "REF .", "The fact that $C_{\\text{stat}}$ is below unity implies that the Pauli-blocking effect of the outgoing quark in the Compton channel is more important than the Bose enhancement effect of the outgoing gluon in the annihilation channel.", "This is consistent, because the Compton process happens more often (due to the combinatorics of the ingoing particles).", "Finally, we obtain the $2\\leftrightarrow 2$ photon production rate from BAMPS including the above explained ingredients in a box calculation.", "As an important numerical check, we compare the numerical results with the analytic expectation by using the exact same matrix elements (using two arbitrary values of $\\kappa $ for illustration) in Fig.", "REF , and find excellent agreement.", "Figure: For two values of κ\\kappa we compare the numerically obtained 2↔22\\leftrightarrow 2 photon rate to the analytic expectation (obtained by using the method from Ref.", ").Motivated by the processes which give contributions to the total photon rate from Refs.", "[70], [71], we include radiative photon processes in BAMPS.", "We restrict ourselves to the simplest bremsstrahlung diagram,Diagrams with more vertices become numerically very elaborate.", "shown in Fig.", "REF with both subdiagrams.", "In Refs.", "[70], [71] it is shown, that at leading-order in the rate, only the self-energy in the form of Fig.", "REF contributes, including a resummation of infinite gluon rungs.", "We want to stick to this picture, and neglect diagrams which would not emerge by cuts of this self-energy, even though in our transport setup those could be substantial.", "To this end, we employ the cutting-rules of Ref.", "[78], and obtain scattering matrix elements.", "Every cut propagator is put on-shell, as well as every opened loop.", "In Fig.", "REF such a cut is shown, for the case of two gluon rungs.", "The loops must be opened by “tics” (see Ref.", "[78]) in every possible way.", "What emerges is exactly the diagrams of Fig.", "REF .", "Note that the cuts of Fig.", "REF produce two on-shell gluons, and one quark line radiating a photon.", "This, and corresponding diagrams with more gluon rungs, represent a sequential scattering with gluons, which is included by default in BAMPS, because the dominating subprocess $q+g\\rightarrow q+g$ was included from the beginning, and the rare radiation of the photon is Compton scattering in this case.", "Note that a (possible) $2\\rightarrow 3$ process like $g+q\\rightarrow q+g+\\gamma $ (with a three gluon vertex), is not included in the set of diagrams resulting from the cuts.", "Ignoring the rigorous LO power counting of Refs.", "[70], [71], and just looking at the number of vacuum QCD vertices, this process could be included and would contribute significantly within BAMPS, because gluons are abundant, especially in the early phase of the QGP.", "This will be investigated in a future study.", "For now we use only one kind of matrix element, motivated by the leading-order picture.", "Figure: An example of the photon self energy and its cuts to obtain scattering matrix elements along the method from Ref. .", "In diagram (a) the dashed line represents one possible cut, and the closed loops must be opened (”tic-ed”) to get a scattering matrix element.", "In (b) the dashed and double dashed lines are possible, topologically different cuts, generating Bremsstrahlung with on-shell gluons of the medium.As we only have vacuum matrix elements, we will insert thermal screening masses by hand into the propagators, as done before in the case of $2\\leftrightarrow 2$ scattering.", "In Appendix  we derive the full squared matrix element $\\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.", "}}$ starting from spinors and propagators without any further approximation.", "This is the radiative matrix element for photons that we will use in BAMPS, using techniques from Ref.", "[57]." ], [ "Interference effects", "Photon radiation from bremsstrahlung processes suffers from the Landau-Pomeranchuk-Migdal (LPM) effect.", "The calculation of the radiative photon production rate in Refs.", "[70], [71] fully includes the interferences among subsequently radiated photons.", "The notion of “destructive interference” of photons is motivated by looking at possible cuts of the retarded photon self energy and the resulting matrix elements.", "They must be summed and squared to obtain the full amplitude.", "In our microscopic description which is based on individual scatterings, we use an effective method to simulate the LPM interference effect.", "Within a transport approach, using individual scatterings for photon production, such interferences are necessarily destroyed, and must be restored by hand.", "At first, we calculate the specific inverse rate $\\lambda ^{\\text{spec}}_{\\text{mfp}}$ of the quark species which appear in the inelastic matrix elements for photon production [this can be seen as a mean-free path (mfp), where only certain scattering processes are included].", "For the calculation of $\\lambda ^{\\text{spec}}_{\\text{mfp}}$ we take solely the specific $2\\leftrightarrow 2$ processes into account which appear as subdiagram before or after the photon is radiated (see Fig.", "REF ).", "These specific processes are: processes 1 $qq\\rightarrow qq$ / $\\bar{q}\\bar{q}\\rightarrow \\bar{q}\\bar{q}$ processes 2 $q\\bar{q}\\rightarrow q\\bar{q}$ Here $q\\ (\\bar{q})$ are quark (antiquark) species, for up, down and strange quarks.", "The corresponding numerical method is explained in Appendix , and a typical process is schematically depicted in Fig.", "REF .", "In Fig.", "REF we show numerical results for the inverse rate (mean-free path) corresponding to these processes separately.", "It depends strongly on the (anti-)quark fugacity and temperature.", "We will come back to the fugacity dependence of the mean-free path and the rate in Sec.", "REF .", "Figure: Here we show a sketch of an LPM interference effect: Due to a short quark mean-free path a subsequent radiation is suppressed.", "Note that this diagram is not used as shown here, we rather evaluate the quark-quark elastic mean-free path dynamically in BAMPS and compare it to the formation time of the photon.", "The photon is produced by the pure bremsstrahlung subdiagram.Next we multiply the amplitude for photon radiation by a Heaviside-function $\\Theta \\left(\\lambda ^{\\text{spec}}_{\\text{mfp}} - \\tau _f \\right)$ which ensures, that the formation time $\\tau _f$ of the radiated photon is smaller than the mean-free path of the radiating quark, $\\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.", "}}\\ \\rightarrow \\ \\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.}}", "\\Theta \\left(\\lambda ^{\\text{spec}}_{\\text{mfp}} - \\tau _f \\right).$ By doing this, we discard photons with such soft $k_\\perp $ (transverse momentum relative to the radiating quark), that the radiating quark could have scattered again within the formation time.", "The $k^2_\\perp $ integration in Eq.", "(REF ) in this case is limited by $k^2_{\\perp ,\\text{min}} = \\left(\\lambda ^{\\text{spec}}_{\\text{mfp}}\\right)^{-2}$ .", "As this procedure reflects the underlying interference effect only incomplete, we must insert a scale factor $K_\\text{inel}$ in front of the matrix element.", "Recall that the current implementation of the LPM effect for radiated gluons in BAMPS is done in a similar way, the only difference is a factor $X_{\\text{LPM}}$ being multiplied to the formation time and a different determination of the mean-free path.", "These differences are motivated by two physical effects: First, radiated gluons suffer from scattering after the radiation process, which dynamically alters their formation time.", "That is why we allow more radiated gluons than would actually be radiated if we required them to be fully formed.", "Second, gluon radiation rates involve far more diagrams (see, e.g., Ref.", "[79]), such that the mean-free path is the total mean-free path $qX\\rightarrow Y$ where $X$ can be a quark or gluon.", "Figure: Numerical results for the specific mean-free path for a quark corresponding to different reactions, depending on temperature and fugacity." ], [ "Fixing the scaling factor for bremsstrahlung", "As mentioned in the previous section, the implementation of radiative photon production is incomplete.", "There are in fact several parts which deviate from the AMY description.", "First, as we include only vacuum matrix elements with Debye screened propagators, we miss the correct treatment of soft momentum transfers.", "In the matrix element there are two propagators (a quark- and a gluon propagator) where we insert Debye or thermal masses by hand and we could in principle tune these Debye masses by multiplying $\\kappa $ -factors as in the $2\\leftrightarrow 2$ case from Sec.", "REF .", "However, it is not clear how and if they should be tuned individually.", "The second simplification is the LPM effect described in the previous section.", "Third, we have the small effect of missing quantum statistics here, too.", "At last, the full AMY rate includes effectively not only the bremsstrahlung process, but also inelastic pair annihilation (a $3\\rightarrow 2$ process), which we do not include here in this study.", "In Refs.", "[70], [71] it is shown that this is a subdominant contribution.", "To cure all these problems, we scale the full matrix element $\\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.}}", "$ with a factor $K_{\\text{inel}}$ .", "Such scaling is the simplest choice, and very common in transport approaches.", "Figure: The equilibrium radiative photon rate weighted by photon energy.For inelastic processes, the total AMY rate $\\mathrm {d}R/\\mathrm {d}E$ diverges for small $E$ , and the integral is ill defined.", "However, for small energies (transverse momentum), experiments do not measure anymore, and, the perturbative AMY description breaks down [80], so that we choose a suitable lower integration limit.", "To be consistent with the $2\\leftrightarrow 2$ photon production we could choose to integrate over $0.15 < E/T < 10$ , and compare the integral $R^{\\text{inel}}$ to the result obtained by integrating the AMY result.", "In this case we obtain $K_{\\text{inel}}=0.79$ .", "Having our application in mind, where we focus on transverse momenta in the range $0.5<p_T/\\mathrm {GeV}<4$ , we translate this at $T\\sim 0.4~\\mathrm {GeV}$ to a sensible integration region of $1<E/T<10$ , where the result is very insensitive to the upper integration limit.", "For the following, we use this integration region, and obtain $K_{\\text{inel}}=0.53$ .", "Using this factor, we make sure that we get (in an equilibrium case) the same number of photons and a similar spectrum in the energy region of interest.", "In Fig.", "REF we show the numerical photon rate compared to the AMY rate, and also its first moment.", "The numerical rate from microscopic scatterings in Fig.", "REF shows a similar slope as the AMY rate in the considered integration region, and the integrals of the curves in the plot are equal.", "The first moment in Fig.", "REF from BAMPS is smaller than the AMY rate by about a factor of $5.2$ , the second moment (not shown) by a factor of $5.3$ .", "In Appendix  we show the corresponding differential cross sections and cross-checks of the kinematics.", "As a note, the thermal photon elliptic flow, being a transfer of flow from a boosted thermal distribution onto photons, is not sensitive to the differential photon rate (because photons are emitted isotropically in the local rest frame)." ], [ "Photon rate at nontrivial quark fugacities", "The photon rate naturally depends on the quark and gluon content of the medium.", "For finite baryon chemical potentials (or quark chemical potential) the rate is modified by the (trivial) statistical factors ($q\\bar{q}$ annihilation and Compton scattering behave differently), but also by other ingredients of the rate, such as the gluon self energies.", "These effects are studied thoroughly in Ref. [81].", "The authors conclude, that the effect of the chemical potential to the photon spectra at RHIC or LHC is small, due to the small baryon chemical potential and the moderate sensitivity of the rates.", "Although we use a simplified diagrammatic setup, the effect of a quark-antiquark number asymmetry is included in the transport approach by default.", "For the present study at high energies, however, the effect is negligible.", "The second, more important characterization of the parton content is the “absolute” fugacity.", "Assuming by the previous argument, that the number of quarks equals the number of antiquarks, we define the gluon (quark) fugacities $\\lambda _g(\\lambda _q)$ as $n_g &= \\lambda _g n_g^{\\text{equilibrium}}\\nonumber \\\\n_q+n_{\\bar{q}} &= \\lambda _q \\left(n_q^{\\text{equilibrium}}+n_{\\bar{q}}^{\\text{equilibrium}} \\right).", "\\nonumber $ Effectively, for the considerations in this section, there is no difference between quark and antiquark.", "Note that the fugacities in heavy-ion collisions are in general time dependent.", "The initial state is still uncertain, especially the quark and gluon content is under debate.", "It is commonly believed, that gluons are saturated or over-saturated [82], and quark-antiquark pairs are not very abundant in the very early phase after the collision [53].", "In Ref.", "[83] an undersaturation of quark-antiquark pairs ($\\lambda _q<1$ ) seems to be favored by data within a rate equation approach.", "However, no precise answer about the fugacity dependence could be given up to now.", "Other studies [84], [82], [85], [86], [87] give slightly different pictures, but we shall not elaborate on this topic here.", "Common ground is a quark fugacity $\\lambda _q$ which is lower than unity and may or may not approach it within the lifetime of the fireball.", "We investigate in the following, how the photon rate behaves for nontrivial quark/gluon fugacities.", "Our arguments are similar to those of Ref. [88].", "Naively, the $2\\leftrightarrow 2$ Compton scattering (quark-antiquark annihilation) rates are proportional to $\\lambda _q\\lambda _g$ ($\\lambda _q\\lambda _q$ ) just by taking the incoming parton distribution functions into account.", "However, the Debye screening prescription from Eq.", "(REF ) lets the quark and gluon fugacities enter one more time into the rate.", "This will scale the rates differently as naively expected.", "In Fig.", "REF we show the fugacity dependence of the $2 \\leftrightarrow 2$ photon production (purple triangles) by comparing the total rate $R$ to the rate at unity fugacity, $R[\\lambda _q]/R[\\lambda _q=1]$ .", "We have computed the Compton scattering and quark-antiquark annihilation rates for several quark fugacities (the gluon fugacity $\\lambda _g$ is unity here), and find a combined scaling as $\\lambda _q^{1.07}$ .", "We conclude that the $2 \\leftrightarrow 2$ rates can be seen as being simply proportional to the quark fugacity.", "The inelastic photon rate will scale naively with $\\lambda _q\\lambda _q$ ; however, our implementation of the LPM effect uses the numerically (i.e., dynamically) evaluated quark mean-free path for specific processes (see Fig.", "REF ), which depends on the average cross sections $\\sigma $ and particle densities $n$ and thus on the (quark) fugacity $\\lambda _q$ as $\\sim 1/(n\\sigma )\\sim 1/(\\sigma \\lambda _q T^3)$ .", "The average cross sections are themselves Debye screened, and decrease for higher fugacities.", "These effects are summarized in Fig.", "REF , where we show the scaling of Debye mass, density, average cross section and mean-free path, for the two processes considered.", "Additionally, the fugacities enter also in the Debye screened gluon propagator.", "In Fig.", "REF we show the scaling of the inelastic photon rate (normalized to the rate at $\\lambda _q=1$ ) with the fugacity and compare with a naive scaling (without the effect from the LPM procedure or Debye screening), $R[\\lambda _q]/R[\\lambda _q=1]=\\lambda _q^{2}$ .", "By fitting a simple power law we find for bremsstrahlung roughly $R\\sim \\lambda _q^{1.36}$ , for $\\lambda _q\\gtrsim 0.3$ .", "Figure: Dependence of the photon rate RR on the quark fugacity λ q \\lambda _q, and the comparison to the naive expectations R∼λ q 1,2 R\\sim \\lambda _q^{1,2} (solid, dotted line).", "The bremsstrahlung rate shows roughly a R∼λ q 1.36 R\\sim \\lambda _q^{1.36} dependence (dashed line, fit), whereas the 2↔22 \\leftrightarrow 2 photon production processes show a behavior R∼λ q 1.07 R\\sim \\lambda _q^{1.07}.Figure: Quark fugacity λ q \\lambda _q scaling of Debye mass m D m_D, quark density nn, average cross section at T=0.4 GeV T=0.4~\\mathrm {GeV}, σ i ≡v rel σ tot ,i(s)\\sigma _i\\equiv \\left\\langle v_\\mathrm {rel}\\sigma _\\mathrm {tot,i}(s) \\right\\rangle (where i=1,2,3i=1,2,3 corresponds to the three different specific processes considered) and the specific inverse rate λ mfp \\lambda _{\\mathrm {mfp}}." ], [ "Box calculation of photon leakeage effect", "To understand the kinetics of photons originating from hard partons qualitatively we use a fixed box with volume $V=L_x \\cdot L_y \\cdot L_z$ , and populate it homogeneously with a thermal distribution of quarks and gluons (temperature $T$ ).", "This distribution can either be at rest with a four-velocity $u^\\mu =(1,0,0,0)$ , or boosted in the $x$ direction, $u^\\mu =(\\gamma ,\\gamma v_x,0,0)$ , such that there is a strong collective flow in the $x$ direction (as seen from the laboratory frame).", "We change the box size to be either very thin, $L_x/L_y \\ll 1$ or cubic, $L_x=L_y=L_z$ .", "Furthermore, we initialize at the geometric center of the box a large amount of “jet”-like particles isotropically with a fixed energy $E_j\\sim 5T-10T$ .", "All particles are allowed to scatter and produce photons, however, when any particle hits the wall, it is deleted.", "We define a transverse momentum, $p_T=\\sqrt{p_x^2+p_y^2}$ .", "Our observable resembles an elliptic flow $v_2$ , but here it is merely a momentum anisotropy, $v_2=\\left\\langle \\frac{p_x^2-p_y^2}{p_T^2} \\right\\rangle _{\\text{average all photons}}.$ To this end, we consider five scenarios: A Cubic box at rest, including jets B Cubic box with flow, without jets C Cubic box with flow, including jets (jet $p_T=10 T$ ) D Thin box, $L_x/L_y \\ll 1$ at rest, including jets (jet $p_T=5T,10 T$ ) E Thin box, $L_x/L_y\\ll 1$ with flow, including jets (jet $p_T=10 T$ ) Evaluating the momentum anisotropy from these scenarios, we plot the results in Fig.", "REF .", "As expected, no flow is visible in the symmetric scenario A.", "In scenario B a thermal, flowing background generates a momentum anisotropy which increases for higher $p_T$ .", "Undisturbed flow from the background is carried over to photons.", "Here we note that, by a simple relativistic effect, the (Lorentz variant) result of Eq.", "(REF ) for produced particles is lower in magnitude than that for the background distribution.", "This effect depends on the boost.", "Including jets, which are isotropically emitted from the center, the flow reduces to zero at exactly the jet energy.", "For Compton scattering and quark-antiquark annihilation a large amount of photons inherit nearly the full momentum from the jets (jet-photon conversion).", "Because the jet momentum is dominant, the momentum anisotropy of these photons is zero, hence the curve of scenario C drops at the jet energy.", "The flow at lower $p_T$ stems from the background flow.", "In scenario D there is no background flow, and no positive $v_2$ contribution.", "The jets, initialized in the middle of the box, traverse it until whichever wall comes first until they are deleted.", "During their traveling path, they can hit a thermal particle and produce a (conversion) photon, with a momentum close to that of the parent jet.", "This is more likely to happen in the (long) $y$ or $z$ direction, than in $x$ , as the box has a small $L_x$ size.", "Most of the photons have larger $p_y$ momenta than $p_x$ , thus the $v_2$ becomes negative (see, e.g., Ref [89] for similar findings).", "We show this effect for two different jet $p_T$ and, clearly, the minimal $v_2$ is reached at exactly the jet $p_T$ .", "This effect can be termed the geometric leakage effect.", "Finally, the combined effect of thermal background flow and jet conversion photons is shown in scenario E: For low $p_T$ there is substantial momentum anisotropy, whereas around the jet $p_T$ the conversion effect dominates and pushes the $v_2$ into the negative region.", "This toy example shows what we can expect in a heavy-ion collision when both, jet particles and thermal flowing particles are present.", "The relative strengths of both effects have to be investigated in a full simulation.", "Figure: Results for the qualitative understanding of elliptic flow of photons originating from flowing thermal background and non thermal \"jet\"-like partons for the 5 scenarios explained in Sec. .", "The thermal medium has a temperature of 0.4 GeV 0.4~\\mathrm {GeV}, and for simplicity photons originate from 2↔22 \\leftrightarrow 2 processes only." ], [ "Jet photon conversion", "To explicitly see how higher energetic partons (“jets”) interact with thermal particles and create a photon, we carry out a simple box calculation, where quark or gluon jets with fixed energy hit particles from a thermal bath.", "In Fig.", "REF we show the resulting photon spectra, normalized by the volume density of jets $n_{\\text{jet}}$ .", "For a gluon jet, the only possible process is Compton scattering.", "It can be seen, that the photon spectrum is peaked at values $E\\sim \\mathcal {O}(T)$ , due to the present channels.", "For gluons we cannot speak of jet-photon conversion.", "Quark jets, interacting only in $2\\leftrightarrow 2$ processes (Compton scattering and quark-antiquark annihilation), have a dominating peak at the jet energy $E_{\\text{jet}}=15~\\mathrm {GeV}$ .", "Due to momentum conservation the direction of the momentum of the photon must be very close to that of the jet quark - this is a true jet-photon conversion.", "The relative strength of the thermal peak at low energies and the peak at the jet energy depends on the ratio $T/E_{\\text{jet}}$ .", "Figure: Spectra of photons which are produced by an incident jet particle with E jet =15 GeV E_{\\text{jet}}=15~\\mathrm {GeV} hitting a thermal bath.", "We show 2↔22\\leftrightarrow 2 and 2→32\\rightarrow 3 contributions separately.For bremsstrahlung we show the result for two different specific mean-free paths (dotted and dash-dotted line).", "The energy of the photon is distributed between the thermal scale and the jet energy scale, and depends on the LPM effect.", "However, in Appendix  we show that the differential cross section is peaked at low transverse momentum, which means the emission is favorably collinear to the jet quark.", "In this case we have a similar effect for the resulting photon as in the jet-photon-conversion case." ], [ "Results", "In the following we show results from realistic simulations of heavy-ion collisions by using the photon production methods explained above within the framework of BAMPS.", "Details concerning the BAMPS setup for heavy-ion collisions can be found in [31], [65], [66], [58].", "The initial geometry of the collisions is governed by a Glauber model [31], [55].", "For the initial parton distribution we use pythia 6.4 [90]; details about the implementation can be found in Ref. [55].", "Because photons are very rare probes, they do not alter the collision dynamics.", "For this reason we use recorded BAMPS events, and sample photons by collisions among the recorded particles.", "This method allows us to enhance the photon cross section by a nearly arbitrary factor and scale the resulting spectra down by this factor (for better statistics).", "We have checked that all our results are independent of these factors.", "The background collision includes the latest improvements from BAMPS, such as the improved Gunion-Bertsch matrix elements for gluon radiation and a pQCD running couplingNote that photon production is independent from the background events, and we chose the coupling to be fixed or running for the photon production cross sections, see also Fig.", "REF .", "[68], [58], [57].", "The evolution of BAMPS runs until the energy density drops locally below $\\epsilon _c=0.6~\\mathrm {GeV}/\\mathrm {fm}^3$ .", "We have checked that the photon spectra are insensitive to this choice, because the rather cool medium in the later stages no longer produces many photons." ], [ "Photon yield from heavy-ion collisions", "At present, BAMPS simulates only the QGP phase of heavy-ion collisions.", "This complicates studies and comparisons with photon data more than for other observables (such as, e.g., heavy quarks, jets, or bulk medium elliptic flow).", "Figure: The p T p_T spectrum of direct photons from the QGP phase of Au + Au collisions at s NN =200 GeV \\sqrt{s_{NN}}=200~\\mathrm {GeV} for 20%-40%20\\%-40\\% most central collisions.", "We show the elastic (magenta dotted) and inelastic (greed dashed) contribution from BAMPS as well as their sum (red solid) in comparison with a recent hydro result (yellow double dashed) from Ref.", "and a result from off-shell transport PHSD .Figure: Same as Fig.", ", but we switch on a running coupling for photon production (green dashed line).In Fig.", "REF we show results for photon spectra in transverse momentum $p_T$ from BAMPS separately for $2\\leftrightarrow 2$ photon production processes (magenta dotted line) and $2\\rightarrow 3$ processes (green dashed line).", "The sum (red solid line) has an important contribution from the inelastic processes, especially at the highest and lowest $p_T$ .", "In Fig.", "REF we show the effect of a running coupling for photon production.", "The momentum transfer of the respective channel serves as scale $Q$ to evaluate the coupling, $\\alpha _s(Q^2)$ , but the coupling constant appearing inside the Debye masses is evaluated at the scale of an effective temperature in the corresponding cell ($Q=2\\pi T_{\\mathrm {eff}}$ ).", "The running coupling changes the slope only slightly, but increases the photon rate by a factor of 2 below $p_T\\lesssim 1.5~\\mathrm {GeV}$ and $1.5$ above $p_T\\gtrsim 1.5~\\mathrm {GeV}$ .", "Other models, such as PHSD [91] and MUSIC [25] produce QGP rates around a factor of five to ten larger in magnitude than our results (for fixed $\\alpha _s$ ), and a significantly steeper slope.", "The quark and gluon fugacities in MUSIC are unity, PHSD states only absolute particle numbers.", "Due to the small yield of photons in the present setup, a possible pre-equilibrium contribution from BAMPS to, e.g., hydrodynamic calculations is negligible.", "As the BAMPS results for photons from the QGP undershoot the hydrodynamic calculations for all $p_T$ , and even hydrodynamics underestimates experimental data, BAMPS can not help in this direction with the present initial state.", "From all the above we see that the initial condition is the main uncertainty, and once more, our results underline the need to understand better the initial quark and gluon content of the fireball (see also Ref. [92]).", "We show the fugacities in BAMPS in Fig.", "REF for the same parameters of the collision.", "Figure: The average quark and gluon fugacities over time in BAMPS for RHIC collisions at s NN =200 GeV \\sqrt{s_{NN}}=200~\\mathrm {GeV} at 20%-40%20\\%-40\\% centrality.", "Shown is the average over only the central cell and a tube of transverse radius 1.5 fm 1.5~\\mathrm {fm} extending in spacetime rapidity -0.5<η s <0.5-0.5<\\eta _s<0.5.We have extracted the quark fugacity by using an effective temperature for two representative geometries, the central cell of the collision and a tube of $1.5~\\mathrm {fm}$ radius and length of one unit in rapidity.", "We remark that, at early times, these equilibrium quantities are only rough estimates of the quark content because the medium is not yet equilibrated.", "As shown in Fig.", "REF the $2 \\leftrightarrow 2$ photon rates scale nearly linearly with the quark fugacity, so that they are strongly affected by the quark fugacities $\\lambda _q\\lesssim 0.2$ at early times in BAMPS.", "The inelastic rate has a more complicated fugacity dependence, such that the photon rate at $\\lambda _q=0.2$ is less than $10\\%$ of the equilibrium rate at $\\lambda _q=1$ .", "The combined effects explain the difference with the other models.", "To see which role is played by the chemically equilibrating medium, we alter the fugacity evolution of the quarks (and thus also the gluons) by tuning arbitrarily the quark-antiquark production cross sectionWe ignore the tuning of the backreaction $q\\bar{q}\\rightarrow gg$ because the purpose of this test is to drive the chemical equilibration faster.", "In the central cell, the quark fugacity even increases above unity for late times and $K_{gg\\rightarrow q\\bar{q}}=100$ .", "by a factor of 10 and 100.", "The resulting fugacity evolution is shown in Fig.", "REF .", "It can be seen, that at around $t=2~\\mathrm {fm/c}$ the quark fugacity increases from $\\lambda _q(t=2~\\mathrm {fm/c})\\approx 0.15$ to $\\lambda _q(t=2~\\mathrm {fm/c})\\approx 0.2$ (for $K_{gg\\rightarrow q\\bar{q}}=10$ ) and $\\lambda _q(t=2~\\mathrm {fm/c})\\approx 0.5$ (for $K_{gg\\rightarrow q\\bar{q}}=100$ ).", "In Fig.", "REF the resulting photon spectra are shown.", "The difference between the three scenarios is moderate, because most of the photons are produced within the first $2~\\mathrm {fm/c}$ .", "The difference in the fugacity is however, much stronger at later times (at $t=4~\\mathrm {fm/c}$ about a factor of five), where not many photons are produced due to the thinner and colder medium.", "This shows, that the quark content at the very initial phase is crucial for photon spectra.", "Because the two other quoted models (MUSIC and PHSD) in Fig.", "REF and Fig.", "REF underestimate the data for RHIC slightly, our results suggest that this problem could be even more severe.", "Figure: Same as Fig.", ", but for artificially increased quark-antiquark production cross section σ gg→qq ¯ =Kσ gg→qq ¯ \\sigma _{gg\\rightarrow q\\bar{q}}=K \\sigma _{gg\\rightarrow q\\bar{q}}, where K=10K=10 (left panel) and K=100K=100 (right panel).", "The photon spectra are mostly sensitive to the early phase, where the notion of fugacity (or temperature) can only be effective.Figure: Same as Fig.", ", but here we change the chemical equilibration during the evolution by artificially increasing the gg→qq ¯gg\\rightarrow q\\bar{q} cross section by a factor of 10 (magenta dotted line) and 100 (green dashed line)." ], [ "Elliptic flow", "Within BAMPS, the event plane is known exactly, because we are dealing only with smooth Glauber initial conditions.", "This is the reason why elliptic flow can be conveniently obtained by averaging $(p_x^2+p_y^2)/p_T^2$ over all particles considered, photons in our case.", "Experimental results of direct photon elliptic flow are a weighted average over all sources of direct photons, weighted by their spectra.", "We can perform weighted averages by taking prompt photons and photons from hadronic scattering from elsewhere, in order to compare with data, but we find it instructive to compare directly the QGP contribution from BAMPS with other studies.", "In Fig.", "REF we show the elliptic flow of photons originating from only $2 \\leftrightarrow 2$ collisions (green upward triangles), only bremsstrahlung (blue squares), and their sum (red points).", "Figure: The elliptic flow of photons from BAMPS for Au + Au collisions at s NN =200 GeV \\sqrt{s_{NN}}=200~\\mathrm {GeV} and 20%-40%20\\%-40\\% centrality.", "Shown are the sum, and the elastic and inelastic contribution separately.The pink downward triangles [93] show elliptic flow of photons induced by jet-plasma interactions within a $(2+1)$ -dimensional hydro model, where a time-dependent jet distribution is assumed and the jet-thermal rate is obtained by integrating separately the $2 \\leftrightarrow 2$ and collinear (bremsstrahlung) rates using a thermal and a jet distribution.", "The final results are obtained by folding these rates over the hydro background.", "The thus-obtained elliptic flow is negative, a robust feature which was also observed in more simple one-dimensional (1D) Bjorken expansion [89], or with different initial conditions.", "Within BAMPS, we do not assume any jet distribution by hand, energetic particles propagate and suffer from energy loss by default.", "As we see in Fig.", "REF , at low $p_T$ the inelastic scattering shows only very little effect from the thermal flow, and its contribution is negative.", "The $2 \\leftrightarrow 2$ photon production at $p_T\\lesssim 1.3~\\mathrm {GeV}$ shows a large flow, translated from the flowing background.", "The maximum magnitude is inline with the hydro result and the PHSD transport model.", "The total photon flow in the QGP from BAMPS is very small in magnitude and negative.", "We estimate the impact of our results when confronted to data by simply replacing our QGP result with that from a complete hydro calculation and compare roughly to experimental data.", "The reweighting of the flow (the strongly flowing hadronic contribution gets more weight, as our QGP yield is lower), will (trivially) enhance the flow at lower $p_T$ .", "At higher $p_T$ we see the diminishing of flow due to the negative $v_2$ from BAMPS.", "Again, bearing in mind the inconsistency, we have added a “pre-equilibrium contribution” from BAMPS to the complete hydro result.", "This, as was the case also for the yield, has only a very small effect." ], [ "Conclusions", "We have implemented photon production cross sections at full leading-order in a dynamical, microscopic transport approach for heavy-ion collisions.", "The conceptual difficulties concerning the rates, which involve in principle infinite scattering amplitudes, could be tackled by tuning the screening mass and fixing overall multiplicative factors.", "Consequently, the analytically fully known leading-order photon production rate has been reproduced by the transport simulation from microscopic scatterings.", "We discussed the Debye mass dependence as well as the fugacity dependence of the photon rate, and found a nontrivial scaling with fugacity, which is different for $2 \\leftrightarrow 3$ and $2 \\leftrightarrow 2$ photon production.", "Having the fugacity dependence of the photon rate under control, we turned to realistic heavy-ion collisions.", "We give results for the direct photon contribution to spectra and elliptic flow from the QGP phase (in this exploratory study we restricted ourselves to RHIC collisions at $\\sqrt{s_{NN}}=200~\\mathrm {GeV}$ ).", "The magnitude of the $p_T$ spectrum naturally depends strongly on the quark content of the medium which in turn is largely influenced by the initial conditions.", "Our implementation of pythia initial conditions combined with the mentioned fugacity dependence has shown the expected smaller yield than hydrodynamic computations which is in complete chemical equilibrium from their initialization time on.", "The $p_T$ spectra from BAMPS are also harder; this is due to the choice of initial condition, but also a distinct feature of the nonequilibrium nature of BAMPS.", "Partons which are not part of a thermal ensemble scatter and make photons and these photons are not expected to show a thermal behavior.", "Especially at higher momenta (between $2-3~\\mathrm {GeV}$ ), the spectrum is harder.", "If yet unknown initial conditions with larger quark content were used in the future, the nonequilibrium photon spectrum would be higher and thus closer to the data.", "In this case one would have a stronger pre-equilibrium contribution, whereas with our present setup the pre-equilibrium contribution is very small.", "A more obvious implication of nonequilibrium photon production can be seen in our results for the elliptic flow.", "Due to microscopic production of photons by partons which are not part of a pure thermal ensemble, the momentum asymmetry of the produced photons is not the result of a simple boosted thermal spectrum.", "Jet-photon conversion, the almost one-to-one transfer of momentum of (usually higher energetic) particles to photons, could have been identified to play an important role.", "We have observed the competing of a thermal flowing medium with positive photon $v_2$ and the nonequilibrium leakage effect with its negative $v_2$ .", "This leakage effect probes the asymmetric geometry of the fireball by the traverse of slightly higher energetic quarks and their conversion into photons, or, radiative but very collinear emission.", "The resulting elliptic flow is dominated by nonthermal emission at higher $p_T$ and strongly negative, and larger but still negative for low $p_T$ , where background flow is more important.", "We believe that other yet unidentified effects solve the photon-puzzle in future.", "Fragmentation photons may play a role as well as the effect of electromagnetic fields on the evolution.", "Both effects can be investigated in microscopic transport models, and with this work we have set the basis." ], [ "Acknowledgements", "We are grateful to J.F.", "Paquet, C. Shen and C. Gale for constant interest in our work, useful discussions and providing data for comparison.", "Furthermore we thank Alexander Rothkopf for fruitful discussions.", "M.G.", "and F.S.", "acknowledge the support from the “Helmhotz Graduate School for Heavy Ion research”.", "The authors are grateful to the Center for Scientific Computing (CSC) Frankfurt for the computing resources.", "This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.", "XZ was supported by the MOST, the NSFC under Grants No.", "2014CB845400, No.", "11275103, No.", "11335005, and No.", "11575092." ], [ "Debye Screening Prescriptions", "The screening masses and thermal quark masses behave very similarly for our purposes.", "They are of order $gT$ but have different prefactors depending on the type of statistics.", "The squared thermal mass for light quarks is defined as $m_{D,q}^2=g^2C_F\\int \\frac{\\mathrm {d}^3 p}{(2\\pi )^3E_p}(f_g+f_q).$ The squared thermal gluon mass (=Debye mass) is defined as $m_{D,g}^2=16\\pi \\alpha _s \\int \\frac{\\mathrm {d}^3 p}{(2\\pi )^3 E_p}(N_c f_g+N_f f_q).$ Using Boltzmann statistic distributions, the squared gluon Debye mass is $m_{D,g}^2=\\frac{8}{\\pi }(N_c+N_f)\\alpha _s T^2\\approx 15.28 \\alpha _s T^2$ whereas the squared thermal quark mass is $m_{D,q}^2=\\frac{1}{9}m_D^2=\\frac{8\\alpha _s T^2}{9\\pi }(N_c+N_f)=\\frac{16}{3\\pi }\\alpha _sT^2\\approx 1.7 \\alpha _s T^2.$ Using quantum statistic distributions, the squared gluon Debye mass is $m_{D,g}^2=\\frac{4\\pi \\alpha _s}{3}\\left(N_c + \\frac{N_f}{2}\\right)T^2 = 6\\pi \\alpha _sT^2 \\approx 18.85 \\alpha _s T^2,$ whereas the squared thermal quark mass is $m_{D,q}^2=\\frac{1}{2}m_\\infty ^2=\\frac{1}{2}\\frac{C_Fg_s^2T^2}{4}=\\frac{2\\pi \\alpha _s}{3}T^2\\approx 2.09 \\alpha _s T^2.$" ], [ "Photon Rates", "The total photon production rate (units $[\\mathrm {energy}^4]$ ) for processes $P+P^\\prime \\rightarrow K+K^\\prime $ , where $K$ is the four-momentum of the photon, can be written as [76], $R&=\\mathcal {N}\\int \\frac{\\mathrm {d}^3 p}{2E_p(2\\pi )^3} \\int \\frac{\\mathrm {d}^3 p^\\prime }{2E_{p^\\prime }(2\\pi )^3} \\int \\frac{\\mathrm {d}^3 k}{2E_k(2\\pi )^3} \\int \\frac{\\mathrm {d}^3 k^\\prime }{2E_{k^\\prime }(2\\pi )^3} (2\\pi )^4\\delta ^{(4)}(P+P^\\prime -K-K^\\prime )\\nonumber \\\\&\\quad \\times \\left|\\mathcal {M}\\right|^2 f(P)f(P^\\prime )\\left(1 \\pm f(K^\\prime ) \\right),$ where $\\mathcal {N}$ is a symmetry factor respecting the electric charges and degeneracies.", "In the case of Compton scattering, the symmetry factor for two flavors is $\\mathcal {N}=320/3$ , for three flavors $\\mathcal {N}=128$ .", "In the case of quark-antiquark annihilation, the symmetry factor for two flavors is $\\mathcal {N}=20$ , for three flavors $\\mathcal {N}=24$ .", "By using an approximation for the case $E\\gg T$ , the differential photon rate can be obtained from the scattering matrix elements $\\mathcal {M}(s,t)$ using [72] $E_k\\frac{\\mathrm {d}R_i}{\\mathrm {d}^3k}=\\frac{\\mathcal {N}_i}{(2\\pi )^6}\\frac{T}{32E_k}e^{-E_k/T}\\int \\limits _0^\\infty \\mathrm {d}s \\frac{1}{s} \\ln \\left\\lbrace \\left(1\\pm e^{-\\frac{s}{4E_kT}} \\right)^{\\pm 1}\\right\\rbrace \\int \\limits _{-s}^{0} \\mathrm {d}t \\left|\\mathcal {M}_i \\right|^2,$ where $s,\\ t$ and $u=-s-t$ are the usual Mandelstam variables.", "However, by using the techniques from Ref.", "[76], the rate can be integrated numerically without the approximation $E\\gg T$ .", "Note that by using Eq.", "(REF ) or (REF ) the matrix element must not diverge for soft momentum transfer.", "These formulas can thus only be used, if either a soft momentum cutoff is applied ($q_{\\text{cut}}$ , as in most previous works, e.g., [76]), or the propagators in the matrix elements are naively screened by using a screening mass.", "This we call Born approximation." ], [ "Algorithm to Determine Specific Mean-free Paths", "The mean-free path is the inverse of the scattering rate per particle $\\lambda _{\\text{mfp}}=R^{-1}$ .", "The inverse rate for scattering of a single particle $q$ within a medium of particle density $n_q$ is $\\lambda _{\\text{mfp,}qq\\rightarrow qq}^q=\\left(n_q \\left\\langle \\sigma ({s})v_{\\text{rel}} \\right\\rangle _\\text{therm} \\right)^{-1},$ where the average is over the thermal ensemble and $v_{\\text{rel}}\\equiv s/(2E_1E_2)$ , where $E_1, E_2$ are the energies of two incoming particles and the Mandelstam variable $s=(P_1+P_2)^2$ is the squared sum over their four-momenta.", "A thermal ensemble allows for the direct calculation of the mean-free path from the thermal ensemble, just given the cross section $\\sigma ({s})$ and the equilibrium density $n_q$ .", "However, we explicitly want to extract the mean-free paths in a chemical and/or kinetically nonequilibrated system.", "For this purpose, we choose all possible scattering partners $i$ in each computational cell and compute their collision probability $P^{i}_{22}$ from Eq.", "(REF ), such that $\\lambda _{\\text{mfp,}qq\\rightarrow qq}^q&=\\left(n_q \\left\\langle \\sigma ({s})v_{\\text{rel}} \\right\\rangle _\\text{therm} \\right)^{-1}\\nonumber \\\\&=N_q\\frac{1}{M} \\sum \\limits _{i=1}^M \\frac{P_{22}^i}{\\Delta t},\\nonumber \\\\&=\\frac{1}{\\Delta V}\\frac{2}{(N_q-1)}\\sum \\limits _{i=1}^M \\frac{\\sigma _i v_{\\text{rel},i}}{\\Delta t},\\nonumber \\\\M&\\equiv \\frac{1}{2}N_q(N_q-1).$ Note, that here $N_q$ is the total number of quarks in the cell with volume $\\Delta V$ , which is the physical number times the number of test particles, $N_{\\text{test}}$ .", "The cross section in Eq.", "(REF ) is divided by $N_{\\text{test}}$ , such that the mean-free path is the physical mean-free path and independent of $N_{\\text{test}}$ .", "For processes $qq\\rightarrow qq$ there are ${N_q \\atopwithdelims ()2}=1/2 N_q(N_q-1)$ possible scattering processes for $N_q$ quarks in the system, and we take numerically the average to get the mean-free path of a quark when considering only scatterings with another quark of the same flavor.", "In a similar way we can compute the mean-free paths for $qq^\\prime \\rightarrow qq^\\prime ,\\ \\bar{q}\\bar{q}^\\prime \\rightarrow \\bar{q}\\bar{q}^\\prime ,\\ q\\bar{q}\\rightarrow q\\bar{q}$ ." ], [ "Bremsstrahlung Diagrams for Quark-Quark Scattering", "In this section we compute the squared matrix element for the $qq\\rightarrow qq\\gamma $ process, shown in Fig.", "REF .", "For this purpose, we label the amplitude of Fig.", "REF with $\\mathcal {M}_a$ , and the one from Fig.", "REF with $\\mathcal {M}_b$ .", "We have to compute $(\\mathcal {M}_a+\\mathcal {M}_b)^\\star \\cdot (\\mathcal {M}_a+\\mathcal {M}_b)$ .", "As customary in scattering theory, the matrix element is given by an average over initial spin, polarization and color states, and a sum over final states." ], [ "Matrix elements", "With the momentum assignment $p_3=p_1+q$ , $p_4=p_2-q-k$ we write down the first matrix element [Fig.", "REF ] by using momentum space Feynman rules: $i\\mathcal {M}_a&=\\bar{u}^w(p_3)(ig)\\gamma ^\\mu \\lambda ^a_{il}u^s(p_1)\\frac{-ig_{\\mu \\nu }\\delta _{ab}}{q^2}\\bar{u}^r(p_4)\\nonumber \\\\&\\quad \\times \\ (ig)\\gamma ^\\nu \\lambda ^b_{mj}\\frac{i(m+{p}_2-{k})}{(p_2-k)^2-m^2}\\nonumber \\\\&\\quad \\times \\ (iQ_{EM})\\gamma ^\\alpha \\epsilon _\\alpha ^\\star (k)u^t(p_2).$ The second matrix element [Fig.", "REF ] is $i\\mathcal {M}_b&=\\bar{u}^w(p_3)(ig)\\gamma ^\\mu \\lambda ^a_{il} u^s(p_1) \\frac{-ig_{\\mu \\nu }\\delta _{ab}}{q^2}\\bar{u}^r(p_4)\\nonumber \\\\&\\quad \\times \\ (iQ_{EM})\\gamma ^\\alpha \\epsilon _\\alpha ^\\star (k)\\nonumber \\\\&\\quad \\times \\frac{i(m+{p}_4+{k})}{(p_4+k)^2-m^2}(ig)\\gamma ^\\nu \\lambda ^b_{mj}u^t(p_2).$ By using the Dirac equation we transform the numerators of the quark propagators in the following way: $({p}_2+m)\\gamma ^\\alpha u(p_2)=2p_2^\\alpha u(p_2)\\nonumber \\\\({p_4}+m)\\gamma ^\\nu u(p_2)=2p_4^\\nu u(p_2),\\nonumber $ and we simplify the denominators, $(p_2-k)^2=-2p_2\\cdot k,\\quad (p_4+k)^2=2p_4\\cdot k.$ Note that, later on, we screen the $t$ -channel quark-propagator in $\\mathcal {M}_a$ by using a Debye mass $m_{D,q}^2$ , $\\frac{1}{-2p_2\\cdot k}\\rightarrow \\frac{1}{-2p_2\\cdot k -m_{D,q}^2},$ and the $s$ -channel propagator in in $\\mathcal {M}_b$ , $\\frac{1}{2p_4\\cdot k}\\rightarrow \\frac{1}{2p_4\\cdot k+m_{D,q}^2}.$ Only at this step we set the masses to zero, $m\\equiv 0 $ .", "The gluon propagator will be screened with the Debye mass $m_{D,g}^2$ , $\\frac{1}{q^2}\\rightarrow \\frac{1}{q^2-m_{D,g}^2}.$" ], [ "Amplitude", "Next we simplify the summed matrix elements, $i\\mathcal {M}_a+i\\mathcal {M}_b&=\\bar{u}^w(p_3)(ig)^2\\gamma ^\\mu \\lambda ^a_{il}\\lambda ^b_{mj} u^s(p_1)\\frac{-ig_{\\mu \\nu }}{q^2}(iQ_{EM})\\bar{u}^r(p_4)\\nonumber \\\\&\\quad \\times \\left[ \\frac{i(\\gamma ^\\nu p_2^\\alpha -\\gamma ^\\nu {k}\\gamma ^\\alpha )}{-2p_2\\cdot k } + \\frac{i(2\\gamma ^\\alpha p_4^\\nu +\\gamma ^\\alpha {k}\\gamma ^\\nu )}{2p_4\\cdot k} \\right]u^t(p_2)\\epsilon _\\alpha ^\\star (k)\\nonumber \\\\&=-ig^2Q_{EM}\\bar{u}^w(p_3)\\gamma _\\nu u^s(p_1)\\frac{\\lambda ^a_{il}\\lambda ^a_{mj}}{q^2}\\bar{u}^r(p_4)\\nonumber \\\\&\\quad \\times \\left[ \\frac{\\gamma ^\\nu {k}\\gamma ^\\alpha -\\gamma ^\\nu p_2^\\alpha }{2p_2\\cdot k } + \\frac{\\gamma ^\\alpha {k}\\gamma ^\\nu +2\\gamma ^\\alpha p_4^\\nu }{2p_4\\cdot k} \\right]u^t(p_2)\\epsilon _\\alpha ^\\star (k).$ This amplitude needs to be squared in the next step, $(i\\mathcal {M}_a+i\\mathcal {M}_b)\\cdot (i\\mathcal {M}_a+i\\mathcal {M}_b)^\\star $ , and then summed over final states and averaged over initial states.", "We define the resulting summed and averaged squared matrix element as $\\overline{\\left|\\mathcal {M}\\right|^2}$ .", "The sum over final photon polarizations reduces to [94] $\\sum \\limits _\\epsilon \\epsilon _\\alpha ^\\star (k)\\epsilon _\\beta (k) \\rightarrow -g^{\\alpha \\beta }.$ The color matrices are (see Ref.", "[94], Eq.", "(17.63)) $\\frac{1}{N_c^2}\\sum \\limits _{\\text{colors}} \\lambda ^a\\lambda ^a\\lambda ^b\\lambda ^b = \\frac{2}{9}.$ The average over initial quark spins and sum over final spins gives a factor $1/4$ , and, by using $\\sum \\limits _{\\text{spin }t}u^t(p)\\bar{u}^t(p)={p},$ we can transform the matrix element into traces, $\\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.}}", "&= \\frac{1}{4}\\frac{2}{9}\\frac{Q_{EM}^2g^4}{q^4}\\text{Tr}\\left\\lbrace {p}_4 \\left[\\frac{-\\gamma ^\\nu {k} \\gamma _\\beta +2\\gamma ^\\nu p_{2,\\beta }}{2p_2\\cdot k}+\\frac{-\\gamma _\\beta {k}\\gamma ^\\nu -2\\gamma _\\beta p_4^\\nu }{2p_4\\cdot k}\\right]\\right.", "\\nonumber \\\\&\\quad \\times {p}_2\\left.\\left[\\frac{\\gamma ^\\beta {k}\\gamma ^\\mu -2\\gamma ^\\mu p_2^\\beta }{2p_2\\cdot k}+\\frac{\\gamma ^\\mu {k}\\gamma ^\\beta +2\\gamma ^\\beta p_4^\\mu }{2p_4\\cdot k}\\right] \\right\\rbrace .$ The gluon momentum squared is $q^2=(p_4-p_2+k)^2$ and the gluon propagator reads, $\\frac{1}{q^4}=\\frac{1}{(2p_4\\cdot k -2k\\cdot p_2-2p_4\\cdot p_2)^2},$ and after screening, $\\frac{1}{q^4}\\rightarrow \\frac{1}{\\left(2p_4\\cdot k -2k\\cdot p_2-2p_4\\cdot p_2-m_{D,g}^2\\right)^2}.$ The trace in Eq.", "(REF ) can be done using the mathematica package FeynCalc 8.2.0 [95], with the result (where we defined the scalar product of four-vectors $(ij)\\equiv p_i\\cdot p_j$ ), $A&\\equiv 2 (25)+m_{D,q}^2\\nonumber \\\\B&\\equiv 2 (45)+m_{D,q}^2\\nonumber \\\\C&\\equiv 4 (45)+m_{D,q}^2\\nonumber \\\\D&\\equiv (35) B^2-2 (34) A (2 (25)-B-m_{D,q}^2) \\nonumber \\\\E&\\equiv (23) A ((25) C+(45) (-B-m_{D,q}^2))+(24) A (2 (34) A+(35) (A+B))+(25) D \\nonumber \\\\F&\\equiv (24) A (A+B)+(25) B^2\\nonumber \\\\G&\\equiv (23) A ((24) B+(45) A)+(34) F\\nonumber \\\\H&\\equiv -2 (23) B+(34) (-B-m_{D,q}^2)+(35) m_{D,q}^2\\nonumber \\\\J&\\equiv (45) H +(24) (35) B+(25) ((34) C+2 (35) (45))\\nonumber \\\\\\overline{\\left|\\mathcal {M}\\right|^2}_{\\text{rad.}}", "&=\\frac{1}{4}\\frac{2}{9}Q_{EM}^2g^4 128\\frac{A ((12) J-2 (13) (24) (45) A)+(14) E +(15) G}{A^2 B^2 (2 (24)+2 (25)-2 (45)+m_{D,g}^2)^2}$ We have checked that the Ward identity is fulfilled." ], [ "Symmetry-factor", "The self energy in Fig.", "REF with the given cut generates the $qq\\rightarrow qq \\gamma $ -process.", "We discuss its multiplicity factor here.", "The photon legs of the self energy can be crossed, which is why the self energy carries a factor of two.", "The four gluon vertices are completely identical.", "Every gluon can be reversed.", "This contributes a factor of four.", "The loop can be opened by tic-ing the upper or lower quark line, which introduces a factor of two.", "In total the symmetry factor is 16." ], [ "Verification of the Bremsstrahlung Process and Kinematics", "To cross-check the inelastic photon production and verify the kinematic integration limits and the limit stemming from the LPM constraint, we show in Fig.", "REF a typical set of sampled photon momenta according to the full bremsstrahlung matrix element.", "Each of the red dots represents one sampled photon for a fixed (but arbitrary) configuration of incoming quark momenta.", "For illustrational purposes (to see the intrinsic asymmetry in $y$ ) we fix the quark line where the photon is emitted, and discard the radiation from the other quark line.", "However, in any real simulation of BAMPS the incoming quarks are randomly taken to be either quark one or two - thus the momentum spectrum will be symmetric in $y$ .", "Omitting the integration over $y$ in Eq.", "(REF ), we compute numerically the differential cross section $\\mathrm {d}\\sigma _{23}/\\mathrm {d}y$ normalized by the total cross section $\\sigma _{\\text{tot}}$ in Fig.", "REF .", "Here the symmetry in $y$ can clearly be seen.", "Omitting the integration over $k_\\perp ^2$ in Eq.", "(REF ), we compute also the differential cross section with respect to $k_\\perp ^2$ , as shown in Fig.", "REF .", "Both figures are done for an arbitrary momentum setup of the incoming quarks, namely $p_1=(2T,0,2T,0), p_2=(2T,0,0,-2T)$ and $T=0.4~\\mathrm {GeV}$ .", "It is clearly visible, that the mean-free path changes the kinematics of the outgoing photon momenta, a larger mean-free path allows more collinear radiation.", "Figure: The exact photon bremsstrahlung matrix element is used to sample photons.", "Their momentum is given in k ⊥ ,q ⊥ ,y,φk_\\perp ,q_\\perp ,y,\\phi -space; here we show several realisations (red dots) as an example.", "The green dashed curves represent the limits.", "The purple and blue dash-dotted lines show the limit from the LPM constraint for larger mean-free paths.", "The asymmetry in yy is forced by using only one fixed quark as the radiating one.Figure: The differential cross section in the rapidity of the radiated photon for various mean-free paths.Figure: The differential cross section in the transverse momentum of the radiated photon for various mean-free paths." ] ]
1612.05811
[ [ "Deep holes and MDS extensions of Reed-Solomon codes" ], [ "Abstract We study the problem of classifying deep holes of Reed-Solomon codes.", "We show that this problem is equivalent to the problem of classifying MDS extensions of Reed-Solomon codes by one digit.", "This equivalence allows us to improve recent results on the former problem.", "In particular, we classify deep holes of Reed-Solomon codes of dimension greater than half the alphabet size.", "We also give a complete classification of deep holes of Reed Solomon codes with redundancy three in all dimensions." ], [ "Introduction", "A deep-hole for a code $\\mathcal {C}$ is a received vector whose distance from $\\mathcal {C}$ attains the maximum possible value viz.", "the covering radius of $\\mathcal {C}$ .", "A $[n,k, \\mathcal {D}]_q$ RS (Reed-Solomon) code $\\mathcal {C}$ consists of codewords $(f(x_1), f(x_2), \\dots , f(x_n))$ as $f(X)$ runs over the set of univariate polynomials of degree at most $k-1$ with $GF(q)$ -coefficients.", "The evaluation set $\\mathcal {D}= \\lbrace x_1, \\dots , x_n\\rbrace $ consists of $n$ distinct and ordered points of $GF(q)$ .", "The covering radius of $\\mathcal {C}$ can be shown to be $n-k$ .", "By a generating polynomial $u(x)$ for a received word $u \\in GF(q)^n$ of $\\mathcal {C}$ , we mean the Lagrange interpolation polynomial of degree at most $n-1$ for the data points $\\lbrace (x_1, u_1), \\dots , (x_n, u_n)\\rbrace $ .", "It was shown in [1] that the problem of determining whether a received word is a deep hole of a given Reed Solomon code is NP-hard.", "Several authors have studied the problem of classifying deep holes of RS codes.", "Cheng and Murray conjectured that: Conjecture 1 (Cheng and Murray [2]) All deep-holes of a $[k,q, \\mathcal {D}=GF(q)]_q$ RS code with $2 \\le k \\le q-2$ have generating polynomials of degree $k$ , except when $q$ is even and $k=q-3$ .", "The exception was not part of [2], but is well known.", "In fact this conjecture is equivalent to a central conjecture in finite geometry (See conjecture 1).", "It may be somewhat surprising to note that the conjecture for $k=q-3$ and $q$ odd, is equivalent to Beniamino Segre's foundational theorem of finite geometry that states – in coding theory terminology– that any 3-dimensional MDS code of length $q+1$ is Reed-Solomon.", "Recently, Zhuang, Cheng and Li obtained the following result: Theorem (Zhuang, Cheng and Li [3]) Let $\\mathcal {C}$ be a $[n,k, \\mathcal {D}]_q$ RS code with $k \\ge \\lfloor (q-1)/2 \\rfloor $ .", "If $q > 2$ is a prime number and then the generating polynomials of deep holes of $\\mathcal {C}$ are $ u(x)=a u_1(X) + u_2(X), \\; a \\in GF(q)^{\\times }, \\, \\text{deg}(u_2) \\le k-1,$ and $u_1(X)$ equals either $X^k$ or the generating polynomial for the data points $\\lbrace (x_i, \\tfrac{1}{x_i - \\delta }) :1 \\le i \\le n\\rbrace $ for some $\\delta \\in GF(q)\\!\\setminus \\!\\mathcal {D}$ .", "The techniques used in [3] necessitate the condition that the alphabet size is an odd prime, and they leave the problem open for $GF(q)$ .", "Our method (which uses the work of Roth and Seroussi [4]) allows us to work with any finite field alphabet.", "The contributions of this work are as follows : Theorem 1 The above theorem of Zhang et al.", "holds not only for $GF(p)$ but for any $GF(q)$ with $q$ odd.", "For $q$ even, the result of the above theorem still holds except when $n=k+3$ .", "In this case $u_1(X)$ may also equal $X^{k+1} + (\\sum _{i=1}^{n} x_i) X^{k}$ in addition to choices for $u_1(X)$ mentioned therein.", "In Theorem REF , we give a geometric interpretation of Theorem REF .", "Our method uses the observation that for a $[n,k,\\mathcal {D}]$ RS code with ($n<q$ ) and a parity check matrix $H$ of $\\mathcal {C}$ , a received vector $u \\in GF(q)^n$ is a deep hole of $\\mathcal {C}$ if and only if the syndrome $Hu$ has the property that the matrix $[H \\,|\\, Hu]$ is an MDS extension of the RS code $\\mathcal {C}^{\\perp }$ by one digit.", "In the case $k \\ge \\lfloor (q-1)/2 \\rfloor $ , the results of Roth and Seroussi [4], allow us to obtain Theorem REF .", "For $[n,k]_q$ RS codes $\\mathcal {C}$ of length $q+1$ (here the evaluation set is $\\mathcal {D}= PG(1,q)$ ), it is not always true that the covering radius of $\\mathcal {C}$ is $n-k$ .", "Sometimes the covering radius equals $n-k-1$ .", "In the latter case the equivalence between deep holes of $\\mathcal {C}$ and MDS extensions of $\\mathcal {C}^{\\perp }$ breaks down.", "In Section , we present some results for RS codes of length $q+1$ .", "In Section we classify deep-holes of a $[n,k]_q$ RS codes with redundancy $n-k=3$ (Theorem REF ).", "We also obtain canonical forms for non GRS 3-dimensional MDS codes extending a GRS code by one digit.", "(Theorem REF )." ], [ "Notation", "Throughout this article, the dimension $k$ of a RS code is in the range $2 \\le k \\le q-1$ .", "As for the length $n$ , we note that the cases where the length $n$ equals $k$ or $k+1$ are uninteresting: In the former case there are no deep holes, and in the later case deep-holes are just those received words which are not codewords.", "Therefore, we impose the condition $k+2 \\le n \\le q+1$ on the length of $\\mathcal {C}$ .", "Generalized Reed Solomon codes (GRS codes) are obtained from RS codes by applying a diagonal Hamming isometry $u \\mapsto \\text{diag}(\\nu _1, \\dots , \\nu _n) u$ of $GF(q)^n$ .", "Clearly the deep holes of the resulting GRS code are obtained from those of the original RS code by applying the same transformation.", "Thus, for the purposes of studying deep holes, it is sufficient to consider only RS codes.", "The dual of a RS code, is a GRS code.", "The definition of a $[n,k, \\mathcal {D}]_q$ RS code is easily extended to allow the evaluation set $\\mathcal {D}$ to be a subset of the projective line $PG(1,q) = GF(q) \\cup \\infty $ .", "Here it is understood that the value of a message polynomial $f(X)$ at $\\infty $ is the coefficient of $X^{k-1}$ .", "The evaluation set $\\mathcal {D}$ of a GRS code is not unique: if $\\varphi (x)= (c+dx)/(a+bx)$ is a fractional linear transformation of $PG(1,q)$ then $\\varphi (\\mathcal {D})$ is also an evaluation set (for example see Proposition 2.5 of [5]).", "Using this freedom we can (and will) always assume $\\mathcal {D}$ to be free of $\\infty $ provided $n \\ne q+1$ .", "Given an evaluation set $\\mathcal {D}= \\lbrace x_1, \\dots ,x_n\\rbrace \\subset GF(q)$ , we associate to $\\mathcal {D}$ some numbers $s_0, \\dots , s_n$ and $\\nu _1, \\dots , \\nu _n$ defined by: $ \\prod _{i=1}^n (X - x_i) = \\sum _{j=0}^n s_j X^{n-j}, \\quad \\nu _j =\\prod _{\\lbrace i: i \\ne j\\rbrace } (x_j - x_i)^{-1}.$ In order to give geometric interpretation to our results, it will be convenient to introduce some terminology.", "We recall that an $[n,k]_q$ MDS code is a $k$ -dimensional code of length $n$ whose minimum distance is $n-k+1$ .", "Equivalently any generator matrix of a $[n,k]_q$ MDS code has the property that all its $k \\times k$ minors are non-zero.", "An $n$ -arc in $PG(k-1,q)$ is a unordered collection of $n$ points of $PG(k-1,q)$ such that any $k \\times n$ matrix whose columns are lifts of the $n$ points of the arc to $GF(q)^n$ , generates a $[n,k]_q$ MDS code.", "Thus there is a bijective correspondence between monomial equivalence classes of $[n,k]_q$ MDS codes and projective equivalence classes of $n$ -arcs in $PG(k-1,q)$ .", "For each element $x \\in GF(q) \\cup \\infty $ we define vectors $c_m(x) \\in GF(q)^m$ by: $ c_{m}(x) = {\\left\\lbrace \\begin{array}{ll} (1,x,x^2, \\dots , x^{m-1})^T &\\text{ if } x \\in GF(q) \\\\(0,0,\\cdots , 0, 1)^T &\\text{ if } x = \\infty .", "\\end{array}\\right.}", "$ We note that a $[n,k,\\mathcal {D}]_q$ RS code with $\\mathcal {D}= \\lbrace x_1, \\dots , x_n\\rbrace \\subset GF(q)$ has a generator and parity check matrix pair given by: rCl Gk(D) = [ck(x1)  |   ck(x2)  |   ... |   ck(xn)] Gk(D) = [1 cn-k(x1)  |   2 cn-k(x2)  |   ... |   n cn-kk(xn)] where $\\nu _i$ are as in (REF ).", "In the case $\\mathcal {D}= \\lbrace x_1, \\dots ,x_q\\rbrace \\cup \\infty $ , the $[q+1,k,\\mathcal {D}]$ RS code $\\mathcal {C}$ has a generator and parity check matrix pair given by: rCl Gk(PG(1,q)) = [ck(x1)  |   ck(x2)  |   ... |   ck(xq)  |   ck()] Gk(PG(1,q)) = [cq+1-k(x1)  |   cq+1-k(x2)  |   ... |   cq+1-k(xq)  |  cq+1-k()].", "The standard RNC (rational normal curve) in $PG(m-1,q)$ consists of $q+1$ points of $PG(m-1,q)$ given by $\\lbrace [c_m(x)] : x \\in GF(q) \\cup \\infty \\rbrace $ .", "(For a nonzero $v \\in GF(q)^m$ , we use the notation $[v] \\in PG(m-1,q)$ to denote the one-dimensional subspace of $GF(q)^m$ represented by $v$ ).", "By a RNC in $PG(m-1,q)$ , we mean the image of the standard RNC under a projective linear transformation of $PG(m-1,q)$ .", "Thus, in the correspondence between arcs and MDS codes, the $n$ -arcs of $PG(k-1,q)$ which correspond to $[n,k]_q$ RS codes, are those arcs which are contained in a RNC.", "A $n$ -arc in $PG(k-1,q)$ is said to be complete if it is not contained in a $n+1$ -arc.", "Equivalently, the corresponding $[n,k]_q$ MDS code cannot be extended to a $[n+1,k]_q$ MDS code.", "Let $\\mathcal {N}_m \\in GF(q)^m$ denote the vector $(0,\\dots ,0,1,0)^T$ .", "For $q$ even, the point $\\mathcal {N}_3 \\in PG(2,q)$ is known as the nucleus of the standard RNC in $PG(2,q)$ .", "Definition 1 For a $[n,k]$ code $\\mathcal {C}$ , we say two received words $u_1, u_2 \\in GF(q)^n$ are coset-equivalent if $u_2 - u_1 \\in \\mathcal {C}$ .", "We say $u_1, u_2$ are equivalent if $u_2 - a u_1 \\in \\mathcal {C}$ for some $a \\in GF(q)^{\\times }$ .", "(In this artcle we denote $GF(q) \\setminus \\lbrace 0\\rbrace $ by $GF(q)^{\\times }$ ).", "In the case $\\mathcal {D}\\subset GF(q)$ , if $u(X)$ is the generating polynomial of a deep hole $u$ , then the generating polynomials of words equivalent to $u$ are $\\lbrace a u(X) +f(X) : a \\ne 0, \\, \\text{deg}(f) <k\\rbrace $ .", "Thus there is a unique $v$ equivalent to $u$ whose generating polynomial is of the form $X^k P_{u}(X)$ with $P_u(X)$ monic of degree at most $n-k-1$ .", "We use the notation $\\rho (\\mathcal {C})$ for the covering radius of a code $\\mathcal {C}$ ." ], [ "Deep holes and MDS extensions", "Let $\\mathcal {C}$ be a $[n,k,\\mathcal {D}]$ RS code.", "If $n \\ne q+1$ , we recall the proof that the covering radius $\\rho (\\mathcal {C})=n-k$ .", "Since $\\rho (\\mathcal {C}) \\le n-k$ for any linear $[n,k]_q$ code, we just need to show there is a received word at a distance of $n-k$ from $\\mathcal {C}$ .", "The word $(x_1^k, \\dots , x_n^k)$ is at a distance $n-k$ from $\\mathcal {C}$ : the vector $(p(x_1)-x_1^k, \\dots , p(x_n)-x_n^k)$ for deg$(p(X)) <k$ has at most $k$ zeros, and there is a $p(X)$ for which this vector has $k$ zeros.", "In the case $n=q+1$ , let $\\mathcal {D}= \\lbrace x_1, \\dots , x_q, \\infty \\rbrace $ with $\\lbrace x_1, \\dots , x_q\\rbrace = GF(q)$ .", "Consider a word of the form $u=(x_1^k, \\dots ,x_q^k,a)$ for any $a \\in GF(q)$ .", "We will show that the distance of $u$ from $\\mathcal {C}$ is $n-k-1$ .", "Let $k \\, {}^\\wedge GF(q)$ denote the subset of $GF(q)$ consists of those elements which can be written as a sum of $k$ distinct elements of $GF(q)$ .", "It is easy to see that $ c \\cdot (k \\, {}^\\wedge GF(q)) = k \\, {}^\\wedge GF(q)$ for all $c \\in GF(q)^{\\times }$ and hence $k \\, {}^\\wedge GF(q) = GF(q)$ .", "Thus, there exist distinct elements $x_1, \\dots , x_k \\in GF(q)$ such that $a = x_1+ \\dots +x_k$ .", "Consider the polynomial $p(X) = X^k - \\prod _{i=1}^k (X - x_i)$ .", "This is a codeword (message word) of $\\mathcal {C}$ and the vector $(p(x_1)-x_1^k, \\dots , p(x_q)-x_q^k, p(\\infty ) - a)$ has exactly $k+1$ zeros.", "This is because $p(\\infty ) = \\sum _{i=1}^k x_i = a$ .", "Thus, we conclude that $\\rho (\\mathcal {C})$ is either $n-k-1$ or $n-k$ .", "We will see in Section that both situations occur.", "Let $G_k(\\mathcal {D})$ and $G_k^{\\perp }(\\mathcal {D})$ be a generator and parity check matrix for $\\mathcal {C}$ as given in () if $n \\ne q+1$ and () if $n=q+1$ .", "For a received word $u$ let $S_{\\mathcal {D}}(u) = G_k^{\\perp }(\\mathcal {D}) u$ be the syndrome of $u$ .", "When $u$ is a non-codeword, we use the term projective syndrome for $[S_{\\mathcal {D}}(u)] \\in PG(n-k-1,q)$ .", "Proposition 1 Let $\\mathcal {C}$ be a $[n,k,\\mathcal {D}]_q$ RS code.", "In case $n=q+1$ , suppose $\\rho (\\mathcal {C})=n-k$ .", "For a received word $u \\in GF(q)^n$ , the augmented matrix: $ G_k^{\\perp }(\\mathcal {D};u):= [ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ] $ generates a $[n+1,n-k]_q$ MDS code if and only if $u$ is deep-hole of $\\mathcal {C}$ .", "Thus $S_{\\mathcal {D}}$ sets up a bijective correspondence between the set of coset-equivalence classes of deep-holes and the set of MDS extensions of the dual RS code $\\mathcal {C}^{\\perp }$ by one digit.", "We recall that the distance of $u$ from $\\mathcal {C}$ is the least number $m$ such that $S_{\\mathcal {D}}(u)$ can be written as a linear combination of some $m$ columns of $G_k^{\\perp }(\\mathcal {D})$ .", "Suppose $\\rho (\\mathcal {C}) = n-k$ (this is automatic if $n \\ne q+1$ ).", "It follows that $u$ is a deep hole of $\\mathcal {C}$ if and only if $[G_k^{\\perp }(\\mathcal {D})\\,|\\, S_{\\mathcal {D}}(u) ]$ generates an $[n+1,k]$ MDS code extending the GRS code $\\mathcal {C}^{\\perp }$ .", "The second assertion follows from the fact that two received words are coset equivalent if and only if their syndromes coincide.", "We need the following result: Theorem (Roth and Seroussi 1986 [4]) Let $\\mathcal {C}$ be a $[n, \\ell ,\\mathcal {D}]$ RS code.", "Suppose $2 \\le \\ell \\le n - \\lfloor (q-1)/2 \\rfloor $ .", "Let $g \\in GF(q)^{\\ell }$ be a vector.", "The augmented matrix $[ G_{\\ell }(\\mathcal {D}) \\,|\\, g]$ generates an $[n+1,\\ell ]_q$ MDS code if and only if: (for $q$ odd) $[g]=[c_{\\ell }(\\delta )]$ for some $\\delta \\in PG(1,q) \\!\\setminus \\!", "\\mathcal {D}$ (for $q$ even) $g$ is either as above, or additionally in case $\\ell =3$ , $[g] = \\mathcal {N}_3$ .", "Under the hypothesis of Proposition REF and $k \\ge \\lfloor (q-1)/2 \\rfloor $ , the matrix $G_k^{\\perp }(\\mathcal {D};u)$ in (REF ) generates a MDS code, if and only if $g=S_{\\mathcal {D}}(u)$ has the form given in the above theorem with $\\ell = n-k$ .", "Thus we have proved: Theorem 2 (Geometric form of Theorem REF ) Under the hypothesis of Proposition REF and and $k \\ge \\lfloor (q-1)/2 \\rfloor $ , a received word $u$ is a deep hole of $\\mathcal {C}$ if and only if: either $[S_{\\mathcal {D}}(u)] = [c_{n-k}(\\delta )]$ lies on the standard RNC in $PG(n-k-1,q)$ for some $\\delta \\in PG(1,q) \\setminus \\mathcal {D}$ .", "or $q$ is even, $n=k+3$ , and $[S_{\\mathcal {D}}(u)]$ is as in part a) or equals the nucleus $\\mathcal {N}_3$ of the standard $RNC$ in $PG(2,q)$ .", "Before, we prove Theorem REF , we need some lemmas.", "We now assume $\\mathcal {C}$ is a $[n,k,\\mathcal {D}]$ RS-code with $\\mathcal {D}\\subset GF(q)$ .", "Given a vector $u \\in GF(q)^n$ , let $u(X)$ be the generating polynomial of $u$ .", "Clearly $u \\mapsto u(X)$ is a linear isomorphism: Lemma 1 A formula for $u(X)$ in terms of $u$ , $s_0, \\dots ,s_n$ and $\\nu _1, \\dots , \\nu _n$ is: $ u(X) = [X^{n-1}, X^{n-2}, \\dots , X, 1] L_n \\, [\\nu _1 c_n(x_1) \\,| \\, \\nu _2 c_n(x_2) \\,| \\, \\dots \\,| \\, \\nu _n c_n(x_n)]\\, u $ where $L_n$ is the $n \\times n$ lower triangular matrix given by $L_{ij} = s_{i-j}$ .", "Moreover, writing $u(X) = u_1(X) + u_2(X)$ where deg$(u_2) <k$ and $u_1$ only contains monomials $X^i$ for $i \\ge k$ , we see that $ u_1(X) = [X^{n-1}, X^{n-2}, \\dots , X^k] L_{n-k} S_{\\mathcal {D}}(u), $ where $L_{n-k}$ is the submatrix of $L_n$ on the first $n-k$ rows and columns.", "The right hand side of (REF ) simplifies to the Lagrange interpolation polynomial $ u(X)= \\sum _{j=1}^n u_j \\nu _j \\prod _{\\lbrace i: i \\ne j\\rbrace } (X - x_i) $ Clearly, $ u_1(X)=[X^{n-1}, X^{n-2}, \\dots , X^k] L_{n-k} \\, G_{k}^{\\perp }(\\mathcal {D}) u $ which is the same as the formula in (REF ).", "Lemma 2 Let $u = (u_1, \\dots , u_n)$ be the vector with $u_i = 1/(x_i - \\delta )$ where $\\delta \\in GF(q) \\setminus \\mathcal {D}$ .", "The generating polynomial of $u$ is $ u(X)= a\\, [X^{n-1}, X^{n-2}, \\dots , X, 1] L_n \\, c_n(\\delta ) $ where $a = -1/ \\prod _{i=1}^n (\\delta - x_i)$ .", "By Lagrange interpolation $ u(X) = \\prod _{i=1}^n \\nu _i (x_i - \\delta )^{-1} \\prod _{\\lbrace \\mu : \\mu \\ne i\\rbrace } (X - x_{\\mu }).", "$ From the definition of the quantitities $\\nu _i$ , it follows that $(X - \\delta ) u(X)$ is the Lagrange interpolation polynomial of degree at most $n$ for the data $ \\lbrace (x,1) : x \\in \\mathcal {D}\\rbrace \\cup \\lbrace (\\delta ,0)\\rbrace $ .", "In other words $ (X - \\delta ) u(X) = 1 +a \\prod _{i=1}^n (X-x_i)= 1+a \\sum _{j=0}^n s_j X^{n-j}.", "$ Using this equation to determine the coefficients of $u(X)$ , we obtain $u(X) = a \\sum _{j=1}^n \\, X^{n-j} (\\sum _{i=0}^{j-1} \\delta ^{j-i-1} s_i), $ which is the same as the formula (REF ) (of Theorem REF ) According to Theorem REF , for $k \\ge \\lfloor (q-1)/2 \\rfloor $ , and $n \\ne k+3$ if $q$ is even, $[S_{\\mathcal {D}}(u)] = [c_{n-k}(\\delta )]$ lies on the standard RNC in $PG(n-k-1,q)$ for some $\\delta \\in PG(1,q) \\setminus \\mathcal {D}$ .", "When $\\delta = \\infty $ , $S_{\\mathcal {D}}(u) =a c_{n-k}(\\infty )$ for some $a \\in GF(q)^{\\times }$ , and thus $L_{n-k} S_{\\mathcal {D}}(u)$ is $a$ times the last column of $L_{n-k}$ which is again $c_{n-k}(\\infty )$ .", "Thus the formula (REF ) implies that $u_1(X) =a X^k$ , as was to be shown.", "In case $\\delta \\in GF(q) \\setminus \\mathcal {D}$ , $S_{\\mathcal {D}}(u) =a c_{n-k}(\\delta )$ for some $a \\in GF(q)^{\\times }$ .", "The generating polynomial $u(X)$ is given by the formula (REF ).", "Comparing with equations (REF ) and (REF ) of Lemma REF , we see that $u(X)$ is of the form $u(X)=b f(X) + g(X)$ , where $b \\in GF(q)^{\\times }$ , deg$(u_2) \\le k-1$ , and $f(X)$ is the generating polynomial for the data points $\\lbrace (x_i, \\tfrac{1}{x_i - \\delta }) :1 \\le i \\le n\\rbrace $ .", "In the case $q$ is even, $n=k+3$ and $k \\ge \\lfloor (q-1)/2 \\rfloor $ , and $[S_{\\mathcal {D}}(u)] \\ne [c_{n-k}(\\delta )]$ for $\\delta \\in PG(1,q) \\setminus \\mathcal {D}$ , we must have $[S_{\\mathcal {D}}(u)]= (0:1:0)$ by Theorem REF .", "It follows from (REF ) that $ u(X) = a [X^{k+2}, X^{k+1}, X^k] ( {\\begin{matrix}1 &0& 0\\\\ s_1& 1& 0\\\\ s_2 &s_1& 1 \\end{matrix}} )\\, ( {\\begin{matrix}0 \\\\1 \\\\0 \\end{matrix}} )+u_2(X)= a(X^{k+1} + s_1 X^k) +u_2(X),$ for some polynomial $u_2$ of degree at most $k-1$ .", "We now give a geometric restatement of the Conjecture REF of Cheng and Murray that we mentioned in Section .", "Since $\\mathcal {D}= GF(q)$ , we note that $PG(1,q) \\setminus \\mathcal {D}= \\lbrace \\infty \\rbrace $ .", "To say that the generating polynomial of a deep hole has degree exactly $k$ is equivalent to the assertion that $[S_{\\mathcal {D}}(u)] = [c_{n-k}(\\infty )]$ (by (REF )).", "Thus Conjecture REF is equivalent to the following conjecture (where $m=q-k$ ) Conjecture ${\\bf 1^{\\prime }}$ For $2 \\le m \\le q-2$ , with the exception of $m=3$ when $q$ is even, any MDS extension (by one digit) of a $m$ dimensional $RS$ code with evaluation set $GF(q)$ must itself be GRS.", "Equivalently, any $(q+1)$ -arc in $PG(m-1,q)$ with $q$ points on a RNC must have all its points on the RNC.", "Corollary 1 (of Theorem REF ) The conjecture of Cheng and Murray holds if $k \\ge \\lfloor (q-1)/2 \\rfloor $ , except when $q$ is even and $k=q-3$ .", "Remark : We strongly believe that Conjecture REF must hold for all $q-2 \\ge k \\ge 2$ , not just $k \\ge \\lfloor (q-1)/2 \\rfloor $ .", "We justify this belief with the next two Propositions.", "Proposition 2 For $q$ odd, Conjecture REF holds for $k=2$ (i.e.", "Conjecture ${\\bf 1^{\\prime }}$ holds for $m = q-2$ ), as it is equivalent to B. Segre's fundamental result [6] that any $q+1$ -arc in $PG(2,q)$ (for odd $q$ ) is a RNC.", "Equivalently, for odd $q$ any $[q+1,3]_q$ MDS code is GRS.", "Let $u$ be a deep hole of a $[q,2]_q$ RS code $\\mathcal {C}$ with $\\mathcal {D}= GF(q)$ .", "According to Proposition REF , the matrix $G_2^{\\perp }(\\mathcal {D};u)$ generates a $[q+1,q-2]_q$ MDS code $\\mathcal {C}_1$ extending the $[q,q-2]_q$ RS code $\\mathcal {C}^{\\perp }$ .", "Consider the $[q+1,3]_q$ MDS code $\\mathcal {C}_1^{\\perp }$ .", "Segre's theorem implies $\\mathcal {C}_1^{\\perp }$ is GRS.", "Since the dual of a GRS code is GRS, it follows that $\\mathcal {C}_1$ is GRS.", "Thus the columns of the matrix $G_2^{\\perp }(\\mathcal {D};u)$ represent the $q+1$ points of some RNC in $PG(q-3,q)$ .", "Now, a RNC in $PG(m-1,q)$ is uniquely determined by any $m+2$ points on it.", "(see [7], or [5] for a purely coding theoretic proof.)", "Since $m+2 = q$ here, and the first $q$ columns of $G_2^{\\perp }(\\mathcal {D};u)$ lie on the standard RNC in $PG(q-3,q)$ , we conclude that the columns of $G_2^{\\perp }(\\mathcal {D};u)$ represent the $q+1$ points of the standard RNC.", "Conversely, we show Segre's theorem is equivalent to Conjecture REF holding for $k=2$ .", "Given a $q+1$ -arc in $PG(2,q)$ , or in other words a $[q+1,3]_q$ MDS code $\\mathcal {C}$ , the dual code $\\mathcal {C}^{\\perp }$ is a $[q+1,q-2]$ MDS code.", "Puncturing $\\mathcal {C}^{\\perp }$ on the last coordinate gives a $[q,q-2]$ MDS code $\\mathcal {C}_1$ .", "The dual to this code is a $[q,2]$ MDS code and 2-dimensional MDS codes are always GRS.", "Thus $\\mathcal {C}_1$ is GRS.", "It follows that $\\mathcal {C}^{\\perp }$ is a $[q+1,q-2]$ MDS code extending a $[q,q-2]$ GRS code.", "Assuming Conjecture REF holds for $k=2$ , i.e Conjecture ${\\bf 1^{\\prime }}$ holds for $m=q-2$ , we deduce that $\\mathcal {C}^{\\perp }$ is GRS.", "Thus $\\mathcal {C}$ is GRS.", "Before, we state our next proposition justifying the remark above, we present another conjecture.", "This conjecture is clearly implied by the MDS conjecture.", "(The MDS conjecture states that the maximum length of a $k$ -dimensional MDS code with $k<q$ is $q+1$ except when $q$ is even and $k=3,q-1$ , in which cases it is $q+2$ .)", "Since the MDS conjecture is widely believed, the same can be said about Conjecture REF .", "We will show that Conjecture REF implies Conjecture REF .", "Conjecture 2 (implied by the MDS conjecture) There is no $[q+2,m]_q$ MDS code extending a $[q+1,m]_q$ RS code, except when $q$ is even and $m=3, q-1$ .", "Equivalently, the RNC in $PG(m-1,q)$ is a complete arc unless $q$ is even and $m=3, q-1$ .", "Proposition 3 Let $2 \\le m \\le q-2$ .", "If Conjecture ${\\bf 1^{\\prime }}$ holds for $m$ , then Conjecture REF holds for $m$ .", "Let $\\mathcal {C}_1$ be a $[q+1,m]_q$ RS code generated by a matrix $G_m= G_m(PG(1,q))$ as in ().", "Let $\\mathcal {C}$ be $[q+2,m]_q$ MDS code generated by the matrix $[G_m \\, |\\, v]$ for some $v \\in GF(q)^m$ .", "Let $\\mathcal {C}_2$ be the $[q+1,m]_q$ MDS code obtained by puncturing $\\mathcal {C}$ on the $q+1$ -th coordinate.", "We note that $\\mathcal {C}_2$ extends a $[q,m,\\mathcal {D}]_q$ RS code with $\\mathcal {D}= GF(q)$ .", "Assuming Conjecture ${\\bf 1^{\\prime }}$ holds for $m$ , it follows that $v = a c_m(\\infty )$ for some $a \\in GF(q)^{\\times }$ , but then the last two colums of the matrix matrix $[G_m \\, |\\, v]$ are linearly dependent, contradicting the MDS property of $\\mathcal {C}$ .", "Thus such a code $[q+2,m]_q$ MDS code $\\mathcal {C}$ does not exist.", "Proposition REF in the next section presents some of the known answers to Conjecture REF .", "Theorem REF can be improved using results of Storme and Szőnyi [8], [9].", "These results state that for $q$ odd sufficiently large and $4 \\le k \\le 0.09q +3.09$ any MDS extension of a $[n,k]$ GRS code with $n \\ge (q+3)/2$ is GRS.", "Similarly for $q$ even sufficiently large, any MDS extension of a $[n,k]$ GRS code with $n \\ge q/2+2$ if $5 \\le k \\le 0.09q +3.59$ or $n \\ge q/2+3$ if $k=4$ , is GRS.", "Thus for such $n,k,q$ the generating functions of deep holes of a $[n,k]_q$ GRS code are as described in Theorem REF .", "Reed Solomon codes of length $q+1$ In this section $\\mathcal {C}$ is a $[q+1,k]_q$ RS code with evaluation set $PG(1,q)$ .", "In terms of arcs, $\\mathcal {C}$ corresponds to a RNC in $PG(k-1,q)$ .", "Let $G_k = G_k(PG(1,q))$ and $G_k^{\\perp } = G_k^{\\perp }(PG(1,q))$ denote the generator and parity check matrix of $\\mathcal {C}$ as given in ().", "As we showed in Section , the covering radius $\\rho (\\mathcal {C})$ satisfies: $ q-k \\le \\rho (\\mathcal {C}) \\le q+1-k.$ It follows that $\\rho (\\mathcal {C}) =q+1-k \\, \\Leftrightarrow $ there exists a vector $u \\in GF(q)^{q+1}$ at a distance of $q+1-k$ from $\\mathcal {C}\\, \\Leftrightarrow $ there exists a vector $v \\in GF(q)^{q+1-k}$ such that the matrix $[G_k^{\\perp } \\,|\\, v]$ generates a MDS code $\\, \\Leftrightarrow $ the RNC in $PG(q-k,q)$ is an incomplete arc.", "This establishes the following theorem due to A.Dür.", "Theorem (Dür 1994 [10]) The covering radius of a $[q+1, k]_q$ RS code $\\mathcal {C}$ is $q-k$ if and only if (any) RNC in $PG(q-k)$ is a complete arc.", "Equivalently there is no MDS extension of $\\mathcal {C}^{\\perp }$ by one digit.", "We can now restate Conjecture REF as follows: (where $k=q+1-m$ ) Conjecture ${\\bf 2^{\\prime }}$ The covering radius of a $[q+1,k]_q$ RS code is $q-k$ except when $q$ is even and $k= 2, q-2$ in which cases it is $q+1-k$ .", "We present some of the known answers to Conjecture REF .", "Proposition 4 Conjecture REF is true for (Roth and Seroussi [4]) $m=2$ and $3 \\le m \\le \\lfloor q/2 \\rfloor + 2$ except $m=3$ when $q$ is even.", "the exceptional cases $m=3, q-1$ with $q$ even.", "(Segre [6]) $m=q-1$ with $q$ odd.", "(Segre [6]) $m=q-2$ with $q$ odd.", "(Segre [11]) $m=q-3$ with $q$ odd.", "(Storme and Thas [12], Storme [13]) $\\lfloor q/2 \\rfloor + 3 \\le m \\le q+3 - 6 \\sqrt{q \\ln q} $ .", "(S. Ball [14]) any $m <q$ if $q$ is a prime.", "(S. Ball , [15]) $m \\le 2 p - 2$ where $q = p^h >p$ and $p$ is prime.", "(Storme and Szőnyi) [8]) $4 \\le m \\le 0.09q+3.09$ with $q$ odd sufficiently large.", "(Storme and Szőnyi) [9]) $q$ even sufficiently large, either $m=4$ or $5 \\le m \\le 0.09q+3.59$ .", "follows from the theorem of Roth and Seroussi above, together with the fact that for $m=2$ every MDS code is GRS.", "follows from the fact that for $q$ even, the matrices: $ H_3=[ c_3(x_1)\\, |\\, \\dots \\,|\\, c_3(x_q) \\,|\\, c_3(\\infty ) \\,|\\, \\mathcal {N}_3]\\;\\text{ and } \\; H_{q-1}= [c_{q-1}(x_1) \\, |\\, \\dots \\, |\\, c_{q-1}(x_q) \\, |\\, \\mathcal {N}_{q-1} \\, |\\, c_{q-1} (\\infty )] $ are parity check matrices of each other, and respectively generate a non-GRS $[q+2,3]$ MDS extension of a $[q+1,3]_q$ GRS code, and a non-GRS $[q+2,q-1]$ MDS extension of a $[q+1,q-1]_q$ GRS code.", "Suppose there is a $[q+2,q-1]$ MDS code.", "Its dual $\\mathcal {C}$ is a $[q+2,3]$ MDS code.", "By the theorem of Segre [6], it follows that the corresponding arc extends the RNC in $PG(2,q)$ contradicting the fact that the RNC in $PG(2,q)$ is a complete arc (Part 1).", "follows from Proposition REF and Proposition REF .", "The result we need from [11] is that any $[q,3]$ MDS code is GRS for $q$ odd (For a proof see [16]).", "It follows that any $[q,q-3]$ MDS code is GRS for $q$ odd.", "Suppose the matrix $[ G_{q-3}(PG(1,q)) \\,| \\, v]$ generates a $[q+2,q-3]$ MDS code.", "Puncturing on the first two coordinates gives a $[q,q-3]$ MDS code, which is GRS.", "The corresponding arc is thus contained in a RNC of $PG(q-4,q)$ .", "This arc has $q-1$ points on the standard RNC.", "Since $q -1 = q-3+2$ , once again appealing to the fact that a RNC in $PG(k-1,q)$ is uniquely determined by any $k+2$ points on it, it follows that $v$ also lies on the standard RNC.", "Thus all $q+2$ columns of $[ G_{q-3}(PG(1,q)) \\,| \\, v]$ lie on the standard RNC in $PG(q-4,q)$ thus contradicting the fact that this matrix generates a MDS code.", ", 9)-10) the RNC is complete in the range indicated as proved in the cited articles.", "-8) follow from the fact that the MDS conjecture holds in the parameter range indicated, as proved in the cited articles.", "We now focus on the problem of classifying the deep holes of $\\mathcal {C}$ in the exceptional cases of Conjecture ${\\bf 2^{\\prime }}$ .", "For $q$ even, $k=2,q-2$ , and $\\mathcal {C}$ a $[q+1,k]_q$ RS code, the matrices $H_{q-1},H_3$ of (REF ) generate MDS codes of length $q+2$ extending $\\mathcal {C}^{\\perp }$ .", "Thus the theorem of Dür above implies that $\\rho (\\mathcal {C})$ is indeed $q+1-k$ .", "We recall that the coset equivalence class of a deep hole $u$ is completely determined by its syndrome $S_{\\mathcal {D}}(u)$ , and the equivalence class of $u$ completely determined by the projective syndrome $[S_{\\mathcal {D}}(u)]$ .", "Theorem 3 Let $\\mathcal {C}$ be a $[q+1,k]_q$ RS code, where $q$ is even and $k = 2, q-2$ .", "Let $u$ be a deep hole of $\\mathcal {C}$ .", "If $k=q-2$ : the projective syndrome $[S_{\\mathcal {D}}(u)] = (0:1:0)$ .", "Thus there is only one equivalence class of deep holes of $\\mathcal {C}$ .", "If $k=2$ : There is a bijective correspondence between the set of equivalence classes of deep holes of $\\mathcal {C}$ and the set of projective equivalence classes of ordered hyperovals of $PG(2,q)$ .", "Part 1): Here $k=q-2$ .", "Part 2) of the theorem of Roth and Seroussi from the previous section states that the code generated by $H_3$ is the only possible MDS extension of a $\\mathcal {C}^{\\perp }$ .", "Thus there is only one equivalence class of deep holes, represented by $[S_{\\mathcal {D}}(u)] = [0:1:0]$ .", "Part 2): Here $k=2$ .", "In this case the generator matrix is $G_2= ( {\\begin{matrix}1 & \\hdots & 1&0 \\\\x_1 & \\hdots & x_q &1 \\end{matrix}} )$ , with $\\lbrace x_1, \\dots , x_q\\rbrace = GF(q)$ .", "Let $x_{q+1} = \\infty $ .", "An ordered hyperoval of $PG(2,q)$ is the ordered set of $q+2$ points of $PG(2,q)$ represented by a the columns of a generator matrix for a $[q+2,3]$ MDS code.", "By a result of B. Segre (see [16]), up to projective equivalence any such hyperoval is represented by a matrix $ G=\\begin{pmatrix}1 & \\hdots & 1&0 &0\\\\x_1 & \\hdots & x_q & 1 & 0 \\\\ u_1 & \\hdots &u_q & 0 &1 \\end{pmatrix}$ with the property that $u_i=0$ if $x_i = 0, \\infty $ and $u_i=1$ when $x_i=1$ , and that $G$ generates a MDS code.", "The condition that $G$ generates a MDS code can be stated as: there are at most two zero entries of $( bx_1+a -u_1, \\dots , bx_q+a-u_q, b)$ for any $(a,b) \\in GF(q)^2$ .", "This is equivalent to $u$ being a deep hole of $\\mathcal {C}$ .", "Since, the equivalence class of a received word $v \\in GF(q)^{q+1}$ of $\\mathcal {C}$ (i.e.", "$\\lbrace au +c : a \\in GF(q)^{\\times }, c \\in \\mathcal {C}\\rbrace $ ) has a unique representative $u$ such that $u_i=0$ if $x_i = 0, \\infty $ and $u_i=1$ when $x_i=1$ , it follows that equivalence classes of deep holes of $\\mathcal {C}$ are in bijective correspondence with projective equivalence classes of ordered hyperovals of $PG(2,q)$ .", "The problem of classifying deep holes of a $[q+1,2]$ RS code for $q$ even, is thus equivalent to the difficult problem of classifying hyperovals of $PG(2,q)$ .", "(See Section 2 of [17] for a survey of this problem).", "The equivalence classes of deep holes $u$ are completely determined by their syndrome $[S_{\\mathcal {D}}(u)] \\in PG(q-2,q)$ .", "Thus, the hyperovals can be studied in terms of possible syndromes $S_{\\mathcal {D}}(u)$ .", "This is done in the work of Storme and Thas [18].", "It is interesting to note that such a syndrome $[S_{\\mathcal {D}}(u)] = (a_0: \\dots : a_{q-2})$ necessarily satisfies $a_0=a_2= \\dots = a_{q-2}=0$ .", "(see Theorem 3.10 of [18]) The problem of classifying deep holes of $\\mathcal {C}$ when $\\rho (\\mathcal {C}) = q-k$ (for example Parts 1), 3)-5) of Proposition REF ) is an open problem (since at least 1991, see Remark 5 of [19]).", "By turning to the syndromes of the deep holes, and setting $m=q-k$ , this problem is equivalent to finding all points of $PG(m,q)$ which are not in the linear span of $m-1$ points of the standard RNC in $PG(m,q)$ .", "We just consider the easiest case of this problem.", "Theorem 4 For $k=q-2$ and $q$ odd, $u=(u_1, \\dots ,u_{q+1})$ is a deep hole of $\\mathcal {C}$ if and only if its projective syndrome $[S_{\\mathcal {D}}(u)]$ does not lie on the standard RNC in $PG(2,q)$ .", "Thus there are exactly $q^2$ equivalence classes of deep holes of $\\mathcal {C}$ .", "Let $m = q-k = 2$ .", "A point of $PG(2,q)$ which is not in the linear span of $m-1 = 1$ points of the standard RNC, is just a point which does not lie on the RNC.", "Classification of deep holes of RS codes of redundancy 3 In this section we will classify deep holes of $[n,k, \\mathcal {D}]_q$ RS codes $\\mathcal {C}$ of redundancy $n-k$ at most 3.", "As remarked earlier, the cases $n-k$ being 0 and 1 are uninteresting: in the former case there are no deep holes, and in the latter case the deep holes are all received words which are not codewords.", "A generator and parity check matrix for $\\mathcal {C}$ is as given in (), ().", "Since the projective syndrome $[S_{\\mathcal {D}}(u)]$ completely determines the equivalence class of a deep hole $u$ , we will focus on determining the possible values for $[S_{\\mathcal {D}}(u)]$ .", "First we consider redundancy 2 case, i.e.", "$[k+2,k, \\mathcal {D}]_q$ RS code $\\mathcal {C}$ with $2 \\le k \\le q-1$ .", "If $k=q-1$ , then the length is $q+1$ , and the theorem of Dür stated in Section , together with the fact that the RNC in $PG(1,q)$ is complete (i.e.", "there are no $[q+2,2]$ MDS codes) implies that $\\rho (\\mathcal {C})=1$ .", "Thus deep-holes of $\\mathcal {C}$ are those received words which are not codewords.", "For $k <q-1$ , the length $k+2 \\le q$ , and hence Proposition REF implies that equivalence classes of deep holes of $\\mathcal {C}$ are in bijective correspondence with $[S_{\\mathcal {D}}(u)] \\in PG(1,q)$ such that $[G_k^{\\perp }(\\mathcal {D}) \\,|\\, S_{\\mathcal {D}}(u)]$ generates a $[k+3,2]$ MDS code.", "Since every 2-dimensional MDS code is GRS, it follows that $[S_{\\mathcal {D}}(u)] \\in PG(1,q) \\setminus \\mathcal {D}$ Now we turn to RS codes of redundancy 3.", "Let $\\mathcal {C}$ be a $[k+3,k,\\mathcal {D}]$ RS code.", "Here $2 \\le k \\le q-2$ .", "We need some preliminary results and some notation.", "Let $\\epsilon $ denote a fixed non-square element of $GF(q)^{\\times }$ when $q$ is odd.", "The group $GL(2,q) = \\lbrace ( {\\begin{matrix}a & b\\\\ c & d \\end{matrix}} ): ad - bc \\ne 0\\rbrace $ acts on $GF(q)^2$ in the standard manner $v \\mapsto gv$ .", "This induces an action of the group $PGL(2,q) = GL(2,q) / \\lbrace \\pm ( {\\begin{matrix}1 & 0\\\\ 0 & 1 \\end{matrix}} )\\rbrace $ on $PG(1,q)$ by $g \\cdot x = (c+d x)/(a + bx)$ .", "Here $x$ denotes $[c_2(x)]$ .", "Consider the action of $GL(2,q)$ on $GF(q)^3$ given by : $g \\cdot \\xi = \\begin{pmatrix}a^2 & 2 a b & b^2\\\\ a c & a d + c b & b d \\\\ c^2 & 2 c d & d^2 \\end{pmatrix}\\xi , \\quad g=( {\\begin{matrix}a & b\\\\ c & d \\end{matrix}} ), \\; \\xi \\in GF(q)^3 $ This induces an action of $PGL(2,q)$ on $PG(2,q)$ (see [5] for details).", "Under this action, it is easy to see that $ g \\cdot [c_3(x)] = [c_3( g \\cdot x)] \\; \\text{(which is $[c_3( \\tfrac{c+d x}{a + bx})]$)}.$ Since $PGL(2,q)$ acts transitively on $PG(1,q)$ it follows that $PGL(2,q)$ acts transitively on the standard RNC in $PG(2,q)$ .", "Thus the standard RNC forms one orbit of the $PGL(2,q)$ action on $PG(2,q)$ .", "We also note that for $q$ even each element of $PGL(2,q)$ fixes the nucleus $(0:1:0)$ , and thus this gives an orbit of size 1.", "Lemma 3 There are 3 orbits for the action of $PGL(2,q)$ on $PG(2,q)$ given by: For $q$ even: i) the standard RNC, ii) the nucleus $(0:1:0)$ , and iii) the orbit of $(1:0:1)$ .", "The stabilizer of $(0:1:0)$ is $PGL(2,q)$ , and the stabilizer of $(1:0:1)$ is $ G_{1} = \\lbrace ( {\\begin{matrix}1+a & a \\\\ a & 1+a \\end{matrix}} ): a \\in GF(q)\\rbrace \\; \\text{isomorphic to the additive group of } \\, GF(q)$ We will denote the union of the two orbits i) and ii) by $\\mathcal {O}_1$ .", "The orbit iii) will be denote $\\mathcal {O}_4$ .", "For $q$ odd: i) the standard RNC, ii) the orbit of $(0:1:0)$ , and iii) the orbit of $(1:0:-\\epsilon )$ .", "The stabilizer of $(0:1:0)$ is $G_0 = \\lbrace x \\mapsto a x^{\\pm 1} : a \\in GF(q)^{\\times }\\rbrace \\; \\text{ isomorphic to the dihedral group of order $2(q-1)$} $ The stabilizer of $(1:0:-\\epsilon )$ is $ G_{\\epsilon }= \\lbrace ( {\\begin{matrix}a & b \\\\ c & d \\end{matrix}} ): ||a+c \\sqrt{\\epsilon }||=1, (b,d)=\\pm (\\epsilon c, a)\\rbrace ,$ isomorphic to the dihedral group of order $2(q+1)$ .", "Here $|| \\cdot || : GF(q^2)^{\\times } \\simeq GF(q)[ \\sqrt{\\epsilon }]^{\\times } \\rightarrow GF(q)^{\\times }$ is the norm.", "We will denote orbits i),ii) and iii) by $\\mathcal {O}_1, \\mathcal {O}_2$ and $\\mathcal {O}_3$ respectively.", "We need to show that the orbits other than the RNC for $q$ odd, and the RNC and its nucleus for $q$ even are as described.", "Let $W$ denote the 3-dimensional space of symmetric bilinear forms on $GF(q)^2$ .", "The group $GL(2,q)$ acts on $W$ by $(g \\cdot B)(v,w) = B(g^{-1} v, g^{-1} w)$ .", "We consider a linear isomorphism: $ \\Phi : GF(q)^3 \\rightarrow W, \\; \\text{given by } \\; \\Phi (M,N,P)(v,w) = v^T ( {\\begin{matrix}P & -N \\\\ -N & M \\end{matrix}} )w.$ The corresponding projective isomorphism will be also denoted $\\Phi : PG(2,q) \\rightarrow \\mathbb {P}W$ .", "For later use, we record the formula $ \\Phi (M,N,P)((1,X)^T, (1,Y)^T) = \\frac{\\text{det} ( {\\begin{matrix}1 & 1 & M \\\\X & Y & N\\\\ X^2 & Y^2 & P \\end{matrix}} )}{Y-X} = MXY - N(X+Y) +P.$ It is easy to check that $\\Phi (g \\cdot \\xi ) = \\text{det}(g)^2 \\,g \\cdot \\Phi (\\xi )$ , and thus at the projective level $\\Phi (g \\cdot [\\xi ]) = g \\cdot [\\Phi ( \\xi )]$ .", "The bilinear form $\\Phi (M,N,P)$ is degenerate if and only if det$( {\\begin{matrix}P & -N \\\\ -N & M \\end{matrix}} )= MP-N^2=0$ .", "Thus $\\Phi $ carries the orbit formed by the standard RNC to the projective space of degenerate symmetric bilinear forms on $GF(q)^2$ .", "For the remaining orbits, it suffices to consider nondegenerate bilinear forms $B$ .", "If $q$ is odd, it is well known that there exists $g \\in GL(2,q)$ such that $(g \\cdot B) (v,w) =v^T ( {\\begin{matrix}0 &1 \\\\1 &0 \\end{matrix}} )w$ or $v^T ( {\\begin{matrix}1 & 0 \\\\ 0 & -\\epsilon \\end{matrix}} )w$ depending on whether or not there is a nonzero vector $v$ with $B(v,v)=0$ (for example see [20]).", "Thus the orbits of $(0:1:0)$ and $(1:0: -\\epsilon )$ are the other orbits.", "The stabilizer of $(0:1:0)$ and $(1:0:-\\epsilon )$ are easy to compute and can also be found in [21].", "For $q$ even, suppose $B(v,v)=0$ for all $v \\in GF(q)^2$ .", "In this case $B(v,w) = \\alpha v^T ( {\\begin{matrix}0 &1 \\\\1 &0 \\end{matrix}} )w$ for some $\\alpha \\in GF(q)^{\\times }$ and thus $B = \\Phi (0:1:0)$ .", "If there exists a vector $v$ such that $B(v,v) \\ne 0$ , we may replace $v$ by $v/ \\sqrt{B(v,v)}$ to achieve $B(v,v)=1$ .", "The nondegeneracy implies there is a vector $w$ with $B(v,w)\\ne 0$ and $B(w,w) \\ne 0$ .", "As above, we can assume $B(w,w)=1$ .", "Thus $B = \\Phi (1:0:1)$ .", "The stabilizer of $(1:0:1)$ is clearly all matrices satisfying $g^Tg=( {\\begin{matrix}1 &0\\\\0&1 \\end{matrix}} )$ , which is as described in the statement.", "The map $( {\\begin{matrix}1+a & a \\\\ a & 1+a \\end{matrix}} )\\mapsto a$ is an isomorphism of $G_{1}$ with the additive group of $GF(q)$ .", "We define some subsets of $PG(2,q)$ associated with an evaluation set $\\mathcal {D}\\subset PG(1,q)$ of size $k+3$ .", "$\\mathcal {O}_1(\\mathcal {D})={\\left\\lbrace \\begin{array}{ll} \\lbrace [c_3(\\delta )]: \\delta \\in PG(1,q) \\setminus \\mathcal {D}\\rbrace &\\text{ if $q$ is odd} \\\\\\lbrace [c_3(\\delta )]: \\delta \\in PG(1,q) \\setminus \\mathcal {D}\\rbrace \\cup (0:1:0) &\\text{ if $q$ is even}.", "\\end{array}\\right.", "}$ $\\mathcal {O}_2(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (0:1:0) : \\bar{g} \\in PGL(2,q)/G_0, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne -y\\rbrace $ for $q$ odd.", "$\\mathcal {O}_3(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (1:0:-\\epsilon ) : \\bar{g} \\in PGL(2,q)/G_{\\epsilon }, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne \\epsilon /y\\rbrace $ for $q$ odd.", "$\\mathcal {O}_4(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (1:0:1) : \\bar{g} \\in PGL(2,q)/G_{1}, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne 1/y\\rbrace $ for $q$ even.", "We note that $\\mathcal {O}_i(\\mathcal {D})$ is a subset of $\\mathcal {O}_i$ .", "The notation of $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ needs some explanantion.", "The notation $PGL(2,q)/G_0$ stands for the left cosets of the stabilizer $G_0$ of $(0:1:0)$ in $PGL(2,q)$ .", "Similarly for $G_{\\epsilon }$ and $G_{1}$ .", "It is easy to show that if a subset $A$ of $PG(1,q)$ has the property that for any pair of distinct elements $a, b \\in A$ , i) $a \\ne -b$ ($q$ odd), or ii) $a \\ne \\epsilon /b$ ($q$ odd), or iii) $a \\ne 1/b$ ($q$ even), then for $g$ in i) $G_{0}$ , ii) $G_{\\epsilon }$ , iii) $G_{1}$ , the sets $gA$ also have the same property.", "Thus $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ are well-defined.", "Moreover, such a set $A$ has size at most i) $(q+3)/2$ , ii) $(q+1)/2$ , iii) $(q+2)/2$ respectively, by a simple application of pigeon hole principle.", "We recall the notation $\\mathcal {D}= \\lbrace x_1, \\dots , x_{k+3}\\rbrace $ .", "The size of these subsets can be expressed as: rCl |O1(D)| = q-k-2   if $q$ is odd, and   q-k-1   if $q$ is even.", "|O2(D)| = | { g G0 PGL(2,q) : gxi - gxj   i j}|.", "|O3(D)| = | { g G PGL(2,q) : gxi / gxj   i j}|.", "|O4(D)| = | { g G1 PGL(2,q) : gxi 1/gxj   i j}|, where $G_0 \\backslash PGL(2,q)$ denotes the set of right cosets of $G_0$ in $PGL(2,q)$ .", "The exact values of the sizes of $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ depends on the configuration of $\\mathcal {D}$ in $PG(1,q)$ , and appears to be a hard problem.", "Theorem 5 The the set of possible values of $[S_{\\mathcal {D}}(u)]$ is: ${\\left\\lbrace \\begin{array}{ll} (0:1:0) &\\text{ if $q$ is even, $k=q-2$ }\\\\\\mathcal {O}_2 \\cup \\mathcal {O}_3 &\\text{ if $q$ is odd, $k=q-2$ } \\end{array}\\right.", "}$ $\\mathcal {O}_1(\\mathcal {D})$ , if $q-3 \\ge k \\ge \\lfloor (q-1)/2 \\rfloor $ .", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_2(\\mathcal {D})$ , if $k=(q-3)/2$ with $q$ odd.", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_2(\\mathcal {D}) \\cup \\mathcal {O}_3(\\mathcal {D})$ , if $2 \\le k \\le (q-5)/2$ with $q$ odd.", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_4(\\mathcal {D})$ , if $2 \\le k \\le (q-4)/2$ with $q$ even.", "First we consider the length $q+1$ case (i.e.", "$k=q-2$ ).", "If $q$ is even, then the only possibility for $[S_{\\mathcal {D}}(u)]$ is the nucleus $(0:1:0)$ by Part 1) of Theorem REF .", "If $q$ is odd, then by Theorem REF , the possibilities for $[S_{\\mathcal {D}}(u)]$ is the complement of the standard RNC in $PG(2,q)$ .", "This proves part 1).", "Now we assume $k \\le q-3$ .", "The length $k+3$ is then at most $q$ and Proposition REF implies that $[ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ]$ generates a $[k+4,3]$ MDS code.", "This always holds if $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_1(\\mathcal {D})$ .", "It remains to consider other possibilities for $[S_{\\mathcal {D}}(u)] $ .", "For $q-3 \\ge k \\ge \\lfloor (q-1)/2 \\rfloor $ , Theorem REF implies that there are no other possibilities.", "This proves part 2).", "We now assume $2 \\le k \\le \\lfloor (q-3)/2 \\rfloor $ .", "Let $B(v,w)$ be the bilinear form on $GF(q)^2$ given by $\\Phi (S_{\\mathcal {D}}(u))$ .", "We are given that $ B((1,x_i),(1,x_j)) \\ne 0, \\; \\forall x_i \\ne x_j \\in \\mathcal {D}.", "$ In case $[S_{\\mathcal {D}}(u)]$ lies on the standard RNC or RNC $\\cup $ its nucleus if $q$ is even, it follows that $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_1(\\mathcal {D})$ , which we have already considered.", "Thus we assume $[S_{\\mathcal {D}}(u)] \\notin \\mathcal {O}_1$ .", "If $q$ is even, that leaves us with $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_4$ .", "Writing $[S_{\\mathcal {D}}(u)] = g^{-1} \\cdot (1:0:1)$ for some $g \\in PGL(2,q)$ , and let $g \\cdot \\mathcal {D}= \\lbrace y_1, \\dots , y_{k+3}\\rbrace $ .", "It follows that: $[ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ] = g^{-1} [ \\mu _1 c_3(y_1) \\,|\\, \\dots \\,|\\, \\mu _{k+3} c_3(y_{k+3}) \\,|\\, (1,0,1)^T], $ for some $\\mu _1, \\dots , \\mu _{k+3} \\in GF(q)^{\\times }$ .", "In this case the condition (REF ) is equivalent to $y_i \\ne 1/y_j$ for $i \\ne j$ .", "(It follows from (REF ) that for $(M,N,P) = (1,0,1)$ the form $Mxy - N(x+y) +P = xy+1= xy-1$ .)", "Hence $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_4(\\mathcal {D})$ .", "This proves part 5).", "Now we turn to the case $q$ odd, and $S_{\\mathcal {D}}(u) = (M,N,P)^T \\notin \\mathcal {O}_1$ .", "In case $(M,N,P) \\in \\mathcal {O}_2$ , let $(M,N,P)^T=g^{-1} \\cdot (0,1,0)^T$ for some $g \\in PGL(2,q)$ , and let $g \\cdot \\mathcal {D}= \\lbrace y_1, \\dots , y_{k+3}\\rbrace $ .", "It follows that: $ [ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ] = g^{-1} [ \\mu _1 c_3(y_1) \\,|\\, \\dots \\,|\\, \\mu _{k+3} c_3(y_{k+3}) \\,|\\, (0,1,0)^T], $ for some $\\mu _1, \\dots , \\mu _{k+3} \\in GF(q)^{\\times }$ .", "In this case the condition (REF ) is equivalent to $y_i \\ne -y_j$ for $i \\ne j$ , because $Mxy - N(x+y) +P = -(x+y)$ .", "Hence $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_2(\\mathcal {D})$ .", "Similarly, if $(M,N,P) \\in \\mathcal {O}_3$ , we get $(M,N,P) \\in \\mathcal {O}_3(\\mathcal {D})$ .", "As mentioned above the set $\\mathcal {O}_3(\\mathcal {D})$ is empty unless $k+3 \\le (q+1)/2$ , thus for $k = (q-3)/2$ , the possibility $S_{\\mathcal {D}}(u) \\in \\mathcal {O}_3$ does not occur.", "This proves parts 3)-4).", "We record the following theorem about canonical forms of non GRS $[n+1,3]$ MDS codes extending a GRS $[n,3]$ code.", "It will be useful to regard two codes $\\mathcal {C}, \\mathcal {C}^{\\prime }$ as diagonally equivalent if there is a diagonal Hamming isometry (a diagonal matrix in $GL(n,q)$ ) which carries $\\mathcal {C}$ to $\\mathcal {C}^{\\prime }$ .", "Note that diagonally equivalent codes are monomially equivalent but the converse is not true in general.", "At the level of arcs, diagonal equivalence yields the notion of ordered arcs, where as monomial equivalence yields the the notion of (unordered) arcs.", "Theorem 6 Let $\\mathcal {C}$ be a non GRS $[n+1,3]$ MDS code extending a $[n,3]$ GRS code $\\mathcal {C}_1$ where $n \\ge 5$ .", "Up to diagonal equivalence, $\\mathcal {C}$ is the code generated by one of the families of matrices $M_1, M_2, M_3$ below.", "Equivalently let $\\mathcal {A}$ be an ordered $n+1$ -arc in $PG(2,q)$ with the first $n$ points (but not the last) on a RNC (where $n \\ge 5$ ), then $\\mathcal {A}$ is projectively equivalent to the ordered arc defined by the columns of one of the families of matrices $M_1, M_2, M_3$ below.", "In the following, $\\mathcal {D}= \\lbrace x_1, \\dots , x_n\\rbrace \\subset PG(1,q)$ denotes a subset of $n \\ge 5$ distinct points satisfying certain conditions.", "$\\mathcal {D}$ satisfies $ x_i \\ne - x_j$ if $i \\ne j$ .", "In this case $n \\le (q+3)/2$ if $q$ is odd and $n \\le q+1$ if $q$ is even.", "$M_1 = \\begin{pmatrix}1 & \\ldots & 1 & 0\\\\x_1 & \\ldots &x_n & 1 \\\\ x_1^2 & \\ldots & x_n^2 & 0 \\end{pmatrix}$ $q$ is odd, $n \\le (q+1)/2$ , and $\\mathcal {D}$ satisfies $ x_i \\ne \\epsilon / x_j$ if $i \\ne j$ .", "$M_2 = \\begin{pmatrix}1 & \\ldots & 1 & 1\\\\x_1 & \\ldots &x_n & 0 \\\\ x_1^2 & \\ldots & x_n^2 & -\\epsilon \\end{pmatrix}$ $q$ is even, $n \\le (q+2)/2$ , and $\\mathcal {D}$ satisfies $ x_i \\ne 1/ x_j$ if $i \\ne j$ .", "$M_3 = \\begin{pmatrix}1 & \\ldots & 1 & 1\\\\x_1 & \\ldots &x_n & 0 \\\\ x_1^2 & \\ldots & x_n^2 & 1 \\end{pmatrix}$ From the fact that a RNC in $PG(2,q)$ is uniquely determined by any 5 points on it, it follows that the matrices $M_i$ above do not generate a GRS code for $n \\ge 5$ (the corresponding arcs do not lie on a RNC).", "To prove that the code $\\mathcal {C}$ in question is diagonally equivalent to the code generated by one of the matrices of the type $M_i$ , let $C_1$ be diagonally equivalent to the code generated by a matrix $G = [c_3(t_1) \\,|\\, \\dots \\,|\\, c_3(t_n)]$ .", "Thus there is a vector $v \\in GF(q)^3$ such that $[G \\,|\\, v]$ generates the non-GRS code $\\mathcal {C}$ .", "The analysis of such matrices $[G \\,|\\, v]$ was carried out in the proof of Theorem REF (see (REF ), (REF )).", "It was shown that there are matrices $P \\in GL(3,q)$ and a diagonal matrix $Q \\in GL(n+1,q)$ such that $P [G \\,|\\, v] Q$ is of the type $M_1, M_2$ or $M_3$ .", "In other words $\\mathcal {C}$ is diagonally equivalent to the code generated by one of the types of matrices $M_i$ .", "We note that two distinct matrices of the type, say $M_2$ may represent the same MDS extension $\\mathcal {C}$ of $\\mathcal {C}_1$ .", "In order to count the diagonal equivalence classes of codes $(\\mathcal {C}_1, \\mathcal {C})$ where $\\mathcal {C}$ is a $[n+1,3]_q$ MDS and non GRS code extending a $[n,3]_q$ RS code $\\mathcal {C}_1$ , we have to factor out the left action of $G_0, G_{\\epsilon }, G_1$ on generator matrices of the type $M_1, M_2, M_3$ .", "It is convenient to use the language of arcs.", "We will now count the number of projective equivalence classes of ordered arcs $(\\mathcal {A}_1, \\mathcal {A})$ where $\\mathcal {A}$ is an ordered $n+1$ -arc not contained in a RNC, but its first $n$ points form the arc $\\mathcal {A}_1$ which is contained in a RNC.", "Let $\\mathcal {M}_i$ be the set of ordered arcs (without using projective equivalence) arising from matrices of the type $M_i$ .", "Let $\\mathcal {G}_i \\subset PGL(2,q)$ be the stabilizer of the point represented by the last column.", "It is easy to see that $\\mathcal {G}_i$ acts freely (i.e.", "without fixed point) on $\\mathcal {M}_i$ .", "This is because the only element of $PGL(2,q)$ which fixes 3 points is the identity transformation.", "The quotient $\\mathcal {G}_i \\backslash \\mathcal {M}_i$ gives the projective equivalence classes of ordered arc pairs $(\\mathcal {A}_1, \\mathcal {A})$ that we are trying to count and which are of type $M_i$ .", "It is straightforward to count the relevant quantities: $|\\mathcal {M}_1|=(q+1)!", "/ (q+1-n)!", "$ if $q$ is even, and $ |\\mathcal {M}_1| = \\frac{\\tfrac{q-1}{2} !", "\\, 2^n}{(\\tfrac{q-1}{2}-n)!}", "+ \\frac{\\tfrac{q-1}{2} !", "\\, 2^n\\, n}{(\\tfrac{q+1}{2}-n)!", "}+\\frac{\\tfrac{q-1}{2} !", "\\, 2^{n-2}\\, n(n-1)}{(\\tfrac{q+3}{2}-n)!}", "\\quad \\text{ if $q$ is odd}.$ Here we use the convention $(-m)!=\\infty $ for natural numbers $m$ .", "We illustrate the method we use to obtain $|\\mathcal {M}_1|$ for $q$ odd.", "The other cases are similar.", "We may write $PG(1,q)$ as the disjoint union of $(q+3)/2$ sets of the form $ \\lbrace \\infty \\rbrace , \\lbrace 0\\rbrace , \\lbrace \\pm \\alpha _1\\rbrace , \\dots , \\lbrace \\pm \\alpha _{(q-1)/2}\\rbrace $ .", "We note that $\\mathcal {M}_1$ consists of $n$ -tuples $(z_1, \\dots ,z_n)$ such that we pick at most one element from each of the $(q+3)/2$ sets above.", "By similar methods, we obtain rCl |M2| = (q+12 !", "2n )/ (q+12-n)!", "|M3| = (q2 !", "2n)/ (q2-n)!+ (q2 !", "2n-1  n)/(q+22-n)!", "The groups $\\mathcal {G}_i$ have been computed previously: $\\mathcal {G}_1$ is $PGL(2,q)$ if $q$ is even and isomorphic to a dihedral group of order $2(q-1)$ for odd $q$ .", "The group $\\mathcal {G}_2$ isomorphic to a dihedral group of order $2(q+1)$ , and the The group $\\mathcal {G}_3$ isomorphic to the additive group $(GF(q),+)$ .", "Thus we obtain that the number of ordered arc pairs $(\\mathcal {A}_1, \\mathcal {A})$ of the type $M_i$ equals: $(q-2)!", "/ (q+1-n)!\\quad $ if $i=1$ and $q$ is even.", "Here $ n\\le q+1$ .", "$ \\tfrac{q-3}{2} !", "\\, 2^{n-4}\\, \\left[ (q+1)(q+3-2n) + n(n-1)\\right]/ (\\tfrac{q+3-2n}{2} \\,!", ")\\quad $ if $i=1$ and $q$ is odd.", "Here $n \\le (q+3)/2$ $\\tfrac{q-1}{2} !", "\\, 2^{n-2} / (\\tfrac{q+1-2n}{2} \\, !", ")\\quad $ if $i=2$ .", "Here $n \\le (q+1)/2$ .", "$\\tfrac{q-2}{2} !", "\\, 2^{n-2} \\, (q+2-n) / (\\tfrac{q+2-2n}{2} !)", "\\quad $ if $i=3$ .", "Here $n \\le (q+2)/2$ .", "Conclusion We solve the problem of classifying deep holes of $[n,k]_q$ RS codes for $k \\ge (q-1)/2$ for non prime $q$ , which was posed as an open problem in the concluding remarks of [3].", "The problem for $k < (q-1)/2$ is open.", "We solve the problem for $n=k+3$ and all $k$ .", "We also solve the problem for $k=2, n=q$ with $q$ odd, by reducing it to Segre's `oval equals conic' theorem.", "For $k=2, n=q+1$ with $q$ even, we show that the problem is equivalent to the difficult problem of classifying hyperovals in projective planes.", "Finally, we obtain canonical forms for $[n+1,3]_q$ MDS but non-GRS codes extending a $[n,3]_q$ GRS code." ], [ "Reed Solomon codes of length $q+1$", "In this section $\\mathcal {C}$ is a $[q+1,k]_q$ RS code with evaluation set $PG(1,q)$ .", "In terms of arcs, $\\mathcal {C}$ corresponds to a RNC in $PG(k-1,q)$ .", "Let $G_k = G_k(PG(1,q))$ and $G_k^{\\perp } = G_k^{\\perp }(PG(1,q))$ denote the generator and parity check matrix of $\\mathcal {C}$ as given in ().", "As we showed in Section , the covering radius $\\rho (\\mathcal {C})$ satisfies: $ q-k \\le \\rho (\\mathcal {C}) \\le q+1-k.$ It follows that $\\rho (\\mathcal {C}) =q+1-k \\, \\Leftrightarrow $ there exists a vector $u \\in GF(q)^{q+1}$ at a distance of $q+1-k$ from $\\mathcal {C}\\, \\Leftrightarrow $ there exists a vector $v \\in GF(q)^{q+1-k}$ such that the matrix $[G_k^{\\perp } \\,|\\, v]$ generates a MDS code $\\, \\Leftrightarrow $ the RNC in $PG(q-k,q)$ is an incomplete arc.", "This establishes the following theorem due to A.Dür.", "Theorem (Dür 1994 [10]) The covering radius of a $[q+1, k]_q$ RS code $\\mathcal {C}$ is $q-k$ if and only if (any) RNC in $PG(q-k)$ is a complete arc.", "Equivalently there is no MDS extension of $\\mathcal {C}^{\\perp }$ by one digit.", "We can now restate Conjecture REF as follows: (where $k=q+1-m$ ) Conjecture ${\\bf 2^{\\prime }}$ The covering radius of a $[q+1,k]_q$ RS code is $q-k$ except when $q$ is even and $k= 2, q-2$ in which cases it is $q+1-k$ .", "We present some of the known answers to Conjecture REF .", "Proposition 4 Conjecture REF is true for (Roth and Seroussi [4]) $m=2$ and $3 \\le m \\le \\lfloor q/2 \\rfloor + 2$ except $m=3$ when $q$ is even.", "the exceptional cases $m=3, q-1$ with $q$ even.", "(Segre [6]) $m=q-1$ with $q$ odd.", "(Segre [6]) $m=q-2$ with $q$ odd.", "(Segre [11]) $m=q-3$ with $q$ odd.", "(Storme and Thas [12], Storme [13]) $\\lfloor q/2 \\rfloor + 3 \\le m \\le q+3 - 6 \\sqrt{q \\ln q} $ .", "(S. Ball [14]) any $m <q$ if $q$ is a prime.", "(S. Ball , [15]) $m \\le 2 p - 2$ where $q = p^h >p$ and $p$ is prime.", "(Storme and Szőnyi) [8]) $4 \\le m \\le 0.09q+3.09$ with $q$ odd sufficiently large.", "(Storme and Szőnyi) [9]) $q$ even sufficiently large, either $m=4$ or $5 \\le m \\le 0.09q+3.59$ .", "follows from the theorem of Roth and Seroussi above, together with the fact that for $m=2$ every MDS code is GRS.", "follows from the fact that for $q$ even, the matrices: $ H_3=[ c_3(x_1)\\, |\\, \\dots \\,|\\, c_3(x_q) \\,|\\, c_3(\\infty ) \\,|\\, \\mathcal {N}_3]\\;\\text{ and } \\; H_{q-1}= [c_{q-1}(x_1) \\, |\\, \\dots \\, |\\, c_{q-1}(x_q) \\, |\\, \\mathcal {N}_{q-1} \\, |\\, c_{q-1} (\\infty )] $ are parity check matrices of each other, and respectively generate a non-GRS $[q+2,3]$ MDS extension of a $[q+1,3]_q$ GRS code, and a non-GRS $[q+2,q-1]$ MDS extension of a $[q+1,q-1]_q$ GRS code.", "Suppose there is a $[q+2,q-1]$ MDS code.", "Its dual $\\mathcal {C}$ is a $[q+2,3]$ MDS code.", "By the theorem of Segre [6], it follows that the corresponding arc extends the RNC in $PG(2,q)$ contradicting the fact that the RNC in $PG(2,q)$ is a complete arc (Part 1).", "follows from Proposition REF and Proposition REF .", "The result we need from [11] is that any $[q,3]$ MDS code is GRS for $q$ odd (For a proof see [16]).", "It follows that any $[q,q-3]$ MDS code is GRS for $q$ odd.", "Suppose the matrix $[ G_{q-3}(PG(1,q)) \\,| \\, v]$ generates a $[q+2,q-3]$ MDS code.", "Puncturing on the first two coordinates gives a $[q,q-3]$ MDS code, which is GRS.", "The corresponding arc is thus contained in a RNC of $PG(q-4,q)$ .", "This arc has $q-1$ points on the standard RNC.", "Since $q -1 = q-3+2$ , once again appealing to the fact that a RNC in $PG(k-1,q)$ is uniquely determined by any $k+2$ points on it, it follows that $v$ also lies on the standard RNC.", "Thus all $q+2$ columns of $[ G_{q-3}(PG(1,q)) \\,| \\, v]$ lie on the standard RNC in $PG(q-4,q)$ thus contradicting the fact that this matrix generates a MDS code.", ", 9)-10) the RNC is complete in the range indicated as proved in the cited articles.", "-8) follow from the fact that the MDS conjecture holds in the parameter range indicated, as proved in the cited articles.", "We now focus on the problem of classifying the deep holes of $\\mathcal {C}$ in the exceptional cases of Conjecture ${\\bf 2^{\\prime }}$ .", "For $q$ even, $k=2,q-2$ , and $\\mathcal {C}$ a $[q+1,k]_q$ RS code, the matrices $H_{q-1},H_3$ of (REF ) generate MDS codes of length $q+2$ extending $\\mathcal {C}^{\\perp }$ .", "Thus the theorem of Dür above implies that $\\rho (\\mathcal {C})$ is indeed $q+1-k$ .", "We recall that the coset equivalence class of a deep hole $u$ is completely determined by its syndrome $S_{\\mathcal {D}}(u)$ , and the equivalence class of $u$ completely determined by the projective syndrome $[S_{\\mathcal {D}}(u)]$ .", "Theorem 3 Let $\\mathcal {C}$ be a $[q+1,k]_q$ RS code, where $q$ is even and $k = 2, q-2$ .", "Let $u$ be a deep hole of $\\mathcal {C}$ .", "If $k=q-2$ : the projective syndrome $[S_{\\mathcal {D}}(u)] = (0:1:0)$ .", "Thus there is only one equivalence class of deep holes of $\\mathcal {C}$ .", "If $k=2$ : There is a bijective correspondence between the set of equivalence classes of deep holes of $\\mathcal {C}$ and the set of projective equivalence classes of ordered hyperovals of $PG(2,q)$ .", "Part 1): Here $k=q-2$ .", "Part 2) of the theorem of Roth and Seroussi from the previous section states that the code generated by $H_3$ is the only possible MDS extension of a $\\mathcal {C}^{\\perp }$ .", "Thus there is only one equivalence class of deep holes, represented by $[S_{\\mathcal {D}}(u)] = [0:1:0]$ .", "Part 2): Here $k=2$ .", "In this case the generator matrix is $G_2= ( {\\begin{matrix}1 & \\hdots & 1&0 \\\\x_1 & \\hdots & x_q &1 \\end{matrix}} )$ , with $\\lbrace x_1, \\dots , x_q\\rbrace = GF(q)$ .", "Let $x_{q+1} = \\infty $ .", "An ordered hyperoval of $PG(2,q)$ is the ordered set of $q+2$ points of $PG(2,q)$ represented by a the columns of a generator matrix for a $[q+2,3]$ MDS code.", "By a result of B. Segre (see [16]), up to projective equivalence any such hyperoval is represented by a matrix $ G=\\begin{pmatrix}1 & \\hdots & 1&0 &0\\\\x_1 & \\hdots & x_q & 1 & 0 \\\\ u_1 & \\hdots &u_q & 0 &1 \\end{pmatrix}$ with the property that $u_i=0$ if $x_i = 0, \\infty $ and $u_i=1$ when $x_i=1$ , and that $G$ generates a MDS code.", "The condition that $G$ generates a MDS code can be stated as: there are at most two zero entries of $( bx_1+a -u_1, \\dots , bx_q+a-u_q, b)$ for any $(a,b) \\in GF(q)^2$ .", "This is equivalent to $u$ being a deep hole of $\\mathcal {C}$ .", "Since, the equivalence class of a received word $v \\in GF(q)^{q+1}$ of $\\mathcal {C}$ (i.e.", "$\\lbrace au +c : a \\in GF(q)^{\\times }, c \\in \\mathcal {C}\\rbrace $ ) has a unique representative $u$ such that $u_i=0$ if $x_i = 0, \\infty $ and $u_i=1$ when $x_i=1$ , it follows that equivalence classes of deep holes of $\\mathcal {C}$ are in bijective correspondence with projective equivalence classes of ordered hyperovals of $PG(2,q)$ .", "The problem of classifying deep holes of a $[q+1,2]$ RS code for $q$ even, is thus equivalent to the difficult problem of classifying hyperovals of $PG(2,q)$ .", "(See Section 2 of [17] for a survey of this problem).", "The equivalence classes of deep holes $u$ are completely determined by their syndrome $[S_{\\mathcal {D}}(u)] \\in PG(q-2,q)$ .", "Thus, the hyperovals can be studied in terms of possible syndromes $S_{\\mathcal {D}}(u)$ .", "This is done in the work of Storme and Thas [18].", "It is interesting to note that such a syndrome $[S_{\\mathcal {D}}(u)] = (a_0: \\dots : a_{q-2})$ necessarily satisfies $a_0=a_2= \\dots = a_{q-2}=0$ .", "(see Theorem 3.10 of [18]) The problem of classifying deep holes of $\\mathcal {C}$ when $\\rho (\\mathcal {C}) = q-k$ (for example Parts 1), 3)-5) of Proposition REF ) is an open problem (since at least 1991, see Remark 5 of [19]).", "By turning to the syndromes of the deep holes, and setting $m=q-k$ , this problem is equivalent to finding all points of $PG(m,q)$ which are not in the linear span of $m-1$ points of the standard RNC in $PG(m,q)$ .", "We just consider the easiest case of this problem.", "Theorem 4 For $k=q-2$ and $q$ odd, $u=(u_1, \\dots ,u_{q+1})$ is a deep hole of $\\mathcal {C}$ if and only if its projective syndrome $[S_{\\mathcal {D}}(u)]$ does not lie on the standard RNC in $PG(2,q)$ .", "Thus there are exactly $q^2$ equivalence classes of deep holes of $\\mathcal {C}$ .", "Let $m = q-k = 2$ .", "A point of $PG(2,q)$ which is not in the linear span of $m-1 = 1$ points of the standard RNC, is just a point which does not lie on the RNC." ], [ "Classification of deep holes of RS codes of redundancy 3", "In this section we will classify deep holes of $[n,k, \\mathcal {D}]_q$ RS codes $\\mathcal {C}$ of redundancy $n-k$ at most 3.", "As remarked earlier, the cases $n-k$ being 0 and 1 are uninteresting: in the former case there are no deep holes, and in the latter case the deep holes are all received words which are not codewords.", "A generator and parity check matrix for $\\mathcal {C}$ is as given in (), ().", "Since the projective syndrome $[S_{\\mathcal {D}}(u)]$ completely determines the equivalence class of a deep hole $u$ , we will focus on determining the possible values for $[S_{\\mathcal {D}}(u)]$ .", "First we consider redundancy 2 case, i.e.", "$[k+2,k, \\mathcal {D}]_q$ RS code $\\mathcal {C}$ with $2 \\le k \\le q-1$ .", "If $k=q-1$ , then the length is $q+1$ , and the theorem of Dür stated in Section , together with the fact that the RNC in $PG(1,q)$ is complete (i.e.", "there are no $[q+2,2]$ MDS codes) implies that $\\rho (\\mathcal {C})=1$ .", "Thus deep-holes of $\\mathcal {C}$ are those received words which are not codewords.", "For $k <q-1$ , the length $k+2 \\le q$ , and hence Proposition REF implies that equivalence classes of deep holes of $\\mathcal {C}$ are in bijective correspondence with $[S_{\\mathcal {D}}(u)] \\in PG(1,q)$ such that $[G_k^{\\perp }(\\mathcal {D}) \\,|\\, S_{\\mathcal {D}}(u)]$ generates a $[k+3,2]$ MDS code.", "Since every 2-dimensional MDS code is GRS, it follows that $[S_{\\mathcal {D}}(u)] \\in PG(1,q) \\setminus \\mathcal {D}$ Now we turn to RS codes of redundancy 3.", "Let $\\mathcal {C}$ be a $[k+3,k,\\mathcal {D}]$ RS code.", "Here $2 \\le k \\le q-2$ .", "We need some preliminary results and some notation.", "Let $\\epsilon $ denote a fixed non-square element of $GF(q)^{\\times }$ when $q$ is odd.", "The group $GL(2,q) = \\lbrace ( {\\begin{matrix}a & b\\\\ c & d \\end{matrix}} ): ad - bc \\ne 0\\rbrace $ acts on $GF(q)^2$ in the standard manner $v \\mapsto gv$ .", "This induces an action of the group $PGL(2,q) = GL(2,q) / \\lbrace \\pm ( {\\begin{matrix}1 & 0\\\\ 0 & 1 \\end{matrix}} )\\rbrace $ on $PG(1,q)$ by $g \\cdot x = (c+d x)/(a + bx)$ .", "Here $x$ denotes $[c_2(x)]$ .", "Consider the action of $GL(2,q)$ on $GF(q)^3$ given by : $g \\cdot \\xi = \\begin{pmatrix}a^2 & 2 a b & b^2\\\\ a c & a d + c b & b d \\\\ c^2 & 2 c d & d^2 \\end{pmatrix}\\xi , \\quad g=( {\\begin{matrix}a & b\\\\ c & d \\end{matrix}} ), \\; \\xi \\in GF(q)^3 $ This induces an action of $PGL(2,q)$ on $PG(2,q)$ (see [5] for details).", "Under this action, it is easy to see that $ g \\cdot [c_3(x)] = [c_3( g \\cdot x)] \\; \\text{(which is $[c_3( \\tfrac{c+d x}{a + bx})]$)}.$ Since $PGL(2,q)$ acts transitively on $PG(1,q)$ it follows that $PGL(2,q)$ acts transitively on the standard RNC in $PG(2,q)$ .", "Thus the standard RNC forms one orbit of the $PGL(2,q)$ action on $PG(2,q)$ .", "We also note that for $q$ even each element of $PGL(2,q)$ fixes the nucleus $(0:1:0)$ , and thus this gives an orbit of size 1.", "Lemma 3 There are 3 orbits for the action of $PGL(2,q)$ on $PG(2,q)$ given by: For $q$ even: i) the standard RNC, ii) the nucleus $(0:1:0)$ , and iii) the orbit of $(1:0:1)$ .", "The stabilizer of $(0:1:0)$ is $PGL(2,q)$ , and the stabilizer of $(1:0:1)$ is $ G_{1} = \\lbrace ( {\\begin{matrix}1+a & a \\\\ a & 1+a \\end{matrix}} ): a \\in GF(q)\\rbrace \\; \\text{isomorphic to the additive group of } \\, GF(q)$ We will denote the union of the two orbits i) and ii) by $\\mathcal {O}_1$ .", "The orbit iii) will be denote $\\mathcal {O}_4$ .", "For $q$ odd: i) the standard RNC, ii) the orbit of $(0:1:0)$ , and iii) the orbit of $(1:0:-\\epsilon )$ .", "The stabilizer of $(0:1:0)$ is $G_0 = \\lbrace x \\mapsto a x^{\\pm 1} : a \\in GF(q)^{\\times }\\rbrace \\; \\text{ isomorphic to the dihedral group of order $2(q-1)$} $ The stabilizer of $(1:0:-\\epsilon )$ is $ G_{\\epsilon }= \\lbrace ( {\\begin{matrix}a & b \\\\ c & d \\end{matrix}} ): ||a+c \\sqrt{\\epsilon }||=1, (b,d)=\\pm (\\epsilon c, a)\\rbrace ,$ isomorphic to the dihedral group of order $2(q+1)$ .", "Here $|| \\cdot || : GF(q^2)^{\\times } \\simeq GF(q)[ \\sqrt{\\epsilon }]^{\\times } \\rightarrow GF(q)^{\\times }$ is the norm.", "We will denote orbits i),ii) and iii) by $\\mathcal {O}_1, \\mathcal {O}_2$ and $\\mathcal {O}_3$ respectively.", "We need to show that the orbits other than the RNC for $q$ odd, and the RNC and its nucleus for $q$ even are as described.", "Let $W$ denote the 3-dimensional space of symmetric bilinear forms on $GF(q)^2$ .", "The group $GL(2,q)$ acts on $W$ by $(g \\cdot B)(v,w) = B(g^{-1} v, g^{-1} w)$ .", "We consider a linear isomorphism: $ \\Phi : GF(q)^3 \\rightarrow W, \\; \\text{given by } \\; \\Phi (M,N,P)(v,w) = v^T ( {\\begin{matrix}P & -N \\\\ -N & M \\end{matrix}} )w.$ The corresponding projective isomorphism will be also denoted $\\Phi : PG(2,q) \\rightarrow \\mathbb {P}W$ .", "For later use, we record the formula $ \\Phi (M,N,P)((1,X)^T, (1,Y)^T) = \\frac{\\text{det} ( {\\begin{matrix}1 & 1 & M \\\\X & Y & N\\\\ X^2 & Y^2 & P \\end{matrix}} )}{Y-X} = MXY - N(X+Y) +P.$ It is easy to check that $\\Phi (g \\cdot \\xi ) = \\text{det}(g)^2 \\,g \\cdot \\Phi (\\xi )$ , and thus at the projective level $\\Phi (g \\cdot [\\xi ]) = g \\cdot [\\Phi ( \\xi )]$ .", "The bilinear form $\\Phi (M,N,P)$ is degenerate if and only if det$( {\\begin{matrix}P & -N \\\\ -N & M \\end{matrix}} )= MP-N^2=0$ .", "Thus $\\Phi $ carries the orbit formed by the standard RNC to the projective space of degenerate symmetric bilinear forms on $GF(q)^2$ .", "For the remaining orbits, it suffices to consider nondegenerate bilinear forms $B$ .", "If $q$ is odd, it is well known that there exists $g \\in GL(2,q)$ such that $(g \\cdot B) (v,w) =v^T ( {\\begin{matrix}0 &1 \\\\1 &0 \\end{matrix}} )w$ or $v^T ( {\\begin{matrix}1 & 0 \\\\ 0 & -\\epsilon \\end{matrix}} )w$ depending on whether or not there is a nonzero vector $v$ with $B(v,v)=0$ (for example see [20]).", "Thus the orbits of $(0:1:0)$ and $(1:0: -\\epsilon )$ are the other orbits.", "The stabilizer of $(0:1:0)$ and $(1:0:-\\epsilon )$ are easy to compute and can also be found in [21].", "For $q$ even, suppose $B(v,v)=0$ for all $v \\in GF(q)^2$ .", "In this case $B(v,w) = \\alpha v^T ( {\\begin{matrix}0 &1 \\\\1 &0 \\end{matrix}} )w$ for some $\\alpha \\in GF(q)^{\\times }$ and thus $B = \\Phi (0:1:0)$ .", "If there exists a vector $v$ such that $B(v,v) \\ne 0$ , we may replace $v$ by $v/ \\sqrt{B(v,v)}$ to achieve $B(v,v)=1$ .", "The nondegeneracy implies there is a vector $w$ with $B(v,w)\\ne 0$ and $B(w,w) \\ne 0$ .", "As above, we can assume $B(w,w)=1$ .", "Thus $B = \\Phi (1:0:1)$ .", "The stabilizer of $(1:0:1)$ is clearly all matrices satisfying $g^Tg=( {\\begin{matrix}1 &0\\\\0&1 \\end{matrix}} )$ , which is as described in the statement.", "The map $( {\\begin{matrix}1+a & a \\\\ a & 1+a \\end{matrix}} )\\mapsto a$ is an isomorphism of $G_{1}$ with the additive group of $GF(q)$ .", "We define some subsets of $PG(2,q)$ associated with an evaluation set $\\mathcal {D}\\subset PG(1,q)$ of size $k+3$ .", "$\\mathcal {O}_1(\\mathcal {D})={\\left\\lbrace \\begin{array}{ll} \\lbrace [c_3(\\delta )]: \\delta \\in PG(1,q) \\setminus \\mathcal {D}\\rbrace &\\text{ if $q$ is odd} \\\\\\lbrace [c_3(\\delta )]: \\delta \\in PG(1,q) \\setminus \\mathcal {D}\\rbrace \\cup (0:1:0) &\\text{ if $q$ is even}.", "\\end{array}\\right.", "}$ $\\mathcal {O}_2(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (0:1:0) : \\bar{g} \\in PGL(2,q)/G_0, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne -y\\rbrace $ for $q$ odd.", "$\\mathcal {O}_3(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (1:0:-\\epsilon ) : \\bar{g} \\in PGL(2,q)/G_{\\epsilon }, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne \\epsilon /y\\rbrace $ for $q$ odd.", "$\\mathcal {O}_4(\\mathcal {D})= \\lbrace \\bar{g} \\cdot (1:0:1) : \\bar{g} \\in PGL(2,q)/G_{1}, x \\ne y \\in g^{-1} \\cdot \\mathcal {D}\\Rightarrow x \\ne 1/y\\rbrace $ for $q$ even.", "We note that $\\mathcal {O}_i(\\mathcal {D})$ is a subset of $\\mathcal {O}_i$ .", "The notation of $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ needs some explanantion.", "The notation $PGL(2,q)/G_0$ stands for the left cosets of the stabilizer $G_0$ of $(0:1:0)$ in $PGL(2,q)$ .", "Similarly for $G_{\\epsilon }$ and $G_{1}$ .", "It is easy to show that if a subset $A$ of $PG(1,q)$ has the property that for any pair of distinct elements $a, b \\in A$ , i) $a \\ne -b$ ($q$ odd), or ii) $a \\ne \\epsilon /b$ ($q$ odd), or iii) $a \\ne 1/b$ ($q$ even), then for $g$ in i) $G_{0}$ , ii) $G_{\\epsilon }$ , iii) $G_{1}$ , the sets $gA$ also have the same property.", "Thus $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ are well-defined.", "Moreover, such a set $A$ has size at most i) $(q+3)/2$ , ii) $(q+1)/2$ , iii) $(q+2)/2$ respectively, by a simple application of pigeon hole principle.", "We recall the notation $\\mathcal {D}= \\lbrace x_1, \\dots , x_{k+3}\\rbrace $ .", "The size of these subsets can be expressed as: rCl |O1(D)| = q-k-2   if $q$ is odd, and   q-k-1   if $q$ is even.", "|O2(D)| = | { g G0 PGL(2,q) : gxi - gxj   i j}|.", "|O3(D)| = | { g G PGL(2,q) : gxi / gxj   i j}|.", "|O4(D)| = | { g G1 PGL(2,q) : gxi 1/gxj   i j}|, where $G_0 \\backslash PGL(2,q)$ denotes the set of right cosets of $G_0$ in $PGL(2,q)$ .", "The exact values of the sizes of $\\mathcal {O}_i(\\mathcal {D}), i=2,3,4$ depends on the configuration of $\\mathcal {D}$ in $PG(1,q)$ , and appears to be a hard problem.", "Theorem 5 The the set of possible values of $[S_{\\mathcal {D}}(u)]$ is: ${\\left\\lbrace \\begin{array}{ll} (0:1:0) &\\text{ if $q$ is even, $k=q-2$ }\\\\\\mathcal {O}_2 \\cup \\mathcal {O}_3 &\\text{ if $q$ is odd, $k=q-2$ } \\end{array}\\right.", "}$ $\\mathcal {O}_1(\\mathcal {D})$ , if $q-3 \\ge k \\ge \\lfloor (q-1)/2 \\rfloor $ .", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_2(\\mathcal {D})$ , if $k=(q-3)/2$ with $q$ odd.", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_2(\\mathcal {D}) \\cup \\mathcal {O}_3(\\mathcal {D})$ , if $2 \\le k \\le (q-5)/2$ with $q$ odd.", "$\\mathcal {O}_1(\\mathcal {D}) \\cup \\mathcal {O}_4(\\mathcal {D})$ , if $2 \\le k \\le (q-4)/2$ with $q$ even.", "First we consider the length $q+1$ case (i.e.", "$k=q-2$ ).", "If $q$ is even, then the only possibility for $[S_{\\mathcal {D}}(u)]$ is the nucleus $(0:1:0)$ by Part 1) of Theorem REF .", "If $q$ is odd, then by Theorem REF , the possibilities for $[S_{\\mathcal {D}}(u)]$ is the complement of the standard RNC in $PG(2,q)$ .", "This proves part 1).", "Now we assume $k \\le q-3$ .", "The length $k+3$ is then at most $q$ and Proposition REF implies that $[ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ]$ generates a $[k+4,3]$ MDS code.", "This always holds if $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_1(\\mathcal {D})$ .", "It remains to consider other possibilities for $[S_{\\mathcal {D}}(u)] $ .", "For $q-3 \\ge k \\ge \\lfloor (q-1)/2 \\rfloor $ , Theorem REF implies that there are no other possibilities.", "This proves part 2).", "We now assume $2 \\le k \\le \\lfloor (q-3)/2 \\rfloor $ .", "Let $B(v,w)$ be the bilinear form on $GF(q)^2$ given by $\\Phi (S_{\\mathcal {D}}(u))$ .", "We are given that $ B((1,x_i),(1,x_j)) \\ne 0, \\; \\forall x_i \\ne x_j \\in \\mathcal {D}.", "$ In case $[S_{\\mathcal {D}}(u)]$ lies on the standard RNC or RNC $\\cup $ its nucleus if $q$ is even, it follows that $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_1(\\mathcal {D})$ , which we have already considered.", "Thus we assume $[S_{\\mathcal {D}}(u)] \\notin \\mathcal {O}_1$ .", "If $q$ is even, that leaves us with $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_4$ .", "Writing $[S_{\\mathcal {D}}(u)] = g^{-1} \\cdot (1:0:1)$ for some $g \\in PGL(2,q)$ , and let $g \\cdot \\mathcal {D}= \\lbrace y_1, \\dots , y_{k+3}\\rbrace $ .", "It follows that: $[ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ] = g^{-1} [ \\mu _1 c_3(y_1) \\,|\\, \\dots \\,|\\, \\mu _{k+3} c_3(y_{k+3}) \\,|\\, (1,0,1)^T], $ for some $\\mu _1, \\dots , \\mu _{k+3} \\in GF(q)^{\\times }$ .", "In this case the condition (REF ) is equivalent to $y_i \\ne 1/y_j$ for $i \\ne j$ .", "(It follows from (REF ) that for $(M,N,P) = (1,0,1)$ the form $Mxy - N(x+y) +P = xy+1= xy-1$ .)", "Hence $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_4(\\mathcal {D})$ .", "This proves part 5).", "Now we turn to the case $q$ odd, and $S_{\\mathcal {D}}(u) = (M,N,P)^T \\notin \\mathcal {O}_1$ .", "In case $(M,N,P) \\in \\mathcal {O}_2$ , let $(M,N,P)^T=g^{-1} \\cdot (0,1,0)^T$ for some $g \\in PGL(2,q)$ , and let $g \\cdot \\mathcal {D}= \\lbrace y_1, \\dots , y_{k+3}\\rbrace $ .", "It follows that: $ [ G_k^{\\perp }(\\mathcal {D}) \\, | \\, S_{\\mathcal {D}}(u) ] = g^{-1} [ \\mu _1 c_3(y_1) \\,|\\, \\dots \\,|\\, \\mu _{k+3} c_3(y_{k+3}) \\,|\\, (0,1,0)^T], $ for some $\\mu _1, \\dots , \\mu _{k+3} \\in GF(q)^{\\times }$ .", "In this case the condition (REF ) is equivalent to $y_i \\ne -y_j$ for $i \\ne j$ , because $Mxy - N(x+y) +P = -(x+y)$ .", "Hence $[S_{\\mathcal {D}}(u)] \\in \\mathcal {O}_2(\\mathcal {D})$ .", "Similarly, if $(M,N,P) \\in \\mathcal {O}_3$ , we get $(M,N,P) \\in \\mathcal {O}_3(\\mathcal {D})$ .", "As mentioned above the set $\\mathcal {O}_3(\\mathcal {D})$ is empty unless $k+3 \\le (q+1)/2$ , thus for $k = (q-3)/2$ , the possibility $S_{\\mathcal {D}}(u) \\in \\mathcal {O}_3$ does not occur.", "This proves parts 3)-4).", "We record the following theorem about canonical forms of non GRS $[n+1,3]$ MDS codes extending a GRS $[n,3]$ code.", "It will be useful to regard two codes $\\mathcal {C}, \\mathcal {C}^{\\prime }$ as diagonally equivalent if there is a diagonal Hamming isometry (a diagonal matrix in $GL(n,q)$ ) which carries $\\mathcal {C}$ to $\\mathcal {C}^{\\prime }$ .", "Note that diagonally equivalent codes are monomially equivalent but the converse is not true in general.", "At the level of arcs, diagonal equivalence yields the notion of ordered arcs, where as monomial equivalence yields the the notion of (unordered) arcs.", "Theorem 6 Let $\\mathcal {C}$ be a non GRS $[n+1,3]$ MDS code extending a $[n,3]$ GRS code $\\mathcal {C}_1$ where $n \\ge 5$ .", "Up to diagonal equivalence, $\\mathcal {C}$ is the code generated by one of the families of matrices $M_1, M_2, M_3$ below.", "Equivalently let $\\mathcal {A}$ be an ordered $n+1$ -arc in $PG(2,q)$ with the first $n$ points (but not the last) on a RNC (where $n \\ge 5$ ), then $\\mathcal {A}$ is projectively equivalent to the ordered arc defined by the columns of one of the families of matrices $M_1, M_2, M_3$ below.", "In the following, $\\mathcal {D}= \\lbrace x_1, \\dots , x_n\\rbrace \\subset PG(1,q)$ denotes a subset of $n \\ge 5$ distinct points satisfying certain conditions.", "$\\mathcal {D}$ satisfies $ x_i \\ne - x_j$ if $i \\ne j$ .", "In this case $n \\le (q+3)/2$ if $q$ is odd and $n \\le q+1$ if $q$ is even.", "$M_1 = \\begin{pmatrix}1 & \\ldots & 1 & 0\\\\x_1 & \\ldots &x_n & 1 \\\\ x_1^2 & \\ldots & x_n^2 & 0 \\end{pmatrix}$ $q$ is odd, $n \\le (q+1)/2$ , and $\\mathcal {D}$ satisfies $ x_i \\ne \\epsilon / x_j$ if $i \\ne j$ .", "$M_2 = \\begin{pmatrix}1 & \\ldots & 1 & 1\\\\x_1 & \\ldots &x_n & 0 \\\\ x_1^2 & \\ldots & x_n^2 & -\\epsilon \\end{pmatrix}$ $q$ is even, $n \\le (q+2)/2$ , and $\\mathcal {D}$ satisfies $ x_i \\ne 1/ x_j$ if $i \\ne j$ .", "$M_3 = \\begin{pmatrix}1 & \\ldots & 1 & 1\\\\x_1 & \\ldots &x_n & 0 \\\\ x_1^2 & \\ldots & x_n^2 & 1 \\end{pmatrix}$ From the fact that a RNC in $PG(2,q)$ is uniquely determined by any 5 points on it, it follows that the matrices $M_i$ above do not generate a GRS code for $n \\ge 5$ (the corresponding arcs do not lie on a RNC).", "To prove that the code $\\mathcal {C}$ in question is diagonally equivalent to the code generated by one of the matrices of the type $M_i$ , let $C_1$ be diagonally equivalent to the code generated by a matrix $G = [c_3(t_1) \\,|\\, \\dots \\,|\\, c_3(t_n)]$ .", "Thus there is a vector $v \\in GF(q)^3$ such that $[G \\,|\\, v]$ generates the non-GRS code $\\mathcal {C}$ .", "The analysis of such matrices $[G \\,|\\, v]$ was carried out in the proof of Theorem REF (see (REF ), (REF )).", "It was shown that there are matrices $P \\in GL(3,q)$ and a diagonal matrix $Q \\in GL(n+1,q)$ such that $P [G \\,|\\, v] Q$ is of the type $M_1, M_2$ or $M_3$ .", "In other words $\\mathcal {C}$ is diagonally equivalent to the code generated by one of the types of matrices $M_i$ .", "We note that two distinct matrices of the type, say $M_2$ may represent the same MDS extension $\\mathcal {C}$ of $\\mathcal {C}_1$ .", "In order to count the diagonal equivalence classes of codes $(\\mathcal {C}_1, \\mathcal {C})$ where $\\mathcal {C}$ is a $[n+1,3]_q$ MDS and non GRS code extending a $[n,3]_q$ RS code $\\mathcal {C}_1$ , we have to factor out the left action of $G_0, G_{\\epsilon }, G_1$ on generator matrices of the type $M_1, M_2, M_3$ .", "It is convenient to use the language of arcs.", "We will now count the number of projective equivalence classes of ordered arcs $(\\mathcal {A}_1, \\mathcal {A})$ where $\\mathcal {A}$ is an ordered $n+1$ -arc not contained in a RNC, but its first $n$ points form the arc $\\mathcal {A}_1$ which is contained in a RNC.", "Let $\\mathcal {M}_i$ be the set of ordered arcs (without using projective equivalence) arising from matrices of the type $M_i$ .", "Let $\\mathcal {G}_i \\subset PGL(2,q)$ be the stabilizer of the point represented by the last column.", "It is easy to see that $\\mathcal {G}_i$ acts freely (i.e.", "without fixed point) on $\\mathcal {M}_i$ .", "This is because the only element of $PGL(2,q)$ which fixes 3 points is the identity transformation.", "The quotient $\\mathcal {G}_i \\backslash \\mathcal {M}_i$ gives the projective equivalence classes of ordered arc pairs $(\\mathcal {A}_1, \\mathcal {A})$ that we are trying to count and which are of type $M_i$ .", "It is straightforward to count the relevant quantities: $|\\mathcal {M}_1|=(q+1)!", "/ (q+1-n)!", "$ if $q$ is even, and $ |\\mathcal {M}_1| = \\frac{\\tfrac{q-1}{2} !", "\\, 2^n}{(\\tfrac{q-1}{2}-n)!}", "+ \\frac{\\tfrac{q-1}{2} !", "\\, 2^n\\, n}{(\\tfrac{q+1}{2}-n)!", "}+\\frac{\\tfrac{q-1}{2} !", "\\, 2^{n-2}\\, n(n-1)}{(\\tfrac{q+3}{2}-n)!}", "\\quad \\text{ if $q$ is odd}.$ Here we use the convention $(-m)!=\\infty $ for natural numbers $m$ .", "We illustrate the method we use to obtain $|\\mathcal {M}_1|$ for $q$ odd.", "The other cases are similar.", "We may write $PG(1,q)$ as the disjoint union of $(q+3)/2$ sets of the form $ \\lbrace \\infty \\rbrace , \\lbrace 0\\rbrace , \\lbrace \\pm \\alpha _1\\rbrace , \\dots , \\lbrace \\pm \\alpha _{(q-1)/2}\\rbrace $ .", "We note that $\\mathcal {M}_1$ consists of $n$ -tuples $(z_1, \\dots ,z_n)$ such that we pick at most one element from each of the $(q+3)/2$ sets above.", "By similar methods, we obtain rCl |M2| = (q+12 !", "2n )/ (q+12-n)!", "|M3| = (q2 !", "2n)/ (q2-n)!+ (q2 !", "2n-1  n)/(q+22-n)!", "The groups $\\mathcal {G}_i$ have been computed previously: $\\mathcal {G}_1$ is $PGL(2,q)$ if $q$ is even and isomorphic to a dihedral group of order $2(q-1)$ for odd $q$ .", "The group $\\mathcal {G}_2$ isomorphic to a dihedral group of order $2(q+1)$ , and the The group $\\mathcal {G}_3$ isomorphic to the additive group $(GF(q),+)$ .", "Thus we obtain that the number of ordered arc pairs $(\\mathcal {A}_1, \\mathcal {A})$ of the type $M_i$ equals: $(q-2)!", "/ (q+1-n)!\\quad $ if $i=1$ and $q$ is even.", "Here $ n\\le q+1$ .", "$ \\tfrac{q-3}{2} !", "\\, 2^{n-4}\\, \\left[ (q+1)(q+3-2n) + n(n-1)\\right]/ (\\tfrac{q+3-2n}{2} \\,!", ")\\quad $ if $i=1$ and $q$ is odd.", "Here $n \\le (q+3)/2$ $\\tfrac{q-1}{2} !", "\\, 2^{n-2} / (\\tfrac{q+1-2n}{2} \\, !", ")\\quad $ if $i=2$ .", "Here $n \\le (q+1)/2$ .", "$\\tfrac{q-2}{2} !", "\\, 2^{n-2} \\, (q+2-n) / (\\tfrac{q+2-2n}{2} !)", "\\quad $ if $i=3$ .", "Here $n \\le (q+2)/2$ ." ], [ "Conclusion", "We solve the problem of classifying deep holes of $[n,k]_q$ RS codes for $k \\ge (q-1)/2$ for non prime $q$ , which was posed as an open problem in the concluding remarks of [3].", "The problem for $k < (q-1)/2$ is open.", "We solve the problem for $n=k+3$ and all $k$ .", "We also solve the problem for $k=2, n=q$ with $q$ odd, by reducing it to Segre's `oval equals conic' theorem.", "For $k=2, n=q+1$ with $q$ even, we show that the problem is equivalent to the difficult problem of classifying hyperovals in projective planes.", "Finally, we obtain canonical forms for $[n+1,3]_q$ MDS but non-GRS codes extending a $[n,3]_q$ GRS code." ] ]
1612.05447
[ [ "Computing wedge probabilities" ], [ "Abstract A new formula for the probability that a standard Brownian motion stays between two linear boundaries is proved.", "A simple algorithm is deduced.", "Uniform precision estimates are computed.", "Different implementations have been made available online as R packages." ], [ "Introduction", "Let $W=\\lbrace W_t\\,,\\;t\\geqslant 0\\rbrace $ be a standard Brownian motion defined on a filtered probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ .", "The probability that $W_t$ remains in the planar region between two linear boundaries $-a_1 t -b_1$ and $a_2 t +b_2$ will be referred to as wedge probability, and denoted by $k(a_1,b_1;a_2,b_2)$ : $k(a_1,b_1;a_2,b_2)=\\mathbb {P}[-a_1 t -b_1 \\leqslant W_t \\leqslant a_2 t + b_2\\,,\\;\\mbox{for all } t\\geqslant 0]\\;.$ It is positive if and only if $a_1,b_1,a_2,b_2$ are all positive, which will be assumed from now on.", "Doob [8] expressed $k(a_1,b_1;a_2,b_2)$ as the sum of a convergent series.", "Since then, Doob's formula has been extended or applied by many authors, including [3], [10], [16], [4], [14], [23].", "One reason for its success is that many boundary crossing problems reduce to computing a wedge probability, through the representation of a certain Gaussian process in terms of $W$ [19], [6].", "The earliest example is the standard Brownian bridge: $\\left\\lbrace B_{t},0\\leqslant t<1\\right\\rbrace \\overset{d}{=}\\left\\lbrace (1-t)W_{t/(1-t)},0\\leqslant t<1\\right\\rbrace \\,.$ From this representation one gets: $\\mathbb {P}[\\sup _{t\\in [0,1]} |B_t|\\leqslant a] = k(a,a;a,a)\\;,$ which is the the distribution function of the test statistic in the Kolmogorov-Smirnov two-sided test, computed by Kolmogorov [17]; see [24] for historical aspects.", "More generally, the probability that a (non necessarily standard) Brownian bridge stays between two linear segments is a wedge probability.", "This remark makes wedge probabilities a building block for more general boundary crossing problems, through the method of piecewise linear approximations.", "The exit probability of a stochastic process from a region of the plane limited by two curves is called Boundary Crossing Probability (BCP).", "Applications of BCP's can be found in many fields, from non-parametric statistics to biology or finance: see [27] and references therein.", "Explicit results are scarse [15].", "A general approximation method has been proposed by Wang and Potzelberger [26] for single boundaries and Novikov et al.", "[18] for double boundaries; see also [21], [7], [27].", "The idea is to replace the two (nonlinear) boundaries by piecewise linear approximations.", "Given its two values at the bounds of an interval, the conditional distribution of $W$ is that of a Brownian bridge on that interval.", "Thus the probability that it stays between two linear segments is a wedge probability.", "Using the independent increment property of $W$ , the probability that the standard Brownian motion stays between two piecewise linear boundaries is written as a multidimensional Gaussian integral, the integrand being a product of wedge probabilities [18].", "The integral can be approximated either as a Gauss-Hermite quadrature [13] or by a Monte Carlo method [20].", "In both cases, the integrand must be repeatedly evaluated, which implies that many wedge probabilities must be calculated for very different sets of values.", "The problem is that in Doob's formula, as well as in all other equivalent formulas published since, the speed of convergence of the series depends on the parameters, and may be very slow for small values.", "This makes the BCP approximation algorithms numerically unstable.", "The key to efficient computation of wedge probabilities has long been available: Jacobi's theta functions and their double expression through Poisson's summation formula.", "Kolmogorov [17] had already given two formulas for $k(a,a;a,a)$ , and remarked the interest for numerical computation: one converges fast for relatively large values of $a$ , the other for relatively small values.", "This is routinely used in statistical softwares implementing the Kolmogorov-Smirnov test.", "The connection of $k(a_1,b_1;a_2,b_2)$ with theta functions has been pointed out by Salminen and Yor [23].", "However, no alternative to Doob's formula has been deduced so far; this is the main contribution of this paper (Proposition REF ).", "The algorithmic consequence is that computing at most three terms of the series either in Doob's formula or in the new alternative suffices to approximate $k(a_1,b_1;a_2,b_2)$ with precision smaller than $10^{-16}$ .", "Uniform bounds on precision and algorithmic consequences are presented in section .", "Several implementations are compared in section .", "An R package wedge has been made available online [9].", "Its companion wedgeParallel permits full use of a multicore structure." ], [ "Alternative to Doob's formula", "Doob [8] (formula (4.3) p. 398) expresses $k(a_1,b_1;a_2,b_2)$ as follows: $k(a_1,b_1;a_2,b_2)= 1 - \\sum _{n=1}^{+\\infty }\\mathrm {e}^{-2A_n}+\\mathrm {e}^{-2B_n}-\\mathrm {e}^{-2C_n}-\\mathrm {e}^{-2D_n}\\;,$ with: $A_n &=& n^2a_2b_2+(n-1)^2a_1b_1+n(n-1)(a_2b_1+a_1b_2)\\;,\\\\B_n &=& (n-1)^2a_2b_2+n^2a_1b_1+n(n-1)(a_2b_1+a_1b_2)\\;,\\\\C_n &=& n^2(a_1b_1+a_2b_2)+n(n-1)a_2b_1+n(n+1)a_1b_2\\;,\\\\D_n &=& n^2(a_1b_1+a_2b_2)+n(n+1)a_2b_1+n(n-1)a_1b_2\\;.$ Here is another expression.", "Proposition 2.1 For $a_1,b_1,a_2,b_2>0$ , denote: $a_\\pm = \\scriptstyle {\\frac{a_1\\pm a_2}{2}}\\displaystyle {\\,;\\;b_\\pm =}\\scriptstyle {\\frac{b_1\\pm b_2}{2}}\\displaystyle {\\,;\\;c = }\\scriptstyle {\\frac{a_1b_1 - a_2b_2}{2}}\\displaystyle {\\,;\\;d = }\\scriptstyle {\\frac{a_1b_2 - a_2b_1}{2}}\\displaystyle {\\;.", "}$ Then: $\\begin{array}{rl}\\displaystyle {k(a_1,b_1;a_2,b_2)=}\\scriptstyle {\\sqrt{\\frac{\\pi }{2a_+b_+}}}\\displaystyle {\\,\\mathrm {e}^{\\frac{d^2}{2a_+b_+}}\\sum _{n=1}^{+\\infty }}&\\displaystyle {\\Big (\\mathrm {e}^{-\\frac{\\pi ^2(2n)^2}{8a_+b_+}}\\big (\\cos ( \\scriptstyle {\\frac{\\pi (2n)d}{2a_+b_+}}\\displaystyle {)-\\cos (}\\scriptstyle {\\frac{\\pi (2n)c}{2a_+b_+}}\\displaystyle {)\\big )}}\\\\&\\displaystyle {+\\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8a_+b_+}}\\big (\\cos (\\scriptstyle {\\frac{\\pi (2n-1)d}{2a_+b_+}}\\displaystyle {)+\\cos (}\\scriptstyle {\\frac{\\pi (2n-1)c}{2a_+b_+}}\\displaystyle {)\\big )\\Big )\\;.", "}}\\end{array}$ Proof: In terms of $a_\\pm $ , $b_\\pm $ , $c$ , $d$ , the expressions of $A_n$ , $B_n$ , $C_n$ , $D_n$ are: $A_n &=&(2n-1)^2 a_+b_+-(2n-1) c+ a_-b_-\\;,\\\\B_n &=&(2n-1)^2 a_+b_++(2n-1) c+ a_-b_-\\;,\\\\C_n &=& (2n)^2 a_+b_+ + 2n d\\;,\\\\D_n &=& (2n)^2 a_+b_+ - 2n d\\;.$ Hence: $\\begin{array}{rl}\\displaystyle {k(a_1,b_1;a_2,b_2)=1+2\\sum _{n=1}^{+\\infty }}&\\displaystyle {\\Big (\\mathrm {e}^{-2(2n)^2a_+b_+}\\cosh (2(2n)d)}\\\\&\\displaystyle {-\\mathrm {e}^{-2a_-b_-} \\mathrm {e}^{-2(2n-1)^2a_+b_+}\\cosh (2(2n-1)c)\\Big )\\;.", "}\\end{array}$ Not meaning to add anything to the “bewildering variety of notations” for theta functions [1], let us denote by $\\theta $ the following function of two complex variables: $\\theta (u,v) = \\sum _{n=-\\infty }^{+\\infty } \\mathrm {e}^{-2n^2 u} \\cosh (2n v)\\;.$ Observe that: $\\theta (u,v+\\scriptstyle {\\frac{\\mathrm {i}\\pi }{2}}\\displaystyle {)} = \\sum _{n=-\\infty }^{+\\infty } (-1)^n \\mathrm {e}^{-2n^2 u} \\cosh (2n v)\\;.$ By Poisson's summation formula (see for instance formula (11) p. 236 of [25]), one gets: $\\theta (u,v) = \\sqrt{\\frac{\\pi }{2u}}\\,\\mathrm {e}^{v^2/(2u)}\\sum _{n=-\\infty }^{+\\infty } \\mathrm {e}^{-\\pi ^2n^2/(2u)}\\cos (\\pi n v/u)\\;,$ and: $\\theta (u,v+\\scriptstyle {\\frac{\\mathrm {i}\\pi }{2}}\\displaystyle {)}= \\sqrt{\\frac{\\pi }{2u}}\\,\\mathrm {e}^{v^2/(2u)}\\sum _{n=-\\infty }^{+\\infty } \\mathrm {e}^{-\\pi ^2(n+\\frac{1}{2})^2/(2u)}\\cos (\\pi (n+\\scriptstyle {\\frac{1}{2}}\\displaystyle {) v/u)\\;.", "}$ From (REF ), (REF ), and (REF ): $k(a_1,b_1;a_2,b_2)&=&\\frac{1}{2}\\big (\\theta (a_+b_+,d)+\\theta (a_+b_+,d+\\scriptstyle {\\frac{\\mathrm {i}\\pi }{2}}\\displaystyle {)}\\big )\\\\&&-\\frac{\\mathrm {e}^{-2a_-b_-}}{2}\\big (\\theta (a_+b_+,c)-\\theta (a_+b_+,c+\\scriptstyle {\\frac{\\mathrm {i}\\pi }{2}}\\displaystyle {)}\\big )\\;.$ Combining four evaluations of $\\theta $ does not quite solve the numerical problem for small values of $a_+b_+$ .", "The terms of the four series need further rearrangement.", "It is obtained observing that: $c^2-4a_-b_-a_+b_+ = d^2\\;.$ Hence: $\\mathrm {e}^{-2a_-b_-}\\mathrm {e}^{\\frac{c^2}{2a_+b_+}} =\\mathrm {e}^{\\frac{d^2}{2a_+b_+}}\\;.$ From there, (REF ) follows.", "$\\square $ As remarked by Salminen and Yor [23], the symmetry and scaling properties of $k$ can be read on Doob's formula.", "They also appear on (REF ): $k(a_1,b_1;a_2,b_2) = k(a_2,b_2;a_1,b_1)=k(b_1,a_1;b_2,a_2)=k(\\frac{a_1}{u},ub_1;\\frac{a_2}{u},ub_2)\\;.$ When slopes or intercepts are equal the expressions are simpler.", "If $a_1=a_2=a$ then $a_+=a$ , $a_-=0$ , $c=-d=a b_-$ , and: $k(a,b_1;a,b_2)&=&1+2\\sum _{n=1}^{+\\infty } (-1)^n\\mathrm {e}^{-2n^2a b_+}\\cosh (2nab_-)\\\\&=&\\sqrt{\\frac{2\\pi }{ab_+}}\\mathrm {e}^{\\frac{ab_-^2}{2b_+}}\\sum _{n=1}^{+\\infty } \\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8ab_+}}\\cos (\\pi (2n-1)\\scriptstyle {\\frac{b_-}{2b_+}}\\displaystyle {)) \\;.", "}$ When both slopes and intercepts are equal one gets: $k(a,b;a,b)&=&1+2\\sum _{n=1}^{+\\infty } (-1)^{n}\\mathrm {e}^{-2n^2ab}\\\\&=&\\sqrt{\\frac{2\\pi }{ab}}\\sum _{n=1}^{+\\infty } \\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8ab}}\\;.$ The first sum is formula (4.3') of [8].", "When slopes equal intercepts, the probability for a standard Brownian bridge to stay in a horizontal band is obtained, i.e.", "formula (4.9) p. 448 of [5].", "If $b_1=a_1$ and $b_2=a_2$ , then $b_\\pm =a_\\pm $ , $c=2a_+a_-$ , $d=0$ , and: $k(a_1,a_1;a_2,a_2)&=&1+\\sum _{n=1}^{+\\infty } 2\\mathrm {e}^{-2(n(a_1+a_2))^2}-\\mathrm {e}^{-(2n(a_1+a_2)-a_1)^2}-\\mathrm {e}^{-(2n(a_1+a_2)-a_2)^2}\\\\&=&\\sqrt{\\frac{\\pi }{2}}\\frac{1}{a_+}\\sum _{n=1}^{+\\infty }\\mathrm {e}^{-\\frac{\\pi ^2(2n)^2}{8a_+^2}}(1-\\cos (\\pi (2n)\\scriptstyle {\\frac{a_-}{a_+}}\\displaystyle {))}\\\\&&\\hspace*{45.52458pt}+\\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8a_+^2}}(1+\\cos (\\pi (2n-1)\\scriptstyle {\\frac{a_-}{a_+}}\\displaystyle {))\\;.", "}$ Finally, the case where all four parameters are equal is the probability for a standard Brownian bridge to stay in a horizontal band centered at 0, i.e.", "the distribution function of the test statistic in the Kolmogorov-Smirnov two-sided test.", "The formulas were originally found by Kolmogorov [17]; Feller [12] gave a simpler proof.", "Both had remarked the double expression coming from theta functions, and its interest for numerical computation.", "$k(a,a;a,a)&=&1+2\\sum _{n=1}^{+\\infty } (-1)^{n}\\mathrm {e}^{-2n^2 a^2}\\\\&=&\\frac{\\sqrt{2\\pi }}{a}\\sum _{n=1}^{+\\infty } \\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8a^2}}\\;.$" ], [ "Algorithm and precision", "Denote by $K_{1,N}$ and $K_{2,N}$ the partial sums up to $N$ in formulas (REF ) and (REF ).", "$K_{1,N}= 1 - \\sum _{n=1}^{N}\\mathrm {e}^{-2A_n}+\\mathrm {e}^{-2B_n}-\\mathrm {e}^{-2C_n}-\\mathrm {e}^{-2D_n}\\;,$ $\\begin{array}{rl}\\displaystyle {K_{2,N} =\\sqrt{\\frac{\\pi }{2a_+b_+}}\\,\\mathrm {e}^{\\frac{d^2}{2a_+b_+}}\\sum _{n=1}^{N}}&\\displaystyle {\\Big (\\mathrm {e}^{-\\frac{\\pi ^2(2n)^2}{8a_+b_+}}\\big (\\cos ( \\scriptstyle {\\frac{\\pi (2n)d}{2a_+b_+}}\\displaystyle {)-\\cos (}\\scriptstyle {\\frac{\\pi (2n)c}{2a_+b_+}}\\displaystyle {)\\big )}}\\\\&\\displaystyle {+\\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8a_+b_+}}\\big (\\cos (\\scriptstyle {\\frac{\\pi (2n-1)d}{2a_+b_+}}\\displaystyle {)+\\cos (}\\scriptstyle {\\frac{\\pi (2n-1)c}{2a_+b_+}}\\displaystyle {)\\big )\\Big )\\;.", "}}\\end{array}$ The question is: for a given set of parameters $a_1,b_1,a_2,b_2$ , which of $K_{1,N}$ and $K_{2,N}$ should be computed, and which value of $N$ ensures a given precision?", "Proposition REF bounds remainders.", "Proposition 3.1 Denote by $R_{1,N}$ and $R_{2,N}$ the remainders: $R_{1,N} = K_{1,\\infty }-K_{1,N}\\quad \\mbox{and}\\quad R_{2,N} = K_{2,\\infty }-K_{2,N}\\;.$ For $N>1$ : $R_{1,N} \\leqslant \\scriptstyle {\\frac{1}{4a_+b_+(N-1)}}\\displaystyle {\\mathrm {e}^{-8a_+b_+(N-1)^2}\\;,}$ $R_{2,N} \\leqslant \\scriptstyle {\\left(\\frac{2}{\\pi }\\right)^{3/2}\\frac{\\sqrt{a_+b_+}}{N}}\\displaystyle { \\mathrm {e}^{2a_+b_+}\\mathrm {e}^{-\\frac{\\pi ^2 N^2}{2a_+b_+}}\\;.", "}$ Proof: For both bounds, the following well known inequality is used: for any positive $u$ , $\\sum _{n=N+1}^{+\\infty } \\mathrm {e}^{-un^2}\\leqslant \\frac{\\mathrm {e}^{-uN^2}}{2uN}\\;.$ In $R_{1,N}$ , all four terms $A_n$ , $B_n$ , $C_n$ , $D_n$ are larger than $4(n-1)^2a_+b_+$ .", "Hence: $R_{1,N} \\leqslant 4 \\sum _{n=N+1}^{+\\infty }\\mathrm {e}^{-8(n-1)^2a_+b_+}\\;,$ from where (REF ) follows by (REF ).", "For $R_{2,N}$ , notice first that $|d|\\leqslant 2\\sqrt{a_+b_+}$ , by Schwarz inequality.", "Hence: $R_{2,N} &\\leqslant &\\sqrt{\\frac{2\\pi }{a_+b_+}} \\mathrm {e}^{2a_+b_+} \\sum _{n=N+1}^{+\\infty }\\mathrm {e}^{-\\frac{\\pi ^2(2n)^2}{8a_+b_+}}+\\mathrm {e}^{-\\frac{\\pi ^2(2n-1)^2}{8a_+b_+}}\\\\&=&\\sqrt{\\frac{2\\pi }{a_+b_+}} \\mathrm {e}^{2a_+b_+}\\sum _{n=2N+1}^{+\\infty }\\mathrm {e}^{-\\frac{\\pi ^2n^2}{8a_+b_+}}\\;.$ Using again (REF ) leads to (REF ).", "$\\square $ As expected, the bound on $R_{1,N}$ decreases with $a_+b_+$ , the bound on $R_{2,N}$ increases.", "Denote by $\\tau _N$ the value of $a_+b_+$ such that both bounds are equal, and by $\\varepsilon _N$ their common value.", "Computing $K_{1,N}$ if $a_+b_+\\geqslant \\tau _N$ and $K_{2,N}$ else ensures $\\varepsilon _N$ precision at least, whatever $a_1,b_1,a_2,b_2$ .", "The values of $\\tau _N$ and $\\varepsilon _N$ for $N=2,\\ldots ,8$ are given in Table REF .", "Table: Threshold and precision per number of terms computed.In particular, for $N=3$ a precision $\\varepsilon _3=1.8~10^{-17}$ is obtained, which is below current machine double precision.", "Thus $N=3$ was chosen as default value in our implementation.", "Computing more than two terms is usually not necessary.", "To illustrate this a Monte Carlo study has been conducted, over $10^6$ four-tuples of independent random values drawn in the interval $[0{,}10]$ with cumulative distribution function $(x/10)^{1/2}$ .", "That choice ensured that wedge probabilities covered the whole range of $[0,1]$ , with higher mass on values close to 0 or 1.", "For both sums the number of terms to convergence was defined as the first value of $N$ such that the remainder is smaller than $\\varepsilon =10^{-16}$ .", "As predicted by Proposition REF , in all $10^6$ cases either $K_{1,N}$ or $K_{2,N}$ reached $\\varepsilon $ precision with $N=3$ terms or less.", "Actually, in $2.5\\%$ of the cases, the result was smaller than $\\varepsilon $ or larger than $1-\\varepsilon $ : no summation was needed.", "In $73.8\\%$ of the cases $N=1$ sufficed to get $\\varepsilon $ precision, and in $20.6\\%$ of the cases $N=2$ terms were necessary; only in $0.56\\%$ of the cases were $N=3$ terms necessary.", "Experimental results evidenced the need for an alternative to $K_{1,N}$ .", "Indeed, in 442 out of the $10^6$ cases, the number of terms to convergence of $K_{1,N}$ was larger than 100, and in 1558 cases it was larger than 50.", "Figure REF presents the numbers of terms to convergence as a function of $\\log (a_+b_+)$ , for all $10^6$ random values, and both $K_{1,N}$ and $K_{2,N}$ .", "As expected, the numbers decrease for $K_{1,N}$ ; they increase for $K_{2,N}$ .", "Figure: Number of terms to convergence as a function of log(a + b + )\\log (a_+b_+)for K 1,N K_{1,N} and K 2,N K_{2,N}, over 10 6 10^6 simulated values ofa 1 ,b 1 ;a 2 ,b 2 a_1,b_1;a_2,b_2.", "The dashed vertical line marks the theoreticalthreshold for N=3N=3 i.e.", "τ 3 =1.136\\tau _3=1.136." ], [ "Implementation", "The calculation of successive terms in $K_{1,N}$ or $K_{2,N}$ is easily vectorized.", "This makes the computation of a vector of wedge probabilities relatively fast in pure R [22].", "Our objective was to explore gains in computing time, using existing R tools.", "The most widely used of these tools is Rcpp [11].", "It uses (usually faster) compiled C++ code, interfaced with the R environment.", "Most computers now have a multicore architecture.", "However by default, both R or Rcpp use only one core.", "Taking full advantage of a multicore stucture can be done for example through RcppParallel [2].", "Numerical experiments have been made using vectors of simulated entries with the same distribution as in section : independent entries on $[0{,}10]$ with cumulative distribution function $(x/10)^{1/2}$ .", "Five implementations were considered: pure R, one-core Rcpp, RcppParallel with 4, 6, and 8 cores.", "Table 2 reports running times on a MacBookPro Retina 15.", "The running time for $10^6$ values in pure R ($0.725$ second), can be considered satisfactory.", "However, the gain in time goes up to twentyfold if an eight-core architecture is used.", "One of the known limitations of vectorized versions in pure R is memory space: ours cannot deal with vectors larger than $10^7$ entries.", "Table: Times in second for calculating wedge probabilities overvectors of size n=10 6 ,10 7 ,10 8 n=10^6,10^7,10^8, in pure R,Rcpp, RcppParallel with 4, 6, 8threads.", "For n=10 8 n=10^8, calculations exceed memory space in pure R.An R package wedge has been made available online [9].", "In order to address installation issues for users not interested by a parallel version, the companion package wedgeParallel has been left as an option." ] ]
1612.05764
[ [ "Stability of Azimuthal-angle Observables under Higher Order Corrections\n in Inclusive Three-jet Production" ], [ "Abstract Recently, a new family of observables consisting of azimuthal-angle generalised ratios was proposed in a kinematical setup that resembles the usual Mueller-Navelet jets but with an additional tagged jet in the central region of rapidity.", "Non-tagged minijet activity between the three jets can affect significantly the azimuthal angle orientation of the jets and is accounted for by the introduction of two BFKL gluon Green functions.", "Here, we calculate the, presumably, most relevant higher order corrections to the observables by now convoluting the three leading-order jet vertices with two gluon Green functions at next-to-leading logarithmic approximation.", "The corrections appear to be mostly moderate giving us confidence that the recently proposed observables are actually an excellent way to probe the BFKL dynamics at the LHC.", "Furthermore, we allow for the jets to take values in different rapidity bins in various configurations such that a comparison between our predictions and the experimental data is a straightforward task." ], [ "Introduction", "One of the most active fields of research in Quantum Chromodynamics (QCD) is the resummation of large logarithms in the center-of-mass energy squared $s$ for processes dominated by the so-called multi-Regge kinematics (MRK).", "To account for these logarithms, one can make use of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework in the leading logarithmic (LLA) [1], [2], [3], [4], [5], [6] and next-to-leading logarithmic (NLLA) approximation [7], [8].", "In inclusive multi-jet production, when the outermost in rapidity jets have a large rapidity difference, we may assume that the process follows the MRK and therefore, the BFKL resummation becomes relevant.", "Figure: Inclusive three-jet production process in multi-Regge kinematics.A classic example is Mueller-Navelet jets [9], that is, the configuration in hadronic colliders with two final state jetsAnother interesting idea, suggested in [10] and investigated in [11], is the study of the production of two charged light hadrons, $\\pi ^{\\pm }$ , $K^{\\pm }$ , $p$ , $\\bar{p}$ , with large transverse momenta and well separated in rapidity.", "which are produced with large and similar transverse momenta, $k_{A,B}$ , and a large rapidity separation $Y=\\ln ( x_1 x_2 s/(k_A k_B))$ .", "$x_{1,2}$ are the longitudinal momentum fractions of the partons that are adjacent to the jets.", "Various works [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] addressing the azimuthalIn this work, we denote the azimuthal angles by $\\theta $ contrary to the usual practice that prefers $\\phi $ .", "angle ($\\theta $ ) profile of the two tagged jets, suggest the presence of important minijet activity populating the rapidity interval which can be taken into account by considering a BFKL gluon Green function connecting the two jets.", "However, it was shown that the azimuthal angle behaviour of the tagged jets is strongly contaminated by collinear effects [27], [28], that have their origin at the $n=0$ Fourier component in $\\theta $ of the BFKL kernel.", "This dependence is mostly canceled if ratios of projections on azimuthal angle observables ${\\cal R}^m_n = \\langle \\cos {(m \\, \\theta )} \\rangle / \\langle \\cos {(n \\, \\theta )} \\rangle $  [27], [28] (where $m,n$ are integers and $\\theta $ is the azimuthal angle between the two tagged jets) are studied.", "It also seems that these offer a clearer signal of BFKL effects than the usual predictions for the behaviour of the hadron structure functions $F_{2,L}$ (well fitted within NLL approaches [29], [30]).", "The confrontation of different NLLA calculations for these ratios ${\\cal R}^m_n$  [31], [32], [33], [34], [35] against experimental data at the LHC has been successful.", "Recently, we proposed new observables for processes at the LHC that may be considered as a generalisation of the Mueller-Navelet jets.", "These processes are inclusive three-jet [36], [37] and four-jet production [38], [39] with the outermost jets widely separated in rapidity $Y$ , whereas any other tagged jet is to be found in more central regions of the detector.", "The main idea behind all this effort is that we need more exclusive final states in order to be able to address a number of theoretical issues, e.g.", "what is the optimal way to implement the running of the strong coupling or could one speak about saturation effects at present energies, etc.", "Investigating more exclusive final states (with more than two jets) although more challenging on a technical level, allows for more complex observables to be defined so that one can finally choose those that encapsulate the essence of these features of MRK that are distinct in the BFKL dynamics only.", "In the remaining of this paper, we will focus only on inclusive three-jet production.", "Figure: A primitive lego plotdepicting a three-jet event.", "k A k_A is a forward jet withlarge positiverapidity Y A Y_A and azimuthal angle θ A \\theta _A, k B k_B is a forward jet withlarge negativerapidity Y B Y_B and azimuthal angle θ B \\theta _B andk J k_J is a central jet withrapidity y J y_J and azimuthal angle θ J \\theta _J.", "The fade-brown areas to the left and righthighlight the regionsin rapidity that are not covered by the standard detectors.The key idea presented in [36] was to get theoretical predictions for the partonic-level ratios ${\\cal R}^{M N}_{P Q} =\\frac{ \\langle \\cos {(M \\, \\theta _1)} \\cos {(N \\, \\theta _2)} \\rangle }{\\langle \\cos {(P \\, \\theta _1)} \\cos {(Q \\, \\theta _2)} \\rangle } \\, ,$ where $\\theta _1$ is the azimuthal angle difference between the first and the second (central) jet, while, $\\theta _2$ is the azimuthal angle difference between the second and the third jet.", "In [37], we presented a first phenomenological analysis at LLA for the respective hadronic-level ratios $R^{M N}_{P Q}$ .", "These were obtained after using collinear factorization to produce the two most forward/backward jets and convoluting the partonic differential cross section, which follows the BFKL dynamics, with collinear parton distribution functions included in the forward “jet vertex\" [41], [42], [43], [44], [45], [46], [47].", "In addition, the two Mueller-Navelet jet-vertices were linked with the centrally produced jet via two BFKL gluon Green functions.", "Finally, we integrated over the momenta of all produced jets, using actual LHC experimental cuts.", "Our predictions in [37], although may in principle be directly compared to experimental data once these are available, do not resolve two issues: (I) They do not offer any estimate of the theoretical uncertainty that comes into play once higher order corrections are considered.", "(II) Since we restricted the central jet to be produced in the middle of the rapidity interval between the outermost jets, one could possibly raise concerns of whether a experimental analysis following the kinematical setup used in Ref.", "[37] is possible at all.", "Here, we address both of these issues.", "To that end, regarding issue (I), one needs to calculate higher order corrections for the ratios at partonic-level.", "This comprises of two steps: considering NLLA corrections to the BFKL kernel and NLO corrections to the jet vertices.", "However, although the corrections to the jet vertices may be in general significant, we expect them not to affect much the azimuthal angle characteristics of the jets which are driven mostly by the minijet activity in the rapidity intervals between the jets.", "Demanding three tagged jets along with central minijets leaves little room for higher order real emission activity near the jet vertices.", "We expect that the higher order virtual corrections to the vertices may be interpreted as K-factor corrections which would cancel out in our observables since we consider ratios.", "We have argued previously that the minijet activity is accounted for by the introduction of the two gluon Green functions.", "Large corrections from LLA to NLLA for the gluon Green function, which is actually a usual outcome in many BFKL-based calculations, could potentially have a strong impact on the ratios and this at any rate needs to be assessed.", "Therefore, in this work we work with NLLANote that from now on, we will refer to the results with NLLA gluon Green functions and LO vertices as NLLA results.", "gluon Green functions and LO jet vertices.", "The answer to issue (II) is, naturally, positive since allowing for the central jet to live in a rapidity range instead of a single point, as long as this range is located generally in the middle of the rapidity interval between the outermost jets, does not affect the values of the generalised rations in Eq.", "REF , as was shown in [36].", "Nevertheless, to avoid any confusion and to have a complete study, in the present work we are also considering cases in which the central jet lives in a rapidity bin of unit width, while the central value of the bin may vary.", "Other potential sources of uncertainty could be due to the particular PDF sets one uses.", "One can still argue though that the uncertainty due to different PDF sets does not need to be ascertained before one has gauged how large are the full beyond the LLA corrections to the partonic-level ratios, since it will be overshadowed by the latter.", "Indeed, from first tries we see no significant difference in the results when we work with different PDF sets and therefore we do not offer any dedicated analysis on that here.", "In the bulk of the paper we present theoretical predictions for the ratios $R^{M N}_{P Q}$ at NLLA and we compare these to the LLA ones.", "In particular, in Section 2 we define the computational framework and our notation for the LLA and NLLA calculations.", "In Section 3 we present results for $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ as a function of the rapidity distance $Y$ between the outermost jets while the central jet is fixed at the middle of this distance, for both $\\sqrt{s}=7$ and $\\sqrt{s}=13$ TeV colliding energies.", "In Section 4, we present the LLA and NLLA results for $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ while the central jet is allowed to take values in the rapidity bin $[-0.5,0.5]$ .", "The results are plotted again as functions of the rapidity interval $Y$ between the outermost jets for $\\sqrt{s}=7$ and $\\sqrt{s}=13$ TeV.", "In Section 5, we do not keep $Y$ fixed at any certain value, instead, we allow for the forward jet to be in the rapidity interval $[3,4.7]$ , for the backward one to be in the symmetric rapidity interval $[-4.7,-3]$ while the rapidity of the central jet takes again values in a bin of unit 1.", "The central value of the bin though, may now take five different values, namely, $\\lbrace -1, -0.5, 0, 0.5, 1\\rbrace $ and we plot both the LLA and NLLA results for $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ as a function of that central value, again for $\\sqrt{s}=7$ and $\\sqrt{s}=13$ TeV.", "We finish our work with Conclusions and Outlook." ], [ "Hadronic inclusive three-jet production in multi-Regge kinematics", "The process under investigation (see Figs.", "REF and REF ) is the production of two forward/backward jets, both characterized by high transverse momenta $\\vec{k}_{A,B}$ and well separated in rapidity, together with a third jet produced in the central rapidity region and with possible associated minijet production.", "This corresponds to ${\\rm proton }(p_1) + {\\rm proton} (p_2) \\rightarrow {\\rm j}(k_A, Y_A) + {\\rm j}(k_J, y_J) + {\\rm j}(k_B,Y_B) + {\\rm minijets}\\;,$ where ${\\rm j}(k_A, Y_A) $ is the forward jet with transverse momentum $k_A$ and rapidity $Y_A$ , ${\\rm j}(k_B, Y_B) $ is the backward jet with transverse momentum $k_B$ and rapidity $Y_B$ and ${\\rm j}(k_J, y_J) $ is the central jet with transverse momentum $k_J$ and rapidity $y_J$ .", "In collinear factorization the cross section for the process (REF ) reads $& \\frac{d\\sigma ^{3-{\\rm jet}}}{dk_A \\, dY_A \\, d\\theta _A \\,dk_B \\, dY_B \\, d\\theta _B \\,dk_J \\, dy_J d\\theta _J} =\\\\ \\nonumber \\hspace{28.45274pt}& \\sum _{r,s=q,{\\bar{q}},g}\\int _0^1 dx_1 \\int _0^1 dx_2\\ f_r\\left(x_1,\\mu _F\\right)\\ f_s\\left(x_2,\\mu _F\\right) \\;d{\\hat{\\sigma }}_{r,s}\\left(\\hat{s},\\mu _F\\right) \\;,$ where the $r, s$ indices specify the parton types (quarks $q = u, d, s, c, b$ ; antiquarks $\\bar{q} = \\bar{u}, \\bar{d}, \\bar{s}, \\bar{c}, \\bar{b}$ ; or gluon $g$ ), $f_{r,s}\\left(x, \\mu _F \\right)$ are the initial proton PDFs; $x_{1,2}$ represent the longitudinal fractions of the partons involved in the hard subprocess; $d\\hat{\\sigma }_{r,s}\\left(\\hat{s}, \\mu _F \\right)$ is the partonic cross section for the production of jets and $\\hat{s} \\equiv x_1x_2s$ is the squared center-of-mass energy of the hard subprocess (see Fig.", "REF ).", "The BFKL dynamics enters in the cross-section for the partonic hard subprocess $d{\\hat{\\sigma }}_{r,s}$ in the form of two forward gluon Green functions $\\varphi $ to be described in a while.", "Using the definition of the jet vertex in the leading order approximation [41], we can present the cross section for the process as $& \\frac{d\\sigma ^{3-{\\rm jet}}}{dk_A \\, dY_A \\, d\\theta _A \\,dk_B \\, dY_B \\, d\\theta _B \\,dk_J \\, dy_J d\\theta _J} =\\nonumber \\\\ \\hspace{28.45274pt}&\\frac{8 \\pi ^3 \\, C_F \\, \\bar{\\alpha }_s^3}{N_c^3} \\,\\frac{x_{J_A} \\, x_{J_B}}{k_A \\, k_B \\, k_J} \\,\\int d^2 \\vec{p}_A \\int d^2 \\vec{p}_B \\,\\delta ^{(2)} \\left(\\vec{p}_A + \\vec{k}_J- \\vec{p}_B\\right) \\,\\nonumber \\\\ \\hspace{28.45274pt}& \\times \\left(\\frac{N_c}{C_F}f_g(x_{J_A},\\mu _F)+\\sum _{r=q,\\bar{q}}f_r(x_{J_A},\\mu _F)\\right) \\,\\nonumber \\\\ \\hspace{28.45274pt}& \\times \\left(\\frac{N_c}{C_F}f_g(x_{J_B},\\mu _F)+\\sum _{s=q,\\bar{q}}f_s(x_{J_B},\\mu _F)\\right)\\nonumber \\\\ \\hspace{28.45274pt}& \\times \\varphi \\left(\\vec{k}_A,\\vec{p}_A,Y_A - y_J\\right)\\varphi \\left(\\vec{p}_B,\\vec{k}_B,y_J - Y_B\\right),$ where $N_c$ is the number of colors in QCD and $C_F$ is the Casimir operator, $C_F = (N_c^2-1)/(2N_c)$ .", "In order to lie within multi-Regge kinematics, we have considered the ordering in the rapidity of the produced particles $Y_A > y_J > Y_B$ , while $k_J^2$ is always above the experimental resolution scale.", "$x_{J_{A,B}}$ are the longitudinal momentum fractions of the two external jets, linked to the respective rapidities $Y_{{A,B}}$ by the relation $x_{{A,B}} = k_{A,B} \\, e^{\\, \\pm \\, Y_{{A,B}}} / \\sqrt{s}$ .", "$\\varphi $ are BFKL gluon Green functions normalized to $ \\varphi \\left(\\vec{p},\\vec{q},0\\right) = \\delta ^{(2)} \\left(\\vec{p} - \\vec{q}\\right)$ and $\\bar{\\alpha }_s$ is defined in terms of the strong coupling as $\\bar{\\alpha }_s = N_c/\\pi \\, \\alpha _s \\left(\\mu _R\\right)$ .", "Building up on the work in Refs.", "[36], [37], we study observables for which the BFKL approach will be distinct from other formalisms and also rather insensitive to possible higher order corrections.", "We focus on new quantities whose associated distributions are different from the ones which characterize the Mueller-Navelet case, though still related to the azimuthal-angle correlations by projecting the differential cross section on the two relative azimuthal angles between each external jet and the central one $\\Delta \\theta _{\\widehat{AJ}} = \\theta _A - \\theta _J - \\pi $ and $\\Delta \\theta _{\\widehat{JB}} = \\theta _J - \\theta _B - \\pi $ (see Fig.", "REF ).", "Taking into account the factors coming from the jet vertices, it is possible to rewrite the projection of the differential cross section on the azimuthal angle differences (Eq.", "(7) in Ref.", "[36] ) in the form $& \\int _0^{2 \\pi } d \\theta _A \\int _0^{2 \\pi } d \\theta _B \\int _0^{2 \\pi }d \\theta _J \\cos {\\left(M \\Delta \\theta _{\\widehat{AJ}} \\right)} \\,\\cos {\\left(N \\Delta \\theta _{\\widehat{JB}} \\right)}\\\\& \\hspace{14.22636pt}\\frac{d\\sigma ^{3-{\\rm jet}}}{dk_A \\, dY_A \\, d\\theta _A \\,dk_B \\, dY_B \\, d\\theta _B \\,dk_J \\, dy_J d\\theta _J} =\\nonumber \\\\& \\hspace{1.9919pt}\\frac{8 \\pi ^4 \\, C_F \\, \\bar{\\alpha }_s^3}{N_C^3} \\,\\frac{x_{J_A} \\, x_{J_B}}{k_A \\, k_B}\\left(\\frac{N_C}{C_F}f_g(x_{J_A},\\mu _F) \\,+\\sum _{r=q,\\bar{q}}f_r(x_{J_A},\\mu _F)\\right) \\,\\nonumber \\\\& \\times \\hspace{1.9919pt}\\left(\\frac{N_C}{C_F}f_g(x_{J_B},\\mu _F)+\\sum _{s=q,\\bar{q}}f_s(x_{J_B},\\mu _F)\\right) \\,\\sum _{L=0}^{N}\\left( \\begin{array}{c}\\hspace{-5.69046pt}N \\\\\\hspace{-5.69046pt}L\\end{array} \\hspace{-5.12128pt}\\right)\\left(k_J^2\\right)^{\\frac{L-1}{2}}\\nonumber \\\\& \\times \\hspace{1.9919pt}\\int _{0}^\\infty d p^2 \\, \\left(p^2\\right)^{\\frac{N-L}{2}} \\,\\int _0^{2 \\pi } d \\theta \\frac{(-1)^{M+N} \\cos { \\left(M \\theta \\right)} \\cos {\\left((N-L) \\theta \\right)}}{\\sqrt{\\left(p^2 + k_J^2+ 2 p k_J \\cos {\\theta }\\right)^{N}}}\\nonumber \\\\& \\times \\hspace{1.9919pt}\\varphi ^{(LLA,NLLA)}_{M} \\left(k_A^2,p^2,Y_A-y_J\\right)\\varphi ^{(LLA,NLLA)}_{N} \\left(p^2+ k_J^2 + 2 p k_J \\cos {\\theta },k_B^2,y_J-Y_B\\right).", "\\nonumber $ In this expression the gluon Green function $\\varphi $ is either at LLA ($\\varphi ^{(LLA)}$ ) or at NLLA ($\\varphi ^{(NLLA)}$ ) accuracy.", "In particular, at LLA we have $\\varphi ^{(LLA)}_{n} \\left(k^2,q^2,y\\right) \\; &= \\;2 \\, \\int _0^\\infty d \\nu \\cos {\\left(\\nu \\ln {\\frac{k^2}{q^2}}\\right)}\\frac{e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\nu \\right) y}}{\\pi \\sqrt{k^2 q^2} }, $ while the LLA BFKL kernel $\\chi _{n} \\left(\\nu \\right) $ reads $\\chi _{n} \\left(\\nu \\right) \\; &= \\; 2\\, \\psi (1) -\\psi \\left( \\frac{1+n}{2} + i \\nu \\right) -\\psi \\left(\\frac{1+n}{2} - i \\nu \\right)$ and $\\psi $ is the logarithmic derivative of Euler's gamma function.", "At NLLA we have $\\varphi ^{(NLLA)}_{n} \\left(k^2,q^2,y\\right)=2 \\int _0^\\infty d \\nu \\cos {\\left(\\nu \\ln {\\frac{k^2}{q^2}}\\right)}\\frac{e^{\\bar{\\alpha }_s \\left( \\chi _{|n|} (\\nu ) + \\bar{\\alpha }_s \\chi _{|n|}^{(1)}(\\nu ) \\right) Y}}{\\pi \\sqrt{k^2 q^2} } \\, , $ where the NLLA contribution $\\chi _{|n|}^{(1)}(\\nu )$ , calculated in [48] (see also [49]), can be presented in the form $\\chi _{n}^{(1)}(\\nu )=-\\frac{\\beta _0}{8\\, N_c}\\left(\\chi _n^2(\\nu )-\\frac{10}{3}\\chi _n(\\nu )-i\\chi ^\\prime _n(\\nu )\\right) + {\\bar{\\chi }_{n}}(\\nu )\\, ,$ with $-4 \\bar{\\chi }_{n}(\\nu ) &=& \\frac{\\pi ^2-4}{3}\\chi _n(\\nu )-6\\zeta (3)-\\chi _n^{\\prime \\prime }(\\nu ) +\\,2\\,\\phi _n(\\nu )+\\,2\\,\\phi _n(-\\nu ) \\nonumber \\\\&& \\hspace{-85.35826pt} +\\frac{\\pi ^2\\sinh (\\pi \\nu )}{2\\,\\nu \\, \\cosh ^2(\\pi \\nu )} \\left(\\left(3+\\left(1+\\frac{n_f}{N_c^3}\\right)\\frac{11+12\\nu ^2}{16(1+\\nu ^2)}\\right)\\delta _{n0}-\\left(1+\\frac{n_f}{N_c^3}\\right)\\frac{(1+4\\nu ^2)\\delta _{n2}}{32(1+\\nu ^2)}\\right),\\hspace{28.45274pt}$ and $\\phi _n(\\nu ) &=& \\sum _{k=0}^\\infty \\frac{(-1)^{k+1}}{k+(n+1)/2+i\\nu }\\left[\\psi ^{\\prime }(k+n+1)-\\psi ^{\\prime }(k+1)\\right.\\nonumber \\\\&&\\hspace{-62.59596pt}\\left.+(-1)^{k+1}(\\beta ^{\\prime }(k+n+1)+\\beta ^{\\prime }(k+1)) -\\frac{(\\psi (k+n+1)-\\psi (k+1))}{k+(n+1)/2+i\\nu }\\right],$ whereas $4 \\beta ^{\\prime }(z) = \\psi ^{\\prime } \\left((z+1)/2 \\right) -\\psi ^{\\prime } \\left(z / 2\\right)$ .", "In order to make an appropriate choice of the renormalization scale $\\mu _R$ , we used the Brodsky-Lepage-Mackenzie (BLM) prescription [50] which is proven a very successful choice for fitting the data in Mueller-Navelet studies [31], [32].", "It consists of using the MOM scheme and choosing the scale $\\mu _R$ such that the $\\beta _0$ -dependence of a given observable vanishes.", "Applying the BLM prescription leads to the modification of the exponent in Eq.", "(REF ) in the following way: $\\bar{\\alpha }_s \\left( \\chi _{|n|} (\\nu ) + \\bar{\\alpha }_s \\chi _{|n|}^{(1)}(\\nu ) \\right) Y \\,\\, \\rightarrow \\,\\,\\bar{\\alpha }_s \\left( \\chi _{|n|} (\\nu ) \\left( 1+ \\frac{\\alpha _s}{\\pi } T \\right) + \\bar{\\alpha }_s \\chi _{|n|}^{(1)}(\\nu ) \\right) Y \\, ,$ where $T&=&T^{\\beta }+T^{\\,\\rm conf}\\;,\\\\T^{\\beta }&=&-\\frac{\\beta _0}{2}\\left( 1+\\frac{2}{3}I \\right)\\;,\\\\T^{conf}&=& \\frac{C_A}{8}\\left[ \\frac{17}{2}I +\\frac{3}{2}\\left(I-1\\right)\\xi +\\left( 1-\\frac{1}{3}I\\right)\\xi ^2-\\frac{1}{6}\\xi ^3 \\right]\\;.$ Here $I=-2\\int _0^1dx\\frac{\\ln \\left(x\\right)}{x^2-x+1}\\simeq 2.3439$ and $\\xi $ is a gauge parameter, fixed at zero.", "Following this procedure, the renormalization scale $\\mu _R$ is fixed at the value $( \\mu _R^{\\rm BLM})^2=k_{A}k_{B}\\ \\exp \\left[\\frac{1+4I}{3}+\\frac{1}{2} \\chi _{n}\\left(\\nu \\right)\\right] \\, .$ In our numerical analysis we consider two cases.", "In one, we set $\\mu _R=\\mu _R^{\\rm BLM}$ only in the exponential factor of the gluon Green function $\\varphi _n$ , while we let the argument of the $\\bar{\\alpha }_s^3$ in Eq.", "REF to be at the `natural' scale $\\sqrt{k_{A}k_{B}}$ , that is, $\\bar{\\alpha }_s^3(\\sqrt{k_{A}k_{B}})$ .", "In the second case, we fix $\\mu _R=\\mu _R^{\\rm BLM}$ everywhere in Eq.", "REF .", "These two cases lead in general to two different but similar values for our NLLA predictions and wherever we present plots we fill the space in between so that we end up having a band instead of a single curve for the NLLA observables.", "The band represents the uncertainty that comes into play after using the BLM prescription since there is no unambiguous way to apply it.", "The experimental observables we initially proposed are based on the partonic-level average values (with $M,N$ being positive integers) ${\\cal C}_{MN} \\, = \\,\\langle \\cos {\\left(M \\left( \\theta _A - \\theta _J - \\pi \\right)\\right)}\\cos {\\left(N \\left( \\theta _J - \\theta _B - \\pi \\right)\\right)}\\rangle && \\\\&&\\hspace{-256.0748pt} = \\frac{\\int _0^{2 \\pi } d \\theta _A d \\theta _B d \\theta _J \\cos {\\left(M \\left( \\theta _A - \\theta _J - \\pi \\right)\\right)} \\cos {\\left(N \\left( \\theta _J - \\theta _B - \\pi \\right)\\right)}d\\sigma ^{3-{\\rm jet}} }{\\int _0^{2 \\pi } d \\theta _A d \\theta _B d \\theta _Jd\\sigma ^{3-{\\rm jet}} },\\nonumber $ whereas, in order to provide testable predictions for the current and future experimental data, we introduce the hadronic-level values $C_{MN}$ after integrating ${\\cal C}_{M,N}$ over the momenta of the tagged jets, as we will see in the following sections.", "From a more theoretical perspective, it is important to have as good as possible perturbative stability in our predictions (see [18] for a related discussion).", "This can be achieved by removing the contribution stemming from the zero conformal spin, which corresponds to the index $n=0$ in Eqs.", "(REF ) and (REF ).", "We, therefore, introduce the ratios $R_{PQ}^{MN} \\, = \\, \\frac{C_{MN}}{C_{PQ}}$ which are free from any $n=0$ dependence, as long as $M, N , P, Q >0$ .", "The postulate that Eq.", "generally describes observables with good perturbative stability is under scrutiny in Sections 3, 4 and 5 where we compare LLA and NLLA results.", "Before we proceed to our numerical results in the next sections, we should give a few details with regard to our numerical computations.", "From all the possible ratios, we have chosen to study the following three: $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ .", "These are enough to have an adequate view of how the generic $R^{MN}_{PQ}$ behaves.", "We computed $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ in all cases almost exclusively in Fortran whereas Mathematica was used mainly for cross-checks.", "The NLO MSTW 2008 PDF sets [51] were used and for the strong coupling $\\alpha _s$ we chose a two-loop running coupling setup with $\\alpha _s\\left(M_Z\\right)=0.11707$ and five quark flavours.", "We made extensive use of the integration routine !Vegas!", "[52] as implemented in the !Cuba!", "library [53], [54].", "Furthermore, we used the !Quadpack!", "library [55] and a slightly modified version of the !Psi!", "[56] routine.", "Figure: YY-dependence of the LLA and NLLAR 22 12 R^{12}_{22}, R 33 12 R^{12}_{33} and R 33 22 R^{22}_{33} at s=7\\sqrt{s} = 7 TeV with y J y_J fixed(left) and the relative NLLA to LLA corrections (right).Figure: YY-dependence of the LLA and NLLAR 22 12 R^{12}_{22}, R 33 12 R^{12}_{33} and R 33 22 R^{22}_{33} at s=13\\sqrt{s} = 13 TeV with y J y_J fixed(left) and the relative NLLA to LLA corrections (right)." ], [ "$R^{12}_{22}$ , {{formula:60613685-3356-457f-934b-e1f1939a1d40}} and {{formula:61e74891-cb2c-40ac-acb0-5e7beabd3999}} with the central jet fixed in rapidity", "In this section, we will present results for three generalised ratios, $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ , assuming that the central jet is fixed in rapidity at $y_J = (Y_A+Y_B)/2$ (see Fig.", "REF ).", "In particular, $&C_{MN} =\\nonumber \\\\&\\int _{Y_A^{\\rm min}}^{Y_A^{\\rm max}} \\hspace{-7.11317pt} dY_A\\int _{Y_B^{\\rm min}}^{Y_B^{\\rm max}} \\hspace{-7.11317pt} dY_B\\int _{k_A^{\\rm min}}^{k_A^{\\rm max}} \\hspace{-7.11317pt} dk_A\\int _{k_B^{\\rm min}}^{k_B^{\\rm max}} \\hspace{-7.11317pt} dk_B\\int _{k_J^{\\rm min}}^{k_J^{\\rm max}} \\hspace{-7.11317pt} dk_J\\delta \\left(Y_A - Y_B - Y\\right) {\\cal C}_{MN},$ where the forward jet rapidity is taken in the range delimited by $0 < Y_A < 4.7$ , the backward jet rapidity in the range $-4.7 < Y_B < 0$ , while their difference $Y \\equiv Y_A - Y_B$ is kept fixed at definite values in the range $5.5 < Y < 9$ .", "We can now study the ratios $R_{PQ}^{MN}(Y)$ in Eq.", "() as functions of the rapidity difference Y between the most forward and the most backward jets for a set of characteristic values of $M, N, P, Q$ and for two different center-of-mass energies: $\\sqrt{s} = 7$ and $\\sqrt{s} = 13$ TeV.", "Since we are integrating over $k_A$ and $k_B$ , we have the opportunity to impose either symmetric or asymmetric kinematic cuts, as it has been previously done in Mueller-Navelet studies.", "Here, and for the rest of the paper, we choose to study the asymmetric cut which presents certain advantages over the symmetric one (see Refs.", "[22], [34]).", "To be more precise, we set $k_A^{\\rm min} = 35$ GeV, $k_B^{\\rm min} = 50$ GeV, $k_A^{\\rm max} = k_B^{\\rm max} = 60$ GeV throughout the paper.", "In order to be as close as possible to the characteristic rapidity ordering of the multi-Regge kinematics, we set the value of the central jet rapidity such that it is equidistant to $Y_A$ and $Y_B$ by imposing the condition $y_J = (Y_A + Y_B)/2$ .", "Moreover, since the tagging of a central jet permits us to extract more exclusive information from our observables, we allow three possibilities for the transverse momentum $k_J$ , that is, $20\\, \\mathrm {GeV} < k_J < 35\\, \\mathrm {GeV}$ (bin-1), $35 \\,\\mathrm {GeV} < k_J < 60\\, \\mathrm {GeV}$ (bin-2) and $60\\, \\mathrm {GeV} < k_J < 120\\, \\mathrm {GeV}$ (bin-3).", "Keeping in mind that the forward/backward jets have transverse momenta in the range $\\left[35 \\,\\mathrm {GeV}, 60 \\,\\mathrm {GeV}\\right]$ , restricting the value of $k_J$ within these three bins allows us to see how the ratio $R_{PQ}^{MN}(Y)$ changes its behaviour depending on the relative size of the central jet momentum when compared to the forward/backward ones.", "Throughout the paper, we will keep the same setup regarding bin-1, bin-2 and bin-3 which roughly correspond to the cases of $k_J$ being `smaller' than, `similar' to and `larger' than $k_A$ , $k_B$ , respectively.", "Finally, apart from the functional dependence of the ratios on $Y$ we will also show the relative corrections when we go from LLA to NLLA.", "To be more precise, we define $\\delta x(\\%) = \\left(\\text{res}^{\\rm (LLA)} - \\frac{\\text{res}^{\\rm (BLM-1)}+\\text{res}^{\\rm (BLM-2)}}{2}\\right) \\frac{1}{ \\text{res}^{\\rm (LLA)}}\\,.$ $\\text{res}^{\\rm (BLM-1)}$ is the BLM NLLA result for $\\mu _R=\\mu _R^{\\rm BLM}$ only in the gluon Green function while the cubed term of the strong coupling in Eq.", "REF actually reads $\\bar{\\alpha }_s^3 = \\bar{\\alpha }_s^3(\\sqrt{k_{A}k_{B}})$ ).", "$\\text{res}^{\\rm (BLM-2)}$ is the BLM NLLA result for $\\mu _R=\\mu _R^{\\rm BLM}$ everywhere in Eq.", "REF , therefore, $\\bar{\\alpha }_s^3 = \\bar{\\alpha }_s^3(\\mu _R^{\\rm BLM})$ , as was previously discussed in Section 2.", "In the following, we present our results for $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ , with $y_J = (Y_A+Y_B)/2$ , collectively in Fig.", "REF ($\\sqrt{s} = 7$ TeV) and Fig.", "REF ($\\sqrt{s} = 13$ TeV), In the left column we are showing plots for $R^{MN}_{PQ}(Y)$ whereas to the right we are showing the corresponding $\\delta x(\\%)$ between LLA and NLLA corrections.", "The LLA results are represented with dashed lines whereas the NLLA ones with a continuous band.", "The boundaries of the band are the two different curves we obtain by the two different approaches in applying the BLM prescription.", "Since there is no definite way to choose one in favour of the other, we allow for any possible value in between and hence we end up with a band.", "In many cases, as we will see in the following, the two boundaries are so close that the band almost degenerates into a single curve.", "The red curve (band) corresponds to $k_J$ bounded in bin-1, the green curve (band) to $k_J$ bounded in bin-2 and finally the blue curve (band) to $k_J$ bounded in bin-3.", "For the $\\delta x(\\%)$ plots we only have three curves, one for each of the three different bins of $k_J$ .", "A first observation from inspecting Figs.", "REF and REF is that the dependence of the different observables on the rapidity difference between $k_A$ and $k_B$ is rather smooth.", "$R^{12}_{22}$ (top row in Figs.", "REF and REF ) at $\\sqrt{s} = 7$ TeV and for $k_J$ in bin-1 and bin-3 exhibits an almost linear behaviour with $Y$ both at LLA and NLLA, whereas at $\\sqrt{s} = 13$ TeV the linear behaviour is extended also for $k_J$ in bin-2.", "The difference between the NLLA BLM-1 and BLM-2 values is small, to the point that the blue and the red bands collapse into a single line which in addition lies very close to the LLA results.", "When $k_J$ is restricted in bin-2 (green curve/band), the uncertainty from applying the BLM prescription in two different ways seems to be larger.", "The relative NLLA corrections at both colliding energies are very modest ranging from close to $1\\%$ for $k_J$ in bin-3 to less than $10\\%$ for $k_J$ in the other two bins.", "$R^{12}_{33}$ (middle row in Figs.", "REF and REF ) compared to $R^{12}_{22}$ , shows a larger difference between BLM-1 and BLM-2 values for $k_J$ in bin-1 and bin-2.", "The `green' corrections lower the LLA estimate whereas the `red' ones make the corresponding LLA estimate less negative.", "The corrections are generally below $20\\%$ , in particular, `blue' $\\sim 5\\%$ , `red' $\\sim 10\\%$ and `green' $\\sim 20\\%$ .", "Finally, $R^{22}_{33}$ (bottom row in Figs.", "REF and REF ) also shows a larger difference between BLM-1 and BLM-2 values for $k_J$ in bin-1 and less so for $k_J$ in bin-2.", "Here, the `red' corrections lower the LLA estimate whereas the `green' ones make the corresponding LLA estimate less negative.", "The corrections are smaller than the ones for $R^{12}_{33}$ and somehow larger than the corrections for $R^{12}_{22}$ , specifically, `blue' $\\sim 5\\%$ , `red' $\\sim 5\\%$ and `green' $\\sim 15\\%$ .", "Noticeably, while for $R^{12}_{22}$ and $R^{12}_{33}$ the corrections are very similar at $\\sqrt{s} = 7$ and $\\sqrt{s} = 13$ TeV, the `green' $R^{22}_{33}$ receives larger corrections at $\\sqrt{s} = 7$ TeV.", "One important conclusion we would like to draw after comparing Figs.", "REF and REF is that, in general, for most of the observables there are no striking changes when we increase the colliding energy from 7 to 13 TeV.", "This indicates that a sort of asymptotic regime has been approached for the kinematical configurations included in our analysis.", "It also tells us that our observables are really as insensitive as possible to effects which have their origin outside the BFKL dynamics and which normally cannot be isolated (e.g.", "influence from the PDFs) with a possible exclusion at the higher end of the plots, when $Y \\sim 8.5-9$ .", "There, some of the observables and by that we mean the `red', `green' or `blue' cases of $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ , exhibit a more curved rather than linear behaviour with $Y$ at $\\sqrt{s} = 7$ TeV." ], [ "$R^{12}_{22}$ , {{formula:563159cb-569e-4fef-870c-60502f4db602}} and {{formula:00dfe655-ed24-4643-9719-76fdc944bc78}} after integration over a central jet rapidity bin", "In this section, everything is kept the same as in Section 3 with the exemption of the allowed values for $y_J$ (see Fig.", "REF ).", "While in the previous section $y_J = (Y_A+Y_B)/2$ , here $y_J$ is not anymore dependent on the rapidity difference between the outermost jets, $Y$ , and is allowed to take values in a rapidity bin around $y_J = 0$ .", "In particular, $-0.5 < y_J < 0.5$ , which in turn means that an additional integration over $y_J$ needs to be considered in Eq.", "REF with $y_J^{\\rm min} = -0.5$ and $y_J^{\\rm max} = 0.5$ : $&C_{MN}^{\\text{integ}} =\\nonumber \\\\&\\int _{y_J^{\\rm min}}^{y_J^{\\rm max}} \\hspace{-7.11317pt} dy_J\\int _{Y_A^{\\rm min}}^{Y_A^{\\rm max}} \\hspace{-7.11317pt} dY_A\\int _{Y_B^{\\rm min}}^{Y_B^{\\rm max}} \\hspace{-7.11317pt} dY_B\\int _{k_A^{\\rm min}}^{k_A^{\\rm max}} \\hspace{-7.11317pt} dk_A\\int _{k_B^{\\rm min}}^{k_B^{\\rm max}} \\hspace{-7.11317pt} dk_B\\int _{k_J^{\\rm min}}^{k_J^{\\rm max}} \\hspace{-7.11317pt} dk_J\\delta \\left(Y_A - Y_B - Y\\right) {\\cal C}_{MN},$ With a slight abuse of notation, we will keep denoting our observables $R_{PQ}^{MN}$ : $R_{PQ}^{MN} \\, = \\, \\frac{C_{MN}^{\\text{integ}}}{C_{PQ}^{\\text{integ}}}\\,.$ Therefore, in Figs.", "REF and REF we still have $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ although here they do contain the extra integration over $y_J$ .", "We notice immediately that Fig.", "REF is very similar to the integrated over $y_J$ observables in Fig.", "REF and the same holds for Figs.", "REF and REF .", "Therefore, we will not discuss here the individual behaviours of $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ with $Y$ , neither the $\\delta x(\\%)$ corrections, since this would only mean to repeat the discussion of the previous section.", "We would like only to note that the striking similarity between Fig.", "REF and Fig.", "REF and between Fig.", "REF and Fig.", "REF was to be expected if we remember that the partonic-level quantities $\\mathcal {R}_{PQ}^{MN}$ do not change noticeably if we vary the position in rapidity of the central jet, as long as the position remains “sufficiently\" central (see Ref. [36]).", "This property is very important and we will discuss it more in the next section.", "Here, we should stress that the observables as presented in this section can be readily compared to experimental data." ], [ "$R^{12}_{22}$ , {{formula:60b768c5-738e-44cd-8dc1-4fcca3aa233d}} and {{formula:3d568c41-7116-41ae-8b89-05da42eb4420}} after integration over a forward, backward and central rapidity bin", "In this section, we present an alternative kinematical configuration (see Fig.", "REF ) for the generalised ratios $R_{PQ}^{MN}$ .", "We do this for two reasons.", "Firstly, to offer a different setup for which the comparison between theoretical predictions and experimental data might be easier, compared to the previous section.", "Secondly, to demonstrate that the generalised ratios do capture the Bethe-Salpeter characteristics of the BFKL radiation.", "The latter needs a detailed explanation.", "Let us assume that we have a gluonic ladder exchanged in the $t$ -channel between a forward jet (at rapidity $Y_A$ ) and a backward jet (at rapidity $Y_B$ ) accounting for minijet activity between the two jets.", "By gluonic ladder here we mean the gluon Green function $\\varphi \\left(\\vec{p}_A,\\vec{p}_B,Y_A - Y_B\\right) $ , where $\\vec{p}_A$ and $\\vec{p}_B$ are the reggeized momenta connected to the forward and backward jet vertex respectively.", "It is known that the following relation holds for the gluon Green function: $\\hspace{-8.5359pt}\\varphi \\left(\\vec{p}_A,\\vec{p}_B,Y_A - Y_B\\right) &=&\\int d^2 \\vec{k} \\,\\varphi \\left(\\vec{p}_A,\\vec{k},Y_A - y\\right)\\varphi \\left(\\vec{k},\\vec{p}_B,y - Y_B\\right).$ In other words, one may `cut' the gluonic ladder at any rapidity $y$ between $Y_A$ and $Y_B$ and then integrate over the reggeized momentum $\\vec{k}$ that flows in the $t$ -channel, to recover the initial ladder.", "Which value of $y$ one chooses to `cut' the ladder at is irrelevant.", "Therefore, observables directly connected to a realisation of the r.h.s of Eq.", "REF should display this $y$ -independence.", "In our study actually, we have a very similar picture as the one described in the r.h.s of Eq.", "REF .", "The additional element is that we do not only `cut' the gluonic ladder but we also `insert' a jet vertex for the central jet.", "This means that the $y$ -independence we discussed above should be present in one form or another.", "To be precise, we do see the $y$ -independence behaviour but now we have to consider the additional constraint that $y$ cannot take any extreme values, that is, it cannot be close to $Y_A$ or $Y_B$ .", "For a more detailed discussion of Eq.", "REF , we refer the reader to Appendix A, here we will proceed to present our numerical results.", "The kinematic setup now is different than in the previous sections.", "We allow $Y_A$ and $Y_B$ to take values such that $(Y_A^{\\text{min}} = 3) < Y_A < (Y_A^{\\text{max}} = 4.7)$ and $(Y_B^{\\text{min}} = -4.7) < Y_B < (Y_B^{\\text{max}} = -3)$ .", "Moreover, we allow for the rapidity of the central jet to take values in five distinct rapidity bins of unit width, that is, $y_i-0.5 < y_J<y_i+0.5$ , with $y_i = \\lbrace -1, -0.5, 0, 0.5, 1\\rbrace $ and we define the coefficients $C_{MN}^{\\rm integ}(y_i)$ as function of $y_i$ : $&C_{MN}^{\\rm integ}(y_i) =\\nonumber \\\\&\\int _{y_i-0.5}^{y_i+0.5} \\hspace{-7.11317pt} dy_J\\int _{Y_A^{\\rm min}}^{Y_A^{\\rm max}} \\hspace{-7.11317pt} dY_A\\int _{Y_B^{\\rm min}}^{Y_B^{\\rm max}} \\hspace{-7.11317pt} dY_B\\int _{k_A^{\\rm min}}^{k_A^{\\rm max}} \\hspace{-7.11317pt} dk_A\\int _{k_B^{\\rm min}}^{k_B^{\\rm max}} \\hspace{-7.11317pt} dk_B\\int _{k_J^{\\rm min}}^{k_J^{\\rm max}} \\hspace{-7.11317pt} dk_J\\,\\,{\\mathcal {C}}_{MN}.$ Again, keeping our notation with regard to the ratios uniform, we continue denoting our observables by $R_{PQ}^{MN}$ but now the ratios are functions of $y_i$ instead of $Y$ : $R_{PQ}^{MN}(y_i) \\, = \\, \\frac{C_{MN}^{\\text{integ}}(y_i)}{C_{PQ}^{\\text{integ}}(y_i)}\\,.$ We present our results in Figs.", "REF and REF .", "We see that indeed, the $y_i$ -dependence of the three ratios is very weak.", "Moreover, the similarity between the $\\sqrt{s} = 7$ TeV and $\\sqrt{s} = 13$ TeV plots is more striking that in the previous sections.", "The relative NLLA to LLA corrections seem to be slightly larger here than in the previous sections.", "We would like to stress once more that the results in this section are readily comparable to the experimental data once the same cuts are applied in the experimental analysis.", "Figure: A primitive lego plotdepicting a three-jet event similar to Fig. .", "Here, however,the rapidity of the central jet can take any value in the distinct ranges y i -0.5<y J <y i +0.5y_i-0.5 < y_J < y_i+0.5, wherey i y_i is the central value of the rapidity bin with y i ={-1,-0.5,0,0.5,1}y_i = \\lbrace -1, -0.5, 0, 0.5, 1\\rbrace .In this figure, y i =-1y_i = -1.Moreover, Y=Y A -Y B Y = Y_A - Y_B is not anymore fixed.", "Instead, the forward jet has a rapidity restricted inthe red bin whereas the backward jet in the yellow bin.Figure: y i y_i-dependence of the LLA and NLLAR 22 12 R^{12}_{22}, R 33 12 R^{12}_{33} and R 33 22 R^{22}_{33}at s=7\\sqrt{s} = 7 TeV (left) and the relative NLLA to LLA corrections (right).Figure: y i y_i-dependence of the LLA and NLLAR 22 12 R^{12}_{22}, R 33 12 R^{12}_{33} and R 33 22 R^{22}_{33}at s=13\\sqrt{s} = 13 TeV (left) and the relative NLLA to LLA corrections (right)." ], [ "Summary & Outlook", "We have presented a first complete phenomenological study beyond the LLA of inclusive three-jet production at the LHC within the BFKL framework, focussing on azimuthal-angle dependent observables.", "We considered two colliding energies, $\\sqrt{s} = 7, 13$ TeV and an asymmetric kinematic cut with respect to the transverse momentum of the forward ($k_A$ ) and backward ($k_B$ ) jets.", "In addition, we have chosen to consider an extra condition regarding the value of the transverse momentum $k_J$ of the central jet, dividing the allowed region for $k_J$ into three sub-regions: $k_J$ smaller than $k_{A,B}$ , $k_J$ similar to $k_{A,B}$ and $k_J$ larger than $k_{A,B}$ .", "For a proper study at full NLLA, one needs to consider the NLO jet vertices and the NLLA gluon Green functions.", "We have argued that we expect the latter to be of higher relevance and we proceed to calculate them using the BLM prescription which has been successful in previous phenomenological analyses.", "We have shown how our observables $R^{12}_{22}$ , $R^{12}_{33}$ and $R^{22}_{33}$ change when we vary the rapidity difference Y between $k_A$ and $k_B$ from 5.5 to 9 units for a fixed $y_J$ and from 6.5 to 9 units for $-0.5 < y_J < 0.5$ .", "We have presented both the LLA and NLLA results along with plots that show the relative size of the NLLA corrections compared to the LLA ones.", "We have also presented an alternative kinematical setup where we allow for $Y_A$ and $Y_B$ to take values such that $3 < Y_A < 4.7$ and $-4.7 < Y_B < -3$ , while the rapidity of the central jet takes values in five distinct rapidity bins of unit width, that is, $y_i-0.5 < y_J<y_i+0.5$ , with $y_i = \\lbrace -1, -0.5, 0, 0.5, 1\\rbrace $ .", "In this alternative setup, we presented our results for $R_{12}^{22}$ , $R_{12}^{33}$ and $R_{22}^{33}$ as functions of $y_i$ .", "The general conclusion is that the NLLA corrections are moderate and our proposed observables exhibit a good perturbative stability.", "Furthermore, we see that for a wide range of rapidities, the changes we notice when going from 7 TeV to 13 TeV are small which makes us confident that these generalised ratios pinpoint the crucial characteristics of the BFKL dynamics regarding the azimuthal behavior of the hard jets in inclusive three-jet production.", "It will be very interesting to compare with possible predictions for these observables from fixed order analyses as well as from the BFKL inspired Monte Carlo !BFKLex!", "[57], [58], [59], [60], [61], [62], [63], [64].", "Predictions from general-purpose Monte Carlos tools should also be welcome.", "It would be extremely interesting to pursue an experimental analysis for these observables using LHC data.", "Acknowledgements GC acknowledges support from the MICINN, Spain, under contract FPA2013-44773-P. ASV acknowledges support from Spanish Government (MICINN (FPA2010-17747,FPA2012-32828)) and, together with FC and FGC, to the Spanish MINECO Centro de Excelencia Severo Ochoa Programme (SEV-2012-0249).", "DGG is supported with a fellowship of the international programme \"La Caixa-Severo Ochoa\".", "FGC thanks the Instituto de Física Teórica (IFT UAM-CSIC) in Madrid for warm hospitality." ], [ "$y_J$ independent integrated distributions", "We show now how Eq.", "20 is fulfilled in our normalisations.", "We introduce the notation $t = \\ln {k^2}$ to write the gluon Green function in the form $\\varphi \\left(t_A,t_B,\\theta _A,\\theta _B,Y\\right) &=& \\frac{e^{-\\frac{t_A+t_B}{2}} }{\\pi ^2}\\sum _{n=-\\infty }^\\infty e^{i n \\left(\\theta _A - \\theta _B\\right)} \\nonumber \\\\& & \\int _0^\\infty d \\nu \\cos {\\left(\\nu \\left(t_A-t_B\\right)\\right)} \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\nu \\right) Y}.$ Making use of $d k = \\frac{1}{2} e^{\\frac{t}{2}} dt$ and $k \\, dk \\, d \\theta = \\frac{e^t}{2} d \\theta $ we then want to show that $\\varphi \\left(t_A,t_B,\\theta _A,\\theta _B,Y\\right) & = &\\int _0^{2 \\pi } d \\theta \\int _{-\\infty }^\\infty dt \\, \\frac{ e^t}{2}\\varphi \\left(t_A,t,\\theta _A,\\theta ,y\\right)\\varphi \\left(t,t_B,\\theta ,\\theta _B,Y-y\\right) \\nonumber \\\\&&\\hspace{-99.58464pt} = \\int _0^{2 \\pi } d \\theta \\int _{-\\infty }^\\infty dt \\, \\frac{ e^t}{2} \\frac{e^{-\\frac{t_A+t}{2}} }{\\pi ^2}\\sum _{m=-\\infty }^\\infty e^{i m \\left(\\theta _A - \\theta \\right)}\\int _0^\\infty d \\nu \\cos {\\left(\\nu \\left(t_A-t\\right)\\right)} \\, e^{\\bar{\\alpha }_s \\chi _{|m|} \\left(\\nu \\right) y} \\nonumber \\\\&&\\hspace{-85.35826pt}\\frac{e^{-\\frac{t+t_B}{2}} }{\\pi ^2}\\sum _{n=-\\infty }^\\infty e^{i n \\left(\\theta - \\theta _B\\right)}\\int _0^\\infty d \\mu \\cos {\\left(\\mu \\left(t-t_B\\right)\\right)} \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\mu \\right) (Y-y)}.$ The integration over $\\theta $ generates a $\\delta _m^n$ contribution: $\\varphi \\left(t_A,t_B,\\theta _A,\\theta _B,Y\\right) &=&\\nonumber \\\\&&\\hspace{-99.58464pt} \\frac{e^{-\\frac{t_A+t_B}{2}} }{\\pi ^3} \\sum _{n=-\\infty }^\\infty e^{i n \\left(\\theta _A - \\theta _B\\right)}\\int _0^\\infty d \\nu \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\nu \\right) y} \\nonumber \\\\&&\\hspace{-99.58464pt}\\int _0^\\infty d \\mu \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\mu \\right) (Y-y)}\\int _{-\\infty }^\\infty dt \\cos {\\left(\\nu \\left(t_A-t\\right)\\right)} \\cos {\\left(\\mu \\left(t-t_B\\right)\\right)}.$ It can be shown that $\\int _{-\\infty }^\\infty dt \\cos {\\left(\\nu \\left(t_A-t\\right)\\right)} \\cos {\\left(\\mu \\left(t-t_B\\right)\\right)} &=& \\nonumber \\\\&&\\hspace{-170.71652pt} \\pi \\Bigg ( \\cos { (\\nu t_A - \\mu t_B)} \\delta (\\nu -\\mu )+ \\cos { ( \\nu t_A+ \\mu t_B)} \\delta (\\nu +\\mu )\\Bigg ),$ which can be used to write Eq.", "25 as $\\varphi \\left(t_A,t_B,\\theta _A,\\theta _B,Y\\right) &=&\\nonumber \\\\&&\\hspace{-99.58464pt} \\frac{e^{-\\frac{t_A+t_B}{2}} }{2 \\pi ^2} \\sum _{n=-\\infty }^\\infty e^{i n \\left(\\theta _A - \\theta _B\\right)}\\int _{-\\infty }^\\infty d \\nu \\int _0^\\infty d \\mu \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\mu \\right) (Y-y)} e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\nu \\right) y} \\nonumber \\\\&&\\hspace{-56.9055pt}\\Bigg ( \\cos { (\\nu t_A - \\mu t_B)} \\delta (\\nu -\\mu )+ \\cos { ( \\nu t_A+ \\mu t_B)} \\delta (\\nu +\\mu )\\Bigg ),$ and, finally, $\\varphi \\left(t_A,t_B,\\theta _A,\\theta _B,Y\\right) &=&\\nonumber \\\\&&\\hspace{-99.58464pt} \\frac{e^{-\\frac{t_A+t_B}{2}} }{\\pi ^2} \\sum _{n=-\\infty }^\\infty e^{i n \\left(\\theta _A - \\theta _B\\right)}\\int _0^\\infty d \\mu \\, e^{\\bar{\\alpha }_s \\chi _{|n|} \\left(\\mu \\right) Y}\\cos { (\\mu (t_A -t_B))},$ which is the same as our initial representation for $\\varphi $ in Eq. 23.", "The relation in Eq.", "24 is remarkable because it holds for any rapidity $y$ ." ] ]
1612.05428
[ [ "Observation of the doubly radiative decay $\\eta^{\\prime}\\to\n \\gamma\\gamma\\pi^0$" ], [ "Abstract Based on a sample of $1.31$ billion $J/\\psi$ events collected with the BESIII detector, we report the study of the doubly radiative decay $\\eta^\\prime\\to \\gamma\\gamma\\pi^0$ for the first time, where the $\\eta^\\prime$ meson is produced via the $J/\\psi\\to \\gamma\\eta^\\prime$ decay.", "The branching fraction of $\\eta^\\prime\\to \\gamma\\gamma\\pi^0$ inclusive decay is measured to be ${\\cal B}(\\eta^\\prime\\to \\gamma\\gamma\\pi^0)_{\\text{Incl.", "}}$ = $(3.20\\pm0.07\\mbox{(stat)}\\pm0.23\\mbox{(sys)})\\times 10^{-3}$, while the branching fractions of the dominant process $\\eta^\\prime\\rightarrow\\gamma\\omega$ and the non-resonant component are determined to be ${\\cal B}(\\eta^\\prime\\to \\gamma\\omega)\\times {\\cal B}(\\omega\\to \\gamma\\pi^0) = (23.7 \\pm1.4\\mbox{(stat)}\\pm1.8\\mbox{(sys)})\\times 10^{-4}$ and ${\\cal B}(\\eta^\\prime\\to \\gamma\\gamma\\pi^0)_{\\text{NR}} = (6.16\\pm0.64\\mbox{(stat)} \\pm0.67\\mbox{(sys)})\\times 10^{-4}$, respectively.", "In addition, the $M^2_{\\gamma\\gamma}$-dependent partial widths of the inclusive decay are also presented." ], [ "Introduction", "The $\\eta ^\\prime $ meson provides a unique stage for understanding the distinct symmetry-breaking mechanisms present in low-energy quantum chromodynamics (QCD) [1], [2], [3], [4], [5] and its decays play an important role in exploring the effective theory of QCD at low energy [6].", "Recently, the doubly radiative decay $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ was studied in the frameworks of the linear $\\sigma $ model (L$\\sigma $ M) and the vector meson dominance (VMD) model [7], [8].", "It has been demonstrated that the contributions from the VMD are dominant.", "Experimentally, only an upper limit of the nonresonant branching fraction of ${\\cal B}(\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0)_{\\text{NR}}<8\\times 10^{-4}$ at the 90% confidence level has been determined by the GAMS-2000 experiment [9].", "In this article, we report the first measurement of the branching fraction of the inclusive $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ decay and the determination of the $M^2_{\\gamma \\gamma }$ dependent partial widths, where $M_{\\gamma \\gamma }$ is the invariant mass of the two radiative photons.", "The inclusive decay is defined as the $\\eta ^\\prime $ decay into the final state $\\gamma \\gamma \\pi ^0$ including all possible intermediate contributions from the $\\rho $ and $\\omega $ mesons below the $\\eta ^\\prime $ mass threshold and the nonresonant contribution from the excited vector meson above the $\\eta ^\\prime $ mass threshold.", "Since the contribution from mesons above the $\\eta ^\\prime $ threshold actually derives from the low-mass tail and looks like a contact term, we call this contribution 'nonresonant'.", "The branching fraction for the nonresonant $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ decay is obtained from a fit to the $\\gamma \\pi ^0$ invariant mass distribution by excluding the coherent contributions from the $\\rho $ and $\\omega $ intermediate states.", "The measurement of the $M^2_{\\gamma \\gamma }$ dependent partial widths will provide direct inputs to the theoretical calculations on the transition form factors of $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ and improve the theoretical understanding of the $\\eta ^\\prime $ decay mechanisms." ], [ "Experimental Details", "The source of $\\eta ^\\prime $ mesons is the radiative $J/\\psi \\rightarrow \\gamma \\eta ^\\prime $ decay in a sample of $1.31\\times 10^{9}$ $J/\\psi $ events [10], [11] collected by the BESIII detector.", "Details on the features and capabilities of the BESIII detector can be found in Ref. [12].", "The response of the BESIII detector is modeled with a Monte Carlo (MC) simulation based on geant4 [13].", "The program evtgen [14] is used to generate a $J/\\psi \\rightarrow \\gamma \\eta ^\\prime $ MC sample with an angular distribution of $1 + \\cos ^2\\theta _\\gamma $ , where $\\theta _\\gamma $ is the angle of the radiative photon relative to the positron beam direction in the $J/\\psi $ rest frame.", "The decays $\\eta ^\\prime \\rightarrow \\gamma \\omega (\\rho )$ , $\\omega (\\rho )\\rightarrow \\gamma \\pi ^0$ are generated using the helicity amplitude formalism.", "For the nonresonant $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ decay, the VMD model [7], [8] is used to generate the MC sample with $\\rho (1450)$ or $\\omega (1650)$ exchange.", "Inclusive $J/\\psi $ decays are generated with kkmc [15] generator; the known $J/\\psi $ decay modes are generated by evtgen [14] with branching fractions setting at Particle Data Group (PDG) world average values [16]; the remaining unknown decays are generated with lundcharm [17]." ], [ "Event Selection and Background Estimation", "Electromagnetic showers are reconstructed from clusters of energy deposits in the electromagnetic calorimeter (EMC).", "The energy deposited in nearby time-of-light (TOF) counters is included to improve the reconstruction efficiency and energy resolution.", "The photon candidate showers must have a minimum energy of 25 MeV in the barrel region ($|\\cos \\theta |<0.80$ ) or 50 MeV in the end cap region ($0.86<|\\cos \\theta |<0.92$ ).", "Showers in the region between the barrel and the end caps are poorly measured and excluded from the analysis.", "In this analysis, only the events without charged particles are subjected to further analysis.", "The average event vertex of each run is assumed as the origin for the selected candidates.", "To select $J/\\psi \\rightarrow \\gamma \\eta ^\\prime $ , $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ $(\\pi ^0\\rightarrow \\gamma \\gamma )$ signal events, only the events with exactly five photon candidates are selected.", "To improve resolution and reduce background, a five-constraint kinematic (5C) fit imposing energy-momentum conservation and a $\\pi ^0$ mass constraint is performed to the $\\gamma \\gamma \\gamma \\pi ^0$ hypothesis, where the $\\pi ^0$ candidate is reconstructed with a pair of photons.", "For events with more than one $\\pi ^0$ candidate, the combination with the smallest $\\chi ^{2}_{5\\mbox{c}}$ is selected.", "Only events with $\\chi ^{2}_{5\\mbox{c}}<30$ are retained.", "The $\\chi ^{2}_{5C}$ distribution is shown in Fig.", "REF with events in the $\\eta ^{\\prime }$ signal region of $|M_{\\gamma \\gamma \\pi ^{0}} - M_{\\eta ^{\\prime }}|<25$  MeV ($M_{\\eta ^{\\prime }}$ is the $\\eta ^\\prime $ nominal mass from PDG [16]).", "In order to suppress the multi-$\\pi ^0$ backgrounds and remove the miscombined $\\pi ^0$ candidates, an event is vetoed if any two of five selected photons (except for the combination for the $\\pi ^0$ candidate) satisfies $|M_{\\gamma \\gamma } - M_{\\pi ^0}|<18$  MeV/c$^2$ , where $M_{\\pi ^0}$ is the $\\pi ^0$ nominal mass.", "After the application of the above requirements, the most energetic photon is taken as the primary photon from the $J/\\psi $ decay, and the remaining two photons and the $\\pi ^0$ are used to reconstruct the $\\eta ^\\prime $ candidates.", "Figure REF shows the $\\gamma \\gamma \\pi ^0$ invariant mass spectrum.", "Figure: Distribution of the χ 5C 2 \\chi ^{2}_{5C} of the 5C kinematic fit for the inclusive η ' \\eta ^{\\prime } decay.", "Dots with error bars aredata; the heavy (black) solid-curve is the sum of signal and expected backgrounds from MC simulations; the light (red) solid-curvesis signal components which are normalized to the fitted yields; the (green) dotted-curve is the class I background; and the (pink)dot-dashed-curve is the class II background.Figure: Results of the fit to M γγπ 0 M_{\\gamma \\gamma \\pi ^0} for the selected inclusive η ' →γγπ 0 \\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0 signal events.The (black) dots with error bars are the data.Detailed MC studies indicate that no peaking background remains after all the selection criteria.", "The sources of backgrounds are divided into two classes.", "Background events of class I are from $J/\\psi \\rightarrow \\gamma \\eta ^\\prime $ with $\\eta ^\\prime $ decaying into final states other than the signal final states.", "These background events accumulate near the lower side of the $\\eta ^\\prime $ signal region and are mainly from $\\eta ^\\prime \\rightarrow \\pi ^0\\pi ^0\\eta $ ($\\eta \\rightarrow \\gamma \\gamma $ ), $\\eta ^\\prime \\rightarrow 3\\pi ^0$ and $\\eta ^\\prime \\rightarrow \\gamma \\gamma $ , as shown as the (green) dotted curve in Fig.", "REF .", "Background events in class II are mainly from $J/\\psi $ decays to final states without $\\eta ^\\prime $ , such as $J/\\psi \\rightarrow \\gamma \\pi ^0\\pi ^0$ and $J/\\psi \\rightarrow \\omega \\eta $ ($\\omega \\rightarrow \\gamma \\pi ^0$ , $\\eta \\rightarrow \\gamma \\gamma $ ) decays, which contribute a smooth distribution under the $\\eta ^\\prime $ signal region as displayed as the (pink) dot-dashed curve in Fig.", "REF .", "Table: Observed η ' \\eta ^\\prime signal yields (N η ' N^{\\eta ^\\prime }) and detection efficiencies (ϵ\\epsilon ) for inclusive η ' →γγπ 0 \\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0, η ' →γω\\eta ^\\prime \\rightarrow \\gamma \\omega (ω→γπ 0 \\omega \\rightarrow \\gamma \\pi ^0), and the nonresonant η ' →γγπ 0 \\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0 decays.", "The measuredbranching fractions c ^c in this work, comparison of values from the PDG  and theoretical predictions are listed.", "Thefirst errors are statistical and the second ones are systematic." ], [ "Signal Yields and Branching Fractions", "A fit to the $\\gamma \\gamma \\pi ^0$ invariant mass distribution is performed to determine the inclusive $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ signal yield.", "The probability density function (PDF) for the signal component is represented by the signal MC shape, which is obtained from the signal MC sample generated with an incoherent mixture of $\\rho $ , $\\omega $ and the nonresonant components according to the fractions obtained in this analysis.", "Both the shape and the yield for the class I background are fixed to the MC simulations and their expected intensities.", "The shape for the class II background is described by a third-order Chebychev polynomial, and the corresponding yield and PDF parameters are left free in the fit to data.", "The fit range is 0.70$-$ 1.10 GeV/c$^2$ .", "Figure REF shows the results of the fit.", "The fit quality assessed with the binned distribution is $\\chi ^2/\\text{n.d.f}=108/95=1.14$ .", "The signal yield and the MC-determined signal efficiency for the inclusive $\\eta ^\\prime $ decay are summarized in Table REF .", "In this analysis, the partial widths can be obtained by studying the efficiency-corrected signal yields for each given $M^2_{\\gamma \\gamma }$ bin $i$ for the inclusive $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ decay.", "The resolution in $M^2_{\\gamma \\gamma }$ is found to be about $5\\times 10^2$  (MeV/c$^2)^2$ from the MC simulation, which is much smaller than $1.0\\times 10^4$  (MeV/c$^2)^2$ , a statistically reasonable bin width, and hence no unfolding is necessary.", "The $\\eta ^\\prime $ signal yield in each $M^2_{\\gamma \\gamma }$ bin is obtained by performing bin-by-bin fits to the $\\gamma \\gamma \\pi ^0$ invariant mass distributions using the fit procedure described above.", "Thus the background-subtracted, efficiency-corrected signal yield can be used to obtain the partial width for each given $M^2_{\\gamma \\gamma }$ interval, where the PDG value is used for the total width of the $\\eta ^{\\prime }$ meson [16].", "The results for $d\\Gamma (\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0)/dM^2_{\\gamma \\gamma }$ in each $M^2_{\\gamma \\gamma }$ interval are listed in Table REF and depicted in Fig.", "REF , where the contributions from each component obtained from the MC simulations are normalized with the yields by fitting to $M_{\\gamma \\pi ^0}$ as displayed in Fig.", "REF .", "Table: Results for dΓ(η ' →γγπ 0 )/dM γγ 2 d\\Gamma (\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0)/dM^2_{\\gamma \\gamma } (in units of keV/(GeV/c 2 ) 2 ^2)^2)for thirteen intervals of M γγ 2 M^2_{\\gamma \\gamma }.", "The first uncertainties are statistical and the second systematic.Figure: Partial width (in keV) versus M γγ 2 M^2_{\\gamma \\gamma } for the inclusive η ' →γγπ 0 \\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0 decay.", "The error includesthe statistic and systematic uncertainties.", "The (blue) histogram is the sum of an incoherent mixture of ρ\\rho -ω\\omega and thenonresonant components from MC simulations; the (back) dotted-curves is ω\\omega -contribution; the (red) dot-dashed-curve isthe ρ\\rho -contribution; and the (green) dashed-curve is the nonresonant contribution.", "All the components are normalized usingthe yields obtained in Fig.", ".Figure: Distribution of the invariant mass M γπ 0 M_{\\gamma \\pi ^0} and fit results in the η ' \\eta ^\\prime mass region.", "The points with errorbars are data; the (black) dotted-curve is from the ω\\omega -contribution; the (red) long dashed-curve is from the ρ\\rho -contribution;the (blue) short dashed-curve is the contribution of ρ\\rho -ω\\omega interference; the (green) long dashed curve is the nonresonance;the (pink) histogram is from the class II background; the (black) short dot-dashed curve is the combinatorial backgrounds ofη ' →γω\\eta ^\\prime \\rightarrow \\gamma \\omega , γρ\\gamma \\rho .", "The (blue) solid line shows the total fit function.Assuming that the inclusive decay $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ can be attributed to the vector mesons $\\rho $ and $\\omega $ and the nonresonant contribution, we apply a fit to the $\\gamma \\pi ^0$ invariant mass to determine the branching fraction for the nonresonant $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ decay using the $\\eta ^\\prime $ signal events with $|M_{\\gamma \\gamma \\pi ^0} - m_{\\eta ^\\prime }|<25$  MeV/c$^{2}$ .", "In the fit, the $\\rho $ -$\\omega $ interference is considered, but possible interference between the $\\omega $ ($\\rho $ ) and the nonresonant process is neglected.", "To validate our fit, we also determine the product branching fraction for the decay chain $\\eta ^\\prime \\rightarrow \\gamma \\omega $ , $\\omega \\rightarrow \\gamma \\pi ^0$ .", "Figure REF shows the $M_{\\gamma \\pi ^0}$ distribution.", "Since the doubly radiative photons are indistinguishable, two entries are filled into the histogram for each event.", "For the PDF of the coherent $\\omega $ and $\\rho $ produced in $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ , we use $[\\varepsilon (M_{\\gamma \\pi ^0})\\times E^3_{\\gamma ^{\\eta ^\\prime }}\\times E^3_{\\gamma ^{\\omega (\\rho )}}\\times |\\text{BW}_{\\omega }(M_{\\gamma \\pi ^0}) + \\alpha e^{i\\theta }\\text{BW}_{\\rho }(M_{\\gamma \\pi ^0})|^2\\times \\text{B}^2_{\\eta ^\\prime }\\times \\text{B}^2_{\\omega (\\rho )}]\\otimes \\text{G}(0, \\sigma )$ , where $\\varepsilon (M_{\\gamma \\pi ^{0}})$ is the detection efficiency determined by the MC simulations; $E_{\\gamma ^{\\eta ^\\prime (\\omega /\\rho )}}$ is the energy of the transition photon in the rest frame of $\\eta ^\\prime $ ($\\omega /\\rho $ ); $\\text{BW}_{\\omega }(M_{\\gamma \\pi ^0})$ is a relativistic Breit-Wigner (BW) function, and $\\text{BW}_{\\rho }(M_{\\gamma \\pi ^0})$ is a relativistic BW function with mass-dependent width [18].", "The masses and widths of the $\\rho $ and $\\omega $ meson are fixed to their PDG values [16].", "$\\text{B}^2_{\\eta ^\\prime (\\omega /\\rho )}$ is the Blatt-Weisskopf centrifugal barrier factor for the $\\eta ^\\prime $ ($\\omega /\\rho $ ) decay vertex with radius $R=0.75$ fm [19], [20], and $\\text{B}^2_{\\eta ^\\prime (\\omega /\\rho )}$ is used to damp the divergent tail due to the factor $E^3_{\\gamma ^{\\eta ^\\prime (\\omega /\\rho )}}$ .", "The Gaussian function $\\text{G}(0, \\sigma )$ is used to parameterize the detector resolution.", "The combinatorial background is produced by the combination of the $\\pi ^0$ and the photon from the $\\eta ^\\prime $ meson, and its PDF is described with a fixed shape from the MC simulation.", "The ratio of yields between the combinatorial backgrounds and the coherent sum of $\\rho $ -$\\omega $ signals is fixed from the MC simulations.", "The shape of the nonresonant signal $\\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0$ is determined from the MC simulation, and its yield is determined in the fit.", "The background from the class I as discussed above is fixed to the shape and yield of the MC simulation.", "Finally, the shape from the class II background is obtained from the $\\eta ^\\prime $ mass sidebands (738$-$ 788 and 1008$-$ 1058 MeV/c$^{2}$ ), and its normalization is fixed in the fit.", "The $M_{\\gamma \\pi ^0}$ mass range used in the fit is 0.20$-$ 0.92 GeV/c$^2$ .", "In the fit, the interference phase $\\theta $ between the $\\rho $ - and $\\omega $ -components is allowed.", "Due to the low statistics of the $\\rho $ meson contribution, we fix the ratio $\\alpha $ of $\\rho $ and $\\omega $ intensities to the value for the ratio of ${\\cal B}(\\eta ^\\prime \\rightarrow \\gamma \\rho )\\cdot {\\cal B}(\\rho \\rightarrow \\gamma \\pi ^0)$ and ${\\cal B}(\\eta ^\\prime \\rightarrow \\gamma \\omega )\\cdot {\\cal B}(\\omega \\rightarrow \\gamma \\pi ^0)$ from the PDG [16].", "Figure REF shows the results.", "The yields for the vector mesons $\\rho $ , $\\omega $ and their interference are determined to be $(183\\pm 15)$ , $(2340\\pm 141)$ , and $(174\\pm 92)$ , respectively.", "The signal yields and efficiencies as well as the corresponding branching fractions for the $\\eta ^\\prime \\rightarrow \\gamma \\omega (\\omega \\rightarrow \\gamma \\pi ^0)$ and nonresonant decays are summarized in Table REF ." ], [ "Systematic Uncertainties", "The systematic uncertainties on the branching fraction measurements are summarized in Table REF .", "The uncertainty due to the photon reconstruction is determined to be 1% per photon as described in Ref. [21].", "The uncertainties associated with the other selection criteria, kinematic fit with $\\chi ^2_{5C}<30$ , the number of photons equal to 5 and $\\pi ^0$ veto ($|M_{\\gamma \\gamma } - M_{\\pi ^0}|>18$  MeV/c$^2$ ) are studied with the control sample $J/\\psi \\rightarrow \\gamma \\eta ^{\\prime }$ , $\\eta ^{\\prime }\\rightarrow \\gamma \\omega $ , $\\omega \\rightarrow \\gamma \\pi ^{0}$ decay, respectively.", "The systematic error in each of the applied selection criteria is numerically estimated from the ratio of the number of events with and without the corresponding requirement.", "The corresponding resulting efficiency differences between data and MC (2.7%, 0.5%, and 1.9% , respectively) are taken to be representative of the corresponding systematic uncertainties.", "In the fit for the inclusive $\\eta ^\\prime $ decay, the signal shape is fixed to the MC simulation.", "The uncertainty due to the signal shape is considered by convolving a Gaussian function to account for the difference in the mass resolution between data and MC simulation.", "In the fit to the $\\gamma \\pi ^{0}$ distribution, alternative fits with the mass resolution left free in the fit and the radius $R$ in the barrier factor changed from 0.75 fm to 0.35 fm are performed, and the changes of the signal yields are taken as the uncertainty due to the signal shape.", "In the fit to the $M_{\\gamma \\gamma \\pi ^{0}}$ distribution, the signal shape is described with an incoherent sum of contributions from processes involving $\\rho $ and $\\omega $ and nonresonant processes obtained from MC simulation, where the nonresonant process is modeled with the VMD model.", "A fit with an alternative signal model for the different components, i.e.", "a coherent sum for the $\\rho $ -, $\\omega $ -components and a uniform angular distribution in phase space (PHSP) for the nonresonant process, is performed.", "The resultant changes in the branching fractions are taken as the uncertainty related to the signal model.", "An alternate fit to the $M_{\\gamma \\pi ^{0}}$ distribution is performed, where the PDF of the nonresonant decay is extracted from the PHSP MC sample.", "The changes in the measured branching fractions are considered to be the uncertainty arising from the signal model.", "In the fit to the $M_{\\gamma \\pi ^{0}}$ distribution, the uncertainty due to the fixed relative $\\rho $ intensity is evaluated by changing its expectation by one standard deviation.", "An alternative fit in which the ratio of yields between combinatorial backgrounds and the coherent sum of $\\rho -\\omega $ signals is changed by one standard deviation from the MC simulation is performed, and the change observed in the signal yield is assigned as the uncertainty.", "A series of fits using different fit ranges is performed and the maximum change of the branching fraction is taken as a systematic uncertainty.", "The uncertainty due to the class I background is estimated by varying the numbers of expected background events by one standard deviation according to the errors on the branching fraction values in PDG [16].", "The uncertainty due to the class II background is evaluated by changing the order of the Chebychev polynomial from 3 to 4 for the fit to the $\\eta ^{\\prime }$ inclusive decay, and varying the ranges of $\\eta ^{\\prime }$ sidebands for the fit to the $\\gamma \\pi ^{0}$ invariant mass distribution, respectively.", "The number of $J/\\psi $ events is $N_{J/\\psi } = (1310.6\\pm 10.5)\\times 10^{6}$  [10], [11], corresponding to an uncertainty of 0.8%.", "The branching fractions for the $J/\\psi \\rightarrow \\gamma \\eta ^\\prime $ and $\\pi ^0\\rightarrow \\gamma \\gamma $ decays are taken from the PDG [16], and the corresponding uncertainties are taken as a systematic uncertainty.", "The total systematic errors are 7.1%, 7.7%, 10.8% for the inclusive decay, $\\omega $ contribution and nonresonant decay, respectively, as summarized in Table REF .", "Table: Summary of relative systematic uncertainties (%\\%) for the branching fraction measurements.", "Here η Incl. '", "\\eta ^\\prime _{\\text{Incl.", "}}, η ω ' \\eta ^{\\prime }_\\omega and η NR ' \\eta ^{\\prime }_{\\text{NR}} represent the inclusive η ' →γγπ 0 \\eta ^\\prime \\rightarrow \\gamma \\gamma \\pi ^0, η ' →γω(ω→γπ 0 \\eta ^\\prime \\rightarrow \\gamma \\omega (\\omega \\rightarrow \\gamma \\pi ^0) and nonresonant decays, respectively." ], [ "Summary", "In summary, with a sample of $1.31\\times 10^{9}$ $J/\\psi $ events collected with the BESIII detector, the doubly radiative decay $\\eta ^{\\prime }\\rightarrow \\gamma \\gamma \\pi ^{0}$ has been studied.", "The branching fraction of the inclusive decay is measured for the first time to be ${\\cal B}(\\eta ^{\\prime }\\rightarrow \\gamma \\gamma \\pi ^{0})_{\\text{Incl.}}", "= (3.20\\pm 0.07\\mbox{(stat)}\\pm 0.23\\mbox{(sys)})\\times 10^{-3}$ .", "The $M^{2}_{\\gamma \\gamma }$ dependent partial decay widths are also determined.", "In addition, the branching fraction for the nonresonant decay is determined to be ${\\cal B}(\\eta ^{\\prime }\\rightarrow \\gamma \\gamma \\pi ^{0})_{\\text{NR}}$ = $(6.16\\pm 0.64\\mbox{(stat)}\\pm 0.67\\mbox{(sys)})\\times 10^{-4}$ , which agrees with the upper limit measured by the GAMS-2000 experiment [9].", "As a validation of the fit, the product branching fraction with the omega intermediate state involved is obtained to be ${\\cal B}(\\eta ^{\\prime }\\rightarrow \\gamma \\omega )\\cdot {\\cal B}(\\omega \\rightarrow \\gamma \\pi ^{0})$ = $(2.37\\pm 0.14\\mbox{(stat)} \\pm 0.18\\mbox{(sys)})\\times 10^{-3}$ , which is consistent with the PDG value [16].", "These results are useful to test QCD calculations on the transition form factor, and provide valuable inputs to the theoretical understanding of the light meson decay mechanisms." ], [ "Acknowledgments", "The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.", "This work is supported in part by National Key Basic Research Program of China under Contract No.", "2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts No.", "11125525, No.", "11235011, No.", "11322544, No.", "11335008, No.", "11335009, No.", "11425524, No.", "11505111, No.", "11635010, No.", "11675184; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); the Collaborative Innovation Center for Particles and Interactions (CICPI); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts No.", "U1232201, No.", "U1332201, No.", "U1532257, No.", "U1532258; CAS under Contracts No.", "KJCX2-YWN29, No.", "KJCX2-YW-N45; 100 Talents Program of CAS; National 1000 Talents Program of China; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts No.", "Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No.", "530-4CDP03; Ministry of Development of Turkey under Contract No.", "DPT2006K-120470; The Swedish Research Council; U.S. Department of Energy under Contracts No.", "DE-FG02-05ER41374, No.", "DE-SC-0010118, No.", "DE-SC-0010504, No.", "DE-SC-0012069; U.S. National Science Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No.", "R32-2008-000-10155-0." ] ]
1612.05721
[ [ "$\\Xi_c \\gamma \\rightarrow\\Xi^\\prime_c$ transition in lattice QCD" ], [ "Abstract We evaluate the electromagnetic $\\Xi_c \\gamma \\rightarrow\\Xi_c^\\prime$ transition on 2+1 flavor lattices corresponding to a pion mass of $\\sim 156$ MeV.", "We extract the magnetic Sachs and Pauli form factors which give the $\\Xi_c$-$\\Xi_c^\\prime$ transition magnetic moment and the decay rates of $\\Xi_c^\\prime$ baryons.", "We did not find a signal for the magnetic form factor of the neutral transition $\\Xi_c^0 \\gamma \\rightarrow\\Xi_c^{\\prime 0}$, which is suppressed by the U-spin flavor symmetry.", "As a byproduct, we extract the magnetic form factors and the magnetic moments of $\\Xi_c$ and $\\Xi_c^\\prime$ baryons, which give an insight to the dynamics of $u/d$, $s$ and $c$ quarks having masses at different scales." ], [ "Introduction", "Recent experimental observations of all the ground-state heavy baryons as predicted by the quark model [1] makes it timely to study the structure and decays of these hadrons with theoretical methods.", "Among heavy baryons, $\\Xi _c$ and $\\Xi _c^\\prime $ are particularly interesting as the three quarks they are composed of ($u$ , $s$ and $c$ ) have different flavors and masses at quite different scales.", "Therefore, these two baryons provide a good laboratory to study the heavy-quark dynamics.", "The neutral $\\Xi _c^0$ (c[sd]) and the positive state $\\Xi _c^+$ (c[su]) have the quantum numbers $J^P =\\frac{1}{2}^+$ and an anti-symmetric flavor wavefunction under interchange of light quarks.", "In group theoretical formalism they are members of the anti 4-plet ($\\overline{4}$ ) of the SU(4) structure.", "$\\Xi _c$ baryon was first observed in hyperon-beam experiment at CERN [2] and later confirmed by Fermilab [3] and CLEO Collaboration [4].", "The average mass reported by PDG is $m_{\\Xi _c^0} = 2470.99^{+0.30}_{-0.50}$  MeV [1].", "The two other baryons with the same quark content and quantum numbers, $J^P =\\frac{1}{2}^+$ , are $\\Xi ^{\\prime 0}_c$ (c{sd}) and $\\Xi ^{\\prime +}_c$ (c{su}), which are located on the second layer of the sextet SU(4) multiplet.", "These two baryons have symmetric flavor wavefunctions under interchange of light quarks.", "They were first observed by CLEO Collaboration [5] and confirmed recently by BABAR [6] and BELLE experiments [7].", "The average mass reported by PDG is $m_{\\Xi ^{\\prime 0}_c} = 2577.9 \\pm 2.9$ MeV [1].", "The mass difference between $\\Xi _c^{\\prime }$ and $\\Xi _c$ first reported by CLEO as $\\Delta M^+ = (107.8 \\pm 1.7 \\pm 2.5)$ MeV/$c^2$ and $\\Delta M^0 = (107.0 \\pm 1.4 \\pm 2.5)$ MeV/$c^2$  [5] is too small for any strong decay occur.", "Therefore the electromagnetic $\\Xi _c^\\prime \\rightarrow \\Xi _c \\gamma $ is the dominant decay mode.", "Studying this electromagnetic transition between different multiplets of SU(4) may shed light on the QCD mechanism governing the charmed baryons.", "The experimental facilities such as LHCb, PANDA, Belle II, BESIII and J-PARC are expected to give more detailed information about spectroscopy, decays and structure of the charmed baryons.", "Concurrently, recent lattice-QCD studies provide a precise determination of their spectroscopy.", "The ground state charmed baryons have been studied both in quenched [8], [9] and full QCD [10], [11], [12], [13].", "In this work, we evaluate the $\\Xi _c \\gamma \\rightarrow \\Xi ^\\prime _c$ transition in 2+1-flavor lattice QCD.", "As a by-product we compute the electromagnetic form factors of $\\Xi _c$ and $\\Xi _c^\\prime $ baryons.", "We make our simulations with near physical light-quark masses which give a pion mass of $\\sim 156$  MeV.", "Using an appropriate ratio of two- and three-point correlation functions, we extract the electric and magnetic form factors which give the decay width of $\\Xi _c^\\prime $ and the magnetic moments of $\\Xi _c$ and $\\Xi _c^\\prime $ .", "A particular emphasis is made on the $\\Xi ^0_c \\gamma \\rightarrow \\Xi ^{0\\prime }_c$ transition which is suppressed by the U-spin flavor symmetry.", "We find no signal on the lattice for this neutral transition.", "The electromagnetic decays of charmed baryons have been previously studied in heavy-hadron chiral perturbation theory [14], [15], [16], [17], , quark models [19], [20], [21], QCD sum rules [22], bag model [23], heavy-quark symmetry [24] and lattice QCD [25].", "Preliminary results of this work are given in Ref. [26].", "We start with the definition of the electromagnetic vector current.", "We use the following electromagnetic current to study the electromagnetic and transition form factors of $\\Xi _c$ and $\\Xi _c^{\\prime }$ baryons: $J_\\mu = \\frac{2}{3} \\overline{c}(x) \\gamma _\\mu c(x) - \\frac{1}{3} \\overline{s}(x) \\gamma _\\mu s(x) + c_\\ell \\overline{\\ell }(x) \\gamma _\\mu \\ell (x), $ where $\\ell $ denotes the flavor of the light quark (u and d) and $c_\\ell $ is its charge ($2/3$ or $-1/3$ ).", "Using this definition of the vector current, we couple the current to each valence quark in the baryon and compute the electromagnetic transition form factors, which is described by the matrix element $\\langle {\\cal B}^\\prime (p^{\\prime },s^{\\prime })|J_\\mu (x_1)|{\\cal B}(p,s) \\rangle = \\overline{u}(p^{\\prime },s^{\\prime }) \\Big [ \\gamma _\\mu F_1(q^2) - \\frac{\\sigma _{\\mu \\nu } q_\\nu }{m_B+m_{B^{\\prime }}} F_2(q^2) \\Big ] u(p,s).$ Here $F_1(q^2),F_2(q^2)$ are the Dirac and Pauli form factors respectively.", "$u(p^{\\prime },s^{\\prime })$ and $u(p,s)$ are the Dirac spinor of the outgoing and incoming baryons with masses $m_{B^{\\prime }}$ and $m_B$ , and $q_\\mu = p^{\\prime }_\\mu - p_\\mu $ is the transferred four momentum with the momentum of the incoming (outgoing) baryon $p$ ($p^{\\prime }$ ).", "The electric and magnetic Sachs form factors are defined in terms of Dirac and Pauli form factors as follows: $G_E(q^2) = F_1(q^2) - \\frac{q^2}{(m_B+m_{B^{\\prime }})^2} F_2(q^2),$ $G_M(q^2) = F_1(q^2) + F_2(q^2).$ We extract the form factors considering the following two-point correlation functions $\\langle F^{{\\cal B}\\,{\\cal B}}(t;p;\\Gamma _4)\\rangle =\\sum \\limits \\limits _{x}^{ }\\,e^{i\\,p\\,x} \\Gamma ^{\\beta \\,\\alpha }_4\\,\\langle \\Omega |T(\\chi _{\\cal B}^{\\beta }(x) \\overline{\\chi }_{\\cal B}^{\\alpha }(0))|\\Omega \\rangle ,$ and the following three-point correlation functions $\\langle F^{{\\cal B}^{\\prime } J_\\mu {\\cal B}}(t_2,t_1;p^{\\prime },p;\\Gamma ) \\rangle =-i\\,\\sum \\limits \\limits _{x_1,x_2}^{ }\\,e^{-i\\,p\\,x_2}\\,e^{i\\,q\\,x_1} \\Gamma ^{\\beta \\,\\alpha }\\,\\langle \\Omega |T(\\chi ^{\\beta }_{{\\cal B}^\\prime }(x_2) J_\\mu (x_1)\\overline{\\chi }^{\\alpha }_{\\cal B}(0))|\\Omega \\rangle ,$ with ${\\cal B},{\\cal B}^{\\prime }\\equiv \\Xi _c$ or $\\Xi _c^\\prime $ .", "Here $t_1$ is the time when the electromagnetic current is inserted, $t_2$ is the time when the final baryon is annihilated and $\\Gamma _4 = \\frac{1}{2}\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix}$ with $\\Gamma _i = \\frac{1}{2} \\begin{bmatrix}\\sigma _i & 0 \\\\0 & 0\\end{bmatrix}$ .", "The baryon interpolating fields are chosen as c'= 12 abc [ ( Ta (C 5) cb )sc + ( sTa (C 5) cb ) c ], c = 16 abc [2 ( sTa (C 5) b )cc + ( sTa (C 5) cb ) c - ( Ta (C 5) cb ) sc ], where $\\ell =u$ for the charged states $\\Xi _c^+$ , $\\Xi _c^{^{\\prime }+}$ , and $\\ell = d$ for the neutral states $\\Xi _c^0$ , $\\Xi _c^{^{\\prime }0}$ .", "The indices $a,b,c$ denote color and the charge conjugation matrix is defined as $C= \\gamma _4 \\gamma _2$ .", "In the broken flavor SU(3) symmetry, there is a mixing between $\\Xi _c$ and $\\Xi _c^\\prime $ baryons.", "Such mixing has been argued to be negligibly small [27], [28], which was also confirmed by lattice simulations [29].", "The reason for $\\Xi _c$ -$\\Xi _c^\\prime $ mixing being so small is the approximate SU(3) flavor symmetry and the heavy-quark spin symmetry, where the quantum numbers of the light degrees of freedom are exactly conserved.", "Therefore in our calculations we neglect the small mixing effects.", "We use the following ratio to eliminate the normalization factors and to extract the baryon electromagnetic form factors $R(t_2,t_1,p^{\\prime },p,\\Gamma ,\\mu ) = &\\frac{\\langle F^{\\cal B^{\\prime } J_\\mu B}_{}(t_2,t_1;p^{\\prime },p;\\Gamma ) \\rangle }{\\langle F^{\\cal B^{\\prime }B^{\\prime }}_\\text{shwl}(t_2;p^{\\prime };\\Gamma _4) \\rangle }\\\\ &\\times \\Bigg [\\frac{\\langle F^{\\cal BB}_\\text{shsh}(t_2-t_1;p;\\Gamma _4) \\rangle \\langle F^{\\cal B^{\\prime }B^{\\prime }}_\\text{shsh}(t_1;p^{\\prime };\\Gamma _4) \\rangle \\langle F^{\\cal B^{\\prime }B^{\\prime }}_\\text{shsh}(t_2;p^{\\prime };\\Gamma _4) \\rangle }{\\langle F^{\\cal B^{\\prime }B^{\\prime }}_\\text{shsh}(t_2-t_1;p^{\\prime };\\Gamma _4) \\rangle \\langle F^{\\cal BB}_\\text{shsh}(t_1;p;\\Gamma _4) \\rangle \\langle F^{\\cal BB}_\\text{shsh}(t_2;p;\\Gamma _4) \\rangle }\\Bigg ]^{\\frac{1}{2}}.$ In the large time limit, $t_2-t_1\\gg a$ and $t_1\\gg a$ , the time dependence of the correlators are eliminated and the ratio in Eq.", "(REF ) reduces to $R(t_2,t_1;p^{\\prime },p;\\Gamma ;\\mu ) \\xrightarrow[t_1\\gg a]{t_2 -t_1 \\gg a} \\Pi (p^{\\prime },p;\\Gamma ;\\mu ).$ The Sachs form factors can be extracted from the final form of the ratio above by choosing specific combinations of the projection matrices $\\Gamma $ and the Lorentz index $\\mu $ : $\\Pi (p^{\\prime },p;\\Gamma _4;\\mu =4) = \\frac{1}{2}\\sqrt{\\frac{(E_{{\\cal B}^{\\prime }}+m_{{\\cal B}^{\\prime }})(E_{\\cal B}+m_{\\cal B})}{E_{\\cal B} E_{{\\cal B}^{\\prime }}}} G_E(q^2),$ $\\Pi (p^{\\prime },p;\\Gamma _j;\\mu =i) = \\frac{\\epsilon _{ijk} q_k}{2} \\sqrt{\\frac{(E_{{\\cal B}}+m_{{\\cal B}})}{E_{\\cal B}E_{{\\cal B}^{\\prime }}(E_{{\\cal B}^{\\prime }}+m_{{\\cal B}^{\\prime }})}} G_M(q^2).$ Note that when the incoming and the outgoing baryon states are identical, i.e.", "${\\cal B}^\\prime ={\\cal B}$ , the electric form factor $G_E (q^2=0)$ gives the electric charge of the baryon.", "As for the magnetic form factor, $G_M(q^2=0)$ gives the magnetic moment of the baryon when ${\\cal B}^\\prime ={\\cal B}$ and it gives the transition magnetic moment when ${\\cal B}^\\prime \\ne {\\cal B}$ ." ], [ "Lattice Setup", "We use gauge configurations generated by the PACS-CS collaboration [30], with $O(a)$ -improved Wilson quark action and the Iwasaki gauge action.", "We run our simulations with the hopping parameter of the light quarks $\\kappa ^\\ell = 0.13781$ which gives a near physical value of pion mass $ m_\\pi \\approx 156$  MeV.", "We use the Clover action also for the strange valence quark and take its hopping parameters to be equal to that of the strange sea quark, $\\kappa ^s_\\text{val}=\\kappa ^s_\\text{sea} = 0.13640$ .", "Further details of the gauge configurations are given in Table REF .", "Table: The details of the gauge configurations we employ .", "N s N_s and N t N_t are the spatial and temporal sizes of the lattice, respectively, N f N_f is the number of flavors, aa is the lattice spacing, LL is the volume of the lattice, β\\beta is the inverse gauge coupling, c sw c_{sw} is the Clover coefficient, κ sea f \\kappa _{sea}^f is the hopping parameter of the quark with flavor ff and m π m_\\pi is the pion mass.As for the charm quarks, we apply a Clover action in the form used by Fermilab [31], MILC Collaborations [32], [33] and in our previous work [25].", "In this action, the Clover coefficients in the action are set to tadpole-improved value $\\frac{1}{u_0^3}$ where $u_0$ is the average link.", "We follow the approach in [34] and estimate $u_0$ to be the fourth root of the average plaquette.", "We use the value of the hopping parameter $\\kappa _c= 0.1246$ as determined in our previous work by tuning the spin-averaged static masses of charmonium and open charm mesons to their experimental values [35].", "In order to increase statistics, we insert positive and negative momenta in all spatial directions and make a simultaneous fit over all data.", "We also take account of current insertion along all spatial directions.", "The source-sink time separation is fixed to 12 lattice units (1.09 fm), which is enough to avoid excited state contaminations for electromagnetic form factors [35].", "We have employed multiple source-sink pairs by shifting them 12 lattice units in the temporal direction.", "All statistical errors are estimated by the single-elimination jackknife analysis.", "We insert momentum up to nine units: $(|p_x|, |p_y|, |p_z|)=$ (0,0,0), (1,0,0), (1,1,0), (1,1,1), (2,0,0), (2,1,0), (2,1,1), (2,2,0), (2,2,1) and average over equivalent momenta.", "We consider point-split lattice vector current $j_\\mu = \\frac{1}{2} [\\overline{q}(x+\\mu ) U_\\mu ^{\\dagger }(1+\\gamma _\\mu )q(x)-\\overline{q}(x)U_\\mu (1-\\gamma _\\mu )q(x+\\mu )],$ which is conserved by Wilson fermions therefore do not need any renormalization.", "We use wall-source/sink method [36] that provides a simultaneous extraction of all spin, momentum and projection components of the correlators.", "On the other hand the wall source/sink is a gauge-dependent object which requires fixing the gauge.", "We fix the gauge to Coulomb, which gives a somewhat better coupling to the ground state than Landau.", "By using the wall method we can first compute the shell and wall propagators regardless of the current and momenta inserted.", "Then we contract the propagators to obtain the three-point correlators.", "Only the connected diagrams are considered in this work.", "Possible effects of the disconnected diagrams are discussed in the following section.", "We performed our computations using a modified version of Chroma software system [37] on CPU clusters and with QUDA [38], [39] for propagator inversion on GPUs." ], [ "Results And Discussion", "First, we discuss our results for the $\\Xi _c$ and $\\Xi _c^{\\prime }$ masses.", "We extract the ground-state masses using the two-point correlation functions in Eq.", "(REF ).", "Our results are given in Table REF , along with their experimental values and those of other lattice collaborations.", "As our results are obtained with near physical values of light-quark masses, we do not make any chiral extrapolations.", "Our results for the baryon masses differ by only  $2\\%$ as compared to the experimental values, while there is a good agreement for the mass splitting $m_{\\Xi _c^{\\prime }} - m_{\\Xi _c}$ .", "Note that it is tempting to attribute this small discrepancy to Clover action we are employing for the charm quarks, however, it has been confirmed in Ref.", "[40] that the mass of the triply charmed $\\Omega _{ccc}$ baryon can be calculated in very good agreement with other lattice determinations using relativistic heavy-quark actions.", "Such small discrepancy may be due to our choosing $\\kappa ^s_\\text{val}=\\kappa ^s_\\text{sea} = 0.13640$ to be consistent with PACS-CS.", "This choice of $\\kappa $ values leads to an overestimation of the $\\Omega \\,{\\rm (sss)}$ mass around 100 MeV as compared to its experimental value [30] and of the Kaon mass [41].", "On the other hand, the form factor determinations are rather insensitive to mild changes in baryon masses at the current precision level and a discrepancy of $2\\%$ can be safely neglected.", "Table: The Ξ c \\Xi _c and Ξ c ' \\Xi _c^{\\prime } masses together with experimental values and those of other lattice collaborations.In this work we focus on the magnetic form factors and we make an extrapolation to zero momentum transfer in order to obtain the magnetic moments.", "While the electric charge $G_E(0)$ can be computed directly with our formulation on the lattice, we cannot make a direct measurement of the magnetic form factor at zero momentum $G_M(0)$ .", "To this end, we use the following dipole form to describe the $Q^2$ dependence of the form factors: $G_{E,M} (Q^2) = \\dfrac{G_{E,M}(0)}{\\left(1+Q^2/\\Lambda _{E,M}^2\\right)^2}.$ The form factors, can be calculated from individual quark contributions by $G_{M} (Q^2) = \\frac{2}{3} G_{M}^c (Q^2) - \\frac{1}{3} G_{M}^s (Q^2) + c_\\ell G_{M}^\\ell (Q^2),$ where $c_\\ell =-1/3$ for the $d$ quark and $c_\\ell =2/3$ for the $u$ quark.", "We combine the individual quark contributions using Eq.", "(REF ) for each momentum transfer $Q^2$ and extrapolate the combined form factor values to $Q^2=0$ .", "Table: Individual contributions of uu/dd, ss and cc quarks to the magnetic form factor at different Q 2 Q^2 values of all transitions we study.In the left two panels of Fig.", "REF and Fig.", "REF , we plot the magnetic form factors $G_M(Q^2)$ of the charged $\\Xi _c^+$ , $\\Xi _c^{\\prime +}$ and neutral $\\Xi _c^0$ , $\\Xi _c^{\\prime 0}$ states as functions of $Q^2$ .", "In the SU(4) limit all individual valence quarks give equal contributions to $\\Xi _c^\\prime $ form factors, similar to proton.", "However, SU(4) symmetry is badly broken and the quark contribution decreases as the quark mass increases.", "This is consistent with what has been observed in previous works on charmed baryons [36], [35], [40], [25].", "This is also evident in Table REF , where we list individual contributions of $u$ /$d$ , $s$ and $c$ separately for the transitions we study at all $Q^2$ .", "The heavy $c$ -quark contribution is one order smaller than those of light $u$ /$d$ and $s$ quarks.", "This dominance of light quarks yields a soft core and the form factor decreases rapidly as $Q^2$ increases.", "Due to flavor asymmetry of $\\Xi _c$ baryon wavefunctions, the $u/d$ - and $s$ -quark contributions cancel each other to a great extent leading to a form factor that is dominantly determined by the $c$ quark.", "This cancellation would be exact in the SU(3) symmetric limit.", "Since the $Q^2$ dependence of the $\\Xi _c$ form factors is controlled by the heavy $c$ quark, it yields a hard core and the form factor decreases less rapidly as $Q^2$ increases.", "Figure: Q 2 Q^2 dependence of the magnetic form factors of Ξ c + γ→Ξ c + \\Xi _c^+ \\gamma \\rightarrow \\Xi _c^+ , Ξ c '+ γ→Ξ c '+ \\Xi _c^{\\prime +} \\gamma \\rightarrow \\Xi _c^{\\prime +} and Ξ c + γ→Ξ c '+ \\Xi _c^+ \\gamma \\rightarrow \\Xi _c^{\\prime +}Figure: Q 2 Q^2 dependence of the magnetic form factors of Ξ c 0 γ→Ξ c 0 \\Xi _c^0 \\gamma \\rightarrow \\Xi _c^0 , Ξ c '0 γ→Ξ c '0 \\Xi _c^{\\prime 0} \\gamma \\rightarrow \\Xi _c^{\\prime 0} and Ξ c 0 γ→Ξ c '0 \\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{\\prime 0}The light $u/d$ - and $s$ -quark contributions to the transition magnetic form factors of $\\Xi _c \\gamma \\rightarrow \\Xi _c^{^{\\prime }}$ are equal in magnitude and opposite in sign.", "On the other hand, the $c$ quark has almost no effect.", "When the quark contributions are combined using the formula in Eq.", "(REF ), the $u$ and $s$ contributions to $\\Xi _c^+ \\gamma \\rightarrow \\Xi _c^{^{\\prime }+}$ are multiplied with electric charges of opposite sign and add constructively.", "In contrast, the neutral transition $\\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{^{\\prime }0}$ is highly suppressed as a result of equal electric charges of the $d$ and $s$ quarks.", "According to conserved U-spin flavor symmetry, which assumes a degeneracy between two equally charged $d$ and $s$ quarks, a transition from $\\Xi _c^0$ to $\\Xi _c^{^{\\prime }0}$ is forbidden.", "Our results are in agreement with what U-spin flavor symmetry predicts.", "As shown in Fig.", "REF the magnetic form factor of $\\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{^{\\prime }0}$ neutral transition is consistent with zero.", "In this work, we neglect the effects of disconnected diagrams, which are noisy and costly to compute.", "Contributions of disconnected diagrams to isovector electromagnetic form factors are usually suppressed.", "We expect the sea-quark effects to be also suppressed in our results.", "In the case of nucleon electromagnetic form factors, the contributions of disconnected diagrams have been found to be approximately $0.5\\%$ of those of connected diagrams [42].", "They may however play an important role for the $\\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{^{\\prime }0}$ transition, where the connected contributions cancel.", "As a rule of thumb, the finite-size effects should be negligible when $m_\\pi L \\ge 4$ .", "The $\\kappa ^\\ell = 0.13781$ configurations that we employ, on the other hand, yield $m_\\pi L = 2.3$ which is below the empirical bound.", "We have, however, confirmed that the finite-size effects on this particular setup are under control for physical quantities related to strange and charmed baryons [40].", "As we have discussed above, the magnetic form factor of $\\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{^{\\prime }0}$ should vanish due to U-spin flavor symmetry which assumes a degeneracy between $d$ - and $s$ -quarks.", "This is realized in our numerical calculations when the $d$ - and $s$ -quark contributions cancel each other so that the magnetic form factor is consistent with zero as shown in Fig.", "REF .", "This indicates that the the finite-size effects on the light quarks are either similar as compared to those of strange and charmed quarks or any unaccounted effect is already hidden in the statistical error of the quantities that we extract.", "Note that, our discussion here gives only a qualitative account of the possible finite-size effects rather than a quantitative estimation, which requires further investigation.", "Table: The combined form factor as obtained using Eq.", "() and extrapolated to Q 2 =0Q^2=0, together with the magnetic moments in units of nuclear magneton.Using the values of the magnetic form factors at $Q^2=0$ in Table REF , we calculate the magnetic moments in nuclear magnetons by using $\\mu _B = G_M(0) (e/2 m_B) = G_M(0) (m_N/m_B) \\mu _N,$ where $m_N$ is the physical nucleon mass and $m_B$ is the baryon mass obtained on the lattice.", "In Table REF , we list the combined form factor as obtained using Eq.", "(REF ) and extrapolated to $Q^2=0$ , that is $G_{M} (0)$ , the magnetic moments calculated using $G_{M} (0)$ in units of nuclear magneton.", "The decay width of $\\Xi _c^\\prime $ baryon is related to the Pauli form factor $F_2(0)$ of $\\Xi _c \\gamma \\rightarrow \\Xi _c^{^{\\prime }}$ : $\\Gamma _{B \\gamma \\rightarrow B^{\\prime } } = \\frac{4~\\alpha |\\vec{q}|^3}{(m_{B^{\\prime }}+m_B)^2}|F_2(0)|^2 \\quad \\text{with}\\quad |\\vec{q}| = \\frac{(m_{B^{\\prime }}^2-m_B^2)}{2 m_{B^{\\prime }}}.$ Since the relation in Eq.", "(REF ) is defined in the continuum, we evaluate it by using the experimental masses of $\\Xi _c$ and $\\Xi _c^{^{\\prime }}$ .", "In order to extract $F_2(0)$ from the Sachs form factors $G_E(Q^2)$ and $G_M(Q^2)$ , we solve the two equations in Eqs.", "(REF ) and (REF ) simultaneously for all lattice data and extrapolate to $Q^2=0$ .", "At zero momentum transfer, $G_E(0)=F_1(0)$ and we can immediately deduce that $F_1(0)$ must have a very small value, if not zero.", "Since $\\Xi _c \\gamma \\rightarrow \\Xi _c^{^{\\prime }}$ cannot occur through electric transition, this implies $G_M(0)\\simeq F_2(0)$ .", "Consistently, we find $F_2(0)=2.036(280)\\quad \\text{for} &\\quad \\Xi _c^+ \\gamma \\rightarrow \\Xi _c^{\\prime +}, \\\\F_2(0)=0.039(46)\\quad \\text{for} &\\quad \\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{\\prime 0}.$ Using the formula in Eq.", "(REF ), we obtain the decay widths of $\\Xi _c$ baryons as follows: $\\Gamma _{\\Xi _c^{\\prime +}}=5.468(1.500)~\\text{keV}, \\quad \\Gamma _{\\Xi _c^{\\prime 0}}=0.002(4)~\\text{keV}.$ The decay width can be translated into a lifetime using $\\tau = \\frac{1}{\\Gamma }$ ; $\\tau _{\\Xi _c^{\\prime +}} = 1.148(322)\\times 10^{-19}~\\text{s}.$ Both neutral and charged transitions of $\\Xi _c \\gamma \\rightarrow \\Xi ^{\\prime }_c $ have been previously studied using QCD sum rules [22], heavy hadron chiral perturbation theory [16], [15], [14], quark model [20], [19] and bag model [23].", "For the charged transition, our lattice results for the transition form factor and decay width are in agreement with those from QCD sum rules [22], while other methods previously used predict higher values.", "In the case of neutral transition, their predictions for the transition form factors are small but finite, while we find no signal on the lattice." ], [ "Summary and Conclusion", "We studied the magnetic form factors of the $\\Xi _c \\gamma \\rightarrow \\Xi ^\\prime _c$ transition and of the $\\Xi _c$ , $\\Xi _c^\\prime $ baryons in 2+1-flavor lattice QCD.", "We have extracted the magnetic Sachs and Pauli form factors which give the $\\Xi _c$ -$\\Xi _c^\\prime $ transition magnetic moment and the decay widths of $\\Xi _c^\\prime $ baryons.", "We determined individual quark contributions to the magnetic moments, which give an invaluable insight to the dynamics of $u/d$ , $s$ and $c$ quarks having masses at different scales.", "In the case of $\\Xi ^{\\prime }_c$ baryons the heavy $c$ -quark contribution is much smaller than those of light $u/d$ and $s$ quarks.", "On the other hand, due to antisymmetric flavor wavefunctions of the $\\Xi _c$ baryons, the $u/d$ - and $s$ -quark contributions cancel each other to a great extent leading to a form factor that is dominantly determined by the $c$ quark.", "We find that the $c$ quark has a negligibly small contribution to the $\\Xi _c \\gamma \\rightarrow \\Xi ^\\prime _c$ transition and, $u/d$ and $s$ quarks contribute with opposite sign.", "Using the Pauli form factor $F_2(Q^2=0)$ , we extracted the decay widths of $\\Xi _c^\\prime $ baryons.", "The decay width of the charged $\\Xi _c^\\prime $ baryon on the lattice is determined as $\\Gamma _{\\Xi _c^{\\prime +}}=5.468(1.500)~\\text{keV}$ and we did not find a signal for the magnetic form factor of the neutral transition $\\Xi _c^0 \\gamma \\rightarrow \\Xi _c^{\\prime 0}$ , which is suppressed by the U-spin flavor symmetry.", "This work is supported in part by The Scientific and Technological Research Council of Turkey (TUBITAK) under project number 114F261 and in part by KAKENHI under Contract Nos.", "25247036, 24250294 and 16K05365.", "This work is also supported by the Research Abroad and Invitational Program for the Promotion of International Joint Research, Category (C) and the International Physics Leadership Program at Tokyo Tech." ] ]
1612.05722
[ [ "Stimulated Emission of Dark Matter Axion from Condensed Matter\n Excitations" ], [ "Abstract We discuss a possible principle for detecting dark matter axions in galactic halos.", "If axions constitute a condensate in the Milky Way, stimulated emissions of the axions from a type of excitation in condensed matter can be detectable.", "We provide general mechanism for the dark matter emission, and, as a concrete example, an emission of dark matter axions from magnetic vortex strings in a type II superconductor are investigated along with possible experimental signatures." ], [ "Introduction", "Recent discovery of the Higgs particle revealed that the standard model of particle physics correctly describes the fundamental constituents of our universe, up to the energy scale $\\sim 1\\, \\textrm {TeV}$ [1].", "However, the current standard cosmology strongly indicates the existence of additional constituents, which are known as the dark matter [2], [3].", "In particular, observations of the rotation velocities of galaxies imply that there exists the dark matter halo around the Milky Way, whose mass density is given by $\\rho _{DM}~ \\simeq ~ 3 \\times 10^{14}~ [\\textrm {eV}/c^2\\, m^3]~ \\simeq ~5 \\times 10^{- 22}~ [\\textrm {kg}/m^3] .$ Although there are various candidates of the dark matter, including the weakly interacting massive particles (WIMPs), the origin and properties of the dark matter remain a mystery, so far.", "(See [4], [5] for the current status of the dark matter research.)", "Among them, the axion is one of the promising candidates for the dark matter in our universe.", "The axion is a hypothetical elementary particle, which gives the most plausible solution to the strong CP puzzle in quantum chromodynamics (QCD) [6], [7].", "Through various experiments and astrophysical observations for axion search, the properties of the axion, such as the mass and coupling strength, are severely constrained [1], [8], [9].", "For example, the typical mass consistent with the experimental constraints can be summarized as $10^{-6}\\, \\left[\\textrm {eV}/c^2\\right]\\, \\lesssim \\, m_{a}\\, \\lesssim \\, 10^{-3}\\, \\left[\\textrm {eV}/c^2\\right],$ for the invisible QCD axion models [10], [11], [12], [13].", "This small mass distinguishes the axions from other WIMPs, whose masses are assumed to be $m_{\\textrm {wimp}} \\gtrsim 10^{9}\\, [\\textrm {eV}/c^{2}]$ .", "In contrast to the standard WIMPs which are usually fermions, axions obey the Bose-Einstein statistics.", "In order to consistently explain the dark matter density in terms of the axion, a large number density of the bosonic axions is required in our galaxy, and the possibility of the Bose-Einstein condensation of dark matter axions due to such a large density has been discussed [14], [15], [16], [17], [18].", "In this paper, we discuss a possible consequence of the Bose-Einstein condensation of the dark matter axions: stimulated emissions of the axions.", "Based on the coupling between the axion field and the electromagnetic field, we discuss the stimulated emissions of the axions from collective excitations in various condensed matter systems.", "In particular, as a concrete example, we investigate the stimulated emission from the magnetic vortex strings in type II superconductors, where a mobile vortex ensemble, such as vortex flow or vortex liquid, is realized near the critical temperature.", "The emission rate of the dark matter axions are estimated, and a possible experimental signature of the emission is discussed." ], [ "Properties of Dark Matter Axions", "In this section, we briefly summarize the fundamental properties of the axion, in particular, the interaction between the axions and the electromagnetic fields.", "We also discuss the properties of the axion which make it to be a good candidate for the (cold) dark matter, with an emphasis on the difference from other dark matter candidates, such as WIMPs." ], [ "Coupling of Axion with Electromagnetic Field", "Since the axion is a (pseudo) Nambu-Goldstone boson [19], [20] of the spontaneous breaking of the Peccei-Quinn (PQ) symmetry [6], [7], the axion field $\\phi _{a}$ couples to other particles through its derivative, originated from a universal coupling, ${\\cal L}_{\\textrm {NG}} \\simeq j_{\\textrm {PQ}}^{\\mu } \\left(\\partial _{\\mu }\\phi _{a}\\right) ,$ where $j_{\\textrm {PQ}}^{\\mu }$ is the conserved current of the PQ-symmetry.", "Although the strength of this coupling is very small for the invisible axion models, various experiments to detect the axions using this type of coupling have been implemented so far [8], [9].", "The axion has an additional coupling in the leading order with the electromagnetic (EM) field, which is a parity-odd coupling originated from the quantum anomaly of the PQ symmetry: ${\\cal L}_{\\textrm {int}} = g_{0}\\,\\phi _{a}\\,\\vec{E}\\cdot \\vec{B} .$ In this paper, we focus on this axion coupling (REF ) and discuss its consequence.The experiments for the axion detection using this coupling have also been performed [8], [9].", "With this coupling, the action for the axion field and EM-field is given by $S &=& \\int \\!\\!", "d^4x~ \\frac{1}{2} \\left[\\frac{\\hbar ^{2}}{c^2} \\left(\\frac{\\partial \\phi _{a}}{\\partial t}\\right)^2- \\hbar ^2 \\left(\\vec{\\nabla } \\phi _{a}\\right)^2 - m_{a}^2 c^2 \\phi _{a}^2\\right] \\nonumber \\\\&+& \\int \\!\\!", "d^4x~ \\left[ \\left( \\frac{\\varepsilon _{0}}{2}\\vec{E}^2 -\\frac{1}{2 \\mu _{0}} \\vec{B}^2 \\right) + g_{0}\\,\\phi _{a}\\,\\vec{E}\\cdot \\vec{B} + \\cdots \\right] ,$ where $m_{a}$ is the mass of the axion and the dots represent higher order terms with respect to the coupling $g_{0}$ .", "The coupling strength $g_{0}$ is constrained to be very small from various experiments and astronomical observations [1], [8], [9], and the current bound is given by $g_{0}~ \\lesssim ~ 10^{-10}~ [\\textrm {GeV}^{-1}]~ \\simeq ~ 10^{-50}~\\left[\\frac{\\textrm {sec}~ C^2}{\\sqrt{\\textrm {kg}~ m}}\\right],$ where the first value is in the natural units, and the second one is in the SI units with $C$ being Coulomb for the electric charge.", "(See the Appendix for the conversion between these unit systems.)", "We optimistically take the value $g_{0} \\simeq 10^{-10} [\\textrm {GeV}^{-1}]$ in this paper.", "Due to this small value, the higher order terms with respect to $g_{0}$ can be completely neglected.", "Although the coupling (REF ) leads to the axion-photon scattering and the axion decay to two photons, the scattering and decay rates are very small and the lifetime of the axion is much longer than the current age of the universe.", "This indicates that the axions produced in the early universe with some mechanism, such as the misalignment of the vacuum angle and the cosmic string decay, can remain for a long time, and these relic axions can be promising candidates for the dark matter in our universe.", "Particularly, the production mechanisms are mainly non-thermal, and the axions, which are produced around the time of the QCD transition, behave as non-relativistic particles from the beginning.", "So, the axion is a good candidate for the cold dark matter [2], [3].", "If the axion accounts for the major part of the dark matter of our universe, the abundance of the axion in our galaxy should explain the mass density (REF ), which is determined by the observation of the rotation curve of various galaxies [4], [5].", "The velocity distribution of the dark matter in the galactic halo has also been discussed, and it is known that the violent relaxation by gravitational attraction [21] leads to the Maxwellian distribution, where the root-mean-squared velocity is given by $v_{a} \\simeq 2.7 \\times 10^5\\, [m/\\textrm {sec}] \\sim 10^{-3}\\, c \\, .$ As noted above, since $v_{a}/c \\sim 10^{-3} \\ll 1$ , the axions in our galaxy behave as non-relativistic particles.", "Although the mass of the axion depends on the models beyond the standard model of the elementary particles, the plausible mass range is discussed from the various viewpoints including cosmology, astronomy and elementary particle physics [8].", "In particular, in order to explain the dark matter abundance in our universe, the expected mass should be in the range (REF ).", "If we take $m_{a} \\simeq 10^{-6}\\, [\\textrm {eV}/c^2]$ as a typical value, we can calculate the number density $n_{a}$ of the dark matter axions in our galaxy, $n_{a} = \\frac{\\rho _{DM}}{m_{a}} \\simeq 3 \\times 10^{20}\\, \\left[m^{-3}\\right] .$" ], [ "Non-Relativistic Limit of Axion Field", "Since the axions behave as non-relativistic particles ($v_{a} \\sim 10^{-3} c$ ), we can reasonably consider the non-relativistic limit of the axion field $\\phi _{a}(x, t)$ .", "At first, we separate the rest mass contribution from the time dependence of the axion field as $\\phi _{a}(x, t) = \\frac{1}{\\sqrt{2\\, m_{a}}}\\left( \\varphi _{a}(x, t)\\, e^{-i \\frac{m_{a} c^2}{\\hbar } t} +\\varphi ^{\\dagger }_{a}(x, t)\\, e^{i \\frac{m_{a} c^2}{\\hbar } t} \\right) ,$ where $\\phi _{a}^{\\dagger } = \\phi _{a}$ is satisfied for the real scalar.", "Inserting this ansatz into the axion part of the Lagrangian (REF ), we obtain the following non-relativistic action $S^{NR}_{a} = \\int \\!\\!", "d^4 x\\, {\\cal L}_{a}^{NR}= \\int \\!\\!", "d^4 x\\, \\left[ i \\hbar ~\\varphi _{a}^{\\dagger } \\frac{\\partial \\varphi _{a}}{\\partial t} - \\frac{\\hbar ^2}{2\\, m_{a}} |\\vec{\\nabla } \\varphi _{a}|^2 \\right] ,$ where the second derivative term with respect to time is discarded.", "This is nothing but the Lagrangian of the Schr$\\ddot{\\text{o}}$ dinger field, which satisfies the non-relativistic Schr$\\ddot{\\text{o}}$ dinger equation.", "The non-relativistic axion field $\\varphi _{a}(x, t)$ becomes a complex scalar field, whose dimension is given by $[L^{-3/2}]$ .", "Here, we define the rest mass frequency as $\\Omega ~ \\equiv ~ \\frac{m_{a} c^2}{\\hbar }~\\simeq ~ 1.5 \\times 10^{9}\\, [\\textrm {sec}^{-1}] ,$ for the typical mass $m_{a} \\simeq 10^{-6}\\, [\\textrm {eV}/c^2]$ .", "In the non-relativistic limit, the axion coupling is also renormalized.", "Inserting the non-relativistic ansatz of the axion field (REF ) into the original coupling (REF ), we obtain the axion coupling in the non-relativistic limit, ${\\cal L}^{NR}_{\\textrm {int}}= g\\, \\left(\\varphi _{a} e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a} e^{i \\Omega t} \\right)\\vec{E}\\cdot \\vec{B} .$ Here, the non-relativistic coupling strength is defined as $g~ \\equiv ~ \\frac{g_{0}}{\\sqrt{2\\, m_{a}}}~\\simeq ~ 10^{-29}\\, \\left[\\frac{\\textrm {sec}~ C^2}{\\textrm {kg}~ \\sqrt{m}}\\right] ,$ where the typical value $m_{a} \\sim 10^{-6} [\\textrm {eV}/c^2]$ is used.", "This is the basic formula describing the interaction between the dark matter axions and excitations in condensed matter systems." ], [ "Dark Matter Axion as Condensate", "Based on the discussion above, we are lead to a natural but somewhat surprising consequence: the dark matter axions can constitute a (Bose-Einstein (BE)) condensate.", "At first, the non-relativistic Lagrangian of the axion (REF ) essentially describes non-interacting particles obeying the Bose statistics.The coupling strength with themselves and other particles are so small that it can be essentially ignored.", "Given the large number density (REF ) and applying the standard argument on the BE condensation to this axion system, the critical temperature is estimated by $k_{B} T_{c} \\simeq \\left(2 \\pi \\hbar ^{2}/m_{a}\\right) n_{a}^{2/3} \\sim 10^{7} [\\textrm {eV}]$ .", "Furthermore, the thermal energy of dark matter axions, $k_{B} T_{a}$ , is estimated from the average velocity (REF ) to be $k_{B} T_{a}\\sim (1/2) m_{a} v_{a}^{2} \\simeq 10^{-13} [\\textrm {eV}]$ .", "Therefore, if the dark matter axions are in thermal equilibrium in our galaxy, the axion field is naturally expected to be in a BE condensate phase at the extreme low temperature $T_{a}~ (\\ll T_{c})$ .", "The possibility of the thermalization of the dark matter axions by the small self-interaction and the gravitational interaction has been recently discussed in the literatures [14], [15], [16], [17], [18], and some consequences from the resulting BE condensation have been explored.", "In this paper, we assume that the dark matter axions do thermalize and form the BE condensate in our galaxy.It has been argued the possibility that miniclusters composed of axions have been formed through the galaxy formation process [22], [23].", "(For the recent discussions, see [18], [24], [25].)", "In this case, we assume that a significant fraction of axions have been thrown off due to the collisions and mergings of such miniclusters, and some fraction of axions forms a smooth halo distribution.", "Assuming that the dark matter axions are in a condensed phase, we can estimate the occupation number ${\\cal N}_{k}$ of the mode with the typical momentum $p_{a} \\simeq m_{a} v_{a} \\sim 10^{-9}\\, [\\textrm {eV}/c]$ .", "The corresponding thermal de Broglie wave length is given by $\\lambda _{a} = 2 \\pi \\hbar /p_{a} \\sim 10^3\\, [m]$ , which characterizes the quantum coherence scale of the BE condensation, as discussed in [18].", "Using this wave length, the typical occupation number at the mode with $k_{a} = 2 \\pi /\\lambda _{a}$ becomes very large : ${\\cal N}_{k} \\simeq n_{a}\\, \\lambda _{a}^{3} = \\left(\\frac{\\rho _{DM}}{m_{a}}\\right)\\cdot \\left(\\frac{2 \\pi \\hbar }{m_{a} v_{a}}\\right)^{3}\\, \\simeq \\, 10^{29} .$ As explained below, this large occupation number ${\\cal N}_{k}$ gives an enhancement factor for the transition probability, which is analogous to the case of the stimulated emission of photons, used in the LASER.", "Note that the occupation number ${\\cal N}_{k}$ is proportional to $m_{a}^{-4}$ .", "In this condensed regime, the non-relativistic axion field $\\varphi _{a}(x, t)$ behaves as a coherent classical field in the leading approximation, $\\langle \\, \\left|\\varphi _{a}\\right|^{2}\\, \\rangle \\simeq n_{a} ~ \\Longrightarrow ~\\varphi _{a} \\simeq \\varphi ^{\\dagger }_{a} \\simeq \\sqrt{n_{a}} \\sim 10^{10}\\, \\left[m^{-3/2}\\right].$ Using this condensed ansatz of the axion field, we can further rewrite the axion coupling in the following form: ${\\cal L}^{BE}_{\\textrm {int}} = g \\left(\\varphi _{a}\\,e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a}\\,e^{i \\Omega t} \\right)\\vec{E}\\cdot \\vec{B} ~\\simeq ~ 2\\,g\\,\\sqrt{n_{a}} \\cos (\\Omega t)\\,\\vec{E}\\cdot \\vec{B}.$ The factor $\\sqrt{n_{a}}$ also leads to an enhancement of the coupling strength $g$ , and gives a time-periodic coupling for collective excitations, as will be seen in the following." ], [ "Excitations Coupled with Axions in Condensed Matter", "Based on the basic interaction (REF ), three types of excitations in condensed matter systems can possibly couple with the dark matter axions.", "Excitations creating the electric field $\\vec{E}$ under the external magnetic field $\\vec{B}^{\\textrm {ex}}$ : For concreteness, we consider the homogeneous external magnetic field in the $z$ -direction, $\\vec{B}^{\\textrm {ex}} = (0, 0, B_{0})$ .", "In the external magnetic field, the excitations creating the $z$ -component of electric field $E_{z}$ can couple with the axions.", "In this case, the basic coupling (REF ) approximately becomes ${\\cal L}^{BE}_{\\textrm {int}} \\simeq g\\,B_{0}\\left(\\varphi _{a}\\,e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a}\\,e^{i \\Omega t} \\right)\\left(\\gamma a + \\gamma ^{*} a^{\\dagger }\\right) .$ Here, the electric field associated with the excitation is given by $E_{z} = \\gamma a + \\gamma ^{*} a^{\\dagger }$ in the linear approximation regime, where $a$ is the annihilation operator for a mode of the excitation and $\\gamma $ is the corresponding coefficient.", "The possible candidates for such excitations in condensed matter systems are plasmons in (semi)condcutors, polaritons in insulators, and vortices in superconductors, and so on.", "Excitations creating the magnetic field $\\vec{B}$ under the external electric field $\\vec{E}^{\\textrm {ex}}$ : Similarly, considering the external electric field in the $z$ -direction, $\\vec{E}^{\\textrm {ex}} = (0, 0, E_{0})$ , the excitation creating the $z$ -component of magnetic field can couple with the axions.", "Then, the basic coupling (REF ) is given by ${\\cal L}^{BE}_{\\textrm {int}} \\simeq g\\,E_{0}\\left(\\varphi _{a}\\,e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a}\\,e^{i \\Omega t} \\right)\\left(\\gamma a + \\gamma ^{*} a^{\\dagger }\\right) ,$ where the magnetic field associated with the excitation is assumed to be $B_{z} = \\gamma a + \\gamma ^{*} a^{\\dagger }$ .", "The candidates are magnons and domain walls in magnetic systems, etc.", "Excitations which can create both the electric and magnetic field: Recent studies on multiferroic and topological materials show that there exist the excitations creating both the electric field in response to an external magnetic field and the magnetic field in response to an external electric field.", "Through the magnetoelectric effect, these excitations have the coupling with the axions in the form of (REF ) or (REF ), depending on the external field.", "The interesting candidates for such excitations are skyrmions in chiral-lattice magnets [26] and condensed matter analogues of the axions in a topological insulator with time-reversal breaking (so-called topological magnetic insulator) [27], [28].A realization of axion electrodynamics is also discussed by using optical lattices [29].", "In the following, we focus on the case of the excitations of type A, since the discussions are almost parallel for those of type B and C. In particular, as a concrete example and an interesting possibility, the magnetic vortex strings in type II superconductors will be discussed thoroughly." ], [ "Axion-Vortex Coupling in Type II Superconductors", "In this section, we discuss the effects of the axion coupling (REF ) in the dynamics of type II superconductors and, in particular, magnetic vortex strings." ], [ "Ginzburg-Landau Description of Superconductor and Magnetic Vortex", "We start with the Ginzburg-Landau (GL) equations, which give the effective description of type II superconductors [30], [31].", "$&& 0 ~=~ \\frac{\\hbar ^2}{2 m^{*}}\\left(\\vec{\\nabla } - i \\frac{e^{*}}{\\hbar }\\vec{A}\\right)^2 \\psi - \\alpha \\,\\psi -\\beta \\, |\\psi |^2\\,\\psi , \\\\&& \\frac{1}{\\mu _{0}} \\left(\\vec{\\nabla }\\times \\vec{B}\\right) ~=~ i \\frac{\\hbar e^{*}}{2 m^{*}}\\left[\\left(\\vec{\\nabla } \\psi ^{\\dagger }\\right) \\psi - \\psi ^{\\dagger }\\left(\\vec{\\nabla } \\psi \\right)\\right] -\\frac{e^{*\\,2}}{m^{*}} |\\psi |^2 \\vec{A} .", "$ Here, we denote the order parameter of superconductors as $\\psi $ , and $e^{*}$ and $m^{*}$ are the electric charge and mass of the order parameter, respectively.", "$\\alpha $ and $\\beta $ correspond to the parameters which can be calculated from the microscopic theory à la BCS.", "From the second equation (), we can read the supercurrent, $\\vec{J}_{s} &=& i \\frac{\\hbar e^{*}}{2 m^{*}} \\left\\lbrace \\left(\\vec{\\nabla } \\psi ^{\\dagger }\\right) \\psi - \\psi ^{\\dagger }\\left(\\vec{\\nabla } \\psi \\right)\\right\\rbrace -\\frac{e^{*\\,2}}{m^{*}} |\\psi |^2 \\vec{A} \\nonumber \\\\&=& \\frac{\\hbar e^{*}}{m^{*}} |f|^2 \\left( \\vec{\\nabla } \\theta - \\frac{e^{*}}{\\hbar } \\vec{A} \\right) ,$ where we insert the decomposition of the order parameter, $\\psi (x) = f(x)\\,e^{i \\theta (x)}$ , for the second equality.", "As is well-known, the GL equations have the magnetic vortex solutions that are topologically stable [32], [33].", "For the later discussion, we briefly summarize the solitonic vortex solution.", "A straight single vortex, which is stretched in the $z$ -direction, is described by the field configuration with the cylindrical symmetry: $\\vec{A}_{\\textrm {v}}(x) = \\left(\\frac{\\hbar }{e^{*}}\\right) \\left(\\frac{\\vec{d}_{z} \\times \\vec{r}}{r}\\right) A(r),\\quad \\psi _{\\textrm {v}}(x) = \\sqrt{\\frac{|\\alpha |}{\\beta }}\\, e^{i \\theta } f(r) .$ Here, $(r, \\theta , z)$ are the cylindrical coordinates, and $\\vec{d}_{z}$ is the unit vector in the $z$ -direction.", "$A(r)$ and $f(r)$ are the functions of the radial coordinate $r$ only, and the flux is normalized to be the minimal flux quantum, $h/e^{*}$ .", "With these ansatze, the GL equations are reduced to the following coupled ordinary differential equations with respect to $r$ : $0 &=& \\xi ^2 \\left[\\frac{1}{r} \\frac{d}{dr}\\left(r \\frac{d f(r)}{dr}\\right) -\\left(\\frac{1}{r}-A(r)^2\\right) f(r) \\right] - f(r) + f(r)^3 , \\\\0 &=& \\frac{d}{dr}\\left[\\frac{1}{r} \\frac{d}{dr}\\Bigl (r A(r) \\Bigr )\\right] - \\frac{\\mu _{0}\\, e^{*\\,2} |\\alpha |}{m^{*} \\beta }\\left(A(r) - \\frac{1}{r}\\right) f(r)^2 .", "$ Here, we define the coherence length for the order parameter: $\\xi = \\sqrt{\\frac{\\hbar ^2}{2 m^{*} |\\alpha |}} .$ With the boundary conditions $f(r) \\rightarrow 1$ and $A(r) \\rightarrow 1/r$ at $r \\rightarrow \\infty $ ,For non-singular solutions, the boundary condition $f(r) \\rightarrow 0$ at $r \\rightarrow 0$ is also imposed.", "we can easily obtain the numerical solutions for these equations, which describe the straight vortex with minimal flux.", "Away from the vortex core ($f(r) \\simeq 1$ ), we can obtain another characteristic length scale, $\\lambda _{L} = \\sqrt{\\frac{m^{*} \\beta }{\\mu _{0}\\, e^{*\\,2} |\\alpha |}} ,$ which defines the penetration depth of the electromagnetic field in superconductors." ], [ "Vortex Dynamics with Axion Coupling", "Now, we consider the interaction between the axion fields $\\varphi _{a}$ and magnetic vortex strings based on the axion coupling (REF ).", "Since the vortex string is a typical example of the excitations of type A, we consider the external magnetic field $\\vec{B}^{\\textrm {ex}}$ in the $z$ -direction, $B^{\\textrm {ex}}_{z} = B_{0} = \\textrm {const.", "}~ (B^{\\textrm {ex}}_{x} = B^{\\textrm {ex}}_{y} = 0)$ .", "Then, the axion coupling becomes ${\\cal L}^{NR}_{\\textrm {int}} \\simeq g\\, B_{0} \\left(\\varphi _{a}\\, e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a}\\,e^{i \\Omega t}\\right) E_{z} .$ Here, we derive the electric field created by the vortex motion in a simple set-up.", "For concreteness, we consider the limit of large penetration depth and small coherence length, compared to the size of a superconducting sample.", "In this situation, an external magnetic field becomes almost homogeneous in the sample and the core of magnetic vortex becomes infinitely thin.In the small $\\xi $ limit, we take the order parameter as $\\psi (x) = f e^{i \\theta (x)}$ with a constant $f$ .", "Outside the core of vortex, the GL equation () and supercurrent (REF ) give $\\vec{A} = \\left(\\hbar /e^{*}\\right) \\vec{\\nabla } \\theta $ .", "From the AC Josephson effect and gauge invariance, an electrostatic potential is also given by $\\Phi = - \\left(\\hbar /e^{*}\\right) \\dot{\\theta }$ .", "Since the motion of the vortex with (small) velocity $\\vec{v}_{L}$ implies a time-dependent phase $\\theta = \\theta (\\vec{x}-\\vec{v}_{L} t)$ , the electric field originated from the vortex motion is obtained as $\\vec{E}_{\\textrm {vor}} = - \\vec{\\nabla } \\Phi - \\frac{\\partial \\vec{A}}{\\partial t} =\\left(\\frac{\\hbar }{e^{*}}\\right)\\left(\\vec{\\nabla } \\frac{\\partial }{\\partial t} - \\frac{\\partial }{\\partial t} \\vec{\\nabla }\\right)\\theta (\\vec{x} - \\vec{v}_{L} t) = - \\left(\\frac{h}{e^{*}}\\right) \\left( \\vec{v}_{L} \\times \\vec{d}\\, \\right)\\delta ^{2}(\\vec{x}) ,$ where the multivalued property of the phase $\\theta (x)$ is used, and $\\vec{d}$ is a unit vector along the vortex axis, and the two-dimensional delta function has the support on the core of vortex.", "As seen from the vortex electric field (REF ), the motion in the plane perpendicular to the vortex axis is important, and we focus on the perpendicular motion in the following.", "Concretely, we regard the vortex string as a point particle on the perpendicular plane with a mass $m_{\\textrm {vor}}$ , by integrating out the spatial coordinate dependence in the Lagrangian.", "Using the formula (REF ) and the homogeneous property (REF ) of $\\varphi _{a}$ , the axion coupling can be written in the minimal coupling form, $L_{\\textrm {int}} = \\int \\!\\!d^{3} x~ {\\cal L}^{NR}_{\\textrm {int}} \\simeq \\left(\\frac{g\\, h\\, \\ell _{\\textrm {vor}}}{e^{*}}\\right) \\left(\\varphi _{a}\\,e^{-i \\Omega t} + \\varphi ^{\\dagger }_{a}\\, e^{i \\Omega t}\\right) \\left(\\vec{B}^{\\textrm {ex}} \\times \\vec{d}\\, \\right) \\cdot \\vec{v}_{L} = \\vec{{\\cal A}}_{\\textrm {eff}}\\cdot \\vec{v}_{L} ,$ where $\\ell _{\\textrm {vor}}$ is the length of the vortex string.", "Here, the effective gauge field $\\vec{{\\cal A}}_{\\textrm {eff}}$ is introduced and its magnitude is given by $\\left|{\\cal A}_{\\textrm {eff}}\\right| \\simeq \\left(\\frac{g\\, h\\,\\left|\\varphi _{a}\\right| B_{0}\\,\\ell _{\\textrm {vor}}}{e^{*}}\\right) .$ The above analysis can be simply extended to the case of general type II superconductors, using the time-dependent Ginzburg-Landau theory [34], [31].", "This term and a standard kinetic term lead to the Lagrangian of a charged particle on the perpendicular plane, coupled with an AC electric field, $L_{p} = \\frac{m_{\\textrm {vor}}}{2}\\,\\vec{v}_{L}^{\\, 2} + \\vec{{\\cal A}}_{\\textrm {eff}}(t)\\cdot \\vec{v}_{L}\\, .$ Thus, our problem on the perpendicular motion of a rigid vortex with the axion coupling (REF ) is reduced to the problem of a charged particle coupled with an AC electric field, whose frequency is $\\Omega \\sim 1\\, [\\textrm {GHz}]$ and magnitude is given by $\\left|\\vec{E}\\right| \\simeq \\left(\\frac{g\\, h\\, \\Omega \\, \\sqrt{n_{a}}\\, B_{0}\\, \\ell _{\\textrm {vor}}}{e^{*}}\\right) ,$ where we have used the ansatz of the BE condensation of the axion field (REF ).", "If we put the parameters $B_{0} \\sim 10\\, [\\textrm {T}]$ , $\\ell _{\\textrm {vor}} \\sim 10^{-4}\\, [m]$ as an example, the drift force by the effective “electric field” can be estimated as $F_{\\textrm {drift}} \\sim 10^{-27}\\, [\\textrm {N}]$ , which is so small that it is hardly detectable.", "Note that, for this coupling to work, we require the existence of the vortices or segments of the vortex which are non-parallel to the external magnetic field $\\vec{B}^{\\textrm {ext}}$ , as is evident from (REF ).", "For this purpose, we consider the following types of configurations of the vortex strings: Tilted vortex strings against a tuned external magnetic field, due to a boundary condition of a small superconducting sample.", "Wavy vortex strings with the non-paralell segments or the vortex strings with the kink-like segments, due to the small tension near the phase transition.", "In the following, we discuss mainly the case of the tilted vortex strings of type (a).", "For the wavy or kink-like vortex strings, the calculations can be done in the same way, except for replacing the length and number of the vortex strings with those of the segments." ], [ "Stimulated Emission of Axions from Vortices", "In this section, we estimate the emission probability of axions from a vortex motion, using the first order perturbation theory, i.e.", "the Fermi's golden rule.", "In order to realize the stimulated emission process by using the coupling (REF ), which is originated from the axion coupling (REF ), we require that there exist the excited states of the vortex dynamics having the energy level $\\Delta E \\sim \\hbar \\Omega $ , which enables the emission of the non-relativistic axions with the energy, $E_{a} \\simeq m_{a} c^2 + \\frac{m_{a}}{2} v_{a}^{2}$ .", "Furthermore, as in the case of the LASER, the non-equilibrium situation between the vortex system and the axions is also required, and the enough number of states of the vortices should be excited from the ground state.", "This implies that the vortex system should have a higher temperature than both the axion temperature $T_{a}$ and $T_{\\textrm {gap}} = \\Delta E/k_{B}$ .", "Therefore, we make the following assumptions on the dynamics of vortices in type II superconductors: The energy spectrum of the vortex dynamics has the excitations of the order of $\\hbar \\Omega \\sim 10^{-6}\\,\\textrm {eV}$ , which corresponds to the energy of the condensed axions.", "The vortices in the superconductor are in thermal equilibrium of the temperature $T_{SC}$ , higher than $T_{a}$ and $T_{\\textrm {gap}}$ , without considering the effect of the dark matter axions.", "To realize the assumption (I), we consider the magnetic vortices in mobile vortex systems where the excitaions with such a small energy level can be realized and observed as a resonance in experiments with microwave around GHz.", "Such mobile vortices can be realized in vortex creep, vortex flow, and vortex liquid in a type of superconductors [35].", "For the assumption (II), we take the typical temperature of a mobile vortex ensemble in superconductors as $T_{SC} \\sim 4\\, K$ ,The experiments on superconductors are often performed at the liquid Helium temperature, $4.2\\, K$ .", "where the thermal energy of the vortex is typically $k_{B} T_{SC} \\simeq 10^{-4}\\,\\textrm {eV}$ .", "It should be noted that the energy spectrum of the vortex dynamics in mobile vortex systems is expected to be broad around $\\Delta E$ , and the emission can take place within a certain range of the axion mass.", "In the second quantized language, the non-relativistic axion fields $\\varphi _{a}$ has the Fourier expansion, $&&\\varphi _{a}(x, t) = \\sum _{k} \\frac{1}{\\sqrt{V_{a}}}\\, a_{k}\\, e^{i \\vec{k}\\cdot \\vec{x} - i \\omega t} , \\\\&&\\left[a_{k}, a^{\\dagger }_{k^{\\prime }}\\right] = \\delta _{k,\\,k^{\\prime }}\\, ,\\quad \\left[a_{k}, a_{k^{\\prime }}\\right] =\\left[a^{\\dagger }_{k}, a^{\\dagger }_{k^{\\prime }}\\right] = 0 .$ Here, $V_{a}$ is the volume of a box where the axion field is defined, and we finally take the limit $V_{a} \\rightarrow \\infty $ (or cosmological scale) as usual.", "Inserting the Fourier expansion (REF ), the axion coupling becomes $&&\\int \\!\\!d^{3}x\\, g \\left(\\varphi _{a} e^{-i \\Omega t} + \\varphi _{a}^{\\dagger }e^{i \\Omega t}\\right) \\vec{E}\\cdot \\vec{B} \\nonumber \\\\&&\\simeq \\left(\\frac{g\\, h\\,B_{0}\\, \\ell _{\\textrm {vor}}}{e^{*}\\, \\sqrt{V_{a}}}\\right)\\sum _{k\\,\\simeq \\, k_{a}}\\left(a_{k}\\, e^{- i \\Omega t} + a^{\\dagger }_{k}\\,e^{i \\Omega t}\\right) \\vec{n}_{A}\\cdot \\vec{v}_{L} \\nonumber \\\\&& \\equiv ~ \\vec{{\\cal A}}_{\\textrm {eff}}(t) \\cdot \\vec{v}_{L} ,$ where $\\vec{n}_{A} = (\\vec{B}^{\\textrm {ex}} \\times \\vec{d}~)/ \\left|B^{\\textrm {ex}}\\right|$ is a unit vector in the direction of the effective gauge field.", "Here, we used the low-energy approximation, $e^{i\\vec{k}\\cdot \\vec{x}} \\sim 1$ and $\\Omega \\gg \\omega $ , and the axion distribution, which is highly concentrated around $k \\simeq k_{a}$ based on the assumption of the axion condensation.", "Since this coupling is the same form as the standard minimal coupling to the EM-field, the interaction Hamiltonian, which contributes the perturbative calculation, simply becomes $H_{\\textrm {int}}(t) = - \\frac{\\vec{{\\cal A}}_{\\textrm {eff}}(t)\\cdot \\vec{p}_{L}}{m_{\\textrm {vor}}} .$ From this interaction Hamiltonian, we can estimate the emission probability within the first-order perturbation, by using the Fermi's golden formula, $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] = \\int \\frac{d^{3} k\\, V_{a}}{\\left(2 \\pi \\right)^{3}}\\left(\\frac{2 \\pi }{\\hbar }\\right) \\left|\\langle i |H_{\\textrm {int}}(\\Omega )|f \\rangle \\right|^2 \\delta (E_{i}-E_{f}\\pm \\hbar \\Omega ) ,$ where the momentum integration is performed over the final states of the emitted axion.", "Since we assume that the axions form the BE condensate, the problem is rather simplified.", "In the condensed phase, the state can be expressed by a coherent state, which is an eigenstate of the annihilation operator: $a_{k} |{\\cal N}_{k}\\rangle \\propto \\sqrt{{\\cal N}_{k}}\\, |{\\cal N}_{k}\\rangle , \\quad \\langle {\\cal N}_{k} | {\\cal N}_{k} \\rangle = 1 .$ From this property, the matrix elements for the axion sector is evaluated as $\\langle i|\\, a_{k}^{\\dagger }\\,|f \\rangle = \\langle {\\cal N}_{k}|\\, a_{k}^{\\dagger }\\,| {\\cal N}_{k} \\rangle \\simeq \\sqrt{{\\cal N}_{k}} ,$ where ${\\cal N}_{k}$ is the occupation number of the axions with a condensed momentum $k_{a}$ .", "Inserting the explicit form of $\\vec{{\\cal A}}_{\\textrm {eff}}(t)$ into the formula (REF ) and taking the emission part proportional to $\\delta (E_{i}-E_{f}-\\hbar \\Omega )$ , the probability becomes $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] = \\left(\\frac{g\\, B_{0}\\, \\ell _{\\textrm {vor}}}{e^{*}\\,m_{\\textrm {vor}}}\\right)^{2} \\hbar \\, {\\cal N}_{k} \\int \\!\\!d^{3}k\\,\\left|\\langle p_{L} \\rangle \\right|^2\\,\\delta (E_{i}-E_{f}-\\hbar \\Omega ) ,$ where the matrix element for the vortex sector is denoted as $\\langle p_{L} \\rangle \\equiv \\langle i, \\textrm {vor} |\\,p_{L}\\,| f, \\textrm {vor}\\rangle $ .", "Since we assume that the temperature of the vortex system $T_{SC} \\sim 4\\, K$ is much higher than the temperature of the condensed axions $T_{a} \\sim 10^{-9}\\, K$ , the dominant process is the axion emission to the condensate.", "Note that the (apparent) dependence of the volume $V_{a}$ is cancelled in the resulting probability formula.", "Further assuming the matrix element $\\langle p_{L} \\rangle $ is independent of the momentum of the emitted axion in the low-energy region, we can perform the momentum integration, $\\int \\!\\!d^{3}k\\, \\delta (E_{i}-E_{f}-\\hbar \\Omega ) = 4 \\sqrt{2}\\,\\pi \\,\\frac{m_{a}^{3/2}\\sqrt{\\hbar \\Omega }}{\\hbar ^{3}} .$ Gathering the above formulas, we have the expression for the emission probability of the axion from a single vortex, $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] = {\\cal N}_{k} \\left(4 \\sqrt{2}\\,\\pi \\, m_{a}^{3/2}\\sqrt{\\hbar \\Omega }\\right)\\left(\\frac{g\\, B_{0}}{\\hbar \\, e^{*}\\, \\rho _{\\textrm {vor}}}\\right)^{2} \\left|\\langle p_{L} \\rangle \\right|^2 ,$ where we defined the vortex mass density $\\rho _{\\textrm {vor}} =m_{\\textrm {vor}}/\\ell _{\\textrm {vor}}\\, [\\textrm {kg}/m]$ .", "Finally, in order to evaluate the matrix element $\\left|\\langle p_{L} \\rangle \\right|^2$ , we consider a simple model of the vortex dynamics, for concreteness.", "The model is described by the Hamiltonian with an effective harmonic potential [35]:Another model can be considered based on the transverse oscillations of the vortex string with a small tension.", "$H_{\\textrm {vor}} = \\frac{p_{L}^2}{2\\, m_{\\textrm {vor}}} + \\frac{m_{\\textrm {vor}}\\, \\Omega ^2}{2} x^2= \\hbar \\Omega \\, a^{\\dagger } a ,$ where the creation and annihilation operators, $\\left[a, a^{\\dagger }\\right] = 1$ , are introduced.", "According to our assumption (I), the energy level should be $\\hbar \\Omega \\simeq m_{a} c^2 \\sim 10^{-6}\\, [\\textrm {eV}]$ .", "Using the relation $p_{L} = \\sqrt{\\frac{m_{\\textrm {vor}}\\,\\hbar \\Omega }{2}}\\left( a + a^{\\dagger } \\right)$ , the matrix element can be estimated as $\\left|\\langle p_{L} \\rangle \\right|^2 \\sim m_{\\textrm {vor}} \\left(\\hbar \\Omega \\right)$ .", "With this estimation, the emission probability from a single vortex (or a single segment of the vortex) becomes $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] \\simeq {\\cal N}_{k} \\left(4 \\sqrt{2}\\,\\pi \\,m_{\\textrm {vor}}\\,\\left(m_{a}\\,\\hbar \\Omega \\right)^{3/2}\\right) \\left(\\frac{g\\, B_{0}}{\\hbar \\, e^{*}\\, \\rho _{\\textrm {vor}}}\\right)^{2} .$ If an experimental sample of a superconductor contains the multiple vortices, the probability is proportional to the number of the vortices (for the tilted vortices) or the number of segments (for the wavy or kink-like vortices), which are denoted as $N_{\\textrm {vor}}$ .", "Thus, the emission probability of the dark matter axions from the vortex ensemble in type II superconductors becomes $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] \\simeq {\\cal N}_{k}\\, N_{\\textrm {vor}} \\left(4 \\sqrt{2}\\,\\pi \\,m_{\\textrm {vor}}\\,\\left(m_{a}\\,\\hbar \\Omega \\right)^{3/2}\\right) \\left(\\frac{g\\, B_{0}}{\\hbar \\, e^{*}\\, \\rho _{\\textrm {vor}}}\\right)^{2} .$ As the typical values of an experimental set-up, using $B_{0} \\sim 10\\, [\\textrm {T}]$ , $\\rho _{\\textrm {vor}} \\sim 10^{6}\\, [\\textrm {eV}/c^2\\, m]$ , $\\ell _{\\textrm {vor}} \\sim 10^{-4}\\, [m]$ , and $N_{\\textrm {vor}} \\sim 10^{5}$ , we can numerically estimate the probability, $\\textrm {Prob.}", "\\simeq 10^{10}\\, [\\textrm {sec}^{-1}] .$ Our result implies that, in an appropriate circumstances, the emission probability of the dark matter axions can be sizable, even if the axion coupling strength $g_{0}$ is quite small.", "For this type of process, the Bose statistics and small mass of the dark matter axions are crucial, which distinguishes from other dark matter candidates, such as WIMPs.", "We comment here the difference from the conventional conversion process of axions into photons with a background magnetic field through the axion coupling.", "In this process, a mode mixing between the coherent axion field and the EM-field is essential, where the both fields are treated as classical waves, and the aforementioned stimulated emission does not play any role [36], [37].", "In our axion emission process, the vortex is a heavy localized object compared to the axion, and thus the spontaneous emission of axions can be discussed in analogy with the case of spontaneous radiation from an atom." ], [ "Possible Signature of Dark Matter Axions", "An immediate consequence from the stimulated emission of dark matter axions is the axion-driven non-equilibrium states of the vortex ensemble in type II superconductors [30], [31].", "Emitted axions from the vortices are almost invisible, because the interactions with other particles are negligibly small.", "However, the vortices consequently lose thermal energy due to the emission of axions, by the amount, $E_{\\textrm {lost}} \\simeq \\textrm {Prob.}", "\\times m_{a} c^2\\, [\\textrm {eV}/\\textrm {sec}]$ .", "This process lowers the (local) temperature of the vortex system, and leads to the non-equilibrium state in the vortex ensemble.", "It should be emphasized that the temperature of the condensate of the dark matter axions is extremely low, $T_{a} \\sim 10^{-9} K$ , and the superconductor with vortices has the higher temperature $T_{\\textrm {SC}}\\sim {\\cal O}(1) K$ .", "Thus, the coupled system of the axions and vortices cannot reach the equilibrium as a whole in the laboratory.", "One possible experimental signature of the axion-driven non-equilibrium can be detected by the temperature measurement of a superconducting sample with mobile vortices in the vortex flow or vortex liquid.", "If the stimulated emission discussed in the last section is indeed realized, the adiabatic superconducting sample lowers its temperature spontaneously, whereas its temperature normally should rise in the absence of such an emission.", "Another possibility is to detect the transport phenomena of the magnetic vortices in type II superconductors, such as the vortex Hall effect from the vortex motions, in a non-equilibrium situation [30], [31], [35]." ], [ "Stimulated Emission from Other Excitations in Condensed Matter", "So far, we have intensively discussed the stimulated emission from the magnetic vortices in superconductors, and estimated the emission rates, as a concrete example.", "As discussed in Section 3, there are other types of excitations which can couple with the dark matter axions via the coupling (REF ).", "If such excitations have the energy level $E = \\hbar \\Omega \\sim 10^{-6}\\, [\\textrm {eV}]$ , the same arguments as the vortex case can be applied straightforwardly.", "At first, we consider an excitation in type A with the external magnetic field $\\vec{B}^{\\textrm {ex}} = (0,\\, 0,\\, B_{0})$ .", "In the linear approximation, the electric field created by such an excitation is represented by $E_{z} = \\gamma \\, a + \\gamma ^{*} a^{\\dagger }$ , with the annihilation and creation operators of the excitation mode.For the vortex case, $\\gamma = \\gamma ^{*} =\\left(\\frac{h}{e^{*}\\,\\rho _{\\textrm {vor}}}\\right)\\sqrt{\\frac{m_{\\textrm {vor}} \\hbar \\Omega }{2}}$ .", "In this case, the stimulated emission rate is generally given by the following formula $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] = {\\cal N}_{k} N_{\\textrm {exc}} \\left(4 \\sqrt{2}\\,\\pi \\, m_{a}^{3/2}\\sqrt{\\hbar \\Omega }\\right) \\left(\\frac{g\\, B_{0}\\, |\\gamma |}{2 \\pi \\hbar }\\right)^2 .$ Here, the number of the excitations in the sample is given by $N_{\\textrm {exc}}$ , and the excitation level is assumed to be $\\hbar \\Omega \\simeq m_{a} c^2$ .", "For the excitations of type B, which create a magnetic field under the background electric field $\\vec{E}^{\\textrm {ext}} = (0,\\, 0,\\, E_{0})$ , similar calculations based on the coupling (REF ) can be applied, and we can obtain the emission rate, $\\textrm {Prob.", "}\\, [\\textrm {sec}^{-1}] = {\\cal N}_{k} N_{\\textrm {exc}} \\left(4 \\sqrt{2}\\,\\pi \\, m_{a}^{3/2}\\sqrt{\\hbar \\Omega }\\right) \\left(\\frac{g\\, E_{0}\\, |\\gamma |}{2 \\pi \\hbar }\\right)^2 ,$ where $N_{\\textrm {exc}}$ is the total number of the excitations in the sample.", "If the excitations of type C in a multiferroic (or topological) material also have the energy spectrum of the order $10^{-6}\\,[\\textrm {eV}]$ , the stimulated emission of axions can be realized with either an external electric field or a magnetic field via the coupling (REF ) or (REF ).", "The emission rate is given by the formulas (REF ) or (REF ) in the same way." ], [ "Summary and Discussion", "We have discussed the coupling between dark matter axions and excitations in various condensed matter systems, and physical consequences from the coupling.", "In particular, we investigated the stimulated emission of dark matter axions, which are expected to take place BE condensates in our galaxy, from the various excitations in condensed matter systems through the axion coupling.", "As a concrete example, the emission from magnetic vortex strings of mobile vortex systems in superconductors was investigated, and a possible experimental signature, that is the spontaneous cooling and the resulting non-equilibrium state of the vortex ensemble, was discussed.", "It is expected that the stimulated emissions possibly give a new strategy to detect dark matter axions.", "For more concrete experimental set-up, further understanding of dynamics of the vortex strings in the mobile vortex systems, such as vortex flow and vortex liquid states, is required, and the investigation of such vortex dynamics is a work in progress.", "Although we focused on the vortex strings in s-wave superconductors in this paper, the axion coupling and emission in the case of vortex strings in (chiral) p-wave supercondutors should also be interesting.", "Generalizations to other dark matter models with axion-like particles are straightforward.", "The emission rates are proportional to $g_{0}^{2}/m_{a}^2$ , where $g_{0}$ and $m_{a}$ are the coupling strength and mass of the dark matter axion, and are enhanced for smaller mass axion if condensed matter excitations have the energy spectrum $\\hbar \\Omega \\sim m_{a} c^2.$Recently, new detection methods of ultralight axion-like particles corresponding to BEC dark matters are proposed.", "See [38], [39], [40] for example.", "There is another interesting problem: The excitation of type C, such as an analogue of the axion in a topological magnetic insulator, has the same effective coupling as the axion coupling.", "Thus the mixing between the dark matter axions and condensed matter axions can occur in principle.", "Physical consequences from such a mixing will be discussed in a future work.", "From the perspective of thermodynamics, the axion condensate plays the role of heat bath of an extremely low temperature for the condensed matter excitations discussed in this paper.", "Therefore, the mechanism investigated in this paper can lead to a novel energy conversion process from the condensed matter excitations.", "Acknowledgements The authors thank J. Lustikova, S. Murakami, K. Sato, Y. Shiomi, and F. Takahashi for useful discussions.", "This work was supported in part by Grant-in Aid for Scientific Research on Innovative Areas \"Nano Spin Conversion Science\" (26103005).", "The work of E. S. was supported in part by ERATO, JST." ], [ "Dimensional Analysis on Axion Coupling", "In this Appendix, we determine the unit (or dimension) of the coupling strength $g_{0}$ between the axion and EM-field.", "In the following, we denote the dimension of the Mass, Length, and Time as $[M]$ , $[L]$ , and $[T]$ , respectively.", "For the units of the EM-field, we take the electric charge as the fourth element of our unit, and denote the dimension of charge as $[Q]$ .", "Here, we take the MKSC units (C represents [Coulomb]).", "Since the dimension of the action $S$ is given by $[S] = \\left[M L^{2} T^{-1}\\right]$ , we can determine the dimension of $\\phi _{a}$ , $[\\phi _{a}] = \\left[M^{-1/2} L^{-3/2}\\right] .$ As is well-known, the dimensions of the EM-field are determined from the equations of motion for the charged particles, $[\\vec{E}] = \\left[M L T^{-2} Q^{-1}\\right], \\qquad [\\vec{B}] = \\left[M T^{-1} Q^{-1}\\right] .$ From these relations, we obtain the dimension of the vector potential, $[\\vec{A}] = \\left[M L T^{-1} Q^{-1}\\right]$ .", "From the above dimensions, we have the dimension of the following term (operator), $[\\phi _{a}\\,\\vec{E}\\cdot \\vec{B}] = \\left[M^{3/2} L^{-1/2} T^{-3} Q^{-2}\\right] .$ Using the $[S] = \\left[M L^{2} T^{-1}\\right]$ , we can determine the dimension of the axion coupling strength, $[g_{0}] = \\left[M^{-1/2} L^{-1/2} T Q^{2}\\right] .$ Thus, the coupling strength $g_{\\textrm {mks}}$ in the MKSC unit becomes $g_{0} = g_{\\textrm {mks}}~ \\left[\\frac{\\textrm {sec}~ C^2}{\\sqrt{\\textrm {kg}~ m}}\\right] .$ The experimental bound (or constraint) of the value of the axion coupling strength is usually discussed in the natural units over the communities of high-energy physics, cosmology and astrophysics.", "To convert the MKSC units into the natural units, the following conversion formulas are useful: $1\\, \\textrm {kg} &=& 5.61 \\times 10^{35}~ [\\textrm {eV}/c^2] , \\nonumber \\\\1\\, m &=& 5.07 \\times 10^{6}~ [\\hbar c/\\textrm {eV}] , \\nonumber \\\\1\\, \\textrm {sec} &=& 1.52 \\times 10^{15}~ [\\hbar /\\textrm {eV}] , \\nonumber \\\\1\\, \\textrm {eV} &=& 1.60 \\times 10^{-19}~ [\\textrm {Joule}] ,$ where $c = 3.00 \\times 10^{8}~m/\\textrm {sec}$ is the light velocity.", "In the natural units, we set $c=\\hbar =\\varepsilon _{0}=\\mu _{0}=1$ .", "The (dimensionless) fine structure constant $\\alpha = e^2/4 \\pi \\varepsilon _{0} \\hbar c = 7.30 \\times 10^{-3}~ (\\sim 1/137)$ gives the elementary electric charge, $e = \\left(4 \\pi \\alpha \\,\\varepsilon _{0} \\hbar c \\right)^{1/2} [C],$ and $e=0.303$ (dimensionless) in the natural units.", "Thus, the axion coupling strength in the natural units $g_{\\textrm {nat}}$ is given by $g_{\\textrm {nat}} \\simeq 3.22 \\times 10^{30} \\times g_{\\textrm {mks}}~ [\\textrm {eV}^{-1}] .$ The upper bounds of this coupling strength originate from various astronomical observations of sun, neutron stars, and supernovae, which lead to $g_{\\textrm {nat}} \\lesssim 10^{-19}\\, [\\textrm {eV}^{-1}] \\quad \\Longrightarrow \\quad g_{\\textrm {mks}} \\lesssim 3 \\times 10^{-50}~ \\left[\\frac{\\textrm {sec}~ C^2}{\\sqrt{\\textrm {kg}~ m}}\\right].$ This means quite tiny coupling strength." ] ]
1612.05406
[ [ "Maximum Matching on Trees in the Online Preemptive and the Incremental\n Dynamic Graph Models" ], [ "Abstract We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models.", "In the {\\em Online Preemptive} model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching.", "On seeing an edge, the algorithm has to either accept or reject the edge.", "If accepted, then the adjacent edges are discarded, and all rejections are permanent.", "In this model, the complexity of the problems is settled for deterministic algorithms.", "Epstein et al.", "gave a $5.356$-competitive randomized algorithm for MWM, and also proved a lower bound of $1.693$ for MCM.", "The same lower bound applies for MWM.", "In this paper we show that some of the results can be improved in the case of trees and some special classes of graphs.", "In the online preemptive model, we present a $64/33$-competitive (in expectation) randomized algorithm for MCM on trees.", "Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a \"worst case\" update time of $O(1)$, in the incremental dynamic graph model, which is $3/2$-approximate (in expectation) on trees, and $1.8$-approximate (in expectation) on general graphs with maximum degree $3$.", "Note that this algorithm works only against an oblivious adversary.", "Hence, we derandomize this algorithm, and give a $(3/2 + \\epsilon)$-approximate deterministic algorithm for MCM on trees, with an amortized update time of $O(1/\\epsilon)$.", "We also present a minor result for MWM in the online preemptive model, a $3$-competitive (in expectation) randomized algorithm on growing trees (where the input revealed upto any stage is always a tree, i.e.", "a new edge never connects two disconnected trees)." ], [ "Introduction", "The Maximum (Cardinality/Weight) Matching problem is one of the most extensively studied problems in Combinatorial Optimization.", "See Schrijver's book [13] and references therein for a comprehensive overview of classic work.", "A matching $M\\subseteq E$ is a set of edges such that at most one edge is incident on any vertex.", "Traditionally the problem was studied in the offline setting where the entire input is available to the algorithm beforehand.", "But over the last few decades it has been extensively studied in various other models where the input is revealed in pieces, like the vertex arrival model (adversarial and random), the edge arrival model (adversarial and random), streaming and semi-streaming models, the online preemptive model, etc.", "[10], [5], [4], [11], [6], [9].", "In this paper, we study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive model, and in the Incremental Dynamic Graph model.", "(Refer Section REF for a comparison between the two models.)", "In the online preemptive model, the edges arrive in an online manner, and the algorithm is supposed to accept or reject an edge on arrival.", "If accepted, the algorithm can reject it later, and all rejections are permanent.", "The algorithm is supposed to always maintain a valid matching.", "There is a $5.828$ -competitive deterministic algorithm due to McGregor [11] for MWM, and a tight lower bound for deterministic algorithms due to Varadaraja [15].", "Epstein et al.", "[5] gave a $5.356$ -competitive randomized algorithm for MWM, and also proved a $1.693$ lower bound on the competitive ratio achievable by any randomized algorithm for MCM.", "No better lower bound is known for MWM.", "In [3], the authors gave the first randomized algorithm with competitive ratio ($28/15$ in expectation) less than 2 for MCM in the online preemptive model, on growing trees (defined in Section REF ).", "In Section , we extend their algorithm to give a $64/33$ -competitive (in expectation) randomized (which uses only two bits of randomness) algorithm for MCM on trees.", "Although the algorithm is an extension of the one for growing trees in [3], it motivates the algorithm (described in Section ) for MCM in the incremental dynamic graph model.", "Note that the adversary presenting the edges in the online preemptive model is oblivious, and does not have access to the random choices made by the algorithm.", "In recent years, algorithms for approximate MCM in dynamic graphs have been the focus of many studies due to their wide range of applications.", "Here [1], [2], [8], [14] is a non-exhaustive list some of the studies.", "The objective of these dynamic graph algorithms is to efficiently process an online sequence of update operations, such as edge insertions and deletions.", "It has to quickly maintain an approximate maximum matching despite an adversarial order of edge insertions and deletions.", "Dynamic graph problems are usually classified according to the types of updates allowed: incremental models allow only insertions, decremental models allow only deletions, and fully dynamic models allow both.", "We study MCM in the incremental model.", "Gupta [7] proved that for any $\\epsilon \\le 1/2$ , there exists an algorithm that maintains a $(1+\\epsilon )$ -approximate MCM on bipartite graphs in the incremental model in an “amortized” update time of $O\\left(\\frac{\\log ^2 n}{\\epsilon ^4}\\right)$ .", "We present a randomized algorithm for MCM in the incremental model with a “worst case” update time of $O(1)$ , which is $3/2$ -approximate (in expectation) on trees, and $1.8$ -approximate (in expectation) on general graphs with maximum degree 3.", "This algorithm works only against an oblivious adversary.", "Hence, we derandomize this algorithm, and give a $(3/2+\\epsilon )$ -approximate deterministic algorithm for MCM on trees, with an amortized update time of $O(1/\\epsilon )$ .", "Note that the algorithm of Gupta [7] is based on multiplicative weights update, and it therefore seems unlikely that a better running time analysis for special classes of graphs is possible.", "We present a minor result in Section , a 3-competitive (in expectation) randomized algorithm (which uses only $O(1)$ bits of randomness) for MWM on growing trees in the online preemptive model.", "Although, growing trees is a very restricted class of graphs, there are a couple of reasons to study the performance of the algorithm on this class of input.", "Firstly, almost all lower bounds, including the one due to Varadaraja [15] for MWM are on growing trees.", "Secondly, even for this restricted class, the analysis is involved.", "We use the primal-dual technique for analyzing the performance of this algorithm, and show that this analysis is indeed tight by giving an example, for which the algorithm achieves the competitive ratio 3.", "We describe the algorithm for general graphs, but are only able to analyze it for growing trees, and new ideas are needed to prove a better bound for general graphs." ], [ "Preliminaries", "We use primal-dual techniques to analyze the performance of all the randomized algorithms described in this paper.", "Here are the well known Primal and Dual formulations of the matching problem.", "Table: NO_CAPTIONFor MCM, $w_e=1$ for any edge.", "Any matching $M$ implicitly defines a feasible primal solution.", "If an edge $e\\in M$ , then $x_e=1$ , otherwise $x_e=0$ .", "Suppose an algorithm outputs a matching $M$ , then let $P$ be the corresponding primal feasible solution.", "Let $D$ denote some feasible dual solution.", "The following claim can be easily proved using weak duality.", "Claim 1 If $D \\le \\alpha \\cdot P$ , then the algorithm is $\\alpha $ -competitive.", "If $M$ is any matching, then for an edge $e$ , $X(M,e)$ denotes edges in $M$ which share a vertex with the edge $e$ .", "We will say that a vertex(/an edge) is covered by a matching $M$ if there is an edge in $M$ which is incident on(/adjacent to) the vertex(/edge).", "We also say that an edge is covered by a matching $M$ if it belongs to $M$ .", "In the online preemptive model, growing trees are trees, such that a new edge has exactly one vertex common with already revealed edges." ], [ "Online Preemptive Model vs. Incremental Dynamic Graph Model", "There are two main differences between these models.", "Firstly, in the online preemptive model, once an edge is rejected/removed from the matching maintained by the algorithm, it cannot be added into its matching, whereas in the incremental dynamic graph model, rejected/removed edges can be added to the matching later on.", "Secondly, there is no restriction on how much time an algorithm in the online preemptive model can use to process a revealed edge, whereas in the incremental dynamic graph model, the algorithm is supposed to process the revealed edge fast.", "The term “fast” is used loosely, and is specific to any problem.", "For example, MCM on general graphs can be found in time $O(m\\sqrt{n})$ when the entire input is available [12].", "But for dynamic graphs, every time an edge is inserted, the algorithm is expected to maintain a matching, approximate if not exact, in time lower than the time required by the optimal offline algorithm for MCM (say, for instance, in $O(\\operatorname{polylog}n)$ amortized time)." ], [ "MCM in the Online Preemptive Model", "In this section, we present a randomized algorithm (that uses only 2 bits of randomness) for MCM on trees in the online preemptive model.", "The algorithm maintains four matchings $M_1,M_2,M_3,M_4$ , and it tries to ensure that a large number of input edges are covered by some or other matchings.", "(Here, the term “large number” is used vaguely.", "Suppose more than four edges are incident on a vertex, then at most four of them will belong to matchings, one to each.)", "One of the four matchings is output uniformly at random.", "A more formal description of the algorithm follows.", "[H] Randomized Algorithm for MCM on Trees Pick $l\\in _{u.a.r.", "}\\lbrace 1,2,3,4\\rbrace $ .", "The algorithm maintains four matchings: $M_1,M_2,M_3,$ and $M_4$ .", "On arrival of an edge $e$ , the processing happens in two phases.", "The Augment phase.", "The new edge $e$ is added to each $M_i$ in which there are no edges adjacent to $e$ .", "The Switching phase.", "For $i=2,3,4$ , in order, $M_i\\leftarrow M_i\\setminus X(M_i,e)\\cup \\lbrace e\\rbrace $ , provided it decreases the quantity $\\sum _{j\\in [4],i\\ne j, |X(M_i\\cap M_j,e)| = |X(M_i,e)|}|M_i\\cap M_j|$ .", "Output $M_l$ .", "Although, $l$ is picked randomly at the beginning of the algorithm, this value is not known to the adversary.", "Note that in the switching phase, the expected size of the matching stored by the algorithm might decrease.", "For example, consider two disjoint edges $e_1$ and $e_2$ that have been revealed.", "Each of them will belong to all four matchings.", "So the expected size of the matching stored by the algorithm is 2.", "Now, if an edge $e$ is revealed between $e_1$ and $e_2$ , then $e$ will be added to $M_2$ and $M_3$ .", "The expected size of the matching is now $1.5$ .", "The important thing to notice here is that the decrease is not too much, and we are able to prove that the competitive ratio of the algorithm still remains below 2.", "We begin with the following observations.", "After an edge is revealed, its end points are covered by all four matchings.", "An edge $e$ that does not belong to any matching has four edges incident on its end points such that each of these edges belongs to a distinct matching.", "This holds when the edge is revealed, and does not change subsequently.", "Every edge is covered by at least three matchings.", "An edge is called internal if there are edges incident on both its end points which belong to some matching.", "An edge is called a leaf edge either if one of its end point is a leaf or if all the edges incident on one of its end points do not belong to any matching.", "An edge is called bad if its end points are covered by only three matchings.", "We begin by proving some properties about the algorithm.", "The key structural lemma that keeps “influences” of bad edges local is given below.", "Lemma 2 At most five consecutive vertices on a path can have bad edges incident on them.", "According to Lemma REF , there can be at most four consecutive internal bad edges or at most five bad leaf edges incident on five consecutive vertices of a path.", "Lemma REF is proved in Appendix .", "Once all edges have been seen, we distribute the primal charge among the dual variables, and use the primal-dual framework to infer the competitive ratio.", "If the end points of every edge are covered with four matchings, then the distribution of dual charge is easy.", "However we do have bad edges, and would like the edges in matchings to contribute more to the end-points of these edges.", "Then, the charge on the other end-point would be less and we need to balance this through other edges.", "Details follow.", "Lemma 3 There exists an assignment of the primal charge to the dual variables such that the dual constraint for each edge $e\\equiv (u,v)$ is satisfied at least $\\frac{33}{64}$ in expectation, i.e.", "$\\mathbb {E}[y_u+y_v]\\ge \\frac{33}{64}$ .", "Root the tree at an arbitrary vertex.", "For any edge $e\\equiv (u,v)$ , let $v$ be the parent vertex, and $u$ be the child vertex.", "The dual variable assignment is done after the entire input is seen, as follows.", "Dual Variable Management: An edge $e$ will distribute its primal weight between its end-points.", "The exact values are discussed below.", "In general, we look to transfer all of the primal charge to the parent vertex.", "But this does not work and we need a finer strategy.", "This is detailed below.", "If $e$ does not belong to any matching, then it does not contribute to the value of dual variables.", "If $e$ belongs to a single matching then, depending on the situation, one of 0, $\\epsilon $ , $2\\epsilon $ , $3\\epsilon $ , $4\\epsilon $ , or $5\\epsilon $ of its primal charge will be assigned to $u$ and the rest will be assigned to $v$ .", "If $e$ belongs to two matchings, then at most $6\\epsilon $ of its primal charge will be assigned to $u$ as required.", "The rest is assigned to $v$ .", "If $e$ belongs to three or four matchings, then its entire primal charge is assigned to $v$ .", "We will show that $y_u+y_v\\ge 2+\\epsilon $ for such an edge, when summed over all four matchings.", "The value of $\\epsilon $ is chosen later.", "For the sake of analysis, if there are bad leaf edges incident on both the end points of an internal edge, then we analyze it as a bad internal edge.", "We need to do this because a bad leaf edge might need to transfer its entire primal charge to the vertex on which there are edges which do not belong to any matching.", "Note that the end points of the internal edge would still be covered by three matchings, even if we consider that the bad leaf edges do not exist on its end points.", "The analysis breaks up into eight cases.", "Case 1.", "Suppose $e$ does not belong to any matching.", "There must be a total of at least 4 edges incident on $u$ and $v$ besides $e$ , each belonging to a distinct matching.", "Of these 4, at least a total of 3, say $e_1$ , $e_2$ , and $e_3$ , must be between some children of $u$ and $v$ , to $u$ and $v$ respectively.", "The edges $e_1$ , $e_2$ , and $e_3$ , each assign a charge of at least $1-5\\epsilon $ to $y_u$ and $y_v$ , respectively.", "Therefore, $y_u+y_v\\ge 3-15\\epsilon \\ge 2+\\epsilon $ .", "Case 2.", "Suppose $e$ is a bad leaf edge that belongs to a single matching, and internal edges are incident on $v$ .", "This implies that there is an edge $e_1$ from a child vertex of $v$ to $v$ , which belongs to single matching, and another edge $e_2$ , also belonging to single matching from $v$ to its parent vertex.", "The edge $e$ assigns a charge of 1 to $y_v$ .", "If $e_1$ assigns a charge of 1 or $1-\\epsilon $ or $1-2\\epsilon $ or $1-3\\epsilon $ or $1-4\\epsilon $ to $y_v$ , then $e_2$ assigns $\\epsilon $ or $2\\epsilon $ or $3\\epsilon $ or $4\\epsilon $ or $5\\epsilon $ respectively to $y_v$ .", "In either case, $y_u+y_v=2+\\epsilon $ .", "The key fact is that $e_1$ could not have assigned $5\\epsilon $ to its child vertex.", "Since, then, by Lemma REF , $e$ cannot be a bad edge.", "Case 3.", "Suppose $e$ is a bad leaf edge that belongs to a single matching, and internal edges are incident on $u$ .", "This implies that there are two edges $e_1$ and $e_2$ from children of $u$ to $u$ , each belonging to a single distinct matching.", "The edge $e$ assigns a charge of 1 to $y_v$ .", "Both $e_1$ and $e_2$ assign a charge of at least $1-4\\epsilon $ to $y_u$ .", "In either case, $y_u+y_v\\ge 3-8\\epsilon \\ge 2+\\epsilon $ .", "The key fact is that neither $e_1$ nor $e_2$ could have assigned more than $4\\epsilon $ to their corresponding child vertices.", "Since, then, by Lemma REF , $e$ cannot be a bad edge.", "Case 4.", "Suppose $e$ is an internal bad edge.", "This implies (by Lemma REF ) that there is an edge $e_1$ from a child vertex of $u$ to $u$ , which belongs to a single matching.", "Also, there is an edge $e_2$ , from $v$ to its parent vertex (or from a child vertex $v$ to $v$ ), which also belongs to a single matching.", "The edge $e$ assigns its remaining charge (1 or $1-\\epsilon $ or $1-2\\epsilon $ or $1-3\\epsilon $ or $1-4\\epsilon $ ) to $y_v$ .", "If $e_1$ assigns a charge of 1 or $1-\\epsilon $ or $1-2\\epsilon $ or $1-3\\epsilon $ or $1-4\\epsilon $ to $y_u$ , then $e_2$ assigns $\\epsilon $ or $2\\epsilon $ or $3\\epsilon $ or $4\\epsilon $ or $5\\epsilon $ respectively to $y_v$ .", "In either case, $y_u+y_v=2+\\epsilon $ .", "The key fact is that $e_1$ could not have assigned $5\\epsilon $ to its child vertex.", "Since, then, by Lemma REF , $e$ cannot be a bad edge.", "Case 5.", "Suppose $e$ is not a bad edge, and it belongs to a single matching.", "Then either there are at least two edges $e_1$ and $e_2$ from child vertices of $u$ or $v$ to $u$ or $v$ respectively, or $e_1$ on $u$ and $e_2$ on $v$ , each belonging to a single matching, or one edge $e_3$ from a child vertex of $u$ or $v$ to $u$ or $v$ , respectively, which belongs to two matchings, or one edge $e_4$ from a child vertex of $u$ or $v$ to $u$ or $v$ , respectively, which belongs to single matching, and one edge $e_5$ from $v$ to its parent vertex which belongs to two matchings.", "In either case, $y_u+y_v\\ge 3-10\\epsilon \\ge 2+\\epsilon $ .", "Case 6.", "Suppose $e$ is a bad edge that belongs to two matchings, and internal edge is incident on $u$ or $v$ .", "This implies that there is an edge $e_1$ , from a child vertex of $u$ to $u$ or from $v$ to its parent vertex which belongs to a single matching.", "The edge $e$ assigns a charge of 2 to $y_v$ , and the edge $e_1$ assigns a charge of $\\epsilon $ to $y_u$ or $y_v$ respectively.", "Thus, $y_u+y_v=2+\\epsilon $ .", "Case 7.", "Suppose $e$ is not a bad edge and it belongs to two matchings.", "This means that either there is an edge $e_1$ from a child vertex of $u$ to $u$ , which belongs to at least one matching, or there is an edge from child vertex of $v$ to $v$ that belongs to at least one matching, or there is an edge from $v$ to its parent vertex which belongs to two matchings.", "The edge $e$ assigns a charge of 2 among $y_u$ and $y_v$ .", "The neighboring edges assign a charge of $\\epsilon $ to $y_u$ or $y_v$ (depending on which vertex it is incident to), yielding $y_u+y_v\\ge 2+\\epsilon $ .", "Case 8.", "Suppose, $e$ belongs to 3 or 4 matchings, then trivially $y_u+y_v\\ge 2+\\epsilon $ .", "From the above cases, $y_v+y_v\\ge 3-15\\epsilon $ and $y_u+y_v\\ge 2+\\epsilon $ .", "The best value for the competitive ratio is obtained when $\\epsilon =\\frac{1}{16}$ , yielding $\\mathbb {E}[y_u+y_v]\\ge \\frac{33}{64}$ .", "Lemma REF immediately implies Theorem REF using Claim REF .", "Theorem 4 Algorithm  is a $\\frac{64}{33}$ -competitive randomized algorithm for finding MCM on trees." ], [ "MCM in the Incremental Dynamic Graph Model", "In this section, we present our main result, a randomized algorithm (that uses only $O(1)$ bits of randomness) for MCM in the incremental dynamic graph model, which is $3/2$ -approximate (in expectation) on trees, and is $1.8$ -approximate (in expectation) on general graphs with maximum degree 3, with $O(1)$ worst case update time per edge.", "It is inspired by the randomized algorithm for MCM on trees described in Section .", "In the online preemptive model, we cannot add edges in the matching which were discarded earlier, which results in the existence of bad edges.", "But in the incremental dynamic graph model, there is no such restriction.", "For some $i\\in [3]$ , let $e\\equiv (u,v) \\in M_i$ be switched out by some edge $e^{\\prime }\\equiv (u,u^{\\prime })$ , i.e.", "$M_i\\leftarrow M_i\\setminus \\lbrace e\\rbrace \\cup \\lbrace e^{\\prime }\\rbrace $ .", "If there is an edge $e^{\\prime \\prime }\\equiv (v,v^{\\prime })\\in M_j$ for $i\\ne j$ , then we can add $e^{\\prime \\prime }$ to $M_i$ if possible.", "Using this simple trick, we get a better approximation ratio in this model, and also, the analysis becomes significantly simpler.", "Details follow.", "[H] Randomized Algorithm for MCM Pick $l \\in _{u.a.r.}", "\\lbrace 1,2,3\\rbrace $ .", "The algorithm maintains three matchings: $M_1,M_2,$ and $M_3$ .", "When an edge $e$ is inserted, the processing happens in two phases.", "The Augment phase.", "The new edge $e$ is added to each $M_i$ in which there are no edges adjacent to $e$ .", "The Switching phase.", "For $i=2,3$ , in order, $M_i\\leftarrow M_i\\setminus X(M_i,e)\\cup \\lbrace e\\rbrace $ , provided it decreases the quantity $\\sum _{j\\in [3],i\\ne j, |X(M_i\\cap M_j,e)| = |X(M_i,e)|}|M_i\\cap M_j|$ .", "For every edge $e^{\\prime }$ discarded from $M_i$ , add edges on the other end point of $e^{\\prime }$ in $M_j$ ($\\forall j\\ne i$ ) to $M_i$ if possible.", "Output the matching $M_l$ on query.", "Note that the end points of every edge will be covered by all three matchings, and hence all three matchings are maximal.", "We again use the primal-dual technique to analyze the performance of this algorithm on trees.", "Lemma 5 There exists an assignment of the primal charge amongst the dual variables such that the dual constraint for each edge $e\\equiv (u,v)$ is satisfied at least $\\frac{2}{3}$ rd in expectation.", "Root the tree at an arbitrary vertex.", "For any edge $e\\equiv (u,v)$ , let $v$ be the parent vertex, and $u$ be the child vertex.", "The dual variable assignment is done at the end of input/on query, as follows.", "If $e$ does not belong to any matching, then it does not contribute to the value of dual variables.", "If $e$ belongs to a single matching, then its entire primal charge is assigned to $v$ as $y_v=1$ .", "If $e$ belongs to two matchings, then its entire primal charge is assigned equally amongst $u$ and $v$ , as $y_u=1$ and $y_v=1$ .", "If $e$ belongs to three matchings, then its entire primal charge is assigned to $v$ as $y_v=3$ .", "The analysis breaks up into three cases.", "Case 1.", "Suppose $e$ does not belong to any matching.", "There must be a total of at least 2 edges incident amongst $u$ and $v$ besides $e$ , each belonging to a distinct matchings, from their respective children.", "Therefore, $y_u+y_v\\ge 2$ .", "Case 2.", "Suppose $e$ belongs to a single matching.", "Then either there is an edge $e^{\\prime }$ incident on $u$ or $v$ which belongs to a single matching, from their respective children, or there is an edge $e^{\\prime \\prime }$ incident on $u$ or $v$ which belongs to two matchings.", "In either case, $y_u+y_v\\ge 2$ .", "Case 3.", "Suppose $e$ belongs to two or three matchings, then $y_u+y_v\\ge 2$ trivially.", "Lemma REF immediately implies Theorem REF using Claim REF .", "Theorem 6 Algorithm  is a $\\frac{3}{2}$ -approximate (in expectation) randomized algorithm for MCM on trees, with a worst case update time of $O(1)$ .", "We also analyze Algorithm  for general graphs with maximum degree 3, and prove the following Theorem.", "Theorem 7 Algorithm  is a $1.8$ -approximate (in expectation) randomized algorithm for MCM on general graphs with maximum degree 3, with a worst case update time of $O(1)$ .", "We use the following Lemma to prove Theorem REF .", "Lemma 8 There exists an assignment of the primal charge amongst the dual variables such that the dual constraint for each edge $e\\equiv (u,v)$ is satisfied at least $\\frac{5}{9}$ th in expectation.", "The dual variable assignment is done at the end of input/or query, as follows.", "If $e$ does not belong to any matching, then it does not contribute to the value of dual variables.", "If $e$ belongs to a single matching, then there are two sub cases.", "W.l.o.g., if $u$ is covered by a single matching, then primal charge $x_e=1$ is divided as $y_u=1/2+\\epsilon $ and $y_v=y_v+1/2-\\epsilon $ .", "If both $u$ and $v$ are covered by at least two matchings, then primal charge $x_e=1$ is divided as $y_u=y_u+1/2$ and $y_v=y_v+1/2$ .", "If $e$ belongs to two or three matchings, then its entire primal charge is divided equally amongst $u$ and $v$ .", "The analysis breaks up into three cases.", "Case 1.", "Suppose $e$ does not belong to any matching.", "Then $u$ and $v$ must be covered by a total of at least 3 matchings (counting multiplicities).", "W.l.o.g., if $u$ is covered by a single matching, then $v$ has to be covered by at least two matchings.", "Hence, $y_u=1/2+\\epsilon $ , and $y_v\\ge 1$ .", "Else, both $u$ and $v$ are covered by at least two matchings, then $y_u\\ge 1$ and $y_v\\ge 1$ .", "Therefore, $y_u+y_v\\ge 3/2+\\epsilon $ .", "Case 2.", "Suppose $e$ belongs to a single matching.", "Then, $y_u+y_v\\ge 1+1/2-\\epsilon +1/2-\\epsilon =2-2\\epsilon \\ge 3/2+\\epsilon $ .", "Case 3.", "Suppose $e$ belongs to two or three matchings, then $y_u+y_v\\ge 3/2+\\epsilon $ trivially.", "The proof of Lemma is complete with $\\epsilon =1/6$ .", "Lemma REF immediately implies Theorem REF using Claim REF ." ], [ "A Deterministic Algorithm", "Note that Algorithm  only works against an oblivious adversary.", "In this section, we derandomize Algorithm  to give a $(3/2+\\epsilon )$ -approximation deterministic algorithm, for MCM on trees, with an amortized update time of $O(1/\\epsilon )$ , for any $\\epsilon \\le 1/2$ .", "[H] Deterministic Algorithm for MCM Let $\\epsilon \\in (0,1/2]$ be some input parameter, and $c=1$ .", "The algorithm maintains four matchings: $M_1,M_2,M_3$ and a support matching $M_4$ .", "On query, output matching $M_c$ .", "When an edge $e$ is inserted, the processing happens in four phases.", "The Augment phase.", "The new edge $e$ is added to each $M_i$ in which there are no edges adjacent to $e$ .", "The Switching phase.", "For $i=2,3$ , in order, $M_i\\leftarrow M_i\\setminus X(M_i,e)\\cup \\lbrace e\\rbrace $ , provided $|X(M_i,e)|=1$ and it decreases the quantity $\\sum _{j\\in [3],i\\ne j}|M_i\\cap M_j|$ .", "For the edge $e^{\\prime }$ that is discarded from $M_i$ , add edges on the other end point of $e^{\\prime }$ in $M_j$ ($\\forall j\\in [4],j\\ne i$ ) to $M_i$ if possible.", "The Support phase.", "If the edge $e$ was not added to any $M_i$ , $\\forall i\\in [3]$ , in the Augment or Switching phase, then add it to the support matching $M_4$ if there are no edges adjacent to it in $M_4$ .", "The ChangeCurr phase.", "If $|M_c|< \\left(|M_i|+|M_j|\\right)/(2(1+\\epsilon ))$ such that $i,j,c\\in [3]$ , and are all distinct, then set $c = k$ if $M_k$ is the matching of maximum size among $M_1,M_2,M_3$ .", "Note that there are two more phases in this algorithm than Algorithm , and there is also a minor modification in the description of switching phase.", "These changes are done to ensure that the size of any matching maintained by the algorithm never decreases(, which helps with the analysis as pointed out later).", "An edge can be added to $M_i$ in the Switching phase only if it has one conflicting edge in $M_i$ .", "But this can result in $M_2$ and $M_3$ not being maximal matchings(, which is again required in the analysis as pointed out later).", "The only way this can happen is if some edge $e$ is not added to any matching in the Augment phase, and later on, after the Switching phase, its end points are not covered by $M_2$ and $M_3$ .", "We add such an edge to the support matching $M_4$ , and this edge is later added to $M_2$ and $M_3$ in the Switching phase, thereby ensuring their maximality.", "With these modifications, the approximation ratio claimed in Theorem REF still holds on average size of matchings $M_1,M_2,M_3$ stored by this algorithm.", "We prove the following theorem for Algorithm REF .", "Theorem 9 Algorithm REF is $\\left(\\frac{3}{2}+\\epsilon \\right)$ -approximate for MCM on trees, with an amortized update time of $O(1/\\epsilon )$ .", "We first prove the approximation ratio, and then argue about the update time per edge.", "Step (3c) in Algorithm REF ensures that at each stage $|M_c|\\ge \\left(|M_i|+|M_j|\\right)/(2(1+\\epsilon ))$ , such that $i,j,c \\in [3]$ , and are all distinct, and $M_c$ is the current matching which will be output by the algorithm on query.", "Let $M$ be the optimum matching at any stage.", "Theorem REF implies that $\\frac{|M_c|+|M_i|+|M_j|}{3} &\\ge \\frac{2}{3} |M|\\\\\\Longrightarrow |M_c|+2(1+\\epsilon )|M_c| &\\ge 2|M|\\\\\\Longrightarrow \\left(\\frac{3}{2}+\\epsilon \\right)|M_c| &\\ge |M|.$ Note that the approximation ratio trivially holds after the first edge is inserted as we set $c=1$ (, and will hold even if we set $c=2$ or 3, because first edge is added to all three matchings $M_1,M_2,M_3$ ).", "In the augment or the switching phase, $O(1)$ time is spent per edge.", "Let $M^{\\prime \\prime }, M_c^{\\prime \\prime },M_i^{\\prime \\prime },M_j^{\\prime \\prime }$ represent the matchings $M,M_c,M_i,M_j$ immediately after $c$ was updated, and let $M^{\\prime }, M_c^{\\prime },M_i^{\\prime },M_j^{\\prime }$ represent the respective matchings immediately after the previous time $c$ was updated.", "In the ChangeCurr phase, at most $2|M^{\\prime \\prime }|$ time is potentially spent (because size of any matching stored by the algorithm is at most $|M^{\\prime \\prime }|$ ), while changing the current matching output by the algorithm.", "But we show that this happens very rarely.", "If $|M^{\\prime \\prime }|\\ge 2|M^{\\prime }|$ , then at least $|M^{\\prime \\prime }|/2$ edges have been inserted between two recent updates of $c$ .", "This implies an amortized update time of $O(1)$ per edge.", "Now suppose $|M^{\\prime \\prime }| <2|M^{\\prime }|$ .", "Immediately after the previous update of $c$ , $|M_c^{\\prime }|\\ge (|M_i^{\\prime }|+|M_j^{\\prime }|)/2$ (because $M_c$ is the maximum size matching among $M_1,M_2,$ and $M_3$ ).", "Just before $c$ is updated in the ChangeCurr phase, $|M_c^{\\prime \\prime }| < (|M_i^{\\prime \\prime }|+|M_j^{\\prime \\prime }|)/(2(1+\\epsilon ))$ .", "So, the change in the value of $|M_c|$ is at most $\\frac{|M_i^{\\prime \\prime }|+|M_j^{\\prime \\prime }|}{2(1+\\epsilon )} - \\frac{|M_i^{\\prime }|+|M_j^{\\prime }|}{2}.$ But this value is at least zero, as the size of any matching can never decrease by the description of the algorithm.", "Hence, $\\frac{|M_i^{\\prime \\prime }|+|M_j^{\\prime \\prime }|}{2(1+\\epsilon )} - \\frac{|M_i^{\\prime }|+|M_j^{\\prime }|}{2} &\\ge 0 \\\\\\Longrightarrow |M_i^{\\prime \\prime }|+|M_j^{\\prime \\prime }| - (1+\\epsilon )(|M_i^{\\prime }|+|M_j^{\\prime }|) &\\ge 0 \\\\\\Longrightarrow (|M_i^{\\prime \\prime }|-|M_i^{\\prime }|)+(|M_j^{\\prime \\prime }|-|M_j^{\\prime }|) &\\ge \\epsilon (|M_i^{\\prime }|+|M_j^{\\prime }|) \\\\\\Longrightarrow (|M_i^{\\prime \\prime }|-|M_i^{\\prime }|)+(|M_j^{\\prime \\prime }|-|M_j^{\\prime }|) &\\ge \\epsilon |M^{\\prime }| &\\dots M_i^{\\prime },M_j^{\\prime }\\text{ are maximal} .$ Thus, along with the fact that $|M^{\\prime \\prime }| <2|M^{\\prime }|$ , $\\Omega (\\epsilon |M^{\\prime \\prime }|)$ edges have been inserted between two recent updates of value of $c$ .", "This implies an amortized update time of $O(1/\\epsilon )$ per edge, and finishes the proof." ], [ "MWM in the Online Preemptive Model", "In this section, we present a randomized algorithm (that uses only $O(1)$ bits of randomness) for MWM in the online preemptive model, and analyze its performance for growing trees.", "The algorithm is motivated by the deterministic algorithm for MWM due to McGregor [11].", "McGregor's algorithm is easy to describe – if the weight of the new edge is more than $(1+\\gamma )$ times the weight of the conflicting edges in the current matching, then evict them and add the new edge.", "The algorithm is $(1+\\gamma )(2+1/\\gamma )$ -competitive, and attains the best competitive ratio of $3+2\\sqrt{2} \\approx 5.828$ for $\\gamma =\\frac{1}{\\sqrt{2}}$ .", "It achieves this competitive ratio for the following example.", "Start by presenting an edge of weight $x_0=1$ to the algorithm.", "This edge will be added to the matching.", "Assume inductively that after iteration $i$ , the algorithm's matching has only the edge of weight $x_i$ .", "In iteration $i+1$ , present an edge of weight $y_{i+1}=(1+\\gamma )x_i$ on one end point of $x_i$ (we slightly abuse the notation here, and say that $x_i$ is also the name of the edge of weight $x_i$ ).", "This edge will not be accepted in the algorithm's matching.", "Give an edge of weight $x_{i+1}=(1+\\gamma )x_i + \\epsilon $ on the other end point of $x_i$ .", "This edge will be accepted in the algorithm's matching, and $x_i$ will be evicted.", "This process terminates for some large $n$ , letting $x_{n+1}=(1+\\gamma )x_n$ .", "The edge of weight $x_{n+1}$ will not be accepted in the algorithm's matching.", "The algorithm will hold only the edge of weight $x_n$ , whereas the optimum matching would include edges of weight $y_1,\\dots ,y_{n+1},x_{n+1}$ .", "It can be easily inferred that this gives the required lower bound on the competitive ratio.", "Notice that the edges presented in the example crucially depended on $\\gamma $ .", "To beat this, we maintain two matching, with $\\gamma $ values $\\gamma _1$ and $\\gamma _2$ respectively, and choose one at random.", "We describe the algorithm next.", "[H] Randomized Algorithm for MWM Maintain two matchings $M_1$ and $M_2$ .", "Let $j=1$ with probability $p$ , and $j=2$ otherwise.", "On receipt of an edge $e$ : For $i=1,2$ , if $w(e)>(1+\\gamma _i)w(X(M_i,e))$ , then $M_i=M_i\\setminus X(M_i,e) \\cup \\lbrace e\\rbrace $ .", "Output $M_j$ .", "Note that we cannot just output the best of two matchings because that could violate the constraints of the online preemptive model." ], [ "Analysis", "We use the primal-dual technique to analyze the performance of this algorithm.", "The primal-dual technique used to analyze McGregor's deterministic algorithm for MWM described in [3] is fairly straightforward.", "However the management becomes complicated with the introduction of randomness, and we are only able to analyze the algorithm in a very restricted class of graphs, which are growing trees.", "Theorem 10 The expected competitive ratio of Algorithm  on growing trees is $\\max \\left\\lbrace \\frac{1+\\gamma _1}{p},\\frac{1+\\gamma _2}{1-p},\\frac{(1+\\gamma _1)(1+\\gamma _2)(1+2\\gamma _1)}{p\\cdot \\gamma _1 + (1-p)\\gamma _2 + \\gamma _1\\gamma _2}\\right\\rbrace ,$ where $p$ is the probability to output $M_1$ .", "We maintain both primal and dual variables along with the run of the algorithm.", "Consider a round in which an edge $e\\equiv (u,v)$ is revealed, where $v$ is the new vertex.", "Before $e$ is revealed, let $e_1$ and $e_2$ be the edges incident on $u$ which belong to $M_1$ and $M_2$ respectively.", "If such an $e_i$ does not exist, then we may assume $w(e_i)=0$ .", "The primal and dual variables are updated as follows.", "$e$ is rejected by both matchings, we set the primal variable $x_e=0$ , and the dual variable $y_v=0$ .", "$e$ is added to $M_1$ only, then we set the primal variable $x_e=p$ , and the dual variable $y_u=\\max (y_u,\\min ((1+\\gamma _1) w(e), (1+\\gamma _2) w(e_2)))$ , and $y_v=0$ ;.", "$e$ is added to $M_2$ only, then we set the primal variable $x_e=1-p$ , and the dual variable $y_u=\\max (y_u,\\min ((1+\\gamma _1)w(e_1),(1+\\gamma _2)w(e)))$ , and $y_v=0$ .", "$e$ is added to both the matchings, then we set the primal variable $x_e=1$ , and the dual variables $y_u=\\max (y_u,(1+\\gamma _1)w(e))$ and $y_v=(1+\\gamma _1)w(e)$ .", "When an edge $e^{\\prime }$ is evicted from $M_1$ (or $M_2$ ), we decrease its primal variable $x_{e^{\\prime }}$ by $p$ (or $(1-p)$ respectively), and the corresponding dual variables are unchanged.", "We begin with three simple observations.", "The cost of the primal solution is equal to the expected weight of the matching maintained by the algorithm.", "The dual variables never decrease.", "Hence, if a dual constraint is feasible once, it remains so.", "$y_u \\ge \\min ((1+\\gamma _1)w(e_1),(1+\\gamma _2)w(e_2))$ .", "The idea behind the analysis is to prove a bound on the ratio of the dual cost and the primal cost while maintaining dual feasibility.", "By Observation 2, to ensure dual feasibility, it is sufficient to ensure feasibility of the dual constraint of the new edge.", "If the new edge $e$ is not accepted in any $M_i$ , then $w(e)\\le \\min ((1+\\gamma _1)w(e_1),(1+\\gamma _2)w(e_2))$ .", "Hence, the dual constraint is satisfied by Observation 3.", "Else, it can be seen that the dual constraint is satisfied by the updates performed on the dual variables.", "The following lemma implies Theorem REF using Claim REF .", "Lemma 11 $\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal}} \\le \\max \\left\\lbrace \\frac{1+\\gamma _1}{p},\\frac{1+\\gamma _2}{1-p},\\frac{(1+\\gamma _1)(1+\\gamma _2)(1+2\\gamma _1)}{p\\cdot \\gamma _1 + (1-p)\\gamma _2 + \\gamma _1\\gamma _2}\\right\\rbrace $ after every round.", "We will use the following simple technical lemma to prove Lemma REF .", "Lemma 12 $\\frac{ax+b}{cx+d}$ increases with $x$ iff $ad-bc\\ge 0$ .", "[Proof of Lemma REF ] There are four cases to be considered.", "If edge $e$ is accepted in $M_1$ , but not in $M_2$ .", "Then $(1+\\gamma _1)w(e_1)< w(e) \\le (1+\\gamma _2)w(e_2)$ .", "By Observation 3, before $e$ was revealed, $y_u\\ge (1+\\gamma _1)w(e_1)$ .", "After $e$ is accepted in $M_1$ , $\\Delta \\text{Primal} =p(w(e)-w(e_1))$ , and $\\Delta \\text{Dual} \\le (1+\\gamma _1)(w(e)-w(e_1))$ .", "Hence, $\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal} }\\le \\frac{(1+\\gamma _1)}{p}.$ If edge $e$ is accepted in $M_2$ , but not in $M_1$ .", "Then $(1+\\gamma _2)w(e_2)< w(e) \\le (1+\\gamma _1)w(e_1)$ .", "By Observation 3, before $e$ was revealed, $y_u\\ge (1+\\gamma _2)w(e_2)$ .", "After $e$ is accepted in $M_2$ , $\\Delta \\text{Primal} =(1-p)(w(e)-w(e_2))$ , and $\\Delta \\text{Dual} \\le (1+\\gamma _2)(w(e)-w(e_2))$ .", "Hence, $\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal} }\\le \\frac{(1+\\gamma _2)}{1-p}.$ If edge $e$ is accepted in both the matchings, and $(1+\\gamma _1)w(e_1) \\le (1+\\gamma _2)w(e_2)$ $ < w(e)$ .", "By Observation 3, before $e$ was revealed, $y_u\\ge (1+\\gamma _1)w(e_1)$ .", "After $e$ is accepted in both the matchings, $\\Delta \\text{Dual} \\le (1+\\gamma _1)(2w(e)-w(e_1))$ .", "The change in primal cost is $\\Delta \\text{Primal} &\\ge w(e)-p\\cdot w(e_1) - (1-p)\\cdot w(e_2) \\\\&\\ge w(e) - p\\cdot w(e_1) - (1-p)\\cdot \\frac{w(e)}{1+\\gamma _2}\\\\&=\\frac{p+\\gamma _2}{1+\\gamma _2}w(e) -p\\cdot w(e_1).\\\\\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal} }&\\le (1+\\gamma _1)\\frac{2w(e)-w(e_1)}{\\frac{p+\\gamma _2}{1+\\gamma _2}w(e) -p\\cdot w(e_1)}.$ By Lemma REF , this value increases, for a fixed $w(e)$ , with $w(e_1)$ if $\\gamma _2\\le \\frac{p}{1-2p}$ , and its worst case value is achieved when $(1+\\gamma _1)w(e_1)=w(e)$ .", "Thus, $\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal} }&\\le (1+\\gamma _1)\\frac{2(1+\\gamma _1)(1+\\gamma _2)-(1+\\gamma _2)}{(p+\\gamma _2)(1+\\gamma _1) -p(1+\\gamma _2)}\\\\&= (1+\\gamma _1)(1+\\gamma _2)\\frac{1+2\\gamma _1}{p\\cdot \\gamma _1 + (1-p)\\gamma _2 + \\gamma _1\\gamma _2}.$ If $e$ is accepted in both the matchings, and $(1+\\gamma _2)w(e_2) \\le (1+\\gamma _1)w(e_1) < w(e)$ .", "By Observation 3, before $e$ was revealed, $y_u\\ge (1+\\gamma _2)w(e_2)$ .", "The following bound can be proved similarly.", "$\\frac{\\Delta \\text{Dual}}{\\Delta \\text{Primal} } &\\le (1+\\gamma _1)(1+\\gamma _2)\\frac{1+2\\gamma _1}{p\\cdot \\gamma _1 + (1-p)\\gamma _2 + \\gamma _1\\gamma _2}.$ The following theorem is an immediate consequence of Theorem REF .", "Theorem 13 Algorithm  is a 3-competitive (in expectation) randomized algorithm for MWM on growing trees, when $p=1/3$ , $\\gamma _1=0$ , and $\\gamma _2=1$ ; and the analysis is tight.", "The input for which Algorithm  is 3-competitive is as follows.", "Start by presenting an edge of weight $x_0=1$ .", "It will be added to both $M_1$ and $M_2$ .", "Assume inductively, that currently both matching only contain an edge of weight $x_i$ .", "Present an edge of weight $y_{i+1}=x_i$ on one end point of $x_i$ .", "This edge will not be accepted in either of the matchings.", "Present an edge of weight $x_{i+1}=2\\cdot x_i + \\epsilon $ on the other end point of $x_i$ .", "This edge will be accepted in both the matchings, and $x_i$ will be evicted.", "For a sufficiently large value $n$ , let $x_{n+1}=x_n$ .", "So edge of weight $x_{n+1}$ will not be accepted in either of the matchings.", "Both the matchings will hold only the edge of weight $x_n$ , whereas the optimum matching would include edges of weight $y_1,\\dots ,y_{n+1},x_{n+1}$ .", "The weight of the matching stored by the algorithm is $2^n$ , whereas the weight of the optimum matching is $\\approx 3\\cdot 2^n$ (we have ignored the $\\epsilon $ terms here).", "This gives the competitive ratio 3.", "Note.", "In the analysis of Algorithm  for growing trees, we crucially use the following fact in the dual variable assignment.", "If an edge $e\\notin M_i$ for some $i$ , then a new edge incident on its leaf vertex will definitely be added to $M_i$ , and it suffices to assign a zero charge to the corresponding dual variable.", "This is not necessarily true for more general classes of graphs, and new ideas are needed to analyze the performance for those classes." ], [ "Acknowledgements", "The first author would like to thank Ashish Chiplunkar for helpful suggestions to improve the competitive ratio of Algorithm , and also to improve the presentation of Section ." ], [ "Proof of Lemma ", "We crucially use the following lemma to prove Lemma REF .", "Lemma 14 (a) If an edge $e$ belongs only to $M_4$ at the end of input, then bad edges cannot be incident on both its end points.", "(b) Also, if an edge $e$ was added to $M_4$ only in the switching phase, then $e$ cannot be a bad edge.", "There are two cases to consider.", "Suppose $e$ was added to $M_4$ only when it was revealed.", "Then on one of its end point either there should be two edges incident (other than $e$ ), such that each of them belongs to a single matching, or there should be one edge which belongs to two matchings.", "In either case, the edges incident on that end point of $e$ should have neighboring edges which belong to some matching (by description of algorithm).", "And hence, these edges cannot be bad.", "Suppose $e$ was added to $M_4$ as well as some other matching when it was revealed.", "If $e$ belonged to three matchings when it was revealed, then its neighboring edge will have its end points covered by at least four matching edges, and this number can never go below four.", "If $e$ belonged to two matchings when it was revealed, then it – (a) either has one neighboring edge which belongs to two matchings, – (b) or one neighboring edge on each of its end points, each belonging to distinct matching, – (c) or two neighboring edges on one of its end points, such that both of them belong to distinct matchings.", "In Case (a), this neighboring edge should have a neighboring edge on its other end point which belongs to some matching, and hence it cannot be a bad edge.", "In Case (b), each of these edge should have at least two neighboring edges of their own on their respective other end point, which belong to certain matching.", "Hence, both these edges cannot be bad.", "In Case (c), both these edges should have neighboring edges of their own on their respective other end point, which belong to certain matching.", "Hence, both these edges cannot be bad.", "For the second part of lemma, if edge $e$ added to $M_4$ in the switching phase, then it means that $e$ will have three neighboring edges $e_1$ ,$e_2$ , and $e_3$ , belonging to $M_1$ , $M_2$ , and $M_3$ , respectively.", "This is because $e$ will be added to $M_4$ in the switching phase only if it is not added to $M_2$ or $M_3$ in the switching phase, which means there are edges which belong only to $M_2$ and $M_3$ respectively.", "[Proof of Lemma REF ] There are two cases to consider.", "Suppose if there is a bad leaf edge $e$ which belongs to $M_4$ .", "If $e$ is added to $M_4$ in the switching phase, then $e$ cannot be a bad edge (by part (b) of Lemma REF ).", "So, $e$ has to be added to $M_4$ in the augment phase for it to be a bad leaf edge in future.", "If $e$ was added to $M_4$ alone when revealed, then it must have neighbors $e_1$ and $e_2$ such that both of them do not belong to $M_4$ .", "Then, they must have had neighboring edges $e_1^{\\prime }$ and $e_2^{\\prime }$ respectively which belonged to $M_4$ (at some stage).", "Suppose $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ ) switches $e_1^{\\prime }$ (and/or $e_2^{\\prime }$ respectively) out of $M_4$ , then $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ respectively) cannot be a bad edge (by part (b) of Lemma REF ).", "Otherwise, the Lemma holds due to part (a) of Lemma REF .", "If $e$ was added to two matchings ($M_4$ being one of them) when it was revealed, and finally has only one internal neighboring edge $e_1$ , then $e_1$ will have a neighboring edge $e_2$ on its other end point.", "Either $e_2$ belongs to $M_4$ or its neighboring edge $e_2^{\\prime }$ on other end point belongs to $M_4$ .", "The lemma holds if finally $e_2^{\\prime }$ belongs to $M_4$ (by part (a) of Lemma REF ) or if finally the neighboring edge $e_2^{\\prime \\prime }$ of $e_2^{\\prime }$ belongs to $M_4$ (by part (b) of Lemma REF ).", "(The proof for this case will also work for the case when $e$ was revealed first as a single disconnected edge, and then $e_1$ was revealed on one of its end points.)", "If $e$ was added to two or three matchings ($M_4$ being one of them) when it was revealed, and finally has two internal neighboring edges $e_1$ and $e_2$ , then $e_1$ and $e_2$ must have neighboring edges $e_1^{\\prime }$ and $e_2^{\\prime }$ respectively which belong to $M_4$ (at some stage).", "Suppose $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ ) switches $e_1^{\\prime }$ (and/or $e_2^{\\prime }$ respectively) out of $M_4$ , then $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ respectively) cannot be a bad edge (by part (b) of Lemma REF ).", "Otherwise, the Lemma holds due to part (a) of Lemma REF .", "Let $e_1$ and $e_2$ be two bad internal edges which do not belong to $M_4$ .", "Then, they must have had neighboring edges $e_1^{\\prime }$ and $e_2^{\\prime }$ respectively which belonged to $M_4$ (at some stage).", "Suppose $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ ) switches $e_1^{\\prime }$ (and/or $e_2^{\\prime }$ respectively) out of $M_4$ , then $e_1^{\\prime \\prime }$ (and/or $e_2^{\\prime \\prime }$ respectively) cannot be a bad edge (by part (b) of Lemma REF ).", "Otherwise, the Lemma holds due to part (a) of Lemma REF ." ] ]
1612.05419
[ [ "Measurement of the linear thermo-optical coefficient of\n Ga$_{0.51}$In$_{0.49}$P using photonic crystal nanocavities" ], [ "Abstract Ga$_{0.51}$In$_{0.49}$P is a promising candidate for thermally tunable nanophotonic devices due to its low thermal conductivity.", "In this work we study its thermo-optical response.", "We obtain the linear thermo-optical coefficient $dn/dT=2.0\\pm0.3\\cdot 10^{-4}\\,\\rm{K}^{-1}$ by investigating the transmission properties of a single mode-gap photonic crystal nanocavity." ], [ "introduction", "Several III-V semiconductors are used for nanophotonic devices, such as GaAs [1], GaP [2], InP [3] for their specific properties such as optical tunability.", "Local tuning of the refractive index has been proposed as a method to create localized resonances in a waveguide [4], and to tune complex localized states [5], [6].", "For local thermal tuning, the high thermal conuctivity of Si or $\\rm {Si_3N_4}$ is unfavorable.", "A potential material for thermally tunable nanophotonic devices is GaInP which became popular during recent years [7], [8], [9], [10], [11].", "Its thermal conductivity [12] is more than 6 times smaller than for Si and $\\rm {Si_3N_4}$ , which in addition to absence of two-photon absorption at 1550 nm and favorable nonlinear properties [10] makes it a promising candidate for thermally tunable photonics where multiple closely spaced elements have to be tuned independently.", "The thermo-optical coefficient of this material has indirectly been obtained before as a result of a series of complex measurements and heat diffusion calculations in Ref.", "Sokolov2015, which implies the need of a more direct measurement.", "To our knowledge the precise value of $dn/dT$ has not been reported in the literature.", "Therefore in this work we investigate the thermal response of a single photonic crystal nanocavity made of Ga$_{0.51}$ In$_{0.49}$ P for a homogeneously heated sample where temperature of the sample is directly measured.", "Our rigorous analysis allows us to obtain the precise value of the thermo-optical coefficient of refractive index for this material." ], [ "Sample and experimental setup", "To experimentally measure the thermo-optical coefficient of the Ga$_{0.51}$ In$_{0.49}$ P a nanophotonic sample containing photonic crystal nanocavities was mounted on a thermally controlled stage, so the sample was homogeneously heated and the temperature of the stage was locked with a precision of $\\pm 0.001\\,\\rm {K}$ .", "The temperature drop between the stage and the sample was measured in a separate run using a PT-100 temperature sensor placed in the position of the photonic crystal.", "It was found that temperature drop varied from 1.1 K to 1.4 K linearly for stage temperatures between 27.5 and 77.5 $\\rm {^oC}$ .", "All temperatures mentioned from here are sample temperatures corrected for this temperature drop and measured with an absolute precision of 0.3 K as verified with a calibrated digital thermometer.", "Figure: Experimental setup.", "Transmission through the sample is measured.", "The sample temperature is locked to ±0.001 o C\\pm 0.001\\, \\rm {^oC} using a temperature controller.", "Inset shows the cavity structure.", "Holes are represented with circles.", "Different colors correspond to different hole shifts.The sample is an air-suspended photonic crystal membrane with a linear array of 10 directly coupled mode-gap photonic crystal nanocavities made of Ga$_{0.51}$ In$_{0.49}$ P. The thickness of semiconductor membrane is 180 nm.", "The structure of a single cavity is presented as the inset in Figure REF .", "It is made in a photonic crystal waveguide with width $W_0=0.98\\sqrt{3}a$ where $a$ =485 nm is a period of a triangular photonic crystal lattice.", "Red holes are shifted away from the waveguide by 6 nm, green - by 4 nm and purple ones are shifted by 2 nm to create a cavity mode.", "Such cavities are known for large experimentally measured Q-factors [13], [14] and therefore they are suitable for precise thermal measurements.", "The first and last cavities in the array are coupled to input and output photonic crystal waveguides with width $W_1=1.1\\sqrt{3}a$ .", "The waveguides are used to launch and collect light from the structure.", "Figure: Transmission spectrum.", "Transmission spectra of the sample at 48.7  o C\\rm {^oC}.", "Dashed line represents Fano lineshape fit.The structure was probed by IR light with a wavelength around 1550 nm from the CW tunable laser.", "The light was coupled into and out-coupled from the structure with polarization-maintaining lensed fibers $\\rm {L_1}$ and $\\rm {L_2}$ with NA=0.55.", "The out-coupled light was collected on an IR photodiode for transmission measurements.", "The sample was kept in nitrogen atmosphere to avoid oxidation [15], [16].", "The resonance frequencies of cavities are perturbed by unavoidable disorder, which breaks the resonance hybridization.", "This normally undesired effect allowed us to pick an isolated single cavity resonance.", "We picked the single resonance corresponding to the $\\rm {3^{rd}}$ cavity in the array, which was verified by our pump line-scan technique [7], [8], [17].", "The transmission spectrum of the resonance at 48.7 $\\rm {^oC}$ is presented in Figure REF .", "The spectrum has a clear Fano-like lineshape due to the interference with transmitted TM light.", "The spectrum is fitted with a Fano lineshape function [18], [19] and a $\\rm {1^{st}}$ order polynomial for the background.", "The lineshape is perfectly described by the fit, so it is used to obtain line parameters.", "The loaded Q-factor of the resonance is $Q=1.6\\pm 0.1\\cdot 10^5$ ." ], [ "Experiment description", "For the measurement of the thermo-optical coefficient the resonance wavelength was measured for several temperatures of the sample ranging from 26.4 to 76 $\\rm {^oC}$ .", "Spectra for all temperatures are presented in Figure REF .", "The resonance experiences a redshift which signifies that the material has a positive thermo-optical response.", "All spectra have a Fano-like line shape.", "There was no systematic change of the Q-factor which signifies that within this temperature range the mode-profile of the cavity does not change.", "In total the resonance redshifts by about 4.5 nm when the temperature rises by approximately 49.7 $\\rm {^oC}$ .", "The dependence of resonance wavelength on the temperature of the sample is presented in Figure REF .", "The resonance wavelength changes linearly with increasing temperature, as shown by the fit.", "Figure: Resonance wavelength versus temperature of the sample.", "The red line represents a line fit of the experimental data.", "The error for temperature is smaller than the datapoint size.When a sample is heated, there is always an extra resonance shift as a result of the evaporation of the water film from the surface of the sample [15] in addition to the redshift due the increase in temperature.", "The exact magnitude of this extra shift depends on details of the surface condition.", "In laser heating experiments with similar temperature changes and resonance shifts we have observed this extra shift to be about 0.6 nm or less.", "As a worst case estimate we add an additional error term to the error of the linear fit which corresponds to the extra change of the resonance wavelength by 0.6 nm when temperature is increased by 49.7 $\\rm {^oC}$ .", "The resulting tuning slope is $d\\lambda /dT=9\\pm 1\\cdot 10^{-2}\\, \\rm {nm/\\,K}$ .", "Using perturbation theory and scaling of Maxwell equations one can get a precise value of linear thermo-optic coefficient $dn/dT$ of the refractive index of Ga$_{0.51}$ In$_{0.49}$ P. To first order, the following equation for relative resonance wavelength shift can be applied [20]: $\\frac{\\Delta \\lambda }{\\lambda }=\\frac{\\Delta \\lambda _n}{\\lambda }+\\frac{\\Delta \\lambda _a}{\\lambda }$ Where $\\Delta \\lambda _n/\\lambda $ and $\\Delta \\lambda _a/\\lambda $ are relative resonance wavelength changes caused by two processes: refractive index increase and sample expansion due to heating.", "The first term is a direct result of the first order perturbation theory [20], while the second term comes as a consequence of scaling nature of Maxwell equations [20]: $\\frac{\\Delta \\lambda _n}{\\lambda }=\\frac{\\Delta n}{n}\\cdot \\frac{\\int \\limits _{mebrane}^{}\\varepsilon |\\mathbf {E}(\\mathbf {r})|^2d\\mathbf {r}}{\\int \\limits _{all}^{}\\varepsilon |\\mathbf {E}(\\mathbf {r})|^2d\\mathbf {r}}=\\frac{1}{n}\\frac{dn}{dT}\\Delta T\\mathcal {E}_m$ $\\frac{\\Delta \\lambda _a}{\\lambda }=\\frac{\\Delta a}{a}=\\alpha _T\\Delta T$ Here $\\Delta n$ is the refractive index change due to temperature increase $\\Delta T$ , $\\varepsilon $ and $n$ are the dielectric constant and refractive index of the membrane material and $\\mathbf {E}(\\mathbf {r})$ is the electric field of the cavity mode.", "$\\mathcal {E}_m$ is the fraction of the electric-field energy inside the membrane.", "In Eq.", "REF we took into account that the change in the refractive index of ambient nitrogen is negligible [21] in comparison to the change of the refractive index of the semiconductor.", "According to our 3D FDTD calculations the fraction of electric-field energy in the membrane is 0.88.", "In Eq.", "REF $\\Delta a/a$ is the relative change of the photonic crystal lattice and $\\alpha _T$ is the thermal expansion coefficient.", "Finally, the $dn/dT$ value can be obtained from: $\\frac{dn}{dT}=\\frac{n}{\\lambda \\mathcal {E}_m}(\\frac{d\\lambda }{dT}-\\alpha _T\\lambda )$ The thermal expansion coefficient [22] $\\alpha _T$ for Ga$_{0.51}$ In$_{0.49}$ P is equal to $5.4\\pm 0.3\\cdot 10^{-6}\\, \\rm {K^{-1}}$ .", "The experimentally measured value of $dn/dT$ is then equal to $dn/dT=2.0\\pm 0.3\\cdot 10^{-4}\\,\\rm {K^{-1}}$ .", "We note that in case of GaInP linear expansion gives a noticeable contribution to the tuning slope of the cavity resonance.", "Without taking into account that effect the value of $dn/dT$ would be about 10% larger.", "In Ref.", "Sokolov2015 we used complex modeling to estimate $dn/dT$ for a locally laser heated membrane, where the temperature was calculated from absolute power, thermal conductivity and absorptivity, thereby introducing many uncertainties.", "In addition, due to the local heating the membrane in that experiment was thermally stressed.", "The present experiment employs a direct temperature measurement with much less uncertain parameter performed on an unstressed membrane." ], [ "Conclusion", "In conclusion, we investigated the thermo-optical effect of the refractive index for Ga$_{0.51}$ In$_{0.49}$ P. Our measurement took place for a freely expanding membrane and we took into account the effect of thermal expansion of the material.", "We found no significant Q-factor change during our measurement which guarantees that the working wavelength of photonic devices based on nanocavities made of Ga$_{0.51}$ In$_{0.49}$ P can be safely biased with temperature within the range of about 5 nm.", "This work enables precise thermal tuning of GaInP-based photonic devices, making GaInP one of the lowest thermal conductivity semiconductors used in photonics.", "Funding.", "European Research Council project (ERC) (279248), Nederlandse Organisatie voor Wetenschappelijk.", "Acknowledgement.", "The authors would like to thank Sanli Faez, Emre Yüce and Willem Vos for helpful discussions and advises and Cornelis Harteveld for technical support." ] ]
1612.05544
[ [ "Flaxion: a minimal extension to solve puzzles in the standard model" ], [ "Abstract We propose a minimal extension of the standard model which includes only one additional complex scalar field, flavon, with flavor-dependent global U(1) symmetry.", "It not only explains the hierarchical flavor structure in the quark and lepton sector (including neutrino sector), but also solves the strong CP problem by identifying the CP-odd component of the flavon as the QCD axion, which we call flaxion.", "Furthermore, the flaxion model solves the cosmological puzzles in the standard model, i.e., origin of dark matter, baryon asymmetry of the universe, and inflation.", "We show that the radial component of the flavon can play the role of inflaton without isocurvature nor domain wall problems.", "The dark matter abundance can be explained by the flaxion coherent oscillation, while the baryon asymmetry of the universe is generated through leptogenesis." ], [ "Introduction", "There are several puzzles in the standard model (SM) of particle physics, which may be solved by new physics models based on (spontaneously broken) symmetries.", "Although one may be able to introduce a new symmetry to solve each puzzle, it is desirable to have a unified picture of those symmetries from the point of view of simplicity and minimality, as we suggest in this paper.", "One of the mysteries of the SM is the hierarchical flavor structure of the Yukawa couplings.", "The Froggatt-Nielsen mechanism is an attractive possibility to explain the quark/lepton mass hierarchy and their mixing matrices [1].", "It introduces a new complex scalar field called flavon, whose vacuum expectation value (VEV) generates the SM Yukawa couplings.", "In this model a global Abelian flavor symmetry U(1)$_F$ is imposed.", "Another puzzle in the SM is the strong CP problem in the quantum chromo dynamics (QCD).", "The Peccei-Quinn (PQ) mechanism [2] utilizes a global U(1) symmetry, U(1)$_{\\rm PQ}$ , to solve it; the pseudo Nambu-Goldstone boson associated with the spontaneous breaking of U(1)$_{\\rm PQ}$ , called axion [3], [4], dynamically cancels the strong CP angle.", "Moreover, the PQ model explains the present dark matter (DM) abundance through the coherent oscillation of the axion field [5] if the breaking scale of U(1)$_{\\rm PQ}$ is at a relevant scale.", "It is also remarkable that the right-handed neutrinos can have large masses through the U(1)$_{\\rm PQ}$ breaking [6].", "Thus tiny left-handed neutrino masses are naturally explained through the seesaw mechanism [7].", "In this paper, we propose a new minimal extension of the SM in which the flavor U(1)$_F$ and the U(1)$_{\\rm PQ}$ are unified.", "We introduce only one additional complex scalar field, flavon, charged under the global U(1)$_F$ whose VEV naturally explains the Yukawa structure.", "As long as this U(1)$_F$ is exact up to the QCD anomaly, its angular component remains nearly massless, which we call flaxion.", "Assuming that U(1)$_F$ is anomalous under SU(3)$_C$ , the flaxion gets the potential after the QCD phase transition as ordinary axion and solves the strong CP problem.", "Similar possibility was already pointed out long ago in Ref.", "[8] followed by several studies [9].", "We use the minimality as our guiding principle and add only one complex scalar field to the SM.", "Then, with such an additional complex scalar field (as well as right-handed neutrinos), we show that it is possible to explain the followings:$1\\endcsname $A similar approach was made in Refs.", "[10], [11] in the framework of KSVZ axion model, although they did not address the flavor structure.", "(1) Yukawa flavor structure, (2) strong CP problem, (3) neutrino masses and mixings, (4) dark matter, (5) baryon asymmetry, and (6) inflation.", "In particular, we point out that a successful inflation takes place by identifying the flavon field as the inflaton.", "By utilizing the idea of attractor inflation [12], we have a phenomenologically viable inflation with successful reheating consistent with leptogenesis [13] and without domain wall nor flaxion isocurvature fluctuation problems.", "We emphasize that our model is more economical than other axion models.", "In the KSVZ axion model [14] heavy vector-like quarks are necessary, while in the DFSZ axion model [15] we need two Higgs doublets.", "In this sense, our model is economical: addition of only one new scalar field is sufficient to explain the flavor structure, solve the strong CP problem and provide a good DM candidate.", "This paper is organized as follows.", "In Sec.", ", we present our model and derive flavon/flaxion coupling to the SM particles.", "Experimental constraints, in particular flavor-violating neutral current (FCNC) processes mediated by the flaxion, are also summarized.", "In Sec.", ", cosmological aspects of the flaxion model is discussed.", "There we show that the flavon field acts as the inflaton through the attractor-type mechanism for flattening the potential without domain wall nor isocurvature problems.", "We conclude in Sec.", "with several remarks." ], [ "Model", "The model we consider is described by the following Yukawa terms in the Lagrangian: $-\\mathcal {L} &= y_{ij}^d\\left( \\frac{\\phi }{M} \\right)^{n_{ij}^d} \\overline{Q}_i H d_{Rj}+ y_{ij}^u\\left( \\frac{\\phi }{M} \\right)^{n_{ij}^u} \\overline{Q}_i \\widetilde{H} u_{Rj} \\nonumber \\\\&+ y_{ij}^l\\left( \\frac{\\phi }{M} \\right)^{n_{ij}^l} \\overline{L}_i H l_{Rj}+ y_{i\\alpha }^\\nu \\left( \\frac{\\phi }{M} \\right)^{n_{i\\alpha }^\\nu } \\overline{L}_i \\widetilde{H} N_{R\\alpha } \\nonumber \\\\&+ \\frac{1}{2} y_{\\alpha \\beta }^N\\left( \\frac{\\phi }{M} \\right)^{n_{\\alpha \\beta }^N} M \\overline{ N_{R\\alpha }^c} N_{R\\beta } +{\\rm h.c.} \\, ,$ where $M$ is a mass scale corresponding to the cut-off scale of this model.", "Here $Q_i$ , $u_{Ri}$ , $d_{Ri}$ , $L_i, e_{Ri}$ , $N_{R\\alpha }$ $(i=13)$ denote the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, right-handed charged lepton and right-handed neutrino, respectively.", "$H$ denotes the SM Higgs doublet, and $\\widetilde{H}=i\\sigma _2 H^*$ .", "Finally, $\\phi $ is a complex scalar field, called flavon, whose VEV $\\left<\\phi \\right>\\equiv v_\\phi $ gives rise to the SM Yukawa couplings [1].", "The hierarchy of the Yukawa coupling constants is explained by the smallness of $\\epsilon $ defined as $\\epsilon \\equiv \\frac{v_\\phi }{M}.$ We assume all $y_{ij} \\sim \\mathcal {O}(1)$ and $\\epsilon \\sim 0.2$ to explain the hierarchical structure of the Yukawa matrix (see App. ).", "For a while we do not specify the number of right-handed neutrinos.", "The minimal number required to reproduce the experimental results is two ($\\alpha =1,2$ ) [16], [17], while we do not exclude the possibility of three right-handed neutrinos ($\\alpha =13$ ).", "After $\\phi $ and $H$ get VEVs, the mass matrices are given by $m_{ij}^d = y_{ij}^d \\epsilon ^{n_{ij}^d}v_{\\rm EW},~~~m_{ij}^u = y_{ij}^u \\epsilon ^{n_{ij}^u}v_{\\rm EW},~~~m_{ij}^l = y_{ij}^l \\epsilon ^{n_{ij}^l}v_{\\rm EW},$ where $\\left<H\\right> \\equiv v_{\\rm EW} = 174$  GeV.$2\\endcsname $The Higgs boson may naturally have mass of $\\sim M$ in this framework.", "The fine-tuning issue to obtain the electroweak scale is not addressed in the present study.", "This model possesses a global chiral U(1) symmetry, which we denote by U(1)$_F$ , under which the flavon is assumed to have a charge $+1$ and the SM Higgs is neutral.", "Denoting the U(1)$_F$ charges of the SM quarks and leptons as $q_{Q_i}, q_{u_i}$ etc., we have the following relations: $& n_{ij}^u = q_{Q_i} - q_{u_j}, \\\\& n_{ij}^d = q_{Q_i} - q_{d_j}, \\\\& n_{ij}^l = q_{L_i} - q_{l_j}, \\\\& n_{i\\alpha }^\\nu = q_{L_i} - q_{N_\\alpha } \\\\& n_{\\alpha \\beta }^N = -q_{N_\\alpha } - q_{N_\\beta }.$ An example for generating the desired quark and lepton masses and the CKM matrix is$3\\endcsname $The charges of the left-handed fields, $q_{Q_i}$ and $q_{L_i}$ , are chosen to approximately reproduce the CKM and MNS matrices.", "The other charges, $q_{u_i}, q_{d_i}, q_{l_i}$ are determined by $n^f_{ii}\\simeq \\log (m^f_i/m^t_i)/\\log \\epsilon $ with $\\epsilon \\simeq 0.23$ .", "(See App. ).", "$\\begin{pmatrix}q_{Q_1} & q_{Q_2} & q_{Q_3} \\\\q_{u} & q_{c} & q_{t} \\\\q_{d} & q_{s} & q_{b}\\end{pmatrix}=\\begin{pmatrix}3 & 2 & 0 \\\\-5 & -1 & 0 \\\\-4 & -3 & -3\\end{pmatrix}, $ and $\\begin{pmatrix}q_{L_1} & q_{L_2} & q_{L_3} \\\\q_{e} & q_{\\mu } & q_{\\tau }\\end{pmatrix}=\\begin{pmatrix}1 & 0 & 0 \\\\-8 & -5 & -3\\end{pmatrix}.", "$ That means $&n_{ij}^u=\\begin{pmatrix}8 & 4 & 3 \\\\7 & 3 & 2 \\\\5 & 1 & 0\\end{pmatrix},~~n_{ij}^d=\\begin{pmatrix}7 & 6 & 6 \\\\6 & 5 & 5 \\\\4 & 3 & 3\\end{pmatrix},~~n_{ij}^l=\\begin{pmatrix}9 & 6 & 4 \\\\8 & 5 & 3 \\\\8 & 5 & 3\\end{pmatrix}.$ Note that for this charge assignment on the lepton doublets, the large $\\nu _\\mu $ –$\\nu _\\tau $ mixing of the neutrino sector is obtained independently of the charges of the right-handed neutrinos [18], [19].", "(See App.", "REF )." ], [ "Flavon interactions", "Now let us see the flavon interactions.", "Expanding the flavon and Higgs as $\\phi = v_\\phi + \\frac{1}{\\sqrt{2}}(s+ia),\\qquad H=\\begin{pmatrix}0 \\\\v_{\\rm EW} + \\frac{h}{\\sqrt{2}}\\end{pmatrix},$ the quark and charged lepton sectors of the Lagrangian (REF ) are written as $-\\mathcal {L} = \\sum _{f=u,d,l}\\left[m_{ij}^f \\left(1+\\frac{h}{\\sqrt{2} v_{\\rm EW}}\\right)+ \\frac{m_{ij}^f n_{ij}^f (s+ia)}{\\sqrt{2} v_\\phi } \\right] \\overline{f_{Li}} f_{Rj} + {\\rm h.c.}$ The mass term and Higgs Yukawa interactions are simultaneously diagonalized by the biunitary transformation $f_{R_j} \\equiv U^f_{ji} f_{R_i}^{\\prime },\\qquad f_{L_i} \\equiv V^f_{ij} f_{L_j}^{\\prime },\\qquad (V^{f \\dagger } m^f U^f)_{ij} = m_i^f \\delta _{ij}\\,,$ but the terms involving the flavon interaction cannot be diagonalized: $-\\mathcal {L} &= \\sum _{f=u,d,l}\\left[m_{i}^f \\left( 1 + \\frac{h}{\\sqrt{2} v_{\\rm EW}} \\right)\\overline{f^{\\prime }_{Li}} f^{\\prime }_{Ri}+\\kappa ^f_{ij} \\frac{s+ia}{\\sqrt{2} v_\\phi }\\,\\overline{f^{\\prime }_{Li}} f^{\\prime }_{Rj}\\right]+ {\\rm h.c.} \\, ,$ where the matrix $\\kappa ^f_{ij}$ is given by $\\kappa ^f_{ij} \\equiv V^{f\\dagger }_{ik} (m_{kn}^f n_{kn}^f) U^f_{nj}.$ Thus the flavon and pseudo-scalar flavon mediate FCNC processes [8], [20], [21].", "The interaction of the pseudo-scalar flavon is then written as $-\\mathcal {L} = \\frac{ia}{\\sqrt{2} v_\\phi } \\sum _{f=u^{\\prime },d^{\\prime },l^{\\prime }}\\left[ \\left(\\kappa ^f_{\\rm H}\\right)_{ij} \\overline{f}_i \\gamma _5 f_j+ \\left(\\kappa ^f_{\\rm AH}\\right)_{ij} \\overline{f}_i f_j \\right], $ where $\\kappa ^f_{\\rm H} = (\\kappa ^f + \\kappa ^{f\\dagger })/2$ and $\\kappa ^f_{\\rm AH} = (\\kappa ^f - \\kappa ^{f\\dagger })/2$ are Hermitian and anti-Hermitian parts of $\\kappa ^f$ , respectively.", "Here it may be useful to rewrite the matrices $\\kappa ^f$ in a simple form.", "First note that, the factor $m_{kn}^f n_{kn}^f$ is expressed in a matrix form as $m_{kn}^f n_{kn}^f &= \\left(\\widehat{q}_{Q} m^f - m^f \\widehat{q}_f\\right)_{kn}\\,, \\quad f=u,d\\\\m_{kn}^l n_{kn}^l &= \\left(\\widehat{q}_{L} m^f - m^l \\widehat{q}_l\\right)_{kn}\\,,$ where $(\\widehat{q}_X)_{ij}=q_{X_i}\\delta _{ij}$ are diagonal matrices.", "Then we obtain $\\kappa ^f_{ij} =\\left(V^{f\\dagger } \\widehat{q}_{Q} V^f \\right)_{ij} m_j^f- m_i^f \\left( U^{f\\dagger } \\widehat{q}_{f} U^f \\right)_{ij}\\,,$ and $&\\left(\\kappa ^f_{\\rm H}\\right)_{ij} = \\frac{1}{2} \\left(V^{f\\dagger } \\widehat{q}_{Q} V^f - U^{f\\dagger } \\widehat{q}_{f} U^f\\right)_{ij} (m_j^f + m_i^f),\\\\&\\left(\\kappa ^f_{\\rm AH}\\right)_{ij} = \\frac{1}{2} \\left(V^{f\\dagger } \\widehat{q}_{Q} V^f + U^{f\\dagger } \\widehat{q}_{f} U^f\\right)_{ij} (m_j^f - m_i^f).$ for $f=u,d$ .", "Expressions for $\\kappa ^l$ are obtained by replacing $\\widehat{q}_Q$ with $\\widehat{q}_L$ ." ], [ "Flaxion as QCD axion", "The interaction between the pseudo-scalar flavon and quarks (REF ) yields the effective axion-gluon-gluon interaction through the triangle anomaly diagram.", "The effective interaction is given by $\\mathcal {L}= \\frac{g_s^2}{32\\pi ^2} \\frac{a}{f_a} G_{\\mu \\nu }^a \\widetilde{G}^{\\mu \\nu a}, $ where $f_a \\equiv \\frac{\\sqrt{2}v_\\phi }{N_{\\rm DW}}= \\frac{\\sqrt{2}\\epsilon M}{N_{\\rm DW}}\\, ,$ with the domain-wall number $N_{\\rm DW}= {\\rm Tr}\\left(2 \\widehat{q}_{Q} - \\widehat{q}_{u} - \\widehat{q}_{d}\\right)= {\\rm Tr} \\left( n^u + n^d\\right)\\, ,$ which corresponds to the number of the minima of the potential.", "In the model of Sec.", "REF , $N_{\\rm DW}=26$ .", "As is well known, after taking the QCD instanton effects into account, the interaction (REF ) results in the axion potential to cancel the strong CP angle at the potential minimum.", "Therefore, we can regard the pseudo-scalar flavon $a$ as the axion that solves the strong CP problem via the PQ mechanism.", "We call $a$ as the flaxion.", "The relation between the flaxion mass and the PQ scale is the same as the ordinary QCD axion [22]: $m_a \\simeq 6\\times 10^{-6}\\,{\\rm eV}\\left(\\frac{10^{12}\\,{\\rm GeV}}{f_a}\\right).$ Except that it has a relatively large domain wall number, its cosmological property is the same as the ordinary invisible QCD axion.", "In particular, the coherent oscillation of the flaxion can be a good DM candidate.", "We will discuss the cosmology of flaxion and the flavon in Sec. .", "The flaxion-photon coupling is also important for low-energy phenomenology.", "The effective Lagrangian is given by $\\mathcal {L} = g_{a\\gamma }\\frac{e^2}{32\\pi ^2} \\frac{a}{f_a} F_{\\mu \\nu }\\widetilde{F}^{\\mu \\nu },$ where [22] $g_{a\\gamma } \\equiv \\frac{2}{N_{\\rm DW}}\\sum _{f=u,d,l} \\left[ N_f {\\rm Tr}\\,(n^f) \\left(q^{\\rm (em)}_f\\right)^2\\right] - \\frac{2(4+z)}{3(1+z)},$ with $z\\equiv m_u/m_d \\simeq 0.56$ , $q^{\\rm (em)}_f$ the electromagnetic charge of quarks and leptons, and $N_f=3$ (1) for quarks (leptons).", "For the model presented in Sec.", "REF , we have $g_{a\\gamma } = 113/39-1.95 \\simeq 0.95$ .", "Thus the prospects for the detection of the flaxion DM are similar to the KSVZ and DFSZ axion model  [23], [24], [25]." ], [ "Constraints on flaxion", "Phenomenological consequences of the flaxion are similar to the DFSZ axion, except that the flaxion has FCNC interactions with the quarks and leptons.", "Here we briefly summarize constraints coming from the flavor-violating process induced by the flaxion and also astrophysical constraints.", "The most stringent bound on $f_a$ may come from the process $K^+\\rightarrow \\pi ^+ a$ mediated by the second term of (REF ).", "In order to evaluate the matrix element of such a process, we adopt $\\langle \\pi (p_\\pi ) | \\overline{s} \\gamma ^\\mu d (x)| K(p_K)\\rangle \\simeq F_1((p_K-p_\\pi )^2) e^{-i(p_K-p_\\pi )\\cdot x} (p_K+p_\\pi )^\\mu $ , with $F_1(0)\\simeq 1$ , which holds in the exact SU(3) flavor symmetry limit.", "Then, the matrix element is given by $\\mathcal {M} =\\frac{\\left(\\kappa ^d_{\\rm AH}\\right)_{12}}{\\sqrt{2}v_\\phi }\\left<\\pi (p_\\pi ) | \\overline{s} d | K(p_K)\\right> =\\frac{\\left(\\kappa ^d_{\\rm AH}\\right)_{12}}{\\sqrt{2}v_\\phi }\\frac{m_K^2-m_\\pi ^2}{m_s-m_d}$ , where we have also used the following relation: $\\partial _\\mu \\left<\\pi (p_\\pi ) | \\overline{s} \\gamma ^\\mu d | K(p_K)\\right> =(m_s-m_d)\\left< \\pi (p_\\pi ) | \\overline{s} d | K(p_K)\\right>$ .", "Consequently, the decay rate is evaluated as $\\Gamma (K^+\\rightarrow \\pi ^+ a) =\\frac{m_K^3}{32\\pi v_\\phi ^2}\\left(1-\\frac{m_\\pi ^2}{m_K^2} \\right)^3\\left|\\frac{(\\kappa ^d_{\\rm AH})_{12}}{m_s-m_d} \\right|^2,$ which gives ${\\rm Br}(K^+\\rightarrow \\pi ^+ a) \\simeq 3\\times 10^{-10}\\left( \\frac{10^{10}\\,{\\rm GeV}}{f_a} \\right)^2\\left( \\frac{26}{N_{\\rm DW}} \\right)^2\\left|\\frac{(\\kappa ^d_{\\rm AH})_{12}}{m_s-m_d} \\right|^2.$ Comparing with the current experimental bound, Br$(K^+\\rightarrow \\pi ^+ a)\\lesssim 7.3\\times 10^{-11}$  [26], the bound on $f_a$ is given by $f_a \\gtrsim 2\\times 10^{10}\\,{\\rm GeV} \\left( \\frac{26}{N_{\\rm DW}} \\right)\\left|\\frac{(\\kappa ^d_{\\rm AH})_{12}}{m_s}\\right|.$ Notice that, because $q_{Q_1}-q_{Q_2}=1$ in order to realize realistic flavor structure (see App.", "), $|(\\kappa ^d_{\\rm AH})_{12}/m_s|\\sim O(\\epsilon )$ (or larger, depending on the U(1)$_F$ charges of the quarks) assuming no accidental cancellation.", "In the near future, it is expected that the NA62 experiment [27] will improve the measurement of $K^+\\rightarrow \\pi ^+ \\bar{\\nu }\\nu $ (and $K^+ \\rightarrow \\pi ^+ a$ ), improving the bound on $f_a$ .", "There are also lepton-flavor violating processes mediated by the flaxion.", "Note that processes including double flaxion vertices such as $\\mu -e$ conversion or $\\mu \\rightarrow 3e$ are highly suppressed.", "On the other hand, the decay of muon including the flaxion as a final state might give a stringent bound.", "The three body decay $\\mu \\rightarrow e a\\gamma $  [28], [29], [21] might be the best to constrain the flaxion coupling to the lepton sector.", "The constraint reads Br$(\\mu \\rightarrow ea\\gamma ) \\lesssim 1.1\\times 10^{-9}$ , which is translated to [28] $f_a \\gtrsim 1\\times 10^8\\,{\\rm GeV} \\left( \\frac{26}{N_{\\rm DW}} \\right)\\left|\\frac{(\\kappa ^l_{\\rm AH})_{12}}{m_\\mu }\\right|.$ On the other hand, the observation of SN1987A event at Kamiokande constrains the flaxion-nucleon coupling, so that the duration of supernova does not change significantly.", "The flaxion-nucleon coupling is given by $\\mathcal {L} = \\sum _{N=p,n}\\frac{C_N m_N}{f_a} ia \\overline{N}\\gamma _5 N,$ where $&C_p \\simeq \\left(\\frac{ (\\kappa ^u_{\\rm H})_{11} }{m_u N_{\\rm DW}}-\\frac{1}{1+z} \\right)\\Delta u + \\left(\\frac{ (\\kappa ^d_{\\rm H})_{11} }{m_d N_{\\rm DW}}-\\frac{z}{1+z} \\right)\\Delta d,\\\\&C_n\\simeq \\left(\\frac{ (\\kappa ^u_{\\rm H})_{11} }{m_u N_{\\rm DW}}-\\frac{1}{1+z} \\right)\\Delta d + \\left(\\frac{ (\\kappa ^d_{\\rm H})_{11} }{m_d N_{\\rm DW}}-\\frac{z}{1+z} \\right)\\Delta u,$ with $\\Delta f$ being the spin content of the nucleon: $S_\\mu \\Delta f \\equiv \\left<N| \\bar{f}\\gamma _\\mu \\gamma _5 f |N\\right>$ .", "They are given by $\\Delta u = 0.85$ and $\\Delta d =-0.41$  [30], resulting in $C_p\\simeq -0.4$ and $|C_n|\\ll |C_p|$ for $N_{\\rm DW}\\gg 1$ .", "The constraint reads [31] $\\frac{f_a}{|C_N|} \\gtrsim 1\\times 10^9\\,{\\rm GeV},$ which is weaker than the constraint from $K^+\\rightarrow \\pi ^+ a$ .", "It is a striking property of the flaxion, which has flavor-violating couplings, that the most stringent lower bound on the PQ scale comes from the flavor physics, not from the SN1987A.", "The flaxion-electron coupling is also constrained by the observations of white dwarf stars so that the cooling of the white dwarf stars due to the flaxion emission does not affect the observed luminosity functions of white dwarf stars too much.", "The constraint reads [32] $f_a \\gtrsim 7\\times 10^7\\,{\\rm GeV}\\left( \\frac{26}{N_{\\rm DW}} \\right)\\left|\\frac{(\\kappa ^l_{\\rm H})_{11}}{m_e}\\right|.$ Observations of horizontal branch stars and red-giant stars also put similar constraints on the flaxion-electron coupling [30].", "Let us also comment on the possible constraint from nucleon decay caused by gauge-invariant baryon- and lepton-number violating higher dimensional operators [33], [34].", "If the cutoff scale of these operators are of order $M$ , these operators are schematically written as $\\mathcal {L} \\sim \\frac{QQQL}{M^2},~~~\\frac{uude}{M^2},~~~\\frac{QQue}{M^2},~~~\\frac{QLud}{M^2}, $ which are multiplied by some powers of $\\phi /M$ to be consistent with U(1)$_F$ symmetry.", "Due to the suppression factor of powers of $\\epsilon = v_\\phi /M$ , the effective cutoff scale of these operators can be much higher than $M$ .", "For the charge assignments of (REF ) and (REF ), the most dangerous operator is the last one in (REF ), which is suppressed only by $\\epsilon ^5$ for the first generation quarks and leptons.", "Therefore, the effective cutoff scale of this operator is $M_{\\rm eff}\\sim \\epsilon ^{-2.5} M \\sim 40\\times M$ and hence we need $M \\gtrsim 5\\times 10^{14}$  GeV to avoid the too rapid proton decay [35].", "This is roughly consistent with the phenomenologically preferred value $M\\sim 10^{14}\\text{--}10^{17}$  GeV, as shown in the next section.", "One should also note that this suppression factor crucially depends on the U(1)$_F$ charge assignments on the quarks and leptons.", "As shown in App.", "REF , we have a freedom of constant shift of all the $(q_{Q_i}, q_{u_i}, q_{d_i})$ without affecting $n^u_{ij}$ and $n^d_{ij}$ .", "Using this freedom, it is possible to suppress all of the operators in (REF ) further.", "Since the Lagrangian (REF ) depends only on the combination $n_{ij}$ , all the phenomenological constraints discussed so far, except for the nucleon decay, remain intact with such a shift of U(1)$_F$ charges." ], [ "Flaxion as dark matter", "Let us discuss cosmological consequences of the present model [36].", "As in the case of ordinary QCD axion, the flaxion starts to oscillate around the minimum of the potential.", "Its present density is given by [37] $\\Omega _a h^2 = 0.18\\,\\theta _i^2 \\left( \\frac{f_a}{10^{12}\\,{\\rm GeV}}\\right)^{1.19},$ where $\\theta _i$ denotes the initial misalignment angle which takes the value $0\\le \\theta _i <2\\pi $ .", "Thus, the flaxion oscillation can be dark matter for $f_a\\sim O(10^{12}\\text{--}10^{15})$  GeV, assuming $\\theta _i\\simeq O(0.01$ –$1)$ .", "As discussed in the previous section, the decay constant of the flaxion is related to the parameters in the flavon potential.", "For $N_{\\rm DW}=26$ and $\\epsilon \\sim 0.2$ , for example, the flaxion dark matter is realized when $v_\\phi \\sim O(10^{13}\\text{--}10^{16})\\ {\\rm GeV}$ and $M\\sim O(10^{14}\\text{--}10^{17})\\ {\\rm GeV}$ ." ], [ "Isocurvature and domain wall problem", "Since the domain wall number is larger than unity, one may require that the U(1)$_F$ symmetry be spontaneously broken during inflation to avoid the serious domain wall problem.", "In this case there is a stringent constraint on the inflation energy scale so that the flaxion does not acquire too large isocurvature fluctuations.", "The recent constraint from the Planck result reads $\\sqrt{\\mathcal {P}_{S}/\\mathcal {P}_\\zeta } \\lesssim 0.18$ with $\\mathcal {P}_{\\zeta } \\simeq 2.2\\times 10^{-9}$ , where $\\mathcal {P}_{S}$ and $\\mathcal {P}_{\\zeta }$ are the dimensionless power spectrum of the (uncorrelated) DM isocurvature and curvature perturbations, respectively [38].", "If the flavon field settles down to the potential minimum during inflation, we have $\\mathcal {P}_S \\simeq \\left( \\frac{H_{\\rm inf}}{\\pi f_a \\theta _i} \\right)^2\\left(\\frac{\\Omega _a h^2}{\\Omega _{\\rm CDM} h^2}\\right)^2,$ where $\\Omega _a h^2$ is given by Eq.", "(REF ) and $\\Omega _{\\rm CDM} h^2\\simeq 0.12$ .", "Thus the inflationary scale is bounded as $H_{\\rm {inf}} \\lesssim 3\\times 10^7\\,{\\rm GeV}\\,\\theta _i^{-1} \\left( \\frac{10^{12}\\,{\\rm GeV}}{f_a} \\right)^{0.19}.$ Notice that this constraint is based on the assumption that the flavon already settles down to its potential minimum during inflation.", "However, the dynamics of the flavon field can be non-trivial during and after inflation and it can significantly modify the constraint.", "Below we see that the flavon itself can play the role of inflaton, avoiding this isocurvature bound." ], [ "Flavon inflation", "So far we have assumed that the inflaton sector is independent of the SM + flavon sector.", "More interestingly, it may be possible to identify the flavon itself as the inflaton.$4\\endcsname $A flavon inflation was considered in Ref.", "[39] in a different context.", "First of all, one should note that large field inflation in which $\\varphi \\equiv \\sqrt{2}{\\rm Re(\\phi )}$ rolls down from $\\varphi \\gg v_\\phi $ would be dangerous, since during the reheating stage the flavon passes through the origin $\\varphi =0$ many times and it leads to the nonthermal symmetry restoration through the parametric resonant enhancement of the flaxion field [40], [41].", "Thus the domain wall problem arises after the QCD phase transition in such a case.", "On the other hand, the small-field inflation in which $\\varphi $ rolls down from near the origin toward the potential minimum may be possible.", "Although there is no domain wall problem in this case, the flavon self coupling constant needs to be very small and also the flaxion isocurvature perturbation tends to be too large because it is enhanced due to the smallness of $\\varphi $ during inflation.", "Here we propose a nonminimal large-field inflation model which avoids these difficulties.", "The idea is to extend the flavon kinetic term to effectively flatten the potential at large field value [42].", "In an extreme case in which the kinetic term has a pole at some field value, the effective potential becomes completely flat around the pole after the canonical normalization, and it leads to a class of large field inflation with best-fit value of the scalar spectral index [12].", "Here, we adopt the following Lagrangian: $\\mathcal {L} = -\\frac{|\\partial \\phi |^2}{\\left(1-\\frac{|\\phi |^2}{\\Lambda ^2}\\right)^2} - \\lambda _\\phi \\left( |\\phi |^2-v_\\phi ^2 \\right)^2.$ After the canonical normalization, the flavon potential may be rewritten as $\\mathcal {L} = -\\frac{(\\partial \\widetilde{\\varphi })^2}{2} - \\lambda _\\phi \\left[ \\Lambda ^2 \\tanh ^2\\left(\\frac{\\widetilde{\\varphi }}{\\sqrt{2}\\Lambda }\\right)-v_\\phi ^2 \\right]^2,$ where $\\frac{\\varphi }{\\sqrt{2}\\Lambda } \\equiv \\tanh \\left(\\frac{\\widetilde{\\varphi }}{\\sqrt{2}\\Lambda }\\right).$ Thus the potential is flat for $\\widetilde{\\varphi }\\gg \\Lambda $ and inflation can take place there.", "If $v_\\phi < \\Lambda < \\sqrt{2}v_\\phi $ , the potential height at large field limit is lower than that at the origin (see Fig.", "REF ), hence the flavon does not pass through the origin after inflation.", "Thus there is no domain wall problem in this case.", "Note that the potential minimum in terms of $\\widetilde{\\varphi }$ is $\\frac{\\left< \\widetilde{\\varphi }\\right>}{\\sqrt{2}} = \\Lambda \\tanh ^{-1}\\left(\\frac{v_\\phi }{\\Lambda }\\right).$ As we will discuss in the following, $\\Lambda $ is found to be of the same order of $v_\\phi $ in the parameter region of our interest, and hence $\\left< \\widetilde{\\varphi }\\right>\\sim O(v_\\phi )$ .", "Figure: Schematic picture of the flavon potential for successful inflation.We also note here that, due to the structure of the kinetic term in the present model, the field $a$ is not canonically normalized.", "The canonically-normalzed flaxion field around the vacuum is $\\tilde{a}\\equiv a/\\Delta $ , where $\\Delta \\equiv 1-v_\\phi ^2/\\Lambda ^2.$ Thus, the flaxion interactions (as well as the decay rate) given in the previous section should take account of the correction factor $\\Delta $ .", "For the case of our interest, however, $\\Delta \\sim O(1)$ and hence the discussion given in the previous section is qualitatively unchanged.", "We can analyze the slow-roll inflation dynamics as usual [43].", "The flavon field value during inflation is calculated as $\\widetilde{\\varphi }_N \\simeq \\frac{\\Lambda }{\\sqrt{2}}\\ln \\left( \\frac{16N_e M_P^2}{\\Lambda ^2-v_\\phi ^2} \\right),$ where $N_e \\sim 50$ – 60 denotes the $e$ -folding number at which the present horizon scale exits the horizon.", "The scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ are given by $n_s \\simeq 1 - \\frac{2}{N_e},~~~~~~~~~~r\\simeq \\frac{4}{N_e^2}\\left( \\frac{\\Lambda }{M_P} \\right)^2.$ Thus the scalar spectral index falls into the Planck best-fit region while the tensor-to-scalar ratio is too small to be detected.", "The dimensionless power spectrum of the curvature perturbation is given by $\\mathcal {P}_\\zeta \\simeq \\frac{N_e^2}{6\\pi ^2}\\frac{\\lambda _\\phi (\\Lambda ^2-v_\\phi ^2)^2}{\\Lambda ^2 M_P^2}.$ In order to reproduce the observed magnitude of the curvature perturbation, $\\mathcal {P}_{\\zeta } \\simeq 2.2\\times 10^{-9}$  [38], $\\lambda _\\phi \\simeq 3\\times 10^{-2}\\left( \\frac{50}{N_e} \\right)^2\\left( \\frac{10^{14}\\,{\\rm GeV}}{\\Lambda } \\right)^2\\left( \\frac{\\Lambda ^2}{\\Lambda ^2-v_\\phi ^2} \\right)^2.$ In order for $\\lambda _\\phi $ to be in the perturbative range, we must have $\\Lambda \\gtrsim 10^{13}\\,$ GeV, meaning $f_a \\sim \\Lambda /N_{\\rm DW} \\gtrsim 5\\times 10^{11}$  GeV.", "This is consistent with the scale inferred from the flaxion DM density (REF ).", "The inflation scale is given by $H_{\\rm inf} \\simeq 5\\times 10^8\\,{\\rm GeV}\\left( \\frac{\\Lambda }{10^{14}\\,{\\rm GeV}} \\right).$" ], [ "Suppression of isocurvature perturbation", "Here we show that the isocurvature perturbation of the flaxion is highly suppressed due to the peculiar structure of the kinetic term.", "We parametrize the complex flavon field as $\\phi = \\varphi e^{i\\Theta }/\\sqrt{2}$ .", "Then, the action for the phase $\\Theta $ is given by $\\mathcal {L} = \\frac{\\Lambda ^2}{4}\\sinh ^2\\left(\\frac{\\sqrt{2}\\widetilde{\\varphi }}{\\Lambda }\\right)\\left(\\partial \\Theta \\right)^2.$ Since $\\widetilde{\\varphi }$ slow-rolls during inflation, we may regard the prefactor in (REF ) as a constant.", "Then, the canonically normalized field during inflation is given by $\\tilde{a}_{\\rm inf} =\\frac{\\Lambda }{\\sqrt{2}}\\sinh \\left(\\frac{\\sqrt{2}\\widetilde{\\varphi }}{\\Lambda }\\right)\\Theta .$ The canonical field $\\tilde{a}_{\\rm inf}$ acquires long-wavelength fluctuations of $H_\\mathrm {inf}/2\\pi $ during inflation, and hence the original phase $\\Theta $ fluctuates as $\\mathcal {P}_{\\delta \\Theta }&\\simeq \\frac{2H_\\mathrm {inf}^2}{\\pi ^2\\Lambda ^2}\\exp \\left(-\\frac{2\\sqrt{2}\\widetilde{\\varphi }_N}{\\Lambda }\\right)\\simeq \\frac{H_\\mathrm {inf}^2}{128\\pi ^2\\Lambda ^2}\\left(\\frac{\\Lambda ^2 - v_\\phi ^2}{N_eM_P^2}\\right)^2,$ where $\\mathcal {P}_{\\delta \\Theta }$ is the power spectrum of $\\delta \\Theta $ .", "(Here, we have used $\\widetilde{\\varphi }\\gg \\Lambda $ during inflation.)", "Since it is related to the fluctuation of the initial misalignment angle as $\\delta \\theta _i =N_\\mathrm {DW}\\delta \\Theta $ , the ratio of the DM isocurvature perturbation to the curvature perturbation is estimated as $\\frac{\\mathcal {P}_S}{\\mathcal {P}_\\zeta }\\simeq \\frac{R_a^2N_\\mathrm {DW}^2}{64\\theta _i^2N_e^4} \\frac{\\left(\\Lambda ^2 - v_\\phi ^2\\right)^2}{M_P^4},$ which is highly suppressed.$5\\endcsname $In this scenario, the spectrum of the flaxion isocurvature fluctuation is blue.", "However, even at the smallest scale the isocurvature perturbation is small enough.", "Thus, the observational bound is safely satisfied and the flaxion can be the dominant component of DM." ], [ "Reheating after flavon inflation", "Finally let us discuss the reheating after flavon inflation.", "There are mainly three decay modes of the flavon: decay into right-handed neutrinos, decay into Higgs bosons and decay into flaxions.", "Other decay modes are suppressed either by the loop factor or the final state fermion masses.", "The flavon partial decay rate into the right-handed neutrino pair is given by $\\Gamma (\\widetilde{\\varphi }\\rightarrow N_{R} N_{R}) \\simeq \\sum _{\\alpha \\beta }\\frac{|y^N_{\\alpha \\beta } n^N_{\\alpha \\beta }\\epsilon ^{n^N_{\\alpha \\beta }-1}|^2}{32\\pi }\\Delta ^2 m_\\varphi ,$ where the flavon mass around the potential minimum is given by $m_\\varphi ^2 = 4\\lambda _\\phi v_\\phi ^2 \\Delta ^2.$ Note that $m_\\varphi \\sim 3\\times 10^{13}\\,{\\rm GeV} (v_\\phi /\\Lambda )$ is almost independent of the overall scale $\\Lambda $ .", "Here we have assumed that the flavon is heavier than the right-handed neutrino: $4\\lambda _\\phi \\Delta ^2 \\gtrsim (y^N_{\\alpha \\alpha }\\epsilon ^{n^N_{\\alpha \\alpha }-1})^2$ .", "The partial decay rate of flavon into the Higgs bosons depend on the additional potential term$6\\endcsname $This term potentially leads to the vacuum decay through the resonant enhancement of the Higgs fluctuation [44], [45], [46], [47].", "However, the same term along with large VEV of $\\phi $ can ensure the absolute stability of the Higgs potential [48].", "Note also that there must be a large bare mass term of the Higgs to cancel the flavon-induced mass term so that it obtains the electroweak scale VEV.", "$V = \\lambda _{\\phi H} |\\phi |^2 |H|^2.$ We find the partial decay rate into the Higgs boson pair as $\\Gamma (\\widetilde{\\varphi }\\rightarrow HH) \\simeq \\frac{\\Delta ^2}{8\\pi }\\frac{\\lambda _{\\phi H}^2 v_\\phi ^2}{m_\\varphi }\\simeq \\frac{1}{32\\pi } \\frac{\\lambda _{\\phi H}^2}{\\lambda _\\phi } m_\\varphi ,$ where we have taken account of the four real degrees of freedom in the SM Higgs doublet.$7\\endcsname $This coupling radiatively affects the flavon potential.", "If it is substantially large and $\\lambda _\\phi $ is too small, the flavon potential can be dominated by the radiatively-induced effective potential.", "On the other hand, the flavon partial decay rate into the flaxion pair is given by $\\Gamma (\\widetilde{\\varphi }\\rightarrow aa) \\simeq \\frac{\\Delta ^2}{32\\pi } \\frac{m_\\varphi ^3}{v_\\phi ^2} \\simeq \\frac{\\lambda _\\phi }{8\\pi } \\Delta ^4 m_\\varphi .$ Thus the total decay width of the flavon is $\\Gamma _{\\widetilde{\\varphi }} \\simeq \\left(\\sum _{\\alpha ,\\beta }\\left|y^N_{\\alpha \\beta } n^N_{\\alpha \\beta }\\epsilon ^{n^N_{\\alpha \\beta }-1}\\right|^2\\frac{\\Delta ^2}{4} + \\frac{\\lambda _{\\phi H}^2}{4\\lambda _\\phi } + \\lambda _\\phi \\Delta ^4\\right)\\frac{m_\\varphi }{8\\pi }.$ This is typically much larger than $H_{\\rm inf}$ and hence the reheating is completed almost instantaneously after inflation.", "Thus the reheating temperature, $T_{\\rm R}$ , can be as high as $10^{12}\\text{--}10^{14}$  GeV in our scenario.", "Flaxions are thermalized through interactions with Higgs and right-handed neutrinos and there is no problem of flaxion dark radiation overproduction.", "Lastly let us discuss thermal leptogenesis in the present scenario.", "The final baryon asymmetry through the leptogenesis from the decay of right-handed neutrinos is given by [49] $\\frac{n_B}{s}\\simeq \\epsilon _1 \\kappa _f \\frac{28}{79}\\left(\\frac{n_{N_1}}{s}\\right)_{\\rm th}\\simeq 1.3\\times 10^{-3} \\epsilon _1 \\kappa _f,$ where $(n_{N_1}/s)_{\\rm th}$ is the abundance of the right-handed neutrino in thermal equilibrium, $\\epsilon _1$ denotes the lepton asymmetry generated by per right-handed neutrino decay and $\\kappa _f$ denotes the efficiency factor.", "The asymmetry parameter is calculated as $\\epsilon _1= \\frac{3}{16\\pi } \\frac{m_{N_1} m_{\\nu _3}}{v_{\\rm EW}^2}\\delta _{\\rm eff}\\simeq 1\\times 10^{-4}\\left( \\frac{m_{N_1}}{10^{12}\\,{\\rm GeV}} \\right)\\left( \\frac{m_{\\nu 3}}{0.05\\,{\\rm eV}} \\right)\\delta _{\\rm eff},$ where $\\delta _{\\rm eff}$ is the effective CP angle which satisfies $\\delta _{\\rm eff} \\le 1$ for $m_{N_1}\\ll m_{N_{2(3)}}$  [50], [51].$8\\endcsname $If the mass of $N_1$ is degenerated with $N_2$ , the asymmetry is enhanced [52].", "This can happen in our case if U(1)$_F$ charges of right-handed neutrinos are the same.", "On the other hand, the efficiency factor $\\kappa _f$ crucially depends on the effective neutrino mass $\\widetilde{m}_{\\nu 1} \\equiv \\sum _k |\\epsilon ^{n^\\nu _{k1}}y_{k1}^\\nu |^2 v_{\\rm EW}^2/m_{N_1}$ .", "In the present scenario, it is roughly given by $\\widetilde{m}_{\\nu 1}\\sim \\sum _k \\epsilon ^{2q_{L_k}} v_{\\rm EW}^2/M\\sim m_{\\nu _3}$ .", "(See App. .)", "This corresponds to a so-called strong washout regime ($\\widetilde{m}_{\\nu 1}\\gtrsim m_*\\simeq 1\\times 10^{-3}~{\\rm eV}$ ), where the efficiency factor is approximately given by $\\kappa _f \\sim 0.02\\times (\\widetilde{m}_{\\nu 1}/0.01~{\\rm eV})^{-1.1}$  [49].", "For $\\widetilde{m}_{\\nu 1}\\sim m_{\\nu _3}\\sim 0.05~{\\rm eV}$ , we obtain $\\kappa _f\\sim 3\\times 10^{-3}$ .", "Therefore, the observed baryon asymmetry $n_B/s\\simeq 9\\times 10^{-11}$ can be obtained for $m_{N_1}\\sim O(10^{12})$  GeV.$9\\endcsname $For this mass scale, none of the charged lepton Yukawa coupling is in equilibrium, and the flavor effect [53] can be neglected.", "This can be obtained, for instance, by taking $q_{N_1}=1-5$ for $M\\sim O(10^{14}\\text{--}10^{17})\\,{\\rm GeV}$ ." ], [ "Conclusions and discussion", "We have shown that a simple QCD axion model in which U(1)$_{\\rm PQ}$ is identified with Abelian flavor symmetry U(1)$_F$ solves and explains puzzles in the SM.", "The model contains only one additional complex scalar and right-handed neutrinos.", "Inflation can successfully happen without domain wall nor isocurvature problems.", "Here are some remarks.", "In this paper, we assume that there is only one Higgs doublet.", "Although this is a minimal choice, if there are additional Higgs doublets, we can assign the $\\mathrm {U}(1)_F$ charges to the Higgses so that $N_\\mathrm {DW} = 1$ .", "As an example, we consider a two Higgs doublet model (2HDM) [54] with the following Yukawa interactions (the so-called type-II or type-Y 2HDM): $-\\mathcal {L} = y_{ij}^d\\left( \\frac{\\phi }{M} \\right)^{n_{ij}^d} \\overline{Q}_i H_d d_{Rj}+ y_{ij}^u\\left( \\frac{\\phi }{M} \\right)^{n_{ij}^u} \\overline{Q}_i H_u u_{Rj}.", "$ If we assign the $\\mathrm {U}(1)_F$ charges $q_{H_u}$ and $q_{H_d}$ on $H_u$ and $H_d$ respectively, we obtain $n_{ij}^d &= q_{Q_i} - q_{d_j} - q_{H_d}, \\\\n_{ij}^u &= q_{Q_i} - q_{u_j} - q_{H_u}.$ We may keep $n_{ij}^f$ the same as those in (REF ) by shifting the charges of the right-handed quarks as $q_{f_i} \\rightarrow q_{f_i} - q_{H_f}$ .$10\\endcsname $If $\\tan \\beta \\equiv \\langle H_u\\rangle /\\langle H_d\\rangle $ is not of order unity, the ratio of the overall normalization of $n_{ij}^d$ and $n_{ij}^u$ can be much different than (REF ).", "Then, the domain wall number is given by $N_\\mathrm {DW}= \\left|{\\rm Tr}\\left(2 \\widehat{q}_{Q} - \\widehat{q}_{u} - \\widehat{q}_{d}\\right)\\right|= \\left|26 + 3\\left(q_{H_u} + q_{H_d}\\right)\\right|.$ Thus, the domain wall number is $N_\\mathrm {DW} = 1$ if we take $q_{H_u} + q_{H_d} = -9$ .", "In this case, there is no cosmological domain wall problem even if the PQ symmetry is restored during inflation, as long as $f_a < (4.6-7.2)\\times 10^{10}\\,$ GeV [55] and hence there can be a variety of cosmological scenarios.", "Although the minimality is lost, it is also easy to embed the theory into the supersymmetry (SUSY) framework.", "We can just interpret the Lagrangian (REF ) as the superpotential written by the chiral superfields.", "Since there are two Higgs doublets in minimal SUSY SM, we can choose the U(1)$_F$ charges so that $N_\\mathrm {DW} = 1$ as just shown above.", "In this case, the $\\mu $ -term can be generated by the superpotetnial $W \\sim (\\phi /M)^9 M H_u H_d$ , which may be compatible with high-scale SUSY scenario in which the soft mass scale is $O(100-1000)\\,$ TeV.", "The potential of the flavon can be generated by introducing $\\bar{\\phi }$ and also a “stabilizer field” $X$ , which have U(1)$_F$ charges $-1$ and 0 respectively, and assume the superpotential $W = \\lambda X (\\phi \\bar{\\phi }- v_\\phi ^2).$ After they get soft SUSY breaking masses, they are stabilized at $\\phi \\sim \\bar{\\phi }\\sim v_\\phi $ .", "Since $\\bar{\\phi }$ is oppositely charged under U(1)$_F$ , it cannot directly couple to SM Yukawa terms.", "Note that, with the present assignments of U(1)$_F$ charges, off-diagonal elements of the squark mass matrix are not suppressed enough to avoid SUSY flavor problem if the mass scale of the SUSY particiles is around TeV.", "Such a problem can be solved by high-scale SUSY or flavor-blind mediation model (like gauge mediation).", "(Otherwise one may adopt a different flavor symmetry to suppress the off-diagonal elements of the sfermion mass matrix.)", "Cosmology of this class of models will be non-trivial due to the presence of sflaxion and flaxino which appear in the flavon supermultiplet, although the detailed investigation is beyond the scope of this paper." ], [ "Acknowledgments", "We thank Natsumi Nagata for helpful discussion.", "This work was supported by the Grant-in-Aid for Scientific Research on Scientific Research A (No.26247038 [KH], No.26247042 [KN], No.16H02189 [KH]), Scientific Research C (No.26400239 [TM]), Young Scientists B (No.26800121 [KN], No.26800123 [KH]) and Innovative Areas (No.26104001 [KH], No.26104009 [KH and KN], No.15H05888 [KN], No.16H06490 [TM]), and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.", "The work of YE was supported in part by JSPS Research Fellowships for Young Scientists and by the Program for Leading Graduate Schools, MEXT, Japan." ], [ "Quark and charged lepton masses and CKM matrix", "For general U(1)$_F$ charge assignments on the quark fields $\\begin{pmatrix}q_{Q_1} & q_{Q_2} & q_{Q_3} \\\\q_{u} & q_{c} & q_{t} \\\\q_{d} & q_{s} & q_{b}\\end{pmatrix},$ with assumption $q_{Q_i} \\ge q_{Q_j} \\ge 0$ and $q_{f_i} \\le q_{f_j}\\le 0$ for $i < j$ , the quark mass matrix (normalized by $v_{\\rm EW}$ ) is expressed and decomposed as $m_{ij}^d &\\sim \\epsilon ^{n_{ij}^d} \\sim V^d {\\rm diag}(m_d) U^{d\\dagger }\\nonumber \\\\&\\sim \\begin{pmatrix}1 &\\epsilon ^{q_{Q_1}-q_{Q_2}} & \\epsilon ^{q_{Q_1}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_2}} & 1 & \\epsilon ^{q_{Q_2}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_3}} & \\epsilon ^{q_{Q_2}-q_{Q_3}} & 1\\end{pmatrix}\\begin{pmatrix}\\epsilon ^{q_{Q_1}-q_d} & 0 & 0\\\\0 & \\epsilon ^{q_{Q_2}-q_s} & 0\\\\0 & 0 & \\epsilon ^{q_{Q_3}-q_b}\\end{pmatrix}\\begin{pmatrix}1 &\\epsilon ^{q_{s}-q_{d}} & \\epsilon ^{q_{b}-q_{d}} \\\\\\epsilon ^{q_{s}-q_{d}} & 1 & \\epsilon ^{q_{b}-q_{s}} \\\\\\epsilon ^{q_{b}-q_{d}} & \\epsilon ^{q_{b}-q_{s}} & 1\\end{pmatrix},$ $m_{ij}^u &\\sim \\epsilon ^{n_{ij}^u} \\sim V^u {\\rm diag}(m_u) U^{u\\dagger }\\nonumber \\\\&\\sim \\begin{pmatrix}1 &\\epsilon ^{q_{Q_1}-q_{Q_2}} & \\epsilon ^{q_{Q_1}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_2}} & 1 & \\epsilon ^{q_{Q_2}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_3}} & \\epsilon ^{q_{Q_2}-q_{Q_3}} & 1\\end{pmatrix}\\begin{pmatrix}\\epsilon ^{q_{Q_1}-q_u} & 0 & 0\\\\0 & \\epsilon ^{q_{Q_2}-q_c} & 0\\\\0 & 0 & \\epsilon ^{q_{Q_3}-q_t}\\end{pmatrix}\\begin{pmatrix}1 &\\epsilon ^{q_{c}-q_{u}} & \\epsilon ^{q_{t}-q_{u}} \\\\\\epsilon ^{q_{c}-q_{u}} & 1 & \\epsilon ^{q_{t}-q_{c}} \\\\\\epsilon ^{q_{t}-q_{u}} & \\epsilon ^{q_{t}-q_{c}} & 1\\end{pmatrix}.$ Thus the CKM matrix is given by $V_{\\rm CKM} = V^{u\\dagger } V^d \\sim \\begin{pmatrix}1 &\\epsilon ^{q_{Q_1}-q_{Q_2}} & \\epsilon ^{q_{Q_1}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_2}} & 1 & \\epsilon ^{q_{Q_2}-q_{Q_3}} \\\\\\epsilon ^{q_{Q_1}-q_{Q_3}} & \\epsilon ^{q_{Q_2}-q_{Q_3}} & 1\\end{pmatrix},$ which depends only on the charges of the left-handed quarks.", "Taking account of $O(1)$ Yukawa couplings, it well reproduces observed values of the CKM matrix elements for $\\begin{pmatrix}q_{Q_1} & q_{Q_2} & q_{Q_3}\\end{pmatrix}=\\begin{pmatrix}q_{Q_3}+3 & q_{Q_3}+2 & q_{Q_3}\\end{pmatrix},$ and $\\epsilon \\simeq 0.23$ .", "Charges of right-handed quarks are chosen so that the quark mass eigenvalues are consistent with observed values: $q_{Q_1}-q_d=7,~~~q_{Q_2}-q_s=5,~~~q_{Q_3}-q_b=3,\\\\q_{Q_1}-q_u=8,~~~q_{Q_2}-q_c=3,~~~q_{Q_3}-q_t=0.$ Still we have a degree of freedom to choose $q_{Q_3}$ , corresponding to the overall constant shift of $(q_{Q_i}, q_{u_i},q_{d_i})$ .$11\\endcsname $In other words, we can arbitrarily add baryon charges to the U(1)$_F$ charges.", "A particular example with $q_{Q_3}=0$ is given in (REF ).", "Similarly, for general U(1)$_F$ charge assignments on the leptons $\\begin{pmatrix}q_{L_1} & q_{L_2} & q_{L_3} \\\\q_{e} & q_{\\mu } & q_{\\tau }\\end{pmatrix},$ with assumption $q_{L_i} \\ge q_{L_j} \\ge 0$ and $q_{f_i} \\le q_{f_j}\\le 0$ for $i < j$ , the charged lepton mass matrix (normalized by $v_{\\rm EW}$ ) is decomposed as $m_{ij}^l &\\sim \\epsilon ^{n_{ij}^l} \\sim V^l {\\rm diag}(m_l) U^{l\\dagger }\\nonumber \\\\&\\sim \\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}\\begin{pmatrix}\\epsilon ^{q_{L_1}-q_e} & 0 & 0\\\\0 & \\epsilon ^{q_{L_2}-q_\\mu } & 0\\\\0 & 0 & \\epsilon ^{q_{L_3}-q_\\tau }\\end{pmatrix}\\begin{pmatrix}1 &\\epsilon ^{q_{\\mu }-q_{e}} & \\epsilon ^{q_{\\tau }-q_{e}} \\\\\\epsilon ^{q_{\\mu }-q_{e}} & 1 & \\epsilon ^{q_{\\tau }-q_{\\mu }} \\\\\\epsilon ^{q_{\\tau }-q_{e}} & \\epsilon ^{q_{\\tau }-q_{\\mu }} & 1\\end{pmatrix}.$ The observed charged lepton masses are reproduced for $\\begin{pmatrix}q_{L_1} & q_{L_2} & q_{L_3}\\end{pmatrix}=\\begin{pmatrix}q_{e}+9 & q_{\\mu }+5 & q_{\\tau }+3\\end{pmatrix}.$ The charges of left-handed leptons are partly constrained from the neutrino mass matrix, as shown below." ], [ "Neutrino masses and mixing", "First let us consider the minimal case of two right-handed neutrinos: $N_\\alpha $ $(\\alpha =1,2)$ .", "For general U(1)$_F$ charge assignments on right-handed neutrinos $(q_{N_1}~q_{N_2})$ , the Dirac- and Majorana-mass matrices of neutrinos are given by $(m^\\nu _D)_{ i\\alpha } \\sim v_{\\rm EW}\\begin{pmatrix}\\epsilon ^{q_{L_1}-q_{N_1}} & \\epsilon ^{q_{L_1}-q_{N_2}} \\\\\\epsilon ^{q_{L_2}-q_{N_1}} & \\epsilon ^{q_{L_2}-q_{N_2}} \\\\\\epsilon ^{q_{L_3}-q_{N_1}} & \\epsilon ^{q_{L_3}-q_{N_2}}\\end{pmatrix},~~~(m^N)_{\\alpha \\beta }\\sim M\\begin{pmatrix}\\epsilon ^{-2q_{N_1}} & \\epsilon ^{-q_{N_1}-q_{N_2}} \\\\\\epsilon ^{-q_{N_1}-q_{N_2}} & \\epsilon ^{-2q_{N_2}}\\end{pmatrix}.$ According to the seesaw mechanism, after integrating out heavy right-handed neutrinos, we obtain the following light neutrino mass matrix: $m^\\nu _{ij} =m^{\\nu }_D \\cdot (m^N)^{-1}\\cdot (m^{\\nu }_D)^T \\sim \\frac{v_{\\rm EW}^2}{M}\\begin{pmatrix}\\epsilon ^{2q_{L_1}} & \\epsilon ^{q_{L_1}+q_{L_2}} & \\epsilon ^{q_{L_1}+q_{L_3}} \\\\\\epsilon ^{q_{L_1}+q_{L_2}} & \\epsilon ^{2q_{L_2}} & \\epsilon ^{q_{L_2}+q_{L_3}} \\\\\\epsilon ^{q_{L_1}+q_{L_3}} & \\epsilon ^{q_{L_2}+q_{L_3}} & \\epsilon ^{2q_{L_3}}\\end{pmatrix}.", "$ It is independent of the charges of right-handed neutrinos.", "Note that since the matrix $m^N$ is rank 2, $m^{\\nu }_{ij}$ must contain one zero eigenvalue.", "It is diagonalized as $m_{ij}^\\nu &\\sim U^\\nu {\\rm diag}(m^\\nu ) (U^{\\nu })^T\\nonumber \\\\&\\sim \\frac{v_{\\rm EW}^2}{M}\\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}\\begin{pmatrix}0 & 0 & 0\\\\0 & \\epsilon ^{2q_{L_2}} & 0\\\\0 & 0 & \\epsilon ^{2q_{L_3}}\\end{pmatrix}\\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}.$ The MNS matrix is given by $U_{\\rm MNS} = U^\\nu V^{l\\dagger } \\sim \\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}.", "$ Therefore, the large $\\nu _\\mu -\\nu _\\tau $ mixing is obtained for $q_{L_2}=q_{L_3}$ .", "A reasonable choice to reproduce the observed MNS matrix is thus $\\begin{pmatrix}q_{L_1} & q_{L_2} & q_{L_3}\\end{pmatrix}=\\begin{pmatrix}q_{L_3}+1 & q_{L_3} & q_{L_3}\\end{pmatrix}.$ For $M \\sim 10^{14}\\text{--}10^{15}$  GeV as a representative value as described in the main text, the observed neutrino mass differences are consistent with $q_{L_3}=0$ .", "This is the one given in (REF ).", "For $M \\sim 10^{16}\\text{--}10^{17}$  GeV, a slightly small Yukawa coupling $y^N\\sim O(0.01)$ is required.", "Note that if $q_{L_3}$ takes a half-integer value, all the lepton and right-handed neutrino charges should also be half-integer.", "Next, let us consider the case of three right-handed neutrinos: $N_\\alpha $ $(\\alpha =1$ –$3)$ .", "For general U(1)$_F$ charge assignments on right-handed neutrinos $(q_{N_1}~q_{N_2}~q_{N_3})$ , the Dirac- and Majorana-mass matrices of neutrinos are given by $&(m^\\nu _D)_{ i\\alpha } \\sim v_{\\rm EW}\\begin{pmatrix}\\epsilon ^{q_{L_1}-q_{N_1}} & \\epsilon ^{q_{L_1}-q_{N_2}} & \\epsilon ^{q_{L_1}-q_{N_3}} \\\\\\epsilon ^{q_{L_2}-q_{N_1}} & \\epsilon ^{q_{L_2}-q_{N_2}} & \\epsilon ^{q_{L_2}-q_{N_3}} \\\\\\epsilon ^{q_{L_3}-q_{N_1}} & \\epsilon ^{q_{L_3}-q_{N_2}} & \\epsilon ^{q_{L_3}-q_{N_3}}\\end{pmatrix},\\\\&(m^N)_{\\alpha \\beta }\\sim M\\begin{pmatrix}\\epsilon ^{-2q_{N_1}} & \\epsilon ^{-q_{N_1}-q_{N_2}} & \\epsilon ^{-q_{N_1}-q_{N_3}} \\\\\\epsilon ^{-q_{N_1}-q_{N_2}} & \\epsilon ^{-2q_{N_2}} & \\epsilon ^{-q_{N_2}-q_{N_3}} \\\\\\epsilon ^{-q_{N_1}-q_{N_3}} & \\epsilon ^{-q_{N_2}-q_{N_3}} & \\epsilon ^{-2q_{N_3}}\\end{pmatrix}.$ The resulting structure of the light neutrino mass matrix after integrating out the heavy right-handed neutrinos is the same as (REF ).", "The MNS matrix is also the same as (REF ).", "Only the difference is that there is no zero mass eigenvalues in the light neutrino mass matrix: $m_{ij}^\\nu &\\sim U^\\nu {\\rm diag}(m^\\nu ) (U^{\\nu })^T\\nonumber \\\\&\\sim \\frac{v_{\\rm EW}^2}{M}\\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}\\begin{pmatrix}\\epsilon ^{2q_{L_1}} & 0 & 0\\\\0 & \\epsilon ^{2q_{L_2}} & 0\\\\0 & 0 & \\epsilon ^{2q_{L_3}}\\end{pmatrix}\\begin{pmatrix}1 &\\epsilon ^{q_{L_1}-q_{L_2}} & \\epsilon ^{q_{L_1}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_2}} & 1 & \\epsilon ^{q_{L_2}-q_{L_3}} \\\\\\epsilon ^{q_{L_1}-q_{L_3}} & \\epsilon ^{q_{L_2}-q_{L_3}} & 1\\end{pmatrix}.$" ] ]
1612.05492
[ [ "An improved solution to geometric distortion using an orthogonal method" ], [ "Abstract The geometric distortion of CCD field of view has direct influence on the positional measurements of CCD observations.", "In order to obtain high precision astrometric results, the geometric distortion should be derived and corrected precisely.", "As presented in our previous work Peng et al.", "(2012), a convenient solution has been carried out and also been made with successful application to Phoebe's observations.", "In order to further improve the solution, an orthogonal method based on the Zernike polynomials is used in this work.", "Four nights of CCD observations including Himalia, the sixth satellite of Jupiter, and open clusters (NGC1664 or NGC2324) on each night have been processed to make an application.", "The observations were obtained from the 2.4 m telescope administered by Yunnan Observatories.", "The catalog UCAC4 was used to match reference stars in all of the CCD frames.", "The ephemeris of Himalia is retrieved from the (IMCCE).", "Our results show that the means of observed minus computed (O-C) positional residuals are -0.034 and -0.026 arcsec in right ascension and declination, respectively.", "The corresponding standard deviations are 0.031 and 0.028 arcsec.", "The measurement dispersion is significantly improved than that by using our previous solution." ], [ "Introduction", "The geometric distortion (called GD hereafter) which exists in both the space telescopes and ground-based telescopes has direct influence on astrometric precision of CCD observations.", "Gilmozzi et al.", "([10]) have found significant GD effects in the WFPC1 and WFPC2 of Hubble Space Telescope (HST).", "A very small field of view of 80$^{\\prime \\prime }$$\\times $ 80$^{\\prime \\prime }$ for each CCD chip of WFPC2 has a maximum GD of about 5 pixels at the edge of its field (Anderson & King [2]).", "The astrometric potential of HST was just developed out after deriving the GD patterns and correcting its effects on positional measurements of planetary satellites (French et al. [7]).", "Anderson et al.", "([1]) also applied the GD solution from HST to the ground-based 2.2 m telescope of ESO, and achieved a precision of $\\sim $ 7 mas.", "In our previous works (Peng & Fan [13]; Peng & Tu [14]; Zhang et al.", "[21]), GD effects of the 2.4 m and 1 m telescopes administered by Yunnan Observatories were first studied.", "As presented in Peng et al.", "([15]), an alternative GD solution which is different from the solution of Anderson & King ([2]) has been carried out and also been made with successful application to Phoebe's observations.", "Since then, we have made several works with the new GD solution (Yang et al.", "[19]; Peng et al.", "[16]; Wang et al.", "[18]; Peng et al. [12]).", "As presented in Peng et al.", "([15]), a dense star field should be observed in an overlapping scheme for deriving the GD patterns.", "As a practice, we may take multiple dithered exposures of the same sky field at different offsets in a pattern of “+\" (Anderson et al.", "[1]) or “#\" (Bellini & Bedin [3]).", "The offsets between any two neighboring CCD frames are about 1 arcmin in right ascension or in declination.", "In this way, a same star would appear in different overlapped CCD frames at different pixel positions for many times.", "According to the illustration showed in Peng et al.", "([15]), an iteration method is used for deriving the GD patterns.", "In each iterative step, GDs of all the star images at different pixel positions could be obtained.", "Then all the GDs could be divided into many equal-area boxes, such as 19$\\times $ 19 for the 2.4 m telescope.", "The average in each box would be indicative of the GD at its center if a gradual variation is assumed for the GD distributions.", "However, the scheme of dividing CCD field of view into many equal-area boxes is in some degree subjected to the distribution of star images.", "The GDs at the centers of some boxes can't be obtained when no star image exists in these areas.", "As such, we try to adopt an orthogonal method presented in Plewa et al.", "([17]).", "A list of twenty orthonormal basis vector fields which are based on the Zernike polynomials were used.", "For a detailed derivation, one can see Zhao & Burge ([22], [23]).", "As showed in Plewa et al.", "([17]), the radio source and massive black hole Sgr A* at the Galactic Center can be placed in the origin of an infrared astrometric reference frame with a precision of $\\sim $ 0.17 mas in position (in 2009) and $\\sim $ 0.07 mas yr$^{-1}$ in velocity, after correcting optical distortion in their NACO imager.", "This precision is a factor of 5 better than the previous results.", "This orthogonal method is used in this work to improve our previous GD solution.", "Specifically, instead of dividing the CCD field of view into many equal-area boxes, GDs of all the star images at different pixel positions in each iteration step were directly fitted by this group of basis vector fields.", "This method doesn't depend on the distribution of star images.", "The contents of this paper are arranged as follows: in Section 2, the CCD observations are described; Section 3 presents the details of deriving GD patterns using the orthogonal method; in Section 4, we show the results and make discussions; and finally, in Section 5, conclusions are drawn." ], [ "CCD Observations", "In order to analyze the improvements which the orthogonal method can obtain, four nights of CCD observations targeting Himalia, the sixth satellite of Jupiter, and open clusters (NGC1664 or NGC2324) were processed.", "These observations were obtained from the 2.4 m telescope (Fan et al.", "[5]) administered by Yunnan Observatories (IAU code O44, longitude E 100$^\\circ $ 1$^{\\prime }$ 51$^{\\prime \\prime }$ , latitude N 26$^\\circ $ 42$^{\\prime }$ 32$^{\\prime \\prime }$ , height 3193 m above sea level).", "The CCD detector used is the Yunnan Faint Object Spectrograph and Camera (YFOSC) instrument.", "Specifications of the 2.4 m telescope and YFOSC are listed in Table 1.", "Table 2 lists distributions of the CCD observations with respect to the observational dates.", "The observational dates were chosen according to the epoch when Jupiter was near its opposition.", "A total of 75 CCD frames of Himalia were obtained, as well as 176 CCD frames of calibration fields which were used for deriving GD patterns.", "The exposure time for each CCD frame is from 20s to 40s, depending on the meteorological conditions.", "Table: Specifications of the 2.4 m telescope administered by Yunnan Observatories and the corresponding CCD detector.Table: CCD observations of Himalia and calibration fields by using the 2.4 m telescope administered by Yunnan Observatories.", "Column 1 shows the observational dates.", "Column 2 lists the open clusters observed.", "Column 3 and Column 4 list the numbers of observations for open clusters and Himalia, respectively.", "The Johnson-I filter was used in all observations." ], [ "Details of Deriving GD patterns", "As presented in Peng et al.", "([15]), an important relationship between the distortions at two different pixel positions for a common star can be derived, if the star was observed in two different CCD frames.", "The GDs in two CCD frames can be expressed as follows if the measured errors are temporarily neglected, $dx_i=\\Delta x_i-\\frac{\\hat{e}_icosD_i}{\\hat{e}_jcosD_j}\\Delta x_j +\\frac{\\hat{e}_icosD_i}{\\hat{e}_jcosD_j}dx_j,$ $dy_i=\\Delta y_i-\\frac{\\hat{e}_i}{\\hat{e}_j}\\Delta y_j +\\frac{\\hat{e}_i}{\\hat{e}_j}dy_j.$ In equations (1a) and (1b), all quantities with the suffix $i$ are associated with the $i$ th CCD frame and the suffix $j$ with the $j$ th CCD frame.", "$\\hat{e}=cos\\varphi /\\rho $ is one of the estimated parameters in four-parameter linear transformation.", "$\\rho $ and $\\varphi $ are the approximate angular extent per pixel and the orientation of CCD chip used.", "$\\Delta x$ and $\\Delta y$ are the differences between the measured pixel location ($x_o$ , $y_o$ ) of a star and the indirectly computed one ($x_c$ , $y_c$ ) of the same star by using the four-parameter linear transformation with estimated parameters.", "The four parameters can be solved by a least-squares fitting.", "$D$ is declination of the tangent point on tangent plane of the celestial sphere for each CCD frame.", "For a definite star, equation (1a) and (1b) can be solved if the star appears in $N$ ($N\\gg 2 $ ) CCD frames with different offsets.", "Then the distortions ($dx_i$ , $dy_i$ ) of the star at different pixel positions in many CCD frames can be obtained.", "Furthermore, for all stars, the distortions at different pixel positions in all CCD frames can be collected.", "These distortions are divided into many equal-area boxes, such as 19$\\times $ 19 for the 2.4 m telescope.", "The average in each box will be indicative of the GD at its center.", "Then the distortions of all star images at their pixel positions can be calculated through bilinear interpolation.", "For more details, one can see Peng et al.", "([15]).", "As mentioned above, the scheme of dividing CCD field of view into many equal-area boxes is subjected to the distribution of star images.", "The GDs at the centers of some boxes which have no star images cannot be obtained.", "Thus we take use of an orthogonal method proposed in Plewa et al.", "([17]) which does not depend on the distribution of star images.", "As analyzed in Plewa et al.", "([17]), twenty orthonormal basis vector fields are needed to fully capture the spatial variability of the image distortion.", "These basis vector fields are derived based on the Zernike polynomials.", "For a detailed derivation, one can see Zhao & Burge ([22], [23]).", "The explicit form of the twenty vector fields are listed in Table 3.", "Table: Explicit form of the distortion model in terms of its basis vector fields.", "For a derivation, one can see Zhao & Burge (2007, 2008).", "Column 1 shows the designation of each vector field.", "Column 2 lists the scale factor which should be multiplied by each vector field.", "The dd parameter in the scale factor represents the number of pixels in each dimension after that the original image pixel array is rescaled.", "Column 3 and column 4 list the components in two dimensions, respectively.As showed in Table 3, in order to apply these twenty basis vector fields in our previous GD solution, there are several steps to be accomplished.", "Firstly, the pixel positions of star images are rescaled that the pixel coordinates become much more smaller than the original ones.", "In such a way, the numerical computations can be more precise.", "Secondly, the scale factors listed in Table 3 can be calculated according to the orthonormality for any two vector fields.", "Thirdly, a least-squares fitting is applied for deriving the coefficients of each vector field.", "Finally, the distortion at any pixel position can be directly calculated by using the vector field of GD which is solved in the previous step.", "Specifically, as illustrated in Zhao & Burge ([22], [23]), the two components of each vector field which are $G_x(x,y)$ and $G_y(x,y)$ listed in Table 3 are defined over a unit circle.", "However, CCD chips are always square or rectangle.", "Thus the transformation from an unit circle to a square or a rectangle must be applied.", "In practice, if $\\vec{B}$ and $\\vec{C}$ are two vector fields defined over an unit circle, we define their inner product as $\\left(\\vec{B},\\vec{C}\\right)=\\frac{1}{\\pi }\\iint \\left(\\vec{B}\\cdot \\vec{C}\\right)dxdy,$ where $\\pi $ is the area of an unit circle.", "Then the inner product of two vector fields which are $\\vec{G}_i$ and $\\vec{G}_j$ ($i,j=1\\sim 20$ ) defined over a square or a rectangle is $\\left(\\vec{G}_i,\\vec{G}_j\\right)=\\frac{1}{A}\\iint \\left(\\vec{G}_i\\cdot \\vec{G}_j\\right)dxdy,$ where $A$ is the area of a square or a rectangle.", "In order to satisfy the orthonormality, the inner product of any two vector fields defined over a square or a rectangle is $\\left(\\vec{G}_i,\\vec{G}_j\\right)=\\delta _{ij}={\\left\\lbrace \\begin{array}{ll}1,\\mbox{ if }i=j \\\\0,\\mbox{ if }i\\ne j.\\end{array}\\right.", "}$ According to equation (4), the scale factors listed in Table 3 can be calculated.", "The values of scale factors depend on the rescaled size of image pixel array.", "In practice, the scale factors according to a square pixel array which has $d$ pixels in each dimension are listed in Table 3.", "An iterative method is used for deriving the vector field of GD.", "Specifically, in a definite iteration step, the GDs of all star images are fitted by the distortion model in Table 3.", "The GD pattern in this iteration step is added to the final GD pattern.", "Then the GDs of all star images in the next iteration step are solved again after GD corrections are made.", "When the values of GD pattern in an iteration step are within 0.01 pixel, the iteration process is stopped.", "After the final vector field of GD is solved, the distortions of all star images at their pixel positions can be directly computed.", "Finally, the GD corrections can be applied." ], [ "Results and Discussions", "The catalog UCAC4 (Zacharias et al.", "[20]) was chosen to match reference stars in all CCD frames.", "The minimum and maximum numbers of UCAC4 reference stars available for astrometric reduction of Himalia are 7 and 18, respectively.", "Observed positions are derived relative to these UCAC4 reference stars by using a plate model with four constants.", "However, this is accurate only after all the astrometric effects, including GD effects, are taken into account (Peng et al. [15]).", "Fig.", "1 shows the GD patterns derived by both the previous and improved solutions, and also differences between the GD patterns.", "One can see that the distributions and variations of GD vectors are more smooth after that the improved solution was used.", "From the first row of Fig.", "1, we can see the inconsistent GD vectors.", "Specifically, the areas marked by red rectangles in the GD pattern on February 8 have GD vectors which are inconsistent with the nearby ones.", "Especially for the top right corner of the GD pattern on February 8, the amount of star images in this area is only five.", "The magnitudes of these stars are between 15$\\sim $ 17.", "Thus measured errors would be the primary source and make the GD values incorrect, especially for faint stars.", "The top left corner of the GD pattern on February 9 has no GD vector, because there are no star images in this area.", "The area marked by red rectangle in the GD pattern on February 10 has no GD vector either.", "However, the corresponding areas in the four GD patterns of second row by using the improved solution have reasonable GD vectors.", "We can clearly see these differences from the third row.", "From the third row of Fig.", "1 we can see that the GDs in most areas have only subtle differences between the GD patterns derived by the previous and improved solution.", "In order to show the improvements made by improved solution, the (O-C) residuals and standard deviations (SDs) of common stars falling into the marked area in the top right corner of GD pattern on February 8 are drawn in Fig.", "2.", "The SD of one definite star is based on its (O-C) residuals in many different CCD frames.", "The selected rectangular pixel area has the range of coordinate x from 0 to 1900 and the range of coordinate y from 1800 to 1900.", "Fig.", "2 shows the details.", "The (O-C) residuals and SDs of some stars in the top right corner of GD pattern on February 8 are significantly improved, because the wrong GD vector in the first row of Fig.", "1 is reasonably calculated in the second row.", "These improvements give proof that the GD solution with the orthogonal method is more suitable for deriving GD patterns.", "In order to check how much improvements on the positional precision could be obtained by using the GD solution with orthogonal method, four nights of CCD observations of Himalia were processed.", "The observed positions of Himalia were compared to the ephemerides retrieved from the IMCCE which include satellite ephemeris by Emelyanov ([4]) and planetary ephemeris INPOP13c (Fienga et al. [6]).", "Fig.", "3 shows the (O-C) residuals of positions of Himalia with respect to the observational epochs.", "Table 4 lists the statistics of (O-C) residuals of Himalia by using both the previous and improved GD solutions.", "We can see that the internal agreement or precision on February 8 has relatively high improvement than the other nights.", "The means of (O-C) residuals for all data after using improved GD solution are -0.034$^{\\prime \\prime }$ and -0.026$^{\\prime \\prime }$ in right ascension and declination, respectively.", "The corresponding standard deviations are 0.031$^{\\prime \\prime }$ and 0.028$^{\\prime \\prime }$ .", "Figure: (O-C) residuals of the topocentric apparent positions of Himalia compared to the ephemeris retrieved from the IMCCE which include satellite theory by Emelyanov () and planetary ephemeris INPOP13c, with respect to the Julian Dates.", "The dark and red points represent the (O-C) residuals by using previous and improved GD solutions, respectively.Table: Statistics of (O-C) residuals of the positions of Himalia by using both the previous and improved GD solutions.", "Column 1 shows the observational dates.", "Column 2 shows which GD solution was used.", "The following columns list the means of (O-C) residuals and their standard deviations (SDs) in right ascension and declination, respectively.", "All units are in arcseconds." ], [ "Conclusions", "In this paper, we improve our previous GD solution by using an orthogonal method based on the Zernike polynomials.", "A total of 75 CCD observations obtained from the 2.4 m telescope administered by Yunnan Observatories were processed.", "The precision of astrometric position of Himalia is significantly better with the improved GD solution.", "The results show that means of (O-C) residuals of Himalia are -0.034$^{\\prime \\prime }$ and -0.026$^{\\prime \\prime }$ in right ascension and declination, respectively.", "The corresponding standard deviations are 0.031$^{\\prime \\prime }$ and 0.028$^{\\prime \\prime }$ .", "As is well known, the new catalog Gaia DR1 (Gaia Collaboration et al.", "[9]) was released on September 14, 2016, after that the Gaia space probe (Gaia Collaboration et al.", "[8]) has been launched on December 19, 2013.", "This catalog represents a huge improvement in the available fundamental stellar data and practical definition of the optical reference frame (Lindegren et al. [11]).", "The unprecedent astrometric precision of reference stars can allow us to obtain quite higher positional precision of targets.", "Our improved GD solution is also useful for astrometric data reduction in the future.", "We acknowledge the support of the staff at the Lijiang 2.4 m telescope.", "Funding for the telescope has been provided by CAS and the People¡¯s Government of Yunnan Province.", "This work is financially supported by the National Natural Science Foundation of China (grant nos.", "U1431227 and 11273014)." ] ]
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