diff --git a/.gitattributes b/.gitattributes index 8015a17cff825c2d49a72618271e88549dd0298c..b2d592944f0e1825d712a9b6b8cd6ae53c7ba186 100644 --- a/.gitattributes +++ b/.gitattributes @@ -218,3 +218,4 @@ data_all_eng_slimpj/shuffled/split/split_finalac/part-02.finalac filter=lfs diff data_all_eng_slimpj/shuffled/split/split_finalac/part-05.finalac filter=lfs diff=lfs merge=lfs -text data_all_eng_slimpj/shuffled/split/split_finalac/part-10.finalac filter=lfs diff=lfs merge=lfs -text data_all_eng_slimpj/shuffled/split/split_finalac/part-01.finalac filter=lfs diff=lfs merge=lfs -text +data_all_eng_slimpj/shuffled/split/split_finalac/part-06.finalac filter=lfs diff=lfs merge=lfs -text diff --git a/data_all_eng_slimpj/shuffled/split/split_finalac/part-06.finalac b/data_all_eng_slimpj/shuffled/split/split_finalac/part-06.finalac new file mode 100644 index 0000000000000000000000000000000000000000..19a994fb516165ac186b25d6ed71c59f47211709 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split/split_finalac/part-06.finalac @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e3a548270b8f64e07e53767d5098b3cefc73ebbf6663b6f2df5bf685a5343d8d +size 12576671385 diff --git a/data_all_eng_slimpj/shuffled/split2/finalzgdh b/data_all_eng_slimpj/shuffled/split2/finalzgdh new file mode 100644 index 0000000000000000000000000000000000000000..8c21229a5a2161cf28f5b480b1b7fe0b27db5110 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzgdh @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nRadiation therapy is one of the main modalities to cure cancer, and is used in over half of cancer treatments, either standalone or in conjunction with another modality, such as surgery or chemotherapy. For intensity-modulated radiation therapy (IMRT), the patient body is irradiated from fixed beam locations around patient body, and the radiation field is modulated at each beam position using multi-leaf collimators (MLC). In IMRT, the optimal choice of beam orientations has a direct impact on the treatment plan quality, influencing the final treatment outcome, hence patient quality of life. Current clinical protocols either have the beam orientations selected by protocol or manually by the treatment planner. Beam orientation optimization (BOO) methods solve for a suitable set of beam angles by solving an objective function to a local minimum. BOO has been studied extensively in radiation therapy procedures, for both coplanar \\citep{Bortfeld1993OptimizationConsiderations,Yan1999,Pugachev2001,Djajaputra2003,Li2004AutomaticAlgorithm,Li2005,Romeijn2005A,Schreibmann2005,Aleman2008,Lim2008,Breedveld2009,Lim2009APlanning,Craft2010,Bangert2010,Breedveld2012,Rocha2013BeamMethod,Yuan2015a,Amit2015,Liu2017a,Cabrera-Guerrero2018ComparingRadiotherapy,Rocha2018,OConnor2018,Cabrera-Guerrero2018,CabreraG.2018}, and noncoplanar \\citep{Pugachev2001,Djajaputra2003,Potrebko2008,Llacer2009,Bangert2010,Breedveld2012,Liu2017a,Yu2018,Yarmand2018EffectiveTherapy,OConnor2018,Yuan2018,Rocha,Bedford2019,Ventura2019ComparisonIMRT} IMRT, or intensity-modulated proton therapy\\citep{Oelfke2001,Gu2018,Shirato2018SelectionTreatment,Gu2019} (IMPT) by researchers in the past three decades. However, BOO has not been widely adopted due to their high computational cost and complexity, since it is a large-scale NP hard combinatorial problem\\citep{Azizi-Sultan2006,Yuan2018}. Despite the extensive research, the lack of practical clinically beam orientation selection algorithms still exists due to the computational and time intensive procedure, as well as the sub-optimality of the final solution, and BOO remains a challenging step of the treatment planning process. \n\nTo measure the quality of the BOO solution, it is necessary to calculate dose influence matrices of each potential beam orientation. Dose influence matrices for one beam associates all the individual beamlets in the fluence map with the voxels of the patient body. This calculation is time consuming and requires a large amount of memory to use in optimization. To mange the limited capacity of computational resources, the treatment planning process, after defining the objective function, is divided into two major steps: 1) find a suitable set of beam orientations, and 2) solve the fluence map optimization problem (FMO)\\citep{CabreraG.2018} of those selected beams. However, these two steps are not independent of each other--the quality of BOO solution can be evaluated only after FMO is solved, and FMO can be defined only after BOO is solved. Due to the non-convexity and large scale of the problem, researchers consider dynamic programming methods by breaking the problem into a sequence of smaller problems. One of the successful algorithms specially for solving complex problems such as BOO is a method known as Column Generation (CG). In the original application of CG into radiotherapy, \\citet{Romeijn2005AModulation} solved a direct aperture optimization (DAO) problem by using CG. \\citet{Dong2013} then proposed a greedy algorithm based on column generation, which iteratively adds beam orientations until the desired number of beams are reached. \\citet{Rwigema20154Toxicity} use CG to find a set of 30 beam orientations to be used in $4\\pi$ treatment planning of stereotactic body radiation therapy (SBRT) for patients with recurrent, locally advanced, or metastatic head-and-neck cancers, to show the superiority of $4\\pi$ treatment plans to those created by volumetric modulated arc therapy (VMAT). \\citet{Nguyen2016} used CG to solve the triplet beam orientation selection problem specific to MRI guided Co-60 radiotherapy. \\citet{Yu2018} used an in-house CG algorithm to solve an integrated problem of beam orientation and fluence map optimization. \n\nHowever, CG is a greedy algorithm that has no optimality guarantee, and typically yields a sub-optimal problem. In addition, CG still takes as much as 10 minutes to suggest a 5 beam plan for prostate IMRT. The aim of this work is to find a method to explore a larger area of the decision space of BOO in order to find higher quality solutions than that of CG in a short amount of time. The proposed method starts with a deep neural network that has been trained using CG as a supervisor. This network can mimic the behavior of CG by directly learning CG's fitness evaluations of the beam orientations in a supervised learning manner. The efficiency of this supervised network, which can propose a set of beam angles that are non-inferior to that of CG, within less than two seconds, is presented in our previous work\\citep{SadeghnejadBarkousaraie2019ATherapy}. Given a set of already selected beams, this network will predict the fitness value of each beam, which is how much the beam will improve the objective function when added in the next iteration.\n\nIn this study, we extend our previous work, and combine this trained supervised learning (SL) network with a reinforcement learning method, called Monte Carlo tree search. We use these fitness values from the SL network as a guidance to efficiently navigate action space of the reinforcement learning tree. Specifically, it provides the probability of selecting a beam in the search space of the tree at each iteration, so that the beam with the better likelihood to improve the objective function has the higher chance of being selected at each step. To evaluate our proposed method, we compare its performance against the state-of-the-art CG. We developed three additional combinations of the guided and random search tree approaches for comparison. \n\\section{Methods}\nThe proposed method has a reinforcement learning structure involving a supervised learning network to guide Monte Carlo tree search to explore the beam orientation selection decision space. This method, guided Monte Carlo tree search (GTS), consists of two main phases: 1) Supervised training a deep neural network (DNN) to predict the probability distribution of adding the next beam, based on patient anatomy, and 2), using this network for a guided Monte Carlo tree search method to explore a larger decision space more efficiently to find better solutions. For the first phase we use the CG implementation for BOO problem, where CG iteratively solves a sequence of Fluence Map Optimization (FMO) problems \\citep{CabreraG.2018} by using GPU-based Chambolle-Pock algorithm \\cite{Chambolle2010}, the results of the CG method are used to trained a supervised neural network. For the second phase, which is the main focus of this work, we present a Monte Carlo Tree Search algorithm, using the trained DNN. Each of these phases are presented in the following sections. \n\n\\FloatBarrier\\subsection{Supervised Learning of the Deep Neural Network\\label{DNN-training}}\n\nWe develop a deep neural network (DNN) model that learns from column generation how to find fitness values for each beam based on the anatomical features of a patient and a set of structure weights for the planning target volume (PTV) and organs-at-risk (OAR). The CG greedy algorithm starts with an empty set of selected beams, calculates the fitness values of each beam based on the optimality condition of the objective function shown in \\autoref{main-obj}.\n\\begin{equation}\n\\min_{x} \\frac{1}{2}\\sum_{\\forall s \\in S}w_s^2 \\|D_{s}x - p\\|_2^2 \\quad s.t.\\: x \\geq 0 \\label{main-obj}\n\\end{equation}\nwhere $w_s$ is the weight for structure s, which are pseudo randomly generated between zero and one during the training process to generate many different scenarios. The value, $p$, is the prescription dose for each structure, which is assigned 1 for the PTV and 0 for OARs. At each iteration of CG, fitness values are calculated based on Karush\u2013Kuhn\u2013Tucker (KKT) conditions\\citep{kuhn1951nonlinear,Karush2014} of a master problem, and they represent how much improvement each beam can make in the objective function value. The beam with the highest fitness value is selected to be added to the selected beam set, $S$. Then, FMO for the selected beams is performed, which affects the fitness value calculations for the next iteration. The process is repeated until the desired number of beams are selected. The supervised DNN learns to mimic this behavior through the training of the DNN is shown in figure \\ref{fig:supervisedStructure}. Once trained, this DNN is capable of efficiently providing a suitable set of beam angles in less than 2 seconds, as opposed to the 360 seconds required to solve the same problem using CG. The details of the DNN structure and its training process is described in our previous work \\cite{SadeghnejadBarkousaraie2019ATherapy}.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=1.\\textwidth]{Supervised-Structure.png}\n \\captionv{15}{Supervised-structure}{Schematic of the Supervised Training Structure to predict Beam Orientation fitness values. Column Generation (CG) as teacher and deep neural network (DNN) as Trainee.\n \\label{fig:supervisedStructure}}\n\\end{figure}\n\nPatient anatomical features include the contoured structures (organs at risk) of the images from patients with prostate cancer and the treatment planning weights assigned to each structure. The images of 70 prostate cancer patients are used for this research, each with 6 contours: planning target volume (PTV), body, bladder, rectum, and left and right femoral heads. From 70 patients, 50 was randomly selected to train the network and 7 for its validation. The remaining 13 patients images is used for testing and applying the Monte Carlo tree search method. \n\n \\FloatBarrier\\subsection{Monte Carlo Tree Search}\nThe pre-trained DNN probabilistically guides the traversal of the branches on the Monte Carlo decision tree to add a new beam to the plan. Each branch of the tree starts from root as an empty set of beams, and continues until it reaches the terminal state. After the exploration of each complete plan (selection of 5 beams in our case), the fluence map optimization problem is solved and and based on that, the probability distribution to select next beam will be updated, using the reward function, in the backpropagation stage. Then, starting from root the exploration of the next plan will begin until the stopping criteria is met. \\autoref{fig:tree_structure} shows an example of a tree search, which has discovered seven plans so far. \n\n\\begin{figure}\n \\qtreecentertrue\n \\Tree[.Root [.${b_1}^{1}$ [.${b_2}^{2}$ [.${b_3}^{3}$ [.${b_4}^{4}$ ${b_5}^{5}$ ]]]\n [.${b_6}^2$ [.${b_7}^3$ [.${b_8}^4$ ${b_9}^5$ ]\n [.${b_{10}}^5$ ${b_{11}}^5$ ] \n ]]]\n [.${b_{12}}^1$ \n [.${b_{13}}^2$ [.${b_{14}}^3$ [.${b_{15}}^4$ ${b_{16}}^5$ ]\n [.${b_{20}}^4$ ${b_{21}}^5$ ${b_{22}}^5$ ]]\n [.${b_{17}}^3$ [.${b_{18}}^4$ ${b_{19}}^5$ ]]\n ]]]\n \\captionv{15}{Short title - can be blank}\n {An example of guided tree search, subscript are order that a node is generated, and superscript is the depth of the node in the tree.\n \\label{fig:tree_structure}}\n \n\\end{figure}\n\\subsubsection{Basics of Monte Carlo Tree Search}\n Monte Carlo Tree Search (MCTS) uses the decision tree to explore the decision space, by randomly sampling from it\\cite{Browne2012}. The search process of MCTS consists of four steps: 1) node selection, 2) expansion, 3) simulation, and 4) back-propagation on the simulation result. To explain these processes in detail, there are some properties that need to be defined first, these definitions are as follows:\n \\begin{description}\n \\item[State of the problem:] include patient's anatomical features and a set of selected beam orientations ($B$). At the beginning of the planning, this set has no member, and it is updated throughout the solution procedure.\n \\item[Actions:] the selection of the next beam orientation to be added to set B, given the state of the problem.\n \\item[Solution or terminal state:] state of the problem in which the number of selected beam orientations (size of $B$) is the same as a predefined number of beams ($N$), chosen by user. At this point, a feasible solution for the problem is generated. \n \n \n \n \\item[Decision Tree:] The solution space of a set of discrete numbers--in this work discrete numbers are the beam orientations--specially with iterative structures, can be defined as a tree, where each node and branch represent the selection of one number or a subset of available numbers, respectively.\n \\item[Node ($Y$):] selection of one potential beam orientations is a node.\n \\item[Root ($O$):] a node with empty set of beam orientations, every solution start from the root.\n \\item[Path:] a unique connected sequence of nodes in a decision tree.\n \\item[Branch ($Q$):] a path originated from Root node. Each branch represents the iterative structure of the proposed method. The length of a branch is the number of nodes in that branch. In this work solution is a branch with size $N+1$. There is only one branch from each root to any node in a tree.\n \\item[Leaf:] last node of a branch. There is no exploration of the tree after a leaf is discovered.\n \\item[Internal node:] any node in a tree except for root and leaves.\n\\end{description}\nThe \\textbf{selection} process in the proposed method is guided by a pre-trained DNN as described in \\autoref{DNN-training}. This DNN is used to probabilistically guide the traversal of the branches on the Monte Carlo decision tree to add a new beam to the plan. At each node--starting by root note--the DNN is called to predict an array of fitness values for each beam orientation($P$). An element of this array $P[i]$ represents the likelihood of the selection of the $i^{th}$ beam orientation. For example, if the number of potential beam orientations is 180, with $2^{\\circ}$ separation, $Y$ would be an array of size 180, and $P[2]$ is the likelihood of selecting beam orientation in $2^{nd}$ position of the potential beam orientations, $P[2]=4^\\circ$.\nThe \\textbf{expansion} process happens after selection process at internal nodes, to further explore the tree and create children nodes. The traversal approach in the proposed method is depth first, which means that the branch of a node, that is visited or created for the first time, continues expanding until there are $N+1$ nodes in a branch. In this case, selection and expansion processes are overlapping because only one child node is created or visited at a time, although a node can be visited multiple times and several children can be generated from one node, except for leaf. The leaf node does not have any children. Nodes in a branch must be unique, it means that a branch of each external node ($Q$) can be expanded only to nodes that are not already in the branch. \nIn fact, beam orientation optimization problem can be defined as a dynamic programming problem with the following formula:\n\\begin{equation}\n G_{k}^{S}=S\\union\\{k\\}\\union{G_{n^*}^{S\\union\\{k\\}}} \\mid n^*=\\argmax_{n>k}V_{G_n^{S\\union\\{k\\}}}\\label{dynamic}\n\\end{equation}\nwhere $S$ is a set of indices for previously selected beams, $k$ is index of currently selected beam and ${G_{n^*}^{S\\union\\{k\\}}}$ is a subset of beams to be selected that has the highest reward value. \n Each $\\bf{simulation}$ consists of iteratively selecting a predefined number of beams ($N$), in this work $N=5$. After the exploration of each complete plan, the fluence map optimization problem is solved and used for the \\textbf{back-propagation} step, which is used to update the probability distribution for beam selection. \n \n \\subsubsection{Main Algorithm}\n The detailed of the guided Monte Carlo tree search algorithm in the form of a pseudo code is provided in Algorithm \\ref{alg:GTS}. Several properties of each node in the proposed tree are being updated after the exploration of final states. To simplify the algorithm, these properties are addressed as variables and the following is a list of them:\n\\begin{description}\n \\item[Cost ($Y_v$):] After a leaf is discovered, an FMO problem associated with the beams of that branch will be solved, the value of the FMO cost function is the value associated with its corresponding leaf. The cost value of all other nodes (other than leaves) in a tree is the average cost of its sub-branches. For example in \\autoref{fig:tree_structure} the cost value of node ${b_1}^1$ is the average cost of nodes ${b_2}^2$ and ${b_6}^2$.\n \\item[probability distribution ($Y_P$):] an array of size 180 (the number of potential beam orientations), where $i^{th}$ element of this array represents the chance of improvement in the current cost value if tree branches out by selecting $i^{th}$ beams. After a node is discovered in the tree, this distribution is populated by using DNN. After the first discovery of a node, $Y_P$ is updated based on the reward values.\n \\item[Reward ($Y_R$):] is a function of the node's cost values and the best cost value ever discovered in the search process. The reward values would be updated after each cost calculation and are calculated and updated by the reward calculation procedure defined in line \\ref{Reward_function} of Algorithm \\ref{alg:GTS}.\n \\item[Depth ($Y_D$):] is simply the number of beam orientations selection after node $Y$ is discovered.\n \\item[Name ($Y_{id}$):] a unique string value as id for each node, this value is the path from root to node $Y$.\n \\item[Beam Set ($Y_B$):] the set of beams selected for a branch started from root and ended in node $Y$.\n \\item[Parent ($Y_{parent}$):] the immediate node before node $Y$ in a branch from root to $Y$, except for root node, all other nodes in a tree have one parent.\n \\item[Children ($Y_{children}$)]: the immediate node(s) of the sub-branches from the node $Y$, except for leaves, all other nodes in a tree have at-least one children.\n\\end{description}\n\n\\begin{algorithm}[htbp]\n \\footnotesize\n \\caption{\\textsc{Select $N$ beam orientations from $M$ candidate beams}}\n \\label{alg:GTS}\n \\begin{algorithmic}[1]\n \\Procedure{Initialization}{}\n \\LState{set selected beam as $B\\gets\\emptyset$ an empty set, best cost value as infinity (${V^*}\\gets{\\infty}$), and best selected beam as ${B^*}\\gets{\\emptyset}$} \n \\LState{create a root node object ($O$) with the following properties:}\n \\LState{\\ \\ $\\qquad$ name($O_{id}\\gets{\\mathtt{Root}}$), probability distribution($O_P\\gets{\\emptyset}$), number of visits($O_Z\\gets{0}$), beam index($O_b$)} \n \\LState{\\ \\ $\\qquad$ reward($O_R\\gets{0}$), cost($O_V\\gets{\\infty}$), depth($O_D\\gets{0}$), parent($O_{parent}\\gets{\\emptyset}$), children($O_{children}\\gets{\\emptyset}$)}\n \\LState {assign root node to current node(${Y^{\\#}}\\gets{O}$)}\n \\LState {given the set $B$ as input to $\\mathtt{DNN}$, predict an array of fitness values and assign it to root node ${Y^{\\#}}_P\\gets{Prd(\\mathtt{DNN},B)}$}\n \\LState {set $stop \\gets{False}$}\n \\EndProcedure\n \\While{$stop$ is ${False}$}\n \t\\LState {choose the next beam index ($b$) using the probability distribution of the current node ${Y^{\\#}}_{P}$} \n \t\\LState {create string name as ${ID} \\gets{{Y^{\\#}}_{id} \\concat {b}}$} \\Comment{$\\concat$: string concatenation\n \t\\LState {update selected beam set $B\\gets{B\\cup{b}}$ }\n \t\\If {$ID \\notin {{Y^{\\#}}_{children}}_{id}$ (current node does not have a child named ${ID}$)}\n \t \\LState {create a new node $Y$ with ${Y}_{id}\\gets{ID}$}\n \t \\LState {${{Y}_{parent}}\\gets{Y^{\\#}}$, ${{Y}_{Z}}\\gets 1$}\n \t \n \t \\LState {predict an array of fitness values $F(\\mathtt{DNN},B)$} \n \t \\LState {assign predicted values to new node ($Y_P\\gets{Prd(\\mathtt{DNN},B)}$)}\n \t \\LState {$Y_D\\gets{{Y^{\\#}}_D + 1}$}\n \t \\LState {set beam index($Y_b\\gets{b}$)}\n \t \\LState {add $Y$ as a new child ${Y^{\\#}}_{children}\\gets{{Y^{\\#}}_{children} \\cup Y}$ }\n \t \\LState {update current node $Y^{\\#}\\gets{Y}$}\n \t\\Else\n \t \\LState {update current node ($Y^{\\#}\\gets{X \\vert\\{ X \\in {Y^{\\#}}_{children} \\& X_{id}=ID}\\}$)}\n \t \\LState {update visit parameter of current node, (${Y^{\\#}}_Z\\gets{{Y^{\\#}}_Z + 1}$)}\n \\EndIf\n \\If{$\\vert B\\vert = N$ or ${Y^{\\#}}_D = N$ \n \\Procedure{Reward Calculations \\label{Reward_function}}{}\n \\LState {solve $FMO$ given set $B$ and save as ${Y^{\\#}}_V\\gets{Fmo(B)}$}\n \\If {${V^*} > {Y^{\\#}}_V$}\n \\LState {set ${V^*} \\gets {Y^{\\#}}_V$, {${B^*}\\gets B$}, {${Y^{\\#}}_R\\gets 1.$}\n \\Else:\n \\LState {${Y^{\\#}}_R \\gets ({{V^*}-{Y^{\\#}}_V})\/{{Y^{\\#}}_V} +0.15$ }\n \\EndIf\n \\While{${Y^{\\#}}_{id}\\ne \\mathtt{Root}$}\n \\LState{$Y^{\\#}\\gets{{Y^{\\#}}_{parent}}$}\n \\LState{${Y^{\\#}}_R \\gets{{\\sum_{{X}\\in {Y^{\\#}}_{children}}{{X_R}}}\/{\\vert {Y^{\\#}}_{children}\\vert}}$}\n \\For {$X\\in {Y^{\\#}}_{children}$}\n \t \\LState {${Y^{\\#}}_D[X_b]\\gets{{Y^{\\#}}_D[X_b]\/X_Z +\\sqrt{\\ln{X_Z}\/{Y^{\\#}}_Z}}$}\n \\If {${Y^{\\#}}_V > X_V$}\n \\LState {${Y^{\\#}}_V\\gets{X_V}$}\n \t \\EndIf\n \\EndFor\n \\EndWhile\n \\EndProcedure\n \\If {stopping criteria is met}\n \\LState {$Stop \\gets{True}$}\n \\Else:\n \\LState {$B \\gets{\\emptyset} , Y^{\\#}\\gets{O}$}\n \\EndIf\n \\EndIf\n \\EndWhile\n \\LState {\\Return ${{V^*},{B^*}}$}\n \\end{algorithmic}\n \\vspace{8pt}\n\\end{algorithm}\n\n\\subsection{Algorithms for performance comparison \\label{fouralgorithms}}\nIn general, four frameworks were designed to show the efficiency of the proposed GTS method compared to others. These models are defined as:\n\\begin{description}\n\\item[Guided Tree Search (GTS)]: As presented in Algorithm \\ref{alg:GTS}, used a pre-trained policy network to guild a Monte-Carlo decision tree.\n\\item[Guided Search (GuidS)]: Used the pre-trained network to search the decision space by iteratively choosing one beam based on the predicted probabilities from the policy network. Unlike GTS, the search policy is not updated as the search progresses. This process is detailed in Algorithm \\ref{alg:GuidS}. \n\\item[Randomly sample Tree Search (RTS)]: This algorithm is simple Monte-Carlo tree search method which starts with a uniform distribution to select beam orientations (randomly select them), and then update the search policy as the tree search progresses. Note that all of tree operations used in GTS is also used in this algorithm, except for having a policy network to guide the tree. This method is proposed to show the impact of using DNN to guild the decision tree.\n\\item[Random Search (RandS)]: This method searches the decision space with uniformly random probability until stopping criteria is met. It randomly selects 5 beam orientations and solves the corresponding FMO problem. The search policy is not updated. Its procedure is close to Algorithm \\ref{alg:GuidS} where the ``Select $B$ using DNN'' procedure is replaced by randomly selecting 5 unique beams.\n\\end{description}\n\n\\begin{algorithm}[tbp]\n\\footnotesize\n\\caption{\\textsc{Guided Search algorithm(GuidS)}}\n\\label{alg:GuidS}\n\\hspace*{\\algorithmicindent} \\textbf{Input:}{ Pre-trained DNN}\n \\begin{algorithmic}[1]\n \\LState {initialize $B$ as an empty array, best cost value as infinity (${V^*}\\gets{\\infty}$), and best selected beam as ${B^*}\\gets{\\emptyset}$}\n \\LState {set current number of beam orientations in $B$ as 0, $N_B \\gets{0}$}\n \\LState {set $stop \\gets{False}$}\n \\While{$stop$ = $False$}\n \\Procedure{Select $B$ using DNN}{}\n \\While{$N_B < N$}\n \t\\LState{predict an array of fitness values $P=F(\\mathtt{DNN},B)$}\n \t\\LState{Select next node ($b$) with the probability of $P(b)$}\n \t\\LState{Update $B$: $B=B\\union\\{b\\}$ and $N_B = N_B +1$}\n \\EndWhile\n \\LState \\Return{$B$}\n \\EndProcedure\n \\LState {solve $FMO$ given set $B$ and save as ${V\\gets{Fmo(B)}}$}\n \\If{$V<{V^*}$}\n \\LState{${V^*} \\gets{V}$ and ${B^*}\\gets{B}$}\n \\EndIf\n \\If {stopping criteria is met}\n \\LState {$stop \\gets{True}$}\n \\Else:\n \\LState {$B \\gets{\\emptyset} , N_B \\gets{0}$}\n \\EndIf\n \\EndWhile\n \\LState{\\Return{${B^*}$ and ${V^*}$}}\n \\end{algorithmic}\n \\vspace{2pt}\n\\end{algorithm}\n\n\\FloatBarrier\\subsection{Data}\nWe used images from 70 patients with prostate cancer, each with 6 contours: PTV, body, bladder, rectum, left femoral head, and right femoral head. Additionally, the skin and ring tuning structures were added during the fluence map optimization process to control high dose spillage and conformity in the body. The patients were divided randomly into two exclusive sets: 1) a model development set, which includes training and validation data, consisting of 57 patients, 50 for training and 7 for validation, for cross-validation method, and 2) a test data set consisting of 13 patients. \nEach patient's data contains 6 contours: PTV, body, bladder, rectum, and left and right femoral heads. Column generation was implemented with a GPU-based Chambolle-Pock algorithm\\citep{Chambolle2010}, a first-order primal-dual proximal-class algorithm, to create 6270 training and validation scenarios (22 5-beam plans for each of 57 patients) and 130 test scenarios (10 5-beam plans for each of 13 test patients). The DNN trained over 400 epochs, each with 2500 steps and batch size of one. \n\nThe performances of four methods GTS, GuidS, RTS, and RandS, explained in section \\autoref{fouralgorithms}, are evaluated. Two of these methods, GTS and GuidS, use the pre-trained DNN as a guidance network. We originally had the images of 70 patients with prostate cancer, and used the images of the 57 of them to train and validate DNN and therefore cannot be used for the testing in this project\\footnote{ To keep the proposed methods completely independent from the dataset used for training DNN.}. There are 13 patients that DNN has never seen before and the images of those patients are used in this project as test set. Multiple scenarios can be generated for each patient, based on the weights assigned to patient's structures for planning their treatments. We semi-randomly generated 10 sets of weights for each patients. In total, we have a total of 130 test plans among the 13 test patients for the comparison. All the tests in this paper were performed on a computer with an Intel Core I7 processor@3.6 GHz, 64 GB memory, and an NVIDIA GeForce GTX 1080 Ti GPU with 11 GB video memory.\n\nThe structure weight selection scheme is outlined by the following process:\n\\begin{enumerate}\n \\item{ In $50\\%$ of the times, a uniform distribution in the range of 0 to 0.1 is used to generate a weight for each OAR separately.} \n \\item {\tIn $10\\%$ of the times, the smaller range of 0 to 0.05 is used to select weights for OARs separately, with uniform distribution.}\n \\item {\tAnd finally in $40\\%$ specific ranges were used for each OAR: Bladder: [0,0.2], Rectum: [0,0.2], Right Femoral Head: [0,0.1], Left Femoral Head: [0,0.1], Shell: [0,0.1] and Skin: [0,0.3]}\n\\end{enumerate}\nThe weights range from 0 to 1. This weighting scheme was found to give a clinically reasonable dose, however, the dose itself may not be approved by the physician for that patient. \n\nFinally, considering only test scenarios, FMO solutions of beam sets generated by CG and by the 4 tree search methods were compared with the following metrics: \n\\begin{description}\n \\item [PTV $\\bf{{D}_{98}}$, PTV $\\bf{{D}_{99}}$:]{The dose that $98\\%$ and $99\\%$, respectively, of the PTV received}\n\t\\item [PTV $\\bf{{D}_{max}}$:]{Maximum dose received by PTV, the value of $D_2$ is considered for this metric}\n\t\\item [PTV Homogeneity:]{$\\frac{{{PTV} {D_2}} - {PTV D_{98}}}{{PTV} {D_{50}}}$ where $PTV D_2$ and $D_{50}$ are the dose received by $2\\%$ and $50\\%$, respectively, of PTV}\n\t\\item [ Paddick Conformity Index (${CI}_{Paddick}$) \\citep{van1997conformation,paddick2000simple}:] {$\\frac{{(V_{PTV} \\cap V_{100\\%Iso})}^2}{V_{PTV} \\times V_{100\\%Iso}}$ where $V_{PTV}$ is the volumne of the PTV and $V_{100\\%Iso}$ is the volume of the isodose region that received $100\\%$ of the dose}\n\t\\item [High Dose Spillage ($\\bf{R_{50}}$):] {$\\frac{V_{50\\%Iso}}{V_{PTV}}$ where $V_{50\\%Iso}$ is the volume of the isodose region that received $50\\%$ of the dose}\n\\end{description}\n\n\n\\begin{figure}[tb]\n \\begin{subfigure}{.47\\textwidth}\n \\centering\n \n \\includegraphics[clip,trim= 2mm 4mm 85mm 3mm,scale=0.178]{Success_Fail_rate_BT_20p.png}\n \\caption{\n \n \\label{fig:success-fail}}\n \\end{subfigure}\n \\begin{subfigure}{.52\\textwidth}\n \\centering\n \n \\includegraphics[clip,trim= 35mm 4mm 2mm 3mm,scale=.178]{Success_Fail_rate_BT_100.png}\n \\caption{\n \n \\label{fig:success-fail-all}}\n \\end{subfigure}\n \\vspace*{-2ex}\n \\captionv{15}{success-fail rate}{The rate at which each method successfully found a solution with lower objective function value than that of CG solution. Each attempt is limited to 1000 seconds. \\ref{fig:success-fail} The percentages of test cases that each method found a solution better than CG solution in at least 1 of their 5 attempts. \\ref{fig:success-fail-all} The percentage of test cases that each method found a solution better than CG solution, averaged over all 5 of their attempts to solve the problem.}\n\\end{figure}\n\n\\section{Results}\nAt each attempt to solve a test scenario, each method is given 1000 seconds to search the solution space. Whenever a method finds a solution better than that of CG, the solution and its corresponding time stamp and the number of total solutions visited by this method are saved. We use these values to analyze the performance of each method. The best solution that is found in each attempt to solve the problem is used as the final solution of that attempt and is used for PTV metrics calculation. The average objective function value of final solutions in five attempts are used for the comparison of the performance of four methods with CG solution.\nAt first we compare the efficiency of the four methods of GTS, GuidS, RTS and RandS. Although the main purpose of these methods are to find a solution better than CG, there were some cases that none of these method could beat CG solution, either CG solution was very close to optimal, or there were several local optimums with wide search space which makes it very difficult to explore it efficiently, this is especially true for RTS and RandS methods. The percentages of total number of attempts that each method could successfully find a solution better than CG in at least one of five attempts and averaged over all attempts to solve the problem are presented in \\autoref{fig:success-fail} and \\ref{fig:success-fail-all}, respectively. Note that for this test the stopping criteria was 1000 seconds of computational time. As we expected, GTS and GuidS that are using the pre-trained DNN performed better than the other two methods. However, there are still cases that they are not able to find a better solution than CG. The maximum number of scenarios that all four methods were successfully find a solution with objective function value better than that of CG solution is 102 out of total 130 test cases (78.46\\%). \n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=.8\\textwidth]{avg_obj_Distance.eps}\n \\vspace*{-3ex}\n \\captionv{15}{Distance measure}{\n The distance measure of the average of the best objective function value found by each method compared to CG solution. The value written inside the box is the median and the value at the top of each box-plot is the mean $pm$ standard deviation of the Distance measure.\n \\label{fig:avg-distance}}\n\\end{figure}\n\nThe domain of the objective function value varies for different test-case scenarios, therefore we introduce \\textbf{Distance} measure as the normalized version of the objective values compared to CG solutions for further comparison. Distance measure is the difference between the objective function value of each method and CG, divided by CG objective value ($Distance=\\frac{CG_{obj}-method_{obj}}{CG_obj}\\times 100$). If a method finds a solution better than CG, $Distance$ measure will be positive, and for cases that a method was not successful to find a solution better than CG solution, this value will be negative. It means a method with largest $Distance$ measure found solutions with better qualities--with an objective value smaller than that of CG--and therefore more efficient in the limited time of 1000 seconds. \\autoref{fig:avg-distance} shows the box plot of Distance measure for GTS, GuidS, RTS, and RandS methods. Based on this figure, the average $Distance$ measure for GTS method is 2.48, which is the highest compared to 1.79 for GuidS, 0.67 for RTS, and 0.81 for RandS.\n\n\\begin{figure}[tb]\n \\begin{subfigure}{0.5\\textwidth}\n \\centering\n \\includegraphics[clip,trim= 10mm 27mm 3mm 1mm, scale=.335]{best_timelimi_Time.eps}\n \\captionv{6}{Best time}{\n The best time to beat CG solution.\n \\label{fig:timelimit-Time}}\n \\end{subfigure}\n \\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[clip,trim= 38.5mm 27mm 0mm 2mm,scale=0.335]{avg_timelimi_Time.eps}\n \\captionv{5}{average time}{The average time to beat CG.\n \\label{fig:timelimit-Time-obj}}\n \\end{subfigure}\n \\vspace*{-4ex}\n \\captionv{15}{Time boxplot}{The computational time comparison between GTS, GuidS, RandS and RTS. The first time that each method beats a column generation solution. For this measure only successful scenarios are considered. Mean, standard deviation(at the top of each box as mean $pm$ standard deviation) and median (in the middle of the box) are calculated for each method.\n \\label{fig:besttime}}\n\\end{figure}\n\nFor the computational time evaluation and comparison methods, only the 102 out of 130 test cases, where all four methods had successfully found a better solution than CG within the 1000 seconds time limit, were considered. The box-plot of the best time needed to beat CG solution is presented in \\autoref{fig:timelimit-Time}. It represents the first time that a method finds a solution better than CG solution within 1000 seconds among all five attempts. The box-plot of the average time to beat CG among all attempts to solve each test-cases is provided in \\autoref{fig:timelimit-Time-obj}. To measure the average time to beat CG, all test-cases and all attempts are considered. The fastest method considering this measure is GTS, the average time to beat CG for GTS method is 195 seconds in best case and 237 seconds in total cases. The second fast method is GuidS with the average time of 227 seconds for best and 268 seconds for total cases. RandS outperforms RTS with best time of 337 seconds compared to 364 seconds of RTS. Interestingly, the average total time of RTS and RandS is less than the average of best cases.\n\n\\begin{table}[tbh]\n\\begin{center}\n\\captionv{15}{One-tailed paired sample t-test for $\\%99$ confidence interval }{One-tailed paired sample t-test to test the average objective function value and $Distance$ measure for every pairs of CG, GTS, GuidS, RTS, and RandS methods, with $\\%99$ confidence intervals. All values in red have p-values greater than 0.01.)\n\\label{table:ttest-objValue}\n}\n \n \\begin{tabular}{lrrrr}\n \\hline\n & \\multicolumn{2}{c}{\\textbf{Objective Value}} & \\multicolumn{2}{c}{\\textbf{Distance($\\bf{\\frac{CG-obj}{CG}}$)}} \\\\\n \\hline\n \\multicolumn{1}{l}{\\textbf{Tested methods}} & \\multicolumn{1}{r}{\\textbf{t-statistic}} & \\multicolumn{1}{r}{\\textbf{p-value}} & \\multicolumn{1}{r}{\\textbf{t-statistic}} & \\multicolumn{1}{r}{\\textbf{ p-value }} \\\\\n \\hline\n \\textbf{CG vs GTS} & 6.267 & 2.54E-09 & {7.683} & {1.72E-12} \\\\\n \\textbf{CG vs GuidS} & 4.940 & 1.19E-06 & {6.393} & {1.37E-09} \\\\\n \\textbf{CG vs RTS} & {\\textbf{\\color{red}0.885}} & {\\textbf{\\color{red}1.89E-01}} & {\\textbf{\\color{red}2.014}} & {\\textbf{\\color{red}2.30E-02}} \\\\\n \\textbf{CG vs RandS} & {\\textbf{\\color{red}1.670}} & {\\textbf{\\color{red}4.87E-02}} & {\\textbf{\\color{red}2.340}} & {\\textbf{\\color{red}1.04E-02}} \\\\\n \\textbf{RandS vs RTS} & {\\textbf{\\color{red}-1.496}} & {\\textbf{\\color{red}6.85E-02}} & {\\textbf{\\color{red}1.096}} & {\\textbf{\\color{red}1.38E-01}} \\\\\n \\textbf{RandS vs GTS} & 6.826 & 1.53E-10 & -11.843 & 1.18E-22 \\\\\n \\textbf{RandS vs GuidS} & 3.873 & 8.51E-05 & -4.839 & 1.83E-06 \\\\\n \\textbf{RTS vs GTS} & 8.245 & 8.18E-14 & -15.271 & 5.25E-31 \\\\\n \\textbf{RTS vs GuidS} & 5.257 & 2.95E-07 & -5.832 & 2.08E-08 \\\\\n \\textbf{GuidS vs GTS} & 2.412 & 8.64E-03 & -3.945 & 6.52E-05 \\\\\n \\hline\n \\end{tabular}%\n \n\\end{center}\n\\end{table}\n\nTo study the statistical significant of GTS method compared to other four methods (GuidS, RTS, RandS, and CG), we use one-tailed paired sample t-test to compare the objective function values of each pair of methods, and Distance measure. The null hypothesis is that the average objective function and $Distance$ measure of all methods are the same. If we show the null hypothesis as $GTS=GuidS=RandS=RTS=CG=0$. The alternative hypothesis can be described as $GTSGuidS>RandS>RTS>CG$ for $Distance$ measure. Ten paired sample t-test are performed for objective function values and $Distance$ measure.These statistics are presented in \\autoref{table:ttest-objValue}. The distribution of Objective value and $Distance$ measures are provided in appendix \\autoref{appendix}.\nAs highlighted by red, all pairs of the three methods of CG, RTS, and RandS have p-values greater than 0.01, and are not significantly different, while the average $Distance$ and objective value measures of GuidS and GTS are significantly different. Based on these results GTS outperforms all other methods significantly while in the second position is GuidS as was expected.\n\n\\begin{table}[tb]\n\\begin{center}\n\\captionv{15}{PTV metrics}{Mean $\\pm$ standard deviation for PTV Statistics, Paddick Conformity Index (${CI}_{Paddick}$), and High Dose Spillage ($R_{50}$) of methods: CG, GTS, GuidS, RTS, and RandS.\n\\label{table:PTV-metrics}\n}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lcccccc}\n\\hline\n\\textbf{Method} & \\textbf{PTV $\\bf{D_{98}}$} & \\textbf{PTV $\\bf{D_{99}}$} & \\textbf{PTV $\\bf{D_{max}}$} & \\textbf{PTV Homogeneity} & \\textbf{$\\bf{{CI}_{Paddick}}$} & \\textbf{$\\bf{R_{50}}$} \\\\\n\\hline\n\\textbf{RandS} & 0.977$\\pm$0.012 & 0.960$\\pm$0.019 & 1.071$\\pm$0.040 & 0.089$\\pm$0.046 & 0.867$\\pm$0.070 & 4.673$\\pm$1.022 \\\\\n\\textbf{RTS} & 0.977$\\pm$0.011 & 0.959$\\pm$0.020 & 1.070$\\pm$0.039 & 0.088$\\pm$0.045 & 0.863$\\pm$0.085 & 4.714$\\pm$1.312 \\\\\n\\textbf{GTS} & 0.977$\\pm$0.011 & 0.960$\\pm$0.019 & 1.070$\\pm$0.040 & 0.089$\\pm$0.045 & 0.874$\\pm$0.061 & 4.569$\\pm$0.994 \\\\\n\\textbf{GuidS} & 0.976$\\pm$0.012 & 0.960$\\pm$0.019 & 1.071$\\pm$0.040 & 0.089$\\pm$0.046 & 0.874$\\pm$0.068 & 4.487$\\pm$0.948 \\\\\n\\textbf{CG} & 0.977$\\pm$0.011 & 0.961$\\pm$0.020 & 1.072$\\pm$0.041 & 0.090$\\pm$0.046 & 0.884$\\pm$0.059 & 4.478$\\pm$0.963 \\\\\n\\hline\n \\end{tabular}%\n }\n\\end{center}\n\\end{table}\n\nPTV statistics, Paddick Conformity Index(${CI}_{Paddick}$) and dose spillage($R_{50}$) of plans generated by CG, GTS, GuidS, RTS, and RandS are presented in \\autoref{table:PTV-metrics}. Note that PTV $D_2$ is used to measure PTV $D_{max}$, as recommended by the ICRU Report 83 \\citep{Hodapp2012TheIMRT}. The plans generated by all methods have very similar PTV coverage. CG plans have the highest ${CI}_{Paddick}$ followed by GTS and GuidS plans. While CG and GuidS plans have the lowest dose spillage value followed by GTS. \n\n\\begin{table}[bth]\n\\begin{center}\n\\captionv{16}{Mean and Max Dose}{The average and maximum fractional dose received by each structure in plans generated by GTS, GuidS, RTS, RandS, and, CG methods, where prescription dose is set to 1. \n\\label{table:maxmeandose}\n\\vspace*{-2ex}\n}\n\\resizebox{1.\\textwidth}{!}{\n\\begin{tabular}{cllllll}\n & & \\multicolumn{5}{c}{\\textbf{Methods}} \\\\\n\\hline\n & \\textbf{Structures} & \\textbf{GTS} & \\textbf{GuidS} & \\textbf{RTS} & \\textbf{RandS} & \\textbf{CG} \\\\\n\\hline\n\\multirow{6}{*}{\\rotatebox{90}{\\textbf{Mean Dose}}} \n & \\textbf{PTV} & 1.039 $\\pm$ 0.025 & 1.039 $\\pm$ 0.025 & 1.038 $\\pm$ 0.024 & 1.039 $\\pm$ 0.025 & 1.040 $\\pm$ 0.025 \\\\\n & \\textbf{Body} & \\textbf{0.037 $\\pm$ 0.012} & \\textbf{0.037 $\\pm$ 0.012} & \\textbf{0.037 $\\pm$ 0.012} & \\textbf{0.037 $\\pm$ 0.012} & 0.038 $\\pm$ 0.013 \\\\\n & \\textbf{Bladder} & 0.207 $\\pm$ 0.125 & 0.207 $\\pm$ 0.122 & 0.207 $\\pm$ 0.126 & 0.206 $\\pm$ 0.122 & \\textbf{0.204 $\\pm$ 0.116} \\\\\n & \\textbf{Rectum} & \\textbf{0.317 $\\pm$ 0.109} & 0.321 $\\pm$ 0.111 & 0.319 $\\pm$ 0.110 & 0.322 $\\pm$ 0.115 & 0.334 $\\pm$ 0.116 \\\\\n & \\textbf{L-femoral} & 0.213 $\\pm$ 0.105 & \\textbf{0.201 $\\pm$ 0.103} & 0.217 $\\pm$ 0.115 & 0.212 $\\pm$ 0.111 & 0.222 $\\pm$ 0.112 \\\\\n & \\textbf{R-femoral} & \\textbf{0.214 $\\pm$ 0.101} & 0.221 $\\pm$ 0.110 & 0.227 $\\pm$ 0.124 & 0.224 $\\pm$ 0.124 & 0.217 $\\pm$ 0.109 \\\\\n\\hline \n\\multirow{6}{*}{\\rotatebox{90}{\\textbf{Max Dose}}} & \\textbf{PTV} & 1.113 $\\pm$ 0.055 & 1.113 $\\pm$ 0.055 & 1.113 $\\pm$ 0.055 & 1.114 $\\pm$ 0.057 & 1.116 $\\pm$ 0.058 \\\\\n & \\textbf{Body} & 1.190 $\\pm$ 0.131 & 1.195 $\\pm$ 0.148 & 1.200 $\\pm$ 0.147 & 1.199 $\\pm$ 0.143 & \\textbf{1.173 $\\pm$ 0.130} \\\\\n & \\textbf{Bladder} & \\textbf{1.094 $\\pm$ 0.046} & 1.094 $\\pm$ 0.048 & 1.096 $\\pm$ 0.050 & \\textbf{1.094 $\\pm$ 0.045} & 1.095 $\\pm$ 0.048 \\\\\n & \\textbf{Rectum} & 1.072 $\\pm$ 0.045 & 1.071 $\\pm$ 0.044 & 1.073 $\\pm$ 0.045 & 1.074 $\\pm$ 0.048 & \\textbf{1.071 $\\pm$ 0.040} \\\\\n & \\textbf{L-femoral} & 0.609 $\\pm$ 0.193 & \\textbf{0.596 $\\pm$ 0.209} & 0.619 $\\pm$ 0.215 & 0.609 $\\pm$ 0.220 & 0.613 $\\pm$ 0.193 \\\\\n & \\textbf{R-femoral} & 0.639 $\\pm$ 0.242 & 0.650 $\\pm$ 0.249 & 0.625 $\\pm$ 0.236 & 0.628 $\\pm$ 0.244 & \\textbf{0.617 $\\pm$ 0.245} \\\\\n\\hline\\\\\n\\end{tabular}%\n}\n\\end{center}\n\\end{table}\n\nThe average and maximum dose received by each structure are provided in \\autoref{table:maxmeandose}, these values reflect the fractional dose in plans generated by each method with the assumption that the prescription dose is one-- e.g. if the prescription dose is 70 Gy, the average dose of 0.207 in the table means 14.47Gy ($0.207\\times 70$) in the prescribed plan. The minimum values in each row are shown as bold numbers for easier interpretation of the table. On average plans generated by GTS have lower mean dose to OARs compared to other methods, while plans generated by CG have the lowest maximum dose to OARs. GTS plans spare rectum and right femoral head better than other methods. Although the average fractional dose to bladder by by GTS plans (0.207) is more than CG plans (0.204), GTS plans have lower maximum fractional dose to bladder (1.094) compared to CG (with maximum of 1.095). GuidS plans have minimum average fractional dose to left-femoral head (0.201) with considerable difference compared to the second best (0.212) of RandS plans. As an example, \\autoref{fig:cgvsGTS} shows the dose-volume and dose-wash of plans generated by GTS and CG for one test-case scenario.\n\n\\begin{figure}[htb]\n \\centering\n \n \n \\includegraphics[width=1\\textwidth]{GTSvsCG.png}\n \\captionv{15}{CG vs GTS}{ GTS generated plan (dashed) vs CG generated plan (solid).\n \\label{fig:cgvsGTS}}\n\\end{figure}\n\n\\section{Discussion}\nIn this research, we propose an efficient beam orientation optimization framework capable of finding a improved solution over CG, in a similar amount time, by utilizing a reinforcement learning structure involving a supervised learning network, DNN, to guide Monte Carlo tree search to explore the beam orientation selection decision space.\n\nAlthough CG is a powerful optimization tool, it is a greedy approach that not only is computationally expensive and time consuming, but it also may get stuck in a local optimum. This is particularly true for highly non-linear optimization problems with lots of local optima, such as BOO. In this work, we tried 4 different approaches: 1) Guided Tree Search (GTS), 2) Guided Search (GuidS), 3) Random Tree Seach (RTS), and 4) Random Search (RandS). It is shown that although the quality of solutions using RandS, RTS and CG were not significantly different, in $50\\%$ of test-cases both RandS and RTS, which have no knowledge of the problem at the beginning of the search, can find solutions better than CG. This shows the high potential of improving the solution found by CG. \n\nWe saw that GTS and GuidS, both performs better than other methods, which is expected because both of these methods are using a prior knowledge (trained DNN) to explore the solution space. GTS even outperforms GuidS on average, since GTS is a combination of GuidS and RTS, means adding a search method to GuidS can improve the quality of the solution. But considering the insignificant difference in the performance of RTS and RandS, adding any search method to GuidS may results in better solutions and it may not be directly related to RTS. This issue will be studied in future researches. The poor performance of RTS may also suggest that using uniform tree search is too slow to converge to the optimal selection of beams.\n\nAlthough GTS performs better than others to find solutions with better objective function values, but the dose spillage metrics, and specifically the average dose received by bladder in GTS plans can be improved further. Considering the success of GTS in reducing the objective function and its potential for further improvement, we will continue exploring new methods and techniques to upgrade the quality of treatment planning with the help of artificial intelligence.\n\nWe should note that CG is a greedy and deterministic algorithm, therefor using CG on the same problem always results in the same solution. This is of completely different nature from our search methods and it may not be fair to compare its performance with the four search procedures, which, given infinite time and resources, can act as brute forced approach and guarantee finding the optimal solution. However, our main goal is to find the best possible solution for BOO problem, and in this work we try to see which search algorithms can find a better solutions and faster. Hence we expect to see search algorithms outperform greedy algorithms. The results showed us that the objective function value of GTS and GuidS CG, RTS and RandS perform similarly. Even though, plans generated by DNN solutions may not be superior to CG, but it can mimic the CG algorithm very efficiently \\citep{SadeghnejadBarkousaraie2019ATherapy} and is a very successful tool to explore the search space as shown by GuidS and GTS, especially when compared to CG , which can easily exhaust the computational resources and is very slow to find one solution for a problems. The good performance of CG compared to RandS and RTS shows how powerful the CG method can be to find a solution, and the success of using DNN to explore the decision space represents the proper knowledge that can be achieved by learning from CG. \n\nFinally, GTS is a problem specific search method that needs to be applied on each test-cases separately. To use the knowledge that we can get from the GTS performance, more advanced reinforcement algorithms can be trained to create a single general knowledge-based method that is not only very powerful to find the best possible solutions, but also very fast for doing so. The advanced reinforcement learning method then can be easily applied on more sophisticated and challenging problems such as Proton and $4\\pi$ radiation therapy. For future studies, we are working to develop a smart, fast and powerful tool to be applied in these problems. \n\n\n\\section{Conclusion}\nIn this study, we proposed a method combined of two main components. First, a supervised deep neural network (DNN) to learn column generation (CG) decision-making process, and predict the fitness values of all candidate beams, beam with maximum fitness value will be chosen to add to the solution. CG, although powerful, is a heuristic, greedy algorithm that cannot guarantee the global optimality of the final solution, and leaves room for improvement. A Monte Carlo guided tree search (GTS) is proposed to see if finding a solution with better objective function in reasonable amount of time is feasible. After the DNN is trained, it is used to generate the beams fitness values for nodes in the decision tree, where each node represents a set of selected beams. Fitness values in each node are normalized and used as probability mass function to help deciding decision tree extension. Later probability distribution of beam selection will be modified by reward function, which is based on the solution of FMO problem FMO for every five selected beams. GTS continues to explore the decision tree for 1000 seconds. Along with GTS, three other approaches are also tested, GuidS which is also using DNN to select beams iteratively, but it does not update the probability distribution of beams during the search process. RTS which is a simple tree search algorithm, starts by randomly sampled from a uniform distribution of beam orientations for each node and continues to update beams probability distribution based on the tree search approach presented for GTS. And finally RandS which is randomly select beams, the most trivial and simple approach. \n\n\\clearpage\n\n\\section{Appendix\\label{appendix}}\n\\vspace*{-10mm}\nAlthough statistically with 130 number of test cases we can assume that approximately our metrics follow normal distribution, but based on the graphs of \\autoref{fig:objVal-distribution}, this assumption may not be practical. Because of this, we introduce $Distance$ measures to normalize our metrics. The probability distribution and cumulative probability mass function of $Distance$ measure are presented in \\autoref{fig:distance-distribution}. By this graph we can verify that the this measure approximately normally distributed with similar standard deviation.\n\\begin{figure}[htb]\n \\begin{subfigure}{\\textwidth}\n \\centering\n \\vspace{12pt}\n \n \\includegraphics[clip,trim=0mm 0mm 103mm 0mm,width=\\textwidth]{All_6method_objval.eps}\n \\captionv{15}{Objective values Distribution}{The distribution of objective values using GTS, GuidS, RTS, RandS, and CG methods.\n \\label{fig:objVal-distribution}}\n \\end{subfigure}\n \\begin{subfigure}{ \\textwidth}\n \\centering\n \n \n \\includegraphics[clip, trim=0mm 0mm 120mm 0mm,width=\\textwidth ]{distribution_distance_avg.eps}\n \\captionv{15}{Distance Distribution}{Distance distribution of methods GTS, GuidS, RTS, and RandS.\n \\label{fig:distance-distribution}}\n \\end{subfigure}\n \\captionv{15}{Distribution}{The distribution of objective values and distance to CG objective values\\label{fig:distribution}}\n\\end{figure}\n\n\\section*{References}\n\\vspace*{-10mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe anti-de Sitter\/conformal field theory (AdS\/CFT) correspondence~\\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj} has provided new insight into the dynamics of strongly coupled, four-dimensional (4D) gauge theories from the remarkable duality with a weakly coupled, five-dimensional (5D) gravitational description. Quantum chromodynamics (QCD), the theory of strong interactions, is a gauge theory that becomes strongly coupled in the infrared (IR); therefore, a 5D gravity dual provides a new framework to study nonperturbative phenomena in QCD. This is done in AdS\/QCD models, where the conformal symmetry is broken at a particular energy scale by an IR brane (hard-wall~\\cite{DaRold:2005zs,Erlich:2005qh}) or gradually via a background scalar field (soft-wall~\\cite{Karch:2006pv}). The hard-wall models provide a reasonable fit to the low energy observables except that the mass spectrum\n$m_n\\sim n$. A much better fit to the mass spectrum occurs in soft-wall models that can mimic the observed Regge trajectories of the QCD hadrons~\\cite{Karch:2006pv,Colangelo:2008us}. While the exact 5D dual of QCD remains elusive, further improvements such as chiral symmetry breaking have been incorporated into AdS\/QCD models~\\cite{Evans:2006ea,Evans:2006dj,Csaki:2006ji,Kwee:2007nq,Gherghetta:2009ac} but still describe only part of the rich\nstructure of QCD.\n\nFurthermore, there have been attempts to study strongly coupled gauge theories at finite temperature. This requires extending the metric to an AdS-Schwarzschild solution, describing an extra-dimensional generalization of a black hole. The thermodynamic description of black holes is well established~\\cite{Bekenstein:1973,Bekenstein:1974,Hawking:1974sw}. Using black-hole solutions, finite-temperature properties of strongly coupled gauge theories can be studied, such as those associated with the deconfinement phase transition and the quark-gluon plasma. Interestingly, experimental evidence confirms that the quark-gluon plasma (QGP) is strongly coupled \\cite{Adams:2005dq, Adcox:2004mh, Arsene:2004fa, Back:2004je}, lending support to the theoretical picture obtained via a 5D dual gravity description.\n\n\nThe simplicity of the soft-wall models in providing a reasonable fit to the mass spectrum, together with \nthe success of the black-hole solutions at finite temperature, suggests that a simple model combining \nboth features can be obtained. In this paper, we provide a finite-temperature description\nof a strongly coupled gauge theory using a soft-wall geometry based on the model in \\cite{Batell:2008zm}.\nA black-hole solution of the 5D Einstein equations is given for the metric and two scalar fields. The black-hole \nmetric is asymptotically AdS with an event horizon located at a finite value of the bulk coordinate. In the \n5D gravity dual, one scalar field plays the role of a string-theory dilaton that is dual to a gluon operator. The other scalar field resembles the \nstring-theory tachyon and is dual to chiral operator.\n\nWe attempt to introduce rigor into the soft-wall thermodynamics, beyond what has been done in \\cite{Herzog:2006ra, Colangelo:2009ra}. The solution outlined here is an expansion only valid in the limit of $z, z_h<1$. Unfortunately, this is the price we pay to study the soft-wall thermodynamics; the equations of motion have no known closed-form solution nor a reasonable numerical solution. \n\nThe thermodynamics resulting from our solution leads to interesting consequences. We show that a nonzero thermal condensate function $\\mathcal{G}$ induces a phase transition \\cite{Gursoy:2008za, Gursoy:2008bu, Gubser:2008ny, Galow:2009kw, Megias:2010ku}. Furthermore, $\\mathcal{G}$ contributes leading-order terms to the soft-wall thermodynamics that are absent in the lattice results \\cite{Boyd:1996bx, Miller:2006hr, Panero:2009tv}. However, qualitatively the two agree. The entropy has the expected behavior at high temperatures, scaling as $T^3$. The speed of sound through the thermal plasma is consistent with the conformal value of $1\/3$, matching the upper bound advocated in \\cite{Cherman:2009tw}. \n\nOur analysis begins in Section \\ref{secDynamicAct}, where we present the thermal AdS and black-hole AdS solution and compute the on-shell action. In Section \\ref{secThermo}, we study the thermodynamics of our solution, including a general expression for the entropy and squared speed of sound. We then calculate the free energy difference by carefully matching our two solutions at the AdS boundary. This enables us to compute the transition temperature between the confined and deconfined phases of the gauge theory in Section \\ref{secCond}. Concluding remarks are given in Section \\ref{secDiscuss}.\n\n\\section{Finite-Temperature Action} \n\\label{secDynamicAct}\n\nWe begin by specifying the 5D action in the string frame and show how it is related to the 5D action in the Einstein frame. Because the equations of motion become simpler, we transform the action to the Einstein frame. The action is then converted to a finite-temperature description by introducing a black hole into the metric. We generalize the AdS solution with two scalar fields~\\cite{Batell:2008zm} to a thermal AdS solution, which is used to compute the \non-shell action. The black-hole solution will be given in Section~\\ref{secbhAdS}.\n\n\\subsection{5D Lagrangian}\nWe start with the string-frame action inspired by the dimensionally-reduced type IIB supergravity action \\cite{Son:2007vk, Batell:2008zm},\n\\begin{eqnarray} \n\\label{equString}\n\\mathcal{S}_{\\rm string} &=&-\\frac{1}{16 \\pi G_5} \\int d^5 x \\sqrt{-g} \\Bigg[{\\rm e}^{-2\\Phi} \n\\Bigg(R + 4\\,g^{MN}\\partial_M\\Phi\\partial_N\\Phi - \\frac{1}{2}\\,g^{MN}\\partial_M \\chi\\partial_N \\chi \n- \\nonumber\\\\ \n&&\\qquad\\qquad\\qquad\\qquad V_S(\\Phi, \\chi)\\Bigg)+\\, {\\rm e}^{-\\Phi}\\mathcal{L}_{{\\rm meson}}\\Bigg] + S_{GH}^{(s)}.\n\\end{eqnarray}\nwhere $\\mathcal{L}_{{\\rm meson}}$ contains all the mass terms not considered in this work, $V_S$ is the string-frame scalar potential, and the indices $M,N = (t,x_{1},x_{2},x_{3},z)$. We work with two scalar fields $\\Phi$ and $\\chi$ where previous models have included only the dilaton \\cite{BallonBayona:2007vp, Gursoy:2008za, Gursoy:2008bu, Alanen:2009na, Alanen:2010tg, Gubser:2008ny, Franco:2009if, Galow:2009kw, Megias:2010ku}. The inclusion of a Gibbons-Hawking term, $\\mathcal{S}_{GH}^{(s)}$, is an attempt to be more rigorous than previous models. The string-frame metric is assumed to have an AdS-Schwarzschild form\n\\begin{equation} \\label{equStringMetric}\nds^2 = g_{MN} dx^M dx^N= \\frac{R^{2}}{z^{2}}\\left(-f(z) dt^{2} + d\\vec{x}^{2} + \\frac{dz^{2}}{f(z)}\\right), \n\\end{equation}\nwhere $f(z)$ determines the location of the black-hole horizon. Furthermore, the string-frame action (\\ref{equString}) contains similarities with noncritical string theory. The $\\Phi$ scalar field behaves like a dilaton, while the $\\chi$ scalar field behaves like a closed-string tachyon field. This suggests that our setup could be the low-energy limit of some underlying string theory.\n\nWhile the string-frame action provides a suitable starting point, it is more practical to do calculations in the Einstein frame. Switching to the Einstein frame involves a simple conformal transformation,\n\\begin{equation} \\label{equconformal}\ng_{MN}^{(s)} = {\\rm e}^{\\frac{4}{3}\\Phi}g_{MN}^{(E)}.\n\\end{equation}\nThe gravity-dilaton-tachyon action in the Einstein frame then becomes\n\\begin{equation} \n\\mathcal{S}_E =-\\frac{1}{16 \\pi G_{5}} \\int d^5x \\sqrt{-g}\\left(R - \\frac{1}{2}(\\partial\\phi)^2 - \\frac{1}{2}(\\partial\\chi)^2- V(\\phi, \\chi)\\right)\n+ S_{GH},\n\\label{equEinstein}\n\\end{equation}\nwhere $\\phi = \\sqrt{8\/3} \\Phi$ and $V = V_S\\,{\\rm e}^{\\frac{4}{3}\\Phi}$. We explicitly define the Gibbons-Hawking term $S_{GH}$ as\n\\begin{equation}\n\\mathcal{S}_{GH} = \\frac{1}{8 \\pi G_5}\\int d^4 x \\sqrt{-\\gamma} K , \\label{GHaction}\n\\end{equation}\nwhere $K=\\gamma^{\\mu\\nu}K_{\\mu\\nu}$ and $\\gamma$ is the four-dimensional induced metric at the AdS boundary. The extrinsic curvature $K_{\\mu\\nu}$ is defined by\n\\begin{equation}\\label{defnintcurv}\n K_{\\mu\\nu} = \\frac{1}{2} n^M \\partial_M \\gamma_{\\mu\\nu}~,\n\\end{equation}\n where the vector $n_{M}$ is the outward directed normal to the boundary and\n\\begin{equation}\ng_{MN}\\,n^{M}n^{N} = 1.\n\\end{equation}\nThe boundary of the AdS$_{5}$ space considered here is the $z=0$ plane, making the normal vector \n\\begin{equation}\n n^M = -\\frac{1}{\\sqrt{g_{zz}}} \\left(\\frac{\\partial}{\\partial z}\\right)^M = \\frac{\\delta^M_z}{\\sqrt{g_{zz}}}.\n \\label{defnormal}\n\\end{equation}\nThe Gibbons-Hawking term does not affect the equations of motion, but will have consequences when considering the free energy and deconfinement temperature.\n\nTo introduce temperature, we compactify the Euclidean time coordinate, \n$\\tau\\equiv i t\\rightarrow it+ \\beta$, where $\\beta=1\/T$ is the inverse temperature. The \nfinite-temperature metric in the Einstein frame then becomes\n\\begin{equation} \n\\label{equEinsteinMetric}\nds^{2}=a(z)^{2}\\left(f(z)d\\tau^2+d\\vec{x}^{2}+\\frac{dz^{2}}{f(z)}\\right),\n\\end{equation}\nwith the finite-temperature action given by\n\\begin{eqnarray}\n\\mathcal{S}_E(\\beta)&=&-\\frac{1}{16 \\pi G_{5}} \\int d^4 x \\int_0^{\\beta} d\\tau \\int dz \\sqrt{-g}\\left(R - \\frac{1}{2}(\\partial\\phi)^2 - \\frac{1}{2}(\\partial\\chi)^2 - V(\\phi, \\chi)\n\\right)\\nonumber \\\\\n&&\\qquad\\qquad +\\frac{1}{8 \\pi G_5}\\int d^3 x \\int_0^{\\beta} d\\tau \\sqrt{-\\gamma} K.\n\\label{equEinstein1}\n\\end{eqnarray}\nThe Einstein frame metric (\\ref{equEinsteinMetric}) and action (\\ref{equEinstein1}) will be \nused to solve the Einstein's equations in two realms, thermal AdS (thAdS) and black-hole AdS (bhAdS). All quantities with a subscript $_{0}$ are associated with the thAdS solution.\n\nWe fix the value of the 5D gravitational coupling $G_{5}$ using the QCD matching found in \\cite{Gursoy:2008za}, making\n\\begin{equation}\nG_{5} = \\frac{45\\pi R^{3}}{16 N_{c}^{2}}.\n\\end{equation}\n\n \n\\subsection{Thermal AdS Solution}\n\\label{secThermalAdS}\n\nWe begin by obtaining a thAdS solution that is effectively equivalent to the zero-temperature case explored in \\cite{Batell:2008zm}, where $f(z)=1$. Assuming the scalar fields are a function of only the $z$ coordinate, the action can be expressed as\n\\begin{eqnarray} \n\\label{equAction1}\n\\mathcal{S}_0(\\delta) &=& -\\frac{N_c^2}{45\\pi^2}\\frac{V}{R^3 T} \\int_{\\delta}^{\\infty}{dz \\sqrt{-g}\\left(R-\\frac{1}{2}g^{55}\\phi'^{2} - \\frac{1}{2}g^{55}\\chi'^{2} - V(\\phi,\\chi)\\right)}\\nonumber\\\\\n && \\qquad\\qquad\\qquad+S_{0,GH},\n\\end{eqnarray}\nwhere $V$ is the spatial three-volume. A UV cutoff at $z=\\delta$ has been introduced to regularize \nany singular behavior at the AdS boundary. \n\n The metric associated with the thAdS solution is\n\\begin{equation} \n\\label{equEmetric}\n ds^2 = a(z)^{2}(d\\tau^2 + d\\vec{x}^2 + dz^2) \\equiv {\\rm e}^{-2 c\\,\\phi(z)}\\frac{R^2}{z^2}(d\\tau^2 + d\\vec{x}^2 + dz^2),\n\\end{equation}\nwhere the constant $c$ depends on the conformal transformation. In our case, $c=1\/\\sqrt{6}$.\n\n Two equations of motion come from the 5D Einsteins equations,\n\\begin{eqnarray}\n12 \\frac{a'^{2}}{a^{2}} - 6 \\frac{a''}{a} &=& \\phi'^{2} + \\chi'^{2}, \\label{equGeneralField} \\\\\n6\\frac{a'^{2}}{a^{2}} + 3 \\frac{a''}{a} &=& -a^{2} V(z), \\label{equGeneralV}\n\\end{eqnarray}\nand two more equations come from the varying of the action,\n\\begin{eqnarray}\na^{2}\\frac{\\partial V}{\\partial \\phi} &=& \\phi'' + 3 \\phi' \\frac{a'}{a}, \\label{equVphi} \\\\ \na^{2}\\frac{\\partial V}{\\partial \\chi} &=& \\chi'' + 3 \\chi' \\frac{a'}{a}, \\label{equVchi}\n\\end{eqnarray}\nwhere prime $(')$ denotes derivatives with respect to $z$. \nUsing (\\ref{equEmetric}), we express (\\ref{equGeneralField}), (\\ref{equGeneralV}), (\\ref{equVphi}), and (\\ref{equVchi}) all in terms of $\\phi$, $\\chi$, and $V\\left(\\phi(z),\\chi(z)\\right)$,\n\\begin{eqnarray}\n\\chi'^{2} &=& \\frac{2\\sqrt{6}}{z}\\phi' + \\sqrt{6} \\phi'', \\label{equGeneralField2}\\\\\nV(z) &=& \\frac{{\\rm e}^{\\frac{2}{\\sqrt{6}} \\phi}}{R^2}\\left(-12 - 3\\sqrt{6} z \\phi' - \\frac{3z^{2}}{2} \\phi'^{2} + \\sqrt{\\frac{3}{2}} z^{2} \\phi''\\right), \\label{equGeneralV2}\\\\\n\\frac{\\partial V}{\\partial \\phi} &=& \\frac{z^{2}{\\rm e}^{\\frac{2}{\\sqrt{6}}\\phi}}{R^{2}} \\left(\\phi'' - \\sqrt{\\frac{3}{2}} \\phi'^{2} - \\frac{3}{z} \\phi'\\right),\\label{equVphi2}\\\\\n\\frac{\\partial V}{\\partial \\chi} &=& \\frac{z^{2}{\\rm e}^{\\frac{2}{\\sqrt{6}}\\phi}}{R^{2}} \\left(\\chi'' - \\sqrt{\\frac{3}{2}} \\phi'\\,\\chi' - \\frac{3}{z} \\chi'\\right),\\label{equVchi2}\n\\end{eqnarray}\nwhere we see that the nonlinear term $\\phi'^{2}$ has conveniently cancelled in (\\ref{equGeneralField2}). Of the four equations, we find that three are independent.\n\nAs shown in \\cite{Batell:2008zm}, the solution in the soft-wall model with a quadratic dilaton gives\n\\begin{eqnarray}\n\\phi(z) &=& \\sqrt{\\frac{8}{3}} \\mu^2 z^2, \\label{equphisol}\\\\\n\\chi(z) &=& 2 \\sqrt{6} \\mu z, \\label{equchisol}\\\\\nV(z) &=& \\frac{{\\rm e}^{\\frac{4}{3} \\mu^2 z^2}}{R^2} \\left(-12 - 20 \\mu^2 z^2 - 16 \\mu^4 z^4\\right), \\label{equpotsol}\n\\end{eqnarray}\nwhere $\\mu$ sets the hadronic mass scale. The quadratic behavior of the $\\phi$ solution (\\ref{equphisol}) leads to a Regge-like hadron mass spectrum $(m_n^2\\sim \\mu^2 n)$. The potential is a function of the $z$ coordinate with no unique solution for $V(\\phi,\\chi)$. However, a potential was found in \\cite{Batell:2008zm},\n\\begin{equation}\\label{equbatellV}\nV(\\phi,\\chi) = \\frac{\\chi}{2}{\\rm e}^{\\frac{\\chi^{2}}{18}} + 2 \\phi^{2}{\\rm e}^{\\frac{2}{\\sqrt{6}} \\phi} - 12 \\left[3 {\\rm e}^{\\frac{\\chi}{36}}-2\\left(1-\\frac{2}{\\sqrt{6}}\\right){\\rm e}^{\\frac{\\phi}{\\sqrt{6}}}\\right]^{2}.\n\\end{equation}\nWe can define an alternative potential that still produces (\\ref{equphisol}), (\\ref{equchisol}), and (\\ref{equpotsol}),\n\\begin{equation}\\label{equkelleyV}\nV_{\\rm alt}(\\phi,\\chi) = \\frac{{\\rm e}^{\\frac{2}{\\sqrt{6}}\\phi}}{R^{2}}\\left(-12 + 4\\sqrt{6} \\phi - \\frac{3}{2}\\chi^{2} - 4\\phi^{2} + \\frac{7}{3\\sqrt{6}}\\phi\\chi^{2}-\\frac{2}{27}\\chi^{4}\\right).\n\\end{equation}\nMore examples of consistent and well-defined potentials written in terms of two scalar fields can be found in \\cite{Kapusta:2010mf}.\n\nPart of the the free energy expression is found by substituting the thAdS solution into (\\ref{equAction1}) and finding the Gibbons-Hawking term. In general, we find the Ricci scalar and extrinsic curvature, \n\\begin{eqnarray} \\label{equRicciS1}\nR &=& -\\frac{8 a''}{a^{3}} - \\frac{4 a'^{2}}{a^{4}},\\\\\n \\gamma^{\\mu\\nu} K_{\\mu\\nu} &\\equiv& K = 4\\frac{a_0'}{a_0^2}.\n\\end{eqnarray}\nSubstituting into (\\ref{GHaction}) gives the on-shell Gibbons-Hawking term for the\nthermal AdS solution,\n\\begin{equation} \\label{equGHterm0}\n\\mathcal{S}_{0,GH} = \\frac{N_c^2V}{45\\pi^2} \\frac{8 a_0^2 a_0'}{R^3 T} .\n\\end{equation}\nthe on-shell action (\\ref{equAction1}) then becomes\n\\begin{equation}\n\\mathcal{S}_0(\\delta) = \\frac{N_c^2\\,V}{15\\pi^2} \\frac{2 a'(\\delta) a^2(\\delta)}{R^3 T}.\n\\label{equS1}\n\\end{equation}\nThe on-shell action is a pure boundary term and strictly depends on the AdS boundary conditions as $\\delta\\rightarrow 0$.\n\n\n\\subsubsection{Field\/Operator Correspondence}\\label{secFOcorr}\n\nAccording to the AdS\/CFT dictionary, a dimension-$\\Delta$ operator in $d$-dimensional gauge theory corresponds to a scalar field in the gravity dual with a mass,\n\\begin{equation}\nm^{2} = \\Delta(\\Delta - d).\n\\end{equation} \nExpanding the potential (\\ref{equbatellV}) from \\cite{Batell:2008zm}, we see that \n\\begin{equation}\\label{equAcase}\nm_{\\phi}^{2}R^{2} = -4, \\quad\\quad\\quad m_{\\chi}^{2}R^{2} = -3,\n\\end{equation}\nsuggesting that $\\phi$ is dual to a dimension-2 operator, and $\\chi$ is dual to a dimension-3 operator. However, the potential (\\ref{equkelleyV}) gives the masses as\n\\begin{equation}\\label{equGcase}\nm_{\\phi}^{2}R^{2} = 0, \\quad\\quad\\quad m_{\\chi}^{2}R^{2} = -3,\n\\end{equation}\nindicating that $\\phi$ and $\\chi$ are dual to a dimension-4 and dimension-3 operator, respectively. It is fairly clear that the chiral operator, $q\\bar{q}$, and the gluonic operator, Tr$\\left[F^{2}\\right]$, are the dimension-3 and dimension-4 operators, but the dimension-2 operator is much less clear. \n\nThe most likely dimension-2 operator candidate is $A_{\\mu}^{2}$, which becomes a local expression in the Laudau gauge, $\\partial^{\\mu}A_{\\mu}=0$. Coupling a source term to $A_{\\mu}^{2}$ makes the theory nonrenormalizable at the quantum level. A quadratic source term can be added to remedy this obstacle, though, this ruins the energy interpretation of the effective action \\cite{Vercauteren:2010rk}. In the context of the AdS\/CFT correspondence, $A_{\\mu}^{2}$ is often understood to convey information about the topological defects in the gravity dual \\cite{Gubarev:2000eu, Gubarev:2000nz}. Much more work concerning $A_{\\mu}^{2}$ has been conducted in \\cite{Dudal:2009tq, Vercauteren:2010cg, Vercauteren:2011ze}.\n\nIn the current soft-wall case, the field\/operator correspondence appears complicated. The ambiguity stems from the fact that the original AdS\/CFT dictionary was formulated considering purely free scalar fields. The potentials (\\ref{equbatellV}) and (\\ref{equkelleyV}) clearly have interaction terms. Resolving the issue of whether interaction terms affect the field\/operator correspondence and determining the interpretation of the dimension-2 operator is the subject of future research. In that spirit, we expand upon the published potential (\\ref{equbatellV}) and assume that the fields $\\chi$ and $\\phi$ correspond to the operators $q\\bar{q}$ and Tr$\\left[F^{2}\\right]$. The significance of our work relies on the fact that the dilaton is dual to \\emph{some} temperature-dependent operator. The identity of that operator is a topic for further research.\n\n\\subsection{Black-Hole AdS solution}\n\\label{secbhAdS}\n\nNext, we consider the black-hole AdS solution that describes a deconfined phase, mimicking a free quark-gluon plasma. \nAssuming the solutions are only a function of the $z$ coordinate, the 5D action associated with the black-hole AdS solution simplifies to\n\\begin{eqnarray} \n\\label{equAction2}\n\\mathcal{S}_{bh}(\\delta) &=& -\\frac{N_{c}^{2}}{45 \\pi^{2}}\\frac{V}{R^3 T(z_{h})} \\int_{\\delta}^{z_h}{dz \\sqrt{-g}\\left(R-\\frac{1}{2}g^{55}\\phi'^{2} - \\frac{1}{2}g^{55}\\chi'^{2} - V(\\phi,\\chi)\\right)} \\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad+S_{bh,GH},\n\\end{eqnarray}\nwhere $z_h$ is the location of the black-hole horizon. \nWe will see that $z_h$ is directly related to the temperature of the gauge theory. We begin with the black-hole metric (\\ref{equEinsteinMetric}) and find four independent equations of motion,\n\\begin{eqnarray}\nf''(z) &=& -3 f'(z)\\frac{a'(z)}{a(z)}, \\label{equbkf}\\\\\n\\phi'(z)^{2} + \\chi'(z)^{2} &=& 12 \\frac{a'(z)^{2}}{a(z)^{2}} - 6 \\frac{a''(z)}{a(z)},\\label{equbkfields}\\\\\na(z)^{2}\\frac{\\partial V}{\\partial \\phi} &=& f(z)\\phi''(z) + f'(z)\\phi'(z) + 3 f(z)\\phi'\\frac{a'(z)}{a(z)}, \\label{equbkVphi}\\\\\na(z)^{2}\\frac{\\partial V}{\\partial \\chi} &=& f(z)\\chi''(z) + f'(z)\\chi'(z) + 3 f(z)\\chi'\\frac{a'(z)}{a(z)}. \\label{equbkVchi}\n\\end{eqnarray}\nUnlike in the thAdS case, the potential is already determined. We must use the potential (\\ref{equbatellV}) to connect the soft-wall action to the free energy investigated in Section \\ref{secThermo}. With four independent equations and four unknown functions, $f(z)$, $a(z)$, $\\phi(z)$, and $\\chi(z)$, we cannot assume a fixed relation between the warp factor $a(z)$ and the dilaton $\\phi(z)$. These quantities must evolve independently as the temperature varies. The system of equations associated with bhAdS are difficult to solve but for the simplest cases. \n\nUsing the series expansions, we construct another solution. We use the thAdS solution of Section \\ref{secThermalAdS} as the starting points for these series expansions,\n\\begin{eqnarray}\na(z) &=& \\frac{R}{z}{\\rm e}^{-\\frac{\\phi}{\\sqrt{6}} + \\sum_{n=2}^{\\infty} m_{n}(\\mathcal{G},z_h) z^{n}}, \\label{equseriesmetric} \\\\\n\\phi(z) &=& \\sqrt{\\frac{8}{3}}\\mu^{2}z^{2} + \\sum_{n=2}^{\\infty} p_{n}({\\cal G},z_h) z^n, \\label{equseriesphi} \\\\\n\\chi(z) &=& \\sum_{n=1}^{\\infty} c_{n}(\\mathcal{G},z_h) z^{n},\\label{equserieschi}\\\\\nf(z) &=& 1 + \\sum_{n=4}^{\\infty} f_{n}(\\mathcal{G},z_{h}) z^{n},\\label{equseriesf}\n\\end{eqnarray}\nwhere we see that one of the black-hole conditions, $f(0)=1$, is automatically satisfied.\nThe condensate function $\\mathcal{G}(z_h)$ plays an important role in the free energy and phase transition of the system.\nBy solving (\\ref{equbkf}), (\\ref{equbkfields}), (\\ref{equbkVphi}), and (\\ref{equbkVchi}) in successive powers of $z$, we find the coefficients up to $n=8$ in terms of $f_{4}(\\mathcal{G},z_h)$. The other black-hole condition, $f(z_h)=0$, determines the final unknown coefficient. We calculate the non-zero coefficients for the metric,\n\\begin{eqnarray}\nm_{2} &=& -\\frac{\\sqrt{6}}{4\\mu^{2}}\\mathcal{G}(z_h),\\\\\nm_{6} &=& \\frac{1}{8\\mu^{4} + 3\\sqrt{6}\\mathcal{G}(z_h)}\\Bigg( -\\frac{8\\mu^{6}}{21}f_{4} + \\frac{3\\sqrt{3}\\mu^{2}}{7\\sqrt{2}}f_{4}\\mathcal{G}(z_h) + \\frac{32\\sqrt{2}\\mu^{6}}{21\\sqrt{3}}\\mathcal{G}(z_h) \\nonumber\\\\\n&& \\quad+ \\frac{45}{56\\mu^{2}}f_{4}\\mathcal{G}(z_h)^{2} + \\frac{131\\mu^{2}}{42}\\mathcal{G}(z_h)^{2} +\\frac{9\\sqrt{3}}{8\\sqrt{2}\\mu^{2}}\\mathcal{G}(z_h)^{3}\\nonumber\\\\\n&&\\quad - \\frac{93}{112\\mu^{6}}\\mathcal{G}(z_h)^{4} \\Bigg),\\\\ \nm_{8} &=& \\frac{1}{8\\mu^{4}+3\\sqrt{6}\\mathcal{G}(z_h)}\\Bigg(-\\frac{4\\mu^{8}}{9}f_{4} + \\frac{11\\mu^{4}}{21\\sqrt{6}}f_{4}\\mathcal{G}(z_h) +\\frac{184\\sqrt{2}\\mu^{8}}{243\\sqrt{3}}\\mathcal{G}(z_h) \\nonumber\\\\\n&&\\quad + \\frac{71}{56}f_{4}\\mathcal{G}(z_h)^{2} + \\frac{2543\\mu^{4}}{1701}\\mathcal{G}(z_h)^{2} + \\frac{1507}{378\\sqrt{6}}\\mathcal{G}^{3} +\\frac{117\\sqrt{3}}{224\\sqrt{2}\\mu^{4}}f_{4}\\mathcal{G}(z_h)^{3}\\nonumber\\\\\n&&\\quad + \\frac{203}{432\\mu^{4}}\\mathcal{G}(z_h)^{4} - \\frac{895}{896\\sqrt{6}\\mu^{8}}\\mathcal{G}(z_h)^{5} \\Bigg),\n\\end{eqnarray}\nthe field $\\phi$,\n\\begin{eqnarray}\np_{4} &=& -\\mathcal{G}(z_h),\\\\ \np_{6} &=& -\\frac{\\mu^{2}}{\\sqrt{6}}f_{4} - \\frac{\\mu^{2}}{3}\\mathcal{G}(z_h) + \\frac{11}{8\\sqrt{6}\\mu^{2}}\\mathcal{G}(z_h)^{2},\\\\\np_{8} &=& \\frac{1}{8\\mu^{4} + 3\\sqrt{6}\\mathcal{G}}\\Bigg(-\\frac{80\\sqrt{2}\\mu^{8}}{21\\sqrt{3}}f_{4} - \\frac{12\\mu^{4}}{7}f_{4}\\mathcal{G}(z_h) - \\frac{1408\\mu^{8}}{567}\\mathcal{G}(z_h) \\nonumber\\\\\n&&\\quad + \\frac{3\\sqrt{6}}{7}f_{4}\\mathcal{G}(z_h)^{2} -\\frac{107\\sqrt{2}\\mu^{4}}{63\\sqrt{3}}\\mathcal{G}(z_h)^{2}-\\frac{29}{27}\\mathcal{G}(z_h)^{3}\\nonumber\\\\\n&&\\quad-\\frac{3083}{1008\\sqrt{6}\\mu^{4}}\\mathcal{G}(z_h)^{4}\\Bigg),\n\\end{eqnarray}\nthe field $\\chi$,\n\\begin{eqnarray}\nc_{1} &=& \\sqrt{24\\mu^{2}+\\frac{9\\sqrt{6}}{\\mu^{2}}\\mathcal{G}(z_h)},\\\\\nc_{3} &=& \\frac{\\frac{\\sqrt{3}}{2\\mu^{4}}\\mathcal{G}(z_h)^{2}-2\\sqrt{2}\\mathcal{G}(z_h)}{\\sqrt{8\\mu^{2}+\\frac{3\\sqrt{6}}{\\mu^{2}}\\mathcal{G}(z_h)}},\\\\\nc_{5} &=& \\frac{1}{\\sqrt{8\\mu^{4}+3\\sqrt{6}\\mathcal{G}(z_h)}}\\Bigg(-\\sqrt{3}\\mu^{3}f_{4} - \\frac{9\\sqrt{2}}{8\\mu}f_{4}\\mathcal{G}(z_h) - 3\\sqrt{2}\\mu^{3}\\mathcal{G}(z_h)\\nonumber\\\\\n&&\\quad - \\frac{5\\sqrt{3}}{4\\mu}\\mathcal{G}(z_h)^{2} + \\frac{9\\sqrt{2}}{8\\mu^{5}}\\mathcal{G}(z_h)\\Bigg),\\\\\nc_{7} &=& \\frac{1}{\\sqrt{8\\mu^{4}+3\\sqrt{6}\\mathcal{G}(z_h)}}\\Bigg(-\\frac{20\\mu^{5}}{7\\sqrt{3}}f_{4} - \\frac{39\\mu}{14\\sqrt{2}}f_{4}\\mathcal{G}(z_h) - \\frac{568\\sqrt{2}\\mu^{5}}{189}\\mathcal{G}(z_h)\\nonumber\\\\\n&&\\quad - \\frac{111\\sqrt{3}}{112\\mu^{3}}f_{4}\\mathcal{G}(z_{4})^{2} - \\frac{1123\\mu}{126\\sqrt{3}}\\mathcal{G}(z_h)^{2} - \\frac{67}{24\\sqrt{2}\\mu^{3}}\\mathcal{G}(z_h)^{3} \\nonumber\\\\\n&&\\quad+ \\frac{151\\sqrt{3}}{224\\mu^{7}}\\mathcal{G}(z_h)^{4} \\Bigg),\n\\end{eqnarray}\nand $f(z)$,\n\\begin{eqnarray}\nf_{4} &=& \\frac{-1}{z_{h}^{4}\\left(1+\\frac{4\\mu^{2}}{3}z_{h}^{2}+\\mu^{4}z_{h}^{4} + \\left(\\frac{\\sqrt{6}}{2\\mu^{2}} + \\frac{\\sqrt{6}}{2}\\right)z_{h}^{2}\\mathcal{G}(z_h) + \\frac{27}{32\\mu^{4}}z_{h}^{4}\\mathcal{G}(z_h)^{2}\\right)},\\\\\nf_{6} &=& \\frac{4\\mu^{2}}{3}f_{4} + \\frac{3}{\\sqrt{6}\\mu^{2}}f_{4}\\mathcal{G}(z_h), \\\\\nf_{8} &=& \\mu^{4} f_{4} + \\frac{\\sqrt{6}}{2}f_{4}\\mathcal{G}(z_h) + \\frac{27}{32\\mu^{4}}f_{4}\\mathcal{G}(z_h)^{2}.\n\\end{eqnarray}\n\nIn general, the coefficients of the $z^n$ terms in $p_{n}$ and $m_{n}$ have direct consequences for the free energy. For coefficients $04$ the function ${\\cal G}(z_h)$ does not affect the free energy. Only the $n=4$ coefficients affect the free energy expression. As we will see in Section~\\ref{secThermo}, the condensate function ${\\cal G}$ then plays a crucial role in giving rise to a finite transition temperature.\n\nAn expression for the on-shell action can be obtained by simplifying (\\ref{equAction2}). The induced metric in this case is\n\\begin{equation}\\label{equinducedbh}\n\\gamma = a(z)^{2}\\left( f(z)d\\tau^{2} + d\\vec{x}^{2} \\right);\n\\end{equation} \ntherefore, the Gibbons-Hawking action term becomes\n\\begin{equation}\\label{equSbhgh}\n\\mathcal{S}_{bh,GH} = \\frac{N_{c}V}{45\\pi^{2}R^{3}T}\\left(8 f \\,a^{2}a' + f' \\,a^{3}\\right).\n\\end{equation} \nGiven that the Ricci tensor is \n\\begin{equation}\nR = -\\frac{4 f a'^{2}}{a^{4}} - \\frac{8 f a''}{a^{3}} - \\frac{8 f'a'}{a^{3}} - \\frac{f''}{a^{2}},\n\\end{equation}\nthe total on-shell action can then be written as\n\\begin{equation}\n\\mathcal{S}_{bh}(\\delta) = \\frac{N_c^2}{45\\pi^2}\\frac{V}{R^3 T} \\left(6 f(\\delta)a'(\\delta) a^2(\\delta) +f'(\\delta) a^3(\\delta) \\right).\n\\label{bhonshell}\n\\end{equation}\nAgain, the on-shell action is a pure boundary term and depends only on the AdS boundary conditions.\nThis action will be used to compute the free energy in Section~\\ref{secThermo}.\n\n\n\\subsubsection{Field\/Operator Correspondence at Finite Temperature}\n\nThe scalar field solutions (\\ref{equseriesphi}) and (\\ref{equserieschi}) correspond to turning on thermal condensates in the gauge theory. To show the relation between ${\\cal G}(z_h)$ and the operator condensates, we assume the kinetic term of a canonically normalized bulk scalar field fluctuation, \ndenoted $\\omega$, to be \n\\begin{equation} \n\\label{equgenbulk}\n\\mathcal{S}_{{\\rm bulk}} = \\frac{1}{2 L^3}\\int d^5 x \\sqrt{-g}\\, \\partial_M \\omega\\partial^M \\omega.\n\\end{equation}\nLet us first consider the scalar field $\\phi$, where the coupling to the four-dimensional boundary operator is\n\\begin{equation} \\label{equgenboundary}\n\\mathcal{S}_{{\\rm boundary}} = \\int{d^{4}x \\,\\omega_{\\phi}\\, {\\rm Tr}(F^{2})}.\n\\end{equation}\nIn terms of the 't Hooft coupling $\\lambda = {\\rm e}^{\\Phi}$, the Yang-Mills field strength is \n\\begin{equation}\\label{equYMcouple}\n\\mathcal{S}_{{\\rm boundary}} = -\\int{d^{4}x\\, \\frac{1}{4\\lambda} {\\rm Tr} (F^{2}) },\n\\end{equation}\nmaking the dilaton fluctuation, \n\\begin{equation}\\label{equfluxphi}\n\\delta \\mathcal{S}_{{\\rm boundary}} = \\frac{1}{4} \\int{ d^4x\\, \\delta\\Phi \\,{\\rm e}^{-\\Phi} {\\rm Tr}(F^{2})}.\n\\end{equation}\nWe are only interested in the fluctuation $\\delta\\Phi = \\Phi - \\Phi_{0}$, which allows one to compute the difference between thermal and vacuum values of $\\langle {\\rm Tr}F^{2} \\rangle$ \\cite{Gursoy:2008za, Megias:2010ku}. From (\\ref{equseriesphi}), we find that $\\delta\\Phi=-\\sqrt{3\/8}{\\cal G}(z_h) z^4$. Recall that the relation between the expectation value of a $\\Delta$-dimensional operator in $d$-dimensional space and the field is\n\\begin{equation}\n\\frac{\\omega}{z^{\\Delta}} \\rightarrow \\frac{\\langle \\mathcal{O} \\rangle}{2\\Delta - d}.\n\\end{equation}\nThe function ${\\cal G}(z_h)$ then relates directly to the gluon condensate,\n\\begin{equation}\n\\langle {\\rm Tr} (F^{a})^{2} \\rangle - \\langle {\\rm Tr} (F^{a})^{2} \\rangle_{0} = \n-\\sqrt{\\frac{2}{3}} \\left(\\frac{32 N_c^2\\lambda}{45\\pi^2}\\right) {\\mathcal G}(z_h). \n\\end{equation} \n\nA similar correspondence can be obtained for the scalar field $\\chi$, where it is tempting to\nrelate $\\chi$ to the three-dimensional operator $q \\bar{q}$. In this case, the coefficient of the \n$z^3$ term in (\\ref{equserieschi}) is proportional to the renormalized chiral condensate, $\\langle q \\bar{q}\\rangle -\\langle q \\bar{q}\\rangle_{0} $. \nFor this correspondence to hold, $\\chi$ would need to be a bifundamental field in the gauge theory. This can be done by \npromoting $\\chi$ to a bifundamental field $\\chi^{ab}$, where $a,b$ are group indices and writing, for example\n\\begin{equation}\n\\langle \\chi^{ab} \\rangle = \\chi(z) \n \\left( \n \\begin{array}{cc} \n 1 & 0 \\\\\n 0 & 1\n\\end{array}\\right),\n\\label{bifundveveqn}\n\\end{equation}\nfor an $SU(2)\\times SU(2)$ symmetry. \n\nUnder this assumption, we can follow the same procedure as for the $\\phi$ field, and derive an operator correspondence for $\\chi$. Assuming a bulk kinetic term of the form (\\ref{equgenbulk}) and boundary operator coupling\n\\begin{equation}\nS_{boundary} = \\int{d^{4}x \\,\\omega_{\\chi}\\,\\bar{q}q},\n\\end{equation}\nwe find that\n\\begin{equation}\n\\langle \\bar{q} q\\rangle - \\langle \\bar{q} q \\rangle_{0} =\n-\\frac{2 N_{c}^{2}}{27\\pi^{2}}\\left(\\frac{\\mathcal{G}(z_h)}{\\mu}-\\frac{5}{8}\\frac{\\mathcal{G}(z_h)^{2}}{\\mu^{5}}\\right).\n\\end{equation}\nThus, we see that the function ${\\cal G}(z_h)$ is intimately related to the thermal condensates \n$\\langle \\bar{q}q \\rangle$ and $\\langle {\\rm Tr} F^2 \\rangle$. In fact, the ratio of the gluon condensate and the\nchiral condensate is given roughly by\n\\begin{equation}\n \\frac{\\langle {\\rm Tr} (F^{a})^{2} \\rangle - \\langle {\\rm Tr} (F^{a})^{2} \\rangle_{0}}\n {\\langle \\bar{q} q\\rangle - \\langle \\bar{q} q \\rangle_{0}} \\approx \\frac{16}{5} \\sqrt{6} \\lambda \\mu,\n\\end{equation}\nand is consistent with that obtained in perturbation theory to first order \\cite{Shifman:1978bx}.\n\n\n\\section{Thermodynamics} \n\\label{secThermo} \n\nHaving determined the black-hole solution of the 5D gravity-dilaton-tachyon system we now\ninvestigate the thermodynamics. We begin with Hawking's black-hole thermodynamics, where the temperature is found,\n\\begin{eqnarray}\nT(z_h) &=& -\\frac{\\partial_{z}f(z)}{4\\pi}\\Big|_{z=z_h} = \\frac{1}{4\\pi \\mathcal{P}(z_h)\\,a(z_h)^{3}}\\nonumber\\\\\n&\\approx& \\frac{2}{\\pi z_{h}}\\left(1-\\frac{1}{\\frac{1}{2} - 3\\mu^{2}z_{h}^{2} - \\frac{9\\sqrt{3}z_{h}^{2}}{4\\sqrt{2}\\mu^{2}} + \\frac{6\\mu^{2}(\\sqrt{6}z_{h}^{4}\\mathcal{G}(z_h)-5)}{12\\mu^{2} + 8\\mu^{4}z_{h}^{2} + 3\\sqrt{6}z_{h}^{2}\\mathcal{G}(z_h)} }\\right). \\label{equexplicitT}\n\\end{eqnarray}\nThe quantity $\\mathcal{P}$ is defined as\n\\begin{equation}\n\\mathcal{P}(z) = \\int_{0}^{z} dx \\,a(x)^{-3}.\n\\end{equation}\nOf course, all thermodynamic quantities will depend on the condensate function, whose behavior we address in Section \\ref{secCond}.\n\n\n\\subsection{Entropy} \\label{subsecEntropy}\nThe entropy is found in the usual way from black-hole thermodynamics, a subject extensively covered in \\cite{Bekenstein:1973,Bekenstein:1974,Hawking:1974sw}. The entropy $S$ is defined as \n\\begin{equation}\nS = \\frac{A_{bh}}{4 G_{5}}.\n\\end{equation}\nAs expected for a relativistic gas at high temperature, the entropy density $s(T)$ behaves as $T^3$ at high temperature. \nIn our model, we can also calculate the subleading temperature behavior and check that it is consistent.\nWe compute the entropy density from the area of the black-hole horizon using the induced metric $\\gamma$. Using the metric (\\ref{equseriesmetric}), we obtain\n\\begin{equation} \\label{equBHarea}\nA_{bh} = \\int d^3x \\,\\sqrt{\\gamma} = \\frac{V\\,R^3}{z_h^3}{\\rm e}^{-3\\left( \\frac{\\phi}{\\sqrt{6}} + m2\\,z_{h}^{2} + m6\\,z_{h}^{6} + m8\\,z_{h}^{8}\\right)} ,\n\\end{equation}\nwhere $V$ is the spatial volume. Using the first few terms in our metric expansion, the entropy density is then given by\n\\begin{equation}\ns = \\frac{4N_{c}^{2}V}{45\\pi}\\left(\\frac{1}{z_{h}^{3}} - \\frac{2\\mu^{2}}{z_h} - \\frac{3\\sqrt{3}\\mathcal{G}}{2\\sqrt{2}\\mu^{2}z_h} + 2\\mu^{4}z_{h} + 2\\sqrt{6} z_{h} \\mathcal{G} + \\frac{27 z_{h}\\mathcal{G}^{2}}{16\\mu^{4}}\\right).\n\\end{equation}\n\n\n\\subsection{Speed of Sound} \\label{secvsound}\nWith the temperature and entropy known, we also determine the speed of sound through the deconfined medium of the gauge theory. The speed of sound $v_s$ characterizes the hydrodynamic evolution of the deconfined, strongly coupled plasma. It has been suggested that $v_s^2$ in holographic models obeys an upper limit of 1\/3 \\cite{Cherman:2009tw}.\nThis can be checked for our solution using the relation\n\\begin{eqnarray} \nv_s^2 &=& \\frac{s \\frac{dT}{dz_h}}{T \\frac{ds}{dz_h}} = \\frac{d\\log{T}}{d\\log s} \\nonumber\\\\\n&=& -1 - \\frac{1}{3}\\frac{d\\log{\\mathcal{P}}}{d\\log{a}}.\n\\end{eqnarray}\nIn our model, the exact form of $v_{s}^{2}$ greatly depends on the form of $\\mathcal{G}$. However, we do confirm that \n\\begin{equation} \n\\frac{d\\log{\\mathcal{P}}}{d\\log{a}} \\rightarrow -4,\n\\end{equation}\nin the high temperature limit, recovering the conformal limit with an upper bound of $v_s^2=1\/3$.\n\n\n\\subsection{Free Energy} \\label{secFEnergy}\n\nThe free energy of the deconfined phase is calculated in two ways. We use thermodynamic identities to define the free energy as\n\\begin{equation}\n\\mathcal{F} = -\\int{S\\, dT}= \\mathcal{F}_{{\\rm min}}-\\int{s\\, V \\frac{dT}{dz_{h}}dz_{h}}. \\label{equfreeEdefine}\n\\end{equation}\nwhere we have found the Bekenstein entropy and have an expression for $T(z_h)$. We have explicitly written the integration constant $\\mathcal{F}_{{\\rm min}}$ since the problem with using (\\ref{equfreeEdefine}) is common among energy definitions, setting the zero point. Because $z_h(T)$ is a multi-valued function, the integral is non-trivial. Calculating the value $\\mathcal{F}_{{\\rm min}}$ generally requires the expression of entropy in the large-$z_h$ region, exactly where our expanded solutions are invalid. While the integral in (\\ref{equfreeEdefine}) is completely calculable, we find the free energy by using the on-shell action. Finding the zero-point energy using actions is much easier; one merely subtracts the background free energy as defined by the thAdS action,\n\\begin{equation}\\label{equfreeEaction}\n\\mathcal{F} = {\\rm lim}_{\\delta\\rightarrow 0} T\\left[\\mathcal{S}_{bh}(\\delta) - \\mathcal{S}_0(\\delta)\\right].\n\\end{equation}\nBoth $\\mathcal{S}_{bh}$ and $\\mathcal{S}_{0}$ have been computed earlier. \n\nBefore evaluating (\\ref{equfreeEaction}), we must properly match the thermal AdS and black-hole AdS metrics at the boundary, $\\delta\\rightarrow 0$ \\cite{Witten:1998zw}. This requires matching the intrinsic geometry of the two solutions at the boundary cut-off \\cite{Gursoy:2008za}, where\n\\begin{eqnarray}\na_0(\\delta) &=& a(\\delta)\\sqrt{f(\\delta)}, \\nonumber\\\\\nV_{0}a_0(\\delta)^{3} &=& V a(\\delta)^{3}.\\label{equmatch}\n\\end{eqnarray}\nIn order for (\\ref{equmatch}) to be satisfied, we must evaluate the thAdS and bhAdS solutions at different cut-off points, $\\tilde{\\delta}$ and $\\delta$ respectively. We find that \n\\begin{equation}\n\\tilde{\\delta} = \\frac{\\sqrt{8\\mu^{2}\\delta^{2} - 2\\sqrt{6}\\mathcal{G}\\delta^{4} + \\frac{3\\sqrt{6}\\mathcal{G}}{\\mu^{2}}\\delta^{2}}}{2\\sqrt{2}\\mu}.\n\\end{equation}\nCombining the matching with (\\ref{equfreeEaction}), we obtain a rather simple expression for the free energy,\n\\begin{equation}\\label{equfreeEmod2}\n\\mathcal{F} = \\frac{N_c^2 V}{45\\pi^2 R^3} {\\rm lim}_{\\delta\\rightarrow 0} \\left(6 f(\\delta) a(\\delta)^{2}a'(\\delta) + f' a(\\delta)^{3} -6 \\sqrt{f(\\delta)} a(\\delta)^4 \\frac{a_0'(\\tilde{\\delta})}{a_0(\\tilde{\\delta})^2}\\right),\n\\end{equation}\nwhich can be reduced to\n\\begin{eqnarray} \\label{equfreeEfG}\n\\mathcal{F} = \\frac{N_c^{2}V}{45\\pi^2}\\left(f_{4}(z_h,\\mathcal{G}) + 2\\sqrt{6} \\mathcal{G} \\right) = \\frac{2N_c^{2}V}{45\\sqrt{6}\\pi^2}\\mathcal{G} -\\frac{1}{4} T S.\n\\end{eqnarray}\nIt should be noted that we also checked that the black-hole energy $E$ satisfies the thermodynamic formula \n$E={\\cal F} + T S$ by computing the ADM energy in Appendix \\ref{appADM}.\n\n\n\n\\section{The Condensate Function and the Phase Transition}\\label{secCond}\n\nAll the thermodynamic relationships rely on the behavior of the condensate function; therefore, we need to find a solution to $\\mathcal{G}$ to evaluate the temperature, entropy, and speed of sound. The free energy of the system gives us enough information to solve for $\\mathcal{G}$. We only need to set (\\ref{equfreeEdefine}) and (\\ref{equfreeEfG}) equal to one another. Taking the derivative with respect to $z_h$ removes any unknown constants, giving\n\\begin{equation}\\label{equdiffforG}\n\\frac{df_{4}}{d z_h} + 2\\sqrt{6}\\frac{d\\mathcal{G}}{d z_h} = s\\,V\\,\\frac{dT}{d z_h},\n\\end{equation}\nor in more simplified terms,\n\\begin{equation} \\label{equdiffG}\n \\frac{d\\mathcal{G}(z_h)}{dz_h} = \\frac{1}{2 \\sqrt{6} \\mathcal{P}(z_h)} \\left(\\frac{a'(z_h)}{a(z_h)} + \\frac{\\mathcal{P}'(z_h)}{4 \\mathcal{P}(z_h)}\\right).\n\\end{equation}\nUnfortunately, (\\ref{equdiffG}) is a stiff equation. One can often find stable solutions to these equations within a certain region, but our case is barely within a region of stability. \n\n We use two methods to solve for $\\mathcal{G}$. First, we introduce a series expansion for $\\mathcal{G}$,\n\\begin{equation}\\label{equGseries}\n\\mathcal{G}(z_h) = \\sum_{j=-\\infty}^{\\infty} g_{j} \\mu^{j+4} z_{h}^{j},\n\\end{equation}\nand expand (\\ref{equdiffG}) in powers of $z_h$. Performing the expansion and matching coefficients, we find that the useful nonzero $g_{j}$'s are\n\\begin{eqnarray}\\label{equGcoeff}\ng_{-2}\\equiv g &=& 1.43290,\\nonumber\\\\\ng_{0} &=& -6.09417.\\nonumber\\\\\n\\end{eqnarray}\nSince the nonlinear nature of (\\ref{equdiffG}) spoils the series solution quite quickly, we find that only the first two terms give an accurate solution for $G$ in the range of $z_h<1$. Using the series expansion to inform the boundary conditions, we then numerically solve for $\\mathcal{G}$ in (\\ref{equdiffG}). However, it is valid within a limited range of $z_{h}$, matches the series solution in that range, and diverges around $z\\sim 0.5$ GeV$^{-1}$. The two solutions are plotted in Figure \\ref{figcondensate}. With the condensate function $\\mathcal{G}$, we can take a second look at the thermodynamics.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.45]{condensate.pdf}\n\\caption{The series and numerical solution for $\\mathcal{G}$ is plotted. We use the series solution for the thermodynamics since the differential equation clearly gives a numerically unstable solution for $z\\ge 0.3$ GeV$^{-1}$.}\n\\label{figcondensate}\n\\end{center}\n\\end{figure}\n\nGiven the expression (\\ref{equseriesf}), the temperature can be written as \n\\begin{eqnarray}\nT(z_h) &\\approx& \\frac{2}{\\pi z_h}\\left(\\frac{16+12\\sqrt{6}g + 27g^{2}}{32+12\\sqrt{6}g + 27g^{2}}\\right) + \\frac{\\mu^{2}z_{h}^{2}}{\\pi(32+12\\sqrt{6}g + 27g^{2})^{2}}\\times\\nonumber\\\\\n&&\\qquad\\Bigg(\\frac{2048}{3}+516\\sqrt{6}g+192 g^{2}+256\\sqrt{6}g_{0} + 1728 g g_{0}\\nonumber\\\\\n&&\\qquad+\\, 216\\sqrt{6} g^{2} g_{0}\\Bigg).\\label{equTexactwithG}\n\\end{eqnarray}\nWe clearly see that the condensate function contributes a finite piece to the temperature expression. Increasing the strength of the condensate produces \n\\begin{equation}\\label{equTlimit}\n\\lim_{g\\rightarrow \\infty}\\,T(z_h)\\rightarrow \\frac{2}{\\pi z_h},\n\\end{equation}\nwhich is a factor of 2 larger than the conformal limit found in \\cite{Gursoy:2008za, Gursoy:2009kk}. We find that other work in this area has assumed that the condensate terms are suppressed logarithmically. However, in the construction of this model, the soft-wall set-up has no natural means of generating the logarithmic suppression in $\\mathcal{G}$. As a result, we see that the condensate contributes to leading-order behavior in the small-$z_h$\/large-$T$ limit.\n\nIt will be useful to have an inverted function $z_{h}(T)$ which we use to transform functions of $z_h$ to functions of $T$. For simplicity, we use only the first-order term,\n\\begin{equation}\\label{equTexpand}\nz_{h}(T) \\approx \\frac{2}{\\pi\\,T}\\left(\\frac{16+12\\sqrt{6}g + 27g^{2}}{32+12\\sqrt{6}g + 27g^{2}}\\right).\n\\end{equation} \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.45]{temp.pdf}\n\\caption{The temperature as a function of the horizon location $z_h$ with and without a condensate \n$\\mathcal{G}$. The inclusion of a condensate in this particular model precludes a second, unstable black-hole \nsolution from developing.}\n\\label{figTemp}\n\\end{center}\n\\end{figure}\n\nAnalytically, the entropy of the deconfined phase can be written in terms of $z_h$, \n\\begin{eqnarray}\ns(z_h) &=& \\frac{4N_c^2}{45\\pi}\\frac{1}{(32+16\\sqrt{6}g+27g^{2})^{2}}\\Bigg[32 -\\frac{86\\sqrt{6}}{7}g +\\frac{279}{14}g^{2} - \\frac{313\\sqrt{3}}{14\\sqrt{2}} \\nonumber\\\\\n&&\\quad + \\frac{3623}{168}g^{4} - \\frac{313}{224\\sqrt{6}}g^{5} + \\frac{90081}{3584}g^{6}\\Bigg] +\\ldots\\nonumber\\\\\n&\\approx& \\frac{1.77}{z_{h}^{3}} - \\frac{25.35\\mu^{2}}{z_{h}} + 156.27 \\mu^{4} z_{h} +\\ldots,\\label{equspecificentropy} \n\\end{eqnarray}\nor in terms of $T$, \n\\begin{equation}\ns(T) \\approx 13.88 T^{3} - 50.35 \\mu^{2}\\,T +\\ldots = C(T) T^{3},\\label{equGEntropy}\n\\end{equation}\nwhere the function $\\mathcal{C}(T)$ modifies the entropy behavior at low temperatures.\nOur result, plotted in Figure \\ref{figEntropy}, agrees qualitatively with the lattice data presented in \\cite{Boyd:1996bx}. The high-temperature limit of $s(T)\/T^3$, however, is shifted from lattice results because of the contribution from the condensate function.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.45]{entropy.pdf}\n\\caption{The entropy density divided by $T^3$ as a function of temperature. At high temperatures the entropy density slowly evolves to the conformal case of $s\\sim T^3$ in each case. }\n\\label{figEntropy}\n\\end{center}\n\\end{figure}\n\nThe speed of sound through the QGP-like thermal phase can be expressed as \n\\begin{eqnarray}\nv_{s}^{2}(z_h) &\\approx& \\frac{1}{3} - 2.718\\mu^{2}z_{h}^{2} + 21.938 \\mu^{4}z_{h}^{4},\\label{equv2zh}\\\\\nv_{s}^{2}(T) &\\approx& \\frac{1}{3} - 0.689\\frac{\\mu^{2}}{T^{2}} + 1.409\\frac{\\mu^{4}}{T^{4}}.\\label{equv2T}\n\\end{eqnarray} \nWe clearly see that (\\ref{equv2zh}) and (\\ref{equv2T}) give the expected $v_{s}^{2} = 1\/3$ in the small-$z$ and large-$T$ limit. The speed of sound through our QGP is plotted in Figure \\ref{figVelo}. As the figure shows, $v_{s}^{2}$ deceases more rapidly with the condensate terms included; however, the difference between the two cases are not as stark as it was in the entropy case. \n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.45]{velocity.pdf}\n\\caption{The squared speed of sound in the strongly coupled plasma as a function of the temperature. The conformal limit of $v_s^2 = \\frac{1}{3}$ is recovered at high temperatures.}\n\\label{figVelo}\n\\end{center}\n\\end{figure}\n\nAn issue arises when we consider the behavior of $\\mathcal{G}$. In Section \\ref{secFOcorr}, we argued that the dual operator corresponding to the dilaton field is either a dimension-2 or a dimension-4 gluon operator. On dimensional grounds, previous work asserts that $\\mathcal{G}\\sim T^{4}$. However, our model clearly shows that any self-consistent solution in the soft-wall model requires that $\\mathcal{G}\\sim T^{2}$. The consequences of the temperature dependence on the field\/operator duality are subjects for further research. \n\n\\subsection{Deconfinement Temperature} \n\\label{secDeTemp}\n\nFinding the deconfinement temperature requires examining the free energy of the deconfined phase in (\\ref{equfreeEfG}). The point at which $\\mathcal{F}=0$ occurs when the black-hole solution becomes more energetically favored than the thermal solution. It is instructive to first consider the case without a condensate. By setting ${\\cal G}(z_h)=0$ in (\\ref{equfreeEfG}), the transition temperature is determined from the condition\n\\begin{equation}\n\\label{equHerzog}\nf_{4}(\\mathcal{G}(z_h),z_h)=0.\n\\end{equation} \nHowever, (\\ref{equHerzog}) has only one possible solution, $z_{h}\\rightarrow\\infty$; therefore, no transition temperature exists since $z_h\\rightarrow\\infty$ occurs in the region of the unstable black hole. This mimics the scenario considered in \\cite{Herzog:2006ra, Colangelo:2009ra}, but we include the full back-reaction of the scalar field and take the Gibbons-Hawking term into account. Therefore, we see that a nonzero condensate is needed to obtain a transition temperature in our model. \n\nUsing the leading behavior and the numerical solution, we find the free energy behavior and plot it in Figure \\ref{figEnergy}. The phase transition occurs at a critical $z_c=0.5262$. Using (\\ref{equexplicitT}) and the first two terms of the expansion for $\\mathcal{G}$, this corresponds to a critical temperature of $T_{c}=919$ MeV. This is much larger than either theoretical reasoning or lattice calculations suggest \\cite{Shifman:1978bx,Shifman:1978bw,Shifman:1978by, Boyd:1996bx, Miller:2006hr}. There are no current plans (that this author can find) for experiments that would produce such high temperatures; however, experimental evidence for QGP has already been detected. The large transition temperature is most likely the product of the crude soft-wall model and the absence of any suppression terms in $\\mathcal{G}$. As $z_h$ increases ($T$ decreases), we see that free energy goes to zero as in \\cite{Gursoy:2008za, Megias:2010ku}; however, the free energy expression is valid below the the region of $z_h\\approx 1$. Thus, the true behavior of the free energy at increasing values of $z_h$ requires more numerical work. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.45]{freeE.pdf}\n\\caption{The free energy $\\cal F$ plotted as a function of $z_h$. Only when including a gluon condensate do we find a solution to $\\mathcal{F} = 0$. The free energy with no gluon condensate approaches zero as $z_h\\rightarrow\\infty$, but never crosses the x-axis.}\n\\label{figEnergy}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion} \n\\label{secDiscuss}\n\nWe have presented a five-dimensional, gravity-dilaton-tachyon, black-hole, dynamical solution that represents a dual description of a strongly coupled gauge theory at finite temperature. The solution generalizes the soft-wall geometry considered in \\cite{Batell:2008zm}, which generates a quadratic dilaton and Regge mass trajectory ($m_n\\sim\\sqrt{n}$). In the string frame, the dilaton and tachyon fields appear to be duals to the gluon and chiral operators, respectively. However, the actual field\/operator correspondence is ambiguous. The black-hole solution describes a deconfined free-gluon phase. A transition from confined matter exists, predicted at a extremely high $T_{c}=919$ MeV. A condensate function is needed for any transition to occur.\n\n Despite being a mere series expansion, we argue that it gives insight into thermodynamics of a strongly coupled gauge theory produced by the soft-wall model. We expected to find two major contributions to the thermodynamics: the underlying conformal limit and condensate terms. Although we considered a finite number of terms in our infinite solution expansion, our calculations suggest that the condensate terms produce convergent quantities, as shown in (\\ref{equTlimit}). With further study and numerical rigor, we believe a viable closed-form, black-hole solution with a lower transition temperature will be found.\n\nWhile our model has some features reminiscent of QCD at finite temperature, it still represents a crude approximation with a number of shortcomings. The bhAdS solution is only valid in the region of small $z_h$. The power-law dependence of the scalar fields and metric does not include the logarithmic corrections needed to suppress condensate terms as in \\cite{Gursoy:2008za, Megias:2010ku}, resulting in noticeable changes in the behaviors of the temperature, entropy, and speed of sound. Furthermore, using the AdS\/CFT dictionary, the dilaton field appears to be dual to a dimension-2 operator, $(A_{\\mu})^{2}$, in contradiction to the standard assumption of a dimension-4 operator, $Tr[F^{2}]$. Nevertheless, our 5D dynamical black-hole solution with two scalar fields provides a toy model to understand the nontrivial properties of strong-coupled gauge theories at finite temperature. \n\n\n\\section*{Acknowledgements} \n\\label{acknowl}\n\nI like to thank Tony Gherghetta for his contributions and advice. I also thank Joe Kapusta for his continuing support and Christopher Herzog for his clarification on calculating deconfinement temperature. I would like to thank Francesco Nitti for a critical reading of an earlier draft \nversion of this manuscript. The work was supported by the US Department of Energy (DOE) under Grant No. DE-FG02-87ER40328 and by the Doctoral Dissertation Fellowship from the Graduate School at University of Minnesota.\n\n\n\n\\vspace{40pt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\@startsection{section}{1}%\n {\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}%\n {\\reset@font\\normalfont\\bfseries\\centering}}\n\\makeatother\n\n\\theoremstyle{remark}\n\\newtheorem{Remark}[equation]{Remark}\n\\newtheorem*{Claim}{Claim}\n\\newtheorem{Question}[equation]{Question}\n 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G}}\n\\newcommand{\\operatornamewithlimits{\\hbox{\\Large$\\boxtimes$}}}{\\operatornamewithlimits{\\hbox{\\Large$\\boxtimes$}}}\n\\newcommand{\\operatorname{Ad}}{\\operatorname{Ad}}\n\\newcommand{\\operatorname{Lie}}{\\operatorname{Lie}}\n\\newcommand{\\operatorname{reg}}{\\operatorname{reg}}\n\\newcommand{\\operatorname{Met}}{\\operatorname{Met}}\n\\newcommand{\\operatorname{Diff}}{\\operatorname{Diff}}\n\\newcommand{\\operatorname{BDiff}}{\\operatorname{BDiff}}\n\\newcommand{{\\mathcal S}}{{\\mathcal S}}\n\\newcommand{{\\mathcal N}}{{\\mathcal N}}\n\\newcommand{\\operatorname{Crit}}{\\operatorname{Crit}}\n\\newcommand{\\operatorname{id}}{\\operatorname{id}}\n\\newcommand{\\operatorname{p}}{\\operatorname{p}}\n\\newcommand{\\operatorname{SW}}{\\operatorname{SW}}\n\\newcommand{\\Z_p}{{\\mathbb Z}_p}\n\\newcommand{\\mathbb X}{\\mathbb X}\n\\begin{document}\n\\title[Smoothability of ${\\mathbb Z}\\times{\\mathbb Z}$-actions on 4-manifolds]{Smoothability of ${\\mathbb Z}\\times{\\mathbb Z}$-actions on 4-manifolds}\n\\author{Nobuhiro Nakamura}\n\\address{Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan}\n\\email{nobuhiro@ms.u-tokyo.ac.jp}\n\\keywords{group action, smoothability, Enriques surface }\n\\subjclass[2000]{Primary 57S05; Secondary: 57M60 57R57 }\n\\begin{abstract}\nWe construct a nonsmoothable ${\\mathbb Z}\\times{\\mathbb Z}$-action on the connected sum of an Enriques surface and $S^2\\times S^2$, such that each of the generators is smoothable.\nWe also construct a nonsmoothable self-homeomorphism on an Enriques surface.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\\label{sec:intro}\nThe purpose of this paper is to prove the existence of a nonsmoothable ${\\mathbb Z}\\times{\\mathbb Z}$-action on a $4$-manifold, such that each of the generators is smoothable: \n\\begin{Theorem}\\label{thm:main}\nLet $X$ be the connected sum of an Enriques surface with $S^2\\times S^2$.\nThen, there exists a pair $(f_1, f_2)$ of self-homeomorphisms of $X$ which has the following properties{\\textup :} \n\\begin{enumerate}\n\\item $f_1$ and $f_2$ commute.\n\\item Each one of $f_1$ and $f_2$ can be smoothed for some smooth structure on $X$.\nHowever, $f_1$ and $f_2$ can not be smoothed at the same time for any smooth structure on $X$.\n\\end{enumerate}\n\\end{Theorem}\nWe also construct a nonsmoothable self-homeomorphism of an Enriques surface.\n\\begin{Theorem}\\label{thm:Enriques}\nThere exists a self-homeomorphism of an Enriques surface $Y$ which is nonsmoothable with respect to any smooth structure on $Y$.\n\\end{Theorem}\nTo prove these results, we modify the argument in \\cite{Nf} which analyses the Seiberg-Witten moduli for families, and give more convenient constraints on diffeomorphisms of $4$-manifolds, and then, construct homeomorphisms which violate the constraints.\n\\begin{Acknowledgements}\nThe author would like to thank M.~Furuta, Y.~Kametani, K.~Kiyono and M.~Ue for helpful discussions and comments on earlier versions of the paper.\nIt is also his pleasure to thank the referee for detailed comments and many valuable suggestions which enable him to make the paper corrected and improved.\n\\end{Acknowledgements}\n\\section{Constraints on diffeomorphisms}\\label{sec:constraint}\nIn this section, we review the paper \\cite{Nf}, and give some modifications of its results. \nIn the paper \\cite{Nf}, the author investigated the Seiberg-Witten moduli of families of $4$-manifolds, and as an application, gave some constraints on diffeomorphisms of $4$-manifolds.\nLet $X$ be a closed oriented smooth $4$-manifold, and $B$ another closed manifold.\nWe assume a family $\\mathbb X$ of $X$ over $B$ is given as a fiber bundle over $B$ whose fibers are diffeomorphic to $X$ as oriented manifolds.\nThe fiber over $b\\in B$ is denoted by $X_b$.\nLet $T(\\mathbb X\/B)$ be the tangent bundle along the fiber of $\\mathbb X$, and assume a metric on $T(\\mathbb X\/B)$ is given.\nIn order to consider the Seiberg-Witten equations on the family $\\mathbb X$, we need a family of $\\Spin^{c}$-structures on $\\mathbb X$.\nOne can obtain such a family of $\\Spin^{c}$-structures if a $\\Spin^{c}$-structure on $T(\\mathbb X\/B)$ is given. \nFor this purpose, we gave somewhat complicated sufficient conditions. (See Proposition 2.1 of \\cite{Nf} and its correction \\cite{Nf-c}.) \nIn order to obtain a more convenient condition, we will take an alternative approach using classifying maps as described in \\cite{Sz}.\n\nLet $\\operatorname{Diff}(X)$ be the group of orientation-preserving diffeomorphisms of $X$.\nThe classifying space $B\\operatorname{Diff}(X)$ classifies families $\\mathbb X\\to B$ as above. \nSuppose a $\\Spin^{c}$-structure $c$ on $X$ is given.\nLet us consider the group ${{\\mathcal S}}(X,c)$ of pairs $(f,u)$, where $f$ is an orientation-preserving diffeomorphism and $u\\colon f^*c\\to c$ is an isomorphism.\nThe corresponding classifying space $B{{\\mathcal S}}(X,c)$ classifies families $\\mathbb X\\to B$ with a $\\Spin^{c}$-structure $\\tilde{c}$ on $T(\\mathbb X\/B)$ such that the restriction of $\\tilde{c}$ to each fiber is isomorphic to $c$.\nWe have the forgetful map $\\Phi\\colon {\\mathcal S}(X,c)\\to\\operatorname{Diff}(X)$. \nIn general, $\\Phi$ is not surjective. \nLet ${\\mathcal N}(X,c)$ be the image of $\\Phi$. Then there is an exact sequence\n$$\n1\\to \\G\\to {\\mathcal S}(X,c)\\to {\\mathcal N}(X,c)\\to 1,\n$$\nwhere $\\G=\\operatorname{Aut}(c)\\cong\\operatorname{Map}(X,S^1)$.\nNote that $B{\\mathcal N}(X,c)$ classifies families $\\mathbb X\\to B$ whose structure groups are included in ${\\mathcal N}(X,c)$.\nThe exact sequence leads to a fibration\n$$\nB\\G\\to B{\\mathcal S}(X,c)\\to B{\\mathcal N}(X,c).\n$$\nSuppose it is given a family $\\mathbb X\\to B$ classified by $\\rho\\colon B\\to B{\\mathcal N}(X,c)$ .\nIf $b_1(X)=0$, then $B\\G$ is homotopic to $\\operatorname{\\C P}^\\infty\\cong BS^1$.\nIn such a case, there is the sole obstruction to lift $\\rho\\colon B\\to B{\\mathcal N}(X,c)$ to $\\tilde{\\rho}\\colon B\\to B{\\mathcal S}(X,c)$ in $H^3(B;{\\mathbb Z})$.\nIn particular, if $\\mathop{\\text{\\rm dim}}\\nolimits B\\leq 2$, then every $\\rho\\colon B\\to B{\\mathcal N}(X,c)$ has a lift $\\tilde{\\rho}\\colon B\\to B{\\mathcal S}(X,c)$.\n\nTwo kinds of families whose structure groups are in ${\\mathcal N}(X,c)$ will be used in the proofs of propositions below.\nThe first is a mapping torus $X_f=(X\\times [0,1])\/f\\to S^1$ defined by a diffeomorphism $f\\colon X\\to X$ satisfying $f^*c\\cong c$.\nThe second is a ``double'' mapping torus $X_{(f_1,f_2)}\\to S^1\\times S^1$ defined by two commutative diffeomorphisms $f_1$ and $f_2$ satisfying $f_1^* c\\cong f_2^*c\\cong c$.\nIf the family $\\mathbb X$ is $X_f$ or $X_{(f_1,f_2)}$ as above, we always have a $\\Spin^{c}$-structure on $T(\\mathbb X\/B)$ by the previous paragraph.\n\nWhen a $\\Spin^{c}$-structure $\\tilde{c}$ on $T(\\mathbb X\/B)$ is given, the Seiberg-Witten moduli space for the family $\\mathbb X$ is given as follows.\nLet us define the bundle of parameters $\\Pi\\to B$ by\n$$\n\\Pi = \\{(g_b,\\mu_b)\\in\\operatorname{Met}(X_b)\\times\\Omega^2(X_b)\\,|\\,*_b\\mu_b=\\mu_b\\},\n$$\nwhere $\\operatorname{Met}(X_b)$ is the space of Riemannian metrics on $X_b$ and $*_b$ is the Hodge star for the metric $g_b$.\nIf we choose a section $\\eta$ of $\\Pi$, then the moduli space for the family $(\\mathbb X,\\tilde{c})$ is defined by\n$$\n\\M(\\mathbb X,\\tilde{c},\\eta)=\\coprod_{b\\in B}\\M(X_b,c_b,\\eta_b),\n$$\nwhere $\\M(X_b,c_b,\\eta_b)$ is the Seiberg-Witten moduli space of the fiber $X_b$ with the $\\Spin^{c}$-structure $c_b=\\tilde{c}|_{X_b}$ for the parameter $\\eta_b=(g_b,\\mu_b)$. \n\nWith these understood, we can modify the results in \\cite{Nf} as follows.\nFor a $\\Spin^{c}$-structure $c$ on $X$, let $L$ be the determinant line bundle of $c$.\nThen the virtual dimension $d(c)$ of the Seiberg-Witten moduli of $(X,c)$ is given by,\n$$\nd(c) =\\frac14(c_1(L)^2-\\operatorname{sign}(X)) - (1+b_+).\n$$\n\\begin{Proposition}\\label{prop:prop1}\nLet $X$ be a closed oriented smooth $4$-manifold with $b_1=0$ and $b_+=1$, $c$ a $\\Spin^{c}$-structure on $X$ with $d(c)=0$, and $f\\colon X\\to X$ an orientation-preserving diffeomorphism. \nIf $f^*c$ is isomorphic to $c$, then $f$ preserves the orientation of $H^+(X;{\\mathbb R})$.\n\\end{Proposition}\n\nThe proof of \\propref{prop:prop1} is given by a slight modification of the proof of Theorem 1.2 of \\cite{Nf}. \nFor reader's convenience, we outline it briefly.\nSuppose a diffeomorphism $f$ satisfying $f^*c\\cong c$ is given, and consider the mapping torus $X_f\\to B=S^1$ by $f$.\nUnder the assumptions of \\propref{prop:prop1}, the moduli space $\\M(X_f,\\tilde{c},\\eta)$ of $X_f$ for a generic choice of $\\eta$ is a compact $1$-dimensional manifold whose boundary points consist of reducibles.\nLet us introduce a vector bundle $H^+_\\eta\\to B$ by $H^+_\\eta = \\coprod_{b\\in B} H^+_{g_b}$, where $H^+_{g_b}$ is the space of $g_b$-self-dual harmonic $2$-forms of $X_b$. \n(Such a bundle $H_\\eta$ is defined for any family.)\nThen, it is proved the number of reducibles is equal modulo $2$ to the number of zeros of a generic section of $H^+_\\eta$.\nIf $f$ reverses the orientation of $H^+(X;{\\mathbb R})$, then $H^+_\\eta$ is a nontrivial real line bundle over $S^1$.\nThis is a contradiction, because the number of boundary points of a compact $1$-dimensional manifold is even.\n\nSimilarly, we can prove the following by modifying the proof of Theorem 1.1 of \\cite{Nf}:\n\\begin{Proposition}\\label{prop:prop2}\nLet $X$ be a closed oriented smooth $4$-manifold with $b_1=0$ and $b_+=2$, and $c$ a $\\Spin^{c}$-structure on $X$ with $d(c)=-1$.\nSuppose a pair $(f_1,f_2)$ of orientation preserving diffeomorphisms on $X$ satisfies the following conditions{\\textup :}\n\\begin{enumerate}\n\\item $f_1$ and $f_2$ commute. \n\\item $f_1$ and $f_2$ preserve the isomorphism class of $c$.\n\\end{enumerate}\nThen, $w_2\\left(H^+_{\\eta}\\right)=0$, where $H^+_{\\eta}$ is the bundle associated to $X_{(f_1,f_2)}$.\n\\end{Proposition}\n\\section{Nonsmoothable self-homeomorphism on Enriques surface}\\label{sec:Enriques}\nThe purpose of this section is to prove \\thmref{thm:Enriques}.\nFirst, note that the Enriques surface can be decomposed into three connected summands {\\it topologically} by a theorem due to Hambleton and Kreck\\cite{HK}.\nIn fact, the following theorem can be proved from Theorem 3 in \\cite{HK} and its proof.\n\\begin{Theorem}[Hambleton-Kreck~\\cite{HK}]\\label{thm:HK}\nThe Enriques surface is homeomorphic to a topological manifold $Y=|E_8|\\# \\Sigma\\#(S^2\\times S^2)$, where $|E_8|$ is the ``$E_8$-manifold'', i.e., the simply-connected closed topological $4$-manifold whose intersection form is the negative definite $E_8$, and $\\Sigma$ is a non-spin rational homology $4$-sphere with fundamental group ${\\mathbb Z}\/2$.\n\\end{Theorem}\n\\begin{Remark}\nNeither $\\Sigma$ nor $|E_8|\\#(S^2\\times S^2)$ is smoothable, because both have nontrivial Kirby-Siebenmann invariants.\n\\end{Remark}\nNow, we will construct a self-homeomorphism of $Y$.\nLet $\\varphi\\colon S^2\\times S^2\\to S^2\\times S^2$ be an orientation-preserving diffeomorphism which has the following properties:\n\\begin{enumerate}\n\\item There is a $4$-ball $B_0\\subset S^2\\times S^2$ such that the restriction of $\\varphi$ to $B_0$ is the identity map on $B_0$. \n\\item $\\varphi$ reverses the orientation of $H^+(S^2\\times S^2;{\\mathbb R})$.\n\\end{enumerate}\nSuch a $\\varphi$ can be easily constructed as follows:\n\\begin{Example}\\label{ex:involution}\nAssume $S^2\\times S^2 = \\operatorname{\\C P}^1\\times\\operatorname{\\C P}^1$. Let $\\varphi_0$ be the automorphism on $\\operatorname{\\C P}^1\\times\\operatorname{\\C P}^1$ defined by the complex conjugation.\nChoose a fixed point $p_0$ of $\\varphi_0$.\nThen, a required $\\varphi$ is obtained by perturbing $\\varphi_0$ around $p_0$ to be the identity on a neighborhood of $p_0$.\n\\end{Example}\n\nLet us define a self-homeomorphism $f$ on $Y$ by $f=\\operatorname{id}_{|E_8|\\#\\Sigma}\\#\\varphi$, where $\\operatorname{id}_{|E_8|\\#\\Sigma}$ is the identity map of $|E_8|\\#\\Sigma$.\n(Note that we can take a connected sum of $\\varphi$ with $\\operatorname{id}_{|E_8|\\#\\Sigma}$ on $B_0\\subset S^2\\times S^2$.)\nNow, we claim that $f$ is nonsmoothable with respect to any smooth structure on $Y$.\n\nTo prove $f$ nonsmoothable, we will temporarily need a {\\it topological} $\\,\\Spin^{c}$-structure on the topological manifold $Y$.\nLet us make a digression for it.\n(A brief description for topological spin structures is found in \\cite{Edmonds}, Section 3. See also \\cite{FQ}, 10.2B.)\nBy Kister-Mazur's theorem, the tangent microbundle $\\tau Y$ determines up to isomorphism the topological ``frame'' bundle $F$ whose structure group $\\operatorname{\\rm STop}(4)$ consists of orientation-preserving homeomorphisms of ${\\mathbb R}^4$ preserving the origin. \nIt is known that the inclusion $\\operatorname{\\rm SO}(4)\\to \\operatorname{\\rm STop}(4)$ induces an isomorphism on $\\pi_1$ and both have trivial $\\pi_0$ and $\\pi_2$ (\\cite{KS},V and \\cite{FQ}, 8.7).\nLet $\\phi \\colon \\operatorname{\\rm SpinTop}(4)\\to \\operatorname{\\rm STop}(4)$ be the unique double covering.\nThen, a topological spin structure on $Y$ is defined as a double covering $\\tilde{F}\\to F$ whose restriction to each fiber is $\\phi$. \nTopological $\\Spin^{c}$-structures are similarly defined by using $\\operatorname{\\rm SpinTop}^c(4) :=\\operatorname{\\rm SpinTop}(4)\\times_{{\\mathbb Z}_2}\\operatorname{\\rm U}(1)\\to \\operatorname{\\rm STop}(4)$.\nThe set of isomorphic classes of topological $\\Spin^{c}$-structures has a principal action of $H^2(Y;{\\mathbb Z})$ as in the case of true $\\Spin^{c}$-structures.\n\\begin{Lemma}\\label{lem:pres}\nLet $c$ be the topological $\\Spin^{c}$-structure on $Y$ whose $c_1(L)$ is a torsion class.\nThen $f^*c$ is isomorphic to $c$.\n\\end{Lemma}\n\\proof\nIn this proof, all spin\/$\\Spin^{c}$-structures are understood as topological ones. \nThe $\\Spin^{c}$-structure $c$ can be identified with the sum of the unique spin structure $c_0$ on $|E_8|\\# (S^2\\times S^2)$ and a $\\Spin^{c}$-structure $c_\\Sigma$ on $\\Sigma$ whose $c_1(L)$ is a torsion class.\nSince $f$ is the identity on $\\Sigma$, $f$ preserves $c_\\Sigma$.\nOn the other hand, since $c_0$ is the unique spin structure on $|E_8|\\# (S^2\\times S^2)$, $f^*c_0\\cong c_0$.\n\\endproof\nLet us prove $f$ nonsmoothable.\nOnce a smooth structure on $Y$ is given, we have a reduction of the topological frame bundle $F$ to the true frame $\\operatorname{\\rm SO}(4)$-bundle, and also a topological $\\Spin^{c}$-structure is reduced to the corresponding true $\\Spin^{c}$-structure.\nSuppose $f$ is smoothed. \nBy \\lemref{lem:pres}, $f^*c$ is isomorphic to $c$ as true $\\Spin^{c}$-structures.\nOn the other hand, $f$ is an orientation-preserving diffeomorphism which reverses the orientation of $H^+(Y)$. \nThis contradicts \\propref{prop:prop1}.\n\\section{Proof of \\thmref{thm:main}}\\label{sec:comm}\nIn this section, we prove \\thmref{thm:main}.\nTo begin with, we collect the ingredients needed for our construction.\nLet $S_0=S^2\\times S^2$, and fix a $4$-ball $B_0^\\prime\\subset S_0$.\nFor $i=1,2$, let $(S_i,\\varphi_i)$ be copies of $(S^2\\times S^2, \\varphi)$, and fix smooth $4$-balls $B_i^\\prime\\subset S_i$ on which $\\varphi_i|_{B_i^\\prime}$ are the identity maps.\nIf we make a connected sum of $S_i$ ($i=0,1,2$) with another manifold, remove $B_i^\\prime$ from $S_i$ and glue it along the boundary to another.\nLet $Z$ be $|E_8|\\# \\Sigma$. \nLater, we will choose $4$-balls $B_0$, $B_1$ and $B_2$ in $Z$ so that\n\\begin{itemize}\n\\item $B_1\\cap B_0=\\emptyset$, $B_1\\cap B_2=\\emptyset$, and\n\\item if we make a connected sum of $Z$ with $S_i$ ($i=0,1,2$), remove $B_i$ from $Z$ and glue $\\overline{Z\\setminus B_i}$ and $\\overline{S_i\\setminus B_i^\\prime}$. \n(The resulting connected sum will be denoted as $Z\\#_{\\partial B_i} S_i$.)\n\\end{itemize}\nLet $E_1$ and $E_2$ be smooth $4$-manifolds homeomorphic to an Enriques surface.\nThe basic idea of our construction is as follows.\nThe connected sum $S_1\\#_{\\partial B_1}Z\\#_{\\partial B_2}S_2$ can be assumed as a connected sum of an Enriques surface with $S^2\\times S^2$ in two ways: $S_1\\#E_1$ and $E_2\\#S_2$.\nThen, commutative two homeomorphisms $f_1$, $f_2$ will be defined by $\\varphi_1\\#\\operatorname{id}_{E_1}$ and $\\operatorname{id}_{E_2}\\#\\varphi_2$,\n\nLet us begin the precise construction.\nChoose a $4$-ball $B_0\\subset Z$ arbitrarily. \nThen $Z\\#_{\\partial B_0}S_0$ is homeomorphic to an Enriques surface.\nFix a homeomorphism $\\bar{h}_1\\colon E_1\\to Z\\#_{\\partial B_0}S_0$.\nNext, choose $B_1$ so that $D_1:=\\bar{h}_1^{-1}(B_1)$ is a {\\it smoothly embedded $4$-ball} in $E_1$. \nTake a smooth connected sum $S_1\\#_{\\partial D_1}E_1$ and a (topological) connected sum $S_1\\#_{\\partial B_1}Z\\#_{\\partial B_0}S_0$ so that a homeomorphism $h_1=\\operatorname{id}_{S_1}\\#\\bar{h}_1\\colon S_1\\#_{\\partial D_1}E_1\\to S_1\\#_{\\partial B_1}Z\\#_{\\partial B_0}S_0$ is defined. \n\nNote that $S_1\\#_{\\partial B_1}Z$ is also homeomorphic to an Enriques surface.\nFix a homeomorphism $\\bar{h}_2\\colon E_2\\to S_1\\#_{\\partial B_1}Z$.\nChoose $B_2$ so that $D_2:=\\bar{h}_2^{-1}(B_2)$ is a {\\it smoothly embedded $4$-ball} in $E_2$. \nTake a smooth connected sum $E_2\\#_{\\partial D_2}S_2$ and a (topological) connected sum $S_1\\#_{\\partial B_1}Z\\#_{\\partial B_2}S_2$ so that a homeomorphism $h_2=\\bar{h}_2\\#\\operatorname{id}_{S_2}\\colon E_2\\#_{\\partial D_2}S_2\\to S_1\\#_{\\partial B_1}Z\\#_{\\partial B_2}S_2$ is defined.\n\n\nDefine the self-diffeomorphism $\\bar{f}_1$ on $S_1\\#_{\\partial D_1}E_1$ by $\\bar{f}_1=\\varphi_1\\#\\operatorname{id}_{E_1}$, and $\\bar{f}_2$ on $E_2\\#_{\\partial D_2} S_2$ by $\\bar{f}_2=\\operatorname{id}_{E_2}\\#\\varphi_2$.\nChoose a homeomorphism $h\\colon S_1\\#_{\\partial B_1}Z\\#_{\\partial B_2}S_2 \\to S_1\\#_{\\partial B_1}Z\\#_{\\partial B_0}S_0$ so that $h|_{S_1\\setminus B_1^\\prime}$ is the identity map.\nVia homeomorphisms $h$, $h_1$ and $h_2$, we obtain self-homeomorphisms $f_1$ and $f_2$ of $X:=S_1\\#_{\\partial B_1}Z\\#_{\\partial B_2}S_2$ induced from $\\bar{f}_1$ and $\\bar{f}_2$, respectively.\nThen each $f_i$ ($i=1,2$) is smoothable for the smooth structure $E_i\\#_{\\partial D_i} S_i$.\nClearly, $f_1$ and $f_2$ commute. \nLet $c$ be the $\\Spin^{c}$-structure on $X$ whose $c_1(L)$ is a torsion class.\nAs in \\lemref{lem:pres}, we can see that $f_1$ and $f_2$ preserve the isomorphism class of $c$.\nHowever, $w_2\\left(H^+_{\\eta}\\right)\\neq 0$ by construction.\nBy \\propref{prop:prop2}, $f_1$ and $f_2$ can not be smoothed at the same time.\nThus, \\thmref{thm:main} is proved.\n\\section{Remarks}\\label{sec:remarks}\nWe give two remarks.\nThe first is on another possibility of application of \\propref{prop:prop2}.\nThe following problem would be interesting: {\\it Find two diffeomorphisms of a smooth manifold homeomorphic to a connected sum of an Enriques surface $E$ with $S^2\\times S^2$ that are simultaneously smoothable, commute up to isotopy, but do not have representatives in their isotopy classes that commute. }\nIf we want to construct such two diffeomorphisms on the {\\it smooth} connected sum $E\\#S^2\\times S^2$, then one of the difficulties would be as follows. \nTo appeal to \\propref{prop:prop2}, one of two diffeomorphisms will be required to reverse the orientation of the $H^+(E)$-part of $H^2(E\\#S^2\\times S^2)$, and it will be easy if we can construct such a diffeomorphism as a connected sum of a diffeomorphism $f$ of $E$ with one of $S^2\\times S^2$. \nHowever, this method is impossible, because \\propref{prop:prop1} prohibits such an $f$.\n\nThe second remark is on a generalization of the construction of the moduli spaces for families. \nIn fact, we can construct the moduli space for a family without a family of $\\Spin^{c}$-structures.\nMore precisely, we claim the following: {\\it When a family $\\mathbb X\\to B$ is classified by $\\rho \\colon B\\to B{\\mathcal N}(X,c)$, we can always construct the moduli space $\\M(\\mathbb X,c)$ for the family $\\mathbb X$, even if $\\rho$ does not have a lift $\\tilde{\\rho}\\colon B\\to B{\\mathcal S}(X,c)$.}\nThe construction is outlined as follows.\nBy taking local trivializations, the family $\\mathbb X$ can be given via transition functions $\\psi_{\\beta\\alpha}\\colon U_\\alpha\\cap U_\\beta \\to {\\mathcal N}(X,c)$ for an appropriate covering $\\{U_\\lambda\\}_{\\lambda\\in\\Lambda}$ of $X$.\nSuppose the intersection of every two members in $\\{U_\\lambda\\}_{\\lambda\\in\\Lambda}$ is contractible.\nThen we can take a lift of each $\\psi_{\\beta\\alpha}\\colon U_\\alpha\\cap U_\\beta \\to {\\mathcal N}(X,c)$ to $\\tilde{\\psi}_{\\beta\\alpha}\\colon U_\\alpha\\cap U_\\beta \\to {\\mathcal S}(X,c)$. In general, such $\\tilde{\\psi}_{\\beta\\alpha}$ do not satisfy the cocycle condition, but satisfy it {\\it up to gauge}, i.e., $\\psi_{\\gamma\\beta}\\psi_{\\beta\\alpha}\\psi_{\\gamma\\alpha}^{-1}$ is a gauge transformation.\nOne can define local families $\\M(U_\\lambda\\times X,c)=\\coprod_{b\\in U_\\lambda}\\M(\\{b\\}\\times X,c)\\to U_\\lambda$ of moduli spaces and attaching maps $\\tilde{\\psi}_{\\beta\\alpha}^*$ between them induced from $\\tilde{\\psi}_{\\beta\\alpha}$. \n(Here, we need a little care on metrics and perturbations.)\nSince the moduli spaces are defined as the quotient spaces divided by the gauge transformations, $\\tilde{\\psi}_{\\beta\\alpha}^*$ satisfy the cocycle condition.\nTherefore, the global family $\\M(\\mathbb X,c)$ can be constructed from the local families $\\M(U_\\lambda\\times X,c)$ via $\\tilde{\\psi}_{\\beta\\alpha}^*$.\nSuch a family $\\M(\\mathbb X,c)$ would be useful for further applications. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nDuring the last years a gradually increasing number of O, early B-type, and WR stars have been investigated \nfor magnetic fields, and as a result, about a dozen magnetic O-type stars \nare presently known (e.g., Hubrig et al.\\ \\cite{Hubrig2008}; Martins et al.\\ \\cite{Martins2010};\nHubrig et al.\\ \\cite{Hubrig2011a}).\nThe recent detections of magnetic fields in \nmassive stars generate a strong motivation to study the correlations between \nevolutionary state, rotation velocity, and surface composition, and to\nunderstand the origin and the role of magnetic fields in massive stars.\n\nThe star $\\zeta$\\,Ophiuchi (=HD\\,149757) of spectral type O9.5V is a\nwell-known rapidly rotating runaway star with extremely interesting\ncharacteristics. It undergoes episodic mass loss seen as emission in\nH$\\alpha$, and it is possible that it rotates with almost break-up\nvelocity with $v$\\,sin\\,$i=400$\\,km\\,s$^{-1}$ (Kambe et al.\\\n\\cite{Kambe1993}). Various studies indicate different types of spectral\nand photometric variability. The UV resonance lines show multiple\ndiscrete absorption components (DAC) in the UV (e.g.\\ Howarth et al.\\\n\\cite{Howarth1984}) and strong line profile variations in optical\nspectra reconciled with traveling sectorial modes of high degree (e.g.\\\nReid et al.\\ \\cite{Reid1993}). Highly precise {\\em MOST}\n(Microvariability and Oscillations of Stars) satellite photometry in\n2004 has yielded at least a dozen significant oscillation frequencies\nbetween 1 and 10 cycles\/day, hinting at a behaviour similar to \n$\\beta$~Cephei-type stars (Walker et al.\\ \\cite{Walker2005}). No\nunambiguous rotation period could be identified in spectroscopic and \nphotometric observations, although Balona \\& Kambe\n(\\cite{BalonaKambe1999})\nfavored a period in the region of 1 cycle\/day.\n\n$\\zeta$\\,Oph is also well-known for its variability in the X-ray band. \nOskinova \\hbox{et~al.\\\/}\\ (\\cite{Oskinova2001}) studied the {\\em ASCA} observations of\n\\mbox{$\\zeta$\\,Oph}\\ that covered just more than the expected rotation period of the\nstar. A clearly detected periodic X-ray flux variability with $\\sim$20\\%\\\namplitude in the {\\em ASCA} passband (0.5-10\\,keV) was reported. A\nperiod of 0.77\\,d was detected and a possible connection with the\nrecurrence time (0.875\\,d$\\pm$0.167\\,d) of the DACs\nin UV spectra of the star was discussed. The DACs in\nthe spectra of O stars are commonly explained by \nlarge-scale structures in the stellar wind, modulated by\nrotation and possibly related to a surface magnetic field \n(Cranmer \\& Owocki \\cite{CranmerOwocki1996}).\nWaldron \\hbox{et~al.\\\/}\\ (in preparation, private communication) found that\n{\\em SUZAKU} data on \\mbox{$\\zeta$\\,Oph}\\ suggest a period of\n$\\sim$0.98\\,d that is consistent but slightly larger than\nthe X-ray periodicities found in {\\em ASCA} data (Oskinova \\hbox{et~al.\\\/}\\ \\cite{Oskinova2001})\nand in {\\em Chandra} HETGS data (Waldron \\cite{Waldron2005}). In addition,\nthe HETGS data appear to indicate an additional cyclic period of\n$\\sim$0.33\\,d in the hard X-ray band ($>$1.2\\,keV). \n\nThe results of our previous studies seem to indicate that the presence\nof a magnetic field is more frequently detected in candidate runaway\nstars than in stars belonging to clusters or associations (Hubrig et\nal.\\ \\cite{Hubrig2011b}; Hubrig et al.\\ \\cite{Hubrig2011a}). The\ncurrently best available astrometric, spectroscopic, and photometric\ndata were used to calculate the kinematical status of magnetic O-type\nstars with previously unknown space velocities. The results suggest\nthat most of the magnetic O-type stars can be considered as candidate\nrunaway stars.\n\nThe available observational material suggests that $\\zeta$\\,Oph is a\nmain sequence single star in the field with runaway characteristics.\nUsually, to explain the origin of massive stars in the field, two\nmechanisms are discussed in the literature. In one scenario, close\nmultibody interactions in a dense cluster cause one or more stars to\nbe scattered out of the region (e.g.\\ Leonard \\& Duncan\n\\cite{LeonardDuncan1990}). For this mechanism, runaways are ejected in\ndynamical three- or four-body interactions. An alternative mechanism\ninvolves a supernova (SN) explosion within a close binary, ejecting\nthe secondary due to the conservation of momentum (Zwicky\n\\cite{Zwicky1957}; Blaauw \\cite{Blaauw1961}). Blaauw\n(\\cite{Blaauw1952}) suggested the origin of $\\zeta$\\,Oph in the\nScorpius OB2 association due to its proper motion vector, which points\naway from the association. More recently, Hoogerwerf et\nal.\\ (\\cite{Hoogerwerf2001}) suggested that the star gained it space\nvelocity of $\\sim$30\\,km\\,s$^{-1}$ in a supernova explosion within a\nclose binary in Upper Scorpius about 1--2\\,Myr ago. The authors\nidentified PSR~B1929+10 as an associated pulsar with a characteristic\nage of $\\sim$3\\,Myr, consistent with the kinematic age of $\\zeta$\\,Oph\nwithin the uncertainties. Tetzlaff et al.\\ (\\cite{Tetzlaff2010})\nreinvestigated the scenario of a binary SN in Upper Scorpius involving\n$\\zeta$\\,Oph and PSR~B1929+10 and concluded that it is very likely\nthat both objects were ejected during the same supernova event. In\ntheir work, the considered association age range implies that the\nprogenitor star of the produced neutron star had a spectral type\nbetween O6\/O7 and O9 with a mass range from 18 to 37\\,\\mbox{$M_\\odot$}{}. The\nX-ray emission of the pulsar seems to be dominated by non-thermal\nradiation processes (e.g.\\ Becker et al.\\ \\cite{Becker2006}). An\narc-like nebula surrounding PSR~B1929+10 and extending up to\n10\\arcsec{} was identified in {\\em Chandra} data and interpreted as a\nbow-shock nebula (Hui \\& Becker \\cite{HuiBecker2008}).\nThe estimated magnetic field strength in the shocked\nregion accounts for $\\sim$75\\,$\\mu$G, while the typical magnetic field\nstrength in the ISM is about 2--6\\,$\\mu$G.\n\nThe presence of a bow-shock nebula has also been detected for $\\zeta$\\,Oph.\nFigure~\\ref{fig:bsh} shows an image based on archival {\\em Spitzer}\nIRAC maps (AOR 17774848). \nRecently, Kobulnicky \\hbox{et~al.\\\/}\\ (\\cite{Kobulnicky2010})\nanalyzed a sample of bow shocks around massive stars in Cygnus-X.\nThey used the analytical description of momentum-driven bow shocks and\ndust\/polycyclic aromatic hydrocarbon emission models to estimate\nstellar mass loss rates from the observed properties of the bow shocks. \nIt was found that mass-loss rates in the range between\n$10^{-7}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}\\ and a few times $10^{-6}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}\\ are required to\ngenerate the bow shocks around typical B2V - O5V type stars. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{zoph-irac.2.ps}\n\\caption{(online colour at: www.an-journal.org) Combined IR {\\em Spitzer} IRAC (3.6\\,\\mbox{$\\mu$m}\\ blue,\n 4.5\\,\\mbox{$\\mu$m}\\ green, 8.0\\,\\mbox{$\\mu$m}\\ red) image of the bow shock around the\n runaway star \\mbox{$\\zeta$\\,Oph}. Archival data have been used.\n Galactic coordinates are shown. The image size is $\\sim 36'\n \\times 31'$. }\n\\label{fig:bsh}\n\\end{figure}\n\nThe mass-loss rate \\mbox{$\\dot{M}$}\\ of this star was empirically obtained from\ndifferent diagnostics by a number of authors. Repolust\n\\hbox{et~al.\\\/}\\ (\\cite{Repolust2004}) fitted the H$\\alpha$ photospheric\n absorption line and derived the upper limit on the \\mbox{$\\zeta$\\,Oph}\\ mass-loss rate\n as $1.8\\times 10^{-7}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}. Fullerton \\hbox{et~al.\\\/}\\ (\\cite{Fullerton2006})\n determined the radio-based mass-loss rate of \\mbox{$\\zeta$\\,Oph}\\ as $1.1\\times\n 10^{-7}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}. The mass-loss rates determined from radio depend on\n the square of the density since the physical mechanism responsible for\n the radio emission is free-free emission. On the other hand,\n Fullerton \\hbox{et~al.\\\/}\\ derive a much smaller mass-loss rate from fitting the\n UV P\\,{\\sc v} resonance doublet, the product of the mass-loss rate\n and the ion fraction of P$^{+4}$ being only $\\dot{M}q({\\rm P^{+4}})\n \\raisebox{-.4ex}{$\\stackrel{<}{\\scriptstyle \\sim}$} 1.3 \\times 10^{-10}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}\\ with $q({\\rm P^{+4}})\\raisebox{-.4ex}{$\\stackrel{<}{\\scriptstyle \\sim}$} 1$. The\n mass-loss rates derived from fitting the wind profiles of UV\n resonance lines depend linearly on the density. To resolve this\n discordance in mass-loss determinations based on $\\rho^2$- and\n $\\rho$-diagnostics, Fullerton \\hbox{et~al.\\\/}\\ suggest that the winds are strongly\n clumped with a volume filling factor of $\\sim$10$^{-3}$--10$^{-5}$.\n Marcolino \\hbox{et~al.\\\/}\\ (\\cite{Marcolino2009}) analyzed optical and UV\n spectra of \\mbox{$\\zeta$\\,Oph}\\ among their sample of O-type dwarfs. They derive\n an upper limit on the mass-loss rate of \\mbox{$\\zeta$\\,Oph}\\ as $1.6 \\times\n 10^{-9}$\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}\\ if the wind was smooth. This value agrees with the\n $\\dot{M}q({\\rm P^{+4}})$ value obtained by Fullerton\n \\hbox{et~al.\\\/}\\ (\\cite{Fullerton2006}).\n\nUsing the example of the O-type supergiant $\\zeta$\\,Puppis, Oskinova\n\\hbox{et~al.\\\/}\\ (\\cite{Oskinova2007}) demonstrated that the discordance of\nmass-loss rates found by Fullerton \\hbox{et~al.\\\/}\\ can be overcome by\naccounting for stellar wind porosity (see also Sundqvist\n\\hbox{et~al.\\\/}\\ \\cite{Sundqvist2010}). It was found for the O5Ia star\n$\\zeta$\\,Puppis that only a moderate reduction of the mass-loss rate\nby a factor of 2--3 (compared to the smooth wind models) is required\nto reproduce both H$\\alpha$ and P\\,{\\sc v} lines. If this result\n holds also for non-supergiant O type stars, the mass-loss rate of\n \\mbox{$\\zeta$\\,Oph}\\ is only a few times lower compared to the radio-based\n mass-loss determined by Fullerton \\hbox{et~al.\\\/}, i.e.\\\n$\\sim$10$^{-7}$\\,\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}. Importantly, this mass-loss rate is in agreement with values \nthat are required to produce bow shocks around O stars \n(Kobulnicky \\hbox{et~al.\\\/}\\ \\cite{Kobulnicky2010}). \n\nAn additional aspect, which may hint at the presence of a magnetic\nfield in runaway stars, is that a number of individual abundance\nstudies indicate nitrogen enrichment in the atmospheres of runaway\nstars (e.g.\\ Boyajian et al.\\ \\cite{Boyajian2005}). Nitrogen\nenrichment was found in $\\zeta$\\,Oph by Villamariz \\& Herrero\n(\\cite{VillamarizHerrero2005}). Recent NLTE abundance analyses (e.g.,\nMorel et al.\\ \\cite{Morel2008}; Hunter et al.\\ \\cite{Hunter2008})\nsuggest that slow rotators have peculiar chemical enrichment such as\nnitrogen excess or boron depletion, and these peculiarities are linked\nto the presence of a magnetic field.\nOn the other hand, Hubrig et al.\\ (\\cite{Hubrig2011c}) showed that\nsome magnetic massive stars previously assumed to be slow rotators,\nare in fact fast rotators, but are viewed close to their rotation\npoles.\n\nTo test the magnetic nature of this particularly interesting rapidly\nrotating runaway star, we acquired spectropolarimetric observations\nwith the low-resolution spectropolarimeter FORS\\,1 at the VLT. In this\nwork we report the first detection of a magnetic field in this star.\n\n\n\n\\section{Magnetic field measurements}\n\\label{magnetic_field}\n\nSpectropolarimetric observations with FORS\\,1 (Appenzeller et al.\\ \\cite{Appenzeller1998})\nwere obtained on 2008 May 23\n(MJD\\,54609.34) using grism 600B and a slit width of 0$\\farcs$4 to achieve a spectral resolving power \nof $R\\approx2000$.\nThe use of the mosaic detector made of \nblue optimized E2V chips and a pixel size of 15\\,$\\mu$m allowed us to cover a large\nspectral range, from 3250 to 6215\\,\\AA{}, which includes all hydrogen Balmer lines \nfrom H$\\beta$ to the Balmer jump.\nSix continuous series of two exposures with durations between 0.3 and 3\\,sec were taken\nat two retarder waveplate setups ($\\alpha=+45^\\circ$ and $-$45$^\\circ$). For all but the first exposure pairs\nwe achieved a signal-to-noise ratio between 1000 and 1500.\nMore details on the observing technique with FORS\\,1 can be \nfound elsewhere (e.g., \nHubrig et al.\\ \\cite{Hubrig2004a}, \\cite{Hubrig2004b}, and references therein).\nThe mean longitudinal \nmagnetic field, $\\left< B_{\\rm z}\\right>$, was derived using \n\n\\begin{equation} \n\\frac{V}{I} = -\\frac{g_{\\rm eff} e \\lambda^2}{4\\pi{}m_ec^2}\\ \\frac{1}{I}\\ \n\\frac{{\\rm d}I}{{\\rm d}\\lambda} \\left, \n\\label{eqn:one} \n\\end{equation} \n\n\\noindent \nwhere $V$ is the Stokes parameter that measures the circular polarisation, $I$ \nis the intensity in the unpolarised spectrum, $g_{\\rm eff}$ is the effective \nLand\\'e factor, $e$ is the electron charge, $\\lambda$ is the wavelength, $m_e$ the \nelectron mass, $c$ the speed of light, ${{\\rm d}I\/{\\rm d}\\lambda}$ is the \nderivative of Stokes $I$, and $\\left$ is the mean longitudinal magnetic \nfield. \nThe longitudinal magnetic field was measured in two ways: using only the absorption hydrogen Balmer \nlines or using the entire spectrum including all available absorption lines.\nWe obtain for the mean longitudinal magnetic field using all available absorption lines \n$\\left< B_z\\right>_{\\rm all}= 141\\pm45$\\,G and for the mean longitudinal magnetic field using \nthe hydrogen Balmer lines $\\left< B_z\\right>_{\\rm hyd}= 123\\pm54$\\,G.\nOur detection using the entire spectrum has a significance of \n3.1$\\sigma$, determined from the formal uncertainties we derive. \nIn the Stokes $V$ spectra distinct Zeeman signatures are well visible at the position \nof hydrogen and metal lines. In Fig.~\\ref{fig:a} we display Stokes $I$ and $V$ spectra\nin the spectral regions around H$\\beta$ and the Na~I doublet. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{hd149b.eps}\n\\includegraphics[width=0.45\\textwidth]{hd149a.eps}\n\\caption{\nStokes $I$ and $V$ spectra of $\\zeta$\\,Oph in the spectral regions around H$\\beta$ (top) and the Na~I doublet\n(bottom). \n}\n\\label{fig:a}\n\\end{figure}\n\nIn Fig.~\\ref{fig:x} we present time series of Stokes $I$ spectra corresponding to \nour data set of six sub-exposures in the region around He~II 4686\\,\\AA{} and He~I 4713\\,\\AA{}.\nAlthough the time lapse between the observations of the first pair and the last pair is only 13 minutes, \nsome small line profile variations in the He I line are already detectable at such short time scales. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{hd149x.eps}\n\\caption{\nSix Stokes $I$ spectra corresponding to six sub-exposures in the region around He~II 4686\\,\\AA{}\nand He~I 4713\\,\\AA{}. Small line profile variations in the He~I line are already detectable on the time\nscale of only 13\\,min.\n}\n\\label{fig:x}\n\\end{figure}\n\nThe only other measurements of the magnetic field in $\\zeta$\\,Oph have been presented by Schnerr et al.\\\n(\\cite{Schnerr2008}) who used the MuSiCoS spectropolarimeter to derive Stokes $I$ and $V$ spectra \nwith the Least Square Deconvolution method. No longitudinal magnetic field was detected in this star\nat a level more than 3$\\sigma$. However, the measurement errors in their observations were in the range\nfrom 700\\,G to more than 3\\,kG. \n\nMassive stars usually end their evolution with a final supernova explosion, producing neutron stars \nor black holes. The initial masses of these stars range from $\\sim$8--10\\,\\mbox{$M_\\odot$}{} to \n100\\,\\mbox{$M_\\odot$}{} or more, which correspond to spectral types earlier than about B2. \nContrary to the case of Sun-like stars, the\nmagnetic fields of stars on the upper main sequence (Ap\/Bp stars),\nwhite dwarfs, and neutron stars are dominated by large spatial\nscales and do not change on yearly time scales. In each of these\nclasses there is a wide distribution of magnetic field strengths,\nbut the distribution of magnetic fluxes appears to be similar in\neach class, with maxima of $\\Phi_\\mathrm{max}=\\pi R^2B\\sim 10^{27-28}\\mathrm{G~cm^2}$\n(Reisenegger \\cite{Reisenegger2001};\nFerrario \\& Wickramasinghe \\cite{FerrarioWickramasinghe2005}), arguing for a fossil field\nwhose flux is conserved along the path of stellar evolution.\nAccording to Reisenegger (\\cite{Reisenegger2009}) the magnetic fluxes have possibly been generated on or \neven before the main-sequence\nstage and then inherited by the compact remnants. \n\nThe magnetic field strength of the pulsar PSR~B1929+10 is\n0.5129$\\times$10$^{12}$\\,Gauss.\nAssuming simple conservation of magnetic flux we obtain\na field strength of just a few Gauss for the more massive pulsar\nprogenitors.\nFor $\\zeta$\\,Oph, which is supposed to be formed in a binary SN, sharing\nthe same environment\nwith the SN progenitor, the expected field strength would be of the\norder of 10\\,G.\nThis value is notably lower than our current measurement, possibly\nindicating that either the magnetic field\nof this middle-aged pulsar has significantly decayed during the few Myrs\nafter the SN explosion or other mechanisms\nplay a role in the generation of magnetic fields in O-type stars.\n\n\n\n\n\\section{Discussion}\n\\label{sect:discussion}\n\n\n$\\zeta$\\,Oph has been extremely well studied in all wavelength ranges, from the X-ray by all major X-ray satellites\n(with the exception of {\\em XMM-Newton}) to the infrared region with {\\em Spitzer}.\nIn view of the detection of a magnetic field on \\mbox{$\\zeta$\\,Oph}\\ reported\nin this work, we review its X-ray properties with the aim to\nunderstand whether the X-ray emission of \\mbox{$\\zeta$\\,Oph}\\ is dominated by magnetic\nor wind instability processes. \n\nBabel \\& Montmerle (\\cite{BabelMontmerle1997}) studied the case of a rotating star with a\ndipole magnetic field sufficiently strong to confine stellar wind.\nThe magnetic field locally dominates the bulk motion of stellar wind,\nwhen the ratio of magnetic to kinetic energy density, $B^2\/\\mu_0\\rho\nv^2 > 1$, where $v$ is the supersonic flow speed. A collision between\nthe wind streams from the two hemispheres in the closed magnetosphere\nleads to a strong shock and X-ray emission.\n\nMHD simulations in the framework of this magnetically confined\nwind shock (MCWS) model were performed by ud-Doula \\& Owocki (\\cite{udDoulaOwocki2002})\nand Gagn{\\'e} \\hbox{et~al.\\\/}\\ (\\cite{Gagne2005}). Using their notation, the wind is\nconfined when \\mbox{$\\eta_\\ast\\equiv\n (R_\\ast^2B^2)(\\dot{M}\\mbox{$v_\\infty$})^{-1} > 1$}. New observations are\nrequired to establish whether the magnetic field of \\mbox{$\\zeta$\\,Oph}\\ is a\ndipole. However, for the purpose of this discussion, let us assume that\nthe field has an average strength of 150\\,G. Using the stellar\nparameters of \\mbox{$\\zeta$\\,Oph}\\ as inferred by Marcolino \\hbox{et~al.\\\/}\\ (\\cite{Marcolino2009}),\nwe estimate $\\eta_\\ast(\\mbox{$\\zeta$\\,Oph})\\sim 10^3$, i.e.\\ the magnetic field should dominate\nthe wind motion up to the Alfv{\\'e}n radius that is located at $\\raisebox{-.4ex}{$\\stackrel{<}{\\scriptstyle \\sim}$}$10\\,$\\mbox{$R_\\ast$}$.\nIn this case, the X-ray emission should mainly originate from the MCWS.\n\nThe MCWS model predicts that the X-ray emitting plasma should be\nlocated at a few \\mbox{$R_\\ast$}\\ from the photosphere; that the X-ray emission\nlines should be narrow; that the X-ray luminosity should be higher and\nthe spectrum harder than in non-magnetic stars; that in \n oblique magnetic rotators the X-ray emission should be modulated\nperiodically as a consequence of the occultation of the hot plasma by\na cool torus of matter, or by the opaque stellar core.\n\nThe lines of He-like ions observed in X-ray spectra are useful to\nderive the location of the line formation region in hot stars \n because forbidden line emission is depressed by ultraviolet\n pumping. The latter depends on the distance to the stellar\n photosphere (Gabriel \\& Jordan \\cite{GabrielJordan1969}; Blumenthal\n\\hbox{et~al.\\\/}{} \\cite{Blumenthal1972}). The Si\\,{\\sc xiii} line observed in the\n{\\em Chandra}\\ HETGS\/MEG spectrum is shown in Fig.~\\ref{fig:sixiii}. \n prominent forbidden line can easily be distinguished in this fugure, while\nnormally forbidden lines are strongly suppressed in the spectra of\nOB-type stars. The presence of the forbidden line implies that the\nline formation region is located far from the photosphere, so that the\nradiative excitation does not lead to the depopulation of the\ncorresponding metastable energy levels. Waldron \\& Cassinelli\n(\\cite{WaldronCassinelli2008}) found that the Si\\,{\\sc xiii} line is\nformed at $1.8 \\pm 0.7$\\,\\mbox{$R_\\ast$}\\ in \\mbox{$\\zeta$\\,Oph}\\ and that other He-like\nlines are formed even further out in the wind. Interestingly, the\n strong forbidden Si\\,{\\sc xiii} line is also observed in the {\\em\n Chandra} spectrum of the magnetic star $\\tau$\\,Sco. Cohen\n \\hbox{et~al.\\\/}\\ (\\cite{Cohen2003}) derive a Si\\,{\\sc xiii} line formaiton\n radius for $\\tau$\\,Sco in the range between 1.1\\,$R_\\ast$ and 1.5\\,$R_\\ast$.\nThese radii of line formation are smaller than those found in the\n prototypical MCWS model object $\\theta^1$\\,Ori\\,C, $3.4 \\pm\n 0.8$\\,\\mbox{$R_\\ast$}\\ (Waldron \\& Cassinelli \\cite{WaldronCassinelli2008}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{SiXIII.ps}\n\\caption{The Si\\,{\\sc xiii} line observed in the spectrum of \\mbox{$\\zeta$\\,Oph}{}\n(co-added MEG $\\pm$1). Vertical dashed lines indicate the rest-frame\nwavelength: $\\lambda_{\\rm R}$ -- resonant line, $\\lambda_{\\rm I}$ -- \nsum of intercombination lines, $\\lambda_{\\rm F}$ -- forbidden line. \nThe rest-frame wavelengths are corrected for the radial velocity \ntaken from Hoogerwerf \\hbox{et~al.\\\/}\\ (\\cite{Hoogerwerf2001}).}\n\\label{fig:sixiii}\n\\end{figure}\n\nOskinova \\hbox{et~al.\\\/}\\ (\\cite{Oskinova2006}) studied the {\\em Chandra}\\ spectrum of \\mbox{$\\zeta$\\,Oph}\\ among\nother O-type stars. They found that the X-ray emission lines in this\nstar are narrow and that the signatures of wind absorption on line\nprofiles are weak. Figure~\\ref{fig:vel} shows the Fe\\,{\\sc xvii} and \nNe\\,{\\sc x} lines as measured by {\\em Chandra}\\ plotted over units of the wind\nterminal velocity, \\mbox{$v_\\infty$}=1550\\,km\\,s$^{-1}$. The lines are only slightly broadened, \nif at all. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{twoline_vel.ps}\n\\caption{Fe\\,{\\sc xvii} (upper panel) and Ne\\,{\\sc x} (lower panel)\n lines observed in the spectrum of \\mbox{$\\zeta$\\,Oph}\\ (co-added MEG $\\pm$1). \nVertical dashed lines indicate the rest-frame wavelength, corrected for the \nradial velocity. }\n\\label{fig:vel}\n\\end{figure}\n\nThe X-ray luminosity of \\mbox{$\\zeta$\\,Oph}, $\\mbox{$L_{\\rm X}$}=1.2\\times 10^{31}$\\,erg\\,s$^{-1}$\nand the ratio $\\mbox{$L_{\\rm X}$}\/\\mbox{$L_{\\rm bol}$}=4\\times 10^{-8}$ are quite usual among late\ntype OV stars (Oskinova \\hbox{et~al.\\\/}\\ \\cite{Oskinova2006}). Adopting the\n mass-loss rate from \\mbox{$\\zeta$\\,Oph}\\ as $\\raisebox{-.4ex}{$\\stackrel{<}{\\scriptstyle \\sim}$} 1.8\\times 10^{-7}$\\,\\mbox{$M_\\odot\\,{\\rm yr}^{-1}$}{},\n Oskinova \\hbox{et~al.\\\/}{} (\\cite{Oskinova2006}) noticed that in \\mbox{$\\zeta$\\,Oph}{} the ratio\n of X-ray to wind mechanical luminosity $L_{\\rm mech}$ ($\\mbox{$\\dot{M}$}\\mbox{$v_\\infty$}^2\/2$), $L_{\\rm\n X}\/L_{\\rm mech} \\raisebox{-.4ex}{$\\stackrel{>}{\\scriptstyle \\sim}$} 8.5\\times 10^{-5}$, is a few times\n higher than in other single O-type stars. This may be related to\n the lower wind opacity in \\mbox{$\\zeta$\\,Oph}, or it may hint at some additional\n mechanism of X-ray generation besides the intrinsic wind shocks.\n\nFrom their analysis of {\\em Chandra}\\ spectra, Zhekov\\& Palla\n(\\cite{ZhekovPalla2007}) derived the differential emission measure (DEM)\nfor \\mbox{$\\zeta$\\,Oph}\\ among other OB stars in their sample. They found that in\n\\mbox{$\\zeta$\\,Oph}\\ the DEM sharply peaks at about 6\\,MK. While this is a\nsignificantly lower temperature than found for the DEM peak in case of\n$\\theta^1$\\,Ori\\,C (50\\,MK), it is higher than found for other stars of\nsimilar spectral types ($\\sim$3\\,MK). Thus, considering the X-ray\ntemperature of \\mbox{$\\zeta$\\,Oph}{}, it is not straightforward to attribute its X-ray\nemission to the MCWS. On the other hand, recent studies of O stars with\ndetected magnetic fields (e.g., HD\\,191612, HD\\,108) show that their\nX-ray properties are diverse (Naz{\\'e} \\hbox{et~al.\\\/}\\ \\cite{Naze2004},\n\\cite{Naze2010b}) and may be difficult to fully reconcile with the\npredictions of the MCWS model. \n\nClearly, new observations are needed to better understand the magnetic field of \\mbox{$\\zeta$\\,Oph}\\\nand its link with the X-ray emission from this star.\n\n{\n\\acknowledgements\nWe would like to thank Y.~Naz\\'e for drawing our attention to this interesting star and the \nanonymous referee for valuable comments.\nLMO acknowledges the financial support from grant number FKZ~50~OR~1101. This work used \narchival data obtained with the Spitzer Space Telescope, which is operated by the Jet \nPropulsion Laboratory, California Institute of Technology, under a contract with NASA. \nWe also used data obtained from the Chandra Data Archive and software provided\nby the Chandra X-ray Center (CXC).}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Notations}\n\nFor integers $a,b\\in \\mathbb{Z}$, we set $[a,b]=\\{x\\in \\mathbb{Z}: a\\leq x\\leq b\\}$. For a real number $x$, we denote by $\\lfloor x\\rfloor$ the largest integer that is less\nthan or equal to $x$, and by $\\lceil x\\rceil$ the smallest integer that is greater than or equal to $x$.\n\n\nLet $R$ be a finite commutative ring with identity $1_R$. The addition and multiplication of $R$ will be denoted as $+$ and $*$ respectively.\n We use idempotent to mean an element $e\\in R$ such that $e*e=e$.\nLet ${\\rm U}(R)$ be the group of all units of $R$.\nThe set ${\\rm Aut}(R)$ of all ring automorphisms $R\\rightarrow R$ forms a group under the operation of composition of functions.\nLet ${\\rm Spec} \\ R$ be the spectrum of $R$, i.e., the set of all prime ideals. Let $\\Psi$ be a subgroup of the group ${\\rm Aut}(R)$.\nThen the group $\\Psi$ acts on the set ${\\rm Spec} \\ R$ given by:\n$\\psi P=\\psi(P)\\in {\\rm Spec} \\ R \\ \\mbox{ where } \\psi\\in \\Psi \\mbox{ and } P\\in {\\rm Spec} \\ R.$\nFor any $P\\in {\\rm Spec} \\ R$, let $$\\mathscr{O}_P=\\{Q\\in {\\rm Spec} \\ R: Q=\\psi P \\mbox{ for some } \\psi\\in \\Psi\\}$$ be the orbit of $P$ under the action of $\\Psi$ on ${\\rm Spec} \\ R$. Note that $$|\\mathscr{O}_P|=\\frac{|\\Psi|}{|{\\rm St}(P)|}$$ where $${\\rm St}(P)=\\{\\psi\\in \\Psi: \\psi(P)=P\\}$$ is the stabilizer of $P$.\n\n\n\nWe also need to introduce notation and terminologies on sequences over rings and follow the notation of A. Geroldinger, D.J. Grynkiewicz and\nothers used for sequences over groups (cf. [\\cite{Grynkiewiczmono}, Chapter 10] or [\\cite{GH}, Chapter 5]). For any nonempty subset $A$ of the ring $R$ (usually $A$ is taken to be $R$ or ${\\rm U}(R)$ in this paper), let ${\\cal F}(A)$\nbe the free commutative monoid, multiplicatively written, with basis\n$A$. We denote multiplication in $\\mathcal{F}(A)$ by the boldsymbol $\\cdot$ and we use brackets for all exponentiation in $\\mathcal{F}(A)$. By $T\\in {\\cal F}(A)$, we mean $T$ is a sequence of terms from $A$ which is\nunordered, repetition of terms allowed. Say\n$$T=a_1a_2\\cdot\\ldots\\cdot a_{\\ell}$$ where $a_i\\in A$ for $i\\in [1,\\ell]$.\nThe sequence $T$ can be also denoted as $T=\\mathop{\\bullet}\\limits_{a\\in A}a^{[{\\rm v}_a(T)]},$ where ${\\rm v}_a(T)$ is a nonnegative integer and\nmeans that the element $a$ occurs ${\\rm v}_a(T)$ times in the sequence $T$. By $|T|$ we denote the length of the sequence, i.e., $|T|=\\sum\\limits_{a\\in A}{\\rm v}_a(T)=\\ell.$ By $\\varepsilon$ we denote the\n{\\sl empty sequence} in ${\\cal F}(A)$ with $|\\varepsilon|=0$. We call $T'$\na subsequence of $T$ if ${\\rm v}_a(T')\\leq {\\rm v}_a(T)\\ \\ \\mbox{for each element}\\ \\ a\\in A,$ denoted by $T'\\mid T,$ moreover, we write $T^{''}=T\\cdot T'^{[-1]}$ to mean the unique subsequence of $T$ with $T'\\cdot T^{''}=T$. We call $T'$ a {\\sl proper} subsequence of $T$ provided that $T'\\mid T$ and $T'\\neq T$. In particular, $\\varepsilon$ is a proper subsequence of every nonempty sequence.\nWe call $T$ a {\\bf $\\Psi$-idempotent-product} sequence provided that there exists $\\psi_1,\\ldots,\\psi_{\\ell}\\in \\Psi$ (not necessarily distinct) such that $\\prod\\limits_{i=1}^{\\ell} \\psi_i(a_i)$ is an idempotent of $R$. We call $T$ a {\\bf $\\Psi$-idempotent-product-free} sequence provided that $T$ contains no nonempty subsequence which is $\\Psi$-idempotent-product.\n\nNote that the restriction $\\psi\\mid {\\rm U}(R)$\nis an automorphism of the group ${\\rm U}(R)$. For convenience, in what follows we still use $\\psi$ instead of $\\psi\\mid {\\rm U}(R)$ to denote the automorphism of the group ${\\rm U}(R)$. Then it does make a sense to say a sequence of terms from ${\\rm U}(R)$ to be $\\Psi-$idempotent-product or $\\Psi-$idempotent-product free.\n\nTo give our main result, we still need the following notation. For any pair of positive integers $(m,h)$, let $${\\rm T}(m; h)={\\rm max} \\sum\\limits_{i=1}^h t_i, \\ \\ \\ \\ \\ \\ (t_i\\geq 0)$$ where\n\\begin{equation}\\label{equation iti1$, there exists some element $c_P\\in P$ such that $(c_P)+P^{{\\rm Ind}(P)}=P$.\n Then\n${\\rm I}_{\\Psi}(R)\\geq {\\rm D}_{\\Psi}({\\rm U}(R))+\\sum\\limits_{P\\in {\\rm Spec}(R)} \\frac\n{T\\left({\\rm Ind}(P); \\ \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}.$\n\\end{lemma}\n\n\n\n\\begin{proof} Denoted\n\\begin{equation}\\label{equation the definition of hole}\n{\\rm G}_P=\\{c\\in P: (c)+P^{{\\rm Ind}(P)}=P \\},\n\\end{equation}\nby the hypothesis of the lemma, we have\n\\begin{equation}\\label{equation holeP is not empty}\n{\\rm G}_P\\neq \\emptyset.\n\\end{equation}\n\n\n\\noindent \\textbf{Claim A.} Let $P\\in {\\rm Spec} \\ R$ and $\\psi\\in \\Psi$. Then,\n\n(i) $\\psi(P)^t=\\psi(P^t)$ for all $t\\geq 0$, and in particular, ${\\rm Ind}(\\psi(P))={\\rm Ind}(P)$;\n\n(ii) $\\psi(G_p)=G_\\psi(P)$.\n\n\\noindent {\\sl Proof of Claim A.}\n\n(i) Trivial.\n\n(ii) Take an arbitrary element $a\\in G_p$. By \\eqref{equation the definition of hole} and Conclusion (i), we have that $(\\psi(a))+\\psi(P)^{{\\rm Ind}(\\psi(P))}=(\\psi(a))+\\psi(P)^{{\\rm Ind}(P)}=(\\psi(a))+\\psi(P^{{\\rm Ind}(P)})=\\psi((a))+\\psi\\left(P^{{\\rm Ind}(P)}\\right)=\\psi\\left((a)+P^{{\\rm Ind}(P)}\\right)=\\psi(P)$. Since $\\psi(a)\\in \\psi(G_p)\\subset \\psi(P)$, it follows that $\\psi(a)\\in G_{\\psi(P)}$. By the arbitrariness of choosing $a$, we have $\\psi(G_p)\\subset G_{\\psi(P)}$. Since $\\psi^{-1}\\in \\Psi$, it follows that $G_{\\psi(P)}=\\psi(\\psi^{-1}(G_{\\psi(P)}))\\subset \\psi(G_{\\psi^{-1}(\\psi(P))})= \\psi(G_P)$, and thus, $\\psi(G_p)=G_{\\psi(P)}$. \\qed\n\n\n\n\n\n\n\\noindent \\textbf{Claim B.} Let $P\\in {\\rm Spec} \\ R$ be such that ${\\rm Ind}(P)>1$. Let $\\ell\\in [1,{\\rm Ind}(P)-1]$ and $a_1,\\ldots,a_{\\ell}\\in G_P$ (not necessarily distinct). Then $\\prod\\limits_{i=1}^{\\ell} a_i \\notin P^{{\\rm Ind}(P)}.$\n\n\n\n\n\\noindent {\\sl Proof of Claim B.} It suffices to consider the case that $\\ell={\\rm Ind}(P)-1.$\n Since $P^ {{\\rm Ind}(P)}\\subsetneq P^ {{\\rm Ind}(P)-1}$, we can take some element $x$ of $P^ {{\\rm Ind}(P)-1}\\setminus P^ {{\\rm Ind}(P)}$. Since $x$ is a finite sum of products of the form $b_{1}*b_{2}*\\cdots *b_{{\\rm Ind}(P)-1}$ where $b_{1}, b_{2},\\ldots, b_{{\\rm Ind}(P)-1}\\in P$, it follows\nthat\n\\begin{equation}\\label{equation sequence over Pwith}\n\\{T\\in {\\cal F}(P): |T|={\\rm Ind}(P)-1 \\mbox{ and } \\pi(T)\\in P^ {{\\rm Ind}(P)-1}\\setminus P^ {{\\rm Ind}(P)}\\}\\neq \\emptyset.\n\\end{equation}\nThen we take a sequence $y_1\\cdot \\ldots\\cdot y_{{\\rm Ind}(P)-1} \\in {\\cal F}(P)$ in the set given as \\eqref{equation sequence over Pwith} such that\nthe number of common terms of sequences $y_1\\cdot \\ldots\\cdot y_{{\\rm Ind}(P)-1}$ and $a_1\\cdot \\ldots\\cdot a_{{\\rm Ind}(P)-1}$ is {\\bf maximal},\nsay\n\\begin{equation}\\label{equation y1=a1,...yt=at}\ny_i=a_i \\ \\mbox{ for each } \\ i\\in [1,s]\n\\end{equation}\n(with $s \\in [0,{\\rm Ind}(P)-1]$ being maximal), moveover, both sequences $y_{s+1}\\cdot\\ldots\\cdot y_{{\\rm Ind}(P)-1}$ and $a_{s+1}\\cdot\\ldots\\cdot a_{{\\rm Ind}(P)-1}$ have no common terms.\nTo prove Claim B, we need only to show that $s={\\rm Ind}(P)-1$.\nAssume to the contrary that $$s<{\\rm Ind}(P)-1.$$\nLet $z=\\prod\\limits_{i\\in [1,{\\rm Ind}(P)-1]\\setminus \\{s+1\\}} y_i.$\nSince $y_{s+1}*z=y_{s+1}*\\prod\\limits_{i\\in [1,{\\rm Ind}(P)-1]\\setminus \\{s+1\\}} y_i=\\prod\\limits_{i\\in [1,{\\rm Ind}(P)-1]} y_i\\notin P^{{\\rm Ind}(P)}$, it follows that\n$y_{s+1}\\in P\\setminus (P^{{\\rm Ind}(P)}: z)$, equivalently,\n \\begin{equation}\\label{equation (PindP:b)capPinP}\n (P^{{\\rm Ind}(P)}: z)\\cap P\\subsetneq P.\n \\end{equation}\nSince $a_{s+1}\\in {\\rm G}_P$, then $(a_{s+1})+P^{{\\rm Ind}(P)}=P$. Combined with\n \\eqref{equation (PindP:b)capPinP}, we conclude that $a_{s+1}\\notin (P^{{\\rm Ind}(P)}: z)$. It follows from \\eqref{equation y1=a1,...yt=at} that\n$\\left(\\prod\\limits_{i\\in [1, s+1]} a_i\\right) * \\left(\\prod\\limits_{i\\in [s+2, {\\rm Ind}(P)-1]} y_i\\right)=a_{s+1}* \\left(\\prod\\limits_{i\\in [1, s]} a_i\\right) * \\left(\\prod\\limits_{i\\in [s+2, {\\rm Ind}(P)-1]} y_i\\right)=a_{s+1}* \\left(\\prod\\limits_{i\\in [1, s]} y_i\\right) * \\left(\\prod\\limits_{i\\in [s+2, {\\rm Ind}(P)-1]} y_i\\right)=a_{s+1}* \\left(\\prod\\limits_{i\\in [1, {\\rm Ind}(P)-1]\\setminus \\{s+1\\}} y_i\\right) =a_{s+1}*z\\notin P^{{\\rm Ind}(P)}$. Obviously, $\\left(\\prod\\limits_{i\\in [1, s+1]} a_i\\right) * \\left(\\prod\\limits_{i\\in [s+2, {\\rm Ind}(P)-1]} y_i\\right)\\in P^{{\\rm Ind}(P)-1}$, and so $\\left(\\prod\\limits_{i\\in [1, s+1]} a_i\\right) * \\left(\\prod\\limits_{i\\in [s+2, {\\rm Ind}(P)-1]} y_i\\right)\\in P^{{\\rm Ind}(P)-1}\\setminus P^{{\\rm Ind}(P)}$ which implies that the sequence $$a_1\\cdot\\ldots\\cdot a_{s+1}\\cdot y_{s+2}\\cdot\\ldots\\cdot y_{{\\rm Ind}(P)-1}\\in \\{T\\in {\\cal F}(P): |T|={\\rm Ind}(P)-1 \\mbox{ and } \\pi(T)\\in P^ {{\\rm Ind}(P)-1}\\setminus P^ {{\\rm Ind}(P)}\\}.$$ Then we derive a contradiction with the choosing of the sequence $y_1\\cdot\\ldots\\cdot y_{{\\rm Ind}(P)-1}$. This proves Claim B. \\qed\n\n\n\nLet $\\mathscr{O}$ be\nan arbitrary orbit of ${\\rm Spec} R$ under the action of $\\Psi$.\nSay,\n\\begin{equation}\\label{equation orbit O}\n\\mathscr{O}=\\{P_1,P_2,\\ldots,P_{h}\\} \\ \\mbox{ where } \\ h=\\frac{|\\Psi|}{|{\\rm St}(P_1)|}.\n\\end{equation}\nFor any nonempty subset $X\\subset [1,h]$, we define\n\\begin{equation}\\label{equation def H(O;X)}\nH_{\\mathscr{O}; X}=\\left\\{a\\in R: a\\equiv 1_R {\\pmod {Q^{{\\rm Ind}(Q)}}} \\mbox{ for all } Q\\in {\\rm Spec} R\\setminus \\{P_i: i\\in X\\}\\right\\}\\cap\\left(\\bigcap\\limits_{i\\in X}{\\rm G}_{P_i}\\right).\n\\end{equation}\nFor any $t\\in [1,h]$, let\n\\begin{equation}\\label{equation def H(O;t)}\n\\mathscr{H}_{\\mathscr{O}; t}=\\bigcup\\limits_{\\stackrel{X\\subset [1,h]}{|X|=t}} H_{\\mathscr{O}; X}.\n\\end{equation}\n\n\nIn the following, we shall give three claims on the properties of $H_{\\mathscr{O}; X}$, $\\mathscr{H}_{\\mathscr{O}; t}$ given as above.\n\n\\noindent\\textbf{Claim C.} For any element $b\\in H_{\\mathscr{O}; X}$, we have $\\{i\\in [1,h]: b\\in P_i\\}=X.$\n\n\\noindent {\\sl Proof of Claim C.} By \\eqref{equation the definition of hole} and \\eqref{equation def H(O;X)}, we see that $b\\in P_i$ for each $i\\in X$. For any $Q\\in {\\rm Spec} R\\setminus \\{P_i: i\\in X\\}$, since $b\\equiv 1_R \\pmod {Q^{{\\rm Ind}(Q)}}$, then $b\\equiv 1_R \\pmod Q$ and so $b\\notin Q$, done. \\qed\n\n\n\n\\noindent \\textbf{Claim D.} $H_{\\mathscr{O}; X}\\neq \\emptyset$ for any nonempty set $X\\subset [1,h]$. In particular, $\\mathscr{H}_{\\mathscr{O}; t}\\neq \\emptyset$ for each $t\\in [1,h]$.\n\n\\noindent {\\sl Proof of Claim D.} \\ By \\eqref{equation holeP is not empty}, we take an element\n\\begin{equation}\\label{equation xi in HolePi}\nc_i\\in {\\rm G}_{P_i} \\mbox{ for each } i\\in X.\n\\end{equation}\n Note that $\\{P^{{\\rm Ind}(P)}: P\\in {\\rm Spec} R\\}$ is a family of ideals which are pairwise coprime. By the Chinese Remainder Theorem, we can find one element $a$ such that\n\\begin{equation}\\label{equation a equiv xi mod piindPi for iin[1,k]}\na\\equiv c_i \\pmod {P_i^{{\\rm Ind}(P_i)}} \\mbox{ for each } i\\in X\n\\end{equation}\nand\n\\begin{equation}\\label{equation a equiv 1mod Q in ClaD}\na\\equiv 1_R \\pmod {Q^{{\\rm Ind}(Q)}} \\mbox{ for each } Q\\in {\\rm Spec} R\\setminus \\{P_i: i\\in X\\}.\n\\end{equation}\nTo derive $H_{\\mathscr{O}; X}\\neq \\emptyset$, by \\eqref{equation a equiv 1mod Q in ClaD}\n we need only to show that $a\\in \\bigcap\\limits_{i\\in X} G_{P_i}$.\n\n\nLet $i\\in X$. It follows from \\eqref{equation a equiv xi mod piindPi for iin[1,k]} that\n\\begin{equation}\\label{equation a=xi+v}\nc_i=a+v_i \\mbox{ for some } v_i\\in P_i^{{\\rm Ind}(P_i)}.\n\\end{equation}\nSince $G_{P_i}\\subset P_i$, it follows from \\eqref{equation xi in HolePi} and \\eqref{equation a=xi+v} that $a\\in P_i$. Then it suffices to show that $(a)+P_i^{{\\rm Ind}(P_i)}=P_i$.\nTake an arbitrary element $b\\in P_i$. By \\eqref{equation the definition of hole} and \\eqref{equation xi in HolePi}, we derive that $b=r_b c_i+u_b$ where $r_b \\in R$ and $u_b\\in P_i^{{\\rm Ind}(P_i)}$. Combined with \\eqref{equation a=xi+v}, we have that $b=r_b(a+v_i)+u_b=r a+(r_b v_i +u_b)\\in (a)+P_i^{{\\rm Ind}(P_i)}$. By the arbitrariness of choosing $b$, we proved $(a)+P_i^{{\\rm Ind}(P_i)}=P_i$ and so $H_{\\mathscr{O}; X}\\neq \\emptyset$. Then $\\mathscr{H}_{\\mathscr{O};t}\\neq \\emptyset$ follows from \\eqref{equation def H(O;t)} trivially. \\qed\n\n\n\n\\noindent \\textbf{Claim E.} For any $t\\in [1,h]$ and any $\\psi\\in \\Psi$, we have $\\psi(\\mathscr{H}_{\\mathscr{O}; t})=\\mathscr{H}_{\\mathscr{O}; t}.$\n\n\\noindent {\\sl Proof of Claim E.} Take an arbitrary element $a\\in \\mathscr{H}_{\\mathscr{O};t}$, equivalently, $a\\in H_{\\mathscr{O}; X}$ for some $X\\subset [1,h]$ with $|X|=t$. Since $\\psi$ acts on ${\\rm Spec} R$, it follows from \\eqref{equation orbit O} that there exists $X'\\subset [1,h]$ of cardinality $|X'|=|X|=t$ such that\n \\begin{equation}\\label{equation psi(Pi)inX=PjinX'}\n \\{\\psi(P_i):i\\in X\\}= \\{P_j:j\\in X'\\},\n \\end{equation}\n and follows from Claim A (i) that \\begin{equation}\\label{equation Qindx=Q'index}\n \\{\\psi(Q)^{{\\rm Ind}(\\psi(Q))}: Q\\in {\\rm Spec} R\\setminus \\{P_i: i\\in X\\}\\}=\\{Q'^{{\\rm Ind}(Q')}: Q'\\in {\\rm Spec} R\\setminus \\{P_j: j\\in X'\\}\\}.\n \\end{equation}\nBy \\eqref{equation def H(O;X)} and Claim A (i), for all $Q\\in {\\rm Spec} R\\setminus \\{P_i: i\\in X\\}$, we have that $\\psi(a)-1_R=\\psi(a)-\\psi(1_R)=\\psi(a-1_R)\\in \\psi(Q^{{\\rm Ind}(Q)})=\\psi(Q)^{{\\rm Ind}(Q)}=\\psi(Q)^{{\\rm Ind}(\\psi(Q))}$, i.e., $\\psi(a)\\equiv 1_R \\pmod {\\psi(Q)^{{\\rm Ind}(\\psi(Q))}}$.\nCombined with \\eqref{equation Qindx=Q'index}, we conclude that $$\\psi(a)\\equiv 1_R \\pmod {Q'^{{\\rm Ind}(Q')}} \\mbox{ for all } Q'\\in {\\rm Spec} R\\setminus \\{P_j: j\\in X'\\}\\}.$$\nSince $a\\in \\bigcap\\limits_{i\\in X}{\\rm G}_{P_i}$, it follows from \\eqref{equation psi(Pi)inX=PjinX'} and Claim B that $\\psi(a)\\in \\psi(\\bigcap\\limits_{i\\in X}{\\rm G}_{P_i})=\\bigcap\\limits_{i\\in X}\\psi({\\rm G}_{P_i})=\\bigcap\\limits_{j\\in X'}{\\rm G}_{P_j}.$ Then we conclude that $\\psi(a)\\in H_{\\mathscr{O}; X'}\\subset \\mathscr{H}_{\\mathscr{O};t}$, completing the proof of Claim D. \\qed\n\nLet $\\mathscr{O}$ be the orbit given as \\eqref{equation orbit O}. By $B_{\\mathscr{O}}$ we denote the sequence associated with the orbit $\\mathscr{O}$ which are given as below.\nBy Claim D, we choose an element\n\\begin{equation}\\label{equation btin H(O,t)}\nb_t\\in \\mathscr{H}_{\\mathscr{O};t} \\mbox{ for each } t\\in [1,h].\n\\end{equation}\nLet $d_1,d_2,\\ldots,d_{h}$ be positive integers such that\n\\begin{equation}\\label{equation sum di=T()}\n\\sum\\limits_{i=1}^h d_i=T({\\rm Ind}(P_1); \\ h)\n\\end{equation}\nand\n\\begin{equation}\\label{equation tdt1$. Then there exists no ideal $A$ of $R$ such that $Q^2\\subsetneq A \\subsetneq Q$. Let $R=D\\diagup J$ where $D$ is a Dedekind domain and $J\\lhd D$.\nLet $\\varphi: D\\rightarrow R$ be the canonical epimorphism. Then\n\\begin{equation}\\label{equation kernal J=}\nJ=\\prod\\limits_{i=1}^t P_i^{\\alpha_i}\n\\end{equation}\n where $t\\geq 1, \\alpha_1,\\ldots, \\alpha_t\\geq 1$, and $P_1,\\ldots,P_t$ are distinct prime ideals of $D$. It follows from \\eqref{equation kernal J=} that\n \\begin{equation}\\label{equation Q=varphi(P)}\n Q=\\varphi(P)\n \\end{equation}\n for some $P\\in {\\rm Spec} D$ with\n \\begin{equation}\\label{equation J subset P}\n J\\subset P.\n \\end{equation}\n Then $P=P_i$ for some $i\\in [1,t]$, say $$P=P_1.$$\n\n \\noindent \\textbf{Claim I.} There exists no ideal $N\\lhd R$ such that $Q^2\\subsetneq N\\subsetneq Q$.\n\n \\noindent {\\sl Proof of Claim I.} Assume to the contrary that there exists some ideal $N\\lhd R$ such that $Q^2\\subsetneq N\\subsetneq Q$.\nBy \\eqref{equation Q=varphi(P)} and \\eqref{equation J subset P}, we have that $\\varphi(P^2+J)=\\varphi(P^2)\\subset \\varphi(P)*\\varphi(P)=Q^2$, and thus, $P^2\\subset P^2+J=\\varphi^{-1}(\\varphi(P^2+J))\\subset \\varphi^{-1}(Q^2)\\subsetneq\\varphi^{-1}(N) \\subsetneq \\varphi^{-1}(Q)=P$. Then we derive a contradiction, since $D$ is a Dedekind domain implying that there exists no ideal $M\\lhd D$ such that $P^2\\subsetneq M\\subsetneq P$. This proves Claim I. \\qed\n\nTake an arbitrary $Q\\in {\\rm Spec R}$ such that ${\\rm Ind}(Q)>1$. Take an element $x\\in Q\\setminus Q^2$.\nSince $Q^2\\subsetneq (x)+Q^2\\subset Q$, it follows from Claim I that $Q=(x)+Q^2$.\nThen we have that\n\\begin{equation}\\label{equation interationforx+q2}\n(x)+Q^{k}=(x)+((x)+Q^2)^k=(x)+\\left(\\sum\\limits_{i=0}^{k-1}(x)^{k-i}*Q^{2i}\\right)+Q^{2k}=(x)+Q^{2k} \\ \\mbox{ for any } \\ k\\geq 1.\n\\end{equation}\nFix an integer $m>\\ln {\\rm Ind}(Q)$. It follows from \\eqref{equation interationforx+q2} that $Q=(x)+Q^2=(x)+Q^4=\\cdots=(x)+Q^{2^m}=(x)+Q^{{\\rm Ind}(Q)}$. Then the conclusion follows from Lemma \\ref{Lemma finite commutative rings} readily.\n\\end{proof}\n\nThen we close the paper with the following two conjectures.\n\n\\begin{conj}\\label{Conjecture} Let $R$ be a finite ring with identity.\nLet $\\Psi$ be a subgroup of the group ${\\rm Aut}(R)$. Then\n${\\rm I}_{\\Psi}(\\mathcal{S}_R)\\geq {\\rm D}_{\\Psi}({\\rm U}(R))+\\sum\\limits_{P\\in spec(R)} \\frac\n{T\\left({\\rm Ind}(P); \\ \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}.\n$\n\\end{conj}\n\n\\begin{conj}\\label{Conjecture} Let $R$ be a finite P.I.R with identity.\nLet $\\Psi$ be a subgroup of the group ${\\rm Aut}(R)$. Then\n${\\rm I}_{\\Psi}(\\mathcal{S}_R)={\\rm D}_{\\Psi}({\\rm U}(R))+\\sum\\limits_{P\\in spec(R)} \\frac\n{T\\left({\\rm Ind}(P); \\ \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}.\n$\n\\end{conj}\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent {\\bf Acknowledgements}\n\nThis work is supported by NSFC (grant no. 11971347).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaeli b/data_all_eng_slimpj/shuffled/split2/finalzzaeli new file mode 100644 index 0000000000000000000000000000000000000000..dc641eea82e7e7b90f446b8b6d0be7a3a90a0109 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaeli @@ -0,0 +1,5 @@ +{"text":"\n\\section{The proposed discriminative residual coding}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[height=0.85in]{figs\/Bitrate.pdf}\n \\caption{The proposed thresholding mechanism uses different quantization schemes depending on the target bitrate.}\n \\label{fig:bitrate}\n \\vspace{-4mm}\n\\end{figure}\n\n\n\nTo improve the coding gain further, we apply \\textit{discriminative} coding to the residual signals $\\bar{\\bm r}_n$, which is the information sent to the receiver in place of the full cepstrum $\\bm{c}_n$. \nDue to the overall smoothness of speech, the prediction in the cepstrum domain results in the residuals that follow a Gaussian distribution with zero mean and small variance. The larger residual values mainly occur in transient events, such as plosives.\nTo fully make use of the residual signal's statistical advantage, we apply \\textit{discriminative} coding that distinguishes the more ``code-worthy\" frames from the rest by thresholding the $L_1$ norm of the residuals. This way, frames with significant residual energy are assigned more bits, and bits assigned to the less significant frames are minimized. \n\n\n\nScanning the entire training set, we define a threshold value $\\theta$ depending on the target bitrate. The quantization process eq.\\eqref{qtz} is therefore expanded to: \n\\begin{equation} \n \\bar{\\bm r} = \\mathcal{Q}({\\bm r}) = \n \\begin{cases}\n \\mathcal{Q}_\\text{L}(\\bm{r}) & \\text{if } ||\\bm r||_1 \\geq \\theta \\\\\n \\mathcal{Q}_\\text{S}(\\bm{r}) & \\text{otherwise},\n \\end{cases} \n\\end{equation}\nwith $\\mathcal{Q}_\\text{L}$ and $\\mathcal{Q}_\\text{S}$ representing quantization schemes with large and small $L_1$ norms that use large and small codebooks, respectively. Particularly, when the target bitrate is extremely low, we \\textit{discard} low $L_1$ norm frames entirely, i.e., $\\mathcal{Q}_\\text{S}(\\bm{r})=\\bm{0}$. \n\nThe thresholding mechanism is illustrated in Fig.\\ref{fig:bitrate}. The low-bitrate scheme ($\\sim$ 0.93 kbps) uses $\\mathcal{Q}_\\text{L}$ for the top 25 \\% residual frames while discarding the rest without any coding. The intermediate bitrate ($\\sim$ 1.47 kbps) case keeps the top 7\\% for the $\\mathcal{Q}_\\text{L}$ quantization and the rest 90\\% for $\\mathcal{Q}_\\text{S}$. The $\\sim$ 2.87 kbps case uses $\\mathcal{Q}_\\text{L}$ quantization for all residual frames with no thresholding. \n\nSimilar to LPCNet's coding scheme, we separately code the first component $\\bm r_0$ of the residuals vector and the rest of dimensions $\\bm r_{1-17}$; Note here that we dropped the frame index $n$ and used subscript to indicate one of the 18 cepstrum coefficients within the frame. Also, since we noticed that $\\bm r_0$ and $\\bm r_{1-17}$ have different $L_1$ norm distributions, we define thresholds and apply discriminative coding to the scalar and vector components independently.\n\n\n\n\n\n\\begin{table}[]\n \\centering\n \\resizebox{\\columnwidth}{!}{%\n \\begin{tabular}{c||c|c||c|c||c|c}\n \n \n Target bitrate (kbps) & \\multicolumn{2}{c||}{$\\sim$ 0.93} & \\multicolumn{2}{c||}{$\\sim$ 1.47} & \\multicolumn{2}{c}{ $\\sim$ 2.87} \\\\\n \\hline\n $\\mathcal{Q}_\\text{L}$ percentage & \\multicolumn{2}{c||}{25 $\\%$} & \\multicolumn{2}{c||}{7 $\\%$} & \\multicolumn{2}{c}{ 100 $\\%$} \\\\\n \\hline\n \\multicolumn{7}{c}{Codebook Size (bits) : Bits-per-frame according to Huffman coding}\\\\\n \\hline\n Stages & 1st & 2nd & 1st & 2nd & 1st & 2nd\\\\\n \\hline\n $\\mathcal{Q}_\\text{L}(\\bm{r}_0)$ & 8 : 7.0 & - & 8 : 7.4 & - &8 : 7.2 & - \\\\\n \\hline\n $\\mathcal{Q}_\\text{S}(\\bm{r}_0)$ & - & - & 4 : 2.9 & - & - & - \\\\\n \\hline\n $\\mathcal{Q}_\\text{L}(\\bm{r}_{1:17})$ & 10 : 9.8 & 10 : 9.9 & 10 : 9.2 & 10 : 9.4 & 10 : 9.2 & 10 : 9.6\\\\\n \\hline\n $\\mathcal{Q}_\\text{S}(\\bm{r}_{1:17})$ & - & - & 9 : 8.0 & - & - & -\\\\\n \\end{tabular}\n }\n \\caption{Codebook sizes and bitrate assignments.}\n \\label{tab:codebooks}\n \\vspace{-3mm}\n\\end{table}\n\nTable \\ref{tab:codebooks} summarizes how we conduct discriminative and multi-stage quantization depending on the target bitrate. For scalar quantization, we use the same codebook size of $2^9=512$ in all $\\mathcal{Q}_\\text{L}$ cases, while only $16$ codewords for $\\mathcal{Q}_\\text{S}$ in the mid-bitrate case or skips coding in the low-bitrate case. All scalar quantizers use a single-stage quantization scheme. \nAs for the VQ for $\\bm{c}_{1:17}$, we employ either one or two-stage quantization for $\\mathcal{Q}_\\text{L}$ with a codebook of size 1024 in each stage; $\\mathcal{Q}_\\text{S}$ cases use a single 512-size codebook or skip coding in the low-bitrate case ($\\sim$0.93). We also estimate the bitrate considering Huffman coding by computing the frequencies $\\bm p$ of all codewords from coding randomly-selected 2-second segments per training samples and derive the average bit-per-frame by $\\sum_i^N \\bm p_ilog_2\\bm p_i$. \nApart from the bits we have stated in the \\tabref{tab:codebooks}, we also need to count in the bits for coding pitch parameters in LPCNet's original way, which takes up 0.275 kbps. We use the bitrates of Huffman coding in the rest of the paper, although it is close to the bitrates based on the codebook. In the $\\sim$0.93kbps case, for example, the target bitrate in our table is calculated by $0.25 \\times(7\\times100+(9.9+9.8 )\\times100) + 275 = 932 \\text{bps}$, given that each frame is for 10 ms (meaning 100 frames per second), and only $25\\%$ of the frames are coded in this example.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this work, we proposed a lightweight, low-latency, low-bitrate speech coding framework. In line with the parametric coding paradigm, we designed a feature predictor to capture the temporal redundancy and reduce the burden of coding raw feature frames. Moreover, we applied the discriminative coding scheme to the residual signal to further improve coding gain.\nWe showed that the proposed combination of predictive coding and discriminative residual coding can be harmonized well with the original LPCNet-based codec by providing a more effective quantization scheme than the original multi-stage VQ. We open-source our codes at \\url{https:\/\/saige.sice.indiana.edu\/research-projects\/predictive-LPCNet}. \n\n\\section{Experiments}\n\\subsection{Data}\nWe use the Librispeech \\cite{PanayotovV2015Librispeech} corpus's \\texttt{train-clean-100} fold for training, and \\texttt{dev-clean} for validation, at 16kHz sampling rate. \n18 Bark-scale cepstral coefficients are produced for each 20 ms frame with an overlap of 10ms. In addition, we extract and quantize the 2-dimensional pitch parameters using LPCNet's open-sourced framework. \n\n\\subsection{Training}\nThe training process consists of three steps and is conducted sequentially: prediction model training, codebook learning, and vocoder training. Hence, the results from the preceding steps will be used in the following training. \nCompared to a potential end-to-end learning approach, our modularized learning can circumvent the issue of dealing with non-differentiable quantization. \n\nBoth the feature predictor and the vocoder will eventually operate in a synthesis mode, where the inputs to the model are the synthesized results from the previous step. Therefore, we add noise to the input during training for a more robust development, as suggested in \\cite{ValinJ2019lpcnetcoding,JinZ2018fftnet}. \nFinally, the vocoder is finetuned with the quantized input features.\n\nCodebook training is based on the residuals ${\\bm{r}}$ produced from the encoder of the feature predictor $\\mathcal{F}_{pred}$. For both vector and scalar codebooks, we run k-means clustering and pick the learned centroids as the codewords. \nWhen generating residuals for codebook training, the encoder skips the quantization step (eq. \\eqref{qtz}) but will consider the residual thresholding. That is, the residual $\\bm r_n$ will be added back to the prediction result $\\hat{\\bm c}_n$ (as in eq. \\ref{add-back}) only if $||\\bm r||_1 \\geq \\theta$. \nWe randomly pick 2-second segments from each utterance in training set to generate the residual vector for codebook training. Codebooks are trained exclusively for each bitrate.\n\n\nThe feature predictor model we used in the experiments contains $0.65$M parameters, and the entire codec, including the LPCNet vocoder, has $2.5$M parameters. Our codec is suitable for the real-time coding task because of the causality preserved in the frame-level prediction. The algorithmic delay of our codec is $75$ ms, to which the LPCNet vocoder contributes $60$ms-latency from its convolution operation. Another $15$ ms-delay comes from our feature predictor, which occurs while waiting for the ground-truth cepstral-frame of $10$ms with an extra $5$ms look-ahead to compute a cepstrum.\n\n\\subsection{Evaluation and baseline}\n\nWe employ two state-of-the-art low-bitrate codecs as baselines, LPCNet at 1.6kbps and Lyra V2 \\footnote{https:\/\/opensource.googleblog.com\/lyra-v2-a-better-faster-and-more-versatile-speech-codec.html} at 3.2kbps. Lyra V2 is an improved version of Lyra\\footnote{https:\/\/ai.googleblog.com\/lyra-new-very-low-bitrate-codec-for.html} \\cite{kleijn2021generative}, integratin SoundStream \\cite{Zeghidour2021soundstream} in its original architectures for a better coding gain.\n\n\nWe perform a MUSHRA test \\cite{mushra} on our codec at three different bitrates and the two baselines. Ten gender-balanced clean utterances from the LibriSpeech \\texttt{test-clean} set are used. The trials also include a hidden reference and a low-pass-filtered anchor at 3.5kHz. Ten speech experts participated in the test, and no one was excluded per the listener exclusion policy.\n\n\\section{Introduction}\n\nA speech codec, in general, comprises modules for speech compression, quantization, and reconstruction. It has been used in various communication and entertainment applications after standardization \\cite{BessetteB2002amrwb, SchroederM1985celp} or open-sourcing \\cite{ValinJM2012opus}. The common goal in speech coding is to achieve the maximum coding gain, i.e., maintaining the perceptual quality and intelligibility of the reconstructed speech signals with a minimum bitrate.\n\nThe involvement of neural networks has greatly benefited the coding trade-off, effectively eliminating the codes' redundancy while improving the reconstruction quality. More recently, the advances of generative models and their applications in speech coding led to a trend in very low-bitrate speech codecs. The first WaveNet-based speech codec \\cite{KleijnW2018wavenet} demonstrates the usage of neural synthesis in both waveform and parametric coding. The latter is more favored in subsequent studies because of its inherent advantages in dealing with very condensed speech features. These neural vocoders work on the decoder side, leveraging the powerful neural synthesis architecture. Their encoding parts are relatively simplified, relying on existing Codec 2 codes \\cite{codec2} as in the original WaveNet-based speech codec \\cite{KleijnW2018wavenet} or the dimension-reduced frequency-domain speech representations, e.g., cepstrum features \\cite{ValinJ2019lpcnet, ValinJ2019lpcnetcoding}, and LPC analysis \\cite{KlejsaJ2019samplernn}. \n\nIn this line of work, the performance bottleneck comes from the very compact codes, leading to poorer reconstruction quality. To mitigate the issue, some efforts apply more complex models in the encoder to improve the quality of the features \\cite{kim2021neurally, yoshimura2018wavenet} or use generative models for post-processing \\cite{zhao2018convolutional, skoglund2019improving, biswas2020audio} at the end of the existing codec to facilitate signal restoration. However, the output performance is still bounded by the quality of the coding features. \n\nEnd-to-end neural codecs that train the encoder, quantizer, and decoder jointly work as an alternative to the low-bitrate generative speech vocoders \\cite{KankanahalliS2018icassp, ZhenK2019interspeech, ZhenK2022taslp, Zeghidour2021soundstream}.\nIn this way, the neural encoder participates in removing the redundancy in the source signal and produces features that are more associated with the decoder, in contrast to the traditional speech features. Various other methods have been developed to improve the quality of the features, regarding robustness \\cite{casebeer2021enhancing, Lim2020robust}, scalability \\cite{jiang2022cross} and the variability issues \\cite{kleijn2021generative}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.8\\columnwidth]{figs\/overview.pdf}\n \\caption{The overview of the proposed neural speech coding system with feature predictors and LPCNet-based vocoder.}\n \\label{fig:overview}\n \\vspace{-6mm}\n\\end{figure}\n\nHowever, end-to-end codecs tend to suffer in very low-bitrates cases ($<$2 kbps) because that requires the coding features to be extremely small and expressive simultaneously. To deal with that, an ultra-low bitrate codec \\cite{SiahkoohiA2022interspeech} borrows the embeddings from a self-supervised training task to increase the expressiveness of the state-of-the-art codec SoundStream's features \\cite{Zeghidour2021soundstream}, and can obtain a decent speech quality with a very low bitrate, $0.6$ kbps.\nTF-Codec \\cite{jiang2022predictive} addresses the problem by reducing the temporal redundancy in the latent features with a predictive model and reports decent reconstruction quality at $1$ kbps. However, both models entail high complexity. Besides, because TF-Codec's prediction model runs on a latent space that requires a specific pair of encoder and decoder, it brings an extra cost for other existing codecs to mount its predictive module directly. \n\n\nIn this paper, we aim at a low-bitrate, low-delay, and low-complexity neural speech codec that utilizes neural feature prediction to reduce the temporal redundancy from the sequence of feature frames.\nWe introduce a gated recurrent unit (GRU)-based \\cite{ChoK2014emnlp} frame-level feature predictor that can forecast the feature vector at time frame $t$ using its preceding frames. Since the decoder also employs the exact feature predictor, it can ``generate\" most of the feature vector at no cost of bitrate, while the imperfectly generated feature vectors are compensated by the coded residual coming from the encoder side. \nAdditionally, we employ \\textit{discriminative coding} in the residual space. This idea is demonstrated in source-aware neural audio coding by distinguishing speech and noise sources in the latent feature space \\cite{YangH2021sanac}. In this paper, we use different entropy coding strategies at each frame depending on the amount of information they carry. Compared to the TF-Codec, our model explicitly codes only the prediction residuals, and the proposed predictive modules are designed to work in combination with existing low-complexity neural codecs. In particular, we are based on the efficient LPCNet-based speech coding framework \\cite{ValinJ2019lpcnetcoding}, and our analysis and model training mainly focuses on the cepstral coefficients.\n\n\n\n\n\n\\section{The proposed predictive coding}\n\\subsection{Overview}\nIn conjunction with the LPCNet's sample-level vocoding, the proposed feature prediction model performs hierarchical prediction: first in the feature space and then in the sample level. As shown in Fig \\ref{fig:overview}, the frame-level feature predictor $\\mathcal{F}_{pred}$ works on both the encoder and decoder sides. The encoder computes and quantizes the frame-level prediction residuals $\\bar{\\bm r}$ and passes them to the decoder. Then, the decoder adds the received residuals to its own feature predictions $\\hat{\\bm c}$ to obtain the recovered frame-level features. The sample-level predictive coding (i.e., the LPCNet vocoder) works only on the decoder that synthesizes waveform samples $\\hat{\\bm s}$ from the recovered features as the codec's output.\n\n\n\n\n\\subsection{The frame-level feature predicion}\n\n\\subsubsection{Feature predictor}\nWe apply a WaveRNN-based model \\cite{KalchbrennerN2018wavernn} to make a frame-level prediction on the 18-dimensional continuous cepstral coefficients. \nWaveRNN explicitly considers the output at time $n-1$ as the estimation of the $n$-th's sample. \nIn our frame-by-frame feature prediction scenario, the recurrent neural network $\\mathcal{H}(\\cdot)$ takes in previous hidden state $\\bm h_{n-1}$ and the previous feature vector $\\bm c_{n-1}$, to predict the next frame $\\hat{\\bm c}_n $. Additionally, we condition the frame-level prediction with pitch parameters $\\bm m_n$ (period and correlation) used in LPCNet. Our model consists of two gated recurrent unit (GRU) layers \\cite{ChoK2014emnlp}, with 384 and 128 hidden units, respectively, followed by a fully connected layer. The feature predictor $\\mathcal{F}_\\text{pred}: \\bm c_{n-1} \\mapsto \\hat{\\bm c}_n$ can therefore be recursively defined as,\n\\begin{equation}\\label{eq:prediction}\n \\bm h_n = \\mathcal{H}(\\bm c_{n-1}, \\bm h_{n-1}, \\bm m_n), \\quad\n \\hat{\\bm c}_n = \\text{tanh}(\\bm W \\bm h_{n}), \n\\end{equation}\nwhere $n$ represents the time-domain index. We found the results are more stable by scaling input and output features to the range of $[-1,1]$. To this end, the output linear layer employs a tanh activation after a linear combination with parameter $\\bm W$. Biases are omitted for brevity.\n\nWe optimize the model by minimizing the mean squared error (MSE) between the prediction and target $\\mathcal{L} = MSE(\\bm c_n, \\hat{\\bm c}_n)$. We chose it over the maximum log-likelihood approach with explicit Gaussian modeling of the features because modeling the cepstrum coefficients with Gaussian distributions was unreliable. \n\n\n\\subsubsection{Feature residual coding}\n\n\nWe employ the predictor in both the encoder and decoder to cover the information that can be inferred from the temporal dependency. Thus, for the decoder to recover the features, it is only necessary to provide the decoder with the residuals between the prediction and ground-truth features. This kind of explicit residual coding can lead to a more efficient coding scheme, given that our predictor model makes reliable predictions, especially in the areas of smooth signals, reducing the entropy of the residual. \n\n\n\nThe primary pipeline for residual coding is then summarized recursively as follows:\n\\begin{align}\n \\text{Encoder:} \\quad \\hat{\\bm c}_n &=\\mathcal{F}_\\text{pred}(\\bar{\\bm{c}}_{n-1}) \\label{enc_pred}\\\\\n \\bm r_n &= \\bm c_n - \\hat{\\bm c}_n\\\\\n \\bar{\\bm r}_n &= \\mathcal{Q}(\\bm r_n)\\quad \\text{(send it to the decoder)} \\label{qtz}\\\\\n \\bar{\\bm c}_n &= \\hat{\\bm c}_n + \\bar{\\bm r}_n \\quad \\text{(input for the next round } n+1 \\text{)} \\label{add-back}\n\\end{align}\n\nThe encoder explicitly computes the residual $\\bm r_n$, and then the quantizer $\\mathcal{Q}(\\cdot)$ converts it into a bitstring $\\bar{\\bm{r}}_n$ as the final code.\nNote that we opt to input the \\textit{noisy feature} $\\bar{\\bm c}$ instead of the original feature $\\bm c$ into the encoder's feature predictor (eq. \\eqref{enc_pred}) in order to match the encoder's output to the decoder's circumstance. Since the decoder does not have access to the original features, it has no choice but to use the noisy ones $\\bar{\\bm c}$ as the predictor's input. Therefore, by repeating the decoder's behavior in the encoder, we aim to guarantee that the residuals provided by the encoder are the accurate compensation for the decoder's feature prediction. \n\nThe decoder first pre-computes the prediction $\\hat{\\bm c}_n$, and then supplement it with the quantized residual $\\bar{\\bm r}_n$ received from the encoder to finalize the feature reconstruction $\\bar{\\bm c}_n$.\n\\begin{align}\n \\text{Decoder:} \\quad \\hat{\\bm c}_n &=\\mathcal{F}_\\text{pred}(\\bar{\\bm{c}}_{n-1})\\\\\n \\bar{\\bm c}_n &= \\hat{\\bm c}_n + \\bar{\\bm r}_n.\n\\end{align}\nWhen running $\\mathcal{F}_\\text{pred}$ in either the encoder or decoder, we starts with zero-initialized input $\\bar{\\bm c}_0 = \\bm 0$ , and iteratively update the input tensor with the model predictions . \n\n\n\n\n\n\n\n\\subsection{The sample-level vocoder: LPCNet}\n\nWe borrow LPCNet to complete time-domain synthesis on the decoder side. LPCNet takes as input pitch parameters $\\bm m_n$ and cepstrum features $\\bm c_n$. Then, it integrates LPC analysis into the neural generative model, i.e., $p_t = \\sum_{\\tau=1}^T a_\\tau \\hat{s}_{t-\\tau}$, which computes the prediction $p_t$ for the sample index $t$ by using $T$ previously estimated samples $\\hat{s}_{t-T:t-1}$. In this way, the burden of spectral envelop modeling is taken away from the neural network. The prediction coefficient $a_\\tau$ is computed only from the 18-band Bark-frequency cepstrum (the transmitted code in the original LPCNet coder \\cite{ValinJ2019lpcnetcoding}), forming a very compact bitstring. \nOn top of the DSP-based linear prediction, LPCNet also employs a WaveRNN network $\\mathcal{G}$ to estimate the prediction residuals ${e}_t$ directly in a causal manner:\n\\begin{equation}\n \\hat{e}_t = \\mathcal{G}(p_t, \\hat{s}_{ 0$ is even, let $k$ be the maximal element of $Y$ and write $Y= Y_0 \\cup \\lbrace k\\rbrace$. Then we obtain \\begin{equation}\n0=\\mathrm{tr}_{\\mathcal{H}}(\\lbrace \\psi_{Y_0}, \\psi_k \\rbrace) = 2\\mathrm{tr}_{\\mathcal{H}}(\\psi_{Y_0} \\psi_k) = 2\\mathrm{tr}_{\\mathcal{H}}(\\psi_{Y}).\n \\end{equation}\n Thus we obtain (\\ref{eq:trpsi0}) for any non-empty set $Y$.\n\\end{proof}\n\nLet the Hamiltonian $H$ be a Hermitian operator. $H$ generates a one parameter family of automorphisms, called time evolution, on $\\mathcal{B}$: defining the Liouvillian \\begin{equation}\n\\mathcal{L} = \\mathrm{i}[H,\\cdot], \\label{eq:Liouvillian}\n\\end{equation}\nfor any $\\mathcal{O}\\in\\mathcal{B}$ we define \\begin{equation}\n|\\mathcal{O}(t)) = \\mathrm{e}^{\\mathcal{L}t} |\\mathcal{O}).\n\\end{equation}\nIn fact, $\\mathrm{e}^{\\mathcal{L}t}$ is unitary, since \\begin{equation}\n(\\mathcal{O}(t)|\\mathcal{O}(t)) = 2^{-N\/2} \\mathrm{tr}_{\\mathcal{H}}\\left( \\left(\\mathrm{e}^{\\mathrm{i}Ht}\\mathcal{O}\\mathrm{e}^{-\\mathrm{i}Ht}\\right)^\\dagger\\left(\\mathrm{e}^{\\mathrm{i}Ht}\\mathcal{O}\\mathrm{e}^{-\\mathrm{i}Ht}\\right)\\right) = 2^{-N\/2}\\mathrm{tr}_{\\mathcal{H}}\\left( \\mathcal{O}^\\dagger \\mathcal{O}\\right) = (\\mathcal{O}|\\mathcal{O}). \\label{eq:Lunitary}\n\\end{equation}\nMore generally, using the cyclic properties of the trace, we conclude that for any $A,B\\in\\mathcal{B}$: \\begin{equation}\n(A|\\mathcal{L}|B) = -(B|\\mathcal{L}|A). \\label{eq:Lantisymmetric}\n\\end{equation}\n\nDefine the projection matrices\\begin{equation}\n\\mathbb{Q}_s |Y) = \\mathbb{I}[|Y| = s] |Y).\n\\end{equation}\nNote that \\begin{equation}\n\\sum_{s=0}^N \\mathbb{Q}_s = 1 \\label{eq:sumPs}\n\\end{equation}\n(with 1 the identity of $\\mathrm{End}(\\mathcal{B})$). We say that the non-null vectors of $\\mathbb{Q}_s$ correspond to \\emph{operators of size $s$}. \nGiven $\\mathcal{O}\\in \\mathcal{B}$, we say that \\begin{equation}\nP_s(\\mathcal{O},t) = \\frac{(\\mathcal{O}(t)| \\mathbb{Q}_s |\\mathcal{O}(t))}{(\\mathcal{O}(t)|\\mathcal{O}(t))} \\label{eq:PsOt}\n\\end{equation}\nis the probability that operator $\\mathcal{O}$ is size $s$ at time $t$. To see that this is a well-defined probability measure on $\\lbrace 0, 1,\\ldots, N\\rbrace$, observe that $\\mathbb{Q}_s$ is positive semidefinite and hence $P_s \\ge 0$; then from (\\ref{eq:sumPs}), \\begin{equation}\n\\sum_{s=0}^N P_s(\\mathcal{O},t) = 1,\n\\end{equation} \nfor any $\\mathcal{O}\\in \\mathcal{B}$ and $t\\in\\mathbb{R}$. For simplicity, we will drop the explicit $\\mathcal{O}$ in $P_s(t)$, as our formalism does not depend on the particular choice of operator.\n\n\n\\subsection{The SYK ensemble}\\label{sec:ensemble}\nThe SYK model corresponds to a random ensemble of Hamiltonians. Define \\begin{equation}\nF := \\lbrace X\\subset V : |X| = q\\rbrace\n\\end{equation}\nto be the set of all subsets of $V$ with exactly $q$ elements.\nFor each $X\\in F$, let $J_X$ be an independent and identically distributed (iid) Rademacher\\footnote{In the physics literature, the random variables $J_X$ are typically taken to be Gaussian. We expect that a very similar result to ours will hold in this case as well, but we found the combinatorial problem discussed in Section \\ref{sec:trans} to be a bit more elegant for Rademacher random variables.} random variable: \\begin{equation}\n\\mathbb{P}\\left[J_X = \\sigma \\right] = \\mathbb{P}\\left[J_X = -\\sigma \\right] = \\frac{1}{2}, \n\\end{equation}\nwhere \\begin{equation}\n\\sigma := \\left[2q\\left(\\begin{array}{c} N-1 \\\\ q-1 \\end{array}\\right) \\right]^{-1\/2}. \\label{eq:sigmadef}\n\\end{equation}\nThe $q$-local SYK model is the random ensemble of Hamiltonians $H$, corresponding to the random Hermitian matrix\n\\begin{equation}\nH := \\mathrm{i}^{q\/2} \\sum_{X\\in F} J_X \\prod_{i \\in X} \\psi_i := \\sum_{X\\in F} H_X.\n\\end{equation}\nThe randomness in the SYK ensemble is essential in our proof of operator growth bounds. Averages over the ensemble of random variables $\\lbrace J_X\\rbrace $ are denoted as $\\mathbb{E}[\\cdots]$, and probability is denoted as $\\mathbb{P}[\\cdots]$, as above.\nWe define $\\mathcal{L}_X := \\mathrm{i}[H_X,\\cdot]$.\n\\begin{prop} \\label{propk}\nIf $\\mathbb{Q}_s|\\mathcal{O}_s) = |\\mathcal{O}_s)$, $X\\in F$, and $\\mathbb{Q}_{s^\\prime} \\mathcal{L}_X|\\mathcal{O}_s) \\ne 0$, then there exists $k\\in\\mathbb{Z}^+$ for which \\begin{equation}\ns^\\prime -s = q+2-4k. \\label{eq:kdef}\n\\end{equation} \nIn particular, \\begin{equation}\n|s^\\prime - s| \\le q-2. \\label{eq:qm2max}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nSince $\\mathcal{L}_X|Y)$ is proportional to $|[\\psi_X,\\psi_Y])$, we analyze when $[\\psi_X,\\psi_Y]\\ne 0$ is possible. Without loss of generality we write \\begin{equation}\nZ = X\\cap Y, \\;\\;\\;\\;\\; V=X-Z, \\;\\;\\;\\;\\; W = Y-Z,\n\\end{equation}in which case it suffices to constrain\n\\begin{equation}\n\\lVert [\\psi_X,\\psi_Y]\\rVert = \\lVert [\\psi_V\\psi_Z,\\psi_W\\psi_Z]\\rVert = \\lVert [\\psi_V\\psi_Z,\\psi_W]\\psi_Z + \\psi_W [\\psi_V,\\psi_Z]\\psi_Z\\rVert.\n\\end{equation}\nBy repeated use of (\\ref{eq:psiYpsii}), if $A\\cap B = 0$, \\begin{equation}\n\\psi_A \\psi_B = (-1)^{|A||B|}\\psi_B\\psi_A, \\label{eq:psiApsiB}\n\\end{equation}\nhence $[\\psi_A,\\psi_B]\\ne 0$ if and only if $|A|$ and $|B|$ are both odd. Since $|V\\cap Z|$ is even, we conclude that $[\\psi_X,\\psi_Y]=0$ unless $|V|$ and $|Z|$ are both odd, in which case $|[\\psi_X,\\psi_Y])$ is proportional to $|\\psi_V\\psi_W)$. If $X\\in F$, then $|X|=q$, and so $\\mathbb{Q}_{s^\\prime}\\mathcal{L}_X\\mathbb{Q}_s|Y)\\ne 0$ only if $|Y|=s$ and \\begin{equation}\ns^\\prime = |X|+|Y|-2|X\\cap Y| = s+q - 2|X\\cap Y|.\n\\end{equation}\nSince $|X\\cap Y|$ is odd, we obtain the desired result.\n\\end{proof}\nSince by definition $s^\\prime > s$, we conclude that \\begin{equation}\n2k-1 < \\frac{q}{2}. \\label{eq:2kminus1}\n\\end{equation}\n\nIt will be useful to define the following partition of the set of all non-trivial operator sizes: \\begin{equation}\n\\lbrace 1,\\ldots ,N \\rbrace = \\bigcup_{l=0}^{N^\\prime} R_l \\label{eq:ldef}\n\\end{equation}\nwhere \\begin{equation}\nN^\\prime := \\left\\lceil \\frac{N-1}{q-2} \\right\\rceil \\label{eq:Nprimedef}\n\\end{equation}\nand \\begin{equation}\nR_l := \\left\\lbrace \\begin{array}{ll} \\lbrace 1\\rbrace &\\ l=0 \\\\ \\lbrace m \\in \\mathbb{Z}: (l-1)(q-2)+1s^\\prime} K_{ s^\\prime s}(t) \\sqrt{P_{s^\\prime}(t)} \\label{eq:prop2}\n\\end{equation}\nAnalogously, there exist functions $K_l: \\mathbb{R} \\rightarrow [-\\mathcal{K}_l, \\mathcal{K}_l]$ such that \\begin{equation}\n\\frac{\\mathrm{d}\\varphi_l}{\\mathrm{d}t} = K_{l-1}(t) \\varphi_{l-1}(t) - K_l(t) \\varphi_{l+1}(t), \\label{eq:prop2dos}\n\\end{equation}\n(recall $l$ was defined in (\\ref{eq:ldef})) so long as \\begin{equation}\n\\mathcal{K}_l = \\max\\left(\\max_{s\\in R_l} \\sum_{s^\\prime \\in R_{l+1}} \\mathcal{K}_{s^\\prime s}, \\max_{s^\\prime \\in R_{l+1}} \\sum_{s\\in R_{l}} \\mathcal{K}_{s^\\prime s} \\right) \\label{eq:Kldef}\n\\end{equation}\nand $K_{-1}(t) = K_{N^\\prime}(t)=0$. These latter restrictions simply restrict the dynamics to operators in blocks $R_0$ to $R_{N^\\prime}$.\n\\end{prop}\n\\begin{proof}\nFor simplicity in this proof, the $t$-dependence of $\\mathcal{O}$ is implicit; without loss of generality, we may take $\\lVert \\mathcal{O}\\rVert = 1$ by (\\ref{eq:Lunitary}). For $s\\in \\lbrace0,\\ldots,N\\rbrace$, let $|\\mathcal{A}_s)$ be a unit norm operator such that \\begin{equation}\n\\mathbb{Q}_s|\\mathcal{O}) = \\sqrt{P_s}|\\mathcal{A}_s),\n\\end{equation}\nand note that if $P_s \\ne 0$, $|\\mathcal{A}_s)$ is unique. Now from (\\ref{eq:Liouvillian}) and (\\ref{eq:PsOt}), \\begin{align}\n\\frac{\\mathrm{d}P_s}{\\mathrm{d}t} &= (\\mathcal{O}| [\\mathbb{Q}_s,\\mathcal{L}] |\\mathcal{O}) = \\sum_{s^\\prime} \\sqrt{P_sP_{s^\\prime}} \\left[(\\mathcal{A}_s|\\mathcal{L}|\\mathcal{A}_{s^\\prime})-(\\mathcal{A}_{s^\\prime}|\\mathcal{L}|\\mathcal{A}_{s})\\right] \\notag \\\\\n&= 2\\sqrt{P_s} \\sum_{s^\\prime < s} K_{ss^\\prime}(t)\\sqrt{P_{s^\\prime}} - 2\\sqrt{P_s} \\sum_{s^\\prime > s} K_{s^\\prime s}(t)\\sqrt{P_{s^\\prime}} \\label{eq:prop2first}\n\\end{align}\nwhere in the first line, we used (\\ref{eq:sumPs}) and in the second line we used (\\ref{eq:Lantisymmetric}) and defined \\begin{equation}\nK_{ss^\\prime}(t) = (\\mathcal{A}_s|\\mathbb{Q}_s\\mathcal{L}\\mathbb{Q}_{s^\\prime}|\\mathcal{A}_{s^\\prime}). \\label{eq:proofKss}\n\\end{equation}\nSince $\\mathrm{d}\\sqrt{P_s} = \\mathrm{d}P_s \/ 2\\sqrt{P_s}$, we obtain (\\ref{eq:prop2}). Combining (\\ref{eq:endBnorm}), (\\ref{eq:calKss}) and (\\ref{eq:proofKss}), we obtain (\\ref{eq:prop2}). \n\nThe analogue result for block probabilities is identically derived. In addition, observe that (\\ref{eq:qm2max}) implies that \\begin{equation}\n\\mathbb{Q}_{l^\\prime} \\mathcal{L} \\mathbb{Q}_l \\ne 0 \\text{ only if } |l^\\prime - l| \\le 1.\n\\end{equation}\nHence we obtain (\\ref{eq:prop2dos}) where \\begin{equation}\nK_l(t) := (\\mathcal{O}(t) | \\mathbb{Q}_{l+1} \\mathcal{L}\\mathbb{Q}_l | \\mathcal{O}(t)) \\le \\lVert \\mathbb{Q}_{l+1}\\mathcal{L}\\mathbb{Q}_l \\rVert := \\mathcal{K}_l.\n\\end{equation}\nUsing (\\ref{eq:endBnorm}): \n\\begin{align}\n\\mathcal{K}_l &= \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\frac{(\\mathcal{O}^\\prime|\\mathbb{Q}_{l+1}\\mathcal{L}\\mathbb{Q}_l|\\mathcal{O})}{\\sqrt{(\\mathcal{O}|\\mathcal{O})(\\mathcal{O}^\\prime|\\mathcal{O}^\\prime)}} \\le \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\sum_{\\substack{ s\\in R_l \\\\s^\\prime \\in R_{l+1} } }\\sqrt{P_s(\\mathcal{O})P_{s^\\prime}(\\mathcal{O}^\\prime)} \\lVert \\mathbb{Q}_{s^\\prime} \\mathcal{L}\\mathbb{Q}_s\\rVert \\notag \\\\\n&\\le \\sup_{\\mathcal{O},\\mathcal{O}^\\prime} \\sum_{\\substack{ s\\in R_l \\\\s^\\prime \\in R_{l+1} } }\\frac{P_s(\\mathcal{O})+P_{s^\\prime}(\\mathcal{O}^\\prime)}{2} \\mathcal{K}_{s^\\prime s}.\n\\end{align}\nA simple identity leads to (\\ref{eq:Kldef}).\n \\end{proof}\n\n(\\ref{eq:prop2dos}) can be interpreted as follows. $\\varphi_l(t)$ are the coefficients of the real-valued quantum wave function $|\\varphi(t)\\rangle$ of an auxiliary quantum mechanical system defined on the Hilbert space \\begin{equation}\n\\mathcal{H}_{\\mathrm{aux}} := \\mathbb{C}^{1+N^\\prime} := \\mathrm{span}\\lbrace |0\\rangle, |1\\rangle, \\ldots, |N^\\prime\\rangle \\rbrace;\n\\end{equation} \nthe latter basis states are defined such that \\begin{equation}\n\\varphi_l(t) := \\langle l|\\varphi(t)\\rangle. \\label{eq:auxvarphil}\n\\end{equation}\nThe auxiliary Hamiltonian is \\begin{equation}\nH_{\\mathrm{aux}}(t) := \\sum_{l=0}^{N^\\prime - 1} \\mathrm{i} K_l(t) \\left( |l\\rangle\\langle l-1| - |l-1\\rangle\\langle l| \\right). \\label{eq:Haux}\n\\end{equation}\nThe Schr\\\"odinger equation for this auxiliary quantum system is (\\ref{eq:prop2dos}). \n\n\\subsection{Lyapunov exponent}\nDefine the operator (block) size distribution \\begin{equation}\n\\mathbb{E}_{\\mathrm{s},t} \\left[ f(l) \\right] := \\sum_{l=0}^{N^\\prime} f(l) P_l(t).\n\\end{equation} A formal definition of the many-body Lyapunov exponent, heuristically defined in (\\ref{eq:introlyapunov}), is given by the growth rate of the logarithm of the average operator size $\\mathbb{E}_{\\mathrm{s},t}[l]$ (recall $l$ was defined in (\\ref{eq:ldef}). This Lyapunov growth is constrained by the following theorem, which is our first main result:\n\\begin{thm} \\label{lyapunovtheor}\nSuppose that there exist $c\\in \\mathbb{R}^+$ and $M \\in \\mathbb{Z}^+$ such that \\begin{equation}\n\\mathcal{K}_l \\le c(l+1)\\;\\;\\;\\; \\text{ if }l \\le M. \\label{eq:Kll}\n\\end{equation}\nThen for any $\\epsilon \\in \\mathbb{R}^+$, the many-body Lyapunov exponent obeys \\begin{equation}\n\\frac{\\log \\mathbb{E}_{\\mathrm{s},t}[l]}{t} := \\lambda(t) \\le 2c (1+\\epsilon) \\label{eq:lambdadef}\n\\end{equation}\nfor times \\begin{equation}\n|t| < \\frac{1}{4c(1+\\mathrm{e})}\\left[ \\log M - 2 - \\log \\log \\frac{N^{\\prime 3}}{2\\epsilon}\\right] . \\label{eq:scramblingtime}\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nWithout loss of generality we assume $t\\ge 0$. We begin with the following lemma. Note that here and below, we write $\\mathbb{E}_{\\mathrm{s},t}$ as $\\mathbb{E}_{\\mathrm{s}}$ for convenience.\n \\begin{lem}\nIf (\\ref{eq:Kll}) holds with $M\\ge N^\\prime-1$, then for $n\\in \\mathbb{Z}^+$, \n\\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[l^n \\right] \\le 4cn (1+\\mathrm{e}) \\left(\\mathbb{E}_{\\mathrm{s}}\\left[l^n \\right] + (\\mathrm{e}n)^n\\right) \\label{eq:ddtln}\n\\end{equation}\nIn the special case $n=1$, the following stronger inequality holds: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[l \\right] \\le c \\left(2\\mathbb{E}_{\\mathrm{s}} \\left[l \\right] + 1\\right). \\label{eq:ddtln1}\n\\end{equation}\\label{lmael}\n\\end{lem}\n\\begin{proof}\nWe begin by using (\\ref{eq:prop2dos}): for any non-decreasing function $f:\\mathbb{Z} \\rightarrow \\mathbb{R}$, \\begin{align}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[f(l) \\right] &= \\sum_{l=0}^{N^\\prime } f(l) \\left(2 \\varphi_l \\frac{\\mathrm{d}\\varphi_l}{\\mathrm{d}t}\\right) = 2\\sum_{l=0}^{N^\\prime} f(l) \\varphi_l \\left[ K_{l-1}\\varphi_{l-1} - K_l \\varphi_l\\right] = 2 \\sum_{l=0}^{N^\\prime- 1} K_l \\varphi_l \\varphi_{l+1} [ f(l+1)-f(l)] \\notag \\\\\n&\\le 2c\\sum_{l=0}^{N^\\prime- 1} \\varphi_l \\varphi_{l+1} (l+1) [ f(l+1)-f(l)] \\le c\\sum_{l=0}^{N^\\prime - 1} (P_l + P_{l+1}) (l+1) [f(l+1)-f(l)]. \\label{eq:fl1fl}\n\\end{align}\nIn particular, choosing $f(l)=l^n$, we may further loosen this inequality using elementary inequalities: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}} \\left[f(l) \\right] \\le 2c \\sum_{l=0}^{N^\\prime -1} P_l \\left( (l+1)^{n+1} - l^{n+1}\\right). \\label{eq:ddtln2}\n\\end{equation}\nNow observe that \\begin{equation}\n(l+1)^{n+1}-l^{n+1} = (n+1) l^n + \\sum_{k=0}^{n-1} \\left(\\begin{array}{c} n+1 \\\\ k \\end{array}\\right) l^k \\le (n+1)l^n + n(n+1) (l+1)^{n-1}.\n\\end{equation}\nNext, note the inequality \\begin{equation}\nn(l+1)^{n-1} < \\mathrm{e} l^n + (\\mathrm{e}n)^n \\label{eq:lma5ineqinterm}\n\\end{equation}\nwhich we derive by multiplying both sides of (\\ref{eq:lma5ineqinterm}) by $l^{-n}$, assuming $l>1$ (the inequality is trivial when $l=0$): \\begin{equation}\n\\frac{n}{l} \\left(1+\\frac{1}{l}\\right)^{n-1} < \\frac{n}{l} \\mathrm{e}^{n\/l} < \\mathrm{e} + \\left(\\frac{\\mathrm{e}n}{l}\\right)^n.\n\\end{equation}\nFor $n\\le l$, the first term on the right hand side is always at least as large as the middle term; for $n> l$, the second term on the right is larger. Combining (\\ref{eq:ddtln2}) and (\\ref{eq:lma5ineqinterm}), we obtain (\\ref{eq:ddtln}). \n\nFor the case $n=1$, we use that $f(l+1)-f(l)=1$. Directly plugging in to (\\ref{eq:fl1fl}) we obtain (\\ref{eq:ddtln1}).\n\\end{proof}\nThe next lemma shows that even when $K_l$ grow faster than (\\ref{eq:Kll}) at large $l$, $P_l(t)$ is very small for $l>M$ at early times. \n\\begin{lem}\\label{lmalarge}\nIf $K_l(t)$ obeys (\\ref{eq:Kll}), then \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{P}_{\\mathrm{s}}[ l>M] \\le 2\\mathrm{e}c^2(M+1)t \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right]. \\label{eq:PslM}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe begin by employing (\\ref{eq:prop2dos}): \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{P}_{\\mathrm{s}}[ l>M] = 2\\sum_{l=M+1}^{N^\\prime} \\varphi_l (K_{l-1} \\varphi_{l-1} - K_{l+1}\\varphi_{l+1}) = 2K_M \\varphi_M \\varphi_{M+1} \\le 2c(M+1)\\varphi_{M+1}. \\label{eq:PsgM}\n\\end{equation}\nIn the last inequality, we used (\\ref{eq:Kll}) along with $\\varphi_l(t)\\le 1$ for any $l$. Hence, to obtain (\\ref{eq:PslM}), it suffices to bound $\\varphi_{M+1}(t)$. \n\nLet $K \\in \\mathbb{R}^{(N^\\prime+1) \\times (N^\\prime+1)}$ correspond to the transition matrix whose entries are \\begin{equation}\nK_{l^\\prime l}(t) = K_l \\mathbb{I}(l=l^\\prime - 1) - K_{l^\\prime}(t) \\mathbb{I}(l^\\prime=l - 1).\n\\end{equation}\n(indices run from $l=0$ to $l=N^\\prime$). Hence $K$ is tridiagonal and antisymmetric. Let us define the orthogonal matrix $U(t,t^\\prime)$ by the differential equation \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} U(t,t^\\prime) = K(t) U(t,t^\\prime) , \\;\\;\\;\\;\\;\\; U(t^\\prime,t^\\prime) = 1.\n\\end{equation}\n$U(t,t^\\prime)$ generates the continuous time quantum walk with transition rates $K_l(t)$.\n\nNext, we define the quantum walk transition matrix $\\widetilde{K}(t)$ as follows: \\begin{equation}\n\\widetilde{K}_{l^\\prime l}(t) := K_l \\mathbb{I}(M>l=l^\\prime - 1) - K_{l^\\prime}(t) \\mathbb{I}(M>l^\\prime=l - 1).\n\\end{equation}\nThis matrix corresponds to excising the sites $l>M$ from the walk. We define an analogous time evolution operator \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\widetilde{U}(t,t^\\prime) = \\widetilde{K}(t) \\widetilde{U}(t,t^\\prime) , \\;\\;\\;\\;\\;\\; \\widetilde{U}(t^\\prime,t^\\prime) = 1.\n\\end{equation}\n\nNow we use the following integral identity:\\footnote{In the physics literature, this is called the integral form of the memory matrix formula \\cite{zwanzig, mori, forster}.} \\begin{align}\n\\varphi_{M+1}(t) &= U_{M+1,0}(t,0) = \\widetilde{U}_{M+1,0}(t,0) + \\int\\limits_0^t \\mathrm{d}t^\\prime \\sum_{l,l^\\prime} U_{M+1,l^\\prime}(t,t^\\prime) (K_{l^\\prime l}(t) - \\widetilde{K}_{l^\\prime l}(t) ) \\widetilde{U}_{l,0}(t^\\prime,0). \\label{eq:memorymatrix}\n\\end{align}\nDue to the fact that $\\widetilde{U}$ does not evolve into sites with $l>M$, we can immediately simplify (\\ref{eq:memorymatrix}):\n\\begin{equation}\n\\varphi_{M+1}(t) = \\int\\limits_0^t \\mathrm{d}t^\\prime \\; K_M(t^\\prime) \\; U_{M+1,M+1}(t,t^\\prime)\\widetilde{U}_{M,0}(t^\\prime,0).\n\\end{equation}\nUsing (\\ref{eq:Kll}) along with orthogonality of $U(t,t^\\prime)$ and the triangle inequality: \\begin{equation}\n\\varphi_{M+1}(t) \\le c(M+1) \\int\\limits_0^t \\mathrm{d}t^\\prime \\; \\widetilde{U}_{M,0}(t^\\prime,0). \\label{eq:memorymatrix2}\n\\end{equation}\n\nWe now recognize that \\begin{equation}\n\\widetilde{U}_{M,0}(t^\\prime,0) =\\widetilde{\\varphi}_M(t^\\prime) \\label{eq:widetildevarphi}\n\\end{equation}\nis the solution to the blocked quantum walk generated by $\\widetilde{K}$. This blocked quantum walk obeys Lemma~\\ref{lmael}; integrating (\\ref{eq:ddtln}), we obtain \\begin{equation}\n\\mathbb{E}_{\\widetilde{\\mathrm{s}}} \\left[ l^n \\right] \\le (\\mathrm{e}n)^n \\left( \\mathrm{e}^{4c(1+\\mathrm{e})nt}-1 \\right).\n\\end{equation}\nHere $\\mathbb{E}_{\\widetilde{\\mathrm{s}}} [\\cdots]$ denotes averages in the probability distribution of the blocked quantum walk. Using Markov's inequality, \\begin{equation}\n\\widetilde{\\varphi}_M(t) \\le \\inf_{n\\in\\mathbb{Z}^+} \\frac{\\mathbb{E}_{\\widetilde{\\mathrm{s}}} \\left[ l^n \\right]}{M^n} < \\inf_{n\\in\\mathbb{Z}^+} \\left(\\frac{\\mathrm{e}^{1+4c(1+\\mathrm{e})t}n}{M} \\right)^n \\le \\exp\\left[1 -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right], \\label{eq:widetildevarphiM}\n\\end{equation}\nwhere in the last step we used the following sequence of inequalities for $z\\in \\mathbb{R}^+$: \\begin{equation}\n\\inf_{n\\in\\mathbb{Z}^+} \\left(\\frac{n}{z}\\right)^n \\le \\left(\\frac{1}{z} \\left\\lfloor \\frac{z}{\\mathrm{e}}\\right\\rfloor \\right)^{\\lfloor z\/\\mathrm{e}\\rfloor} \\le \\exp \\left[ - \\left\\lfloor \\frac{z}{\\mathrm{e}}\\right\\rfloor\\right] < \\exp \\left[ 1- \\frac{z}{\\mathrm{e}}\\right] . \n\\end{equation}\n\nCombining (\\ref{eq:PsgM}), (\\ref{eq:memorymatrix2}), (\\ref{eq:widetildevarphi}) and (\\ref{eq:widetildevarphiM}), and using the fact that our bound on $\\widetilde{\\varphi}_M(t)$ is a monotonically increasing function of time, we obtain (\\ref{eq:PslM}).\n\\end{proof}\n\nThe last step is to combine (\\ref{eq:ddtln1}) with Lemma \\ref{lmalarge} to bound the true Lyapunov exponent. Defining the non-decreasing functions \\begin{subequations}\\begin{align}\nf_>(l) &:= (l-M) \\mathbb{I}[l>M], \\\\\nf_<(l) &:= l - f_>(l),\n\\end{align}\\end{subequations} we write \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[l] = \\mathbb{E}_{\\mathrm{s}}[ f_<(l) + f_>(l)]\n\\end{equation}\nand bound each piece separately. Using the fact that (\\ref{eq:PslM}) is an increasing function of $t$: \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[f_>(l)] \\le (N^\\prime - M) \\mathbb{P}_{\\mathrm{s}}[l>M] \\le 2\\mathrm{e}c^2(M+1)N^\\prime t^2 \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right].\n\\end{equation}\nThen using (\\ref{eq:fl1fl}) and $f_<(l+1) \\le f_<(l) + 1$: \\begin{equation}\n\\frac{\\mathrm{d}}{\\mathrm{d}t} \\mathbb{E}_{\\mathrm{s}}[f_<(l)] \\le c \\sum_{l=0}^{N^\\prime } (2l+1) P_l(t) = c\\left(2\\mathbb{E}_{\\mathrm{s}} \\left[l \\right] + 1\\right).\n\\end{equation}\nWe conclude that \\begin{equation}\n\\mathbb{E}_{\\mathrm{s}}[l] \\le \\frac{\\mathrm{e}^{2ct}-1}{2} + 2\\mathrm{e}c^2(M+1)N^\\prime t^2 \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right].\n\\end{equation}\nUsing the definition of $\\lambda(t)$ in (\\ref{eq:lambdadef}) and the concavity of the logarithm, along with $\\log x < x$: \\begin{equation}\n\\lambda(t) \\le 2c \\left[ 1 + \\mathrm{e}(M+1)N^\\prime c t \\exp\\left[ -M \\mathrm{e}^{-2-4c(1+\\mathrm{e})t} \\right] \\right].\n\\end{equation}\n\nLet us define \\begin{equation}\nt = \\frac{\\log M - 2 - r}{4c(1+\\mathrm{e})}.\n\\end{equation}\nThen, using $M+1 \\le 2N^\\prime$ and $\\log M \\le N^\\prime$: \\begin{equation}\n\\frac{\\lambda}{2c} \\le 1 + \\frac{\\mathrm{e} N^{\\prime 3} }{2(1+\\mathrm{e})} \\exp\\left[- \\mathrm{e}^r \\right] < 1 + \\frac{ N^{\\prime 3} }{2} \\exp\\left[- \\mathrm{e}^r \\right].\n\\end{equation}\nDemanding that the inequality in (\\ref{eq:lambdadef}) holds and solving for $r$, we obtain (\\ref{eq:scramblingtime}).\n\\end{proof}\n\nThis theorem can be interpreted as follows. For any $0<\\kappa<1$, define the operator scrambling time \\begin{equation}\nt_{\\mathrm{s},\\kappa} = \\inf \\left\\lbrace t\\in \\mathbb{R}^+ : \\mathbb{E}_{\\mathrm{s},t}[l] \\ge \\kappa N^\\prime \\right\\rbrace .\n\\end{equation} \n(Recall that $N^\\prime$ is the maximal value of $l$, as defined in (\\ref{eq:Nprimedef})). It was conjectured in \\cite{sekino} that a quantum ``scrambling time\" $t_{\\mathrm{s}} = \\mathrm{\\Omega}(\\log N)$ would necessarily grow at least logarithmically with the number of degrees of freedom in any system with few-body interactions. For example, in the SYK model, we would demand that $q$ is finite. In recent years, this operator scrambling time has become the preferred definition of scrambling in the physics literature, though this is likely out of convenience \\cite{lucas1805}.\nTheorem \\ref{lyapunovtheor} implies that $t_{\\mathrm{s},\\kappa} = \\mathrm{O}(\\log N)$, as summarized in the following corollary:\n\\begin{cor} \\label{corolscramble}\nIf (\\ref{eq:Kll}) holds for $M = N^\\alpha$ for $0<\\alpha<1$, then there exists an $N$-independent $b \\in \\mathbb{R}^+$ for which \\begin{equation}\nt_{\\mathrm{s},\\kappa } \\ge b \\log N \\label{eq:scrambleb}\n\\end{equation}\nfor $N>N_0$, for some finite $N_0\\in \\mathbb{Z}^+$.\n\\end{cor}\n\\begin{proof}\nThere exists an $N_*\\in\\mathbb{Z}^+$ such that \\begin{equation}\n\\frac{\\alpha \\log N_* }{8c(1+\\mathrm{e})} < \\frac{1}{4c(1+\\mathrm{e})} \\left[ \\alpha \\log N_* - 2 -\\log\\log N_*^{\\prime 3}\\right].\n\\end{equation}\nSuppose that $N>N_*$. Using Theorem \\ref{lyapunovtheor}, we conclude that at time $t=t_{\\mathrm{s}}$, where $t_{\\mathrm{s}}$ is given by (\\ref{eq:scrambleb}) where \\begin{equation}\nb :=\\frac{\\alpha }{8c(1+\\mathrm{e})} ,\n \\end{equation}\n \\begin{equation}\n \\mathbb{E}_{\\mathrm{s}}[l] \\le \\exp\\left[ \\frac{3\\alpha \\log N}{8(1+\\mathrm{e})}\\right] = N^{3\\alpha\/8(1+\\mathrm{e})}\n \\end{equation}\n We conclude that the corollary holds so long as $N_0$ is chosen such that $\\kappa N_0 > N_0^{3\\alpha\/8(1+\\mathrm{e})}$ and $N_0 \\ge N_*$.\n\\end{proof}\n\nWe emphasize that the results of this section are completely general, and apply to a large family of models beyond the SYK model, as soon as (\\ref{eq:Kll}) can be proved.\n\n\\section{Operator growth in the SYK ensemble}\n\n\\subsection{Bounding the transition rates}\\label{sec:trans}\nWhat remains is to show that (\\ref{eq:Kll}) holds in the SYK ensemble, with very high probability, at large $N$. Proving this fact constitutes the second main result of this paper. The result is summarized in the following theorem:\n\\begin{thm}\\label{theorSYK}\nLet $\\kappa \\in \\mathbb{R}^+$ and $\\theta \\in \\mathbb{R}^+$ obey \\begin{subequations}\\begin{align}\n2\\kappa \\log N + 2 &< \\sqrt{N}, \\label{eq:kappasqrtNN} \\\\\n2(q-2) &< N^{\\kappa\\theta} - 1, \\\\\n2q (1+\\sqrt{N})&<(q\\kappa \\log N - 1) \\sqrt{N} . \\label{eq:annoying} \n\\end{align}\\end{subequations}\nLet us also assume that \\begin{equation}\nq < \\frac{N}{2}. \\label{eq:qlessN2}\n\\end{equation}Then, in the SYK model introduced in Section \\ref{sec:ensemble}, with probability at least \\begin{equation}\n\\mathbb{P}_{\\mathrm{success}} \\ge 1 - \\frac{2(q-2)}{N^{\\kappa \\theta}-1},\n\\end{equation}\n(\\ref{eq:Kll}) is obeyed with \n\\begin{subequations}\\label{eq:theorcm}\\begin{align}\n c &= \\mathrm{e}^{\\theta + 1\/\\kappa} \\left[\\sqrt{\\frac{2(q-2)}{q}} \\left(1-\\frac{2\\theta}{5\\kappa \\sqrt{N}\\log N}\\right)^{-1} + \\frac{8}{q^{q-9\/2}N^{1\/4}} \\left(\\frac{4\\theta}{5\\kappa \\log N} \\right)^{(q-3)\/2} \\right] , \\\\\n M &= \\left\\lfloor \\frac{\\theta}{5\\kappa} \\frac{\\sqrt{N}}{q^3\\log N} \\right\\rfloor -1 .\n\\end{align}\\end{subequations}\n\\end{thm}\n\\begin{proof}\nOur strategy will be to work primarily with $\\mathcal{K}_{s^\\prime s}$. At the very end of the calculation, we will use (\\ref{eq:Kldef}) to bound $\\mathcal{K}_l$. We begin with the following proposition:\n\\begin{prop}\nDefine the symmetric and positive semidefinite matrix \\begin{equation}\nM_{s^\\prime s} = \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}} \\mathbb{Q}_{s^\\prime}\\mathcal{L}\\mathbb{Q}_s. \\label{eq:MssDef}\n\\end{equation}\nIf the maximal eigenvalue of $M_{s^\\prime s}$ is $\\mu_{s^\\prime s}$, \\begin{equation}\n\\mathcal{K}_{s^\\prime s} = \\sqrt{\\mu_{s^\\prime s}}. \\label{eq:Kmu}\n\\end{equation} \\label{propmuss}\n\\end{prop}\n\\begin{proof}\n Let $\\mathcal{O}\\in \\mathcal{B}$ obey $\\lVert \\mathcal{O}\\rVert = 1$, and define \\begin{equation}\n|\\mathcal{O}^\\prime) = \\mathbb{Q}_{s^\\prime}\\mathcal{L}\\mathbb{Q}_s|\\mathcal{O}).\n\\end{equation} From (\\ref{eq:endBnorm}) and (\\ref{eq:calKss}), we see that $\\mathcal{K}_{s^\\prime s}$ is simply the maximal length of the vector $|\\mathcal{O}^\\prime)$. Now observe that $M_{s^\\prime s}$ gives us a very simple way of measuring the length of $|\\mathcal{O}^\\prime)$. Therefore,\n \\begin{equation}\n\\mathcal{K}_{s^\\prime s}^2 = \\sup_{\\mathcal{O}\\in\\mathcal{B}} (\\mathcal{O}^\\prime | \\mathcal{O}^\\prime ) = \\sup_{\\mathcal{O}\\in\\mathcal{B}} (\\mathcal{O}|M_{s^\\prime s}|\\mathcal{O}) = \\mu_{s^\\prime s},\n\\end{equation}\nwhere for the last equality we used a variational principle which holds for a symmetric matrix.\n\\end{proof}\n\nDenote \\begin{equation}\nC_s := \\frac{N!}{s!(N-s)!},\n\\end{equation}\nand observe that $M_{s^\\prime s} \\in \\mathbb{R}^{C_s\\times C_s}$ is a positive semidefinite random matrix. From Markov's inequality, for any $p\\in\\mathbb{Z}^+$, \\cite{furedi} \n\\begin{equation}\n\\mathbb{P}\\left[\\mu_{s^\\prime s} \\ge a \\right] \\le \\frac{\\mathbb{E}\\left[ \\mu_{s^\\prime s}^p\\right]}{a^p}\\le \\frac{\\mathbb{E}\\left[ \\mathrm{tr}(M_{s^\\prime s}^p)\\right]}{a^p}. \\label{eq:markov4}\n\\end{equation}\nWe will choose $p=\\mathrm{O}(\\log C_s)$, so that the number of eigenvalues $C_s$ accounted for in the trace is irrelevant ($C_s^{1\/p} \\rightarrow 1$). Importantly, at finite size $s$, $p = \\mathrm{O}(s\\log N)$. We will see that this is sufficiently small to make bounding $\\mu_{s^\\prime s}$ analytically tractable. \n\n \nHence, let us define \\begin{equation}\nB_{s^\\prime s}^{(p)} := \\mathbb{E}\\left[\\mathrm{tr}\\left(M^p_{s^\\prime s}\\right)\\right]. \\label{eq:BssDef}\n\\end{equation}\nWe analyze the average $\\mathbb{E}[\\cdots]$ over the random variables $J_X$ by converting it to a combinatorial problem. To do so, let us write out \\begin{equation}\nB^{(p)}_{s^\\prime s} = \\mathbb{E}\\left[ \\sum_{X_1,\\ldots, X_p, Y_1,\\ldots,Y_p \\in F} \\sum_{Z\\subseteq V} (Z| \\prod_{i=1}^p \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z) \\right] \\label{eq:explicittrace}\n\\end{equation}\nwhere the sum over $Z$ is a sum over the basis of Proposition \\ref{prop1}, without loss of generality. We now read (\\ref{eq:explicittrace}) from right to left, starting with $\\mathbb{Q}_s|Z)$, which restricts the subset $Z\\subseteq V$ to have exactly $s$ elements: $|Z|=s$, and $Z=\\lbrace i_1,i_2,\\ldots, i_s\\rbrace$. We draw a graph $G$ which we associate to $\\mathbb{Q}_s |Z)$: \\begin{equation} \\label{eq:minimalgraph}\n\\begin{tikzpicture}\n\\draw (-0.5,0) node[left] {$\\mathbb{Q}_s |i_1\\cdots i_s) \\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (0.8, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (0.8, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\draw (1.4, 0.25) node {$\\cdots$};\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $i_1$};\n\\draw (0.8, 0.35) node[above] {\\color{blue} \\footnotesize $i_2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $i_s$};\n\\end{tikzpicture}\n\\end{equation}\nwhere the (blue) circles denote the fermions, and the orange triangle is a ``root\" to the graph -- it has edges drawn to the $s$ original fermions in the operator $\\mathbb{Q}_s |Z)$ (recall that $|Z)$ corresponds to a product operator). We have written a $\\sim$ in (\\ref{eq:minimalgraph}) because we will not bother to keep track of an overall sign in the vector, although its orientation in $\\mathcal{B}$ and its dependence on any random variables $J_X$ are each important. For simplicity, let us assume that $s^\\prime=s+q-2$. Without loss of generality, we assume that $Y_p \\cap Z = \\lbrace i_s\\rbrace$, $Y=\\lbrace i_s,j_1,\\ldots, j_{q-1}\\rbrace$. Then, we draw \n\\begin{equation} \\label{eq:graphtree}\n\\begin{tikzpicture}\n\\draw (-0.5, 0.3) node[left] {$\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s|Y)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (0.8, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25) -- (3, 0.25) -- (2.2, 0.85);\n\\draw (2.7, 0.85) -- (3,0.25) -- (3.7, 0.85);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (0.8, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\fill[color=blue] (2.2, 0.85) circle (3pt);\n\\fill[color=blue] (2.7, 0.85) circle (3pt);\n\\fill[color=blue] (3.7, 0.85) circle (3pt);\n\\fill[color=red] (2.9, 0.15) rectangle (3.1, 0.35);\n\\draw (1.4, 0.25) node {$\\cdots$};\n\\draw (3.2, 0.85) node {$\\cdots$};\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $i_1$};\n\\draw (0.8, 0.35) node[above] {\\color{blue} \\footnotesize $i_2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $i_s$};\n\\draw (2.2, 0.95) node[above] {\\color{blue} \\footnotesize $j_1$};\n\\draw (2.7, 0.95) node[above] {\\color{blue} \\footnotesize $j_2$};\n\\draw (3.7, 0.95) node[above] {\\color{blue} \\footnotesize $j_{q-1}$};\n\\draw (3, 0.15) node[below] {\\color{red} \\footnotesize $Y_p$};\n\\end{tikzpicture}.\n\\end{equation}\nThe way to read this graph is as follows: the coupling (factor, drawn as a red square) $Y_p$ connected to the fermion $i_s$, and spawned the fermions $j_1,\\ldots, j_{q-1}$. Each fermion (circle) with an odd degree is present in the operator; those with an even degree are not present. Because of the projectors $\\mathbb{Q}_{s^\\prime}$ and $\\mathbb{Q}_s$, we had to start with an operator of size $s$ and add exactly $q-2$ net fermions. From (\\ref{eq:psiApsiB}), we know that this vector is proportional to one of our simple basis vectors (a product operator), which is why we can simply draw the graph (so long as we neglect the proportionality coefficient). The fermions do not directly connect to each other, but rather connect through the factors. \n\nLet us continue and study the operator $\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z)$. It is easiest to first illustrate the possibilities with a simple example. Consider the theory with $s=3$, $s^\\prime=5$, $q=4$. Let us first consider the theory where $X_p=Y_p = \\lbrace 3,4,5,6\\rbrace$ and $Z=\\lbrace 1,2,3\\rbrace$. Then we draw \n\\begin{equation}\\label{eq:Xpexample1}\n\\begin{tikzpicture}\n\\draw (-0.5,0) node[left] {$\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_p}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s |Z)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (1.15, 0.25);\n\\draw (1.15, -0.4) -- (2, 0.25);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (1.15, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $1$};\n\\draw (1.15, 0.35) node[above] {\\color{blue} \\footnotesize $2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $3$};\n\\end{tikzpicture}\n\\end{equation}\nwhere the absence of the factor $Y_p$ reminds us that since $J_{Y_p}$ has appeared twice in the sequence, this sequence is non-trivial under random averaging. We neglect to draw any fermion or factor which has degree zero, which is why the fermions 4, 5 and 6 are not shown. \n\nHowever, suppose instead $X_p = \\lbrace 2,4,5,7\\rbrace$. In this case, \n\\begin{equation}\\label{eq:Xpexample2}\n\\begin{tikzpicture}\n\\draw (-0.5,0.3) node[left] {$\\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_p}\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s |Z)\\sim$};\n\\draw (1.15, -0.4) -- (0.3, 0.25);\n\\draw (1.15, -0.4) -- (1.15, 0.25) -- (1.8, 1.1) -- (1, 1.1);\n\\draw (1.15, -0.4) -- (2, 0.25) -- (3, 0.25) -- (2.3, 0.85);\n\\draw (3, 0.85) -- (3,0.25) -- (3.7, 0.85);\n\\draw (3, 0.85) -- (1.8, 1.1) -- (2.3, 0.85);\n\\fill[color=orange] (1,-0.5) -- ++(60:0.25) -- ++(-60:0.25) -- cycle;\n\\fill[color=blue] (0.3, 0.25) circle (3pt);\n\\fill[color=blue] (1.15, 0.25) circle (3pt);\n\\fill[color=blue] (2, 0.25) circle (3pt);\n\\fill[color=blue] (2.3, 0.85) circle (3pt);\n\\fill[color=blue] (3, 0.85) circle (3pt);\n\\fill[color=blue] (3.7, 0.85) circle (3pt);\n\\fill[color=blue] (1,1.1) circle (3pt);\n\\fill[color=red] (2.9, 0.15) rectangle (3.1, 0.35);\n\\fill[color=red] (1.7, 1) rectangle (1.9, 1.2);\n\\draw (0.3, 0.35) node[above] {\\color{blue} \\footnotesize $1$};\n\\draw (1.15, 0.35) node[above] {\\color{blue} \\footnotesize $2$};\n\\draw (2, 0.35) node[above] {\\color{blue} \\footnotesize $3$};\n\\draw (2.3, 0.95) node[above] {\\color{blue} \\footnotesize $4$};\n\\draw (3, 0.95) node[above] {\\color{blue} \\footnotesize $5$};\n\\draw (3.7, 0.95) node[above] {\\color{blue} \\footnotesize $6$};\n\\draw (1,1.2) node[above] {\\color{blue} \\footnotesize $7$};\n\\draw (1.8, 1.2) node[above] {\\color{red} \\footnotesize $X_p$};\n\\draw (3, 0.15) node[below] {\\color{red} \\footnotesize $Y_p$};\n\\end{tikzpicture}\n\\end{equation}\nBecause the factors $X_p \\ne Y_p$, we must draw both of them, together with an edge to all vertices\/fermions $i\\in X_p$ or $Y_p$. We can only remove a factor when that exact factor shows up a second time. And if a factor shows up a third time, it is redrawn in, and so on. Note that the only factors that $X_p$ can be are those which destroy 3 fermions and create one, in this simple example. \n\nIt is straightforward to generalize these rules, which we summarize one more time. If the next factor in the sequence is present in the existing factor graph, that factor is deleted along with its edges to $q$ fermions. Any fermions which subsequently have degree zero are removed. If the factor is new, we draw that factor and $q$ edges to its fermions. The number of odd degree fermions in each graph is fixed by the projectors to alternate between $s$ and $s^\\prime$. \n\n\nOur next goal is to throw away detailed information about what specific factors and fermions appeared, and to only keep track of the sequence of graphs. Let $\\mathcal{G}$ be the space of all graphs, modulo graph isomorphism. Two graphs $G_1$ and $G_2$ are isomorphic if and only if there is a permutation on fermions $\\pi \\in \\mathrm{S}^V$ such that $\\pi \\cdot G_1 = G_2$ (the group action on fermions and factors is canonical, while the root is invariant). We define $G_\\triangle$ to be the unique (up to isomorphism) element of $\\mathcal{G}$ with zero factors and $s$ fermions connected to the root, as in (\\ref{eq:minimalgraph}). Let $\\mathcal{G}_-$ be the subset of $\\mathcal{G}$ consisting of $s$ odd degree fermions and no more than $p$ factors, and $\\mathcal{G}_+$ be the subset of $\\mathcal{G}$ with $s^\\prime$ odd degree fermions, subject to the constraint that any graph in $\\mathcal{G}_+$ or $\\mathcal{G}_-$ can be reached by adding and removing factors to $G_\\triangle$, according to the rules above, and with no intermediate graphs containing more than $p$ factors. \n\nDefine $\\langle G_2| \\mathcal{N}_+|G_1 \\rangle$ to be the number of factors $X$ which can be added (or removed) to any fixed graph $G$ isomorphic to $G_1 \\in \\mathcal{G}_-$, to create any graph isomorphic to $G_2 \\in \\mathcal{G}_+$. Similarly, we define $\\langle G_1| \\mathcal{N}_-|G_2 \\rangle$ to be the number of factors which take a fixed graph isomorphic to $G_2\\in \\mathcal{G}_+$ to any graph isomorphic to $G_1 \\in \\mathcal{G}_-$. We interpret $\\mathcal{N}_+ : \\mathbb{Z}^{\\mathcal{G}_+ \\times \\mathcal{G}_-} \\rightarrow \\mathbb{Z}$ and $\\mathcal{N}_- : \\mathbb{Z}^{\\mathcal{G}_- \\times \\mathcal{G}_+} \\rightarrow \\mathbb{Z}$ as integer-valued matrices, using the angle bra-ket notation to denote matrix elements. Many of these matrix elements are zero. $\\mathcal{N}_+$ and $\\mathcal{N}_-$ are both non-negative matrices.\n\n\\begin{prop}\n\\label{propcomb}\nIf $G_0 = G_p = G_\\triangle$, then \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le \\left(4\\sigma^2\\right)^p C_s \\sum_{ \\substack{G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- \\\\H_1,\\ldots, H_p \\in \\mathcal{G}_+ } }\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle . \\label{eq:combineq}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof} We expand out the sums in (\\ref{eq:explicittrace}) over all possible couplings. Using the algorithm described above to associate a graph with each of the $2p+1$ operators $\\mathbb{Q}_s|Z)$, $\\mathbb{Q}_{s^\\prime}\\mathcal{L}_{Y_p}\\mathbb{Q}_s|Z)$, etc., we may convert the sequence of factors \\begin{equation}\n\\mathcal{Y}_Z := (Y_p, X_p, Y_{p-1},X_{p-1},\\ldots, Y_1,X_1)_Z \\label{eq:calYZ}\n\\end{equation}\nread right to left, along with the initial operator $|Z)$, into a sequence of $2p+1$ graphs \\begin{equation}\n\\mathcal{Z}(\\mathcal{Y}_Z) := (G_0, H_1, G_1, H_2,\\ldots, H_p, G_p) \\label{eq:calZdef}\n\\end{equation}\nwith $G_0=G_\\triangle$. Due to the projectors $\\mathbb{Q}_s$ and $\\mathbb{Q}_{s^\\prime}$, any sequence $\\mathcal{Y}$ which (before disorder averaging) is not zero must map to a sequence $\\mathcal{Z}(\\mathcal{Y}_Z)$ in which $G_i \\in \\mathcal{G}_-$ and $H_i \\in \\mathcal{G}_+$. When calculating $B_{s^\\prime s}^{(p)}$, we require that $G_p = G_\\triangle$; otherwise, there is a coupling which appears an odd number of times in $\\mathcal{Y}$, so the disorder average of that sequence vanishes. We define \\begin{equation}\n\\mathcal{Z}_{s^\\prime s}^{(p)} := \\lbrace (G_\\triangle, H_1, G_1, H_2,\\ldots, H_p, G_\\triangle) : H_i \\in \\mathcal{G}_+,G_i \\in \\mathcal{G}_- \\rbrace. \n\\end{equation} Since only one factor can be added or removed in each step, it is not possible to have more than $p$ factors in any graph in $\\mathcal{Z}(\\mathcal{Y}_Z)$. We define the equivalence relation $\\mathcal{Y}_1 \\sim \\mathcal{Y}_2$ if and only if $\\mathcal{Z}(\\mathcal{Y}_1) = \\mathcal{Z}(\\mathcal{Y}_2)$, and denote $\\mathcal{Y}_{1} \\in \\mathcal{Z}(\\mathcal{Y})$.\n\nWe then write \\begin{equation}\nB_{s^\\prime s}^{(p)} = \\sum_{Z\\subseteq V} \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} (Z| \\prod_{i=1}^p \\mathbb{Q}_s \\mathcal{L}^{\\mathsf{T}}_{X_i}\\mathbb{Q}_{s^\\prime} \\mathcal{L}_{Y_i}\\mathbb{Q}_s |Z).\n\\end{equation}\nDue to the Rademacher distribution on the random variables $J_X$, the expectation value has become trivial, encoded in the fact that the graph sequence $\\mathcal{Z}$ ends at $G_\\triangle$. For other distributions on $J_X$, the sum above must be weighted in a more complicated way when the sum involves $\\mathbb{E}[J_X^{2k}]$ for $k>1$. We now apply the triangle inequality together with $(Z|Z)=1$: \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le \\sum_{Z\\subseteq V : |Z|=s} \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\prod_{i=1}^p \\lVert \\mathcal{L}_{X_i}\\rVert \\lVert \\mathcal{L}_{Y_i}\\rVert = \\sum_{\\mathcal{Z} \\in \\mathcal{Z}_{s^\\prime s}^{(p)}} \\sum_{Z\\subseteq V : |Z|=s} \\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\left(2\\sigma\\right)^{2p}.\n\\end{equation}\nIn the last step, we used (\\ref{eq:psiApsiB}) along with the fact that we may exchange the first two sums, whose summands are independent. It now remains to evaluate each sum in turn. By definition of $\\mathcal{N}_+$ and $\\mathcal{N}_-$: \\begin{equation}\n\\sum_{\\mathcal{Y}_Z\\in \\mathcal{Z}} \\left(2\\sigma\\right)^{2p} = \\left(4\\sigma^2\\right)^p \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle,\n\\end{equation}\nusing (\\ref{eq:calZdef}) and $G_0=G_\\triangle$. By permutation symmetry, \\begin{equation}\n\\sum_{Z\\subseteq V : |Z|=s}\\left(4\\sigma^2\\right)^p \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle = \\left(4\\sigma^2\\right)^pC_s \\prod_{i=1}^p \\langle G_i |\\mathcal{N}_-|H_i\\rangle \\langle H_i | \\mathcal{N}_+ |G_{i-1}\\rangle\n\\end{equation}\nHence we obtain (\\ref{eq:combineq}).\n\\end{proof}\n\n\\begin{prop}\n\\label{propsym}\nLet $\\mathcal{Z} = (G_\\triangle, H_1, G_1, \\ldots, G_{p-1}, H_p, G_\\triangle) \\in \\mathcal{Z}_{s^\\prime s}^{(p)}$. Then \\begin{equation}\n\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle = \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle .\\label{eq:symid}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nPick any $Z\\subseteq V$ with $|Z|=s$. Given a sequence of factors $\\mathcal{Y}_Z$, given by (\\ref{eq:calYZ}), with $\\mathcal{Y}_Z \\in \\mathcal{Z}$, define the reversed sequence \\begin{equation}\n\\mathcal{Y}_Z^{\\mathrm{r}} := (X_1,Y_1,\\ldots, X_p, Y_p)_Z.\n\\end{equation}\nwhich corresponds to factors in (\\ref{eq:explicittrace}) read left to right instead. By construction, $\\mathcal{Y}_Z^{\\mathrm{r}} \\in \\mathcal{Z}^{\\mathrm{r}}$, defined by \\begin{equation}\n\\mathcal{Z}^{\\mathrm{r}} := (G_\\triangle, H_p, G_{p-1}, \\ldots, G_1, H_1, G_\\triangle).\n\\end{equation}\nClearly, $\\left(\\mathcal{Y}_Z^{\\mathrm{r}}\\right)^{\\mathrm{r}} = \\mathcal{Y}_Z$ and $\\left(\\mathcal{Z}^{\\mathrm{r}}\\right)^{\\mathrm{r}} = \\mathcal{Z}$. As each sequence $\\mathcal{Y}_Z$ has a unique reverse $\\mathcal{Y}_Z^{\\mathrm{r}}$, \\begin{equation}\n\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle = \\sum_{\\mathcal{Y}_Z \\in \\mathcal{Z}} 1 = \\sum_{\\mathcal{Y}_Z^{\\mathrm{r}} \\in \\mathcal{Z}^{\\mathrm{r}}} 1 = \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle ,\n\\end{equation}\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lma12}\nDefine a transfer matrix $\\mathcal{M}_{s^\\prime s}^{(p)}\\in \\mathbb{R}^{\\mathcal{G}_s^{(p)} \\times \\mathcal{G}_s^{(p)}} $ component wise as \\begin{equation} \n\\langle G_1|\\mathcal{M}_{s^\\prime s}^{(p)} |G_2\\rangle = \\sum_{H\\in \\mathcal{G}_{s^\\prime}^{(p)}} \\langle G_1|\\mathcal{N}_-|H\\rangle\\langle H|\\mathcal{N}_+|G_2\\rangle.\n\\end{equation}\nThen if $\\nu_{s^\\prime s}^{(p)}$ is the maximal (left or right) eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$,\n\\begin{equation}\nB_{s^\\prime s}^{(p)} \\le C_s \\left(\\nu_{s^\\prime s}^{(p)}\\right)^p. \\label{eq:Bfinalbound}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nRewriting (\\ref{eq:combineq}) in terms of the transfer matrix: \\begin{equation}\nB_{s^\\prime s}^{(p)} \\le \\langle G_\\triangle | \\left(\\mathcal{M}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle.\n\\end{equation}\nNow, letting $G_0=G_p=G_\\triangle$, and using the property that \\begin{equation}\n\\sum_{H_1,\\ldots, H_p \\in \\mathcal{G}_+}\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle =\\sum_{H_1,\\ldots, H_p \\in \\mathcal{G}_+} \\prod_{i=1}^p \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle \\label{eq:presymmetrizing}\n\\end{equation} which follows from Proposition \\ref{propsym},\n\\begin{align}\nB_{s^\\prime s}^{(p)} &\\le \\left(4\\sigma^2\\right)^p C_s \\sum_{ \\substack{G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_s^{(p)} \\\\H_1,\\ldots, H_p \\in \\mathcal{G}_{s^\\prime}^{(p)} } }\\prod_{i=1}^p \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle \\notag \\\\\n&= C_s \\sum_{ G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- }\\prod_{i=1}^p \\left[4\\sigma^2 \\sum_{H_i \\in \\mathcal{G}_+} \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle\\right] \\notag \\\\\n&= C_s \\sum_{ G_1,\\ldots, G_{p-1} \\in \\mathcal{G}_- }\\prod_{i=1}^p \\left[4\\sigma^2 \\sqrt{\\sum_{H_i \\in \\mathcal{G}_+} \\langle G_i | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i-1}\\rangle} \\sqrt{\\sum_{H_i \\in \\mathcal{G}_+} \\langle G_{i-1} | \\mathcal{N}_-|H_i\\rangle\\langle H_i|\\mathcal{N}_+|G_{i}\\rangle} \\right] \\notag \\\\\n&= C_s \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle \\label{eq:symmetrizing}\n\\end{align}\nwhere we have defined the symmetrized transfer matrix \\begin{equation}\n\\langle G_1|\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} |G_2\\rangle := \\sqrt{\\langle G_1|\\mathcal{M}_{s^\\prime s}^{(p)} |G_2\\rangle\\langle G_2|\\mathcal{M}_{s^\\prime s}^{(p)} |G_1\\rangle}.\n\\end{equation}\nIn the third line of (\\ref{eq:symmetrizing}), we have used the distributive property along with (\\ref{eq:presymmetrizing}) and the trivial identity that $x=y$ implies $\\sqrt{xy}=x$.\n\nSince $\\mathcal{M}_{s^\\prime s}^{(p)}$ is non-negative, $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ is a symmetric and positive semidefinite and acts on a finite dimensional vector space. Let $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ be its maximal eigenvalue. We conclude (for example, using elementary variational methods) that, since $\\langle G_\\triangle | G_\\triangle \\rangle = 1$, \\begin{equation}\n \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)} \\right)^p |G_\\triangle\\rangle \\le \\left(\\widetilde{\\nu}_{s^\\prime s}^{(p)}\\right)^p. \\label{eq:nutildebound}\n\\end{equation}\n\nIt remains to relate $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ to $\\nu_{s^\\prime s}^{(p)}$. By definition of the set $\\mathcal{G}_-$ (and the fact it has a finite number of elements), for any two graphs $G_{1,2} \\in \\mathcal{G}_-$, there exists an integer $n<\\infty$ such that \\begin{equation}\n\\langle G_1 | \\left(\\mathcal{M}_{s^\\prime s}^{(p)}\\right)^n |G_2\\rangle > 0.\n\\end{equation}\nThis identity follows from the fact that there exist a sequence of factors from $G_\\triangle$ to $G_1$ and $G_2$, as well as the reverse sequences from $G_2$ or $G_1$ to $G_\\triangle$. Hence $\\mathcal{M}_{s^\\prime s}^{(p)}$ is an irreducible non-negative matrix, and it follows that $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ is also irreducible. By the Perron-Frobenius Theorem \\cite{meyer}: (\\emph{1}) $\\mathcal{M}_{s^\\prime s}^{(p)}$ has a maximal eigenvector $|\\phi \\rangle$ and $\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}$ has a maximal eigenvector $|\\widetilde{\\phi} \\rangle$, which each obey \\begin{equation}\n\\langle G_\\triangle|\\phi\\rangle \\ne 0 \\text{ and } \\langle G_\\triangle|\\widetilde{\\phi}\\rangle \\ne 0; \\label{eq:phiGne0}\n\\end{equation}\n(\\emph{2}) $\\nu_{s^\\prime s}^{(p)}$ and $\\widetilde{\\nu}_{s^\\prime s}^{(p)}$ are non-degenerate; (\\emph{3}) as stated in the lemma, $\\nu_{s^\\prime s}^{(p)}$ is the maximal left and maximal right eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$. As $\\mathbb{R}^{\\mathcal{G}_-}$ is a finite dimensional vector space, (\\ref{eq:phiGne0}) implies that\n\\begin{align}\n\\nu_{s^\\prime s}^{(p)} = \\lim_{n\\rightarrow \\infty} \\frac{\\log \\langle G_\\triangle | \\left(\\mathcal{M}_{s^\\prime s}^{(p)}\\right)^n |G_\\triangle\\rangle }{n}= \\lim_{n\\rightarrow \\infty} \\frac{\\log \\langle G_\\triangle | \\left(\\widetilde{\\mathcal{M}}_{s^\\prime s}^{(p)}\\right)^n |G_\\triangle\\rangle }{n} = \\widetilde{\\nu}_{s^\\prime s}^{(p)}. \\label{eq:nunutilde}\n\\end{align}\nCombining (\\ref{eq:nutildebound}) and (\\ref{eq:nunutilde}) we obtain (\\ref{eq:Bfinalbound}).\n\\end{proof}\n\nUsing Lemma \\ref{lma12}, we now begin to bound the maximal eigenvalue of the matrix $M_{s^\\prime s}$ defined in (\\ref{eq:MssDef}). Let \\begin{equation}\np = \\lceil \\kappa s \\log N \\rceil, \\label{eq:pkappa}\n\\end{equation}\nwhere the parameter $\\kappa \\in \\mathbb{R}^+$ will be O(1). We first combine (\\ref{eq:markov4}) and (\\ref{eq:BssDef}). Using the inequality \\begin{equation}\nC_s > \\left(\\frac{\\mathrm{e}N}{s}\\right)^s,\n\\end{equation}\nwe find that since $N\\ge 4$ and $s\\ge 1$: \\begin{equation}\n\\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}\\right] \\le \\left(\\frac{\\mathrm{e}^{-2\/\\kappa}}{\\mathrm{e}^\\theta} \\right)^p \\left(\\frac{\\mathrm{e}N}{s}\\right)^s = \\left(\\frac{\\mathrm{e}^{-2\/\\kappa}}{\\mathrm{e}^\\theta} \\left(\\frac{\\mathrm{e}N}{s}\\right)^{1\/\\kappa \\log N} \\right)^p \\le \\mathrm{e}^{-p\\theta} = \\frac{1}{N^{\\kappa\\theta s}}.\n\\end{equation}\nMoreover, since there are at most $2(q-2)$ non-vanishing $\\mathcal{K}_{s^\\prime s}$ coefficients involving a fixed operator size, we conclude that \\begin{align}\n\\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}, \\text{ for any } s, s^\\prime \\right] &\\le \\sum_{|s^\\prime -s| \\le q-2} \\mathbb{P}\\left[\\mu_{s^\\prime s}> \\nu_{s^\\prime s}^{(p)} \\mathrm{e}^{\\theta + 2\/\\kappa}\\right] \\le 2(q-2) \\sum_{s=1}^N \\frac{1}{N^{\\kappa\\theta s}} \\notag \\\\\n&< \\frac{2(q-2)}{N^{\\kappa\\theta}-1} =1- \\mathbb{P}_{\\mathrm{success}}.\n\\end{align}\nHence, with probability $\\mathbb{P}_{\\mathrm{success}}$, we may assume that $\\mu_{s^\\prime s} \\le \\mathrm{e}^{\\theta + 2\/\\kappa} \\nu_{s^\\prime s}^{(p)}$.\n\nOf course, it remains to bound $\\nu_{s^\\prime s}^{(p)}$, which we do in the following lemma:\n\n\\begin{lem}\\label{lma13}\nIf $p$ is given by (\\ref{eq:pkappa}), $k$ is defined in (\\ref{eq:kdef}), and we assume (\\ref{eq:kappasqrtNN}), then \\begin{equation}\n\\nu_{s^\\prime s}^{(p)} < \\left( \\frac{s+q-2}{q-1} \\frac{2s}{q} \\left(\\frac{2q^2s}{N}\\right)^{2k-2}+ \\frac{2^q (q-1)! (s+q)^{q-1}}{N^{(q-2)\/2}} \\right) \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{\\sqrt{N}}\\right]. \\label{eq:lma13}\n\\end{equation}\n\n\\end{lem}\n\\begin{proof}\nLet us interpret $\\nu_{s^\\prime s}^{(p)}$ as the maximal left eigenvalue of $\\mathcal{M}_{s^\\prime s}^{(p)}$. Defining $\\mathbb{R}^{\\mathcal{G}_-}_+$ as the set of all vectors with strictly positive entries, we begin by invoking the Collatz-Wielandt bound \\cite{meyer}: \\begin{equation}\n\\nu_{s^\\prime s}^{(p)} = \\inf_{|\\phi\\rangle \\in \\mathbb{R}^{\\mathcal{G}_-}_+} \\sup_{G\\in \\mathcal{G}_-} \\frac{\\langle \\phi| \\mathcal{M}_{s^\\prime s}^{(p)} | G\\rangle}{\\langle \\phi|G\\rangle}. \\label{eq:collatzwieland}\n\\end{equation}\nClearly, we can bound $\\nu_{s^\\prime s}^{(p)}$ by simply guessing any $|\\phi\\rangle \\in \\mathbb{R}^{\\mathcal{G}_-}_+$. We choose \\begin{equation}\n\\langle \\phi |G\\rangle = N^{-|V\\cap G|\/2}\n\\end{equation}\nwhere $|V\\cap G|$ denotes the number of fermions (of non-zero degree!) in the graph $G$. \n\nNow let us write out \\begin{equation}\n\\sup_{G\\in \\mathcal{G}_-} \\frac{\\langle \\phi| \\mathcal{M}_{s^\\prime s}^{(p)} | G\\rangle}{\\langle \\phi|G\\rangle} = \\sup_{G\\in \\mathcal{G}_-} 4\\sigma^2\\sum_{H^\\prime \\in \\mathcal{G}_+, H\\in \\mathcal{G}_-} N^{(|G\\cap V| - |H\\cap V|)\/2} \\langle H|\\mathcal{N}_-|H^\\prime\\rangle\\langle H^\\prime | \\mathcal{N}_+|G\\rangle . \\label{eq:supGsum}\n\\end{equation}\nGiven graphs $G$, $H$ and $H^\\prime$, let us define the following four parameters: \\begin{subequations}\\begin{align}\na_+ := |(H^\\prime - H^\\prime \\cap G)\\cap V|, \\\\ \na_- := |(H - H^\\prime \\cap H)\\cap V|, \\\\ \nb_+ := |(G - H^\\prime \\cap G)\\cap V|, \\\\ \nb_- := |(H^\\prime - H^\\prime \\cap H)\\cap V|.\n\\end{align}\\end{subequations}\n$a_+$ and $a_-$ are the number of \\emph{new} fermions added to the graph in the first and second step respectively; $b_+$ and $b_-$ represent the number of fermions \\emph{removed} from the graph in the first and second step respectively. Note the following constraints: \\begin{subequations}\\label{eq:abbounds}\\begin{align}\n0\\le a_+ \\le q+1-2k, \\\\\n0 \\le a_- \\le 2k-1, \\\\\n0 \\le b_+ \\le 2k-1, \\\\\n0 \\le b_- \\le q+1-2k.\n\\end{align}\\end{subequations} \\begin{equation}\n|G\\cap V| - |H\\cap V| = b_+ + b_- - a_+ - a_-. \\label{eq:GVHVab}\n\\end{equation}\nLastly, note that $a_+$ and $a_-$ are non-negative if and only if a factor is added to the graph, and $b_+$ and $b_-$ are non-negative if and only if a factor is removed from the graph in that step.\n\nThere are four possible kinds of sequences of $H$ and $H^\\prime$, corresponding to whether a factor is added (A) or removed (R) from the graph in each step: RR, AA, RA, AR. Because we keep the starting graph fixed, and sum over all possible ways to add or remove factors to the graph, we can efficiently overestimate the sum over all possible modifications to the graph with fixed $a_\\pm$ or $b_\\pm$. Let \\begin{equation}\nv=|G\\cap V|\n\\end{equation}\nto be the number of vertices in $G$. In the first step, the number of ways to add a factor is \\begin{equation}\nN_{\\mathrm{A}}(a_+) := \\sum_{H^\\prime \\in \\mathcal{G}_+ : |H^\\prime \\cap V| = a_+ + |G\\cap V|} \\langle H^\\prime|\\mathcal{N}_+|G\\rangle = \\left(\\begin{array}{c} N - v \\\\ a_+ \\end{array}\\right) \\left(\\begin{array}{c} s \\\\ 2k-1 \\end{array}\\right) \\left(\\begin{array}{c} v-s \\\\ q+1-2k-a_+ \\end{array}\\right),\n\\end{equation}\nwhere the first combinatorial factor is the choice of $a_+$ distinct fermions to add to the graph, the second is the number of $(2k-1)$-tuples of the $s$ odd degree fermions present in the graph, and the third is the number $(q+1-2k-a_+)$-tuples of even degree fermions to add an extra edge to. If instead we remove a factor, we find \\begin{equation}\nN_{\\mathrm{R}}(b_+) \\le \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s}{b_+} \\right\\rfloor &\\ b_+ > 0 \\\\ p &\\ b_+ = 0 \\end{array}\\right.,\n\\end{equation}\nwhere the first line corresponds to the maximal number of factors that can have $b_+ > 0$ odd degree fermions, and the second line is a crude bound: we can remove no more factors than the maximal number $p$ allowed in any graph in $\\mathcal{G}_-$.\n\nNext we look at the AA sequences, where two factors are added sequentially. Here we find \\begin{equation}\nN_{\\mathrm{AA}}(a_+, a_-) := N_{\\mathrm{A}}(a_+) \\left(\\begin{array}{c} N - v - a_+ \\\\ a_- \\end{array}\\right) \\left(\\begin{array}{c} s+q+2-4k \\\\ q+1-2k \\end{array}\\right) \\left(\\begin{array}{c} v+a_+-s-q-2+4k \\\\ 2k-1-a_- \\end{array}\\right).\n\\end{equation}\nIn the RA sequences, we find \\begin{equation}\nN_{\\mathrm{RA}}(b_+, a_-) \\le N_{\\mathrm{R}}(b_+) \\times \\left(\\begin{array}{c} N - v + b_+ \\\\ a_- \\end{array}\\right) \\left(\\begin{array}{c} s+q+2-4k \\\\ q+1-2k \\end{array}\\right) \\left(\\begin{array}{c} v-b_+-s-q-2+4k \\\\ 2k-1-a_- \\end{array}\\right).\n\\end{equation}\nFor the RR sequences, we find \\begin{equation}\nN_{\\mathrm{RR}}(b_+,b_-) \\le N_{\\mathrm{R}}(b_+) \\times \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s+q+2-4k}{b_-} \\right\\rfloor &\\ b_- > 0 \\\\ p-1 &\\ b_- = 0 \\end{array}\\right.,\n\\end{equation}\nwhile for the AR sequences: \\begin{equation}\nN_{\\mathrm{AR}}(a_+,b_-) \\le N_{\\mathrm{A}}(a_+) \\times \\left\\lbrace \\begin{array}{ll} \\displaystyle \\left\\lfloor \\dfrac{s+q+2-4k}{b_-} \\right\\rfloor &\\ b_- > 0 \\\\ p &\\ b_- = 0 \\end{array}\\right..\n\\end{equation}\n\nNow we must perform the sum over $a_\\pm$ and $b_\\pm$ in (\\ref{eq:supGsum}). We start with the sum over AA sequences, where we will crudely bound the six distinct choose functions for convenience. Using (\\ref{eq:GVHVab}), \\begin{align}\n\\sum_{a_+=0}^{q+1-2k} \\sum_{a_-=0}^{2k-1} \\frac{N_{\\mathrm{AA}}(a_+, a_-) }{N^{(a_+ + a_-)\/2} }&< \\sum_{a_+=0}^{q+1-2k} \\sum_{a_-=0}^{2k-1} \\frac{N^{a_+\/2} s^{2k-1} v^{q+1-2k-a_+}}{a_+! (2k-1)!(q+1-2k-a_+)!} \\times \\frac{N^{a_-\/2} (s+q)^{q+1-2k} (v+q)^{2k-1-a_-}}{a_-! (q+1-2k)! (2k-1-a_-)!} \\notag \\\\\n&< \\frac{(s+q)^q}{(q+1-2k)!(2k-1)!}\\frac{\\left(\\sqrt{N}+v\\right)^{q+1-2k}}{(q+1-2k)!} \\frac{\\left(\\sqrt{N}+v+q\\right)^{2k-1}}{(2k-1)!} \\notag \\\\\n&< \\frac{(s+q)^q \\left(\\sqrt{N}+v+q\\right)^q }{(q+1-2k)!^2(2k-1)!^2}. \\label{eq:lastsum1}\n\\end{align}\nNext, we bound the RA sequences. For convenience, we may just use $N_{\\mathrm{R}}(b_+) \\le p$: \\begin{align}\n\\sum_{b_+=0}^{2k-1} \\sum_{a_-=0}^{2k-1} \\frac{N_{\\mathrm{RA}}(a_+, a_-) }{N^{( a_- - b_+)\/2} }&< \\sum_{b_+=0}^{2k-1} \\sum_{a_-=0}^{2k-1} p N^{b_+\/2} \\frac{N^{a_-\/2} (s+q)^{q+1-2k} v^{2k-1-a_-} }{a_-! (q+1-2k)! (2k-1-a_-)!} \\notag \\\\\n&< \\frac{p N^{(2k-1)\/2}}{1-N^{(1-2k)\/2}} \\frac{(s+q)^{q+1-2k}}{(q+1-2k)!} \\frac{\\left(\\sqrt{N} + v\\right)^{2k-1}}{(2k-1)!} \\notag \\\\\n&< \\frac{p (s+q)^{q+1-2k}}{(1-N^{-1\/2})(q+1-2k)!(2k-1)!} \\left(N + v\\sqrt{N}\\right)^{q\/2} \\label{eq:lastsum2}\n\\end{align}\nSimilarly, \\begin{align}\n\\sum_{b_+=0}^{2k-1} \\sum_{b_-=0}^{q+1-2k} N^{(b_+ + b_-)\/2} N_{\\mathrm{RR}}(b_+, b_-) < p(p-1) \\frac{N^{q\/2}}{(1-N^{-1\/2})^2} \\label{eq:lastsum3}\n\\end{align}\nLastly, and most importantly, we bound the AR sequences. In this sum, we will split off the $b_- = q+1-2k$ contribution, and bound that more carefully: \\begin{align}\n\\sum_{b_-=0}^{q+1-2k} \\sum_{a_+=0}^{q+1-2k} \\frac{N_{\\mathrm{AR}}(a_+, b_-)}{N^{(a_+ - b_-)\/2}} &< \\left(p + \\sum_{b_-=1}^{q+1-2k} N^{b_-\/2} \\left\\lfloor \\frac{s+q+2-4k}{b_-} \\right\\rfloor \\right)\\left( \\sum_{a_+=0}^{q+1-2k} \\frac{N^{a_+\/2} s^{2k-1} v^{q+1-2k-a_+}}{a_+! (2k-1)!(q+1-2k-a_+)!} \\right) \\notag \\\\\n&< \\left(\\frac{p N^{(q-2k)\/2}}{1-N^{-1\/2}} + N^{(q+1-2k)\/2} \\left\\lfloor \\frac{s+q+2-4k}{q+1-2k} \\right\\rfloor \\right) \\frac{s^{2k-1}}{(2k-1)!} \\frac{\\left(\\sqrt{N}+v\\right)^{q+1-2k} }{(q+1-2k)!}\\notag \\\\\n&< \\left(\\frac{pN^{-1\/2}}{1-N^{-1\/2}} +\\left\\lfloor \\frac{s+q+2-4k}{q+1-2k} \\right\\rfloor \\right) \\frac{s^{2k-1}}{(2k-1)!} \\frac{\\left(N+v\\sqrt{N}\\right)^{q+1-2k}}{(q+1-2k)!}. \\label{eq:lastsum4}\n\\end{align}\n\nThe combinatorial bounds above, by construction, did not depend on the initial graph $G$. Therefore, combining (\\ref{eq:lastsum1}), (\\ref{eq:lastsum2}), (\\ref{eq:lastsum3}) and (\\ref{eq:lastsum4}), and employing (\\ref{eq:sigmadef}), (\\ref{eq:collatzwieland}) and (\\ref{eq:supGsum}), along with \\begin{equation}\nv < qp + s\n\\end{equation}\nand other simple inequalities, we obtain \n\\begin{align}\n\\nu_{s^\\prime s}^{(p)} &< \\nu_1 + \\nu_2\n\\end{align}\nwhere \\begin{subequations}\\begin{align}\n\\nu_1 &= \\left(\\frac{N+(qp+s)\\sqrt{N}}{N-q}\\right)^{q+1-2k} \\left(\\frac{q}{N-q}\\right)^{2k-2} \\frac{2s^{2k-1}}{q(2k-1)!} \\left(\\frac{pN^{-1\/2}}{1-N^{-1\/2}} + \\frac{s+q+2-4k}{q+1-2k} \\right) \\\\\n\\nu_2 &= \\frac{(q-1)!}{(N-q)^{(q-2)\/2}}\\left(\\frac{\\sqrt{N}+(p+1)q+s}{\\sqrt{N-q}}\\right)^q \\left(\\frac{p}{1-N^{-1\/2}} + \\frac{(s+q)^{q+1-2k}}{(2k-1)!(q+1-2k)!}\\right)^2.\n\\end{align}\\end{subequations}\nNow we simplify. Using that $N-q > \\frac{1}{2}N$ from (\\ref{eq:qlessN2}), along with $p<1+\\kappa s \\log N$,\n\\begin{align}\n\\nu_1 &< \\left(1 + 2\\frac{q + s\\sqrt{N}(1+q\\kappa \\log N) + q\\sqrt{N}}{N}\\right)^{q+1-2k} \\left(\\frac{2q}{N}\\right)^{2k-2} \\frac{2s^{2k-1}}{q(2k-1)!} \\left(\\frac{s+q-2}{q+1-2k} + \\frac{2(p+1)}{\\sqrt{N}} \\right) \\notag \\\\\n&< \\frac{s+q-2}{q+1-2k} \\frac{2s}{q} \\left(\\frac{2qs}{N}\\right)^{2k-2} \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{\\sqrt{N}}\\right].\n\\end{align}\nIn the second line, we used (\\ref{eq:annoying}), $(s+q-2)\\ge q+1-2k$, and $1+x < \\mathrm{e}^x$, to simplify further. Next, \\begin{align}\n\\nu_2 &< \\frac{2^{(q-2)\/2}(q-1)!}{N^{(q-2)\/2}} \\left(\\sqrt{2} \\left(1 + \\frac{s+2q+q\\kappa s \\log N}{\\sqrt{N}} \\right)\\right)^q\\left(\\frac{2\\kappa s \\log N}{\\sqrt{N}} + \\frac{(s+q)^{q+1-2k}}{(2k-1)!(q+1-2k)!} \\right) \\notag \\\\\n&< \\frac{2^q (q-1)! (s+q)^{q+1-2k}}{N^{(q-2)\/2}}\\exp\\left[ \\frac{2q^2 \\kappa s \\log N}{\\sqrt{N}}\\right] \n\\end{align}\nwhere in the second line we used the fact that $2\\kappa s\\log N + 2 < \\sqrt{N}(s+q)^{q+1-2k}$, since $q+1-2k>0$ and we assumed (\\ref{eq:kappasqrtNN}). Making a few final simplifications, we obtain (\\ref{eq:lma13}).\n\\end{proof}\n\nThe last step to prove our theorem is to simply bound $\\mathcal{K}_l$. Recall the relation between $l$ and $s$ defined in (\\ref{eq:ldef}). With probability no smaller than $\\mathbb{P}_{\\mathrm{success}}$ we may invoke Lemma \\ref{lma13} as a bound on every $\\mu_{s^\\prime s}$. Hence from Proposition~\\ref{propmuss} and (\\ref{eq:Kldef}), \\begin{align}\n\\mathcal{K}_l &\\le \\max \\left \\lbrace \\max_{s\\in R_l} \\sum_{s^\\prime \\in R_{l+1}} \\sqrt{\\mathrm{e}^{\\theta + 2\/\\kappa}\\nu_{s^\\prime s}^{(p)}}, \\max_{s^\\prime \\in R_{l+1}} \\sum_{s\\in R_{l}} \\sqrt{\\mathrm{e}^{\\theta + 2\/\\kappa}\\nu_{s^\\prime s}^{(p)}} \\right\\rbrace \\notag \\\\\n&< \\mathrm{e}^{\\theta\/2+1\/\\kappa} \\sum_{k=1}^{q\/2-1} \\sqrt{\\nu_{1+(q-2)l + q+2-4k, 1+(q-2)l}^{(p)}} \\notag \\\\\n&<\\mathrm{e}^{\\theta\/2+1\/\\kappa} \\left[ \\sqrt{\\frac{s+q-2}{q-1} \\frac{2s}{q}} \\dfrac{1}{\\displaystyle 1 -\\frac{2q^2s}{N}} + q\\sqrt{\\frac{2^q (q-1)! (s+q)^{q-1}}{N^{(q-2)\/2}}} \\right] \\exp\\left[ \\frac{5q^2 \\kappa s \\log N}{2\\sqrt{N}}\\right] \\notag \\\\\n&< \\mathrm{e}^{\\theta\/2+1\/\\kappa} \\left[ \\sqrt{\\frac{2(1+(q-2)(l+1))(1+(q-2)l)}{q(q-1)}} \\dfrac{1}{\\displaystyle 1 -\\frac{2q^3(l+1)}{N}} + \\frac{2^{q} q! (l+1)^{(q-1)\/2}}{N^{(q-2)\/4}} \\right] \\notag \\\\ \n&\\;\\;\\;\\;\\;\\; \\times \\exp\\left[ \\frac{5q^3 \\kappa (l+1) \\log N}{2\\sqrt{N}}\\right] \\notag \\\\\n&1$) is not sharply peaked around the mean value which we have overestimated. Instead, the distribution of eigenvalues is highly peculiar, with only a small fraction of eigenvalues, which we conjecture is $\\mathrm{O}(N^{1-s})$ for $s<(q-2)M$, within an O(1) factor of $\\mathcal{K}_{s^\\prime s}$. We conjecture that the maximal eigenvector of $M_{s^\\prime s}$ is dominated by treelike factor graphs, analogous to (\\ref{eq:graphtree}), with $\\mathrm{O}(s\/q)$ leaves attached to a root which connects to a single fermion. These are precisely the graphs associated to a growing operator which started from a single fermion. Indeed, explicit calculations confirm that such treelike graphs have significantly larger weight in $\\mathrm{tr}(M_{s^\\prime s}^p)$ for $p=\\kappa s \\log N$. It appears as though the fastest growing operators of average size $\\bar s$ is a single fermion operator $\\psi_1(t)$, evolved to an appropriate time $t$. It would be interesting if this set of conjectures can be proven or disproven.\n\n\\subsection{Comparison with perturbation theory}\\label{sec:pert}\nLet us now compare our bounds to prior calculations in the SYK model using perturbation theory. First, let us discuss the Lyapunov exponent as $N\\rightarrow \\infty$. We have found that \\begin{equation}\n\\lambda \\le 2\\sqrt{\\frac{2(q-2)}{q}},\n\\end{equation}\na slight improvement over \\cite{chen1}. It is known analytically that \\cite{stanford1802} \\begin{equation}\n\\lambda_{\\mathrm{perturbative}} = 2 \\;\\;\\; (q=\\infty),\n\\end{equation}\nimplying that our result has over estimated the true value by a factor of $\\sqrt{2}$. \n\n\\cite{stanford1802} also argued that the block probabilities $P_l(t)$ took the form \\begin{subequations}\\label{eq:SYKopprob}\\begin{align}\nP_0(t) &\\approx 1 - \\frac{4}{q} \\log \\cosh t + \\cdots , \\\\ \nP_l(t) &\\approx \\frac{2}{lq} (\\tanh t)^{2l} + \\cdots.\\;\\;\\;\\; (l>0)\n\\end{align}\\end{subequations}\nat leading order in a large $N$ and large $q$ expansion (with no bound on the subleading corrections, denoted above as $\\cdots$). It is interesting to compare this with the following result: \\begin{prop}\nConsider the quantum walk of $|\\varphi(t)\\rangle$ generated by (\\ref{eq:Haux}) with \\begin{equation}\nK_l(t) = c(l+1), \\label{eq:Klcl2}\n\\end{equation}\non the half-line where $N^\\prime = \\infty$. Then \\begin{equation}\nP_l(t) = (\\tanh (ct))^{2l} \\mathrm{sech}^2 (ct). \\label{eq:optqw}\n\\end{equation} \\label{propquantumwalk}\n\\end{prop}\n\\begin{proof}\nWithout loss of generality, we rescale time so that $c=1$. Then, we repackage (\\ref{eq:prop2dos}) using generating functions: \\begin{equation}\nG(z,t) := \\sum_{l=0}^\\infty z^{l+1} P_l(t),\n\\end{equation}\nso that (\\ref{eq:prop2dos}) with (\\ref{eq:Klcl2}) implies that \\begin{equation}\n\\frac{\\partial G}{\\partial t} = z^2 \\frac{\\partial G}{\\partial z} - z \\frac{\\partial}{\\partial z} \\left(\\frac{G}{z}\\right) = \\left(z^2-1\\right)\\frac{\\partial G}{\\partial z} + \\frac{G}{z}.\n\\end{equation}\nThis equation is solved by the method of characteristics. The characteristic curves $z(t)$ solve the differential equation \\begin{equation}\n\\frac{\\mathrm{d}z}{\\mathrm{d}t} = \\left(1-z^2\\right).\n\\end{equation}\nWith initial condition $z(0)=r$, we find \\begin{equation}\nt = \\frac{1}{2} \\log \\frac{(1-r)(1+z)}{(1+r)(1-z)},\n\\end{equation}\nor \\begin{equation}\nr = \\frac{z\\cosh t - \\sinh t}{\\cosh t - z\\sinh t}.\n\\end{equation}\nSolving the equation \\begin{equation}\n\\frac{\\partial G(r,t)}{\\partial t} = \\frac{G}{z}\n\\end{equation}\nwith $G(r,0)=r$ (corresponding to $P_0(0)=1$): \\begin{equation}\n\\log \\frac{G}{r} = \\int\\limits_0^t \\mathrm{d}t^\\prime \\frac{\\cosh t^\\prime + r\\sinh t^\\prime}{\\sinh t^\\prime + r\\cosh t^\\prime} = \\log \\frac{\\sinh t + r\\cosh t }{r}.\n\\end{equation}\nThus, \\begin{equation}\nG(z,t)= \\frac{z\\mathrm{sech}t}{(1 - z\\tanh t)},\n\\end{equation}\nwhich leads to (\\ref{eq:optqw}) upon Taylor expanding and employing (\\ref{eq:varphisqrt}).\n\\end{proof}\n\nSome of the discrepancy between (\\ref{eq:optqw}) and (\\ref{eq:SYKopprob}) can be accounted for by our sloppy overestimate of $K_l(t)$ in the SYK model. In particular, a more careful analysis demonstrates that $K_0(t) \\lesssim \\sqrt{2\/q}$ and $K_l(t) \\lesssim l$. However, this slow first step does not change our estimate for the Lyapunov exponent.\n\nThis result is highly suggestive that the qualitative structure of the growing operator distribution in the SYK model, calculated perturbatively, is not substantially modified by non-perturbative physics. Rather it appears quite similar to an ``optimal\" quantum walk that locally maximizes the transition rates from one operator size to the next.\\footnote{However, such an ``optimal\" quantum walk likely does not maximize the probability of large size operators, and perhaps does not even optimize the time-dependent average size.} This may imply some universality to the patterns of operator growth in random regular $q$-local quantum systems. If such universality exists, it may have interesting implications for quantum gravity.\n\n\n\n\n\\section{Conclusion}\nIn this paper, we have proven the fast scrambling conjecture in the SYK model with a finite but large number $N$ of degrees of freedom. While this result is not physically surprising, it is pleasing to have a mathematically careful derivation of this result. We also expect that the methods developed here will lead to further advances in our technology for bounding quantum information dynamics and operator growth \\cite{chen1, chen2} beyond the Lieb-Robinson theorem \\cite{liebrobinson, hastings}.\n\nWe would like to say that our demonstration of the robustness of operator growth to non-perturbative physics in at least one holographic model is a signature that the bulk geometry is semiclassical and that non-perturbative fluctuations in quantum gravity are provably mild. Unfortunately, this remains a conjecture, as the emergent geometry arises at finite temperature. It would be interesting if our methods can be generalized to finite temperature states. \n\nLastly, we expect these techniques are useful for designing and constraining toy models of quantum gravity which can be experimentally studied using quantum simulation \\cite{Garttner2017, Li2017}. At the very least, any tentative model must reproduce the exponential growth in operator size which is a hallmark of particles falling towards black hole horizons. Our methods will not only bound the Lyapunov exponent of any proposed model, but also check whether the full time evolution of the operator size distribution could be consistent with a theory of quantum gravity.\n\n\\section*{Acknowledgments}\nThis work was supported by the University of Colorado.\n\n \n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe concept of homomorphisms of $(m, n)$-colored-mixed graph was introduced by J. Nes\\v{e}t\\v{r}il and A. Raspaud~\\cite{MNCM}\nin order to generalize homomorphisms of $k$-edge-colored graphs and oriented graphs.\n\nAn \\emph{$(m, n)$-colored-mixed graph} $G=(V, A_1, A_2,\\cdots, A_m, E_1, E_2,\\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges.\nWe do not allow two arcs or edges to have the same endpoints.\nThe case $m=0$ and $n=1$ corresponds to simple graphs, $m=1$ and $n=0$ to oriented graphs and $m=0$ and $n=k$ to $k$-edge-colored graphs. For the case $m=0$ and $n = 2$\n($2$-edge-colored graphs) we refer to the two types of edges as \\emph{blue} and \\emph{red} edges.\n\nA \\emph{homomorphism} from an $(m, n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a mapping $\\varphi:V(G) \\rightarrow V(H)$\nsuch that every edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation).\nIf $G$ admits a homomorphism to $H$, we say that $G$ is \\emph{$H$-colorable} since this homomorphism can be seen as a coloring of the vertices of $G$\nusing the vertices of $H$ as colors. The edges and arcs of $H$ (and their colors) give us the rules that this coloring must follow.\nGiven a class of graphs $\\mathcal{C}$, a graph is \\emph{$\\mathcal{C}$-universal} if for every graph $G \\in \\mathcal{C}$ is $H$-colorable.\nThe class $P_g^{(m, n)}$ contains every planar $(m, n)$-colored-mixed graph with girth at least $g$.\n\nIn this paper, we consider some planar $P_g^{(m, n)}$-universal graphs with $k$ vertices.\nThey are depicted in Figures~\\ref{fig:t_oriented} and~\\ref{fig:t_2edge}.\nThe known results about this topic are as follows.\n\n\\begin{theorem}\\label{thm:known}{\\ }\n\\begin{enumerate}\n\\item $K_4$ is a planar $P^{(0,1)}_3$-universal graph. This is the four color theorem.\n\\item $K_3$ is a planar $P^{(0,1)}_4$-universal graph. This is Gr\u00f6tzsch's Theorem \\cite{grotzsch}.\n\\item $\\overrightarrow{C_6^2}$ is a planar $P_{16}^{(1,0)}$-universal graph~\\cite{P10}.\n\\end{enumerate}\n\\end{theorem}\n\nOur first result shows that, in addition to the case of $(0,1)$-graphs covered by Theorems~\\ref{thm:known}.1 and~\\ref{thm:known}.2,\nour topic is actually restricted to the cases of oriented graphs (i.e., $(m,n)=(1,0)$) and 2-edge-colored graphs (i.e., $(m,n)=(0,2)$).\n\n\\begin{theorem}\\label{thm:Pmn}\nFor every $g\\ge3$, there exists no planar $P_g^{(m,n)}$-universal graph if $2m+n\\ge3$.\n\\end{theorem}\n\nAs Theorems~\\ref{thm:known}.1 and~\\ref{thm:known}.2 show for $(0,1)$-graphs, there might exist a trade-off between minimizing the girth $g$ and the number\nof vertices of the universal graph, for a fixed pair $(m,n)$.\nFor oriented graphs, Theorem~\\ref{thm:known}.3 tries to minimize the girth.\nFor oriented graphs and 2-edge-colored graphs, we choose instead to minimize the number of vertices of the universal graph.\n\n\\begin{theorem}\\label{thm:positive}{\\ }\n\\begin{enumerate}\n\\item $\\overrightarrow{T_5}$ is a planar $P_{28}^{(1,0)}$-universal graph on 5 vertices.\n\\item $T_6$ is a planar $P_{22}^{(0, 2)}$-universal graph on 6 vertices.\n\\end{enumerate}\n\\end{theorem}\n\nThe following results shows that Theorem~\\ref{thm:positive} is optimal in terms of the number of vertices of the universal graph.\n\n\\begin{theorem}\\label{thm:negative}{\\ }\n\\begin{enumerate}\n\\item For every $g\\ge3$, there exists an oriented bipartite cactus graph (i.e., $K_4^-$ minor-free graph) with girth at least $g$ and oriented chromatic number at least 5.\n\\item For every $g\\ge3$, there exists a 2-edge-colored bipartite outerplanar graph (i.e., $(K_4^-,K_{2,3})$ minor-free graph) with girth at least $g$ that does not map to a planar graph with at most 5 vertices.\n\\end{enumerate}\n\\end{theorem}\n\nMost probably, Theorem~\\ref{thm:positive} is not optimal in terms of girth. The following constructions give lower bounds on the girth.\n\n\\begin{theorem}\\label{thm:ce}{\\ }\n\\begin{enumerate}\n\\item There exists an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\\overrightarrow{T_5}$.\n\\item There exists a 2-edge-colored planar graph with girth $11$ that does not map to $T_6$.\n\\item There exists a 2-edge-colored bipartite planar graph with girth $10$ that does not map to $T_6$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{figure}[H]\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\includegraphics{t_oriented}\n \\caption{The $P_{28}^{(1,0)}$-universal graph overrightarrow$(T_5)$.\\label{fig:t_oriented}}\n\n\\end{center}\n\\end{minipage}\\hfill\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\includegraphics{t_2edge}\n \\caption{The $P_{22}^{(0,2)}$-universal graph $T_6$.\\label{fig:t_2edge}}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nNext, we obtain the following complexity dichotomies:\n\n\\begin{theorem}\\label{thm:NPC}{\\ }\n\\begin{enumerate}\n\\item For any fixed girth $g\\ge 3$, either every graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$ or it is NP-complete\nto decide whether a graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$.\nEither every bipartite graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $\\overrightarrow{T_5}$.\n\\item Either every graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a graph in $P_g^{(1,0)}$ maps to $T_6$.\nEither every bipartite graph in $P_g^{(0,2)}$ maps to $T_6$ or it is NP-complete to decide whether a bipartite graph in $P_g^{(1,0)}$ maps to $T_6$.\n\\end{enumerate}\n\\end{theorem}\n\nFinally, we can use Theorem~\\ref{thm:NPC} with the non-colorable graphs in Theorem~\\ref{thm:ce}.\n\n\\begin{corollary}\\label{cor:cor}{\\ }\n\\begin{enumerate}\n\\item Deciding whether a bipartite graph in $P_{14}^{(1,0)}$ maps to $\\overrightarrow{T_5}$ is NP-complete.\n\\item Deciding whether a graph in $P_{11}^{(0,2)}$ maps to $T_6$ is NP-complete.\n\\item Deciding whether a bipartite graph in $P_{10}^{(0,2)}$ maps to $T_6$ is NP-complete.\n\\end{enumerate}\n\\end{corollary}\n\nA 2-edge-colored path or cycle is said to be \\emph{alternating} if any two adjacent edges have distinct colors.\n\n\\begin{proposition}[folklore]\\label{prop:3n-6}{\\ }\n\\begin{itemize}\n\\item Every planar simple graph on $n$ vertices has at most $3n-6$ edges.\n\\item Every planar simple graph satisfies $(\\mad(G)-2)\\cdot(g(G)-2)<4$.\n\\end{itemize}\n\\end{proposition}\n\n\\section{Proof of Theorem~\\ref{thm:positive}}\nWe use the discharging method for both results in Theorem~\\ref{thm:positive}. The following lemma will handle the discharging part.\nWe call a vertex of degree $n$ an $n$-vertex and a vertex of degree at least $n$ an $n^+$-vertex.\nIf there is a path made only of $2$-vertices linking two vertices $u$ and $v$, we say that $v$ is a weak-neighbor of $u$.\nIf $v$ is a neighbor of $u$, we also say that $v$ is a weak-neighbor of $u$. We call a (weak-)neighbor of degree $n$ an $n$-(weak-)neighbor.\n\n\\begin{lemma}\\label{lem:discharge}\nLet $k$ be a non-negative integer.\nLet $G$ be a graph with minimum degree 2 such that every 3-vertex has at most $k$ 2-weak-neighbors and every path contains at most $\\tfrac{k+1}2$ consecutive 2-vertices.\nThen $\\mad(G)\\ge2+\\tfrac2{k+2}$. In particular, $G$ cannot be a planar graph with girth at least $2k+6$.\n\\end{lemma}\n\n\\begin{proof}\nLet $G$ be as stated. Every vertex has an initial charge equal to its degree. Every $3^+$-vertex gives $\\tfrac1{k+2}$ to each of its 2-weak-neighbors.\nLet us check that the final charge $ch(v)$ of every vertex $v$ is at least $2+\\tfrac2{k+2}$.\n\\begin{itemize}\n \\item If $d(v)=2$, then $v$ receives $\\tfrac1{k+2}$ from both of its 3-weak-neighbors. Thus $ch(v)=2+\\tfrac2{k+2}$.\n \\item If $d(v)=3$, then $v$ gives $\\tfrac1{k+2}$ to each of its 2-weak-neighbors. Thus $ch(v)\\ge3-\\tfrac{k}{k+2}=2+\\tfrac2{k+2}$.\n \\item If $d(v)=d\\ge4$, then $v$ has at most $\\tfrac{k+1}2$ 2-weak-neighbors in each of the $d$ incident paths.\n Thus $ch(v)\\ge d-d\\paren{\\tfrac{k+1}2}\\paren{\\tfrac1{k+2}}=\\tfrac d2\\paren{1+\\tfrac1{k+2}}\\ge2+\\tfrac2{k+2}$.\n\\end{itemize}\nThis implies that $mad(G)\\ge2+\\frac2{k+2}$.\nFinally, if $G$ is planar, then the girth of $G$ cannot be at least $2k+6$, since otherwise $(\\mad(G)-2)\\cdot(g(G)-2)\\ge\\paren{2+\\tfrac2{k+2}-2}\\paren{2k+6-2}=\\paren{\\tfrac2{k+2}}\\paren{2k+4}=4$, which contradicts Proposition~\\ref{prop:3n-6}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:positive}.1}\nWe prove that the oriented planar graph $\\overrightarrow{T_5}$ on 5 vertices from Figure~\\ref{fig:t_oriented} is $P_{28}^{(1,0)}$-universal by contradiction.\nAssume that $G$ is an oriented planar graphs with girth at least $28$ that does not admit a homomorphism to $\\overrightarrow{T_5}$\nand is minimal with respect to the number of vertices.\nBy minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $\\overrightarrow{T_5}$-coloring of $G-v$ can be extended to $G$.\nSimilarly, $G$ does not contain the following configurations.\n\n\\begin{itemize}\n\\item A path with 6 consecutive 2-vertices.\n\\item A $3$-vertex with at least 12 2-weak-neighbors. \n\\end{itemize}\n\nSuppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6u_7$ such that the degree of $u_i$ is two for $1\\le i\\le6$.\nBy minimality of $G$, $G-{u_1,u_2,u_3,u_4,u_5,u_6}$ admits a $\\overrightarrow{T_5}$-coloring $\\varphi$.\nWe checked on a computer that for any $\\varphi(v_0)$ and $\\varphi(v_6)$ in $V\\paren{\\overrightarrow{T_5}}$\nand every possible orientation of the 7 arcs $u_iu_{i+1}$, we can always extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction.\n\nSuppose that $G$ contains a 3-vertex $v$ with at least 12 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$\nand let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$.\nWithout loss of generality and by the previous discussion, we have $5\\ge l_1\\ge l_2\\ge l_3$ and $l_1+l_2+l_3\\ge12$.\nSo we have to consider the following cases:\n\\begin{itemize}\n\\item\\textbf{Case 1:} $l_1=5$, $l_2=5$, $l_3=2$.\n\\item\\textbf{Case 2:} $l_1=5$, $l_2=4$, $l_3=3$.\n\\item\\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=4$.\n\\end{itemize}\n\nBy minimality, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $\\overrightarrow{T_5}$-coloring $\\varphi$.\nLet us show that in all three cases, we can extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$ to get a contradiction.\n\nWith an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\\varphi(u)$ by a path\nmade of $l$ 2-vertices ($0\\le l\\le5$) then $v$ can be colored in:\n\n\\begin{itemize}\n\\item at least 1 color if $l=0$,\n\\item at least 2 colors if $l=1$,\n\\item at least 2 colors if $l=2$ (the sets $\\acc{c, d, e}$ and $\\acc{b, c, d}$ are the only sets of size 3 that can be forbidden from $v$),\n\\item at least 3 colors if $l=3$, \n\\item at least 4 colors if $l=4$ and\n\\item at least 4 colors if $l=5$ (only the sets $\\acc{b}$, $\\acc{c}$, and $\\acc{e}$ can be forbidden from $v$).\n\\end{itemize}\n\nIn Case 1, $u_3$ forbids at most 3 colors from $v$ since $l_3=2$. If it forbids less than $3$ colors,\nwe will be able to find a color for $v$ since $u_1$ and $u_2$ forbid at most 1 color from $v$. The only sets of 3 colors that $u_3$ can forbid are $\\acc{b,c,d}$ and $\\acc{c, d, e}$.\nSince $u_1$ and $u_2$ can each only forbid $b$, $c$ or $e$, we can always find a color for $v$.\n\nIn Case 2, $u_1$ and $u_2$ each forbid at most one color and $u_3$ forbids at most $2$ colors so there remains at least one color for $v$.\n\nIn Case 3, $u_1$, $u_2$, and $u_3$ each forbid at most one color, so there remains at least two colors for $v$.\n\nWe can always extend $\\varphi$ into a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction.\n\nSo $G$ contains at most 5 consecutive 2-vertices and every 3-vertex has at most 11 2-weak-neighbors.\nUsing Lemma~\\ref{lem:discharge} with $k=11$ contradicts the fact that the girth of $G$ is at least 28.\n\n\\subsection{Proof of Theorem~\\ref{thm:positive}.2}\nWe prove that the 2-edge-colored planar graph $T_6$ on 6 vertices from Figure~\\ref{fig:t_2edge} is $P_{22}^{(0,2)}$-universal by contradiction.\nAssume that $G$ is a 2-edge-colored planar graphs with girth at least $22$ that does not admit a homomorphism to $T_6$ and is minimal with respect to the number of vertices.\nBy minimality, $G$ cannot contain a vertex $v$ with degree at most one since a $T_6$-coloring of $G-v$ can be extended to $G$.\nSimilarly, $G$ does not contain the following configurations.\n\n\\begin{itemize}\n\\item A path with 5 consecutive 2-vertices.\n\\item A $3$-vertex with at least 9 2-weak-neighbors. \n\\end{itemize}\n\nSuppose that $G$ contains a path $u_0u_1u_2u_3u_4u_5u_6$ such that the degree of $u_i$ is two for $1\\le i\\le5$.\nBy minimality of $G$, $G-{u_1, u_2, u_3, u_4, u_5}$ admits a $T_6$-coloring $\\varphi$.\nWe checked on a computer that for any $\\varphi(v_0)$ and $\\varphi(v_6)$ in $V(T)$ and every possible colors of the 6 edges $u_iu_{i+1}$,\nwe can always extend $\\varphi$ into a $T_6$-coloring of $G$, a contradiction.\n\nSuppose that $G$ contains a 3-vertex $v$ with at least 9 2-weak-neighbors. Let $u_1$, $u_2$, $u_3$ be the $3^+$-weak-neighbors of $v$\nand let $l_i$ be the number of common 2-weak-neighbors of $v$ and $u_i$, i.e., $2$-vertices on the path between $v$ and $l_i$.\nWithout loss of generality and by the previous discussion, we have $4\\ge l_1\\ge l_2\\ge l_3$ and $l_1+l_2+l_3\\ge 9$. So we have to consider the following cases:\n\n\\begin{itemize}\n\\item\\textbf{Case 1:} $l_1=3$, $l_2=3$, $l_3=3$.\n\\item\\textbf{Case 2:} $l_1=4$, $l_2=3$, $l_3=2$.\n\\item\\textbf{Case 3:} $l_1=4$, $l_2=4$, $l_3=1$.\n\\end{itemize}\n\nBy minimality of $G$, the graph $G'$ obtained from $G$ by removing $v$ and its 2-weak-neighbors admits a $T_6$-coloring $\\varphi$.\nLet us show that in all three cases, we can extend $\\varphi$ into a $T_6$-coloring of $G$ to get a contradiction.\n\nWith an extensive search on a computer we found that if a vertex $v$ is connected to a vertex $u$ colored in $\\varphi(u)$\nby a path $P$ made of $l$ 2-vertices ($0\\le l\\le 4$) then $v$ can be colored in:\n\n\\begin{itemize}\n\\item at least 1 color if $l=0$ (the sets ${a, c, d, e, f}$ and ${b, c, d, e, f}$ of colors are the only sets of size 5 that can be forbidden from $v$\nfor some $\\varphi(u)\\in T$ and edge-colors on $P$),\n\\item at least 2 colors if $l=1$ (the sets ${a, b, c, f}$ and ${b, c, e, f}$ are the only sets of size 4 that can be forbidden from $v$),\n\\item at least 3 colors if $l=2$ (the sets ${b, c, f}$, ${c, e, f}$ and ${d, e, f}$ are the only sets of size 3 that can be forbidden from $v$),\n\\item at least 4 colors if $l=3$ (the set ${c, b}$ is the only set of size 2 that can be forbidden from $v$), and\n\\item at least 5 colors if $l=4$ (the sets ${c}$ and ${f}$ are the only sets of size 1 that can be forbidden from $v$).\n\\end{itemize}\n\nSuppose that we are in Case 1. Vertices $u_1$, $u_2$, and $u_3$ each forbid at most 2 colors from $v$ since $l_1=l_2=l_3=3$.\nSuppose that $u_1$ forbids 2 colors. It has to forbid colors $c$ and $f$ (since it is the only pair of colors that can be forbidden by a path made of 3 2-vertices).\nIf $u_2$ or $u_3$ also forbids 2 colors, they will forbid the exact same pair of colors. We can therefore assume that they each forbid 1 color from $v$.\nThere are 6 available colors in $T_6$, so we can always find a color for $v$ and extend $\\varphi$ to a $T_6$-coloring of $G$, a contradiction.\nWe proceed similarly for the other two cases.\n\nSo $G$ contains at most 4 consecutive 2-vertices and every 3-vertex has at most 8 2-weak-neighbors.\nThen Lemma~\\ref{lem:discharge} with $k=8$ contradicts the fact that the girth of $G$ is at least 22.\n\n\\section{Proof of Theorem~\\ref{thm:negative}.1}\nWe construct an oriented bipartite cactus graph with girth at least $g$ and oriented chromatic number at least 5. Let $g'$ be such that $g'\\ge g$ and $g'\\equiv4\\pmod{6}$.\nConsider a circuit $v_1,\\cdots,v_{g'}$. Clearly, the oriented chromatic number of this circuit is 4 and the only tournament on 4 vertices it can map to\nis the tournament $\\overrightarrow{T_4}$ induced by the vertices $a$, $b$, $c$, and $d$ in $\\overrightarrow{T_5}$.\nNow we consider the cycle $C=w_1,\\cdots,w_{g'}$ containing the arcs $w_{2i-1}w_{2i}$ with $1\\le i\\le g'\/2$, $w_{2i+1}w_{2i}$ with $1\\le i\\le g'\/2-1$, and $w_{g'}w_1$.\n\nSuppose for contradiction that $C$ admits a homomorphism $\\varphi$ such that $\\varphi(w_1)=d$.\nThis implies that $\\varphi(w_2)=a$, $\\varphi(w_3)=d$, $\\varphi(w_4)=a$, and so on until $\\varphi(w_{g'})=a$.\nSince $\\varphi(w_{g'})=a$ and $\\varphi(w_1)=d$, $w_{g'}w_1$ should map to $ad$, which is not an arc of $\\overrightarrow{T_4}$, a contradiction.\n\nOur cactus graph is then obtain from the circuit $v_1,\\cdots,v_{g'}$ and $g'$ copies of $C$ by identifying every vertex $v_i$ with the vertex $w_1$ of a copy of $C$.\nThis cactus graph does not map to $\\overrightarrow{T_4}$ since one of the $v_i$ would have to map to $d$ and then the copy of $C$ attached to $v_i$ would not be $\\overrightarrow{T_4}$-colorable.\n\n\\section{Proof of Theorem~\\ref{thm:negative}.2}\nWe construct a 2-edge-colored bipartite outerplanar graph with girth at least $g$ that does not map to a 2-edge-colored planar graph with at most 5 vertices.\nLet $g'$ be such that $g'\\ge g$ and $g'\\equiv2\\pmod{4}$. Consider an alternating cycle $C=v_0,\\cdots,v_{g'-1}$.\nFor every $0\\le i\\le g'-3$, we add $g'-2$ 2-vertices $w_{i,1},\\cdots,w_{i,g'-2}$ that form the path $P_i=v_iw_{i,1}\\cdots w_{i,g'-2}v_{i+1}$\nsuch that the edges of $P_i$ get the color distinct from the color of the edge $v_iv_{i+1}$. Let $G$ be the obtained graph.\nThe 2-edge-colored chromatic number of $C$ is 5.\nSo without loss of generality, we assume for contradiction that $G$ admits a homomorphism $\\varphi$ to a 2-edge-colored planar graph $H$ on 5 vertices.\nLet us define $\\mathcal{E}=\\bigcup_{i\\texttt{ even}}\\varphi(v_i)$ and $\\mathcal{O}=\\bigcup_{i\\texttt{ odd}}\\varphi(v_i)$.\nSince $C$ is alternating, $\\varphi(v_i)\\ne\\varphi(v_{i+2})$ (indices are modulo $g'$). Since $g'\\equiv2\\pmod{4}$, there is an odd number of $v_i$ with an even (resp. odd) index.\nThus, $\\abs{\\mathcal{E}}\\ge3$ and $\\abs{\\mathcal{O}}\\ge3$. Therefore we must have $\\mathcal{E}\\cap\\mathcal{O}\\ne\\emptyset$.\n\nNotice that every two vertices $v_i$ and $v_j$ in $G$ are joined by a blue path and a red path such that the lengths of these paths have the same parity as $i-j$.\nThus, the blue (resp. red) edges of $H$ must induce a connected spanning subgraph of $H$. Since $|V(H)|=5$, $H$ contains at least 4 blue (resp. red) edges.\nSince red and blue edges play symmetric roles in $G$ and since $|E(H)|\\le9$ by Proposition~\\ref{prop:3n-6}, we assume without loss of generality that $H$ contains exactly 4 blue edges.\nMoreover, these 4 blue edges induce a tree. In particular, the blue edges induce a bipartite graph which partitions $V(H)$ into 2 parts.\nThus, every $v_i$ with even index is mapped into one part of $V(H)$ and every $v_i$ with odd index is mapped into the other part of $V(H)$.\nSo $\\mathcal{E}\\cap\\mathcal{O}=\\emptyset$, which is a contradiction.\n\n\\section{Proof of Theorem~\\ref{thm:Pmn}}\nLet $T$ be a $P_g^{(m, n)}$-universal planar graph for some $g$ that is minimal with respect to the subgraph order.\n\nBy minimality of $T$, there exists a graph $G \\in P_g^{(m, n)}$ such that every color in $T$ has to be used at least once to color $G$.\nWithout loss of generality, $G$ is connected, since otherwise we can replace $G$ by the connected graph obtained from $G$\nby choosing a vertex in each component of $G$ and identifying them. We create a graph $G'$ from $G$ as follows:\n\nFor each edge or arc $uv$ we create $4m+n$ paths starting at $u$ and ending at $v$ made of vertices of degree 2:\n\n\\begin{itemize}\n\\item For each type of edge, we create a path made of $g-1$ edges of this type.\n\n\\item For each type of arc, we create two paths made of $g-1$ arcs of this type such that the paths alternate between forward and backward arcs.\nWe make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path.\n\n\\item Similarly, for each type of arc we create two paths made of $g$ arcs of this type such that the paths alternate between forward and backward arcs.\nWe make the paths such that $u$ is the tail of the first arc of one path and the head of the first arc of the other path.\n\\end{itemize}\n\nNotice that $G'$ is in $P_g^{(m, n)}$ and thus admits a homomorphism $\\varphi$ to $T$.\nSince $G$ is connected and every color in $T$ has to be used at least once to color $G$, we can find for each pair of vertices and $(c_1, c_2)$ in $T$\nand each type of edge a path $(v_1, v_2,\\cdots, v_l)$ in $G'$ made only of edges of this type such that $\\varphi(v_1)=c_1$ and $\\varphi(v_l)=c_2$. \\newline\n\nThis implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of edge, there exists a walk from $c_1$ to $c_2$ made of edges of this type.\nTherefore, for $1\\le j\\le n$, the subgraph induced by $E_j(T)$ is connected and contains all the vertices of $T$.\nSo $E_j(T)$ contains a spanning tree of $T$. Thus $T$ contains at least $|V(T)|-1$ edges of each type.\\newline\n\nSimilarly, we can find for each pair of vertices $(c_1, c_2)$ in $T$ and each type of arc a path of even length $(v_1, v_2,\\cdots, v_{2l-1})$ in $G'$ made only of arcs of this type,\nstarting with a forward arc and alternating between forward and backward arcs such that $\\varphi(v_1)=c_i$ and $\\varphi(v_l)=c_2$.\nWe can also find a path of the same kind with odd length.\\newline\n\nThis implies that for every pair of vertices $(c_1, c_2)$ in $T$ and each type of arc there exist a walk of odd length and a walk of even length\nfrom $c_1$ to $c_2$ made of arcs of this type, starting with a forward arc and alternating between forward and backward arcs.\nLet $p$ be the maximum of the length of all these paths. Given one of these walks of length $l$, we can also find a walk of length $l+2$\nthat satisfies the same constraints by going through the last arc of the walk twice more.\nTherefore, for every $l\\ge p$, every pair of vertices $(c_1, c_2)$ in $T$, and every type of arc,\nit is possible to find a homomorphism from the path $P$ of length $l$ made of arcs of this type, starting with a forward arc and alternating\nbetween forward and backward arcs to $T$ such that the first vertex is colored in $c_1$ and the last vertex is colored in $c_2$.\\newline\n\nWe now show that this implies that $|A_j(T)|\\ge2|V(T)|-1$ for $1\\le j\\le m$.\nLet $P$ be a path $(v_1, v_2,\\cdots, v_p, v_{p+1})$ of length $p$ starting with a forward arc and alternating between forward and backward arcs of the same type.\nWe color $v_1$ in some vertex $c$ of $T$. Let $C_i$ be the set of colors in which vertex $v_i$ could be colored.\nWe know that $C_1=c$ and $C_2$ is the set of direct successors of $c$. Set $C_3$ is the set of direct predecessors of vertices in $C_2$ so $C_1\\subseteq C_3$ and,\nmore generally, $C_i \\subseteq C_i+2$. Let $uv$ be an arc in $T$. If $u\\in C_i$ with $i$ odd, then $v\\in C_{i+1}$.\nIf $v\\in C_i$ with $i$ even then $u\\in C_{i+1}$. We can see that $uv$ is capable of adding at most one vertex to a $C_i$ (and every $C_j$ with $j\\equiv i\\mod 2$ and $i\\le j$).\nWe know that $C_{p+1}=V(T)$ hence $T$ contains at least $2|V(T)|-1$ arcs of each type.\\newline\n\nTherefore, the underlying graph of $T$ contains at least $m\\paren{2|V(T)|-1}+n\\paren{|V(T)|-1}=\\paren{2m+n}|V(T)|-m-n$ edges, which contradicts Proposition~\\ref{prop:3n-6} for $2m+n\\ge3$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.1}\nWe construct an oriented bipartite 2-outerplanar graph with girth $14$ that does not map to $\\overrightarrow{T_5}$.\n\nThe oriented graph $X$ is a cycle on 14 vertices $v_0,\\cdots,v_{13}$ such that the tail of every arc is the vertex with even index, except for the arc $\\overrightarrow{v_{13}v_0}$.\nSuppose for contradiction that $X$ has a $\\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$.\nThe directed path $v_{12}v_{13}v_0$ implies that $h(v_{12})\\ne h(v_0)$.\nIf $h(v_0)=a$, then $h(v_1)\\in\\acc{b,c}$ and $h(v_2)=a$ since $h(v_2)\\ne b$. By contagion, $h(v_0)=h(v_2)=\\cdots=h(v_{12})=a$, which is a contradiction. Thus $h(v_0)\\ne a$.\nIf $h(v_0)=c$, then $h(v_1)=d$ and $h(v_2)=c$ since $h(v_2)\\ne b$. By contagion, $h(v_0)=h(v_2)=\\cdots=h(v_{12})=c$, which is a contradiction. Thus $h(v_0)\\ne c$.\nSo $h(v_0)\\not\\in\\acc{a,b,c}$, that is, $h(v_0)\\in\\acc{d,e}$. Similarly, $h(v_{12})\\in\\acc{d,e}$.\nNotice that $\\overrightarrow{T_5}$ does not contain a directed path $xyz$ such that $x$ and $z$ belong to $\\acc{d,e}$.\nSo the path $v_{12}v_{13}v_0$ cannot be mapped to $\\overrightarrow{T_5}$.\nThus $X$ does not have a $\\overrightarrow{T_5}$-coloring $h$ such that no vertex with even index maps to $b$.\n\nConsider now the path $P$ on 7 vertices $p_0,\\cdots,p_6$ with the arcs $\\overrightarrow{p_1p_0}$, $\\overrightarrow{p_1p_2}$, $\\overrightarrow{p_3p_2}$, $\\overrightarrow{p_4p_3}$, $\\overrightarrow{p_5p_4}$, $\\overrightarrow{p_5p_6}$. It is easy to check that there exists no $\\overrightarrow{T_5}$-coloring $h$ of $P$ such that $h(p_0)=h(p_6)=b$.\n\nWe construct the graph $Y$ as follows: we take 8 copies of $X$ called $X_{\\texttt{main}}$, $X_0$, $X_2$, $X_4$, $\\cdots$, $X_{12}$.\nFor every couple $(i,j)\\in\\acc{0,2,4,6,8,10,12}^2$, we take a copy $P_{i,j}$ of $P$, we identify the vertex $p_0$ of $P_{i,j}$\nwith the vertex $v_i$ of $X_{\\texttt{main}}$ and we identify the vertex $p_6$ of $P_{i,j}$ with the vertex $v_j$ of $H_i$.\n\nSo $Y$ is our oriented bipartite 2-outerplanar graph with girth $14$. Suppose for contradiction that $Y$ has a $\\overrightarrow{T_5}$-coloring $h$.\nBy previous discussion, there exists $i\\in\\acc{0,2,4,6,8,10,12}$ such that the vertex $v_i$ of $X_{\\texttt{main}}$ maps to $b$.\nAlso, there exists $j\\in\\acc{0,2,4,6,8,10,12}$ such that the vertex $v_j$ of $X_i$ maps to $b$.\nSo the corresponding path $P_{i,j}$ is such that $h(p_0)=h(p_6)=b$, a contradiction. Thus $Y$ does not map to $\\overrightarrow{T_5}$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.2}\nWe construct a 2-edge-colored 2-outerplanar graph with girth $11$ that does not map to $T_6$.\nWe take 12 copies $X_0,\\cdots,X_{11}$ of a cycle of length $11$ such that every edge is red.\nLet $v_{i,j}$ denote the $j^{\\text{\\tiny th}}$ vertex of $X_i$.\nFor every $0\\le i\\le 10$ and $0\\le j\\le 10$, we add a path consisting of 5 blue edges between $v_{i,11}$ and $v_{j,i}$.\n\nNotice that in any $T_6$-coloring of a red odd cycle, one vertex must map to $c$.\nSo we suppose without loss of generality that $v_{0,11}$ maps to $c$.\nWe also suppose without loss of generality that $v_{0,0}$ maps to $c$.\nThe blue path between $v_{0,11}$ and $v_{0,0}$ should map to a blue walk of length 5 from $c$ to $c$ in $T_6$.\nSince $T_6$ contains no such walk, our graph does not map to $T_6$.\n\n\\section{Proof of Theorem~\\ref{thm:ce}.3}\nWe construct a 2-edge-colored bipartite 2-outerplanar graph with girth $10$ that does not map to $T_6$.\nBy Theorem~\\ref{thm:negative}.2, there exists a bipartite outerplanar graph $M$ with girth at least $10$\nsuch that for every $T_6$-coloring $h$ of $M$, there exists a vertex $v$ in $M$ such that $h(v)=c$.\n\nLet $X$ be the graph obtained as follows. Take a main copy $Y$ of $M$.\nFor every vertex $v$ of $Y$, take a copy $Y_v$ of $M$. Since $Y_v$ is bipartite, let $A$ and $B$ the two independent sets of $Y_v$.\nFor every vertex $w$ of $A$, we add a path consisting of 5 blue edges between $v$ and $w$.\nFor every vertex $w$ of $B$, we add a path consisting of 4 edges colored (blue, blue, red, blue) between $v$ and $w$.\n\nNotice that $X$ is indeed a bipartite 2-outerplanar graph with girth $10$.\nWe have seen in the previous proof that $T_6$ contains no blue walk of length 5 from $c$ to $c$.\nWe also check that $T_6$ contains no walk of length 4 colored (blue, blue, red, blue) from $c$ to $c$.\nBy the property of $M$, for every $T_6$-coloring $h$ of $X$, there exist a vertex $v$ in $Y$ and a vertex $w$ in $Y_v$ such that $h(v)=h(w)=c$.\nThen $h$ cannot be extended to the path of length 4 or 5 between $v$ and $w$.\nSo $X$ does not map to $T_6$.\n\n\n\\section{Proof of Theorem~\\ref{thm:NPC}.1}\nLet $g$ be the largest integer such that there exists a graph in $P_g^{(1,0)}$ that does not map to $\\overrightarrow{T_5}$.\nLet $G\\in P_g^{(1,0)}$ be a graph that does not map to $\\overrightarrow{T_5}$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.\n\nLet $G'$ be obtained from $G$ by removing an arbitrary arc $v_0v_3$ and adding two vertices $v_1$ and $v_2$ and the arcs $v_0v_1$, $v_2v_1$, $v_2v_3$.\nBy minimality, $G'$ admits a homomorphism $\\varphi$ to $\\overrightarrow{T_5}$. Suppose for contradiction that $\\varphi(v_2)=c$. This implies that $\\varphi(v_1)=\\varphi(v_3)=d$.\nThus $\\varphi$ provides a $\\overrightarrow{T_5}$-coloring of $G$, a contradiction. So $\\varphi(v_2)\\ne c$ and, similarly, $\\varphi(v_2)\\ne e$.\n\nGiven a set $S$ of vertices of $\\overrightarrow{T_5}$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\\in V(H)$ such that \nfor every vertex $x\\in V\\paren{\\overrightarrow{T_5}}$, we have $x\\in S$ if and only if there exists a $\\overrightarrow{T_5}$-coloring $\\varphi$ of $H$ such that $\\varphi(v)=x$.\nThus, with the graph $G'$ and the vertex $v_2$, we force a non-empty set $\\mathcal{S}\\subset V\\paren{\\overrightarrow{T_5}}\\setminus\\acc{c,e}=\\acc{a,b,d}$.\n\nWe use a series of constructions in order to eventually force the set $\\acc{a,b,c,d}$ starting from $\\mathcal{S}$.\nRecall that $\\acc{a,b,c,d}$ induces the tournament $\\overrightarrow{T_4}$.\nWe thus reduce $\\overrightarrow{T_5}$-coloring to $\\overrightarrow{T_4}$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\\cite{GO15}.\n\nThese constructions are summarized in the tree depicted in Figure~\\ref{fig:oriented}. The vertices of this forest contain the non-empty subsets of $\\acc{a,b,d}$ and a few other sets.\nIn this tree, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every arc has a label indicating the construction that is performed.\nIn every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$ and we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.\n\n\\begin{figure}[htpb]\n\\begin{center}\n \\includegraphics[scale=0.8]{oriented}\n \\caption{Forcing the set $\\acc{a,b,c,d}$.\\label{fig:oriented}}\n\\end{center}\n\\end{figure}\n\n\\begin{itemize}\n\\item Arcs labelled \"out\": The set $S_2$ is the out-neighborhood of $S_1$ in $\\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $vw$.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Arcs labelled \"in\": The set $S_2$ is the in-neighborhood of $S_1$ in $\\overrightarrow{T_5}$. We construct $H_2$ from $H_1$ by adding a vertex $w$ and the arc $wv$.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Arc labelled \"Z\": Let $g'$ be the smallest integer such that $g'\\ge g$ and $g'\\equiv4\\pmod{6}$. We consider a circuit $v_1,\\cdots,v_{g'}$.\nFor $2\\le i\\le g'$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_2$. Let $\\varphi$ be any $T_6$-coloring of $H_2$.\nBy construction, $\\acc{\\varphi(v_2),\\cdots,\\varphi(v_{g'})}\\subset S_1=\\acc{a,b,d}$.\nA circuit of length $\\not\\equiv0\\pmod{3}$ cannot map to the 3-circuit induced by $\\acc{a,b,d}$, so $\\varphi(v_1)\\in\\acc{c,e}$.\nIf $\\varphi(v_1)=c$ then $\\varphi(v_2)=d$ and if $\\varphi(v_1)=e$ then $\\varphi(v_2)=a$. Thus $S_2=\\acc{ad}$.\n\\end{itemize}\n\n\\section{Proof of Theorem~\\ref{thm:NPC}.2}\nLet $g$ be the largest integer such that there exists a graph in $P_g^{(0,2)}$ that does not map to $T_6$.\nLet $G\\in P_g^{(0,2)}$ be a graph that does not map to $T_6$ and such that the underlying graph of $G$ is minimal with respect to the homomorphism order.\n\nLet $G'$ be obtained from $G$ by subdividing an arbitrary edge $v_0v_3$ twice to create the path $v_0v_1v_2v_3$\nsuch that the edges $v_0v_1$ and $v_1v_2$ are red and the edge $v_2v_3$ gets the color of the original edge $v_0v_3$.\nBy minimality, $G'$ admits a homomorphism $\\varphi$ to $T_6$.\nSuppose for contradiction that $\\varphi(v_1)=f$. This implies that $\\varphi(v_0)=\\varphi(v_2)=b$. Thus $\\varphi$ provides a $T_6$-coloring of $G$, a contradiction.\n\nGiven a set $S$ of vertices of $T_6$, we say that we force $S$ if we specify a graph $H$ and a vertex $v\\in V(H)$ such that \nfor every vertex $x\\in V(T_6)$, we have $x\\in S$ if and only if there exists $T_6$-coloring $\\varphi$ of $H$ such that $\\varphi(v)=x$.\nThus, with the graph $G'$ and the vertex $v_1$, we force a non-empty set $\\mathcal{S}\\subset V(T_6)\\setminus\\acc{f}=\\acc{a,b,c,d,e}$.\n\nRecall that the core of a graph is the smallest subgraph which is also a homomorphic image.\nWe say that a subset $S$ of $V(T_6)$ is \\emph{good} if the core of the subgraph induced by $S$ is isomorphic\nto the graph $T_4$ which is a a clique on 4 vertices such that both the red and the blue edges induce a path of length $3$.\nWe use a series of constructions in order to eventually force a good set starting from $\\mathcal{S}$.\nWe thus reduce $T_6$-coloring to $T_4$-coloring, which is NP-complete for subcubic bipartite planar graphs with any given girth~\\cite{MO17}.\n\nThese constructions are summarized in the forest depicted in Figure~\\ref{fig:2edge}.\nThe vertices of this forest are the non-empty subsets of $\\acc{a,b,c,d,e}$ together with a few auxiliary sets of vertices containing $f$.\nIn this forest, an arc from $S_1$ to $S_2$ means that if we can force $S_1$, then we can force $S_2$. Every set with no outgoing arc is good.\nWe detail below the construction that is performed for each arc. In every case, we suppose that $S_1$ is forced on the vertex $v$ of a graph $H_1$\nand we construct a graph $H_2$ that forces $S_2$ on the vertex $w$.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[scale=0.8]{2edge}\n \\caption{Forcing a good set.\\label{fig:2edge}}\n\\end{center}\n\\end{figure}\n\n\\begin{itemize}\n\\item Blue arcs: The set $S_2$ is the blue neighborhood of $S_1$ in $T_6$. We construct $H_2$ from $H_1$ by adding a vertex $w$ adjacent to $v$ such that $vw$ is blue.\nThus, $S_2$ is indeed forced on the vertex $w$ of $H_2$.\n\\item Red arcs: The set $S_2$ is the red neighborhood of $S_1$ in $T_6$. The construction is as above except that the edge $vw$ is red.\n\\item Dashed blue arcs: The set $S_2$ is the set of vertices incident to a blue edge contained in the subgraph induced by $S_1$ in $T_6$. We construct $H_2$ from two copies of\n$H_1$ by adding a blue edge between the vertex $v$ of one copy and the vertex $v$ of the other copy. Then $w$ is one of the vertices $v$.\n\\item Dashed red arcs: The set $S_2$ is the set of vertices incident to a red edge contained in the subgraph induced by $S_1$ in $T_6$.\nThe construction is as above except that the added edge is red.\n\\item Arc labelled \"X\": Let $g'=2\\ceil{g\/2}$. We consider an even cycle $v_1,\\cdots,v_{g'}$ such that $v_1v_{g'}$ is red and the other edges are blue.\nFor every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$. We thus obtain the graph $H_2$ and we set $w=v_1$.\nLet $\\varphi$ be any $T_6$-coloring of $H_2$. In any $T_6$-coloring of $H_2$, the cycle $v_1,\\cdots,v_{g'}$ maps to a 4-cycle with exactly one red edge contained\nin the subgraph of $T_6$ induced by $S_1=\\acc{a,b,c,d,e}$. These 4-cycles are $aedb$ with red edge $ae$ and $cdba$ with red edge $cd$.\nSince $w$ is incident to the red edge in the cycle $v_1,\\cdots,v_{g'}$, $w$ can be mapped to $a$, $e$, $c$, or $d$ but not to $b$. Thus $S_2=\\acc{a,c,d,e}$.\n\\item Arc labelled \"Y\": We consider an alternating cycle $v_0,\\cdots,v_{8g-1}$. \nFor every vertex $v_i$, we take a copy of $H_1$ and we identify its vertex $v$ with $v_i$.\nWe obtain the graph $H_2$ by adding the vertex $x$ adjacent to $v_0$ and $v_{4g+2}$ such that $xv_0$ and $xv_{4g+2}$ are blue. We set $w=v_0$.\nIn any $T_6$-coloring $\\varphi$ of $H_2$, the cycle $v_1,\\cdots,v_{g'}$ maps to the alternating $4$-cycle $acde$ contained in $S_1=\\acc{a,c,d,e}$ such that $\\varphi(v_i)=\\varphi(v_{i+4\\pmod{8g}})$.\nSo, a priori, either $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{a,d}$ or $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{c,e}$.\nIn the former case, we can extend $\\varphi$ to $H_2$ by setting $\\varphi(x)=b$. In the latter case, we cannot color $x$ since $c$ and $e$ have no common blue neighbor in $T_6$.\nThus, $\\acc{\\varphi(v_0),\\varphi(v_{4g+2})}=\\acc{a,d}$ and $S_2=\\acc{a,d}$.\n\\end{itemize}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Comparing results of \\nameA and \\nameB}\nIn this section, we provide results from our analysis of measurements generated\nby the \\nameA and \\nameB platforms. In particular, we use measurements from each\nplatform to (1) understand which countries are the least free -- \\ie have the\nhighest amounts of censorship and (2) to demonstrate challanges in finding\nground truth when conducting censorship measurements. We use these results\nto demonstrate how the \\nameA and \\nameB platforms can provide complementary\ninsights into censor behavior.\n\n\\subsection{Identifying the least free countries}\nUsing the tests described in Section \\ref{sec:platforms}, we now report the\ncountries found to be the least free based on measurements obtained from the\n\\nameA and \\nameB platforms.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[h]{0.45\\textwidth}\n\\includegraphics[width=1\\textwidth]\n{plots\/iclab_overall_censorship_stats\/out_top_6_url_method_combs_no_null-crop.pdf}\n\\end{subfigure}\n\n\\begin{subfigure}[h]{0.45\\textwidth}\n\\includegraphics[width=1\\textwidth]\n{plots\/ooni_overall_censorship_stats\/top_6_censorship_with_errors-crop.pdf}\n\\end{subfigure}\n\n\\caption{The six most censored countries according to measurements from \\nameA\n(top) and \\nameB (bottom).}\n\\label{fig:iclab-ooni-least-free}\n\\end{figure}\n\nFigure \\ref{fig:iclab-ooni-least-free} illustrates the fraction of URLs that were\ncensored in each of the six least free countries -- Iran (IR), Saudi Arabia\n(SA), India (IN), Cyprus (CY), China (CN), and Russia (RU) -- based on tests\nconducted by the \\nameA platform; and Iran (IR), Saudi Arabia\n(SA), India (IN), Greece (GR), Qatar (QA), and Turkey (TR) according to \\nameB's\nmeasurements. We note that the remaining countries only displayed marginal\namounts of censorship -- \\ie under 5\\% of all tested URLs were censored.\nWe find that both platforms find the most censorship in Iran, Saudi Arabia,\nand India.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]\n{plots\/ooni-iran\/ooni-iran}\n\\caption{The number of domains seen blocked in Iran spiked in \\nameB measurement\ndata in fall of 2015, several months before the election.}\n\\label{fig:iranblocking}\n\\end{figure}\n\nIn the specific case of Iran, both the \\nameA and \\nameB platforms show\ncomparably high levels of censorship. We see that both platforms are able to\ndetect the large fraction of blockpages served by Iranian censors and that in\naddition to identifying the Iran blockpage, the TTL and RST anomaly detectors\nin the \\nameA platform are also triggered. We attribute the extremely high\nlevels of blocking observed to the fact that the measurements from both\nplatforms were carried out around the same time period as the Iranian\nparliamentary elections. We investigate further using past data from the \\nameB\nplatform and confirm that in October 2015, four months before the elections, a\nsharp rise in censorship was observed. This is illustrated in Figure\n\\ref{fig:iranblocking}. The URLs tested by the \\nameB platform during this time\nincluded content relating to political news and speech, social media,\ncensorship circumvention tools, and pornography.\n\nAnalyzing the results for Saudi Arabia and India we find that measurements\nperformed on the \\nameA platform see significantly less information controls\nthan on the \\nameB platform. In particular, while measurements from the \\nameA\nplatform detected a number of blockpages and RSTs in each of these countries, we\nfind that it did not encounter the large number of DNS anomalies and incomplete\nTCP connections that are observed by the \\nameB platform. We attribute this to\nthe fact that measurements from the \\nameB platform are usually obtained from\nresidential networks where covert censorship is observed (\\ie censorship without\nexplicitly serving a blockpage).\n\nFinally, the results from both platforms also provide an insight into the\ncensorship infrastructure in place in each country. The presence of a single\ndominant method of censorship in Iran and Saudi Arabia are indicative of the\npresence of a central censorship apparatus, while the case of India -- where\nmultiple equally dominant methods are observed -- is indicative of censorship\nbeing implemented by local ISPs rather than the central government.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ |l|c|l|c| }\n \\hline\n URL & \\# of VPs & URL & \\# of VPs \\\\\n \\hline\n battle.net & 1459 & uol.com.br & 842 \\\\\n 163.com & 1417 & alibaba.com & 748 \\\\\n baidu.com & 1350 & yahoo.com & 700 \\\\\n hao123.com & 1333 & directrev.com & 564 \\\\\n youth.cn & 918 & roblox.com & 415 \\\\\n \\hline\n\\end{tabular}\n\\caption{Websites with the highest number of TCP \\texttt{RST} packets in \\nameA.}\n\\label{tab:rsts}\\end{center}\n\\end{table}\n\n\\subsection{The elusive ground truth}\nGround truth plays a crucial role in analyzing censorship measurement data, and\nthere are several challenges associated with gathering ground-truth censorship data\nat scale. Comparing measurement data collected in the field against a baseline \ncollected in well-provisioned network settings (\\ie in the lab) helps delineate \ncensorship from server-side blocking caused by VPN blocking or automated measurements \nnot looking like real user traffic.\nTable \\ref{tab:rsts} shows websites with the highest number of TCP \\texttt{RST} \npackets in their streams across \\nameA's vantage points, pointing to possible \nserver-side blocking.\n\nAmong the list of websites with many observed \\texttt{RST}s are several\nwebsites hosted in China (\\eg \\texttt{163.com} and \\texttt{baidu.com}) that\nexhibit anomalous TCP behaviors when queried by \\nameA -- \\ie the IPID values \nfrom \\texttt{SYN} and \\texttt{SYN-ACK} packets are different from the rest of \nthe packets receieved, and sequence numbers overlapping between packets.\n\nThis anomaly is hard to distinguish from anomalous traits that are caused by the\nGreat Firewall of China. Similarly, gaming websites \\texttt{roblox.com} and \\texttt{battle.net}\naggressively block VPN users, while \\texttt{yahoo.com} and \\texttt{directrev.com} (an ad\nmarketplace website) do the same to a lesser degree. Other websites\ncan also respond unexpectedly (\\eg due to server misconfiguration) and\ntrigger false alarms. As an example, the Iranian retail\nwebsite \\texttt{digikala.com} shows sequence number anomalies as tested by 587 of\n\\nameA's vantage points, but is not censored in any of them.\n\n\\nameB faces a similar problem in determining ground truth. Many of the `control'\nmeasurements used by the local client to determine what sites should look like\nare conducted through Tor. In practice, many websites either fully deny, or\ndisplay substantially different content to visitors through the Tor network,\nmaking it difficult for the probe to determine if the local result is correct\nor not.\n\nAn additional challenge arises due to websites that are suddenly unavailable for\nnon-censorship reasons -- \\eg a dead website with a registered domain and\nunavailable webserver. For these cases, the \\nameA platform verifies if the\nwebpage could be loaded from any one of its other vantage points. If the page\nwas unable to be loaded successfully, it is discarded from the test outputs.\nFigure \\ref{fig:iclab-dead-sites} illustrates the URLs that were censored in the\n20 least free countries. We observe several vertical bands in this figure. These\nare indicative of dead websites, ones which could not be loaded from any vantage\npoint.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]\n{plots\/url_scatter\/t-top_20_url_vs_country-crop.pdf}\n\\caption{URLs censored in the 20 least free countries according to \\nameA.\nColors are indicative of the type of blocking observed (see Figure\n\\ref{fig:iclab-ooni-least-free}). The black vertical lines show websites\nthat are either no longer available or have blocked access to all of\nthe \\nameA vantage points.}\n\\label{fig:iclab-dead-sites}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this paper we presented the fundamental design decisions faced by\ndevelopers of large-scale longitudinal censorship measurement platforms --\nwhere to obtain vantage points, how to use these vantage points to collect\nmeasurements, what measurements to collect, and how to analyze them. We then\ndescribed the decisions that were made in development of the \\nameA and \\nameB\nplatform and their influence on the measurements obtained by these platforms.\n\nIn particular, we find that the \\nameA platform is able to provide a more\nreliable and global picture of censorship by harnessing dedicated global VPN\ninfrastructures in addition to on-the-ground volunteers. However, we also find\nthat this dependence on VPNs can result in measurements being carried out\non vantage points further away from residential networks which impacts the\nconclusions drawn from the platform. For example, the \\nameA platform sees\nsignificantly less censorship than the \\nameB platform in India and Saudi\nArabia. To this extent, it is important to work with representatives in affected\ncountries, and the responsive nature of \\nameB has been successful in gaining\nsupport to measure several important political events.\nThe challenge of obtaining representative, global, reliable, and response\nmeasurements remains a goal we continue to aspire to.\n\nIn addition, we showed how the results obtained from each of these platforms can\nbe used to provide a deeper insight into understanding regional censorship at a\nglobal scale. By analyzing the types of censorship observed in several countries\nwe were able to identify characteristics of the implemented censorship apparatus\n-- \\ie results obtained by both platforms suggest the presence of a\ndecentralized censorship infrastructure in India and a mostly centralized\ninfrastructure in Iran and Saudi Arabia.\n\nFinally, our current investigation also uncovers open challenges that remain in\nbeing able to distinguish censorship from anomalies that arise from phenomena\nsuch as misconfigured webservers, network outages, end-point discrimination,\nand unresponsive websites. Our platforms plan on addressing these limitations\nin future work.\n\n\n\\section{Design Decisions} \\label{sec:challenges}\n\nWe now discuss the choices of\n(1) where\nmeasurements are run,\n(2) how measurements are run,\nand\n(3) how data is intepreted .\nThese three design decisions are central to\nthe design of a global censorship measurement platform.\n\n\\subsection{Where measurements are run} \nThere are two options when considering\nvantage points: crowd-sourced and dedicated infrastructure vantage points.\nPlatforms using a crowd-sourced approach rely on volunteers running measurement\nsoftware. They have the ability to turn citizens in any location into vantage\npoints. Dedicated infrastructure, on the other hand are \ndistributed and operated exclusively for the platform. Both approaches have their own\nbenefits and drawbacks.\n\n\\myparab{Cost and availability.} A hurdle in setting up a\ndedicated infrastructure is distributing infrastructure globally.\nHowever, once this infrastructure is in place, it \n has the capability of performing on-demand measurements; limited only\nby the reliability and uptime of the infrastructure. Crowd-sourced platforms, \non the other hand,\nincur no setup cost\nbut are dependent on the availability of volunteers to execute\nmeasurements. As a consequence,\ncrowd-sourced\nplatforms are unable to provide a reliable flow of measurements from a region.\n\n\\myparab{Representativeness and diversity of measurements.}\nCrowd-sourced platforms\nhave \nthe potential to obtain a view of the Internet from a wide variety of networks\n(\\eg residential, academic, and corporate).\nIn contrast, dedicated infrastructure faces an uphill battle of distributing devices \n or may leverage existing infrastructure (\\eg academic networks, or dedicated\nhosting networks). \nVantage point location can \n impact conclusions drawn from their measurements. As an example,\nmeasurements conducted from the UK academic network (JANET) do not \nobserve the \n``Great Firewall of Cameron'' \\cite{GFC}, since they are placed\noutside of its purview. Crowd-sourced platforms can also leverage \npublic interest and news coverage \nto introduce additional vantage points.\n\n\\myparab{Safety and risk.} Information controls\nmeasurements using humans in the field poses a significant and hard to quantify risk. In\nmany regions (\\eg Syria) this risk has been determined to be too high for \nvolunteers. Such risks impede crowd-sourced measurements, while infrastructure \n(such as VPN or hosting networks) allow for\nmeasurements while posing little to no risk to users.\n\n\n\\subsection{Measurement autonomy} \nA censorship measurement platform\ncan either \nuse a \ncentral server to schedule experiments, or leave these tasks to each vantage point. This\ndichotomy has an impact on several capabilities of the platform.\n\n\\myparab{Time and context sensitive measurements.} The time and political\ncontext of measurements are important for understanding evolving and\nabrupt policy changes. As an example, during the rise of ISIS in 2014, the\nIndian government blocked (and subsequently unblocked) access to 32 websites\nincluding GitHub, Vimeo, and PasteBin for propagating ``Anti India content''\n\\cite{ZDNet-IndiaDoT}. Centrally controlled measurement platforms have the\nadvantage of being able to evolve existing and schedule new measurements \n in response to changing political and social situations. Locally controlled\nplatforms, however, do not have this capability. Instead they are dependent on\nthe update schedule of the local vantage point.\n\n\n\\myparab{Infrastructure requirements.} The ability to remotely schedule \nexperiments and aggregate data centrally allows for the use of\ncomputationally constrained infrastructure, not needing technically savvy local\nmaintenance efforts. This comes with the cost of bandwidth requirements\nassociated with shipping unprocessed data to a central\nserver. Locally controlled platforms require local management of the platform\ninfrastructure to ensure up-to-date experiments, higher computational\ncapabilities for processing gathered data, and lower bandwidth for communicating\nprocessed results of measurements.\n\n\\subsection{Gathering and interpreting data}\n A censorship measurement\nplatform must specify data collected and how it will identify censorship \nin this\ndata.\n\n\\myparab{Type and quantity of data gathered.} A platform may record packet\ncaptures of entire tests or selectively gather data such as packet headers and\nresponses. While complete packet captures are ideal\nfor deep aposteriori analysis and to identify censorship not\nvisible at the application layer. However, they require root privileges, high storage \nand bandwidth \nrequirements, and may accidentally collect data of other system users.\n\n\\myparab{Identifying censorship events.} Another challenge that arises\nduring the processing of gathered data is defining when ``censorship'' has occurred. \nThis task is complicated by \nstrange protocol implementations (\\eg load\nbalancers that cause gaps in TCP sequence numbers), server side blocking~\\cite{torndss16}, and \nregular network failures.\n\n\\section{Introduction} \\label{sec:introduction}\n\nThe last five years have cemented the Internet as critical infrastructure for\ncommunication. In particular, it has demonstrated high\nutility for citizens and political activists to obtain accurate information,\norganize political movements, and express dissent. This fact has not gone\nunnoticed, with governments clamping down on this medium \\via censorship and\ninformation controls.\nConsequently, there has been a surge of interest in measuring various aspects\nof online information controls. More specifically, data obtained\nfrom such measurements has been used by (1) political activists to understand\nthe motivation for and the impact of such government policies \nand\n(2) researchers to build safer and more secure censorship circumvention tools\nby understanding the techniques used to implement these policies \\cite{tor-pt}.\n\nWhile there have been numerous efforts to characterize\nonline information controls \\cite{Chaabance-IMC14, censmon, Roberts2011a,\nWright2011a, Aryan2013, Aceto2015a}, the data gathered or\nused by these measurements have limited scope due to the specificity of\nlocations and time-periods considered. In order to gain a nuanced understanding\nof the evolution of Internet censorship, in terms of policy and techniques, a\nmeasurement platform needs to be able to gather longitudinal data\nfrom a diverse set of regions while performing accurate analysis using robust \nand well specified techniques. We present and compare two such platforms --\n\\nameA and \\nameB -- that represent different points in the censorship\nmeasurement design space.\n\n\nIn this paper, we first identify three primary design decisions made in \nthe development of censorship measurement platforms. Then, we describe how \n\\nameA and \\nameB address these decisions, \nwhile\nconsidering the impact of these decisions on the measurement results produced by the systems. \n Finally, we show how \\nameA and \\nameB, when used together, provide a unique \n insight into the current state of information\ncontrols around the globe.\n\n\n\\section{The \\nameA and \\nameB Platforms}\\label{sec:platforms}\n\nIn this section we describe the design decisions made during the development of\nthe \\nameA and \\nameB platforms.\n\n\\subsection{Vantage points}\nThe most fundamental difference between the \\nameA and \\nameB platforms is the \napproach each system takes to recruit vantage points. \n\\nameA relies on a dedicated infrastructure to perform measurements.\nThis allows measurements that require permissions that may not be compatible with \nsoftware to be run on end-user systems. As a consequence, the system has thus far \nfocused on deployment on VPN vantage points and a limited deployment of \n Raspberry Pi's installed with \\nameA software.\nIn contrast, \\nameB takes a lighter-weight software-based approach. \nand assumes some amount of technical savvy on the \npart of volunteers.\nThis leads to differences in the\navailability and representativeness of measurements from each platform.\n In Figure \\ref{fig:choropleth}, we see that as\na result of the decision to use VPN end points, \\nameA is\nable to provide vantage points for measurements in significantly more countries\nthan \\nameB (151 for \\nameA and 46 for \\nameB in the last 100\ndays\\footnote{Since its release in 2012, \\nameB has received nearly 10M\nmeasurements from volunteers in 95 countries.}). However, we found that that \\nameB's crowd-sourced model is\nable to provide more AS-level diversity -- \\ie \\nameB provides vantage points\nfrom an average of 3.15 different networks (ASes) in each country, compared to\n\\nameA's average of 1.46 networks per country. \n\n\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[h]{0.5\\textwidth}\n\\includegraphics[trim=0cm 0.125cm 0cm 0cm, clip=true,width=\\textwidth]\n{figures\/combined-choro-sol.pdf}\n\\caption{Global availability of measurements.}\n\\label{fig:choropleth}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.5\\textwidth}\n\\includegraphics[trim=0cm 0.125cm 0cm 0cm, clip=true,width=\\textwidth]\n{figures\/country-availability.pdf}\n\\caption{Country-level temporal availability. Green and red indicate \nthe availability of measurement data from \\nameA and \\nameB, respectively. %\nBlack indicates the availability of measurements from\nboth platforms from the same region on the given day.}\n\\label{fig:country-availability-heat}\n\\end{subfigure}\n\\caption{{Availability of measurements from the \\nameA and \\nameB platforms in\nthe last 100 days.}}\n\\label{fig:vantage-points}\n\\end{figure}\n\n\n\n\\subsection{System architecture}\nIn terms of system architectures, \\nameA uses a central controller to schedule\nexperiments while \\nameB processes data both on the vantage point and inside of\nits data processing pipeline.\nThis introduces several key differences in how each platform\nhandles its vantage points.\n\n\\myparab{Measurement scheduling.}\n\\nameA takes a centralized approach to scheduling experiments, leveraging a single \nserver that is able to schedule experiments on all deployed nodes (\\eg VPNs, Raspberry Pis) or \na subset thereof (e.g, a given country). This facilitates the execution of ongoing or one-off measurements.\nIn contrast, \\nameB takes a decentralized approach. Recommended measurements\nare hard-coded into the \\nameB platform source-code and require vantage points\n(technically savvy volunteers) to regularly download updates in order to execute\nnew measurements. Repetition of measurements is dependent on individual\nvolunteer availability. Volunteers\nalso have the option to add their own tests and modify inputs to existing tests\n(\\eg they may change the set of URLs being used by a test).\n\n\\myparab{Performing measurements.}\n\\nameA and \\nameB also differ in their approach to performing measurements. \n\\nameA takes a ``simple node'' approach, with nodes largely being responsible for collecting \ndata and transmitting it back to the central server for later analysis. This lowers the \ncomputational requirement of the vantage points but increased demands on bandwidth. \nIn contrast, \\nameB performs measurements and analysis on the device and ships \nprocessed data back to a central server. Importantly, \\nameB allows volunteers to\nopt-out of submitting measurement reports to the \\nameB publishing server, while \\nameA \ntakes an informed approach with vantage points opting into participate in the system.\n\nFigure \\ref{fig:country-availability-heat} shows the impacts of these decisions on the platforms.\n\\nameB has a core set of vantage points that continuously measure and a few opportunistic \nmeasurements. \\nameA on the other hand exhibits large coordinated testing as a result \nof its VPN vantage points. \n\n\\subsection{Tests and analysis}\n\nBoth platforms perform a battery of tests to identify censors that may be\nblocking or manipulating content. The \\nameA test infrastructure is extensible,\nallowing new tests to be scheduled on vantage points without the need for\nupdating their software. In addition to custom tests, the \\nameA platform\nperiodically schedules a baseline test on each vantage point. This baseline\nexperiment tests connectivity to a set of URLs that are composed of the Alexa\nTop 500 websites and a country-specific list of potentially blocked URLs\n(obtained from the CitizenLab). In contrast, tests on the \\nameB platform are\nnot scheduled remotely and new tests need to be obtained by software updates.\nExisting tests, however, do not require software updates to evolve the list of\ndomains that they test connectivity to. The default experiments included in the\n\\nameB platform test connectivity to the global and country-specific lists of\npotentially blocked URLs (also obtained from the CitizenLab).\n\nIn terms of analysis, the \\nameA platform does not perform analysis on the\nvantage points, rather it leaves all post-processing to the centralized servers.\nThis allows \\nameA to perform retroactive analysis on existing results. The\n\\nameB platform, on the other hand, performs data analysis on the vantage\npoints. This allows independent and private deployments by in-country watchdog\ngroups. \n\nWe now briefly describe the tests conducted to identify censorship by each\nplatform. \n\n\\myparab{DNS anomaly detection.}\nFor each URL to be tested on a given vantage point, the \\nameA and \\nameB vantage\npoints perform DNS name resolution queries for the domain name associated with\nthat URL using both the default DNS resolver configured on the machine as well\nas Google's DNS at \\texttt{8.8.8.8}. The \\nameA platform concludes that an\nanomaly (\\eg DNS injection, tampering, \\etc) has occurred if a second DNS\nresponse is received within 2 seconds of the first. The \\nameB platform on\nthe other hand, makes several requests at once and does not wait between\nrequests. Requests are also made to control resolver that binds to a non\nstandard DNS port. The client is able to report failures to resolve directly,\nand resolutions are included in the generated report to allow further analysis\nby the central analysis infrastructure.\n\n\\myparab{HTTP tampering, proxy, and blockpage detection.}\nFor each URL to be tested on a given vantage point, the \\nameA and \\nameB\nvantage points issue HTTP GET requests and record received responses, with\n\\nameA to follow redirects. The responses received from these tests are\nprocessed to identify blockpages and evidence of HTTP tampering. The \\nameA\nplatform uses regular expression pattern matching to identify known blockpages\nand responses obtained by the same test executed from a censor-free vantage\npoint in the US to identify instances of content manipulation. The \\nameB\nplatform uses meta-data (\\eg status codes, response sizes, \\etc) obtained from a\nTor control channel to identify HTTP tampering. Additionally, the \\nameB\nplatform is also able to detect the presence of HTTP proxies. It does this by\ngenerating malformed HTTP requests that cause proxies on the vantage point\nnetwork to reveal their presence (\\eg by modifying the malformed headers). The\ndata processing pipeline is then capable of identifying specific types of proxy\nsoftware based on known fingerprints.\n\n\\myparab{TLS man-in-the-middle detection.}\nThe \\nameA platform also performs tests on HTTPS compatible URLs. For each such\nURL, a TLS handshake is performed and all server certificates that are received\nare checked for validity. If they are found to have been expired or signed by an\nuntrusted certificate authority, a TLS anomaly is reported.\n\n\\myparab{Sequence number, TTL, and RST anomaly detection.}\nFor each of the above tests, the \\nameA platform analyzes raw data (packet\ncaptures) of TCP streams to identify inconsistent sequence number and TTL values\nin packet headers. Additionally, the presence of pre-mature RST packets is also\nrecorded. If any of these are identified, the \\nameA platform reports anomalies\nthat may be the result of a censor injecting packets into a TCP stream. \n\n\\myparab{TCP connectivity test.}\nThe \\nameB platform attempts to establish TCP connections to a specified set of\nhosts to validate that the handshake can be completed and detect instances of\nIP level blocking. In this test, the vantage point attempts to establish a\nconnection to the end host directly and also via a Tor control channel. If the\ncontrol channel succeeds, while the direct connection fails, an anomaly is\nreported.\n\n\\myparab{Circumvention protocol tests.}\nFinally, the \\nameB platform also includes a set of tests designed to detect the\navailability of several circumvention system by (1) mimicking the protocols\ninvolved and (2) by launching bundled instances of the actual tools and checking\nwhether they are able to successfully complete connections. Currently the test\nconsiders the connectivity of all Tor pluggable transports (scramblesuit, meek,\nfteproxy, obfsproxy versions 2 to 4), Psiphon, Lantern, and the OpenVPN protocol.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\nScalar mesons are still a puzzle in light meson spectroscopy: they have complex structure,\nand there are too many states to be accommodated within the quark model without difficulty~\\cite{polosa}. \nIn particular, the structure of the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave is a longstanding problem. In recent years many experiments have performed \naccurate studies of the decays of heavy-flavored hadrons producing a $K \\pi$ system in the final state.\nThese studies include searches for \\ensuremath{C\\!P}\\xspace violation~\\cite{cp}, and searches for, and observation of, new exotic resonances~\\cite{zs} and\ncharmed mesons~\\cite{bs}.\nHowever, the still poorly known structure of \nthe $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave is a source of large systematic uncertainties.\nThe best source of information on the scalar structure of the $K \\pi$ system comes from the LASS experiment, which studied the reaction $\\mbox{${K^{-}}$} p \\mbox{$\\rightarrow$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{+}}$} n$~\\cite{lass_kpi}.\nPartial wave analysis of the $K \\pi$ system reveals a large contribution from the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude over the mass\nrange studied.\nIn the description of the $I=1\/2$ scalar amplitude up to a $K \\pi$ mass of about 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ the $K^*_0(1430)$ resonant amplitude is added coherently to an effective-range\ndescription in such a way that the net amplitude\nactually decreases rapidly at the resonance mass. The $I=1\/2$ $\\mathcal{S}$-wave amplitude representation is given explicitly in Ref.~\\cite{babar_z}.\nIn the LASS analysis, in the region above 1.82 \\mbox{${\\mathrm{GeV}}\/c^2$}, the $\\mathcal{S}$-wave suffers from a two-fold ambiguity, but in both solutions it is understood in terms of the presence of a $K^*_0(1950)$ resonance. It should be noted that the extraction of the $I=1\/2$ $\\mathcal{S}$-wave amplitude is complicated by the presence of an $I=3\/2$ contribution. \n\nFurther information on the $K \\pi$ system has been extracted from Dalitz plot analysis of the decay $D^+ \\mbox{$\\rightarrow$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{+}}$}$ where, in order to fit \nthe data, the presence of an additional resonance, the $\\kappa(800)$, was claimed~\\cite{aitala}. Using the same data, a Model Independent Partial Wave Analysis (MIPWA)\nof the $K \\pi$ system was developed for the first time~\\cite{aitala1}.\nThis method allows the amplitude and phase of the $K \\pi$ $\\mathcal{S}$-wave to be extracted as\n functions of mass (see also Refs.~\\cite{cleo} and ~\\cite{focus}). However in these analyses the phase space is limited to mass values less than 1.6 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ due to the kinematical limit imposed by the $D^+$ mass.\nA similar method has been used to extract the $\\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ $\\mathcal{S}$-wave amplitude in a Dalitz plot analysis of $D^+_s \\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$} \\mbox{${\\pi^{+}}$}$~\\cite{marco}.\n\nIn the present analysis, we consider three-body \\ensuremath{\\eta_c}\\xspace decays to $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ and obtain new information on the $K \\pi$ $I=1\/2$ $\\mathcal{S}$-wave amplitude extending up to a mass of 2.5 \\mbox{${\\mathrm{GeV}}\/c^2$}. We emphasize that, due to isospin conservation in the \\ensuremath{\\eta_c}\\xspace hadronic decay to $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$,\nthe $(K \\pi)$ amplitude must have $I=1\/2$ , and there is no $I=3\/2$ contribution.\nThe {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ experiment first performed a Dalitz plot analysis of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\eta$ using an isobar model~\\cite{etakk}. The analysis reported the first observation of $K^*_0(1430) \\mbox{$\\rightarrow$} K \\eta$, and observed that \\ensuremath{\\eta_c}\\xspace decays to three pseudoscalars are dominated by intermediate\nscalar mesons. A previous search for charmonium resonances decaying to \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace in two-photon interactions is reported in Ref.~\\cite{kkpipipi0}. \nWe continue these studies of \\ensuremath{\\eta_c}\\xspace decays and extract the $K \\pi$ $\\mathcal{S}$-wave amplitude by performing a MIPWA of both \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace final states. \n\nWe describe herein studies of the $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ system produced in two-photon interactions.\nTwo-photon events in which at least one of the interacting photons is not quasi-real are\nstrongly suppressed by the selection \ncriteria described below. This implies that the allowed $J^{PC}$ values of\nany produced resonances are $0^{\\pm+}$, $2^{\\pm+}$, $3^{++}$, $4^{\\pm+}$...~\\cite{Yang}. \nAngular momentum conservation, parity conservation, and charge conjugation\ninvariance imply that these quantum numbers also apply to\nthe final state except that the $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ state cannot be in a $J^P=0^+$ state.\n\nThis article is organized as follows. In Sec.\\ II, a brief description of the\n{\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector is given. Section III is devoted to the event reconstruction and data selection of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace system. In Sec.\\ IV, we describe studies of efficiency and resolution,\nwhile in Sec.\\ V we describe the MIPWA. In Secs. VI and VII we perform Dalitz plot analyses of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decays. Section VIII is devoted to discussion of the measured $K \\pi$ $\\mathcal{S}$-wave amplitude, and finally results are summarized in Sec.~IX.\n\n\\section{The {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector and dataset}\n\nThe results presented here are based on data collected\nwith the {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector\nat the PEP-II asymmetric-energy $e^+e^-$ collider\nlocated at SLAC, and correspond \nto an integrated luminosity of 519~\\mbox{${\\mathrm{fb}^{-1}}$}~\\cite{luminosity} recorded at\ncenter-of-mass energies at and near the $\\Upsilon (nS)$ ($n=2,3,4$)\nresonances. \nThe {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ detector is described in detail elsewhere~\\cite{BABARNIM}.\nCharged particles are detected, and their\nmomenta are measured, by means of a five-layer, double-sided microstrip detector,\nand a 40-layer drift chamber, both operating in the 1.5~T magnetic \nfield of a superconducting\nsolenoid. \nPhotons are measured and electrons are identified in a CsI(Tl) crystal\nelectromagnetic calorimeter. Charged-particle\nidentification is provided by the measurement of specific energy loss in\nthe tracking devices, and by an internally reflecting, ring-imaging\nCherenkov detector. Muons and \\KL\\ mesons are detected in the\ninstrumented flux return of the magnet.\nMonte Carlo (MC) simulated events~\\cite{geant}, with reconstructed sample sizes \nmore than 10 times larger than the corresponding data samples, are\nused to evaluate the signal efficiency and to determine background features. \nTwo-photon events are simulated using the GamGam MC\ngenerator~\\cite{BabarZ}.\n\n\\section{ {\\boldmath Reconstruction and selection of $\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} events}\n\nTo study the reaction\n\\begin{equation}\n\\gamma \\gamma \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}\n\\end{equation}\nwe select events in which the $e^+$ and $e^-$ beam particles are scattered \nat small angles, and hence are undetected in the final state. \nWe consider only events for which the number of well-measured charged-particle tracks with\ntransverse momentum greater than 0.1~\\mbox{${\\mathrm{GeV}}\/c$}\\ is exactly equal to four, and for which there are no more than five photon candidates\nwith reconstructed energy in the electromagnetic calorimeter greater than 100 MeV.\nWe obtain $K^0_S \\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ candidates by means of a vertex fit of pairs of oppositely charged tracks which\nrequires a $\\chi^2$ fit probability greater than 0.001. Each \\KS candidate is then combined with \ntwo oppositely charged tracks, and fitted to a common vertex, with the requirements that the fitted vertex be within the\n$e^+ e^-$ interaction region and have a $\\chi^2$ fit probability greater than 0.001.\nWe select kaons and pions by applying high-efficiency particle identification criteria.\nWe do not apply any particle identification requirements\nto the pions from the \\KS decay.\nWe accept only $K_S^0$ candidates with decay lengths from the main vertex of the event greater than 0.2 cm, and\nrequire $\\cos \\theta_{\\KS}>0.98$, where $\\theta_{\\KS}$ is defined as the angle between the \\KS momentum direction and the\nline joining the primary and the \\KS vertex.\nA fit to the $\\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$}$ mass spectrum using a linear function for the background and a Gaussian\nfunction with mean $m$ and width $\\sigma$ gives $m=497.24$ \\mbox{${\\mathrm{MeV}}\/c^2$}\\ and $\\sigma=2.9$ \\mbox{${\\mathrm{MeV}}\/c^2$}. We select the $\\KS$ signal region to be within\n$\\pm 2 \\sigma$ of $m$ and reconstruct the \\KS 4-vector by adding the three-momenta of the pions and computing the energy using the \\KS PDG mass value~\\cite{pdg}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig1.eps}\n\\caption{Distributions of \\mbox{$p_T$}\\xspace\\ for $\\gamma \\gamma \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$. The data are shown as (black) points with error bars,\nand the signal MC simulation as a (red) histogram; the vertical dashed line indicates the selection applied to select two-photon events.}\n\\label{fig:fig1}\n\\end{center}\n\\end{figure}\nBackground arises mainly from random combinations of particles from\n\\ensuremath{e^+e^-}\\xspace\\ annihilation, from other two-photon processes, and from events with initial-state photon radiation (ISR). The ISR \nbackground is dominated by $J^{PC}=1^{--}$ resonance production~\\cite{isr}.\nWe discriminate against \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace events produced via ISR by requiring $\\ensuremath{{\\rm \\,mm}}\\xspace\\equiv(p_{\\ensuremath{e^+e^-}\\xspace}-p_{\\mathrm{rec}})^2>10$~GeV$^2$\/$c^4$, where\n$p_{\\ensuremath{e^+e^-}\\xspace}$ is the four-momentum of the initial state and $p_{\\mathrm{rec}}$ is the four-momentum of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace system. \n\nThe \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum shows a prominent \\ensuremath{\\eta_c}\\xspace signal.\nWe define \\mbox{$p_T$}\\xspace\\ as the magnitude of the vector sum of the transverse momenta, in the \\ensuremath{e^+e^-}\\xspace\\ rest frame, of the final-state particles with respect to the beam axis.\nSince well-reconstructed two-photon events are expected to have low values of \\mbox{$p_T$}\\xspace, we optimize the selection as a function of this variable.\nWe produce $\\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ mass spectra with different \\mbox{$p_T$}\\xspace selections and fit the mass spectra to extract the number of \\ensuremath{\\eta_c}\\xspace signal events ($N_s$) and the number\nof background events below the \\ensuremath{\\eta_c}\\xspace signal ($N_b$). We then compute the purity, defined as $P = N_s\/(N_s + N_b)$, and the significance $S = N_s\/\\sqrt{N_s + N_b}$. To obtain the best significance with the highest purity, \nwe optimize the selection by requiring the maximum value of the product of purity and significance, $P \\cdot S$, and find that this corresponds to the requirement $\\mbox{$p_T$}\\xspace<0.08~\\mbox{${\\mathrm{GeV}}\/c$}$.\n\nFigure~\\ref{fig:fig1} shows the measured \\mbox{$p_T$}\\xspace\\ distribution in comparison to the corresponding \\mbox{$p_T$}\\xspace\\ distribution obtained from simulation of the signal process.\nA peak at low \\mbox{$p_T$}\\xspace\\ is observed indicating\nthe presence of the two-photon process. The shape of the peak agrees well with that seen in the MC simulation. \nFigure~\\ref{fig:fig2} shows the $\\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ mass spectrum in the \\ensuremath{\\eta_c}\\xspace mass region. A clear \\ensuremath{\\eta_c}\\xspace signal over a background of about 35\\% can be seen, together with a residual \\ensuremath{{J\\mskip -3mu\/\\mskip -2mu\\psi\\mskip 2mu}}\\xspace signal. Information on the fitting procedure is given at the end of Sec. IV.\nWe define the \\ensuremath{\\eta_c}\\xspace signal region as the range 2.922-3.039~\\mbox{${\\mathrm{GeV}}\/c^2$}\\ ($m(\\ensuremath{\\eta_c}\\xspace) \\pm 1.5 \\ \\Gamma$), which contains 12849 events with a purity of\n$(64.3 \\pm 0.4)$\\% . \nSideband regions are defined by the ranges 2.785-2.844~\\mbox{${\\mathrm{GeV}}\/c^2$} \\ and 3.117-3.175~\\mbox{${\\mathrm{GeV}}\/c^2$}\\ (within 3.5-5 $\\Gamma$), respectively as indicated (shaded) in\nFig.~\\ref{fig:fig2}.\n\nDetails on data selection, event reconstruction, resolution, and efficiency measurement for the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decay can be found in Ref.~\\cite{etakk}.\nThe \\ensuremath{\\eta_c}\\xspace signal region for this decay mode contains 6710 events with a purity of $(55.2 \\pm 0.6)$\\%. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig2.eps}\n\\caption{The $\\KS \\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ mass spectrum in the \\ensuremath{\\eta_c}\\xspace mass region after requiring $\\mbox{$p_T$}\\xspace<0.08~\\mbox{${\\mathrm{GeV}}\/c$}$. The solid curve shows the total fitted function,\nand the dashed curve shows the fitted background contribution. The shaded areas show signal and sideband regions.}\n\\label{fig:fig2}\n\\end{center}\n\\end{figure}\n\n\\section{Efficiency and resolution}\n\nTo compute the efficiency, MC signal events are generated using a detailed detector simulation~\\cite{geant} in which the \\ensuremath{\\eta_c}\\xspace decays uniformly in phase space.\nThese simulated events are reconstructed and analyzed in the same manner as data. The efficiency is computed as the ratio of \nreconstructed to generated events. \nDue to the presence of long tails in the Breit-Wigner (BW) representation of the resonance, we apply \nselection criteria to restrict the generated events to the \\ensuremath{\\eta_c}\\xspace mass region. \nWe express the efficiency as a function of the invariant mass, $m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$~\\cite{conj}, and $\\cos \\theta$, where $\\theta$ is the angle, in the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ \nrest frame, between the directions of the \\mbox{${K^{+}}$}\\ and the boost from the $\\KS \\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ rest frame.\n\nTo smooth statistical fluctuations, this efficiency map is parametrized as follows.\nFirst we fit the efficiency as a function of $\\cos \\theta$ in separate intervals of $m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$, using \nLegendre polynomials up to $L=12$:\n\\begin{equation}\n\\epsilon(\\cos\\theta) = \\sum_{L=0}^{12} a_L(m) Y^0_L(\\cos\\theta),\n\\end{equation}\nwhere $m$ denotes the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ invariant mass.\nFor each value of $L$, we fit the mass dependent coefficients $a_L(m)$ with a seventh-order polynomial in $m$.\nFigure~\\ref{fig:fig3} shows the resulting fitted efficiency map $\\epsilon(m,\\cos \\theta)$.\nWe obtain $\\chi^2\/N_{\\rm cells}=217\/300$ for this fit, where $N_{\\rm cells}$ is the number of cells in the efficiency map.\nWe observe a significant decrease in\nefficiency in regions of $\\cos\\theta \\sim \\pm 1$ due to the impossibility of reconstructing $\\ensuremath{K^\\pm}\\xspace$ mesons with laboratory momentum\nless than about 200~\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace, and \\mbox{${\\pi^{\\pm}}$} and $\\KS(\\mbox{$\\rightarrow$} \\mbox{${\\pi^{+}}$} \\mbox{${\\pi^{-}}$})$ mesons with laboratory momentum less than about 100~\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace (see Fig. 9 of Ref.~\\cite{babar_z}). These effects result from energy loss in the beampipe and inner-detector material.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{fig3.eps}\n\\caption{Fitted detection efficiency in the $\\cos \\theta \\ vs. \\ m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$ plane. Each interval shows the average value of the fit for that region.}\n\\label{fig:fig3}\n\\end{center}\n\\end{figure}\n\nThe mass resolution, $\\Delta m$, is measured as the difference between the generated and reconstructed \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace invariant-mass values.\nThe distribution has a root-mean-squared value of 10 \\mbox{${\\mathrm{MeV}}\/c^2$}, and is parameterized by the sum of a Crystal Ball~\\cite{cb} and a Gaussian function. \nWe perform a binned fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum in data using the following model. The background is described by a second-order polynomial, and the \\ensuremath{\\eta_c}\\xspace resonance is represented by a nonrelativistic BW function convolved with the resolution function. \nIn addition, we allow for the presence of a residual \\ensuremath{{J\\mskip -3mu\/\\mskip -2mu\\psi\\mskip 2mu}}\\xspace contribution modeled with a Gaussian function. Its parameter values are fixed to those obtained from a fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum for the ISR data sample obtained by requiring\n$\\lvert \\ensuremath{{\\rm \\,mm}}\\xspace \\rvert<1 \\ {\\rm GeV}^2\/c^4$.\nThe fitted \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum is shown in Fig.~\\ref{fig:fig2}. We obtain the following \\ensuremath{\\eta_c}\\xspace parameters:\n\\begin{equation}\n \\begin{split}\n m=2980.8 \\pm 0.4 \\ \\mbox{${\\mathrm{MeV}}\/c^2$}, \\ \\Gamma=33 \\pm 1 \\ \\mbox{${\\mathrm{MeV}}$},\\\\\n \\ N_{\\ensuremath{\\eta_c}\\xspace}=9808 \\pm 164,\n \\end{split}\n\\end{equation}\nwhere uncertainties are statistical only. Our measured mass value is 2.8 \\mbox{${\\mathrm{MeV}}\/c^2$}\\ lower than the world average~\\cite{pdg}.\nThis may be due to interference between the \\ensuremath{\\eta_c}\\xspace amplitude and that describing the background in the signal region~\\cite{bes_int}.\n\n\\section{Model Independent Partial Wave Analysis}\n\nWe perform independent MIPWA of the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plots in the \\ensuremath{\\eta_c}\\xspace mass region using unbinned maximum likelihood fits.\nThe likelihood function is written as\n\\begin{eqnarray}\n\\mathcal{L} = \\nonumber\\\\ \n \\prod_{n=1}^N&\\bigg[&f_{\\rm sig}(m_n) \\epsilon(x'_n,y'_n)\\frac{\\sum_{i,j} c_i c_j^* A_i(x_n,y_n) A_j^*(x_n,y_n)}{\\sum_{i,j} c_i c_j^* I_{A_i A_j^*}} \\nonumber\\\\\n& &+(1-f_{\\rm sig}(m_n))\\frac{\\sum_{i} k_iB_i(x_n,y_n,m_n)}{\\sum_{i} k_iI_{B_i}}\\bigg]\n\\end{eqnarray}\n\\noindent where\n\\begin{itemize}\n\\item $N$ is the number of events in the signal region;\n\\item for the $n$-th event, $m_n$ is the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace or the \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace invariant mass;\n\\item for the $n$-th event, $x_n=m^2(K^+ \\mbox{${\\pi^{-}}$})$, $y_n=m^2(\\KS \\mbox{${\\pi^{-}}$})$ for \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace; $x_n=m^2(K^+ \\mbox{${\\pi^{0}}$})$, $y_n=m^2(K^- \\mbox{${\\pi^{0}}$})$ for $\\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$; \n\\item $f_{\\rm sig}$ is the mass-dependent fraction of signal obtained from the fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace or \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace mass spectrum;\n\\item for the $n$-th event, $\\epsilon(x'_n,y'_n)$ is the efficiency parametrized as a function of $x'_n=m(\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$})$ for \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and $x'_n=m(\\mbox{${K^{+}}$} \\mbox{${K^{-}}$})$ for \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace, and $y'_n=\\cos \\theta$ (see Sec. IV);\n\\item for the $n$-th event, the $A_i(x_n,y_n)$ describe the complex signal-amplitude contributions;\n\\item $c_i$ is the complex amplitude for the $i$-th signal component; the $c_i$ parameters are allowed to vary during the fit process;\n\\item for the $n$-th event, the $B_i(x_n,y_n)$ describe the background probability-density functions assuming that interference between signal and background amplitudes can be ignored;\n\\item $k_i$ is the magnitude of the $i$-th background component; the $k_i$ parameters are obtained by fitting the sideband regions;\n\\item $I_{A_i A_j^*}=\\int A_i (x,y)A_j^*(x,y) \\epsilon(x', y')\\ {\\rm d}x{\\rm d}y$ and \n$I_{B_i}~=~\\int B_i(x,y) {\\rm d}x{\\rm d}y$ are normalization\n integrals. Numerical integration is performed on phase space generated events with \\ensuremath{\\eta_c}\\xspace signal and background generated according to the experimental distributions. In case of MIPWA or when resonances have free parameters, integrals are re-computed at each minimization step.\n Background integrals and fits dealing with amplitudes having fixed resonance parameters are computed only once. \n\\end{itemize}\nAmplitudes are described along the lines described in Ref.~\\cite{kopp}.\nFor an \\ensuremath{\\eta_c}\\xspace meson decaying into three pseudoscalar mesons via an intermediate\n resonance $r$ of spin $J$ (i.e. $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} C r$, $r \\mbox{$\\rightarrow$} A B$), each amplitude\n $A_i(x,y)$ is represented by the product of a complex Breit-Wigner (BW)\n function and a real angular distribution function represented by\n the spherical harmonic function $\\sqrt{2 \\pi} Y_J^0({\\rm cos} \\theta)$; $\\theta$\n is the angle between the direction of $A$, in the rest frame of $r$,\n and the direction of $C$ in the same frame. This form of the angular\n dependence results from angular momentum conservation in the rest\n frame of the \\ensuremath{\\eta_c}\\xspace, which leads to the production of $r$ with helicity 0.\n\n It follows that\n\\begin{equation}\n A_i(x,y) = BW(M_{AB}) \\sqrt{2 \\pi} Y_J^0({\\rm cos} \\theta).\n\\label{eq:spin}\n\\end{equation}\n\n The function $BW(M_{AB})$ is a relativistic BW function of the form\n \\begin{equation}\n BW(M_{AB}) = \\frac{F_{\\eta_c} F}{M_r^2 - M_{AB}^2 - i M_r \\Gamma_{\\rm tot}(M_{AB})}\n\\end{equation}\n where $M_r$ is the mass of the resonance $r$, and $\\Gamma_{\\rm tot}(M_{AB})$ is\n its mass-dependent total width. In general, this mass dependence\n cannot be specified, and a constant value should be used. However,\n for a resonance such as the $K^*_0(1430)$, which is approximately elastic,\n we can use the partial width $\\Gamma_{AB}$, and specify the mass-dependence\n as:\n\\begin{equation}\n\\Gamma_{AB} = \\Gamma_r \\left(\\frac{p_{AB}}{p_r}\\right)^{2J+1} \\left(\\frac{M_r}{M_{AB}}\\right)F^2\n\\end{equation}\nwhere\n\\begin{equation}\np_{AB} = \\frac{\\sqrt{\\left(M_{AB}^2-M_A^2-M_B^2\\right)^2-4M_A^2M_B^2}}{2M_{AB}}.\n\\end{equation}\nand $p_r$ is the value of $p_{AB}$ when $M_{AB}=M_r$. \n\nThe form factors $F_{\\ensuremath{\\eta_c}\\xspace}$ and $F$ attempt to model the underlying quark structure of the parent particle and the intermediate\nresonances. We set $F_{\\ensuremath{\\eta_c}\\xspace}$ to a constant value, while for $F$ we use Blatt-Weisskopf penetration factors~\\cite{blatt} (Table~\\ref{tab:tab_blatt}), that depend on a single parameter $R$ representing the meson ``radius'', for which we assume $R=1.5 \\ \\mbox{${\\mathrm{GeV}}$}^{-1}$.\nThe $a_0(980)$ resonance is parameterized as a coupled-channel Breit-Wigner function whose parameters are taken from Ref.~\\cite{cbar}.\n\n\\begin{table}\n \\caption{Summary of the Blatt-Weisskopf penetration form factors.}\n \\label{tab:tab_blatt}\n\\begin{center}\n \\begin{tabular}{cc}\n \\hline \\\\ [-2.3ex]\nSpin & $F$ \\\\\n\\hline \\\\ [-2.3ex]\n0 & $1$ \\\\\n&\\\\\n1 & {\\Large $\\frac{\\sqrt{1+(R_r p_r)^2}}{\\sqrt{1+(R_r p_{AB})^2}}$} \\\\\n&\\\\\n2 & {\\Large $\\frac{\\sqrt{9+3(R_r p_r)^2+(R_r p_r)^4}}{\\sqrt{9+3(R_r p_{AB})^2+(R_r p_{AB})^4}}$} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nTo measure the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave we make use of a MIPWA technique first described in Ref.~\\cite{aitala1}.\nThe $K \\pi$ $\\mathcal{S}$-wave, being the largest contribution, is taken as the reference amplitude. \nWe divide the $K \\pi$ mass spectrum into 30 equally-spaced mass intervals 60 \\mbox{${\\mathrm{MeV}}$}\\ wide, and \nfor each interval we add to the fit two new free parameters,\nthe amplitude and the phase of the $K \\pi$ $\\mathcal{S}$-wave in that interval. \nWe fix the amplitude to 1.0 and its phase to $\\pi\/2$ at an arbitrary point in the mass spectrum, for which we choose interval 14, corresponding to a mass of 1.45 \\mbox{${\\mathrm{GeV}}\/c^2$}. The number of associated free parameters is therefore 58.\n\nDue to isospin conservation in the hadronic $\\eta_c$ and $K^*$ decays, the $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitudes are combined with positive signs, and so \ntherefore are symmetrized with respect to the two $K^* \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ modes. In particular we write the $K \\pi$ $\\mathcal{S}$-wave amplitudes as\n\n\\begin{equation}\n A_{\\mathcal{S}\\myhyphen\\rm{wave}} = \\frac{1}{\\sqrt{2}}(a_j^{K^+ \\pi^-}e^{i\\phi_j^{K^+ \\pi^-}} + a_j^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}e^{i\\phi_j^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}}),\n \\label{eq:amp}\n\\end{equation}\n\n\\noindent where $a^{K^+ \\pi^-}(m)=a^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}(m)$ and $\\phi^{K^+ \\pi^-}(m) = \\phi^{\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\pi^-}(m)$, for $\\eta_c \\mbox{$\\rightarrow$} \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0 \\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$~\\cite{conj} and\n\n\\begin{equation}\n A_{\\mathcal{S}\\myhyphen\\rm{wave}} = \\frac{1}{\\sqrt{2}}(a_j^{K^+ \\pi^0}e^{i\\phi_j^{K^+\\pi^0}} + a_j^{K^- \\pi^0}e^{i\\phi_j^{K^- \\pi^0}}),\n\\end{equation}\nwhere $a^{K^+ \\pi^0}(m)=a^{K^- \\pi^0}(m)$ and $\\phi^{K^+ \\pi^0}(m) = \\phi^{K^- \\pi^0}(m)$, for \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace.\nFor both decay modes the bachelor kaon is in an orbital $\\mathcal{S}$-wave with respect to the relevant $K \\pi$ system, and so does not affect\nthese amplitudes. The second amplitude in Eq.(\\ref{eq:amp}) is reduced because the $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0$ is observed as a $\\KS$, but the same reduction\nfactor applies to the first amplitude through the bachelor $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace^0$, so that the equality of the three-body amplitudes is preserved.\n\nOther resonance contributions are described as above. The $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ contribution is symmetrized in the same way as the $\\mathcal{S}$-wave amplitude.\n\nWe perform MC simulations to test the ability of the method to find the correct solution.\nWe generate \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace event samples which yield reconstructed samples having the same size as the data sample, according to arbitrary mixtures of resonances, and extract the $K \\pi$ $\\mathcal{S}$-wave using the MIPWA method. We find that the fit is able to extract correctly\nthe mass dependence of the amplitude and phase.\n\nWe also test the possibility of multiple solutions by starting the fit from random values or\nconstant parameter values very far from the solution found by the fit.\nWe find only one solution in both final states and conclude that the fit converges to give the correct $\\mathcal{S}$-wave behaviour for\ndifferent starting values of the parameters.\n\nThe efficiency-corrected fractional contribution $f_i$ due to resonant or non-resonant contribution $i$ is defined as follows:\n\\begin{equation}\nf_i = \\frac {|c_i|^2 \\int |A_i(x_n,y_n)|^2 {\\rm d}x {\\rm d}y}\n{\\int |\\sum_j c_j A_j(x,y)|^2 {\\rm d}x {\\rm d}y}.\n\\end{equation}\nThe $f_i$ do not necessarily sum to 100\\% because of interference effects. The uncertainty for each $f_i$ is evaluated by propagating the full covariance matrix obtained from the fit.\n\nWe test the quality of the fit by examining a large sample of MC events at the generator level weighted \nby the likelihood fitting function and by the efficiency.\nThese events are used to\ncompare the fit result to the Dalitz plot and its projections with proper normalization.\nIn these MC simulations we smooth the fitted $K \\pi$ $\\mathcal{S}$-wave amplitude and phase by means of a cubic spline.\nWe make use of these weighted events to compute a \\mbox{2-D} $\\chi^2$ over the Dalitz plot. For this purpose, we divide the Dalitz plot into a grid of $25 \\times 25$\ncells and consider only those containing at least five events. We compute \n$\\chi^2 = \\sum_{i=1}^{N_{\\rm cells}} (N^i_{\\rm obs}-N^i_{\\rm exp})^2\/N^i_{\\rm exp}$, where $N^i_{\\rm obs}$ and $N^i_{\\rm exp}$ are event yields from data and simulation, respectively.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=12cm]{fig4.eps}\n\\caption{Dalitz plot for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ events in the signal region.}\n\\label{fig:fig4}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig5.eps}\n\\caption{The $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ Dalitz plot projections on (a) $m^2(\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$})$, (b) $m^2(\\KS \\mbox{${\\pi^{\\pm}}$})$, and (c) $m^2(\\KS \\ensuremath{K^\\pm}\\xspace)$. The superimposed curves result from the MIPWA described in the text. The shaded regions show the\nbackground estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig5}\n\\end{center}\n\\end{figure*}\n\n\\section{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} }\n\nFigure~\\ref{fig:fig4} shows the Dalitz plot for the candidates in the \\ensuremath{\\eta_c}\\xspace signal region, and Fig.~\\ref{fig:fig5} shows the corresponding Dalitz plot projections. Since the width of the \\ensuremath{\\eta_c}\\xspace meson is $32.3 \\pm 1.0$ MeV, no mass constraint can be applied.\n\nThe Dalitz plot is dominated by the presence of horizontal and vertical uniform bands at the position of the \\ensuremath{K^*_0(1430)}\\xspace resonance. We also observe\nfurther bands along the diagonal. Isospin conservation in \\ensuremath{\\eta_c}\\xspace decay requires that the $(K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace)$ system have I=1, so that these structures\nmay indicate the presence of $a_0$ or $a_2$ resonances. Further narrow bands are observed at the position\nof the $K^*(892)$ resonance, mostly in the $\\KS \\mbox{${\\pi^{-}}$}$ projection; these components are consistent with originating from background, as will be shown.\n\nThe presence of background in the \\ensuremath{\\eta_c}\\xspace signal region requires precise study of its structure. This can be achieved by\nmeans of the data in the \\ensuremath{\\eta_c}\\xspace sideband regions, for which the Dalitz plots are shown in Fig.~\\ref{fig:fig6}.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig6.eps}\n\\caption{Dalitz plots for the $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ sideband regions: (a) lower, (b) upper.}\n\\label{fig:fig6}\n\\end{center}\n\\end{figure*}\n\nIn both regions we observe almost uniformly populated resonant structures mostly in the $\\KS \\mbox{${\\pi^{-}}$}$ mass, especially in the regions corresponding to the $K^*(892)$`<\nand $K^*_2(1430)$ resonances. The resonant structures in $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ mass are weaker. The three-body decay of a pseudoscalar meson into a spin-one or spin-two resonance yields a non-uniform distribution (see Eq.~\\ref{eq:spin}) in the relevant resonance band on the Dalitz plot. The presence of uniformly populated bands in the $K^*(892)$ and $K^*_2(1430)$ mass regions, indicates that these\nstructures are associated with background. Also, the asymmetry between the two $K^*$ modes in background \nmay be explained as being due to interference between the $I = 0$ and $I = 1$ isospin configurations for the $K^*(\\mbox{$\\rightarrow$} K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ final\nstate produced in two-photon fusion. \n\nWe fit the \\ensuremath{\\eta_c}\\xspace sidebands using an incoherent sum of amplitudes, which includes contributions from the $a_0(980)$, $a_0(1450)$, $a_2(1320)$, $K^*(892)$, $K^*_0(1430)$, $K^*_2(1430)$, $K^*(1680)$, and $K^*_0(1950)$ resonances. To better constrain the sum of the fractions to one, we make use of the channel likelihood method~\\cite{chafit} and include resonances\nuntil no structure is left in the background and an accurate description of the Dalitz plots is obtained.\n\nTo estimate the background composition in the \\ensuremath{\\eta_c}\\xspace signal region we perform a linear mass dependent interpolation of the\nfractions of the different contributions, obtained from the fits to the sidebands, and normalized using the results from the fit to the \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace mass spectrum. The estimated background contributions are indicated by the shaded regions in Fig.~\\ref{fig:fig5}.\n\n\\subsection{MIPWA of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$}}\n\nWe perform the MIPWA including the resonances listed in Table~\\ref{tab:tab1}. In this table, and in the remainder of the paper,\nwe use the notation $(K \\pi) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ or $K^* \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ to represent the corresponding symmetrized amplitude.\nAfter the solution is found we test for\nother contributions, including spin-one resonances, but these are found to be consistent with zero, and so are not included.\nThis supports the observation that\nthe observed \\ensuremath{K^*(892)}\\xspace structures originate entirely from background.\nWe find a dominance of the $K \\pi$ $\\mathcal{S}$-wave amplitude, with small contributions from $a_0 \\pi$ amplitudes and a\nsignificant $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ contribution.\n\n\\begin{table*}\n\\caption{Results from the $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ MIPWA. Phases are determined relative to the $(K\\pi \\ \\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude which is fixed to $\\pi\/2$ at 1.45 \\mbox{${\\mathrm{GeV}}\/c^2$}.}\n\\label{tab:tab1}\n\\begin{center}\n \\begin{tabular}{|l | r@{}c@{}r | r@{}c@{}r | r@{}c@{}r | r@{}c@{}r|}\n \\hline \\\\ [-2.3ex]\n & \\multicolumn{6}{c|} {$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} & \\multicolumn{6}{c|}{ \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace } \\cr\n \\hline \\\\ [-2.3ex]\n Amplitude & \\multicolumn{3}{c|} {Fraction (\\%)} & \\multicolumn{3}{c|}{Phase (rad)} & \\multicolumn{3}{c|}{Fraction (\\%)} & \\multicolumn{3}{c|}{Phase (rad)}\\cr\n \\hline \\\\ [-2.3ex]\n$(K\\pi \\ \\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 107.3 $\\pm$ & \\, 2.6 $\\pm$ & \\, 17.9 & & fixed & & 125.5 $\\pm$ & \\, 2.4 $\\pm$ & \\, 4.2 & & fixed & \\cr\n$a_0(980) \\pi$ & 0.8 $\\pm$ & \\, 0.5 $\\pm$ & \\, 0.8 & 1.08 $\\pm$ & \\, 0.18 $\\pm$ & \\, 0.18 & 0.0 $\\pm$ & \\, 0.1 $\\pm$ & \\, 1.7 & & - & \\cr\n$a_0(1450) \\pi$ & 0.7 $\\pm$ & \\, 0.2 $\\pm$ & \\, 1.4 & 2.63 $\\pm$ & \\, 0.13 $\\pm$ & \\, 0.17 & 1.2 $\\pm$ & \\, 0.4 $\\pm$ & \\, 0.7 & 2.90 $\\pm$ & \\, 0.12 $\\pm$ & \\, 0.25\\cr\n$a_0(1950) \\pi$ & 3.1 $\\pm$ & \\, 0.4 $\\pm$ & \\, 1.2 & $-$1.04 $\\pm$ & \\, 0.08 $\\pm$ & \\, 0.77& 4.4 $\\pm$ & \\, 0.8 $\\pm$ & \\, 0.8& $-$1.45 $\\pm$ & \\, 0.08 $\\pm$ & \\, 0.27\\cr\n$a_2(1320) \\pi$& 0.2 $\\pm$ & \\, 0.1 $\\pm$ & \\, 0.1 & 1.85 $\\pm$ & \\, 0.20 $\\pm$ & \\, 0.20 & 0.6 $\\pm$ & \\, 0.2 $\\pm$ & \\, 0.3& 1.75 $\\pm$ & \\, 0.23 $\\pm$ & \\, 0.42\\cr\n$K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 4.7 $\\pm$ & \\, 0.9 $\\pm$ & \\, 1.4 & 4.92 $\\pm$ & \\, 0.05 $\\pm$ & \\, 0.10 & 3.0 $\\pm$ & \\, 0.8 $\\pm$ & \\, 4.4 & 5.07 $\\pm$ & \\, 0.09 $\\pm$ & \\, 0.30\\cr\n \\hline \\\\ [-2.3ex]\nTotal & 116.8 $\\pm$ & \\, 2.8 $\\pm$ & \\, 18.1 & & & & 134.8 $\\pm$ & \\, 2.7 $\\pm$ & \\, 6.4 & & & \\cr\n$-$ $2\\log {\\cal L}$ & \\multicolumn{3}{c|} {$-$4314.2} & & & & \\multicolumn{3}{c|}{$-$2339} & & & \\cr\n$\\chi^2\/N_{\\rm cells}$ & \\multicolumn{3}{c|} {301\/254=1.17} & & & & \\multicolumn{3}{c|}{283.2\/233=1.22} & & & \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe table lists also a significant contribution from the $a_0(1950) \\pi$ amplitude, where $a_0(1950)^+ \\mbox{$\\rightarrow$} \\KS \\mbox{${K^{+}}$}$ is a new\nresonance. \nWe also test the spin-2 hypothesis for this contribution by replacing the amplitude for $a_0 \\mbox{$\\rightarrow$} K^0_S K^+$ with an $a_2 \\mbox{$\\rightarrow$} K^0_S K^+$ amplitude with parameter values left free in the fit.\nIn this case no physical solution is found inside the allowed ranges of the parameters, and the additional contribution is found consistent with zero. This new state has isospin one, and the spin-0 assignment is preferred over that of spin-2.\n\nA fit without this state gives a poor description of the high mass $\\KS \\mbox{${K^{+}}$}$ projection, as can be seen in Fig.~\\ref{fig:fig7}(a). We obtain $-2\\log {\\cal L} = -$4252.9 and\n$\\chi^2\/N_{\\rm cells}=1.33$ for this fit.\nWe then include in the MIPWA a new scalar resonance decaying to $\\KS \\mbox{${K^{+}}$}$ with free parameters. \nWe obtain \n$\\Delta (\\log {\\cal L})=61$ and $\\Delta \\chi^2=38$ for an increase of four new parameters.\nWe estimate the significance for the $a_0(1950)$ resonance using the fitted \nfraction divided by its statistical and systematic errors added in quadrature, and obtain $2.5\\sigma$.\nSince interference effects may also contribute to the significance, this procedure gives a conservative estimate.\nThe systematic uncertainties associated with the $a_0(1950)$ state are described below.\nThe fitted parameter values for this state are given in Table~\\ref{tab:tab2}.\nWe note that we obtain $\\chi^2\/N_{\\rm cells}=1.17$ for this final fit, indicating good description of the data.\nThe fit projections on the three squared masses from the MIPWA are shown in Fig.~\\ref{fig:fig5}, and they indicate that the description of the data is quite good.\n\n\\begin{table}\n\\caption{Fitted $a_0(1950)$ parameter values for the two \\ensuremath{\\eta_c}\\xspace decay modes.}\n\\label{tab:tab2}\n\\begin{center}\n\\begin{tabular}{|l|c|r@{}c@{}r|}\n \\hline \\\\ [-2.3ex]\nFinal state & Mass (\\mbox{${\\mathrm{MeV}}\/c^2$}) & \\multicolumn{3}{c|}{Width (\\mbox{${\\mathrm{MeV}}$})}\\cr\n\\hline \\\\ [-2.3ex]\n$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ & 1949 $\\pm$ 32 $\\pm$ 76 & 265 $\\pm$ & \\, 36 $\\pm$ & \\,110\\cr\n\\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace & 1927 $\\pm$ 15 $\\pm$ 23 & 274 $\\pm$ & \\, 28 $\\pm$ & \\, 30 \\cr\n\\hline \\\\ [-2.3ex]\nWeighted mean & 1931 $\\pm$ 14 $\\pm$ 22 & 271 $\\pm$ & \\, 22 $\\pm$ & \\, 29\\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig7.eps}\n\\caption{The mass projections (a) $\\KS \\ensuremath{K^\\pm}\\xspace$ from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and (b) $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$. The histograms show the MIPWA fit projections with (solid, black) and without (dashed, red) the presence of the $a_0(1950)^+ \\mbox{$\\rightarrow$} \\KS \\ensuremath{K^\\pm}\\xspace$ resonance. The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig7}\n\\end{center}\n\\end{figure*}\n\nWe compute the uncorrected Legendre polynomial moments $\\langle Y^0_L \\rangle$ in each $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$, $\\KS \\mbox{${\\pi^{-}}$}$ and $\\KS \\mbox{${K^{+}}$}$ mass interval by weighting each event by the relevant $Y^0_L(\\cos \\theta)$ function.\nThese distributions are shown in Fig.~\\ref{fig:fig8} as functions of $K \\pi$ mass after combining $\\mbox{${K^{+}}$} \\mbox{${\\pi^{-}}$}$ and $\\KS \\mbox{${\\pi^{-}}$}$, and in Fig.~\\ref{fig:fig9} as functions of $\\KS \\mbox{${K^{+}}$}$ mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimental distributions. We observe good agreement for all the distributions, which indicates that the fit is able to reproduce the local structures apparent in the Dalitz plot.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig8.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ as functions of $K \\pi$ mass, and combined for $\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}$ and $\\KS \\mbox{${\\pi^{\\mp}}$}$; the superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig8}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig9.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ as a function of $\\KS \\ensuremath{K^\\pm}\\xspace$ mass, the superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig9}\n\\end{center}\n\\end{figure*}\n\nWe compute the following systematic uncertainties on the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude and phase. The different contributions are added in quadrature.\n\\begin{itemize}\n\\item{} Starting from the solution found by the fit, we generate MC simulated events which are fitted using a MIPWA. In this way we estimate the bias introduced by the fitting method.\n\\item{} The fit is performed by interpolating the $K \\pi$ $\\mathcal{S}$-wave amplitude and phase using a cubic spline.\n\\item{} We remove low-significance contributions, such as those from the $a_0(980)$ and $a_2(1320)$ resonances.\n\\item{} We vary the signal purity up and down according to its statistical uncertainty.\n\\item{} The effect of the efficiency variation as a function of $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace \\pi$ mass is evaluated by computing separate efficiencies\n in the regions below and above the $\\eta_c$ mass.\n\\end{itemize}\n\nThese additional fits also allow the computation of systematic uncertainties on the amplitude fraction and phase values, as well as on the parameter values for the $a_0(1950)$ resonance; these are summarized in Table~\\ref{tab:a0_sys}. In the evaluation of overall systematic uncertainties, all effects are assumed to be uncorrelated, and are added in quadrature.\n\nThe measured amplitude and phase values of the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave as functions of mass obtained from the MIPWA of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ are shown in Table~\\ref{tab:tab6}. Interval 14 of the $K \\pi$ mass contains the fixed amplitude and phase values.\n\n\\begin{table*}\n \\caption{Systematic uncertainties on the $a_0(1950)$ parameter values from the two \\ensuremath{\\eta_c}\\xspace decay modes.}\n \\label{tab:a0_sys}\n \\begin{center}\n \\begin{tabular}{|l|r|r|r|r|r|r|}\n \\hline \\\\ [-2.3ex]\n & \\multicolumn{3}{c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} & \\multicolumn{3}{c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$} \\cr\n \\hline \\\\ [-2.3ex] \n Effect & Mass & Width & Fraction (\\%) & Mass & Width & Fraction (\\%)\\cr\n & (\\mbox{${\\mathrm{MeV}}\/c^2$}) & (MeV) & & (\\mbox{${\\mathrm{MeV}}\/c^2$}) & (MeV) & \\cr \n \\hline \\\\ [-2.3ex]\nFit bias & 11 & 22 & 0.5 & 1 & 10 & 0.5 \\cr\nCubic spline & 24 & 79 & 0.6 & 14 & 9 & 0.2\\cr\nMarginal components & 70 & 72 & 0.0 & 2 & 8 & 0.3 \\cr\n\\ensuremath{\\eta_c}\\xspace purity & 3 & 16 & 1.0 & 18 & 26 & 0.4 \\cr\nEfficiency & 11 & 8 & 0.2 & 1 & 15 & 0.2 \\cr\n \\hline \\\\ [-2.3ex]\nTotal & 76 & 110 & 1.3 & 23 & 30 & 0.8\\cr\n\\hline\n \\end{tabular}\n \\end{center} \n\\end{table*}\n\n\\begin{table*}\n \\caption{Measured amplitude and phase values for the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave as functions of mass obtained from the MIPWA of $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\eta_c \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$. The first error is statistical, the second systematic. The amplitudes and phases in the mass interval 14\n are fixed to constant values.}\n\\label{tab:tab6}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n \\hline \\\\ [-2.3ex]\n \\multicolumn{2}{|c}{} & \\multicolumn{2}{|c}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} & \\multicolumn{2}{|c|}{$\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$} \\cr\n \\hline \\\\ [-2.3ex]\nN & $K \\pi$ mass & Amplitude & Phase (rad) & Amplitude & Phase (rad)\\cr\n \\hline \\\\ [-2.3ex]\n 1 & 0.67 & $ \\ 0.119 \\pm 0.100 \\ \\pm 0.215$ & $ \\ \\ 0.259 \\pm 0.577 \\ \\pm 1.290$ & $ \\ 0.154 \\pm 0.350 \\ \\pm 0.337$ & $ \\ \\ 3.786 \\pm 1.199 \\ \\pm 0.857$ \\cr\n 2 & 0.73 & $ \\ 0.103 \\pm 0.043 \\ \\pm 0.113$ & $ -0.969 \\pm 0.757 \\ \\pm 1.600$ & $ \\ 0.198 \\pm 0.124 \\ \\pm 0.216$ & $ \\ \\ 3.944 \\pm 0.321 \\ \\pm 0.448$ \\cr\n 3 & 0.79 & $ \\ 0.158 \\pm 0.086 \\ \\pm 0.180$ & $ \\ \\ 0.363 \\pm 0.381 \\ \\pm 1.500$ & $ \\ 0.161 \\pm 0.116 \\ \\pm 0.098$ & $ \\ \\ 1.634 \\pm 0.584 \\ \\pm 0.448$ \\cr\n 4 & 0.85 & $ \\ 0.232 \\pm 0.128 \\ \\pm 0.214$ & $ \\ \\ 0.448 \\pm 0.266 \\ \\pm 1.500$ & $ \\ 0.125 \\pm 0.118 \\ \\pm 0.031$ & $ \\ \\ 3.094 \\pm 0.725 \\ \\pm 0.448$ \\cr\n 5 & 0.91 & $ \\ 0.468 \\pm 0.075 \\ \\pm 0.194$ & $ \\ \\ 0.091 \\pm 0.191 \\ \\pm 0.237$ & $ \\ 0.307 \\pm 0.213 \\ \\pm 0.162$ & $ \\ \\ 0.735 \\pm 0.326 \\ \\pm 0.255$ \\cr\n 6 & 0.97 & $ \\ 0.371 \\pm 0.083 \\ \\pm 0.129$ & $ \\ \\ 0.276 \\pm 0.156 \\ \\pm 0.190$ & $ \\ 0.528 \\pm 0.121 \\ \\pm 0.055$ & $ -0.083 \\pm 0.178 \\ \\pm 0.303$ \\cr\n 7 & 1.03 & $ \\ 0.329 \\pm 0.071 \\ \\pm 0.102$ & $ \\ \\ 0.345 \\pm 0.164 \\ \\pm 0.273$ & $ \\ 0.215 \\pm 0.191 \\ \\pm 0.053$ & $ \\ \\ 0.541 \\pm 0.320 \\ \\pm 0.638$ \\cr\n 8 & 1.09 & $ \\ 0.343 \\pm 0.062 \\ \\pm 0.062$ & $ \\ \\ 0.449 \\pm 0.196 \\ \\pm 0.213$ & $ \\ 0.390 \\pm 0.146 \\ \\pm 0.046$ & $ \\ \\ 0.254 \\pm 0.167 \\ \\pm 0.144$ \\cr\n 9 & 1.15 & $ \\ 0.330 \\pm 0.070 \\ \\pm 0.081$ & $ \\ \\ 0.687 \\pm 0.167 \\ \\pm 0.221$ & $ \\ 0.490 \\pm 0.135 \\ \\pm 0.089$ & $ \\ \\ 0.618 \\pm 0.155 \\ \\pm 0.099$ \\cr\n 10 & 1.21 & $ \\ 0.450 \\pm 0.059 \\ \\pm 0.042$ & $ \\ \\ 0.696 \\pm 0.156 \\ \\pm 0.226$ & $ \\ 0.422 \\pm 0.092 \\ \\pm 0.102$ & $ \\ \\ 0.723 \\pm 0.242 \\ \\pm 0.267$ \\cr\n 11 & 1.27 & $ \\ 0.578 \\pm 0.048 \\ \\pm 0.112$ & $ \\ \\ 0.785 \\pm 0.208 \\ \\pm 0.358$ & $ \\ 0.581 \\pm 0.113 \\ \\pm 0.084$ & $ \\ \\ 0.605 \\pm 0.186 \\ \\pm 0.166$ \\cr\n 12 & 1.33 & $ \\ 0.627 \\pm 0.047 \\ \\pm 0.053$ & $ \\ \\ 0.986 \\pm 0.153 \\ \\pm 0.166$ & $ \\ 0.643 \\pm 0.106 \\ \\pm 0.039$ & $ \\ \\ 1.330 \\pm 0.264 \\ \\pm 0.130$ \\cr\n 13 & 1.39 & $ \\ 0.826 \\pm 0.047 \\ \\pm 0.105$ & $ \\ \\ 1.334 \\pm 0.155 \\ \\pm 0.288$ & $ \\ 0.920 \\pm 0.153 \\ \\pm 0.056$ & $ \\ \\ 1.528 \\pm 0.161 \\ \\pm 0.160$ \\cr\n \\textcolor{red}{14} & \\textcolor{red}{1.45} & \\textcolor{red}{$ \\ 1.000 $} & \\textcolor{red}{$ \\ 1.570 $} & \\textcolor{red}{$ \\ 1.000 $} & \\textcolor{red}{$ \\ 1.570 $} \\cr\n 15 & 1.51 & $ \\ 0.736 \\pm 0.031 \\ \\pm 0.059$ & $ \\ \\ 1.918 \\pm 0.153 \\ \\pm 0.132$ & $ \\ 0.750 \\pm 0.118 \\ \\pm 0.076$ & $ \\ \\ 1.844 \\pm 0.149 \\ \\pm 0.048$ \\cr\n 16 & 1.57 & $ \\ 0.451 \\pm 0.025 \\ \\pm 0.053$ & $ \\ \\ 2.098 \\pm 0.202 \\ \\pm 0.277$ & $ \\ 0.585 \\pm 0.099 \\ \\pm 0.047$ & $ \\ \\ 2.128 \\pm 0.182 \\ \\pm 0.110$ \\cr\n 17 & 1.63 & $ \\ 0.289 \\pm 0.029 \\ \\pm 0.065$ & $ \\ \\ 2.539 \\pm 0.292 \\ \\pm 0.180$ & $ \\ 0.366 \\pm 0.079 \\ \\pm 0.052$ & $ \\ \\ 2.389 \\pm 0.230 \\ \\pm 0.213$ \\cr\n 18 & 1.69 & $ \\ 0.159 \\pm 0.036 \\ \\pm 0.089$ & $ \\ \\ 1.566 \\pm 0.308 \\ \\pm 0.619$ & $ \\ 0.312 \\pm 0.074 \\ \\pm 0.043$ & $ \\ \\ 1.962 \\pm 0.195 \\ \\pm 0.150$ \\cr\n 19 & 1.75 & $ \\ 0.240 \\pm 0.034 \\ \\pm 0.067$ & $ \\ \\ 1.962 \\pm 0.331 \\ \\pm 0.655$ & $ \\ 0.427 \\pm 0.093 \\ \\pm 0.063$ & $ \\ \\ 1.939 \\pm 0.150 \\ \\pm 0.182$ \\cr\n 20 & 1.81 & $ \\ 0.381 \\pm 0.031 \\ \\pm 0.059$ & $ \\ \\ 2.170 \\pm 0.297 \\ \\pm 0.251$ & $ \\ 0.511 \\pm 0.094 \\ \\pm 0.063$ & $ \\ \\ 2.426 \\pm 0.156 \\ \\pm 0.277$ \\cr\n 21 & 1.87 & $ \\ 0.457 \\pm 0.035 \\ \\pm 0.085$ & $ \\ \\ 2.258 \\pm 0.251 \\ \\pm 0.284$ & $ \\ 0.588 \\pm 0.098 \\ \\pm 0.080$ & $ \\ \\ 2.242 \\pm 0.084 \\ \\pm 0.210$ \\cr\n 22 & 1.93 & $ \\ 0.565 \\pm 0.042 \\ \\pm 0.067$ & $ \\ \\ 2.386 \\pm 0.255 \\ \\pm 0.207$ & $ \\ 0.729 \\pm 0.114 \\ \\pm 0.095$ & $ \\ \\ 2.427 \\pm 0.098 \\ \\pm 0.254$ \\cr\n 23 & 1.99 & $ \\ 0.640 \\pm 0.044 \\ \\pm 0.055$ & $ \\ \\ 2.361 \\pm 0.228 \\ \\pm 0.092$ & $ \\ 0.777 \\pm 0.119 \\ \\pm 0.075$ & $ \\ \\ 2.306 \\pm 0.102 \\ \\pm 0.325$ \\cr\n 24 & 2.05 & $ \\ 0.593 \\pm 0.046 \\ \\pm 0.065$ & $ \\ \\ 2.329 \\pm 0.235 \\ \\pm 0.268$ & $ \\ 0.775 \\pm 0.134 \\ \\pm 0.075$ & $ \\ \\ 2.347 \\pm 0.107 \\ \\pm 0.299$ \\cr\n 25 & 2.11 & $ \\ 0.614 \\pm 0.057 \\ \\pm 0.083$ & $ \\ \\ 2.421 \\pm 0.230 \\ \\pm 0.169$ & $ \\ 0.830 \\pm 0.134 \\ \\pm 0.078$ & $ \\ \\ 2.374 \\pm 0.105 \\ \\pm 0.199$ \\cr\n 26 & 2.17 & $ \\ 0.677 \\pm 0.067 \\ \\pm 0.117$ & $ \\ \\ 2.563 \\pm 0.218 \\ \\pm 0.137$ & $ \\ 0.825 \\pm 0.140 \\ \\pm 0.070$ & $ \\ \\ 2.401 \\pm 0.127 \\ \\pm 0.189$ \\cr\n 27 & 2.23 & $ \\ 0.788 \\pm 0.085 \\ \\pm 0.104$ & $ \\ \\ 2.539 \\pm 0.228 \\ \\pm 0.241$ & $ \\ 0.860 \\pm 0.158 \\ \\pm 0.123$ & $ \\ \\ 2.296 \\pm 0.131 \\ \\pm 0.297$ \\cr\n 28 & 2.29 & $ \\ 0.753 \\pm 0.097 \\ \\pm 0.125$ & $ \\ \\ 2.550 \\pm 0.234 \\ \\pm 0.168$ & $ \\ 0.891 \\pm 0.167 \\ \\pm 0.133$ & $ \\ \\ 2.320 \\pm 0.131 \\ \\pm 0.273$ \\cr\n 29 & 2.35 & $ \\ 0.646 \\pm 0.096 \\ \\pm 0.118$ & $ \\ \\ 2.315 \\pm 0.241 \\ \\pm 0.321$ & $ \\ 0.994 \\pm 0.202 \\ \\pm 0.076$ & $ \\ \\ 2.297 \\pm 0.153 \\ \\pm 0.197$ \\cr\n 30 & 2.41 & $ \\ 0.789 \\pm 0.184 \\ \\pm 0.187$ & $ \\ \\ 2.364 \\pm 0.336 \\ \\pm 0.199$ & $ \\ 0.892 \\pm 0.322 \\ \\pm 0.098$ & $ \\ \\ 2.143 \\pm 0.292 \\ \\pm 0.393$ \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n \n\\subsection{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$} using an isobar model}\n\nWe perform a Dalitz plot analysis of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace using a standard isobar model, where all resonances\nare modeled as BW functions multiplied by the corresponding angular functions. In this case the $K \\pi$ $\\mathcal{S}$-wave is represented by a superposition of interfering $K^*_0(1430)$, \\ensuremath{K^*_0(1950)}\\xspace, non-resonant (NR), and possibly $\\kappa(800)$ contributions.\nThe NR contribution is parametrized as an amplitude that is constant in magnitude and phase\nover the Dalitz plot.\nIn this fit the $K^*_0(1430)$ parameters\nare taken from Ref.~\\cite{etakk}, while all other parameters are fixed to PDG values. We also add the $a_0(1950)$\nresonance with parameters obtained from the MIPWA analysis.\n\n\n\nFor the description of the \\ensuremath{\\eta_c}\\xspace signal, amplitudes are added one by one to ascertain the associated increase of the likelihood value and decrease of the \\mbox{2-D} $\\chi^2$. \nTable~\\ref{tab:tab4} summarizes the fit results for the amplitude fractions and phases.\nThe high value of $\\chi^2\/N_{\\rm cells}=1.82$ (to be compared with $\\chi^2\/N_{\\rm cells}=1.17$) indicates a poorer description of the data than that obtained with the MIPWA method. Including the $\\kappa(800)$ resonance does not improve the fit quality. If included, it gives a fit fraction of $(0.8 \\pm 0.5)$\\%.\n\nThe Dalitz plot analysis shows a dominance of scalar meson amplitudes, with small contributions from spin-two resonances. The $K^*(892)$ contribution is consistent with originating entirely from background. Other spin-1 $K^*$ resonances have been included in the fit, \nbut their contributions have been found to be\nconsistent with zero. We note the presence of a sizeable non-resonant contribution. However, in this case the sum of the fractions is significantly lower than 100\\%, indicating important interference effects.\nFitting the data without the NR contribution gives a much poorer description, with $-2\\log {\\cal L}=-$4115 and $\\chi^2\/N_{\\rm cells}=2.32$.\n\nWe conclude that the $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ Dalitz plot is not well-described by an isobar model in which the $K \\pi$ $\\mathcal{S}$-wave is modeled as a superposition of Breit-Wigner functions. A more complex approach is needed, and the MIPWA is able to describe\nthis amplitude without the need for a specific model.\n\n\\begin{table}\n\\caption{Results from the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace Dalitz plot analysis using an isobar model. The listed uncertainties are statistical only.}\n\\label{tab:tab4}\n\\begin{center}\n \\begin{tabular}{|l|r@{}c|r@{}c|}\n \\hline \\\\ [-2.3ex]\n Amplitude & \\multicolumn{2}{c|}{Fraction \\%} & \\multicolumn{2}{c|}{Phase (rad)}\\cr\n \\hline \\\\ [-2.3ex]\n$K^*_0(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 40.8 $\\pm$ & \\, 2.2 & 0. & \\cr\n$K^*_0(1950) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 14.8 $\\pm$ & \\, 1.7 & $-$1.00 $\\pm$ & \\, 0.07 \\cr\nNR & 18.0 $\\pm$ & \\, 2.5 & 1.94 $\\pm$ & \\, 0.09 \\cr\n$a_0(980) \\pi$ & 10.5 $\\pm$ & \\, 1.2 & 0.94 $\\pm$ & \\, 0.12 \\cr\n$a_0(1450) \\pi$ & 1.7 $\\pm$ & \\, 0.5 & 2.94 $\\pm$ & \\, 0.13 \\cr\n$a_0(1950) \\pi$ & 0.7 $\\pm$ & \\, 0.2 & $-$1.76 $\\pm$ & \\, 0.24 \\cr\n$a_2(1320) \\pi$ & 0.2 $\\pm$ & \\, 0.2 & $-$0.53 $\\pm$ & \\, 0.42 \\cr\n$K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ & 2.3 $\\pm$ & \\, 0.7 & $-$1.55 $\\pm$ & \\, 0.11 \\cr\n \\hline \\\\ [-2.3ex]\nTotal & 88.8 $\\pm$ & \\, 4.3 & & \\cr\n$-2\\log {\\cal L}$ & $-$4290.7 & \\, & & \\cr\n$\\chi^2\/N_{\\rm cells}$ & 467\/256=1.82 & \\, & & \\cr\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Dalitz plot analysis of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$}\\ }\n\nThe \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plot~\\cite{etakk} is very similar to that for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ decays. It is dominated by uniformly populated bands at\nthe $K^*_0(1430)$ resonance position in $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ mass squared. It also shows a broad diagonal structure indicating\nthe presence of $a_0$ or $a_2$ resonance contributions. The Dalitz plot projections are shown in Fig.~\\ref{fig:fig10}.\n\nThe \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace Dalitz plot analysis using the isobar model has been performed already in Ref.~\\cite{etakk} . It was found that\nthe model does not give a perfect description of the data. In this section we obtain a new measurement\nof the $K \\pi$ $\\mathcal{S}$-wave by making use of the MIPWA method. In this way we also perform a cross-check of the results\nobtained from the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace analysis, since analyses of the two $\\ensuremath{\\eta_c}\\xspace$ decay modes should give consistent results, given\nthe absence of I=3\/2 $K \\pi$ amplitude contributions.\n\n\\subsection{MIPWA of {\\boldmath$\\protect \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace$}}\n\nWe perform a MIPWA of \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decays using the same model and the same mass grid as for \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace.\nAs for the previous case we obtain a better description of the data if we include an additional $a_0(1950)$ resonance, whose parameter values are listed in Table~\\ref{tab:tab2}. We observe good agreement between the parameter values obtained from the two \\ensuremath{\\eta_c}\\xspace decay modes.\nThe table also lists parameter values obtained as the weighted mean of the two measurements.\nTable~\\ref{tab:tab1} gives the fitted fractions from the MIPWA fit.\n\nWe obtain a good description of the data, as evidenced by the value $\\chi^2\/N_{\\rm cells}=1.22$, and observe\nthe $a_0(1950)$ state with a significance of $4.2\\sigma$.\nThe fit projections on the $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$, $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, and $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ squared mass distributions are shown in Fig.~\\ref{fig:fig10}. As previously, there is\na dominance of the ($K \\pi$ $\\mathcal{S}$-wave) $\\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude, with a significant $K^*_2(1430) \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ amplitude, and small contributions from $a_0 \\pi$ amplitudes. We observe good agreement between fractions and relative phases of the\namplitudes between the \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace decay modes.\nSystematic uncertainties are evaluated as discussed in Sec. VI.A.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig10.eps}\n\\caption{The $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ Dalitz plot projections, (a) $m^2(\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$})$, (b) $m^2(\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$})$, and (c) $m^2(\\mbox{${K^{+}}$} \\mbox{${K^{-}}$})$. The superimposed curves result from the MIPWA described in the text. The shaded regions show the\nbackground estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.}\n\\label{fig:fig10}\n\\end{center}\n\\end{figure*}\n\nWe compute the uncorrected Legendre polynomial moments $\\langle Y^0_L \\rangle$ in each $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$, $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ mass interval by weighting each event by the relevant $Y^0_L(\\cos \\theta)$ function.\nThese distributions are shown in Fig.~\\ref{fig:fig11} as functions of $K \\pi$ mass, combined for $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, and in Fig.~\\ref{fig:fig12} as functions of $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimental distributions. We observe good agreement for all the distributions, which indicates that also in this case the fit is able to reproduce the local structures apparent in the Dalitz plot.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig11.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ as functions of $K \\pi$ mass, combined for $\\mbox{${K^{+}}$} \\mbox{${\\pi^{0}}$}$ and $\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$. The superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig11}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig12.eps}\n\\caption{Legendre polynomial moments for $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ as a function of $\\mbox{${K^{+}}$} \\mbox{${K^{-}}$}$ mass. The superimposed curves result from the Dalitz plot fit described in the text.}\n\\label{fig:fig12}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{fig13.eps}\n\\caption{The $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude (a) and phase (b) from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ (solid (black) points) and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace\n (open (red) points); only statistical uncertainties are shown. The dotted lines indicate the $K \\eta$ and $K \\eta'$ thresholds.}\n\\label{fig:fig13}\n\\end{center}\n\\end{figure*}\n\n\\section{The $I=1\/2$ {\\boldmath$\\protect K \\pi \\ \\mathcal{S}$}-wave amplitude and phase}\n\nFigure~\\ref{fig:fig13} displays the measured $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude and phase from both $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and \\ensuremath{\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$}\\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}}\\xspace.\nWe observe good agreement between the amplitude and phase values obtained from the two measurements.\n\nThe main features of the amplitude (Fig.~\\ref{fig:fig13}(a)) can be explained by the presence of a clear peak related to the $K^*_0(1430)$ resonance\nwhich shows a rapid drop around 1.7 \\mbox{${\\mathrm{GeV}}\/c^2$}, where a broad structure is present which can be related to the $K^*_0(1950)$ resonance.\nThere is some indication of feedthrough from the $K^*(892)$ background.\nThe phase motion (Fig.~\\ref{fig:fig13}(b)) shows the expected behavior for the resonance phase, which varies by about $\\pi$ in the $K^*_0(1430)$ resonance region. The phase shows a drop around 1.7 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ related to interference with the $K^*_0(1950)$ resonance.\n\nWe compare the present measurement of the $K \\pi$ $\\mathcal{S}$-wave amplitude from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ with measurements\nfrom LASS~\\cite{lass_kpi} in Fig.~\\ref{fig:fig14}(a)(c) and E791~\\cite{aitala1} in Fig.~\\ref{fig:fig14}(b)(d).\nWe plot only the first part of the LASS measurement since it suffers from\na two-fold ambiguity above the mass of 1.82 \\mbox{${\\mathrm{GeV}}\/c^2$}.\nThe Dalitz plot fits extract invariant amplitudes. Consequently, in Fig.~\\ref{fig:fig14}(a), the LASS $I=1\/2$ $K \\pi$ scattering amplitude\nvalues have been multiplied by the factor $m(K \\pi)\/q$ to convert to invariant amplitude, and normalized so as to equal\nthe scattering amplitude at 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ in order to facilitate comparison to the \\ensuremath{\\eta_c}\\xspace results. Here $q$ is the momentum of either meson in the $K \\pi$ rest frame. For better comparison, the LASS absolute phase measurements have been displaced by $-0.6$ rad before plotting them in Fig.~\\ref{fig:fig14}(c).\nIn Fig.~\\ref{fig:fig14}(b) the E791 amplitude has been obtained by multiplying the amplitude $c$ in Table III of Ref.~\\cite{aitala1} by the Form Factor $F_D^0$, for which the mass-dependence is motivated by theoretical speculation. This yields amplitude values corresponding to the E791 Form Factor having value 1, as for the \\ensuremath{\\eta_c}\\xspace analyses. In Fig.~\\ref{fig:fig14}(d), the E791 phase measurements have been displaced by $+0.9$ rad, again in order to facilitate comparison to the \\ensuremath{\\eta_c}\\xspace measurements.\n\nWhile we observe similar phase behavior\namong the three measurements up to about 1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}, we observe striking differences in the mass dependence of the amplitudes.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=18cm]{fig14.eps}\n\\caption{The $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude measurements from $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ compared to the (a) LASS and (b) E791 results: the corresponding $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave phase measurements compared to the\n (c) LASS and (d) E791 measurements. \n Black dots indicate the results from the present analysis; square (red) points indicate the LASS or E791 results. The LASS data are plotted in the region having only one solution.}\n\\label{fig:fig14}\n\\end{center}\n\\end{figure*}\n\n\\section{Summary}\n\nWe perform Dalitz plot analyses, using an isobar model and a MIPWA method, of data on the decays $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$, where the \\ensuremath{\\eta_c}\\xspace mesons are produced in two-photon interactions in the {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}\\ experiment at SLAC. \nWe find that, in comparison with the isobar models examined here, an improved description of the data is obtained by using a MIPWA method.\n\nWe extract the $I=1\/2$ $K \\pi$ $\\mathcal{S}$-wave amplitude and phase and find good agreement between the measurements\nfor the two $\\eta_c$ decay modes.\nThe $K \\pi$ $\\mathcal{S}$-wave is dominated by the presence of the $K^*_0(1430)$ resonance which is observed as a clear peak\nwith the corresponding increase in phase of about $\\pi$ expected for a resonance. A broad structure in the 1.95 \\mbox{${\\mathrm{GeV}}\/c^2$}\\ mass region indicates\nthe presence of the $K^*_0(1950)$ resonance.\n\nA comparison between the present measurement and previous experiments indicates a similar trend for the phase up to a mass of\n1.5 \\mbox{${\\mathrm{GeV}}\/c^2$}. The amplitudes, on the other hand, show very marked differences.\n\nTo fit the data we need to introduce a new $a_0(1950)$ resonance in both $\\eta_c \\mbox{$\\rightarrow$} \\ensuremath{\\KS\\ensuremath{K^\\pm}\\xspace \\mbox{${\\pi^{\\mp}}$}}\\xspace$ and $\\ensuremath{\\eta_c}\\xspace \\mbox{$\\rightarrow$} \\mbox{${K^{+}}$} \\mbox{${K^{-}}$} \\mbox{${\\pi^{0}}$}$ decay modes, and their associated parameter values are in good agreement. The weighted averages for\nthe parameter values are:\n\n\\begin{equation}\n \\begin{split}\n m(a_0(1950))=1931 \\pm 14 \\pm 22 \\ {\\rm MeV}\/c^2, \\\\\n \\Gamma(a_0(1950))= 271 \\pm 22 \\pm 29 \\ {\\rm MeV}\n \\end{split}\n\\end{equation}\n\n\\noindent with significances of 2.5$\\sigma$ and 4.2$\\sigma$ respectively, including systematic uncertainties.\nThese results are, however, systematically limited, and more detailed studies of the $I=1$ $K \\kern 0.2em\\overline{\\kern -0.2em K}{}\\xspace$ $\\mathcal{S}$-wave will be\nrequired in order to improve the precision of these values.\n\n\n\\section{Acknowledgements}\nWe are grateful for the \nextraordinary contributions of our PEP-II\\ colleagues in\nachieving the excellent luminosity and machine conditions\nthat have made this work possible.\nThe success of this project also relies critically on the \nexpertise and dedication of the computing organizations that \nsupport {\\em B}{\\footnotesize\\em A}{\\em B}{\\footnotesize\\em AR}.\nThe collaborating institutions wish to thank \nSLAC for its support and the kind hospitality extended to them. \nThis work is supported by the\nUS Department of Energy\nand National Science Foundation, the\nNatural Sciences and Engineering Research Council (Canada),\nthe Commissariat \\`a l'Energie Atomique and\nInstitut National de Physique Nucl\\'eaire et de Physique des Particules\n(France), the\nBundesministerium f\\\"ur Bildung und Forschung and\nDeutsche Forschungsgemeinschaft\n(Germany), the\nIstituto Nazionale di Fisica Nucleare (Italy),\nthe Foundation for Fundamental Research on Matter (The Netherlands),\nthe Research Council of Norway, the\nMinistry of Education and Science of the Russian Federation,\nMinisterio de Economia y Competitividad (Spain), and the\nScience and Technology Facilities Council (United Kingdom).\nIndividuals have received support from \nthe Marie-Curie IEF program (European Union), the A. P. Sloan Foundation (USA) \nand the Binational Science Foundation (USA-Israel).\nThe work of A. Palano and M. R. Pennington was supported (in part) by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.\n\n\\renewcommand{\\baselinestretch}{1}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzavos b/data_all_eng_slimpj/shuffled/split2/finalzzavos new file mode 100644 index 0000000000000000000000000000000000000000..8a73c4dc66ba8b838300e75afc790a53a325675f --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzavos @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWehrl proposed \\cite{W} \na hybrid between quantum mechanical and classical entropy\nthat enjoys monotonicity, strong subadditivity and positivity -- physically\ndesirable properties, some of which both kinds of entropy lack \\cite{B,E}. \nThis new entropy is the ordinary Shannon entropy of\nthe probability density provided by the lower\nsymbol of the density matrix. \n\nFor a quantum mechanical system with density matrix $\\rho$, Hilbert space\n$\\H$,\nand a family of normalized\ncoherent states $|z\\rangle$,\nparametrized symbolically by\n$z$ and satisfying\n$\\int dz \\, |z\\rangle\\langle z| = {\\bf 1}$ (resolution of identity),\nthe Wehrl entropy is\n\\begin{equation}\nS_W(\\rho) = -\\int dz \\, \\langle z|\\rho|z\\rangle \\ln \\langle z|\\rho|z\\rangle . \\label{W0}\n\\end{equation}\nLike quantum mechanical entropy,\n$S_Q = -\\tr \\rho\\ln\\rho $,\nWehrl entropy is always non-negative, in fact\n$S_W > S_Q \\geq 0$.\nIn view of this inequality it is interesting to ask for the minimum\nof $S_W$ and the corresponding minimizing density matrix.\nIt follows from concavity of $-x \\ln x$ that a minimizing\ndensity matrix must be a pure state, \\emph{i.e.}, $\\rho = |\\psi\\rangle\\langle\\psi|$\nfor a normalized vector $|\\psi\\rangle \\in \\H$ \\cite{A}. (Note that\n$S_W(|\\psi\\rangle\\langle\\psi|)$ depends on $|\\psi\\rangle$ and is non-zero,\nunlike the quantum entropy which is of course zero for pure states.)\n\nFor Glauber coherent states Wehrl conjectured \\cite{W} and Lieb proved\n\\cite{A}\nthat the minimizing state $|\\psi\\rangle$ is again a coherent\nstate. It turns out that all Glauber coherent states\nhave Wehrl entropy one, so Wehrl's conjecture can be written as follows:\n\\begin{thm}[\\rm Lieb] \\label{thmWL}\nThe minimum of $S_W(\\rho)$ for states in $\\H = L^2(\\Re)$\nis one,\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|)\n= -\\int dz \\, |\\langle\\psi|z\\rangle|^2 \\ln |\\langle\\psi|z\\rangle|^2 \\geq 1 , \\label{W1}\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{thm}\nTo prove this, Lieb used a clever combination of the sharp\nHausdorff-Young inequality \\cite{Y1,Y3,LL}\nand the sharp Young inequality \\cite{Y2,Y1,Y3,LL}\nto show that\n\\begin{equation}\ns \\int dz \\, |\\langle z|\\psi\\rangle|^{2 s} \\leq 1, \\quad s \\geq 1, \\label{W2}\n\\end{equation}\nagain with equality if and only if $|\\psi\\rangle$ is a coherent state.\nWehrl's conjecture follows from this in the limit $s \\rightarrow 1$\nessentially because\n(\\ref{W1}) is the derivative of (\\ref{W2}) with respect to $s$ at $s=1$.\nAll this easily generalizes to $L^2(\\Re^n)$ \\cite{A,Y4}.\n\nThe lower bound on the Wehrl entropy is related to\nHeisenberg's uncertainty principle \\cite{AH,G} and it has been speculated that\n$S_W$ can be used to measure uncertainty due to both quantum and thermal\nfluctuations \\cite{G}.\n\nIt is very surprising that `heavy artillery' like the sharp constants\nin the mentioned inequalities are needed in Lieb's proof. To elucidate\nthis situation, Lieb suggested \\cite{A} studying the analog of Wehrl's\nconjecture for Bloch coherent states $|\\Omega\\rangle$, where one should\nexpect significant \nsimplification since these are finite dimensional Hilbert spaces.\nHowever, no progress has been made, not even for a single spin, even though\nmany attempts have been made \\cite{B}. Attempts to proceed again \nalong the lines of Lieb's original proof have failed to provide a sharp\ninequality and the direct computation of the entropy and related integrals,\neven numerically, was unsuccesful \\cite{S}.\n\nThe key to the recent progress is a geometric representation of a\nstate of spin~$j$ as $2j$ points on a sphere.\nIn this representation the expression\n$|\\langle\\Omega|\\psi\\rangle|^2$ factorizes into a product\nof $2j$ functions $f_i$ on the sphere, \nwhich measure the square chordal distance\nfrom the antipode of the point parametrized by\n$\\Omega$ to each of the $2j$ points on the sphere.\nLieb's conjecture,\nin a generalized form analogous to (\\ref{W2}),\nthen looks like the quotient of two H\\\"older inequalities\n\\begin{equation}\n\\frac{|\\!|f_1 \\cdots f_{2j}|\\!|_s}{|\\!|f_1 \\cdots f_{2j}|\\!|_1}\n\\leq \n\\frac{\\prod_{i=1}^{2j}|\\!|f_i|\\!|_{2js}}{\\prod_{i=1}^{2j} |\\!|f_i|\\!|_{2j}},\n\\label{holder}\n\\end{equation}\nwith the one with the higher power winning against the other one.\nWe shall give a group theoretic proof of this inequality\nfor the special case $s \\in \\N$ in theorem~\\ref{natural}.\n\nIn the geometric representation the \nWehrl entropy of spin states finds a direct physical\ninterpretation: It is the classical entropy of a single particle on a sphere\ninteracting via Coulomb potential with $2j$ fixed sources; $s$ plays the\nrole of inverse temperature.\n\nThe entropy integral (\\ref{W0}) can now be done because \n$|\\langle\\Omega|\\psi\\rangle|^2$ factorizes\nand one finds a formula for the Wehrl entropy of any state.\nWhen we evaluate the entropy explicitly for states of spin 1,\n3\/2, and 2 we find surprisingly simple expressions solely in\nterms of the square chordal distances between the points on the\nsphere that define the given state. \n\nA different, more group theoretic approach seems to point \nto a connection between\nLieb's conjecture and the norm of certain spin $j s$ states with\n$1 \\leq s \\in \\Re$ \\cite{J}.\nSo far, however, this has only been useful for proving the analog\nof inequality (\\ref{W2}) for $s \\in \\N$.\n\nWe find that a proof of Lieb's conjecture for low spins can be reduced\nto some beautiful spherical geometry, \nbut the unreasonable difficulty of a complete proof\nis still a great puzzle; its resolution may very well lead to\ninteresting mathematics and perhaps physics.\n\n\\section{Bloch coherent spin states}\n\nGlauber coherent states \n$|z\\rangle = \\pi^{-\\frac{1}{4}} e^{-(x-q)^2\/2} e^{ipx}$,\nparametrized by $z = (q + i p)\/\\sqrt{2}$ and with measure\n$dz = dp dq\/2\\pi$,\nare usually introduced as\neigenvectors of the annihilation operator\n$a = (\\hat x + i \\hat p)\/\\sqrt{2}$, \n$a|z\\rangle = z |z\\rangle$,\nbut the same states can also be\nobtained by the action of the Heisenberg-Weyl group\n$H_4 = \\{a^\\dagger a, a^\\dagger, a, I\\}$\non the extremal state\n$|0\\rangle = \\pi^{-\\frac{1}{4}} e^{-x^2\/2}$.\nGlauber coherent states are thus elements of the coset\nspace of the Heisenberg-Weyl group\nmodulo the stability subgroup $U(1)\\otimes U(1)$ that leaves the extremal state\ninvariant. (See \\emph{e.g.}\\ \\cite{C1} and references therein.)\nThis construction easily generalizes\nto other groups, in particular to SU(2), where it gives\nthe Bloch coherent spin states \\cite{BC} that we are interested in:\nHere the Hilbert space can be any one of the finite dimensional spin-$j$\nrepresentations $[j] \\equiv \\C^{2j+1}$ of SU(2), \n$j = {1\\over 2}, 1, \\frac{3}{2}, \\ldots$,\nand\nthe extremal state for each $[j]$ is the\nhighest weight vector $|j,j\\rangle$. The stability subgroup is U(1)\nand the coherent states are thus \nelements of the sphere $S_2 = $SU(2)\/U(1);\nthey can be labeled by\n$\\Omega = (\\theta,\\phi)$ and are\nobtained from $|j,j\\rangle$ by rotation:\n\\begin{equation}\n|\\Omega\\rangle_j = \\R_j(\\Omega) |j,j\\rangle. \\label{Om}\n\\end{equation}\nFor spin $j = \\frac{1}{2}$ we find\n\\begin{equation}\n|\\omega\\rangle = p^{\\frac{1}{2}} e^{-i{\\phi\\over 2}} |\\U\\rangle \n + (1-p)^{\\frac{1}{2}} e^{i{\\phi\\over 2}} |\\D\\rangle , \\label{coh}\n\\end{equation}\nwith $p \\equiv \\cos^2\\frac{\\theta}{2}$.\n(Here and in the following $|\\omega\\rangle$ is short for the spin-$\\frac{1}{2}$ coherent\nstate $|\\Omega\\rangle_\\frac{1}{2}$; $\\omega = \\Omega = (\\theta,\\phi)$. \n$|\\U\\rangle \\equiv |\\frac{1}{2},\\frac{1}{2}\\rangle$ and\n$|\\D\\rangle \\equiv |\\frac{1}{2},-\\frac{1}{2}\\rangle$.)\nAn important observation for what follows is that\nthe product of two coherent states for the same $\\Omega$\nis again a coherent state:\n\\begin{eqnarray} \n|\\Omega\\rangle_j \\otimes |\\Omega\\rangle_{j'} \n\t& = & (\\R_j \\otimes \\R_{j'})\\, (|j,j\\rangle \\otimes |j',j'\\rangle) \\nonumber \\\\\n\t& = & \\R_{j+j'}\\, |j+j',j+j'\\rangle \t \n\t\\; = \\; |\\Omega\\rangle_{j+j'} .\t \n\\end{eqnarray}\nCoherent states are in fact the only states for which\nthe product of a spin-$j$ state with a spin-$j'$ state is\na spin-$(j+j')$ state and not a more general element of\n$[j+j'] \\oplus \\ldots \\oplus [\\,|j - j'|\\,]$.\nFrom this key property\nan explicit representation for Bloch coherent states of higher spin \ncan be easily derived:\n\\begin{eqnarray}\n|\\Omega\\rangle_j & = & \\left(|\\omega\\rangle\\right)^{\\otimes 2j} \n \\; = \\; \\left(p^{\\frac{1}{2}} e^{-i{\\phi\\over 2}} |\\U\\rangle \n + (1-p)^{\\frac{1}{2}} e^{i{\\phi\\over 2}} |\\D\\rangle\\right)^{\\otimes 2j} \\nonumber \\\\\n\t & = & \\sum_{m=-j}^j {2 j \\choose j + m}^{\\frac{1}{2}}\n\t p^{j+m\\over 2} (1-p)^{j-m\\over 2} \n\t e^{-i m {\\phi\\over 2}} |j,m\\rangle. \\label{Coh}\n\\end{eqnarray}\n(The same expression can also be obtained directly from (\\ref{Om}), see\n\\emph{e.g.}\\ \\cite[chapter 4]{C2}.)\nThe coherent states as given are normalized $\\langle\\Omega|\\Omega\\rangle_j =1$\nand satisfy\n\\begin{equation}\n(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\Omega\\rangle_j\\langle\\Omega|_j = P_j , \\qquad \\mbox{(resolution of\nidentity)} \\label{project}\n\\end{equation}\nwhere $P_j = \\sum |j,m\\rangle\\langle j,m|$ is the projector onto $[j]$.\nIt is not hard to compute the Wehrl entropy for a coherent state\n$|\\Omega'\\rangle$: Since the integral over the sphere is invariant under rotations\nit is enough to consider the coherent state $|j,j\\rangle$; then use\n$|\\langle j,j|\\Omega\\rangle|^2 = |\\langle\\U\\!\\!|\\omega\\rangle|^{2\\cdot 2j} = p^{2j}$\nand $d\\Omega\/4\\pi = -dp\\,d\\phi\/2\\pi$, where \n$p =\\cos^2 \\frac{\\theta}{2}$ as above, to obtain\n\\begin{eqnarray}\nS_W(|\\Omega'\\rangle\\langle\\Omega'|) \n& = & -(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\Omega'\\rangle|^2 \\ln |\\langle\\Omega|\\Omega'\\rangle|^2 \\nonumber \\\\\n& = & -(2j+1) \\int_0^1 dp \\, p^{2j} \\, 2j \\ln p \\, = \\, \\frac{2j}{2j+1}.\n\\end{eqnarray}\nSimilarly, for later use,\n\\begin{equation}\n(2js+1) \\int \\frac{d\\Om}{4\\pi} |\\langle\\Omega'|\\Omega\\rangle|^{2s} = (2js+1) \\int_0^1 dp \\, p^{2js} = 1.\n\\end{equation}\nAs before the density matrix that minimizes $S_W$\nmust be a pure state $|\\psi\\rangle\\langle\\psi|$. \nThe analog of theorem~\\ref{thmWL} for spin states is:\n\\begin{conj}[\\rm Lieb] \\label{conject1}\nThe minimum of $S_W$ for states in $\\H = \\C^{2j+1}$\nis $2j\/(2j+1)$,\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) \n= -(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^2 \\ln |\\langle\\Omega|\\psi\\rangle|^2\n\\geq \\frac{2j}{2j+1},\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{conj}\n\n\\noindent\n\\emph{Remark:} For spin 1\/2 this is an identity because\nall spin~1\/2 states are coherent states. The first non-trivial case\nis spin $j=1$.\n\n\\section{Proof of Lieb's conjecture for low spin}\n\nIn this section we shall geometrize the description of spin states, use this\nto solve the entropy integrals\nfor all spin and prove Lieb's conjecture for low spin by actual computation\nof the entropy.\n\n\\begin{lemma}\nStates of spin $j$ are in one to one correspondence to $2j$ points\non the sphere $S_2$: With $2j$ points,\nparametrized by $\\omega_k = (\\theta_k, \\phi_k)$, $k = 1, \\ldots , 2j$,\nwe can associate a state\n\\begin{equation}\n|\\psi\\rangle = c^\\frac{1}{2} P_j (|\\omega_1\\rangle \\otimes \\ldots \\otimes |\\omega_{2j}\\rangle) \\; \\in \\; [j] ,\n\\label{psiprod}\n\\end{equation}\nand every state $|\\psi\\rangle \\in [j]$ is of that form. (The spin-$\\frac{1}{2}$ states\n$|\\omega_k\\rangle$ are given by (\\ref{coh}),\n$c^\\frac{1}{2} \\neq 0$ fixes the\nnormalization of $|\\psi\\rangle$, and $P_j$ is the projector onto spin~$j$.)\n\\label{sphere}\n\\end{lemma}\n\n\\noindent \\emph{Remark:} \nSome or all of the points may coincide.\nCoherent states are exactly\nthose states for which all points on the sphere coincide. \n$c^\\frac{1}{2} \\in \\C$ may contain an (unimportant) phase\nthat we can safely ignore in the following.\nThis representation is unique up to permutation\nof the $|\\omega_k\\rangle$. The $\\omega_k$ may be found by looking at\n$\\langle\\Omega|\\psi\\rangle$ as a function of $\\Omega = (\\theta,\\psi)$:\nthey are the antipodal points to the zeroes of this function. \n\n\\noindent {\\sc Proof}:\nRewrite (\\ref{Coh}) in complex coordinates for $\\theta\\neq 0$\n\\begin{equation}\nz = \\left(\\frac{p}{1-p}\\right)^\\frac{1}{2} e^{i\\phi} = \\cot\\frac{\\theta}{2} e^{i\\phi}\n\\end{equation}\n(stereographic projection)\nand contract it with $|\\psi\\rangle$ to find\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle = \\frac{e^{-ij\\phi}}{(1+z\\bar z)^j} \n\\sum_{m=-j}^{j_{\\mbox{\\tiny max}}} \n\t {2 j \\choose j + m}^{\\frac{1}{2}} z^{j+m} \\psi_m , \\label{poly}\n\\end{equation}\nwhere $j_{\\mbox{\\scriptsize max}}$ is the largest value of $m$ for which\n$\\psi_m$ in the expansion\n$|\\psi\\rangle = \\sum\\psi_m |m\\rangle$\nis nonzero. This is a polynomial of degree\n$j+j_{\\mbox{\\scriptsize max}}$ in $z \\in \\C$ and can thus be factorized:\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle = \n\\frac{ e^{-ij\\phi} \\psi_{j_{\\mbox{\\tiny max}}} }{ (1+z\\bar z)^j } \n\\prod_{k=1}^{j+j_{\\mbox{\\tiny max}}} (z - z_k) . \\label{fact}\n\\end{equation}\nConsider now the spin~$\\frac{1}{2}$ states\n$|\\omega_k\\rangle = (1+z_k \\bar z_k)^{-\\frac{1}{2}}(|\\U\\rangle - z_k |\\D\\rangle)$\nfor $1 \\leq k \\leq j+j_{\\mbox{\\tiny max}}$ and\n$|\\omega_m\\rangle = |\\D\\rangle$ for $j+j_{\\mbox{\\tiny max}} < m \\leq 2j$. According\nto (\\ref{poly}):\n\\begin{equation}\n\\langle\\omega|\\omega_k\\rangle = \\frac{e^{-\\frac{i\\phi}{2}}}{(1+z\\bar z)^\\frac{1}{2} \n(1+ z_k\\bar z_k)^\\frac{1}{2}}(z - z_k) , \n\\qquad \\langle\\omega|\\omega_m\\rangle = \n\\frac{e^{-\\frac{i\\phi}{2}}}{(1+z\\bar z)^\\frac{1}{2}},\n\\end{equation}\nso by comparison with (\\ref{fact}) and with an appropriate constant\n$c$\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle \n= c^\\frac{1}{2} \\langle\\omega|\\omega_1\\rangle \\cdots \\langle\\omega|\\omega_{2j}\\rangle\n= c^\\frac{1}{2} \\langle\\Omega|\\omega_1\\otimes\\ldots\\otimes\\omega_{2j}\\rangle.\n\\label{fac}\n\\end{equation}\nBy inspection we see that this expression is still valid when\n$\\theta = 0$ and with the help of (\\ref{project}) we can complete the\nproof the lemma.$\\blob$\\\\[1ex]\nWe see that the geometric representation of spin states leads to a\nfactorization of $\\langle\\Omega|\\psi\\rangle|^2$. In this representation we can\nnow do the entropy integrals, essentially because the logarithm becomes a\nsimple sum.\n\n\\begin{thm} \\label{theorem}\nConsider any state $|\\psi\\rangle$ of spin $j$. According to\nlemma~\\ref{sphere}, it can be written as\n$|\\psi\\rangle = c^\\frac{1}{2} P_j (|\\omega_1\\rangle \\otimes \\ldots \\otimes |\\omega_{2j}\\rangle).$\nLet $\\R_i$ be the rotation that turns $\\omega_i$ to the `north pole',\n$\\R_i|\\omega_i\\rangle = |\\U\\rangle$, let $|\\psi^{(i)}\\rangle = \\R_i|\\psi\\rangle$,\nand let $\\psi_m^{(i)}$ be the coefficient of $|j,m\\rangle$ in the expansion\nof $|\\psi^{(i)}\\rangle$,\nthen the Wehrl entropy is:\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) =\n\\sum_{i=1}^{2j} \\sum_{m=-j}^{j} \\left(\\sum_{n=0}^{j-m}\n\\frac{1}{2j+1-n}\\right) |\\psi_m^{(i)}|^2 - \\ln c . \\label{formula}\n\\end{equation}\n\\end{thm}\n\n\\noindent \\emph{Remark:}\nThis formula reduces the computation of the Wehrl entropy of any\nspin state to its factorization in the sense of lemma~\\ref{sphere},\nwhich in general requires the solution of a\n$2j$'th order algebraic equation. This may explain why previous \nattempts to do the entropy integrals have failed.\nThe $n=0$ terms in the expression for the entropy\nsum up to $2j\/(2j+1)$, the entropy of a coherent state, \nand Lieb's conjecture can be thus be written\n\\begin{equation}\n\\ln c \\leq \\sum_{i=1}^{2j} \\sum_{m=-j+1}^{j-1} \\left(\\sum_{n=1}^{j-m}\n\\frac{1}{2j+1-n}\\right) |\\psi_m^{(i)}|^2.\n\\end{equation}\nNote that $\\psi^{(i)}_{-j} = 0$ by construction of\n$|\\psi^{(i)}\\rangle$: $\\psi^{(i)}_{-j}$ contains a factor\n$\\langle\\downarrow|\\U\\rangle$.\\\\\nA similar calculation gives\n\\begin{equation}\n\\ln c = 2j + \\int\\frac{d\\Om}{4\\pi} \\, \\ln|\\langle\\Omega|\\psi\\rangle|^2 .\n\\end{equation}\n\n\\noindent\n{\\sc Proof}:\nUsing lemma~\\ref{sphere}, (\\ref{project}),\nthe rotational invariance of the\nmeasure and the inverse Fourier transform in $\\phi$ we find\n\\begin{eqnarray}\n\\lefteqn{S_W(|\\psi\\rangle\\langle\\psi|) \\; = \\;\n{ -(2j+1)}\n\\int\\frac{d\\Om}{4\\pi} |\\langle\\Omega|\\psi\\rangle|^2 \\sum_{i=1}^{2j} \\ln |\\langle\\omega|\\omega_i\\rangle|^2\n- \\ln c} \\nonumber \\\\\n&& = { -(2j+1)} \\sum_{i=1}^{2j} \\int\\frac{d\\Om}{4\\pi}\n|\\langle\\Omega|\\psi^{(i)}\\rangle|^2 \\ln |\\langle\\omega|\\U\\rangle|^2 - \\ln c \\nonumber \\\\\n&& = {\\scriptstyle -(2j+1)} \\sum_{i=1}^{2j}\\sum_{m=-j}^j |\\psi^{(i)}_m|^2\n{\\scriptstyle {2j \\choose j + m}}\n\\int_0^1 dp \\, \n p^{j+m} (1 \\! - \\! p)^{j-m} \\ln p - \\ln c.\n\\end{eqnarray}\nIt is now easy to do the remaining $p$-integral by partial integration\nto proof the theorem.$\\blob$\\\\[1ex]\nLieb's conjecture for low spin can be proved with the help of\nformula (\\ref{formula}). For spin 1\/2 there is\nnothing to prove, since all states of spin 1\/2 are coherent states.\nThe first nontrivial case is spin 1:\n\n\\begin{cor}[\\rm spin 1]\nConsider an arbitrary state of spin 1. Let\n$\\mu$ be the square of the\nchordal distance between the two points on the sphere of radius~$\\frac{1}{2}$\nthat represent this state. It's Wehrl entropy is given by\n\\begin{equation}\nS_W(\\mu) = \\frac{2}{3} + c\\cdot\\left(\\frac{\\mu}{2} + \\frac{1}{c} \\ln\n\\frac{1}{c}\\right) , \\label{entropy1}\n\\end{equation}\nwith\n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{\\mu}{2}.\n\\end{equation}\nLieb's conjecture holds for all states of spin 1:\n$S_W(\\mu) \\geq 2\/3 = 2j\/(2j+1)$ with equality for $\\mu = 0$, \\emph{i.e.}\\ for\ncoherent states.\n\\end{cor}\n\n\\noindent {\\sc Proof}:\nBecause of rotational invariance we can assume without loss of generality\nthat the first point is at the `north pole' of the sphere and that\nthe second point is parametrized as $\\omega_2 = (\\tilde\\theta, \\tilde\\phi = 0)$,\nso that\n$\\mu = \\sin^2\\frac{\\tilde\\theta}{2}$ . Up to normalization (and an irrelevant\nphase)\n\\begin{equation}\n|\\tilde\\psi\\rangle = P_{j=1}|\\U\\otimes\\tilde\\omega\\rangle\n\\end{equation}\nis the state of interest. But from (\\ref{coh})\n\\begin{equation}\n|\\U\\otimes\\tilde\\omega\\rangle = (1-\\mu)^\\frac{1}{2}|\\U\\;\\U\\rangle + \\mu^\\frac{1}{2}|\\U\\;\\D\\rangle.\n\\end{equation}\nProjecting onto spin 1 and inserting the normalization constant $c^\\frac{1}{2}$\nwe find\n\\begin{equation}\n|\\psi\\rangle = c^\\frac{1}{2}\\left((1-\\mu)^\\frac{1}{2} |1,1\\rangle + \\mu^\\frac{1}{2}\n\\frac{1}{\\sqrt{2}}|1,0\\rangle\\right). \\label{state}\n\\end{equation}\nThis gives (ignoring a possible phase) \n\\begin{equation}\n1 = \\langle\\psi|\\psi\\rangle = c\\left(1 - \\mu + \\frac{\\mu}{2}\\right) = c\\left(1 -\n\\frac{\\mu}{2}\\right) \\label{cvalue}\n\\end{equation}\nand so $1\/c = 1 - \\mu\/2$. Now we need to compute the components\nof $|\\psi^{(1)}\\rangle$ and $|\\psi^{(2)}\\rangle$. Note that\n$|\\psi^{(1)}\\rangle = |\\psi\\rangle$ because $\\omega_1$ is already pointing to\nthe `north pole'. To obtain $|\\psi^{(2)}\\rangle$ we need to rotate point 2\nto the `north pole'. We can use the remaining rotational freedom\nto effectively exchange the two points, thereby recovering the original\nstate $|\\psi\\rangle$. The components of \nboth $|\\psi^{(1)}\\rangle$ and $|\\psi^{(2)}\\rangle$ can thus be read off (\\ref{state}):\n\\begin{equation}\n\\psi_1^{(1)} = \\psi_1^{(2)} = c^\\frac{1}{2}(1-\\mu)^\\frac{1}{2} ,\n\\qquad \\psi_0^{(1)} = \\psi_0^{(2)} = c^\\frac{1}{2} \\mu^\\frac{1}{2}\/\\sqrt{2}.\n\\end{equation}\nInserting now $c$, $|\\psi_1^{(1)}|^2 = |\\psi_1^{(2)}|^2 = c(1-\\mu)$,\nand $|\\psi_0^{(1)}|^2 = |\\psi_0^{(2)}|^2 = c \\mu\/2$ into (\\ref{formula})\ngives the stated entropy.\n\nTo prove Lieb's conjecture for states of spin~1 we use\n(\\ref{cvalue}) to show that\nthe second term in (\\ref{entropy1}) is always non-negative and zero only for\n$\\mu = 0$, \\emph{i.e.}\\ for a coherent state. This follows from\n\\begin{equation}\n\\frac{c \\mu}{2} - \\ln c \\geq \\frac{c \\mu}{2} + 1 - c = 0\n\\end{equation}\nwith equality for $c=1$ which is equivalent to $\\mu=0$ .$\\blob$\n\\vspace{2ex}\n\\begin{figure}\n\\begin{center}\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\begin{picture}(30.00,21.00)(10,5)\n\\put(10.00,0.00){\\line(5,1){25.00}}\n\\put(35.00,5.00){\\line(-4,3){20.00}}\n\\put(15.00,20.00){\\line(-1,-4){5.00}}\n\\put(9.00,0.00){\\makebox(0,0)[rc]{$1$}}\n\\put(15.00,21.00){\\makebox(0,0)[cb]{$3$}}\n\\put(36.00,5.00){\\makebox(0,0)[lc]{$2$}}\n\\put(11.00,10.00){\\makebox(0,0)[rc]{$\\mu$}}\n\\put(26.00,13.00){\\makebox(0,0)[lb]{$\\epsilon$}}\n\\put(27.00,2.00){\\makebox(0,0)[ct]{$\\nu$}}\n\\end{picture}\n\\end{center}\n\\caption{Spin~3\/2}\n\\label{three}\n\\end{figure}\n\\begin{cor}[\\rm spin 3\/2]\nConsider an arbitrary state of spin 3\/2. Let $\\epsilon$, $\\mu$, $\\nu$\nbe the squares of the chordal distances between the three points on\nthe sphere of radius $\\frac{1}{2}$ that represent this state (see figure~\\ref{three}). \nIt's Wehrl entropy\nis given by\n\\begin{equation}\nS_W(\\epsilon,\\mu,\\nu) = \\frac{3}{4} + \nc\\cdot\\left(\\frac{\\epsilon+\\mu+\\nu}{3} - \\frac{\\epsilon\\mu + \\epsilon\\nu + \\mu\\nu}{6}\n+\\frac{1}{c} \\ln\\frac{1}{c} \\right) \\label{spin32}\n\\end{equation}\nwith \n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{\\epsilon+\\mu+\\nu}{3} .\n\\end{equation}\nLieb's conjecture holds for all states of spin 3\/2:\n$S_W(\\epsilon,\\mu,\\nu) \\geq 3\/4 = 2j\/(2j+1)$ with equality for \n$\\epsilon = \\mu = \\nu = 0$, \\emph{i.e.}\\ for\ncoherent states.\n\\end{cor}\n\n\\noindent {\\sc Proof}: The proof is similar to the spin~1 case, but the\ngeometry and algebra is more involved.\nConsider a sphere of radius ${1\\over 2}$, with points 1, 2, 3 on its surface,\nand two planes through its center; the first plane\ncontaining\npoints 1 and 3, the second plane containing points 2 and 3. The intersection\nangle $\\phi$\nof these two planes satisfies\n\\begin{equation}\n 2\\cos\\phi \\sqrt{\\epsilon\\mu(1-\\epsilon)(1-\\mu)} = \\epsilon + \\mu - \\nu - 2\\epsilon\\mu .\n\\label{phi}\n\\end{equation}\n$\\phi$ is the azimuthal angle of point 2, if point 3 is at the `north pole' of\nthe sphere and point 1 is assigned zero azimuthal angle.\n\nThe states $|\\psi^{(1)}\\rangle$,\n$|\\psi^{(2)}\\rangle$, and $|\\psi^{(3)}\\rangle$ all have one point at the\nnorth pole of the sphere. It is enough to compute the values of\n$|\\psi_m^{(i)}|^2$ for\none $i$, the other values can be found by appropriate permutation of\n$\\epsilon$, $\\mu$, $\\nu$. (Note that we make no restriction on\nthe parameters $0\\leq \\epsilon$, $\\mu$, $\\nu \\leq 1$ other than that they are\nsquare chordal distances between three points on a sphere of\nradius $\\frac{1}{2}$.)\nWe shall start with $i = 3$: Without loss of generality\nthe three points can be parametrized as $\\omega^{(3)}_1 = (\\tilde\\theta,0)$,\n$\\omega^{(3)}_2 = (\\theta,\\phi)$, and $\\omega^{(3)}_3 = (0,0)$\nwith $\\mu = \\sin^2{\\tilde\\theta\\over 2}$ and $\\epsilon = \\sin^2{\\theta\\over 2}$.\nCorresponding spin-$\\frac{1}{2}$ states are\n\\begin{eqnarray}\n|\\omega^{(3)}_1\\rangle & = & (1-\\mu)^\\frac{1}{2}|\\U\\rangle + \\mu^\\frac{1}{2}|\\D\\rangle ,\\label{om1}\\\\\n|\\omega^{(3)}_2\\rangle & = & (1-\\epsilon)^\\frac{1}{2} e^{-i\\phi\\over 2}|\\U\\rangle \n+ \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2}|\\D\\rangle , \\label{om2}\\\\\n|\\omega^{(3)}_3\\rangle & = & |\\U\\rangle , \\label{om3} \n\\end{eqnarray}\nand up to normalization, the state of interest is\n\\begin{eqnarray}\n|\\tilde\\psi^{(3)}\\rangle \n& = & P_{j=3\/2} |\\omega^{(3)}_1\\otimes\\omega^{(3)}_2\\otimes\\omega^{(3)}_3\\rangle \\nonumber \\\\\n& = & (1-\\epsilon)^\\frac{1}{2} (1-\\mu)^\\frac{1}{2} e^{-i\\phi\\over 2}\n |{3\\over 2},{3\\over 2}\\rangle \\nonumber \\\\\n&& + \\left( (1-\\mu)^\\frac{1}{2} \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2} \n + \\mu^\\frac{1}{2} (1-\\epsilon)^\\frac{1}{2} e^{-i\\phi\\over 2} \\right) \n { {1 \\over \\sqrt{3}}} |{3\\over 2},{1\\over 2}\\rangle \\nonumber \\\\\n&& + \\mu^\\frac{1}{2} \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2} \n { {1 \\over \\sqrt{3}}} |{3\\over 2},-{1\\over 2}\\rangle .\n\\end{eqnarray}\nThis gives \n\\begin{eqnarray}\n|\\tilde\\psi^{(3)}_{3\\over 2}|^2 & = & (1-\\epsilon)(1-\\mu),\\\\\n|\\tilde\\psi^{(3)}_{1\\over 2}|^2 & = & {1 \\over 3}\\left(\n\\epsilon(1-\\mu) + \\mu(1 - \\epsilon) + 2 \\sqrt{\\epsilon\\mu(1-\\mu)(1-\\epsilon)} \\cos\\phi\\right) \\nonumber \\\\\n& = & {2\\over 3}\\epsilon(1-\\mu) + {2\\over 3}\\mu(1 - \\epsilon) -{\\nu\\over 3}, \\\\\n|\\tilde\\psi^{(3)}_{-{1\\over 2}}|^2 & = & {\\epsilon \\mu\\over 3},\n\\end{eqnarray}\nand\n$|\\tilde\\psi^{(3)}_{-{3\\over 2}}|^2 = 0$. The sum of these expressions\nis\n\\begin{equation}\n{1\\over c} = \\langle\\tilde\\psi|\\tilde\\psi\\rangle =\n1 - {\\epsilon + \\mu + \\nu \\over 3} ,\n\\end{equation}\nwith $0 < 1\/c \\leq 1$.\nThe case $i=1$ is found by exchanging $\\mu \\leftrightarrow \\nu$ (and also\n$3 \\leftrightarrow 1$, $\\phi \\leftrightarrow -\\phi$).\nThe case $i=2$ is found by permuting\n$\\epsilon\\rightarrow\\mu\\rightarrow\\nu\\rightarrow\\epsilon$ (and also $1 \\rightarrow 3\n\\rightarrow 2 \\rightarrow 1$).\nUsing (\\ref{formula}) then gives the stated entropy.\n\nTo complete the proof Lieb's conjecture for all states of \nspin~$3\/2$ we need to show\nthat the second term in (\\ref{spin32}) is always non-negative and zero\nonly for $\\epsilon=\\mu=\\nu=0$.\nFrom the inequality $(1-x)\\ln(1-x) \\geq -x + x^2\/2$ for $0 \\leq x < 1$,\nwe find\n\\begin{equation}\n{1\\over c}\\ln{1\\over c} \\geq -{\\epsilon+\\mu+\\nu\\over 3} + {1\\over 2}\\left(\n{\\epsilon+\\mu+\\nu\\over 3}\\right)^2 ,\n\\end{equation}\nwith equality for $c=1$. Using the inequality between algebraic and geometric\nmean it is not hard to see that\n\\begin{equation}\n\\left({\\epsilon+\\mu+\\nu\\over 3}\\right)^2 \\geq {\\epsilon\\mu + \\nu\\epsilon + \\mu\\nu \\over 3}\n\\end{equation}\nwith equality for $\\epsilon=\\mu=\\nu$. Putting everything together and inserting\nit into (\\ref{spin32}) we have, as desired, $S_W \\geq 3\/4$ with equality\nfor $\\epsilon=\\mu=\\nu=0$, \\emph{i.e.}\\ for coherent states.$\\blob$\n\\vspace{1ex}\n\\begin{figure}\n\\begin{center}\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\begin{picture}(30.00,36.00)(10,7)\n\\put(10.00,15.00){\\line(5,1){25.00}}\n\\put(35.00,20.00){\\line(-4,3){20.00}}\n\\put(15.00,35.00){\\line(-1,-4){5.00}}\n\\put(10.00,15.00){\\line(3,-2){15.00}}\n\\put(25.00,5.00){\\line(2,3){10.00}}\n\\put(9.00,15.00){\\makebox(0,0)[rc]{$1$}}\n\\put(15.00,36.00){\\makebox(0,0)[cb]{$3$}}\n\\put(36.00,20.00){\\makebox(0,0)[lc]{$2$}}\n\\put(25.00,4.00){\\makebox(0,0)[ct]{$4$}}\n\\put(11.00,25.00){\\makebox(0,0)[rc]{$\\mu$}}\n\\put(26.00,28.00){\\makebox(0,0)[lb]{$\\epsilon$}}\n\\put(19.00,24.00){\\makebox(0,0)[lb]{$\\gamma$}}\n\\put(27.00,17.00){\\makebox(0,0)[ct]{$\\nu$}}\n\\put(17.00,9.00){\\makebox(0,0)[rt]{$\\alpha$}}\n\\put(31.00,11.00){\\makebox(0,0)[rt]{$\\beta$}}\n\\put(15.00,35.00){\\line(1,-3){10.00}}\n\\end{picture}\n\\end{center}\n\\caption{Spin~2}\n\\end{figure}\n\\begin{cor}[\\rm spin 2]\nConsider an arbitrary state of spin 2. Let $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$\nbe the squares of the chordal distances between the four points on\nthe sphere of radius $\\frac{1}{2}$ that represent this state\n(see figure). It's Wehrl entropy\nis given by\n\\begin{equation}\nS_W(\\epsilon,\\mu,\\nu,\\alpha,\\beta) = \\frac{4}{5} + c \\cdot \\left( \\sigma + \\frac{1}{c}\n\\ln\\frac{1}{c}\\right), \\label{S2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{1}{4}\\sum\\lipic\n+\\frac{1}{12}\\sum\\papic \\label{c2}\n\\end{equation}\nand\n\\begin{equation}\n\\sigma = \\frac{1}{12}\\left(-\\frac{1}{2}\\sum\\trpic\n-\\frac{5}{3}\\sum\\papic-\\sum\\wepic+3\\sum\\lipic\n\\right)\n\\end{equation}\nwith\n\\begin{equation}\n\\sum\\trpic \\equiv \\alpha\\mu\\nu+\\epsilon\\beta\\nu+\\epsilon\\mu\\gamma+\\alpha\\beta\\gamma,\n\\end{equation}\n\\begin{equation}\n\\sum\\papic \\equiv \\alpha\\epsilon+\\beta\\mu+\\gamma\\nu,\n\\qquad\n\\sum\\lipic \\equiv \\alpha+\\beta+\\gamma+\\mu+\\nu+\\epsilon,\n\\end{equation}\n\\begin{equation}\n\\sum\\wepic \\equiv \\alpha\\mu+\\alpha\\nu+\\mu\\nu+\\beta\\epsilon\n+\\beta\\nu+\\epsilon\\nu+\\epsilon\\gamma+\\epsilon\\mu+\\mu\\gamma\n+\\alpha\\beta+\\alpha\\gamma+\\beta\\gamma.\n\\end{equation} \\label{spin2}\n\\end{cor}\n\n\\noindent \\emph{Remark:} The fact that the four points lie on the surface\nof a sphere imposes a complicated constraint on the parameters\n$\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$. Although we have convincing numerical evidence\nfor Lieb's conjecture for spin~2,\nso far a rigorous proof has been limited to\ncertain symmetric configurations\nlike equilateral triangles with centered fourth point ($\\epsilon=\\mu=\\nu$ and\n$\\alpha=\\beta=\\gamma$), and squares ($\\alpha=\\beta=\\epsilon=\\mu$ and\n$\\gamma=\\nu$). It is not hard to find values of the parameters\nthat give values of $S_W$ below the entropy for coherent states, but they\ndo \\emph{not} correspond to any configuration of points on the sphere,\nso in contrast to spin 1 and spin 3\/2\nthe constraint is now important.\n$S_W$ is concave in each of the parameters $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$.\n\n\\noindent {\\sc Proof}: The proof is analogous to the spin~1 and spin~3\/2\ncases but the geometry and algebra are considerably more complicated,\nso we will just give a sketch. Pick four points on the sphere,\nwithout loss of generality parametrized as $\\omega_1^{(3)} =(\\tilde\\theta,0)$,\n$\\omega_2^{(3)} =(\\theta,\\phi)$, $\\omega_3^{(3)} = (0,0)$,\nand $\\omega_4^{(3)} =(\\bar\\theta,\\bar\\phi)$. Corresponding spin $\\frac{1}{2}$ states\nare $|\\omega_1^{(3)}\\rangle$, $|\\omega_2^{(3)}\\rangle$, $|\\omega_3^{(3)}\\rangle$,\nas given in (\\ref{om1}), (\\ref{om2}), (\\ref{om3}), and\n\\begin{equation}\n|\\omega_4^{(3)}\\rangle = (1-\\gamma)^\\frac{1}{2} e^{-i\\bar\\phi \\over 2} |\\U\\rangle\n+ \\gamma^\\frac{1}{2} e^{i\\bar\\phi \\over 2} |\\D\\rangle.\n\\end{equation}\nUp to normalization, the state of interest is\n\\begin{equation}\n|\\tilde\\psi^{(3)}\\rangle =\nP_{j=2} |\\omega_1^{(3)} \\otimes \\omega_2^{(3)} \\otimes\\omega_3^{(3)} \\otimes\\omega_4^{(3)}\\rangle.\n\\end{equation}\nIn the computation of $|\\tilde\\psi^{(3)}_m|^2$ we encounter\nagain the angle $\\phi$, compare (\\ref{phi}),and two new\nangles $\\bar\\phi$ and\n$\\bar\\phi -\\phi$.\nLuckily both can again be expressed as angles between planes that\nintersect the circle's center and we have\n\\begin{eqnarray}\n2 \\cos\\bar\\phi\\sqrt{\\mu\\gamma(1-\\mu)(1-\\gamma)} & = & \n\\mu + \\gamma - \\alpha - 2\\mu\\gamma, \\\\\n2 \\cos(\\bar\\phi-\\phi)\\sqrt{\\epsilon\\gamma(1-\\epsilon)(1-\\gamma)}\n& = & \\gamma + \\epsilon - \\beta - 2\\gamma\\epsilon,\n\\end{eqnarray}\nand find $1\/c = \\sum_m |\\tilde\\psi^{(3)}_m|^2$ as given in (\\ref{c2}).\nBy permuting the parameters $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$ appropriately we can derive expressions for the remaining\n$|\\tilde\\psi^{(i)}_m|^2$'s and then compute $S_W$ (\\ref{S2}) with the\nhelp of $(\\ref{formula})$.$\\blob$\n\n\\section{Higher spin}\n\nThe construction outlined in the proof of corollary~\\ref{spin2}\ncan in principle also be applied to states of higher spin, but\nthe expressions pretty quickly become quite unwieldy.\nIt is, however, possible to use theorem~\\ref{theorem} to show that\nthe entropy is extremal for coherent states:\n\n\\begin{cor}[\\rm spin $j$]\nConsider the state of spin $j$ characterized by $2j -1$ coinciding\npoints on the sphere and a $2j$'th point, a small (chordal) distance\n$\\epsilon^\\frac{1}{2}$ away from them. The Wehrl entropy of this small deviation\nfrom a coherent state, up to third order in $\\epsilon$, is\n\\begin{equation}\nS_W(\\epsilon) = {2j\\over 2j+1} + {c \\over 8 j^2} \\epsilon^2 \\quad + {\\cal O}[\\epsilon^4] ,\n\\end{equation}\nwith\n\\begin{equation}\n{1 \\over c} = 1 - {2j - 1 \\over 2j} \\epsilon \\quad \\mbox{(exact)} .\n\\end{equation}\n\\end{cor}\n\nA generalized version of Lieb's conjecture, analogous to (\\ref{W2}), \nis \\cite{A}\n\\begin{conj} \\label{conject2}\nLet $|\\psi\\rangle$ be a normalized state of spin $j$, then\n\\begin{equation}\n(2j s + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^{2s} \\leq 1 , \\quad s > 1 ,\n\\label{norms}\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{conj}\n\n\\noindent \\emph{Remark:} \nThis conjecture is equivalent to the ``quotient of two H\\\"older inequalities\"\n(\\ref{holder}).\nThe original conjecture~\\ref{conject1} \nfollows from it in the limit $s \\rightarrow 1$.\nFor $s=1$ we simply get the norm of the spin $j$ state $|\\psi\\rangle$,\n\\begin{equation}\n(2j + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle_j|^2 = | P_j | \\psi\\rangle|^2 ,\n\\label{norm}\n\\end{equation}\nwhere $P_j$ is the projector onto spin $j$.\nWe have numerical evidence for low spin\nthat an analog of conjecture~\\ref{conject2}\nholds in fact for a much larger class of convex functions than\n$x^s$ or $x \\ln x$.\n\n\nFor $s \\in \\N$ there is a surprisingly simple group theoretic argument\nbased on (\\ref{norm}):\n\\begin{thm}\nConjecture~\\ref{conject2} holds for $s \\in \\N$. \\label{natural}\n\\end{thm}\n\n\\noindent \\emph{Remark:} For spin 1 and spin 3\/2 (at $s=2$) this was\nfirst shown by Wolfgang Spitzer by direct computation of the\nintegral.\n\n\\noindent {\\sc Proof}: Let us consider\n$s=2$, $|\\psi\\rangle \\in [j]$ with $||\\psi\\rangle|^2 = 1$,\nrewrite (\\ref{norms}) as follows\nand use (\\ref{norm})\n\\begin{eqnarray}\n\\lefteqn{(2j\\cdot 2 + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^{2\\cdot 2}} \\nonumber \\\\\n&& = (2(2j)+ 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega\\otimes\\Omega|\\psi\\otimes\\psi\\rangle|^2 \n= |P_{2j} |\\psi\\otimes\\psi\\rangle |^2.\n\\end{eqnarray}\nBut $|\\psi\\rangle\\otimes|\\psi\\rangle \\in [j]\\otimes[j] = [2j]\\oplus[2j-1]\\oplus\\ldots\n\\oplus[0]$, so $|P_{2j} |\\psi\\otimes\\psi\\rangle |^2 < ||\\psi\\otimes\\psi\\rangle |^2 = 1$\nunless $|\\psi\\rangle$ is a coherent state, in which case\n$|\\psi\\rangle\\otimes|\\psi\\rangle \\in [2j]$ and we have equality. The proof for\nall other $s \\in \\N$ is completely analogous.$\\blob$\\\\[1ex]\nIt seems that there should also be a similar group theoretic\nproof for all real, positive $s$ related to (infinite dimensional)\nspin~$js$ representations of su(2) (more precisely: sl(2)). \nThere has been some progress and it is now clear that there will\nnot be an argument as simple as the one given above \\cite{J}.\nCoherent states of the form discussed in \\cite{C3} (for the hydrogen atom)\ncould be of importance here, since they easily generalize to non-integer\n`spin'.\n\nTheorem~\\ref{natural} provides a quick, crude, lower limit on the\nentropy:\n\n\\begin{cor}\nFor states of spin $j$\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) \\geq \\ln{4j+1\\over 2j+1} > 0.\n\\end{equation}\n\\end{cor}\n\n\\noindent {\\sc Proof}: This follows from Jensen's inequality and\nconcavity of $\\ln x$:\n\\begin{eqnarray}\nS_W(|\\psi\\rangle\\langle\\psi|) & = &\n-{\\textstyle (2j+1)}\\int\\frac{d\\Om}{4\\pi}\\, |\\langle\\Omega|\\psi\\rangle|^2 \\ln |\\langle\\Omega|\\psi\\rangle|^2 \\nonumber \\\\\n& \\geq & -\\ln\\left({\\textstyle (2j+1)} \n\\int\\frac{d\\Om}{4\\pi}\\, |\\langle\\Omega|\\psi\\rangle|^{2\\cdot 2}\\right) \\nonumber \\\\\n& \\geq & -\\ln{2j+1 \\over 4j + 1} .\n\\end{eqnarray}\nIn the last step we have used theorem~\\ref{natural}.$\\blob$\\\\[1ex]\nWe hope to have provided enough evidence to convince the\nreader that it is reasonable to expect that\nLieb's conjecture is indeed true for all spin. All cases\nlisted in Lieb's original article, $1\/2$, $1$, $3\/2$, are now\nsettled --\nit would be nice if someone\ncould take care of the remaining ``dot, dot, dot\" \\ldots \n\n\\section*{Acknowledgments}\n\nI would like to thank Elliott Lieb \nfor many discussions, constant support, encouragement, and for reading the\nmanuscript.\nMuch of the early work was done in collaboration with Wolfgang Spitzer.\nTheorem~\\ref{natural} for spin 1 and spin 3\/2 at $s=2$ is due to him and\nhis input was crucial in eliminating many other plausible approaches.\nI would like to thank him for many discussions and excellent team work.\nI would like to thank Branislav Jur\\v co for joint work on the\ngroup theoretic aspects of the problem and stimulating discussions\nabout coherent states.\nIt is a pleasure to thank Rafael Benguria, Almut Burchard, Dirk Hundertmark,\nLarry Thomas, and Pavel Winternitz for many valuable discussions.\nFinancial support by the Max Kade Foundation is gratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\n\n\nIn recent years, significant success has been achieved in applying Reinforcement Learning (RL) to different real-world scenarios \\cite{chenTopKOffPolicyCorrection2019, tangDeepValuenetworkBased2019, gauciHorizonFacebookOpen2019}. \nOffline Reinforcement Learning (ORL) \\cite{levineOfflineReinforcementLearning2020} is pioneering a real-world adaptation of RL, focusing on algorithms that can learn a policy from a fixed, previously recorded dataset, without having to interact with the environment during training.\nMany recent community efforts were focused on advanced benchmarks \\cite{fuD4RLDatasetsDeep2021,gulcehreRLUnpluggedSuite2021} and nuanced algorithm comparisons \\cite{brandfonbrenerOfflineRLOffPolicy2021, qinNeoRLRealWorldBenchmark2021}. \n\nHowever, it is widely recognized that even deep online RL is plagued with evaluation issues, such as insufficient account for statistical variability \\cite{agarwalDeepReinforcementLearning2022}, or sensitivity to the choice of hyperparameters \\cite{hendersonDeepReinforcementLearning2018}. The latter problem is one of the reasons why the results of many deep RL methods are usually reported for a narrow range of values.\n\nWhen it comes to deep ORL algorithms, the choice of hyperparameters also plays a major role in the final performance \\cite{wuBehaviorRegularizedOffline2019}. While many in the community note that the whole training and evaluation pipeline needs improvement \\cite{fuBenchmarksDeepOffPolicy2021}, the comparisons of new deep ORL algorithms are still mostly done through online performance reports on the best set of hyperparameters \\cite{kostrikovOfflineReinforcementLearning2021, fujimotoMinimalistApproachOffline2021}.\n\nIn this paper, we argue that the hyperparameter search should not be ignored in the deep offline RL setting, demonstrating that the conclusions about the algorithms change when we control for the number of trained policies deployed online\\footnote{Code is available at \\href{https:\/\/tinkoff-ai.github.io\/eop\/}{tinkoff-ai.github.io\/eop}}. To this end, we introduce the notion of an online budget, i.e. the number of policies deployed online, and suggest to use an entire pipeline similar to the one in \\cite{paineHyperparameterSelectionOffline2020} for reporting results: training, offline selection, and online evaluation, but one where the selection is done by uniform sampling. As we will demonstrate, this decision allows us to re-use the Expected Validations Performance (EVP) \\cite{dodgeShowYourWork2019} technique from the NLP field to get reliable estimates of expected maximum performance under different online budgets from just one round of hyperparameter search.\n\n\n\n\n\n\\begin{figure*}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\textwidth]{pipeline2.pdf}}\n\\caption{\\textbf{Left}: A widespreaded approach for reporting deep offline RL results, commonly known as online policy selection. \\textbf{Right}: Full deep offline RL evaluation pipeline. We argue for reporting results under the second pipeline with varying sizes of the online evaluation budget $B$. Note that selecting hyperparameters that perform best overall on the online policy selection tasks from the same domain \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021} can also be put to the right pipeline as a special case.}\n\\label{fig:eval_pipelines}\n\\end{center}\n\\vskip -0.2in\n\\end{figure*}\n\n\\textbf{Our Contributions} Here, we list the main contributions of our work:\n\\begin{itemize}\n \\item We demonstrate that the preference between deep offline RL algorithms is budget-dependent. We stress that this is more critical for offline settings than for online ones, and that current evaluation methodology does not account for such dependence.\n \\item We propose to use Expected Validation Performance \\cite{dodgeShowYourWork2019}, a technique actively employed in NLP, for reliable comparison of deep offline RL algorithms under varying online evaluation budgets. To stress the online nature of comparison (in opposition to validation), we refer to it as Expected Online Performance (EOP). This tool can take the both major components of deep offline RL into account: the offline policy selection (OPS) method as well as online evaluation budget. Furthermore, it can be applied without additional computational expenses.\n \\item Using the proposed tool, we also demonstrate that Behavioral Cloning \\cite{pomerleauEfficientTrainingArtificial1991} is often more favorable under a limited evaluation budget.\n \\item In addition, EOP can be applied to comparisons of OPS methods. Using EOP, we illustrate that their preference is also budget-dependent.\n\\end{itemize}\n\nIn the end, we also discuss how the proposed solution relates to the recently introduced Active-OPS \\cite{konyushkovaActiveOfflinePolicy2021} and deployment-constrainted RL setup \\cite{matsushimaDeploymentEfficientReinforcementLearning2020}.\n\n\\section{Background}\n\\subsection{Offline RL}\n\nReinforcement learning (\\textit{RL}) is a framework for solving sequential decision-making problems. It is typically formulated as a Markov Decision Process (MDP) over a 5-tuple $(S, A, P, r, \\gamma)$, with action space $A$, state space $S$, transition dynamics $P$, reward function $r$, and discount factor $\\gamma$. The goal of the learning agent is to obtain a policy $\\pi(s, a)$ that maximizes the expected discounted return $E_{\\pi}[\\sum_{t=0}^{\\infty}\\gamma^{t}r_{t+1}]$ through interaction with the MDP.\n\nIn \\textit{Offline RL}, also known as Batch RL, instead of obtaining data and learning via environment interactions, the agent solely relies on a static dataset $D$ that was collected under some unknown behavioral policy (or a mixture of policies) $\\pi_{\\mu}$. This setting is considered more challenging, as the agent loses its ability for exploration \\cite{levineOfflineReinforcementLearning2020} and is faced with the problem of extrapolation error -- being unable to correct its estimation inaccuracies when selected actions are not present in the training dataset \\cite{fujimotoBenchmarkingBatchDeep2019}.\n\n\\subsection{Offline RL Evaluation}\n\n\\begin{table*}[h]\n\\caption{\\textbf{Best final performance of deep offline RL algorithms if they were evaluated under a different number of policies deployed online} for Hopper-v3 environment. This table highlights that the usage of different online evaluation budgets ($B$) may lead to different conclusions on preference between algorithms. $N$ is the total number of hyperparameters evaluated for a specific algorithm.}\n\\vspace{0.15in}\n\\centering\n\\begin{tabular}{lccccccc|cr}\n\\toprule\nAlgorithm & $B = 1$ & $B = 2$ & $B = 3$ & $B = 4$ & $B = 8$ & $B = 15$ & $B = 30$ & Final & $N$\n\\\\\n\\midrule\n{BC} & \\textbf{1794} & \\textbf{2057} & \\textbf{2179} & - & - & - & - & 2179 & 3 \\\\\n{CQL} & 1773 & 1954 & 2072 & \\textbf{2161} & \\textbf{2391} & \\textbf{2603} & \\textbf{2832} & \\textbf{2832} & 30 \\\\\n{PLAS} & 1475 & 1833 & 1996 & 2096 & 2316 & 2507 & - & 2507 & 15 \\\\\n{BCQ} & 1325 & 1605 & 1742 & 1826 & 1986 & - & - & 2062 & 12\\\\\n{CRR} & 1013 & 1339 & 1477 & 1545 & 1636 & - & - & 1668 & 12 \\\\\n{BREMEN} & 883 & 1148 & 1318 & 1439 & 1691 & - & - & 1795 & 12 \\\\\n{MOPO} & 11 & 18 & 24 & 30 & 46 & 63 & 78 & 78 & 30 \\\\\n\\bottomrule\n\\end{tabular}\\\\\n\\label{table:online-budget-dependence-hopper}\n\\end{table*}\n\nTraining and evaluation of deep offline RL algorithms is still in active development, and various authors approach it in different ways by simplifying the genuine offline setting \\cite{gulcehreRLUnpluggedSuite2021}. At the core of the simplification are two primary issues: (1) unlimited amount of online evaluations available, and therefore (2) sidestepping offline policy selection. For example, it is common to report the maximum performance for the best set of hyperparameters (Figure \\ref{fig:eval_pipelines}, Left). Moreover, in many cases, the number of search trials is not made explicit \\cite{kumarConservativeQLearningOffline2020}.\n\nTo eliminate these simplifications, we adhere to a more general setup for training and evaluating offline RL algorithms similar to \\citet{paineHyperparameterSelectionOffline2020} in order to satisfy hard offline constraints (Figure \\ref{fig:eval_pipelines}, Right).\n\nFirst, the dataset $D$ is randomly split trajectory-wise into training $D_{T}$ and validation $D_{V}$ subsets accordingly. Then a sequence of hyperparameter assignments $(h_{1}, h_{2}, ..., h_{N})$ is sampled for running an algorithm of interest, resulting in a sequence of policies $(\\pi_{1}, \\pi_{2}, ..., \\pi_{N})$. Note that at this stage, we do not know how good these policies are.\n\nThen, $B \\leq N$ of policies are arbitrarily chosen for online evaluation, which we refer to as an \\textit{online evaluation budget}. In the most restricted offline RL setting, $B = 1$. However, the generalization to $B > 1$ is justified by the online evaluation budget being conditioned on the relevant decision-making problem and the available resources.\n\nTo choose policies for online evaluation, offline policy selection (OPS) methods can be used. In specific domains, like recommender systems, policies can be picked based on established offline metrics, e.g. Recall, computed on the validation $D_{V}$ dataset \\cite{xinSelfSupervisedReinforcementLearning2020a}. However, such metrics do not always exist, and it is often necessary to rely on general methods \\cite{voloshinEmpiricalStudyOffPolicy2020, fuBenchmarksDeepOffPolicy2021} or proxy tasks \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021}.\n\n\\section{Online Evaluation Budget Matters}\n\\label{sec:online-eval-matters}\n\nAs can be seen in Figure \\ref{fig:eval_pipelines}, Left, using online policy selection makes the evaluation budget $B$ equivalent to the number of hyperparameter search trials $N$. On the other hand, \\citet{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021} search for the best set of hyperparameters using proxy tasks, but the online evaluation budget on the target task is $B = 1$.\n\nMeanwhile, there is a whole spectrum of values in-between that could be relevant not only for a specific problem, but for a specific context. By context we mean a certain space of resources (computational resources, robotics hardware, time constraints, online testing capacity). Here, a practitioner may work on the same problem, but have a lower or bigger amount of resources available for online evaluation. \n\nTherefore, a natural question to ask when analysing results of deep ORL algorithms is \"Will the conclusions about the algorithms change, if I have a lower or higher online evaluation budget than the one reported in the paper?\". Unfortunately, current evaluation and report methodologies do not provide an answer, and the dependence between varied online evaluation budget and the resulting performance of the algorithms is left unreported. \n\nTo address this issue independently, it is necessary to access detailed experimental results showing which hyperparameters resulted in which performance. While some authors open-source such data \\cite{qinNeoRLRealWorldBenchmark2021}, it is not a common practice to do so.\n\nTo demonstrate that the conclusions about algorithm preference are dependent on the available online evaluation budget, we rely on open-sourced\\footnote{Note that there is a discrepancy in open-sourced and reported results. There is additional data on CRR and BREMEN, but data on MB-PPO is not provided.} results by \\citet{qinNeoRLRealWorldBenchmark2021}. For each algorithm in Table \\ref{table:online-budget-dependence-hopper}, we compute the expected maximum performance under uniform policy selection (i.e. the policies are chosen at random) given a specific online evaluation budget $B$. The final column is what would be reported in the paper, demonstrating that CQL \\cite{kumarConservativeQLearningOffline2020} significantly outperforms its competitors. However, in budgets up to 4, Behavioral Cloning performs the best. Also, note that the preference between CRR \\cite{wangCriticRegularizedRegression2020} and BREMEN \\cite{matsushimaDeploymentEfficientReinforcementLearning2020} is reversed starting from the budget of 8.\n\n\\section{Accounting for the Budget \\raisebox{-0.32ex}{\\includegraphics[scale=0.08]{bank_emoji.png}}}\n\n\\begin{figure*}[!h]\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/Walker2d-v3_1000_low_uniform.pdf}}\n \\caption{Walker2d, Low-1000}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/Hopper-v3_1000_medium_uniform.pdf}}\n \\caption{Hopper, Medium-1000}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/HalfCheetah-v3_1000_high_uniform.pdf}}\n \\caption{HalfCheetah, High-1000}\n \\end{subfigure}\n \\caption{\\textbf{Expected Online Performance graphs under uniform offline policy selection} on \\citet{qinNeoRLRealWorldBenchmark2021} data. The proposed EOP graph clearly demonstrates that preference between algorithms is budget-dependent. Furthermore, it highlights that BC is often more favorable to offline RL algorithms under limited online evaluation budgets. The X-axis denotes the number of policies deployed online, and Y-axis refers to the normalized performance. Note that the number of estimates for a concrete algorithm is upper-bounded by the total number of hyperparameter assignments ($N$) evaluated for this algorithm. Shadowed area represents one standard deviation.}\n \n \\label{fig:eop_neorl}\n\\end{figure*}\n\nIn the previous section, we demonstrated multiple model comparisons where authors would have reached a different conclusion if they had used a smaller (or bigger) online evaluation budget. To resolve this issue, we use a tool from the NLP field, Expected Validation Performance (EVP) \\cite{dodgeShowYourWork2019}, that can be adapted for enhancing the quality of experimental reports in a deep ORL setting.\n\n\\subsection{Expected Online Performance}\n\\label{sec:eop_math}\n\nHere, we give a detailed description for EVP, reframed for an offline RL setting, and with the computational budget replaced by an online evaluation budget (typically, $B \\ll N$). We refer to this approach as Expected Online Performance (EOP).\n\nHaving all $N$ policies evaluated online after hyperparameter search, we want an estimate of the expected maximum performance, given that we could deploy only $1 \\leq B \\leq N$ policies out of $N$. \n\\newpage\nThe parameters of interest to us are $\\theta_{1}$,...,$\\theta_{B}$, where \n\\begin{equation} \\label{eq:expected_online_performace}\n \\theta_{b} := E[max(V(\\pi_{1}),...,V(\\pi_{b}))] = E[V_{(b:b)}]\n\\end{equation}\nfor $1 \\leq b \\leq B$ and $V$ is an random variable (RV) representing the result of online evaluation.\nIn other words, $\\theta_{b}$ is the expected value of the $b^{th}$ order statistic for a sample of size $b$. The $i^{th}$ order statistic $V_{(i:b)}$ is an RV representing the $i^{th}$ smallest value if the RVs were sorted.\n\nOriginally, EVP operates over one stage -- hyperparameter value selection. But in an ORL setting, there is also a second stage -- policy selection (see Figure \\ref{fig:eval_pipelines}, Right). To account for this discrepancy, we note that uniformly sampled hyperparameter values and then uniformly sampled policies result in the probability of a policy being selected for online evaluation proportional to the probability of its hyperparameters being used. Virtually, that makes $\\theta_{b}$ be based on a sample size $b$ drawn independent and identically distributed. \n\nIn this case, the estimator proposed in \\citet{dodgeShowYourWork2019} can be readily applied. The derivation is similar to \\citet{tangShowingYourWork2020}:\n\\begin{equation}\n\\begin{aligned}\n Pr[V_{(b:b)} < v] & = Pr[V(\\pi_{1}) \\leq v \\wedge ... \\wedge V(\\pi_{b}) \\leq v] \\\\\n & = \\prod_{i=1}^{b} Pr[V(\\pi_{i}) \\leq v],\n\\end{aligned}\n\\end{equation}\nwhich we denote as $F^{b}(v)$. Then\n\\begin{equation} \\label{eq:expected_online_performace_cdf}\n \\theta_{b} = E[V_{(b:b)}] = \\int_{-\\inf}^{\\inf} vdF^{b}(v).\n\\end{equation}\nWithout loss of generality, assume $V(\\pi_{1}) \\leq ... \\leq V(\\pi_{N})$. To approximate the Cumulative Distribution Function (CDF), use Empirical Cumulative Distribution Function (ECDF)\n\\begin{equation}\n \\hat{F}^{b}_{N}(v) = (\\frac{1}{N}\\sum_{i=1}^{N}I[V(\\pi_{i}) \\leq v])^{b}\n\\end{equation}\nTo arrive at the final estimator, replace CDF with an ECDF in Equation \\ref{eq:expected_online_performace_cdf}\n\\begin{equation}\n \\hat{\\theta}_{b} = \\int_{-\\inf}^{\\inf} vd\\hat{F}^{b}_{N}(v)\n\\end{equation}\nwhich, by definition, evaluates to\n\\begin{equation} \\label{eq:expected_online_performace_uniform_selection}\n \\hat{\\theta}_{b} = \\frac{1}{N}\\sum_{i=1}^{N}V(\\pi_{i})(\\hat{F}^{b}_{N}(V(\\pi_{i})) - \\hat{F}^{b}_{N}(V(\\pi_{i-1}))\n\\end{equation}\n\nTo summarize, $\\hat{\\theta_{b}}$ corresponds to the estimated expected maximum performance given that (1) hyperparameters were randomly sampled from a pre-defined grid, (2) we could deploy $1 \\leq b \\leq B$ policies out of $N$ for online evaluation, and (3) these $b$ policies were picked by uniform policy selection.\n\nThe major advantage of this estimator is that, if our evaluation methodology satisfies all three conditions described above, then the computation within a single round of hyperparameter search is sufficient to construct a reliable estimate of\nexpected online performance for different values of $b$, without requiring any further experimentation \\cite{dodgeShowYourWork2019}.\nMoreover, for a compact presentation, we can plot a graph over the entire range of values for $b$, demonstrating the dependence between the final performance and online evaluation budget (Figure \\ref{fig:eop_neorl}).\n\nNote, that there are alternative estimators for the quantity of interest \\cite{tangShowingYourWork2020}. However, \\citet{dodgeExpectedValidationPerformance2021} compared different approaches for estimating the expected maximum and found that the employed estimator \\cite{dodgeShowYourWork2019} is favored amongst existing approaches in terms of both MSE criterion and a percent of incorrect conclusions.\n\n\n\\subsection{Target Metric}\n\n\\begin{figure*}[!h]\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures\/citylearn_999_medium_uniform.pdf}}\n \\caption{CityLearn, Medium-1000}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures\/finrl_999_high_uniform.pdf}}\n \\caption{FinRL, High-1000}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures\/industrial_999_medium_uniform.pdf}}\n \\caption{Industrial Benchmark, Medium-1000}\n \\end{subfigure}\n \\caption{\\textbf{Expected Online Performance graphs under uniform offline policy selection.} The proposed EOP graph clearly demonstrates that preference between algorithms is budget dependent. Furthermore, it highlights that BC is often more favorable to offline RL algorithms under limited online evaluation budgets. Shadowed area represents one standard deviation.}\n \n \\label{fig:eop_other_domains}\n\\end{figure*}\n\nThe target metric can be represented by any convenient measure used in literature, e.g., absolute policy performance or policy performance normalized by an expert \\cite{fuD4RLDatasetsDeep2021}. In our case studies (Section \\ref{sec:case_studies}), we rely on a modified version of the latter. The main motivation behind this modification is that the original metric is normalized by the value provided by a domain-specific expert \\cite{fuD4RLDatasetsDeep2021}. However, the final results of offline RL algorithms are highly dependent on training data, and expecting to achieve expert performance while training on data from weak policies can be too optimistic. Therefore, we normalize by the performance of the best policy (as there can be multiple) that collected the training data.\n\n\\subsection{Online Evaluation Budget}\n\nThe original EVP makes it possible to use various quantities as an argument to the target metric, e.g. number of hyperparameters enumerated or training time. Similarly, for EOP in the deep ORL setting, several options can be used, such as the number of trajectories, number of timesteps, or number of policies. We suggest using the latter option, and equate the online evaluation budget $B$ with the number of policies deployed for online evaluation. This choice provides researchers with the flexibility of defining their own amount of computation for getting reliable estimates for policy values.\n\n\n\n\n\n\n\\subsection{Beyond Uniform Policy Selection}\n\n\\begin{figure*}[!h]\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_cql_citylearn_9999_low.pdf}}\n \\caption{CQL, CityLearn, Low-10000}\n \\label{fig:eop_ops_a}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_td3+bc_citylearn_9999_low.pdf}}\n \\caption{TD3+BC, CityLearn, Low-10000}\n \\label{fig:eop_ops_b}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.32\\textwidth}\n \\centering\n \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_cql_finrl_999_low.pdf}}\n \\caption{CQL, FinRL, Low-1000}\n \\label{fig:eop_ops_c}\n \\end{subfigure}\n \\caption{\\textbf{Inverse of Normalized Regret@K over varied online evaluation budgets}. The suggested method can also be used for comparing offline policy selection methods. A similar pattern emerges -- preference for OPS methods is also budget-dependent. Furthermore, uniform policy selection can be competitive under low budgets, and action difference performs reasonably well across a range of environments, dataset sizes, and policy levels. Shadowed area represents one standard deviation.}\n \\label{fig:eop_ops}\n\\end{figure*}\n\nIn Section \\ref{sec:eop_math} we outlined an estimator of expected maximum performance under varied online evaluation budgets. Although, an assumption was made that besides randomly sampled hyperparameters, the policies are also selected uniformly. However, comparing deep ORL methods with different OPS methods is also of interest.\n\nSince the policies selected under an arbitrary strategy (e.g. Fitted-Q Evaluation \\cite{leBatchPolicyLearning2019}) are generally not i.i.d, the plug-in estimator derived above would be invalid. However, one can still estimate the parameters of interest using a vanilla average estimator.\n\\begin{equation}\n \\hat{\\theta_{b}} = \\frac{1}{M}\\sum^{M}_{r=1}max(V^{r}_{1}, ..., V^{r}_{b})\n\\end{equation}\nwhere $M$ is a number of hyperparameter search rounds with offline policy selection, $V^{r}_{i}$ corresponds to an RV representing the result of online evaluation for the $i^{th}$ selected policy in round $r$. Note that this estimator loses the appealing computational side of EVP, and requires multiple runs of hyperparameter search with offline policy selection. This makes it more expensive in terms of computing time.\n\nIn one of our case studies (Section \\ref{sec:case_ops}), we demonstrate how one could use EOP for comparing not only deep ORL algorithms, but OPS methods as well.\n\n\\section{Case Studies}\n\\label{sec:case_studies}\n\nTo further demonstrate the use of the proposed technique and to identify whether online evaluation budget changes the preference between deep ORL algorithms besides the environment analyzed in Section \\ref{sec:online-eval-matters}, we consider several case studies covering a range of decision-making problems: robotics, finances, and energy management. \n\n\\subsection{NeoRL, Robotic Tasks}\n\\label{sec:neorl_robotic_tasks}\n\n\nContinuing the closer look at the results presented in \\citet{qinNeoRLRealWorldBenchmark2021} from Section \\ref{sec:online-eval-matters}, we build Expected Online Performance graphs for other robotics environments (Figure \\ref{fig:eop_neorl}). This once again confirms that the preference is budget-dependent. Moreover, this is consistent across environments, dataset sizes, and policy levels (for a complete set of graphs, check the Appendix). \nThere are many cases when the conclusion changes with a budget. For example, in the Walker-2d environment, CQL is clearly preferred to PLAS \\cite{zhouPLASLatentAction2020a}, but only up to the 9-10 policies available for online evaluation. Another example can be seen in Figure \\ref{fig:eop_neorl}b for the Hopper environment: Behavioral Cloning significantly outperforms its competitiors at low budgets, but loses to PLAS at higher ones. The same holds for the HalfCheetah environment, where CQL starts to prevail at higher budgets.\n\nAs the online budget is upper-bound by the total number of enumerated hyperparameters, it is clear (Figure \\ref{fig:eop_neorl}) that different algorithms were tuned more or less excessively. This results in more optimistic results for one algorithm, and in more pessimistic for another. Consider BCQ \\cite{leBatchPolicyLearning2019} in Figure \\ref{fig:eop_neorl}a. It is tuned up to 13 hyperparameter assignments showing the best result, but as long as a competing algorithm, PLAS, is tuned for 14 and more assignments, it starts to outperform BCQ. An even more vivid example is depicted in Figure \\ref{fig:eop_neorl}c, where MOPO \\cite{yuMOPOModelbasedOffline2020} starts to outperform both BREMEN and CRR at 2x more hyperparameters tested. Note that reporting just one policy value (either using online policy selection or proxy tasks) hides this issue.\n\n\n\\subsection{NeoRL and Other Domains}\n\\label{sec:neorl_other_tasks}\n\nTo validate that our findings hold outside of the open-sourced experimental results provided by \\citet{qinNeoRLRealWorldBenchmark2021}, and to cover a wider range of decision-making problems, we benchmark CQL \\cite{kumarConservativeQLearningOffline2020}, TD3+BC \\cite{fujimotoMinimalistApproachOffline2021}, and BC on the CityLearn \\cite{vazquez-canteliCityLearnV1OpenAI2019}, FinRL \\cite{liuFinRLDeepReinforcement2020}, and Industrial Benchmark \\cite{heinBenchmarkEnvironmentMotivated2017} environments\\footnote{Detailed descriptions of the environments and algorithms used can be found in the Appendix \\ref{appendix:sec:hyperparams}, \\ref{appendix:sec:envs_bases_datasets}.}. In addition, we make sure that the hyperparameter search budgets are equal for all the algorithms to avoid the issue described in the previous section. The hyperparameter grids were deferred to the Appendix \\ref{appendix:sec:hyperparams}. We average mean returns over 100 evaluation trajectories and 3 seeds.\n\nIn Figure \\ref{fig:eop_other_domains}, we see that Behavioral Cloning is quite competitive against both CQL and TD3+BC under limited online evaluation budgets. This akin to the results we observed in robotics environments (Figure \\ref{fig:eop_neorl}), suggesting that BC is often more preferable in restricted settings to deep ORL algorithms.\n\n\n\\subsection{Offline Policy Selection}\n\\label{sec:case_ops}\n\nAs the EOP incorporates offline policy selection, we can also use it to compare how well OPS methods perform against each other.\nTo do so, we use inverse of normalized Regret@K (in our case at $B$) as a target metric. This allows us to answer the following question: \"If we were able to run policies corresponding to $k$ hyperparameter settings in the actual environment and get reliable estimates for their values that way, how far would the best in the set we picked be from the best of all hyperparameter settings considered?\" \\cite{paineHyperparameterSelectionOffline2020}. But instead of reporting one value of $k$ as in \\citet{paineHyperparameterSelectionOffline2020}, we can easily report on all the values of $B$. Moreover, the estimator outlined in Section \\ref{sec:eop_math} can be used for presenting the results of uniform policy selection.\n\nWe do not aim to benchmark and compare the entire myriad of offline policy selection approaches, but rather to demonstrate how one can use the proposed tool for such purposes. To do so, we test several methods on the environments from the previous section, namely $V(s_0)$ using FQE, TD-Error, and Action Difference \\cite{leBatchPolicyLearning2019,hussenotHyperparameterSelectionImitation2021}.\n\nThe results can be found in Figure \\ref{fig:eop_ops}. First, we observe a pattern similar to the one described in Sections \\ref{sec:online-eval-matters}, \\ref{sec:neorl_robotic_tasks}, \\ref{sec:neorl_other_tasks}: \\textit{preference between offline policy selection methods is also budget-dependent}. Therefore, it is not enough to report the result of such methods under just one selected threshold.\nSecond, there is no clear winner among all setups, as even TD-Error may sometimes perform good (Figure \\ref{fig:eop_ops_a}, for more -- check the Appendix). However, Action Difference often performs reasonably well across many dataset sizes and policy levels in Industrial Benchmark and CityLearn environments (Figures \\ref{fig:eop_ops_a}, \\ref{fig:eop_ops_b}). We hypothesize that such a selection method can serve as a post-training conservative regularizator (i.e. picking policies that are more similar to the behavioral ones), but that requires further investigation.\nAnd the last notable observation is that uniform policy selection can be competitive to other considered methods, especially when the online evaluation budget is limited. Sometimes it can even perform the best among the entire set of methods (Figure \\ref{fig:eop_ops_a}).\n\n \n\\section{Related Work}\nTo the best of our knowledge, the closest concept to our work is deployment-constrained RL \\cite{matsushimaDeploymentEfficientReinforcementLearning2020, suMUSBOModelbasedUncertainty2021}. The core idea of this setting is to consider the number of policies deployed online and to reuse the data for iterative training from such deployments. \\citet{matsushimaDeploymentEfficientReinforcementLearning2020,suMUSBOModelbasedUncertainty2021} propose new algorithms that are especially suited for this setting, claiming that they are more deployment-efficient. However, they also relied on an extensive hyperparameter search reporting for the best set of hyperparameters. This hides the actual number of policies evaluated online, while our approach prevents that. An interesting direction for future work would be to adapt the EOP for this iterative setting as well.\n\n\\citet{konyushkovaActiveOfflinePolicy2021} formulated a new problem, which is an extension to OPS, Active Offline Policy Selection (A-OPS). The major difference is to allow for an OPS method to have a feedback loop from newly trained policies deployed online, and re-adjust which policy should be run next. While this is an important step forward, EOP can actually subsume A-OPS as one of the OPS methods, since sequential policy testing is allowed. Furthermore, our paper aims to consolidate all the parts of a deep ORL pipeline, while \\citet{konyushkovaActiveOfflinePolicy2021} focuses on a new problem.\n\n\nRecently, \\citet{agarwalDeepReinforcementLearning2022} scrutinized the evaluation methodology of deep RL algorithms, and advocated for a set of statistical tools to be employed for more reliable comparison. However, \\citet{agarwalDeepReinforcementLearning2022} focuses on reliable evaluation \\textit{after hyperparameter tuning}, while our work highlights its importance in offline deep RL setting, and argues that results should be reported under varied hyperparameter tuning capacity when comparing deep ORL algorithms.\n\n\\citet{brandfonbrenerQuantileFilteredImitation2021} noted the extensive online evaluation budgets used in recent works on deep ORL. To address this issue, comparison between algorithms was made under a small hyperparameter tuning budget ($B=4$). However, evaluating only under a limited budget may not be enough. Our paper demonstrates that the preference between algorithms can be budget-dependent, requiring evaluation under various budgets.\n\nThere is a sizeable body of work on Offline Policy Evaluation \\cite{voloshinEmpiricalStudyOffPolicy2020, fuBenchmarksDeepOffPolicy2021} and, specifically, on Offline Policy Selection \\cite{paineHyperparameterSelectionOffline2020, hussenotHyperparameterSelectionImitation2021, yangOfflinePolicySelection2020}. This work does not aim to compare or benchmark these types of methods, but to provide a procedure for comparing deep ORL algorithms with OPS (and promote the usage of simple uniform selection) for achieving reliable conclusions. In addition, we demonstrate that a similar pattern, online-budget dependence, is relevant for OPS methods as well.\n\nBehavioral Cloning is typically reported in deep ORL papers as a baseline, and many papers claim to beat this baseline \\cite{kumarConservativeQLearningOffline2020, fujimotoMinimalistApproachOffline2021, kumarStabilizingOffPolicyQLearning2019, leBatchPolicyLearning2019, wangCriticRegularizedRegression2020, wuBehaviorRegularizedOffline2019}. However, when considering learning from human demonstrations, \\citet{mandlekarWhatMattersLearning2021} demonstrated the superiority of BC over deep ORL algorithms, especially when recurrence is employed in network architecture. Reinforcing the effectiveness of BC, this work suggests that BC is preferable not only in settings with human demonstrations, but across a diverse range of decision-making problems, given that the online evaluation budget is severely limited.\n\n\n\n\n\n\\section{Closing Remarks}\n\n\n\n\n\n\n\n\n\\textbf{Motivation}:\nWhile a lot of RL community efforts are focused on offline RL datasets' general evaluation, this paper questions the methods' performance under the entire spectrum of their hyperparameters and available resources.\nAs different problems and contexts may allow for different online evaluation budgets, we argue that they can also make different optimal solutions possible within these constraints.\nWe hope that the proposed evaluation technique and our findings will encourage the ORL community to report performance results under different online evaluation budgets.\n\n\n\\textbf{Limitations}:\nAlthough this work emphasizes the importance of several online evaluations, it does not investigate the possible evaluation risks. Many real-world applications have critical corner cases, especially in the autonomous driving, healthcare, and finance domains \\cite{10.1007\/s10664-020-09881-0}. We note that the EOP has limited applicability in such risk-sensitive scenarios due to its focus on maximum performance.\n\n\n\\textbf{Opportunities}:\nEOP proposes a unified methodology for finding the best-performing setup under different online budget constraints. Unlike Deep Learning domains, the deep ORL evaluation is still in active development. The possibility of having a standard performance evaluation report opens avenues for adopting more precise methods for different tasks and contexts or creating online budget-dependent ORL algorithms.\n\n\n\n\n\n\n\\section{Conclusion}\n\nA lot of community effort was recently devoted to developing new algorithms and datasets, while noting that the whole evaluation pipeline is still to be improved upon \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021}. In this work, we demonstrated (Section \\ref{sec:online-eval-matters}) one of the problems with such pipelines, and proposed a technique named EOP (Section \\ref{sec:eop_math}) to address it when presenting the results of deep ORL algorithms. \nSeveral empirical results were found (Section \\ref{sec:case_studies}): (1) Behavioral Cloning is often more\nfavorable under a limited evaluation budget, (2) Online Policy Selection method preferences are also budget-dependent.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nQR models have become increasingly popular since the seminal work of\n\\cite{koenker1978regression}. In contrast to the mean regression\nmodel, QR belongs to a robust model family, which can give an\noverall assessment of the covariate effects at different quantiles\nof the outcome \\citep{koenker2005quantile}. In particular, we can\nmodel the lower or higher quantiles of the outcome to provide a\nnatural assessment of covariate effects specific for those\nregression quantiles. Unlike conventional models, which only address\nthe conditional mean or the central effects of the covariates, QR\nmodels quantify the entire conditional distribution of the outcome\nvariable. In addition, QR does not impose any distributional\nassumption on the error, except requiring the error to have a zero\nconditional quantile. The foundations of the methods for independent\ndata are now consolidated, and some statistical methods for\nestimating and drawing inferences about conditional quantiles are\nprovided by most of the available statistical programs (e.g., R,\nSAS, Matlab and Stata). For instance, just to name a few, in the\nwell-known R package \\verb\"quantreg()\" is implemented a variant of\nthe \\cite{barrodale1977algorithms} simplex (BR) for linear\nprogramming problems described in \\cite{koenker1987algorithm}, where\nthe standard errors are computed by the rank inversion method\n\\citep{koenker2005quantile}. Another method implemented in this\npopular package is Lasso Penalized Quantile Regression (LPQR),\nintroduced by \\cite{tibshirani1996regression}, where a penalty\nparameter is specified to determine how much shrinkage occurs in the\nestimation process. QR can be implemented in a range of different\nways. \\cite{koenker2005quantile} provided an overview of some\ncommonly used quantile regression techniques from a\n\"classical\" framework.\n\n\\cite{kottas2001bayesian} considered\nmedian regression from a Bayesian point of view, which is a special case of quantile regression,\nand discussed non-parametric modeling for the error distribution\nbased on either P\\'{o}lya tree or Dirichlet process priors. Regarding\ngeneral quantile regression, \\cite{yu2001bayesian} proposed a\nBayesian modeling approach by using the ALD,\n\\cite{kottas2009bayesian} developed Bayesian semi-parametric models\nfor quantile regression using Dirichlet process mixtures for the\nerror distribution, {\\cite{geraci07} studied quantile regression\nfor longitudinal data using the ALD.} Recently, \\cite{kozumi2011gibbs} developed\na simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the ALD.\n\nAn interesting aspect to be considered in statistical modelling is\nthe diagnostic analysis. This can be carried out by conducting an\ninfluence analysis for detecting influential observations. One of\nthe most technique to detect influential observations is the\ncase-deletion approach. The famous approach of Cook (1977) has been\napplied extensively to assess the influence of an observation in\nfitting a statistical model; see \\cite{cook82} and the references\ntherein. It is difficult to apply this approach directly to the QR\nmodel because the underlying observed-data likelihood function is\nnot differentiable at zero. \\cite{zhu2001case} presents an approach\nto perform diagnostic analysis for general statistical models that\nis based on the Q-displacement function. This approach has been\napplied successfully to perform influence analysis in several\nregression models, for example, \\cite{xie2007case} considered in\nmultivariate $t$ distribution,\n\\cite{Matos.Lachos.Bala.Labra.2012} obtained case-deletion measures\nfor mixed-effects models following the \\cite{zhu2001case}'s approach\nand in \\cite{Zeller.Labra.Lachos.Balakrishnan.2010} we can see\nsome results about local influence for mixed-effects models\nobtained by using the Q-displacement function.\n\nTaking advantage of the likelihood structure imposed by the ALD, the\nhierarchical representation of the ALD, we develop here an\nEM-type algorithm for obtaining the ML estimates at the $p$th level,\nand by simulation studies our EM algorithm outperformed the\ncompeting BR and LPQR algorithms, where the standard error is\nobtained as a by-product. Moreover, we obtain case-deletion\nmeasures for the QR model. Since QR methods complement and improve\nestablished means regression models, we feel that the assessment of\nrobustness aspects of the parameter estimates in QR is also an\nimportant concern at a given quantile level $p\\in(0, 1)$.\n\n\n\n\\indent The rest of the paper is organized as follows. Section 2\nintroduces the connection between QR and ALD as well as outlining\nthe main results related to ALD. Section 3 presents an EM-type\nalgorithm to proceed with ML estimation for the parameters at the\n$p$th level. Moreover, the observed information matrix is derived.\nSection \\ref{Sec Diagnostic} provides a brief sketch of the case-deletion method\nfor the model with incomplete data, and also develop a methodology pertinent to the ALD. Sections \\ref{sec application}\nand \\ref{sec simulation study} are dedicated to the analysis of real\nand simulated data sets, respectively. Section 6 concludes with a\nshort discussion of issues raised by our study and some possible\ndirections for the future research.\n\n\\section{The quantile regression model} \\label{sec tCR}\nEven though considerable amount of work has been done on regression models and their extensions, regression models by using asymmetric Laplace distribution have received little attention in the literature. Only recently, the a study on quantile regression model based on asymmetric Laplace distribution\n was presented by Tian et al. (2014) who a derived several interesting\nand attractive properties and presented an EM algorithm. Before\npresenting our derivation, let us recall firstly the definition of\nthe asymmetric Laplace distribution and after this, we will\npresent the quantile regression model.\n\n\\subsection{Asymmetric Laplace distribution}\nAs discussed in \\cite{yu2001bayesian}, we say that a random variable\nY is distributed as an ALD with location parameter $\\mu$, scale\nparameter $\\sigma>0$ and skewness parameter $p\\in (0,1)$, if its\nprobability density function (pdf) is given by\n\\begin{equation}\\label{pdfAL}\nf(y|\\mu,\\sigma,p)=\\frac{p(1-p)}{\\sigma}\\exp\\Big\n\\{-\\rho_p\\big(\\frac{y-\\mu}{\\sigma}\\big)\\Big\\},\n\\end{equation}\nwhere $\\rho_p(.)$ is the so called check (or loss) function defined\nby $\\rho_p(u)=u(p-\\mathbb{I}\\{u<0\\})$, with $\\mathbb{I}\\{.\\}$\ndenoting the usual indicator function. This distribution is\ndenoted by $ALD(\\mu,\\sigma,p)$. It is easy to see that\n$W=\\rho_p\\big(\\frac{Y-\\mu}{\\sigma}\\big)$ follows an exponential\ndistribution $\\exp(1)$.\n\nA stochastic representation is useful to obtain some properties of\nthe distribution, as for example, the moments, moment generating function (mgf), and estimation algorithm. For the ALD \\cite{kotz2001laplace}, \\cite{kozobowski00} and \\cite{zhou13} presented the following stochastic representation:\nLet $U\\sim\n{\\exp}(\\sigma)$ and $Z\\sim N(0,1)$ be two independent random variables. Then,\n$Y\\sim ALD(\\mu,\\sigma,p)$ can be represented as\n\\begin{equation}\\label{st-ALD}\nY\\buildrel d\\over=\\mu+\\vartheta_p U+\\tau_p\\sqrt{\\sigma U} Z,\n\\end{equation}\nwhere $\\vartheta_p=\\frac{1-2p}{p(1-p)}$ and\n$\\tau^2_p=\\frac{2}{p(1-p)}$, and $\\buildrel d\\over=$\ndenotes equality in distribution.\n\\indent Figure \\ref{fig:ald} shows how the skewness of the ALD\nchanges with altering values for $p$. For example, for $p=0.1$\nalmost all the mass of the ALD is situated in the right tail. For $p=0.5$, both tails of the ALD have equal mass and the distribution then equals the more common double exponential\ndistribution. In contrast to the normal distribution with a\nquadratic term in the exponent, the ALD is linear in the exponent.\nThis results in a more peaked mode for the ALD together with thicker\ntails. On the other hand, the normal distribution has heavier\nshoulders compared to the ALD.\n\\begin{figure}[!htb]\n\\begin{center}\n{\\includegraphics[scale=0.6]{ald.ps\n\\caption{Standard asymmetric Laplace density \\label{fig:ald}}}\n\\end{center}\n\\end{figure}\nFrom (\\ref{st-ALD}), we have the hierarchical representation of the ALD, see \\cite{lum12}, given by\n\\begin{eqnarray}\nY|U=u&\\sim& N(\\mu+\\vartheta_p u,\\tau^2_p\\sigma u ),\\label{hierar1}\\\\\nU&\\sim& exp(\\sigma)\\label{hierar2}.\n\\end{eqnarray}\nThis representation will be useful for the implementation of the EM algorithm. Moreover, since $Y|U=u\\sim N(\\mu+\\vartheta_p u,\\tau^2_p\\sigma u )$, then one can derive easily the pdf of $Y$. That is, the pdf in ( \\ref{pdfAL}) can be expressed as\n\\begin{equation}\\label{pdfALs}\nf(y|\\mu,\\sigma,p)=\\frac{1}{\\sqrt{2\\pi}}\\frac{1}{\\tau_p\\sigma^{\\frac{3}{2}}} \\exp\\Big(\\frac{\\delta(y)}{\\gamma}\\Big)\nA(y),\n\\end{equation}\nwhere $\\delta(y)=\\frac{|y-\\mu|}{\\tau_p\\sqrt{\\sigma}}$, $\\gamma=\\sqrt{\\frac{1}{\\sigma}\\big(2+\\frac{\\vartheta_p^2}{\\tau^2_p}\\big)}=\\frac{\\tau_p}{2\\sqrt{\\sigma}}$ and $A(y)=2\\Big(\\frac{\\delta(y)}{\\gamma}\\Big)^{1\/2} K_{1\/2}\\big(\\delta(y)\\gamma\\big)$, with $K_{\\nu}(.)$ being the modified Bessel function of the third kind. It easy to see that\nthat the conditional distribution of $U$, given $Y=y$, is\n$U|(Y=y)\\sim GIG (\\frac{1}{2},\\delta,\\gamma)$. Here, $GIG(\\nu, a, b)$ denotes the Generalized Inverse\nGaussian (GIG) distribution; see \\cite{barndorff2001non} for more details. The pdf of GIG distribution is given by\n$$h(u|\\nu,a,b)=\\frac{(b\/a)^{\\nu}}{2K_{\\nu}(ab)}u^{\\nu-1}\\exp\\Big\\{-\\frac{1}{2}\\big(a^2\/{u}+b^2u\\big)\\Big\\},\\,\\,u>0,\\,\\,\\,\\,\\nu\\in \\mathbb{R},\\,\\,a,b>0.$$\nThe moments of $U$ can be expressed as\n$$E[U^k]=\\left(\\frac{a}{b}\\right)^{k}\\frac{K_{\\nu+k}(ab)}{K_{\\nu}(ab)},\\,\\,\\ k\\in \\mathbb{R}.$$\nSome properties of the Bessel function of the third kind\n$K_{\\lambda}(u)$ that will be useful for the developments\nhere are: (i) $K_{\\nu}(u)=K_{-\\nu}(u)$; (ii)\n$K_{\\nu+1}(u)=\\frac{2\\nu}{u}K_{\\nu}(u)+K_{\\nu-1}(u)$; (iii) for\nnon-negative integer $r$,\n$K_{r+1\/2}(u)=\\sqrt{\\frac{\\pi}{2u}}\\exp(-u)\\sum_{k=0}^r\n\\frac{(r+k)!(2u)^{-k}}{(r-k)!k!}$. A special case is $K_{1\/2}(u)=\n\\sqrt{\\frac{\\pi}{2u}}\\exp(-u)$.\\\\\n\n\n\\subsection{ Linear quantile regression}\n\nLet $y_i$ be a response variable and $\\mathbf{x}_i$ a $k\\times 1$\nvector of covariates for the $i$th observation, and let\n$Q_{y_i}(p|\\mathbf{x}_i)$ be the $p$th $(0 < p < 1)$ quantile regression\nfunction of $y_i$ given $\\mathbf{x}_i$, $i=1,\\ldots,n$ . Suppose that the relationship\nbetween $Q_{y_i}(p|\\mathbf{x}_i)$ and $\\mathbf{x}_i$ can be modeled as $Q_{y_i}(p|\\mathbf{x}_i)\n= \\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p$, where $\\mbox{${\\bm \\beta}$}_p$ is a vector $(k\\times 1)$\nof unknown parameters of interest. Then, we consider the quantile\nregression model given by\n\\begin{equation}\\label{QRmodel}\ny_i = \\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p + \\epsilon_i,\\,\\,\\,i=1,\\ldots,n,\n\\end{equation}\nwhere $\\epsilon_i$ is the error term whose distribution (with density,\nsay, $f_p(.)$) is restricted to have the $p$th quantile equal to\nzero, that is, $\\int^{0}_{-\\infty}f_p(\\epsilon_i)d\\epsilon_i=p$. The error density $f_p(.)$ is often left unspecified in the classical literature. Thus, quantile regression estimation for\n$\\mbox{${\\bm \\beta}$}_p$ proceeds by minimizing\n\\begin{eqnarray}\\label{lossEq}\n\\widehat{\\mbox{${\\bm \\beta}$}}_p=arg\\,\\, min_{\\mbox{${\\bm \\beta}$}_{p}} \\sum^n_{i=1}\n\\rho_p\\big({y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p}\\big),\n\\end{eqnarray}\nwhere $\\rho_p(.)$ is as in (\\ref{pdfAL}) and $\\widehat{\\mbox{${\\bm \\beta}$}}_p$ is\nthe quantile regression estimate for $\\mbox{${\\bm \\beta}$}_p$ at the $p$th\nquantile. The special case $p=0.5$ corresponds to median\nregression. As the check function is not differentiable at zero,\nwe cannot derive explicit solutions to the minimization problem.\nTherefore, linear programming methods are commonly applied to obtain\nquantile regression estimates for $\\mbox{${\\bm \\beta}$}_p$. A connection between\nthe minimization of the sum in (\\ref{lossEq}) and the\nmaximum-likelihood theory is provided by the ALD; see \\cite{geraci07}. It is also true that under\nthe quantile regression model, we have\n\\begin{equation}\\label{Wi}\nW_i=\\frac{1}{\\sigma}\\rho_p\\big(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p\\big)\\sim \\exp(1).\n\\end{equation}\nThe above result is useful to check the model in practice, as\nwill be seen in the Application Section.\n\nNow, suppose $y_1, \\ldots,y_n$ are independent observations such as $Y_i \\sim ALD (\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p,\\sigma,p),$\n$i=1,\\ldots,n$. Then, from (\\ref{pdfALs}) the\nlog--likelihood function for $\\mbox{${ \\bm \\theta}$}=(\\mbox{${\\bm \\beta}$}_p^{\\top},\\sigma)^{\\top} $ can be expressed as\n\\begin{equation}\\label{likel}\n\\ell(\\mbox{${ \\bm \\theta}$})=\\sum^n_{i=1} \\ell_i(\\mbox{${ \\bm \\theta}$}),\n\\end{equation}\nwhere\n$\\ell_i(\\mbox{${ \\bm \\theta}$})=c-\\frac{3}{2}\\log{\\sigma}+\\frac{\\vartheta_p}{\\tau_p^2\\sigma}(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)+\\log(A_i)$,\nwith $c$ is a constant does not depend on $\\mbox{${ \\bm \\theta}$}$ and\n$A_i=2\\big({\\frac{\\delta_i}{\\gamma}}\\big)^{1\/2}K_{1\/2}(\\lambda_i)=\\frac{\\sqrt{2\\pi}}{\\gamma}\\exp(-\\lambda_i),$\nwith $\\delta_i=\\delta(y_i)={|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p|}\/{\\tau_p\\sqrt{\\sigma}})$ and $\\lambda_i=\\delta_i\\gamma$.\n\nNote that if we consider $\\sigma$ as a nuisance parameter, then the\nmaximization of the likelihood in (\\ref{likel}) with respect to the\nparameter $\\mbox{${\\bm \\beta}$}_p$ is equivalent to the minimization of the\nobjective function in (\\ref{lossEq}). and hence the relationship\nbetween the check function and ALD can be used to reformulate the QR\nmethod in the likelihood framework.\n\nThe log--likelihood function is not differentiable at zero.\nTherefore, standard procedures the estimation can not be\ndeveloped following the usual way. Specifically, the standard\nerrors for the maximum likelihood estimates is not based\non the genuine information matrix. To overcome this\nproblem we consider the empirical information matrix as will\nbe described in the next Subsection.\n\n\n\n\n\n\\subsection{Parameter estimation via the EM algorithm}\nIn this section, we discuss an estimation method for QR based on\nthe EM algorithm to obtain ML estimates. Also, we consider the\nmethod of moments (MM) estimators,which can be effectively used as\nstarting values in the EM algorithm.\nHere, we show how to employ the EM algorithm\nfor ML estimation in QR model under the ALD. From the hierarchical\nrepresentation (\\ref{hierar1})-(\\ref{hierar2}), the QR model in\n(\\ref{QRmodel}) can be presented as\n\\begin{eqnarray}\nY_i|U_i=u_i&\\sim& N(\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p+\\vartheta_p\\label{repHier1}\nu_i,\\tau_p^2\\sigma u_i),\\\\ U_i&\\sim&\n\\exp(\\sigma),\\,\\,\\,\\,i=1,\\ldots,n,\\label{repHier2}\n\\end{eqnarray}\nwhere $\\vartheta_p$ and $\\tau_p^2$ are as in (\\ref{st-ALD}). This\nhierarchical representation of the QR model is convenient to describe\nthe steps of the EM algorithm. Let $\\mathbf{y} = (y_1, \\ldots, y_n)$ and $\\mathbf{u} = (u_1,\\ldots , u_n)$ be\nthe observed data and the missing data, respectively. Then, the\ncomplete data log-likelihood function of\n$\\mbox{${ \\bm \\theta}$}=(\\mbox{${\\bm \\beta}$}^{\\top}_p,\\sigma)^{\\top}$, given $(\\mathbf{y},\\mathbf{u})$,\nignoring additive constant terms, is given by\n$\\ell_{c}(\\mbox{${ \\bm \\theta}$}|{\\bf y},\\mathbf{u})=\\sum_{i=1}^{n}\\ell_{c}(\\mbox{${ \\bm \\theta}$}|y_i,u_i)$,\nwhere\n\\begin{eqnarray*}\n\\ell_{c}(\\mbox{${ \\bm \\theta}$}|y_i,u_i) =-\\frac{1}{2} \\log( 2\\pi\\tau_p^2)\n -\\frac{3}{2} \\log(\\sigma) -\\frac{1}{2}\\log (u_i) - \\frac{1}{2\\sigma\\tau_p^2}\n{u^{-1}_i}(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p-\\vartheta_p u_i)^2 -\n\\frac{1}{\\sigma} u_i,\n\\end{eqnarray*}\nfor $i=1,\\ldots,n$. In what follows the superscript $(k)$ indicates\nthe estimate of the related parameter at the stage $k$ of the\nalgorithm. The E-step of the EM algorithm requires evaluation of the\nso-called Q-function $Q(\\mbox{${ \\bm \\theta}$}|\\mbox{${ \\bm \\theta}$}^{(k)}) =\n\\textrm{E}_{\\scriptsize \\mbox{${ \\bm \\theta}$}^{(k)}}[\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{y},\\mathbf{u})|{\\bf y}, \\mbox{${ \\bm \\theta}$}^{(k)}]$, where\n$\\textrm{E}_{\\scriptsize \\mbox{${ \\bm \\theta}$}^{(k)}}[.]$ means that the\nexpectation is being effected using $\\mbox{${ \\bm \\theta}$}^{(k)}$ for $\\mbox{${ \\bm \\theta}$}$.\nObserve that the expression of the Q-function is completely\ndetermined by the knowledge of the expectations\n\\begin{eqnarray} \\label{weith}\n{\\cal E}_{ s i}(\\mbox{${ \\bm \\theta}$}^{(k)}) = \\textrm{E}[U^s_i |y_i, \\mbox{${ \\bm \\theta}$}^{(k)}],\\,\\,\\, s=-1,1,\n\\end{eqnarray}\nthat are obtained of properties of the $GIG(0.5, a, b)$ distribution. Let $\\mbox{${ \\bm \\xi}$}^{(k)} _{s}=\\big( {\\cal\nE}_{s1}(\\mbox{${ \\bm \\theta}$}^{(k)}), \\ldots, {\\cal E}_{sn}(\\mbox{${ \\bm \\theta}$}^{(k)})\n\\big)^{\\top}$ be the vector that contains all quantities defined\nin (\\ref{weith}). Thus, dropping unimportant constants, the\nQ-function can be written in a synthetic form as\n$Q(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\sum_{i=1}^{n}Q_i(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})$,\nwhere {\\small{\n\\begin{eqnarray} \\label{eqn qfunction}\nQ_i(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}}) =\n-\\frac{3}{2}\\log\\sigma-\\frac{1}{2\\sigma\\tau_p^2}\n\\left[ {\\cal\nE}_{-1i}(\\mbox{${ \\bm \\theta}$}^{(k)})(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)^2-2(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)\\vartheta_p+\\frac{1}{4}{\\cal E}_{1\ni}(\\mbox{${ \\bm \\theta}$}^{(k)})\\tau_p^4\\right]. \\, \\, \\, \\, \\,\n\\end{eqnarray}}}\nThis quite useful expression to implement the M-step, which consists of maximizing it over $\\mbox{${ \\bm \\theta}$}$. So the EM algorithm can be summarized as follows:\\\\\n\\noindent \\emph{E-step}: Given $\\mbox{${ \\bm \\theta}$}=\\mbox{${ \\bm \\theta}$}^{(k)}$, compute ${\\cal E}_{si}(\\mbox{${ \\bm \\theta}$}^{(k)})$ through of the relation\n\\begin{equation}\\label{deltai}\n{\\cal E}_{ si}(\\mbox{${ \\bm \\theta}$}^{(k)}) =E[U^s_i|y_i,\\mbox{${ \\bm \\theta}$}^{(k)}]=\\left(\\frac{\\delta^{(k)}_i}{\\gamma^{(k)}}\\right)^{s}\\frac{K_{1\/2+s}\\big(\\lambda^{(k)}_i\\big)}{K_{1\/2}\\big(\\lambda^{(k)}_i\\big)}, s=-1,1,\n\\end{equation}\nwhere\n$\\delta^{(k)}_i=\\frac{|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}^{(k)}_p|}{\\tau_p\\sqrt{\\sigma^{(k)}}}$, $\\gamma^{(k)}=\\frac{\\tau_p}{2\\sqrt{\\sigma^{(k)}}}$ and\n$\\lambda^{(k)}_i={\\delta^{(k)}_i\\gamma^{(k)}}$;\\\\\n\\noindent{\\em M-step}: Update ${\\mbox{${ \\bm \\theta}$}}^{(k)}$ by\nmaximizing $Q(\\mbox{${ \\bm \\theta}$}|\\mbox{${ \\bm \\theta}$}^{(k)})$ over $\\mbox{${ \\bm \\theta}$}$,\nwhich leads to the following expressions\n{\\small{\n\\begin{eqnarray*}\n{\\mbox{${\\bm \\beta}$}}^{(k+1)}_p&=&\\left(\\mathbf{X}^{\\top} D(\\mbox{${ \\bm \\xi}$}^{(k)} _{-1})\\mathbf{X} \\right)^{-1}\\mathbf{X}^{\\top}\\big(D(\\mbox{${ \\bm \\xi}$}^{(k)} _{-1})\\mathbf{Y}-\\vartheta_p {\\bf 1}_n\\big),\\, \\, \\\\\n{\\sigma}^{(k+1)}&=&\\frac{1}{3n\\tau^2_p}\\Big[Q(\\mbox{${\\bm \\beta}$}^{(k+1)}, \\mbox{${ \\bm \\xi}$}_{-1}^{(k)})-2{\\bf 1}^{\\top}_n(\\mathbf{Y}-\\mathbf{X}\n\\mbox{${\\bm \\beta}$}^{(k+1)})\\vartheta_p+\\frac{\\tau_p^4}{4}{\\bf\n1}^{\\top}_n\\mbox{${ \\bm \\xi}$}^{(k)} _{1}\\Big],\n\\end{eqnarray*}}}\nwhere $D(\\mathbf{a})$ denotes the diagonal matrix, with the diagonal\nelements given by $\\mathbf{a}=(a_1,\\ldots,a_p)^{\\top}$ and\n$Q(\\mbox{${\\bm \\beta}$}, \\mbox{${ \\bm \\xi}$}_{-1})= (\\mathbf{Y}-\\mathbf{X} \\mbox{${\\bm \\beta}$})^{\\top}D(\\mbox{${ \\bm \\xi}$}_{-1}) (\\mathbf{Y}-\\mathbf{X}\\mbox{${\\bm \\beta}$})$. A similar expression for $\\mbox{${\\bm \\beta}$}^{(k+1)}_p$ is obtained in \\cite{tian13}.\nThis process is iterated until some distance involving two\nsuccessive evaluations of the actual log-likelihood $\\ell(\\mbox{${ \\bm \\theta}$})$,\nlike $||\\ell({\\mbox{${ \\bm \\theta}$}}^{(k+1)})-\\ell({\\mbox{${ \\bm \\theta}$}}^{(k)})||$ or\n$||\\ell({\\mbox{${ \\bm \\theta}$}}^{(k+1)})\/\\ell({\\mbox{${ \\bm \\theta}$}}^{(k)})-1||$, is small\nenough. This algorithm is implemented as part of the R package\n\\verb\"ALDqr()\", which can be downloaded at not cost from the\nrepository CRAN. Furthermore, following the results given in\n\\cite{Yu2005}, the MM estimators for $\\mbox{${\\bm \\beta}$}_p$ and $\\sigma$ are\nsolutions of the following equations:\n\\begin{eqnarray}\\label{ab;m}\n\\widehat{\\mbox{${\\bm \\beta}$}}_{pM}=\\big(\\mathbf{X}^{\\top}\n\\mathbf{X}\\big)^{-1}\\mathbf{X}^{\\top}\\big(\\mathbf{Y}-\\widehat{\\sigma}_M \\vartheta_p{\\bf\n1}_n\\big)\\, \\, \\, \\,{\\rm and} \\, \\,\n\\widehat{\\sigma}_{M}=\\displaystyle \\frac{1}{n}\\sum_{i=1}^n\n\\rho_p\\big(y_i-\\mathbf{x}_i^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}}_{pM}\\big),\n\\end{eqnarray}\nwhere $\\vartheta_p$ is as (\\ref{st-ALD}). Note that the\nMM estimators do not have explicit closed form and numerical\nprocedures are needed to solve these non-linear equations. They can be used as initial values in the iterative procedure for computing the ML estimates based on the EM-algorithm.\nStandard errors for the maximum likelihood estimates is based on\nthe empirical information matrix, that according to \\cite{meilijson89} formula, is defined as\n\\begin{eqnarray}\\label{Imatrix }\n\\mathbf{L}(\\mbox{${ \\bm \\theta}$})=\\sum_{j=1}^{n}\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$})\\textbf{s}^\\top(y_j|\\mbox{${ \\bm \\theta}$})-n^{-1}\\textbf{S}(y_j|\\mbox{${ \\bm \\theta}$})\\textbf{S}^\\top(y_j|\\mbox{${ \\bm \\theta}$}),\n\\end{eqnarray}\nwhere $\\textbf{S}(y_j|\\mbox{${ \\bm \\theta}$})=\\sum_{j=1}^{n}\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$})$. It is noted from the result of \\cite{louis82} that the individual score can be determined as\n$\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$}) ={\\partial \\log f(y_j|\\mbox{${ \\bm \\theta}$})}\/{\\partial \\mbox{${ \\bm \\theta}$}} = E\\Big({\\partial \\ell_{c_j}(\\mbox{${ \\bm \\theta}$}|y_j, u_i)}\/{\\partial \\mbox{${ \\bm \\theta}$}} | y_j,\\mbox{${ \\bm \\theta}$}\\Big)$. Asymptotic\nconfidence intervals and tests of the parameters at the $p$th\nlevel can be obtained assuming that the ML estimator\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$ has approximately a normal multivariate distribution. \\\\\n\nFrom the EM algorithm, we can see that $ {\\cal E}_{-1 i}(\\mbox{${ \\bm \\theta}$}^{(k)})$ is inversely proportional to $d_i=|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}^{(k)}_p|\/\\sigma$. Hence, $u_i(\\mbox{${ \\bm \\theta}$}^{(k)})= {\\cal E}_{-1 i}(\\mbox{${ \\bm \\theta}$}^{(k)})$ can be\ninterpreted as a type of weight for the $i$th case in the estimates\nof $\\mbox{${\\bm \\beta}$}^{(k)}_p$, which tends to be small for outlying\nobservations. The behavior of these weights can be used as tools for\nidentifying outlying observations as well as for showing that we are considering a robust approach, as will be seen in Sections 4 and 5.\n\n\\section{Case-deletion measures} \\label{Sec Diagnostic}\nCase-deletion is a classical approach to study the effects of\ndropping the $i$th case from the data set. Let $\\mathbf{y}_{c}=(\\mathbf{y},\\mathbf{u})$ be the augmented data set, and a quantity\nwith a subscript ``$[i]$'' denotes the original one with the $i$th\nobservation deleted. Thus, The complete-data log-likelihood function\nbased on the data with the $i$th case deleted will be denoted by\n$\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{y}_{c[i]})$. Let\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top}_{p[i]},\n\\widehat{\\sigma^2}_{[i]})^{\\top}$ be the maximizer of the function\n{{ $Q_{[i]}(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\n\\textrm{E}_{\\scriptsize{\\widehat{\\mbox{${ \\bm \\theta}$}}}}\\left[\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{Y}_{c[i]})|\\mathbf{y}\n\\right] $}}, where $\\widehat{\\mbox{${ \\bm \\theta}$}}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top},\n\\widehat{\\sigma^2})^{\\top}$ is the ML estimate of $\\mbox{${ \\bm \\theta}$}$. To\nassess the influence of the $i$th case on $\\widehat{\\mbox{${ \\bm \\theta}$}}$, we\ncompare the difference between $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ and\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$. If the deletion of a case seriously influences\nthe estimates, more attention needs to be paid to that case.\nHence, if $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ is far from $\\widehat{\\mbox{${ \\bm \\theta}$}}$\nin some sense, then the $i$th case is regarded as influential. As\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ is needed for every case, the required\ncomputational effort can be quite heavy, especially when the sample\nsize is large. Hence, To calculate the case-deletion estimate $\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}$ of $\\mbox{${ \\bm \\theta}$}$, \\citep[see][]{zhu2001case} proposed the following one-step approximation based on the Q-function,\n\\begin{eqnarray}\\label{theta1}\n\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}= \\widehat{\\mbox{${ \\bm \\theta}$}}+ \\big\\{\n-\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{eqn Hessian Matrix and Grad}\n\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\displaystyle\\frac{\\partial^2\nQ(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})}{\\partial\\mbox{\\mbox{${ \\bm \\theta}$}}\\partial{\\mbox{${ \\bm \\theta}$}}^{\\top}}\n\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}} \\,\\,\\,\\, \\textrm{and}\n\\,\\,\\,\\,\n \\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\displaystyle\\frac{\\partial{{Q}_{[i]}(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\\partial{\\mbox{${ \\bm \\theta}$}}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}},\n\\end{eqnarray}\nare the Hessian matrix and the gradient vector evaluated at\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$, respectively. The Hessian matrix\nis an essential element in the method developed by \\cite{zhu2001case} to obtain the measures for\ncase-deletion diagnosis. For developing the case-deletion measures, we have to obtain the elements in (\\ref{theta1}), $\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ and $\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$. These formulas can be obtained quite easily from (\\ref{eqn qfunction}):\n\n\\begin{enumerate}\n\\item[1.] The components of $\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ are\n{\\small{\n\\begin{eqnarray*}\n\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})&=& \\frac{ \\partial{Q_{[i]}({\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\\partial{\\mbox{${\\bm \\beta}$}}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}} ={{\\frac{1}{\\widehat{\\sigma}}}} E_{1[i]}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\,\\, \\,\n\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\n\\frac{\\partial{Q_{[i]}({\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\n\\partial{\\sigma}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}}=-\\frac{1}{2\\widehat{\\sigma^2}}\nE_{2[i]}, \\label{sigma}\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray}\n E_{1[i]} & = & \\frac{1}{\\tau_{p}^{2}} \\sum_{j\\neq i} \\left[{\\cal E}_{-1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}})\\mathbf{x}_{j}-\\mathbf{x}_{j}\\vartheta_{p} \\right] \\,\\,\\,\\, \\textrm{ and} \\label{eqn E1i} \\\\\nE_{2[i]} & = & \\sum_{j\\neq i} \\left[3\n\\widehat{\\sigma}-\\frac{1}{\\tau_{p}^{2}} {\\cal\nE}_{-1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}_p})^{2}-2(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}_p})\\vartheta_{p}\n+ \\frac{1}{4}{\\cal\nE}_{1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})\\tau_{p}^{4}\\right].\n\\label{eqn E2i}\n\\end{eqnarray}}}\n\\item[2.] The elements of the second order partial derivatives of $Q(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ evaluated at $\\widehat{\\mbox{${ \\bm \\theta}$}}$ are\n{{\n\\begin{eqnarray*}\n\\ddot{Q}_{\\beta}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})& = & -\\frac{1}{\\widehat{\\sigma}\\tau_p^{2}} \\mathbf{X}^{\\top}D\\big(\\mbox{${ \\bm \\xi}$} _{-1}^{(k)}\\big)\\mathbf{X},\\nonumber\\\\\n\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\}&\n=& \\frac{3}{4\\widehat{\\sigma^2}} - \\frac{1}{2\\widehat{\\sigma^3}\\tau^2_p}\\Big[Q\\big(\\mbox{${\\bm \\beta}$},\\mbox{${ \\bm \\xi}$}^{(k)} _{-1}\\big)-2{\\bf 1}^{\\top}_n(\\mathbf{Y}-\\mathbf{X}\n\\mbox{${\\bm \\beta}$})\\vartheta_p+\\frac{\\tau_p^4}{4}{\\bf\n1}^{\\top}_n\\mbox{${ \\bm \\xi}$}^{(k)} _{1}\\Big]\n\\end{eqnarray*}}}\nand $\\ddot{Q}_{\\beta \\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\}\n= {\\bf 0}$.\n\\end{enumerate}\nIn the following result, we will obtain the\none-step approximation of\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top}_{p[i]},\n\\widehat{\\sigma}_{[i]})^{\\top}$, $i=1,\\ldots,n$ based on\n(\\ref{theta1}), viz., the relationships between the parameter\nestimates for the full data set and the data with the $i$th case\ndeleted.\n\\begin{theorem} \\label{the;1}\nFor the QR model defined in (\\ref{repHier1}) and (\\ref{repHier2}), the\nrelationships between the parameter estimates for full data set and\nthe data with the $i$th case deleted are as follows:\n\\begin{eqnarray*}\\label{aprox1}\n\\widehat{\\mbox{${\\bm \\beta}$}}^1_{p[i]}&=& \\widehat{\\mbox{${\\bm \\beta}$}}_p+ \\tau_p^{2} \\big(\\mathbf{X}^{\\top}D\\big(\\widehat{\\mbox{${ \\bm \\xi}$}}_{-1}\\big)\\mathbf{X}\\big)^{-1} \\textbf{E}_{1[i]}\\,\\,\\, \\, \\,{\\rm and }\\,\\, \\, \\, \\,\n\\widehat{\\sigma^2}^1_{[i]}= \\widehat{\\sigma^2} - \\frac{1}{2\\widehat{\\sigma^2}}\\Big(\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\Big)^{-1} E_{2[i]},\n\\end{eqnarray*}\nwhere $\\textbf{E}_{1[i]}$ and $E_{2[i]}$ are as in (\\ref{eqn E1i}) and (\\ref{eqn E2i}), respectively.\n\\end{theorem}\nTo asses the influence of the $i$th case on the ML estimate\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$, we compare $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$\nand $\\widehat{\\mbox{${ \\bm \\theta}$}}$ based on metrics, proposed by \\cite{zhu2001case}, for measuring the\ndistance between $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ and $\\widehat{\\mbox{${ \\bm \\theta}$}}$. For that, we consider here the following;\n\n\\begin{enumerate}\n\\item {\\it Generalized Cook distance}:\n\\begin{equation}\\label{GCD}\nGD_i=(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}-\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{ -\\ddot{\nQ}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}-\\widehat{\\mbox{${ \\bm \\theta}$}}),\n\\quad i=1,\\ldots,n.\n\\end{equation}\nUpon substituting (\\ref{theta1}) into (\\ref{GCD}), we obtain the\napproximation\n$$GD^1_i=\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\nAs $\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ is a diagonal matrix, one can obtain easily a type of Generalized Cook distance for parameters $\\mbox{${\\bm \\beta}$}$ and $\\sigma$, respectively, as follows\n$$GD^1_i(\\mbox{${\\bm \\beta}$})=\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}_{\\beta}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\n$$GD^1_i(\\sigma)=\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\n\n\\item {\\it Q-distance}: This measure of the influence of the $i$th case is based on the\n$Q$-distance function, similar to the likelihood distance $LD_i$\n\\citep{cook82}, defined as\n\\begin{equation}\\label{QD}\nQD_i=2\\big\\{Q(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})-Q(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}.\n\\end{equation}\nWe can calculate an approximation of the likelihood displacement\n$QD_i$ by substituting (\\ref{theta1}) into (\\ref{QD}), resulting\nin the following approximation $QD^{1}_i$ of $QD_i$:\n\\begin{equation*}\\label{QD1}\nQD^1_i=2\\big\\{Q(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})-Q(\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}.\n\\end{equation*}\n\\end{enumerate}\n\n\\section{Application}\\label{sec application}\n\nWe illustrate the proposed methods by applying them to the\nAustralian Institute of Sport (AIS) data, analyzed by Cook and\nWeisberg (1994) in a normal regression setting. The data set\nconsists of several variables measured in $n=202$ athletes (102\nmales and 100 females). Here, we focus on body mass index (BMI),\nwhich is assumed to be explained by lean body mass (LBM) and gender\n(SEX). Thus, we consider the following QR model:\n$$BMI_i=\\beta_0+\\beta_1 LBM_i+\\beta_2 SEX_i+\\epsilon_i,\\,\\,\\,\\,\\,i=1,\\ldots,202,$$\nwhere $\\epsilon_i$ is a zero $p$ quantile. This model can be fitted\nin the R software by using the package \\verb\"quantreg()\", where one\ncan arbitrarily use the BR or the LPQR algorithms. In order to\ncompare with our proposed EM algorithm, we carry out quantile\nregression at three different quantiles, namely $p= \\{0.1, 0.5,\n0.9\\}$ by using the ALD distribution as described in Section 2. The\nML estimates and associated standard errors were obtained by using\nthe EM algorithm and the observed information matrix described in\nSubsections 2.3, respectively. Table \\ref{table.application}\ncompares the results of our EM, BR and the LPQR estimates under the\nthree selected quantiles. The standard error of the LPQR estimates\nare not provided in the R package \\verb\"quantreg()\" and are not\nshown in Table \\ref{table.application}. From this table we can see\nthat estimates under the three methods only exhibit slight\ndifferences, as expected. However, the standard errors of our EM\nestimates are smaller than those via the BR algorithm. This suggests\nthat the EM algorithm seems to produce more accurate estimates of\nthe regression parameters at the $p$th level.\n\\begin{table}[ht!]\n\\centering\n\\caption{AIS data. Results of the parameter estimation via EM, Barrodale and Roberts (BR) and Lasso Penalized Quantile Regression (LPQR) algorithms for three selected quantiles.}\n{\\small{\n\\begin{tabular}{ccccccc}\n\n \\hline\\hline\n \\multicolumn{2}{c}{} & \\multicolumn{2}{c}{EM} & \\multicolumn{2}{c}{BR} & \\multicolumn{1}{c}{LPQR} \\\\\n\n $p$ & Parameter & MLE & SE & Estimative & SE &Estimative \\\\\n \\hline\n 0.1 & $\\beta_{0}$ & 9.3913 & 0.7196 & 9.3915 & 1.2631 & 9.8573 \\\\\n & $\\beta_{1}$ & 0.1705 & 0.0091 & 0.1705 & 0.0160 & 0.1647 \\\\\n & $\\beta_{2}$ & 0.8312 & 0.2729 & 0.8209 & 0.4432 & 0.6684 \\\\\n & $\\sigma$ & 0.2617 & 0.0252 & 1.0991 & ------ & 1.0959 \\\\\n \\hline\n 0.5 & $\\beta_{0}$ & 7.6480 & 0.8717 & 7.6480 & 1.1120 & 7.6480 \\\\\n & $\\beta_{1}$ & 0.2160 & 0.0116 & 0.2160 & 0.0159 & 0.2160 \\\\\n & $\\beta_{2}$ & 2.2499 & 0.3009 & 2.2226 & 0.4032 & 2.2226 \\\\\n & $\\sigma$ & 0.6894 & 0.0590 & 0.6894 & ------ & 0.6894 \\\\\n \\hline\n 0.9 & $\\beta_{0}$ & 5.8000 & 0.5887 & 5.8000 & 1.6461 & 6.0292 \\\\\n & $\\beta_{1}$ & 0.2700 & 0.0084 & 0.2700 & 0.0256 & 0.2678 \\\\\n & $\\beta_{2}$ & 3.9596 & 0.1937 & 3.9658 & 0.6203 & 3.8271 \\\\\n & $\\sigma$ & 0.3391 & 0.0258 & 1.2677 & ------ & 1.2767 \\\\\n \\hline\\hline\n\n\n\\end{tabular}\n}}\n \\label{table.application}%\n\\end{table}%\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.7]{perfilThetaAIS2.eps}~\n\\caption{AIS data: ML estimates and $95\\%$ confidence intervals for\nvarious values of $p$. \\label{fig:2b}}\n\\end{center}\n\\end{figure}\nTo obtain a more complete picture of the effects, a series of QR\nmodels over the grid $p=\\{0.1, 0.15,\\ldots, 0.95\\}$ is estimated.\nFigure \\ref{fig:2b} gives a graphical summary of this analysis. The\nshaded area depicts the $95\\%$ confidence interval from all the\nparameters. From Figure \\ref{fig:2b} we can observe some interesting\nevidences which cannot be detected by mean regression. For example,\nthe effect of the two variables (LBM and gender) become stronger for\nthe higher conditional quantiles, indicating that the BMI are\npositively correlated with the quantiles. The robustness of the\nmedian regression $(p=0.5)$ can be assessed by considering the\ninfluence of a single outlying observation on the EM estimate of\n$\\mbox{${ \\bm \\theta}$}$. In particular, we can assess how much the EM estimate of\n$\\mbox{${ \\bm \\theta}$}$ is influenced by a change of $\\delta$ units in a single\nobservation $y_{i}$. Replacing $y_{i}$ by\n$y_{i}(\\delta)=y_{i}+\\delta sd(\\mathbf{y})$, where $sd(.)$ denotes\nthe standard deviation. Let $\\widehat{\\beta}_{j}(\\delta)$ be the EM\nestimates of $\\beta_j$ after contamination, $j=1,2,3$. We are\nparticularly interested in the relative changes\n$|(\\widehat{\\beta}_{j}(\\delta)-\\widehat{\\beta}_{j})\/\\widehat{\\beta}_{j}|$.\nIn this study we contaminated the observation corresponding to\nindividual $\\left\\{\\#146\\right\\}$ and for $\\delta$ between 0 and 10.\nFigure \\ref{fig:change} displays the results of the relative\nchanges of the estimates for different values of $\\delta$. As\nexpected, the estimates from the median regression model are less\naffected by variations on $\\delta$ than those of the mean\nregression. Moreover, Figure \\ref{fig:2c} shows the Q-Q plot and envelopes for mean and\nmedian regression, which are obtained based on the distribution of\n$W_i$, given in (\\ref{Wi}), that follows $\\exp(1)$ distribution. The lines in these figures represent the 5th\npercentile, the mean and the $95$th percentile of $100$ simulated\npoints for each observation. These figures clearly show that the\nmedian regression\ndistribution provides a better-fit than the standard mean regression to the AIS data set.\\\\\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.35]{meanVsmedianbeta0.ps}~\\includegraphics[scale=0.35]{meanVsmedianbeta1.ps}~\\includegraphics[scale=0.35]{meanVsmedianbeta2.ps}\n\\caption{Percentage of change in the estimation of $\\beta_0$,\n$\\beta_1$ and $\\beta_2$ in comparison with the true value, for\nmedian $(p=0.5)$ and mean regression, for different contaminations\n$\\delta$. \\label{fig:change}}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!tb]\n\\begin{center}\n\\includegraphics[scale=0.55]{envelopesAIS.ps}~\n\\caption{AIS data: Q--Q plots and simulated envelopes for mean and\nmedian regression.\\label{fig:2c}}\n\\end{center}\n\\end{figure}\nAs discussed at the end of Section 2.3 the estimated distance\n$\\widehat{d}_i=|y_i-\\mathbf{x}^{\\top}_i\\widehat{\\mbox{${\\bm \\beta}$}}_p|\/\\widehat{\\sigma}$\ncan be used efficiently as a measure to identify possible outlying\nobservations. Figure \\ref{fig:mahal}(left panel) displays the index\nplot of the distance $d_i$ for the median regression model\n$(p=0.5)$. We see from this figure that observations {\\#75, \\#162,\n\\#178 and \\#179} appear as possible outliers. From the EM-algorithm,\nthe estimated weights $u_i(\\widehat{\\mbox{${ \\bm \\theta}$}})={\\cal E}_{\nsi}(\\widehat{\\mbox{${ \\bm \\theta}$}})$ for these observations are the smallest ones\n(see right panel in Figure \\ref{fig:mahal}), confirming the\nrobustness aspects of the maximum likelihood estimates against\noutlying observations of the QR models. Thus, larger $d_i$ implies a\nsmaller $u_i(\\widehat{\\mbox{${ \\bm \\theta}$}})$, and the estimation of $\\mbox{${ \\bm \\theta}$}$\ntends to give smaller weight to outlying observations in the sense\nof the distance $d_i$.\n\nFigure \\ref{fig:1b} shows the estimated quartiles of two levels of\ngender at each LBM point from our EM algorithm along with the\nestimates obtained via mean regression. From this figure we can see\nclear attenuation in $\\beta_1$ due to the use of the median\nregression related to the mean regression. It is possible to observe\nin this figure some atypical individuals that could have an\ninfluence on the ML estimates for different values of quantiles. In\nthis figure, the individuals $\\#75, ~\\#130, ~\\#140 ~\\#162, ~\\#160$\nand $\\#178$ were marked since they were detected as potentially\ninfluential.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.47]{di.ps}~\\includegraphics[scale=0.5]{weights_di.ps}\n\\caption{AIS data: Index plot of the distance $d_i$ and the\nestimated weights $u_i$.\\label{fig:mahal}}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.5]{Female.ps}~\\includegraphics[scale=.5]{Man.ps}\n\\caption{AIS data: Fitted regression lines for the three selected\nquantiles along with the mean regression line. The influential\nobservations are numbered. \\label{fig:1b}}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\centering \\hspace {1cm}\\centering \\\\\n\\includegraphics[scale=0.35]{gdi01.ps}~\\includegraphics[scale=0.35]{gdi05.ps}~\\includegraphics[scale=0.35]{gdi09.ps}\\\\\n\\includegraphics[scale=0.35]{QDip01.ps}~\\includegraphics[scale=0.35]{QDip05.ps}~\\includegraphics[scale=0.35]{QDip09.ps}\\\\\n\\caption{Index plot of (first row) approximate likelihood distance\n$GD^1_i$. (second row). Index plot of approximate likelihood\ndisplacement $QD^1_i$. The influential observations are\nnumbered.\\label{pert3} }\n\\end{center}\n\\end{figure}\n\n\n\nIn order to identify influential observations at different quantiles\nwhen some observation is eliminated, we can generate graphs of the\ngeneralized Cook distance $GD^{l}_i$, as explained in Section\n\\ref{Sec Diagnostic}. A high value for $GD^{l}_{i}$ indicates that\nthe $i$th observation has a high impact on the maximum likelihood\nestimate of the parameters. Following \\cite{Barros}, we can use\n$2(p+1)\/n$ as benchmark for the $GD^{l}_{i}$ at different quantiles.\nFigure \\ref{pert3} (first row) presents the index plots of\n$GD^{l}_i$. We note from this figure that, only observation $\\#140$\nappears as influential in the ML estimates at $p=0.1$ and\nobservations $\\#75, \\#178$ as influential at $p=0.5$, whereas\nobservations $\\#75,\\#162, \\#178$ and $\\#179$ appear as influential\nin the ML estimates at $p=0.9$. Figure \\ref{pert3} (second row)\npresents the index plots of $QD^1_i$. From this figure, it can be\nnoted that observations $\\#76,\\#130, \\#140$ appear to be influential\nat $p=0.1$, whereas observations $\\#75,\\#162$ and $\\#178$ seem to be\n influential in the ML estimates at $p=0.1$, and in addition\nobservation $\\#179$ appears to be influential at $p=0.9$.\n\n\n\\section{Simulation studies} \\label{sec simulation study}\nIn this section, the results from two simulation studies are\npresented to illustrate the performance of the proposed method.\n\n\\subsection{Robustness of the EM estimates (Simulation study 1)}\n\nWe conducted a simulation study to assess the performance of the\nproposed EM algorithm, by mimicking the setting of the AIS data by\ntaking the sample size $n = 202$. We simulated data from the model\n\n\\begin{equation}\ny_{i}=\\beta_{1} + \\beta_{2}x_{i2} + \\beta_{3}x_{i3} + \\epsilon_{i},\n\\,\\,\\,\\,\\,\\,\\,\\, i=1,\\ldots,202, \\label{simulation_1}\n\\end{equation}\nwhere the $x_{ij}'s$ are simulated from a uniform distribution\n(U(0,1)) and the errors $\\epsilon_{ij}$ are simulated from four\ndifferent distributions: $(i)$ the standard normal distribution\n$N(0,1)$, $(ii)$ a Student-t distribution with three degrees of\nfreedom, $t_{3}(0,1)$, $(iii)$ a heteroscedastic normal\ndistribution, $(1+x_{i2})N(0,1)$ and, $(iv)$ a bimodal mixture\ndistribution $0.6t_3(-20,1)+0.4t_3(15,1)$. The true values of the\nregression parameters were taken as $\\beta_1=\\beta_2=\\beta_3=1$. In\nthis way, we had four settings and for each setting we generated\n$10000$ data sets.\n\n\nOnce the simulated data were generated, we fit a QR model, with $p=\n0.1,\\, 0.5$ and $0.9$, under Barrodale and Roberts (BR), Lasso\n(Lasso) and EM algorithms by using the \"quantreg()\" package and our\n\\verb\"ALDqr()\" package, from the R language, respectively. For the\nfour scenarios, we computed the bias and the square root of the\nmean square error (RMSE), for each parameter over the $M=10,000$\nreplicas. They are defined as:\n\\begin{eqnarray}\nBias(\\gamma) &=& \\overline{\\widehat{\\gamma}}-\\gamma\\label{bias} \\,\\, {\\rm and}\\, \\, \\, \\,\nRMSE(\\gamma) = \\sqrt{SE(\\gamma)^2 + Bias(\\gamma)^2}\\label{EQM}\n\\end{eqnarray}\nwhere $\\overline{\\widehat{\\gamma}} =\n\\frac{1}{M}\\sum_{i=1}^{M}\\widehat{\\gamma}_i$ and $SE(\\gamma)^2 =\n{\\frac{1}{M-1}\\sum_{i=1}^{M}\\lp\\widehat{\\gamma}_i -\n\\overline{\\widehat{\\gamma}}\\rp^2},$ with $\\gamma =\n\\beta_1,\\beta_2,\\beta_3$ or $\\sigma$, $\\widehat{\\gamma}_i$ is the\nestimate of $\\gamma$ obtained in replica $i$ and $\\gamma$ is the\ntrue value. Table \\ref{table.simul1} reports the simulation results for $p =\n0.1,\\, 0.5$ and $0.9$. We observe that the EM yields lower biases\nand RMSE than the other two estimation methods under all the\ndistributional scenarios. This finding suggests that the EM would\nproduce better results than other alternative methods typically used\nin the literature of QR models.\n\\begin{table}[htbp!]\n \\centering\n \\caption{Simulation study. Bias and root mean-squared\n error (RMSE) of $\\mbox{${\\bm \\beta}$}$ under different error distributions. The estimates under Barrodale and Roberts (BR)\nand Lasso (Lasso) algorithms were obtained by the \"quantreg()\"\npackage from the R language.} {\\footnotesize{\n\\begin{tabular}{lccccccc}\n\n \\hline\\hline\n \\multicolumn{2}{c}{} & \\multicolumn{2}{c}{$\\beta_{1}$} & \\multicolumn{2}{c}{$\\beta_{2}$} & \\multicolumn{2}{c}{$\\beta_{3}$} \\\\\n\n \\hline\n Method & $p$ & Bias & RMSE & Bias & RMSE & Bias & RMSE \\\\\n \\hline\n $\\epsilon \\sim N(0,1)$ & \\multicolumn{7}{c}{} \\\\\n BR & 0.1 & -1.2639 & 1.3444 & 0.0076 & 0.5961 &-0.0030 & 0.5934 \\\\\n & 0.5 & 0.0064 & 0.3376 & -0.0048 & 0.4390 &-0.0051 & 0.4453 \\\\\n & 0.9 & 1.2640 & 1.3460 & 0.0030 & 0.6051 & 0.0069 & 0.6039 \\\\\n\n LPQR & 0.1 & -0.9664 & 1.0464 & -0.3072 & 0.6165 &-0.3110 & 0.6187 \\\\\n & 0.5 & 0.1474 & 0.3628 & -0.1463 & 0.4534 &-0.1462 & 0.4576 \\\\\n & 0.9 & 1.5901 & 1.6460 & -0.3164 & 0.6173 &-0.3076 & 0.6179 \\\\\n\n EM & 0.1 & -1.2551 & 1.3362 & -0.0055 & 0.5964 &-0.0090 & 0.6020 \\\\\n & 0.5 & 0.0040 & 0.3286 & -0.0050 & 0.4332 &-0.0031 & 0.4363 \\\\\n & 0.9 & 1.2694 & 1.3484 & -0.0071 & 0.6019 &-0.0120 & 0.5955 \\\\\n \\hline\n $\\epsilon \\sim t_{3}(0,1)$ & \\multicolumn{7}{c}{} \\\\\n BR & 0.1 & -1.2446 & 1.3364 & -0.0290 & 0.6274 &-0.0313 & 0.6259 \\\\\n & 0.5 & 0.1049 & 0.4870 & 0.1213 & 0.6714 & 0.1123 & 0.6708 \\\\\n & 0.9 & 2.3618 & 2.8408 & 1.0056 & 2.4928 & 0.9459 & 2.4332 \\\\\n\n LPQR & 0.1 & -0.9315 & 1.0219 & -0.3478 & 0.6422 &-0.3412 & 0.6354 \\\\\n & 0.5 & 0.3007 & 0.5410 & -0.0928 & 0.6310 &-0.0831 & 0.6237 \\\\\n & 0.9 & 3.0443 & 3.2880 & 0.1911 & 1.6375 & 0.2231 & 1.6601 \\\\\n\n EM & 0.1 & -1.2287 & 1.3213 & -0.0402 & 0.6209 &-0.0374 & 0.6265 \\\\\n & 0.5 & 0.0965 & 0.4866 & 0.1352 & 0.6789 & 0.1304 & 0.6758 \\\\\n & 0.9 & 2.3781 & 2.8459 & 0.9464 & 2.4082 & 0.9264 & 2.4167 \\\\\n \\hline\n $\\epsilon \\sim (1+x_{2})N(0,1)$ & \\multicolumn{7}{c}{} \\\\\n BR & 0.1 & -1.2869 & 1.4256 & 0.0130 & 0.8706 &-1.2554 & 1.5381 \\\\\n & 0.5 & -0.0051 & 0.4468 & 0.0049 & 0.6336 & 0.0061 & 0.6509 \\\\\n & 0.9 & 1.2868 & 1.4259 & 0.0018 & 0.8686 & 1.2307 & 1.5256 \\\\\n\n LPQR & 0.1 & -1.1393 & 1.2272 & -0.3694 & 0.7773 &-1.1450 & 1.2756 \\\\\n & 0.5 & 0.1834 & 0.4520 & -0.1906 & 0.6193 &-0.1963 & 0.6304 \\\\\n & 0.9 & 1.6972 & 1.7933 & -0.3621 & 0.7925 & 0.7494 & 1.1587 \\\\\n\n EM & 0.1 & -1.2772 & 1.4140 & 0.0051 & 0.8646 &-1.2341 & 1.5195 \\\\\n & 0.5 & 0.0954 & 0.4892 & 0.1289 & 0.6724 & 0.1316 & 0.6694 \\\\\n & 0.9 & 1.2599 & 1.3987 & 0.0076 & 0.8723 & 1.2488 & 1.5315 \\\\\n \\hline\n $\\epsilon \\sim 0.6t_3(-20,1)+0.4t_3(15,1)$ & \\multicolumn{7}{c}{} \\\\\n BR & 0.1 & -1.2350 & 1.3268 & -0.0395 & 0.6160 &-0.0396 & 0.6192 \\\\\n & 0.5 & 0.1029 & 0.4896 & 0.1214 & 0.6780 & 0.1212 & 0.6741 \\\\\n & 0.9 & 2.3857 & 2.8737 & 0.9657 & 2.4574 & 0.9558 & 2.4585 \\\\\n\n LPQR & 0.1 & -0.9664 & 1.0464 & -0.3072 & 0.6165 &-0.3110 & 0.6187 \\\\\n & 0.5 & 0.1474 & 0.3628 & -0.1463 & 0.4534 &-0.1462 & 0.4576 \\\\\n & 0.9 & 1.5901 & 1.6460 & -0.3164 & 0.6173 &-0.3076 & 0.6179 \\\\\n\n EM & 0.1 & -0.9327 & 1.0201 & -0.3491 & 0.6433 &-0.3355 & 0.6372 \\\\\n & 0.5 & 0.2880 & 0.5343 & -0.0745 & 0.6216 &-0.0717 & 0.6159 \\\\\n & 0.9 & 3.0624 & 3.3102 & 0.1702 & 1.6627 & 0.2221 & 1.6575 \\\\\n \\hline\\hline\n\n\\end{tabular}\n}}\n\n \\label{table.simul1}%\n\\end{table}%\n\n\n\\subsection{Asymptotic properties (Simulation study 2)} \\label{sec simulation study 2}\n\nWe also conducted a simulation study to evaluate the finite-sample\nperformance of the parameter estimates. We generated artificial\nsamples from the regression model (\\ref{simulation_1}) with\n$\\beta_1=\\beta_2=\\beta_3=1$ and $x_{ij}\\sim U(0,1)$. We chose\nseveral distributions for the random term $\\epsilon_i$ a little\ndifferent than the simulation study 1, say, $(i)$ normal\ndistribution $N(0,2)$ (N1), $(ii)$ a Student-t distribution\n$t_{3}(0,2)$ (T1), $(iii)$ a heteroscedastic normal distribution,\n$(1+x_{i2})N(0,2)$ (N2) and, $(iv)$ a bimodal mixture distribution\n$0.6t_3(-20,2)+0.4t_3(15,2)$ (T2). Finally, the sample sizes were\nfixed at $n = 50, 100, 150, 200, 300,$ $400, 500, 700$ and $800$.\n\nFor each combination of parameters and sample sizes, $10000$ samples\nwere generated under the four different situations of error\ndistributions (N1, T1, N2, T2). Therefore, 36 different simulation\nruns are performed. Once all the data were simulated, we fit the QR\nmodel with $p=0.5$ and the bias (\\ref{bias}) and the square root of\nthe mean square error (\\ref{EQM}) were recorded. The results are\nshown in Figure \\ref{fig:77a}. We can see a pattern of convergence\nto zero of the bias and MSE when $n$ increases. As a general rule,\nwe can say that bias and MSE tend to approach to zero when the\nsample size increases, indicating that the estimates based on the\nproposed EM-type algorithm do provide good asymptotic properties.\nThis same pattern of convergence to zero is repeated considering\ndifferent levels of the quantile $p$.\n\n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.35]{bias1_beta1.ps}~\\includegraphics[scale=0.35]{RMSE1_beta1.ps}\\\\\n\\includegraphics[scale=0.35]{bias2_beta2.ps}~\\includegraphics[scale=0.35]{RMSE1_beta2.ps}\\\\\n\\includegraphics[scale=0.35]{bias3_beta3.ps}~\\includegraphics[scale=0.35]{RMSE1_beta3.ps}\\\\\n\\caption{Simulation study 2. Average bias (first column) and average\nMSE (second column) of the estimates of $\\beta_1$,$\\beta_2$,\n$\\beta_3$ with $p=0.5$ (median regression), where $N1=N(0,2)$,\n$T1=t_3(0,2)$, $N2=(1+x_2)N(0,2)$ and\n$T2=0.6t_3(-20,2)+0.4t_3(15,2)$ .\\label{fig:77a}}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\nWe have studied a likelihood-based approach to the estimation of the\nQR based on the asymmetric Laplace distribution (ALD). By utilizing\nthe relationship between the QR check function and the ALD, we cast\nthe QR problem into the usual likelihood framework. The mixture\nrepresentation of the ALD allows us to express a QR model as a\nnormal regression model, facilitating the implementation of an EM\nalgorithm, which naturally provides the ML estimates of the model\nparameters with the observed information matrix as a by product. The\nEM algorithm was implemented as part of the R package \\textit{ALDqr()}. We hope that by making the code of our method\navailable, we will lower the barrier for other researchers to use\nthe EM algorithm in their studies of quantile regression. Further,\nwe presented diagnostic analysis in QR models, which was based on\nthe case-deletion technique suggested by \\cite{zhu2001case} and\n\\cite{ZhuLee2001}, which are the counterparts for missing data\nmodels of the well-known ones proposed by \\cite{cook77} and\n\\cite{cook86}. The simulation studies demonstrated the superiority\nof the proposed methods to the existing methods, implemented in the\npackage \\verb\"quantreg()\". We applied our methods to a real data set\n(freely downloadable from R) in order to illustrate how the\nprocedures can be used to identify outliers and to obtain robust ML\nparameter estimates. From these results, it is encouraging that the\nuse of ALD offers a better alternative in the analysis of QR models.\n\nFinally, the proposed methods can be extended to a more general\nframework, such as, censored (Tobit) regression models, measurement\nerror models, nonlinear regression models, stochastic volatility\nmodels, etc and should yield satisfactory results at the expense of\nadditional complexity in implementation. An in-depth investigation\nof such extensions is beyond the scope of the present paper, but\nthese are interesting topics for further research.\n\n\n\n\\section*{Acknowledgements}\nThe research of V. H. Lachos was supported by Grant 305054\/2011-2\nfrom Conselho Nacional de Desenvolvimento Cient\\'{\\i}fico e Tecnol\\'{o}gico\n(CNPq-Brazil) and by Grant 2014\/02938-9 from Funda\\c c\\~ao de\nAmparo \\`{a} Pesquisa do Estado de S\\~ao Paulo (FAPESP-Brazil).\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{1}\n\nThe low dimensional microscopic dynamics of heat conduction has\nbeen an attractive question since early year last century. Much\nmore attention has been payed to this problem in the last two\ndecades due to the dramatic achievement in the application of\nminiaturized devices \\cite{Forsman,Taillefer,Tighe,Hone,Kim,Zhang}\nwhich can be described by 1D or 2D models. More and more numerical\ncalculations are focused on the minimal requirements for a\ndynamical model where Fourier's law holds or not\n\\cite{Lepri_1,Casati,Posch,Kaburaki,Lepri_2,Alonso_1,Moasterio,Li_2,Li_3,Alonso_2}.\nA convergent heat conductivity was shown in ding-a-ling model\n\\cite{Casati,Posch} which is chaotic. The studies on Lorentz gas\nmodel \\cite{Alonso_1,Moasterio} (the circular scatters are\nperiodically placed in the channel) of which the Lyapunov exponent\nis nonzero gave a finite heat conductivity which fulfils the\nFourier law explicitly. Hence, chaos was ever regarded as an\nindispensable factor to normal heat conduction. Whereas, the FPU\nmodel \\cite{Kaburaki,Lepri_2} indicated that the chaotic behavior\nis not sufficient to arrive at normal heat conduction. Recently, a\nseries of billiard gas models \\cite{Li_2,Li_3,Alonso_1,Alonso_2}\nwere devoted to explore the normal heat conduction of quasi 1D\nchannels with zero Lyapunov exponent. However, the role of chaos\nin heat conduction has not been well understood. Additionally, the\nexponential stability and instability frequently coexist in the\nscatters of real system. Thus the model with various degree of\nchaos deserves further investigation from the microscopic point of\nview, and it will be also interesting to explore non-equilibrium\nstationary states and to determine the steady temperature field.\n\nIn this paper, we focus on the quasi one-dimensional gas model\nwhich is closer to the real system. The scatters in our model are\nthe isosceles right triangle with a segment of circle substituting\nfor the right angle. In this case the edges of scatters are the\ncombination of line and a quarter of circle. Such a channel is of\nchaos, which indicates exponential instability of microscopic\ndynamics. Our paper is organized as follows. In section \\ref{2},\nwe introduce the model and investigate the degree of chaos for\nvarious channels with different arc-radius and channel height. In\nsection \\ref{3}, we study the heat transport behavior and the\ncorresponding diffusive behavior by changing the radius of top arc\nand channel height. In section \\ref{4}, we investigate the\nnon-equilibrium stationary state and determine the steady\ntemperature field numerically. We also analyze the dependence on\ndiffusion exponent $\\beta$ and system size $N$ of temperature\nprofile theoretically. In section \\ref{5}, we discuss the relation\nbetween our work and others and summarize our main conclusions.\n\n\n\\section{the model}\\label{2}\n\nWe consider a billiard gas channel with two parallel walls and a\nseries of scatters. The channel consists of $N$ replicate cells of\nlength $l$ and height $h$, and each cell is placed with two\nscatters as shown in Fig. \\ref{fig:systematic}. The scatter's\ngeometry is an isosceles right triangle of hypotenuse $a$ whose\nvertex angle is replaced by a segment of circle with radius $R$\nwhich is tangential to the two sides of the triangle. At the two\nends of the channel are two heat baths with temperature $T_{L}$\nand $T_{R}$. Noninteracting particles coming from these heat baths\nare scattered by the walls and the straight lines as well as the\narcs of the scatters in the channel.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.46\\textwidth]{fig_1}%\n\\caption{\\label{fig:systematic} The channel with $N$ replicate\ncells. Here, $l=2.2$, $a=1.2$, $h$ changes from $1.0$ to $0.27$,\nand $R$ from $0$ to $0.848528$ to ensure a quarter of circle\nalways.}\n\\end{figure}\n\nFor such a channel, the degree of chaos can be characterized\nqualitatively by Poincare surface of section (SOS) \\cite{Vega}.\nSuppose we take out one unit cell from the channel and close the\ntwo ends by straight walls. Then the problem becomes a billiard\nproblem. A particle moves within the cell and makes elastic\ncollision with the mirror-like boundary. We investigate the\nsurfaces of section ($s$, $v_\\tau$) under different initial\nconditions. $s$ is the length along the billiard boundary from the\ncollision point to the reference point. $v_\\tau$ is the tangential\ncomponent of velocity with respect to the boundary at that point.\nThe filling behavior of phase space shown in Fig. \\ref{fig:sos}\nindicates the degree of chaos. In case I, it's non chaotic. The\nsurface of section is regular and periodic as shown in Fig.\n\\ref{fig:sos}(a). As the radius $R$ is increased from (a) to (e),\nthe motion becomes more complex and the map becomes dense with\npoints except some regular islands. In Fig. \\ref{fig:sos}(f), the\nregular parts disappear which indicates strong chaos.\n\n\\begin{figure}[h]\n\\includegraphics[1.5cm,1cm][8cm,12cm]{fig_2}\n\\caption{\\label{fig:sos} Poincare surface-of-section of the\nbilliard problem. The billiard starts with an incident angle $0.8$\nand unit velocity. (a) $R=0$, $h=1.0$; (b) $R=0.001$, $h=1.0$; (c)\n$R=0.015$, $h=1.0$; (d) $R=0.1$, $h=1.0$; (e) $R=0.848528$,\n$h=1.0$; (f) $R=0.848528$, $h=0.27$. }\n\\end{figure}\n\n\n\\section{the heat transport and diffusion behavior}\\label{3}\n\nTo study the heat conduction of the model, the heat flux is\ninvestigated firstly. In calculating the heat flux, we follow Ref.\n\\cite{Alonso_1}. For simplicity, the particles from the two heat\nbaths are supposed to have definite velocities $\\sqrt{2T_{L}}$ and\n$\\sqrt{2T_{R}}$ respectively \\cite{Li_2}. We consider one particle\ncolliding with a heat bath during a period of simulating time. The\nenergy exchange $(\\Delta$E$)_{j}$ at the $j$th collision with the\nheat bath is defined as\n\\begin{equation}\n(\\Delta E)_{j}=E_{h}-E_{p},\n\\end{equation}\nwhere $E_{h}$ denotes for the energy of particle taken from the\nbath and $E_{p}$ for that carried in the channel. For $M$\ncollisions between the particle and the bath wall during the\nsimulation time $t$, the heat flux is given by\n\\begin{equation}\nJ_{1} (N)=\\frac{\\sum_{j=1}^M(\\Delta E)_{j}}{t}.\n \\label{eq:flux}\n\\end{equation}\n\nAs there is one heat carrier in each cell and the channel has $N$\nreplicas, there are $N$ particles in the whole channel. Summing\nover the heat flux of $N$ heat carriers, we have $J_{N} (N)=N\nJ_{1} (N)$. Meanwhile, the Fourier's law reads\n\\begin{equation}\nJ_{N} (N)=-\\kappa\\frac{d T}{d x}=\\kappa\\frac{T_{L}-T_{R}}{Nl},\n\\label{eq:fourier}\n\\end{equation}\nwhere $\\kappa$ refers to the heat conductivity which is determined\nby Eqs. (\\ref{eq:flux}) and (\\ref{eq:fourier}),\n\\begin{equation}\n\\kappa\\thicksim N^{2}J_{1} (N).\n\\end{equation}\n\nWe consider various cases by changing the radius $R$ of the top\narc of the scatters to investigate their effects on heat\nconduction. The heat flux of a single particle versus system size\nshown in Fig. \\ref{fig:heatflow} are four typical cases. Namely,\ncase I: the $ {}_{\\blacksquare}$ studies for $R=0,\\,h=1.0$; case\nI\\!I: the $\\circ$ for $R=0.001,\\,h=1.0$; case I\\!I\\!I: the\n$\\vartriangle$ for $R=0.848528,\\,h=1.0$, and case I\\!V: the\n$\\triangledown$ for $R=0.848528,\\,h=0.27$, respectively. The total\ncell numbers are chosen as $N=20$, $40$, $80$, $160$, $320$, $640$\nand $1280$ respectively. After a sufficient long period of\nsimulation time, the heat flux approaches to a constant value.\nClearly, the value of heat flux decreases with increasing $R$ for\nthe same size. Remarkably, there is $20$ times difference of heat\nflux between case I\\!I\\!I and case I\\!V, which indicates that\nsmaller height suppresses the heat flux greatly. Thus, it appears\nthat the value of heat flux can be adjusted in this way in\ndesigning heat-control devices. Furthermore, our calculations show\nthat the heat flux dependence on $N$ exhibits faint non-linearity\nalthough the curve looks linear for all cases except case I\\!V in\nthe log-log scale .\n\n\\begin{figure}[h]\n\\includegraphics[2cm,1cm][8cm,6cm]{fig_3}\n\\caption{\\label{fig:heatflow} The heat flux of a single particle\nversus system size($N=20$, $40$, $80$, $160$, $320$, $640$ and\n$1280$) with the divergence exponent of heat conductivity\n$\\alpha=0.721$, $0.526$, $0.101$, $0.009$ for four typical cases\nrespectively (left panel). The ratio of heat flux\n$J_{1}(N)\/J_{1}(2N)$ for different system sizes (right panel). }\n\\end{figure}\n\nIn order to observe the deviation from the line, which arises from\nthe finite-size effect, we calculate the ratio of heat flux versus\nsystem size for various radius. The data for the aforementioned\nfour cases are plotted in the right panel of Fig.\n\\ref{fig:heatflow}, from which one can see that both the\nincreasing of system size and of the arc radius bring the ratio an\nupward tendency to the value $4$ which ensures the Fourier law. In\ncase I where $R=0$, there is only a slight increase for the ratio\naround $2.4$. Whereas, the ratio rises drastically along with the\nincreasing of systems size even if the radius $R$ is merely\n$0.001$ (the case I\\!I). When $R=0.848528$ (the case I\\!I\\!I), the\nscatters become a full segment of quarter circle. The ratio also\nrises drastically at first and gradually after $N>160$ in this\ncase. In both cases I\\!I and I\\!I\\!I, it seemly approaches to\ndistinct asymptotic values which are all different from that for\nnormal conduction. This implies that smaller degree of chaos is\ninsufficient to bring about a normal heat conduction although the\nincreasingly chaotic degree makes the divergent exponent of heat\nconduction smaller. In case I\\!V, we maintain the scatters at\nradius $R=0.848528$ and reduce the height $h$ from $1.0$ to\n$0.27$. In this strongly chaotic case, the ratio fluctuates around\nthe value $4$ (dotted line) which means that Fourier law is\nobeyed.\n\nIt is known that the normal heat conduction happens when\n$\\alpha=0$, which indicates the heat conductivity is independent\nof system size, and the anomalous heat conduction corresponds to\nthe case of $\\alpha>0$. The heat conductivity $\\kappa$ we\ncalculated can be given by $\\kappa\\thicksim N^{\\alpha}$ with\n$\\alpha \\gtrsim 0$ despite the heat-flux ratio has a different\nincrease in asymptotic value for all cases (except case I\\!V).\n\nWe calculate $\\alpha$ at the range of system sizes $N$ from $20$\nto $1280$ by averaging over many realizations for various radius\n$R$ at fixed channel height $h=1.0$, and the plot of the\ndependence of $\\alpha$ on $R$ is shown in Fig. \\ref{fig:alpha}(a).\nOne can see that $\\alpha$ descends from $0.721$ through $0.526$ to\n$0.092$ if $R$ increases from $0$ to $0.848528$ for a fixed height\n$h=1.0$. Clearly, the $\\alpha$ descends rapidly for small radius\n({\\it e.g. } $R=0.001$ in case I\\!I) and slowly for larger ones. This\nillustrates that the appearance of arc on the top of the scatter\nsuppresses the divergent exponent $\\alpha$ drastically. If the\nchannel height $h$ for fixed $R=0.848528$ is changed from $1.0$ to\n$0.27$, the $\\alpha$ is found to diminish to $0.009$. Therefore\nthe $\\kappa$ appears to be independent of system size and the\nFourier law holds in this case.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.52\\textwidth]{fig_4}\n\\caption{\\label{fig:alpha} (color on line) (a) Conductivity\ndivergence exponent $\\alpha$ versus circular radius $R$. The\n${}_{^\\blacksquare}$, refers to the magnitude of $\\alpha$ for\n$h=1.0$, $R=0,\\,0.001,\\,0.05,\\,0.1,\\,0.2,\\,0.4,\\,0.6,$ and\n$0.848528$; the $\\blacktriangle$ for $h=0.5$ and $R=0.848528$; the\n$\\bigstar$ for $h=0.27$ and $R=0.848528$ has the value of $0.009$.\n(b) The relation between $\\beta$ and $\\alpha$, where the circle is\nthe numerical result, the red line is of $\\alpha=2-2\/\\beta$\n\\cite{Li_4} and the dashed line is the result of Ref.\n\\cite{Denisov}. (c) Log-log plot of mean square displacement\n$\\langle x(t)^{2}\\rangle$ versus time $t$. The curves from top to\nbottom on the right correspond to cases I, I\\!I, I\\!I\\!I and I\\!V\nrespectively. The ensemble has $10^5$ particles starting from the\ncenter of the channel at time $t=0$ where $x=0$ with the unit\nvelocity and random direction.}\n\\end{figure}\n\nSince the characteristic of heat transport is found being closely\nrelated to the diffusion\nbehavior\\cite{Alonso_2,Li_2,Li_3,Li_4,Li_5,Denisov}, we\ninvestigate the diffusion property for the above cases\nsubsequently. For a particle starting at the origin at time $t=0$\nand diffusing along $x$ direction, the mean square displacement\n$\\langle(x(t)-x(0))^{2}\\rangle$ characterizes its diffusion\nbehavior. For normal diffusion, the Einstein relation of\n$\\langle(x(t)-x(0))^{2}\\rangle = Dt$ holds, where $D$ is diffusion\ncoefficient. If the mean square displacement does not grow\nlinearly in time, {\\it {i.e.}}, $\\langle(x(t)-x(0))^{2}\\rangle =\nDt^{\\beta}$, we refer to anomalous diffusion. Recently, the\nconnection between anomalous diffusion and corresponding heat\nconduction in 1D system was discussed hotly\n\\cite{Li_4,Li_5,Denisov}. We plot the mean square displacement\nversus time $t$ in Fig. \\ref{fig:alpha}(c) for the aforementioned\nfour cases. Note that $10^{5}$ particles were put at the center of\nthe channel where $x=0$ with unit velocity and random direction in\nthe simulations. The top solid line and the bottom short-dash-dot\nline are precisely straight in the whole simulation period\n($t=10^5$), which correspond to the case I (non-chaotic) and case\nI\\!V (strong chaotic), respectively. We obtain $\\beta=1.628$ for\nthe case I which corresponds to $\\alpha=0.721$, and $\\beta=1.001$\nto $\\alpha=0.009$ for the case I\\!V. Beyond these two cases do the\ncurves keep asymptotically linear at large time $t$ with diffusion\nexponent $\\beta$ between the values of above two cases. The best\nfits of the slope give $\\beta=1.357$ which corresponds to\n$\\alpha=0.526$ for case I\\!I and $\\beta=1.050$ to $\\alpha=0.101$\nfor case I\\!I\\!I, respectively. The relation between divergent\nexponent $\\alpha$ and diffusion exponent $\\beta$ fits the relation\nof $\\alpha=2-2\/\\beta$ proposed by Li and Wang in Ref. \\cite{Li_4},\nas is plotted in Fig. \\ref{fig:alpha}(b). Whereas Denisov {\\it et\nal.} presented another connection of $\\alpha$ with $\\beta$ on the\nbasis of the L$\\acute{e}$vy walk model \\cite{Denisov}. More\ndetails about the origin of the discrepancy between above two\nrelations can be found in Ref. \\cite{Li_5}.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.54\\textwidth]{fig_5}\n\\caption{\\label{fig:MFP} (color on line) (a), (b), (c) and (d) The\nPDFs of the flight distance $|\\delta x|$ between two consecutive\ncollisions for above four cases I, I\\!I, I\\!I\\!I and I\\!V\nrespectively. $N$ represents the system size. Note that the\nlog-log scale is used in (b) and the inset of (d). In the latter\ncase (d), Gaussian distribution (red dashed-line) is in comparison\nto the numerical PDF. (e) The typical trajectory with periodicity\nin case of $R=0$ , $h=1.0$.}\n\\end{figure}\n\nAs different diffusion behaviors are likely related to the\ntrajectory characteristics of the particle propagation, we\ninvestigate the PDF $\\psi(|\\delta x|)$ of the flight distance\n$|\\delta x|$ in $x$-direction between two consecutive collisions\nwith the scatters. After a long time for adequate collisions in\nthe channel, the PDF for aforementioned fore cases, shown in Fig.\n\\ref{fig:MFP}(a) to (d) respectively, take on completely different\nforms for different cases. In case I, the discrete values of\nprobability indicate that the trajectories are abundant of\nperiodicity, which is almost alike for larger system size. The\nmaximum value of PDF appears when $|\\delta x|=0.447$, and the\ntypical trajectory is plotted in Fig. \\ref{fig:MFP}(e) which shows\nexplicitly that the parallel passage makes the periodical\ntrajectory possible, and the particles are easier to propagate\nalong the channel with fewer collisions. It is superdiffusion in\nthis case. In case I\\!I, only smaller system size has the explicit\nperiodicity. With the system size growing, the periodicity is\ndestroyed by the collisions with the segment of circle time and\ntime. The PDF gets smoother in this case. In case I\\!I\\!I, the\nperiodicity happens only for large flight distance $|\\delta x|$\nwith very small number of families. Note that the maximum of PDF\nis corresponding to the value $|\\delta x|$ of $2.2$ which is just\nthe length of a cell, and the PDF decays in power law. In this\ncase it requires more collisions and takes more time for the\nparticles to escape a certain region. Thus the propagation is\nsuppressed but is still of superdiffusion. The normal diffusion\ntakes place when the particles are scattered by sufficiently large\ndensity of hyperbolic scatters (case I\\!V). Consequently, the\nstrong chaos presents the trajectory of heat carriers with more\naperiodicity. The PDF takes on its characteristic form which has a\nGaussian tail as shown in the inset of Fig. \\ref{fig:MFP}(d).\n\nThus, the propagation modes are responsible for the diffusion\nbehavior. The abundance of aperiodicity of trajectory is the\ncharacteristic of chaotic channel and may also play an crucial\nrole in the normal diffusion. In other words, if the trajectory in\na certain system emerges the aperiodicity due to some other\nmechanisms, such as in polygonal billiard gas model\n\\cite{Alonso_2}, the normal diffusion behaviors may happen.\n\n\n\\section{The calculation of temperature field }\\label{4}\n\nWe calculate the temperature field following the approach proposed\nin Ref. \\cite{Alonso_1}. The temperature of $i$th cell is defined\nby averaging the kinetic energy over all visits into the cell\n\\begin{equation}\nT_{i}==\\frac{\\displaystyle\\sum_{j=1}^{m}t_{j}E_{ij}}{\\displaystyle\\sum_{j=1}^{m}t_{j}},\n\\end{equation}\nwhere $t_{j}$ denotes for the time spent within the cell in the\n$j$th visit, and $m$ for the total number of visit. For\nsufficiently large $m$ we expect a steady temperature profile, and\nthis is indeed verified in our calculations for totally $10^{10}$\nvisits. The temperature profiles we obtained are plotted in Fig.\n\\ref{fig:temp}. It is worthwhile to point out that the steady\ntemperature profiles between non-chaotic and chaotic system are\nquite different in thermodynamics limit, which is due to the\ndifferent diffusion behaviors as shown in Fig. \\ref{fig:alpha}(c).\nAs case I is non-chaotic and has uniform diffusion exponent\n$\\beta$, the temperature profiles keep almost the similar shape\nfor different system sizes. At the two ends of channel there are\nlarge temperature jumps which play an important role in the\nFourier transport and dynamics of the system \\cite{Aoki}. These\njumps arise from the boundary heat resistance which usually\nappears when there is a heat flux across the interface of the two\nadjacent materials. In case I\\!I and I\\!I\\!I which are chaotic and\nhave asymptotically decreasing $\\beta$ versus time $t$, there also\nexists the boundary heat resistance. Unlike in case I, the\ntemperature jump here is smaller and diminishes when the system\nsize grows. For larger size is there almost no temperature jump\nwhich corresponds to a nearly linear temperature profile. Both the\nlarger system size $N$ and the arc radius lead to the increase of\nchaos degree which is responsible for the decrease of diffusion\nexponent $\\beta$ ($\\geqslant 1$). In case I\\!V, which is strong\nchaotic, the temperature profiles are almost linear for various\nsystem sizes we considered, corresponding to the normal heat\nconduction.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.48\\textwidth]{fig_6}\n\\caption{\\label{fig:temp} (color on line) Numerical results of\ntemperature profiles for $T_{L}=1.0$, $T_{R}=0.9$ and sizes\n$N=20$(solid), $40$(dash), $80$(dot), $160$(dash dot), $320$(dash\ndot dot), $640$(short dash) and $1280$(short dot), respectively.\nThe four panels refer to (I)$R=0,\\,h=1.0$; (I\\!I)\n$R=0.001,\\,h=1.0$; (I\\!I\\!I) $R=0.848528,\\,h=1.0$, and (I\\!V)\n$R=0.848528,\\,h=0.27$. The red lines correspond to the best fits\nfor the numerical temperature profile at $N=1280$ with Eq.\n(\\ref{eq:temp}), giving the analytical values $\\beta$ with $1.65$,\n$1.34$, $1.06$ and $1.00$ for above four cases respectively. }\n\\end{figure}\n\nWe estimate the temperature profiles from the average point of\nview. Considering the incident particles from the left heat bath\n(where $x=0$) propagating along the x-axis to the right end, we\nsuppose that a reflecting boundary is placed at the origin of the\nx-axis and an absorbing one at the other end. When\n$2\\geqslant\\beta\\geqslant 1$, we assume that the mean density\n$n_{L}(x)$ of the particles at site $x$ in the steady state is\nproportional to $(1-x)^{\\gamma}$ with\n$\\gamma=(2\/\\beta-1)\\beta^{3\/2}$, where we set $x=i\/N$. Under this\nassumption, we have $n_{L}(x)\\sim 1-x$ \\cite{Alonso_1} when\n$\\beta=1$ (normal diffusion) and $n_{L}(x)\\sim const$ when\n$\\beta=2$ (ballistic diffusion). The conservation of particle\nnumber requires\n\\begin{equation}\nn_{L}(x)\\sim \\frac{(1-x)^{\\gamma}}{D_{L}},\n\\end{equation}\nwhere $D_{L}$ are the diffusion coefficient. Likewise, for a\nparticle propagating from right to left,we have\n\\begin{equation}\nn_{R}(x)\\sim \\frac{x^{\\gamma}}{D_{R}},\n\\end{equation}\n\nWe assume $D_{L}=T_{L}^{\\beta\/2}$ and $D_{R}=T_{R}^{\\beta\/2}$.\nThus, if $2\\geqslant\\beta\\geqslant 1$, the temperature is given by\n\\begin{eqnarray}\n&&T(x)=\\frac{T_{L}n_{L}(x)+T_{R}n_{R}(x)}{n_{L}(x)+n_{R}(x)}\\nonumber\\\\\n&&\n=\\frac{T_{L}T_{R}^{\\beta\/2}(1-x)^{\\gamma}+T_{L}^{\\beta\/2}T_{R}x^{\\gamma}}\n{T_{L}^{\\beta\/2}x^{\\gamma}+T_{R}^{\\beta\/2}(1-x)^{\\gamma}}.\n\\label{eq:temp}\n\\end{eqnarray}\nwhere $\\gamma=(2\/\\beta-1)\\beta^{3\/2}$.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.4\\textwidth]{fig_7}\n\\caption{\\label{fig:comp}(color on line) Numerical results of\ntemperature profile in comparison to the analytical results. The\nnumerical temperature profiles with $R=0.848528, N=40$; $R=0.4,\nN=40$; $R=0.2, N=40$; $R=0.1, N=80$; $R=0.05, N=160$ at $h=1.0$\nand $R=0.848528, N=20$ at $h=0.5$ almost share the same shape. The\nred line is the plot of Eq.(\\ref{eq:temp}) with $\\beta=1.25$. }\n\\end{figure}\n\nAs shown in Figs. \\ref{fig:temp} and \\ref{fig:comp}, the\nanalytical results (in red lines) are in good agreement with the\nnumerical ones for all the cases except those at the two ends of\nthe channel for superdifusion cases. These deviations are likely\ndue to the different boundary conditions we used. Furthermore, the\nvalue of $\\beta$, obtained by the best fits for the numerical\ntemperature profile at $N=1280$ with Eq. (\\ref{eq:temp}), agrees\nwith the simulating result greatly for aforementioned four cases.\nOne can see clearly from Eq.(\\ref{eq:temp}) that temperature\nprofiles are closely related to the diffusion exponent $\\beta$,\nnamely, the case with smaller diffusion exponent tends to have\nsmaller temperature jump. Accordingly, it is not unexpected that\ndifferent chaotic cases may share the same temperature profile if\nthey have the identical diffusion exponent $\\beta$. As shown in\nFig. \\ref{fig:comp}, the case with smaller diffusion exponent\nrequires smaller system size for achieving the same temperature\nprofile. Moreover, our calculations show that the results of\nEq.(\\ref{eq:temp}) are consistent with the numerical ones even in\nlarger temperature gradient. Thus, the temperature profile is\nmostly dependent on the diffusion behavior which is remarkably\naffected by the finite-size effect for chaotic cases of\nsuperdiffusion.\n\n\n\\section{discussion and conclusion}\\label{5}\n\nIn summary, we have investigated the role that the chaos plays in\nthe heat conduction by billiard gas channel. We have demonstrated\nthat the degree of dynamical chaos is enhanced by increasing the\narc radius or the system size for chaotic channel, and the mass\nand heat transport behavior is significantly related to the degree\nof dynamical chaos of a channel. The stronger the chaos is, the\ncloser to normal transport behaviors the model seems to be.\nFurthermore, our numerical results of two exponents $\\alpha$ and\n$\\beta$ for both non-chaotic and chaotic cases when\n$\\beta\\geqslant 1$ satisfies the formula $\\alpha=2-2\/\\beta$\n\\cite{Li_4}. We also discussed the microscopic dynamics by the PDF\nof flight distance in $x$-direction. It seems that aperiodicity of\ntrajectory plays an important role in diffusion behavior. Finally,\nour results showed that the temperature jumps at both ends of the\nchannel depend mostly on the diffusion property for both\nnon-chaotic and chaotic channels, and the finite-size effect is\nmore crucial for chaotic ones.\n\nAs is known that the billiard gas model is applicable for\ncapturing the underlying dynamics of particles without\ninteraction. It is therefore worthwhile to discuss the relation\nbetween our work and others in this field.\n\nAlonso {\\it et al. } \\cite{Alonso_1} investigated 1D Lorentz gas model\nfull of periodically distributed half circular scatters. By\ndefining the heat conductivity and temperature field as\nstatistical average over time on the hypothesis of local thermal\nequilibrium, the Fourier law holds in this case, and a linear\ngradient is given for quite small temperature difference . Our\nwork starts from the same approach but different scatter geometry\nis taken into account. Thus it is not surprising that our work has\nsome overlap with theirs in spirit. However, we pay much more\nattention to the role played by different degree of dynamical\nchaos in heat conduction. As a result, our intensive calculations\nextended the results in Ref. \\cite{Alonso_1} and concluded that\nonly sufficient strong chaos results in the normal diffusion,\nthus, the normal heat transport.\n\nLi {\\it et al. } \\cite{Li_2} presented the dependence of heat conductivity\non system size and the temperature profile in channel with zero\nLyapunov exponent where the right triangle scatters are\nperiodically distributed. In this case, the exponent stability\nleads to abnormal transport behavior. Clearly, their result is the\nnon-chaotic limit of our model.\n\n\n\\section*{Acknowledgement}\n\nWe would like to thank B. Li for providing Ref. \\cite{Li_5} prior\nto publication and helpful discussion. The work is supported by\nNSFC No.10225419 and 90103022.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{}\n\n\nWe present an analysis of the average spectral properties of ~12,000 SDSS quasars as a function of accretion disc inclination, as measured from the equivalent width of the [O III] 5007\\AA\\ line. The use of this indicator on a large sample of quasars from the SDSS DR7 has proven the presence of orientation effects on the features of UV\/optical spectra, confirming the presence of outflows in the NLR gas and that the geometry of the BLR is disc-like. Relying on the goodness of this indicator, we are now using it to investigate other bands\/components of AGN. Specifically, the study of the UV\/optical\/IR SED of the same sample provides information on the obscuring \"torus\". The SED shows a decrease of the IR fraction moving from face-on to edge-on sources, in agreement with models where the torus is co-axial with the accretion disc. Moreover, the fact we are able to observe the broad emission lines also in sources in an edge-on position, suggests that the torus is rather clumpy than smooth as in the Unified Model. The behaviour of the SED as a function of EW[OIII] is in agreement with the predictions of the clumpy torus models as well.\n\n\n\n\n\\tiny\n \\fontsize{8}{11}\\helveticabold { \\section{Keywords:} galaxies: active, galaxies: nuclei, galaxies: Seyfert, quasars: emission lines, quasars: general}\n\\end{abstract}\n\n\\section{Introduction}\n\nThe fact we are not able to spatially resolve the inner regions of Active Galactic Nuclei (AGN), combined with their axisymmetric geometry \\citep{AntonucciMiller1985, Antonucci1993}, can make it difficult to interpret their emissions.\nThe Unified Model predicts the orientation to be one of the main drivers of the diversification in quasars spectra. For this reason, an indicator of the inclination of the source with respect the line of sight of the observer is essential to get further in studying these objects.\n\nDespite several quasars properties have been found to provide information on the inclination of the inner nucleus \\citep{WillsBrowne1986,WillsBrotherton1995,Boroson2011,Decarli2011,VanGorkom2015}, we still lack an univocal measurement of this quantity. This problem is even harder when dealing with not-jetted objects, the most among quasars \\citep[$>90\\%$,][]{Padovani2011}, for which we can not rely on the presence of the strongly collimated radio-jets, directed perpendicularly to the accretion disc.\n\nIn order to give a more accurate description of the components surrounding the central engine and to understand where are the boundaries between one and another, we need to know which components are being intercepted by our line of sight.\n\nAssuming that some of these inner components are characterised by a spherical geometry can often simplify the scenario, while at the same time misleading us. We use the emission lines coming from the Broad Line Region (BLR) to give an estimate of the mass of the central Super Massive Black Hole (SMBH), but in doing that we do not take properly into consideration the geometry of the BLR, i.e we use an average \\emph{virial factor} $f$ to account for the uknown in the geometry and kinematics of the emitting region and we overlook the effects of orientation on the emission lines \\citep{JarvisMcLure2006,Shen2013}. These measurements can then be used in turn to examine the relations between the SMBH and their host galaxies - one of the few tools available to understand the connection between structures on such different spatial scales - their uncertainties affecting these studies \\citep[e.g. ][]{ShenKelly2010}.\nIf the BLR is characterised by a non-spherical geometry we are sistematically underestimating the BH masses in all the sources in which the velocity we intercept, i.e. the line width we measure, is only a fraction of the intrinsic velocity of the emitting gas orbiting around the SMBH.\nThe inclination of the source with respect to the line of sight is therefore crucial to both the understanding of how the nuclei work and how they affect the formation and evolution of galaxies in the Universe.\nIn this proceeding we show recent results on the optical spectra and we present a preliminary result on the Spectral Energy Distribution of quasars, that we obtained using the EW[OIII] as an orientation indicator.\n\n\n\\section{Orientation effects on emission features}\n\n\\subsection{Optical spectra}\nBased on the properties of the [OIII] $5007$\\AA~ line - negligibly contaminated by non-AGN processes and coming from the Narrow Line Region (NLR), whose dimensions ensure the isotropy of the emissions \\citep{Mulchaey1994, Heckman2004} - and on the strong anisotropy of the continuum emitted by the optically-thick\/geometrically-thin accretion disc \\citep{ShakuraSunyaev1973}, we proposed the equivalent width (EW) of the [OIII] line, the ratio between the two luminosities, as an indicator of quasars orientation \\citep{Risaliti2011,Bisogni2017}. \n\nIn \\cite{Risaliti2011} we examined the distribution of the observed EW[OIII] in a large sample of quasars from the SDSS DR5 ($\\sim 6000$) and verified the presence of an orientation effect: the distribution shows a power law tail at the high EW[OIII] values that can not be ascribable to the intrinsic differences in the NLR among different objects, i.e. the intrinsic EW[OIII] distribution, the one we would observe if all the sources were seen in a face-on position.\nThe observed EW[OIII] distribution is a convolution of the intrinsic properties of the NLR emissions in the different objects, such as the ionising continuum and the covering factor of the clouds, and the effects due to their inclination angle. \n\nIn \\cite{Bisogni2017} we selected a larger sample of objects from the SDSS DR7 ($\\sim 12000$), this time with the aim of looking for evidences of orientation effects in the optical spectra.\nWe split our sample in six bins of EW[OIII], each one corresponding to an inclination angle range. Within each bin, the spectra were stacked in order to produce a master spectrum.\nWe then analysed both the broad and the narrow emission lines as a function of EW[OIII], i.e of the inclination angle, finding orientation effects on both of them.\nFig. \\ref{fig1} shows the presence of orientation effects on the broad component of H$\\beta$: the width of the broad line, either represented by the line dispersion $\\sigma$, the Full Width Half Maximum (FWHM) or the Inter-Percentile Velocity width (IPV), increases steadily when we move from low to high EW[OIII]. We found the same result for the other broad lines examined (H$\\alpha$ and MgII, see \\cite{Bisogni2017} for more details). This behaviour is what is expected if the BLR geometry is disc-like and we are moving from sources in a face-on position to sources in an edge-on position.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=18cm]{FWHM_sigma_IPV_Hb.pdf}\n\\end{center}\n\\caption{Full Width Half Maximum, Inter-Percentile Velocity width and dispersion $\\sigma$ as a function of the EW[OIII] for the H$\\beta$ line. All these quantities, describing the rotational velocity of the gas orbiting around the central SMBH, increase moving from low to high EW[OIII] as expected if the BLR is disc-shaped and we are moving from face-on to edge-on positions.}\\label{fig1}\n\\end{figure}\n\n\n\n\nAs for the narrow emission lines, we examined the [OIII] $\\lambda 5007$\\AA, the most prominent among them in the optical spectrum. This line is known to be contaminated by emissions coming from non-virialized gas, i.e. not orbiting around the central SMBH, but outflowing perpendicularly to the accretion disc \\citep{Heckman1984, Boroson2005}. \nIf the EW[OIII] is a good indicator of the inclination of the source, we should see the blue component of the line, emitted by outflowing gas, decreasing both in intensity and in velocity shift with respect to the nominal wavelength of the emission moving to high EW[OIII] values. Going from face-on to edge-on position in fact we are not intercepting anymore the outflow perpendicular to the accretion disc. This behaviour is found in the [OIII] line profile (Fig. \\ref{fig2}).\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=18cm]{profile+vel_OIII.pdf}\n\\end{center}\n\\caption{Left panel: [OIII] $5007$\\AA~ line profile as a function of EW[OIII]. The blue component of the line decreases moving from low to high EW[OIII]. Right panel: velocity shift of the blue component of the [OIII] line with respect to the velocity of the [OII] $\\lambda 3727$ line, accounting for the systemic velocity of the host galaxy. The shift is decreasing (in modulus) when moving from low to high EW[OIII]. Both the trends are expected if we are moving from face-on to edge-on positions, where the outflow velocity component of the gas is not intercepted anymore by the line of sight.}\\label{fig2}\n\\end{figure}\n\nWe want to stress that within each EW[OIII] bin, therefore within each inclination angle range, the population of quasars is characterised by different SMBH masses, luminosities and accretion rates. These properties are considered among the main drivers of the variance in quasars spectra \\citep{Marziani2003,Zamanov2002,ShenHo2014}. In our study however, as it is designed, even if the effects produced by these properties are present, they are diluited in the stacked representative spectra. \nAs a confirmation that the orientation, even if not the only driver, plays a major role in the variance of quasars spectra, we found a clear trend of the Eigenvector 1, i.e. the anticorrelation between the FeII and [OIII] emissions intensity that \\cite{BorosonGreen1992} identify as the main responsible for quasars spectral variance, with the EW[OIII].\nSpecifically, when we move from low to high EW[OIII], i.e. from low to high inclination angles, we see the [OIII] intensity increasing, while FeII emission becomes less and less intense. This can be explained in terms of orientation: the BLR shares the same anisotropy of the accretion disc and therefore the intensity of its emissions, in this case FeII, is decreased by a factor $\\cos \\theta$, i.e. decreases when moving to edge-on sources. On the other hand the [OIII] line appears as more evident in edge on positions because the luminosity of the continuum emitted by the accretion disc is decreased by the factor $\\cos \\theta$.\n\n\n\\subsection{Infrared emissions}\n\n\nThe observed EW[OIII] distribution has implications for the obscuring component as well.\n\nThe torus is depicted in the Unified Model as a smooth and toroidal structure that can reach $\\sim 1- 10$ pc in size \\citep{Burtscher2013}. If this is true then, there is a maximum inclination angle beyond which we are not able anymore to intercept the emissions coming from the very inner components, such as the continuum emitted by the accretion disc and the broad lines emitted by the BLR.\nIn this case, however, the observed EW[OIII] distribution would drop when the line of sight is starting to intercept the torus. This is not what we observe: the power law keeps going very steadily to the highest EW[OIII] values.\nMoreover, we are intercepting broad emission lines in positions corresponding to high inclination angles.\nBoth these facts are not compatible with the torus being a smooth structure and rather suggest a clumpy structure.\n\nTo test the indicator and exploit its potential, we are now interested in investigating the infrared emissions.\n \nWe then collect photometric data for the same sample in the UV, optical and IR band to study the Spectral Energy Distributions (SED) of the sources.\n\n\\section{Sample and data analysis}\nFor the sample of $\\sim 12000$ objects we selected from the SDSS DR7 the following photometric data are available:\n\n\\begin{itemize}\n\\item[-] Far Ultra Violet and Near Ultra Violet bands from \\emph{Galaxy Evolution Explorer} (\\emph{GALEX}) DR5 \\citep{Bianchi2011}.\n\\item[-] \\emph{ugriz} SDSS photometric data from \\cite{Shen2011}.\n\\item[-] The J, H and K bands from the \\emph{Two Micron All-Sky Survey} (2MASS) \\citep{Skrutskie2006}.\n\\item[-] The $3.4$, $4.6$, $12$ and $22\\,\\mu$m photometric data from the \\emph{Wide-field Infrared urvey Explorer} (\\emph{WISE}) \\citep{Wright2010}.\n\\end{itemize}\n\nWe first correct all the magnitudes for Galactic extinction using the maps from the \\cite{Schlegel1998}. Then, for each EW[OIII] bin, we use the same approach as for the optical spectroscopic data: we rest-frame the data according to the sources redshift and then we perform a stacking of the interpolated SED in order to produce a master SED, on which we can examine the effects produced by the orientation.\nBefore stacking them, we normalise each SED by dividing for the value of $\\nu L_{\\nu}$ at $\\lambda=15 \\, \\mu$m, a reference wavelength in the mid-infrared spectral range, the band in which the torus emits. In doing that, we are normalizing the individual SED for the intrinsic differences of the torus and of other components in the different objects, such as the size and covering factor of the obscuring region, its distance from the central engine and the properties of the ionising continuum, whose emission is being reprocessed by the torus. This makes us able to compare the average behaviour of the obscuring structure at different inclinations with respect to the line of sight of the observer. The final SED are shown in Fig. \\ref{fig3}.\n\n\\section{Results and discussion: implications for the obscuring torus}\n\nIn the optical stacks corresponding to the highest inclination angles (high EW[OIII]) we are able to detect emissions from the BLR.\nThis evidence implies three possible scenarios: the absence of the torus, a torus that is mis-aligned with respect the plane of the accretion disc (and of the BLR) and a clumpy torus.\nThe first scenario is ruled out by the fact that the IR emission is clearly visible in the SED of the sample (Fig. \\ref{fig3}).\nAs for the second one, we see the IR emission in the stacked SED decreasing progressively as a function of the indicator, defined through the anisotropic properties of the emission coming from the accretion disc itself. If torus and accretion disc were not co-axial, we would not see such an orderly behaviour.\n\nThe only scenario we are left with is therefore a clumpy torus, leading to a differention between type 1 and type 2 AGN due only to the photon escaping probability \\citep{Elitzur2008}.\nDue to the selection we performed (i.e. we selected blue objects, and verified that the continuum in our stacked optical spectra was not experiencing any reddening, see \\cite{Bisogni2017}), when we are looking at sources with a high EW[OIII], i.e. with a high inclination angle, we are dealing with type 1 sources in which the BLR is intercepted through the dusty clouds of the torus.\n\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{.5\\linewidth}\n\\centering\\includegraphics[scale=0.6]{SED_stacks.pdf\n\\end{minipage}%\n\\begin{minipage}[b]{.5\\linewidth}\n\\centering\\includegraphics[scale=0.575]{SED_zoom_torus_stacks.pdf\n\\end{minipage}\n\\caption{(Left panel) Spectral Energy Distributions for the six EW[OIII] bins, corresponding to different inclination angle ranges. The master SED for a EW[OIII] bin was realized as follows: the photometric data for each source from the \\emph{GALEX}, SDSS, 2MASS and \\emph{WISE} surveys were corrected for Galactic reddening, rest-framed, interpolated on a common grid and normalized to the $\\nu L_{\\nu}$ value corresponding to the reference wavelength ($15\\mu$m); for every channel in the grid we then selected the median value. (Right panel) Total flux for the six EW[OIII] bins in the spectral range in which the emission coming from the torus is predominant. SED corresponding to low EW[OIII] values are characterised by a shallower decrease in the emission at the shorter IR wavelengths with respect to the longer ones, while in the case of high EW[OIII] the decrease is steeper. This behaviour is in agreement with the clumpy torus models (see text for details).}\\label{fig3}\n\\end{figure}\n\nWe can compare our results with the clumpy models in literature \\citep{Nenkova2008a, Nenkova2008b} that examine the infrared emission of the torus as a function of the inclination angle with respect to the observer.\nIf the torus is a clumpy structure, what we expect is that the IR emission at shorter wavelengths decreases progressively more than the ones at longer wavelengths when we are reaching edge-on position, due to the combination of an increasing number of clouds intercepted by the line of sight and of a higher absorption at the shorter than at the longer wavelengths \\citep{Nenkova2008b}.\n\nThe behaviour of the stacked SED as a function of EW[OIII] confirms this scenario. At low EW[OIII] (low inclination angle) we are able to intercept the IR emissions coming from the inner clouds of the torus, that are directly illuminated by the ionising continuum, while at high EW[OIII] (high inclination angle) the IR emission coming from the inner clouds is shielded and we can detect it only after it is absorbed by the clouds in the outskirts of the torus. This produces the decrease in the flux at the shorter wavelengths, that becomes progressively more important for stacks corresponding to higher inclinations.\n\nAs a final verification that our results are not biased by any characteristics of the sample, we made the following checks:\n1.) since our sample contains non-jetted as well as jetted quasars, the most extreme among them (blazars) could contaminate the part of the SED pertaining to the torus emission. We verified that the sub-sample composed by non-jetted quasars only gives the same result as the complete sample.\\\\\n2.) $\\sim 50\\%$ of the objects in our sample has a redshift $z>0.47$. This is the critical value beyond which the normalisation flux at $15\\,\\mu$m is retrieved through an extrapolation rather than an interpolation of the SED.\nWe verified that the analysis on the $z<0.47$ and $z>0.47$ sub-samples does not give different results. The only differences in the $z<0.47$ ($z>0.47$) sub-sample we recognise with respect to the complete sample SED are: a lower (higher) luminosity in Big Blue Bump (accretion disc), due to the fact that our sample is on average more luminous at higher redshifts, and a higher (lower) emission in the optical\/NIR band, due to a higher contribution from the host galaxy for sources at lower redshift. We conclude that the extrapolation of the $15\\,\\mu$m flux in $z>0.47$ sources does not affect our results.\n\n\n\\section{Conclusions}\n\nIn this proceeding we summarise the results of the analysis on the optical spectra and we present the preliminary results of the analysis on the infrared emission of $12000$ sources of the SDSS DR7 as a function of the EW[OIII], a new orientation indicator.\nWe find that:\n\\begin{itemize}\n\\item[-] the BLR shares the same geometry of the accretion disc; we are intercepting the intrinsic velocity of the orbiting gas only when we are looking at sources in edge-on positions. If not properly taken into account, the orientation effects affecting the broad emissions lead to an underestimation of the SMBH virial masses in every position but the edge-on ones.\n\\item[-] the presence of outflowing gas in the NLR is clearly seen in the profile of the [OIII] $\\lambda 5007$\\AA~ as a function of the inclination angle. The blue component decreases both in intensity and in the shift with respect to the reference wavelength moving from face-on to edge-on positions.\n\\item[-] the preliminary analysis of the SED reveals a stronger decrease in the IR emission corresponding to the shorter wavelengths with respect to the longer ones when moving from low to high EW[OIII] values, as expected in the theoretical models for clumpy tori when moving from low to high inclination angles.\n\\end{itemize}\n\nFurther analysis is needed in order to investigate properly the emission coming from the torus. Starting from these first results, we are in the process of performing a SED fitting for each source in the sample with \\emph{AGNfitter} \\citep{CalistroRivera2016}. We will then be able to repeat the analysis on the representative SED, this time having also information on the single components contributing to the total emission.\n\n\nWe will also investigate the sources in our sample for which multiple observations are available (e.g. Stripe82, new BOSS spectra) in order to look for evidences of \\emph{changing look} behaviours as a function of the EW[OIII]. If our interpretation of the data implying a clumpy structure for the torus is correct, we expect to see some of the sources that were included in our sample as Type 1 AGN changing to Type 2 objects at a different epoch. This behaviour is expected more frequently for sources with a high EW[OIII], where the orientation effect is dominant, but it is not excluded even for sources with low EW[OIII] values.\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\n\\section*{Funding}\nSupport for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number AR7-18013 X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. E.L. is supported by a European Union COFUND\/Durham Junior Research Fellowship (under EU grant agreement no. 609412).\n\n\n\\bibliographystyle{frontiersinSCNS_ENG_HUMS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbobo b/data_all_eng_slimpj/shuffled/split2/finalzzbobo new file mode 100644 index 0000000000000000000000000000000000000000..876f4964204d364d3de9ee79b3399ae663130e70 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzbobo @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\begin{definition}\\label{def1.0}\nFor $\\lambda\\in \\mathbb{R}$ we denote by $e_\\lambda(x):=e^{2\\pi i\\lambda\\cdot x}$. We say that a finite Borel measure $\\mu$ on $\\mathbb{R}$ is {\\it spectral} if \nthere exists a set $\\Lambda$ such that the family of exponential functions $E(\\Lambda):=\\{e_\\lambda : \\lambda\\in\\Lambda\\}$ is an orthogonal basis for $L^2(\\mu)$. We call $\\Lambda$ a {\\it spectrum} for $\\mu$. If $E(\\Lambda)$ is an orthogonal set then we say that $\\Lambda$ is {\\it orthogonal}. \n\n\nWe say that a bounded Borel subset $\\Omega$ of $\\mathbb{R}$ is {\\it spectral} if the restriction of the Lebesgue measure to $\\Omega$ is a spectral measure.\n\n We say that a finite subset $A$ of $\\mathbb{R}$ is {\\it spectral} if the counting measure on $A$ is a spectral measure. \n\\end{definition}\n\n\nSpectral sets have been introduced in relation to the Fuglede conjecture \\cite{Fug74}:\n\\begin{conjecture}\nA bounded Borel subset $\\Omega$ of $\\mathbb{R}$ is spectral if and only if it tiles $\\mathbb{R}$ by translations, i.e., there exists a set $\\mathcal T$ in $\\mathbb{R}$ such that $\\{\\Omega+t : t\\in\\mathcal T\\}$ is a partition of $\\mathbb{R}$ (up to Lebesgue measure zero). \n\\end{conjecture}\n\nThe conjecture can be formulated in any dimension, but it is known to be false in both directions for dimensions 3 or higher \\cite{Tao04,FaMaMo06}. In dimensions 1 and 2, as far as we know at the moment of writing this paper, the conjecture is open in both directions. \n\n\nIn \\cite{JoPe98}, Jorgensen and Pedersen have constructed a new example of a spectral measure, a fractal one. Their construction is based on a scale 4 Cantor set, where the first and and third intervals are kept and the other two are discarded. The appropriate measure for this set is the Hausdorff measure $\\mu_4$ of dimension $\\frac{\\ln 2}{\\ln 4}=\\frac12$. They proved that this measure is spectral with spectrum \n\\begin{equation}\n\\Lambda:=\\left\\{\\sum_{k=0}^n 4^kl_k: l_k\\in\\{0,1\\}, n\\in\\mathbb{N}\\right\\}.\n\\label{eqspmu4}\n\\end{equation}\n\n\n\nMany other examples of fractal measures have been constructed since, see e.g. \\cite{MR1785282,LaWa02,DJ06,DJ07d}, and many other spectra can be constructed for the same measure, see e.g., \\cite{DHS09}. Among other things, we will show that the spectrum $\\Lambda$ in \\eqref{eqspmu4} tiles $\\mathbb{Z}$ by translations. \n\n\n\n\n\n\n\n\nA large class of examples of spectral measures is based on affine iterated function systems. \n\n\\begin{definition}\\label{def1.1}\nLet $R$ be an integer $R\\geq 2$. We call $R$ the {\\it scaling factor}. Let $B\\subset\\mathbb{Z}$, $0\\in\\mathbb{Z}$, $N:=\\#B$. We define the affine iterated function system\n$$\\tau_b(x)=R^{-1}(x+b),\\quad(x\\in\\mathbb{R},b\\in B).$$\nBy \\cite{Hut81} there exist a unique compact set $X_B$ called {\\it the attractor} of the IFS $\\{\\tau_b\\}$, such that \n$$X_B=\\cup_{b\\in B}\\tau_b(X_B).$$\nThe set $X_B$ can be described using the base $R$ representation of real numbers, with digits in $B$:\n$$X_B=\\left\\{\\sum_{k=1}^\\infty R^{-k}b_k : b_k\\in B\\right\\}.$$\nAlso by \\cite{Hut81}, there exists a unique Borel probability measure $\\mu_B$ on $\\mathbb{R}$ that satisfies the invariance equation \n\\begin{equation}\n\\mu_B(E)=\\frac{1}{N}\\sum_{b\\in B}\\mu_B(\\tau_b^{-1}E)\\mbox{ for all Borel subsets $E$ of $\\mathbb{R}$}.\n\\label{eq1.1.1}\n\\end{equation}\nEquivalently, for all continuous compactly supported functions $f$:\n\\begin{equation}\n\\int f\\,d\\mu_B=\\frac1N\\sum_{b\\in B}\\int f\\circ\\tau_b\\,d\\mu_B.\n\\label{eq1.1.2}\n\\end{equation}\nThe measure $\\mu_B$ is called {\\it the invariant measure} of the IFS $\\{\\tau_b\\}$. \nIn addition the measure $\\mu_B$ is supported on the attractor $X_B$.\n\\end{definition} \n\n\\begin{definition}\\label{def1.1i}\nLet $L\\subset\\mathbb{Z}$, $0\\in L$. We say that $(B,L)$ is a {\\it Hadamard pair with scaling factor $R$} if $\\#L=\\#B=N$ and the matrix \n$$\\frac{1}{\\sqrt N}\\left(e^{2\\pi i R^{-1}b\\cdot l}\\right)_{b\\in B,l\\in L}$$\nis unitary. We call this matrix {\\it the matrix associated with $(B,L)$}.\n\nWe define the function \n\\begin{equation}\nm_B(x)=\\frac{1}{N}\\sum_{b\\in B}e^{2\\pi ib\\cdot x},\\quad (x\\in\\mathbb{R})\n\\label{eq1.1i.1}\n\\end{equation}\n\nGiven a Hadamard pair $(B,L)$ we say that a finite set of points $\\{x_0,\\dots,x_{r-1}\\}$ in $\\mathbb{R}$ is a {\\it cycle} for $L$ if there exist $l_0,\\dots,l_{r-1}$ in $L$ such that \n$$\\frac{x_0+l_0}{R}=x_1,\\dots,\\frac{x_{r-2}+l_{r-2}}{R}=x_{r-1},\\frac{x_{r-1}+l_{r-1}}{R}=x_0.$$\nWe call $l_0,\\dots,l_{r-1}$ {\\it the digits of this cycle.}\nWe say that this cycle is {\\it extreme} for $(B,L)$ if \n$$|m_B(x_k)|=1\\mbox{ for all }k\\in\\{0,\\dots,r-1\\}.$$\nThe points $\\{x_i\\}$ are called {\\it (extreme) cycle points}.\n\\end{definition}\n\nWhen $(B,L)$ is a Hadamard pair with scaling factor $R$, then the measure $\\mu_B$ is always spectral and a spectrum can be constructed using digits in $L$ and extreme cycles. \n\n\\begin{theorem}\\label{th1.2}\\cite{DJ06}\nIf $(B,L)$ is a Hadamard pair then $\\mu_B$ is a spectral measure with spectrum $\\Lambda(L)$ where $\\Lambda$ is the smallest set which contains $-C$ for all cycles $C$ for $L$ which are extreme for $(B,L)$, and with the property that $R\\Lambda(L)+L\\subset \\Lambda(L)$. \n\\end{theorem}\n\nThis spectrum can be described in terms of base $R$ representations of integers using only digits in $L$. \n\n\\begin{definition}\\label{def1.2}\nLet $L$ be a set of integers. We say that an integer $x$ can be represented in base $R$ using digits in $L$ if there exist integers $x_0,x_1,\\dots$, with $x_0=x$ and digits $l_0,l_1,\\dots$ in $L$ such that \n$$x_k=Rx_{k+1}+l_k\\mbox{ for all $k\\geq0$}.$$\nWe call $l_0l_1\\dots$ a {\\it representation} of $x$ in base $R$. \n\\end{definition}\n\n\n\n\n\\begin{proposition}\\label{pr0.1.6}\nLet $(B,L)$ be a Hadamard pair. Assume in addition that all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then the spectrum $\\Lambda(L)$ defined in Theorem \\ref{th1.2} is the set of integers which can be represented in base $R$ using digits in $L$.\n\\end{proposition}\n\n\n\nNext we turn our attention to finite spectral subsets of $\\mathbb{Z}$. The variant of the Fuglede conjecture for such sets is that a finite subset $A$ of $\\mathbb{Z}$ is spectral if and only if it tiles $\\mathbb{Z}$ by translations. In \\cite{CoMe99}, Coven and Meyerowitz proposed a characterization of sets that tile integers by translations, in terms of cyclotomic polynomials. \n\\begin{definition}\\label{def2.1.1}\nLet $A$ be a finite multiset of nonnegative integers, by multiset we mean that some elements $a\\in A$ might be counted with multiplicity $m_a$. We define the polynomial corresponding to $A$ by \n$$A(x)=\\sum_{a\\in A}m_ax^a.$$\n\nFor $s\\in\\mathbb{N}$, we denote by $\\Phi_s(x)$ the $s$-th cyclotomic polynomial. We denote by $S_A$ the set of all prime powers such that the $s$-th cyclotomic polynomial divides $A(x)$. \n\n\nWe say that the set $A$ (without any multiplicity) {\\it satisfies the Coven-Meyerowitz property (or shortly, $A$ has the CM-property) }if the following two conditions are satsisfied:\n\\begin{enumerate}\n\t\\item[(T1)] $A(1)=\\prod_{s\\in S_A}\\Phi_s(1).$\n\t\\item[(T2)] If $s_1,\\dots,s_m\\in S_A$ are powers of distinct primes then $\\Phi_{s_1\\dots s_m}(x)$ divides $A(x)$.\n\\end{enumerate} \n\\end{definition}\n\nCoven and Meyerowitz proved in \\cite{CoMe99} that a set with the CM-property tiles $\\mathbb{Z}$ by translations and they conjectured that the reverse is also true, and proved the conjecture in some special cases (when the size of the set has at most two prime factors). \\mathcal{L} aba proved in \\cite{Lab02} that the CM-property also implies that the set is spectral. Combining these results we show that the tiling sets and spectra fit together nicely in a {\\it complementary pair}. We are also interested in the extreme cycles due to their importance for the spectra of fractal measures. \n\n\n\\begin{definition}\\label{def2.0}\nLet $A,A'$ be two subsets of $\\mathbb{R}$. We say that $A$ and $A'$ {\\it have disjoint differences} if $(A-A)\\cap(A'-A')=\\{0\\}$. In this case we denote by \n$A\\oplus A'=\\{a+a' : a\\in A, a'\\in A'\\}$; we use the sign $\\oplus$ to indicate that the sets have disjoint differences; equivalently, for any $x\\in A+A'$ there exist unique $a\\in A$ and $a'\\in A'$ such that $x=a+a'$; equivalently, the sets $A+a'$, $a'\\in A'$ are disjoint. \n\\end{definition}\n\n\\begin{definition}\\label{def2.1}\nLet $R\\in\\mathbb{Z}$, $R\\geq 2$. Let $(B,L)$ and $(B',L')$ be two Hadamard pairs with scaling factor $R$, $\\#B=N$, $\\#B'=N'$, not necessarily equal. We say that the two Hadamard pairs are {\\it complementary} if the following conditions are satisfied: \n\\begin{enumerate}\n\t\\item $B\\oplus B'$ and $L\\oplus L'$ are complete sets of representatives $\\mod R$.\n\t\\item The extreme cycles for $(B,L)$ and the extreme cycles for $(B',L')$ are contained in $\\mathbb{Z}$.\n\t\\item The greatest common divisor of the points in $B\\oplus B'$ is 1.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\\begin{theorem}\\label{th0.2.1.2}\nLet $B$ a finite set of nonnegative integers with $\\gcd(B)=1$ and which satsifies the Coven-Meyerowitz property. Let $R$ be the lowest common multiple of the elements in $S_B$. Then there exist finite sets $B',L,L'$ of nonnegatve integers such that \n\\begin{enumerate}\n\t\\item $(B,L)$ and $(B',L')$ are complementary Hadamard pairs (relative to the number $R$).\n\t\\item $B'$ satisfies the Coven-Meyerowitz property.\n\\end{enumerate}\n\\end{theorem}\n\nOnce we have two complementary Hadamard pairs $(B,L)$, $(B',L')$ with scaling factor $R$, we can construct the two fractal measures $\\mu_B$ and $\\mu_{B'}$ with spectra $\\Lambda(L)$ and $\\Lambda(L')$ respectively. The next theorem shows that the convolution of the two measures $\\mu_B$ and $\\mu_{B'}$ is the Lebesgue measure on a tile of $\\mathbb{R}$, it is also the invariant measure $\\mu_{B\\oplus B'}$ for the affine IFS associated to scaling by $R$ and digits $B\\oplus B'$. The two spectra always have disjoint differences and moreover, under some restrictions on the encodings of the extreme cycles for $(B\\oplus B',L\\oplus L')$, $(B,L)$ and $(B',L')$, the two sets complement each other, in the sense that $\\Lambda(L)$ tiles $\\mathbb{Z}$ with $\\Lambda(L')$.\n\n\n\n\\begin{definition}\\label{def2.2}\nLet $(B,L)$ and $(B',L')$ be complementary Hadamard pairs with scaling factor $R$. We define the maps $p:L\\oplus L'\\rightarrow L$ and $p':L\\oplus L'\\rightarrow L'$ by \n$$p(l+l')=l,\\quad p'(l+l')=l'\\mbox{ for all }l\\in L ,l'\\in L'.$$\nFor a sequence $a_0a_1\\dots$ of digits in $L\\oplus L'$ we define \n$$p(a_0a_1\\dots)=p(a_0)p(a_1)\\dots,\\quad p'(a_0a_1\\dots)=p'(a_0)p'(a_1)\\dots.$$\n\\end{definition}\n\n\\begin{theorem}\\label{th2.3}\nLet $(B,L)$ and $(B',L')$ be complementary Hadamard pairs with scaling factor $R$. Let $\\Lambda(L)$ be the set of integers that can be represented in base $R$ using digits from $L$, and similarly for $\\Lambda(L')$.\n\\begin{enumerate}\n\t\\item The measure $\\mu_{B\\oplus B'}$ is the Lebesgue measure on the attractor $X_{B\\oplus B'}$ and has spectrum $\\mathbb{Z}$. Moreover $X_{B\\oplus B'}$ is translation congruent to $[0,1]$, i.e., there exists a measurable partition $\\{A_n\\}_{n\\in\\mathbb{Z}}$ of $[0,1]$ such that $\\{A_n+n\\}_{n\\in\\mathbb{Z}}$ is a partition of $X_{B\\oplus B'}$. \n\t\\item The measure $\\mu_{B\\oplus B'}$ is the convolution of the measures $\\mu_B$ and $\\mu_{B'}$. \n\t\\item The set $\\Lambda(L)$ is a spectrum for $\\mu_B$ and the set $\\Lambda(L')$ is a spectrum for $\\mu_{B'}$.\n\t\\item The sets $\\Lambda(L)$ and $\\Lambda(L')$ have disjoint differences. \t\n\t\\item The set $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$ if and only if for any digits $a_0\\dots a_{r-1}$ of an extreme cycle for $(B\\oplus B',L\\oplus L')$, the sequence $p(a_0\\dots a_{r-1})$ consists of the digits of an extreme cycle for $(B,L)$ and the sequence $p'(a_0\\dots a_{r-1})$ consists of the digits of an extreme cycle for $(B',L')$. \n\tThe equality $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$ means that $\\Lambda(L)$ tiles $\\mathbb{Z}$ by $\\Lambda(L')$.\n\\end{enumerate}\n\n\\end{theorem}\n\n\n\n\n\n\nNext, we focus on sets $B$ of small size: 2,3,4,5 and investigate when such a set is spectral and when a Hadamard pair with scaling factor $R$ can be complemented. We base our results on the classification of Hadamard matrices of size 2,3,4,5. For size $\\#B=2,3,4$ this is fairly simple, see \\cite{TaZy06}. For size 5, the problem becomes more complicated but it was solved by Haagerup \\cite{Haa97}.\n\n\\begin{definition}\nA $N\\times N$ matrix $H$ is called a {\\it Hadamard matrix} if it is unitary and all its entries have the same absolute value $\\frac{1}{\\sqrt N}$. Two Hadamard matrices $H$, $H$' are said to be {\\it equivalent} if one can be obtained from the other after permutations of row and columns and multiplication of rows and columns by complex numbers of absolute value 1; formally: there exist permutation $\\pi$ and $\\rho$ of the set $\\{1,\\dots,N\\}$ and complex numbers $c_1,\\dots,c_N,d_1,\\dots,d_N$ on the unit circle $\\mathbb{T}=\\{z: |z|=1\\}$ such that \n$$H_{ij}'=c_id_jH_{\\pi(i)\\rho(j)},\\quad(i,j\\in\\{1,\\dots,N\\}).$$\nThe matrix of the Fourier transform on $\\mathbb{Z}_N$, $\\frac{1}{\\sqrt{N}}(e^{2\\pi i\\frac{jk}{N}})_{j,k=0}^{N-1}$ is called the {\\it standard Hadamard matrix}.\n\n\\end{definition}\n\n\n\\begin{theorem}\\label{th1.15}(See \\cite{TaZy06,Haa97})\nLet $N=2,3$ or $5$. Any Hadamard matrix of size $N$ is equivalent to the standard Hadamard matrix. If $N=4$, any $4\\times 4$ Hadamard matrix is equivalent to one of the following form:\n\\begin{equation}\\frac12\n\\begin{pmatrix}\n1&1&1&1\\\\\n1&-1&\\rho&-\\rho\\\\\n1&-1&-\\rho&\\rho\\\\\n1&1&-1&-1\n\\end{pmatrix}\n\\label{eqmat4}\n\\end{equation}\nfor some $\\rho\\in\\mathbb{T}$.\n\\end{theorem}\n\nAs far as we know, there is no classification for Hadamard matrices of size 6 or higher. Beauchamp and Nicoar\\u a gave a classification of {\\it self-adjoint} $6\\times 6$ Hadamard matrices in \\cite{MR2398121}.\n\n\n\nA Hadamard matrix is said to be in de-phased form if its first row and column contain only the number $1$.\n\n\n\\begin{corollary} \\label{perm}\nLet $N=2$, $3$, $4$, or $5$. Any two Hadamard matrices $A$ and $B$ of size $N$ in de-phased form which are equivalent are also equivalent via permutations only, that is, there are permutation matrices $P_1$ and $P_2$ such that $A=P_1 B P_2$.\n\n\\end{corollary}\n\\begin{definition}\\label{def0.3.1}\nLet $B$ be a finite spectral subset of $\\mathbb{R}$ with spectrum $\\Lambda$, $\\#B=\\#\\Lambda=:N$. The matrix \n$$\\frac{1}{\\sqrt{N}}(e^{2\\pi ib\\cdot\\lambda})_{b\\in B,\\lambda\\in\\Lambda}$$\nis a Hadamard matrix and we called it {\\it the Hadamard matrix associated to $B$ and $\\Lambda$}.\n\\end{definition}\n\nThis enables us to describe the spectral sets of size $2,3,4,5$.\n\n\\begin{theorem} \\label{standard}\nLet $B \\subset \\mathbb{Z}$ have $N$ elements and spectrum $\\Lambda$. Assume $0$ is in $B$ and $\\Lambda$. Suppose the Hadamard matrix associated to $(B,\\Lambda)$ is equivalent to the standard $N$ by $N$ Hadamard matrix. Then $B$ has the form $B=d B_0$ where $d$ is an integer and $B$ is a complete set of residues modulo $N$ with $\\gcd(B)=1$. In this case any such spectrum $\\Lambda$ has the form $\\Lambda = \\frac{1}{R} f L_0$ where $f$ and $R$ are integers, $L_0$ is a complete set of residues modulo $N$ with greatest common divisor one, and $R=NS$ where $S$ divides $df$ and $\\frac{df}{S}$ is mutually prime with $N$. The converse also holds.\n\n\\end{theorem}\n\n\n\n\n\n\n\\begin{corollary}\\label{pr0.1}A set $B \\subset \\mathbb{Z}$ with $|B|=N=2$, $3$, or $5$, where $0 \\in B$ is spectral if and only if $B= N^k B_0$ where $k$ is a positive integer and $B_0$ is a complete set of residues modulo $N$.\n\\end{corollary}\n\n\n\nWe can also describe all possible Hadamard pairs of size 2,3,4,5.\n\n\n\\begin{theorem}\\label{thha4}\nLet $B$ be spectral with spectrum $\\Lambda$ and size $N=4$. Assume $0$ is in both sets. Then there exists a set of integers $L$, containing $0$, and an integer scaling factor $R$ so that $\\Lambda= \\frac{1}{R} L$.\n\n$(B,L)$ is a Hadamard pair (each containing $0$) of integers of size $N=4$, with scaling factor $R$, if and only if $R=2^{C+M+a+1} d$, $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$, and $L=2^M \\{0, n_1, n_1 + 2^a n_2, 2^a n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ is a positive integer, $C$ and $M$ are non-negative integers, and $d$ divides $c_1 n$, $c_3 n$, $n_2 c$, and $n_3 c$, where $c$ is the greatest common divisor of the $c_k$'s and similarly for $n$.\n\\end{theorem}\n\nUsing the classification of Hadamard matrices of small dimension we can also show that Hadamard pairs of size 2,3,4,5 can always be complemented. We can give a more general result:\n\n\\begin{theorem}\\label{th0.4a}\nLet $(B,L)$ be a Hadamard pair of integers of size $N$ (containing zero as their first element), with scaling factor an integer $R$, where the matrix associated with $(B,L)$ is equivalent to the $N\\times N$ standard Hadamard matrix. Assume that all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then $(B,L)$ has a complementary Hadamard pair of integers.\n\\end{theorem}\n\n\n \n\n\n\n\n\\begin{theorem}\\label{th0.5a}\nLet $(B,L)$ be a Hadamard pair of size $|B|=|L|=2,3,4$ or $5$, with scaling factor $R$, and assume all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then $(B,L)$ has a complementary Hadamard pair.\n\\end{theorem}\n\nThe cases 2,3,5 follow imediately from Theorem \\ref{th0.4a} since the Hadamard matrix associated to the pair $(B,L)$ has to be equivalent to the standard one. For size 4, the situation is different. \n\nA useful tool for our construction of Hadamard pairs is the following proposition, which is closely relation to Di\\c{t}\\u a's construction of Hadamard matrices (see e.g. \\cite{TaZy06}):\n\n\\begin{proposition} \\label{prHP}\n Let $(B,L)$ and $(F,G)$ be Hadamard pairs of integers with the same scaling factor $R$ and such that $b\\cdot g$ is a multiple of $R$ for every $g\\in G$ and $b\\in B$. Then $(B\\oplus F, L \\oplus G)$ is a Hadamard pair with scaling factor $R$.\n \\end{proposition}\n \n\n\nFinally, we study spectral sets with Lebesgue measure as part of the original Fuglede conjecture. A wonderful result due to Iosevich and Kolountzakis \\cite{IoKo12} states that the spectrum $\\Lambda$ of a bounded spectral subset $\\Omega$ of $\\mathbb{R}$ has to be periodic. More precisely\n\\begin{theorem}(\\cite{IoKo12}) Let $\\Omega$ be a bounded Borel subset of $\\mathbb{R}$ with Lebesgue measure $|\\Omega|=1$. If $\\Omega$ is spectral with spectrum $\\Lambda$ then $\\Lambda$ is periodic, i.e., there exists $p>0$ such that $\\Lambda+p=\\Lambda$; moreover the period $p$ is an integer. \n\\end{theorem}\n\n\n\\begin{definition}\\label{def0.7}\nLet $p\\in\\mathbb{N}$, we say that {\\it spectral implies tile for period $p$} if every bounded Borel subset $\\Omega$ of $\\mathbb{R}$, with Lebesgue measure $|\\Omega|=1$ and which has a spectrum $\\Lambda$ of period $p$, is also a tile. \n\\end{definition}\n\nIn the original paper \\cite{Fug74}, Fuglede proved that his conjecture is true in the case when the spectrum or the tiling set is a lattice. This corresponds to the case of period equal to 1. We prove that spectral implies tile for periods 2,3,4,5. \n\n\n\\begin{theorem}\\label{th0.9}\nSpectral implies tile for period $2$, $3$, $4$, or $5$. \n\\end{theorem}\n\nWe end the paper with some examples to illustrate our results. Example \\ref{ex4.1} shows that in the well known Jorgensen Pedersen example, of a scale 4 Cantor set, the spectrum $\\Lambda$ described in \\eqref{eqspmu4} tiles $\\mathbb{Z}$ with translations and the tiling set is the spectrum of a complementary fractal measure. \n\n\n\\section{Proofs and other results}\n\n\n\\subsection{Spectra of fractals and base $R$ representations of integers}\n\n\\begin{proposition}\\label{pr1.2}\nLet $d$ be the greatest common divisor of the points in $B$. Let $M=\\max\\{ l: l\\in L\\}$, $m=\\min\\{ l: l\\in L\\}$. Then for every extreme cycle point $x$ for $(B,L)$ we have $x\\in\\frac1d\\mathbb{Z}$ and $\\frac m{R-1}\\leq x\\leq \\frac{M}{R-1}$. \n\\end{proposition}\n\n\\begin{proof}\nSince $|m_B(x)|=1$ and $0\\in B$, using the triangle inequality we obtain that $e^{2\\pi i bx}=1$ for all $b\\in B$. Therefore $bx\\in\\mathbb{Z}$ for all $b\\in B$. \nThis implies that $dx\\in\\mathbb{Z}$ so $x\\in\\frac1d\\mathbb{Z}$. \n\nLet $x=x_0, x_1,\\dots, x_{r-1}$ be a cycle for $L$, with digits $l_0,\\dots, l_{r-1}$. Then we have \n$$\\frac{x_0+l_0+Rl_1+\\dots+R^{r-1}l_{r-1}}{R^r}=x_0\\mbox{ so }x_0=\\frac{l_0+Rl_1+\\dots+R^{r-1}l_{r-1}}{R^r-1}$$\nwhich implies\n$$x_0\\leq \\frac{M\\frac{R^{r}-1}{R-1}}{R^r-1}=\\frac{M}{R-1},$$\nand similarly for the lower bound. \n\\end{proof}\n\n\\begin{proposition}\\label{pr1.3}\nLet $L$ be a complete set of representatives $\\mod R$. Then every integer $x$ has a unique representation in base $R$ using digits in $L$. Moreover any such representation $l_0l_1\\dots$ is eventually periodic, i.e., there exists $n_0\\geq0$ and $r\\geq 1$ such that $l_{n+r}=l_n$ for all $n\\geq n_0$.\n\\end{proposition}\n\n\\begin{proof}\nLet $x\\in\\mathbb{Z}$ and $x_0=x$. Since $L$ is a complete set of representatives $\\mod R$, there is a unique $l_0\\in L$ and some $x_1\\in\\mathbb{Z}$ such that $x_0=Rx_1+l_0$. By induction, we obtain the sequence $\\{x_n\\}$ and $\\{l_n\\}$ in a unique way. We have to show that the sequence $\\{l_n\\}$ is eventually periodic. \nLet $M=\\max\\{|l| : l\\in L\\}$. By induction we can show that, for all $n\\in\\mathbb{N}$,\n$$x_n=\\frac{x_0-l_0-Rl_1-\\dots-R^{n-1}l_{n-1}}{R^n}.$$\nThis implies that for $n$ large enough such that $|x_0\/R^n|\\leq 1$ we have \n$$| x_n|\\leq 1+M\\left(\\frac{1}R+\\dots+\\frac{1}{R^{n-1}}\\right)\\leq1+ \\frac M{R-1}.$$\nSo $x_n$ lies in a compact interval from some point on. But $x_n$ is also an integer so the numbers $x_n$ take finitely many values. Therefore there exists $n_0\\geq 0$ and $r\\geq 1$ such that $x_{n_0}=x_{n_0+r}$. This implies that $l_{n_0}=l_{n_0+r}$ and $x_{n_0+1}=x_{n_0+r+1}$. By induction $l_{n}=l_{n+r}$ for all $n\\geq n_0$. \n\\end{proof}\n\n\\begin{definition}\\label{def1.4}\nIf $L$ is a complete set of representatives $\\mod R$, we write $x=l_0l_1\\dots$ if $l_0l_1\\dots$ is the base $R$ representation of $x$ using digits in $L$. For $l_0,\\dots, l_{r-1}$ in $L$ we denote by $\\uln{l_0\\dots l_{r-1}}$ the periodic sequence $l_0\\dots l_{r-1}l_0\\dots l_{r-1}\\dots$. If $x=\\uln{l_0\\dots l_{r-1}}$ for some $l_0,\\dots,l_{r-1}\\in L$, we say that $x$ has a {\\it periodic} representation in base $R$ using digits in $L$.\n\nWe say that $l_0\\dots l_{r-1}$ is a {\\it cycle} for $L$ if there exists an integer that has base $R$ representation equal to $\\uln{l_0\\dots l_{r-1}}$.\n\\end{definition}\n\n\n\\begin{proposition}\\label{pr1.5}\nIf $\\{x_0,\\dots,x_{r-1}\\}$ is a cycle for $L$ with digits $l_0,\\dots, l_{r-1}$, and if $x_0\\in\\mathbb{Z}$ then $x_1,\\dots, x_{r-1}\\in\\mathbb{Z}$ and the points $-x_0,\\dots,-x_{r-1}$ have periodic expansions in base $R$ using digits in $L$:\n\\begin{equation}\n-x_0=\\uln{l_0\\dots l_{r-1}},\\quad -x_1=\\uln{l_1\\dots l_{r-1}l_0},\\quad\\dots\\quad, -x_{r-1}=\\uln{l_{r-1}l_0\\dots l_{r-2}}.\n\\label{eq1.5.1}\n\\end{equation}\nConversely, if $-x_0\\in\\mathbb{Z}$ has a periodic expansion in base $R$ using digits in $L$ , $-x_0=\\uln{l_0\\dots l_{r-1}}$, and we define \n$$x_1=-\\,\\uln{l_1\\dots l_{r-1}l_0},\\quad\\dots\\quad, x_{r-1}=-\\,\\uln{l_{r-1}l_0\\dots l_{r-2}},$$\nthen $\\{x_0,\\dots, x_{r-1}\\}$ is a cycle for $L$ contained in $\\mathbb{Z}$. \n\n\\end{proposition}\n\n\\begin{proof}\nIf $\\{x_0,x_1\\dots,x_{r-1}\\}$ is a cycle for $L$ with digits $l_0,\\dots, l_{r-1}$ then $-x_{r-1}=R(-x_0)+l_{r-1}$ so $x_{r-1}\\in\\mathbb{Z}$. By induction, all points in this cycle are in $\\mathbb{Z}$. We have also $-x_0=R(-x_1)+l_0$, $-x_1=R(-x_2)+l_1,\\dots$. This shows that $-x_0=\\uln{l_0\\dots,l_{r-1}}$, $-x_1=\\uln{l_1\\dots l_{r-1}l_0}$, etc.\n\nFor the converse, if $-x_0=\\uln{l_0\\dots l_{r-1}}$ then $-x_0=R(-x_1)+l_0$ and the number $-x_1$ will have the representation $\\uln{l_1\\dots l_{r-1}l_0}$. \nThis implies also $(x_0+l_0)\/R=x_1$. The rest follows by induction.\n\\end{proof}\n\n\n\\begin{myproof}[Proof of Proposition \\ref{pr0.1.6}]\nFirst, note that the points in $L$ are incongruent $\\mod R$. Indeed if $l-l'=Rk$ for some $k\\in\\mathbb{Z}$, then from the unitarity of the matrix in Definition \\ref{def1.1i} we have\n$$0=\\frac1N\\sum_{b\\in B}e^{2\\pi i R^{-1}b\\cdot(l-l')}=\\frac{1}{N}\\sum_{b\\in B}e^{2\\pi i b\\cdot k}=1,$$\na contradiction.\nFrom Proposition \\ref{pr1.5} we see that for any extreme cycle point $x$, the point $-x$ has a periodic representation using digits in $L$. Also, if $x$ is an integer that has a periodic representation using digits in $L$, then $-x$ is an cycle point in $\\mathbb{Z}$. Since $-x$ is in $\\mathbb{Z}$ it follows that $m_B(-x)=1$ so $-x$ is an extreme cycle point for $(B,L)$. \n\nThis implies that the set $\\Lambda'$ of integers that can be represented in base $R$ using digits in $L$, contains $-C$ for all extreme cycles $C$. We show that $R\\Lambda'+L\\subset \\Lambda'$. \nIf $x\\in\\Lambda'$, $x=l_0l_1\\dots$ then $Rx+l_{-1}=l_{-1}l_0l_1\\dots$ so $Rx+l_{-1}\\in \\Lambda'$ for any $l_{-1}\\in L$. \n\nThe minimality of $\\Lambda(L)$ implies that $\\Lambda(L)\\subset\\Lambda'$. To obtain the converse inclusion, take \n$x\\in\\Lambda'$. With Proposition \\ref{pr1.3}, $x$ has an eventually periodic expansion \n$x=k_0\\dots k_{n-1}\\uln{l_0\\dots l_{r-1}}$. If $c=\\uln{l_0\\dots l_{r-1}}$ then $x=k_0+R k_1+\\dots+R^{n-1} k_{n-1}+R^n c$. We have that $-c$ is an extreme cycle point so $c$ is in $\\Lambda(L)$. By the invariance of $\\Lambda(L)$ we get that $x\\in\\Lambda(L)$. So $\\Lambda'\\subset\\Lambda$. \n\\end{myproof}\n\n\n\n\n\n\n\n\n\\subsection{The Coven-Meyerowitz property and complementary Hadamard pairs}\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.2.1.2}]\nThe hard part of this theorem was covered in \\cite[Theorem A]{CoMe99} where the tiling property is proved, i.e., the existence of the set $B'$ such that $B\\oplus B'=\\mathbb{Z}_R$, and in \\cite[Theorem 1.5]{Lab02} where it is shown the spectral property, i.e., the existence of the set $L$. We will include parts of their proofs here to be able to get some more information. \n\n\nThe set $B'$ is defined as follows: first, define the polynomial $B'(x)=\\prod \\Phi_s(x^{t(s)})$ where the product is take over all the prime power factors of $R$ which are not in $S_A$ and $t(s)$ is the largest factor of $R$ which is prime to $s$. It is shown in \\cite{CoMe99} that this polynomial has coefficients $0$ or $1$, therefore it corresponds to a set $B'$, and $B\\oplus B'=\\mathbb{Z}_R$ (addition modulo $R$). \n\nTake a number $s$ that appears in the product that defines $B'(x)$. Since $s$ is a prime power, say $s=p^\\alpha$, the cyclotomic polynomial is of the form $\\Phi_s(x)=1+x^{p^{\\alpha-1}}+x^{2p^{\\alpha-1}}+\\dots+x^{(p-1)p^{\\alpha-1}}$ (see e.g. \\cite[Lemma 1.1]{CoMe99}). Hence\n$$\\Phi_s(x^{t(s)})=1+x^{p^{\\alpha-1}t(s)}+x^{2p^{\\alpha-1}t(s)}+\\dots+x^{(p-1)p^{\\alpha-1}t(s)}.$$\n\n So all the coefficients are nonnegative for all the factors that appear in this product. Therefore, there are no cancelations. This implies that $x^{p^{\\alpha-1}t(s)}$ appears with a positive coefficient in $B'(x)$. So $p^{\\alpha-1}t(s)$ is in $B'$. The greatest common divisor of the elements in $B'$ must divide $p^{\\alpha-1}t(s)$ which divides $st(s)$, and by the definition of $t(s)$ this will divide $R$. \n \n Therefore we have that $\\gcd(B')$ divides $R$. We will use this property to show that all the extreme cycles for the Hadamard pair $(B',L')$ are in $\\mathbb{Z}$.\n \n \n Since $B\\oplus B'=\\mathbb{Z}_R$, we have by \\cite[Lemma1.3]{CoMe99} that for any prime power $s$ that divides $R$, the cyclotomic polynomial $\\Phi_s(x)$ divides $B(x)$ or $B'(x)$. Then, with \\cite[Lemma 2.1]{CoMe99}, we obtain that $S_B$ and $S_{B'}$ are disjoint sets whose union is the set of all prime power factors of $R$, and also \n $B'(1)=\\prod_{s\\in S_{B'}}\\Phi_s(1)$, so the (T1) property is satsisfied by $B'$.\n \n To see that the (T2) property is satsisfied by $B'$ we follow again the proof of \\cite[Theorem A]{CoMe99}: it is shown there that if $s=s_1\\dots s_m$ is a product of distinct prime power factors of $R$ and $s_i$ is not in $S_B$, then $\\Phi_s(x)$ divides $\\Phi_{s_i}(t(x))$ (\\cite[Lemma 1.1.(6)]{CoMe99}) so it divides $B'(x)$. So, if all $s_1,\\dots,s_m$ are in $S_{B'}$, then they will not be in $S_B$ so $\\Phi_s(x)$ will divide $B'(x)$, which proves (T2). Hence $B'$ has the CM-property.\n \n Since $\\gcd(B)=1$, we have also $\\gcd(B\\oplus B')=1$.\n \n \n Now we take care of the spectral part. We use the proof of \\cite[Theorem 1.5]{Lab02}. The set $L$ will contain all sums of the form $R\\cdot \\sum_{s\\in S_B}\\frac{k_s}{s}$ where $s\\in S_B$, $s=p^\\alpha$ for some $\\alpha>0$, $p$ prime and $k_s\\in\\{0,\\dots,p-1\\}$. Since $B$ satsifies the CM-property, it is shown in \\cite{Lab02} that $(B,L)$ is a Hadamard pair. Obviously $L$ is a subset of $\\mathbb{Z}$, since the elements of $S_B$ divide $R$. Similarly we can construct $L'$ for $B'$, since we showed that $B'$ has the CM-property. \n \n Next we show that $L\\oplus L'=\\mathbb{Z}_R$. We have $|L|\\cdot |L'|=|B|\\cdot|B'|=R$. \n We prove that $(L-L)\\cap (L'-L')=\\{0\\}$ (in $\\mathbb{Z}_R$). Suppose we have \n \\begin{equation}\\label{eqdisj2.1}\nR\\cdot\\sum_{s\\in S_B}\\frac{k_s-l_s}{s}=R\\cdot \\sum_{s\\in S_{B'}}\\frac{k_s-l_s}{s}\\quad\\mod R,\n\\end{equation}\nwhere $k_s$, $l_s$ for $s$ in either $S_B$ or $S_{B'}$ are as above. We proved above that $S_B$ and $S_{B'}$ are disjoint and their union consists of all prime power factors of $R$. \n \n Take some prime $p$ that divides $R$ and let $s=p^\\alpha$ be the largest power that divides $R$. Then $s$ appears in one of the sums in \\eqref{eqdisj2.1}, and $R\\cdot\\frac{k_s-l_s}{s}$ is not divisible by $p$ unless $k_s=l_s$. For all the other elements $s'\\in S_B\\cup S_{B'}$ the numbers $R\\cdot\\frac{k_{s'}-l_{s'}}{s'}$ are divisible by $p$. Therefore, the equality \\eqref{eqdisj2.1} implies that $k_s=l_s$. By induction we assume that $k_s=l_s$ for all $s$ in $S_B$ or $S_{B'}$ of the form $s=p^\\beta$ with $1\\leq \\gamma\\leq \\beta\\leq\\alpha$. Then consider $s=p^{\\gamma-1}$, which is in either $S_B$ or $S_{B'}$. Then $R\\frac{k_s-l_s}{s}$ is not divisible by $p^{\\alpha-\\gamma+2}$ unless $k_s=l_s$ and for the other $s'\\in S_B\\cup S_{B'}$ for which $k_s\\neq l_s$, the number $R\\frac{k_{s'}-l_{s'}}{s'}$ is divisible by $p^{\\alpha-\\gamma+2}$. Using equation \\eqref{eqdisj2.1} we obtain that $k_s=l_s$. Therefore for all the powers $s$ of $p$ that appear in either $S_B$ or $S_{B'}$ we have $k_s=l_s$. Since the prime $p$ was an arbitrary prime factor of $R$, we get that $k_s=l_s$ for all $s\\in S_B\\cup S_{B'}$. Hence $(L-L)\\cap(L'-L')=\\{0\\}$ in $\\mathbb{Z}_R$. This means that the map from $L\\times L'$ to $\\mathbb{Z}_R$, $(l,l')\\mapsto l+l'\\mod R$ is injective. But since $|L\\times L'|=R$ the map will be also surjective so $L\\oplus L'=\\mathbb{Z}_R$.\n \n It remains to deal with the extreme cycles. By Proposition \\ref{pr1.2}, since $\\gcd(B)=1$, we have that the extreme cycles for $(B,L)$ are in $\\mathbb{Z}$. \n We also proved above that $d':=\\gcd(B')$ divides $R$. By Proposition \\ref{pr1.2} any extreme cycle for $(B',L')$ is contained in $\\frac1{d'}\\mathbb{Z}$. Take the first two points in such a cycle $x_0=\\frac{k_0}{d'}$, $x_1=\\frac{k_1}{d'}$, and for some $l_0'\\in L'$:\n $$\\frac{\\frac{k_0}{d'}+l_0'}{R}=\\frac{k_1}{d'}.$$\n Then $$\\frac{k_0}{d'}+l_0'=R\\cdot\\frac{k_1}{d'}.$$\n But since $R$ is divisble by $d'$, it follows that $R\\cdot\\frac{k_1}{d'}$ is in $\\mathbb{Z}$; also $l_0'$ is in $\\mathbb{Z}$, so $x_0=\\frac{k_0}{d'}$ is in $\\mathbb{Z}$. Thus all extreme cycles for $(B',L')$ are contained in $\\mathbb{Z}$.\n \n \n \n \n \n\\end{myproof}\n\n\\begin{myproof}[Proof of Theorem \\ref{th2.3}]\nSince $L\\oplus L'$ and $B\\oplus B'$ are complete sets of representatives $\\mod R$, they form a Hadamard pair. With Proposition \\ref{pr1.2}, we have that all extreme cycles for $(B\\oplus B', L\\oplus L')$ are contained in $\\mathbb{Z}$. With Propositions \\ref{pr1.3} and \\ref{pr0.1.6} we see that $\\mathbb{Z}$ is the spectrum for $\\mu_{B\\oplus B'}$. Using the results from \\cite{DJ11} we obtain that $\\mu_{B\\oplus B'}$ is the Lebesgue measure on its support $X_{B\\oplus B'}$ and $X_{B\\oplus B'}$ is translation congruent to $[0,1]$. \n\n\nFor (ii), note that \n$m_{B\\oplus B'}=m_B\\cdot m_{B'}$. According to \\cite{DJ06}, the Fourier transforms of the measures are \n$$\\widehat\\mu_{B\\oplus B'}(x)=\\prod_{k=1}^\\infty m_{B\\oplus B'}(R^{-k}x)$$\nand similarly for $\\mu_B$ and $\\mu_{B'}$ and the products are uniformly and absolutely convergent on compact sets. Therefore $\\widehat\\mu_{B\\oplus B'}=\\widehat\\mu_B\\cdot\\widehat\\mu_{B'}$ which implies that $\\mu_{B\\oplus B'}=\\mu_B*\\mu_{B'}$.\n\n\n(iii) follows from Proposition \\ref{pr0.1.6}.\n\nFor (iv), let $\\lambda=l_0l_1\\dots$, $\\lambda'=l_0'l_1'\\dots$, $\\gamma=k_0k_1\\dots$, $\\gamma'=k_0'k_1'\\dots$ such that $\\lambda-\\gamma=\\lambda'-\\gamma'$. Reducing $\\mod R$, we obtain $l_0-k_0\\equiv l_0'-k_0'\\mod R$. This implies $l_0+k_0'\\equiv l_0'+k_0\\mod R$, but since $L\\oplus L'$ is a complete set of representatives $\\mod R$, we get $l_0+k_0'=l_0'+k_0$ so $l_0-k_0=l_0'-k_0'$. Since $L$ and $L'$ have disjoint differences, it follows that $l_0=k_0$ and $l_0'=k_0'$. Then, by induction $l_n=k_n$ and $l_n'=k_n'$ for all $n$, so $\\lambda=\\gamma$ and $\\lambda'=\\gamma'$. \n\n\nFor (v), take an extreme cycle for $(B\\oplus B',L\\oplus L')$ with digits $a_0\\dots a_{r-1}$. Then $x=\\uln{a_0\\dots a_{r-1}}$ is a point in $\\mathbb{Z}=\\Lambda(L)\\oplus \\Lambda(L')$. Thus $x=l_0l_1\\dots + l_0'l_1'\\dots$. This implies that $a_0\\equiv l_0+l_0'\\mod R$. Since $L\\oplus L'$ is a complete set of representatives $\\mod R$, this means that $a_0=l_0+l_0'$ and $l_0=p(a_0)$, $l_0'=p'(a_0)$. By induction $l_n=p(a_n)$ and $l_n'=p'(a_n)$ for all $n$. So $l_0l_1\\dots=p(\\uln{a_0\\dots a_{r-1}})$ and $l_0'l_1'\\dots=p'(\\uln{a_0\\dots a_{r-1}})$, so $p(a_0\\dots a_{r-1})$ contains the digits of an extreme cycle for $(B,L)$ and similarly for $p'$.\n\nFor the converse, note that $R(\\Lambda(L)\\oplus\\Lambda(L'))+L\\oplus L'\\subset \\Lambda(L)\\oplus \\Lambda(L')$. So, by Proposition \\ref{pr0.1.6} and Proposition \\ref{pr1.5}, it is enough to show that $\\Lambda(L)\\oplus\\Lambda(L')$ contains all points $\\uln{a_0\\dots a_{r-1}}$ where $a_0\\dots a_{r-1}$ are the digits of an extreme cycle for $(B\\oplus B',L\\oplus L')$. But the hypothesis implies that $p(\\uln{a_0\\dots a_{r-1}})$ represents a point in $\\Lambda(L)$ and $p'(\\uln{a_0\\dots a_{r-1}})$ represents a point in $\\Lambda(L')$. One can easily see that \n$$\\uln{a_0\\dots a_{r-1}}=p(\\uln{a_0\\dots a_{r-1}})+p'(\\uln{a_0\\dots a_{r-1}})$$\nbecause the two sides are congruent $\\mod R^n$ for all $n$. This implies that $\\uln{a_0\\dots a_{r-1}}\\in\\Lambda(L)\\oplus\\Lambda(L')$.\n\n\\end{myproof}\n\n\n\\begin{proposition}\\label{pr3.4}\nLet $R$ be an integer $R\\geq 2$. Let $B,B'$ finite sets of integers such that $B\\oplus B'=\\{0,1,\\dots,R-1\\}$. Then $\\mu_B*\\mu_{B'}$ is the Lebesgue measure on $[0,1]$. If $\\Lambda$ is an orthogonal set for $\\mu_B$ and $\\Lambda'$ is an orthogonal set for $\\mu_{B'}$ then $\\Lambda$ and $\\Lambda'$ have disjoint differences.\n\\end{proposition}\n\n\\begin{proof}\nThe proof that $\\mu_B*\\mu_{B'}$ is the Lebesgue measure on $[0,1]$ is the same as the proof of Theorem \\ref{th2.3}(i) and (ii), the attractor of the IFS associated to $B\\oplus B'=\\{0,\\dots,R-1\\}$ is $[0,1]$. \nTake $\\lambda\\neq\\gamma$ in $\\Lambda$ and $\\lambda'\\neq\\gamma'$ in $\\Lambda'$ such that $\\lambda-\\gamma=\\lambda'-\\gamma'$. Since $e_\\lambda$ is orthogonal to $e_\\gamma$, we have that $\\widehat\\mu_B(\\lambda-\\gamma)=0$. Also $\\widehat\\mu_{B'}(\\lambda'-\\gamma')=0$. But $\\widehat\\mu_B$ and $\\widehat\\mu_{B'}$ can be extended to entire functions \n$$\\widehat\\mu_B(z)=\\int e^{-2\\pi itz}\\,\\mu_B(t),\\quad(z\\in\\mathbb{C})$$\nand similarly for $\\mu_B'$. Their product is the Fourier transform of the Lebesgue measure on $[0,1]$ which is \n$$\\widehat\\mu_{B\\oplus B'}(z)=e^{-\\pi i z}\\frac{\\sin\\pi z}{\\pi z}.$$\nThe zeros of $\\widehat\\mu_{B\\oplus B'}$ on $\\mathbb{R}$ have multiplicity one. But the relations above shows that $\\lambda-\\gamma=\\lambda'-\\gamma'$ is zero of multiplicity at least 2 for $\\widehat\\mu_B\\cdot\\widehat\\mu_{B'}=\\widehat\\mu_{B\\oplus B'}$. This contradiction proves the conclusion.\n\\end{proof}\n\n\n\n\\subsection{Finite sets of size 2,3,4,5}\n\n\n\\begin{remark}\\label{remf1}\nWe will often ignore the multiplicative constant $\\frac{1}{\\sqrt{N}}$ for Hadamard matrices. So, when we say that some number $z$ with $|z|=1$ is an entry in a Hadamard matrix, we actually mean that $\\frac{1}{\\sqrt{N}}z$ is.\n\n\n\nWe also note here that many times the study of a spectral set $B$ in $\\mathbb{Z}$ with spectrum $\\Lambda$ in $\\mathbb{R}$ can be reduced to the study of Hadamard pairs, so we can assume in addition that $\\Lambda$ has the form $\\Lambda=\\frac{1}{R}L$ for some $R$ integer and $L$ in $\\mathbb{Z}$. First, we examine what happens if $\\Lambda$ has only rational numbers. \n\n\nIf $\\beta$ is a finite subset of the rational numbers, and $\\Lambda$ is a spectrum of rational numbers for $\\beta$, then $B,L$ is a Hadamard pair with scaling factor $RQ$, where $R$ is the least common multiple of the denominators of the numbers in $\\Lambda$ and $Q$ the least common multiple of the the denominators of the numbers in $\\beta$, and $L=R \\Lambda$, $B=Q \\beta$.\n\n\n\nIndeed, the matrix associated with $(\\beta,\\Lambda)$ is unitary, and therefore so is the matrix associated with $(Q \\beta, R \\Lambda)$ with scaling factor $QR$.\n\n\n\nNow assume $\\beta$ is a finite subset of the rational numbers and $\\Lambda$ is a spectrum of real numbers for $\\beta$. If the unitary matrix associated with $(\\beta, \\Lambda)$ has at least one column which contains only roots of unity, then $\\Lambda$ must contain only rational numbers, because the entries of that column are $e^{2 \\pi i b \\lambda_j}$ for all $\\lambda_j \\in \\Lambda$. Thus, whenever we know such a thing about the columns of the Hadamard matrices for a certain size $N=\\#B$, we know from the above theorem that when considering spectra (for finite sets of integers), it is sufficient to consider Hadamard pairs. For example, this property holds true of $N=2$, $3$, $4$, and $5$. \n\nFor the remainder of the section we assume Hadamard pairs $(B,L)$ are such that $B$ and $L$ each contain $0$ as their first element, and due to the above notions we restrict our attention to Hadamard pairs which are subsets of $\\mathbb{Z}$.\n\nIt is clear that the first row and first column of a Hadamard matrix associated to such a Hadamard pair must contain only $1$'s (ignoring the multiplicative constant $\\frac1{\\sqrt N}$). Therefore, when we consider Hadamard matrices in this section, we consider only the ones which are in \"de-phased\" form, i.e. their first row and column contains only $1$'s. For any Hadamard matrix $H$ there are diagonal matrices $D_1$ and $D_2$ so that $D_1 H D_2$ is de-phased (see e.g. \\cite{TaZy06}), so we lose no generality in dealing with matrices in this way. In addition, we only consider one ordering of the rows and the columns of a Hadamard matrix, for changing the ordering of the rows and the columns corresponds to changing the ordering of the elements in $B$ and $L$. Since we know the equivalence classes of the Hadamard matrices for $N=2$, $3$, $4$, and $5$, and we know by Corollary \\ref{perm} that everything in those equivalence classes are permutation equivalent, we lose no generality in dealing with Hadamard matrices in this way.\n\n\n\n\\end{remark}\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{standard}]\n\nFirst, we need some lemmas.\n\n\\begin{lemma}\nLet $H$, $H'$ be two equivalent Hadamard matrices whose first rows and columns are constant $\\frac{1}{\\sqrt{N}}$. Then there exist permutations $\\pi$, $\\psi$ of $\\{1,...,N\\}$ such that\n\\begin{equation}\n{H'}_{j,k} = \\frac{H_{\\pi (1) \\psi (1)} H_{\\pi (j) \\psi (k)} }{\\sqrt{N} H_{\\pi (j) \\psi (1)} H_{\\pi (1) \\psi (k)} } .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $H$ and $H'$ are equivalent, there are permutations $\\pi$ and $\\psi$ and constants $c_1,...,c_N$, $d_1,...,d_N \\in \\mathbb{T}$ (the unit circle) such that\n\\begin{equation}\nH'_{j,k} = c_j d_k H_{\\pi (j) \\psi (k)} .\n\\end{equation}\nSince $H'_{j,1} = \\frac{1}{\\sqrt{N}} = c_j d_1 H_{\\pi (j) \\psi (1)} $, we obtain $c_j = \\frac{1}{\\sqrt{N} d_1 H_{\\pi (j) \\psi (1)} } $. Similarly, $d_k = \\frac{1}{\\sqrt{N} c_1 H_{\\pi (1) \\psi (k)} } $. Since $H'_{1,1} = \\frac{1}{\\sqrt{N}} = c_1 d_1 H_{\\pi (1) \\psi (1)}$, we obtain\n\\begin{equation}\n{H'}_{j,k} = \\frac{ H_{\\pi (j) \\psi (k)} }{{N} c_1 d_1 H_{\\pi (j) \\psi (1)} H_{\\pi (1) \\psi (k)} } ,\n\\end{equation}\nand the result follows.\n\n\n\\end{proof}\n\n\\begin{lemma}\\label{lem2.8}\nLet $H$ be a Hadamard matrix whose first row and columns are constant $\\frac{1}{\\sqrt{N}}$. Suppose $H$ is equivalent to the standard Hadamard matrix of size $N$. Then this matrix is permutation equivalent to the standard Hadamard matrix.\n\n\\end{lemma}\n\n\\begin{proof}\n Using the previous lemma, we find permutations $\\tau,\\psi$ of $\\{0,1,2,\\dots,N-1\\}$ such that\n\\begin{equation}\nH_{j,k} = \\frac{e^\\frac{2 \\pi i \\tau (j) \\psi (k) }{N} e^\\frac{2 \\pi i \\tau (1) \\psi (1) }{N} }{\\sqrt{N} e^\\frac{2 \\pi i \\tau (1) \\psi (k) }{N} e^\\frac{2 \\pi i \\tau (j) \\psi (1) }{N} } = \\frac{1}{\\sqrt{N}} e^\\frac{2 \\pi i ( \\tau (j) - \\tau (1) )( \\psi (k) - \\psi (1) }{N} .\n\\end{equation}\nNow notice that modulo $N$, the functions $\\tau'(j)= \\tau(j)-\\tau(1)$ and $\\psi'(k)=\\psi(k)-\\psi(1)$ are permutations of $\\{0,1,2,\\dots,N-1\\}$. Thus $H$ is permutation equivalent to the standard Hadamard matrix.\n\n\n\\end{proof}\n\nNow assume that $B \\subset \\mathbb{Z}$ with spectrum $\\Lambda$ has $N$ elements, and $0$ is in both sets, and the matrix associated with $B$ and $\\Lambda$ is equivalent to the standard Hadamard matrix of size $N$. If the greatest common divisor $d$ of $B$ is $1$, we may perform our calculations on the sets $\\frac{1}{d} B$ and $d \\Lambda$, which have the same associated matrix. Therefore, we assume without loss of generality that the greatest common divisor of $B$ is $1$. \n\nWe apply the lemma above, and relabel the elements in $B$, so that $C$ is a permutation of $B$ and $\\Gamma$ is a permutation of $\\Lambda$, with elements $c_0=0,c_1,\\dots,c_{N-1}$ and $\\gamma_0=0,\\gamma_1,\\dots,\\gamma_{N-1}$ respectively, and the matrix associated to $C$ and $\\Gamma$ is the standard Hadamard matrix of size $N$. From the second row of this matrix, we obtain (here $i$ is the complex number, not an index)\n\\begin{equation}\ne^{2 \\pi i c_j \\gamma_1 } = e^{ 2 \\pi i j\/N } ,\n\\end{equation}\nso $c_j \\gamma_1 = \\frac{j}{N} + m_j$ for some integers $m_j$. Then we write $\\gamma_1 = \\frac{z_1}{z_2}$ in lowest terms, as it is a rational number. Thus $c_j \\frac{z_1}{z_2} = \\frac{j+N m_j}{N} $. Taking $j=1$, we find that $z_2$ is divisible by $N$, so we let $z_2 = N z_3$. Thus $c_j z_1 = (j+N m_j) z_3$. Thus, since $z_1$ and $z_3$ are mutually prime, $z_3$ divides $c_j$ for all $j$. Since we know the greatest common divisor of $C$ is one, $z_3 = 1$. Thus $c_j z_1 \\equiv j$ modulo $N$, so $C$ is a complete set of residues modulo $N$. Therefore, so is $B$.\n\nTo prove that we can take $\\gcd(B_0)=1$, suppose $\\gcd(B_0)=e$. Then $\\frac{1}{e}B_0$ is again a complete set of representatives modulo $N$; indeed, if $\\frac{b_1}{e}\\equiv\\frac{b_2}{e}\\mod N$ then $b_1\\equiv b_2\\mod N$ so $b_1=b_2$. Also $\\gcd(B_0)=1$ and we can write $B=dB_0=de\\frac{1}{e}B_0$. \n\nNow we consider $\\Lambda$. Examining the second column of the standard matrix, we find that $e^{2 \\pi i c_1 \\gamma_k} = e^{2 \\pi i k\/N}$. Therefore $\\gamma_k$ is rational for all $k$, so $\\Lambda$ is a set of rational numbers. Let $R$ be their lowest common denominator. Then $\\Lambda = \\frac{1}{R} L$ where $L$ is a set of integers containing zero. Thus $L$ is spectral with spectrum $\\frac{1}{R} B$. So $L = f L_0$ where $L_0$ is a complete set of residues modulo $N$ with greatest common divisor one.\n\nWe now have that $(B,L)$ is a Hadamard pair with scaling factor $R$, whose matrix $H$ is equivalent (and thus permutation equivalent) to the standard Hadamard matrix. We assume without loss of generality that $H$ is the standard Hadamard matrix (after changing the order of the elements in $B$ and $L$). We let the elements in $B_0$ and $L_0$ be $b_j$ and $l_k$ respectively, $b_0=l_0=0$. Then\n\\begin{equation}\ne^{\\frac{2 \\pi i d f b_j l_k}{R}} = e^{\\frac{2 \\pi i j k}{N}}.\n\\end{equation}\nThus there are integers $m_{j,k}$ such that\n\\begin{equation}\nNdfb_j l_k = R(jk+ Nm_{j,k}) .\n\\end{equation}\nLetting $j=k=1$, we have that $N$ divides $R$ and thus $R=NS$. Thus\n\\begin{equation}\ndfb_j l_k = S(jk+ Nm_{j,k}) .\n\\end{equation}\nThus $S$ divides $dfW$ where $W$ is the product of the greatest common divisors of $B_0$ and $L_0$, and thus $W=1$. Therefore $S$ divides $df$, so $df=St$. Thus\n\\begin{equation}\ntb_j l_k = jk+ Nm_{j,k} .\n\\end{equation}\nThus $tb_1 l_1 = 1+ Nm_{j,k}$ so $t=\\frac{df}{S}$ is mutually prime with $N$.\n\nConversely, let $B= d B_0$, $L= f L_0$ and $R = NS$ where $S$ divides $df$ and that quotient $t$ is mutually prime with $N$. Assume $B_0$ and $L_0$ are complete sets of residues modulo $N$. Since $t$ is mutually prime with $N$, $t B_0$ is a complete set of residues modulo $N$. Reorder $t B_0$ and $L_0$ from least to greatest modulo $N$. Then the matrix associated with $B$ and $L$ with scaling factor $R$ has entries\n\\begin{equation}\ne^{\\frac{2 \\pi i d f b_j l_k}{R} } = e^{\\frac{2 \\pi i t b_j k}{N} } = e^{\\frac{2 \\pi i j k}{N} }.\n\\end{equation}\nThus the matrix associated with $B,L$ with scaling factor $R$ is equivalent to the standard Hadamard matrix of size $N$, so $B,L$ is a Hadamard pair with scaling factor $R$. The same reasoning applies to any spectrum of rational numbers which meets the criteria.\n\n\\end{myproof}\n\n\nThe above classifies the Hadamard pairs for a certain class which contains all Hadamard pairs of size $N=2$, $3$, and $5$. More specifically, we have the next item.\n\n\\begin{myproof}[Proof of Corollary \\ref{pr0.1}]\nAll Hadamard matrices of size $2$, $3$ and $5$ are equivalent to the respective standard Hadamard matrices of those sizes, so by Theorem \\ref{standard} the spectral sets are in the form $B=H_0 B_0$. We know $B_0$ contains $0$, and that $B_0$ is a complete set of residues modulo $N$. We let $H_0 = q N^k$ with $q$ not divisible by $N$. Then, since $N$ is prime in this special case, $q$ is an automorphism of the integers modulo $N$, so $N^k q B_0 = N^k B_1$ where $B_1=qB_0$ is a complete set of residues modulo $N$ which contains $0$.\n\n\\end{myproof}\n\n\n\n\n\n\n\nWe now move on to $N=4$, a case where there are other types of Hadamard matrices.\n\n\n\\begin{myproof}[Proof of Corollary \\ref{perm}]\nFor $N$ equal to $2$, $3$, or $5$, all dephased Hadamard matrices are equivalent to the standard one, so by Lemma \\ref{lem2.8} they are permutation equivalent to it.\n\nLet $N=4$. Let $A$ and $B$ be equivalent de-phased Hadamard matrices, where\n$$A=\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&q&-q\\\\\n 1&-1&-q&q\\\\\n 1&1&-1&-1\n\\end{pmatrix}.\n$$\nWe shall prove that $B$ is permutation equivalent to $A$.\nBefore we proceed, let us prove a lemma:\n\\begin{lemma}\\label{lem3i.1}\nIf the numbers $\\alpha,\\beta,\\gamma\\in\\mathbb{T}=\\{z\\in\\mathbb{C} : |z|=1\\}$ satisfy the relation\n\\begin{equation}\n1+\\alpha+\\beta+\\gamma=0,\n\\label{eq3i.1.1}\n\\end{equation}\nthen one of them must be $-1$.\n\\end{lemma}\n\n\\begin{proof}\nTake the conjugate in \\eqref{eq3i.1.1} and multiply by $\\alpha\\beta\\gamma$: $\\alpha\\beta\\gamma+\\alpha\\beta+\\alpha\\gamma+\\beta\\gamma=0$. Multiply \\eqref{eq3i.1.1} by $\\alpha\\beta$: $\\alpha\\beta+\\alpha^2\\beta+\\alpha\\beta^2+\\alpha\\beta\\gamma=0$. Now substract these two reltations to obtain\n$0=\\alpha\\beta^2-\\beta\\gamma+\\alpha^2\\beta-\\alpha\\gamma=(\\beta+\\alpha)(\\alpha\\beta-\\gamma)$ so\n$\\alpha+\\beta=0$ or $\\alpha\\beta=\\gamma$. Similarly, by symmetry, we obtain $\\alpha+\\gamma=0$ or $\\alpha\\gamma=\\beta$. Also $\\beta+\\gamma=0$ or $\\beta\\gamma=\\alpha$.\n\n\nIf $\\alpha+\\beta=0$ then, using \\eqref{eq3i.1.1}, we get that $\\gamma=-1$. Therefore, if one of these relations is true, then the lemma is proved. If none is true, then we must have $\\alpha\\beta=\\gamma$ and $\\alpha\\gamma=\\beta$ and $\\beta\\gamma=\\alpha$. Multiply them: $\\alpha^2\\beta^2\\gamma^2=\\alpha\\beta\\gamma$, so $\\alpha\\beta\\gamma=1$. Multiply the first relation by $\\gamma$, we obtain that $\\gamma^2=1$. Similarly $\\alpha^2=1$ and $\\beta^2=1$. So $\\alpha,\\beta,\\gamma=\\pm 1$. We cannot have all of them $1$, because of \\eqref{eq3i.1.1}, therefore one has to be -1.\n\\end{proof}\n\nTherefore the matrix $B$ has the number negative one in every row and every column. \nThus each row and column has a 1 and -1. Consider now the other entries of the matrix which are not $\\pm 1$. If we fix one, denote it by $t$ then the other non $\\pm1$ entries which lie on the same row or column will have to be $-t$, because of \\eqref{eq3i.1.1}. Using the same procedure we can fill out some more entries by $t$ and all the entries of the matrix are completely determined in this way. Now suppose we have two rows such that the entries 1 and $-1$ do not match, for example $(1, -1, \\ast,\\ast)$ and $(1,\\ast, -1,\\ast)$. Then\nthe two rows will be of the form $(1,-1, t,-t)$ and $(1,-t,-1,t)$. By orthogonality, we get $0=1+\\cj t-t-t\\cj t=\\cj t-t$. So $t$ has to be real so $t=\\pm1$.\n\nIf we have two rows such that the $\\pm1$ entries match, for example $(1,-1,t,-t)$ and $(1,-1, -t, t)$, then the last row is forced to be $(1,1,-1,-1)$.\nThus, in both cases, one of the rows has to have two ones and two $-1$. Similarly, one of the columns has the same property. Thus $B$ is permutation equivalent to a matrix of the form:\n\n\n$$C=\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&t&-t\\\\\n 1&-1&-t&t\\\\\n 1&1&-1&-1\n\\end{pmatrix}.\n$$\nTherefore $A$ is equivalent to $C$. Now let us prove another lemma. This lemma is found in \\cite{TaZy06}, where it is attributed to Haagerup \\cite{Haa97}, though it does not appear in its present form in \\cite{Haa97}.\n\n\\begin{lemma}\\label{haag}\nLet $H$ be a Hadamard matrix and consider the set $T(H) = \\{ H_{j,k} \\overline{H_{n,k}} H_{n,m} \\overline{H_{j,m}} \\}$. If $A$ and $B$ are equivalent Hadamard matrices, $T(A) = T(B)$. The set $T$ is called the invariants of a Hadamard matrix. \n\n\n\\end{lemma}\n\n\\begin{proof}\nAssume $A$ and $B$ are permutation equivalent. Then, since they have the same entries, $T(A) = T(B)$. Then it is sufficient to prove that if $A$ and $C$ are equivalent via diagonal matrices $X$ and $Y$, so that $A = X C Y$, then they have the same invariants. Note that $A_{j,k} = X_{j,j} C_{j,k} Y_{k,k}$. We compute the elements of $T(A)=T(XCY)$:\n\\begin{equation}\nA_{j,k} \\overline{A_{n,k}} A_{n,m} \\overline{A_{j,m}} = X_{j,j} C_{j,k} Y_{k,k} \\overline{X_{n,n} C_{n,k} Y_{k,k}} X_{n,n} C_{n,m} Y_{m,m} \\overline{X_{j,j} C_{j,m} Y_{m,m}}\n\\end{equation}\nSimplifying the right hand side, we then obtain the desired result as $X_{q,q} \\overline{X_{q,q}} =1$.\n\n\\end{proof}\n\nNow notice that the above lemma implies that if two Hadamard matrices are equivalent and de-phased, they have the same elements (we let $j$ and $k$ be any numbers and the rest of the indices be $1$). Therefore, examining $A$ and $C$, we can see that $t = \\pm q$, so $A$ and $C$ are permutation equivalent. But $B$ and $C$ are permutation equivalent, so $A$ and $B$ are permutation equivalent. \n\n\n\n\n\\end{myproof}\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{thha4}]\n\nWe first prove that the spectra can be decomposed as $\\frac1R L$. We know from Theorem \\ref{th1.15} and Corollary \\ref{perm} that the matrix associated with $B$ and $\\Lambda$ is permutation equivalent with a matrix that has a $-1$ in every row except the first. Therefore for each non-zero element $\\lambda_k$ of $\\Lambda$ there is a $j$ such that $b_j \\in B$ and $e^{2 \\pi i b_j \\lambda_k} = -1$. Therefore since $b_j$ are integers, $\\Lambda$ is a set of rational numbers. So we let $\\Lambda = \\frac{1}{R} L$ where $L$ is a set of integers containing $0$, and we have the result.\n\n Using Theorem \\ref{th1.15} and Corollary \\ref{perm} we have that the matrix $H:=\\frac1{\\sqrt{4}}\\left(e^{2\\pi ib\\lambda}\\right)_{b\\in B,\\lambda\\in\\Lambda}$ is of the form given in \\eqref{eqmat4}, after some pemutations of $B$ and $\\Lambda$. This means, upon some relabelling, that we have for some $\\lambda\\in\\Lambda$ and $B=\\{0,b_1,b_2,b_3\\}$: $e^{2\\pi ib_1\\lambda}=1$, $e^{2\\pi i b_2\\lambda}=-1$, $e^{2\\pi ib_3\\lambda}=-1$. Therefore $b_1\\lambda=k_1, b_2\\lambda=\\frac{2k_2+1}2,b_3\\lambda=\\frac{2k_3+1}2$ for some $k_1,k_2,k_3\\in\\mathbb{Z}$.\n\nWe can write $b_1=2^{a_1}c_1$, $b_2=2^{a_2}c_2$, $b_3=2^{a_3}c_3$ with $a_1,a_2,a_3\\geq0$ in $\\mathbb{Z}$ and $c_1,c_2,c_3$ odd. We get that $\\frac{2^{a_1}c_1}{2^{a_2}c_2}=\\frac{2k_1}{2k_2+1}$ so $2^{a_1}c_1(2k_2+1)=2^{a_2+1}k_1c_2$. This implies that $a_1\\geq a_2+1$.\n\nAlso $\\frac{2^{a_2}c_2}{2^{a_3}c_3}=\\frac{2k_2+1}{2k_3+1}$ so $2^{a_2}c_2(2k_3+1)=2^{a_3}c_3(2k_2+1)$, which implies that $a_2=a_3$.\n\nSince $B$ is spectral iff $\\frac{1}{2^{a_2}}B$ is spectral, we can assume, without loss of generality, dividing by $\\frac{1}{2^{a_1}}$, that $B$ is of the form\n$$B=\\{0,2^ac_1,c_2,c_3\\},$$\nwith $a\\geq 1$, $c_1,c_2,c_3$ odd.\n\nSince every row has a $-1$, there is a $\\lambda_2\\in\\Lambda$ such that $e^{2\\pi i 2^a c_1\\lambda_2}=-1$. Therefore $2^{a+1}c_1\\lambda_2=2m+1$ for some $m\\in\\mathbb{Z}$. So $\\lambda_2=\\frac{2m+1}{2^{a+1}c_1}$. The other two entries on the column of $\\lambda_2$ must be opposite:\n$$e^{2\\pi i c_2\\frac{2m+1}{2^{a+1}c_1}}=-e^{2\\pi i c_3\\frac{2m+1}{2^{a+1}c_1}},$$\nwhich means that\n$$\\frac{2m+1}{2^ac_1}c_2=\\frac{2m+1}{2^ac_1}c_3+2q+1,$$\nfor some $q\\in\\mathbb{Z}$. Then $(2m+1)c_2=(2m+1)c_3+2^ac_1(2q+1)$. This implies that $c_3-c_2=2^ad$ for some odd number $d$. This proves that $B$ has the given form.\n\n\n\nWe have proved that $B$ (containing $0$) is a set of integers with $N=4$ elements is spectral if and only if it is of the form given in the theorem.\n\nAssume now $(B,L)$ is a Hadamard pair with scaling factor $R$. Then, since $B$ and $L$ are both spectral sets of integers, we must have $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$ and $L=2^M \\{0, n_1, n_1 + 2^K n_2, 2^K n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ and $K$ are positive integers, and $C$ and $M$ are non-negative integers.\n\nRecall the Hadamard matrix for $N=4$,\n$$\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&e^{\\pi i q}&-e^{\\pi i q}\\\\\n 1&-1&-e^{\\pi i q}&e^{\\pi i q}\\\\\n 1&1&-1&-1\n\\end{pmatrix}.\n$$\nHere $q$ is any rational number, though we will see that not all rational numbers correspond to a Hadamard pair. We do not yet know which elements (other than $0$) in $B$ and $L$ are associated with which rows and columns. \n\nFirst we shall prove that the odd elements in $\\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$ can not be associated with the entry $+1$ in the matrix above. Let us assume for contradiction's sake that the elements $2^C g \\in B$ and $2^M f \\in L$ are associated with the matrix entry $1$, where $g$ is odd. Then $\\text{exp} \\left( \\frac{2 \\pi i 2^{C+M} g f}{R} \\right) = 1$, so $\\frac{2 \\pi i 2^{C+M} g f}{R} = 2 \\pi i Z $ for some integer $Z$. Then, the matrix entry associated with $2^{C+a} c_1 \\in B$ and $2^M f \\in L$ must be $-1$, as $0$ is associated with $1$ and $-1$ are the only other entries in that column. Then $\\text{exp} \\left( 2 \\pi i \\frac{2^{a+M+C} c_1 f}{R} \\right) = -1 = \\text{exp} \\left( 2 \\pi i \\frac{2^{a+M+C} c_1 f g}{R g} \\right)$. Substituting, $-1 = \\text{exp} \\left( 2 \\pi i \\frac{2^{a} c_1 Z}{ g} \\right)$. Since $g$ is odd, this is impossible. Therefore, the first non-zero element of $B$ (in our current ordering) must be associated with the matrix element $1$ that is not in the first row or column. By similar reasoning, so must the last element of $L$.\n\nTherefore, the first non-zero element of $B$ is associated with the second column of the matrix, as depicted above, and the last element of $L$ is associated with the last row of the matrix. In making these statements we make use of the fact that changing the order of the elements in a set which is part of a Hadamard pair permutes the columns or rows of the associated matrix and vice versa, and that therefore it is sufficient to consider the order of the rows and columns of $A$ as depicted above.\n\nNow we shall show $K=a$. We have, from the second column and last row: $\\text{exp} (\\pi i) = \\text{exp} \\left( \\frac{2 \\pi i 2^{C+M+a} c_1 g}{R} \\right) = \\text{exp} \\left( \\frac{2 \\pi i 2^{C+M+K} n_3 f}{R} \\right)$, where $g$ and $f$ are odd. Thus $R$ has a power of $2$ exactly equal to both $1+C+M+a$ and $1+C+M+K$, so $K=a$.\n\nWe now also know that $R= 2^{C+M+a+1} d$, where $d$ is odd. Let $c$ be the greatest common divisor of the $c_k$'s and $n$ that of the $n_k$'s. Examining column two in the matrix above, we can see that for every $2^Mg$ in $L$, we have $\\exp\\left(\\frac{\\pi i c_1g}{d}\\right)=\\pm1$. Therefore, $d$ must divide $c_1 g$. Thus, since $d$ is odd, $d$ divides $c_1 n_1$, then it divides $c_1 n_2$ and $c_1n_3$. Therefore $d$ divides $c_1 n$. Similarly, from the last row we have that $d$ divides $n_3 c$. From the third column and the last column, since the corresponding entries are equal or opposite, we get that $\\text{exp} \\left( 2 \\pi i \\frac{2^{a+C} c_3 l_j}{R} \\right) = \\pm 1$ for all $l_j \\in L$. Therefore, since $d$ is odd, $d$ must divide $c_3 l_j$ for every $l_j \\in L$, so as before $d$ must divide $c_3 n$. Similarly, comparing the second and third rows, we have that $d$ divides $n_2 c$.\nThus, we have that $B$, $L$, and $R$ are as stated.\n\nConversely, it is easy to check that such a $B$, $L$, and $R$ lead to the Hadamard matrix above.\n\n\n\n\\end{myproof}\n\nThis gives a complete classification of Hadamard pairs of integers when $N=4$. \n\n\\begin{remark}\nThere are Hadamard matrices that do not correspond to Hadamard pairs.\n\n\nConsider the case when $q=0$ in the construction above for Hadamard matrices where $N=4$, which corresponds to the matrix\n$$\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&1&-1\\\\\n 1&-1&-1&1\\\\\n 1&1&-1&-1\n\\end{pmatrix} .\n$$\nAssume this matrix has a Hadamard pair, so it can be written as above. Consider the matrix element associated with $c_2$ and $n_1$. We have from the proof of Theorem \\ref{thha4} that $k=\\frac{c_2 n_1}{2^a d}$, where $k$ is an integer (so that the matrix entry is $-1$ or $1$), but $c_2$ and $n_1$ are odd and the denominator of the right hand side is even, so no Hadamard pair of integers has this as the associated matrix. Therefore no Hadamard pair of rational numbers has this as the associated matrix, and therefore, since every column contains an $R$th root of unity for some integer $R$, no set of integers $B$ and set of real numbers $\\Lambda$ has this as the associated matrix.\n\nAt this point we recall that the Hadamard matrices for $N=6$ are not completely classified. The above example suggests a question: what are the Hadamard matrices for $N=6$ that arise from Hadamard pairs? We do not yet know how to answer this question.\n\\end{remark}\n\n\n\n\\subsection{Spectral sets in $\\mathbb{R}$}\n\n\n\\begin{lemma}\\label{lem0.8}\nLet $p\\in \\mathbb{N}$. Assume the following statement is true: for every set $\\Gamma=\\{\\lambda_0=0,\\lambda_1,\\dots,\\lambda_{p-1}\\}$ in $\\mathbb{R}$, which has a spectrum of the form $\\frac1p A$ with $A\\subset\\mathbb{Z}$, there exists a subset $\\mathcal T$ of $\\mathbb{Z}$ such that for any spectrum of $\\Gamma$ of the form $\\frac1pA'$ with $A'\\subset\\mathbb{Z}$, the set $A'$ tiles $\\mathbb{Z}$ by $\\mathcal T$. \n\nThen spectral implies tile for period $p$. \n\n\\end{lemma}\n\n\\begin{proof}\nThe result follows from \\cite{DJ12}.\n\n\\end{proof}\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.9}] We use Lemma \\ref{lem0.8}.\n\nFor $p=2$, take a set $\\Gamma=\\{0,\\lambda\\}$ which has a spectrum of the form $\\frac12A$ with $A\\subset\\mathbb{Z}$. Using a translation we can assume $0\\in A$, so $A=\\{0,b\\}$ with $b\\in\\mathbb{Z}$. Write $b=2^ac$ with $a\\geq 0$, $c$ odd. Since $\\frac12 A$ is a spectrum for $\\Gamma$, the matrix $\\frac1{\\sqrt2}(e^{2\\pi i\\lambda a})_{\\lambda\\in\\Lambda,a\\in A}$ is unitary and the first row is $\\frac{1}{\\sqrt2}(1,1)$ and the second is $\\frac{1}{\\sqrt2}(1, e^{2\\pi i\\lambda\\frac12 2^a c})$. Therefore $e^{2\\pi i\\lambda\\frac12 2^ac}=-1$, hence $\\frac12 2^ac\\lambda=\\frac12+k$ for some $k\\in\\mathbb{Z}$. Thus $\\lambda=\\frac{1+2k}{2^ac}$. \n\nNow take another spectrum of the same form $\\frac12A'$ with $A'=\\{0,2^{a'}c'\\}$. Then $\\lambda=\\frac{1+2k'}{2^{a'}c}$ with $k'\\in\\mathbb{Z}$. This implies that \n$2^{a'}c'(1+2k')=2^ac(1+2k)$. Since $c$ and $c'$ are odd this means that $a=a'$. So the number $a$ depends only on $\\Gamma$, not on the choice of the spectrum $\\frac12A$. \n\nIf a set $A$ is of the form $\\{0,2^ac\\}$ with $a\\geq0$, $c$ odd then $A$ tiles $\\mathbb{Z}$ by $\\mathcal T:=\\{0,1,\\dots, 2^a-1\\}\\oplus 2^{a+1}\\mathbb{Z}$. Indeed $\\{0,c\\}\\oplus 2\\mathbb{Z}=\\mathbb{Z}$ so $2^a\\{0,c\\}\\oplus 2^{a+1}\\mathbb{Z}=2^a\\mathbb{Z}$ so $A\\oplus 2^a\\mathbb{Z}\\oplus \\{0,1,\\dots, 2^a-1\\}=\\mathbb{Z}$. Since $\\mathcal T$ depends only on $a$ and not on $c$, hence it depends only on $\\Gamma$ and not on the choice of the spectrum $\\frac12A$, it follows that the hypothesis of Lemma \\ref{lem0.8} are satisfied for $p=2$ and therefore spectral implies tile for period 2. \n\n\nFor $p=3$, $\\Gamma=\\{0,\\lambda_1,\\lambda_2\\}$ which has a spectrum of the form $\\frac13A$ with $A\\subset\\mathbb{Z}$. Again, we can assume all the spectra contain $0$. \nThen $A$ is also spectral with spectrum $\\frac13\\Gamma$. From Corollary \\ref{pr0.1} we see that $A=3^aB$ with $a\\geq0$ and $B$ a complete set of representatives modulo 3. We claim that the number $a$ depends only on $\\Gamma$, not on the choice of the spectrum $\\frac13A$. As we see from the proof of Corollary \\ref{pr0.1}, the first row of the matrix $(e^{2\\pi i\\lambda b})_{\\lambda\\in\\Gamma,b\\in\\frac13A}$ is $(1,1,1)$ and the other two have the entries $\\{1,e^{2\\pi i\/3},e^{4\\pi i\/3}\\}$. This means that there is a $b_1\\in B$ such that $e^{2\\pi i \\lambda_1\\frac13 3^ab_1}=e^{2\\pi i\/3}$ and $b_1\\not\\equiv 0\\mod 3$.\nThen $\\frac13 3^ab_1\\lambda_1=\\frac13+k$ for some $k\\in\\mathbb{Z}$, so $\\lambda_1=\\frac{1+3k}{3^ab_1}$. \n\nNow take anothe spectrum $\\frac13A'$ with $A'=3^{a'}B'$. We get $\\lambda_1=\\frac{1+3k'}{3^{a'}b_1'}$ for some $k'\\in\\mathbb{Z}$, $b_1'\\in B'$, $b_1'\\not\\equiv0\\mod 3$. Then $3^{a'}b_1'(1+3k)=3^ab_1(1+3k')$. Since $b_1',b_1$ are not divisible by 3, it follows that $a=a'$. \n\nA set of the form $3^aB$ where $a\\geq 0$ and $B$ is a complete set of representatives modulo 3 tiles $\\mathbb{Z}$ by $\\mathcal T:=\\{0,1,\\dots,3^a-1\\}\\oplus 3^{a+1}\\mathbb{Z}$. Indeed \n$B\\oplus 3\\mathbb{Z}=\\mathbb{Z}$, which implies that $3^aB\\oplus 3^{a+1}\\mathbb{Z}=3^a\\mathbb{Z}$, so $3^aB\\oplus 3^{a+1}\\mathbb{Z}\\oplus\\{0,1,\\dots,3^a-1\\}=\\mathbb{Z}$.\n\nSince $\\mathcal T$ depends only on $\\Gamma$, Lemma \\ref{lem0.8} shows that spectral implies tile for period 3. \n\n\n\nFor $p=4$, $\\Gamma=\\{0,\\lambda_1,\\lambda_2, \\lambda_3 \\}$ with spectrum of the form $\\frac14A$ with $A\\subset\\mathbb{Z}$. We assume all the spectra contain $0$.\nThen $A$ is also spectral with spectrum $\\frac14\\Gamma$. From Theorem \\ref{thha4} we have $A=2^m \\{0, 2^a c_1, c_2, c_2 +2^a c_3 \\}$ where all $c_i$ are odd, $m$ and $a$ are integers, and $a$ is positive.\nIn the present case, up to permutation of rows and columns, all Hadamard matricies are equivalent to the following (we omit the constant $\\frac{1}{2}$):\n$$\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&e^{\\pi i q}&-e^{\\pi i q}\\\\\n 1&-1&-e^{\\pi i q}&e^{\\pi i q}\\\\\n 1&1&-1&-1\n\\end{pmatrix},\n$$\nwhere $q$ is a real (even rational) number.\nWithout loss of generality, we assume that $\\lambda_1$ is associated to the first column of the matrix. Then we deduce that $2^{m+a} c_1$ is associated to the last row of the matrix (see the proof of Theorem \\ref{thha4}). Hence, $e^{2 \\pi i \\frac14 \\lambda_1 2^{m+a} c_1}=1$. Thus, from the second column and either (or both) the second or third row, $e^{2 \\pi i \\frac14 \\lambda_1 2^{m} c_2}=-1$. Thus $\\lambda_1 2^{m-1}c_2$ is odd. Thus, since $c_2$ is odd, $\\lambda_1$ determines $m$. From the third column and last row, we obtain $e^{2 \\pi i \\frac14 \\lambda_2 2^{m+a} c_1}=-1$. Then $\\lambda_2 2^{m+a-1}c_1$ is odd. Thus, $\\lambda_2$ determines $m+a$. This means that $\\Gamma$ determines $m$ and $a$.\n\nTherefore, in our calculations we take $A$ as above with $a$ and $m$ fixed. It remains to show the existence of a tile $\\mathcal T$ for $A$ that depends only on $a$ and $m$, which will show by Lemma \\ref{lem0.8} that spectral implies tile for period 4.\n\nWe shall turn our attention to the simpler problem of finding a tile dependent only on $a$ for $A_0 = \\{0, 2^a c_1, c_2, c_2 +2^a c_3\\}$. We consider this set, modulo $2^{a+1}$. We have representatives for $0$, $2^a$, an odd number, and $2^a$ plus that odd number. We consider $T_0 = \\{0,2,4,6,\\dots,2^a -2\\}$. We notice that $T_0 \\oplus A_0 = \\mathbb{Z} (\\mod 2^{a+1})$. Hence, $T_0 \\oplus A_0 \\oplus 2^{a+1} \\mathbb{Z} = \\mathbb{Z}$, so we have a tile for $A_0$. We notice that $2^m T_0 \\oplus 2^m A_0 \\oplus 2^m 2^{a+1} \\mathbb{Z} = 2^m \\mathbb{Z}$, so $\\{0,1,\\dots,2^m -1\\} \\oplus 2^m T_0 \\oplus 2^{m+a+1} \\mathbb{Z} \\oplus A = \\mathbb{Z}$. Therefore, $\\mathcal T = \\{0,1,\\dots,2^m -1\\} \\oplus 2^m T_0 \\oplus 2^{m+a+1} \\mathbb{Z}$ is a tile for $A$ which depends only on $a$ and $m$, so spectral implies tile for period 4.\n\n\n\n\n\n For $p=5$, $\\Gamma = \\{0, \\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4\n \\}$, with spectrum $\\frac{1}{5} B$ where $B$ is a set of integers\n containing $0$. Then $B$ is spectral with spectrum $\\frac{1}{5}\n \\Gamma$. Then, by Corollary \\ref{pr0.1}, $B=5^a \\{0,b_1,b_2,b_3,b_4\\}$ where\n $\\{0,b_1,b_2,b_3,b_4\\}$ is a complete set of residues modulo $5$ and\n $a$ is a non-negative integer. We shall show that the number $a$\n depends only on $\\Gamma$. \\\\\n With Lemma \\ref{lem2.8} the matrix associated with $(B,\\frac15\\Gamma)$ is (after some relabeling of $B,\\Gamma$):\n $$\\begin{pmatrix}\n 1&1&1&1&1\\\\\n 1&w&w^2&w^3&w^4\\\\\n 1&w^2&w^4&w&w^3\\\\\n 1&w^3&w&w^4&w^2 \\\\\n 1&w^4&w^3&w^2&w \n\\end{pmatrix},\n$$\nwhere $w=e^{\\frac{2 \\pi i}{5}}$. Select $j$ so that $b_j \\equiv 1 (\\mod 5)$. Now select $k$ so that $e^{\\frac{2 \\pi i}{5}} = e^{\\frac{2 \\pi i b_j \\lambda_k}5}$. Then $e^{\\frac{2 \\pi i}{5}} = e^{\\frac{2 \\pi i 5^a \\lambda_k}5}$. Thus, $a$ depends only on $\\Gamma$. Thus, it remains to show the existence of a tile $\\mathcal T$ for $B$ that depends only on $a$. \\\\\nSince $B_0=\\{0,b_1,b_2,b_3,b_4\\}$ is a complete set of residues modulo $5$, a tile for $B_0$ is $T_0 = 5\\mathbb{Z}$. Therefore, a tile for $B$ is $\\mathcal T = \\{0,1,\\dots,5^a -1\\} \\oplus 5^{a+1} \\mathbb{Z}$. This tile depends only on $a$, so spectral implies tile for period $5$.\n\n\\end{myproof}\n\n\n\\subsection{Complementing Hadamard pairs}\nNow we would like to find complementary Hadamard pairs whenever possible for the cases $N=2,3,4,5$ that we have been exploring.\n\\begin{myproof}[Proof of Proposition \\ref{prHP}] One can check this directly, by verifying the orthogonality of the rows, but we show that we are in a particular case of a more general construction of Hadamard matrices. \n\n We shall prove that the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$ can be obtained by Di\\c t\\u a's construction (see e.g. \\cite{TaZy06}), and is therefore a Hadamard matrix. Di\\c ta's construction is a generalization of the fact that the tensor product of Hadamard matrices is a Hadamard matrix: Let $A$ be a Hadamard matrix and $\\{Q_1, \\dots, Q_k\\}$ be (possibly different) Hadamard matrices. Let $\\{E_1, E_2, \\dots, E_k\\}$ be unitary diagonal matrices whose first element is $1$, and where $E_1$ is the identity. Then the following is a Hadamard matrix:\n $$D = \\begin{pmatrix}\n A_{1,1} E_1 Q_1 & A_{1,2} E_2 Q_2 & \\dots & A_{1,k} E_k Q_k \\\\\n . & . & . & . \\\\\n A_{k,1} E_1 Q_1 & A_{k,2} E_2 Q_2 & \\dots & A_{k,k} E_k Q_k \\\\\n\\end{pmatrix} .\n$$\nConsider one way to write the matrix elements of the tensor product of matrices of size $N$:\n\\begin{equation}\n(A\\otimes B)_{\\alpha, \\beta} = A_{j,l} B_{m,n},\n\\end{equation}\nwhere $\\alpha = N(j-1) +m $ and $\\beta = N(l-1) +n$. As one varies $n$, $m$, $j$, and $l$, one obtains the elements of $(A\\otimes B)$. We generalize this formula to fit Di\\c{t}\\u a's construction, and assume $A$ is size $J$ and the $Q$s and $E$s are size $N$:\n\\begin{equation}\nD_{\\alpha, \\beta} = A_{j,l} \\left( E_l Q_l \\right)_{m,n} ,\n\\end{equation}\nwhere $\\alpha = J(j-1) +m $ and $\\beta = J(l-1) +n$. As before one varies the indexes on the right to obtain the entries in $D$. We notice that the $E$s are diagonal matrices, and therefore \n\\begin{equation} \\label{dita}\nD_{\\alpha, \\beta} = A_{j,l} ( E_l )_{m,m} (Q_l )_{m,n} .\n\\end{equation}\n\n\nNow consider the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$. Let $B_j \\in B$, $L_l \\in L$, $F_m \\in F$, $G_n \\in G$. Thus $j$ and $l$ range from $1$ to, say, $J$, and $n$ and $m$ range from $1$ to, say, $N$. We have\n\\begin{equation}\nX_{\\alpha, \\beta} =\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j+F_m)(L_l+G_n)\\right) \\right)_{j,l,m,n}.\n\\end{equation}\nThe interaction of the indexes on the left and right depends on the way we organize $B\\oplus F$ and $ L \\oplus G$. We shall choose to organize $B\\oplus F$ in such a way that the first $N$ elements of the set are given by $B_1 + F_m$, for $1\\leq m \\leq N$, and so on. We shall do the same things with $L \\oplus G$, fix $L$ first and vary $G$. In this way, we have determined that, as in the constructions above, $\\alpha = J(j-1) +m $ and $\\beta = J(l-1) +n$, and thus by varying $j$, $l$, $m$, and $n$, we obtain $X$. \n\n\nFrom the hypothesis, $\\texttt{exp}\\left( \\frac{2 \\pi i}{R}B_j G_n\\right)=1$ for $B_j\\in B$, $G_n\\in G$. Thus we have\n\n\\begin{equation}\nX_{\\alpha, \\beta}=\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{j,l,m,n}.\n\\end{equation}\nWe arrange the indices:\n\\begin{equation} \nX_{\\alpha, \\beta}=\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\right)_{j,l} \\left(\\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\right)_{l,m} \\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{m,n}.\n\\end{equation}\nThis is exactly like Di\\c{t}\\u{a}'s construction \\eqref{dita}: the role of the constants $(E_l)_{m,m}$ are played by the constants $\\left(\\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\right)_{l,m}$, and when $l$ or $m$ are $1$ this is indeed $1$, and otherwise they are roots of unity as required. In addition the matrices $\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\right)_{j,l}$ and $\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{m,n}$ are Hadamard matrices. Thus, the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$ is a Hadamard matrix, so they are a Hadamard pair.\n\\end{myproof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.4a}]\nAs in Theorem \\ref{standard}, we have that $B=h_0 N^f \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$ and $L=h_1 N^g \\{0 = l_0, l_1 , \\dots ,l_{N-1} \\}$ for some non-negative integers $f$ and $g$ and positive integers $h_0$ and $h_1$ not divisible by $N$, and $\\{ b_i \\}$ and $\\{ l_i \\}$ are complete sets of residues modulo $N$. Here we have decomposed the greatest common divisors of $B$ and $L$ into powers of $N$, and other numbers. Also $R=NS$ where $S$ divides $N^{f+g}h_0h_1$ and $Z_2:=N^{f+g}h_0h_1\/S$ is prime with $N$. We then have $R= N^{f+g+1} h_0 h_1 \/ Z_2$, where $N$ and $Z_2$ are mutually prime.\n\nFirst, we further rearrange things. Notice that since $R$ is an integer, $Z_2$ must divide $h_0 h_1$. We can write $h_0 = w_0 z_0$ and $h_1 = w_1 z_1$, $Z_2=z_0 z_1$. Then $z_0$ and $z_1$ are mutually prime with $N$. Therefore we may rewrite $B$ in the following way: $B=w_0 z_0 N^f \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$, where, since $z_0$ and $N$ are mutually prime, $z_0 \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$ is a complete set of residues modulo $N$. Thus we let $B=w_0 N^f \\{0 = B_0, B_1 , \\dots ,B_{N-1} \\}$, $L=w_1 N^g \\{0 = L_0, L_1 , \\dots ,L_{N-1} \\}$, where $\\{B_k \\}$ and $\\{L_j \\}$ are complete sets of residues modulo $N$, and $R=N^{f+g+1} w_0 w_1$.\n\n\n\n\nLet $B' = T_0 \\oplus T_1 \\oplus T_2 \\oplus T_3$ and $L' = U_0 \\oplus U_1 \\oplus U_2 \\oplus U_3$, where\n\\begin{equation}\nT_0 = \\{ 0,1,2,\\dots,w_0 -1 \\} ; U_0 = \\{ 0,1,2,\\dots,w_1 -1 \\}\n\\end{equation}\n\\begin{equation}\nT_1 = \\{ 0, w_0, 2w_0, \\dots, (N^f -1) w_0 \\} ; U_1 = \\{ 0, w_1, 2w_1, \\dots, (N^g -1) w_1 \\}\n\\end{equation}\n\\begin{equation}\nT_2 = \\{ 0, w_0 N^{f+1} , \\dots, (N^g - 1) w_0 N^{f+1} \\}; U_2 = \\{ 0, w_1 N^{g+1} , \\dots, (N^f -1) w_1 N^{g+1} \\}\n\\end{equation}\n\\begin{equation}\nT_3 = \\{ 0, w_0 N^{f+g+1} , \\dots, (w_1 - 1) w_0 N^{f+g+1} \\}; U_3 = \\{ 0, w_1 N^{f+g+1} , \\dots, (w_0 -1) w_1 N^{f+g+1} \\}\n\\end{equation}\n\n\nWe shall show that $B' , L'$ is the desired complementary Hadamard pair.\n\n\nFirst, we show that $B \\oplus B' = \\mathbb{Z} (\\mod R)$, and likewise for $L$ and $L'$. Notice that $B \\oplus T_1 = w_0 (\\mathbb{Z} (\\mod N^{f+1}))$. Then, $B \\oplus T_0 \\oplus T_1 = \\mathbb{Z} (\\mod N^{f+1} w_0)$. Then, $B \\oplus T_0 \\oplus T_1 \\oplus T_2 = \\mathbb{Z} (\\mod N^{f+g+1} w_0)$. Lastly, $B \\oplus T_0 \\oplus T_1 \\oplus T_2 \\oplus T_3 = \\mathbb{Z} (\\mod N^{f+g+1} w_0 w_1)$, and we are done. Similar reasoning applies to $L'$.\n\nNow we show that $B' , L'$ are a Hadamard pair with scaling factor $R$. By performing a few cancelations, we notice that $T_0 , U_3$ is a Hadamard pair with scaling factor $R$. Similarly, so is $T_1 , U_2$. In addition, notice that $t_1 u_3$ is a multiple of $R$ for every $t_1 \\in T_1 , u_3 \\in U_3 $. Thus, by Proposition \\ref{prHP}, $T_0 \\oplus T_1 , U_3 \\oplus U_2 $ is a Hadamard pair with scaling factor $R$. Similarly, so is $T_2 \\oplus T_3 , U_1 \\oplus U_0$. Now notice that $tu$ is a multiple of $R$ for every $t \\in T_2 \\oplus T_3 , u \\in U_3 \\oplus U_2 $. Thus, by Proposition \\ref{prHP}, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\nNow we show that $\\text{gcd} (B\\oplus B') = 1$. If $w_0 >1$, $1\\in B'$. If not, if $f=0$, $N \\in B'$ and $B$ contains an element of the form $Nk+1$ so $\\gcd(B\\oplus B')=1$; if $f>0$ then $1\\in B'$, so we are done.\n\nNow we show that the extreme cycles for $B',L'$ are contained in $\\mathbb{Z}$. If $f>0$, $1\\in B'$, so we are done (by Proposition \\ref{pr1.2}). If not, $\\text{gcd}(B')$ divides $w_0 N$, so the extreme cycle points are in $\\mathbb{Z} \/ w_0 N$. Consider two such points, $x$ and $y$, where $x=\\frac{y+l}{R}$ for some $l\\in L'$. Upon multiplying by $R$, we notice that the left hand side is an integer, as is $l$, so $y$ is an integer, and we are done.\n\nThus, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\n\\end{myproof}\n\nDue to the above theorem, we have a complementary Hadamard pair for every Hadamard pair when $N=2$, $3$, and $5$, whenever such a thing is possible. We turn our attention to the case $N=4$.\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.5a}]\nThe cases of size 2,3,5 are covered by Theorem \\ref{th0.4a} so we considered the case of size 4. As above, we have that $R=2^{C+M+a+1} d$, $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$, and $L=2^M \\{0, n_1, n_1 + 2^a n_2, 2^a n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ is a positive integer, $C$ and $M$ are non-negative integers, and $d$ divides $c_1 n$, $c_3 n$, $n_2 c$, and $n_3 c$, where $c$ is the greatest common divisor of the $c_k$'s and similarly for $n$.\n\nWe begin by constructing sets $B'$ and $L'$ such that $B\\oplus B' = L\\oplus L' = \\mathbb{Z} (\\mod R)$. Let $B' = T_0 \\oplus T_1 \\oplus T_2 \\oplus T_3$ and $L' = U_0 \\oplus U_1 \\oplus U_2 \\oplus U_3$, where\n\\begin{equation}\nT_0 = 2^{C+1} \\{0,1,2,\\dots2^{a-1} -1 \\} ; U_0 = 2^{M+1} \\{0,1,2,\\dots2^{a-1} -1 \\} ;\n\\end{equation}\n\\begin{equation}\nT_1 =\\{0,1,2,\\dots2^{C} -1 \\} ; U_1 = \\{0,1,2,\\dots2^{M} -1 \\}\n\\end{equation}\n\\begin{equation}\nT_2 = 2^{a+C+1} \\{0,1,2,\\dots2^{M} -1 \\} ; U_2 = 2^{a+M+1} \\{0,1,2,\\dots2^{C} -1 \\}\n\\end{equation}\n\\begin{equation}\nT_3 = U_3 = 2^{a+M+C+1} \\{0,1,2,\\dots, d -1 \\} .\n\\end{equation}\nNotice that $\\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\} \\oplus \\{0,2,4,\\dots,2^a -2 \\} = \\mathbb{Z}(\\mod2^{a+1})$. Then $$B\\oplus T_0 = 2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\} \\oplus 2^C \\{0,2,4,\\dots,2^a -2 \\} = 2^C \\mathbb{Z}_{2^{a+1}}.$$ Thus $B \\oplus T_0 \\oplus T_1 = \\mathbb{Z} (\\mod 2^{a+C+1})$. Therefore, $B \\oplus B' = \\mathbb{Z}(\\mod R)$. Similar logic applies to $L$ and $L'$.\n\nNow we must show $B',L'$ are a Hadamard pair with scaling factor $R$. Consider the polynomial\n\\begin{equation}\nB'(z) \\equiv \\sum_{b' \\in B'} z^{b'} .\n\\end{equation}\nSince $B'$ is a direct sum of sets, we have\n\\begin{equation}\nB'(z) = \\sum_{t_0 \\in T_0} z^{t_0} \\sum_{t_1 \\in T_1} z^{t_1} \\sum_{t_2 \\in T_2} z^{t_2} \\sum_{t_3 \\in T_3} z^{t_3} .\n\\end{equation}\nNow we let $p_n (z) = \\sum_{k=0}^{n-1} z^{k}$. Then, rewriting the product that is $B'(z)$, we have\n\\begin{equation}\nB'(z) = p_{2^{a-1}} (z^{2^{C+1}}) p_{2^{C}} (z) p_{2^{M}} (z^{2^{a+C+1}}) p_{d} (z^{2^{a+M+C+1}}) .\n\\end{equation}\nNow let $l_1 ' \\neq l_2 ' \\in L'$. We would like to show that if $q = l_1 ' - l_2 '$ then $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$. This in turn would imply that the matrix associated with $B',L'$ and scaling factor $R$ is unitary and thus, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\nAny difference $q$ of distinct elements in $L'$ can be written\n\\begin{equation} \\label{425}\nq= q_1 + 2^{M+1} q_2 + 2^{a+M+1} q_3 + 2^{a+M+C+1} q_4 ,\n\\end{equation}\nwhere $q_1 \\in \\pm \\{0,1,\\dots,2^M -1 \\}$, $q_2 \\in \\pm \\{0,1,\\dots,2^{a-1} -1 \\}$, \n$q_3 \\in \\pm \\{0,1,\\dots,2^C -1 \\}$, and $q_4 \\in \\pm \\{0,1,\\dots,d -1 \\}$, and at least one $q_j$ is non-zero.\n\nNotice that since $p_n (z) (z-1) = z^n - 1$, the zeroes of $p_n$ are exactly the $n$th roots of unity other than $1$. We shall use this to prove by cases that $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$ for any $q \\in L'$.\n\nNow assume $q \\neq 0$ modulo $d$. Notice $$p_d \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right)^{2^{a+C+M+1}} \\right) = p_d \\left( \\text{exp} \\left( \\frac{2 \\pi i}{d} q \\right) \\right)=0,$$ and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0.$ Thus, we may assume $q= 0$ modulo $d$, and thus we let $q = q_0 d$.\n\nNext assume $q \\neq 0$ modulo $2^M$. Then since $d$ is odd, the same is true of $q_0$. Notice $$p_{2^M} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q_0 d \\right)^{2^{a+C+1}} \\right) = p_{2^M} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^M} q_0 \\right) \\right)=0,$$ and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$. Thus, we may assume $q = 0$ modulo $2^M d$, so we let $q=q_a 2^M d$. Then, from \\eqref{425}, we can see that $q_1 = 0$, and thus $2^{M+1} d$ divides $q$. Thus we let $q= q_b 2^{M+1} d$.\n\nNext assume $q_b \\neq 0$ modulo $2^{a-1}$. Then $p_{2^{a-1}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q_b 2^{M+1} d \\right)^{2^{C+1}} \\right) = p_{2^{a-1}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^{a-1}} q_b \\right) \\right)=0$, and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$. Thus we may assume $q = 0$ modulo $2^{M+a} d$, so examining \\eqref{425}, we see that $q_2=0$. Thus $2^{M+a+1} d$ divides $q$, so we let $q= q_w 2^{M+a+1} d$.\n\nNow assume $q_w \\neq 0$ modulo $2^{C}$. Then $p_{2^{C}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q_w 2^{M+a+1} d \\right) \\right) = p_{2^{C}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^{C}} q_w \\right) \\right)=0$, and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$.\n\nThus $q$ must be a multiple of $R$, otherwise $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$. But a difference of distinct elements in $L'$ contains no such thing, so $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R} q \\right) \\right)=0$ and thus $B',L'$ are a Hadamard pair with scaling factor $R$. \n\nNext we must show that the greatest common divisor of elements in $B \\oplus B'$ is one. If $C>0$, this is true because $1\\in T_1$. If $C=0$ then $B$ contains an odd number and since $\\gcd(T_2)$ divides $2^{a+C+1}$ we get that $\\gcd(B\\oplus B')=1$. \n\nLastly, we must show that the extreme cycles for $B',L'$ are contained in $\\mathbb{Z}$. By construction, the greatest common divisor of $B'$ divides $R$. Therefore all the extreme cycle points must be in $\\mathbb{Z} \/ R$. Consider two such points, $\\frac{x}{R}$ and $\\frac{y}{R}$, consecutive in the cycle. Then we have $\\frac{x}{R} = \\frac{l'+\\frac{y}{R}}{R}$ for some $l' \\in L'$. Multiplying both sides by $R$, we can see that the left hand side is an integer. Therefore, so is the right hand side, so $\\frac{y}{R}$ must be an integer. But $y$ was arbitrary, so we are done.\n\n\\end{myproof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Examples}\n\nIn the following examples, we will frequently refer to the extreme cycles of $L\\oplus L'$. It is to be understood that these cycles are extreme for $(B \\oplus B',L\\oplus L')$ with scaling factor $R$, where, since we are dealing with complementary Hadamard pairs, the greatest common divisor of $B \\oplus B'$ is $1$. We also refer to the digits of a cycle as the cycle itself. \n\n\\begin{example}\\label{ex4.1}\nLet $R=4$, $B=\\{0,2\\}$, $B'=\\{0,1\\}$. Then $\\mu_B$ is the 4-Cantor measure defined in \\cite{JoPe98} and $\\mu_{B'}$ is a contraction by 2 of this measure. Let $L=\\{0,3\\}$ and $L'=\\{0,2\\}$. We check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nIt is easy to check that $(B,L)$ and $(B',L')$ are Hadamard pairs and $B\\oplus B'$ and $L\\oplus L'$ are complete sets of representatives $\\mod 4$. \nBy Proposition \\ref{pr1.2}, the extreme cycles for $(B,L)$ are contained in $\\frac12\\mathbb{Z}\\cap [0,1]$. We can check the points $\\{0,1\/2,1\\}$ one by one and \nwe see that the extreme cycles are $\\{0\\}$ with digits $\\underline 0$ and $\\{1\\}$ with digits $\\uln3$. \n\n\n\nFor $(B',L')$, the extreme cycles are contained in $\\mathbb{Z}\\cap[0,2\/3]$. So we have only one extreme cycle $\\{0\\}$ with digits $\\uln0$. \n\nThus, the condition (ii) in Definition \\ref{def2.1} is satisfied. Condition (iii) is also satisfied. So we have complementary Hadamard pairs. \n\nSince $B\\oplus B'=\\{0,1,2,3\\}$, the attractor $X_{B\\oplus B'}$ is the unit interval $[0,1]$ and $\\mu_{B\\oplus B'}$ is the Lebesgue measure on the unit interval. Therefore the convolution of the measures $\\mu_B$ and $\\mu_{B'}$ is the Lebegue measure on the unit interval. \n\nNext, we find the extreme cycles for $L\\oplus L'=\\{0,2,3,5\\}$. These are contained in $\\mathbb{Z}\\cap[0,5\/3]$. We have $\\frac{1+3}{4}=1$. So the only extreme cycles are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln3$. Since $p(3)=3$ and $p'(3)=0$ and $\\uln3$ is a cycle for $L$ and $\\uln0$ is a cycle for $L'$, Theorem \\ref{th2.3} (v) implies that the spectrum $\\Lambda(L)$ for $\\mu_B$ tiles $\\mathbb{Z}$ by the spectrum $\\Lambda(L')$ for $\\mu_{B'}$.\n\nNote that $\\Lambda(L)$ contains negative numbers: for example $-1$ has the representation $\\uln3$, $-4$ has the representation $0\\uln3$.\n\n\n\nTake now $L=\\{0,1\\}$ and $L'=\\{0,6\\}$. One can check as above that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. The extreme cycle for $(B,L)$ is $\\{0\\}$ with digits $\\underline 0$ and the extreme cycles for $(B',L')$ are $\\{0\\}$ with digits $\\underline 0$ and $\\{2\\}$ with digits $\\underline 6$. \n\nThe spectrum $\\Lambda(L)$ for $\\mu_B$ is the one described in \\eqref{eqspmu4}. We have $L\\oplus L'=\\{0,1,6,7\\}$. The extreme cycles for $(B\\oplus B', L\\oplus L')$ are $\\{0\\}$ with digits $\\underline 0$ and $\\{ 2\\}$ with digits $\\underline 6$. Since $p(\\underline 6)=\\underline 0$ which is an extreme cycle for $(B,L)$ and $p'(\\underline 6)=\\underline 6$ which is an extreme cycle for $(B',L')$, it follows that $\\Lambda(L)$ tiles $\\mathbb{Z}$ with $\\Lambda(L')$.\n\\end{example}\n\n\\begin{example}\\label{ex4.2}\nLet $R=4$, $B=\\{0,2\\}$, $B'=\\{0,1\\}$, $L=\\{0,1\\}$ , $L'=\\{0,2\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nThe only extreme cycle for $(B,L)$ and $(B',L')$ is $\\{0\\}$. The spectra $\\Lambda(L)$ and $\\Lambda(L')$ are contained in $\\mathbb{N}$. \nSince $L\\oplus L'=\\{0,1,2,3\\}$ there is a non-trivial extreme cycle for $L\\oplus L'$, $1=\\frac{1+3}4$. Therefore we have that $\\uln3$ is an extreme cycle for $L\\oplus L'$. \nBut $p(\\uln3)=\\uln1$ and $p'(\\uln3)=\\uln2$ and these are not extreme cycles for $L$ and $L'$ respectively. \n\\end{example}\n\n\\begin{example}\\label{ex4.3}\nLet $R=6$, $B=\\{0,1,2\\}$, $B'=\\{0,3\\}$, $L=\\{0,2,10\\}$ , $L'=\\{0,1\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nThe extreme cycles for $(B,L)$ are $\\{0\\}$ with digits $\\uln0$ and $\\{2\\}$ with digits $\\uln{(10)}$. $(B',L')$ has only the trivial cycle.\n\nWe consider $L\\oplus L' = \\{0,1,2,3,10,11\\}$. The extreme cycles are are $\\{0\\}$ with digits $\\uln0$ and $\\{2\\}$ with digits $\\uln{(10)}$. It is clear that $p(\\uln0)$ and $p'(\\uln0)$ are cycles for $L$ and $L'$ respectively. Notice that $p(\\uln{(10)})=\\uln{(10)}$, which is a cycle for $L$, and $p'(\\uln{(10)})=\\uln0$, which is a cycle for $L'$. Therefore, by theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nIf we replace $10$ in $L$ by $4$, we still have complementary Hadamard pairs, but the extreme cycles for $L\\oplus L' = \\{0,1,2,3,4,5\\}$ are different, and now all the extreme cycles for $(B,L)$ and $(B',L')$ are trivial.\nFor $L\\oplus L'$ we still have $\\{0\\}$ with digits $\\uln0$, and now the other cycle is $\\{1\\}$ with digits $\\uln5$. Since $p(\\uln5)=\\uln4$ is not a cycle for $L$ and $p'(\\uln5)=\\uln1$ is not a cycle for $L'$, we have by Theorem \\ref{th2.3} that $\\Lambda(L)\\oplus\\Lambda(L') \\neq \\mathbb{Z}$. \n\\end{example}\n\n\\begin{example}\nLet $R=8$, $B=\\{0,3,4,7\\}$, $B'=\\{0,2\\}$, $L=\\{0,3,4,7\\}$ , $L'=\\{0,2\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. The matrix associated with $(B,L)$ and scaling factor $R$ is interesting: it is\n$$\\begin{pmatrix}\n 1&1&1&1\\\\\n 1&-1&e^{\\pi i \/4}&-e^{\\pi i \/4}\\\\\n 1&-1&-e^{\\pi i \/4}&e^{\\pi i \/4}\\\\\n 1&1&-1&-1\n\\end{pmatrix}.\n$$\nThe extreme cycles for $(B,L)$ are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln{7}$. $(B',L')$ has only the trivial cycle.\n \nWe consider $L\\oplus L' = \\{0,2,3,4,5,6,7,9\\}$. The cycles are are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln{7}$. It is clear that $p(\\uln0)$ and $p'(\\uln0)$ are cycles for $L$ and $L'$ respectively. Notice that $p(\\uln{7})=\\uln{7}$, which is a cycle for $L$, and $p'(\\uln{7})=\\uln0$, which is a cycle for $L'$. Therefore, by Theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nIf we replace $2$ in $L'$ by $14$, we still have complementary Hadamard pairs, but the extreme cycles for $L\\oplus L' = \\{0,3,4,7,14,17,18,21\\}$ are different. The extreme cycles for $(B,L)$ are unchanged. The extreme cycles for $(B',L')$ are now $\\{0\\}$ with digits $\\uln{0}$ and $\\{2\\}$ with digits $\\uln{(14)}$.\n\nFor $L\\oplus L'$ we still have $\\{0\\}$ with digits $\\uln0$, and now we also have $\\{1\\}$ with digits $\\uln7$, $\\{2\\}$ with digits $\\uln{(14)}$, and $\\{3\\}$ with digits $\\uln{(21)}$. We have $p(\\uln7)=\\uln7$, which is an extreme cycle for $(B,L)$, and $p'(\\uln7)=\\uln0$, which is an extreme cycle for $(B',L')$. We also have $p(\\uln{14})=\\uln0$, which is an extreme cycle for $(B,L)$, and $p'(\\uln{14})=\\uln{(14)}$, which is an extreme cycle for $(B',L')$. Finally, we have $p(\\uln{21})=\\uln7$, which is an extreme cycle for $(B,L)$, and $p'(\\uln{21})=\\uln{(14)}$, which is an extreme cycle for $(B',L')$. Therefore, by Theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nSo $\\Lambda(L)$ tiles with two very different tiling sets $\\Lambda(\\{0,2\\})$ and $\\Lambda(\\{0,14\\})$. \n\n\n\\end{example}\n\n\\begin{acknowledgements}\nThis work was partially supported by a grant from the Simons Foundation (\\#228539 to Dorin Dutkay).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{alpha}\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe Standard Model (SM) can explain a multitude of observations. However, several phenomena still require explanations, {\\it e.g.}~the existence and nature of dark matter, the matter--antimatter asymmetry of the universe, and the origin of neutrino masses. \n\nA popular model to explain these beyond the SM physics is minimal supersymmetry (MSSM). Although the MSSM addresses issues of fine-tuning in the Higgs mass and there are dark matter candidates in MSSM, it has been constrained stringently by LHC searches \\cite{CMSsusy, ATLASsusy}. The MSSM currently also lacks a simple mechanism to generate neutrino masses as well as the baryon asymmetry of the universe. As such, it is necessary to consider supersymmetric models beyond the MSSM. \n\nOne extension of the MSSM that addresses these questions is the $R$-symmetric MSSM \\cite{HALL1991289}. In $R$-symmetric MSSM the superpartners are charged under a global $U(1)_R$ symmetry while their SM counterparts are neutral. While this global symmetry is unbroken, gauginos cannot be Majorana particles. Additional adjoint fields with opposite $U(1)_R$ charge, with respect to gauginos, are introduced so that gauginos can acquire Dirac masses \\cite{FAYET1975104, FAYET1976135}. $R$-symmetric MSSM addresses SUSY $CP$ and flavor problems by forbidding one-loop diagrams mediated by Majorana gauginos as well as forbidding left-right sfermion mixing \\cite{Kribs:2007ac, Dudas:2013gga}. Apart from gauginos, this model requires electroweak (EW) partners for higgsinos with an opposite $R$-charge so that a $\\mu$-term is allowed. It was shown that the scalar components of these new superfields can help to have a first-order EW phase transition \\cite{Fok:2012fb}. Moreover, new interactions can bring in new sources of \\emph{CP} violation. Hence this model can potentially explain the baryon asymmetry of the universe\\footnote{Another mechanism for generating the baryon asymmetry in such models is oscillations and out-of-equilibrium decays of a pseudo-Dirac bino \\cite{Ipek:2016bpf}.} \\cite{Fok:2012fb}. Furthermore, Dirac gluinos make the fine-tuning problem milder in $R$-symmetric MSSM \\cite{Kribs:2007ac, Fox:2002bu}. \n\nThe global $U(1)_R$ symmetry is broken because the gravitino acquires a mass. Consequently small $U(1)_R$-breaking Majorana masses for gauginos will be generated through anomaly mediation \\cite{Randall:1998uk, Giudice:1998xp, ArkaniHamed:2004yi}. Since the $U(1)_R$ symmetry is only approximate, gauginos in $R$-symmetric MSSM are pseduo-Dirac fermions, having both Dirac and Majorana masses. \n\nThe LHC phenomenology of $R$-symmetric MSSM is different than minimal SUSY models. For example in $R$-symmetric MSSM the supersymmetric particles need to be produced in particle--antiparticle pairs since the initial SM state is $U(1)_R$ symmetric. Furthermore some production channels for supersymmetric particles are not available due to the $U(1)_R$ symmetry. Hence collider limits on $R$-symmetric MSSM tend to be less stringent than the ones on MSSM, see \\emph{e.g.}, \\cite{Frugiuele:2012kp, Alvarado:2018rfl,Diessner:2017ske,Kalinowski:2015eca}. \n\nIn this work we study the LHC phenomenology of a version of the $R$-symmetric MSSM in which the $U(1)_R$ symmetry is elevated to $U(1)_{R-L}$, where $L$ is the lepton number. We give the details of the model in Section~\\ref{sec:model}. It has been shown that in this model the pseudo-Dirac bino can play the role of right-handed neutrinos \\cite{Coloma:2016vod} and light Majorana neutrino masses are generated via an inverse-seesaw mechanism. The smallness of the light neutrino masses is given by a hierarchy between the source of $U(1)_R$-breaking, namely the gravitino mass $m_{3\/2}$, and the messenger scale $\\Lambda_M$. As benchmark points this requires $m_{3\/2}\\sim$~10~keV and $\\Lambda_M\\sim$~100~TeV. \n\nThe mixing between electroweak gauginos and the SM neutrinos allows the gauginos to decay to gauge bosons and leptons, which can remove the usual $\\slashed{E}_{\\mathrm{T}}$ signature associated with SUSY searches. In cases where the lepton is a neutrino there is still $\\slashed{E}_{\\mathrm{T}}$ in the event but the kinematics are different from typical weak scale SUSY models.\nWe use current searches for jets+$\\slashed{E}_{\\mathrm{T}}$ at the LHC with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=36~{\\rm fb}^{-1}$ to find the constraints on squark and bino masses in this model. We focus on the parameter region with $100~{\\rm GeV}<\\ensuremath{M_{\\tilde{B}}}<\\ensuremath{M_{\\tilde{q}}}$. We also forecast our results for $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~{\\rm fb}^{-1}$. The analysis is described in Section~\\ref{sec:LHCpheno}. Our results are shown in Figure~\\ref{fig:Plotexcatlas} and our conclusions are given in Section~\\ref{sec:conclusions}.\n\n\n\\section{Model}\n\\label{sec:model}\n\nIn this section we review the model that was considered in \\cite{Coloma:2016vod}. This is an extension of $U(1)_R$--symmetric SUSY models \\cite{Kribs:2007ac} where, instead of the $R$ symmetry, the model has a global $U(1)_{R-L}$ symmetry. The field content and the $U(1)_R$ and $U(1)_{R-L}$ charges of the relevant superfields are given in Table \\ref{table:fields}. Note that in the rest of the text we use $U(1)_R$ instead of $U(1)_{R-L}$ whenever the distinction is not important.\n\n$U(1)_R$--symmetric SUSY is an extension of the MSSM in which the superpartners of the SM particles are charged under a global $U(1)$ symmetry. The SM particles are not charged under this symmetry. This model was introduced \\cite{Kribs:2007ac} to solve the SUSY \\emph{CP} and flavor problems. Due to the $U(1)_R$ symmetry, Majorana masses for the gauginos are forbidden as well as left-right mixing of sfermions. Hence, \\emph{e.g.}, one-loop diagrams that would generate a large electric dipole moment for fermions are suppressed, solving the SUSY \\emph{CP} problem. Similar arguments follow for the flavor problem.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nSuperfields\t&\t$U(1)_R$\t&\t$U(1)_{R-L}$ \\\\\n\\hline\\hline\n$Q, U^c, D^c$\t&\t 1\t&\t1 \\\\\n$L$\t&\t1\t&\t0 \\\\\n$E^c$\t&\t1\t&\t2\t\\\\\t\n\\hline\n$H_{u,d}$\t&\t0\t&\t0\t\\\\\n$R_{u,d}$\t&\t2\t&\t2\t\\\\\n\\hline\n$W_{\\tilde{B},\\tilde{W},\\tilde{g}}$\t&\t1\t&\t1\t\\\\\n$\\Phi_{S,T,\\mathcal{O}}$\t&\t0\t&\t0\t\\\\\n\\hline\ngravitino\/goldstini & 1 & 1 \\\\ \n\\hline\n\\end{tabular}\n\\caption{The relevant field content of the model. (SM charges are not shown.) $\\Phi_{S,T,\\mathcal{O}}$ are superfields which has the same SM charges as $W_{\\tilde{B},\\tilde{W},\\tilde{g}}$ and their fermionic components, $S, T,\\mathcal{O}$ are the Dirac partners of the bino, wino and the gluino respectively. The fermionic components of the superfields $R_{u,d}$ are the Dirac partners of the Higgsinos $\\tilde{h}_{u,d}$.} \\label{table:fields}\n\\end{table}\n\nPhenomenologically one novel aspect of $U(1)_R$--symmetric SUSY is that gauginos in this model are not Majorana fermions, since they are charged under a global symmetry, but are instead Dirac particles. In order to make gauginos into Dirac fermions, adjoint fields are added for each SM gauge field with opposite $U(1)_R$ charges \\cite{Fox:2002bu}. These new fields, $\\Phi_{S,T,\\mathcal{O}}$ are called singlino, tripletino and octino respectively and their fermionic components, $S,T,\\mathcal{O}$, become the Dirac partners of the bino, weakino and the gluino\\footnote{Dirac gauginos have been studied in the literature extensively. See, \\emph{e.g.}, \\cite{Benakli:2010gi, Benakli:2008pg, Goodsell:2012fm}}. The Dirac nature of gauginos means that $t$-channel gluino exchange diagrams which contribute to squark pair production are suppressed and the rate for squark production at the LHC is reduced, allowing for lighter squarks. Such models have been dubbed ``super-safe\" \\cite{Kribs:2012gx}. Furthermore, the minimal incarnation of Dirac gauginos, supersoft SUSY breaking \\cite{Fox:2002bu}, has only a $D$-term spurion leading to improved renormalization properties. Sfermions only receive finite contributions to their mass, rather than logarithmically divergent as in $F$-term breaking scenarios. The ratio between gluino and squark masses is also larger, $m_{\\tilde{g}}\/m_{\\tilde{f}}\\sim 5-10$, than in alternative scenarios. \n\nThe unbroken $R$-symmetry forbids the usual $\\mu$-term. In order to give mass to higgsinos, superfields $R_{u,d}$, with the same SM charge as the Higgs superfield but opposite $U(1)_R$ charges are added. While the usual two Higgs doublets $H_{u,d}$ acquire vacuum expectation values (vev), $R_{u,d}$ do not.\n\nHere we will be considering several sources of SUSY breaking, both $F$- and $D$-term. We envision two sectors each of which separately break SUSY. The first contains both a $D$-term and $F$-term spurion, of comparable size, and is coupled to the fields in the supersymmetric standard model. The second is not coupled to the standard model, except through gravity and has a higher SUSY breaking scale than the first sector. We are agnostic as to whether this is in $F$- or $D$-terms, or both, and parametrize the SUSY breaking simply as $F_2$. This second sector will raise the mass of the gravitino ($m_{3\/2}$) and provide an additional goldsitino with tree-level mass $2 m_{3\/2}$ \\cite{Cheung:2010mc,Cheung:2011jq}.\n\n\n\n\n\n\\subsection{SUSY breaking and superpartner masses}\n\nWe focus for now on the effects of the SUSY breaking that is communicated non-gravitationally to the SM. SUSY is broken in a hidden sector which communicates with the visible sector at the messenger scale $\\Lambda_M$. SUSY breaking is incorporated via the spurions,\n\\begin{align}\nW'_\\alpha = \\theta_\\alpha D~,\\qquad X=\\theta^2 F~.\n\\end{align}\nWe assume that $X$ transforms non-trivially under some symmetry of the SUSY-breaking sector so that gauginos do not have Majorana masses of the form $\\int d^2\\theta (X\/\\Lambda_M)W_\\alpha W^\\alpha$, where $W^\\alpha$ is a SM gauge field strength superfield. $W'_\\alpha$ is the field strength of a hidden $U(1)'$ which gets a $D$-term vev. In this case, Dirac gaugino masses come from the supersoft term \\cite{Fox:2002bu}\n\\begin{align}\\label{eq:supersoftop}\n\\int d^2\\theta\\, \\frac{\\sqrt{2} c_i}{\\Lambda_M} W'_\\alpha W_i^\\alpha\\Phi_i~,\n\\end{align}\nwhere $c_i$ is a dimensionless coefficient, that we take to be $\\mathcal{O}(1)$, and $i=\\tilde{B}, \\tilde{W}, \\tilde{g}$. This operator can be generated by integrating out messenger fields, of mass $\\sim \\Lambda_M$, charged under both the SM and the $U(1)'$. The Dirac mass of the gaugino is $M_i=c_i D\/\\Lambda_M$. \nThe operator (\\ref{eq:supersoftop}) also gives a mass to the scalar adjoint, while leaving the pseudoscalar massless\\footnote{Pseudoscalars in the extended superpartners can acquire masses through another soft term of the form $\\int d^2\\theta \\frac{W'_\\alpha W^{'\\alpha}}{\\Lambda_M^2}\\Phi_i^2$~\\cite{Fox:2002bu}.}, and introduces a trilinear coupling between the scalar adjoint, the SM and the $D$-term. At one loop a scalar charged under gauge group $i$ receives a \\emph{finite} soft mass from the gaugino\n\\begin{equation}\nm^2 = \\frac{C_i \\alpha_i \\left(M_i\\right)^2}{\\pi}\\log 4~,\n\\end{equation}\nwhere $C_i$ is the quadratic Casimir of the scalar and we have assumed the scalar adjoint only receives a mass from (\\ref{eq:supersoftop}). We will be interested in a spectrum with the bino in the $\\mathcal{O}(100~{\\rm GeV} - {\\rm TeV})$ mass range and the squarks in the same range, but heavier than the bino. If the sfermion masses are entirely from the supersoft operator this means the right-handed sleptons would be below the LEP bound. Thus, at least for the right handed sleptons, we include additional sources of SUSY breaking through the operator\n\\begin{equation}\n\\int d^4\\theta\\, \\frac{X^\\dagger X}{\\Lambda_M^2}c_{ij} \\Psi^\\dagger_i \\Psi_j~,\n\\end{equation} \nwith $\\Psi_i$ a right handed lepton superfield. We assume $F\\sim D$ and $c_{ij}\\sim 1$.\nThe squarks can be heavier than the bino from the finite supersoft contributions alone, as long as the gluino is sufficiently heavy, in the multi-TeV mass range. \n\n\nAs all global symmetries, $U(1)_R$ is broken due to gravity. Anomaly mediation \\cite{Randall:1998uk} generates a Majorana mass for the gauginos proportional to the gravitino mass, $m_{3\/2}$, \n\\begin{align}\nm_i=\\frac{\\beta(g)}{g}m_{3\/2}~,\n\\end{align}\nwhere $\\beta(g)$ is the beta function for the appropriate SM gauge coupling $g$. The gravitino picks up mass from all sources of SUSY breaking, $m_{3\/2}^2 = \\sum_i (F_i^2 + D_i^2\/2)\/\\sqrt{3} M_{\\rm Pl}^2$. We assume that the messenger scale $\\Lambda_M$ is below the Planck scale and thus $m_i\\ll M_i$. We ignore the small anomaly mediated corrections to scalar masses.\nNote that $U(1)_R$-breaking Majorana masses for the Dirac partners, $\\tilde{m}_i \\Phi_i \\Phi_i$, could also be generated. We assume these are much smaller than the Dirac gaugino masses as well. (For LHC studies we will set the Majorana masses to zero.)\nDue to the small anomaly-mediated Majorana gaugino masses, the gauginos are pseudo-Dirac particles. \n\n\n\n\n\\subsection{Neutrino masses}\n\\label{sec:neutrinomasses}\n\nIt has been shown in \\cite{Coloma:2016vod} that the operators,\n\\begin{equation}\n\\frac{f_i}{\\Lambda_M^2}\\int d^2\\theta\\, W'_\\alpha W_{\\tilde{B}}^\\alpha H_u L_i \\ \\ \\ \\text{and}\\ \\ \\ \\frac{d_i}{\\Lambda_M}\\int d^4\\theta\\, \\phi^\\dagger \\Phi_S H_u L_i\n\\end{equation}\n(where $\\phi=1+\\theta^2 m_{3\/2}$) can generate two non-zero neutrino masses through the Inverse Seesaw mechanism\n\\cite{Mohapatra:1986aw,Mohapatra:1986bd}, with the bino--singlino pair acting as a pseudo-Dirac right-handed neutrino. These operators can be generated by integrating out two pairs of gauge singlets $N_i,N_i'$, with R-charge 1 and lepton number $\\mp 1$.\n\n\nOnce the Higgs acquires a vev the neutrino-bino mass mixing matrix, in the basis $(\\nu_i, \\tilde{B}, S)$, is \n\\begin{equation}\n\\mathbb{M} =\t\\begin{pmatrix}\n\t\t\t0_{3\\times3}\t&\t\\mathbf{Y}v\t&\t\\mathbf{G}v \\\\\n\t\t\t\\mathbf{Y}^Tv\t&\tm_{\\tilde{B}}\t&\tM_{\\tilde{B}}\t\t\\\\\n\t\t\t\\mathbf{G}^Tv\t&\tM_{\\tilde{B}}\t\t\t&\tm_S\n\t\t\t\\end{pmatrix}~, \\label{eq:Mneutrino}\n\\end{equation}\t\t\t\nwith $Y_i= f_i M_{\\tilde{B}}\/\\Lambda_M$ and $G_i = d_i m_{3\/2}\/\\Lambda_M$. The mass matrix $\\mathbb{M}$ has an Inverse Seesaw structure with $\\mathbf{G}\\ll \\mathbf{Y}$. The light neutrino masses do not depend on the Dirac bino mass and at normal ordering they are given by\n\\begin{align}\nm_1 = 0,~~~m_2=\\frac{m_{3\/2}\\, v^2}{\\Lambda_M^2}(1-\\rho),~~~ m_3=\\frac{m_{3\/2}\\, v^2}{\\Lambda_M^2}(1+\\rho)~,\n\\end{align}\nwhere $\\rho = \\hat{\\mathbf{Y}}\\cdot\\hat{\\mathbf{G}}$, which is determined by the neutrino mass splittings to be $\\simeq0.7$. We ignore the small corrections, $\\mathcal{O}(m_S\/M_D)$, due to Majorana masses.\nParametrically the neutrino masses are\n \\begin{align}\n m_\\nu\\simeq (2-20)\\times10^{-2}~{\\rm eV}\\left(\\frac{m_{3\/2}}{10~{\\rm keV}}\\right)\\left(\\frac{100~{\\rm TeV}}{\\Lambda_M}\\right)^2~.\n \\end{align}\nTo recover the correct neutrino mixing matrix, and by setting all phases in the neutrino sector to zero, $\\mathbf{Y}$ and $\\mathbf{G}$ must have the approximate form\n\\begin{equation}\n\\mathbf{Y}\\simeq \\frac{M_{\\tilde{B}}}{\\Lambda_M}\n\\begin{pmatrix}\n0.35 \\\\\n0.85 \\\\\n0.35\n\\end{pmatrix},\\quad\n\\mathbf{G}\\simeq \\frac{m_{3\/2}}{\\Lambda_M}\n\\begin{pmatrix}\n-0.06 \\\\\n0.44 \\\\\n0.89\n\\end{pmatrix}~. \\label{eq:YG}\n\\end{equation}\n\nLow-energy searches for lepton flavor violation place strong constraints on these couplings. The strongest current constraint comes from ${\\rm Br}(\\mu\\to e\\gamma)$ \\cite{TheMEG:2016wtm} and places a lower bound on the messenger scale $\\Lambda_M>30$~TeV, independent of $M_{\\tilde{B}}$ or $m_{3\/2}$. Future experiments, \\emph{e.g.} Mu2e \\cite{Bartoszek:2014mya}, will probe $\\Lambda_M\\sim100$~TeV. We use the word ``bi$\\nu$o\" from now on to refer to the pseudo-Dirac bino in order to emphasize that it is involved in neutrino-mass generation.\n\n\n\\subsection{Neutralino mixing}\n\nIn $R$-symmetric models the Higgs sector is extended by two additional $SU(2)$ doublets, $R_{u,d}$, that do not acquire a vev. Once electroweak symmetry is broken there is mixing between $R_{u,d}$ and the adjoint fermions, $S$ and $T$, in addition to the usual wino-bi$\\nu$o mixing. However, the neutrinos only mix with the bi$\\nu$o. Significant neutralino mixing only changes the collider phenomenology and does not affect the generation of neutrino masses, which happens only through bi$\\nu$o--neutrino mixing. We follow \\cite{Kribs:2008hq} to investigate the neutralino mixing in this model.\n\nThe relevant part of the superpotential for neutralino mixing is \n\\begin{align}\\label{eq:Wneutralino}\n\\mathcal{W}=\\mu_u H_u R_u + \\mu_d H_d R_d + \\Phi_S \\left( \\lambda^u_{\\tilde{B}} H_u R_u + \\lambda^d_{\\tilde{B}} H_d R_d \\right) + \\Phi_T \\left( \\lambda^u_{\\tilde{W}} H_u R_u + \\lambda^d_{\\tilde{W}} H_d R_d \\right).\n\\end{align}\nAfter EW symmetry breaking, together with Dirac gaugino masses, kinetic terms and ignoring the small Majorana gaugino masses, (\\ref{eq:Wneutralino}) generates the neutralino mass matrix \n\\begin{align} \\label{eq:Mneutralino}\n\\mathbb{M}_N=\\begin{pmatrix}\n\t\tM_{\\tilde{B}}\t&\t0\t&\t\\frac{g_Y v_u}{\\sqrt{2}}\t&\t-\\frac{g_Y v_d}{\\sqrt{2}} \\\\\n\t\t0\t\t\t&\tM_{\\tilde{W}}\t&\t-\\frac{g_2 v_u}{\\sqrt{2}}\t&\t\\frac{g_2 v_d}{\\sqrt{2}} \\\\\n\t\t\\frac{\\lambda_{\\tilde{B}}^u v_u}{\\sqrt{2}}\t&\t-\\frac{\\lambda_{\\tilde{W}}^u v_u}{\\sqrt{2}}\t&\t\\mu_u\t&\t0\t\\\\\n\t\t-\\frac{\\lambda_{\\tilde{B}}^d v_d}{\\sqrt{2}}\t&\t\\frac{\\lambda_{\\tilde{W}}^d v_d}{\\sqrt{2}}\t&\t0\t&\t\\mu_d\n\\end{pmatrix}\n\\end{align}\nin the basis $( \\tilde{B},\\tilde{W}, \\tilde{R}_u, \\tilde{R}_d )\\times ( S,T,\\tilde{h}_u, \\tilde{h}_d )$, where $\\tilde{R}_{u,d}$ are the fermionic components of the superfield $R_{u,d}$ (see Table \\ref{table:fields}). Here $v_{u,d}\\equiv \\langle H_{u,d}\\rangle$ are the up\/down-type Higgs vevs defined as $v_u^2+v_d^2=v^2\/2 \\simeq(174~{\\rm GeV})^2$ and $M_{\\tilde{B},\\tilde{W}}$ are the bi$\\nu$o and wino Dirac masses defined in (\\ref{eq:supersoftop}).\n\nThe neutralino mass matrix $\\mathbb{M}_N$ has a rather simple form due to the Dirac nature of gauginos. It further simplifies for large $\\tan\\beta \\equiv v_u\/v_d$. In this limit\n\\begin{align}\n\\mathbb{M}_N\\simeq\\begin{pmatrix}\n\t\tM_{\\tilde{B}}\t&\t0\t&\t\\frac{g_Y v}{2}\t&\t0 \\\\\n\t\t0\t\t\t&\tM_{\\tilde{W}}\t&\t-\\frac{g_2 v}{\\sqrt{2}}\t\t&\t0 \\\\\n\t\t\\frac{\\lambda_{\\tilde{B}}^u v}{2}\t&\t-\\frac{\\lambda_{\\tilde{W}}^u v}{2}\t&\t\\mu_u\t&\t0\t\\\\\n\t\t0\t&\t0\t&\t0\t&\t\\mu_d\n\\end{pmatrix}.\n\\end{align}\nIt can immediately be seen that one of the states, with mass $\\mu_d$, decouples. Furthermore, in the limit where $\\lambda_{\\tilde{B}}^u = \\lambda_{\\tilde{W}}^u=0$, there is no mixing between the bi$\\nu$o, weakino and the Higgsinos. For simpicity, we assume a hierarchy $\\mu>M_{\\tilde{B},\\tilde{W}}$ and work in this limit, where the lightest neutralino is a pure bi$\\nu$o. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.7\\textwidth]{spectrum_new.pdf}\n\\caption{Approximate spectrum of particles in the model described in Section \\ref{sec:model}.}\\label{fig:spectrum}\n\\end{figure}\n\n\\subsection{Gravitino\/Goldstino dark matter}\n\\label{sec:goldstini}\n\nAs discussed in Section~\\ref{sec:model}, the model has two independent sectors that break supersymmetry. The breaking at the lower scale involves a $D$-term spurion but for the purposes of the discussion here it is sufficient to parametrize the two breaking scales as $\\tilde{F}_{(1,2)}$, with $\\tilde{F}^2=F^2+D^2\/2$ and $\\tilde{F}_{2}>\\tilde{F}_{1}$. A viable neutrino mass spectrum, and the spectrum of superparticles we are interested in, is achieved with $F_1\\sim (10\\ensuremath{\\rm~TeV}\\xspace)^2$ and $F_2\\sim (10^4\\ensuremath{\\rm~TeV}\\xspace)^2$, as shown in Figure~\\ref{fig:spectrum}.\n\nSince there are two independent sources of SUSY breaking, there are two goldstini \\cite{Cheung:2010mc,Cheung:2011jq}, of which one linear combination is eaten by the gravitino to have mass $m_{3\/2} = \\sqrt{(\\tilde{F}_{1})^2 + (\\tilde{F}_{2})^2}\/\\sqrt{3}M_{\\mathrm{Pl}}\\sim 10\\ensuremath{\\rm~keV}\\xspace$, while the other is twice as heavy, at tree level. Furthermore, the couplings of the uneaten goldstini are enhanced relative to the gravitino's by a factor of $\\tilde{F}_2\/\\tilde{F}_1$. \n\nBoth the gravitino, $\\tilde{G}$, and the golstino, $\\zeta$, are lighter than the other $R$-symmetry-odd particles. The goldstino can decay into a gravitino and SM particles, \\emph{e.g.} $\\zeta\\rightarrow \\tilde{G} \\psi\\bar{\\psi}$. The lifetime for this process is \n\\begin{equation}\n\\tau_{\\zeta\\rightarrow \\tilde{G}\\psi\\bar{\\psi}} \\sim \\frac{9\\pi^3}{4} \\frac{M_{\\mathrm{Pl}}^4}{m_{3\/2}^5}\\left(\\frac{\\tilde{F}_1}{\\tilde{F}_2}\\right)^2~.\n\\end{equation}\nFurthermore, even though the gravitino in this model is the LSP, it can decay into neutrinos and photons via the neutrino-bi$\\nu$o mixing. The gravitino lifetime is $\\tau = \\Gamma^{-1} \\sim \\frac{M_{\\rm Pl}^2}{\\theta^2 m_{3\/2}^3}\\sim 10^{39}$~s for $m_{3\/2}\\sim10$~keV and the bi$\\nu$o-neutrino mixing angle $\\theta \\sim Y v\/M_{\\tilde{B}}\\sim 10^{-3}$. Thus, for the range of parameters we are interested in, both the gravitino and goldstino are cosmologically stable and may contribute to dark matter.\n\nIt has been shown that a gravitino with mass $O(1-10~{\\rm keV})$ could be a warm dark matter candidate \\cite{Takayama:2000uz, Gorbunov:2008ui, Cheung:2011nn, Monteux:2015qqa}. The parameter region we study in this model suggests that gravitino could be a dark matter candidate if $T_{\\rm reh} \\sim O({\\rm TeV})$. However, with the same parameters, goldstino would be overproduced since its couplings are enhanced by a factor of $\\tilde{F}_2\/\\tilde{F}_1$. The abundance of goldstino depends on the production mechanism and $T_{\\rm reh}$. (Depending on the masses of the gravitino, goldstino and other sparticles, the dominant production channel can be either decays or scatterings.) If $T_{\\rm reh}< M_{\\tilde{B},\\tilde{q}}$, the sparticle abundance, and hence the abundance of goldstinos, will be suppressed. One expects a range of reheat temperatures where there will be just enough goldstino\/gravitino to make up the correct dark matter abundance. Finding this range requires detailed calculations for allowed range of sparticle masses. We leave this for future work.\n\n\n\n\n\\section{LHC phenomenology}\\label{sec:LHCpheno}\n\nIn this section we recast current LHC searches to find the constraints on the model described in Section \\ref{sec:model}. In order to make the LHC analysis more tractable, we assume the following mass hierarchy for the SUSY particles (see Figure \\ref{fig:spectrum}). \n\\begin{itemize}\n\\item Gravitino is the LSP with $m_{3\/2}\\sim O(10~ {\\rm keV})$.\n\\item Next-to-lightest supersymmetric particle (NLSP) is a pure bi$\\nu$o, and the other neutralinos are decoupled. Note that there are two physical bi$\\nu$o states with masses $M_{\\tilde{B}}\\pm \\frac{m_{\\tilde{B}}+m_S}{2}$. For simplicity we take the Majorana masses to be zero in the LHC analysis. Hence the physical bi$\\nu$o mass is $M_{\\tilde{B}}$. \n\\item Squarks are degenerate and heavier than the bi$\\nu$o. We do not apply any flavor tags in the analyses and only consider the first two generations of squarks, which gives a conservative estimate for the rate. \n\\item Slepton masses are of the same order as squarks, and slepton production is irrelevant.\n\\item As expected for an $R$-symmetric model, the gluino and charginos are considerably heavier than the sfermions and the squark production cross section is reduced due to the suppressed $t$-channel gluino contribution. \n\\end{itemize}\n\nThe LHC phenomenology of this model should be compared to both models with right-handed neutrinos and to the MSSM. \n\\begin{enumerate}\n\\item In models with right-handed neutrinos that address the origin and size of the neutrino masses, the SM singlets are only produced in EW processes via their mixing with the SM neutrinos. Due either to small mixing angles between the right-handed neutrinos and the SM neutrinos or to large right-handed neutrino masses, their production rates are greatly suppressed at the LHC. However, the bi$\\nu$o can be produced in decays of colored particles in the model we consider. Hence this is a neutrino-mass model that can currently be probed at the LHC. \n\n\\item In this model all supersymmetric particles need to be produced in sparticle--antisparticle pairs due to the $U(1)_R$ symmetry. (At 13~TeV LHC, the main $\\tilde{q}\\tilde{q}^\\dagger$-production channel is gluon fusion.) Furthermore, some sparticle-production channels, \\emph{e.g.} t-channel gluino exchange, are not present due again to the $U(1)_R$ symmetry. Hence it is expected that the constraints on this model are weaker than the ones on MSSM \\cite{Kribs:2012gx}. Furthermore in this model the lightest neutralino, namely the bi$\\nu$o, decays promptly and produces a combination of jets, leptons and missing energy.\n\\end{enumerate}\n\n\n\\subsection{Expected signals and search strategies}\n\\label{sec:signals}\n\nDue to the sparticle spectrum we assume, bi$\\nu$os are predominantly produced via squark decays with $Br(\\tilde{q}\\to q \\tilde{B}^\\dagger)=1$. The bi$\\nu$o subsequently decays through one of four possible modes: (i) $\\tilde{B}\\to\\tilde{G}\\gamma $; (ii) $\\tilde{B}\\to W^-\\ell^+$; (iii) $\\tilde{B}\\to Z\\bar{\\nu}$; and \n(iv) $\\tilde{B}\\to h\\bar{\\nu}$. The first decay mode is strongly suppressed by the Planck mass, $\\Gamma(\\tilde{B}\\to\\tilde{G}\\gamma)\\sim \\frac{M_{\\tilde{B}}^5}{M_{\\rm Pl}^2 m_{3\/2}^2}\\sim 10^{-8}$~eV. The rest of the decay modes are only suppressed by the neutrino-bi$\\nu$o mixing angle and their branching ratios are approximately equal to 1\/3. (Note that due to the $U(1)_{R-L}$ symmetry, $\\tilde{B}\\to W^+\\ell^-$ decay is not allowed.) The total decay width of the bi$\\nu$o is $\\Gamma_{tot}\\sim M_{\\tilde{B}}Y^2\\sim M_{\\tilde{B}}^3\/\\Lambda_M^2\\sim O(10~{\\rm MeV})$ for $M_{\\tilde{B}}=500~$GeV and $\\Lambda_M=100~$TeV. Hence, it decays promptly to final states, which include a combination of jets, leptons and missing energy. We show some of the final states with large branching fractions in Table \\ref{table:BR} and Figures \\ref{fig:signals1}-\\ref{fig:signals2}.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nSignal\t&\tBranching fraction\t&\tLHC searches \\\\\n\\hline\\hline \n$6 j + \\slashed{E}_{\\mathrm{T}}$\t&\t20$\\%$\t&\tATLAS \\cite{ATLAS-CONF-2017-022}, CMS \\cite{Khachatryan:2016kdk, Khachatryan:2016epu, Sirunyan:2017cwe} \\\\\n\\hline\n$6 j + 1\\ell + \\slashed{E}_{\\mathrm{T}}$\t&\t15$\\%$\t&\tATLAS \\cite{Aad:2016qqk, Aaboud:2016lwz}, CMS \\cite{Khachatryan:2016epu, Khachatryan:2016iqn} \\\\\n\\hline\n$4 j + 2\\ell + \\slashed{E}_{\\mathrm{T}}$\t&\t6$\\%$\t&\tATLAS \\cite{Aaboud:2016zpr, Aaboud:2016qeg}, CMS \\cite{Khachatryan:2016iqn, Khachatryan:2017qgo} \\\\\n\\hline\n$4 j + \\slashed{E}_{\\mathrm{T}}$\t&\t5$\\%$\t&\tATLAS \\cite{ATLAS-CONF-2017-022}, CMS \\cite{Khachatryan:2016kdk, Khachatryan:2016epu, Sirunyan:2017cwe} \\\\\n\\hline\n$6 j + 2\\ell$\t&\t3$\\%$\t&\tATLAS \\cite{Aaboud:2016qeg}, CMS \\cite{CMS-PAS-EXO-17-003}\\\\\n\\hline\n\\end{tabular}\n\\caption{Some of the signals that are produced by bi$\\nu$o production and subsequent decays in the model described in Section~\\ref{sec:model} with their branching fractions and relevant LHC searches. Here the leptons $\\ell = e,\\mu$ and $j=u,d,s,c$.} \\label{table:BR}\n\\end{table}\n\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\includegraphics[width=\\textwidth]{signal6j.pdf}\\label{fig:6jets}\n \\caption{6 jets + missing energy}\n \\end{subfigure}%\n ~\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\includegraphics[width=\\textwidth]{signal4j.pdf}\\label{fig:4jets}\n \\caption{4 jets + missing energy}\n \\end{subfigure}\n \\caption{Final states with jets and missing energy. We recast current SUSY searches at ATLAS and CMS for this signal.} \\label{fig:signals1}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\begin{subfigure}[b]{0.5\\textwidth}\n \\includegraphics[width=\\textwidth]{signal6j1lep.pdf}\\label{fig:4jets}\n \\caption{6 jets + 1 lepton + missing energy}\n \\end{subfigure}%\n ~\n \\begin{subfigure}[b]{0.5\\textwidth}\n \\includegraphics[width=\\textwidth]{signal6j2lep.pdf}\\label{fig:6jets2lep}\n \\caption{6 jets + 2 leptons}\n \\end{subfigure}\n \\caption{Final states with leptons and missing energy. Leptoquark searches are recast for these signals.} \\label{fig:signals2}\n\\end{figure*}\n\n\nWe emphasize the importance of final states with leptons, \\emph{e.g.} $6j+2\\ell$ and $6j + 1\\ell+\\slashed{E}_{\\mathrm{T}}$, as smoking-gun signals in determining if bi$\\nu$o is the source of neutrino mass generation, see Fig.\\ref{fig:signals2}. The bi$\\nu$o-neutrino mixing angle is $\\theta_i \\simeq \\frac{Y_i v}{M_{\\tilde{B}}}$ where $Y_i$ is given in (\\ref{eq:YG}). The branching ratio of bi$\\nu$o into different lepton species is fully determined by the neutrino mixing parameters. For example, in searches for first- and second-generation leptoquarks, relative rates of $ee:\\mu\\mu = 1:16$ and $e\\nu:\\mu\\nu= 1:2$ are expected\\footnote{Note that these branching fractions are given for the case where the phases in the PMNS matrix are set to zero. The matrix elements, hence the branching ratios, will change for non-zero phases~\\cite{Gavela:2009cd}.}. \n\n\\subsection{Analysis}\n\nOur model is implemented in \\feynrules{} \\cite{Alloul:2013bka} and the events are generated with \\madgraph{5} \\cite{Alwall:2014hca}, using \\pythia{8} \\cite{Sjostrand:2014zea} for parton shower and hadronization, and \\delphes{} \\cite{deFavereau:2013fsa} for detector simulation at $\\sqrt{s}=13$ TeV and $\\mathcal{L}=36~\\text{fb}^{-1}$. We use the default settings for jets in \\madgraph{5} with $R=0.4, ~p_{Tj}>20$ GeV and $|\\eta_j|<5$. We generate signal events for bi$\\nu$os in the mass range $100~{\\rm GeV}<\\ensuremath{M_{\\tilde{B}}}<\\ensuremath{M_{\\tilde{q}}}$ with a common squark mass for first and second generation squarks, $200~{\\rm GeV}<\\ensuremath{M_{\\tilde{q}}}<1200~$GeV, in 50~GeV mass increments. We set all other sparticle masses to 10~TeV such that they are decoupled. As the bi$\\nu$o mass gets closer to the squark mass, the computational time required to generate events increases. Hence, we do not consider splittings smaller than $25\\ensuremath{\\rm~GeV}\\xspace$, {\\it i.e.}\\ $\\ensuremath{M_{\\tilde{q}}}-\\ensuremath{M_{\\tilde{B}}} \\ge 25~\\ensuremath{\\rm~GeV}\\xspace$. For $\\ensuremath{M_{\\tilde{B}}}\\lesssim 90~$GeV, the gauge bosons are off-shell and the phase space and the energy distribution of the final states are different. We leave a study of light bi$\\nu$os to future work and focus on $\\ensuremath{M_{\\tilde{B}}}>100~$GeV. \n \nWe find that currently the most constraining search is the jets$ + \\slashed{E}_{\\mathrm{T}}$ final state due its large branching ratio and the integrated luminosity used in available analyses. At the partonic level there are processes leading to 6q$ + \\slashed{E}_{\\mathrm{T}}$ and 4q$ + \\slashed{E}_{\\mathrm{T}}$ final states, see Figure~\\ref{fig:signals2}. We analyze this search in detail and use it to constrain the parameter space of the bi$\\nu$o model.\n\n\nWe use the $m_{\\rm eff}$-based analysis given by ATLAS \\cite{ATLAS-CONF-2017-022}. The observable $m_{\\rm eff}$ is defined as the scalar sum of the transverse momenta of the leading jets and missing energy, $\\slashed{E}_{\\mathrm{T}}$. Taken together with $\\slashed{E}_{\\mathrm{T}}$, $m_{\\rm eff}$ strongly suppresses the multijet background. There are 24 signal regions in this analysis. These regions are first divided according to jet multiplicities (2-6 jets).\n Signal regions with the same jet multiplicity are further divided according to the values of $m_{\\rm eff}$ and the $\\slashed{E}_{\\mathrm{T}}\/m_{\\text{eff}}$ or $\\slashed{E}_{\\mathrm{T}}\/\\sqrt{H_T}$ thresholds. \n In each signal region, different thresholds are applied on jet momenta and pseudorapidities to reduce the SM background.\nConstraints on the smallest azimuthal separation between $\\slashed{E}_{\\mathrm{T}}$ and \n the momenta of any of the reconstructed jets further reduces the multi-jet background. Two of the signal regions require two large radius jets and in all signal regions the required jet momentum is $p_T>50~$GeV and missing energy $\\slashed{E}_{\\mathrm{T}}>250~$GeV. \n The thresholds on the observables which characterize the signal regions have been chosen to target models with squark or gluino pair production and direct decay of squarks\/gluinos or one-step decay of squark\/gluino via an intermediate chargino or neutralino.\n\nIn order to identify the allowed parameter points we compare the signal cross section to the measured cross section limits at 95$\\%$ C.L. in all 24 signal regions using the code from \\cite{Asadi:2017qon}. If the signal cross section of a parameter point exceeds the measured cross section at 95$\\%$ C.L. in at least one bin we take this parameter point to be ruled out. \n \nWe also analyze the expected exclusion limits at the end of LHC Run 3 with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~\\text{fb}^{-1}$, by rescaling with the luminosity the expected number of signal and background events, as given in~\\cite{ATLAS-CONF-2017-022}. In order to obtain the allowed parameter region at a high-luminosity LHC we use the median expected exclusion significance \\cite{Kumar:2015tna}\n\\begin{align}\nZ_{exc}=\\Big[2\\left(s-b \\log\\left(\\frac{b+s+x}{2b}\\right)-\\frac{b^2}{\\Delta_b^2}\\log\\left(\\frac{b-s+x}{2b}\\right)\\right)-(b+s-x)(1+\\tfrac{b}{\\Delta_b^2})\n\\Big]^{1\/2}~,\n\\end{align}\nwith\n\\begin{align}\nx=\\big[(s+b)^2-4sb\\tfrac{\\Delta_b^2}{(b+\\Delta_b^2)}\\big]^{1\/2}~,\n\\end{align}\nwhere $s$ is the signal, $b$ is background and $\\Delta_b$ is the uncertainty on the background prediction. For a 95$\\%$ C.L. median exclusion, we require $Z_{exc}>1.645$. We assume, as a conservative estimate, that the relative background uncertainty after $300~\\text{fb}^{-1}$ remains the same as it is now, as presented in \\cite{ATLAS-CONF-2017-022}. The estimate that $\\Delta_b\/b$ is constant could be improved upon, especially if the background is estimated from data in sidebands.\n\n\n\\subsection{Results and discussion}\n\\label{sec:results}\n\nWe show 95\\% exclusion limits on squark and bi$\\nu$o masses for current and forecasted searches in Figure~\\ref{fig:Plotexcatlas}.\nWe find that squarks heavier than 950~GeV are not excluded for any bi$\\nu$o mass by current LHC data with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=36~{\\rm fb}^{-1}$. In the mass regions we analyzed, bi$\\nu$o masses 100--150~GeV are not currently excluded for squark masses above 350~GeV, as the resulting jet momenta and missing energy do not pass the search cuts. We also project limits for a high-luminosity LHC with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~{\\rm fb}^{-1}$. This forecast shows that even with this luminosity upgrade, as long as the same cuts are used in the analysis, the LHC can probe squark masses up to 1150~GeV. However, bi$\\nu$o masses lighter than 150~GeV for $\\ensuremath{M_{\\tilde{q}}}>800$~GeV will still be allowed. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Plotexcfore300.pdf\n\\caption{\\label{fig:Plotexcatlas} Current and forecasted 95\\% exclusion limits from searches for jets$+ \\slashed{E}_{\\mathrm{T}}$ final state in the squark mass ($\\ensuremath{M_{\\tilde{q}}}$) -- bi$\\nu$o mass ($\\ensuremath{M_{\\tilde{B}}}$) plane. The dark red region is excluded by our recast of the ATLAS analysis \\cite{ATLAS-CONF-2017-022} which uses $\\mathcal{L}=36~{\\rm fb}^{-1}$ data at $\\sqrt{s}=$~13~TeV. The red dashed line shows a forecast for $\\mathcal{L}=300~\\text{fb}^{-1}$ at $\\sqrt{s}=$~13~TeV. We have not analyzed the parameter ranges in the gray striped regions, which correspond to regions where bi$\\nu$o is heavier than squarks or the bi$\\nu$o is lighter than the SM gauge bosons.}\n\\end{figure}\n\nWe find that constraints in the parameter region we considered come from (4-6)-jet signal regions with small to medium values of $m_{\\rm eff}$ for $\\mathcal{L}=36~{\\rm fb}^{-1}$. For our forecast to $\\mathcal{L}=300~{\\rm fb}^{-1}$, we find that constraints for $\\ensuremath{M_{\\tilde{q}}}<800$~GeV mostly come from a 4-jet region with small $m_{\\rm eff}$ while a 6-jet region with medium $m_{\\rm eff}$ is most constraining for $\\ensuremath{M_{\\tilde{q}}}>800$~GeV.\n\n\n\nIn discussing our results we emphasize the differences between this model and some other ($R$-symmetric) SUSY models. \n\\begin{enumerate}\n\\item In this model the gluinos are heavy and decoupled. Furthermore due to the $R$-symmetry some important squark production channels are not allowed. Hence, compared to the MSSM~\\cite{SUSYxsec}, the squark--antisquark production cross-section is $O(0.1)$ smaller. \n\\item Due to the sparticle spectrum we assume, squarks decay to a quark and the lightest neutralino 100\\% of the time. The lightest neutralino, which we take to be purely bi$\\nu$o, decays promptly to gauge bosons and leptons due to a broken $U(1)_{R-L}$ symmetry. In comparison to MSSM scenarios where the missing energy is carried by the neutralino, in this model there would be cascade decays and the missing energy is carried by light neutrinos. \n\\item Similarly, due to the large number of jets and how the missing energy is distributed in this model, constraints on squark and bi$\\nu$o masses are expected to be different than some other $R$-symmetric models, \\emph{e.g.} \\cite{Kribs:2012gx}. Although it is not straightforward to make a direct comparison, we point out that in \\cite{Kribs:2012gx} the LSP is massless and the most constraining signal region contains only 2 jets whereas in this model the constraining signal regions contain 4-6 jets. The authors in \\cite{Kribs:2012gx} mention as the LSP mass is increased to 300~GeV, all constraints disappear. However, note that, even with a finite mass, the LSP in that work does not decay. We emphasize that we do not consider the region where $\\ensuremath{M_{\\tilde{B}}}<100~$GeV. In this region the bi$\\nu$o would decay via off-shell gauge or Higgs bosons. Due to the low mass of the bi$\\nu$o, final states may not pass the missing energy and jet momentum cuts in the current analysis. We leave an analysis of this region to future work.\n\n\\item The closest study to ours is done in \\cite{Frugiuele:2012kp}. In addition to some technical differences between the two models, in~\\cite{Frugiuele:2012kp} the authors fix the lightest neutralino mass to be 1~TeV while we do a scan over both the squark and the bi$\\nu$o masses. In \\cite{Frugiuele:2012kp} the limit on the squark mass is found to be $\\ensuremath{M_{\\tilde{q}}}\\simeq 650~$GeV by using an ATLAS jets+$\\slashed{E}_{\\mathrm{T}}$ analysis \\cite{Aad:2012hm} at $\\sqrt{s}=$~7~TeV with $\\mathcal{L}=4.7~{\\rm fb}^{-1}$ data. In our work we do not consider the region where $\\ensuremath{M_{\\tilde{B}}}> \\ensuremath{M_{\\tilde{q}}}$. In this region the bi$\\nu$o decays off-shell and it is expected that energy will be distributed to jets and missing energy democratically. \n We expect the bound on the squark masses coming from the ATLAS anaylsis we use~\\cite{ATLAS-CONF-2017-022} to be similar to that given by our most constraining signal region, $5j+\\slashed{E}_{\\mathrm{T}}$, {\\it i.e.}~$\\ensuremath{M_{\\tilde{q}}}> 950$~GeV.\n\\end{enumerate}\n\nWe also analyzed the final state with $6 j + 2\\ell$, which is a possible smoking gun signature for this model as the branching fractions of the bi$\\nu$o to different lepton families is fully determined by the neutrino mixing parameters. We recast the CMS leptoquark analysis \\cite{CMS-PAS-EXO-17-003}, which looks for a final state of two muons and two jets produced in the decay of a leptoquark pair. We find that this analysis currently has a very small exclusion power due to the small signal-to-background ratio ($S\/B\\sim 10^{-2}$). \n\n\\section{Conclusions}\\label{sec:conclusions}\n\nLHC constraints on sparticle masses in the MSSM are becoming more and more stringent. Avoiding these strong experimental constraints and keeping superpartners light often leads to considering extensions of the MSSM. These extensions are characterized either by adding additional operators ({\\it e.g.}~R-parity violation) or adding additional fields ({\\it e.g.}~Dirac gauginos). We studied one such extension, with additional fields, which allows for a global $U(1)_{R-L}$ symmetry on the supersymmetric sector. This leads to phenomenology associated with both R-parity violation and Dirac gauginos. It was previously shown in \\cite{Coloma:2016vod} that the role of right-handed neutrinos can be played by one of these Dirac gauginos, the pseudo-Dirac bi$\\nu$o, and that the observed neutrino mass spectrum can be achieved. \n\nWe considered a scenario where the lightest neutralino is a pure bi$\\nu$o, and this state is the lightest SM superpartner. The squarks, which have a QCD production cross section, decay to the bi$\\nu$o. The mixing of this state with SM neutrinos means that it in turn can decay, despite the presence of a $U(1)_{R-L}$ symmetry. The bi$\\nu$o decays to a combination of quarks, leptons and missing energy. We investigated the LHC constraints on this model and found the strongest comes from a recast of the most recent ATLAS analysis with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=36~{\\rm fb}^{-1}$, see Figure~\\ref{fig:Plotexcatlas}. The constraints go up to only $\\ensuremath{M_{\\tilde{q}}}=950~$GeV and squarks as light as 350~GeV are allowed for $\\ensuremath{M_{\\tilde{B}}}=100-150~$GeV. We also forecast constraints for $\\mathcal{L}=300~{\\rm fb}^{-1}$ at $\\sqrt{s}=13~$TeV and show that high-luminosity LHC can probe up to $\\ensuremath{M_{\\tilde{q}}}=1150~$GeV if the same cuts for the jets+$\\slashed{E}_{\\mathrm{T}}$ analysis are used. However, even with the high-luminosity, low bi$\\nu$o masses cannot be excluded. The flavor of the charged lepton in the bi$\\nu$o decay depends upon the neutrino mixing parameters and thus the LHC is potentially sensitive to parameters in the neutrino sector, for instance through flavor ratios in leptoquark searches. \n\n\n\nWhile our analysis indicates that, in these models, the squarks may be as light $950$ GeV for any bi$\\nu$o mass, and as light as 350~GeV for bi$\\nu$o between 100--150 GeV, it is intriguing to wonder if they can be even lighter. We have not investigated the bounds for bi$\\nu$o mass below $100$ GeV, nor the region with $M_{\\tilde{B}}> M_{\\tilde{q}}$. It is also an interesting question to understand what are the ideal set of cuts for the jet+$\\slashed{E}_{\\mathrm{T}}$ final state to probe this model. Most importantly, the smoking-gun signals involving lepton final states need careful attention to find the best discovery path for this model. In a separate direction, the viability of gravitino\/goldstino dark matter in this model requires detailed calculations of their production mechanisms given the sparticles masses allowed by LHC data. \n\n\n\n\\section*{Acknowledgements}\n\nWe thank Pilar Coloma for her collaboration in the early stages of this work. We are grateful to Angelo Monteux for sharing his analysis code with us as well as his help with running the code. SI acknowledges support from the University Office of the President via a UC Presidential Postdoctoral fellowship and partial support from NSF Grant No.~PHY-1620638. This work was performed in part at Aspen Center for Physics, which is supported by NSF grant PHY-1607611. JG has received funding\/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896.\nPF was supported by the DoE under contract number DE-SC0007859 and Fermilab, operated by Fermi Research Alliance, LLC under\ncontract number DE-AC02-07CH11359 with the United States Department of Energy.\n\n\n\\bibliographystyle{JHEP}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSuppose that $\\ell \\geq 5$ is prime. \nMany papers \\cite{Ono-Skinner} \\cite{Bruinier} \\cite{Bruinier-Ono} \\cite{Ahlgren-Boylan} \\cite{Ahlgren-Boylan2} \\cite{Ahlgren-Choi-Rouse} \\cite{Ahlgren-Rouse}\nstudy half-integral weight modular forms with few non-vanishing coefficients modulo $\\ell$ and give applications for divisibility properties of the algebraic parts of the central critical values of modular $L$-functions and the orders of Tate-Shafarevich groups of elliptic curves.\n\nThese results are modulo $\\ell$ analogues of a theorem of Vign\\`{e}ras in characteristic $0$. \nIf $\\lambda$ is a non-negative integer and $N$ is a positive integer with $4\\mid N$, let $M_{\\lambda+\\frac{1}{2}}(\\Gamma_1(N))$ be the space of modular forms of weight $\\lambda+\\frac{1}{2}$ (in the sense of \\cite{Shimura}) on $\\Gamma_1(N)$. \nVign\\`{e}ras \nproved that a form $F(z) \\in M_{\\lambda+\\frac{1}{2}}(\\Gamma_1(N))$ whose coefficients are supported on finitely many square classes of integers is a linear combination of single-variable theta series. The precise result is below (Bruinier \\cite{Bruinier-Vigneras} gave a different proof of this theorem ).\n\n \\begin{theorem}{\\cite{Vigneras}}\\label{thm:Vigneras}\nSuppose that $\\lambda \\geq 0$ is an integer, that $N$ is a positive integer with $4 \\mid N$, and that $F(z) \\in M_{\\lambda+ \\frac{1}{2}}(\\Gamma_1(N))$. If there exist finitely many square-free integers $t_1$,$t_2$,...,$t_m$ for which\n\n\\[\nF(z)=\\sum_{i=1}^{m}\\sum_{n=0}^{\\infty}a(t_{i}n^{2})q^{t_{i}n^{2}} , \\ \\ \\ q= e^{2\\pi i z}\n\\]\n then $\\lambda=0 \\text{ or } 1$ and $F(z)$ is a linear combination of theta series.\n \\end{theorem}\n \n A recent result of Bella\\\"{\\i}che, Green and Soundararajan \\cite{Belliache-Green-Sound} implies for any half-integral weight modular form that the number of coefficients $\\leq X$ which do not vanish modulo $\\ell$\nis $\\gg\\frac{\\sqrt X}{\\log\\log X}$. It is natural to suspect that the only half-integral weight forms for which the number of non-vanishing coefficients is close to this lower bound are\nthose which are supported on finitely many square classes modulo $\\ell$. Forms of half-integral weight on $\\operatorname{SL}_2(\\mathbb{Z})$ whose coefficients are sparse modulo $\\ell$ play an important role in the recent work of Ahlgren, Beckwith and Raum \\cite{Scarcity} on scarcity of congruences for the partition function. \n \n Ahlgren, Choi and Rouse \nproved a modulo $\\ell$ analogue of Theorem~\\ref{thm:Vigneras} for forms $f(z)$ in the Kohnen plus-space $S_{\\lambda+\\frac{1}{2}}^{+}(\\Gamma_{0}(4))$. Their main theorem was the following.\n \n \\begin{theorem}{\\cite{Ahlgren-Choi-Rouse}}\\label{Ahlgren-Choi-RouseMainTheorem}\n Suppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime $v$ of $K$ above $\\ell$. Let $\\mathcal{O}_{v}$ denote the ring of $v$-integral elements of $K$. Suppose that $f \\in S^{+}_{\\lambda+\\frac{1}{2}}(\\Gamma_0(4)) \\cap \\mathcal{O}_{v}[[q]]$ satisfies\n \\[\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^{2})q^{t_{i}n^{2}} \\not \\equiv 0 \\pmod{v},\n \\]\n where each $t_{i}$ is a square-free positive integer. If $\\lambda+\\frac{1}{2} < \\ell(\\ell+1+\\frac{1}{2})$, then $\\lambda$ is even and\n \\[\n f \\equiv a(1)\\sum_{n=1}^{\\infty}n^{\\lambda}q^{n^{2}} \\pmod{v}.\n \\]\n \\end{theorem}\n \n \n In this paper, we study the analogous question for half-integral weight modular forms on $\\operatorname{SL}_{2}(\\mathbb{Z})$. Before we state our main result, we introduce some notation. If $\\lambda \\geq 0$ is an integer, $N$ is a positive integer, and $\\nu$ is a multiplier system on $\\Gamma_0(N)$ in weight $\\lambda+\\frac{1}{2}$, we denote by $S_{\\lambda+\\frac{1}{2}}(N, \\nu)$ the space of cusp forms of weight $\\lambda+\\frac{1}{2}$ and multiplier $\\nu$ on $\\Gamma_0(N)$ (details will be given in the next section). Let $\\nu_{\\eta}$ be the multiplier for the Dedekind eta function defined in \\eqref{etamultiplier}. With this notation, we prove the following theorem.\n\n\n \n \\begin{theorem}\\label{thm:main}\n Suppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime $v$ of $K$ above $\\ell$. \n Let $\\mathcal{O}_{v}$ denote the ring of $v$-integral elements in $K$. Suppose that $\\lambda$ is a non-negative integer satisfying $\\lambda +\\frac{1}{2} < \\frac{\\ell^{2}}{2}$. Suppose that $r$ is a positive integer with $(r,6)=1$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu^{r}_{\\eta}) \\cap \\mathcal{O}_{v}[[q^{\\frac{r}{24}}]]$ satisfies\n \n \\[\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^2)q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod v,\n \\]\n where each $t_{i}$ is a square-free positive integer. Then one of the following is true.\n \n \\begin{enumerate}\n \\item \n $f \\equiv a(1)\\displaystyle \\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)n^{\\lambda}q^{\\frac{n^2}{24}} \\pmod v $.\n\nIn this case, $r \\equiv 1 \\pmod{24}$ and $\\lambda$ is even.\n \n \\item\n $f \\equiv \\displaystyle a(\\ell)\\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod v $.\n \n In this case, \n $r \\equiv \\ell \\pmod{24}$ and $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\n \\item\n $f \\equiv a(1)\\displaystyle \\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)n^{\\lambda}q^{\\frac{n^2}{24}}+a(\\ell)\\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod v $, \n where $a(1) \\not \\equiv 0 \\pmod{v}$ and $a(\\ell) \\not \\equiv 0 \\pmod{v}$.\nIn this case, $r \\equiv \\ell \\equiv ~1 \\pmod{24}$ and $\\lambda \\equiv \\frac{\\ell -1}{2} \\pmod{\\ell-1}$. \n \\end{enumerate}\n \n \\end{theorem}\n\n \\begin{remark}\n For an example of case $(1)$ of Theorem~\\ref{thm:main}, let $\\ell \\geq 5$ be prime and $\\lambda$ be a nonnegative integer. \n Lemma~\\ref{Lemma1} below implies that there exists a form $f \\in S_{(\\frac{\\lambda}{2})(\\ell+1)}(1,\\nu_{\\eta})$ such that $f\\equiv\\Theta^{\\frac{\\lambda}{2}}(\\eta)\\pmod{\\ell}$, where $\\Theta$ is the Ramanujan $\\Theta$-operator defined in \\eqref{RamanujanTheta}. We have\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{\\ell}.\n \\]\n For an example of case $(2)$ of Theorem~\\ref{thm:main}, set $f=\\eta^{\\ell}$. Since $\\eta^{\\ell}(z) \\equiv \\eta(\\ell z) \\pmod{\\ell} $, we have\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{\\ell}.\n \\]\n For an example of case $(3)$, suppose that $\\ell$ is a prime such that $\\ell \\equiv 1 \\pmod{24}$. Lemma~\\ref{Lemma1} implies that there exists a form $g \\in S_{(\\frac{\\ell-1}{4})(\\ell+1)+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that \n $g \\equiv \\Theta^{\\frac{\\ell-1}{4}}(\\eta) \\pmod{\\ell}$. Set $f=24^{\\frac{\\ell-1}{2}}g+\\eta^{\\ell}E_{\\ell-1}^{\\frac{\\ell-1}{4}}$ . We have\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\frac{\\ell-1}{2}}q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{\\ell}.\n \\]\n For this example, note that $\\frac{\\ell-1}{4}(\\ell+1) \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n \\end{remark}\n \\begin{remark}\n The upper bound on $\\lambda$ is sharp. For an example which illustrates this, set $f=\\eta^{\\ell^{2}}$. Then\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell^{2}n^{2}}{24}} \\pmod{\\ell}.\n \\]\n Note that we have $\\lambda+\\frac{1}{2}=\\frac{\\ell^{2}}{2}$ in this case.\n\\end{remark}\n The paper is organized as follows. In Section $2$, we give some background results on modular forms of integral and half-integral weight. In Section $3$, we prove some preliminary results. In Section $4$, we make a preliminary reduction for the proof of Theorem~\\ref{thm:main}, and in Section $5$, we prove the theorem.\n \\section{Background}\nSuppose that $k \\in \\frac{1}{2}\\mathbb{Z}$, that $N$ is a positive integer, and that $\\chi$ is a Dirichlet character modulo $N$. For a function $f(z)$ on the upper half plane and \n\\[\n \\gamma =\\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\operatorname{GL}_{2}^{+}(\\mathbb{Q}),\n\\]\nwe have the weight $k$ slash operator \n\\[\nf(z)\\big|_k \\gamma := \\operatorname{det}(\\gamma)^{\\frac{k}{2}}(cz+d)^{-k}f\\(\\frac{az+b}{cz+d}\\).\n\\]\n\nSuppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime $v$ of $K$ above $\\ell$.\nLet $\\mathcal{O}_{v}$ be the ring of $v$-integral elements of $K$.\nIf $\\nu$ is a multiplier system on $\\Gamma_0(N)$, \nwe denote by $M_{k}(N,\\nu)$, $S_{k}(N,\\nu)$\nand $M_{k}^{!}(N, \\nu)$ \nthe spaces of modular forms, cusp forms, \nand weakly holomorphic modular forms \nof weight $k$ and multiplier \n$\\nu$ on $\\Gamma_0(N)$ whose Fourier coefficients are in $\\mathcal{O}_{v}$.\nWhen $k$ is an integer and the multiplier $\\nu$ is trivial, we write $M_{k}(N)$, $S_{k}(N)$\nand $M_{k}^{!}(N)$.\nForms in these spaces satisfy the transformation law\n\n\\[\nf \\big|_k \\gamma= \\nu(\\gamma)f \\ \\ \\ \\text{ for } \\ \\ \\ \\gamma = \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\Gamma_0(N)\n\\]\nand the appropriate conditions at the cusps of $\\Gamma_0(N)$. \n\n\nThroughout, let $q:=e(z)=e^{2 \\pi i z}$. \nWe define the eta function by\n\\[\n\\eta(z):= q^{\\frac{1}{24}}\\prod_{n=1}^{\\infty}(1-q^{n})\n\\]\nand the theta function by\n\\[\n\\theta(z):= \\sum_{n=-\\infty}^{\\infty}q^{n^2}.\n\\]\nThe eta function has a multiplier $\\nu_{\\eta}$ satisfying\n\n\\[\n\\eta(\\gamma z)=\\nu_{\\eta}(\\gamma)(cz+d)^{\\frac{1}{2}}\\eta(z), \\ \\ \\ \\ \\ \\gamma= \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\operatorname{SL}_{2}(\\mathbb{Z});\n\\]\nthroughout, we choose the principal branch of the square root. For $c>0$, we have the formula \\cite[~$\\mathsection$$4.1$]{Knopp}\n\n\\begin{equation}\\label{etamultiplier}\n\\nu_{\\eta}(\\gamma)=\n \\begin{cases} \n \\(\\frac{d}{c}\\)e\\(\\frac{1}{24}((a+d)c-bd(c^2-1)-3c )\\), & \\text{if } c \\text{ is odd,} \\\\\n\\(\\frac{c}{d}\\)e\\(\\frac{1}{24}((a+d)c-bd(c^2-1)+3d-3-3cd)\\) & \\text{if } c \\text{ is even}.\n\\end{cases}\n\\end{equation}\nFor the multiplier of the theta function we have\n\\[\n\\nu_{\\theta}(\\gamma):= (cz+d)^{-\\frac{1}{2}}\\frac{\\theta(\\gamma z)}{\\theta(z)}=\\(\\frac{c}{d}\\)\\epsilon_{d}^{-1}, \\ \\ \\ \\ \\ \\gamma= \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\Gamma_0(4),\n\\]\nwhere\n\\[\n\\epsilon_{d}=\n\\begin{cases}\n1, & \\text{if } d \\equiv 1 \\pmod{4}, \\\\\ni, & \\text{if } d \\equiv 3 \\pmod{4}.\n\\end{cases}\n\\]\n\n\nIn the next several paragraphs, we follow the exposition in \\cite{Scarcity}.\nIf $f \\in M_{k}(N,\\chi\\nu_{\\eta}^{r})$, then $\\eta^{-r}f \\in M^{!}_{k-\\frac{r}{2}}(N,\\chi)$.\nThis implies that $f$ has a Fourier expansion of the form\n\n\\begin{equation}\\label{lemma4.2}\nf = \\sum_{n \\equiv r (24)}a(n)q^{\\frac{n}{24}}.\n\\end{equation} \nThese facts together imply the following lemma.\n\\begin{lemma}\\label{integerhalfinteger}\nSuppose that $0 0$ such that $\\ell j+r \\equiv 0 \\pmod{24}$, and define \n\\[\nh:= \\eta^{\\ell j}g \\in S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1).\n\\]\nSuppose that $x \\in \\mathcal{O}_{v}$ and that $\\sigma \\in \\operatorname{Gal}(K\/\\mathbb{Q})$ is a Frobenius automorphism for the prime $v$. Then we have $x^{\\sigma} \\in \\mathcal{O}_{v}$ and \n\n\\[\nx^{\\sigma} \\equiv x^{\\ell} \\pmod{v}.\n\\]\nNote that $\\sigma$ preserves the space $S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1)$. Since $U_{\\ell}$ acts as \n$T(\\ell, \\lambda+\\frac{\\ell j}{2}+\\frac{1}{2},1)$ modulo $v$, we see that $\\bar{h\\sl U_{\\ell}} \\in \\bar{S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1)}$. We have\n\\[\n\\bar{h^{\\sigma}}=(\\bar{h\\sl U_{\\ell}})^{\\ell}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we know that there exists an integer $\\beta \\geq 0$ such that\n\n\\[\nk:= \\omega(\\bar{h\\sl U_{\\ell}})=\\frac{1}{\\ell}\\omega(\\bar{h^{\\sigma}})=\\frac{1}{\\ell}\\(\\lambda-\\beta(\\ell-1)+\\frac{\\ell j}{2}+\\frac{1}{2}\\).\n\\]\nTherefore, arguing as in the proof of Lemma~\\ref{Lemma1}, we can find a form\n$H \\in S_{k}(1)$ such that $\\bar{H}=\\bar{h\\sl U_{\\ell}}=\\bar{\\eta^{j}(g\\sl U_{\\ell}})$ and\n$f:= \\frac{H}{\\eta^{j}} \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1)$. Then, we see that $f \\in S_{k-\\frac{j}{2}}(1,\\nu_{\\eta}^{r\\ell})$, and we have $\\bar{f}=\\bar{g\\sl U_{\\ell}}$. \nThe lemma follows since $k-\\frac{j}{2} \\leq \\frac{1}{\\ell}(\\lambda+\\frac{1}{2})$.\n\n\\end{proof}\n\n\\section{Preliminary Reduction}\nBefore proving Theorem~\\ref{thm:main}, we reduce the number of square classes on which our forms may be supported and the number of multipliers which we must consider.\n\n\\begin{proposition}\\label{Proposition1}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime above $\\ell$. Suppose that $\\lambda$ is a non-negative integer, that $r$ is a positive integer with $(r,6)=1$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$. Further, suppose that\n\n\\begin{equation}\\label{finitelymanysquareclasses}\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^2)q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod v,\n \\end{equation}\n where each $t_{i}$ is a square-free positive integer. Then\n \\begin{equation}\\label{twosquareclasses}\n f \\equiv \\sum_{n=1}^{\\infty}a(n^2)q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \n \\pmod v.\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFix an $i \\in \\{1,...,m \\}$. We may assume that there exists an integer $n_{i}$ for which $a(t_{i}n_{i}^{2}) \\not \\equiv 0 \\pmod{v}$. Recalling our notation \\eqref{twist} and the facts \\eqref{chi TRIV} and \\eqref{chi P}, we follow the argument in the proof of Lemma $4.1$ of \\cite{Ahlgren-Boylan} to find primes $p_{1},...,p_{n} \\geq 5$, each relatively prime to $n_{i}t_{i}\\ell$ and a form\n\\[\nG_{i} \\in S_{\\lambda+\\frac{1}{2}}(p_{1}^{2}\\cdots p_{n}^{2},\\nu_{\\eta}^{r})\n\\]\nsatisfying\n\\[\nG_{i} \\equiv \\sum_{(n,\\prod p_{j})=1}a(t_{i}n^{2})q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nNote that \n\n\\[\nG_{i}^{24} \\in S_{24\\lambda+12}(p_{1}^{2},...,p_{s}^{2}).\n\\]\nSince \n\\[\nG_{i}^{24} \\equiv \\sum_{n=1}^{\\infty}b(t_{i}n)q^{t_{i}n} \\pmod{v}\n\\]\nfor some coefficients $b(t_{i}n)$, we can apply \nthe following result\nto conclude that $t_{i}=1 \\text{ or } \\ell$.\n\n\\begin{theorem}{\\cite[Thm 3.1]{Ahlgren-Choi-Rouse}}\\label{thm:ACR THM 3.1}\nSuppose that $K$ is a number field and that $v$ is a prime above $\\ell$ with ring of $v$-integral elements $\\mathcal{O}_{v}$. Suppose that $k$ is positive integer and that \n\\[\nf=\\sum_{n=1}^{\\infty}a(n)q^{n} \\in S_{2k}(\\Gamma_0(N)). \n\\]\nIf $t > 1$ satisfies $(t,\\ell N)=1$ and\n\\[\nf \\equiv \\displaystyle \\sum_{n=1}^{\\infty}a(tn)q^{tn} \\pmod{v},\n\\]\nthen $f \\equiv 0 \\pmod v$.\n\\end{theorem}\n\n\\end{proof}\nThe next result reduces the number of multipliers which we must consider.\n\n\\begin{lemma}\\label{lemmatwomultipliers}\n\nSuppose that $r$ is a positive integer with $(r,6)=1$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ satisfies \\eqref{twosquareclasses}. Then we have\n\\begin{equation}\\label{twomultipliers}\nr \\equiv 1 \\pmod{24} \\ \\ \\ \\ \\text{ or } \\ \\ \\ \\ r \\equiv \\ell \\pmod{24}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $f$ satisfies \\eqref{twosquareclasses}, it follows that either $a(n^{2}) \\neq 0$ or $a(\\ell n^{2}) \\neq 0$ for some positive integer $n$. It follows from \\eqref{lemma4.2} and the fact that $r^{2} \\equiv 1 \\pmod{24}$ whenever $(r,6)=1$ that we have \\eqref{twomultipliers}. \n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{thm:main}}\nThe proof of Theorem~\\ref{thm:main} will proceed in several steps. We first consider the case when $r \\equiv 1 \\pmod{24}$ and $\\lambda$ is even.\n\n\\begin{theorem}\\label{Theorem1}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime above $\\ell$. Suppose that $\\lambda$ is a non-negative integer and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta})$ has the form \\eqref{twosquareclasses}. If $\\lambda$ is even and $\\lambda < 2\\ell^{2}+\\ell-1$, then\n\n\\[\n\\sum_{\\ell \\nmid n}a(n^{2})q^{\\frac{n^{2}}{24}} \\equiv a(1)\\sum_{\\ell \\nmid n}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nDefine $\\bar{\\lambda}:= \\lambda \\pmod{\\ell-1}$. By Lemma~\\ref{Lemma1}, we have forms $g(z) \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1,\\nu_{\\eta})$ and \n$h(z) \\in S_{(\\ell+1)\\frac{\\bar{\\lambda}+2}{2}+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that\n\n\\[\ng=\\sum_{n=1}^{\\infty}c(n)q^{\\frac{n}{24}} \\equiv \\Theta(f) \\equiv \\sum_{n=1}^{\\infty}\\frac{n^2}{24}a(n^2)q^{\\frac{n^2}{24}} \\pmod{v}\n\\]\nand\n\\[\nh=\\sum_{n=1}^{\\infty}b(n)q^{\\frac{n}{24}} \\equiv 24^{\\frac{\\bar{\\lambda}}{2}}a(1)\\Theta^{\\frac{\\bar{\\lambda}+2}{2}}(\\eta) \\equiv a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)\\frac{n^{\\bar{\\lambda}+2}}{24}q^{\\frac{n^2}{24}} \\pmod{v}.\n\\]\n\nIt suffices to show that $g \\equiv h \\pmod{v}$. To this end, we make use of this theorem, which follows from an argument of Bruinier and Ono \\cite[Thm 3.1]{Bruinier-Ono} (see \\cite[Thm 2.1]{Ahlgren-Choi-Rouse}).\n\n\\begin{theorem}\\label{Bruinier-Ono}\nSuppose that $N$ is a positive integer with $4 \\mid N$. Suppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$. \nSuppose that $\\lambda$ is a non-negative integer and that $r$ is a positive integer with $(r,6)=1$. Suppose that\n\n\\[\nf(z)=\\sum_{n=1}^{\\infty}a(n)q^{n} \\in S_{\\lambda+\\frac{1}{2}}(N,\\chi\\(\\tfrac{12}{\\bullet}\\) \\nu_{\\theta}^{r}),\n\\]\nthat $\\ell \\nmid N$, and that $p \\nmid N\\ell$ is prime. If there exists $\\epsilon_{p} \\in \\{ \\pm 1\\}$ such that\n\n\\[\nf(z) \\equiv \\sum_{\\( \\frac{n}{p}\\) \\in \\{0, \\epsilon_{p}\\}}a(n)q^{n} \\pmod v,\n\\]\nthen we have\n\\[\n(p-1)f(z)\\sl T\\(p^2,\\lambda +\\mfrac{1}{2},\\chi\\) \\equiv \\epsilon_{p}\\chi(p)\\(\\tfrac{(-1)^{\\lambda}}{p}\\)(p^{\\lambda}+p^{\\lambda-1})(p-1)f(z) \\pmod v.\n\\]\n\\end{theorem}\n\n\n\n\nBy \\eqref{passing to Shimura's space}, we can apply Theorem~\\ref{Bruinier-Ono} to $g(24z)$ to conclude that, for odd primes $p \\geq 5$ with $p \\not \\equiv 0,1 \\pmod{\\ell}$, we have\n\n\\begin{equation}\\label{applicationofBruinierOno}\ng(24z) \\sl T\\(p^2,\\lambda+\\ell+1+\\mfrac{1}{2},1\\) \\equiv \\(\\frac{12}{p}\\)(p^{\\bar{\\lambda}+2}+p^{\\bar{\\lambda}+1})g(24z) \\pmod{v}.\n\\end{equation}\nSuppose that $n$ is a positive integer satisfying $(n,6)=1$ which is divisible only by primes $p \\not \\equiv 0,1 \\pmod{\\ell}$.\n If $p$ is such a prime, write $n=p^{a}n_{0}$ if $p^{a} \\mid \\mid n$.\nThe definition of the Hecke operator on $S_{\\lambda+\\ell+1+\\frac{1}{2}}(576,(\\frac{12}{\\bullet})\\nu_{\\theta}^{r})$ implies that we have\n\\[\nc(n^2p^2)+p^{\\bar{\\lambda}+1}\\(\\frac{12n^2}{p}\\)c(n^2)+p^{2\\bar{\\lambda}+3}c\\(\\frac{n^2}{p^2}\\) \\equiv \\(\\frac{12}{p}\\)(p^{\\bar{\\lambda}+2}+p^{\\bar{\\lambda}+1})c(n^2) \\pmod{v},\n\\]\nand an induction argument on $a$ then implies that\n\\[\nc(p^{2a}n_{0}^{2}) \\equiv \\(\\frac{12}{p}\\)^{a}p^{a(\\bar{\\lambda}+2)}c(n_{0}^{2}) \\pmod{v}.\n\\]\n\nThus, we have\n\\[\nc(n^{2}) \\equiv \\(\\frac{12}{n}\\)n^{\\bar{\\lambda}+2}c(1) \\equiv \\(\\frac{12}{n}\\)\\frac{n^{\\bar{\\lambda}+2}}{24}a(1) \\equiv b(n^2) \\pmod{v}.\n\\]\nThis shows that the coefficients $c(n^{2})$ and $b(n^{2})$ agree whenever $n$ is a positive integer such that $(n,6)=1$ which is divisible only by primes $p \\geq 5$ with $p \\not \\equiv 0,1 \\pmod{\\ell}$. \n\n\nNow define\n\\[\nk:= \\max\\{\\lambda+\\ell+1,(\\ell+1)\\frac{\\bar{\\lambda}+2}{2}\\}. \n\\]\nThese numbers agree modulo $\\ell-1$ by virtue of $\\lambda$ being even, so by multiplying $g$ or $h$ by an appropriate power of $E_{\\ell-1} \\equiv 1 \\pmod{\\ell}$, we see that there exist forms $g_{1}$ and $h_{1}$ in $S_{k+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that $g_{1} \\equiv g \\pmod{v}$ and $h_{1} \\equiv h \\pmod{v}$. \nThus, to prove the theorem, it suffices to show that $g_{1} \\equiv h_{1} \\pmod{v}$.\nNote that $c(n^2) \\equiv b(n^2) \\equiv 0 \\pmod{v}$ for positive integers $n$ such that $(n,6) \\neq 1$ by \\eqref{lemma4.2}, and that $c(n^2)$ and $b(n^2)$ vanish modulo $\\ell$ whenever $n$ is divisible by $\\ell$. This implies that $c(n) \\equiv b(n) \\pmod{v}$ whenever $n < (2\\ell+1)^{2}$.\n Thus,\n$\\eta^{-1}(g_{1}-h_{1}) \\in M_{k}(1)$ is of the form\n\n\\[\n\\eta^{-1}(g_{1}-h_{1}) \\equiv cq^{\\frac{\\ell^{2}+\\ell}{6}}+ \\cdots \\pmod{v}\n\\]\nfor some $c \\in \\mathcal{O}_{v}$. By arguing as in the proof of Lemma~\\ref{Lemma1}, we may assume that $\\eta^{-1}(g_{1}-h_{1}) \\in S_{k}(1)$. To prove that $g_{1} \\equiv h_{1} \\pmod{v}$, it suffices to show by \\cite[Thm 1]{Sturm} that \n\n\\[\n\\frac{k}{12} < \\frac{\\ell^{2}+\\ell}{6}.\n\\]\nSince $\\bar{\\lambda} < \\ell$, we have\n\n\\[\n\\frac{(\\ell+1)(\\bar{\\lambda}+2)}{24}<\\frac{\\ell^{2}+\\ell}{6}.\n\\]\nSince $\\lambda < 2\\ell^{2}+\\ell-1$, we have\n\\[\n\\frac{\\lambda+\\ell+1}{12}<\\frac{\\ell^{2}+\\ell}{6}.\n\\]\nThe result follows.\n\\end{proof}\n\nWe now consider what happens when $\\lambda$ is odd.\n\n\\begin{proposition}\\label{oddcases}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$. Suppose that $\\lambda$ is a non-negative integer, that $r$ is a positive integer with $(r,6)=1$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ has the form\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nIf $\\lambda$ is odd, then $\\Theta(f) \\equiv 0 \\pmod{v}$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose by way of contradiction that $\\Theta(f) \\not \\equiv 0 \\pmod{v}$. By Lemma~\\ref{Lemma1}, there exists $g \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1,\\nu_{\\eta}^{r})$ such that\n\n\\[\ng \\equiv \\sum_{n=1}^{\\infty} \\frac{n^{2}}{24}a(n^{2})q^{\\frac{n^{2}}{24}} \\not \\equiv 0 \\pmod{v},\n\\]\nso there exists $n_{0}$ such that $a(n_{0}^{2}) \\neq 0$.\nBy \\eqref{lemma4.2}, we have $r \\equiv 1 \\pmod{24}$. By Lemma~\\ref{integerhalfinteger}, we then have $\\eta^{-1}f \\in M_{\\lambda}(1)=\\{0\\},$ which is a contradiction. \nThus, $\\Theta(f) \\equiv 0 \\pmod{v}$.\n\\end{proof}\n\nWe require one more result before proving Theorem~\\ref{thm:main}.\n\n\\begin{proposition}\\label{Spicy}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field which is Galois over $\\mathbb{Q}$, and that $v$ is a prime of $K$ above $\\ell$. \nSuppose that $r$ is a positive integer with $(r,6)=1$ and that $g \\in S_{\\lambda'+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ satisfies \n\n\\[\n g \\equiv \\sum_{n=1}^{\\infty}a(n^2)q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod v. \n \\]\nIf $\\lambda' < \\frac{\\ell-1}{2}$, then $\\lambda'=0$, $r=1$, and $g=c\\eta$ for \nsome $c \\in \\mathcal{O}_{v}$. \n\\end{proposition}\n\n\\begin{proof}\nFirst assume that $\\lambda'=0$. Assume without loss of generality that $0 \\omega(\\bar{GE_{\\ell-1}^{\\lambda'-1}})$ since $\\omega(GE_{\\ell-1}^{\\lambda'-1}) \\leq 2\\lambda'$ and $\\lambda'$ is even. This would imply that\n$\\omega(\\bar{H})=\\omega(\\bar{\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes\\(\\frac{12}{\\bullet}\\)\\)})= \\ell(\\lambda'-1)+\\lambda'+1$. This contradicts the fact that $\\omega(\\bar{H})$ is a multiple of $\\ell$.\nThus, $a(1) \\equiv 0 \\pmod{\\ell}$. By \\eqref{*}, we have $\\Theta(g) \\equiv 0 \\pmod{v}$. The result now follows as in the odd case.\n\n\\end{proof}\n\nNow we prove Theorem~\\ref{thm:main}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$.\nWe may assume that $K$ is Galois over $\\mathbb{Q}$. \nSuppose that $r$ is a positive integer satisfying $(r,6)=1$, that $\\lambda$ is a non-negative integer satisfying $\\lambda+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ has the property that\n\n\\[\nf \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^{2})q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nBy Proposition~\\ref{Proposition1} and Lemma~\\ref{lemmatwomultipliers}, we may assume that\n\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod{v}\n\\]\nand that either $r \\equiv 1 \\pmod{24}$ or $r \\equiv \\ell \\pmod{24}$. So, we need only consider the cases when $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta})$ and when $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$ with $\\ell \\not \\equiv 1 \\pmod{24}$.\n\nSuppose that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta})$. Assume that $\\lambda$ is even. If $\\lambda=0$, then $f=c\\eta$ for some $c \\in \\mathcal{O}_{v}$. This implies that\n\\[\nf = a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{n^{2}}{24}},\n\\] \nwhich has the form of case $(1)$ of Theorem~\\ref{thm:main}, so assume that $\\lambda > 0$.\nTheorem~\\ref{Theorem1} then implies that\n\\begin{equation}\\label{**}\n\\Theta^{\\ell-1}(f)=\\sum_{\\ell \\nmid n}a(n^{2})q^{\\frac{n^{2}}{24}} \\equiv a(1) \\sum_{\\ell \\nmid n}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\end{equation}\nDefine $\\bar{\\lambda}:=\\lambda \\pmod{\\ell-1}.$ By Lemma~\\ref{Lemma1}, we have\n\\begin{equation}\\label{R.L.C.}\n\\bar{\\Theta^{\\ell-1}(f)}=\\bar{24^{\\frac{\\bar{\\lambda}}{2}}a(1)\\Theta^{\\frac{\\bar{\\lambda}}{2}}(\\eta)} \\in \\bar{S_{\\frac{\\bar{\\lambda}}{2}(\\ell+1)+\\frac{1}{2}}(1,\\nu_{\\eta})}.\n\\end{equation}\nThe fact that\n\\[\n\\(\\frac{\\bar{\\lambda}}{2}\\)(\\ell+1)+\\frac{1}{2} < \\frac{\\ell^{2}}{2}\n\\]\nimplies that $\\bar{f-\\Theta^{\\ell-1}(f)} \\in \\bar{S_{\\lambda'+\\frac{1}{2}}(1,\\nu_{\\eta})}$, where $\\lambda'+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$. Since\n\\[\nf-\\Theta^{\\ell-1}(f) \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2}n^{2})q^{\\frac{\\ell^{2} n^{2}}{24}} \\pmod{v},\n\\]\nwe apply Lemma~\\ref{Lemma2} to conclude that there exists $g \\in S_{\\lambda^{*}+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$ \nwith $\\lambda^{*}+\\frac{1}{2} < \\frac{\\ell}{2}$ satisfying\n\\[\ng \\equiv \\(f-\\Theta^{\\ell-1}(f)\\)\\sl U_{\\ell} \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2} n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nIf $g \\equiv 0 \\pmod{v}$, then\n\\[\nf \\equiv \\Theta^{\\ell-1}(f)\\pmod{v},\n\\]\nand, by \\eqref{**}, this proves that\n\\[\nf \\equiv a(1) \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\]\nThis has the form of case $(1)$ of Theorem~\\ref{thm:main}.\n\nIf $g \\not \\equiv 0 \\pmod{v}$, then Proposition~\\ref{Spicy} implies that $\\lambda'=0$ and $g=c\\eta$ for some $c \\in \\mathcal{O}_{v}$, which means that\n\n\\[\nf-\\Theta^{\\ell-1}(f) \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nThus, we have\n\n\\begin{equation}\\label{deletthis}\nf \\equiv a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}}+a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\end{equation}\n\nSince $g \\not \\equiv 0 \\pmod{v}$, we have $a(\\ell) \\not \\equiv 0 \\pmod{v}$. Proposition~\\ref{Spicy} applied to $f$ implies that $\\ell \\equiv 1 \\pmod{24}$. \nIf $a(1) \\equiv 0 \\pmod{v}$, then \\eqref{deletthis} has the form of case $(2)$ of Theorem~\\ref{thm:main}.\nThis is equivalent to the congruence\n\n\\[\nf \\equiv a(\\ell)\\eta^{\\ell} \\pmod{v}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we have $\\lambda \\equiv \\omega(\\bar{\\eta^{-1}f}) \\equiv \\omega(\\bar{\\eta^{\\ell-1}}) \\equiv \\frac{\\ell-1}{2}\\pmod{\\ell-1}$.\n\nIf $a(1) \\not \\equiv 0 \\pmod{v}$, then \\eqref{deletthis} has the form of case $(3)$ of Theorem~\\ref{thm:main}. This is equivalent to the congruence\n\n\n\\[\nf \\equiv 24^{\\frac{\\lambda}{2}}a(1)\\Theta^{\\frac{\\lambda}{2}}(\\eta)+a(\\ell)\\eta^{\\ell} \\pmod{v}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we have $\\omega(\\bar{\\eta^{-1}f}) \\equiv \\omega(\\bar{\\eta^{-1}\\Theta^{\\frac{\\lambda}{2}}(\\eta)}) \\equiv \\lambda \\pmod{\\ell-1}$. This implies that\n$\\omega(\\bar{\\eta^{\\ell-1}}) \\equiv \\lambda \\pmod{\\ell-1}$. Since $\\omega(\\bar{\\eta^{\\ell-1}})=\\frac{\\ell-1}{2}$, we have $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\n \n Now assume that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta})$ and that $\\lambda$ is odd. By Proposition~\\ref{oddcases}, we have\n \\[\n f \\equiv \\sum a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n \\] \nBy Lemma~\\ref{Lemma2} (since $\\lambda+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$), there exists $g \\in S_{\\lambda'+\\frac{1}{2}}(1, \\nu_{\\eta}^{\\ell})$ such that $g \\equiv f\\sl U_{\\ell} \\pmod{v}$ and $\\lambda'+\\frac{1}{2} < \\frac{\\ell}{2}$. \nProposition~\\ref{Spicy} implies that $\\lambda'=0$ and $g =c\\eta$ for some $c \\in \\mathcal{O}_{v}$. Thus,\n\\[\nf \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nSince $f \\not \\equiv 0 \\pmod{v}$ has a Fourier expansion of the form $\\eqref{lemma4.2}$, we have \n$r \\equiv \\ell \\equiv 1 \\pmod{24}$\nin this case. As above, this implies that $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$, which implies that $\\lambda$ is even. This is a contradiction, so $\\lambda$ cannot be odd in this case.\n\nFinally, suppose that $\\ell \\not \\equiv 1 \\pmod{24}$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$. \nBy \\eqref{lemma4.2}, we have\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nBy Lemma~\\ref{Lemma2}, there is $g \\in S_{\\lambda'+\\frac{1}{2}}(1,\\nu_{\\eta})$ with $\\lambda'+\\frac{1}{2}< \\frac{\\ell}{2}$ such that $g \\equiv f\\sl U_{\\ell} \\pmod{v}$.\nProposition~\\ref{Spicy} implies that $\\lambda'=0$ and that $g=c\\eta$ for some $c \\in \\mathcal{O}_{v}$. Thus,\n\\[\nf \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nThis has the form of case $(2)$ of Theorem~\\ref{thm:main}. As above, we have $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\\end{proof}\n\n \\section{Acknowledgements}\nThe author would like to thank Scott Ahlgren for suggesting this project and for advice and guidance for this work. The author would also like to thank the referee for carefully reading this manuscript and making helpful comments which improved its exposition. Finally, the author would like to thank the Graduate College Fellowship program at the University of Illinois at Urbana-Champaign and the Alfred P. Sloan Foundation for their generous research support. \n \n\n \n \n \\bibliographystyle{amsalpha}\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nThe transition from carbon-based electricity production to renewable sources provides pressing arguments for investing in more transmission capacity between European countries \\cite{kristiansen2018}. Renewable power sources, such as wind farms and photovoltaic panels are often more efficiently placed at certain geographical locations with, e.g., higher levels of wind or sunshine than others. A well-developed international transmission network can help achieve a system where production can take place where it is opportune, and consumption where it is needed \\cite{UN2006multi}. Furthermore, renewable sources typically have higher levels of uncertainty in the associated production levels \\cite{konstantelos2014valuation}. International transmission lines can help balance the power production among geographical regions and thus help mitigate production uncertainty through geographical diversification \\cite{hasche2010general}.\n\nAn important obstacle to achieving the desired interconnected transmission network is the fact that the welfare benefits and costs associated with transmission expansions are often unevenly distributed among countries \\cite{mezHosi2016model}. In fact, situations can arise in which an investment in transmission capacity that is beneficial from a Europe-wide system perspective is detrimental to the economic welfare in an individual country. If this negatively affected country is one of the countries hosting the proposed transmission expansion, then it may block the investment and as a consequence, hurt the system as a whole \\cite{huppmann2015national}. This issue motivates the need for compensation mechanisms that distribute welfare gains in order to convince all countries to follow through on the transmission expansion plan and thus help achieve the system optimum \\cite{olmos2018transmission}.\n\nWe consider welfare compensation mechanisms in international transmission expansion planning (TEP) under uncertainty. In the literature, several studies have been performed that determine compensation amounts that should be paid to compensate countries \\textit{in expectation} \\cite{hogan2018primer,jansen2015alternative,konstantelos2017integrated,kristiansen2018}. However, by restricting attention to expected values, important effects resulting from uncertainty are neglected. The \\textit{actual} benefits\/costs depend on realization of uncertain elements, such as renewable production levels and electricity prices at different nodes. Hence, if uncertainty is ignored when constructing compensation mechanisms, countries run the risk that in some scenarios, they compensate more than they benefit or are not compensated enough to cover their welfare loss. \n\nIn this paper, we explore the potential of different compensation mechanisms to mitigate the risk associated with investing in a new transmission line. We use a fixed, lump sum payment as a benchmark and investigate whether other compensation mechanisms can perform better in terms of the risk faced by the countries involved as a consequence of the investment in the new transmission line. One alternative mechanism is proposed in the literature \\cite{kristiansen2018}, in the form of power purchase agreements (PPAs). PPAs are contracts that essentially give a certain country a virtual, fixed price at which it can trade in the power market \\cite{kristiansen2018}. Deviations of the spot price from this fixed price are then used to determine the compensation amount to be paid or received. We show that in a stochastic setting only one country can receive a PPA. This is in contrast with the deterministic case, in which multiple PPAs can be constructed, one for each country, such that they balance each other exactly.\n\nBesides these two mechanisms from the literature, we also propose two novel compensation mechanisms. We aim to construct compensation mechanisms that achieve risk mitigation by using scenario information to determine the compensation amount. For this purpose, we propose to base the compensations on economic measures related to the economic value of the proposed transmission line investment in the realized scenario. Specifically, we use the amount of flow through the new transmission line and its economic value as measures to base our novel compensation mechanisms on.\n\nWe compare the different compensation mechanisms numerically, using a case study of the Northern-European electricity market, focusing on a new transmission line between Norway and Germany. We test the ability of each compensation mechanism to mitigate the risk associated with the transmission expansion investment for each of the affected countries. Here, we consider risk both in terms of the variability of the welfare effect of the investment and in terms of the expected welfare loss as a result of the investment. We run experiments in two settings: a setting with bilateral compensations between Norway and Germany only and a setting with multilateral compensations between all countries that are significantly affected by the proposed investment. \n\nIn both cases, we show that a theoretically ideal mechanism, that equally shares welfare benefits in every scenario, significantly outperforms lump sum payments in terms of its ability to mitigate risk of the countries involved. This demonstrates the potential for risk mitigation through alternative compensation mechanisms. Out of the other compensation mechanisms, our novel value-based mechanism appears to perform best. In particular, it consistently outperforms the lump sum payments. Hence, we show that risk mitigation can actually be achieved by using scenario-dependent compensation mechanisms. Finally, for PPAs, the question of which country receives the PPA turns out to be crucial: a Germany-based PPA performs bad for both Norway and Germany, while a Norway-based PPA shifts risk from Germany to Norway, compared to the lump sum.\n\nThe remainder of this paper is structured as follows. In Section~\\ref{sec:literature_review} we review the literature on welfare distribution in TEP. Next, in Section~\\ref{sec:illustrative_examples} we provide simple, illustrative examples that motivate the need for welfare compensation. In Section~\\ref{sec:TEP} we present and solve a transmission expansion model in the Northern-European market. The resulting optimal transmission expansion plan serves as a test environment for our investigation into different welfare mechanisms in Section~\\ref{sec:compensation_mechanisms}. In this section, which constitutes the core of the paper, we discuss existing compensation mechanisms, propose several novel mechanisms, and test their performance in the case study from Section~\\ref{sec:TEP}. Section~\\ref{sec:conclusion} concludes the paper. Finally, Section~\\ref{sec:mathematical_model} and Section~\\ref{sec:data} in the appendix contain a description of the mathematical model formulation and the data used, respectively, in the TEP model of Section~\\ref{sec:TEP}. \n\n\\section{Literature review}\n\\label{sec:literature_review}\n\nTEP is an active topic of research within the field of operations research \\cite{kristiansen2018}. Much of the literature is aimed at developing methods to find good candidates for TEP investments \\cite{latorre2003,hemmati2013comprehensive}. Typically, finding an optimal combination of investments is a challenging task, both from a modeling and a computational point of view. One major difficulty is that many relevant parameters, such as demand levels and renewable energy production, may be uncertain, necessitating stochastic models \\cite{Zhao2009flexible}. Furthermore, investments are typically of a discrete nature, which introduces the need for integer decision variables \\cite{alguacil2003,delatorre2008}. Finally, in order to properly model the effect of the transmission expansion on the power market, explicit modeling of the market participants through equilibrium models may be required \\cite{gabriel2012}. All these factors make TEP a challenging area of research; see \\cite{mahdavi2018transmission} for a recent review of the literature on TEP.\n\nIn this paper we are not mainly interested in finding the best transmission expansion plan, however, but in making this plan practically \\textit{achievable} by constructing welfare-sharing mechanisms that allocate costs and benefits in such a way that all the relevant actors are willing to follow through on the proposed expansion plan. In the literature, most effort in this direction has been spent on creating mechanisms to share the \\textit{investment costs} \\cite{erli2005transmission,konstantelos2017integrated,nylund2014regional,roustaei2014transmission}. Currently, the most-used cost-sharing strategy is the \\textit{equal share principle} \\cite{huang2016mind}. According to this principle, the investment costs are split equally between the two countries hosting a new cable. Before the year 2016, all except for two transmission expansion projects in the EU followed this principle \\cite{huang2016mind}. Recently, though, there has been a trend towards the so-called \\textit{beneficiaries pay principle} \\cite{ACER,FERC2012order1000}. According to this principle, each country should pay a share of the investment cost proportional to its benefit from the investment. A specific allocation method that follows this principle is the \\textit{net positive benefit differential} \\cite{hogan2018primer, konstantelos2017}. The hope is that such a benefit-based allocation method leads to better incentives and ultimately more investments that are beneficial to the system as a whole.\n\nOne shortcoming of these cost allocation methods, however, is that they ignore potential \\textit{welfare losses} as a result of transmission expansion. As discussed in the introduction, even if the investment costs are zero, some countries might be worse off as a result of transmission expansion, which is especially problematic if one of these countries is hosting the proposed transmission expansion. Recently, a few authors have recognized this problem and proposed methods to share welfare gains and losses \\cite{konstantelos2017integrated,kristiansen2018,churkin2019can,churkin2021review}. These authors take a cooperative game theory perspective and propose compensation amounts between countries, based on different conceptions of fairness, e.g., the net postitive benefit differential \\cite{konstantelos2017integrated}, the Shapley value \\cite{kristiansen2018}, or the nucleolus \\cite{churkin2019can}. \n\nTo the best of our knowledge, all papers in this literature consider a single compensation amount based on the \\textit{expected} welfare benefits to the various countries. That is, they ignore the possibility of compensations varying per scenario and the resulting potential for risk mitigation, as discussed in the introduction. Rather, most papers seem to implicitly assume a lump sum payment.\n\nOne alternative mechanism has been proposed in the literature: power purchase agreements (PPAs) \\cite{kristiansen2018}. Such a PPA is a contract based on a fixed price $\\pi^{\\text{PPA}}$ for purchasing power for each country. The country then pays an amount proportional to the net flow into the country times the difference between the spot price and the predetermined price $\\pi^{\\text{PPA}}$ to a fund. If the price $\\pi^{\\text{PPA}}$ would be less beneficial to the country than the actual spot price at which it trades, then the country pays into the fund. In the reverse case it receives money from the fund. The PPA prices $\\pi^{\\text{PPA}}$ are determined up front such that in expectation, each country receives their ``fair'' share of the total welfare. This PPA-based mechanism does depend on the \\textit{actual} behavior of the system (in terms of flows and prices). Hence, this mechanism could potentially be able to mitigate risk resulting from the transmission expansion faced by the relevant countries. Although Kristiansen et al. \\cite{kristiansen2018} propose to use PPAs as a welfare compensation mechanism, they don't recognize their potential for risk mitigation. In fact, the authors assume a deterministic setting. In Section~\\ref{sec:compensation_mechanisms} we show that the construction of PPAs is fundamentally different in a stochastic setting.\n\nIn the remainder of this paper, we contribute to this literature by testing the performance of various compensation mechanisms in terms of their ability to mitigate the risk of investing in a new transmission cable for the countries involved. We consider both mechanisms from the literature (lump sum payments and PPAs) and novel mechanisms (based on the flow through the new transmission line and its economic value).\n\n\n\\section{Illustrative examples} \\label{sec:illustrative_examples}\n\nIn this section we present two simple, illustrative examples to motivate our study of welfare compensation mechanisms. In Section~\\ref{subsec:two-node} we consider a two-node system and show that in the absence of other nodes, there is no need for welfare distribution. In Section~\\ref{subsec:three-node} we extend the model to a three-node system and provide an example in which transmission expansion is desirable from a system point of view, but negatively affects one of the hosting countries. This motivates the need for compensation mechanisms studied in this paper.\n\n\n\n\n\\subsection{Two-node system: basics} \\label{subsec:two-node}\n\nFirst we consider a network consisting of two country nodes, as illustrated in Figure~\\ref{subfig:network_configurations_2_nodes}. Each node $i=1,2$ contains power suppliers and consumers, represented by a linear supply and demand curve $S_i(\\pi_i)$ and $D_i(\\pi_i)$, respectively. We assume perfect competition with all market participants being price takers. Moreover, we assume that that initially, there is no transmission capacity between the two nodes. Hence, initially every node is an independent market and initial equilibrium prices $\\pi^*_1$ and $\\pi^*_2$ occur where the supply and demand curves meet. See Figure~\\ref{fig:surplus_two_nodes} for an illustration. The green and red areas represent the consumer surplus (CS) and producer surplus (PS), respectively, in each node. Note that $\\pi^*_1 > \\pi^*_2$, indicating that power is more scarce in node 1 as compared to node 2. \n\n\n\\begin{figure}[h]\n \\centering\n \\begin{subfigure}[b]{0.25\\columnwidth}\n \\centering\n \\input{Figures\/Network2Nodes.tex}\n \\caption{}\n \\label{subfig:network_configurations_2_nodes}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.5\\columnwidth}\n \\centering\n \\input{Figures\/Network3NodesRadial.tex}\n \\caption{}\n \\label{subfig:network_configurations_3_nodes}\n \\end{subfigure}\n \\caption{Network configurations used in the examples in Section~\\ref{sec:illustrative_examples}.}\n \\label{fig:network_configurations}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\input{Figures\/Surplus2Nodes.tex}\n \\caption{Surpluses in the two-node example in Section~\\ref{subsec:two-node}.}\n \\label{fig:surplus_two_nodes}\n\\end{figure}\n\n\nBefore introducing the possibility of transmission expansion, it is useful to define the import\/export curve corresponding to node 1 and 2. We define the import curve for node 1 as $I_1(\\pi_1) := D_1(\\pi_1) - S_1(\\pi_1)$ and the export curve for node 2 as $E_2(\\pi_2) := S_2(\\pi_1) - D_2(\\pi_2)$. Here, $I_1(\\pi_1)$ represents the amount that node 1 is willing to import at price $\\pi_1$, while $E_2(\\pi_2)$ is the amount that node 2 is willing to export at price $\\pi_2$. See Figure~\\ref{fig:intput_output} for an illustration of the import\/export graph corresponding to the example in Figure~\\ref{fig:surplus_two_nodes}. On the horizontal axis we have the variable $f$, representing flow from node 2 to 1 (i.e., import to 1\/export from 2). Observe that, by definition, $I_1(\\pi^*_1) = 0$ and $E_2(\\pi^*_2) = 0$. Moreover, the figure shows that if transmission capacity between node 1 and 2 were unlimited, the combined market would clear at a common price $\\pi_1 = \\pi_2 = \\bar{\\pi}$ with an associated flow of $\\bar{f}$. \n\\begin{figure}[h]\n \\centering\n \\input{Figures\/Capacity2Nodes.tex}\n \\caption{Import\/export graph for the two-node example in Section~\\ref{subsec:two-node}.}\n \\label{fig:intput_output}\n\\end{figure}\n\n\nNow consider a social planner that can invest in transmission capacity $x$ between node 1 and 2. We assume that transmission capacity $x$ can be any non-negative real number and that the marginal cost of investment $C$ is constant. For a given investment $x$, the transmission system operator (TSO) will earn congestion rent: the TSO buys power at the low-price node and sells at the high-price node. It earns an amount $CR = (\\pi_1 - \\pi_2) f$, where $f$ again represents the flow from node 2 to 1. If $x \\geq \\bar{f}$, then the transmission capacity constraint $f \\leq x$ is be non-binding and we obtain the common price $\\bar{\\pi}$ and flow $\\bar{f}$ described above. Now suppose that $x < \\bar{f}$. Then, the transmission capacity constraint is binding and a price difference will arise between the two nodes. An illustration is given in Figures~\\ref{fig:surplus_two_nodes} and \\ref{fig:intput_output} for a capacity of $x^\\prime$. The associated prices, demand and supply levels, and flow are denoted by a prime. \n\nThe welfare gains associated with this capacity are visualized in Figure~\\ref{fig:surplus_two_nodes}. The welfare gain to node 1 is represented by the triangle defined by the three black dots in the left graph, and similarly for node 2. The congestion rent is also part of the welfare gain, and is represented by the rectangle in the left graph. Note that it is not necessarily allocated to a specific region, although it is shown in the diagram representing node 1 in the picture. Equivalently, the welfare effects are visualized in Figure~\\ref{fig:intput_output}. Here, the upper triangle represents the welfare gain to node 1 and has the same area as the left triangle in Figure~\\ref{fig:surplus_two_nodes}. The lower triangle represents the welfare gain to node 2 and has the same area as the right triangle in Figure~\\ref{fig:surplus_two_nodes}. Finally, the rectangle represents the congestion rent and has the same size as the rectangle in Figure~\\ref{fig:surplus_two_nodes}. \n\nFigure~\\ref{fig:intput_output} reveals how much a welfare-maximizing social planner should invest. The welfare gains are represented by the colored trapezoid. Clearly, the marginal welfare gains are decreasing in $x$. The marginal cost is constant at $C$. If $\\pi_1^* - \\pi_2^* \\leq C$, then an investment of $x=0$ is optimal. In the more interesting case where $\\pi_1^* - \\pi_2^* > C$, it is optimal to invest an amount $x$ such that $\\pi_1 - \\pi_2 = C$, and congestion rent equals investment cost. Hence, as expected from economic theory, in the social optimum marginal revenue equals marginal cost. \n\nLooking at the welfare effects of transmission expansion, we see that in each node there are winners and losers. In node 1 consumers benefit from the lower price, while producers are hurt. In node 2 the effects are reversed. However, it is clear from Figure~\\ref{fig:intput_output} that the total welfare effects of transmission expansion, represented by the areas of the two triangles, are non-negative for both nodes. In the next subsection we will see that the latter result need not hold for systems with more than two nodes.\n\n\n\\subsection{Three-node system: welfare issues} \\label{subsec:three-node}\n\nNext, we consider a network consisting of three nodes, illustrated in Figure~\\ref{subfig:network_configurations_3_nodes}. The purpose is to provide an example in which transmission expansion would be beneficial for the three-node system as a whole, but one of the nodes would be worse off than without the added transmission capacity. In particular, we show that this can be the case for one of the nodes that is hosting the new transmission capacity, which means that they will be able to block the transmission expansion and thus, hurt the system as a whole.\n\nThe three-node example we consider is designed to be the simplest possible example that manifests the welfare loss problem outlined in the previous paragraph. Let node 1 be a supply node, meaning that it has only supply, represented by a linear supply curve, and no demand. In contrast, let node 2 and 3 be demand nodes, meaning that they have only demand, represented by a linear demand curve, and no supply. The supply and demand curves are given by $S_1(\\pi_1) = \\pi_1$, $D_2(\\pi_2) = 6 - \\pi_2$, $D_3(\\pi_3) = 6 - \\pi_3$. As in the two-node setting, we assume perfect competition and price-taking behavior for all market participants.\n\nWe consider an initial situation in which there is a transmission line between node 1 and 2 of unlimited capacity and no transmission lines between other pairs of nodes. In this situation, node 1 and 2 form a single market with a common price $\\pi$ and an equilibrium is found where the supply curve of node 1 meets the demand curve of node 2: $\\pi_1 = \\pi_2 = \\pi^* = 3$, $s_1^* = d_2^* = q^* = 3$. This situation is illustrated in the top graph in Figure~\\ref{fig:surplus_three_nodes}. Note that the import\/export graphs are given by $I_2 = D_2$ and $E_1 = S_1$. The green area represents the consumer surplus in node 2, while the red are represents the producer surplus in node 1. The welfare distribution in this old situation is described in the top half of Table~\\ref{tab:welfare_distribution}.\n\n\n\\begin{figure}[t]\n \\centering\n \\input{Figures\/Surplus3Nodes.tex}\n \\caption{Import\/output graph for the three-node example in Section~\\ref{subsec:three-node}}\n \\label{fig:surplus_three_nodes}\n\\end{figure}\n\n\nNext, suppose there is the possibility of opening a transmission line between node 2 and 3. Suppose the cost of building this line is zero and it has unbounded capacity. If the line is built, then node 2 and 3 can be seen as a single demand node and together, node 1, 2 and 3 form a single, joint market, without any constraints on transmission between nodes. Hence, we can find the market equilibrium by aggregating the two demand curves to obtain the joint demand curve $D(\\pi) = D_2(\\pi) + D_3(\\pi) = 12 - 2\\pi$. The new equilibrium is found at $\\pi_1 = \\pi_2 = \\pi_3 = \\pi^* = 4$, and $s_1 = 2 d_2 = 2 d_3 = q^* = 4$. The new situation is illustrated in the bottom graph in Figure~\\ref{fig:surplus_three_nodes}. Note that the import\/export graphs are given by $I_{2,3} = D$ and $E_1 = S_1$. The joint consumer surplus in node 2 and 3 is represented by the green area, while the producer surplus in node 1 is given by the red area. \n\nThe welfare distribution in the new situation is described in Table~\\ref{tab:welfare_distribution}. Note that the total welfare of the entire system has increased. This is as expected, since the new situation has fewer (transmission) constraints than the old situation. However, observe that the total welfare of node 2 has decreased. The new connection with the demand node 3 has made power scarcer and thus, increased the price. This higher price hurts the consumers in node 2, while there are no producers in node 2 to benefit from the higher price. Hence, node 2 is worse off with the new connection than without. Importantly, node 2 is one of the two end points of the cable and hence, it can be expected to be able to block the connection. Moreover, in practice, a direct connection between node 1 and 3 might be infeasible (or very expensive) for geographical reasons.\n\nIn this situation, node 1 and 3 may decide to spend part of their welfare gains to compensate node 2 for its welfare loss. Indeed, they have sufficient welfare gains (3.5 for node 1 and 2 for node 3) to compensate the welfare loss of 2.5 in node 2. Importantly, such a compensation may help achieve the social optimum, which is desirable from a system point of view. This example motivates our investigation into welfare compensation schemes in the remainder of this paper.\n\n\\begin{table}[t]\n \\centering\n \\begin{tabular}{@{}llrrrr@{}}\n \\toprule\n \\textbf{} & \\textbf{} & \\textbf{1} & \\textbf{2} & \\textbf{3} & \\textbf{System} \\\\ \\midrule\n \\multirow{3}{*}{Old situation} & CS & 0 & 4.5 & 0 & 4.5 \\\\\n & PS & 4.5 & 0 & 0 & 4.5 \\\\ \\cmidrule(l){2-6} \n & TW & 4.5 & 4.5 & 0 & 9 \\\\ \\midrule\n \\multirow{3}{*}{New situation} & CS & 8 & 0 & 0 & 8 \\\\\n & PS & 0 & 2 & 2 & 4 \\\\ \\cmidrule(l){2-6} \n & TW & 8 & 2 & 2 & 12 \\\\ \\bottomrule\n \\end{tabular}%\n \\caption{Welfare distribution -- in terms of consumer surplus (CS), producer surplus (PS), and total surplus (PS) -- in the three-node example of Section~\\ref{subsec:three-node}.}\n \\label{tab:welfare_distribution}\n\\end{table}\n\n\n\n\n\\section{Transmission expansion planning}\n\\label{sec:TEP}\n\nIn this section we use a TEP model to find a system-optimal transmission expansion plan in a case study of the Northern European power market. The results from this model are used in Section~\\ref{sec:compensation_mechanisms} to investigate the performance of various welfare compensation mechanisms. The goal here is not to find the best possible transmission expansion plan in real life; more sophisticated models exist in the literature that are likely more capable for that purpose (see, e.g., \\cite{mahdavi2018transmission}). Instead, the goal is to find a reasonable candidate transmission expansion plan that can serve as a basis for an analysis of different welfare compensation mechanisms.\n\nThe mathematical model can be described as a mathematical program with equilibrium constraints (MPEC), in which a social planner determines the optimal transmission expansion plan from a system welfare point of view, while taking the optimal subsequent behavior of all market participants into account through optimality conditions in the form of equilibrium constraints. A full description of the mathematical model is given in Section~\\ref{sec:mathematical_model} in the appendix.\n\nOur mathematical model is used to analyze a case study of the Northern European power market, focusing on possible investment in a new transmission line between the price zones NO2 and DE, representing part of Norway and the whole of Germany, respectively. Figure~\\ref{fig:LineDiagram} provides a schematic overview of the power system used in the case study. We allow for investments in transmission cables within Norway, too (i.e., between all zones colored red). However, we disallow investments in any other cables, in order to be able to isolate the effects of investments in the new NO2-DE line.\n\n\n\\begin{figure}[t]\n\\centering\n\\input{Figures\/NodeMap.tex}\n\\caption{Line diagram of the power system in the case study.}\n\\label{fig:LineDiagram}\n\\end{figure}\n\nThe data used to parametrize the model is based on historical data on investments in and operation of the Northern European power market. A detailed description of the data used in the case study can be found in Section~\\ref{sec:data} in the appendix.\n\n\n\\subsection{Results}\n\\label{subsec:results}\nWe first consider a benchmark setting in which we disallow investment in the NO2-DE line, but allow for investments within Norway. In this benchmark a social planner invests in 221 MW of extra capacity in the NO1-NO2 line. This comes at an annualized investment cost of 8.0 million euros.\n\nNext, we allow investment in the NO2-DE line, as well as the other lines within Norway. We find that a social planner would invest in a capacity expansion of 4147 MW in the NO2-DE cable. This comes at an annualized investment cost of 294 million euros per year. As expected, this is equal to the expected yearly congestion rent earned on the new line. Besides investment in the NO2-DE line, a social planner would also invest in 464 MW of extra capacity the NO1-NO2 cable, at an annual investment cost of 16.9 million euros. Note that this investment is larger than in the benchmark setting. The rationale behind this additional investment is to export the electricity produced in NO1 (or imported into NO1) through NO2 to DE and the rest of Europe.\n\nCompared to the benchmark, the social planner solution leads to a system-wide social welfare increase of 84.6 million euros per year. This constitutes a return on investment of 27.9\\%. Hence, from a system point of view, it is highly desirable to invest in a new transmission cable. However, some countries -- Germany in particular -- do not profit from the extra capacity and might block the investment plan.\n\nThe welfare effects of the transmission investment are illustrated in Figure~\\ref{fig:welfare_country}. Three countries benefit significantly from the investment: Norway, Austria, and France. In Norway, producers benefit from their additional capacity to export electricity to mainland Europe, while in Austria and France, consumers benefit from the resulting lower prices. On the other hand, two countries are significantly negatively affected: Germany and Denmark. In both countries, producers suffer from the lower prices as a consequence of cheap Norwegian electricity entering the market. This price drop results in both lower profit margins and loss of sales: some of the demand in Germany will be satisfied through Norwegian power entering the country through the new NO2-DE line. One benefit of this, however, is that the power produced in Norway is more than 95\\% renewable \\cite{norwayministryofpetroliumenergy2016}, while in Germany it is only about 45\\% \\cite{germanumweltsbundesamt2021}. Hence, the new line can indirectly contribute to German sustainability goals. \n\nNevertheless, the negative welfare impact of the capacity expansion in Germany may well be an obstacle for realization of the transmission expansion. Since Germany is one of the hosting countries, it can block the investment if it deems the new line to be detrimental to its welfare. However, since the system welfare effects are positive, it is in principle possible to construct compensation mechanisms that result in a net welfare gain for Germany. Notably, since the net welfare gain in Norway (88.2 million annually) exceeds the net welfare loss in Germany (85.1 million annually), a bilateral compensation scheme between these two countries is a viable option.\n\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=0.75]{Figures\/plot_welfare_country_adjusted.png}\n \\caption{Total welfare effects of the NO2-DE cable per country}\n \\label{fig:welfare_country}\n\\end{figure}\n\n\n\n\\section{Compensation mechanisms}\n\\label{sec:compensation_mechanisms}\n\nIn this section we investigate different compensation mechanisms that may be used to compensate countries for welfare losses resulting from transmission expansion investments. First, in Section~\\ref{subsec:incentive_effects} we shortly discuss potential incentive effects of compensation mechanisms. Next, in Section~\\ref{subsec:mechanisms} we list and discuss a number of existing compensation mechanisms and we propose two novel mechanisms. We discuss the rationale behind them and their expected performance in a stochastic setting. Finally, in Section~\\ref{subsec:performance} we numerically test the performance of the different mechanisms using the case study from Section~\\ref{sec:TEP}.\n\n\\subsection{Incentive effects}\n\\label{subsec:incentive_effects}\n\nOne important potential effect of compensation mechanisms is that they might skew incentives of actors in the power market. That is, compensation mechanisms may create incentives for certain actors in the power market to change their behavior in order to receive a larger compensation amount. Importantly, such incentive distortions deviate from ``pure'', market-based incentives and hence, may steer the market equilibrium away from a perfect market equilibrium. This may well have negative effects on the total welfare in the system as a whole. In this paper, we steer away from this issue as much as possible, but a short discussion is in place.\n\nIn practice, the question whether compensation mechanisms affect incentives depends on \\textit{who receives the compensation}. However, in the literature this issue seems to have been ignored. Welfare compensations are modeled as transfers of money between \\textit{countries}, but what agent within a country should receive the money is typically not specified. Given the fact that the proposed compensations are meant for compensating the \\textit{total welfare} in a country, it seems most reasonable to assume that they are paid between \\textit{governments}, which represent the countries as a whole. Since governments do not trade in power markets directly, it seems to be safe to assume that compensations between governments do not skew incentives of any market players. \n\nHowever, in practice, governments may decide to use the compensations to, e.g., change taxes\/subsidies on power, in order to compensate the groups within the country that are affected by the transmission line investment (e.g., consumers or producers). In this case, the compensation may skew incentives of market players. Moreover, if the tax\/subsidy change \\textit{depends on the actual amount of compensation received} by the government, then not only the expected amount of compensation, but also the particular compensation \\textit{mechanism} may skew incentives of market players. \n\nIn this paper, we assume that governments do not use compensations in a way that affects the incentives of players in the energy market. This assumption is in line with the paradigm of avoiding protection and aiming for perfect competition, on which the European power market is founded \\cite{olmos2018transmission}. It allows us to steer away from the issue of incentive distortions as much as possible and investigate the \\textit{risk} effects of compensation mechanisms in isolation. We believe that the topic of incentive distortions resulting from compensation mechanisms is a complicated and interesting topic in its own right and deserves attention in future research.\n\n\n\\subsection{Mechanisms}\n\\label{subsec:mechanisms}\n\n\n\\subsubsection{Lump sum payment: issues}\n\nThe most straightforward compensation mechanism is a lump sum payment. This consists of a fixed payment from one country to another. It has the benefits of being very simple and completely predictable. However, in the presence of uncertainty, a lump sum has some drawbacks. In a stochastic setting, the \\textit{actual} welfare effect resulting from investment in the new line is uncertain. Hence, the lump sum payment should be based on the \\textit{expected} welfare effect. However, there might well be a discrepancy between this expected welfare effect and its actual, realized value. As a result, there may be scenarios in which the lump sum compensation to Germany is not enough or in which Norway must compensate Germany, even though in reality it does not profit from the new line. The potential of such scenarios might make countries hesitant to accept the lump sum mechanism.\n\nMore generally speaking, uncertainty about the actual welfare effects introduces \\textit{risk} for the countries involved. A lump sum mechanism ignores this risk completely by focusing on only the expected value. Other mechanisms might be able to deal with risk in a smarter way. Ideally, a mechanism compensates countries more in scenarios in which they are hurt more by the new line and vice versa. This would reduce the risk of the countries involved, potentially making them more willing to accept the compensation mechanism.\n\nIn order to be able to construct a risk-mitigating mechanism, a few conditions need to hold. Firstly, the new line's welfare effects in the compensating countries and compensated countries should be negatively correlated, such that they can share their risk between scenarios. For the relevant countries in our case study, Norway and Germany, this correlation turns out to be $-0.34$; see Table~\\ref{tab:corr_coalition}. This moderately negative correlation suggests that there is indeed a potential for risk sharing, although probably only to a modest degree. The relationship between the welfare effects in Norway and Germany is presented in more detail in the scatter plot in Figure~\\ref{fig:scatter_NO_DE_welfare_delta}. Note that just over half of the dots, representing scenarios, are to the top-right of the red 45 degree line. In those cases, there is enough total benefit in both countries combined that could theoretically be shared such that neither country suffers from the new line. For the other half of the scenarios this is not possible, however. This again suggests that there is indeed room for risk sharing, although to a limited degree.\n\nSecondly, for a compensation mechanism to be able to exploit such a negative correlation, it should \\textit{depend on the scenario}. That is, the compensation amount should be different under different circumstances. Evidently, a lump sum payment lacks this property, so alternatives are warranted.\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between the welfare effects of the new transmission line in different countries.}\n\\label{tab:corr_coalition}\n\\begin{tabular}{llllll}\n\\toprule\n & \\textbf{NO} & \\textbf{AT} & \\textbf{FR} & \\textbf{DE} & \\textbf{DK} \\\\ \\midrule\n\\textbf{NO} & 1.00 & 0.29 & 0.23 & -0.34 & -0.63 \\\\\n\\textbf{AT} & 0.29 & 1.00 & 0.98 & -0.27 & -0.63 \\\\\n\\textbf{FR} & 0.23 & 0.98 & 1.00 & -0.31 & -0.57 \\\\\n\\textbf{DE} & -0.34 & -0.27 & -0.31 & 1.00 & 0.36 \\\\\n\\textbf{DK} & -0.63 & -0.63 & -0.57 & 0.36 & 1.00 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\n\n\\subsubsection{Power purchase agreement}\nOne scenario-dependent compensation mechanism proposed in the literature \\cite{kristiansen2018} is a so-called power purchase agreement (PPA). This entails giving a certain country, say country A, a virtual, fixed price $\\pi^{\\text{PPA}_A}$ for importing\/exporting power through the new transmission line. Then, after trading at the spot price, the country is compensated such that it is as though the country had traded at the PPA price $\\pi^{\\text{PPA}_A}$. Mathematiaclly, we define the compensation to country $A$ in scenario $\\omega$ by \n\\begin{align*}\n C_{A}^{\\text{PPA}_A,\\omega} = \\sum_{t \\in \\mathcal{T}} f_{AB}^{\\omega,t} (\\pi^{\\text{PPA}_A} - \\pi_A^{\\omega,t}).\n\\end{align*}\nAssuming country $A$ tends to export (i.e., typically $f_{AB}^{\\omega,t}$ is positive), a low PPA price yields a negative compensation to country A, while a high PPA price yields a positive compensation. The reverse holds if country $A$ tends to import. More general definitions of PPAs exist, which give a country a virtual, fixed price for its trade through \\textit{all} transmission cables it is connected to \\cite{kristiansen2018}. However, we find it unreasonable to expect that a country would be willing to essentially take over all of another country's price risk in a compensation scheme for a single transmission line investment. Hence, we choose to restrict the definition to focus on trade through the new transmission line only. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.4]{Figures\/scatter_NO_DE_welfare_Delta.png}\n \\caption{Scatter plot of the effect of the new line on the welfare in Norway and Germany. The dashed red line represents all points for which the aggregated welfare effect for Norway and Germany is zero.}\n \\label{fig:scatter_NO_DE_welfare_delta}\n\\end{figure}\n\n\nOne important property of a PPA is that it is connected to a specific country. This means that one country gets a PPA, i.e., trades at a virtual fixed price, while the other country pays the difference. In a deterministic setting it is possible to construct two PPAs, one for each country, such that the compensation to be paid to the first country according to its PPA exactly matches the compensation to be paid by the second country according to its PPA. This property is called \\textit{budget balancedness} \\cite{narahari2014game}. A multi-country PPA mechanism is proposed in \\cite{kristiansen2018}. However, in a stochastic setting, a multi-country PPA mechanism cannot be constructed, as this would require different PPA prices for every scenario. We illustrate this in the following example.\n\n\\begin{example}\nConsider a network consisting of two countries, A and B, with no initial transmission capacity. There is a proposed investment in a cable of capacity 10 between A and B. We consider two simplified, equally likely, one-period scenarios. In scenario 1 (indicated by the superscript 1), the flow through the new cable is $f_{AB}^1 = 10$, the prices in A and B are $\\pi_A^1 = 1$, $\\pi_B^1 = 2$, and the welfare effects of the new cable are $\\Delta \\text{TW}_A^1 = -10$ and $\\Delta \\text{TW}_B^1 = 20$. In scenario 2, we have $f_{AB}^2 = 10$, $\\pi_A^2 = 1$, $\\pi_B^2 = 3$, $\\Delta \\text{TW}_A^2 = -10$, and $\\Delta \\text{TW}_B^1 = 40$. Before realization of the uncertainty, we need to construct two PPAs that share the welfare gains\/losses equally in expectation. Note that $\\mathbb{E}[\\Delta \\text{TW}_A] = -10$ and $\\mathbb{E}[\\Delta \\text{TW}_B] = +30$. Hence, on average, A should receive a compensation $C_A$ of 20 by B. For A, this yields the following equation for its PPA price:\n\\begin{align*}\n \\mathbb{E}[C_A^{\\text{PPA}_A}] = \\mathbb{E}[f_{AB} (\\pi^{\\text{PPA}_A} - \\pi_A)] = 20,\n\\end{align*}\nwhich, after some calculation, yields $\\pi^{\\text{PPA}_A} = 3$. Similarly, for country B we have the equation\n\\begin{align*}\n \\mathbb{E}[C_B^{\\text{PPA}_B}] = \\mathbb{E}[f_{BA} (\\pi^{\\text{PPA}_B} - \\pi_A)] = -20,\n\\end{align*}\nwhich yields $\\pi^{\\text{PPA}_B} = 4.5$. We observe that using these PPA prices, the compensation received by A equals the compensation paid by B, i.e., $\\mathbb{E}[C_A^{\\text{PPA}_A}] = -\\mathbb{E}[C_B^{\\text{PPA}_B}]$. So buget balancedness holds in expectation. \n\nNow consider a single scenario, e.g., scenario 1. For this scenario we have $C_A^{\\text{PPA}_A,1} = f^1_{AB} (\\pi^{\\text{PPA}_A} - \\pi^1_A) = 10 (3 - 1) = 20$. For B, we have $C_B^{\\text{PPA}_B,1} = f^1_{BA} (\\pi^{\\text{PPA}_B} - \\pi^1_B) = - 10 (4.5 - 2) = - 25$. Note that $C_A^{\\text{PPA}_A,1} \\neq - C_B^{\\text{PPA}_B,1}$, so the compensations do not add to zero. Similarly, for scenario 2 we observe $C_A^{\\text{PPA}_A,2} = 20$ and $C_B^{\\text{PPA}_B,2} = -15$, which also do not add to zero. We conclude that in a stochastic setting, budget balancedness generally does not hold for PPAs. \\hfill \\qedsymbol\n\\end{example}\n\nAs a consequence of this lack of budget balancedness in a stochastic setting, we must pick one country that receives the PPA, i.e., that gets to trade at a virtual, fixed price. This is also the reason for the subscript $A$ in $C_i^{\\text{PPA}_A}$, $i=A,B$, and $\\pi^{\\text{PPA}_A}$. Now, it is not hard to show that a Norway-based PPA yields larger compensations if $\\pi^{\\text{NO2}}$ is higher. Similarly, a Germany-based PPA yields larger compensations if $\\pi^{\\text{DE}}$ is higher. In Table~\\ref{tab:price_correlations} we observe that in both these situations, Norway tends to profit more from the new line, while Germany suffers more. Hence, in situations where higher compensations are desired, both PPAs indeed yield higher compensations. This gives us some confidence that the PPAs might be able to succeed in mitigating risk for the countries involved. Based on the fact that the correlations with the NO2 price are stronger than those with the DE price, we also expect the Norway-based PPA to perform better than the Germany-based PPA.\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between welfare effect of the new transmission line in Norway and Germany and different price measures.}\n\\label{tab:price_correlations}\n\\begin{tabular}{llll}\n\\toprule\n& \\textbf{$\\pi^{\\text{NO2}}$} & \\textbf{$\\pi^{\\text{DE}}$} \\\\ \\midrule\n\\textbf{$\\Delta \\text{TW}_{\\text{NO}}$} & 0.23 & 0.52 \\\\\n\\textbf{$\\Delta \\text{TW}_{\\text{DE}}$} & -0.70 & -0.13 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\\subsubsection{Novel mechanisms}\n\\label{subsubsec:novel_mechanisms}\n\nBesides the lump sum and PPAs, we investigate the potential of other, novel compensation mechanisms. We aim for risk-mitigating mechanisms. Ideally, the mechanism is such that the compensation amount mimics the relative welfare effects in the countries.\n\nOne possibility would be to \\textit{compute} the actual welfare effects using an economic model (such as the one presented in this paper) and base the compensation on this value. The advantage of such a model is that it is likely as close to compensating actual welfare effects as we can plausibly get, and hence, has the greatest potential for risk sharing. Specifically, we define the \\textit{ideal} mechanism as the mechanism that directly shares the welfare benefits from the new transmission line among the participating countries in every scenario, according to some distribution rule represented by the coefficients $\\lambda_i \\geq 0$, $i \\in I$, with $\\sum_{i \\in I} \\lambda_i = 1$. That is, for every scenario $\\omega \\in \\Omega$, the compensation to country $i$ in the set $I$ of participating countries is given by\n\\begin{align*}\n C^{\\text{ideal},\\omega}_i = \\lambda_i \\bigg(\\sum_{j \\in I} \\Delta \\text{TW}^\\omega_j\\bigg) - \\Delta \\text{TW}^\\omega_i.\n\\end{align*}\nThe coefficients $\\lambda_i$, $i \\in I$, should be chosen in such a way that in expectation, the total welfare gains are distributed according to the predetermined distribution rule (e.g., the Shapley value or an equal-share principle).\n\nThere are a number of downsides to this theoretically ideal mechanism, though. The main issue is that the compensation amounts depend on the welfare benefits to the countries \\textit{as computed by the model}. Inevitably, however, the model will be imperfect and thus, the compensation levels are not ``correct''. Hence, conflicts may arise over the model to be used, which may undermine trust between the parties involved. Moreover, even if the model were perfect, it might be seen as a black box by non-experts; a simpler mechanism might be preferred. \n\nTo construct other novel, risk-sharing compensation mechanisms, we take the following approach. We search for economic measures that (1) depend on the scenario, (2) relate to the new transmission line, and (3) are correlated with the welfare effects in the hosting countries. If we find such measures, then we can base a compensation mechanism on them.\n\nBased on the first two conditions listed above, we propose two candidate measures: the amount of flow through the new line (referred to as \\textit{flow}), and the economic value of the flow through the new line (referred to as \\textit{flow value}). We define flow and flow value such that flow in the direction $\\text{NO2}\\to\\text{DE}$ is counted as positive and flow in the opposite direction as negative. Moreover, note that the flow value is ambiguous: in periods in which the line is congested (i.e., flow equals transmission capacity), there is a price differential between the two connected nodes. In such cases, we compute the flow value by using the \\textit{average} of the NO2 price and the DE price.\n\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between two flow-based measures and the welfare effect of the new transmission line in different countries.}\n\\label{tab:corr_country_measure}\n\\begin{tabular}{llllll}\n\\toprule\n\\textbf{} & \\textbf{NO} & \\textbf{AT} & \\textbf{FR} & \\textbf{DE} & \\textbf{DK} \\\\ \\midrule\n\\textbf{flow} & 0.40 & -0.20 & -0.25 & 0.45 & 0.12 \\\\\n\\textbf{flow value} & 0.73 & 0.29 & 0.26 & -0.18 & -0.29 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\nTo investigate the potential of these three candidate measures, we compute for each measure its correlations with the welfare effects of the new line in the hosting countries; see Table~\\ref{tab:corr_country_measure}. Ideally, these correlations are strong and of opposite sign. We observe that flow value is positively correlated with the welfare effect of the new line in all three benefiting countries (Norway, Austria, and France), while negatively correlated with the welfare effect in the suffering countries (Germany and Denmark). Zooming in on the two hosting countries, we observe that it is particularly strongly correlated with the welfare effect in Norway. Moreover, in Figure~\\ref{fig:scatter_flow_value_vs_NO_DE_welfare_delta} we observe that high values for the flow value correspond to scenarios in which Norway benefits most from the new line, while Germany tends to suffer. Based on these observations, flow value appears to be a good candidate measure to base a compensation mechanism on. \n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.4]{Figures\/scatter_flow_value_vs_NO_DE_welfare_delta.png}\n \\caption{Scatter plot of the total value (using the hourly average of the NO2 and DE price) of the flow through the NO2-DE line and the welfare effect of the new line for Norway and Germany. Every dot\/cross represents a scenario.}\n \\label{fig:scatter_flow_value_vs_NO_DE_welfare_delta}\n\\end{figure}\n\nFor flow, on the other hand, the signs of the correlations do not line up with the sign of the expected welfare effect in each country. Moreover, the strongest correlation has a magnitude of 0.45, which is less than then 0.73 we observe for flow value. Hence, flow seems to be a weaker candidate measure to base a compensation mechanism on. Nevertheless, for the sake of completeness we will keep it in our list of candidates and test its performance rigorously.\n\nBased on these two measures, we propose the following novel compensation mechanisms: the \\textit{flow-based} compensation mechanism, under which country $i$ receives a compensation amount of\n\\begin{align*}\n C_i^{\\text{flow},\\omega} = \\alpha_i \\sum_{t \\in \\mathcal{T}} f_{\\text{NO2-DE}}^{\\omega,t},\n\\end{align*}\nfor some $\\alpha_i \\in \\mathbb{R}$, and the \\textit{value-based} compensation mechanism, under which country $i$ receives\n\\begin{align*}\n C_i^{\\text{value},\\omega} = \\beta_i \\sum_{t \\in \\mathcal{T}} f_{\\text{NO2-DE}}^{\\omega,t} \\cdot \\bar{\\pi}^{\\omega,t},\n\\end{align*}\nfor some $\\beta_i \\in \\mathbb{R}$, where $\\pi_t := \\frac{1}{2} ( \\pi^{NO2}_t + \\pi^{DE}_t )$. The values of $\\alpha_i$ and $\\beta_i$, respectively, are chosen in such a way that on average, every country receives a compensation based on some predetermined rule (e.g., the Shapley value). In both cases, to achieve budget balancedness, the values of $\\alpha$ and $\\beta$, respectively, over all countries $I$ participating in the compensation scheme should add to zero, i.e., $\\sum_{i \\in I} \\alpha_i = 0$ and $\\sum_{i \\in I} \\beta_i = 0$.\n\n\n\\subsection{Performance}\n\\label{subsec:performance}\nWe now numerically investigate the performance of the various compensation mechanisms on the case study from Section~\\ref{sec:TEP}. We implemented the following compensation mechanisms: no compensation, lump-sum compensation, a Norway-based PPA, a Germany-based PPA, a flow-based mechanism, a value-based mechanism, and a theoretically ideal model-based mechanism. In order to have a fair comparison, we parametrized each compensation mechanism in such a way that it yields the same expected compensation amount. The amount is such that the expected net welfare effect of the new line after compensation is equal in both countries. Incidentally, this coincides with the Shapley value proposed in \\cite{kristiansen2018}. Since the expected compensation amount is equal among the mechanisms, a risk neutral country should be indifferent about them.\n\nHowever, the performance of the mechanisms may vary in terms of the resulting risk that the countries are exposed to. We assume all countries are risk-averse and hence, we prefer mechanisms that reduce the amount of risk faced by each country as a consequence of the transmission investment. Note that the term ``risk'' is ambiguous; it is sometimes interpreted as analogous to ``variability'', and sometimes as ``likelihood and\/or magnitude of losses'' \\cite{rockafellar2007coherent}. In our analysis we don't choose one interpretation over the other, but include measures corresponding to both interpretations of risk.\n\n\\subsubsection{Bilateral compensation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.75]{Figures\/boxplot_comp_adjusted.png}\n \\caption{Boxplot of the compensation amount paid by Norway to Germany under various compensation mechanisms. The green dashed lines indicate averages, while the orange solid lines indicate medians.}\n \\label{fig:boxplot_comp}\n\\end{figure}\n\nFirst, we consider a setting with bilateral compensations between the two hosting countries Norway and Germany. We start by analyzing the compensation amount itself under various mechanisms. In Figure~\\ref{fig:boxplot_comp} we present a boxplot of the compensation paid by Norway to Germany. Note that indeed, the lump sum leads to a fixed compensation amount, while the other mechanisms vary per scenario. The PPAs show a significantly lager variability than the flow- and value-based mechanisms. Hence, the latter two are more predictable, which may be seen as an advantage.\n\nMore important, however, is how the different mechanisms translate into net welfare effects of the new line. In particular, we are interested in the corresponding risk faced by each country. We first focus on variability-based measures of risk. For this purpose, we refer to the boxplots in Figure~\\ref{fig:boxplot_DE_NO} and the first two columns of Table~\\ref{tab:compensation_statistics}. Note that the theoretical ideal mechanism is able to significantly reduce these measures of variability, though not completely. This is due to the fact that there is variability in the combined welfare effects of the new line, aggregated over both countries. Hence, we should not expect to be able to eliminate all variability using compensation mechanisms, although the ideal mechanism shows that significant reduction is theoretically possible.\n\nNext, we observe that a Germany-based PPA increases the risk faced by both countries, even that of Germany. Hence, the variability in the the compensation amount observed in Figure~\\ref{fig:boxplot_comp} does not cancel out the variability in the welfare effects before compensation. This may be caused by the fact that the correlations between the welfare effects of the new line in Norway and Germany and the price in Germany are not large enough. \n\nA Norway-based PPA, however, reduces the risk faced by Germany significantly, to a level close to the theoretical ideal mechanism, while increasing that faced by Norway. Here, at least for Germany, the correlation between the welfare effect in Germany and the NO2 price is apparently large enough for the variability in the compensation amount to cancel out variability in the welfare effect of the new cable in Germany. For Norway, however, the corresponding effect is not strong enough, which might be explained by the weaker correlation between the NO2 price and the welfare effect of the new line on Norway, observed in Table~\\ref{tab:price_correlations}. Hence, the net effect of the Norway-based PPA is that it shifts risk from Germany to Norway. This may or may not be desirable, depending on the risk preferences of the two countries. \n\nFinally, we observe that the value-based mechanism outperforms all other measures for both countries, except for the Norway-based PPA, which is the best option for Germany. Notably, the value-based mechanism succeeds in reducing the amount of risk faced by both Norway and Germany, compared to a lump sum. Hence, it succeeds in its purpose of risk mitigation. However, the improvement over the lump sum is only modest. Comparing the value-based mechanism to the theoretical ideal, we see that there is still a significant potential for improvement.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Various measures capturing the level of risk faced by Germany and Norway as a result of the new line, after compensation by different mechanisms. The measures are: standard deviation of the compensation amount, standard deviation of the net total welfare effect, probability of welfare loss, expected welfare loss, CVaR of welfare loss (with parameter 0.80). All numbers (except for the percentages) are in millions of euros annually.}\n\\label{tab:compensation_statistics}\n\\begin{tabular}{l|l|ll|ll|ll|ll}\n\\toprule\nmechanism & \\textbf{std(C)} & \\multicolumn{2}{c|}{\\textbf{std($\\Delta \\text{NTW}$)}} & \\multicolumn{2}{c|}{\\textbf{P(L)}} & \\multicolumn{2}{c|}{\\textbf{E{[}L{]}}} & \\multicolumn{2}{c}{$\\textbf{CVaR}_{\\textbf{80}}\\textbf{(L)}$} \\\\ \n & & DE & NO & DE & NO & DE & NO & DE & NO \\\\ \\midrule\nno comp. & 0.0 & 124.8 & 188.8 & 80\\% & 43\\% & 109.1 & 40.5 & 212.1 & 140.4 \\\\\nlump sum & 0.0 & 124.8 & 188.8 & 63\\% & 47\\% & 49.2 & 78.6 & 125.4 & 227.1 \\\\\nPPA DE & 130.5 & 156.4 & 212.1 & 57\\% & 53\\% & 58.9 & 82.5 & 145.2 & 275.5 \\\\\nPPA NO & 211.2 & 94.9 & 209.7 & 37\\% & 53\\% & 34.8 & 87.2 & 150.8 & 272.0 \\\\\nflow & 22.4 & 136.4 & 181.1 & 57\\% & 43\\% & 52.4 & 76.0 & 144.8 & 230.8 \\\\\nflow value & 19.3 & 122.8 & 175.2 & 57\\% & 47\\% & 48.1 & 73.9 & 131.2 & 213.8 \\\\\nideal\t & 130 & 93.6 & 93.6 & 47\\% & 47\\% & 38.0 & 38.0 & 124.8 & 124.8 \\\\\n\\bottomrule\n\\end{tabular}%\n\\end{table*}\n\n\\begin{figure}[h]\n \\centering\n \\begin{subfigure}[b]{\\columnwidth}\n \\centering\n \\includegraphics[scale=0.75]{Figures\/boxplot_DE_adjusted.png}\n \\caption{Germany}\n \\label{subfig:boxplot_DE}\n \\end{subfigure}\\\\\n \\begin{subfigure}[b]{\\columnwidth}\n \\centering\n \\includegraphics[scale=0.75]{Figures\/boxplot_NO_adjusted.png}\n \\caption{Norway}\n \\label{subfig:boxplot_NO}\n \\end{subfigure}\n \\caption{Boxplots of the net welfare effects of the new line for Germany and Norway under various compensation mechanisms. The green dashed lines indicate averages, while the orange solid lines indicate medians.}\n \\label{fig:boxplot_DE_NO}\n\\end{figure}\n\n\n\n\nNext, in the remaining columns of Table~\\ref{tab:compensation_statistics} we consider loss-oriented measures. We compute the probability of a net welfare loss, the expected net welfare loss (only scenarios with losses contribute to this value; gains are regarded as zeros), and the $80\\%$ conditional value at risk (CVaR) \\cite{rockafellar2002conditional} of the net welfare loss, representing the expected value of the $20\\%$ worst cases. We observe that the theoretical ideal achieves in mitigating risk significantly, although not completely. The fact that it cannot completely eliminate the risk of loss was already argued in Section~\\ref{subsec:mechanisms} and illustrated by Figure~\\ref{fig:scatter_NO_DE_welfare_delta}. However, Table~\\ref{tab:compensation_statistics} shows that significant improvements are possible, at least in theory. \n\nTurning to the other mechansims, we observe that the Germany-based PPA performs bad on all fronts. For the Norway-based PPA we see a similar performance as before (good for Germany, bad for Norway), except according to the CVaR measure: in terms of CVaR it performs relatively bad for Germany as well. The interpretation is that although the expected losses to Germany are small, the very worst scenarios are worse than under other mechanisms. Next, the value-based mechanism outperforms all other mechanisms (except the theoretical ideal) according to most measures. It outperforms the flow-based mechanism on all measures except the probability of a loss in Norway. Moreover, it outperforms the lump sum on all but one measure: CVaR for Germany. Hence, again, it seems to succeed in its purpose of mitigating risk for most countries. However, comparing it with the theoretical ideal mechanism, there is again significant room for improvement.\n\n\n\n\n\nWe can draw the following conclusions. Firstly, the theoretical ideal mechanism shows that there is indeed a significant potential for mitigating risk by using scenario-dependent compensations. Secondly, a Germany-based PPA performs by far the worst of all measures considered. It increases the risk faced by the countries compared to a lump sum. However, a Norway-based PPA does yield good results for Germany. It basically transfers part of the risk from Germany to Norway. Such behavior might or might not be desirable, depending on the relative risk preferences of both countries. Next, the value-based mechanism consistently outperforms the flow-based mechanism. Hence, including the additional price information allows the mechanism to succeed better in risk sharing. \n\nFinally, the value-based mechanism outperforms most other mechanisms according to most measures. In particular, it outperforms the lump sum on all measures except CVaR for Germany. Hence, it seems to succeed in achieving what it was constructed to do: reducing the risk faced by both hosting countries. However, the value-based mechanism only improves upon a lump sum by a modest degree. Comparing this with the theoretical ideal mechanism, which performs much better, we conclude that there is still significant room for improvement. In particular, a mechanism based on some measure exhibiting stronger correlations with the welfare effects in Norway and Germany than flow value does in Table~\\ref{tab:corr_country_measure}, might achieve higher levels of risk mitigation.\n\n\n\n\\subsubsection{Multilateral compensation}\n\nIn practice, countries hosting a planned transmission expansion might want to involve other countries that are affected by the planned investment. One reason for this may be to avoid conflict that might hurt future cooperation. In our case study in particular, there are three other countries that are significantly affected by the proposed transmission expansion: Austria, France, and Denmark. Even though the welfare gain for Norway is sufficient to compensate Germany on its own, the majority of the welfare gains actually falls on other countries, most notably Austria and France. Moreover, besides Germany, Denmark is another high-production country that is hurt by the cheap Norwegian power entering the European market. Hence, involving all these five countries in a compensation scheme might be a desirable course of action.\n\nGiven this motivation, we now investigate the performance of different compensation mechanisms in a five-country coalition consisting of the countries mentioned above. We compare lump sum payments to the flow- and value-based mechanisms proposed in Section~\\ref{subsubsec:novel_mechanisms}. Note that the definition of PPAs is unclear in this multilateral setting. For instance, suppose we give Norway a PPA, then it is unclear how we should use this to determine the compensation paid from Austria to Denmark. For this reason, we disregard PPAs in this analysis. Finally, in practice, the share of the benefits allocated to each country is a topic for negotiations. In this case study, we assume an equal-share principle, which entails that every mechanism is parametrized such that the expected net total welfare effect of the new transmission line after compensation is equal in all five countries (25.6 million euros annually).\n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\caption{Various measures capturing the level of risk faced by each coalition country as a result of the new line, after compensation by different mechanisms. All numbers (except for the percentages) are in millions of euros annually.}\n\\label{tab:compensation_statistics_coalition}\n\\begin{tabular}{cllllllll}\n\\toprule\n\\multicolumn{1}{l}{\\textbf{}} &\n \\textbf{mechanism} &\n \\textbf{std(C)} &\n \\textbf{std($\\Delta\\text{NTW}$)} &\n \\textbf{P(L)} &\n \\textbf{E{[}L{]}} &\n \\textbf{$\\text{CVaR}_{\\text{0.8}}\\text{(L)}$} \\\\ \\midrule\n\\multirow{4}{*}{NO} & no comp. & 0.0 & 188.8 & 43\\% & 40.5 & 140.4 \\\\\n & lump sum & 0.0 & 188.8 & 43\\% & 67.7 & 203.1 \\\\\n & flow & 16.2 & 183.0 & 43\\% & 66.2 & 205.8 \\\\\n & flow value & 14.0 & 178.9 & 43\\% & 64.5 & 193.5 \\\\ \n & ideal & 159.8& 43.2 & 27\\% & 7.0 & 33.0 \\\\ \\midrule\n\\multirow{4}{*}{AT} & no comp. & 0.0 & 79.7 & 0\\% & 0.0 & 0.0 \\\\\n & lump sum & 0.0 & 79.7 & 40\\% & 15.5 & 61.4 \\\\\n & flow & 23.0 & 87.2 & 47\\% & 20.1 & 72.4 \\\\\n & flow value & 19.9 & 76.3 & 43\\% & 15.3 & 50.3 \\\\\n & ideal & 63.5 & 43.2 & 27\\% & 7.0 & 33.0 \\\\ \\midrule\n\\multirow{4}{*}{FR} & no comp. & 0.0 & 54.2 & 0\\% & 0.0 & 0.0 \\\\\n & lump sum & 0.0 & 54.2 & 30\\% & 8.0 & 38.2 \\\\\n & flow & 11.5 & 58.1 & 37\\% & 9.7 & 42.2 \\\\\n & flow value & 9.9 & 52.5 & 37\\% & 7.3 & 31.5 \\\\ \n & ideal & 47.8 & 43.2 & 27\\% & 7.0 & 33.0 \\\\ \\midrule\n\\multirow{4}{*}{DE} & no comp. & 0.0 & 124.8 & 80\\% & 109.1 & 212.1 \\\\\n & lump sum & 0.0 & 124.8 & 60\\% & 34.2 & 101.4 \\\\\n & flow & 28.6 & 140.0 & 53\\% & 40.3 & 128.2 \\\\\n & flow value & 24.7 & 122.8 & 53\\% & 34.9 & 110.9 \\\\ \n & ideal & 125.4& 43.2 & 27\\% & 7.0 & 33.0 \\\\ \\midrule\n\\multirow{4}{*}{DK} & no comp. & 0.0 & 36.1 & 97\\% & 60.2 & 112.7 \\\\\n & lump sum & 0.0 & 36.1 & 20\\% & 5.4 & 27.0 \\\\\n & flow & 22.1 & 44.6 & 30\\% & 9.0 & 39.4 \\\\\n & flow value & 19.1 & 35.7 & 27\\% & 5.4 & 25.5 \\\\\n & ideal & 70.1 & 43.2 & 27\\% & 7.0 & 33.0 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table*}\n\nIn Table~\\ref{tab:compensation_statistics_coalition} we present performance measures for the various compensation mechanisms. First, we observe that under all mechanisms (except the theoretical ideal), Norway and Germany face by far the largest levels of risk. This can be understood by the fact that since they host the new cable, the cable affects these countries most directly. Next, we observe that the value-based mechanism outperforms the flow-based mechanism on almost all measures for all countries. This is in line with our results in the bilateral case and can again be explained by the more desirable correlations observed in Table~\\ref{tab:corr_country_measure} \n\nThe question remains which of the remaining two mechanisms, lump sum and the value-based mechanism, is preferable. In terms of a variability-based definition of risk, the value-based mechanism is clearly the winner. It has a smaller standard deviation of net total welfare effects than the lump sum for all countries. Hence, it indeed succeeds in risk mitigation. Next, in terms of a loss-based definition of risk, the results are again in favor of the flow-value based mechanism, though less strongly so. In terms of expected loss and CVaR of loss, all countries except for Germany are better off with the value-based mechanism. The reason for underperformance for Germany may be the weak correlation between flow value and the welfare effect of the new line in Germany; see Table~\\ref{tab:corr_country_measure}.\n\nOverall, we conclude that the value-based mechanism consistently outperforms the flow-based mechanism and also outperforms the lump sum in terms of most measures for most countries. All in all, the value-based mechanism seems the most promising in terms of mitigating risk of the countries involved. However, comparing it to the theoretical ideal mechanism, there is again significant room for improvement.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe investigated the potential of existing and novel welfare compensation mechanisms in TEP under uncertainty. The simplest existing mechanism, lump sum payments, does not take uncertainty into account at all. Other mechanisms, under which the compensation amount depends on the scenario, can potentially exploit negative correlations between the welfare effects of a new transmission line in benefiting and suffering countries in order to mitigate the risk of all parties involved. \n\nWe conducted numerical experiments in a case study of the Northern-European power sector. We find a system-optimal investment in a new transmission line between Norway and Germany that benefits Norway but hurts Germany in terms of expected total welfare. We considered both bilateral compensations (between Norway and Germany alone) and multilateral compensations (involving other affected countries, too). In both settings, we observed that a theoretically ideal, model-based mechanism can significantly reduce the levels of risk faced by the countries involved. This highlights the relevance of our research and the potential of mitigating risk by using scenario-dependent compensation mechanisms.\n\nWe analyzed one scenario-dependent mechanism from the literature: PPAs. We first show that in a stochastic setting, budget balancedness does not hold for PPAs. Hence, one should select a single country that receives the PPA. Moreover, in the numerical experiments we observed that a Germany-based PPA \\textit{increases} risk for both countries, while a Norway-based PPA shifts risk from Germany to Norway. We conclude that when considering a PPA, one should be careful in selecting the country at which the PPA is based.\n\nWe also tested two novel mechanisms, based on the flow through the new line and its economic value. In both the bilateral and multilateral setting, our novel value-based mechanisms performs best in terms of mitigating risk for the countries involved. It appears to do so by successfully exploiting negative correlations in the welfare effects of the new transmission line between benefiting and suffering countries. In particular, the value-based mechanism outperforms the lump sum payment. However, the improvement is only moderate. Comparing it with the theoretically ideal mechanism, we see that there is still significant potential for improvement. We expect that the level of outperformance may be higher if the negative correlations mentioned above are stronger or if the correlations between the economic value of flow through the new transmission line and the welfare effects of this line in the neighboring countries are stronger. \n \nWhile we deem the value-based mechanism most promising, its performance may well depend on the specific practical setting at hand. Therefore, in practical situations, we suggest to run an analysis like the one presented in this paper before choosing a particular compensation mechanism. Note that if the proposed transmission expansion plan is found by running a TEP model, then the proposed analysis can be performed by a simple extension of this model. One might use this paper as a blueprint for such an analysis. In any case, we suggest to include our value-based mechanism as one of the candidate compensation mechanisms.\n\nFuture research might focus on further investigating the performance of various compensation mechanisms in settings beyond the case study investigated in our paper. In particular, it would be interesting to test our hypothesis that the value-based mechanism performs better in situations with a strong correlation between the economic value of the flow through the new transmission line and its welfare effects in the neighboring countries. In addition, novel mechanisms may be developed based on measures exhibiting such strong correlations, or mixtures of several mechanisms may be tested. Moreover, it would be interesting to investigate the performance of different compensation mechanisms in a setting with investments in multiple transmission lines simultaneously, rather than the single-line setting used in this paper.\n\nAnother avenue for future research might be to investigate the incentive-distorting effects of welfare compensations and various mechanisms in particular, as sketched in Section~\\ref{subsec:incentive_effects}. It would be interesting to see how much welfare compensations may cause the system to deviate from the equilibrium and what the welfare effects of these deviations are. Such an analysis might be able to identify the types of government interventions (taxes\/subsidies) and compensation mechanisms that minimize this issue of incentive distortions.\n\n\n\\paragraph{Acknowledgements}\nWe want to thank THEMA Consulting for the data they shared with us.\n\n\n\\begin{appendices}\n\n\\section{Mathematical model}\n\\label{sec:mathematical_model}\n\nOur TEP model consists of two levels. In the lower level, the producers, consumers and the TSO act in the electricity market. We assume that each of these actors maximizes their own surplus and we assume perfect competition with all actors being price takers. The equilibrium problem arising from these interacting optimization problems can be formulated as a mixed-complementarity program (MCP) consisting of the Karush-Kuhn-Tucker (KKT) conditions of each actor's optimization problem, combined with a market clearing condition. In the upper level, a social planner is endowed with the task of deciding the transmission expansion investment levels. We assume that the social planner acts as a Stackelberg leader that tries to maximize total welfare of the entire system, while taking the optimal decisions of the followers into account. Together, the bi-level model constitutes an MPEC.\n\n\\subsection{Notation}\n\\noindent\\textbf{Sets:}\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$\\mathcal{N}$ & Set of nodes (indexed by $n$) \\\\\n$\\mathcal{L}$ & Set of lines (indexed by $l$) \\\\\n$\\mathcal{G}_n$ & Set of generators in node $n$ (indexed by $g$) \\\\\n$\\mathcal{R}_n$ & Set of renewables in node $n$ (indexed by $r$) \\\\\n$\\mathcal{S}$ & Set of seasons (indexed by $s$) \\\\\n$\\mathcal{T}$ & Set of time periods (indexed by $t$) \\\\\n$\\mathcal{T}_S$ & Set of time periods in season $s$ (indexed by $t$) \\\\\n$\\Omega$ & Set of scenarios (indexed by $\\omega$) \\\\\n\\end{tabularx}\n\\\\\n\n\\noindent\\textbf{Parameters:}\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$P_\\omega$ & Probability for scenario $\\omega$ \\\\\n$I^{R}_{\\omega rt}$ & Production profile for renewable $r$ in scenario $\\omega$ in time period $t$ \\\\\n$G^{R}_{r}$ & Installed capacity for renewable $r$ $[\\SI{}{\\mega\\watt}]$\\\\\n$C^{I,R}_{r}$ & Investment cost for renewable $r$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$C^{I,G}_{g}$ & Investment cost for generator $g$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$C^{G}_{gt}$ & Marginal cost for generator $g$ in time period $t$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$G^{Max}_{g}$ & Installed generation capacity for generator $g$ [$\\SI{}{\\mega\\watt}$]\\\\\n$Q^{Max}_{\\omega gs}$ & Production limit for generator $g$ in scenario $\\omega$ in season $s$ [$\\SI{}{\\mega\\watt h}$]\\\\\n$A_{nl}$ & Node-line incidence matrix entry for node $n$ and line $l$\\\\\n$C^{I,L}_{l}$ & Investment cost for line $l$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n\\end{tabularx}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$F^{Max}_{l}$ & Maximum line capacity for line $l$ [$\\SI{}{\\mega\\watt}$]\\\\\n$D^{A}_{\\omega nt}$ & Slope of inverse demand function for node $n$ in scenario $\\omega$ in time period $t$\\\\\n$D^{B}_{\\omega nt}$ & Constant of inverse demand function for node $n$ in scenario $\\omega$ in time period $t$\\\\\n\\end{tabularx}\n\\\\\n\n\\noindent\\textbf{Variables:}\n\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$y^R_{r}$ & Capacity expansion for renewable $r$ [$\\SI{}{\\mega\\watt}$] \\\\\n$y_{g}$ & Capacity expansion for generator $g$ [$\\SI{}{\\mega\\watt}$] \\\\\n$q_{\\omega gt}$ & Production for generator $g$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$f_{\\omega lt}$ & Flow in line $l$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$x_{l}$ & Capacity expansion for line $l$ [$\\SI{}{\\mega\\watt}$] \\\\\n$d_{\\omega nt}$ & Demand in node $n$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$\\pi_{\\omega nt}$ & Price in node $n$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$] \n\\end{tabularx}\n\n\n\n\\subsection{Lower-level problem}\n\\label{subsec:lower-level_problem}\nIn this subsection we describe the lower-level equilibrium problem. This equilibrium problem can be represented as an MCP consisting of the KKT conditions of the optimization problems of all the actors in the market, combined with a market clearing constraint. We will not present the KKT conditions explicitly, but simply state each actor's optimization problem.\n\n\n\\subsubsection{Conventional energy producer problem}\n\\label{subsubsec:producer_problem}\nEvery conventional generating unit (power plant) is modeled as an independent generating company maximizing its operational profits minus investment costs. Each generating company can freely choose its generation levels to maximize their profits. Moreover, it has the possibility to invest in additional generating capacity if needed. Eq.~\\eqref{opt:GenCoProblem} describes the optimization problem for generator $g \\in \\mathcal{G}_n$ located in node $n \\in \\mathcal{N}$, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, s \\in \\mathcal{S}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{q_{\\omega gt}, y_{g}}\n{\\sum_{\\omega \\in \\Omega}\\sum_{t \\in \\mathcal{T}}P_\\omega\\left(\\pi_{\\omega nt}-C^G_{gt}\\right)q_{\\omega gt}-C^{I,G}_{g}y_g\\label{opt:GenCoObjective}}\n{\\label{opt:GenCoProblem}}\n{}\n\\addConstraint{q_{\\omega gt}}{\\leq G^{Max}_{g}+y_{g}\\label{opt:GenCoProdCap}}\n\\addConstraint{\\sum_{t\\in\\mathcal{T}_S}q_{\\omega gt}}{\\leq Q^{Max}_{\\omega gs}\\label{opt:GenCoEnergyLimit}}\n\\addConstraint{q_{\\omega gt}, y_{g}}{\\geq 0\\label{opt:GenCoLargerThanZero}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:GenCoObjective} consists of maximizing the expected revenue minus the investment costs in new generating capacity. Operating costs and investment costs are both assumed to be linear. Eq.~\\eqref{opt:GenCoProdCap} states that the production must be no more than the existing generating capacity plus the invested generating capacity. Eq.~\\eqref{opt:GenCoEnergyLimit} states that the total production in a season must be no more than the available quantity, which may vary per season. Finally, Eq.~\\eqref{opt:GenCoLargerThanZero} states that the generation and investment quantities must be non-negative. \n\n\n\n\\subsubsection{Renewable energy producer problem}\n\\label{subsubsec:RES_problem}\nThe renewable energy companies want maximize their profit earned from generating and selling renewable energy. In contrast with the conventional power producers, they are not able to freely choose their production levels; these are determined by the wind and solar profile. They can choose to invest in additional generating capacity, though. The optimization problem for renewable energy producer $r \\in \\mathcal{R}_n$ located in node $n \\in \\mathcal{N}$ is given by Eq.~\\eqref{opt:RESProblem}.\n\\begin{maxi!}[1]\n{y^R_{r}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{t\\in\\mathcal{T}}P_\\omega\\pi_{\\omega nt}\\left(G^R_{r}+y^R_{r} \\right) I^R_{\\omega rt}-C^{I,R}_r y^R_r\\label{opt:RESObjective}}\n{\\label{opt:RESProblem}}\n{}\n\\addConstraint{y_{r}^R}{\\geq 0\\label{opt:RESConNonneg}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:RESObjective} consists of maximizing the expected revenue minus the investments in new renewable capacity. Investment costs are assumed to be linear. Eq.~\\eqref{opt:RESConNonneg} states that capacity investment is non-negative.\n\n\\subsubsection{Consumer problem}\n\\label{subsubsec:consumer_problem}\nThe consumers aim to satisfy their demand for power at the lowest possible cost. In other words, they want to maximize the consumer surplus in each node $n$. Assuming a linear demand function, Eq.~\\eqref{opt:ConsumerProblem} represents the maximization problem for the consumers located in node $n$.\n\n\\begin{maxi!}[1]\n{d_{\\omega nt}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{t\\in\\mathcal{T}}P_\\omega\\left(\\frac{1}{2}D^A_{\\omega nt}d_{\\omega nt}+D^B_{\\omega nt}-\\pi_{\\omega nt}\\right)d_{\\omega nt}\\label{opt:ConsumerObjective}}\n{\\label{opt:ConsumerProblem}}\n{}\n\\end{maxi!}\nEq.~\\eqref{opt:ConsumerObjective} is the objective function for the consumers, representing the consumer surplus.\n\n\n\\subsubsection{TSO Problem}\n\\label{subsubsec:TSO_problem}\nWe assume a single TSO that operates all lines. The TSO maximizes the expected congestion rent for all lines, and sets the line flows accordingly. Recall that we assume that the TSO is a price taker. Hence, it will not use its market power and it will basically act as a dummy player (this is confirmed by our finding that in the equilibrium solution the TSO makes zero profit in expectation). The optimization problem for the TSO is given by Eqs.~\\eqref{opt:TSOProblem}, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, l \\in \\mathcal{L}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{f_{\\omega lt}, x_{l}}\n{-\\sum_{\\omega \\in \\Omega}\\sum_{i \\in \\mathcal{N}}\\sum_{l\\in \\mathcal{L}}\\sum_{t \\in \\mathcal{T}}P_{\\omega}A_{nl}f_{\\omega lt}\\pi_{\\omega nt}\\label{opt:TSOObjective}}\n{\\label{opt:TSOProblem}}\n{}\n\\addConstraint{f_{\\omega lt}}{\\leq F^{Max}_{l}+x_{l}\\label{opt:TSOflowPos}}\n\\addConstraint{f_{\\omega lt}}{\\geq -F^{Max}_{l}-x_{l}\\label{opt:TSOflowNeg}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:TSOObjective} consists of the expected congestion rent earned from all lines. Eqs.~\\eqref{opt:TSOflowPos} and \\eqref{opt:TSOflowNeg} state that the flow in a line must not exceed the current capacity plus the invested capacity. Note that the optimization model can be equivalently separated into optmization problems for each line. Hence, the assumption of a single TSO is without loss of generality.\n\n\n\\subsubsection{Market clearing}\nThe market clearing constraint is used to connect the market actors' decisions together. It guarantees that the market clears, i.e., that supply meets demand. It is given by\n\n\\begin{equation}\n d_{\\omega nt}+\\sum_{l\\in \\mathcal{L}}A_{nl}f_{\\omega lt}=\\sum_{g\\in \\mathcal{G}_n}q_{\\omega gt}+\\sum_{r\\in \\mathcal{R}_n}\\left(G^R_{r}+y^R_{r}\\right)I^R_{\\omega rt} \\label{opt:MarketClearing} \n\\end{equation}\nIn particular, Eq.~\\eqref{opt:MarketClearing} states that for a given node, the sum of demand and net outgoing flows must be equal to the total amount of power generated from both conventional and renewable sources. The market price is given by the dual variable $\\pi_{\\omega nt}$ corresponding to this constraint.\n\n\\subsection{Upper-level problem} \\label{subsec:upper-level_problem}\nThe upper-level problem consists of the social planner's transmission expansion problem. The social planner chooses the investment levels $x_l$, $l \\in \\mathcal{L}$, that maximize the expected net total welfare in the system. Here, we define net total welfare as gross total welfare (consisting of producer surplus, consumer surplus, and congestion rent) minus investment cost (which are assumed to be linear). The social planner acts as a Stackelberg leader that takes the other actors' optimal responses into account. \n\nLet $TW(x,q,y,y^R,d,f,\\pi)$ denote the expected gross total welfare corresponding to the decision vectors $x,q,y,y^R,d,f$ and price vector $\\pi$. That is, $TW$ is equal to the sum of the objective functions of the optimization problems of all producers, consumers, and the TSO. Let $KKT_1, \\ldots, KKT_4$ denote the sets of KKT conditions corresponding to the problems in Eqs.~\\eqref{opt:GenCoProblem}-\\eqref{opt:TSOProblem}, respectively. Then, the social planner's problem can be described as an MPEC of the form\n\n\\begin{maxi!}[1]\n{x_{l} \\geq 0}\n{TW(x,q,y,y^R,d,f,\\pi) - \\sum_{l\\in\\mathcal{L}}C^{I,L}_l x_l}\n{\\label{opt:GovProblem}}\n{\\label{opt:GovObjective}}\n\\addConstraint{KKT_1, \\ldots, KKT_4}{\\label{opt:GovKKT}}\n\\addConstraint{\\text{Market clearing condition Eq. } \\eqref{opt:MarketClearing}}{\\label{opt:GovMarketClearing}}\n\\end{maxi!}\nEq.~\\eqref{opt:GovObjective} represents the social planner's objective, consisting of maximizing the net total welfare. Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} are the equilibrium constraints arising from the lower-level MCP.\n\n\\subsubsection{Quadratic programming reformulation}\nThe social planner's optimization problem, given by the MPEC Eq.~\\eqref{opt:GovProblem}, can be reformulated as a single quadratic program. To show this, first define $TW^*(x)$ as the value of $TW(x,q,y,y^R,d,f,\\pi)$ at an optimal solution to the lower-level MCP defined by Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} (this value is unique, as we will prove shortly). Then, the social planner's problem consists of maximizing net total welfare:\n\n\\begin{maxi!}[1]\n{x_{l} \\geq 0}\n{TW^*(x) - \\sum_{l\\in\\mathcal{L}}C^{I,L}_l x_l}\n{\\label{opt:GovProblemReformulated}}\n{}\n\\end{maxi!}\n\n\n\nSimilar to the classical result by \\cite{samuelson1952}, it can be shown that the lower-level MCP is equivalent to a central planner quadratic optimization problem in which total welfare is maximized. The proof of this equivalence, which we omit for brevity, is through the observation that the KKT conditions to the quadratic program, which are necessary and sufficient, are equivalent to the mixed-complementarity Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} defining the lower-level MCP. Observe that $TW^*(x)$ denotes the optimal value of the quadratic program, which proves that this value is indeed unique. \n\nSubstituting the lower-level quadratic programming formulation for $TW^*(x)$ into Eq.~\\eqref{opt:GovProblemReformulated}, and combining the upper and lower level decisions into a single optimization problem yields a quadratic programming reformulation for the social planner's MPEC. It is given by Eq.~\\eqref{opt:CentralPlanner}, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, g \\in \\mathcal{G}_n, r \\in \\mathcal{R}_n, n \\in \\mathcal{N}, l \\in \\mathcal{L}, s \\in \\mathcal{S}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{\\substack{q_{\\omega gt}, y_{g} \\\\ y^R_{r} d_{\\omega nt} \\\\ f_{\\omega lt}, x_{l}}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{i\\in\\mathcal{N}}\\sum_{t\\in\\mathcal{T}}P_\\omega\\left(\\frac{1}{2}D^A_{\\omega nt}d_{\\omega nt}+D^B_{\\omega nt}\\right)d_{\\omega nt}}{\\label{opt:CentralPlanner}}{} \\nonumber\n\\breakObjective{-\\sum_{\\omega\\in\\Omega}\\sum_{g\\in\\mathcal{G}_n}\\sum_{t\\in\\mathcal{T}}P_\\omega C^G_{gt}q_{\\omega gt} -\\sum_{g\\in\\mathcal{G}_n}C^{I,G}_{g}y_g} \\nonumber\n\\breakObjective{-\\sum_{r\\in\\mathcal{R}_n}C^{I,R}_ry^R_r-\\sum_{l\\in\\mathcal{L}}C^{I,L}_lx_l \\label{opt:CentralPlannerObj3}} \n\\addConstraint{q_{\\omega gt}}{\\leq G^{Max}_{g}+y_{g} \\label{opt:CentralPlannerConFirst}}\n\\addConstraint{\\sum_{t\\in\\mathcal{T}_S}q_{\\omega gt}}{\\leq Q^{Max}_{\\omega gs}}\n\\addConstraint{d_{\\omega nt}+\\sum_{l\\in \\mathcal{L}}A_{nl}f_{\\omega lt}}{=\\sum_{g\\in \\mathcal{G}_n}q_{\\omega gt}+\\sum_{r\\in \\mathcal{R}_n}\\left(G^R_{r}+y^R_{r}\\right)I^R_{\\omega rt}}\n\\addConstraint{f_{\\omega lt}}{\\leq F^{Max}_{l}+x_{l}}\n\\addConstraint{f_{\\omega lt}}{\\geq -F^{Max}_{l}-x_{l}}\n\\addConstraint{q_{\\omega gt}, d_{\\omega nt}}{\\geq 0}\n\\addConstraint{y_{g}, y^R_{r}, x_{l}}{\\geq 0 \\label{opt:CentralPlannerConLast}}\n\\end{maxi!}\nHere, the objective function in Eq.~\\eqref{opt:CentralPlannerObj3} consists of the sum of the objective functions of all market participants' optimization problems less investment cost in new transmission lines. The constraints in Eqs.~\\eqref{opt:CentralPlannerConFirst}--\\eqref{opt:CentralPlannerConLast} are a concatenation of the constraints from all market actors' individual optimization problems and the market clearing constraint.\n\n\n\n\n\\section{Data}\n\\label{sec:data}\nThe model presented in Section~\\ref{sec:mathematical_model} is solved using a power system consisting of Norway, Germany and their neighbouring regions. In this section we describe the data used to parametrize the model.\n\nMarginal investment costs of generators and transmission lines are estimated using data on historical investments \n\\cite{Blumsac2022Basic,schroder2013current,NorNed2008longest,Statnett2013Cooperation,Powerengineeringinternational2012Germany,Northconnect2022FAQ} and annualized using economic lifetimes based on \\cite{schroder2013current} using an assumed interest rate of 4\\%. The estimates are given in Table~\\ref{tab:InvestmentCosts in sec:Results}. Since our main concern is the interaction between Norway and Germany, all capacity expansions in transmission and generation in and between other countries than Norway and Germany are fixed at zero.\n\n\\begin{table}[t]\n\\centering\n\\caption{Marginal annualized investment costs [\\SI{}{\\text{\\EUR{}}\\per\\mega\\watt}] for generating technologies and transmission lines.}\n\\label{tab:InvestmentCosts in sec:Results}\n\\begin{tabular}{@{}lrllr@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Generators}} & \\multicolumn{1}{c}{\\textbf{}} & \\multicolumn{2}{c}{\\textbf{Lines}} \\\\ \\midrule\nCoal & 136,559 & & NO1-NO2 & 36,399 \\\\\nLignite & 140,826 & & NO1-NO3 & 56,874 \\\\\nCCGT & 70,413 & & NO1-NO5 & 52,324 \\\\\nOther gas & 38,407 & & NO2-NO5 & 53,461 \\\\\nSolar & 66,224 & & NO5-NO3 & 79,623 \\\\\nWind & 78,735 & & NO3-NO4 & 130,809 \\\\\n & & & NO2-DE & 70,864 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nInitial generation and transmission capacities are based on data from the EMPIRE model in \\cite{backe2022empire}, the ENTSO-E Transparicy platform\n\\cite{EntsoE-Capacities} and from privately communicated data from THEMA Consulting Group. In line with policy goals of phasing out or reducing production from nuclear plants, some nations have their nuclear plant capacities reduced or completely removed. Germany and Belgium have their nuclear capcities completely removed, while France have a reduced capacity in order to mimic normal operating conditions as well as future capacity goals. Using plant efficiencies, together with privately communicated fuel- and $\\mathrm{CO_2}$-price time series from THEMA Consulting Group, we have approximated seasonal marginal operational costs for the thermal power plants. These costs are given in Table \\ref{tab:MarginalCostGen}.\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Marginal costs [\\SI{}{\\text{\\EUR{}}\\per\\mega\\watt}] for generating technologies.}\n\\label{tab:MarginalCostGen}\n\\begin{tabular}{@{}lrrrr@{}}\n\\toprule\n \\multicolumn{1}{c}{\\textbf{}}& \\multicolumn{1}{c}{\\textbf{Winter}}&\\multicolumn{1}{c}{\\textbf{Spring}}&\\multicolumn{1}{c}{\\textbf{Summer}}&\\multicolumn{1}{c}{\\textbf{Autumn}}\n \\\\ \\midrule\nCoal & 35,2 & 29,8 & 30,1 & 35,4 \\\\\nLignite & 38,9 & 32,9 & 33,3 & 39,4 \\\\\nCCGT & 41 & 33,8 & 35,2 & 39,3 \\\\\nGas & 63,1 & 52,1 & 54,2 & 60,4 \\\\\nNuclear & 15 & 15 & 15 & 15 \\\\\nHydro & 0 & 0 & 0 & 0 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nScenario-specific data are sampled from time series from NordPool \\cite{Nordpool2022Historical} and Open Power Systems Data \\cite{openpowersystemsdata2022timeseries}. The linear demand curves are constructed from demand- and price data. Assuming that inverse demand is written as $\\pi = ad + b$, the parameters are calculated as\n\\begin{align*}\n a = \\frac{1}{\\varepsilon} \\frac{|P|}{D}, \\qquad b = \\left(1 - \\frac{1}{\\varepsilon}\\right) |P|,\n\\end{align*}\nwith $P$ and $D$ being the historical price and demand for a particular hour, respectively, and a price elasticity of demand of $\\varepsilon = -0.05$, which is in line with estimates in, e.g., \\cite{matar2018households}. We use the absolute value $|P|$ to account for historical hours with negative prices. This method ensures that demand is downward sloping with an elasticity close to \\SI{-0.05}{} in cases when model outcomes are close to historical outcomes. Production profiles for solar and wind plants scaled as a factor between 0 and 1 are found from \\cite{RenewablesNinja}.\n\nThe problem instance consists of a total of 30 scenarios. Each scenario is sampled randomly with historical data from the years 2013-2017. We sample hourly data from consecutive one-week periods, one for each of the four seasons. In this way, each scenario gets four 168-hour seasons resulting in a total of 672 hours in each scenario.\n\n\\end{appendices}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\nA typical source for the construction of a Kasparov $K$-homology\ncycle is an elliptic differential complex. If the elliptic complex\nis equivariant with respect to the action of a group $\\mathsf{G}$, and\nif moreover the group action satisfies an additional conformality\nproperty (see below), then one can obtain an element of equivariant $K$-homology. But if\n$\\mathsf{G}$ is a semisimple Lie group of rank greater than one, non-trivial examples of such complexes cannot\nexist (\\cite{Puschnigg}). This paper describes a means of constructing an equivariant\n$K$-homology class from the\nBernstein-Gelfand-Gelfand complex for\n$\\SL(3,\\mathbb{C})$---a differential complex which is neither elliptic nor\nconformal, but which satisfies some weaker (`directional') form of\nthese conditions.\n\nThe motivation for this construction comes from the Baum-Connes\nconjecture. Although an understanding of the conjecture is not\nessential to this paper, it is useful for perspective. The conjecture asserts that for a second countable locally compact group $\\mathsf{G}$, the \n {\\em assembly map}\n$$\n \\mu_{\\Gamma} : K^\\Gamma(\\underline{E}\\Gamma) \\to K(C^*_r\\Gamma).\n$$\nis an isomorphism, thus giving a `topological computation' of the $K$-theory of the reduced group $C^*$-algebra. For a fuller description of the conjecture and its many consequences, we refer the reader to the expository article \\cite{HigsonICM} and the foundational paper \\cite{BCH}.\n\nThe conjecture has been proven for a wide class of groups, amongst which we mention in particular the discrete subgroups of simple Lie groups of real rank one. A notable unknown, however, is the group $\\SL(3,\\mathbb{Z})$. More broadly, the conjecture is unknown for general discrete subgroups of semisimple Lie groups of rank greater than one.\n\nFor subgroups of rank one semisimple groups $\\mathsf{G}$, the proofs in each case centre on a canonical idempotent $\\gamma$ in the representation ring $R(\\mathsf{G}):=KK^\\mathsf{G}(\\mathbb{C},\\mathbb{C})$. (See \\cite{Kas-Lorentz} for $\\mathsf{G}=\\SO_0(n,1)$, \\cite{JK} for $\\mathsf{G}=\\SU(n,1)$, \\cite{Julg}\\footnote{The first proof of Baum-Connes for discrete subgroups of $\\Sp(n,1)$ was due to V.~Lafforgue, but used a somewhat different approach.} for $\\Sp(n,1)$). For our purposes, the most convenient way to describe this idempotent $\\gamma$ is via the following fact.\n\n\\begin{theorem}[Kasparov]\n\\label{thm:split_surjection}\nLet $\\mathsf{G}$ be a semisimple Lie group and $\\mathsf{K}$ a maximal compact subgroup. The restriction map $R(\\mathsf{G}) \\to R(\\mathsf{K})$ is a split surjection of rings.\n\\end{theorem}\n\nThe unit in $R(\\mathsf{K})$ is the class of the trivial $\\mathsf{K}$-representation, and its image under the splitting is an idempotent in $R(\\mathsf{G})$. This is $\\gamma$. \n\nIf $\\gamma=1\\in R(\\mathsf{G})$ then the restriction map is an isomorphism. In this case, the `Dirac-dual Dirac method' of Kasparov implies that the Baum-Connes conjecture holds for all discrete subgroups of $\\mathsf{G}$. This is the approach taken in the papers cited above, although in the case of $\\Sp(n,1)$ a weaker notion of `triviality' for $\\gamma$ must be used.\n\nThe idempotent $\\gamma$ was originally defined via equivariant $K$-homology for the proper $\\mathsf{G}$-space $\\mathsf{G\/K}$ (\\cite{Kas88}). In the rank-one proofs mentioned above, however, $\\gamma$ is more conveniently constructed using the compact space $\\mathsf{G\/B}$, where $\\mathsf{B}$ is a minimal parabolic subgroup. This can be explained by the fact that the induced representations from $\\mathsf{B}$ give a natural topological parameterization of (the relevant subset of) representations of $\\mathsf{G}$, namely the generalized principal series, including the complementary series. It is also pertinent that $\\mathsf{B}$ is amenable, so itself satisfies Baum-Connes.\n\nIt is instructive to consider the construction of $\\gamma$ in the simple example $\\mathsf{G}=\\SL(2,\\mathbb{C})$. One begins with the Dolbeault complex for the homogeneous space $\\mathsf{G}\/\\mathsf{B} \\cong \\CC \\mathrm{P}^1$:\n$$\n \\Omega^{0,0}\\CC \\mathrm{P}^1 \\xrightarrow{\\overline{\\partial}} \\Omega^{0,1}CP^1.\n$$ \nThis is a $\\mathsf{G}$-equivariant elliptic complex. Importantly, though, $\\CC \\mathrm{P}^1$ does not admit a $\\mathsf{G}$-invariant Riemannian metric. The action is conformal (with respect to the natural $\\mathsf{K}$-equivariant metric), and the translation representation of $\\mathsf{G}$ on $L^2\\Omega^{0,\\bullet}\\CC \\mathrm{P}^1$ can be made unitary by the introduction of a scalar Radon-Nikodym factor. But the operator $D := \\overline{\\partial} + \\overline{\\partial}^*$ will not be $\\mathsf{G}$-equivariant, not even in the weak sense of defining an unbounded equivariant Fredholm module. Somewhat magically though, replacing $D$ by its operator phase results in a bounded equivariant Fredholm module. For this to work it is crucial that the $\\mathsf{G}$-action is conformal\\footnote{In general, the conformality requirement is even stronger: the ratio of the Radon-Nikodym factors in degrees $p$ and $p+1$ must be independent of $p$. We will not explain this further.} on the Hermitian bundles $\\Omega^{0,p}\\CC \\mathrm{P}^1$.\n\nIn order to maintain this crucial conformality property for the other rank one cases, one must use increasingly complicated subellitpic differential complexes --- the Rumin complex for $\\SU(n,1)$; a quaternionic analogue thereof for $\\Sp(n,1)$ --- and corresponding nonstandard pseudodifferential calculi. We remark that $K$-homological constructions using even nonstandard pseudodifferential calculi typically result in finitely-summable Fredholm modules. Puschnigg \\cite{Puschnigg} has shown that simple Lie groups of higher rank do not admit any nontrivial finitely summable Fredholm modules.\n\nThis motivates our construction using the Bernstein-Gelfand-Gelfand (`BGG') complex.\n\n\\begin{theorem}[Bernstein-Gelfand-Gelfand]\nLet $\\mathsf{G}$ be a complex semisimple group and $\\mathsf{B}$ a minimal parabolic subgroup. For any finite dimensional holomorphic representation $V$ of $\\mathsf{G}$ there is a differential complex, consisting of direct sums of homogeneous line bundles over $\\mathsf{G\/B}$ and $\\mathsf{G}$-equivariant differential operators between them, which resolves $V$.\n\\end{theorem}\n\nThe bundles in each degree here are not conformal, but their component line bundles are individually conformal. (Trivially, any group action on a Hermitian line bundle is conformal.) The question is whether this structure is enough to produce an element of equivariant $K$-homology. In this paper, we answer this question affirmatively in the case of $\\mathsf{G}=\\SL(3,\\mathbb{C})$. We thereby obtain an explicit construction of the splitting map $ R(\\mathsf{K}) \\to R(\\mathsf{G})$, and in particular a construction of $\\gamma$, which factors through $KK^\\mathsf{G}( C(\\mathsf{G\/B}), \\mathbb{C})$.\n\nThe construction is based upon harmonic analysis of $\\SU(3)$ rather than some nonstandard pseudodifferential calculus. An indication of the difficulties of a purely pseudodifferential approach is given in Chapter 5 of \\cite{Yuncken-thesis}. In fact, our construction could be made without any reference to pseudodifferential operators at all, though pseudodifferential theory has become so central to index theory that to do so might seem somewhat eccentric.\n\nMuch of the required harmonic analysis has been developed in \\cite{Yuncken:PsiDOs} in the broader context of $\\SU(n)$ ($n\\geq2$). We expect that the results of this paper should be extendable the groups $\\SL(n,\\mathbb{C})$, and indeed to complex semisimple groups in general. The main technical difficulty in the case of $\\SL(n,\\mathbb{C})$ is an appropriate version of the the operator partition of unity of Lemma \\ref{lem:operator_po1} of this paper. For general semisimple groups, the required directional harmonic analysis is yet to be developed.\n\n\n\nAs for the Baum-Connes Conjecture itself, it is known that $\\gamma\\neq1$ for any group $\\mathsf{G}$ which has Kazhdan's property $T$. Therefore,\na direct translation of Kasparov's method cannot prove the\nBaum-Connes conjecture for simple Lie groups of rank greater than\none---some subtle variation of Kasparov's argument would be\nrequired. Nevertheless, it is expected that the present construction\nwill be useful for further study of the Baum-Connes conjecture.\n\n\n\\medskip\n\nLet us now describe the BGG complex in more detail. In fact, knowledge of the cohomological version of the BGG complex is unnecessary for the present paper, since our $K$-homological version will be produced from scratch. \nBut it is such a strong motivation that it is worth spending some time explaining it.\n\nFinite dimensional holomorphic representations of $\\mathsf{G}$ are parameterized by their highest weights. Let $V^\\lambda$ denote the representation with highest weight $\\lambda$. Any weight $\\mu$ of $\\mathsf{G}$ extends to a holomorphic character of $\\mathsf{B}$ (see Section \\ref{sec:homogeneous_vector_bundles}), and we denote by $\\Lhol{\\mu}$ the corresponding induced holomorphic line bundle over $\\scrX:=\\mathsf{G\/B}$. The Borel-Weil Theorem states that $V^\\lambda$ is equivariantly isomorphic to the space of global holomorphic sections of $\\Lhol{\\lambda}$. \n\nRecall that the Weyl group $\\Lie{W}$ is a group of reflections on the weight space. It is generated by the {\\em simple reflections}---reflections in the walls orthogonal to a choice of simple roots for $\\mathsf{G}$. Word length in these generators defines a length function $l:\\Lie{W}\\to\\mathbb{N}$. We need the {\\em shifted action} of the Weyl group defined by the formula $w\\star \\mu := w(\\mu+\\rho) - \\rho$, where $\\rho$ is the half-sum of the positive roots. Bernstein, Gelfand and Gelfand \\cite{BGG} showed that there is a holomorphic $\\mathsf{G}$-equivariant differential operator from $\\Lhol{\\mu}$ to $\\Lhol{\\nu}$ if and only if $\\mu=w\\star\\lambda$ and $\\nu = w'\\star\\lambda$ for some dominant weight $\\lambda$ and some $w,w'\\in\\Lie{W}$ with $l(w')\\geql(w)$. What is more, these operators can be assembled into an exact complex as follows\\footnote{Strictly speaking, Bernstein, Gelfand and Gelfand made a homological complex by assembling intertwiners between Verma modules. What we are calling the BGG complex here is a dual cohomological complex. See the appendix of \\cite{CSS} for an explanation of this. }. One defines the degree $p$ cocycle space $C^p := \\bigoplus_{l(w) = p} C^\\infty(\\scrX, \\Lhol{w\\star \\lambda})$. The collection of equivariant differential operators between any $\\Lhol{w\\star\\lambda}$ and $\\Lhol{w'\\star\\lambda}$ with $l(w)=p$, $l(w')=p+1$ defines a matrix of operators $C^p \\to C^{p+1}$. With an appropriate choice of signs these operators resolve the Borel-Weil inclusion $V^\\lambda \\hookrightarrow C^\\infty(\\scrX;\\Lhol{\\lambda})$.\n\n\n\n\nIn the case of $\\SL(3,\\mathbb{C})$, we get a complex\n\\begin{equation}\n\\label{eq:BGG_resolution}\n \\xymatrix@!C=7.2ex{\n & C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_1}\\star\\lambda}) \\ar[rr]\\ar[ddrr]\n \\ar@{.}[dd]|-{\\bigoplus}\n && C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\star\\lambda}) \\ar[dr]\n \\ar@{.}[dd]|-{\\bigoplus}\n \\\\\n V^\\lambda \\hookrightarrow C^\\infty(\\scrX;\\Lhol{\\lambda})\\ar[ur]\\ar[dr]\\quad\\quad\n &&&&\\quad\\quad C^\\infty(\\scrX;\\Lhol{w_\\rho\\star\\lambda}) \\\\\n & C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_2}\\star\\lambda})\\ar[rr]\\ar[uurr]\n && C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\star\\lambda})\\ar[ur] \\\n }\n\\end{equation}\nwhere $\\alpha_1$, $\\alpha_2$ and $\\rho = \\alpha_1+\\alpha_2$ are the positive roots, and $\\reflection{\\alpha}$ denotes the reflection in the wall orthogonal to $\\alpha$.\n\nIn this paper, we define a `normalized', {\\em i.e.}, $L^2$-bounded, version of this complex which is analogous to the equivariant Fredholm module constructed above from the Dolbeault complex of $\\CC \\mathrm{P}^1$.\n\n\\medskip\n\nTo complete this overview, we give a very brief description of the harmonic analysis upon which our $K$-homological BGG construction is based. The space $\\scrX:= \\mathsf{G\/B}$ is the complete flag variety of $\\mathbb{C}^3$. Corresponding to the simple roots $\\alpha_1$ and $\\alpha_2$, there are $\\mathsf{G}$-equivariant fibrations $\\scrX\\to\\scrX[i]$ $(i=1,2)$ where $\\scrX[1]$ and $\\scrX[2]$ are the Grassmannians of lines and planes in $\\mathbb{C}^3$. As described in \\cite{Yuncken:PsiDOs}, associated to each of these fibrations is a $C^*$-algebra $\\scrK[\\alpha_i]$ of operators on the $L^2$-section space of any homogeneous line bundle over $\\scrX$. This algebra contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the given fibration. A key property is that the intersection $\\scrK[\\alpha_1]\\cap \\scrK[\\alpha_2]$ consists of compact operators. Ultimately, this allows us to apply the Kasparov Technical Theorem to construct a Fredholm module from the normalized BGG operators.\n\n\n\n\n\n\n\n\\medskip\n\nThe structure of the paper is as follows. Section \\ref{sec:preliminaries} gives the background on the structure theory of the semisimple Lie group $\\mathsf{G}=\\SL(3,\\mathbb{C})$, the flag variety $\\scrX$ and its homogeneous line bundles, mainly for the purpose of setting notation. \n\nIn Section \\ref{sec:harmonic_analysis} we review the $C^*$-algebras $\\scrK[\\alpha_i]$ of \\cite{Yuncken:PsiDOs} and their relation to longitudinal pseudodifferential operators on the flag variety $\\scrX$. We also prove two important new results concerning these algebras. For the sake of stating these results elegantly, it is convenient to place the $C^*$-algebras $\\scrK[\\alpha_i]$ in the context of $C^*$-categories (see Section \\ref{sec:categories} for details).\n\n\n\\begin{theorem}\n\\label{thm:harmonic_analysis_results}\nLet $E$, $E'$ be $\\mathsf{G}$-homogeneous line bundles over $\\scrX$. Let $\\scrA$ denote the simultaneous multiplier category of $\\scrK[\\alpha_1]$ and $\\scrK[\\alpha_2]$ (see Definition \\ref{def:A}).\n\\begin{enumerate}\n\\item The translation operators $g: \\LXE \\to \\LXE$ belong to $\\scrA$, for all $g\\in\\mathsf{G}$.\n\\item If $T:\\LXE \\to L^2(\\scrX;E')$ is a longitudinal pseudodifferential operator of order zero tangent to one of the fibrations $\\scrX\\to\\scrX[i]$ ($i=1,2$), then $T\\in\\scrA$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{thm:harmonic_analysis_results}(i) is proven in Section \\ref{sec:principal_series}. Part (ii) is restated in Theorem \\ref{thm:PsiDOs_in_A}. The proof requires some lengthy computations in $\\SU(3)$ harmonic analysis which are presented in Appendix \\ref{sec:PsiDOs_in_A}. \n\nIn Section \\ref{sec:construction}, we combine the above results to construct an element of $KK^\\mathsf{G}(C(\\scrX),\\mathbb{C})$ from the BGG complex. We also explain why this yields the splitting of the restriction morphism $R(\\mathsf{G}) \\to R(\\mathsf{K})$.\n\n\\medskip\n\nPart of this work appeared in the author's doctoral dissertation\n\\cite{Yuncken-thesis}. I would like to thank my thesis adviser, Nigel Higson. I would also like to thank Erik Koelink for\nseveral informative conversations.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Notation and Preliminaries}\n\\label{sec:preliminaries}\n\n\\subsection{Lie groups}\n\\label{sec:Lie_groups}\n\nThroughout this paper $\\mathsf{G}$ will denote the group $\\SL(3,\\mathbb{C})$. We fix notation for the following subgroups: $\\mathsf{K}=\\SU(3)$, its maximal compact subgroup; $\\mathsf{H}$, the Cartan subgroup of diagonal matrices; $\\mathsf{A}$, the subgroup of diagonal matrices with positive real entries; $\\mathsf{M} = \\mathsf{H}\\cap\\mathsf{K}$, the maximal torus of $\\mathsf{K}$; $\\mathsf{N}$, the subgroup of upper triangular unipotent matrices; and $\\mathsf{B}=\\mathsf{MAN}$ the subgroup of upper triangular matrices. Their Lie algebras are denoted $\\mathfrak{g}$, $\\mathfrak{k}$, $\\mathfrak{h}$, $\\mathfrak{a}$, $\\mathfrak{m}$, $\\mathfrak{n}$ and $\\mathfrak{b}$. \n\nWe use $V^\\dagger$ to denote the dual of a complex vector space $V$. \nWe make the usual identifications of the complexifications $\\mathfrak{m}_\\mathbb{C}$ and $\\mathfrak{a}_\\mathbb{C}$ with $\\mathfrak{h}$ by extending the inclusions $\\mathfrak{a},\\mathfrak{m}\\hookrightarrow\\mathfrak{h}$ to $\\mathbb{C}$-linear maps. We thereby identify characters of $\\mathsf{A}$ and $\\mathsf{M}$ with elements of $\\mathfrak{h}^\\dagger$. Characters of $\\mathfrak{h}$ will be denoted by $\\chi = \\chi_{\\Lie{M}} \\oplus \\chi_{\\Lie{A}}$, where $\\chi_{\\Lie{M}}$ and $\\chi_{\\Lie{A}}$ are the restrictions of $\\chi$ to $\\mathfrak{m}$ and $\\mathfrak{a}$, respectively. The corresponding group character of $\\mathsf{H}$ will be denoted $e^\\chi$. The weight lattice in $\\mathfrak{m}_\\mathbb{C}^\\dagger \\cong \\mathfrak{h}^\\dagger$ will be denoted by $\\Lambda_W$.\n\nThe set of roots of $\\mathsf{K}$ is denoted $\\Delta$. We fix the notation\n$$\nX_{\\alpha_1} = \\left(\\begin{array}{ccc} 0&1&0\\\\0&0&0\\\\0&0&0 \\end{array}\\right),~\nX_{\\alpha_2} = \\left(\\begin{array}{ccc} 0&0&0\\\\0&0&1\\\\0&0&0 \\end{array}\\right),~\nX_\\rho = \\left(\\begin{array}{ccc} 0&0&1\\\\0&0&0\\\\0&0&0 \\end{array}\\right)~\n\\in \\mathfrak{k}_\\mathbb{C} \\cong \\mathfrak{g},\n$$\nwhich are root vectors for the roots $\\alpha_1$, $\\alpha_2$ and $\\rho:=\\alpha_1+\\alpha_2$. We fix these as our set of positive roots $\\Delta^+$, so $\\Sigma := \\{\\alpha_1, \\alpha_2\\}$ is the set of simple roots. For each $\\alpha\\in\\Delta^+$, $Y_\\alpha$ will denote the transpose of $X_\\alpha$. We abbreviate $X_{\\alpha_i}$ and $Y_{\\alpha_i}$ to $X_i$ and $Y_i$, whenever convenient.\n\nWe put $H_i := [X_i, Y_i] \\in \\mathfrak{m}_\\mathbb{C}$. The elements $X_i, Y_i, H_i$ span a Lie subalgebra isomorphic to $\\mathfrak{sl}(2,\\mathbb{C})$, which we denote by $\\mathfrak{s}_i$. We also put\n$$\n H_1' := \\left(\\begin{array}{ccc} 1&0&0\\\\0&1&0\\\\0&0&-2 \\end{array}\\right), \\qquad\n H_2' := \\left(\\begin{array}{ccc} -2&0&0\\\\0&1&0\\\\0&0&1 \\end{array}\\right) \\qquad \n \\in \\mathfrak{m}_\\mathbb{C} \\cong \\mathfrak{h},\n$$\nso that for fixed $i=1,2$, $H_i$ and $H_i'$ span $\\mathfrak{h}$ and $H_i'$ commutes with $\\mathfrak{s}_i$.\n\n\nThe Weyl group of $\\mathsf{G}$ is $\\mathsf{W}\\cong S_3$. We let $\\reflection{\\alpha}$ denote the reflection in the wall orthogonal to the root $\\alpha$. The {simple reflections} $w_{\\alpha_1}$ and $w_{\\alpha_2}$ are generators of $W$, and the minimal word length in these generators defines the length function $l$ on $W$. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Homogeneous vector bundles}\n\\label{sec:homogeneous_vector_bundles}\n\nThroughout, $\\scrX$ will denote the homogeneous space $\\scrX=\\mathsf{G}\/\\mathsf{B} = \\mathsf{K}\/\\mathsf{M}$.\n\nLet $\\chi = \\chi_{\\Lie{M}}\\oplus\\chi_{\\Lie{A}}$ be a character of $\\mathfrak{h}$. As usual, we extend it trivially on $\\mathfrak{n}$ to a character of $\\mathfrak{b}$. We use $L_\\chi$ to denote the $\\mathsf{G}$-homogeneous line bundle over $\\scrX$ which is induced from $\\chi$. That is, continuous sections of $L_\\chi$ are identified with $\\mathsf{B}$-equivariant functions on $\\mathsf{G}$ as follows:\n\\begin{multline}\n \\CXL[\\chi] = \\{ s:\\mathsf{G}\\to\\mathbb{C} \\text{ continuous }\\; | \\; s (gman) = e^{\\chi_{\\Lie{M}}}(m^{-1})e^{\\chi_{\\Lie{A}}}(a^{-1}) s (g) \n \\\\\n \\forall g\\in\\mathsf{G}, m\\in\\mathsf{M}, a\\in\\mathsf{A}, n\\in\\mathsf{N} \\}.\n\\label{eq:B-equivariance}\n\\end{multline}\nThe $\\mathsf{G}$-action on sections is by left translation: $g'\\cdots(g) := s({g'}^{-1}g)$.\nRestricting to $\\mathsf{K}$, we have the `compact picture' of $\\CXL[\\chi]$:\n\\begin{multline}\n \\CXL[\\chi] \\cong \\{ s:\\mathsf{K} \\to \\mathbb{C} \\text{ continuous } \\; | \\; s(km) = e^{\\chi_{\\Lie{M}}}(m^{-1})s(k) \n \\\\\n \\forall k\\in\\mathsf{K}, m\\in\\mathsf{M} \\}.\n\\label{eq:M-equivariance}\n\\end{multline}\nNote that, as a $\\mathsf{K}$-homogeneous bundle, $L_\\chi$ depends only on $\\chi_{\\Lie{M}}$.\n\nThe compact picture gives a Hermitian metric on $L_\\chi$. Specifically, the pointwise inner product of sections is given by\n$$\n \\ip{s_1(k),s_2(k)} = \\overline{s_1(k)} s_2(k) \\quad \\in C(\\scrX).\n$$\nThe $L^2$-section space $\\LXL[\\chi]$ is the completion of $\\CXL[\\chi]$ with respect to the inner product\n\\begin{equation}\n\\label{eq:inner_product}\n \\ip{s_1,s_2} = \\int_\\mathsf{K} \\overline{s_1(k)} s_2(k) \\, dk.\n\\end{equation}\n\n\n\nSome cases warrant special notation. If $\\mu$ is a weight for $\\mathsf{K}$, we let $\\Lhol{\\mu}$ denote the holomorphic line bundle $L_{\\mu\\oplus\\mu}$. We also let $E_\\mu$ denote the `unitarily induced' bundle $L_{\\mu\\oplus\\rho}$. On $E_\\mu$ the translation action $U_\\mu:\\mathsf{G} \\to \\scr{L}(\\LXE[\\mu])$ is a {unitary} representation. These will be the main focus of our attention.\n\nRestricting $U_\\mu$ to $\\mathsf{K}$, $\\LXE[\\mu]$ becomes a subrepresentation of the left regular representation $\\mathsf{K}$. If $R$ denotes the \\emph{right} regular representation, then the equivariance condition of Equation \\eref{eq:M-equivariance} becomes $R(m) s = e^{-\\mu}(m) s$ for all $m\\in\\mathsf{M}$. Infinitesimally,\n\\begin{eqnarray}\n \\LXE[\\mu] &=& \\{s\\in L^2(\\mathsf{K}) \\; | \\; R(M) s = -\\mu(M) s \\text{ for all } M\\in\\mathfrak{m} \\} \\nonumber \\\\\n &=& p_{-\\mu} L^2(\\mathsf{K}),\n\\label{eq:m-equivariance}\n\\end{eqnarray}\nwhere $p_{-\\mu}$ denotes the orthogonal projection onto the $(-\\mu)$-weight space of the {\\em right} regular representation of $\\mathsf{K}$ on $L^2(\\mathsf{K})$.\n\nLet $\\chi$, $\\chi'$ be characters of $\\mathsf{B}$. If $f\\in\\CXL[\\chi'-\\chi]$ then pointwise multiplication by $f$, denoted $\\multop{f}$, maps $\\CXL[\\chi]$ to $\\CXL[\\chi']$. This gives a $\\mathsf{G}$-equivariant bundle isomorphism $\\End(L_\\chi,L_{\\chi'}) \\cong L_{\\chi'-\\chi}$. In particular, $\\End(E_\\mu, E_{\\mu'}) \\cong L_{(\\mu'-\\mu)\\oplus0}$ for any weights $\\mu$, $\\mu'$. Moreover, for any $f\\in\\CXL[(\\mu'-\\mu)\\oplus0]$,\n\\begin{equation}\n\\label{eq:covariance_of_multops}\n U_{\\mu'}(g) \\multop{f} U_\\mu(g^{-1}) = \\multop{g\\cdot f}.\n\\end{equation}\nIn this picture, a locally trivializing partition of unity on $E_\\mu$ takes the following form.\n\n\\begin{lemma}\n\\label{lem:partition_of_unity}\n\nFor any weight $\\mu$, there exists a finite collection of continuous sections $\\varphi_1,\\ldots,\\varphi_n \\in \\CXL[\\mu\\oplus0]$ such that $\\sum_{j=1}^n \\multop{\\varphi_j} \\multop{\\overline{\\varphi_j}} = 1$.\n\n\\end{lemma}\n\n\\begin{proof}\nLet $f_1,\\ldots,f_n\\in \\CX$ be a partition of unity subordinate to a locally trivializing cover of $E_\\mu$. Composing $f_j^{\\frac{1}{2}}$ with the corresponding local trivialization $L_0 \\xrightarrow{\\cong} L_{\\mu\\oplus0}$ gives the sections $\\varphi_j$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Parabolic subgroups and equivariant fibrations}\n\\label{sec:parabolic_subgroups}\n\nLet $\\mathsf{P}$ be a parabolic subgroup, $\\mathsf{B}\\leq \\mathsf{P} \\leq \\mathsf{G}$, with Lie algebra $\\mathfrak{p}$. Let $S\\subseteq \\Sigma$ be the set of simple roots $\\alpha$ such that the root space $\\mathfrak{g}_{-\\alpha}$ is contained in $\\mathfrak{p}$. This set classifies $\\mathsf{P}$, and we therefore introduce the notation \n$$\n \\mathsf{P}_{\\Sigma} := \\mathsf{G}, \\quad\n \\mathsf{P}_{\\{\\alpha_1\\}} := \\left\\{ \\smatrix{ *&*&*\\\\ *&*&*\\\\0&0&* } \\right\\}, \\quad\n \\mathsf{P}_{\\{\\alpha_2\\}} := \\left\\{ \\smatrix{ *&*&*\\\\ 0&*&*\\\\0&*&* } \\right\\}, \\quad\n \\mathsf{P}_{\\emptyset} := \\mathsf{B}.\n$$\n(Here $*$ denotes possibly nonzero entries.) We will simplify this by writing $\\mathsf{P}_i := \\mathsf{P}_{\\{\\alpha_i\\}}$ whenever convenient.\n\nFor $i=1,2$, let $\\scrX[i] := \\mathsf{G}\/\\mathsf{P}_i$. The natural maps $\\fibration[i] : \\scrX \\to \\scrX[i]$ are equivariant fibrations with fibres $\\mathsf{P}_i\/\\mathsf{B} \\cong \\CC \\mathrm{P}^1$. We will denote the corresponding foliations of $\\scrX$ by $\\foliation[i]:= \\ker D\\fibration[i]$.\n\n\nDenote the compact part of $\\mathsf{P}_S$ by $\\mathsf{K}_S := \\mathsf{P}_S \\cap \\mathsf{K}$. Explicitly,\n\\begin{eqnarray*}\n \\mathsf{K}_\\Sigma &:=& \\mathsf{K}, \\\\\n \\mathsf{K}_1 &:=& \\mathsf{P}_1 \\cap \\mathsf{K} \n = \\left\\{ \\ulmatrix{A}{0}{0}{0&0&z} \\bigg| \\quad\n A\\in\\mathrm{U}(2), ~ z = (\\det A)^{-1} \\right\\}, \\\\\n \\mathsf{K}_2 &:=& \\mathsf{P}_2 \\cap \\mathsf{K} \n = \\left\\{ \\drmatrix{z&0&0}{0}{0}{A} \\bigg| \\quad\n A\\in\\mathrm{U}(2), ~ z = (\\det A)^{-1} \\right\\}, \\\\\n \\mathsf{K}_\\emptyset &:=& \\mathsf{M}.\n\\end{eqnarray*}\nThen $\\scrX[i]=\\mathsf{K}\/\\mathsf{K}_i$ ($i=1,2$).\n\nThe complexified Lie algebra $(\\mathfrak{k}_i)_\\mathbb{C}$ of $\\mathsf{K}_i$ decomposes as $\\mathfrak{s}_i\\oplus\\mathfrak{z}_i$, where $\\mathfrak{s}_i := \\vspan{X_i, H_i, Y_i} \\cong \\mathfrak{sl}(2,\\mathbb{C})$ and $\\mathfrak{z}_i := \\vspan{H_i'}\\subset \\mathfrak{m}_\\mathbb{C}$. (Notation as in Section \\ref{sec:Lie_groups}.) For the sake of fixing notation, we recall the representation theory of $\\mathfrak{s}_i \\cong \\mathfrak{sl}(2,\\mathbb{C})$. The weights of $\\mathfrak{sl}(2,\\mathbb{C})$ are parameterized by the integers. The restriction of a weight $\\mu$ of $\\mathsf{K}$ to a weight of $\\mathfrak{s}_i$ is $\\mu_i := \\mu(H_i) \\in \\mathbb{Z}$. The dominant weights are the nonnegative integers $\\mathbb{N}$. \n\nLet $X,H,Y\\in\\mathfrak{sl}(2,\\mathbb{C})$ be the basis elements corresponding to $X_i, H_i. Y_i\\in\\mathfrak{s}_i$. The irreducible representation of $\\mathfrak{sl}(2,\\mathbb{C})$ with highest weight $\\delta\\in\\mathbb{N}$ will be denoted $V^\\delta$. It has an orthonormal basis of weight vectors $\\{ e_\\delta, e_{\\delta-2}, \\ldots, e_{-\\delta+2}, e_{-\\delta} \\}$, such that\n\\begin{eqnarray}\n\\label{eq:X-formula}\n X\\cdot e_j &=& \\frac{1}{2} \\sqrt{(\\delta-j)(\\delta+j+2)} \\,e_{j+2} \\\\\n\\label{eq:H-formula}\n H\\cdot e_j &=& j\\, e_{j} \\\\\n\\label{eq:Y-formula}\n Y\\cdot e_j &=& \\frac{1}{2} \\sqrt{(\\delta-j+2)(\\delta+j)} \\, e_{j-2} \n\\end{eqnarray} \n\n\n\n\\subsection{Harmonic analysis}\n\\label{sec:harmonic_notation}\n\nFor any compact group $\\mathsf{C}$, we will use $\\irrep{C}$ to denote the set of irreducible representations of $\\mathsf{C}$, often referred as {\\em $\\mathsf{C}$-types}. For any unitary representation $\\pi$ of $\\mathsf{C}$, we use $V^\\pi$ to denote its representation space, and $\\pi^\\dagger$ to denote its contragredient representation.\n\n\nFor a representation $\\pi$ of $\\mathsf{K}=\\SU(3)$ and elements $\\xi\\in V^\\pi$, $\\eta^\\dagger\\in V^{\\pi\\dagger}$, we use $\\matrixunit{\\eta^\\dagger}{\\xi}$ to denote the matrix unit $\\matrixunit{\\eta^\\dagger}{\\xi}(k) := (\\eta^\\dagger, \\pi(k)\\xi)$. \nRecall the Peter-Weyl isomorphism\n\\begin{eqnarray*}\n \\bigoplus_{\\pi\\in\\irrep{K}} V^{\\pi\\dagger} \\otimes V^\\pi & \\cong & L^2(\\mathsf{K}) \\\\\n \\eta^\\dagger \\otimes \\xi & \\mapsto & (\\dim V^\\pi)^\\frac{1}{2} \\matrixunit{\\eta^\\dagger}{\\xi}.\n\\end{eqnarray*}\nwhich intertwines $\\bigoplus \\pi$ and $\\bigoplus\\pi^\\dagger$ with the left and right regular representations, respectively.\nIf $p_\\mu$ denotes the projection onto the $\\mu$-weight space of a representation then from Equation \\ref{eq:m-equivariance},\n$$\n \\LXE[\\mu] \\cong \\bigoplus_{\\pi\\in\\irrep{K}} V^{\\pi\\dagger} \\otimes p_{-\\mu}V^\\pi .\n$$\n\n\n\n\n\n\\section{Harmonic analysis on the flag variety}\n\\label{sec:harmonic_analysis}\n\n\\subsection{Harmonic $C^*$-categories}\n\\label{sec:harmonic_decompositions}\n\\label{sec:categories}\n\n\nWe will make much use of the results of \\cite{Yuncken:PsiDOs} regarding harmonic analysis on flag manifolds for $\\SL(n,\\mathbb{C})$. In this section, we review the major definitions and results of that paper. Because we are only interested in $n=3$ here, we will simplify the notation somewhat. \n\nLet $\\mathsf{K}'$ be a closed subgroup of $\\mathsf{K}=\\SU(3)$. Let $\\scr{H}$ be a Hilbert space equipped with a unitary representation of $\\mathsf{K}$. For $\\sigma\\in\\irrep{K}'$, we let $p_\\sigma$ denote the orthogonal projection onto the $\\sigma$-isotypical subspace of $\\mathcal{H}$ (with representation restricted to $\\mathsf{K}'$). \nIf $F\\subseteq \\irrep{K}'$ is a set of $\\mathsf{K}'$-types, we let $p_F := \\sum_{\\sigma\\inF} p_\\sigma$. \n\nWe are particularly interested in the four subgroups $\\mathsf{K} \\geq \\mathsf{K}_1, \\mathsf{K}_2, \\geq \\mathsf{M}$ above. Note that the isotypical subspaces of $\\mathsf{M}$ are the weight spaces.\n\nIf $\\mathsf{K}''$ is a subgroup of $\\mathsf{K}'$, then the isotypical projections of $\\mathsf{K}'$ and $\\mathsf{K}''$ commute. In particular, the isotypical projections of $\\mathsf{K}$, $\\mathsf{K}_1$ and $\\mathsf{K}_2$ commute with the weight-space projections. These isotypical projections can therefore be restricted to any weight-space of a unitary $\\mathsf{K}$-representation.\n\n\\begin{definition}\nA {\\em harmonic $\\mathsf{K}$-space} $H$ is a direct sum of weight spaces of unitary $\\mathsf{K}$-representations: $H = \\bigoplus_{k} p_{\\mu_k} \\mathcal{H}_k$ for some weights $\\mu_k$ and unitary $\\mathsf{K}$-representations on $\\mathcal{H}_k$.\n\nA harmonic $\\mathsf{K}$-space $H$ is called {\\em finite multiplicity} if for every $\\pi\\in\\irrep{K}$, $p_\\pi H$ is finite dimensional.\n\\end{definition}\n\n\\begin{example}\n\\label{ex:finite_multiplicities}\nThe (right) regular representation is a finite multiplicity harmonic $\\mathsf{K}$-space by the Peter-Weyl Theorem, as is $\\LXE[\\mu]$ for any weight $\\mu$. More generally, any homogeneous vector bundle $E$ over $\\scrX$ decomposes equivariantly into line bundles, so $\\LXE$ is a harmonic $\\mathsf{K}$-space. \n\\end{example}\n\n\n\n\n\n\\begin{definition}\n\\label{def:A_K}\nLet $S\\subseteq\\Sigma$. Let $A:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces. For $\\sigma',\\sigma\\in\\irrep{K}_S$, let $A_{\\sigma'\\sigma} := p_\\sigma' A p_\\sigma$, so that $(A_{\\sigma'\\sigma})$ is the matrix decomposition of $A$ with respect to the decompositions of $H,H'$ into $\\mathsf{K}_S$-types.\n\\begin{enumerate}\n\\item We say $A$ is {\\em $\\mathsf{K}_S$-harmonically proper} if the matrix $(A_{\\sigma'\\sigma})$ is row- and column-finite, {\\em i.e.}, if for every $\\sigma \\in \\irrep{K}_S$, there are only finitely many $\\sigma'\\in \\irrep{K}_S$ for which either $A_{\\sigma'\\sigma}$ or $A_{\\sigma\\sigma'}$ is nonzero.\n\\item We say $A$ is {\\em $\\mathsf{K}_S$-harmonically finite} if the matrix $(A_{\\sigma'\\sigma})$ has only finitely many nonzero entries.\n\\end{enumerate}\n\nDefine $\\scrA[S](H,H')$, resp.~$\\scrK[S](H,H')$, to be the operator-norm closure of the $\\mathsf{K}_S$-harmonically proper, resp.~$\\mathsf{K}_S$-harmonically finite, operators from $H$ to $H'$. \n\\end{definition}\n\n\n\nIf $H=H'$, we write $\\scrA[S](H)$ and $\\scrK[S](H)$ for $\\scrA[S](H,H)$ and $\\scrK[S](H,H)$, respectively. These are $C^*$-subalgebras of the algebras $\\scr{L}(H)$ of bounded operators on $H$. Letting $H$ and $H'$ vary, we consider $\\scrA[S]$ and $\\scrK[S]$ as defining $C^*$-categories of operators between harmonic $\\mathsf{K}$-spaces.\nWe also use $\\scrK$ and $\\scr{L}$ to denote the $C^*$-categories of compact operators and bounded operators, respectively, between Hilbert spaces.\n\n\n\\begin{lemma}[\\thmcitemore{Yuncken:PsiDOs}{Lemma 3.2}]\n\nIf $S\\subseteq S' \\subseteq \\Sigma$ then $\\scrK[S'] \\subseteq \\scrK[S]$.\n\n\\end{lemma}\n\n\nThe following two results are restatements of Lemmas 3.4 and 3.5 of \\cite{Yuncken:PsiDOs}.\n\n\\begin{proposition}\n\\label{prop:KS_equivalent_conditions}\nLet $K:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces. The following are equivalent:\n\\begin{enumerate}\n\\item $K \\in \\scrK[S]$,\n\\item For any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|p_F^\\perp K\\|<\\epsilon$ and $\\|K p_F^\\perp \\| < \\epsilon$.\n\\item For any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|K - p_F K p_F \\|<\\epsilon$.\n\\end{enumerate}\n\\end{proposition}\n\n\nIf $A$ and $K$ are bounded linear operators, we say $K$ is {\\em right-composable} for $A$ if the codomain of $K$ is the domain of $A$. {\\em Left-composability} is defined similarly.\n\n\\begin{proposition}\n\\label{prop:AS_equivalent_conditions}\nLet $A:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces. The following are equivalent:\n\\begin{enumerate}\n\\item $A \\in \\scrA[S]$,\n\\item For any $\\sigma\\in\\irrep{K}_S$, and any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|p_F^\\perp A p_\\sigma\\|<\\epsilon$ and $\\| p_\\sigma A p_F^\\perp \\| < \\epsilon$.\n\\item For any $\\sigma\\in\\irrep{K}_S$, $Ap_\\sigma$ and $p_\\sigma A$ are in $\\scrK[S]$.\n\\item $A$ is a two-sided multiplier of $\\scrK[S]$, meaning that $AK\\in\\scrK[S]$ for all right-composable $K\\in\\scrK[S]$, and $KA\\in\\scrK[S]$ for all left-composable $K\\in \\scrK[S]$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\n\nWe now describe some considerable simplifications from \\cite{Yuncken:PsiDOs} in the case of homogeneous vector bundles for $\\SU(3)$.\n\n\\begin{lemma}\n\\label{lem:degeneracy}\nLet $E$, $E'$ be $\\mathsf{K}$-homogeneous vector bundles over $\\scrX$, and put $H=\\LXE$, $H'= \\LXEprime$. Then $\\scrK[\\Sigma](H,H') = \\scrK(H,H')$ and $\\scrA[\\Sigma](H,H') = \\scrK[\\emptyset](H,H') = \\scrA[\\emptyset](H,H') = \\scr{L}(H,H')$.\n\n\\end{lemma}\n\n\\begin{proof}\nSince $H$ and $H'$ are direct sums of finitely many weight spaces for the right regular representation of $\\mathsf{K}$, any bounded operator from $H$ to $H'$ is $\\mathsf{M}$-harmonically finite. Hence, $\\scrK[\\emptyset](H, H') = \\scrA[\\emptyset](H,H') = \\scr{L}(H,H')$.\n\nLemma 3.3 of \\cite{Yuncken:PsiDOs} shows that $\\scrK[\\Sigma](H,H')= \\scrK(H,H')$. By Proposition \\ref{prop:AS_equivalent_conditions} above, any bounded operator $A:H\\to H'$ is in $\\scrA[\\Sigma]$.\n\\end{proof}\n\n\nThe only nontrivial cases, then, are $\\scrK[\\{\\alpha_i\\}]$ and $\\scrA[\\{\\alpha_i\\}]$, which we abbreviate as $\\scrK[\\alpha_i]$ and $\\scrA[\\alpha_i]$.\n\n\n\\begin{definition}\n\\label{def:A}\nAs in \\cite{Yuncken:PsiDOs}, we put $\\scrA := \\cap_{S\\subseteq\\Sigma} \\scrA[S]$, the simultaneous multiplier category of all $\\scrK[S]$ ($S\\subseteq\\Sigma$). \nNote, though, that by Lemma \\ref{lem:degeneracy} this reduces to $\\scrA(H,H') = \\scrA[\\alpha_1](H,H') \\cap \\scrA[\\alpha_2](H,H')$ when $H$, $H'$ are $L^2$-section spaces of homogeneous vector bundles.\n\\end{definition}\n\n\nIn the generality of \\cite{Yuncken:PsiDOs}, it is necessary to adjust the operator spaces $\\scrK[S]$ by defining $\\scrJ[S] := \\scrK[S] \\cap \\scrA$. The next lemma shows that this is not necessary for the current application.\n\n\n\n\\begin{lemma}\n\\label{lem:Ji_is_Ki}\n\nWith $H,H'$ as in Lemma \\ref{lem:degeneracy}, $\\scrK[\\alpha_i](H,H') \\subseteq \\scrA(H,H')$, for $i=1,2$.\nThus, $\\scrJ[\\alpha_i](H,H') = \\scrK[\\alpha_i](H,H')$.\n\n\n\\end{lemma}\n\n\n\\begin{proof}\nLet $i=1$. \nIt is immediate that $\\scrK[\\alpha_1](H,H')\\subseteq \\scrA[\\alpha_1](H,H')$. Lemma 5.4 of \\cite{Yuncken:PsiDOs} implies that on $H$ and $H'$, $p_{\\sigma_1} p_{\\sigma_2}$ is compact for any $\\sigma_1\\in\\irrep{K}_1$ and $\\sigma_2\\in\\irrep{K}_2$. Thus, if $K:H\\to H'$ is $\\mathsf{K}_1$-harmonically finite, then $K p_{\\sigma_2}\\in\\scrK(H,H') \\subseteq \\scrK[\\alpha_2](H,H')$. By Proposition \\ref{prop:AS_equivalent_conditions}, $K\\in\\scrA[\\alpha_2](H,H')$. Taking the norm-closure, $\\scrK[\\alpha_1](H,H')\\subseteq \\scrA[\\alpha_2](H,H')$, which proves the result. The case $i=2$ is analogous.\n\\end{proof}\n\nWe therefore avoid the notation $\\scrJ[\\alpha_i]$ altogether.\n\n\n\\begin{theorem}[\\thmcitemore{Yuncken:PsiDOs}{Theorem 1.11}]\n\\label{thm:lattice_of_ideals}\nLet $E$ be a $\\mathsf{K}$-homogeneous vector bundle over $\\scrX$, and $H:=\\LXE$. Then\n\\begin{enumerate}\n\\item $\\scrK[\\alpha_i](H)$ is an ideal in $\\scrA(H)$, for $i=1,2$.\n\\item $\\scrK[\\alpha_1](H) \\cap \\scrK[\\alpha_2](H) = \\scrK(H)$.\n\\end{enumerate}\n\n\\end{theorem}\n\n\\begin{lemma}[\\thmcitemore{Yuncken:PsiDOs}{Lemma 8.1}]\n\\label{lem:mult_ops_in_A}\n Let $\\mu$, $\\nu$ be weights. For any $f\\in\\CXE[\\mu-\\nu]$, the multiplication operator $\\multop{f}:\\LXE[\\nu] \\to \\LXE[\\mu]$ is in $\\scrA$.\n\\end{lemma}\n\n\\begin{remark}\n\\label{rmk:mult_ops_in_compact_picture}\nLemma \\ref{lem:mult_ops_in_A} depends on $\\mathsf{K}$-equivariant structure only, so that $f$ may be (the restriction to $\\mathsf{K}$ of) a section of $L_{(\\mu-\\nu)\\oplus\\chi_{\\Lie{A}}}$ for any $\\chi_{\\Lie{A}}\\in\\mathfrak{m}_\\mathbb{C}^\\dagger$.\n\\end{remark}\n\n\n\\subsection{Principal series representations}\n\\label{sec:principal_series}\n\n\n\n\nThe purpose of this section is to prove the following important fact, the first of two rather technical harmonic analysis results. \n\n\n\\begin{proposition}\n\\label{prop:U(g)_in_A}\nLet $\\mu\\in\\Lambda_W$. For any $g\\in \\mathsf{G}$, $U_\\mu(g) \\in\\scrA(\\LXE[\\mu])$.\n\\end{proposition}\n\n\nWe will use the notation for the elements of $\\mathfrak{k}_\\mathbb{C}$ from Section \\ref{sec:Lie_groups}, noting that the elements $X_\\alpha$, $Y_\\alpha$ ($\\alpha\\in\\Delta^+$) and $H_i$, $H_i'$ (for either $i=1$ or $2$) form a basis for $\\mathfrak{g}$. We let $X_\\alpha^\\dagger, Y_\\alpha^\\dagger, H_i^\\dagger, {H_i'}^\\dagger$ denote the dual basis elements of $\\mathfrak{g}^\\dagger$. We also recall the notation $\\matrixunit{\\eta^\\dagger}{\\xi}$ for matrix units.\n\n\n\\begin{lemma}\n\\label{lem:a-action}\nLet $A\\in\\mathfrak{a}$. Let $\\pi\\in\\irrep{K}$ and $\\eta^\\dagger \\in V^{\\pi\\dagger}$, $\\xi \\in (V^\\pi)_{-\\mu}$. Then $U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi} = \\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}$, where\n$$\n \\Xi(\\xi) := \\rho(H_i)\\xi\\otimes H_i^\\dagger + \\rho(H_i')\\xi \\otimes {H_i'}^\\dagger \n + \\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\pi(X_{\\alpha})\\xi\\otimes X_\\alpha^\\dagger \n \\quad \\in V^\\pi \\otimes \\mathfrak{g}^\\dagger.\n$$\n\\end{lemma}\n\nNote that $\\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}$ is a matrix unit for the \\emph{non-irreducible} representation $\\pi\\otimes\\Ad^\\dagger$, hence a sum of matrix units for the irreducible components of $\\pi\\otimes\\Ad^\\dagger$.\n\n\\begin{proof}\nDefine functions $\\kappa$, $\\mathsf{a}$, $\\mathsf{n}$ on $\\mathsf{G}$ using the Iwasawa decomposition:\n$$\n g =: \\kappa(g) \\mathsf{a}(g) \\mathsf{n}(g) \\in \\mathsf{KAN}, \\qquad \\text{for $g\\in\\mathsf{G}$}.\n$$\nThe derivatives ${D} \\kappa_e$, ${D} \\mathsf{a}_e$ and ${D} \\mathsf{n}_e$ at the identity are the ($\\mathbb{R}$-linear) projections of $\\mathfrak{g}$ onto the components of the decomposition $\\mathfrak{g}=\\mathfrak{k\\oplus a\\oplus n}$. If $P\\in\\mathfrak{g}$, let us write $P= P_+ + P_0 + P_-$ where $P_+$, $P_0$, $P_-$ are strictly upper-triangular, diagonal, and strictly lower-triangular, respectively. If $P$ is self-adjoint, the $\\mathfrak{k\\oplus a\\oplus n}$ decomposition of $P$ is $P=(-P_+ + P_-) \\oplus P_0 \\oplus 2P_+$. Thus,\n\\begin{eqnarray}\n\\label{eq:k-derivative}\n {D} \\kappa_e (P) &=& \\left(-\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) X_\\alpha\\otimes X_\\alpha^\\dagger \\right) P, \\\\\n\\label{eq:a-derivative}\n {D} \\mathsf{a}_e (P) &=& \\big( H_i \\otimes H_i^\\dagger + H_i'\\otimes {H_i'}^\\dagger \\big) P.\n\\end{eqnarray}\n\n\nFor $a\\in\\mathsf{A}$, $k\\in\\mathsf{K}$, \n$$\n a^{-1}k = kk^{-1}a^{-1}k = k\\,\\kappa(k^{-1}a^{-1}k) \\mathsf{a}(k^{-1}a^{-1}k) \\mathsf{n}(k^{-1}a^{-1}k).\n$$\nIn order to describe the $\\mathsf{G}$-action on a $\\mathsf{K}$-matrix unit, one must extend $\\matrixunit{\\eta^\\dagger}{\\xi}$ to a $\\mathsf{B}$-equivariant function on $\\mathsf{G}$. Equation \\eref{eq:B-equivariance}) gives\n\\begin{eqnarray}\n U_{\\mu}(a)\\matrixunit{\\eta^\\dagger}{\\xi}(k)\n &:=& \\matrixunit{\\eta^\\dagger}{\\xi}(a^{-1} k) \\nonumber\\\\\n &=& e^{\\rho}(\\mathsf{a}(k^{-1}ak)) \\matrixunit{\\eta^\\dagger}{\\xi}(k\\, \\kappa(k^{-1}a^{-1}k)) \\nonumber\\\\\n &=& e^{\\rho}(\\mathsf{a}(k^{-1}ak)) \\big( \\eta^\\dagger, \\pi(k) \\pi(\\kappa(k^{-1}a^{-1}k)) \\xi\\big).\n \\label{eq:extension_to_G}\n\\end{eqnarray}\nLet $a=\\exp(tA)$, and take the derivative with respect to $t$ at $t=0$:\n\\begin{equation*}\n\\label{eq:UA1}\n U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi}(k)\n = \\rho({D} \\mathsf{a}_e(\\Ad k^{-1}(A))) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big)\n - \\big( \\eta^\\dagger, \\pi(k) \\pi({D} \\kappa_e(\\Ad k^{-1}(A)) \\xi\\big).\n\\end{equation*}\nSince $\\Ad k^{-1} (A)$ is self-adjoint, Equations \\eref{eq:k-derivative} and \\eref{eq:a-derivative} give\n\\begin{eqnarray*}\n \\lefteqn{U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi}(k) } \\quad \\\\\n &=& \\rho(H_i) \\big(H_i^\\dagger, \\Ad k^{-1}(A)\\big) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big)\n + \\rho(H_i') \\big({H_i'}^\\dagger, \\Ad k^{-1}(A)\\big) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big) \\\\\n &&\\qquad +\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\big( \\eta^\\dagger, \\pi(k) \\pi(X_\\alpha) \\big(X_\\alpha^\\dagger, \\Ad k^{-1}(A)\\big) \\xi\\big) \\\\\n &=& \\big( A, \\Ad^\\dagger k (H_i^\\dagger) \\big) \\big( \\eta^\\dagger, \\pi(k) \\rho(H_i)\\xi\\big)\n + \\big( A, \\Ad^\\dagger k ({H_i'}^\\dagger) \\big) \\big( \\eta^\\dagger, \\pi(k) \\rho(H_i')\\xi\\big) \\\\\n && \\qquad +\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\big(A, \\Ad^\\dagger k (X_\\alpha^\\dagger)\\big)\\big( \\eta^\\dagger, \\pi(k) \\pi(X_\\alpha) \\xi\\big) \\\\\n &=& \\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}(k).\n\\end{eqnarray*}\n\n\\end{proof}\n\nRecall the decomposition $(\\mathfrak{k}_i)_\\mathbb{C} = \\mathfrak{s}_i \\oplus \\mathfrak{z}_i$ of Section \\ref{sec:parabolic_subgroups}. Let $\\mu\\in\\Lambda_W$. Since $\\mathfrak{z}_i\\subseteq\\mathfrak{h}$, the action of $\\mathfrak{z}_i$ on the $(-\\mu)$-weight space of any $\\mathsf{K}$-representation is completely determined by $\\mu$. Thus, the $\\mathsf{K}_i$-isotypical subspaces of $\\LXE[\\mu] $ are the $\\mathfrak{s}_i$-isotypical subspaces. Moreover, since $\\LXE[\\mu]$ has $\\mathfrak{s}_i$-weight $-\\mu_i := -\\mu(H_i)$, the $\\mathsf{s}_i$-types which occur must have highest weights $|\\mu_i|, |\\mu_i|+2, \\ldots$\n\nIn what follows, we fix $i=1$ or $2$ and let $\\sigma_l$ denote the $\\mathfrak{s}_i$-type with highest weight $l\\in\\mathbb{N}$. We abbreviate $p_l := p_{\\sigma_l}$. Note that $p_l=0$ on $\\LXE[\\mu]$ if $l \\not\\equiv \\mu_i \\pmod{2}$ or $l < |\\mu_i|$.\nThe next lemma shows that $U_\\mu(A)$ is tridiagonal with respect to $\\mathsf{K}_i$-types, and that the off-diagonal entries have at most linear growth.\n\n\n\\begin{lemma}\n\\label{lem:tridiagonal}\nFix $\\mu\\in\\Lambda_W$ and let $A\\in \\mathfrak{a}$. There exists a constant $C>0$ such that for any $m,l \\in \\mathbb{N}$,\n\\begin{equation*}\n \\begin{array}{rcll}\n \\| p_m U_\\mu(A) p_l \\| &=& 0 &\\text{if $|m-l|>2$}, \\\\\n \\| p_m U_\\mu(A) p_l \\| &\\leq& C(l+1) \\qquad &\\text{if $|m-l|=2$}.\n \\end{array}\n\\end{equation*}\n\n\\end{lemma}\n\n\n\\begin{proof}\nLet us take $i=1$, with the case of $i=2$ being entirely analogous.\nSuppose $\\matrixunit{\\eta^\\dagger}{\\xi} \\in p_l \\LXE[\\mu]$, which is to say that $\\eta^\\dagger\\in V^{\\pi\\dagger}$, $\\xi \\in (V^\\pi)_{\\sigma_l}$ for some $\\pi\\in\\irrep{K}$. By Lemma \\ref{lem:a-action}, we need to understand the decomposition of $\\Xi(\\xi)$ into $\\mathfrak{s}_1$-types.\n\nThe adjoint representation of $\\mathfrak{g}$ decomposes into the $\\mathfrak{s}_1$-representations\n$$\n \\vspan{X_1, H_1, Y_1}, \\quad \\vspan{H_1'}, \\quad \\vspan{X_2, X_3}, \\quad \\vspan{Y_2, Y_3},\n$$\nand $\\mathfrak{g}^\\dagger$ decomposes dually. We break up the expression for $\\Xi(\\xi)$ into corresponding parts.\n\nFirstly, $H_1'$ has trivial $\\mathfrak{s}_1$-type, so $\\rho(H_1')\\xi\\otimes{H_1'}^\\dagger$ has $\\mathfrak{s}_i$-type $l$. Next, note that the vector $X_2 \\otimes X_2^\\dagger + X_3 \\otimes X_3^\\dagger \\in \\mathfrak{g}\\otimes\\mathfrak{g}^\\dagger$ also has trivial $\\mathfrak{s}_1$-type, since it corresponds to the identity map on the subrepresentation $\\vspan{X_2, X_3}$. \nThe map\n\\begin{eqnarray*}\n V^\\pi \\otimes \\mathfrak{g}\\otimes \\mathfrak{g}^\\dagger &\\to& V^\\pi \\otimes \\mathfrak{g}^\\dagger \\\\\n \\zeta \\otimes Z \\otimes Z^\\dagger &\\mapsto& \\pi(Z)\\zeta \\otimes Z^\\dagger\n\\end{eqnarray*}\nis a morphism of $\\mathsf{K}$-representations, in particular of $\\mathsf{s}_i$-representations, so $\\pi(X_2)\\xi \\otimes X_2^\\dagger + \\pi(X_3) \\xi \\otimes X_3^\\dagger$ also has $\\mathfrak{s}_1$-type $l$. Similarly, $-\\pi(Y_2)\\xi \\otimes Y_2^\\dagger - \\pi(Y_3) \\xi \\otimes Y_3^\\dagger$ has $\\mathfrak{s}_1$-type $l$.\n\nThus, all the off-diagonal components of $U_\\mu(A)$ are due to the components\n\\begin{equation}\n\\label{eq:off-diagonal_terms}\n \\Xi_1(\\xi) := \\rho(H_1)\\xi \\otimes H_1^\\dagger + \\pi(X_1)\\xi \\otimes X_1^\\dagger - \\pi(Y_1) \\xi \\otimes Y_1^\\dagger\n\\end{equation}\nof $\\Xi(\\xi)$. The coadjoint representation of $\\mathfrak{s}_1$ on $\\vspan{X_1^\\dagger, H_1^\\dagger, Y_1^\\dagger}$ has highest weight $2$, so the fusion rules for $\\SU(2)$-representations imply that \\eref{eq:off-diagonal_terms} contains $\\mathfrak{s}_i$-types $l-2, l, l+2$ only. \n\nIt remains to prove the norm estimate on the off-diagonal terms. By Equations \\eref{eq:X-formula}--\\eref{eq:Y-formula},\n\\begin{equation*}\n\\begin{array}{rclcl}\n \\|\\rho(H_1) \\xi\\| &=& 2\\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|, \\\\ \\\\\n \\|\\pi(X_1) \\xi \\| &=& \\frac{1}{2} \\sqrt{(l-\\mu_i)(l+\\mu_i+2)} \\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|, \\\\ \\\\\n \\|\\pi(Y_1) \\xi \\| &=& \\frac{1}{2} \\sqrt{(l-\\mu_i+2)(l+\\mu_i)} \\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|,\n\\end{array}\n\\end{equation*}\nso the norm of $\\Xi_1(\\xi)$ is bounded by $C_0(l+1) \\|\\xi\\|$ for some constant $C_0$.\nWe need to convert this into a bound on the norm of the matrix units. \n\nDecompose $\\pi\\otimes \\Ad^\\dagger$ into irreducible $\\mathsf{K}$-subrepresentations. Suppose $\\pi'$ is an irreducible subrepresentation of $\\pi\\otimes\\Ad^\\dagger$. By orthogonality of characters, $\\pi$ is a subrepresentation of $\\pi'\\otimes \\Ad$. Therefore $\\dim \\pi \\leq \\dim (\\pi'\\otimes\\Ad) = 8\\dim \\pi' $, so that $\\dim \\pi' \\geq \\frac{1}{8} \\dim \\pi$. This also shows that the number of irreducible components of $\\pi\\otimes\\Ad^\\dagger$ is at most $64$. \n\nFor each irreducible subrepresentation $\\pi'$ of $\\pi\\otimes\\Ad^\\dagger$, let $y_{\\pi'}^\\dagger$ denote the ${\\pi'}^\\dagger$-component of $\\eta^\\dagger\\otimes A$, and $x_{\\pi'}$ the $\\pi'$-component of $\\Xi_1(\\xi)$.\nWe get\n\\begin{eqnarray*}\n \\| p_{l\\pm2} U_\\mu(A) p_l \\matrixunit{\\eta^\\dagger}{\\xi} \\|^2\n & \\leq & \\| \\matrixunit{\\eta\\otimes A}{\\Xi_1(\\xi)} \\|^2 \\\\\n & = & \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\|y_{\\pi'}^\\dagger\\|^2 \\|x_{\\pi'}\\|^2 \\\\\n &\\leq& \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\| \\eta^\\dagger \\otimes A \\|^2 \\| \\Xi_1(\\xi) \\|^2 \\\\\n &\\leq& \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\|\\eta^\\dagger\\|^2 \\|A\\|^2 C_0^2(l+1)^2 \\|\\xi\\|^2 \\\\\n &\\leq& \\|A\\|^2 C_0^2 (l+1)^2 \\sum_{\\pi'} \\frac{8}{\\dim \\pi} \\| \\eta^\\dagger\\|^2 \\| \\xi \\|^2 \\\\\n &\\leq& 8.64.\\|A\\|^2 C_0^2 (l+1)^2 \\| \\matrixunit{\\eta^\\dagger}{\\xi} \\|^2.\n\\end{eqnarray*}\nPutting $C= \\sqrt{512}\\, \\|A\\| \\,C_0$ gives the result.\n\n\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:U(g)_in_A}]\nWe need to show $U_\\mu(g)\\in\\scrA[\\alpha_i]$ for $i=1,2$.\nFor $k\\in\\mathsf{K}$, the left translation action $U_\\mu(k)$ commutes with the decomposition into right $\\mathsf{K}_i$-types, so that $U_\\mu(k) \\in \\scrA[\\alpha_i]$ trivially. By the $\\mathsf{KAK}$-decomposition, it suffices to prove the proposition for $g=a\\in\\mathsf{A}$. \n\n\nWe continue with the notation of the previous lemma. Put $P_m:= \\sum_{j=0}^m p_j$. We will show that for any $l\\in\\mathbb{N}$ and any $\\epsilon>0$, there exists $m\\in\\mathbb{N}$ such that\n$\\| P_m^\\perp U_\\mu(a) p_l\\| <\\epsilon$ and $\\| p_l U_\\mu(a) P_m^\\perp\\| <\\epsilon$, from which Lemma \\ref{prop:AS_equivalent_conditions} gives $U_\\mu(a)\\in\\scrA[\\alpha_i]$. \n\n\nLet $A\\in\\mathfrak{a}$ such that $e^A = a$. Define $\\phi:\\mathbb{N}\\to [0,1]$ by\n$$\n \\phi(n) := \\begin{cases}\n 1, & n\\leq l, \\\\\n \\max\\,\\{0, 1-\\frac{\\epsilon^2}{4C} \\log(n+3) \\}, & n>l,\n \\end{cases}\n$$\nwhere $C$ is the constant of the previous lemma. Define $\\Phi:= \\sum_{n\\in\\mathbb{N}} \\phi(n) p_n$, an operator on $\\LXE[\\mu]$ which is scalar on each $\\mathsf{K}_i$-type.\n\nWe now decompose $U_\\mu(A)$ into its diagonal and off-diagonal components. For convenience of notation, we put $U:= U_\\mu(A)$, then write $U=U_-+U_0+U_+$, where\n$$\n U_- = \\sum_{n=2}^\\infty p_{n-2}U p_n, \\qquad\n U_0 = \\sum_{n=0}^\\infty p_n U p_n, \\qquad\n U_+ = \\sum_{n=0}^\\infty p_{n+2} U p_n.\n$$\nThe diagonal component $U_0$ commutes with $\\Phi$. On the other hand,\n\\begin{eqnarray*}\n \\| [p_{n-2} U p_n, \\Phi] \\| \n &=& \\| (\\phi(n) - \\phi(n-2))\\, p_{n-2} U p_n \\| \\\\\n &\\leq& \\frac{\\epsilon^2}{4C}(\\log(n+3)-\\log(n+1)) \\\\\n &\\leq& \\frac{\\epsilon^2}{2C} \\frac{1}{(n+1)} \\\\\n &\\leq& \\frac{\\epsilon^2}{2},\n\\end{eqnarray*}\nby Lemma \\ref{lem:tridiagonal}. Thus,\n$$\n \\| [U_-, \\Phi] \\| = \\sup_{n\\in\\mathbb{N}} \\| [U_{n-2,n}, \\Phi] \\| \\leq \\frac{1}{2} \\epsilon^2.\n$$\nSimilarly, $\\| [U_+, \\Phi] \\| \\leq \\frac{1}{2} \\epsilon^2$. Therefore, $\\| [U_\\mu(A), \\Phi] \\| \\leq \\epsilon^2$.\n\nLet $s \\in p_l \\LXE[\\mu]$ have norm one. Put $s_t := U_\\mu(e^{tA})s$ for $0\\leq t \\leq 1$. Then\n$$\n | \\frac{d}{dt} \\ip{ \\Phi s_t, s_t } |\n = | \\ip{ \\Phi U_\\mu(A) s_t, s_t } + \\ip{ \\Phi s_t, U_\\mu(A) s_t } |\n = | \\ip{ [\\Phi, U_\\mu(A)] s_t, s_t } |\n \\leq \\epsilon^2,\n$$\nfor all $t$. Therefore,\n\\begin{eqnarray*}\n |\\ip{\\Phi s_1, s_1}| &=& \\left| \\ip{\\Phi s_0, s_0} + \\int_{t'=0}^1 \\frac{d}{dt} \\ip{\\Phi s_t, s_t} \\, dt' \\right|\\\\\n &\\geq& 1 - \\epsilon^2. \n\\end{eqnarray*}\nLet $m$ be the smallest integer for which $\\phi(m)=0$. Put $v := P_m s_1$ and $w:= P_m^\\perp s_1$. Then $\\|v\\|^2 + \\|w\\|^2 =1$, but also\n$$\n \\| v\\|^2 > \\ip{\\Phi v,v} = \\ip{\\Phi v,v} + \\ip{\\Phi w,w} = \\ip{\\Phi s_1, s_1} \\geq 1 -\\epsilon^2.\n$$\nIt follows that $ \\|w\\| < \\epsilon$, {\\em ie}, $\\|P_m^\\perp U_\\mu(a) s \\| < \\epsilon$. Since $s\\in p_l\\LXE[\\mu]$ was arbitrary, $\\|P_m^\\perp U_\\mu(a) p_l\\| < \\epsilon$. \n\nReplacing $a$ with $a^{-1}$, there exists $m'\\in\\mathbb{N}$ such that $\\| P_{m'}^\\perp U_\\mu(a^{-1}) p_l \\| < \\epsilon$. Thus, after enlarging $m$ to be at least $m'$, we have\n$$\n \\| p_l U_\\mu(a) P_{m}^\\perp \\| = \\| P_{m}^\\perp U_\\mu(a^{-1}) p_l \\| < \\epsilon.\n$$\n\n\\end{proof}\n\n\n\nIn fact, Proposition \\ref{prop:U(g)_in_A} holds for any generalized principal series representation. Although we don't actually need this here, it is now trivial to prove.\n\n\\begin{corollary}\nFor any $\\mathsf{G}$-homogeneous line bundle $\\LXL[\\chi]$ over $\\scrX$, the translation operators $s \\mapsto g\\cdot s$ belong to $\\scrA$.\n\\end{corollary}\n\n\\begin{proof}\nLet $\\chi=\\chi_{\\Lie{M}}\\oplus\\chi_{\\Lie{A}}$. A computation of the form of Eq.~\\eref{eq:extension_to_G}\ngives \n$$\n g\\cdots(k) = e^{\\chi_{\\Lie{A}}}(\\mathsf{a}(k^{-1}gk)) s (k\\, \\kappa(k^{-1}g^{-1}k)),\n$$\nfor any $k\\in\\mathsf{K}$, while\n$$\n U_{\\chi_{\\Lie{M}}}(g)s(k) = e^{\\rho}(\\mathsf{a}(k^{-1}gk)) s (k\\, \\kappa(k^{-1}g^{-1}k)).\n$$\nNote that $\\mathsf{a}(m^{-1}gm) = \\mathsf{a}(g)$ for any $m\\in\\mathsf{M}$, $g\\in\\mathsf{G}$.\nTherefore, $g\\cdots = \\multop{f} U_{\\chi_{\\Lie{M}}}(g) s$, where $f(k) := e^{\\chi_{\\Lie{A}}-\\rho}(\\mathsf{a}(k^{-1}gk))$ is in $C(\\mathsf{K\/M}) = C(\\scrX)$. Since $\\multop{f}$ and $U_{\\chi_{\\Lie{M}}}(g)$ are in $\\scrA$, we are done.\n\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Longitudinal pseudodifferential operators}\n\\label{sec:PsiDOs}\n\n\nLet $X\\in\\mathfrak{k}_\\mathbb{C}$ be a root vector, of weight $\\alpha$. Via the right regular representation, $X$ defines a left $\\mathsf{K}$-invariant differential operator on $C^\\infty(\\mathsf{K})$. \nFor each weight $\\mu$, $X$ maps $p_{-\\mu} L^2(\\mathsf{K})$ to $p_{-\\mu+\\alpha} L^2(\\mathsf{K})$, so it defines a $\\mathsf{K}$-invariant differential operator\n$$\n X : \\LXE[\\mu] \\to \\LXE[\\mu-\\alpha].\n$$\nThe principal symbol of this differential operator is a $\\mathsf{K}$-equivariant linear map from the cotangent bundle $T^*\\scrX \\cong K\\times_M (\\mathfrak{k\/m})^*$ to $\\End( E_\\mu, E_{\\mu-\\alpha}) \\cong E_{-\\alpha}$. (Here $(\\mathfrak{k\/m})^*$ denotes the {\\em real} dual of $\\mathfrak{k\/m}$.) By equivariance, this map is determined by its value on the cotangent fibre at the identity coset $\\coset{e}\\in\\scrX$, which is\n\\begin{eqnarray}\n\\label{eq:diff_op_symbol}\n \\Symbol (X) : T^*_{\\coset{e}}\\scrX = (\\mathfrak{k\/m})^* \n & \\to & \\mathbb{C} \\\\\n \\xi & \\mapsto & \\xi(X). \\nonumber\n\\end{eqnarray}\n\n\n\nIf $X\\in(\\mathfrak{k}_i)_\\mathbb{C}$ ($i=1$ or $2$), then the differential operator $X:\\CinftyXE[\\mu] \\to \\CinftyXE[\\mu-\\alpha]$ is tangential to the foliation $\\foliation[i]$ of Section \\ref{sec:parabolic_subgroups}. We will refer to such an operator as an $\\foliation[i]$-longitudinal differential operator. Its {longitudinal principal symbol} is the $\\mathsf{K}$-equivariant map $\\Symbol[i]: \\foliation[i]^* \\to E_{-\\alpha}$ which, at the identity coset, is given by\n\\begin{eqnarray*}\n \\Symbol[i] : (\\foliation[i]^*)_{\\coset{e}} = (\\mathfrak{k}_i \/ \\mathfrak{m})^* &\\to& \\mathbb{C} \\\\\n \\xi &\\mapsto & \\xi(X).\n\\end{eqnarray*}\n\nAn $\\foliation[i]$-longitudinal differential operator is \\emph{longitudinally elliptic} if its longitudinal principal symbol is invertible off the zero section of $T^*\\foliation[i]$. Note that $X_i = -\\frac{1}{2}(X_i' + \\sqrt{-1} \\, X_i'') \\in (\\mathfrak{k}_i)_\\mathbb{C}$ where\n$$\n X_i' = \\smatrix{ 0 & -1 \\\\ 1 & 0 }, \\quad X_i'' = \\smatrix{ 0 & \\sqrt{-1} \\\\ \\sqrt{-1} & 0 }\n$$\nspan $\\mathfrak{k}_i\/\\mathfrak{m}$, so that $X_i$ is $\\foliation[i]$-longitudinally elliptic. Similarly, $Y_i$ is $\\foliation[i]$-longitudinally elliptic. Moreover, $X_i$ and $Y_i$ are formal adjoints. We shall use $X_i$, $Y_i$ also to denote their closures as unbounded operators on the $L^2$-section spaces.\n\nFix $\\mu\\in\\Lambda_W$. Let $E := E_\\mu \\oplus E_{\\mu-\\alpha_i}$, and define $D_i := \\smatrix{0&Y_i\\\\X_i&0}$ on $\\LXE$. The $\\mathfrak{s}_i$-isotypical subspaces of $\\LXE$ are eigenspaces for $D_i$, and by the representation theory of $\\mathfrak{s}_i$---specifically Equations \\eref{eq:X-formula} and \\eref{eq:Y-formula}---its spectrum is discrete.\n\n\n\\medskip\n\nFor the definition and basic properties of longitudinal pseudodifferential operators, we refer the reader to \\cite{MS-GAFS}\\footnote{In this reference, they are called {tangential} pseudodifferential operators.}. If $E$, $E'$ are vector bundles over $\\scrX$, we denote the set of $\\foliation[i]$-longitudinal pseudodifferential operators of order at most $p$ by $\\PsiDO[i]^p(E,E')$. If $E=E'$, we abbreviate this to $\\PsiDO[i]^p(E)$.\n\nLet $C(S^*\\foliation[i]; \\End(E))$ denote the algebra of continuous sections of the pullback of $\\End(E)$ to the cosphere bundle of the foliation $\\foliation[i]$. The longitudinal principal symbol map $\\Symbol[i]:\\PsiDO[i]^0(E) \\to C(S^*\\foliation[i]; \\End(E))$ extends to the operator-norm closure $\\overline{\\PsiDO[i]^{0}}(E)$, and we have Connes' short exact sequence,\n\\begin{equation}\n\\label{eq:symbol-sequence}\n \\xymatrix{\n 0 \\ar[r] &\n \\overline{\\PsiDO[i]^{-1}}(E) \\ar[r] &\n \\overline{\\PsiDO[i]^{0}}(E) \\ar[r]^-{\\Symbol[i]} &\n C(S^*\\foliation[i]; \\End(E)) \\ar[r] &\n 0.\n }\n\\end{equation}\n \n\n\nFor any closed, densely defined, unbounded operator $T$ between Hilbert spaces, we let $\\Ph T$ denote the phase in the polar decomposition: $T = (\\Ph{T}) |T|$. We also use $\\Ph{z}$ to denote the phase of a complex number $z\\in\\mathbb{C}^\\times$.\n\n\n\\begin{lemma}\n\\label{lem:F_in_PsiDO}\nFor any weight $\\mu$, $\\Ph{X_i}:\\LXE[\\mu] \\to \\LXE[\\mu-\\alpha_i]$ and $\\Ph{Y_i}:\\LXE[\\mu-\\alpha_i] \\to \\LXE[\\mu]$ are $\\foliation[i]$-longitudinal pseudodifferential operators. Their longitudinal principal symbols at the identity coset are\n\\begin{eqnarray*}\n \\Symbol[i] (\\Ph{X_i}) (\\xi) &=& \\Ph{(\\xi(X_i))}, \\\\\n \\Symbol[i] (\\Ph{Y_i}) (\\xi) &=& \\Ph{(\\xi(Y_i))} \\;=\\; \\overline{\\Ph{(\\xi(X_i))}}.\n\\end{eqnarray*}\nfor $ \\xi$ in the unit sphere of $(\\mathfrak{k}_i\/\\mathfrak{m})^* \\cong (\\foliation[i]^*)_{\\coset{e}}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $E:= E_\\mu\\oplus E_{\\mu-\\alpha_i}$.\nFix $\\epsilon>0$ such that $\\Spec(D_i) \\cap (-\\epsilon,\\epsilon) = \\{0\\}$. Let $f:\\mathbb{R}\\to[-1,1]$ be smooth with $f(0)=0$ and $f(x) = \\sign(x)$ for all $|x|\\geq\\epsilon$. \nA fibrewise application of \\cite[Theorem 1.3]{Taylor} shows that $ f(D_i) = \\Ph{D_i} \\in \\PsiDO[i]^0(\\scrX;E)$. Moreover the proof of the theorem shows that its full symbol has an asymptotic expansion with leading term $f(\\Symbol[i] D_i)$. Note that\n$$\n (\\Symbol[i] D_i)(\\xi) = \\smatrix{ 0 & \\xi(X_i) \\\\ \\overline{\\xi(X_i)} &0 }\n$$\nhas spectrum $\\{\\pm | \\xi(X_i) | \\}$, so if $\\xi$ is large enough that $|\\xi(X_i)|>\\epsilon$, then\n$$\n f (\\Symbol[i] D_i)(\\xi) = \\Ph{(\\Symbol[i] D_i(\\xi))} \n = \\smatrix{ 0 & \\Ph{(\\xi(X_i))} \\\\ \\Ph{(\\overline{\\xi(X_i)})} &0 }.\n$$\nThis is radially constant on $(\\mathfrak{k}_i\/\\mathfrak{m})^*$ for $|\\xi(X_i)|>\\epsilon$. The principal symbol is the limit at the sphere at infinity. \n\\end{proof}\n\n\n\\begin{theorem}\n\\label{prop:PsiDOs_in_K}\n\\label{thm:PsiDOs_in_A}\nLet $E, E'$ be $\\mathsf{K}$-homogeneous vector bundles over $\\scrX$. Then \n\\begin{enumerate}\n\\item $\\PsiDO[i]^{-1}(E,E') \\subseteq \\scrK[\\alpha_i]$,\n\\item $\\PsiDO[i]^{0}(E,E') \\subseteq \\scrA$,\n\\end{enumerate}\n\\end{theorem}\n\nPart (i) is proven in Proposition 1.12 of \\cite{Yuncken:PsiDOs}. It is also shown there that $\\PsiDO[i]^0(E,E') \\subseteq \\scrA[i]$. The more difficult question of showing $\\PsiDO[i]^0(E,E') \\subseteq \\scrA[j]$ for $j\\neq i$ requires some lengthy computations in noncommutative harmonic analysis. In order not to disrupt the flow of ideas too severely, we have presented the proof in Appendix \\ref{sec:PsiDOs_in_A}.\n\nAs an indication of the subtleties involved, we remark that the longitudinally elliptic differential operator $X_1$ is {\\em not} an unbounded multiplier of $\\scrK[\\alpha_2]$. To see this, note that $(1+X_1^*X_1)^{-\\frac{1}{2}} \\in \\PsiDO[1]^{-1}(E_\\mu) \\subseteq \\scrK[\\alpha_1]$. Since $\\scrK[\\alpha_1].\\scrK[\\alpha_2] \\subseteq \\scrK$, the range of $(1+X_1^*X_1)^{-\\frac{1}{2}}$ as a multiplier of $\\scrK[\\alpha_2]$ is not dense. Thus, $X_1$ is not regular with respect to $\\scrK[\\alpha_2]$ (see \\cite[Chapter 10]{Lance}). Hence, proving that $\\Ph{X_1}$ multiplies $\\scrK[2]$ can not be achieved by direct functional calculus.\n\n\\begin{lemma}\n\\label{lem:F_f_commute}\nLet $i=1,2$ and let $\\mu, \\nu$ be weights. For any $f\\in\\CXE[\\nu-\\mu]$, the diagram\n$$ \n \\xymatrix{\n \\LXE[\\mu] \\ar[r]^{M_f} \\ar[d]_{\\Ph{X_i}} &\n \\LXE[\\nu] \\ar[d]^{\\Ph{X_i}} \\\\\n \\LXE[\\mu-\\alpha_i] \\ar[r]_{M_f} &\n \\LXE[\\nu-\\alpha_i]\n }\n$$\ncommutes modulo $\\scrK[\\alpha_i]$.\n\\end{lemma}\n\n\\begin{remark}\n\\label{rem:commutator_notation}\nWe abbreviate this result by writing $[\\Ph{X_i}, M_s] \\in \\scrK[\\alpha_i]$. By taking adjoints, we also have $[\\Ph{Y_i}, M_s] \\in \\scrK[\\alpha_i]$. \n\\end{remark}\n\n\\begin{proof}\nAs an element of $C(S^*\\foliation[i];E_{\\alpha_i})$, the principal symbol of $\\Ph{X_i}:\\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is independent of the weight $\\mu$. Thus, the above diagram commutes at the level of principal symbols.\n\\end{proof}\n\n\n\n\n\n\\section{The normalized BGG complex}\n\\label{sec:construction}\n\n\n\\subsection{$\\mathsf{G}$-continuity}\n\\label{sec:G-continuity}\n\nBefore embarking on the main construction, we need to make some remarks regarding the issue of $\\mathsf{G}$-continuity. Recall that a bounded operator $A$ between unitary $\\mathsf{G}$-representations is $\\mathsf{G}$-continuous if the map $g \\mapsto g .A. g^{-1}$ is continuous in the operator-norm topology. \n\nRather than burden the notation with extra decorations, we choose to make the convention that {\\bf throughout this section, we use $\\scrK[\\alpha_i]$ ($i=1,2$) to denote its $C^*$-subcategory of $\\mathsf{G}$-continuous elements.} \n\nThis is reasonable, since almost every operator we deal with is $\\mathsf{G}$-continuous. From \\cite{AS4}, we know that for any homogeneous vector bundles $E$, $E'$ over $\\scrX$, the set of longitudinal pseudodifferential operators $\\overline{\\PsiDO[i]^0}(E,E,')$ consists of $\\mathsf{G}$-continuous operators. This includes continuous multiplication operators, in the sense of Section \\ref{sec:homogeneous_vector_bundles} (which are $\\mathsf{G}$-continuous for much simpler reasons).\nThe notable exceptions, of course, are the representations $U_\\mu(g)$ of the group elements themselves.\n\nIn the majority of instances, where $\\mathsf{G}$-continuity is a trivial consequence of the above remarks, we will not make specific mention of it in the proofs.\n\n\n\\subsection{Intertwining operators}\n\n\n\nLet $\\mu$, $\\mu'$ be weights for $\\mathsf{K}= \\SU(3)$. It is well known that the principal series representations $U_\\mu$ and $U_{\\mu'}$ are unitarily equivalent if and only if $\\mu'=w\\cdot\\mu$ for some Weyl group element $w\\in W$. \nWhen $w=\\reflection{\\alpha_i}$ is a simple reflection corresponding to the root $\\alpha_i$, there is a very concise formula for the intertwining operator. \n\n\n\\begin{proposition}\n\\label{prop:intertwiner_formula}\nLet $\\mu$, $\\mu'$ be weights with $\\mu'=\\reflection{\\alpha_i}\\mu$, so that $\\mu-\\mu' = n\\alpha_i$ for some $n\\in\\mathbb{Z}$. If $n>0$, the operator $(\\Ph{X_i})^n: \\LXE[\\mu] \\to \\LXE[\\mu']$ intertwines $U_\\mu$ and $U_\\mu'$. If $n<0$, then $(\\Ph{Y_i})^n: \\LXE[\\mu] \\to \\LXE[\\mu']$ is an intertwiner.\n\\end{proposition}\n\n\n\nThis is essentially the formula given by Duflo in \\cite[Ch.~III]{Duflo}. However, Duflo's formulation is sufficiently different that we feel a brief comparison is worthwhile. \n\n\\begin{proof}\nWe follow the notation for $\\mathfrak{sl}(2,\\mathbb{C})$-representations from the end of Section \\ref{sec:parabolic_subgroups}. Note that Equations \\eref{eq:X-formula} and \\eref{eq:Y-formula} imply that\n$(\\Ph{X})e_j = e_{j+2}$ and $(\\Ph{Y}) e_j = e_{j-2}$. Secondly, with $w=\\smatrix{0&-1\\\\1&0}$, \n\\begin{equation}\n\\label{eq:Phase(X)_vs_w}\n\\begin{array}{rcccl}\n (\\Ph{X})^j \\cdot e_{-j} &=& e_j& =& (-1)^{\\frac{1}{2}(\\delta+j)} w\\cdot e_{-j}, \\\\\n (\\Ph{Y})^j \\cdot e_{j} &= &e_{-j}& =& (-1)^{\\frac{1}{2}(\\delta-j)} w\\cdot e_{j},\n\\end{array}\n\\end{equation}\nfor any $j\\geq 0$. (See \\cite[\\S{}III.3.5]{Duflo}.)\n\nRecall that the restriction of $\\mu$ to a weight of $\\mathfrak{s}_i$ is $\\mu_i := \\mu(H_i)\\in\\mathbb{Z}$. The hypotheses of the proposition are equivalent to saying $\\mu_i = -\\mu'_i = n$.\n\nFirst consider the case $n>0$. Let $A=A(w_i, \\mu,0):\\LXE[\\mu] \\to \\LXE[\\mu']$ be the intertwiner of \\cite[\\S{}III.3.1]{Duflo}. The action of $A$ upon matrix units is given in \\cite[\\S{}III.3.3 and \\S{}III.3.9]{Duflo} as follows. Let $\\pi\\in\\irrep{K}$, $\\eta^\\dagger\\in V^{\\pi\\dagger}$, $\\xi\\in p_{-\\mu}(V^\\pi)$ and suppose that $\\xi$ lies in an irreducible $\\mathfrak{s}_i$-subrepresentation of $V^\\pi$ with highest weight $\\delta$. Then, in the notation of Section \\ref{sec:harmonic_notation}, $A:\\matrixunit{\\eta^\\dagger}{\\xi} \\mapsto \\matrixunit{\\eta^\\dagger}{\\xi'}$ where\n\\begin{eqnarray*}\n\\label{eq:Duflo-formula}\n \\xi' &=& (-1)^{\\frac{1}{2}(\\delta + |\\mu_i|)} |\\mu_i|^{-1} \\pi(w_i)\\xi \\\\\n &=& |\\mu_i|^{-1} (\\Ph{X_i})^n \\xi.\n\\end{eqnarray*}\nHence, $A=|\\mu_i|^{-1} (\\Ph{X_i})^n:\\LXE[\\mu] \\to \\LXE[\\mu']$, where $X_i$ here denotes the right regular action. Thus, $(\\Ph{X_i})^n$ differs from $A$ by the positive scalar $|\\mu_i|=n$.\n\nThe case $n<0$ follows since $\\Ph{Y_i} = \\Ph{X_i}^*$.\n\n\\end{proof}\n\n\n\n\\medskip\n\nWe now recap the directed graph structure which underlies the BGG complex. For our $K$-homological purposes, it will be convenient to make an undirected graph, or more accurately, to include also the reversal of each edge. \n\nAs before, if $\\alpha$ is a positive root, we use $\\reflection{\\alpha}\\in\\Lie{W}$ to denote the reflection in the wall orthogonal to $\\alpha$. For $w,w'\\in\\Lie{W}$, we write $\\edge[\\alpha]{w}{w'}$ if $w'=\\reflection{\\alpha}w$ and $l(w') = l(w) \\pm 1$. We will write $\\edge{w}{w'}$ if $\\edge[\\alpha]{w}{w'}$ for some $\\alpha\\in\\Delta^+$. An edge $\\edge[\\alpha]{w}{w'}$ will be called {\\em simple} if $\\alpha$ is a simple root.\n\nFor $\\mathsf{G}=\\SL(3,\\mathbb{C})$, this yields the graph\n\\begin{equation}\n\\label{eq:Weyl_graph}\n \\xymatrix{\n & \\stackrel{\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[rr]^{\\rho} \\ar@{<->}[ddrr]^(0.7){\\alpha_2} \n && \\stackrel{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[dr]^{\\alpha_2} \\\\\n \\stackrel{1}{\\bullet} \\ar@{<->}[ur]^{\\alpha_1} \\ar@{<->}[dr]_{\\alpha_2} \n &&&& \\stackrel{w_\\rho}{\\bullet} \\\\\n & \\stackrel{\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[rr]_{\\rho} \\ar@{<->}[uurr]_(0.7){\\alpha_1} \n && \\stackrel{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[ur]_{\\alpha_1} \n }\n\\end{equation}\n\n\n\n\n\n\\begin{definition}\n\\label{def:intertwiners}\nFix a dominant weight $\\lambda$. If $\\edge[\\alpha_i]{w}{w'}$ is a simple edge, we denote by $\\intertwiner{\\lambda}{w}{w'}$ the intertwining operator of Lemma \\ref{prop:intertwiner_formula}:\n$$\n \\intertwiner{\\lambda}{w}{w'} := \n \\begin{cases} \n (\\Ph{X_i})^n & \\text{if $n\\geq0$},\\\\\n (\\Ph{Y_i})^{-n} & \\text{if $n\\leq0$},\n \\end{cases} \n$$\nwhere $w\\lambda - w'\\lambda = n\\alpha_i$.\nThese will be referred to as {\\em simple intertwiners}. Note that $\\intertwiner{\\lambda}{w'}{w} = \\intertwiner{\\lambda}{w}{w'}^*$.\n\nFor the non-simple edges, we define intertwiners as compositions of simple intertwiners:\n\\begin{eqnarray}\n \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} &:=& \n \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\nonumber \\\\\n \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}} &:=& \n \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}.\n \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_1}}.\n \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{1}, \n \\label{eq:non-simple_intertwiners}\n\\end{eqnarray}\nand \n$\\intertwiner{\\lambda}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}} := \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}^*$, $\\intertwiner{\\lambda}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}} := \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}^*$.\n\\end{definition}\n\n\\begin{remark}\n\\label{rmk:commuting_diagram_of_intertwiners}\nDuflo's intertwiners form a commuting diagram of the form\n\\begin{equation}\n\\label{eq:simple_intertwiners}\n \\xymatrix@!C=5ex{\n & \\LXE[\\reflection{\\alpha_1}\\lambda] \\ar[ddrr] && \\LXE[\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\lambda] \\ar[dr] \\\\\n \\LXE[\\lambda] \\ar[ur] \\ar[dr] &&&& \\LXE[w_\\rho\\lambda] \\\\\n & \\LXE[\\reflection{\\alpha_2}\\lambda] \\ar[uurr] && \\LXE[\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\lambda] \\ar[ur] \n }\n\\end{equation}\nSince the simple intertwiners $\\intertwiner{\\lambda}{w}{w'}$ defined here are positive scalar multiples of Duflo's, \nthe corresponding diagram of intertwiners $\\intertwiner{\\lambda}{w}{w'}$ commutes up to some positive scalar. But $\\intertwiner{\\lambda}{w}{w'} = (\\Ph{X_i})^n$ is unitary, so that scalar is $1$. The non-simple intertwiners defined by Equation \\eref{eq:non-simple_intertwiners} are precisely those that complete \\eref{eq:simple_intertwiners} to a commuting diagram of the form \\eref{eq:Weyl_graph}.\n\\end{remark}\n\n\n\\begin{definition}\nDefine $\\scrK[\\rho] := \\scrK[\\alpha_1] + \\scrK[\\alpha_2]$. That is, $\\scrK[\\rho](H,H') := \\scrK[\\alpha_1] (H,H') + \\scrK[\\alpha_2] (H,H') $ for any harmonic $\\mathsf{K}$-spaces $H$, $H'$. Following the convention of Section \\ref{sec:G-continuity}, we are including the condition of $\\mathsf{G}$-continuity in this definition.\n\n\\end{definition}\n\n\n\\begin{lemma}\n\\label{lem:intertwiners_in_A}\nLet $\\lambda$ be a dominant weight.\n\\begin{enumerate}\n\\item For each $\\edge{w}{w'}$, $\\intertwiner{\\lambda}{w}{w'} \\in \\scrA$.\n\\item If $\\edge[\\alpha]{w}{w'}$, then $[\\intertwiner{\\lambda}{w}{w'}, \\multop{f}] \\in \\scrK[\\alpha]$ for any $f\\in C(\\scrX)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nPart {(i)} is immediate from Theorem \\ref{thm:PsiDOs_in_A}. If $\\alpha$ is a simple root, then {(ii)} follows from Lemma \\ref{lem:F_f_commute}. For $\\alpha=\\rho$, there are four intertwiners to be checked. The following calculation is representative of all of them:\n\\begin{eqnarray*}\n[\\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} , \\multop{f}] &=& \n [\\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}, \\multop{f}].\n \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n &&\\quad + \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n [ \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}, \\multop{f} ].\n \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n &&\\quad + \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n [ \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1}, \\multop{f} ] . \\\\\n &\\in& \\scrK[\\alpha_1] + \\scrK[\\alpha_2] + \\scrK[\\alpha_1] = \\scrK[\\rho].\n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\\subsection{Normalized BGG operators}\n\n\n\n\\begin{definition}\n\\label{def:shifted_action}\nDefine the {\\em shifted action} of the Weyl group on weights by $w\\star \\mu := w(\\mu+\\rho) -\\rho$.\n\\end{definition}\n\nFrom now on, $\\lambda$ will denote a dominant weight.\n\n\\begin{definition}\n\\label{def:BGG_operators}\nIf $\\edge[\\alpha_i]{w}{w'}$ is a simple edge, then $w\\star \\lambda - w'\\star \\lambda = n\\alpha_i$ for some $n\\in\\mathbb{Z}$. We define the {\\em normalized BGG operator} $\\BGG{\\lambda}{w}{w'} : \\LXE[w\\star \\lambda] \\to \\LXE[w'\\star \\lambda]$ by\n$$\n \\BGG{\\lambda}{w}{w'} := \n \\begin{cases} \n (\\Ph{X_i})^n & \\text{if $n\\geq0$},\\\\\n (\\Ph{Y_i})^{-n} & \\text{if $n\\leq0$}.\n \\end{cases} \n$$\nwhere $w\\star\\lambda-w'\\star\\lambda = n \\alpha_i$.\n\nFor the non-simple arrows, define\n\\begin{eqnarray*}\n \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} &:=& \n \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n \\BGG{\\lambda}{1}{\\reflection{\\alpha_2}}.\n \\BGG{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}} &:=& \n \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}.\n \\BGG{\\lambda}{1}{\\reflection{\\alpha_1}}.\n \\BGG{\\lambda}{\\reflection{\\alpha_2}}{1} \\\\\n \\BGG{\\lambda}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}} &:=& \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}^* \\\\ \n \\BGG{\\lambda}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}} &:=& \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}^*.\n\\end{eqnarray*}\n\\end{definition}\n\nObviously, the definitions of the normalized BGG operators $\\BGG{\\lambda}{w}{w'}$ are identical to the definitions of the intertwining operators $\\intertwiner{\\lambda+\\rho}{w}{w'}$, except that the weights of the principal series representations on which they act differ by the shift of $\\rho$. The next few lemmas describe the consequences of this. To begin with, we have an exact analogue of Lemma \\ref{lem:intertwiners_in_A}, with essentially identical proof.\n\n\\begin{lemma}\n\\label{lem:BGG_in_A}\nLet $\\lambda$ be a dominant weight.\n\\begin{enumerate}\n\\item For each arrow $\\edge{w}{w'}$, $\\BGG{\\lambda}{w}{w'} \\in \\scrA$.\n\\item If $\\edge[\\alpha]{w}{w'}$, then $[\\BGG{\\lambda}{w}{w'}, \\multop{f}] \\in \\scrK[\\alpha]$ for any $f\\in C(\\scrX)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:po1_definition}\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be such that $\\sum_{j=1}^k |\\varphi_j|^2 =1$, as in Lemma \\ref{lem:partition_of_unity}. If $\\edge[\\alpha]{w}{w'}$, then\n$$\n \\BGG{\\lambda}{w}{w'} \n \\equiv \\sum_{j=1}^k \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_j} \n \\pmod{\\scrK[\\alpha]}. \n$$\n\\end{lemma}\n\n\\begin{proof}\nThis is an immediate consequence of Lemma \\ref{lem:F_f_commute}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:F2-1_in_K}\nIf $\\edge[\\alpha]{w}{w'}$, then $\\BGG{\\lambda}{w'}{w} \\BGG{\\lambda}{w}{w'}-1 \\in \\scrK[\\alpha]$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be as in the previous lemma. By Lemmas \\ref{lem:po1_definition} and \\ref{lem:intertwiners_in_A},\n\\begin{eqnarray*}\n \\BGG{\\lambda}{w'}{w} \\BGG{\\lambda}{w}{w'} \n &\\equiv& \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w'}{w'} \\multop{\\varphi_j \\overline{\\varphi_{j'}}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_{j'}} \\pmod{\\scrK[\\alpha]}\\\\\n &\\equiv& \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w'}{w} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_j \\overline{\\varphi_{j'}}}\\multop{\\varphi_{j'}} \\pmod{\\scrK[\\alpha]}\\\\\n &=& \\sum_{j,j'} \\multop{\\overline{\\varphi_j} \\varphi_j \\overline{\\varphi_{j'}} \\varphi_{j'}} \\\\\n &=& 1.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:diagram_commutes}\nThe diagram of normalized BGG operators\n\\begin{equation}\n\\label{eq:BGG_diagram}\n \\xymatrix@!C=5ex{\n & \\LXE[\\reflection{\\alpha_1}\\star\\lambda] \\ar@{<->}[ddrr] \\ar@{<->}[rr] && \\LXE[\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\star\\lambda] \\ar@{<->}[dr] \\\\\n \\LXE[\\lambda] \\ar@{<->}[ur] \\ar@{<->}[dr] &&&& \\LXE[w_\\rho\\star\\lambda] \\\\\n & \\LXE[\\reflection{\\alpha_2}\\star\\lambda] \\ar@{<->}[uurr] \\ar@{<->}[rr] && \\LXE[\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\star\\lambda] \\ar@{<->}[ur] \n }\n\\end{equation}\ncommutes modulo $\\scrK[\\rho]$.\n\\end{lemma}\n\n\\begin{proof}\nFor adjacent edges $w\\xleftrightarrow{\\alpha} w' \\xleftrightarrow{\\alpha'} w''$, a calculation analogous to that of the previous proof gives\n\\begin{equation*}\n \\BGG{\\lambda}{w'}{w''} \\BGG{\\lambda}{w}{w'}\n \\equiv \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\left(\\intertwiner{\\lambda+\\rho}{w'}{w''} \\intertwiner{\\lambda+\\rho}{w}{w'}\\right) \\multop{\\varphi_{j} \\overline{\\varphi_{j'}}}\\multop{\\varphi_{j'}} \n \\pmod{\\scrK[\\alpha']}.\n\\end{equation*}\nNote that $\\scrK[\\alpha']\\subseteq\\scrK[\\rho]$. The commutativity of \\eref{eq:BGG_diagram} modulo $\\scrK[\\rho]$ is therefore a consequence of the commutativity of the corresponding diagram of intertwiners $\\intertwiner{\\lambda+\\rho}{w}{w'}$ (Remark \\ref{rmk:commuting_diagram_of_intertwiners}).\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:F_g_commute}\nLet $\\edge[\\alpha]{w}{w'}$. For any $g\\in G$, \n\\begin{equation}\n\\label{eq:F_g_commute}\n U_{w'\\star\\lambda}(g) \\BGG{\\lambda}{w}{w'} U_{w\\star\\lambda}(g^{-1}) - \\BGG{\\lambda}{w}{w'} \\; \\in \\;\n \\scrK[\\alpha] \n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{proof}\nWe first note that if $A$ is a $\\mathsf{G}$-continuous operator, then so is $g.A.g^{-1}$.\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be as in Lemma \\ref{lem:po1_definition}. Then,\n\\begin{eqnarray}\n\\lefteqn{U_{w'\\star\\lambda}(g) \\BGG{\\lambda}{w}{w'} U_{w\\star\\lambda}(g^{-1})} \\qquad \\nonumber\\\\\n &\\equiv& \\sum_{j} U_{w'\\star\\lambda}(g) \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_j} U_{w\\star\\lambda}(g^{-1}) \\nonumber \\pmod{\\scrK[\\alpha]} \\\\\n &=& \\sum_{j} U_{w'\\star\\lambda}(g) \\multop{\\overline{\\varphi_j}}U_{w'(\\lambda+\\rho)}(g^{-1}) \\intertwiner{\\lambda+\\rho}{w}{w'} U_{w(\\lambda+\\rho)}(g) \\multop{\\varphi_j} U_{w\\star\\lambda}(g^{-1}) \\nonumber \\\\\n &=& \\sum_{j} \\multop{\\overline{g\\cdot \\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{g\\cdot \\varphi_j}.\n \\label{eq:F_g_commute_computation}\n\\end{eqnarray}\nSince $\\sum_{j=1}^k |g\\cdot \\varphi_j|^2 = 1$, Lemma \\ref{lem:po1_definition} shows that \\eref{eq:F_g_commute_computation} equals $\\BGG{\\lambda}{w}{w'}$ modulo $\\scrK[\\alpha]$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Construction of the gamma element}\n\\label{sec:gamma}\n\nFix a dominant weight $\\lambda$. Let $H_\\lambda := \\bigoplus_{w\\in\\Lie{W}} \\LXE[w\\star \\lambda]$. For each $w\\in\\Lie{W}$, let $\\component{w}$ denote the orthogonal projection onto the summand $\\LXE[w \\star \\lambda]$ of $H_\\lambda$. We put a grading on $H_\\lambda$ by declaring $\\LXE[w\\star \\lambda]$ to be even or odd according to the parity of $l(w)$. \n\nFor $f\\in C(\\scrX)$, $M_f$ will denote the multiplication operator on $H_\\lambda$, acting diagonally on the summands. We let $U$ denote the diagonal representation $\\oplus_{w\\in\\Lie{W}} U_{w\\star\\lambda}$ of $\\mathsf{G}$. For each $\\edge{w}{w'}$, we extend the normalized BGG operator $\\BGG{\\lambda}{w}{w'}:\\LXE[w\\star\\lambda] \\to \\LXE[w'\\star\\lambda]$ to an operator $\\BGGextended{\\lambda}{w}{w'}:H_\\lambda \\to H_\\lambda$ by defining it to be zero on the components $\\LXE[w''\\star \\lambda]$ with $w''\\neq w$. \n\nFor the remainder of this section, we use $\\scrK[\\alpha]$, $\\scrA$, $\\scrK$, $\\scr{L}$ to denote $\\scrK[\\alpha](H_\\lambda)$, $\\scrA (H_\\lambda) $, $\\scrK (H_\\lambda) $, $\\scr{L} (H_\\lambda)$.\n\n\n\\begin{lemma}[Kasparov Technical Theorem]\n\\label{lem:KTT}\nThere exist positive $\\mathsf{G}$-continuous operators $N_1,N_2\\in \\scr{L}$ with the following properties:\n\\begin{enumerate}\n\\item $N_1^2+N_2^2 = 1$,\n\\item $N_i \\cdot\\scrK[\\alpha_i] \\subseteq \\scrK$ for each $i=1,2$,\n\\item $N_i$ commutes modulo compact operators with\n\\begin{itemize}\n \\item $\\multop{f}$ for all $f\\in C(\\scrX)$,\n \\item $U(g)$ for all $g\\in\\mathsf{G}$,\n \\item the normalized BGG operators $\\BGGextended{\\lambda}{w}{w'}$, for all $\\edge{w}{w'}$,\n\\end{itemize}\n\\item $N_i$ commutes on the nose with $U(k)$ for all $k\\in\\mathsf{K}$,\n\\item $N_i$ commutes on the nose with the projections $\\component{w}$ for all $w\\in\\Lie{W}$, {\\em i.e.}, $N_i$ is diagonal with respect to the direct sum decomposition of $H_\\lambda$.\n\\end{enumerate}\n\\end{lemma}\n\nNote also that $N_1$ and $N_2$ commute, by \\emph{(i)}.\n\n\\begin{proof}\nSee \\cite[Theorem 20.1.5]{Blackadar}. The $\\mathsf{K}$-invariance of (iv) is obtained by averaging over the $\\mathsf{K}$-translates $U(k)\\, N_i\\, U(k^{-1})$ of $N_i$. Also, the operators $\\sum_w \\pm Q_w$ (taking all possible choices of signs) form a finite group of unitaries, so that a similar averaging trick gives property (v).\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{lem:operator_po1}\nThere exist mutually commuting operators $\\poI{w}{w'} \\in \\scr{L}$, indexed by the edges of the graph \\eref{eq:Weyl_graph}, with the following properties:\n\\begin{enumerate}\n\\item $\\poI{w}{w'} = \\poI{w'}{w}$\n\\item If $\\edge[\\alpha]{w}{w'}$ for $\\alpha\\in\\{\\alpha_1,\\alpha_2,\\rho\\}$, then $\\poI{w}{w'} \\scrK[\\alpha] \\subseteq \\scrK$.\n\\item If $w\\leftrightarrow w' \\leftrightarrow w''$ with $w\\neq w''$ then $\\poI{w'}{w''}\\poI{w}{w'} \\scrK[\\rho] \\subseteq \\scrK$.\n\\item For any $w,w''\\in\\Lie{W}$, $\\sum_{w'} \\poI{w'}{w''}\\poI{w}{w'} =\\delta_{w,w''}$, where the sum is over $w'$ such that $w \\leftrightarrow w' \\leftrightarrow w''$.\n\\item $\\poI{w}{w'}$ satisfies {(iii)}, {(iv)} and{(v)} of Lemma \\ref{lem:KTT}.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{remark}\nTo clarify a possibly misleading notational point, $\\poI{w}{w'}$ does not designate an operator between $\\LXE[w\\star\\lambda]$ and $\\LXE[w'\\star\\lambda]$. Rather it is an operator on $H_\\lambda$ which we will use to modify the operator $\\BGG{\\lambda}{w}{w'}$.\n\\end{remark}\n\n\n\\begin{proof}\nWith $N_1, N_2$ as in the previous lemma, assign operators $\\poI{w}{w'}$ to each arrow as follows:\n$$\n \\xymatrix@!C{\n & \\stackrel{\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[rr]^{-N_1N_2} \\ar@{<->}[ddrr]^(0.7){-N_2^2} \n && \\stackrel{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[dr]^{-N_2} \\\\\n \\stackrel{1}{\\bullet} \\ar@{<->}[ur]^{N_1} \\ar@{<->}[dr]_{N_2} \n &&&& \\stackrel{w_\\rho}{\\bullet} \\\\\n & \\stackrel{\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[rr]_{N_1N_2} \\ar@{<->}[uurr]_(0.7){N_1^2} \n && \\stackrel{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[ur]_{N_1} \n }\n$$\nThe asserted properties can be easily checked using the properties of $N_1$ and $N_2$ from Lemma \\ref{lem:KTT} and the diagram \\eref{eq:Weyl_graph}. It is worth noting particularly that $N_1N_2$ multiplies $\\scrK[\\rho]$ into the compact operators. \n\\end{proof}\n\n\\begin{definition}\n\\label{def:Fredholm_components}\nDefine $\\poIBGG{\\lambda}{w}{w'} := \\poI{w}{w'} \\BGGextended{\\lambda}{w}{w'}$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:compact_commutators}\nFor any $\\edge{w}{w'}$,\n\\begin{enumerate}\n\\item $\\poIBGG{\\lambda}{w}{w'} - \\poIBGG{\\lambda}{w'}{w}^*\\in \\scrK$.\n\\item $[\\poIBGG{\\lambda}{w}{w'}, M_f] \\in \\scrK$, for any $f\\in C(\\scrX)$,\n\\item $U(g)\\,\\poIBGG{\\lambda}{w}{w'}\\,U(g^{-1}) - \\poIBGG{\\lambda}{w}{w'} \\in \\scrK$, for any $g\\in\\mathsf{G}$,\n\\item $\\poIBGG{\\lambda}{w}{w'}$ is $\\mathsf{K}$-invariant, {\\em ie}, $[\\poIBGG{\\lambda}{w}{w'}, U(k)] =0$, for any $k\\in\\mathsf{K}$,\n\\item $\\poIBGG{\\lambda}{w}{w'}$ is $\\mathsf{G}$-continuous.\n\\end{enumerate}\nAlso,\n\\begin{enumerate}\n\\setcounter{enumi}{5}\n\\item For any $w,w''\\in\\Lie{W}$, $\\left( \\sum_{w'} \\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w}{w'} \\right) \\equiv \\delta_{w,w''} \\component{w} \\pmod{\\scrK}$, where the sum is over $w'\\in\\Lie{W}$ such that $w\\leftrightarrow w' \\leftrightarrow w''$.\n\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\edge[\\alpha]{w}{w'}$. By definition, $\\BGG{\\lambda}{w}{w'} = \\BGG{\\lambda}{w'}{w}^*$, so $\\poIBGG{\\lambda}{w}{w'} - \\poIBGG{\\lambda}{w'}{w}^* = [\\poI{w}{w'},\\BGGextended{\\lambda}{w}{w'}]$, which proves (i). \n\nSince $\\poI{w}{w'}$ commutes modulo compacts with multiplication operators,\n$$\n [ \\poIBGG{\\lambda}{w}{w'} , \\multop{f}]\n \\equiv \\poI{w}{w'} [\\BGGextended{\\lambda}{w}{w'}, \\multop{f}] \\pmod{\\scrK} \n$$\nBy Lemma \\ref{lem:BGG_in_A}, the latter is in $\\poI{w}{w'}\\scrK[\\alpha] \\subseteq \\scrK$, which proves (ii). Similarly, for (iii),\n\\begin{multline*}\n U(g)\\,\\poIBGG{\\lambda}{w}{w'}\\,U(g^{-1}) - \\poIBGG{\\lambda}{w}{w'} \\\\\n \\equiv \\poI{w}{w'} \\big( U(g)\\,\\BGGextended{\\lambda}{w}{w'}\\,U(g^{-1}) - \\BGGextended{\\lambda}{w}{w'} \\big)\\pmod{\\scrK}\n\\end{multline*}\nand the latter is in $\\poI{w}{w'} \\scrK[\\alpha] \\subseteq \\scrK$ by Lemma \\ref{lem:F_g_commute}.\n\n\nFor any weight $\\mu$, the differential operator $X_i: \\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is $\\mathsf{K}$-invariant. Likewise for its essential adjoint $Y_i:\\LXE[\\mu-\\alpha_i] \\to \\LXE[\\mu]$. Hence, $\\Ph{X_i}:\\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is $\\mathsf{K}$-equivariant. The normalized BGG operators $\\BGG{\\lambda}{w}{w'}$ are compositions of such operators, and $\\poI{\\lambda}{w}{w'}$ is $\\mathsf{K}$-invariant by definition. This proves (iv).\n\nOnce again, $\\mathsf{G}$-continuity is trivial.\n\nWe prove (vi) in two separate cases. Firstly, suppose $w=w''$. For any $w'$ with $\\edge{w}{w'}$, Lemma \\ref{lem:F2-1_in_K} implies that $\\BGGextended{\\lambda}{w'}{w}\\BGGextended{\\lambda}{w}{w'} \\equiv \\component{w} \\pmod{\\scrK[\\alpha]}$. By Lemma \\ref{lem:operator_po1}(iv),\n\\begin{eqnarray*}\n \\sum_{w'} \\poIBGG{\\lambda}{w'}{w} \\poIBGG{\\lambda}{w}{w'}\n &\\equiv& \\sum_{w'} \\poI{w'}{w}\\poI{w}{w'} \\BGGextended{\\lambda}{w'}{w}\\BGGextended{\\lambda}{w}{w'}\n \\pmod{\\scrK} \\\\\n &\\equiv& \\sum_{w'} \\poI{w'}{w}\\poI{w}{w'} \\component{w} \\pmod{\\scrK} \\\\\n &=& \\component{w}.\n\\end{eqnarray*}\nIf $w\\neq w'$, the result is trivial unless there exists at least one $w'$ such that $w \\leftrightarrow w' \\leftrightarrow w''$. If such a $w'$ exists, Lemma \\ref{lem:diagram_commutes} implies that the products $\\BGG{\\lambda}{w'}{w''} \\BGG{\\lambda}{w}{w'}$ are independent of this intermediate vertex $w'$, modulo $\\scrK[\\rho]$. Let us fix one such product and denote it temporarily by $\\BGG{\\lambda}{w\\to\\cdot}{w''}$. Then by Lemma \\ref{lem:operator_po1}(iv),\n\\begin{eqnarray*}\n \\sum_{w'} \\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w}{w'}\n &\\equiv& \\sum_{w'} \\poI{w'}{w''}\\poI{w}{w'} \\BGGextended{\\lambda}{w'}{w''}\\BGGextended{\\lambda}{w}{w'}\n \\pmod{\\scrK} \\\\\n &\\equiv& \\left(\\sum_{w'} \\poI{w'}{w}\\poI{w}{w'}\\right) \\BGGextended{\\lambda}{w\\to \\cdot}{w''} \\pmod{\\scrK} \\\\\n &=& 0.\n\\end{eqnarray*}\n\n\n\\end{proof}\n\n\n\\begin{definition}\nDefine $\\Fredholm{\\lambda} :=\\sum \\poIBGG{\\lambda}{w}{w'}$, where the sum is over all directed edges in the graph \\eref{eq:Weyl_graph}.\n\\end{definition}\n\n\n\\begin{theorem}\n\\label{thm:main_theorem}\nThe operator $\\Fredholm{\\lambda}\\in \\scr{L}$ defines an element $\\Kcycle{\\lambda} \\in K^\\mathsf{G}(C(\\scrX),\\mathbb{C})$. That is, \n\\begin{enumerate}\n\\item $\\Fredholm{\\lambda}$ is {\\em odd} with respect to the grading of $H_\\lambda$,\n\\item $\\Fredholm{\\lambda} - \\Fredholm{\\lambda}^* \\in \\scrK$,\n\\item $\\Fredholm{\\lambda}^2 - 1 \\in \\scrK$,\n\\item $[\\Fredholm{\\lambda}, M_f] \\in \\scrK$, for any $f\\in C(\\scrX)$,\n\\item $[\\Fredholm{\\lambda}, U(g)] \\in \\scrK$, for any $g\\in\\mathsf{G}$,\n\\item $\\Fredholm{\\lambda}$ is $\\mathsf{G}$-continuous,\n\\end{enumerate}\nMoreover, $\\Fredholm{\\lambda}$ is $\\mathsf{K}$-invariant: $[\\Fredholm{\\lambda},U(k)] = 0$ for all $k\\in\\mathsf{K}$.\n\\end{theorem}\n\n\\begin{proof}\nThis is mostly immediate from the previous lemma. To be explicit about the proof of (iii), Lemma \\ref{lem:compact_commutators}(vi) gives\n\\begin{eqnarray*}\n \\Fredholm{\\lambda}^2 \n &=& \\!\\!\\sum_{w,w',w''\\in\\Lie{W} \\atop w\\leftrightarrow w' \\leftrightarrow w''} \\!\\!\\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w'}{w''} \\\\\n &\\equiv& \\sum_{w} \\component{w} \\pmod{\\scrK} \\\\\n &=& 1.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{definition}\nLet $\\pi_\\lambda$ denote the irreducible representation of $\\mathsf{K}$ with highest weight $\\lambda$. Define a homomorphism of abelian groups \n\\begin{eqnarray*}\n \\theta:R(\\mathsf{K}) &\\to& KK^\\mathsf{G}(C(\\scrX),\\mathbb{C}) \\\\ \n ~ [\\pi_\\lambda] &\\mapsto& \\Kcycle{\\lambda}.\n\\end{eqnarray*}\n\\end{definition}\n\nLet $\\iota:\\mathbb{C}\\to C(\\scrX)$ denote the $G$-equivariant $C^*$-morphism induced by the map of $\\scrX$ to a point.\n\n\\begin{theorem}\n\\label{thm:splitting}\nThe map $\\iota^*\\circ\\theta:R(\\mathsf{K})\\to R(\\mathsf{G})$ is a ring homomorphism which splits the restriction homomorphism $\\Res_\\mathsf{K}^\\mathsf{G}:R(\\mathsf{G}) \\to R(\\mathsf{K})$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\lambda$ be a dominant weight. We have that $\\Res^\\mathsf{G}_\\mathsf{K}\\iota^*\\circ\\theta([\\pi_\\lambda])$ is the $\\mathsf{K}$-index of $\\Fredholm{\\lambda}$. Since $\\Fredholm{\\lambda}$ is $\\mathsf{K}$-equivariant, it decomposes as a direct sum of operators on the $\\mathsf{K}$-isotypical subspaces of $H_\\lambda$, each of which is finite dimensional (Example \\ref{ex:finite_multiplicities}). The $\\mathsf{K}$-index of $\\Fredholm{\\lambda}$ is the sum of the indices of each component.\n\nTo compute this index, we compare with the classical BGG complex. Let $\\mu:= w\\star\\lambda$ be in the shifted Weyl orbit of $\\lambda$. The induced bundle $E_\\mu$ of our normalized $BGG$-complex and the holomorphic bundle $\\Lhol{\\mu}$ of the classical BGG complex \\eref{eq:BGG_resolution} are identical as $\\mathsf{K}$-homogeneous line bundles. The classical BGG resolution is exact and $\\mathsf{K}$-equivariant, so exact in each $\\mathsf{K}$-type. It follows that the index of $\\Fredholm{\\lambda}$ is $[\\pi_\\lambda]$. Thus the composition $\\Res_\\mathsf{K}^\\mathsf{G}\\circ\\iota^*\\circ\\theta$ is the identity on $R(\\mathsf{K})$.\n\nBy Theorem \\ref{thm:split_surjection}, $\\Res_\\mathsf{K}^\\mathsf{G}: \\gamma R(\\mathsf{G}) \\to R(\\mathsf{K})$ is a ring isomorphism, so it suffices to show that the image of $\\iota^*\\theta$ is in $\\gamma R(\\mathsf{G})$. Using \\cite[Theorem 3.6(1)]{Kas88}, we have\n\\begin{eqnarray*}\n\\gamma \\cdot (\\iota^*\\Kcycle{\\lambda}) \n &=& \\iota^* \\otimes_{C(\\mathsf{G\/B})} (1_{C(\\mathsf{G\/B})}\\otimes\\gamma)\n \\otimes_{C(\\mathsf{G\/B})} \\Kcycle{\\lambda} \\\\\n &=& \\iota^* \\otimes_{C(\\mathsf{G\/B})} (\\Ind_\\mathsf{B}^\\mathsf{G} \\Res_\\mathsf{B}^\\mathsf{G}\\gamma) \\otimes_{C(\\mathsf{G\/B})} \\Kcycle{\\lambda}. \n\\end{eqnarray*}\nSince $\\mathsf{B}$ is amenable, $\\gamma$ restricts to the unit in $R(\\mathsf{B})$, so $\\gamma \\cdot (\\iota^* \\Kcycle{\\lambda}) = \\iota^* \\Kcycle{\\lambda}$.\n\n\\end{proof}\n\n\n\n\n\\begin{corollary}\n\\label{cor:gamma}\n $\\gamma = [(H_0, U, \\Fredholm{0})] \\in R(\\mathsf{G})$.\n\\end{corollary}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcmxq b/data_all_eng_slimpj/shuffled/split2/finalzzcmxq new file mode 100644 index 0000000000000000000000000000000000000000..74e638c6fa845e31641fada7224789caa5ace047 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcmxq @@ -0,0 +1,5 @@ +{"text":"\\section{#1}\n}\n\\def\\hskip 1pt\\vrule width1pt\\hskip 1pt{\\hskip 1pt\\vrule width1pt\\hskip 1pt}\n\\def{\\mathsf b}{{\\mathsf b}}\n\\def\\mathsf P{\\mathsf P}\n\\def{\\mathbb H}{{\\mathbb H}}\n\\def{\\mathbb N}{{\\mathbb N}}\n\\def{\\mathbb R}{{\\mathbb R}}\n\\def{\\mathbb C}{{\\mathbb C}}\n\\def{\\mathbb Z}{{\\mathbb Z}}\n\\def{\\mathbb P}{{\\mathbb P}}\n\\def{{\\rm I}\\kern-1pt\\Pi}{{{\\rm I}\\kern-1pt\\Pi}}\n\\def{\\mathbb S}{{\\mathbb S}}\n\\def\\mathfrak{S}{\\mathfrak{S}}\n\\def\\mathbb T{\\mathbb T}\n\\def\\mathcal{T}{\\mathcal{T}}\n\\def\\alpha{\\alpha}\n\\def\\b #1;{{\\bf #1}}\n\\def{\\bf x}{{\\bf x}}\n\\def{\\bf k}{{\\bf k}}\n\\def{\\bf y}{{\\bf y}}\n\\def\\mathbf{u}{\\mathbf{u}}\n\\def{\\bf w}{{\\bf w}}\n\\def{\\bf z}{{\\bf z}}\n\\def{\\bf h}{{\\bf h}}\n\\def\\mathfrak{m}{\\mathfrak{m}}\n\\def\\mathbf{j}{\\mathbf{j}}\n\\def\\mathbf{r}^*{\\mathbf{r}^*}\n\\def{\\mathbf 0}{{\\mathbf 0}}\n\\def\\epsilon{\\epsilon}\n\\def\\lambda{\\lambda}\n\\def\\phi{\\phi}\n\\def\\varphi{\\varphi}\n\\def\\sigma{\\sigma}\n\\def\\mathbf{t}{\\mathbf{t}}\n\\def\\Delta{\\Delta}\n\\def\\delta{\\delta}\n\\def{\\cal F}{{\\cal F}}\n\\def{\\cal O}{{\\cal O}}\n\\def{\\cal P}{{\\cal P}}\n\\def{\\mathcal M}{{\\mathcal M}}\n\\def{\\mathcal C}{{\\mathcal C}}\n\\def\\CR{{\\mathcal C}}\n\\def{\\mathcal N}{{\\mathcal N}}\n\\def{\\cal Z}{{\\cal Z}}\n\\def{\\cal A}{{\\cal A}}\n\\def{\\cal V}{{\\cal V}}\n\\def{\\cal E}{{\\cal E}}\n\\def{\\mathcal H}{{\\mathcal H}}\n\\def{\\bf W}{{\\bf W}}\n\\def{\\mathbb W}{{\\mathbb W}}\n\\def\\mathbb{Y}{\\mathbb{Y}}\n\\def{\\bf Y}{{\\bf Y}}\n\\def{\\widetilde\\epsilon}{{\\widetilde\\epsilon}}\n\\def{\\widetilde E}{{\\widetilde E}}\n\\def{\\overline{B}}{{\\overline{B}}}\n\\def\\ip#1#2{{\\langle {#1}, {#2}\\rangle}}\n\\def\\derf#1#2{{#1}^{(#2)}}\n\\def\\node#1#2{x_{{#1},{#2}}}\n\\def\\mathop{\\hbox{{\\rm ess sup}}}{\\mathop{\\hbox{{\\rm ess sup}}}}\n\\def{{\\bf e}_{q+1}}{{{\\bf e}_{q+1}}}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\eref#1{(\\ref{#1})}\n\\def\\displaystyle{\\displaystyle}\n\\def\\cfn#1{\\chi_{{}_{ #1}}}\n\\def\\varrho{\\varrho}\n\\def\\intpart#1{{\\lfloor{#1}\\rfloor}}\n\\def\\largeint#1{{\\lceil{#1}\\rceil}}\n\\def\\mathbf{ \\hat I}{\\mathbf{ \\hat I}}\n\\def\\mbox{{\\rm dist }}{\\mbox{{\\rm dist }}}\n\\def\\mbox{{\\rm Prob }}{\\mbox{{\\rm Prob }}}\n\\def\\donchitre#1#2{\\vskip 6.5cm\\noindent\n\\parbox[t]{1in}{\\special{eps:#1.eps x=6.5cm y=5.5cm}}\n\\hbox to 7cm{}\\parbox[t]{0.0cm}{\\special{eps:#2.eps x=6.5cm y=5.5cm}}}\n\\def\\chitra#1{\\vskip 9.5cm\\noindent\n\\hskip2in{\\special{eps:#1.eps x=8.5cm y=8.5cm}}\n}\n\\defFrom\\ {From\\ }\n\\def|\\!|\\!|{|\\!|\\!|}\n\\def{\\mathbb X}{{\\mathbb X}}\n\\def{\\mathbb B}{{\\mathbb B}}\n\\def\\mbox{\\mathsf{span }}{\\mbox{\\mathsf{span }}}\n\\def\\bs#1{{\\boldsymbol{#1}}}\n\\def\\bs{\\omega}{\\bs{\\omega}}\n\\def\\bs{\\theta}{\\bs{\\theta}}\n\\def\\mathbf{t}^*{\\mathbf{t}^*}\n\\def\\mathbf{x}^*{\\mathbf{x}^*}\n\\def\\mathsf{dist }{\\mathsf{dist }}\n\\def\\mathsf{supp\\ }{\\mathsf{supp\\ }}\n\\def\\corr#1{{\\color{red} {#1}}}\n\\title{Deep nets for local manifold learning}\n\\author{Charles K. Chui\\thanks{Department of Statistics, Stanford University, Stanford, CA 94305. The research of this author is supported by ARO Grant W911NF-15-1-0385.\n\\textsf{email:} ckchui@stanford.edu. }\\ \n\\ and\nH.~N.~Mhaskar\\thanks{Department of Mathematics, California Institute of Technology, Pasadena, CA 91125;\nInstitute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711. The research of this author is supported in part by ARO Grant W911NF-15-1-0385.\n\\textsf{email:} hrushikesh.mhaskar@cgu.edu. } }\n\\date{}\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nThe problem of extending a function $f$ defined on a training data ${\\mathcal C}$ on an\nunknown manifold ${\\mathbb X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For ${\\mathbb X}$ embedded in a high dimensional ambient Euclidean space ${\\mathbb R}^D$, a deep learning algorithm is developed for finding a local coordinate system for the manifold \\textbf{without eigen--decomposition}, which reduces the problem to the classical problem of function approximation on a low dimensional cube. Deep nets (or multilayered neural networks) are proposed to accomplish this approximation scheme by using the training data. Our methods do not involve such optimization techniques as back--propagation, while assuring optimal (a priori) error bounds on the output in terms of the number of derivatives of the target function. In addition, these methods are universal, in that they do not require a prior knowledge of the smoothness of the target function, but adjust the accuracy of approximation locally and automatically, depending only upon the local smoothness of the target function. Our ideas are easily extended to solve both the pre--image problem and the out--of--sample extension problem, with a priori bounds on the growth of the function thus extended. \n\\end{abstract}\n\n\\bhag{Introduction}\nMachine learning is an active sub--field of Computer Science on algorithmic development for learning and making predictions based on some given data, with a long list of applications that range from computational finance and advertisement, to information retrieval, to computer vision, to speech and handwriting recognition, and to structural healthcare and medical diagnosis. In terms of function approximation, the data for learning and prediction can be formulated as $\\{({\\bf x}, f_{\\bf x})\\}$, obtained with an unknown probability distribution. Examples include: the Boston housing problem (of predicting the median price $f_{\\bf x}$ of a home based on some vector ${\\bf x}$ of 13 other attributes \\cite{vapnik2013nature}) and the floor market problem \\cite{tiao1989model, chakraborty1992forecasting} (that deals with the indices of the wheat floor pricing in three major markets in the United States). For such problems, the objective is to predict the index $f_{\\bf x}$ in the next month, say, based on a vector ${\\bf x}$ of their values over the past few months. Other similar problems include the prediction of blood glucose level $f_{\\bf x}$ of a patient based on a vector ${\\bf x}$ of the previous few observed levels \\cite{sergei, mnpdiabetes}, and the prediction of box office receipts ($f_{\\bf x}$) on the date of release of a movie in preparation, based on a vector ${\\bf x}$ of the survey results about the movie \\cite{sharda2002forecasting}. It is pointed out in \\cite{multilayer, mauropap, compbio} that all the pattern classification problems can also be viewed fruitfully as problems of function approximation. While it is an ongoing research to allow non--numeric input ${\\bf x}$ (e.g., \\cite{treepap}), we restrict our attention in this paper to the consideration of ${\\bf x}\\in{\\mathbb R}^D$, for some integer $D\\ge 1$.\n\n\nIn the following discussion, the first component ${\\bf x}$ is considered as input, while the second component $f_{\\bf x}$ is considered the output of the underlying process. The central problem is to estimate the conditional expectation of $f_{\\bf x}$ given ${\\bf x}$. Various statistical techniques and theoretical advances in this direction are well--known (see, for example \\cite{vapnik1998statistical}). In the context of neural and radial--basis--function networks, an explicit formulation of the input\/output machines was pointed out in \\cite{girosi1990networks,girosi1995regularization}. More recently, the nature of deep learning as an input\/output process is formulated in the same way, as explained in \\cite{lecun2015deep, poggio_deep_net_2015}. To complement the statistical perspective and understand the theoretical capabilities of these processes, it is customary to think of the expected value of $f_{\\bf x}$, given ${\\bf x}$ , as a function $f$ of ${\\bf x}$. The question of empirical estimation in this context is to carry out the approximation of $f$ given samples $\\{({\\bf x}, f({\\bf x}))\\}_{{\\bf x}\\in{\\mathcal C}}$, where ${\\mathcal C}$ is a finite \\emph{training data} set. In practice, because of the random nature of the data, it may be possible that there are several pairs of the form $({\\bf x}, f_{\\bf x})$ in the data for the same values of ${\\bf x}$. In this case, a statistical scheme, such as some kind of averaging of $f_{\\bf x}$ being the simplest one, can be used to obtain a desired value $f({\\bf x})$ for the sample of $f$ at ${\\bf x}$, ${\\bf x}\\in{\\mathcal C}$. From this perspective, the problem of extending $f$ from the traning data set ${\\mathcal C}$ to ${\\bf x}$ not in ${\\mathcal C}$ in machine learning is called the \\emph{generalization problem}.\n\nWe will illustrate this general line of ideas by using neural networks as an example. To motivate this idea, let us first recall a theorem originating with Kolmogorov and Arnold \\cite[Chapter~17, Theorem~1.1]{lorentz_advanced}. According to this theorem, there exist \\emph{universal} Lipschitz continuous functions $\\phi_1,\\cdots,\\phi_{2D+1}$ and \\emph{universal} scalars $\\lambda_1,\\cdots,\\lambda_D\\in (0,1)$, for which every continuous function $f : [0,1]^D\\to {\\mathbb R}$ can be written as \n\\begin{equation}\\label{kolmtheo}\nf({\\bf x})=\\sum_{j=1}^{2D+1}g\\left(\\sum_{k=1}^D \\lambda_k\\phi_j(x_k)\\right), \\qquad {\\bf x}=(x_1,\\cdots,x_D)\\in [0,1]^D,\n\\end{equation}\nwhere $g$ is a continuous function that depends on $f$. In other words, for a given $f$, only one function $g$ has to be determind to give the representation formula \\eref{kolmtheo} of $f$. \n\nA neural network, used as an input\/output machine, consists of an input layer, one or more hidden layers, and an output layer. Each hidden layer consists of a number of neurons arranged according to the network architecture. Each of these neurons has a local memory and performs a simple non--linear computation upon its input. The input layer fans out the input ${\\bf x}\\in{\\mathbb R}^D$ to the neurons at the first hidden layer. The output layer typically takes a linear combination of the outputs of the neurons at the last hidden layer. \nThe right hand side of \\eref{kolmtheo} is a neural network with two hidden layers. The first contains $D$ neurons, where the $j$--th neuron computes the sum $\\sum_{k=1}^D \\lambda_k\\phi_j(x_k)$. The next hidden layer contains $2D+1$ neurons each evaluating the function $g$ on the \noutput of the $j$--th neuron in the first hidden layer. The output layers takes the sum of the results as indicated in \\eref{kolmtheo}.\n\n\nFrom a practical point of view, such a network is clearly hard to construct, since only the existence of the functions $\\phi_j$ and $g$ is known, without a numerical procedure for computing these. In the early mathematical development of neural networks during the late 1980s and early 1990s, instead of finding these functions for the representation of a given continuous function $f$ in \\eref{kolmtheo}, the interest was to study the existence and characterization of \\emph{universal} functions $\\sigma :{\\mathbb R}\\to{\\mathbb R}$, called \\emph{activation functions} of the neural networks, such that each neuron evaluates the activation function upon an affine transform of its input, and the network is capable of approximating any desired real-valued continuous target function $f: K\\to{\\mathbb R}$ arbitrarily closely on $K$, where $K\\subset {\\mathbb R}^D$ is any compact set.\n\nFor example, a neural network with one hidden layer can be expressed as a function\n\\begin{equation}\\label{onehiddenlayer}\n\\mathcal{N}({\\bf x})=\\mathcal{N}_n(\\{{\\bf w}_k\\}, \\{a_k\\}, \\{b_k\\};{\\bf x})= \\sum_{k=1}^n a_k\\sigma({\\bf w}_k\\cdot {\\bf x} +b_k), \\qquad {\\bf x}\\in{\\mathbb R}^D.\n\\end{equation}\nHere, the hidden layer consists of $n$ neurons, each of which has a local memory. The local memory of the $k$--th neuron contains the \\emph{weights} ${\\bf w}_k\\in {\\mathbb R}^D$, and the \\emph{threshold} $b_k\\in{\\mathbb R}$. Upon receiving the input ${\\bf x}\\in{\\mathbb R}^D$ from the input later, the $k$--th neuron evaluates $\\sigma({\\bf w}_k\\cdot{\\bf x}+b_k)$ as its output, where $\\sigma$ is a non--linear activation function. The output layer is just one circuit where the coefficients $\\{a_k\\}$ are stored in a local memory, and the evaluates the linear combination as indicated in \\eref{onehiddenlayer}. Training of this network in order to learn a function $f$ on a compact subset $K\\subset{\\mathbb R}^D$ to an accuracy of $\\epsilon>0$ involves finding the parameters $\\{a_k\\}$, $\\{{\\bf w}_k\\}$, $\\{b_k\\}$ so that \n\\begin{equation}\\label{universal appr} \n\\max_{{\\bf x}\\in K}|f({\\bf x})-\\mathcal{N}({\\bf x})|<\\epsilon.\n\\end{equation}\nThe most popular technique for doing this is the so called back--propagation, which seeks to find these quantities by minimizing an error functional usually with some regularization parameters. \nWe remark that the number $n$ of \\emph{neurons} in the approximant \\eref{onehiddenlayer} must increase, if the tolerance $\\epsilon >0$ in the approximation\nof the target function $f$ is required to be smaller. \n\n\nFrom a theoretical perspective, the main attraction of neural networks with one hidden layer is their \\emph{universal approximation property} as formulated in \\eref{universal appr}, which overshadows the properties of their predecessors, namely: the perceptrons \\cite{minsky1988perceptrons}. In particular, the question of finding sufficient conditions on the actvation function $\\sigma$ that ensure the universal approximation property was investigated in great detail by many authors, with emphasis on the most popular \\emph{sigmoidal function}, defined by the property $\\sigma(t)\\to 1$ for $t\\to\\infty$ and $\\sigma(t)\\to 0$ for $t\\to -\\infty$. For example, Funahashi \\cite{funahashi1989} applied some discretization of an integral formula from \\cite{irie1988} to prove the universal approximation property for some sigmoidal function $\\sigma $. A similar theorem was proved by Hornik, Stinchcombe, White \\cite{hornik1989} by using the Stone--Weierstrass theorem, and another by Cybenko \\cite{cybenko1989} by applying the Hahn--Banach and Riesz Representation theorems. A constructive proof via approximation by ridge functions was given in our paper \\cite{chuili1992}, with algorithm for implementation presented in our follow-up work \\cite{chui1993realization}. A complete characterization of which activation functions are allowed to achieve the universal approximation property was given later in \\cite{mhasmich, leshnolinpinkus}. \n\nHowever, for neural networks with one hidden layer, one of the severe limitations to applying training algorithms based on optimization, such as back--propagation or those proposed in the book \\cite{vapnik1998statistical} of Vapnik, is that it is neccessary to know the number of neurons in $\\mathcal{N}$ in advance. \nTherefore, one major problem in the 1990s, known as the \\emph{complexity problem}, was to estimate the number of neurons required to approximate a function to a desired accuracy. In practice, this gives rise to a trade-off: to achieve a good approximation, one needs to have a large number of neurons, which makes the implementation of the training algorithm harder.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn this regard, nearly a century of research in approximation theory suggests that the higher the order of smoothness of the target function, the smaller the number of neurons should be, needed to achieve the desired accuracy. There are many different definitions of smoothness that give rise to different estimates. For example, under the condition that the Fourier transform of the target function $f$ satisfies $\\displaystyle\\int_{{\\mathbb R}^D} |\\bs{\\omega}\\hat{f}(\\bs{\\omega})|d\\bs{\\omega}<\\infty$, Barron \\cite{barron1993} proved the existence of a neural network with ${\\cal O}(\\epsilon^{-2})$ neurons that gives an $L^2([0,1]^D)$ error of ${\\cal O}(\\epsilon)$. While it is interesting to note that this number of neurons is essentially independent of the dimension $D$, the constants involved in the ${\\cal O}$ term as well as the number of derivatives needed to ensure the condition on the target function may increase with $D$. Several authors have subsequently improved upon such results under various conditions on the activation function as well as the target function so as to ensure that the constants depend polynomially on $D$ (e.g., \\cite{kurkova1, kurkova2, tractable} and references therein).\n\nThe most commonly understood definition of smoothness is just the number of derivatives of the target function. It is well--known from the theory of $n$-widths that if $r\\ge 1$ is an integer, and the only a priori information assumed on the unknown target function is that it is $r$--times continuously differentiable function, then a stable and uniform approximation to within $\\epsilon$ by neural networks must have at least a constant multiple of $\\epsilon^{-D\/r}$ neurons. \nIn \\cite{optneur}, we gave an explicit construction for a neural network that achieves the accuracy of $\\epsilon$ using ${\\cal O}(\\epsilon^{-D\/r})$ neurons arranged in a single hidden layer. It follows that this suffers from a curse of dimensionality, in that the number of neurons increases exponentially with the input dimension $D$. Clearly, if the smoothness $r$ of the function increases linearly with $D$, as it has to in order to satisfy the condition in \\cite{barron1993}, then this bound is also ``dimension independent''.\n\nWhile this is definitely unavoidable for neural networks with one hidden layer, the most natural way out is to achieve local approximation; i.e., given an input ${\\bf x}$, construct a network with a uniformly bounded number of neurons that approximates the target function with the optimal rate of approximation near the point ${\\bf x}$, preferably using the values of the function also in a neighborhood of ${\\bf x}$. Unfortunately, this can never be achieved as we proved in \n\\cite{chui1994neural}. Furthermore, we have proved in \\cite{ chui1996limitations} that even if we allow each neuron to evaluate its own activation function, this local approximation fails. Therefore the only way out is to use a neural network with more than one hidden layer, called \\emph {deep net} (for deep neural network). Indeed, local approximation can be achieved by a deep net as proved in our papers \\cite{multilayer, mhaskar1993neural}. In this regard, it is of interest to point out that an adaptive version of \\cite{multilayer, mhaskar1993neural} was derived in \\cite{lermont} for prediction of time series, yielding as much as 150\\% improvement upon the state--of--the--art at that time, in the study of the floor market problem. \n\nOf course, the curse of dimensionality is inherent to the problem itself, whether with one or more hidden layers. Thus, while it is possible to construct a deep net to approximate a function at each point arbitrarily closely by using a uniformly bounded number of neurons, the uniform approximation on an entire compact set, such as a cube, would still require an approximation at a number of points in the cube, and this number increasing exponentially with the input dimension. Equivalently, the effective number of neurons for approximation on the entire cube is still exponentially increasing with the input dimension.\n\n\nIn addition to the high dimensionality, another difficulty in solving the function approximation problem is that the data may be not just high dimensional but unstructured and sparse. A relatively recent idea which has been found very useful in applications, in fact, too many to list exhaustively, is to consider the points ${\\bf x}$ as being sampled from an unknown, low dimensional sub--manifold ${\\mathbb X}$ of the ambient high dimensional space ${\\mathbb R}^D$. The understanding of the geometry of ${\\mathbb X}$ is the subject of the bulk of modern research in the area of diffusion geometry. An introduction to this subject can be found in the special issue \\cite{achaspissue} of Applied and Computational Harmonic Analysis. \nThe basic idea is to construct the so--called diffusion matrix from the data, and use its eigen--decomposition for finding local coordinate charts and other useful aspects of the manifold. The convergence of the eigen--decomposition of the matrices to that of the Laplace--Beltrami and other differential operators on the manifold is discussed, for example, in \\cite{belkinfound, lafon, singer}. It is shown in \\cite{jones2008parameter, jones2010universal} that some of the eigenfunctions on the manifolds yield a local coordinate chart on the manifold. In the context of deep learning, this idea is explored as a function approximation problem in \\cite{cohen_diffusion_net2015}, where a deep net is developed in order to learn the coordinate system given by the eigenfunctions. \n\nOn the other hand, while much of the research in this direction is focused on understanding the data geometry, the theoretical foundations for the problems of function approximation and harmonic analysis on such data--defined manifold are developed extensively in \\cite{mauropap, frankbern, modlpmz, eignet, heatkernframe, chuiinterp}. The theory is developed more recently for kernel construction on directed graphs and analysis of functions on changing data in our paper \\cite{tauberian}. However, a drawback of the approach based on data--defined manifolds, known as the out--of--sample extension problem, is that since the diffusion matrix is constructed entirely using the available data, the whole process must be done again if new data become available. A popular idea is then to extend the eigen--functions to the ambient space by using the so called Nystr\\\"om extension \\cite{coifman2006geometric}. \n\nThe objective of this present paper is to describe a deep learning approach to the problem of function approximation, using three groups of networks in the deep net. The lowest layer accomplishes dimensionality reduction by learning the local coordinate charts on the unknown manifold \\textbf{without using any eigen--decomposition}. Having found the local coordinate system, the problem is reduced to the classical problem of approximating a function on a cube in a relatively low dimensional Euclidean space. For the next two layers, we may now apply the powerful techniques from approximation theory to approximate the target function $f$, given the samples on the training data set ${\\mathcal C}$. We describe two approaches to construct the basis functions using multi--layered neural networks, and to construct other networks to use these basis functions in the next layer to accomplish the desired function approximation. \n\nWe summarize some of the highlights of our paper. \n\\begin{itemize}\n\\item We give a very simple learning method for learning the local coordinate chart near each point. The subsequent approximation process is then entirely local to each coordinate patch.\n\\item Our method allows us to solve the pre--image problem easily; i.e., to generate a point on the manifold corresponding to a given local coordinate description.\n\\item The learning method itself \\textbf{does not involve} any optimization based technique, except probably for reducing the noise in the values of the function itself.\n\\item We provide optimal error bounds on approximation based on the smoothness of the function, while the method itself \\emph{does not require an a priori knowledge of such smoothness.}\n\\item Our methods can solve easily the \nout--of--sample extension problem. Unlike the Nystr\\\"om extension process, our method does not require any elaborate construction of kernels defined on the ambient space and commuting with certain differential operators on the unknown manifold.\n\\item Our method is designed to control the growth of the out--of--sample extension in a tubular neighborhood of ${\\mathbb X}$, and is local to each coordinate patch.\n\\end{itemize}\n\n This paper is organized as follows. In Section~\\ref{mainsect}, we describe the main ideas in our approach. The local coordinate system is described in detail in Section~\\ref{loccordsect}. Having thus found a local coordinate chart around the input, the problem of function approximation reduces to the classical one. In Section~\\ref{locbasissect}, we demonstrate how the popular basis functions used in this theory can be implemented using neural networks with one or more hidden layers. The function approximation methods which work with unstructured data without using optimization are described in Section~\\ref{approxsect}. In Section~\\ref{extsect}, we explain how our method can be used to solve both the pre--image problem and the out--of--sample extension problem.\n\n\n\n\\bhag{Main ideas and results}\\label{mainsect}\nThe purpose of this paper is to develop a deep learning algorithm to learn a function $f:{\\mathbb X}\\to{\\mathbb R}$, where ${\\mathbb X}$ is a \n$d$ dimensional compact Riemannian sub--manifold of a Euclidean space ${\\mathbb R}^D$, with $d\\ll D$, given \\emph{training data} of the form $\\{({\\bf x}_j, f({\\bf x}_j))\\}_{j=1}^M$, ${\\bf x}_j\\in{\\mathbb X}$. It is important to note that ${\\mathbb X}$ itself is not known in advance; the points ${\\bf x}_j$ are known only as $D$--dimensional vectors, presumed to lie on ${\\mathbb X}$. In Sub--section~\\ref{ideasect}, we explain our main idea briefly. In Sub--section~\\ref{loccordsect}, we derive a simple construction of the local coordinate chart for ${\\mathbb X}$. In Sub--section~\\ref{locbasissect}, we describe the construction of a neural network with one or more hidden layers to implement two of the basis functions used commonly in function approximation. While the well known classical approximation algorithms require a specific placement of the training data, one has no control on the location of the data in the current problem. In Section~\\ref{approxsect}, we give algorithms suitable for the purpose of solving this problem.\n\n\n\\subsection{Outline of the main idea}\\label{ideasect}\n\nOur approach is the following.\n\\begin{enumerate}\n\\item \\label{loccoord} ${\\mathbb X}$ is a finite union of local coordinate neighborhoods, and ${\\bf x}$ belongs to one of them, say $\\mathbb{U}$. We find a local coordinate system for this neighborhood in terms of Euclidean distances on ${\\mathbb R}^D$, say $\\Phi : \\mathbb{U}\\to [-1,1]^d$, where $d$ is the dimension of the manifold. Let ${\\bf y}=\\Phi({\\bf x})$, and with a relabeling for notational convenience, $\\{{\\bf x}_j\\}_{j=1}^K$ be the points in $\\mathbb{U}$, ${\\bf y}_j=\\Phi({\\bf x}_j)$. This way, we have reduced the problem to approximating $g=f\\circ\\Phi :[-1,1]^d\\to{\\mathbb R}$ at ${\\bf y}$, given the values\n$\\{({\\bf y}_j,g({\\bf y}_j))\\}_{j=1}^K$, where $g({\\bf y}_j)=f({\\bf x}_j)$. We note that $\\{{\\bf y}_j\\}$ is a subset of the unit cube of low dimensional Euclidean space, representing a local coordinate patch on ${\\mathbb X}$. Thus, the problem of approximation of $f$ on this patch is reduced that of approximation of $g$, a well studied classical approximation problem.\n\\item \\label{spline} We will summarize the solution to this problem using neural networks with one or more hidden layers, e.g., an implementation of multivariate tensor product spline approximation using multi--layerd neural network. \n\\end{enumerate}\nThus, the layers of our deep learning networks will have three main layers.\n\\begin{enumerate}\n\\item The bottom layer receives the input ${\\bf x}$, figures out which of the points ${\\bf x}_j$ are in the coordinate neighborhood of ${\\bf x}$, and computes the local coordinates ${\\bf y}$, ${\\bf y}_j$.\n\\item The next several layers compute the local basis functions necessary for the approximation, for example, the $B$--splines and their translates using the multi--layered neural network as in \\cite{multilayer}.\n\\item The last layer receives the data $\\{({\\bf y}_j,g({\\bf y}_j))\\}_{j=1}^K$, and computes the approximation described in Step~\\ref{spline} above.\n\\end{enumerate}\n\n\\subsection{Local coordinate learning}\\label{loccordsect}\nWe assume that $1\\le d\\le D$ are integers, ${\\mathbb X}$ is a \n$d$ dimensional smooth, compact, connected, Riemannian sub--manifold of a Euclidean space ${\\mathbb R}^D$, with geodesic distance $\\rho$. \n\nBefore we discuss our own construction of a local coordinate chart on ${\\mathbb X}$, we wish to motivate the work by describing a result from \\cite{jones2010universal}. Let $\\{\\lambda_k^2\\}_{k=0}^\\infty$ be the sequence of eigenvalues of the (negative of the) Laplace--Beltrami operator on ${\\mathbb X}$, and for each $k\\ge 0$, $\\phi_k$ be the eigenfunction corresponding to the eigenvalue $\\lambda_k^2$.\nWe define a formal ``heat kernel'' by\n\\begin{equation}\\label{heatkerndef}\nK_t({\\bf x},{\\bf y})=\\sum_{k=0}^\\infty \\exp(-\\lambda_k^2t)\\phi_k({\\bf x})\\phi_k({\\bf y}).\n\\end{equation} \nThe following result is a paraphrasing of the heat triangulation theorem proved in \\cite[Theorem~2.2.7]{jones2010universal} under weaker assumptions on ${\\mathbb X}$.\n\n\\begin{theorem}\\label{jmstheo}{\\rm (cf. \\cite[Theorem~2.2.7]{jones2010universal})}\n Let ${\\bf x}_0^*\\in{\\mathbb X}$. There exist constants $R>0$, $c_1,\\cdots, c_6>0$ depending on ${\\bf x}_0^*$ with the following property. Let $\\mathbf{p}_1,\\cdots,\\mathbf{p}_d$ be $d$ linearly independent vectors in ${\\mathbb R}^d$, and ${\\bf x}_j^*\\in{\\mathbb X}$ be chosen so that ${\\bf x}_j^*-{\\bf x}_0^*$ is in the direction of $\\mathbf{p}_j$, $j=1,\\cdots,d$, and \n$$\nc_1R\\le \\rho({\\bf x}_j^*,{\\bf x}_0^*)\\le c_2R, \\qquad j=1,\\cdots,d,\n$$ \nand $t=c_3R^2$. Let $B\\subset{\\mathbb X}$ be the geodesic ball of radius $c_4R$, centered at ${\\bf x}_0^*$, and \n\\begin{equation}\\label{jmsphidef}\n\\Phi_{\\mbox{jms}}({\\bf x})=R^d(K_t({\\bf x},{\\bf x}_1^*), \\cdots, K_t({\\bf x},{\\bf x}_d^*)), \\qquad {\\bf x}\\in B.\n\\end{equation}\nThen\n\\begin{equation}\\label{jmsdistpreserve}\n\\frac{c_5}{R}\\rho({\\bf x}_1,{\\bf x}_2)\\le \\|\\Phi_{\\mbox{jms}}({\\bf x}_1)-\\Phi_{\\mbox{jms}}({\\bf x}_2)\\|_d\\le \n\\frac{c_6}{R}\\rho({\\bf x}_1,{\\bf x}_2), \\qquad {\\bf x}_1,{\\bf x}_2\\in B.\n\\end{equation}\n\\end{theorem}\nSince the paper \\cite{jones2010universal} deals with a very general manifold, the mapping $\\Phi_{\\mbox{jms}}$ is not claimed to be a diffeomorphism, although it is obviously one--one on $B$.\n\nWe note that even in the simple case of a Euclidean sphere, an explicit expression for the heat kernel is not known. In practice, the heat kernel has to be approximated using appropriate Gaussian networks \\cite{lafon}. In this section, we aim to obtain a local coordinate chart that is computed directly in terms of Euclidean distances on ${\\mathbb R}^D$, and depends upon $d+2$ trainable \nparameters. The construction of this chart constitutes the first hidden layer of our deep learning process. As explained in the introduction, once this chart is in place, the question of function extension on the manifold reduces locally to the well studied problem of function extension on a $d$ dimensional unit cube. \n\nTo describe our constructions,we first develop some notation.\n\nIn this section, it is convenient to use the notation ${\\bf x}=(x^1,\\cdots,x^D)\\in{\\mathbb R}^D$ rather than ${\\bf x}=(x_1,\\cdots,x_D)$, which we will use in the rest of the sections. If $1\\le d\\le D$ is an integer, and ${\\bf x}\\in{\\mathbb R}^d$, $\\|{\\bf x}\\|_d$ denotes the Euclidean norm of ${\\bf x}$. If ${\\bf x}\\in{\\mathbb R}^D$, we will write $\\pi_c({\\bf x})=(x^1,\\cdots,x^d)$, $\\|{\\bf x}\\|_d=\\|\\pi_c({\\bf x})\\|_d$. If ${\\bf x}\\in{\\mathbb R}^d$, $r>0$,\n$$\nB({\\bf x},r)=\\{{\\bf y}\\in{\\mathbb R}^d : \\|{\\bf x}-{\\bf y}\\|_d\\le r\\}.\n$$\n\nThere exists $\\delta^*>0$ with the following properties. The manifold is covered by finitely many geodesic balls such that for the center ${\\bf x}_0^*\\in{\\mathbb X}$ of any of these balls, there exists a diffeomorphism, namely, the exponential coordinate map $u=(u^1,\\cdots,u^D)$ from $B_d(0,\\delta^*)$ to the geodesic ball around ${\\bf x}_0^*=u(0)$ \\cite[p.~65]{docarmo_riemannian}. If $J$ is the Jacobian matrix for $u$, given by $J_{i,j}({\\bf y})=D_iu^j({\\bf y})$, ${\\bf y}\\in B_d(0,\\delta^*)$, then \n\\begin{equation}\\label{finaljacobiatzero}\nJ(0)=[I_d|0_{d,D-d}].\n\\end{equation}\nFurther, there exists $\\kappa>0$ (independent of ${\\bf x}^*$) such that\n\\begin{equation}\\label{jacobimodcont}\n\\|J(\\mathbf{q})-J(0)\\|\\le \\kappa\\|\\mathbf{q}\\|_d, \\qquad \\mathbf{q}\\in B_d(0,\\delta^*).\n\\end{equation}\nLet $\\eta^*:=\\min(\\delta^*, 1\/(2\\kappa))$. Then \\eref{jacobimodcont} implies that \n\\begin{equation}\\label{jacobinormest}\n1\/2\\le 1-\\kappa\\|\\mathbf{q}\\|_d\\le \\|J(\\mathbf{q})\\|\\le 1+\\kappa\\|\\mathbf{q}\\|_d\\le 2, \\qquad \\mathbf{q}\\in B_d(0,\\eta^*).\n\\end{equation}\nIn turn, this leads to\n\\begin{equation}\\label{rhoeuccomp1}\n(1\/2)\\rho(u(\\mathbf{p}),u(\\mathbf{q}))\\le \\|\\mathbf{p}-\\mathbf{q}\\|_d\\le 2\\rho(u(\\mathbf{p}), u(\\mathbf{q})), \\qquad \\mathbf{p},\\mathbf{q}\\in B_d(0,\\eta^*).\n\\end{equation}\n\n\nLet ${\\bf x}_\\ell^*=u(\\mathbf{q}_\\ell)$, $\\ell=1,\\cdots,d$, be chosen with the following properties:\n\\begin{equation}\\label{nbdcond}\n\\|\\mathbf{q}_\\ell\\|_d\\le \\eta^*, \\qquad \\ell=1,\\cdots,d,\n\\end{equation}\nand, with the matrix function $U$ defined by \n\\begin{equation}\\label{umatrixdef}\nU_{i,j}(\\mathbf{q})=u^i(\\mathbf{q})-({\\bf x}_j^*)^i,\n\\end{equation}\n we have\n\\begin{equation}\\label{indepcond}\n\\|J(0)U(0){\\bf y}\\|_d\\ge \\gamma>0, \\qquad \\|{\\bf y}\\|_d= 1.\n\\end{equation}\nAny set $\\{{\\bf x}_\\ell^*\\}$ with these properties will be called \\emph{coordinate stars} around ${\\bf x}^*$. We note that the matrix $J(0)U(0)$ has columns given by $\\pi_c({\\bf x}^*-{\\bf x}_j^*)$, $j=1,\\cdots,d$, and hence, can be computed without reference to the map $u$. Let \n\\begin{equation}\\label{betastardef}\n\\beta^* := (1\/2)\\min\\left(\\frac{1}{2\\kappa}, \\delta^*,\\frac{\\gamma}{8\\sqrt{d}}\\right).\n\\end{equation} \n\n\\begin{theorem}\\label{loccordtheo}\nLet $\\Psi(\\mathbf{q}):=(\\|u(\\mathbf{q})-u(\\mathbf{q}_\\ell)\\|^2_D)_{\\ell=1}^d\\in{\\mathbb R}^d$. Then \\\\\n{\\rm (a)} $\\Psi$ is a diffeomorphism on $B_d(0,2\\beta^*)$. If $\\mathbf{p}, \\mathbf{q}\\in B_d(0,2\\beta^*)$, ${\\bf x}=u(\\mathbf{p})$, ${\\bf y}=u(\\mathbf{q})$, then\n\\begin{equation}\\label{distortest}\n\\frac{\\gamma}{2}\\rho({\\bf x},{\\bf y})\\le \\|\\Psi(\\mathbf{p})-\\Psi(\\mathbf{q})\\|_d \\le 32\\sqrt{d}\\eta^*\\rho({\\bf x},{\\bf y}).\n\\end{equation}\n{\\rm (b)} The function $\\Psi$ is a diffeomorphism from $B_d(0,\\beta^*)$ onto $B_d(\\Psi(0),\\beta^*)$. \n \\end{theorem}\n \n\\begin{rem}\\label{maptocuberem}\n{\\rm\nLet $\\mathbb{B}=u(B_d(0,\\beta^*))\\subset {\\mathbb X}$ be a geodesic ball around ${\\bf x}_0^*$. For ${\\bf x}\\in \\mathbb{B}$, we define\n$$\n\\phi({\\bf x})=\\Psi(u^{-1}({\\bf x}))=(\\|{\\bf x}-{\\bf x}_\\ell^*\\|_D^2).\n$$\nThen Theorem~\\ref{loccordtheo}(b) shows that \n$\\phi$ is a diffeomorphism from $\\mathbb{B}$ onto $B_d(\\Psi(0),\\beta^*)$. Since $\\Psi(0)=(\\|{\\bf x}_0^*-{\\bf x}_\\ell^*\\|_D^2)$, \n$$\n\\Phi({\\bf x})=\\frac{\\sqrt{d}}{\\beta^*}(\\phi({\\bf x})-\\Psi(0)), \\qquad {\\bf x}\\in \\mathbb{B}\n$$\nmaps $\\mathbb{B}$ diffeomorphically onto $B_d(0,\\sqrt{d})\\supset [-1,1]^d$. Let $\\mathbb{U}=\\Phi^{-1}([-1,1]^d)$. Then $\\mathbb{U}$ is a neighborhood of ${\\bf x}_0^*$ and $\\Phi$ maps $\\mathbb{U}$ diffeomorphically onto $[-1,1]^d$. We oberve that ${\\mathbb X}$ is a union of finitely many neighborhoods of the form $\\mathbb{U}$, so that any ${\\bf x}\\in{\\mathbb X}$ belongs to at least one such neighborhood. Moreover, $\\Phi({\\bf x})$ can be computed entirely in terms of the description of ${\\bf x}$ in terms of its $D$--dimensional coordinates. \\hfill$\\Box$\\par\\medskip\n}\n\\end{rem}\n\n\\begin{rem}\\label{trainrem}\n{\\rm The trainable parameters are thus $\\beta^*$, and the points ${\\bf x}_0^*, \\cdots, {\\bf x}_d^*$. Since $\\|J(0)\\|=1$, the condition \\eref{indepcond} is satisfied if ${\\bf x}_\\ell^*-{\\bf x}_0^*$ are along linearly independent directions as in Theorem~\\ref{jmstheo}.\n\\hfill$\\Box$\\par\\medskip}\n\\end{rem}\n\n\\begin{rem}\\label{networkimprem}\n{\\rm\nSince the mapping $\\Phi$ in Remark~\\ref{maptocuberem} is a quadratic polynomial in ${\\bf x}$, it can be implemented as a neural network with a single hidden layer using the activation function given in \\eref{requdef} as described in Sub--Section~\\ref{polysubsect}.\n}\n\\end{rem}\n\n\\begin{uda}\\label{helixexample}\n{\\rm\nLet $0