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+{"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 $GTS<GuidS<RandS<RTS<CG$ for the objective value parameters and $GTS>GuidS>RandS>RTS>CG$ 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 $0<n<4$, the free energy contains divergences of the power $n$, while for $n>4$ 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"}}
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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<B_{\\rm z}\\right>, \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 $\\left<B_{\\rm z}\\right>$ 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 iti<dh}\n\\sum\\limits_{i=1}^d i t_i< d m \\mbox{ for all } d=1,2,\\ldots,h.\n\\end{equation}\nIn particular, ${\\rm T}(1,h)=0$ for any positive integer $h$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Results}\n\n\nWe start with the following easy proposition.\n\n\\begin{prop} For any integer $h\\geq 1$, there exist $t_1,\\ldots,t_{h}\\geq 0$\nsuch that $\\sum\\limits_{i=1}^d t_i=T(m,h)$ and \\eqref{equation iti<dh} holds,\nwhere $t_1,\\ldots,t_{h}$ satisfy the following recurrence relation:\n\n(i) $t_1=m-1$;\n\n(ii) $t_d=\\left\\lfloor\\frac{(dm-1)-\\sum\\limits_{i=1}^{d-1} i t_i}{d}\\right\\rfloor$ for all $d=2,\\ldots,h$.\n\\end{prop}\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{Lemma finite commutative rings} \\ Let $R$ be a finite commutative ring with identity.\n Let $\\Psi$ be a subgroup of the group ${\\rm Aut}(R)$. Suppose that for every prime ideal $P$ with ${\\rm Ind}(P)>1$, 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 tdt<n IndP1}\n\\sum\\limits_{i=1}^n i d_i<n \\ {\\rm Ind}(P_1)  \\mbox{ for all } n=1,2,\\ldots,h.\n\\end{equation}\n Then we set\n\\begin{equation}\\label{equation sequence BO}\nB_{\\mathscr{O}}=b_1^{[d_1]}\\cdot\\ldots\\cdot b_h^{[d_h]}.\n\\end{equation}\nOne thing worth remarking is that all the quantities $h,d_1,d_2,\\ldots,d_{h}$ in \\eqref{equation sequence BO} varies with $\\mathscr{O}$ taking distinct orbits, and thus,  the length  $|B_{\\mathscr{O}}|$ varies accordingly. Precisely, we have the following.\n\n\n\\noindent \\textbf{Claim F.} $|B_{\\mathscr{O}}|=\\sum\\limits_{P\\in \\mathscr{O}} \\frac\n{T\\left({\\rm Ind}(P); \\  \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}$.\n\n\\noindent {\\sl Proof of Claim F.} Note that ${\\rm St}(P_1)=\\cdots={\\rm St}(P_h)$. By Claim A, we have ${\\rm Ind}(P_1)=\\cdots={\\rm Ind}(P_h)$. Then it follows from \\eqref{equation orbit O}, \\eqref{equation sum di=T()} and \\eqref{equation sequence BO} that $|B_{\\mathscr{O}}|=T({\\rm Ind}(P_1); \\  h)=\\sum\\limits_{i=1}^h \\frac{T({\\rm Ind}(P_1); \\  h)}{h}=\\sum\\limits_{i=1}^h \\frac{T({\\rm Ind}(P_1); \\  \\frac{|\\Psi|}{|{\\rm St}(P_1)|})}{\\frac{|\\Psi|}{|{\\rm St}(P_1)|}}=\\sum\\limits_{i=1}^h \\frac{T({\\rm Ind}(P_i); \\  \\frac{|\\Psi|}{|{\\rm St}(P_i)|})}{\\frac{|\\Psi|}{|{\\rm St}(P_i)|}}=\\sum\\limits_{P\\in \\mathscr{O}} \\frac\n{T\\left({\\rm Ind}(P); \\  \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}$. This proves Claim F.\n\\qed\n\n\n\\noindent \\textbf{Claim G.} For any nonempty subsequence $W$ of $B_{\\mathscr{O}}$, say $W=a_1\\cdot\\ldots\\cdot a_{\\ell}$, and for any $\\psi_1,\\ldots,\\psi_{\\ell}\\in \\Psi$ (not necessarily distinct), there exists $r\\in [1,h]$ such that $\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i)\\in P_r\\setminus P_r^{{\\rm Ind}(P_r)}$.\n\n\n\\noindent {\\sl Proof of Claim G.} By \\eqref{equation btin H(O,t)}, \\eqref{equation sequence BO} and  Claim E, we have that \\begin{equation}\\label{equation psim(am)in i=1tohH}\n\\psi_m(a_m)\\in \\bigcup\\limits_{i=1}^{h}\\mathscr{H}_{\\mathscr{O};i} \\mbox{ for each } m\\in [1,\\ell].\n\\end{equation}\nLet\n$n=\\sharp\\{s\\in [1,h]: \\prod\\limits_{m=1}^{\\ell}\\psi_m(a_m)\\in P_s\\}.$ Obviously, $1\\leq n\\leq h$.\nBy arranging the indices of $(P_1,\\ldots,P_h)$ if necessary, we can assume without loss of generality that\n\\begin{equation}\\label{equation productin[1ton]}\n\\{s\\in [1,h]: \\prod\\limits_{m=1}^{\\ell}\\psi_m(a_m)\\in P_s\\}=[1,n].\n\\end{equation}\nThen\n\\begin{equation}\\label{equation each in at most n primes}\n\\{s\\in [1,h]: \\psi_m(a_m)\\in P_s\\}\\subset [1,n] \\  \\mbox{ for each } m\\in [1,\\ell].\n\\end{equation}\nBy \\eqref{equation psim(am)in i=1tohH}, \\eqref{equation each in at most n primes} and Claim C, we derive that\n\\begin{equation}\\label{equation psimamini=1tonHOi}\n\\psi_m(a_m)\\in \\bigcup\\limits_{t=1}^{n}\\mathscr{H}_{\\mathscr{O};t} \\mbox{ for each } m\\in [1,\\ell].\n\\end{equation}\nBy  \\eqref{equation tdt<n IndP1}, \\eqref{equation sequence BO}, \\eqref{equation each in at most n primes}, \\eqref{equation psimamini=1tonHOi}, Claim C and Claim E, we conclude that\n$$\\begin{array}{llll}\n& &\\sum\\limits_{s=1}^{n} \\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in P_s\\}\\\\\n&=&\\sum\\limits_{s=1}^{h} \\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in P_s\\}\\\\\n&=&\\sum\\limits_{m=1}^{\\ell} \\sharp\\{s\\in [1,h]: \\psi_m(a_m)\\in P_s\\}\\\\\n&=& \\sum\\limits_{t=1}^{n} (t  \\times \\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in \\mathscr{H}_{\\mathscr{O};t}\\})\\\\\n&=&\\sum\\limits_{t=1}^n  (t  \\times \\sharp\\{m\\in [1,\\ell]: a_m\\in \\mathscr{H}_{\\mathscr{O};t}\\})\\\\\n&\\leq & \\sum\\limits_{t=1}^n  t d_t\\\\\n&<& n \\ {\\rm Ind}(P_1).\\\\\n\\end{array}$$\nCombined with Claim A, we derive that there exists\n\\begin{equation}\\label{equation rin[1ton]}\nr\\in [1,n]\n\\end{equation} such that $\\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in P_r\\}<{\\rm Ind}(P_1)={\\rm Ind}(P_r)$. Moreover, since $P_r$ is prime, it follows from \\eqref{equation productin[1ton]} that\n$\\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in P_r\\}\\geq 1,$ and so, $$\\sharp\\{m\\in [1,\\ell]: \\psi_m(a_m)\\in P_r\\}\\in [1, {\\rm Ind}(P_r)-1].$$\nBy arranging the indices of $[1,\\ell]$ if necessary, we can assume without loss of generality that there exists some\n\\begin{equation}\\label{equation uin[1,indpr-1]}\nu\\in [1,{\\rm Ind}(P_r)-1]\n\\end{equation}\nsuch that\n\\begin{equation}\\label{equation twocasepsim(am)}\n\\begin{array}{llll} &\\left\n\\{\\begin{array}{llll}\n               \\psi_m(a_m)\\in P_r, & \\mbox{ if } m=1,\\ldots,u; \\\\\n               \\psi_m(a_m)\\notin P_r, & \\mbox{ if } m=u+1,\\ldots,\\ell. \\\\\n                       \\end{array}\n           \\right. \\\\\n\\end{array}\n\\end{equation}\nBy \\eqref{equation def H(O;X)}, \\eqref{equation def H(O;t)}, \\eqref{equation each in at most n primes},  \\eqref{equation twocasepsim(am)} and Claim C, we conclude that\n$$\\begin{array}{llll} &\\left\n\\{\\begin{array}{llll}\n               \\psi_m(a_m)\\in {\\rm G}_{P_r}, & \\mbox{ if } m=1,\\ldots,u; \\\\\n               \\psi_m(a_m)\\equiv 1_R \\pmod{P_r^{{\\rm Ind}(P_r)}}, & \\mbox{ if } m=u+1,\\ldots,\\ell. \\\\\n                       \\end{array}\n           \\right. \\\\\n\\end{array}$$\nIt follows that $$\\prod\\limits_{m=1}^{\\ell} \\psi_m(a_m)=(\\prod\\limits_{m=1}^{u} \\psi_m(a_m))\\cdot (\\prod\\limits_{m=u+1}^{\\ell} \\psi_m(a_m))\\equiv (\\prod\\limits_{m=1}^{u} \\psi_m(a_m))\\cdot 1_R=\\prod\\limits_{m=1}^{u} \\psi_m(a_m) \\pmod{P_r^{{\\rm Ind}(P_r)}},$$\nand\nfollows from \\eqref{equation uin[1,indpr-1]} and Claim B that $$\\prod\\limits_{m=1}^{u} \\psi_m(a_m) \\not\\equiv 0\\pmod{P_r^{{\\rm Ind}(P_r)}}.$$\nTherefore, it follows from \\eqref{equation productin[1ton]} and \\eqref{equation rin[1ton]} that $\\prod\\limits_{m=1}^{\\ell} \\psi_m(a_m)\\in P_r\\setminus P_r^{{\\rm Ind}(P_r)}$. This proves Claim G.\n\\qed\n\n\nTake a $\\Psi$-product-one free sequence $V$ of terms from the group ${\\rm U}(R)$ with length\n\\begin{equation}\\label{equation |V|_D-1}\n|V|={\\rm D}_{\\Psi}({\\rm U}(R))-1.\n\\end{equation}\nSuppose that $\\mathscr{O}_1,\\ldots,\\mathscr{O}_k$ are all distinct orbits of ${\\rm Spec}  R$ under the action of $\\Psi$. Let\n\\begin{equation}\\label{equation T=Bo1BokV}\nT= B_{\\mathscr{O}_1}\\cdot\\ldots\\cdot B_{\\mathscr{O}_k}\\cdot V.\n\\end{equation}\n\n\n \\noindent \\textbf{Claim H.} The sequence $T$ is $\\Psi$-idempotent-product free.\n\n\\noindent {\\sl Proof of Claim H.} Suppose to the contrary that there exists a nonempty subsequence $T'$ of $T$, say $T'=a_1\\cdot \\ldots\\cdot a_{\\ell}$,  such that $T'$ is a $\\Psi$-idempotent-product sequence. Since $1_R$ is the unique idempotent  in ${\\rm U}(R)$ and V is $\\Psi$-product-one free, it follows  that at least one term of the sequence $B_{\\mathscr{O}_1}\\cdot\\ldots\\cdot B_{\\mathscr{O}_k}$ appears in the sequence $T'$. Then by rearranging the indices $i\\in [1,\\ell]$ we may assume without loss of generality that\n$$a_1\\cdot\\ldots\\cdot a_s\\mid B_{\\mathscr{O}_1},$$\n$$a_{s+1}\\cdot\\ldots\\cdot a_{\\ell} \\mid B_{\\mathscr{O}_2}\\cdot\\ldots\\cdot B_{\\mathscr{O}_k}\\cdot V,$$\nwhere $1\\leq s\\leq \\ell$.\nSince  $T'$ is a $\\Psi$-idempotent-product sequence, there exists $\\psi_1,\\ldots,\\psi_{\\ell}\\in \\Psi$ such that $\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i)$ is an idempotent. By Claim G, there exists some prime ideal $P\\in\\mathscr{O}_1$ such that  $\\prod\\limits_{i=1}^{s}\\psi_i(a_i)\\in P\\setminus P^{{\\rm Ind}(P)}$. By \\eqref{equation def H(O;X)}, we see that  $\\psi_i(a_i)\\equiv 1_R \\pmod {P^{{\\rm Ind}(P)}}$ or $\\psi_i(a_i)\\in {\\rm U}(R)$ according to $a_i\\mid B_{\\mathscr{O}_2}\\cdot\\ldots\\cdot B_{\\mathscr{O}_k}$ or $a_i\\mid V$ respectively, where $i\\in [s+1,\\ell]$.  Then $\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i)\\in P\\setminus P^{{\\rm Ind}(P)}$ still holds. This is a contradiction with $\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i)$ being idempotent, since  $\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i)=(\\prod\\limits_{i=1}^{\\ell}\\psi_i(a_i))^{{\\rm Ind}(P)}\\in P^{{\\rm Ind}(P)}$. This proves Claim H. \\qed\n\nBy \\eqref{equation |V|_D-1}, \\eqref{equation T=Bo1BokV}, Claim F and Claim H,  we conclude that ${\\rm I}_{\\Psi}(\\mathcal{S}_R)\\geq 1+|T|=1+|V|+\\sum\\limits_{i=1}^k |B_{\\mathscr{O}_i}|={\\rm D}_{\\Psi}({\\rm U}(R))+\\sum\\limits_{i=1}^k\\sum\\limits_{P\\in \\mathscr{O}_i} \\frac\n{T\\left({\\rm Ind}(P); \\  \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}\n={\\rm D}_{\\Psi}({\\rm U}(R))+\\sum\\limits_{P\\in {\\rm Spec}(R)} \\frac\n{T\\left({\\rm Ind}(P); \\  \\frac{|\\Psi|}{|{\\rm St}(P)|}\\right)} {\\frac{|\\Psi|}{|{\\rm St}(P)|}}.$ This complete the proof of the lemma. \\end{proof}\n\n\n\n\n\\begin{theorem}\\label{Theorem finite commutative rings} \\ Let $R$ be a finite commutative principal ideal ring with identity. Let $\\Psi$ 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{theorem}\n\nIt is not hard to check that the following proposition holds, i.e., the bound in Theorem \\ref{Theorem finite commutative rings} is best possible in general.\n\n\\begin{prop} Let $L$ be a finite commutative local P.I.R, and let $R=L\\times L$. Let $\\varphi:R\\rightarrow R$ be an automorphism given as $\\varphi:(a,b)\\mapsto (b,a)$ for any $(a,b)\\in R$. Let $H=\\langle \\varphi\\rangle$ be the group of automorphisms of order two generated by $\\varphi$. 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{prop}\n\n\n\n\n\n\n\nIt is shown \\cite{Hungerford} that the following proposition holds. For the reader's convenience, we provide a short proof of this proposition.\n\n\\begin{prop} Any finite commutative principal ideal ring $R$ is a quotient ring some P.I.D.\n\\end{prop}\n\n\\begin{proof} By the fundamental theorem for Noetherian rings, we know that $R\\cong R_1\\times \\cdots\\times R_k$ where $R_1,\\ldots, R_k$  are finite commutative local rings.  It is shown (see \\cite{Hungerford}, Definition 9 and Corollary 11) that a finite commutative local P.I.R is a quotient ring of some P.I.D. Say $D_i$ is a P.I.D, $J_i\\lhd D_i$ and $R_i\\cong D_i\\diagup J_i$ where $i\\in [1,k]$. Then $R\\cong (D_1\\diagup J_1)\\times \\cdots\\times (R_k\\diagup J_k)\\cong (D_1\\times\\cdots\\times D_k)\\diagup (J_1\\times\\cdots\\times J_k)$, where $(J_1\\times\\cdots\\times J_k)\\lhd (D_1\\times\\cdots\\times D_k)$. Note that $D_1\\times\\cdots\\times D_k$ is a P.I.D. The conclusion is proved.\n\\end{proof}\n\nSince a P.I.D is a Dedekind domain, therefore, any finite P.I.R is a quotient ring of some Dedekind domain.\nIn the following, we shall show the result of Theorem \\ref{Theorem finite commutative rings} holds true for a more general setting, i.e., for finite quotient rings of any Dedekind domain.\n\n\\begin{theorem}\\label{Theorem Dedekind rings} \\ Let $R$ be a finite quotient ring of some Dedekind domain.\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{theorem}\n\n\\begin{proof} Let $Q\\in {\\rm Spec} R$ be such that ${\\rm Ind}(Q)>1$. 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"}}